Collin. Foundations for Microwave Engineering

A C LA S S l C REISSUE Foundations for Microwave Engineering SECOND EDITION Robert E. Collin Foundations for Microwave Engineering SECOND EDITION D o n a l d G. Dudley, Series Editor muiaUomjor Microwatt Engineering. Second Edition, covers the major topics of microwave engineering- Its presentation defines the accepted standard for both advanced undergraduate and graduate level courses on microwave engineering. An essential reference book lor the practicing microwave engineer, it features.• Planar transmission lines, as well as an appendix that describes in detail conformal mapping methods for their analysis and attenuation characteristics. • Small aperture coupling and its application in practical components such as directional couplers and cavity coupling • Printed circuit components with an emphasis on techniques such as even and odd mode analysis and the use of symmetry properties • Microwave linear amplifier and oscillator design using solid-state circuits such as varactor devices and transistors. Foimdationi for Microwave Engineering, Second Edition, has extensive coverage of transmission lines, waveguides, microwave circuit theory, impedance matching, and cavity resonators. It devotes an entire chapter to fundamental microwave tubes, as well as other chapters on periodic structures, microwave filters, small signal solid-state microwave mplifier and oscillator design, and negative resistance devices and circuits. Completely updated in 1992, it is being eissued by the IEEE Press in response to requests from our many members, who found u an invaluable textbook and an enduring reference for practicing microwave engineers. <oul ihe Author Robert E. Collin is the author or coauthor ol more than 150 technical papers and five books on electromagnetic nSeory and applications. His classic text, Field Theory of Guided Waves, is also a volume in the series. Professor Collin has had a long and distinguished academic career at Case Western Reserve University. In addition to his professorial duties, he has served as chairman of the Department of Electrical Engineering and as interim dean of engineering. Professor Collin is a life fellow of the IEEE and a member of the Microwave Theory and Techniques Society and the Antennas and Propagation Society (APSI He is a member of the U.S. Commission B of URSI and a member of the Geophysical Society. Other honors include the Diekman Award from Case Western Reserve University for distinguished graduate teaching, the IEEE APS Distinguished Career Award (1992), the IEEE Schelkunoff Prize Paper Award (1992). the IEEE Electromagnetics Award (1998), and an IEEE Third Millennium Medal in 2000. In 1990 Professor Collin was elected to the National Academy of Engineering. The IEEE Press Series on Electromagnetic Wave Theory offers outstanding coverage of the field. It consists of new titles of contemporary interest, as well as reissues and revisions of recognized classics by established authors and researchers. The series emphasizes works of long-term archival significance in electromagnetic waves and applications. Designed specifically for graduate students, researchers and practicing engineers, the series provides affordable volumes that explore and explain electromagnetic waves beyond the undergraduate level. • • ISBN D-7flD3-bD31-l Visit wftww.wiley.com/ieee >W1LEY'INTERSCIEN' 780780"36rmr)" An IEEE Press Classic Reissue FOUNDATIONS FOR MICROWAVE ENGINEERING SECOND EDITION IEEE PRESS SERIES ON ELECTROMAGNETIC WAVE THEORY The IEEE Press Series on Electromagnetic Wave Theory consists of new titles as well as reprintings and revisions of recognized classics that maintain long-term archival significance in electromagnetic waves and applications. Scries Editor Donald G. Dudley University of Arizona Advisory Board Robert E. Collin Case Western Reserve University Akira Ishimaru University of Washington D. S. Jones University of Dundee Associate Editors ELECTROMAGNETIC THEORY, SCATTERING, AND DIFFRACTION INTEGRAL EQUATION METHODS Ehud Heyman Tel-Aviv University Donald R. Wilson University of Houston DIFFERENTIAL EQUATION METHODS ANTENNAS. PROPAGATION, AND MICROWAVES Andreas C. Cangellans University of A rizona David R. Jackson University of Houston BOOKS IN THE IEEE PRESS SERIES ON ELECTROMAGNETIC WAVE THEORY Christopoulos. C, The Transmission-Line Modeling Methods: TLM Clemmow, R C. The Plane Wave Spectrum Representation of Electromagnetic Fields Collin. R. E.. Field Theory of Guided Waves, Second Edition Collin, R. E.. Foundations for Microwave Engineering, Second Edition Dudley. D. G., Mathematical Foundations for Electromagnetic Theory Elliot. R. S., Electromagnetics: History, Theory, and Applications Felsen. L. B.. and Marcuvitz. N.. Radiation and Scattering qf Waves Harrington. R. F, Field Computation by Moment Methods Hansen et aL, Plane-Wave Theory of Time-Domain Fields: Near-Field Scanning Applications Ishimaru. A., Wave Propagation and Scattering in Random Media Jones, D. S.. Methods in Electromagnetic Wave Propagation. Second Edition Lindell. 1. V., Methods for Electromagnetic Field Analysis Peterson el al.. Computational Methods for Electromagnetics Tai. C. T, Generalized Vector and Dyadic Analysis: Applied Mathematics in Field Theory Tai. C. T. Dyadic Green Functions in Electromagnetic Theory. Second Edition Van Bladel, J., Singular Electromagnetic Fields and Sources Volakis et al., Finite Element Method for Electromagnetics: Antennas, Micmwave Circuits, and Scattering Applications Wail. J.. Electromagnetic Waves in Stratified Media An IEEE Press Classic Reissue FOUNDATIONS FOR MICROWAVE ENGINEERING SECOND EDITION IEEE Press Series on Electromagnetic Wave Theory Robert E. Collin Professor of Electrical Engineering Case Western Reserve University Cleveland, OH IEEE Antennas & Propagation Society, Sponsor IEEE Microwave Theory and Techniques Society, Sponsor The Institute of Electrical and Electronics Engineers. Inc., New York WlLEY'INTERSCIENCE A JOHN WILEY & SONS, INC.. PUBLICATION © 2001 THE INSTITUTE OF ELECTRICAL AND ELECTRONICS ENGINEERS, INC. 3 Park Avenue, 17th Floor, New York, NY 10016-5997 Published by John Wiley & Sons, Inc., Hoboken, New Jersey. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, 97S-750-8400, fax 978-750-4470, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken. NJ 07030, (201) 748-6011, fax (201) 748-6008, e-mail: [email protected]. For general information on our other products and services please contact our Customer Care Department within the U.S. at 877-762-2974, outside the U.S. at 317-572-3993 or fax 317-572-4002. Printed in the United States of America 1 0 9 8 7 6 5 4 3 2 ISBN 0-7803-6031-1 Library of Congress Cataloging-in-Publication Data Collin. Robert E. Foundations lor microwave engineering / Robert E. Collin.- 2nd ed. p. cm. - (IEEE Press series on electromagnetic wave theory) Originally published : New York : McGraw Hill, cl992. "An IEEE Press classic reissue." Includes bibliographical references and index. ISBN 0-7803-6031-1 I. Microwave devices. I. Title. II. Series. TK7876 .C645 2000 621.381'3--dc21 00-053874 FOREWORD TO THE REISSUED EDITION The purpose of the IEEE Press Series on Electromagnetic Wave Theory is to publish books of long-term archival significance in electromagnetics. Included are new titles as well as reprints and revisions of recognized classics. The book Foundations for Microwave Engineering, by Robert E. Collin, is by any measure such a classic. The original edition of the book appeared in 1966 and remained in print until the appearance of the second edition in 1992, a span of 26 years. In the second edition. Professor Collin completely updated and modernized his book to include the many advances that had occurred in microwave engineering since the appearance of the original edition. That the second edition has gone out of print has caused concern among many of my colleagues in the IEEE Antennas and Propagation Society (APS) and the IEEE Microwave Theory and Techniques Society (MTT). We at the IEEE Press are delighted to be able lo overcome this difficulty by introducing a reprint of the second edition into our Series on Electromagnetic Wave Theory. The book is a thorough and in-depth exposition on microwave engineering. Furthermore, it will make an excellent companion to Professor Collin's book, Field Theory- of Guided Waves, also included in the series. Professor Collin has been a valued colleague for many years. He is the author or coauthor of five books and more than 150 technical papers. His contributions to electromagnetics span a wide range of subjects and have brought him international respect and many awards. Among these are election to the National Academy of Engineering, the IEEE Electromagnetics Field Award, the IEEE/APS Distinguished Career Award, an IEEE/APS Schelkunoff Prize Paper Award, and the IEEE Third Millennium Medal. It is with pleasure that I welcome this book into the series. Donald G. Dudley University of Arizona Series Editor IEEE Press Series on Electromagnetic Wave Theory IEEE Press 445 Hoes Lane, P.O. Box 1331 Piscataway, NJ 08855-1331 IEEE Press Editorial Board Robert J. Herrick, Editor in Chief M.Akay J. B. Anderson P. M. Anderson J. E. Brewer M.Eden M. E. El-Hawary R. F. Hoyt S. V. Kartalopoulos D. Kirk M.S.Newman M. Padgett W.D. Reeve G. Zobrist Kenneth Moore, Director of IEEE Press Catherine Faduska. Senior Acquisitions Editor Linda Matarazzo. Associate Acquisitions Editor Marilyn G. Catis, Marketing Manager Mark Morrell, Associate Production Editor IEEE Antennas & Propagation Society. Sponsor AP-S Liaison to IEEE Press, Robert Mailloux IEEE Microwave Theory and Techniques Society, Sponsor MTT-S Liaison to IEEE Press, Karl Varian Cover design: William T. Donnelly, WT Design CONTENTS Preface 1 1.1 1.2 1.3 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 •2.11 2.12 3 3.1 3.2 Introduction XV 1 Microwave Frequencies Microwave Applications Microwave Circuit Elements and Analysis References 1 3 6 16 Electromagnetic Theory 17 Maxwell's Equations Constitutive Relations Static Fields Wave Equation Energy and Power Boundary Conditions Plane Waves Plane Waves in Free Space Reflection from a Dielectric Interface 1. Parallel Polarization 2. Perpendicular Polarization Reflection from a Conducting Plane Potential Theory Derivation of Solution for Vector Potential Lorentz Reciprocity Theorem Problems References 17 23 28 31 33 39 44 44 49 49 52 53 56 59 62 65 70 Transmission Lines and Waveguides 71 Part 1 Waves on Transmission Lines Waves on An Ideal Transmission Line Terminated Transmission Line: Resistive Load 72 72 78 vii ViU COOTENTS 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14 3.15 3.16 3.17 3.18 3.19 3.20 3.21 Capacitive Termination Steady-State Sinusoidal Waves Waves on a Lossy Transmission Line Loss-Free Transmission Line Low-Loss Transmission Line . Terminated Transmission Line: Sinusoidal Waves Terminated Lossy Line Part 2 Field Analysis of Transmission Lines Classification of Wave Solutions TEM Waves TE Waves TM Waves Transmission Lines (Field Analysis) Lossless Transmission Line Transmission Line with Small Losses Transmission-Line Parameters In homogeneously Filled Parallel-Plate Transmission Line Low-Frequency Solution High-Frequency Solution Planar Transmission Lines Microstrip Transmission Line Low-Frequency Solutions Microstrip Attenuation High-Frequency Properties of Microstrip Lines Attenuation Coupled Microstrip Lines Strip Transmission Lines Attenuation Coupled Strip Lines Coplanar Transmission Lines Attenuation High-Frequency Dispersion P a r t 3 Rectangular and Circular Waveguides Rectangular Waveguide TE Waves Power Attenuation Dominant TE 1 0 Mode TM Modes 82 85 86 88 89 89 94 96 96 99 100 102 104 104 108 112 117 121 123 125 130 136 153 158 163 164 170 171 173 175 178 180 180 181 182 186 187 190 193 Circular Waveguides TM Modes TE Modes Wave Velocities Phase Velocity Group Velocity Energy-Flow Velocity Ridge Waveguide Fin Line Problems References 194 194 196 198 199 200 204 205 208 210 219 CHAPTER 1 INTRODUCTION The purpose of this introductory chapter is to provide a short, and admittedly incomplete, survey of what the microwave engineering field encompasses. Section 1.2 presents a brief discussion of many of the varied and sometimes unique applications of microwaves. This is followed by a third section in which an attempt is made to show in what ways microwave engineering differs from the engineering of communication systems at lower frequencies. In addition, a number of microwave devices are introduced to provide examples of the types of devices and circuit elements that are examined in greater detail later on in the text. 1 MICROWAVE FREQUENCIES The descriptive term microwaves is used to describe electromagnetic waves with wavelengths ranging from 1 cm to 1 m. The corresponding frequency range is 300 MHz up to 30 GHz for 1-cm-wavelength waves. Electromagnetic waves with wavelengths ranging from 1 to 10 mm are called millimeter waves. The infrared radiation spectrum comprises electromagnetic waves with wavelengths in the range 1 am (10 6 m) up to 1 mm. Beyond the infrared range is the visible optical spectrum, the ultraviolet spectrum, and finally x-rays. Several different classification schemes are in use to designate frequency bands in the electromagnetic spectrum. These classification schemes are summarized in Tables 1.1 and 1.2. The radar band classification came into use during World War II and is still in common use today even though the new military band classification is the recommended one. In the UHF band up to around a frequency of 1 GHz, most communications circuits are constructed using lumped-parameter circuit compoI CONTENTS 4 4.1 4.2 *4.3 *4.4 4.5 4.6 4.7 4.8 4.9 *4.10 *4.11 *4.12 *4.13 5 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 Circuit Theory for Waveguiding Systems Equivalent Voltages and Currents Impedance Description of Waveguide Elements and Circuits One-Port Circuits Lossless One-Port Termination Foster's Reactance Theorem Even and Odd Properties of Z m N-Port Circuits Proof of Symmetry for the Impedance Matrix Proof of Imaginary Nature of [Z] for a Lossless Junction Normalized Impedance and Admittance Matrices Two-Port Junctions Some Equivalent Two-Port Circuits Scattering-Matrix Formulation Symmetry' of Scattering Matrix Scattering Matrix for a Lossless Junction Scattering Matrix for a Two-Port Junction Transmission-Matrix Representation Voltage-Current Transmission Matrix Wave-Amplitude Transmission Matrix Signal Flow Graphs Generalized Scattering Matrix for Power Waves Excitation of Waveguides Probe Coupling in a Rectangular Waveguide Radiation from Linear Current Elements Radiation from Current Loops Waveguide Coupling by Apertures Aperture in a Transverse Wall Aperture in Broad Wall of a Waveguide Problems References IX 220 221 224 224 228 230 232 233 235 236 237 238 245 248 250 251 254 257 257 259 260 268 276 276 286 290 294 Impedance Transformation and Matching Smith Chart Impedance Matching with Reactive Elements Single-Stub Matching Double-Stub Matching Network Triple-Stub Tuner Impedance Matching with Lumped Elements Circuit Q and Bandwidth Design of Complex Impedance Terminations Invariant Property of Impedance Mismatch Factor Waveguide Reactive Elements Shunt Inductive Elements Shunt Capacitive Elements Waveguide Stub Tuners 304 308 309 312 317 319 325 330 334 339 340 341 342 X CONTENTS 5.9 5.10 5.11 5.12 5.13 •5.14 5.15 5.16 *5.17 *5.18 *5.19 6 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11 Quarter-Wave Transformers Theory of Small Reflections Approximate Theory for Multisection Quarter-Wave Transformers Binomial Transformer Chebyshev Transformer Chebyshev Transformer (Exact Results) Filter Design Based on Quarter-Wave-Transformer Prototype Circuit Junction Capacitance and Length Compensation Tapered Transmission Lines Exponential Taper Taper with Triangular Distribution Synthesis of Transmission-Line Tapers Chebyshev Taper Exact Equation for the Reflection Coefficient Problems References 343 347 348 350 352 356 360 365 370 372 372 373 380 383 387 393 Passive Microwave Devices 394 Terminations Variable Short Circuit Attenuators Electronically Controlled Attenuators Phase Shifters Rotary Phase Shifter Electronically Controlled Phase Shifters Directional Couplers Directional-Coupler Designs Coupled-Line Directional Couplers Branch-Line Directional Coupler Lange Directional Coupler Hybrid Junctions Magic T Hybrid Ring Power Dividers Microwave Propagation in Ferrites Faraday Rotation Microwave Devices Employing Faraday Rotation Gyrator Isolator Resonance Isolator Circulators Three-Port Circulator Field Analysis of Three-Port Circulator Other Ferrite Devices Problems References 394 395 397 400 404 404 409 413 416 427 432 434 435 435 437 442 450 460 464 464 466 467 468 471 473 476 476 479 CONTENTS 7 7.1 7.2 7.3 7.4 7.5 7.6 *7.7 *7.8 *7.9 *7.10 8 8.1 8.2 8.3 8.4 8.5 8.6 *8.7 8.8 8.9 8.10 *8.11 8.12 8.13 XI Electromagnetic Resonators 481 Resonant Circuits Transmission-Line Resonant Circuits Series Resonance; Short-Circuited Line Open-Circuited Line Antiresonance Microstrip Resonators Circular Disk Resonator Microwave Cavities Rectangular Cavity Cylindrical Cavity Dielectric Resonators Equivalent Circuits for Cavities Aperture-Coupled Cavity Loop-Coupled Cavity Field Expansion in a General Cavity Cavity Field Expansions in Terms of Short-Circuit Modes Electric Field Expansion Orthogonality Properties Magnetic Field Expansion Orthogonality Properties Relationship between E„ and H„ Modes Oscillations in a Source-Free Cavity Cavity with Lossy Walls Degenerate Modes Excitation of Cavities Cavity Perturbation Theory Problems References 481 485 485 487 488 490 496 500 500 504 508 517 517 523 525 527 528 529 531 531 532 533 534 536 538 541 545 548 Periodic Structures and Filters 550 Capacitively Loaded Transmission-Line-Circuit Analysis Wave Analysis of Periodic Structures Periodic Structures Composed of Unsymmetrical Two-Port Networks Terminated Periodic Structures Matching of Periodic Structures k0-fl Diagram Group Velocity and Energy Flow Floquet's Theorem and Spatial Harmonics Periodic Structures for Traveling-Wave Tubes Periodic Structures for Millimeter-Wave Traveling-Wave Tubes Sheath Helix Some General Properties of a Helix Introduction to Microwave Filters Image-Parameter Method of Filter Design 551 557 559 560 563 564 566 569 571 Wl 580 583 585 587 Xli CONTENTS 8.14 8.15 8.16 8.17 8.18 8.19 8.20 8.21 8.22 8.23 9 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 9.10 9.11 9.12 9.13 10 10.1 10.2 Filter Design by Insertion-Loss Method Specification of Power Loss Ratio Maximally Flat Filter Characteristic Chebyshev Filter Some Low-Pass-Filter Designs Frequency Transformations Frequency Expansion Low-Pass to High-Pass Transformation Low-Pass to Bandpass Transformation Period Bandpass Mapping Impedance and Admittance Inverters A Microstrip Half-Wave Filter Microstrip Parallel Coupled Filter Quarter-Wave-Coupled Cavity Filters Direct-Coupled Cavity Filters Other Types of Filters Problems References 591 592 593 593 595 598 599 599 600 602 603 617 626 635 639 642 642 647 Microwave Tubes 648 Introduction Electron Beams with dc Conditions Ion-Neutralized Beam Beam with Axially Confined Flo\y Brillouin Flow Space-Charge Waves on Beams with Confined Flow Space-Charge Waves on Unfocused Beams Ac Power Relations Velocity Modulation Two-Cavity Klystron Excitation of a Cylindrical Cavity Cavity Excitation by a Velocity-Modulated Beam Reflex Klystron Magnetron O-Type Traveling-Wave Tube AZ-Type Traveling-Wave Tube Gyrotrons Field-Particle Interaction in a Gytotron Other Types of Microwave Tubes Problems References 648 650 650 651 652 654 661 667 670 678 679 683 686 690 692 699 701 703 708 709 712 Solid-State Amplifiers 713 Bipolar Transistors Transistor Biasing Field-Effect Transistors FET Biasing 716 720 721 724 CONTENTS xiii 10.3 10.4 10.5 10-6 10.7 10.8 10.9 10.10 10.11 10.12 11 11.1 11.2 11.3 11.4 11.5 11.6 12 12.1 12.2 12.3 12.4 12.5 12.6 Circle-Mapping Properties of Bilinear Transformations Microwave Amplifier Design Using S , , Parameters Amplifier Power Gain Derivation of Expressions for Gain Amplifier Stability Criteria Conditionally Stable Devices Constant Power-Gain Circles Properties of the Constant Gain Circles Stable Devices Unstable Devices Basic Noise Theory Filtered Noise Noise in Active Devices Noisy Two-Port Networks Low-Noise Amplifier Design Noise Figure Noise Figure for Cascaded Stages Constant Noise-Figure Circles Constant Mismatch Circles Constant Input Mismatch Circle Output Impedance-Mismatch Circle Microwave Amplifier Design Single-Stage Amplifier Design Design of Second Stage for a Two-Stage Amplifier Other Aspects of Microwave Amplifier Design Problems References 725 726 728 730 735 740 744 746 746 750 760 762 765 766 767 768 770 772 776 778 780 780 781 788 793 795 798 Parametric Amplifiers 799 p-n Junction Diodes Manley-Rowe Relations Linearized Equations for Parametric Amplifiers Parametric Up-Converter Negative-Resistance Parametric Amplifier Noise Properties of Parametric Amplifiers Problems References 800 B(M 807 809 814 821 829 830 Oscillators and Mixers 831 Gunn Oscillators Gunn Oscillator Circuits IMP ATT Diodes Transistor Oscillators Three-Port Description of a Transistor Oscillator Circuits Oscillator Design 837 840 851 HV CONTENTS Y2.7 12.8 12.9 12.10 12.11 Mixers Linear Mixer Operation Nonlinear Mixer Operation Mixer Noise Figure Balanced Mixers Other Types of Mixers Mixer Analysis Using Harmonic Balancing Problems References 856 861 862 864 865 868 869 873 875 Appendixes I Useful Relations from Vector Analysis 876 1.1 1.2 1.3 1.4 Vector Algebra Vector Operations in Common Coordinate Systems Rectangular Coordinates Cylindrical Coordinates Spherical Coordinates Vector Identities Green's Identities 876 877 877 877 878 879 880 II Bessel Functions 881 II. 1 II.2 Ordinary Bessel Functions Modified Bessel Functions References 881 883 885 III Conformal Mapping Techniques 886 Conformal Mapping Elliptic Sine Function Capacitance between Two Parallel Strips Strip Transmission Line Conductor Loss Conductor Losses for a Microstrip Transmission Line Attenuation for a Coplanar Line 886 889 892 896 898 903 905 Physical Constants and Other Data 911 Physical Constants Conductivities of Materials Dielectric Constants of Materials Skin Depth in Copper 911 912 912 912 Index 913 111.1 111.2 111.3 111.4 111.5 111.6 111.7 IV IV. 1 IV.2 IV.3 IV.4 PREFACE The first edition of Foundations for Microwave Engineering was published in 1966. The text has remained continuously in use since that time, but it has become clear that it no longer gives an adequate account of modern microwave engineering practice. Since the publication of the first edition there has been a dramatic advance in the microwave field brought about by the development of solid state transistors that can provide amplification and signal generation well into the millimeter wavelength region. Along with the widespread use of solid state devices, compatible transmission line structures and passive components were developed that could be integrated with the solid state devices into compact miniaturized microwave systems. These developments made it mandatory that the text be thoroughly revised if it were to continue serving the needs of the student and the practicing microwave engineer. In the revised addition I have adhered to the same general philosophy that governed the preparation of the first edition. Fundamental principles are stressed and complete derivations are provided for all significant formulas and relationships. All important fundamental concepts and principles are covered to the extent possible within a text of reasonable size. The applications of basic theory and principles are illustrated through detailed analysis of a large number of important components that find widespread use in practical microwave systems. Chapter 1 is an updated introductory chapter. Chapter 2 is essentially the same as in the original edition and provides a comprehensive summary of basic electromagnetic theory that is needed as background for proper understanding of the rest of the text. Many students will already have knowledge of this material before they pursue a course in microwave engineering. For these students, Chapter 2 will serve as a concise reference or review of familiar material. Chapter 3 is very different from that in the first edition. The first part of this chapter provides a more basic introduction to waves on transmission xv XVI PREFACE lines using distributed circuit models. The propagation of pulse signals is also covered. The second part of this chapter is a long section covering the characteristics of planar transmission lines, such as microstrip lines, coupled microstrip lines, strip lines, and coplanar lines or waveguides. The treatment is considerably broader than what is available in any other current text. Most of the formulas for the quasi-TEM mode parameters are derived using conformal mapping methods in a new Appendix III and are not just quoted from the literature. Several new formulas for attenuation have been derived as well as suitable modifications of existing formulas to account for anisotropic substrates. The last part of the chapter covers the basic properties of rectangular and circular waveguides, as in the original edition. Chapter 4 develops the basic microwave circuit theory and includes detailed discussions of the impedance, admittance and scattering matrix descriptions of microwave junctions. New material has been added on signal flow graphs and the generalized scattering matrix for power waves. The material on small aperture coupling has been updated to include radiation reaction that will account for power transmission through an aperture and thereby lead to physically meaningful equivalent circuits for small apertures. Chapter 5 treats a number of topics related to impedance matching and transformations. The old topic of impedance matching with lumped reactive elements has been revived because this is now frequently used in microwave integrated circuits. The design of complex load terminations has also been included because this is required for microwave solid state amplifier design. The available power at any point in a lossless reciprocal network is an invariant quantity. This concept is explained in terms of the impedance mismatch factor. The invariance of the impedance mismatch factor places an important constraint on the design of interstage matching networks in a microwave amplifier and is used in Chapter 10 in the design of microwave amplifiers. The last part of Chapter 5 discusses multisection quarter-wave transformers and tapered transmission lines. A new example of a microstrip half-wave filter design based on the quarter-wave transformer as a prototype circuit has been included. A variety of passive components are described along with detailed analysis in Chapter 6. In addition to those components described in the original edition, new material has been added on coupled-microstrip-line directional couplers, the branch-line coupler, hybrid junctions, and the Wilkinson power divider. New material on electronic controlled attenuators and phase shifters has also been added. Chapter 7 on resonators has been expanded to include new material on microstrip resonators and dielectric resonators. The old material on Fabry-Perot resonators has been deleted in order to make room for a short section on cavity perturbation theory. Chapter 8 on periodic structures and filters now includes a detailed treatment of gap-coupled and edge-coupled microstrip filters. The treatment PREFACE XVII of admittance and impedance inverters was rewritten in order to more fully explain the use of inverters in filter design. Apart from a brief discussion of gyratron tubes, Chapter 9 on microwave tubes remains essentially the same as in the first edition. The old Chapter 10 on masers has been replaced by a new chapter on microwave solid state amplifier design. This chapter gives a complete discussion of the scattering matrix approach to small signal narrow band amplifier design. The treatment is self-contained and all important relations for gain, stabihty, and low noise design are derived. A design strategy for low noise single stage and double stage amplifiers is developed along with considerations for the necessary tradeoffs that must be made between input and output VSWR, gain, low noise figure, and stability. The original Chapter 11 on parametric amplifiers has been retained without any change. A new Chapter 12 on oscillators and mixers has been added. This chapter is of limited scope because of the need to keep the overall length of the text within reasonable bounds. Solid state oscillators using Gunn devices and IMPATT diodes are described in a qualitative way only. An introduction to transistor oscillator design based on small signal scattering matrix parameters is provided. Included in this discussion is the relationship between the two-port and three-port scattering matrix description of a transistor because this is needed in order to efficiently analyze the effect of an impedance inserted in series with one of the transistor leads for feedback purposes, Many textbooks provide introductory treatments of diode mixers without any consideration of the embedding network. Such treatments do not provide a good understanding of diode mixers because it is the impedance properties of the embedding network that determine the diode voltages at the various harmonic frequencies. The introductory treatment of diode mixers in Chapter 12 does include the embedding network and this should provide the student with a more complete understanding of mixer analysis and design. The last part of the chapter describes the harmonic balancing method for the analysis of mixers. I have tried to provide a broad, comprehensive, and self-contained treatment of the fundamental theory and principles, and the methods of analysis and design that are the foundations for microwave engineering. There are, of course, limitations because all books must have a finite length. Many references have been included for the benefit of the reader who wishes to pursue a given topic in greater depth or refer to the original papers that a lot of the material has been based on. This text, in many respects, is a compilation of the work of a great many people. Unfortunately, it has not been possible to always give proper credit to those who were the originators of new concepts and the inventors of new devices. It is my belief that the revised edition will prove to be useful for both senior elective as well as beginning graduate level courses in microwave engineering, and will also serve as a useful reference source on fundamental 2 FOUNDATIONS FOR MICROWAVE ENGINEERING TABLE 1.1 Frequency band designation Frequency band 3-30 kHz 30-300 kHz 300-3.000 kHz Designation Typical service Very low frequency (VLF) Low frequency (LF) Medium frequency (MF) Navigation, sonar 3-30 MHz High frequency (HF) 30-300 MHz Very" high frequency (VHF) 300-3,000 MHz Ultrahigh frequency (UHF> 3-30 GHz Superhigh frequency (SHF) 30-300 GHz Extreme high frequency (EHF) TABLE 1.2 Microwave frequency b a n d designation Microwave band designation Frequency Old New 500-1.000 MHz 1-2 GHz 2 - 3 GHz 3 - 4 GHz 4 - 6 GHz 6 - 8 GHz 8-10 GHz 10-12.4 GHz 12.4-18 GHz 18-20 GHz 20-26.5 GHz 26.5-40 GHz VHF L S C D E F G H I J J J s G c xX Ku K K Ka 8 K Radio beacons, navigational aids AM broadcasting, maritime radio. Coast Guard communication. direction finding Telephone, telegraph, and facsimile; shortwave international broadcasting; amateur radio; citizen's band; ship-to-coast and shipto-aircraft communication Television. FM broadcast. air-traffic control, police. taxicab mobile radio, navigational aids Television, satellite communication. radiosonde, surveillance radar, navigational aids Airborne radar, microwave links, common-carrier land mobile communication, satellite communication Radar, experimental xviii PREFACE principles for the practicing microwave engineer. There is clearly much more material in the revised edition than can be covered in a one semester course. The last four chapters alone would provide sufficient material for a one semester course on active microwave circuits. As an instructor I have always believed thai it was very important to fully understand where formulas came from and how they are derived in order to present the material to students in a meaningful way. It is for this reason that I have attempted to make the text self-contained. In presenting many of the topics to undergraduate students 1 will only outline the basic approach used and will omit the details. It is my hope that other instructors will also view the detailed derivations that are provided in the text as a useful source of information in preparing a microwave engineering course and not as material that must always be presented in class. A number of topics that can be omitted in an undergraduate course are identified by a star. The problems based on these sections are also identified by a star. In recent years the microwave engineering course that I have taught to seniors at Case Western Reserve University has drawn heavily on the material in Chapters 3 through 5, which is very basic core material. In addition, topics have been selected from Chapters 6 and 7 on components and resonators in order to illustrate the application of basic microwave circuit theory. The last quarter of the semester has been generally devoted to microwave solid state amplifier design along with a brief coverage of oscillators and mixers. A better selection of problems and a solutions manual has been prepared for the revised edition. Over the past several years I have also prepared a number of short stand alone computer programs that provide useful tools to remove the drudgery of solving many of the homework problems. These programs are included on a floppy disk along with user instructions as part of the solutions manual. The programs cover the calculation of the characteristics of various planar transmission lines, including attenuation; the cutoff frequency, propagation constant, and attenuation of the dominant mode in rectangular and circular waveguides; impedance transformation along a transmission line; input and output impedances, admittances, and reflection coefficients for a linear two-port, which can be described in terms of impedance, admittance, or scattering matrix parameters; double-stub and lumped element impedance matching with frequency scans; two-port and three-port scattering matrix parameters for a transistor; and a rather long program that implements a design strategy for low noise one- and two-stage microwave amplifiers with various imposed constraints. Students have generally found these programs to be of significant help in problem solving. They have enjoyed working with the microwave amplifier design program. Without a computer program, the design of a microwave amplifier using potentially unstable devices and subject to various constraints on gain, noise figure, and input and output VSWR, is not feasible for students to carry out. The scope of each program PREFACE XiX has been purposefully limited in order to ensure that the student will be fully aware of the solution strategy involved. Many users of the first edition have provided me with helpful comments on the original material. In addition, I have received many helpful comments and suggestions from the following reviewers of the materia! for the revised edition. They are Chin-Lin Chen, Purdue University; M. Yousif El-Ibiary, University of Oklahoma; Irving Kaufman, Arizona State University; Stuart Long, University of Houston; Glenn S. Smith, Georgia Institute of Technology; and Robert J. Weber. Iowa State University. For the most part then - suggestions and recommendations have been incorporated. The new material for the revised edition was typed by Sue Sava. I would like to acknowledge the professional skill with which she prepared this material as well as her willingness to rearrange her schedule so as to meet various deadlines. The last acknowledgment is to my wife Kathleen, who was willing to give up many other activities so that the revision could be carried out. Her encouragement and support of the project never faltered, and without it the revision could not have been undertaken. Robert E. Collin INTRODUCTION 3 nenta. In the frequency range from 1 up to 100 GHz. lumped circuit elements are usually replaced by transmission-line and waveguide components. Thus by the term microwaue engineering we shall mean generally the engineering and design of information-handling systems in the frequency range from 1 to 100 GHz corresponding to wavelengths as long as 30 cm and as short as 3 mm. At shorter wavelengths we have what can be called optical engineering since many of the techniques used are derived from classical optical techniques. The characteristic feature of microwave engineering is the short wavelengths involved, these being of the same order of magnitude as the circuit elements and devices employed. The short wavelengths involved in turn mean that the propagation time for electrical effects from one point in a circuit to another point is comparable with the period of the oscillating currents and charges in the system. As a result, conventional low-frequency circuit analysis based on Kirchhoffs laws and voltage-current concepts no longer suffices for an adequate description of the electrical phenomena taking place. It is necessary instead to c a n y out the analysis in terms of a description of the electric and magnetic fields associated with the device. In essence, it might be said, microwave engineering is applied electromagnetic fields engineering. For this reason the successful engineer in this area must have a good working knowledge of electromagnetic field theory. There is no distinct frequency boundary at which lumped-parameter circuit elements must be replaced by distributed circuit elements. With modern technological processes it is possible to construct printed-circuit inductors that are so small that they retain their lumped-parameter characteristics at frequencies as high as 10 GHz or even higher. Likewise, optical components, such as parabolic reflectors and lenses, are used to focus microwaves with wavelengths as long as 1 m or more. Consequently, the microwave engineer will frequently employ low-frequency lumped-parameter circuit elements, such as miniaturized inductors and capacitors, as well as optical devices in the design of a microwave system. MICROWAVE APPLICATIONS The great interest in microwave frequencies arises for a variety of reasons. Basic among these is the ever-increasing need for more radio-frequencyspectrum space and the rather unique uses to which microwave frequencies can be applied. When it is noted that the frequency range 10 9 to 10 12 Hz contains a thousand sections like the frequency spectrum from 0 to 10 9 Hz, the value of developing the microwave band as a means of increasing the available usable frequency spectrum may be readily appreciated. At one time (during World War II and shortly afterward), microwave engineering was almost synonymous with radar (flAdio Detection And Ranging) engineering because of the great stimulus given to the development of microwave systems by the need for high-resolution radar capable of 4 FOUNDATIONS FOR MICROWAVE ENGINEERING detecting and locating enemy planes and ships. Even today radar, in its many varied forms, such as missile-tracking radar, fire-control radar, weather-detecting radar, missile-guidance radar, airport traffic-control radar, etc., represents a major use of microwave frequencies. This use arises predominantly from the need to have antennas that will radiate essentially all the transmitter power into a narrow pencil-like beam similar to that produced by an optical searchlight. The ability of an antenna to concentrate radiation into a narrow beam is limited by diffraction effects, which in turn are governed by the relative size of the radiating aperture in terms of wavelengths. For example, a parabolic reflector-type antenna produces a pencil beam of radiated energy having an angular beam width of 140°/(Z)/A 0 ), where D is the diameter of the parabola and A 0 is the wavelength. A 90-cm (about 3 ft) parabola can thus produce a 4.7° beam at a frequency of 10'" Hz, i.e., at a wavelength of 3 cm. A beam of this type can give reasonably accurate position data for a target being observed by the radar. To achieve comparable performance at a frequency of 100 MHz would require a 300-ft parabola, a size much too large to be carried aboard an airplane. In more recent years microwave frequencies have also come into widespread use in communication links, generally referred to as microwave links. Since the propagation of microwaves is effectively along line-of-sight paths, these links employ high towers with reflector or lens-type antennas as repeater stations spaced along the communication path. Such links are a familiar sight to the motorist traveling across the country because of their frequent use by highway authorities, utility companies, and television networks. A further interesting means of communication by microwaves is the use of satellites as microwave relay stations. The first of these, the Telstar, launched in July 1962, provided the first transmission of live television programs from the United States to Europe. Since that time a large number of satellites have been deployed for communication purposes, as well as for surveillance and collecting data on atmospheric and weather conditions. For direct television broadcasting the most heavily used band is the C band. The up-link frequency used is in the 5.9- to 6.4-GHz band and the receive or down-link frequency band is between 3.7 and 4.2 GHz. For home reception an 8-ft-diameter parabolic reflector antenna is commonly used. A second frequency band has also been allocated for direct television broadcasting. For this second band the up-link frequency is in the 14- to 14.5-GHz range and the down-link frequencies are between 10.95 and 11.2 GHz and 11.45 and 11.7 GHz. In this band a receiving parabolic antenna with a 3-ft diameter is adequate. At the present time this frequency band is not being used to any great extent in the United States. It is more widely used in Europe and Japan. Terrestrial microwave links have been used for many years. The TD-2 system was put into service in 1948 as part of the Bell Network. It operated in the 3.7- to 4.2-GHZ band and had 480 voice circuits, each occupying a INTRODUCTION 5 3.1-kHz bandwidth. In 1974, the TN-1 system operating in the 10.7- to 11.7-GHz band was put into operation. This system had a capacity of 1,800 voice circuits or one video channel with a 4.5-MHz bandwidth. Since that time the use of terrestrial microwave links has grown rapidly. At the present time most communication systems are shifting to the use of digital transmission, i.e., analog signals are digitized before transmission. Microwave digital communication system development is progressing rapidly. In the early systems simple modulation schemes were used and resulted in inefficient use of the available frequency spectrum. The development of 64-state quadrature amplitude modulation (64-QAM) has made it possible to transmit 2,016 voice channels within a single 30-MHz RF channel. This is competitive with FM analog modulation schemes for voice. The next step up is the 256-QAM system which is under development. For the ready processing and handling of a modulated carrier, modulation sidebands can be only a few percent of the carrier frequency. It is thus seen that the carrier frequency must be in the microwave range for efficient transmission of many television programs over one link. Without the development of microwave systems, our communications facilities would have been severely overloaded and totally inadequate for present operations. Even though such uses of microwaves are of great importance, the applications of microwaves and microwave technology extend much further, into a variety of areas of basic and applied research, and including a number of diverse practical devices, such as microwave ovens that can cook a small roast in just a few minutes. Some of these specific applications are briefly discussed below. Waveguides periodically loaded with shunt susceptance elements support slow waves having velocities much less than the velocity of light, and are used in linear accelerators. These produce high-energy beams of charged particles for use in atomic and nuclear research. The slow-traveling electromagnetic waves interact very efficiently with charged-particle beams having the same velocity, and thereby give up energy to the beam. Another possibility is for the energy in an electron beam to be given up to the electromagnetic wave, with resultant amplification. This latter device is the traveling-wave tube, and is examined in detail in a later chapter. Sensitive microwave receivers are used in radio astronomy to detect and study t h e electromagnetic radiation from the sun and a number of radio stars that emit radiation in this band. Such receivers are also used to detect the noise radiated from plasmas (an approximately neutral collection of electrons and ions, e.g., a gas discharge). The information obtained enables scientists to analyze and predict the various mechanisms responsible for plasma radiation. Microwave radiometers are also used to map atmospheric temperature profiles, moisture conditions in soils and crops, and for other remote-sensing applications as well. Molecular, atomic, and nuclear systems exhibit various resonance phenomena under the action of periodic forces arising from an applied 6 FOUNDATIONS FOR MICROWAVE ENGINEERING electromagnetic field. Many of these resonances occur in the microwave range; hence microwaves have provided a very powerful experimental probe for the study of basic properties of materials. Out of this research on materials have come many useful devices, such as some of the nonreciprocal devices employing ferrites, several solid-state microwave amplifiers and oscillators, e.g., masers, and even the coherent-light generator and amplifier (laser). The development of the laser, a generator of essentially monochromatic (single-frequency) coherent-light waves, has stimulated a great interest in the possibilities of developing communication systems at optical wavelengths. This frequency band is sometimes referred to as the ultramicrowave band. With some modification, a good deal of the present microwave technology can be exploited in the development of optical systems. For this reason, familiarity with conventional microwave theory and devices provides a good background for work in the new frontiers of the electromagnetic spectrum. The domestic microwave oven operates at 2,450 MHz and uses a magnetron tube with a power output of 500 to 1000 W. For industrial heating applications, such as drying grain, manufacturing wood and paper products, and material curing, the frequencies of 915 and 2,450 MHz have been assigned. Microwave radiation has also found some application for medical hyperthermia or localized heating of tumors. It is not possible here to give a complete account of all the applications of microwaves that are being made. The brief look at some of these, as given above, should convince the reader that this portion of the radio spectrum offers many unusual and unique features. Although the microwave engineering field may now be considered a mature and well-developed one, the opportunities for further development of devices, techniques, and applications to communications, industry, and basic research are still excellent. 1.3 MICROWAVE CIRCUIT ELEMENTS AND ANALYSIS At frequencies where the wavelength is several orders of magnitude larger than the greatest dimensions of the circuit or system being examined, conventional circuit elements such as capacitors, inductors, resistors, electron tubes, and transistors are the basic building blocks for the information transmitting, receiving, and processing circuits used. The description or analysis of such circuits may be adequately carried out in terms of loop currents and node voltages without consideration of propagation effects. The time delay between cause and effect at different points in these circuits is so small compared with the period of the applied signal as to be negligible. It might be noted here that an electromagnetic wave propagates a distance of one wavelength in a time interval equal to one period of a sinusoidally INTRODUCTION 7 time-varying applied signal. As a consequence, when the distances involved are short compared with a wavelength A0 (A0 = velocity of light/frequency), the time delay is not significant. As the frequency is raised to a point where the wavelength is no longer large compared with the circuit dimensions, propagation effects can no longer be ignored. A further effect is the great relative increase in the impedance of connecting leads, terminals, etc., and the effect of distributed (stray) capacitance and inductance. In addition, currents circulating in unshielded circuits comparable in size with a wavelength are very effective in radiating electromagnetic waves. The net effect of all this is to make most conventional low-frequency circuit elements and circuits hopelessly inadequate at microwave frequencies. If a rather general viewpoint is adopted, one may classify resistors, inductors, and capacitors as elements that dissipate electric energy, store magnetic energy, and store electric energy, respectively. The fact that such elements have the form encountered in practice, e.g., a coil of wire for an inductor, is incidental to the function they perform. The construction used in practical elements may be considered just a convenient way to build these devices so that they will exhibit the desired electrical properties. As is well known, many of these circuit elements do not behave in the desired manner at high frequencies. For example, a coil of wire may be an excellent inductor at 1 MHz, but at 50 MHz it may be an equally good capacitor because of the predominating effect of interturn capacitance. Even though practical lowfrequency resistors, inductors, and capacitors do not function in the desired manner at microwave frequencies, this does not mean that such energy-dissipating and storage elements cannot be constructed at microwave frequencies. On the contrary, there are many equivalent inductive and capacitive devices for use at microwave frequencies. Their geometrical form is quite different, but they can be and are used for much the same purposes, such as impedance matching, resonant circuits, etc. Perhaps the most significant electrical difference is the generally much more involved frequency dependence of these equivalent inductors and capacitors at microwave frequencies. Low-frequency electron tubes are also limited to a maximum useful frequency range bordering on the lower edge of the microwave band. The limitation arises mainly from the finite transit time of the electron beam from the cathode to the control grid. When this transit time becomes comparable with the period of the signal being amplified, the tube ceases to perform in the desired manner. Decreasing the electrode spacing permits these tubes to be used up to frequencies of a few thousand megahertz, but the power output is limited and the noise characteristics are poor. The development of new types of tubes for generation of microwave frequencies was essential to the exploitation of this frequency band. Fortunately, several new principles of operation, such as velocity modulation of the electron beam and beam interaction with slow electromagnetic waves, were discovered that enabled the necessary generation of microwaves to be carried out. 8 FOUNDATIONS FOR MICROWAVE ENGINEERING (a) id) (c) F I G U R E 1.1 Some common transmission lines, (a) Two-conductor line; (b) coaxial line; (c) shielded strip line. These fundamental principles with applications are discussed in a later chapter. For low-power applications microwave tubes have been largely replaced by solid-state devices, such as transistors and negative resistance diodes. However, for high-power applications microwave tubes are still necessary. One of the essential requirements in a microwave circuit is the ability to transfer signal power from one point to another without radiation loss. This requires the transport of electomagnetic energy in the form of a propagating wave. A variety of such structures have been developed that can guide electromagnetic waves from one point to another without radiation loss. The simplest guiding structure, from an analysis point of view, is the transmission line. Several of these, such as the open two-conductor line, coaxial line, and shielded strip line, illustrated in Fig. 1.1, are in common use at the lower microwave frequencies. At the higher microwave frequencies, notably at wavelengths below 10 cm, hollow-pipe waveguides, as illustrated in Fig. 1.2, are often preferred to transmission lines because of better electrical and mechanical properties. The waveguide with rectangular cross section is by far the most common type. The circular guide is not nearly as widely used. Ul (*) (<r) F I G U R E 1.2 Some common hollow-pipe waveguides, ( a ) Rectangular guide; (6) circular guide; (c) ridge guide. INTRODUCTION 9 The ridge-loaded rectangular guide illustrated in Fig. 1.2c is sometimes used in place of the standard rectangular guide because of better impedance properties and a greater bandwidth of operation. In addition to these standard-type guides, a variety of other cross sections, e.g., elliptical, may also be used. Another class of waveguides, of more recent origin, is surface waveguides. An example of this type is a conducting wire coated with a thin layer of dielectric. The wire diameter is small compared with the wavelength. Along a structure of this type it is possible to guide an electromagnetic wave. The wave is bound to the surface of the guide, exhibiting an amplitude decay that is exponential in the radial direction away from the surface, and hence is called a surface wave. Applications are mainly in the millimeter-wavelength range since the field does extend a distance of a wavelength or so beyond the wire, and this makes the effective guide diameter somewhat large in the centimeter-wavelength range. A disadvantage of surface waveguides and open-conductor transmission lines is that radiation loss occurs whenever other obstacles are brought into the vicinity of the guide. The development of solid-state active devices, such as bipolar transistors and, more notably, field-effect transistors (FET), has had a dramatic impact on the microwave engineering field. With the availability of microwave transistors, the focus on waveguides and waveguide components changed to a focus on planar transmission-line structures, such as microstrip lines and coplanar transmission lines. These structures, shown in Fig. 1.3, can be manufactured using printed-circuit techniques. They are compatible with solid-state devices in that it is easy to connect a transistor to a microstrip circuit but difficult to incorporate it as part of a waveguide circuit. By using gallium-arsenide material it has been possible to design field-effect transistors that provide low noise and useful amplification at millimeter wavelengths. At the lower microwave frequencies hybrid integrated microwave circuits are used. In hybrid circuit construction the transmission lines and transmission-line components, such as matching elements, are manufactured first and then the solid-state devices, such as diodes and transistors, are soldered into place. The current trend is toward the use of monolithic microwave integrated circuits (MMIC) in which both the transmission-line circuits and active devices are fabricated on a single chip. A variety of broadband MMIC amplifiers have been designed. The development of MMIC circuits for operation at frequencies up to 100 GHz is well under way. A unique property of the transmission line is that a satisfactory analysis of its properties may be carried out by treating it as a network with distributed parameters and solving for the voltage and current waves that may propagate along the line. Other waveguides, although they have several properties similar to transmission lines, must be treated as electromagnetic boundary-value problems, and a solution for the electromagnetic fields must be determined. Fortunately, this is readily accomplished for the common 10 FOUNDATIONS FOR MICROWAVE ENGINEERING Ground plane (a) (6) F I G U R E 1.3 (a) microstrip transmission line; (b) coplanar transmission line. waveguides used in practice. For waveguides it is not possible to define unique voltage and current that have the same significance as for a transmission line. This is one of the reasons why the field point of view is emphasized at microwave frequencies. Associated with waveguides are a number of interesting problems related to methods of exciting fields in guides and methods of coupling energy out. Three basic coupling methods are used: (1) probe coupling, (2) loop coupling, and (3) aperture coupling between adjacent guides. They are illustrated in Fig. 1.4, and some of them are analyzed later. These coupling (a) (/>) (c) F I G U R E 1.4 Basic methods of coupling energy into and out of waveguides, ( a ) Probe coupling; (6) loop coupling; (c) aperture coupling. rNTRODUCTION 11 FIGURE 1.5 Waveguide-to-coaxial-line transitions that use probe coupling as shown in Fig. 1.4a. (Photograph courtesy of Ray Moskaluk, Hewlett. Packard Company.) devices are actually small antennas that radiate into the waveguide. A photograph of a waveguide-to-coaxial -line transition is shown in Fig. 1.5. Inductive and capacitive elements take a variety of forms at microwave frequencies. Perhaps the simplest are short-circuited sections of transmission line and waveguide. These exhibit a range of susceptance values from minus to plus infinity, depending on the length of the line, and hence may act as either inductive or capactive elements. They may be connected as either series or shunt elements, as illustrated in Fig. 1.6. They are commonly referred to as stubs and are widely used as impedance-matching elements. In a rectangular guide thin conducting windows, or diaphragms, as illustrated in Fig. 1.7, also act as shunt susceptive elements. Their (a) (b) (c) FIGURE 1.6 Stub-type reactive elements, ( a ) Series element; (6) shunt element; (c) waveguide stub. 12 FOUNDATIONS FOR MICROWAVE ENGINEERING Shunt susceptive elements in a waveguide. (a) Inductive window; (6) capacitive window. F I G U R E 1.8 Cylindrical cavity aperture coupled to rectangular waveguide. inductive or capacitive nature depends on whether there is more magnetic energy or electric energy stored in local fringing fields. Resonant circuits are used bath at low frequencies and at microwave frequencies to control the frequency of an oscillator and for frequency filtering. At low frequencies this function is performed by an inductor and capacitor in a series or parallel combination. Resonance occurs when there are equal average amounts of electric and magnetic energy stored. This energy oscillates back and forth between the magnetic field around the inductor and the electric field between the capacitor plates. At microwave frequencies the LC circuit may be replaced by a closed conducting enclosure, or cavity. The electric and magnetic energy is stored in the field within the cavity. At an infinite number of specific frequencies, the resonant frequencies, there are equal average amounts of electric and magnetic energy stored in the cavity volume. In the vicinity of any one resonant frequency, the input impedance to the cavity has the same properties as for a conventional LC resonant circuit. One significant feature worth noting is the very much larger Q values t h a t may be obtained, these being often in excess of 10 4 , as compared with those obtainable from low-frequency LC circuits. Figure 1.8 illustrates a cylindrical cavity that is aperture coupled to a rectangular waveguide. Figure 1.9 is a photograph of a family of waveguide low-pass filters. The theory and design of microwave filters is given in Chap. 8. A photograph of a family of waveguide directional couplers is shown in Fig. 1.10. The design of directional couplers is covered in Chap. 6. The photograph in Fig. 1.11 shows a family of coaxial-line GaAs diode detectors. When a number of microwave devices are connected by means of sections of transmission lines or waveguides, we obtain a microwave circuit. The analysis of the behavior of such circuits is carried out either in terms of equivalent transmission-line voltage and current waves or in terms of the amplitudes of the propagating waves. The first approach leads to an equivalent-impedance description, and the second emphasizes the wave nature of the fields and results in a scattering-matrix formulation. Both approaches are used in this book. Since transmission-line circuit analysis forms the basis, either directly or by analogy, for the analysis of all microwave circuits, F I G U R E 1.9 A family of waveguide low-pass filters for various microwave frequency bands. (Photographs courtesy of Ray Moskaluk, Hewlett Packard Company.) F I G U R E 1.10 A family of waveguide directional couplers for various microwave frequency bands. (Photographs courtesy of Ray Moskaluk, Hewlett Packard Company.) 13 14 FOUNDATIONS FOR MICROWAVE ENGINEERING F I G U R E 1.11 Coaxial-line GaAs diode detectors for various microwave frequency bands. (Photographs courtesy of Ray Moskaluk, Hewlett Packard Company.) a considerable amount of attention is devoted to a fairly complete treatment of this subject early in the text. This material, together with the field analysis of the waves that may propagate along waveguides and that may exist in cavities, represents a major portion of the theory with which the microwave engineer must be familiar. The microwave systems engineer must also have some understanding of the principles of operation of various microwave tubes, such as klystrons, magnetrons, and traveling-wave tubes, and of the newer solid-state devices, such as masers, parametric amplifiers, and microwave transistors. This is required in order to make intelligent selection and proper use of these devices. In the text sufficient work is done to provide for this minimum level of knowledge of the principles involved. A treatment that is fully adequate for the device designer is very much outside the scope of this book. Solid-state oscillators for use as local oscillators in receiver front ends have largely replaced the klystron. Solid-state oscillators for low-power transmitters are also finding widespread use. Thus the future for microwave engineering is clearly in the direction of integrated solid-state circuits and the development of the necessary passive components needed in these circuits, which are also compatible with the fabrication methods that are used. In the light of the foregoing discussion, it should now be apparent that the study of microwave engineering should include, among other things, at least the following: 1. 2. 3. 4. 5. 6. 7. 8. Electromagnetic theory Wave solutions for transmission lines and waveguides Transmission-line and waveguide circuit analysis Resonators and slow-wave structures Microwave oscillators and amplifiers Antennas Microwave propagation Systems considerations INTRODUCTION 15 F I G U R E 1.12 A microwave network analyzer used to measure scattering matrix parameters. (Photographs courtesy of Ray Moskaluk, Hewlett Packard Company.) Apart from the last three, these are the major topics covered in the text. It is not possible to discuss in any great detail more than a few of the many microwave devices available and in current use. Therefore only a selected number of them are analyzed, to provide illustrative examples for the basic theory being developed. The available technical literature may be, and should be, consulted for information on devices not included here. Appropriate references are given throughout the text. The number of topics treated in this text represents a good deal more than can be covered in a one-semester course. However, rather than limit t h e depth of treatment, it was decided to separate some of the more specialized analytical treatments of particular topics from the less analytical discussion. These specialized sections are marked with a star, and can be eliminated in a first reading without significantly interrupting the continuity of the text.t The student or engineer interested in the design of microwave devices, or in a fuller understanding of various aspects of microwave theory, is advised to read these special sections. As in any engineering field, measurements are of great importance in providing the link between theory and practice at microwave frequencies. I Problems based on material in these sections are also marked by a star. 16 FOUNDATIONS FOR MICROWAVE ENGINEERING Space does not permit inclusion of the subject of microwave measurements in this text. A number of excellent texts devoted entirely to microwave measurements are available, and the reader is referred to them. There are a variety of commercially available instruments that enable microwave measurements to be carried out automatically with computer control. The photograph in Fig. 1.12 shows a network analyzer equipped to measure the scattering-matrix parameters of a microwave device. The scattering-matrix parameters, as a function of frequency, can be displayed on a Smith chart. The scattering-matrix parameters are commonly used in place of the usual impedance and admittance parameters to characterize a microwave device and are described in Chap. 4. REFERENCES 1. Historical Perspectives of Microwave Technology, IEEE Trans., vol. MTT-32, September, 1984, Special Centennial Issue. 2. Kraus, J. D.. "Antennas," 2nd ed., McGraw-HiJJ Book Company. New Yoi-li, 1988. 3. Collin, R. E.: "Antennas and Radiowave Propagation," McGraw-Hill Book Company, New York, 1985. 4. Stutsman, W. L., and G. A. Thiele: "Antenna Theory and Design," John Wiley & Sons, Inc., New York, 1981. 5. Elliott, R. S.: "Antenna Theory and Design." Prentice-Hall, Inc., Englewood Cliffs, N.J., 1981. 6. Balanis, C. A: "Antenna Theory, Analysis, and Design," Harper & Row Publishers, Inc., New York, 1982. 7. Pratt, T., and C. W. Bostian: "Satellite Communications," John Wiley & Sons, New York, 1986. 8. Ivanek, F. (ed.): "Terrestrial Digital Microwave Communications," Artech House Books, Norwood, Mass., 1989. 9. Skolnik, M. I.: "Introduction to Radar Systems." McGraw-Hill Book Company, New York, 1962. 10. Montgomery, C. G.: "Technique of Microwave Measurements," McGraw-Hill Book Company, New York, 1947. 11. Ginzton, E. L.: "Microwave Measurements," McGraw-Hill Book Company, New York, 1957. 12. Bailey, A. E. (ed.): "Microwave Measurement." Peter Peregrinus. London, 1985. 13. Okress, E. C: "Microwave Power Engineering," Academic Press, New York, 1968. 14. Ulaby, F. T., R. K. Moore, and A. K. Fung: "Microwave Remote Sensing: Active and Passive. Microwave Remote Sensing, Fundamentals and Radiometry," vol. 1, AddisonWesley, Reading, Mass.. 1981. CHAPTER 2 ELECTROMAGNETIC THEORY MAXWELL'S EQUATIONS Electric and magnetic fields that vary with time are governed by physical laws described by a set of equations known collectively as Maxwell's equations- For the most part these equations were arrived at from experiments carried out by several investigators. It is not our purpose here to justify the basis for these equations, but rather to gain some understanding of their physical significance and to learn how to obtain solutions of these equations in practical situations of interest in the microwave engineering field. The electric field f and magnetic field SB are vector fields and in general have amplitudes and directions that vary with the three spatial coordinates x, y, z and the time coordinate tf\ In mks units, which are used throughout, the electric field is measured in volts per meter and the magnetic field in webers per square meter. Since these fields are vector fields, the equations governing their behavior are most conveniently written in vector form.t The electric field g and magnetic field & are regarded as fundamental in that they give the force on a charge q moving with velocity v; that is, F = 9(f + v X ^ ) (2.1) tBoIdface script type is used to represent vector fields having arbitrary time dependence. Boldface roman type is used later for the phasor representation of fields having sinusoidal time dependence. t i t is assumed that the reader is familiar with vector analysis. However, for convenient reference, a number of vector formulas and relations are summarized in App. I. 17 18 FOUNDATIONS FOR MICROWAVE ENGINEERING FIGURE 2.1 DIustration of Faraday's law. where F is the force in newtons, g is the charge measured in coulombs, and v is the velocity in meters per second. This force law is called the Lorentz force equation. In addition to the % and 3S fields, it is convenient to introduce two auxiliary field vectors, namely, the electric displacement 91 and the magnetic intensity %?. These are related to % and £8 through the electric and magnetic polarization of material media, a topic covered in the next section. In this section we consider fields in vacuum, or free space, only. In this case the following simple relationships hold: 1 %f =—SS (2.2a) Ma 3 = eor (2.26) where /x0 = 4TT X 10 ~ 7 H / m and is called the permeability of vacuum, and e 0 = 1 0 - 9 / 3 6 i r = 8.854 X 1 0 " 1 2 F / m and is known as the permittivity of vacuum. One of the basic laws of electromagnetic phenomena is Faraday's law, which states that a time-varying magnetic field generates an electric field. With reference to Fig. 2.1, let C denote an arbitrary closed curve that forms the boundary of a nonmoving surface S. The time rate of change of total magnetic flux through the surface S is d(js& • dS)/dt. According to Faraday's law, this time rate of change of total magnetic flux is equal to the negative value of the total voltage measured around C. The later quantity is given by -# C JT • d\. Hence the mathematical statement of Faraday's law is < £ r - d l = - — [&-dS T (2.3) c St Js The line integral of £ around C is a measure of the circulation, or "curling up," of the electric field in space. The time-varying magnetic field may be properly regarded as a vortex source that produces an electric field having nonzero curl, or circulation. Although (2.3) is in a form that is readily interpreted physically, it is not in a form suitable for the analysis of a physical problem. What is required is a differential equation that is equivalent to (2.3). This equation may be obtained by using Stokes' theorem from vector analysis, which states that the line integral of a vector around a closed contour C is equal to the integral of the normal component of the ELECTROMAGNETIC THEORY 19 curl of this vector over any surface having C as its boundary. The curl of a vector is written V X S" (App. I), and hence (2.1) becomes d i*-d\= f V x g • dS = - — [a-dS at Jfi J* Since S is completely arbitrary, the latter two integrals are equal only if 9M V X*= - — (2.4) which is the desired differential equation describing Faraday's law. The curl is a measure of the circulation of a vector field at a point. Helmholtz's theorem from vector analysis states that a vector field is completely denned only when the curl, or circulation, of the field, and also its divergence, are given at every point in space. Now the divergence (or convergence) of field lines arises only if a proper source (or sink) is available. The electric field, in addition to having a curl produced by the vortex source -BSS/dt, has a divergence produced by electric charge. Gauss' law states that the total flux of 9i = e 0 f from a volume V is equal to the net charge contained within V. If p represents the charge density in coulombs per cubic meter, Gauss' law may be written as 6e0&-dS= f PdV (2.5) r J S v This equation may be converted to a differential equation by using the divergence theorem to give 6 e0g • d S = f V • s0& dV= { pdV JS Jy Jy Since V is arbitrary, it follows that V • €(,£"= V -as =p (2.6) where V -31 is the divergence of 9>, that is, a measure of the total outward flux of 9) from a volume element, divided by the volume of the element, as this volume shrinks to zero. Since both the curl and divergence of the electric field are now specified, this field is completely determined in terms of the two sources, HSff/dt and p. To complete the formulation of electromagnetic phenomena, we must now relate the curl and divergence of the magnetic field to their sources. The vortex source that creates the circulation, or curl, of the magnetic field -** is the current. By current is meant the total current density, the conduction current density f measured in amperes per square meter, the displacement current density d^/Ht, and the convection current pv consisting of charge in motion if present. Convection current is not included in this chapter. However, in the chapter dealing with microwave tubes, convection current plays a central role and is discussed in detail there. The displacement current density flS/dt was first introduced by Maxwell, and leads to 20 FOUNDATIONS FOR MICROWAVE ENGINEERING the possibility of wave motion, as will be seen. Mathematically, the circulation of 3f around a closed contour C bounding a surface S as in Fig. 2.1 is given by r c <>a> r &>%• • d\ =J / — • dS +J I/-clS ~c S M s Application of Stokes' law to the left-hand side yields I Js V xjr-dS= /J — s ot • dS+ \jr V (2 7} ' dS from which it may be concluded that (131 V X / = - + / (2.8) Since magnetic charge, as the dual of electric charge, does not exist in nature, it may be concluded that the divergence of 38 is always zero; i.e., the flux lines of 38 are always closed since there are no charges for them to terminate on. Thus the net flux of 38 through any closed surface S is always zero; i.e., just as much flux enters through the surface as leaves it. Corresponding to (2.5) and (2.6), we thus have i*m dS = 0 V-& = 0 (2.9) (2.10) Conduction current, of density f, is the net flow of electric charge. Since charge is conserved, the total rate of flow of charge out of a volume V is equal to the time rate of decrease of total charge within V, as expressed fay the equation <(>S- dS= - - / PdV (2.11) This is the continuity equation, and it may be converted to a differential equation by using the divergence theorem in the same manner as was done to derive (2.6) from (2.5). It is readily found that V -f + ^ = 0 (2.12) ot This equation may also be derived from (2.8) and (2.6). Since the divergence Of the curl of any vector is identically zero, the divergence of (2.8) yields <9V -3J ° = ^ - + V^ Using (2.6) converts this immediately into the continuity equation (2.12). If the displacement current density d3/tit had not been included as part of the total current density on the right-hand side of (2.8), that equation would i ELECTROMAGNETIC THEORY 21 have led to the conclusion that V •/ = 0, a result inconsistent with the continuity equation unless the charge density was independent of time. In summary, the four equations, known as Maxwell's equations, that describe electromagnetic phenomena in vacuum are dSS V x f = - — (2.13a) at ~>9> V X ^ = - + / (2.136) V-&=p (2.13c) V-^= 0 (2.13d) where in (2.136) the convection current pv has not been included. The continuity equation may be derived from (2.136) and (2.13c), and hence contains no additional information. Although -d£J8/dt may be regarded as a source for i>, and H3i/dt as a source of %", the ultimate sources of an electromagnetic field are the current f and charge p. For time-varying fields, that charge density p which varies with time is not independent of f since it is related to the latter by the continuity equation. As a consequence, it is possible to derive the time-varying electromagnetic field from a knowledge of the current density / alone. It is not difficult to show in a qualitative way that (2.13a) and (2.136) lead to wave propagation, i.e., to the propagation of an electromagnetic disturbance through space. Consider a loop of wire in which a current varying with time flows as in Fig. 2.2. The conduction current causes a circulation, or curling, of the magnetic field around the current loop as in Fig. 2.2a (for clarity only a few flux lines are shown). The changing magnetic field in t u r n creates a circulating, or curling, electric field, with field lines that encircle the magnetic field lines as in Fig. 2.26. This changing electric field creates further curling magnetic field lines as in Fig. 2.2c, and so forth. The net result is the continual growth and spreading of the electromagnetic field into all space surrounding the current loop. The (al (b) T> drfi (rrh n x <*• lfl — -^y g 3C F I G U R E 2.2 The growth or generation of an electromagnetic wave from a current loop. 22 FOUNDATIONS FOR MICROWAVE ENGINEERING disturbance moves outward with the velocity of light. A little thought will show that the same characteristic mutual effect between two quantities must always exist for wave motion. That is, quantity A must be generated by quantity B, and vice versa. For example, in an acoustical wave the excess pressure creates a motion of the adjacent air mass. The motion of the air mass by virtue of its inertia in turn creates a condensation, or excess pressure, farther along. The repetition of this process generates the acoustical wave. For the most part, as at lower frequencies, it is sufficient to consider only the steady-state solution for the electromagnetic field as produced by currents having sinusoidal time dependence. The time derivative may then be eliminated by denoting the time dependence of all quantities as eJal and representing all field vectors as complex-phasor space vectors independent of time. Boldface roman type is used to represent these complex-phasor space vectors. For example, the mathematical representation for the electric field ^(x,y, z, t) will be E{x,y,z)eJ"''. Each component of E is in general complex, with a real and imaginary part; thus E = a r ( £ , r + jExl) + a v ( £ v r +jEyt) + a , ( E „ +jEzi) (2.14) where the subscript r refers to the real part and the subscript / refers to the imaginary part. Each component is allowed to be complex in order to provide for an arbitrary time phase for each component. This may be seen by recalling the usual method of obtaining g" from its phasor representation. That is, by definition, Z(x,y,z,t) = R e [ E ( x , y , 2 ) e ^ ' ] Thus Ex = (2.15) Re[(Exr+jExl)e->«'] = V E xr + Wi COS(iOt + <J>) where <b = t a n " H E x i / E x r ) . Unless E x had both an imaginary part jExi a real part Exr, the arbitrary phase angle <t> would not be present. As general rule, the time factor eJmt will not be written down when the phase representation is used. However, it is important to remember both the that such a time dependence is implied and also the rule (2.15) for obtair the physical field vector from its phasor representation. The real imaginary parts of the space components of a vector should not be confus with the space components; for example, Exr and Exi are not two components of E x since the component a x E x is always directed along the axis in space, with the real and imaginary parts simply accounting for arbitrary time phase or origin. A further point of interest in connection with the phasor represent tion is the method used for obtaining the time-average value of a fie quantity. ELECTROMAGNETIC THEORY 23 For example, if r= a J .E 1 cos(«j« + 6-i) + a.yE^cos(o}t + <£2) + a z E 3 cos(w< + 0 3 ) the time-average value of \<g I is 1 fT — I % -gdt 1 r = — f [Efcos2(cot + <t>x) + Ef cos 2 («* + <f>2) +E$cos(wt 4- <f>3)] dt = ±(E? + Ei + EZ) (2.16) where T is the period, equal to 2ir/w. The same result is obtained by simply taking one-half of the scalar, or dot, product of E with the complex conjugate E*; thus ! * £ = |E • E* = | [ ( J £ , + E* ) + ( 4 , . + 2 $ ) + (C + * S ) ] (2-17) since E , E * = ( E , r +./E_ r ,XE Ir - . / « * ) = E* r + g £ , etc. This is equal to (2.16), since £? = E%. + E*„ etc. By using the phasor representation, the time derivative d/dt may be replaced by the factor jm since deJ"'/dt =jweJ"1'. Hence Maxwell's equations, with steady-state sinusoidal time dependence, become V X E - -jcoB (2.18a) VxH=Ja,D + J (2.186) V-D = p V-B = 0 (2.18c) • (2.l8aT) CONSTITUTIVE RELATIONS In material media the auxiliary field vectors & and 91 are defined in terms of the polarization of the material and the fundamental field quantities 38 and g. The relationships of & to 38 and of & to f are known as constitutive relations, and must be known before solutions for Maxwell's equations can be found. Consider first the electric case. If an electric field & is applied to a material body, this force results in a distortion of the atoms or molecules in such a manner as to create effective electric dipoles with a dipole moment 3* per unit volume. The total displacement current is the sum of the vacuum displacement current de0g/dt and the polarization current d3"/dt. To avoid accounting for the polarization current d9"/U explicitly, the 24 FOUNDATIONS FOR MICROWAVE ENGINEERING +</ i<- p = qx -1 _-?/» FIGURE 2.3 Model for determining the polarization of an atom. ii>) lei displacement vector 3> is denned as 91 = e0g +&> (2.19) whence the total displacement current density can be written as 33i/dt. For a great many materials the polarization & is in the direction of the electric field <g*, although rarely will £P have the same time phase as &. A simple classical model will serve to illustrate this point. Figure 2.3a shows a model of an atom consisting of a nucleus with charge q surrounded by a spherically symmetrical electron cloud of total charge -q. The application of a field I? displaces the electron cloud an effective distance A: as in Fig. 2.36. This displacement is resisted by a restoring force kx proportional to the displacement (Prob. 2.1). In addition, dissipation, or damping, effects are present and result in an additional force, which we shall assume to be proportional to the velocity. If m is the effective mass of the electron cloud, the dynamical equation of motion is obtained by equating the sum of the inertial force md2x/dt2, viscous force tni> dx/dt, and restoring force kx to the applied force -q%\ thus d2x dx m—nr 2 + mv—r + kx = -c dt dt (2.20) When I? = Ex cos <ot, the solution for x is of the form x = -A cosicot + <£)• If E x cos wi is represented by the phasor Ex, and x by the phasor X, the solution for X is readily found to be J\. — <> . • —o) m +j(oum + k and hence x = Re(Xejul) =Acos(o>t + 4>) where (q/m)Ex [ ( y - ^ ) 2 + <oV>]1/2 (l)U d> — t a n O) and we have replaced k/m by w 0 . 2 2 — CUQ ELECTROMAGNETIC THEOBY 25 The dipole moment is px, where q*Ex px = -qx = — >,[{OJ2-O>1) - oil) TTTj cos(a>* + *) +- (2.21) o For N such atoms per unit volume the polarization per unit volume is <PX = Npx and the displacement &x is given by Nq2Ex 3X = e0Ex. cos tot H r p ^ cos( (ot + 4>) mUaj2 - (olf + w V j This equation may also be put into the following form: 2 . ®* = EX where 2 [eQ(v 2 Q 6 = tan 2 2 - w ) + Nq /m\ 2 ( w 2 ~G>22)\ , ,2 > ' / 2 + (a>ve0) cos(tot - B) i <.,..\2 +(U>1>) Oil' —5 5 - tan wl - o>i _ (2.22) 0)U —5 = —-= m% - or + Nqz/e0m Two points are of interest in connection with (2.22). One is the linear relationship between & and §*, and hence between S> and £\ The second is the phase lag in 3 relative to W whenever damping forces are present. The phase difference between £?, W, and 3 makes it awkward to handle the relations between these quantities unless phasor representation is used. In phasor representation (2.21) and (2.22) become 92£, (a>'o — (2.23) w2 + j<>iv)m e 0 ( w o - a,2 +jwf) + Nq2/m Dx = _H 2 io0 — to [ H -^—Ex (2.24) + ja)v In general, for linear media, we may write P = *o*,E (2.25) where x e xs a complex constant of proportionality called the electric susceptibility. The equation for D becomes D = « 0 E + P = e 0 ( l +Xe)E = eE = ere0E = (e'-je")E (2.26) where e = e 0 (l + %e) is called the permittivity, and e r = e / e 0 , the dielectric constant of the medium. Note that e is complex whenever damping effects are present and that the imaginary part is always negative. A positive imaginary part would imply energy creation instead of energy loss. [The reader may verify from (2.22) that 0 is always positive.] Loss in a dielectric material may also occur because of a finite conductivity a. The two mechanisms are indistinguishable as far as external effects 26 FOUNDATIONS FOR MICROWAVE ENGINEERING related to power dissipation are concerned. The curl equation for H may be written as V X H = jw(e'-je")E + <TE where J = crE is the conduction current density in the material. We may also write V X H =jio e'-j[e" + - E = jwe'E + (we" + cr)E (2.27) where by e" + cr/io may be considered as the effective imaginary part of the permittivity, or we" + a as the total effective conductivity. The loss tangent of a dielectric medium is defined by tan <5, = we + cr (2.28) Any measurement of tan 5, always includes the effects of finite conductivity cr. At microwave frequencies, however, we" is usually much larger than a because of the large value of w. Materials for which P is linearly related to E and in the same direction as E are called linear isotropic materials. Nonlinear effects generally occur only for very large applied fields, and as a consequence are rarely encountered in microwave work. However, nonisotropic material is of some importance. If the crystal structure lacks spherical symmetry such as that in a cubic crystal, it may be anticipated that the polarization per unit volume will depend on the direction of the applied field. In Fig. 2.4 a two-dimensional sketch of a crystal lacking cubic symmetry is given. The polarization produced when the field is applied along the x axis may be greater than that when the field is applied along the y or z axis because of the greater ease of polarization along the x axis. In this case we must write DT = exxEx Dy = eyyEy D, = e E (2.29) where exx, eyy and ezz are, in general, all different. The dielectric constants e rx = e*xAo> e rv = evyAo> *r* = e « A o a™ known as the principal dielectric constants, and the material is said to be anisotropic. If the coordinate system used had a different orientation with respect to the crystal structure, CBT-- —&"- 6---4>- - & ' F I G U R E 2.4 A noncubic crystal exhibiting anisotropic effects. ELECTROMAGNETIC THEORY 27 the relation between D and E would become Dx = exxEx + exyEy + exzE2 Dy = eyxEx + eyyEy + zyzEz Dt = e!XEx + ezyEy + ezzEz or in matrix form, \D A D > \£xx = Dz f .v* e exy eyy «l \EX % *y* (2.30) E; ". Only for a particular orientation of the coordinate system does (2.30) reduce to (2.29). This particular orientation defines the principal axis of the medium. For anisotropic media the permittivity is referred to as a tensor permittivity (a tensor of rank 2 may be represented by a matrix). For the most part the materials dealt with in this text are isotropic. Nevertheless, an awareness of the existence of anisotropic media and of the nature of the constitutive relations for such media is important. For the magnetic case, H is defined by the constitutive relation € . « MoH € *m = B - /x 0 M (2.31) where M is the magnetic dipole polarization per unit volume. For most materials (ferromagnetic materials excluded), M is linearly related to B and hence to H. By convention this is expressed by the equation M = *mH (2.32) where x„, is called the magnetic susceptibility. Substituting (2.32) into (2.31) gives B = Mo(M + H) = M o ( l + A - m ) H = AtH (2.33) where fj. = fi0(l + xm) is called the permeability. As in the electric case, damping forces cause p. to be a complex parameter with a negative imaginary part; that is, M = M ~JfJ-"- Also, there are magnetic materials that are anisotropic; in particular, ferrites are anisotropic magnetic materials of great usefulness at microwave frequencies. These exhibit a tensor permeability of the following form: Mi [M] = -VM2 0 jfJ-2 Mi 0 0 1 0 (2.34) Ma when a static magnetic field is applied along the axis for which the permeability is fj.3. A discussion of ferrites and their uses is presented later; so further comments on their anisotropic properties is deferred until then. In Sec. 2.1 care was taken to write Maxwell's equations in a form valid not only in vacuum but also in material media. Thus (2.13) and (2.18) are 28 FOUNDATIONS FOR MICROWAVE ENGINEERING valid in general, but with the constitutive relations of this section replacing the free-space relations (2.2). Note, however, that it is not possible to write, in general, constitutive relations of the form & = eg", £& = p.St, when 2 and &, and likewise S8 and St, are not in time phase. For arbitrary time dependence we must write instead 3 = e„l? + 9", .<% = n0(M' +J?) and relate & and Jt to S and St through the dynamical equation of motion governing the polarization mechanism. This difficulty may be circumvented by using the phasor representation for which relations such as D = eE are perfectly valid because the complex nature of e accounts for the difference in time phase.! It should be pointed out, however, that for many materials used at frequencies up to and including microwaves, the losses are so small that 3 and %, and also St and .5?, are very nearly in time phase. In such cases constitutive relations such as 3 = ei>, 9S = p.St apply with negligible error. Significant departure in time phase between 3 and W or £8 and St occurs only in the vicinity of a natural resonance frequency of the equation of motion for the polarization. 2.3 STATIC FIELDS For electric and magnetic fields that are independent of time, the electric and magnetic fields are not coupled, and likewise the current and charge are not coupled. Putting all time derivatives equal to zero in (2.13) yields^ VxE = 0 (2.35a) V • eE = p (2.356) VXH = J (2.36a) V•B = 0 (2.366) V• J= 0 (2.36c) The last equation is the continuity equation for the special case dp/dt = 0. The static electric field has zero curl, or circulation, and this means that the line integral of E around any arbitrary closed contour is zero. This property is just the condition that permits E to be derived from the gradient of a scalar potential function Q> that is, since V X V<J> is identically zero, we may put E=-VO (2.37) t T h e situation here is like that encountered in ac circuit analysis, where in phasor notation the voltage V equals the current / multiplied by the impedance Z; t h a t is, V = IZ. An Ohm's law of this sort cannot be written for the physical voltage and current, for if V= Re(Ve J "') = Vcos uti, then J* = rM/e-""') = [V/(R2 + X2)>/z]coa(.ojt - rf.), where i> = t a n - \X/R1Clearly, V cannot be equated to J multiplied by a constant because of the difference in phase. ±For static fields we are using boldface roman type to represent the physically real vector fields- ELECTROMAGNETIC THEORY 29 • »/ x=a <j> = 0 F I G U R E 2.5 A simple potential problem. / = 0 Substituting (2.37) into (2.356) and assuming that e is a constant independent of the coordinates give -V-E=V24> = -- (2.38) This equation is known as Poisson's equation. When p = 0, Laplace's equation V2<t> = 0 (2.39) is obtained. The basic field problem in electrostatics is to solve Poisson's or Laplace's equation for a potential function $ that satisfies specified boundary conditions. As a simple example consider two infinite conducting planes at x = 0, a, as in Fig. 2.5. Let charge be distributed with a density p = pQx between the two plates.t It is required to find a <1> which is a solution of Poisson's equation and which equals zero on the plane x = 0 and V on the plane x = a. The potential will depend on x only; so (2.38) becomes d2* 'dxT •Pa' Integrating this equation twice gives <t> = - p 0 x 3 / 6 e 0 + C,* + C 2 . Imposing the boundary conditions at x = 0, a yields 0 = C 2 , Poa t T h e example is somewhat artificial since the assumed charge distribution is not a stable one; i.e., the electric field it produces would cause the charge distribution to change. 30 FOUNDATIONS FOR MICROWAVE ENGINEERING and hence C2 = 0, C, = V/a + p 0 a 2 / 6 e 0 . The solution for <t> is thus p0x3 p a2x 4> = _ —— + _0 6e 0 6e 0 V + % a The electric field between the two plates is E = -V4> = - a , — = a i)x ' P o * 2 _ Po^_ _ V\ i 2e 0 6e 0 aJ The solution for the electrostatic field is greatly facilitated by introduction of the scalar potential 4>. For the same reason it is advantageous to introduce a potential function for the solution of magnetostatic problems. Since B always has zero divergence, it may be derived from the curl of a vector potential A; that is, B = VxA (2.40) This makes the divergence of B vanish identically because the divergence the curl of a vector is identically zero. Using (2.40) in (2.36a) and assuming that p. is constant yields the equation A vector identity of use here i s V X V x A = V V - A - V 2 A . The divergence of A may be placed equal to zero without affecting the value of B derived from the curl of A, and hence the equation for A is V2A= - M J (2.41) This equation is a vector Poisson's equation. In rectangular coordinates, (2.41) represents three scalar Poisson's equations, the first being *'K = -fiJK (2.42) In a curvilinear coordinate system, such as a cylindrical coordinate system, (2.41) cannot be written in such a simple component form. The reason ' that, for example, V 2 a r A r does not equal a r V 2 A r because, even though the unit vector a r is of constant length, its orientation varies from point to point since it is always directed along the radius vector from the origin to the point under consideration. The evaluation of V2A in curvilinear coordinates is made by using the vector identity quoted above to give V2A = W • A - V X V X A. These latter operations are readily carried out. The interest in static field solutions at microwave frequencies arises because the field distribution over a cross-sectional plane of a transmission line is a static field distribution and because static field solutions are good approximate solutions to the actual fields in the vicinity of obstacles that a r small compared with the wavelength. The potential theory introduced above may be extended to the time-varying case also, and this is done in a following section. ELECTROMAGNETIC TH EORY 31 WAVE EQUATION For convenience, the two curl equations are repeated here: V x r = ——- (2.43a) at 33! V X / = — (2.436) at where it is assumed for the present that the current density J is zero in the region of interest. These equations, together with the assumed constitutive relations S = eg, £8 = \x&, may be combined to obtain a separate equation for each field. The curl of (2.43a) is VxV Xf = = -p. dt at Using (2.436) and expanding V X V x ? now yields d2g v v - r - v 2 r = -/«•-, Since p is assumed zero and e is taken as a constant, V • ^ == 0, and we obtain V 2 f - lie—£ = 0 (2.44) which is a three-dimensional wave equation. The velocity of propagation v is equal to ( p . e ) _ 1 / 2 . In free space v is equal to the velocity of light c. To illustrate the nature of the solutions of (2.44), consider a case where g has only an x component and depends only on the z coordinate. In this instance d2Zx d2Zx Any function of the form f(z - vt) is a solution of this equation since dz2 dt2 d(vtf and hence d2f i o 2f _ This solution is illustrated in Fig. 2.6 and clearly represents a disturbance propagating in the positive z direction with velocity v. An equally vahd solution is f(z + vt) and represents a disturbance propagating in the negative z direction. 32 FOUNDATIONS FOR MICROWAVE ENGINEERING /(/ nz-vt Az-vt2) FIGURE 2.6 Propagation of a disturbance f(z - vt). By eliminating the electric field, it is readily found that the magnetic field %" also satisfies the wave equation (2.44). In practice, however, we solve the wave equation for either «? or ^ and then derive the other field by using the appropriate curl equation. When constitutive relations such as 9> = e& and £8 =» (i*? cannot be written, the polarization vectors & and J! must be exhibited explicitly in Maxwell's equations. Wave equations for f and 2? may still be derived, but 9> and Jt will now enter as equivalent sources for the field (which they actually are). The derivation is left as a problem at the end of this chapter. For harmonic time dependence, the equation obtained in place of (2.44) is V 2 E + A2E = 0 2 (2.45) 2 where fe «• co fie. This equation is referred to as the Helmholtz equation, or reduced wave equation. The constant k is called the wave number and may be expressed in the form . to f 2v (2-46) k = toy fie = — = 2ir— *• v v A where the wavelength A is equal to v/f. In free space the wave number will be written as k0, and is equal to wy^o^o = 2ir/A 0 . The magnetic field H, as may be surmised, satisfies the same reduced wave equation. In a medium with finite conductivity <r, a conduction current / = a£ will exist, and this results in energy loss because of Joule heating. The wave equation in media of this type has a damping term proportional to cr and the first time derivative of the field. In metals, excluding ferromagnetic materials, the permittivity and permeability are essentially equal to their free-space values, at least for frequencies up to and including the microwave range. T h u s Maxwell's curl equations become V x r = - Mo AT V X / = en— + •7F ** ELECTROMAGNETIC THEORY 33 Elimination of X" in the same manner as before leads to the following wave equation for &: V2i? - n0cr— - n0e0^- = 0 (2.47) The magnetic field -#* also satisfies this equation. For the time-harmonic case damping effects enter in through the complex nature of e and /x, and hence the wave number k. It should be recalled here that, as shown by (2.27), a finite conductivity cr is equivalent to an imaginary term in the permittivity e. In the present case the equivalent permittivity is e = e 0 — jcr/u> and the Helmholtz equation is V 2 E + w2ji0ej 1 -j— IE = 0 I wf0; (2.48) In metals the conduction current o-E is generally very much larger than the displacement current we 0 E, so that the latter may be neglected. For example, a is equal to 5.8 X 10 7 S / m for copper, and at a frequency of 10 1 0 Hz, a/e 0 = 0.55, which is much smaller than tr. Only for frequencies in the optical range will the two become comparable. Thus (2.47) may be simplified to V2g- - M o <r— = 0 (2.49) °-E = 0 (2.50) and (2.48) reduces to V 2 E->M 0 Equation (2.49) is a diffusion equation similar to that which governs the flow of heat in a thermal conductor. ENERGY AND POWER When currents exist in conductors as a result of the application of a suitable potential source, energy is expended by the source in maintaining the currents. The energy supplied by the source is stored in the electric and magnetic fields set up by the currents or propagated (radiated) away in the form of an electromagnetic wave. Under steady-state sinusoidal time-varying conditions, the time-average energy stored in the electric field is 1 1 r r W = R e - / E - D * d V = -/c'E-E*rfV 4 •'v 4 -V If e is a constant and real, (2.51a) becomes We=^-(E-E*dV 4 Jv (2.51a) (2.516) 34 FOUNDATIONS FOR MICROWAVE ENGINEERING The time-average energy stored in the magnetic field is given by 1 l t r W„, = Re - / H* • B dV = - / ii'H • H* dV 4 Jv (2.52a) 4 J\f which, for ii real and constant, becomes Wm=jfH-H*dV (2.526) These expressions for W e and W m are valid only for nondispersive media, i.e., media for which e and ti can be considered independent of w in the vicinity of the angular frequency w with which the fields vary. In general, when the losses are small, so that e" •« e' and /x" <sc it, we have 1 , dwe' Wc= - / E - E * - -dV 4 -V <'a> 1 - (2.53a) SOJU! Wm=- / H - H * — d V (2.536) for the time-average stored electric and magnetic energy. The above equations for the time-average energy in a dispersive medium may be established by considering a classical model of the polarization mechanism similar to that discussed in Sec, 2.2. In a unit volume let the effective oscillating charge of the dipole distribution be —a with an effective mass m. Let the damping force be equal to mu times the velocity of the charge. This damping force takes account of collision effects and loss of energy by radiation from the oscillating charge. The equation of motion for the polarization charge displacement u is d2u du 2 dt dt where u is parallel to the direction of the field ?\ In this equation k is elastic constant giving rise to the restoring force. This constant arises from the Coulomb forces acting on the displaced charge, and hence is of electrical origin. The dipole polarization 3" is -qu, and the polarization current f = d.9>/dt. Introducing the polarization current into the equation of motion gives m dfB q 1 dt mu o/ k rt p q J This equation is formally the same as that which describes the current in a series LCR circuit with an applied voltage 7? equal to W and with m, mv q2 q2 Q k An equivalent circuit describing the polarization is illustrated in Fig. 2.7. ELECTROMAGNETIC THEORY Jp ° r L R $VW' WV 35 c FIGURE 2.7 Equivalent circuit For polarization current. a time dependence e-""' is assumed and phasor notation is used, R-JX J EY E P- - WTX^ where Y is the input admittance and X = c>L - 1 / w C Since P = e0xeE and J p =j(oP, we see that -X-jR ft>e0A-e = <»e0{x'e -JX'e) = ~JY = R2 + X 2 and hence "We = D -X 2 2, Y 2i R Tx (2.54a) The time-average power loss associated with the polarization is the same as the power loss in R in the equivalent circuit. This is given by 1 R Pi = __^E*-^—-, = -EE*o>eQx"e (2-55) pp* 2 R2 + X2 per unit volume. This equation shows that (Df 0 ^ = we" is an equivalent conductance. The time-average energy stored in the system is of two forms. First there is the kinetic energy of motion, that is, \m(du/dt)2 averaged over a cycle, and this is equal to the magnetic energy stored in the inductor in the equivalent circuit. This time-average kinetic energy per unit volume is given by I Um - -Upj; = -EE--^~—-2 2 4" 4~ R ~X'< (2.56a) The second form of stored energy is the potential energy associated with the charge displacement. The time-average value of this energy is equal to the time-average electric energy stored in the capacitor C in the equivalent circuit, and is given by The total time-average energy stored per unit volume is U = U m + Ue. Note 36 FOUNDATIONS FOR MICROWAVE ENGINEERING that U is not given by \EE*e0x'e- The latter expression gives = 4 l/w2C - L -EK fl2 + X2' = u-u or the difference between the potential and kinetic energy stored. To obtain an expression for the total stored energy, note that L + l/<o2C d do) R 2 +X 2 2 R +X 2 2X- \" 1 - R 2 + X2 For a low-loss system, R2 <s X2, and we then have 1 - 2X2/(R2 + X2) — 1; so d I -X 2 L + l/w2C d d^xWTx -(<»«OA£) = R2 + X 2 Multiplying this expression by \EE* now gives the total time-average energy stored, as comparison with (2.56a) and (2.56b) shows. Thus the final expression for the time-average electric energy stored in a volume V is given by the volume integral of U = Ue + U„ plus the free-space energy density e 0 (E - E * ) / 4 a n d i s W. 4 -L E • E E*\dV fl «>toX'e dV iu> 1 ,d(oe' = - / E -E* dV 4 -fy do) since e = e 0 (l + x'*)- This equation is the result given earlier by (2.53a). A similar type of model may be used to establish (2.536) for average stored magnetic energy. It should be pointed out that under ti varying conditions the average stored energy associated with either electric or magnetic polarization includes a kinetic-energy term. This term is negh" gible at low frequencies and also when e' and /j.' are essentially independent of ft> for the range of a> of interest. When this energy is not negligible, the modified expressions for stored energy must be used. Although (2.53) is more general than (2.51) and (2.52), we shall, in | majority of instances, use the latter equations for the stored energy. thereby tacitly assume that we are dealing with material that is nondisf sive or very nearly so. The time-average power transmitted across a closed surface S is g" by the integral of the real part of one-half of the normal component of tt ELECTROMAGNETIC THEORY 37 complex Poynting vector E X H*; that is, 1 , P = R e - ^ E X H * • dS (2.57) The above results are obtained from the interpretation of the complex Poynting vector theorem, which may be derived from Maxwell's equations as follows: If the divergence of E X H*, that is, v" • E X H*. is expanded, we obtain V • E X H* = (V X E) • H* - (V x H*) • E From Maxwell's equations V X E = -jwB and V X H* = - j w D * + J*, and hence V • E X H* = - j w B • H* +jwD* • E - E • J* The integration of this equation throughout a volume V bounded by a closed surface S gives the complex Poynting vector theorem; i.e., ! i If - I V • E X H* dV = 2T -A E X H* • rfS S 2 JM 2%(1) to , = -j(B -H* 2A ~JwV ~ 1 -E-l)*)dV- - , 22 J\rV E -J*dV (2.58a) where the divergence theorem has been used on the left-hand side integral. The above result may be rewritten as 1 , , j B • H* E • D* \ i , +-/E-J*rfV 2 -'v (2.586) where -dS is a vector element of area directed into the volume V. If the medium in V is characterized by parameters e =* e' — je", y. = M' ~ ./M". anc ^ conductivity o-, the real and imaginary parts of (2.58) may be equated to give 1 Re-0EXH* 2% -(-dS) v = -f(M"H-H* 2 V 1 t + - aE-E*dV 2 J v\ (2.59a) V j\t 1 , , / H • H* I m - f h E x H * - ( - d S ) = 2ft> U' 7— -* 2. 's +e"E-E*)dV 4 E • E* \ \ dV (2.596) 4 / Equation (2.59a) is interpreted to state that the real electromagnetic power transmitted through the closed surface S into V is equal to the power loss produced by conduction current o-E, resulting in Joule heating plus the 38 FOUNDATIONS FOR MICROWAVE ENGINEERING power loss resulting from polarization damping forces. Note that we" could be interpreted as an equivalent conductance, as pointed out in Sec. 2.2. equation also shows that /i" and e" must be positive in order to represeenergy loss, and hence the imaginary parts of e and /J. must be negativEquation (2.596) states t h a t the imaginary part of the complex rate energy flow into V is equal to 2w times the net reactive energy W m — stored in the magnetic and electric fields in V. The complex Poynting ve~ theorem is essentially an energy-balance equation. A result analogous to the above may be derived for a conventio network, and serves to demonstrate the validity of the interpretation of (2.58). Consider a simple series RLC circuit as in Fig. 2.7. If the current in the circuit is / and the applied voltage is V, the complex input power given by 1 1 1 -VI* = -ZII* = - f f * R +ja,L— 2 2 2 The time-average power loss in R, magnetic energy stored in the field around L, and electric energy stored in the field associated with C are given, respectively, by P, = -RII* Wm = 1 11* W„ = 4- w2C -jUl* since the voltage across C is I/coC. Hence ^VI*=±ZII*~P, + 2jco(Wm-We) which has the same interpretation as (2.58). This equation may also solved for the impedance Z to give z= (2.60) \II* and provides a general definition of the impedance of a network in terms of the associated power loss and stored reactive energy. The factor \II* in the denominator serves as a normalization factor, and is required in order to make Z independent of the magnitude of the current at the input to the network. In the case of a general time-varying field, an expansion of V • W X # and substitution from Maxwell's equations (2.13) lead to the following Poynting vector theorem for general time-varying fields: , / ax + e0^'— dt M + &-— dt +&-S\dV ELECTROMAGNETIC THEORY 39 Since ^ • dJC/M = \ fl( J? • ^)/dt, etc., and the electric and magnetic polarization currents are fp = d&>/dt, fm = ix0(d^/dt), we have + \y-{f+Jrp)+%--fm\dV (2.61) where - r f S is an element of surface area directed into V. This equation states that the rate of energy flow into V is equal to the time rate of change of the free-space field energy stored in V plus the rate of energy dissipation in Joule heating arising from the conduction current f and, in addition, the instantaneous rate of energy supplied in maintaining the polarization currents. If J! and 21?, and also & and #\ are in phase, there is no energy loss associated with the polarization currents. If these quantities are not in phase, some energy dissipation takes place, leading to increased heating of the material. If the susceptibilities \ e an^ X m c a n D e considered as constants, so that d&>/dt = e0xe(d^/dt) and djt/dt = xJ-d&Zat), then (2.61) becomes ^rx^.(-rfS) = - / v ( ^ - + ^^)^+/vf.^v(2.62) which is the usual form of the Poynting vector theorem. The first term on the right is now interpreted as the instantaneous rate of change of the total electric and magnetic energy stored in the volume V. The susceptibilities can usually be considered as true constants whenever the inertia] and damping forces are small compared with the elastic restoring force in the dynamical equation describing the polarization. For example, with reference to (2.54a), this is the case when k is much greater than tomi' or o)2m, that is, when 1/ioC is large compared with wL and R, so that <rt = c = q - BOUNDARY CONDITIONS In order to find the proper and unique solutions to Maxwell's equations for situations of practical interest (these always involve material bodies with boundaries), a knowledge of the behavior of the electromagnetic field at the boundary separating material bodies with different electrical properties is required. From a mathematical point of view, the solution of a partial differential equation, such as a wave equation, in a region V is not unique unless boundary conditions are specified, i.e., thfe behavior of the field on the boundary of V. Boundary conditions play the same role in the solution of 40 FOUNDATIONS FOR MICROWAVE ENGINEERING FIGURE 2.8 A cylindrical cavity partially filled with a dielectric medium. partial differential equations that initial conditions play in the solution of the differential equations that govern the behavior of electric circuits. As an example, consider the problem of finding a solution to Maxwell's equations inside a cylindrical cavity partially filled with a dielectric medium of permittivity e, as in Fig. 2.8. In practice, the solution is obtained by finding general solutions valid in the two regions labeled /?, and R2. These general solutions must satisfy prescribed conditions on the metallic boundaries and in addition contain arbitrary amplitude constants that can be determined only from a knowledge of the boundary conditions to be applied at the air-dielectric boundary separating regions ff, and R2The integral form of Maxwell's equations provides the most convenient formulation in order to deduce the required boundary conditions. Consider two media with parameters e^itj an^ e2,n2> ^ ' n Fig. 2.9a. If there is no surface charge on the boundary, which is the usual case for nonconducting media, the integral of the displacement flux over the surface of the small "coin-shaped" volume centered on the boundary as in Fig. 2.96 gives, in the limit as h tends to zero, lim (ft D • d S = D2n AS D l n A S = 0 fc-»0'S D2„ = Dln = n D 2 = n D , or (2.63) £u 3 ^ (a) &l le) FIGURE 2.9 Boundary between two different media. ELECTROMAGNETIC THEORY 41 where n denotes the normal component. The limit h —» 0 is taken so that the flux through the sides of the coin-shaped region vanishes. Equation (2.63) simply states that the displacement flux lines are continuous in the direction normal to the boundary. A similar result clearly must hold also for the magnetic flux lines since V • B = 0, and hence, by analogy, n-B2 = n-B1 (2.64) To obtain boundary conditions on the tangential components of the electric field E and magnetic field H, the circulation integrals for E and H are used. If for the contour C in Fig. 2.9c, the width h is made to approach zero, the magnetic flux flowing through this contour vanishes and lim (f) E • d l = lim - ja> J( B • rfS = 0 h k-0~C — 0 S = E.2I M - Elt M or Eu = E2l (2.65) For the same contour C the total displacement current directed through the contour vanishes as h -* 0, so that lim <£ H • dl = lim [jut [ D • rfs] = 0 h-n~c h--o\ Js 1 = (H2l~Hu)M or H2I = Hu (2.66) where t denotes the components tangential to the boundary surface. These latter relations state that the components of E and H tangent to the boundary are continuous across the boundary; i.e., the tangential components on adjacent sides of the boundary are equal at the boundary surface. For the boundary conditions at the surface separating a good conductor (any metal) and free space or air, some simplification is possible. As shown in a later section, the electromagnetic field can penetrate into a conductor only a minute distance at microwave frequencies. The field amplitude decays exponentially from its surface value according to e"u/i', where u is the normal distance into the conductor measured from the surface, and <5, is called the skin depth. The skin depth is given by 1/2 S.= (2.67) W/lff For copper (a = 5.8 X 10 7 S / m ) at a frequency of 10 10 Hz, the skin depth is 6.6 X 10~ 5 cm, truly a very small distance. Likewise, the current J = trE is concentrated near the surface. As the conductivity is made to approach infinity, 8 S approaches zero and the current is squeezed into a narrower and narrower region and in the limit a —* *> becomes a true surface current. Since the skin depth is so small at microwave frequencies for metals, the approximation of infinite conductivity may be made with negligible error (an 42 FOUNDATIONS FOR MICROWAVE ENGINEERING £T = aJ, 0 = 0 F I G U R E 2.10 Boundary- of a perfect conductor. exception is when attenuation is to be calculated, since then infinite conductivity implies no loss). For infinite conductivity the field in the conductor must be zero. Since the flux lines of B are continuous and likewise since the tangential component of E is continuous across the boundary, it is necessary that n • B = 0 (2.68a) E, = n X E = 0 (2.686) at t h e surface of a perfect conductor. This same argument cannot be applied to the normal component of D and the tangential component of H because, as noted above, a surface current J s will exist on the surface in the limit a -* ». Applying Maxwell's equation j> H • d\ =jojjV • dS + fj • dS to the contour C illustrated in Fig. 2.10 gives lim <#H • d\ = H. M = lim (jwD • dS + lim [j • d S h-0J h-OJr = lim hJM h-0J JC M A-0 or in vector form, n X H = J, (2.68c) Note that the field in the conductor goes to zero, that the total displacement current through C vanishes as h. -» 0, but that hj tends to the limiting value J s as the conductivity is made infinite and h is made to approach zero. Associated with the surface current is a charge of density p s on which the normal displacement flux lines terminate. Hence, at the surface of a perfect conductor, n • D = D„ = Ps (2.68rf) When it is desired to take into account the large but finite conductivity (as would be the case in attenuation calculations), an impedance boundary Er^ECTROMAGNETrCTHEORy 43 condition may be used with little error. The metallic surface exhibits a surface impedance Zm, with equal resistive and inductive parts, given by 1 +j Z m = —— <rds (2.69) At the surface a surface current exists, and the relation between this and the electric field tangent to the surface is E, = Z,„JS (2.70) Note that the tangential electric field cannot be zero for finite conductivity, although it may be very small. Now n X H = J s , so that E, = ZmJ, = Zmn X H (2.71) From (2.69) it is seen that the resistive part of the surface impedance is equal to the dc resistance per square of a unit square of metal of thickness 5 S . In a later section the above results are verified; so further comments are reserved until then. In practice, it suffices to make the tangential components of the fields satisfy the proper boundary conditions since, when they do, the normal components of the fields automatically satisfy their appropriate boundary conditions. The reason is that when the fields are a solution of Maxwell's equations, not all the components of the field are independent. For example, when the tangential part of the electric field is continuous across a boundary, the derivatives of the tangential component of electric field with respect to coordinates on the boundary surface are also continuous. Thus the curl of the electric field normal to the surface is continuous, and this implies continuity of the normal component of B. More specifically, if the xy plane is the boundary surface and Ex, E y are continuous, then i>Ex/dx, 9Ex/dy, 9Ey/dx, and dEy,/rfy are also continuous. Hence -jtoB, = SE /Hx - dEx/dy is continuous. For the same reasons continuity of the tangential components of H ensures the continuity of the normal component of D across a boundary. In addition to the boundary conditions given above, a boundary condition must be imposed on the field solutions at the edge of a conducting body such as a wedge. The edge condition requires that the energy stored in the field in the vicinity of an edge of a conducting body be finite. This limits the maximum rate at which the field intensities can increase as the edge is approached.t A detailed analysis shows that at the edge of a two-dimensional perfectly conducting wedge with an internal angle </>, the field components normal to the edge must not increase any faster than r", where r is t J . Meixner, The Behavior of Electromagnetic Fields at Edges, N.Y. Univ. Inst. Math. Set. Res. Rept., vol EM-72. December, 1954. The theory is also discussed in R. E. Collin, "Field Theory of Guided Waves." chap. 1, IEEE Press. Piscataway, N.J.. 1991. revised edition. 44 FOUNDATIONS FOR MICROWAVE ENGINEERING the perpendicular radial distance away from the edge and n TT 2TT - <f> where the integer n must be chosen so that a is greater than or equal to — A at least. When solving for fields in an infinite region of space, the behavior of the field at infinity must also be specified. This boundary condition is called a radiation condition, and requires that the field at infinity be a wave propagating a finite amount of energy outward, or else t h a t the field vanish so fast that the energy stored in the field and the energy flow at infinity are zero. 2.7 PLANE WAVES In this section and the two following ones we shall introduce wave solutions by considering plane waves propagating in free space and reflection of a plane wave from a boundary separating free space and a dielectric, or conducting, medium. The latter problem will serve to derive the boundary conditions given by (2.68) to (2.71) in the preceding section. P l a n e Waves in Free Spaee The electric field is a solution of the Helmholtz equation r>2E d2E d2E This vector equation holds for each component of E, so that d2E; d2Ei d2Et i = x,y,z + 9z* + **£,- = 0 3x' + —s9? (2. The standard procedure for solving a partial differential equation is the method of separation of variables. However, this method does not work for all types of partial differential equations in all various coordinate systems, and when it does not work, a solution is very difficult, if not impossible, to obtain. For the Helmholtz equation the method of separation of variables does work in such common coordinate systems as rectangular, cylindrical, and spherical. Hence this method suffices for the class of problems discussed in this text. The basic procedure is to assume for the solution a product ot functions each of which is a function of one coordinate variable onlySubstitution of this solution into the partial differential equation then separates the partial differential equation into three ordinary differentia1 equations which may be solved by standard means. ELECTROMAGNETIC THEORY 45 In the present case let E x = f(x)g(y)h(z). Substituting this expression into (2.72) gives ghf" + fhg" + fgh" + klfgh = Q where the double prime denotes the second derivative. Dividing this equation by fgh gives f" + g" + h" 7 7 7T+*" = 0 (2 73 - > Each of the first three terms in (2.73), such as f"/f, is a function of a single independent variable only, and hence the sum of these terms can equal a constant -kl only if each term is constant. Thus (2.73) separates into three equations: f" f d*f . Xi+**V=0 g" g h" h 2 dg — + , ^ = 0 d*h — +**/, = < ) (2.74) where kx, ft", k\ are called separation constants. The only restriction so far on k\, ky, k'l is t h a t their sum must equal &jj, that is, k\ + k'\ + k\ = k\ (2.75) so that (2.73) will be satisfied. Equations (2.74) are simple-harmonic differential equations with exponential solutions of the form e±jk'x, e±jk'y, e ' jk=*. As one suitable solution for E x we may therefore choose Ex = Ae-Jk''-Jk>y-jk<! (2.76) where A is an amplitude factor. This solution is interpreted as the x component of a wave propagating in the direction specified by the propagation vector k = aA + aA + a A (*•"") because the scalar product of k with the position vector r = axx + a v y + atz equals kxx + kyy + kzz and is k0 times the perpendicular distance from the origin to a plane normal to the vector k, as illustrated in Fig. 2.11. The k vector may also be written as k — nk0, where n is a unit vector in the direction of k and k0 is the magnitude of k by virtue of (2.75). Although (2.76) gives a possible solution for Ex, this is not the complete solution for the electric field. Similar solutions for E v and E, may be found. The three components of E are not independent since the divergence relation V • E = 0 must hold in free space. This constraint means 46 FOUNDATIONS FOB MICROWAVE F.NOINF.ERING FIGURE2.il Wustration of plane normal to vector k. that only two components of E can have arbitrary amplitudes. However, for V • E to vanish everywhere, all components of E must have the same spatial dependence, and hence appropriate solutions for E y and E, are Ey = Be~ikr E2 = Ce-jkr with B and C amplitude coefficients. Let E„ be the vector axA + a v B + a z C ; then the total solution for E may be written in vector form as E = E„e--' k - r (2.78) The divergence condition gives V-E = V •E0e^k-r = E„ • V V ee" Jk ^ k r = - / k • Ene--'k'r = 0 or k - E0 = 0 (2.79) since v"e~- = — j k e " - , as may be verified by expansion in rectangular coordinates. The divergence condition is seen to constrain the amplitudes A, B, C so that the vector E 0 is perpendicular to the direction of propagation as specified by k. The solution (2.78) is called a uniform plane wave since the constant-phase surfaces given by k • r = const are planes and the field E does not vary on a constant-phase plane. The solution for H is obtained from Maxwell's equation /kr ,kr V X E = -V'toAt0H which gives 1 H 1 .. V X E0e-ik'r = -kx E„e^k-r = 1 E0 x Ve-Jkv Jw^o nXE (Vfl0 <"Mo n X E yonXE (2.80) where Y 0 = v/en/Mo has the dimensions of an admittance and is called the intrinsic admittance of free space. The reciprocal Z 0 = 1/Y0 is called the ELECTROMAGNETIC THEORY 47 F I G U R E 2.12 Space relationship between E, H, and n in a TEM intrinsic impedance of free space. Note that H is perpendicular to E and to n, and hence both E and H lie in the constant-phase planes. For this reason this type of wave is called a transverse electromagnetic wave (TEM wave). The spatial relationship between E, H, and n is illustrated in Fig. 2.12. The physical electric field corresponding to the phasor representation (2.78) is E = Re(E0e-jk-r+J°") = E 0 cos(k • * - « * ) (2.81) where, for simplicity, E 0 has been assumed to be real. The wavelength is the distance the wave must propagate to undergo a phase change of 2TT. If we let A0 denote the wavelength in free space, it follows that |k|A 0 = k0A0 = 2TT so that 2- I k0 = o>VMoeo = ~ = (2.82) This result is the familiar relationship between wavelength A0, frequency f = co/2rr, and velocity c in free space. A wavelength in a direction other than that along the direction of propagation n may also be denned. For example, along the direction of the x axis the wavelength is 2TT A = (2.83) *7 and since k x is less than k0, A^ is greater than A0. The phase velocity is the velocity with which an observer would have to move in order to see a constant phase. From (2.81) it is seen that the phase of E is constant as long as k • r - wt is constant. If the angle between k and r is 8, then k • r - cot ~ k0r cos 6 — cot. Differentiating the relation k0rcos 0 - cot = const dr gives ~dt co " fcocos0 (2.84) for the phase velocity u p in the direction r. Along the direction of propagation, cos 6 = 1 and vp = co/k0 = c. In other directions, the phase velocity is 48 FOUNDATIONS FOR MICROWAVE ENGINEERING F I G U R E 2.13 A wave propagating obliquely to the u axis. greater than c. These results may be understood by reference to Fig. 2.13. When the wave has moved a distance A 0 along the direction n, the constant-phase-plane intersection with the u axis has moved a distance \ u = A o sec0 along the direction u. For this reason the wavelength and phase velocity along u are greater by a factor sec 0 than the corresponding quantities measured along the direction of propagation n. The time-average rate of energy flow per unit area in the direction n is given by P = | R e E X H* • n = | Re y~0E X (n X E*) • n = \Y0E0 • Eg (2.85) The time-average energy densities in the electric and magnetic fields of a TEM wave are, respectively, U. = ~E • E* = ^ E 0 • E* U„ = ^H • H* = ^ y ( ? ( n X E) - ( n x E*) = ^ E 4 4 4 0 • E* = U e and are seen to be equal. Since power is a flow of energy, the velocity v g energy propagation is such that (Ue+Um)vg=P Iz-u—u ' o ^ o ' —u E0 YQu /o (2 E • E c 0- - C 0 0E * " " ' " K T K T " i , 0„ E . ' B Thus, for a TEM wave in free space, the energy in the field is transport* with a velocity c = 3 X 10 s m / s , which is also the phase velocity. Since phase velocity is independent of frequency, a modulated carrier or sign 31 will have all its frequency components propagated with the same velocity c Hence the signal velocity is also the velocity of light c. Later on, in the study of waveguides, situations arise where the phase velocity is dependent on 8 agafrequency and consequently is not equal to the velocity of energy propaS tion or the signal velocity. [ ELECTROMAGNETIC THEORY 49 REFLECTION FROM A DIELECTRIC INTERFACE In Fig. 2.14 the half-space z > 0 is filled with a dielectric medium with permittivity e (dielectric constant er = e/e0; index of refraction 77 = y ^ X A TEM wave is assumed incident from the region 2 < 0. Without loss in generality, the xy axis may be oriented so that the unit vector n : specifying the direction of incidence lies in the xz plane. It is convenient to solve this problem as two special cases, namely (1) parallel polarization, where the electric field of the incident wave is coplanar with n, and the interface normal, i.e., lies in the xz plane, and (2) perpendicular polarization, where the electric field of the incident wave is perpendicular to the plane of incidence as defined by n, and the interface normal, i.e., along the y axis. An incident TEM wave with arbitrary polarization can always be decomposed into a linear sum of perpendicular and parallel polarized waves. The reason for treating the two polarizations separately is that the reflection and transmission coefficients, to be defined, are different for the two cases. Parallel Polarization Let the incident TEM wave be E,- = E 1 e-- y *° n '" r H, = y o n , X E, (2.87) where E a lies in the xz plane. Part of the incident power will be reflected, and the remainder will be transmitted into the dielectric medium. Let the reflected TEM wave be E r = E2e-jk°n*r Hr = y 0 n 2 x E r (2.88) where n 2 and E 2 are to be determined. In the dielectric medium the solution for a TEM wave is the same as that in free space, but with e 0 replaced by e. Thus, in place of A0 = co^fi0e0 and Y0 = ^e0/n0, the parameters k = oijn0e = -qk0 and Y = yje/n0 = 1^0 are used, where 77 = yjer is the index of refraction. The transmitted wave in the dielectric may be A F I G U R E 2.14 Plane wave incident on a dielectric interface. 50 FOUNDATIONS FOR MICROWAVE ENGINEERING expressed by E, = E 3 e ^ " ^ r H r = Yn3 X E, (2.89) with E 3 and n 3 as yet unknown. The boundary conditions to be applied are the continuity of the tangential components of the electric and magnetic fields at the interface p)ane z = 0. These components must be continuous for all values of x and y on the 2 = 0 plane, and this is possible only if the fields on adjacent sides of the boundary have the same variation with x and y. Hence we must have k0nlx = k0n2)l = knax = Tlk0n3x (2.90) i.e., the propagation phase constant along x must be the same for all waves. Since nly was chosen as zero, it follows that n2y = n3 = 0 also. The unit vectors n , , n 2 , n 3 may be expressed as n , = a , sin 0, + a z cos 0, n 2 = a , sin 02 + a . cos 62 n 3 = ax sin 03 + a z cos &3 Equation (2.90) gives or ei = B2 (2.93 which is the well-known SnelPs law of reflection; in addition, (2.90) gives sin6l, = 7) sin0., (2.92) which is also a well-known result specifying the angle of refraction 8 3 in terms of the angle of incidence Bx and the index of refraction 77. The incident electric field Es has components Elx = Ex cos &v Elz = - E , sin0! since n1 • Er must equal zero. Note that Ex is used to denote the mi tude of the vector E x . Since the incident electric field has no y component, the reflected and transmitted electric fields also have zero y components.! Expressing all fields in component form, i.e., E2x = E 2 cos 0 2 E2, = E2 sin 02, E3l = E3 cos 63, EZz = -Es sin 03, and imposing the boundary condition of continuity of the x component at 2 = 0 yields the relation E, cos 6, + E2 cos 02 = E3 cos 83 f If the reflected and transmitted electric fields were assumed to have a v component, boundary conditions which must apply would show that these are, indeed, zero. ELECTROMAGNETIC THEORY l/er , or _ S i n 2 e 51 \ (El + E2)cosex = Erfl - sin 2 0 3 = E3 (2.93) V by using (2.91) and (2.92). Apart from the propagation factor, the magnetic field is given by H, = * > ! X E , H2 = = Y0ay( -nlxEu + nl2Eu) = y 0 a y E , -Y0ayE2 H3=7ay£3 and has only a y component. Continuity of this magnetic field at the boundary requires that Y0(El-E2) = YE, = vY0E3 (2.94) If a reflection coefficient l\ and a transmission coefficient 7, are introduced according to the following relations: amplitude of reflected electric field amplitude of incident electric field E2 Ex amplitude of transmitted electric field E-, = —(2 956) 1 amplitude of incident electric field Ex then the boundary conditions (2.93) and (2.94) may be expressed as 7 = (Cr-sin20,)1/a 1 + T, = 7, — y— 77 cos 0 , 1 - Vx = 777, (2.96a ) v ; (2.966) These equations may be solved to give the Fresnel reflection and transmission coefficients for the case of parallel polarization, namely, fe. - sin 2 0.) " - er cos 0, r T, = -^ Kr^ i 2 (er-sin 0,) +ercos0, 2?7 cos 0j T . = . 2„ t ll//2, (er - s i n 0 , ) + e r cos0 x (2.97a) (2-976) An interesting feature of r\ is that it vanishes for an angle of incidence 0 t = 0ft, called the Brewster angle, where, from (2.97a), e r - s i n 2 0fc = e 2 c o s 2 0 b or sin06= I *r )l/2 (2.98) 52 FOUNDATIONS FOR MICROWAVE ENGINEERING ^.g F I G U R E 2.15 Modulus of reflection coefficient at a dielectric in terface for er = 2.56, 11", I parallel polarization, |r I perpendicular polarization. At this particular angle of incidence all the incident power is transmitted into the dielectric medium. In Fig. 2.15 the reflection coefficient ["", is plotted as a function of 0, for polystyrene, for which er = 2.56. 2 Perpendicular Polarization For perpendicular polarization the roles of electric and magnetic fields are interchanged so that the electric field has only a y component. The fields may, however, still be expressed in the form given by (2.87) to (2.89), but with E[, E 2 , and E 3 having y components only. As in the previous case, the boundary conditions must hold for all values of x and y on the z = 0 plane. Therefore Snell's laws of reflection and refraction again result; i.e., (2.91) and (2.92) must be satisfied. In place of the boundary conditions (2.93) and (2.94), we have £, + E, = E.3 (2.99a) Y 0 ( £ , - E^cosOt = YE 3 cos0 3 (2.996) Introducing the following reflection and transmission coefficients: r2 = into (2.99) yields (2.100a) 1 + T2 = T2 i - r2 = T2 (e.-sin2^) 1/2 (2.1006) COS0, The Fresnel reflection and transmission coefficients for the case of perpendicular polarization thus are ,1/2 cost?! - (e f - sin 2 *^) r2 = 2cos0, T2 = (2.101a) ( e , - s i n 2 0 i ) ' / 2 + cos0! U r - s i n 2 0 ! ) 1 / 2 + cos0, (2.1016) ELECTROMAGNETIC THEORY 53 A notable difference for this case is the nonexistence of a Brewster angle for which T 2 vanishes. For comparison with the case of parallel polarization, F 2 is plotted in Fig. 2.15 for er = 2.56. REFLECTION FROM A CONDUCTING PLANE The essential features of the behavior of the electromagnetic field at the surface of a good conductor may be deduced from an analysis of the simple problem of a TEM wave incident normally onto a conducting plane. The problem is illustrated in Fig. 2.16, which shows a medium with parameters e, ix, a filling the half-space z > 0. Let the electric field be polarized along the x axis so that the incident and reflected fields may be expressed as E; = E1axe~jk°' H, = E J = rE.a i a i 're+ ** Hr= Y0E}aye-jk»z (2.1026) +J -Y0rE1ave *»* where T is the reflection coefficient. In the conducting medium the conduction current trE is much greater than the displacement current jcoeE, so that Helmholtz's equation reduces to (2.50); i.e., V 2 E -jwnaE = 0 The transmitted field is a solution of 8s 712 Jz -JO>H<T\E, = 0 since no x or y variation is assumed. The solution for a wave with an x component only and propagating in the z direction is E, = £ , a . r e f o.^o (2.103a) f, p. o" F I G U R E 2.16 A TEM wave incident normally on a conducting plane. 54 FOUNDATIONS FOR MICROWAVE ENGINEERING with a corresponding magnetic field 1 H.---J where VXE,= - y = (ja>ii<r) y BvE3e-" * +J = —— 1/2 (2.1036) (2.104) and the skin depth S s = (w/xir/2) 1 / 2 . The propagation constant y = a + /« has equal phase and attenuation constants. In the conductor the fields decay by an amount e~l in a distance of one skin depth 8S, which is a very small distance for metals at microwave frequencies (about 10 ~ 5 cm). The intrinsic impedance of the metal is Zm, where 'in (jwn*)l/2 T8S (2.105) and is very small compared with the intrinsic impedance Z 0 = (fi0/e0)s/2 of free space. For example, for copper at 10" MHz, Z,„ = 0.026(1 + j) ft as compared with 377 ft for Z 0 . Note that (2.1036) may be written as H,=:—ayE3e->* = Ym8ivE3e-v* which shows that the ratio of the magnitudes of the electric field to the magnetic field for a TEM wave in a conductor is the intrinsic impedance Z m . Returning to the boundary-value problem and imposing the boundary conditions of continuity of tangential fields at the boundary plane 2 = 0 give (l + r)E1 = Ea = TE1 (1 - F)F 0 JJ, = H, = YmE3 = YmTEx (2.106a) (2.1066) where E3/E1 = T, the transmission coefficient. Solving (2.106) for the reflection coefficient V and T yields f = Zm ~Z° Zm + Z 0 T= 1 + r = (2.107c) 2Zm (2.1076) zm+zQ Since \Zm\ is small compared with Z 0 , the reflection coefficient I" is almost equal to - 1 and the transmission coefficient T is very small. Almost all the incident power is reflected from the metallic boundary. As the conductivity a- is made to approach infinity, the impedance Z m approaches zero and in the limit T = -1 and T = 0. Hence, for a perfect conductor, the tangential electric field at the surface is zero and the tangential magnetic field has a value equal to twice that of the incident wave. ELECTROMAGNETIC THEORY The current density in the conductor is J = o-E, = aTE^a^e total current per unit width of conductor along y is yz 55 . The ,» ,=» aTE,ar Js= / Jdz = o - T ^ a , / e~yz dz = A/m J J o o y This result may also be expressed in the following form: 2aZ2mEl < (Zm+Z0)j<on*' (2.108) by substituting for T from (2.1076) and replacing y by jiofi/Zm from (2.105). As <T -* », the hmiting value of Js becomes 2E. J, = - s - a , = 2 y „ £ i a , (2.109) since Z m -* 0 and <rZ*t -»yw/x. This current exists only on the surface of the conductor since, as a -» oe, the skin depth 5, -» 0; t h a t is, the field decays infinitely fast with distance into the conductor. When a is infinite, T = -1 and the total tangential magnetic field at the surface is 2 Y u E , a v and equal in magnitude to J s . In vector form the boundary conditions at the surface of a perfect conductor are thus seen to be nXE = 0 (2.110a) n x H = J, (2.1106) where n is a unit outward normal at the conductor surface. For finite conductivity the current density at the surface is crT£, and the magnetic field at the surface is YmTE1. In terms of these quantities the total current per unit width may be expressed as <TTE, y aZm y In other words, the total current per unit width is equal to the tangential magnetic field at the surface. The time-average power transmitted into the conductor per unit area is given by the real part of one-half of the complex Poynting vector at the surface, and is P, = | R e E X H* • a 2 = ±TT*E1Ef Re Y m = \TT*ExEfads (2.111) The reader may readily verify that this is equal to the result obtained from a volume integral of J • J * ; that is, £o Jo Equation (2.111) may be simplified with little error by making the following 56 FOUNDATIONS FOR MICROWAVE ENGINEERING approximation: 4<rZ.„Z* aTT* (Zm+Z0)(Zra+Z0)* 4aZ„,Z* 8 Z02 a8*Z* whence (2.111) becomes 1 P (2Y0E1)(2YuEt) <~2 ^ (2112> Note that 2YQEl is the value of the magnetic field, tangent to the surface that would exist if a were infinite. Hence an excellent approximate technique for evaluating power loss in a conductor is to find the tangential magnetic field, say H n that would exist for a perfect conductor, and then compute the power loss according to the relation P, = \ Re(H,H?Zm) = ±Re(JsJ:Zm) (2.113) This procedure is equivalent to assuming that the metal exhibits a surface impedance Z„, and the current is essentially the same as that which would exist for infinite conductivity. The procedure outlined above for power-loss calculations is widely used in microwave work. Although the derivation was based on a consideration of a very special boundary-value problem, the same conclusions result for more complex structures such as conducting spheres and cylinders. In general, the technique of characterizing the metal by a surface impedance Z„, and assuming that the current J, is the same as that for infinite conductivity is valid as long as the conductor surface has a radius of curvature at least a few skin depths in magnitude. 2.10 POTENTIAL THEORY The wave solutions presented in the previous sections have all been sourcefree solutions; i.e., the nature of the sources giving rise to the field was not considered. When it is necessary to consider the specific field generated by a given source, as in antenna problems, waveguide and cavity coupling, etc., this is greatly facilitated by introducing an auxiliary vector potential function A. As will be seen, the vector potential A is determined by the current source, and the total electromagnetic field may be derived from A. Since V • B = 0 always, this condition will hold identically if B is expressed as the curl of a vector potential A since the divergence of the cur of a vector is identically zero. Thus let B = VXA j (2.114) The assumed time dependence e "' is not written out explicitly in (2.114' ELECTROMAGNETIC THEORY 57 since this is a phasor representation. The curl equation for E gives V X E = -v'wB = -jcoV X A or V X (E + > A ) = 0 Now the curl of the gradient of a scalar function <I> is identically zero; so the general integral of the above equation is E + jtoA = -V4> or E = -juA - V<f> (2.115) Substituting this expression into the V X H equation gives V x H = - V X V x A =jaieE + J = w2cA -ju>eV<$ + J (2.116) Up to this point the divergences of A and V<t> have not been specified [note that (2.114) specifies the curl of A only]. Therefore a relation between V • A and $ may be chosen so as to simplify (2.116). Expanding V X V X A to give VV • A - V2A enables us to write (2.116) as VV • A - V2A = k2A - j ' w e / i V * + /xJ where k2 = w2p.e. If now the following condition is specified: VV • A = -jo>e/iV<i> or V - A = -jcop.e<P (2.117) V2A + k2A = -IMJ (2.118) this equation simplifies to Thus A is a solution of the inhomogeneous Helmholtz equation, the current J being the source term. The condition imposed on V • A and <t> in (2.117) is called the Lorentz condition in honor of the man first to propose its use. In the preceding derivation three of Maxwell's equations have been used and are therefore satisfied. The fourth equation, V • D = p, must also hold, and this will be shown to be the case provided the Lorentz condition is obeyed. Hence the three equations (2.114), (2.115), and (2.118), together with the Lorentz condition (2.117), are fully equivalent to Maxwell's equations. To verify the equation V • D = p, take the divergence of (2.115) to obtain V - e E = - y W V • A - e V2<t> (2.119) where e is a constant. Using the Lorentz condition yields V -A V • D = -jcoeV • A - V 2 — : = -JtO/J. 1 JOJ/l 1 V • (V 2 A + A2A) = - — V • J JIO by using (2.118) and noting t h a t V2V • A = V • V2A; that is, these differential operators commute. Now V • J = -jcop from the continuity equation; 58 FOUNDATIONS FOR MICROWAVE ENGINEERING so we obtain VD = - — (-jcop)=p If, instead of eliminating <$> in (2.119), V • A is eliminated by use of the Lorentz condition, we get V • D = p = -jcue{ -jojfie<l>) - e V2<t> V2<1> + £2<T> = — or (2.120) Hence the scalar potential <P is a solution of the inhomogeneous scalar Helmholtz equation, with the charge density p as a source term. For the time-varying field, J and p are not independent, and hence the field can be determined in terms of A and J alone. The scalar potential can always be found from the Lorentz relation, and p from the continuity equation, but explicit knowledge of these is not required in order to solve radiation problems. For convenience, the pertinent equations are summarized here: B = V X A (2.121a) E = -jioA - V4> = -j'aiA + VV • A 2 k A + VV • A V2A + k2A = - M J (2.1216) (2.121c) where the Lorentz condition was used to eliminate V<P in (2.1216). Note that, in rectangular coordinates, (2.121c) is three scalar equations of the form V2AX + k2Ax = -fiJx 2 but that, in other coordinate systems, V A must be expanded according to the relation V2A = V V - A - V x V x A . The simplest solution to (2.121c) is that for an infinitesimal current element J(x',y', z') = J ( r ' ) located at the point x',y', z', as specified by the position vector r' = axx' + ayy' + azz', as in Fig. 2.17. This solution is A ( x , y , z ) = A ( r ) = — J ( r ' ) - — dV 4ir (2.1& ti *,y.z) F I G U R E 2.17 Coordinates used to describe vector potential a current sheet. ELECTROMAGNETIC THEORY 59 where R = |r — r ' | is the magnitude of the distance from the source point to the field point at which A is evaluated; i.e., R=[(x-x')2 (y-y'f + .1/2 (z-z,)2\ + and J(r')rfV' is the total source strength. In terms of this fundamental solution, the vector potential from a general current distribution may be obtained by superposition. Thus, adding up all the contributions from each infinitesimal current element gives ,-jkR -J*|r- A(r) = — / J(x\y,z')—-dx-dy'dz'= — / J ( r ' ) 4ir JV R 4-TT Jy dV \r — r I (2.123) where the integration is over the total volume occupied by the current. Note that the solution for A as given by (2.122) is a spherical wave propagating radially outward from J and with an amplitude falling off as 1/R. The solution (2.123) is a superposition of such elementary spherical waves. *2.11 DERIVATION OF SOLUTION FOR VECTOR POTENTIAL In this section a detailed derivation of the solution to the inhomogeneous Helmholtz equation for a unit current source is given. A unit source is a source of unit strength, localized at a point in space (a familiar example is a point charge). Such a unit source in a three-dimensional space is a generalization of a unit current impulse localized at a time /' along the time coordinate. A current pulse is represented by the Dirac delta function 8(t - t') in circuit theory, where 8(t - t') has the property 8(t -t') = 0 t * f (2.124a) and at t = t' it becomes infinite but is integrable to give ( '5(t - t')dt = 1 (2.1246) A further property is that, for any function f(t) which is continuous at t\ I' +r nt)8(t-?)dt=nn (2.124c) J 4' _ - This result follows since r can be chosen so small that, in the interval t' - T < t < t' + T, the function fit) differs by a vanishing amount from fit') since f(t) is continuous at t'. Hence (2.124c) may be written as f(f)[ by virtue of (2.1246). T 8(t-t')dt=f(t') 60 FOUNDATIONS FOR MICROWAVE ENGINEERING As the preceding discussion has shown, the delta function is a conve nient mathematical way to represent a source of unit strength localized at point along a coordinate axis, in the above example along the time axis. I n an AT-dimensional space a product of N delta functions, one for each coordinate, may be used to represent a unit source. Thus, in three dimen sions, a unit source is represented by 8(x-x-)8(y-y')8(z-z-) = S(r - r ' ) (2.125) where Sir — r ' ) is an abbreviated notation for the product of the three one-dimensional delta functions. The source function Sir — r') has the following properties: 5(r-r')=0 r#r' (2.126a) (J /F(r)8(r-r')iV-P<-") *' " V (2.126c) -V \0 r' not in V where F is an arbitrary vector (or scalar) function that is continuous at r', that is. at x',y',z'. These properties follow from the properties of the one-dimensional delta functions that make up Sir — r'X A unit current source directed along the unit vector a at r' may be expressed as J = a6Xr — r')- The vector potential is a solution of V2A + k2A = -/xa<5(r - r ' ) (2.127) Since the current is in the direction a, the vector potential must also be in this direction, and hence A = A a . Equation (2.127) may therefore be written as a scalar equation: V2A + k2A = -ti8(r - r') (2.128) At all points r # r', A is a solution of V2A + k2A = 0 (2.129) If the source point r' is considered as the origin in a spherical coordinate system, then, since no angle variables occur in the source term in (2.128), the solution for A must have spherical symmetry about the source point r • Thus, in terms of the spherical radial coordinate R = | r - r' I, which is the radial distance from the origin at r', (2.129) is a function of R only and may be written as R2 dR\ dR) 2 d A 2 dA 2 _ + n / n j +k A=0 (2.1« K dR R dR 2 after expressing the independent part of V in spherical coordinates. o r 2 ELECTROMAGNETIC THEORY anticipation of a spherical-wave solution, let A = f(R)e in (2.130) leads to the following equation for f(R): d2 f / 2 \df jkR 61 . Substitution 2jk which is readily verified to have the solution f «= C/R, where C is an arbitrary constant. Consequently, the solution to (2.129) is A = Ce~jkR/R. This solution is singular at R = 0, and the singularity must correspond to that of the source term at this point. To determine the constant C, integrate (2.128) throughout a small sphere of radius r 0 centered on r' and use the delta-function property (2.1266) to obtain 2 [2vT f\v A J '0 k2A)R2 sine dedcbdR + 0 - J v I (V 2 A + k2A) dV = J v -n[8(r- r')dV= -u Now the integral of the term k2R2A, which is proportional to R2, will vanish as r 0 tends to zero. Hence, for sufficiently small r 0 , f VlAdV = -V -a Since V2A = V • VA, the divergence theorem may be used to give j V2AdV = <p VA • dS = j) VA • a r r 0 2 rfft since dS = arr£ dfl, where dil is an element of solid angle. Since A is a function of R only, VA = ar(dA/dR), and hence „ t 9A dA nr r2(h VA • ar d f l = r 0 2 6 —- rffl = ^r2— = ~n •'S •'S«•« "•" Evaluating dA/dR for R = r0 shows that <?A 4ir/ 2 "° 5fl /£ = e~-'*ro \ 4irCr - * r2o in the limit as r 0 tends to zero. Hence 4irC = jx, or C = U / 4 T T , in order for the singularity in the solution for A to correspond to that for a unit source. The above solution for the vector potential from a unit source, namely, e-jklr-r'< M A(r) = : -a 47r | r - r | (2.131) is clearly a function of both the source point and field point. Since (2.131) is the solution for a unit source, it is often called a Green's function and 62 FOUNDATIONS FOR MICROWAVE ENGINEERING denoted by the symbol G as G ( r l r ' ) = G ( r | r ' ) a = G ( x , y , z\x',y\ z')a = H e -y*|r-r'| 7T* AIT \T (2. 132, because, by definition, a Green's function is the solution of a different equation for a unit source. The vector potential from a general current distribution may now expressed in the form ,-/*fr-r'i A(r) = — / J ( r ' ) - dV = f J ( r ' ) G ( r | r ' ) dV (2. 133) v r — r -V since any current distribution J may be considered as a sum of weight unit sources. ATT 2.12 J LORENTZ RECIPROCITY THEOREM The Lorentz reciprocity theorem is one of the most useful theorems in £ solution of electromagnetic problems, since it may be used to deduce number of fundamental properties of practical devices. It provides the basi for demonstrating the reciprocal properties of microwave circuits and showing that the receiving and transmitting characteristics of antennas are the same. It also may be used to establish the orthogonality properties of the modes that may exist in waveguides and cavities.t Another impo: use is in deriving suitable field expansions (analogous to a Fourier seri expansion) for the fields radiated or coupled into waveguides and cavities probes, loops, or coupling apertures. To derive the theorem, consider a volume V bounded by a clo: surface S as in Fig. 2.18. Let a current source J, in V produce a fie: E,, H , , while a second source J 2 produces a field E 2 , H 2 . These fields satisfy Maxwell's equations; so V x E , = -j'w/iH, V X H, =jcoeEi + J, V x E 2 = -jojfiH2 V X H2 = j W E 2 + J2 Expanding the relation V • (E, X H 2 - E 2 X H , ) and using Maxwell's equs tion show that V -(E, X H 2 - E 2 X H , ) = ( F x E , ) - H 2 - ( V X H 2 ) -E, - (V xE2) • H, + ( V X H , ) • (2.134) - J 2 ' E j + «Ji • E 2 t i n any waveguide or cavity an infinite number of field solutions are possible. Any one sow is called a mode for the same reason that the various solutions for vibrating strings membranes are called modes. Orthogonality of modes is discussed in Sec. 3.14. . ELECTROMAGNETIC THEORY 63 FIGURE 2.18 Illustration of the Lorentz reciprocity theorem. Integrating both sides over the volume V and using the divergence theorem give ( V - ( E , x H 2 - E 2 X H . ) d V = Td>(E. X H 2 - E , x H , | -ndS v s J = J / ' ( E 2 - J j - E , - J2)dV v (2.135) where n is the unit outward normal to S. Equation (2.135) is the basic form of the Lorentz reciprocity theorem.! For a number of typical situations that occur, the surface integral vanishes. If S is a perfectly conducting surface, then n X E, = n X E, = 0 on S. Since E l X H 2 • n = (n X E,) • H 2 , etc., the surface integral vanishes in this case. If the surface S is characterized by a surface impedance Z„„ then, according to (2.71), E/=-Z„,nxH or n X E = -Zmn X (n X H) [note that in (2.71) n points into the region occupied by the field, and hence the minus sign is used here, since n is directed out of V]. Consequently, ( n x E , ) - H 2 - ( n x E 2 ) • H, = - Z m [ n X ( n X B J ] • H 2 + Z m [ n x (n x H 2 ) ] • H t = Zm(n X H.,) • (n x H , ) - Z m ( n X H , ) • ( n x H 2 ) = 0 and the surface integral vanishes again. t i n anisotropic media with nonsymmetric permittivity or permeability tensors, a modified form must be used. See, for example, R. F. Harrington and A. T. Villeneuve, Reciprocity Relations for Gyrotropic Media, IRE Trans., vol. MTT-6, pp. 308-310. July, 1958. 64 FOUNDATIONS FOR MICROWAVE ENGINEERING Another example where the surface integral vanishes is when <J • chosen as a spherical surface at infinity for which n = a r . The radiated fi i"8 at infinity is a spherical TEM wave for which H = YarXE= I-I arxE Therefore ( n x E.) • H 2 - ( n x E 2 ) • H , = Y ( a r x E,) • ( a r X E 2 ) -F(arXE2) •(arXEl)=0 and the surface integral vanishes. Actually, for any surface S which encloses all the sources for the field the surface integral will vanish. This result may be seen by applying (2.135) to the volume V bounded by S and the surface of a sphere of infinite radius. There are no sources in this volume, and since the surface integral over the surface of the sphere with infinite radius is zero, we must have, from (2.135), 6T (Ej X H 2 - E 2 X H J • ( - n ) dS = 0 s = (f) ( E , X H 2 - E 2 X B y • n dS ~s Hence the surface integral taken over any closed surface S surrounding all the sources vanishes. When the surface integral vanishes, (2.135) reduces to (Er J2dV= (E2- J,dV (2.136) If J, and J 2 are infinitesimal current elements, then E1(r2)-J2(r2)=E2(r1)-J1(r1) (2-13?) which states that the field E x produced by J x has a component along J2 t h a is equal to the component along J, of the field radiated by J 2 when Ji a" J 2 have unit magnitude. The form (2.137) is essentially the reciprocity principle used in circuit analysis except that E and J are replaced by voltage V and current /. The applications of the reciprocity theorem * illustrated at various points throughout the text and hence are not discussed further at this time. ELECTROMAGNETIC THEORY 65 2 . 1 . An atom of atomic number Z has a nuclear charge Ze and Z electrons revolving around it. As a model of this atom, consider the nucleus as a point charge and treat the electron cloud as a total charge - Z e distributed uniformly throughout a sphere of radius r0. When an external field E is applied, the nucleus is displaced an amount x. Show that a restoring force x(Ze)2/4wrye0 is produced and must be equal to ZeE. Thus show that the induced dipole moment is p = 4—e0rftE and is linearly related to E. 2.2. In a certain material the equation of motion for the polarization is d.96 V— + u>%& = 2e 0 <ogr d'l.9> IF where W is the total field in the dielectric. Find the relation between & and W when r = Re(£e-""") and E is real. If w 0 = 1 0 " and v = 1 0 " \ over what frequency range can a relationship such as 2 = eg = e,,^ +.9" be written if it is assumed that t h e criterion to be used is that the phase difference between $ and % should not exceed 5°? Plot the magnitude and phase angle of the dielectric constant er = e/e0 = (e' - je")/e() as a function of at. 2.3. A dielectric material is characterized by a matrix (tensor) permittivity e 7 3 -2v^5 3 7 -2v/0" -2/6T5 -2v / 0!5 10 xz «0 yy <=y? — f.~ 4 when referred to the xyz coordinate frame. If the coordinate axis is rotated into the principal axis u, v, w, the permittivity is exhibited in diagonal form: H= s«« 0 0 0 em 0 0 0 Find the principal axis and the values of the principal dielectric constants euu/e0, etc. Hint: By definition, along a principal axis a scalar equation such as A, = euuEu holds. In general, if D is directed along a principal axis, then \DX] Dy D> e 0 4 7 3 -2vfr5 3 7 -2v^5 \E, =A E y E, B, -2V/0JT] E -2/61* 10 \ '~ E or in words, when D is directed along a principal axis, it is related to E by a scalar constant A. The above constitutes a set of three homogeneous equations, of which the first is 66 FOUNDATIONS FOR MICROWAVE ENGINEERING Verify that, for a solution, the following determinant must vanish: 7 - 4A/e 0 3 -2v / 0!5 3 7 - 4A/e 0 -2v/olf -2\Z6~lf -2v^5" 10 - 4A/« 0 = 0 This cubic equation gives three roots for A, which may be identified « W * a » e w w F o r ^ o n e r o o t . s a y euw t h e components of a vector dir~ along the corresponding principal axis are proportional to the cofactors above determinant. This type of problem is called a matrix eigenvalue prob The A's are the eigenvalues. Answer: = 2« n 3e„ Unit vectors along the principal axis are *« - 0.5a, 0.5av - \fo.5a. a „ = 0.5a + 0.5a y + '0.5 a , = / 0 . 5 a. - / 0 . 5 a , 2.4. In the interior of a medium with conductivity a and permittivity e, free c is distributed with a density p0(x,y,z) at time t = 0. Show that the decays according to Poe -*/» e a Evaluate the relaxation time T for copper for which a = 5.8 x 10 7 S / m , e j Find r for sea water also for which <r = 4 S / m and € = 80« 0 . If the relaxati time is short compared with the period of an applied time-harmonic field, th will be negligible accumulation of free charge and V • D may be assumed to zero. What is the upper frequency limit for which this is true in the case copper and sea water, i.e., the frequency for which T is equal to the period? Hint: Use the continuity equation, Ohm's law, and the divergence tion for D. 2.5. Show that, when the relaxation time for a material is small compared with period of the time-harmonic field, the displacement current may be neglected comparison with the conduction current. 2.6. Consider two concentric spheres of radii a and b. The outer sphere is kept at potential V, and the inner sphere at zero potential. Solve Laplace's equation spherical coordinates to find the potential and electric field between the spn Take b > a. 2.7. Solve Laplace's equation to find the potential and electric field between coaxial cylinders of radii a and b if the center cylinder is kept at a potenti and the outer cylinder at zero potential. Take b > a. 2.8. Derive (2.45) from (2.18). 2.9. Derive (2.47). 2.10. Express the scalar Helmholtz equation V 2 0 + fc'-ty = o in cylindrical coo nates. If </- = f(<t>)gir)h(z), find the differential equations satisfied by /• and h. ELECTROMAGNETIC THEORY 67 2.11. When material polarization £P and Jt are explicitly taken into account, show that the wave equations satisfied by % and 2? are Si* -VV-*+nae0—t--Vx<?-VxS V2^r-Moe0—i-= a2g <12&> df d 1 dt at at at e0 VV -g> e0 Note that V -3S = 0; so V • JT = -V -JT and V -3 = p; so V • €„£• = p - V • .9". Examination of the source terms in the above equations shows that d<P/at is a polarization current analogous to conduction current f. 2.12. Derive (2.62). 2.13. Between two perfectly conducting coaxial cylinders of radii a and b, b > a, the electromagnetic field is given by E = arE0r~ V * ° 2 H = &d,Y0E0r-1e-J''°2 where k 0 = oi(ji 0 € 0 ) 1 '' 2 , Y 0 = ( e 0 / / x 0 ) 1 / 2 . Find the potential difference between the cylinders and the total current on the inner and outer cylinders. Express the power in terms of the voltage and current, and show that it is equal to that computed from an integration of the complex Poynting vector over the coaxial-line cross section. Show that the characteristic impedance of the line is V/I = ( Z n / 2 i 7 ) l n ( 6 / a ) = 601n(6/a), where V is the voltage and / is the total current on one cylinder. 2.14. A round wire of radius r 0 much greater than the skin depth <5S has a uniform electric field E applied in the axial direction at its surface. Use the surface-impedance concept to find the total current on the wire. Show that the ratio of the ac impedance of the wire to the dc resistance is Evaluate this ratio for copper at f = 10 6 Hz for a = 5.8 X 10 7 S / m , r„ = 0.1 cm, ix = n0. 2.15. The half-space z > 0 is filled with a material with permittivity e„ and permeability ix * /x0. A parallel polarized plane TEM wave is incident at an angle 0„ as in Fig. 2.14. Find the reflection and transmission coefficients for the electric field. Does a Brewster angle exist for which the reflection coefficient vanishes? 2.16. Repeat Prob. 2.15 for the case of a perpendicular polarized incident wave. Does a Brewster angle exist? If so, obtain an expression for it. 2.17. The half-space z > 0 is filled with a material with permeability M and permittivity e. When a plane wave is incident normally on this material, show that the reflection and transmission coefficients are Z - Z0 r = -—=? z 0 where Z = (ix/e)l/1, component only. 2Z T= i +r= z +z 0 Z0 = (n0/e0)1/2. Choose an electric field with an x 68 FOUNDATIONS FOR MICROWAVE ENGINEERING 2.18. The half-space z > 0 is filled with a material of permittivity e 2 and fi = ix0. A second sheet with permittivity e, is placed in front. A plane wavnl incident normally on the structure from the left, as illustrated. Verify that th* reflection coefficient at the first interface vanishes if e l = ( € 2 « 0 ) I / 2 and tl thickness d = j A 0 ( e 0 / e i ) , / B . The electric field may be assumed to have an component only. The matching layer is known as a quarter-wave transform (actually an impedance transformer). This matching technique is used reduce reflections from optical lenses and is called lens blooming, or coated lenses. ffc. FIGURE P2.18 2.19. In terms of the vector potential A from a short current element A?J 0 a, located at the origin, show that the radiated electric and magnetic fields are H E = — + r 0 &2 jZ0 2-n- ku — ja^sinffe ijk0 -F + -r IQAZJZQ/ 4IT k0 0 r Jk r " arcosfle J*o r2 Jk r « — ) a,, sin fl e - j * o r Hint: Use (2.122) and (2.121), and express A as components in a spherical coordinate system r, 0, <#. Note that az = ar cos ff - a„ sin B. 2.20. A dielectric may be characterized by its dipole polarization P per unit volume If p = J = 0 and P is taken into account explicitly, show that, if a vectoi potential A is introduced according to B = V X A, then A is a solution of v"2A + k%A = -j<onnP and that the fields are given by VV • A + k'lA B = V XA E = -7<OMoeo Note that a Lorentz condition is used. Thus an electric dipole P is equiv" to a current element ; w P , or alternatively, a current element «J && considered as an electric dipole P = J//'«». ELECTROMAGNETIC THEORY 69 2 FIGURE P2.21 2.21. A small current loop constitutes a magnetic dipole M = / S a , where / is the current, S the area of the loop, and a a vector normal to the plane of the loop and pointing in the direction that a right-hand screw, rotating in the direction of the current, would advance. The field radiated by such a current loop, with linear dimensions much smaller than a wavelength, may be obtained by a potential theory analogous to that given in Prob. 2.20 by treating the loop as a magnetic dipole M. Thus replace B by fiQH + # „ M in Maxwell's equation and treat M as a source term. Since p is zero, V • D = 0, and this permits D to be expressed as D = -V X A m , where A m is a magnetic-type vector potential. By paralleling the development in the text for the potential A, show that the following relations are obtained: v 2A ' m + k'fAm = D- H -jiuix0e0M -VxA,. klAm + VV • A, Hence, for a z-directed magnetic dipole at the origin, A„, = jtofi0e0M <-j>i<ir 4-n-r from which the fields are readily found. Note that in this problem M represents the magnetic dipoJe source density in Maxwell's equations, but in the solution for the vector potential it represents the total magnetic dipole strength. It would have been more consistent to use M S(r - r') to represent the source density, where <5(r - r') is the three-dimensional Dirac delta function which has the property / S ( r - r-)dV'= 1 rinV Jv 2.22. Consider an arbitrary current element J, in front of a perfectly conducting plane. This current radiates a field E, having zero tangential components on the conducting plane. Use the Lorentz reciprocity theorem to show that a current J 3 parallel to the conducting plane and an infinitesimal distance in front of it does not radiate. 70 FOUNDATIONS FOR MICROWAVE ENGINEERING REFERENCES 1. Kraus, J. D.: ••Electromagnetics," 3rd ed., McGraw-Hill Book Company, New York, 1 9 ^ 2. Hayt, W. H., Jr.: "Engineering Electromagnetics," 5th ed., McGraw-Hill Book Compan New York, 1989. *' 3. Schwarz, S. E.: "Electromagnetics for Engineers," Holt. Rinehart, and Winston, In, Philadelphia, 1990. 4. Johnk, C. T. A.: "Engineering Electromagnetic Fields and Waves." 2nd ed., John Wilev & Sons, Inc., New York, 1988. 5. Wait, J. R.: "Electromagnetic Wave Theory." Harper & Row Publishers, Inc., New Y0I^ 1985. 6. Stratton, J. A. "Electromagnetic Theory," McGraw-Hill Book Company. New York, 194] 7. Shen, L. C, and J. A. Kong: "Applied Electromagnetism," Brooks-Cole, Calif., 1983. CHAPTER 3 TRANSMISSION LINES AND WAVEGUIDES This chapter is a long one and for this reason has been divided into three parts, namely: P a r t 1—Waves on transmission lines Part 2—Field analysis of transmission lines Part 3—Rectangular and circular waveguides The three parts are closely related but independent with the exception of Sec. 3.7, which is needed as an introduction to both Parts 2 and 3. With the exception of this section, the three parts can be studied independently and in any order. In Part 1 we give an introduction to waves on transmission lines using a distributed-circuit model of the transmission line. By using the distributed-circuit model, we are able to study the excitation and propagation of current and voltage waves on a transmission line without the need to invoke Maxwell's equations. The electrical characteristics of a transmission line such as the propagation constant, attenuation constant, characteristic impedance, and the distributed-circuit parameters can only be determined from a knowledge of the fields surrounding the transmission line. Thus in Part 2 we carry out a detailed field analysis of transmission lines. This part also includes an extensive discussion of planar transmission-line structures such as the microstrip line. 71 72 FOUNDATIONS FOR MICROWAVE ENGINEERING Part 3 presents the theory for waves in hollow rectangular and circular waveguides (pipes). In the beginning section of Part 2, we show that Maxwell's equations can be separated into equations that describe threa types of waves. These are transverse electromagnetic waves (TEM) tran verse electric (TE), and transverse magnetic (TM) waves. The TEM wave i the principal wave that can exist on a transmission line. The TE and TM waves are characterized by having no axial component of electric and magnetic field respectively. The TE and TM waves are the fundamental wave types that can exist in hollow-pipe waveguides. Hollow-pip e waveguides do not support TEM waves. The ability to reduce Maxwell's equations into three set of equations, one set for each wave type, facilitates the analysis of transmission lines and waveguides. Thus this decomposition of Maxwell's equations is carried out in the first section of Part 2. PARTI W A V E S O N T R A N S M I S S I O N LINES In this section we introduce the topic of voltage and current waves on a two-conductor transmission line by using a distributed-circuit model of the transmission line. This allows us to explore a number of fundamental properties of one-dimensional waves without having to consider the electromagnetic fields in detail. The distributed-circuit-model approach has limitations and in general must be replaced by a detailed solution for the electromagnetic field associated with the guiding structure if we want to determine the distributed-circuit parameters. The field analysis of transmission lines is presented in Part 2. 3.1 WAVES ON AN IDEAL TRANSMISSION LINE In Fig. 3.1a we show a two-conductor transmission line consisting of parallel round conductors (wires). The conductors will be assumed to perfect, i.e., have infinite conductivity. The conductors extend from z = « infinity, thus forming a semiinfinite transmission line. At 2 = 0 a volt generator with internal resistance R g is connected to the transmission The generator produces a voltage 7'git) that is impressed across the tr mission line. If the generator is switched on at time t = 0, a current will begin to flow into the upper conductor. A return current S\t> then flow on the lower conductor since current flow through the gener must be continuous. The return current is produced by the action t&j electric field established between the two conductors. Since the transffl'1 line is semiinfinite in length, there is no direct conducting path between upper and lower conductors. However, there is a distributed capacitan TRANSMISSION LINES AND WAVEGUIDES 73 Ml) * * .S{t) (a) ./{z.D • IfoTP 1 lb) (d ~Jr(z.t)+~dz F I G U R E 3.1 ( a ) An ideal two-conductor transmission line connected to a voltage generator; (b) equivalent circuit of a differential section of the transmission line with no loss; (c) equivalent circuit seen by the generator. per meter between the two conductors; so we have a capacitive or displacement current flowing from the upper conductor to the Jower conductor. The electric current results in a magnetic field around the conductors and consequently the transmission line will also have a distributed series inductance L per meter. We can model a differentia] section dz of this transmission line as a series inductance Ldz and a shunt capacitance Cdz as shown in Fig. 3.16, If the conductors had finite conductivity, we would also need to include a series resistance in the equivalent circuit of a differential section. However, we are assuming that the conductors are perfect; so the series distributed resistance R per meter is zero. Since electrical effects propagate with a finite velocity v (the speed of light in vacuum), it should be clear that the voltage 7'(z,t) and current J{z, t) at some arbitrary point z on the transmission line will be zero until a time z/v has elapsed after switching the generator on. We will show that the generator launches voltage and current waves on the transmission line that propagate with a finite velocity. The equations that describe these waves are established by applying Kirchhoff s circuit laws to the equivalent circuit of a differential section of the transmission line, along with a specification of the terminal relationships (boundary conditions) that must hold at the generator end. At some arbitrary point z on the transmission line, let the voltage and current be given by 7/'{z, t), ^ ( 2 , t). At a differential distance dz further 74 FOUNDATIONS FOR MICROWAVE ENGINEERING along, the voltage and current have changed by small amounts (dV/dz) dz and (d.f/dz)dz; so the output voltage and current at z + dz will be J(z + dz,t) =T(z,t) dz dz d.y(z,t) ; -dz dz The sum of all potential drops around the circuit must be zero; so we have djr BV -%-'+ Ldz— + T+ — dz = 0 dt dz .7{z + dz,t) =.f(z,t) + dV{z,t) d.S(z,t) (3.1a) dz dt The sum of currents flowing into the output node must also be zero; so we can write — or Li ~ dT J - CdzdS(z,t) dt dJ S - —- dz = 0 dz df'(z,t) (3.16) dt dz These two partial differential equations describe the relationship between the voltage and current waves on the transmission line. We can obtain an equation for the voltage 7/(z,t) by differentiating (3.1a) with respect to z and using (3.1b) to eliminate the current; thus or d2<r(z,t) dz* d*s ' = -L- dzdt 2< d T(z,t) or dz' In a similar way we obtain - LC ( dH' -L\-Cdt' = d2V'(z,t) ^r^- = 0 dt' d2jr{z,t) dz* LC dt2 =0 (3.2a) (3.26) The product LC has the dimensions of one over velocity squared. These equations are one-dimensional wave equations and describe waves propag 8 ing with a velocityt P-JLr {LC t F o r an ideal transmission line in air. v = c -* 3 x 10 m / s , t h e velocity of light. TRANSMI&SION LINES AND WAVEGUIDES 75 Consider the equation 1 d2V dV We can readily show that any two arbitrary functions of the form f+(t z/v) and f~(t + z/v) are solutions of this equation. If we let w = t — z/v then we have df+(t-z/v)^ _ 9f{w) dz dz a*r{u>) and For d2r/dt2 _ dz2 df*(w) dw dw l'T(w) dz v dw i *T(">) v2 dw2 we get d^f'/dw2. Consequently, a2f 1 d2f+ d2r I 1 dz2 v2 dt2 dw2 I v2- -v^2 =o so f*U ~ z/v) is clearly a solution of the one-dimensional wave equation, as is f~(t + z/v). The function f*(t - z/v) is the same as the function f*it) but delayed in time by an amount z/v which equals the distance z divided by the velocity of propagation v. We interpret this solution as a wave propagating in the positive z direction and identify this solution with a superscript + sign. The other solution represents a wave propagating in the -z direction and is identified by the - sign as a superscript. The general solution for the voltage waves on the transmission line is T\z>t) = V-r[t-Z-^ + V r i ( + ^j (3.4) where V + and V' are amplitude constants. By using (3.16) we see that ay — = dz JirJf , St -clvl+— at+ v~-at If we assume that S is of the form s{z)=rr(t~Z-)-rr(t+zthen " I(r£+J^ dz v \ at at upon using af* df* d(t + z/v) _ l af az a(t + z/v) az v at An examination of these equations shows that the assumed solution for 76 FOUNDATIONS FOB MICROWAVE ENGINEERING J'i.z, t) is compatible with that for the voltage p'Cztt) if we choose r= vcv^ /-= ucv- The parameter vC has the dimensions of an admittance and is also equa] tn C/ \lLC = ^C/L . The characteristic admittance Yc of the transmission lin e is defined by this parameter. The reciprocal parameter is called the charan teristic or surge impedance of the transmission line. It is given by -^=yr (3. By using this parameter the solution for the current waves on the transmission line can be expressed in the form V- i z\ V- I z The negative sign preceding the wave with amplitude V~ indicates a reversal in the direction of current flow for the wave propagating in the ~z direction. For the transmission-line circuit shown in Fig. 3.1, the generator launch voltage and current waves propagating in the +2 direction. Since the transmission line extends to infinity, no waves propagating in the -2 direction will be present. Later on we will consider a transmission line that is terminated at z => I with either a resistance, capacitance, or a combination of these elements. Waves propagating in both the +z and -z directions will then exist. For the present case the voltage and current waves on the transmission line are assumed to be nzj) = v+r[t-z-) s(z,t) = rr[t- + + withV =7 Z c At the generator end z = 0 the terminal conditions require that •S(0,t) =,Se where S B is the current supplied by the generator. These terminal cor tions can be expressed in the form «z"r(0 V* + v+r <'> TRANSMISSION LINES AND WAVEGUIDES 77 from which we find t h a t V (3.7) ^<> = Y±R^ The voltage wave launched on the transmission line is thus given by ^^-z^A'-^) (3 8a) - with a corresponding current wave •^•<>-vnr^-£ (3.86) At any point on the transmission line, the voltage waveform is the same as that produced by the generator but delayed in time and reduced in amplitude by the factor Z,./(Rg + Zc). The voltage reduction is the usual voltage t = z/v fa) -+-z S XJ -*-£ X7 X7 w *? z = vt F I G U R E 3.2 Time-distance and distance-time plots of voltage waveform '/^(z, t) on a transmission line for a single-cycle sinusoidal generator voltage pulse. 78 FOUNDATIONS FOR MICROWAVE ENGINEERING division factor associated with the equivalent circuit shown in Fig. 3.2C p the infinite line the generator sees only an equivalent impedance Z c equal t the transmission-line characteristic impedance. In Fig. 3.2a we show a time-distance plot of the voltage waveform on transmission line for the case when the generator produces a single cycle r a sinusoidal waveform, i.e., 7g(t) = V 0 s i n / 0 < t < 2TT Figure 3.2a shows ^ ( z , / ) as a function of t at various distances z, while Fig. 3-26 shows 7''(z,t) as a function of z for various values of t. In the latter plot note that the leading edge of the waveform is the initial voltage produced by the generator at time t = 0 and hence the waveform appears reversed when plotted as a function of z. 3.2 TERMINATED TRANSMISSION LINE: RESISTIVE LOAD In Fig. 3.3 we show a transmission line terminated at a distance / from generator in a load resistance RL. At the load end the terminal conditions are ^ ( M ) = n ^ £ # i s{i,t)=.yL (3-9° (3.S (6) (c) F I G U R E 3.3 The terminated transmission TRANSMISSION LINES AND WAVEGUIDES 79 If we choose R L equal to the characteristic impedance ZL., then 2^ = JrLRi=Jri.Zc. For a wave propagating in the +z direction, ^(z,t) = ZeJ(z,t\ so that at z = /, T{l,t) = Z^l.t), which satisfies the load terminal condition. Thus by choosing R L = Z c the forward propagating wave will be completely absorbed by the load resistor and no reflected wave will be generated at the load end. Thus, in order to avoid a reflected wave, such as a reflected pulse in a digital circuit application, the transmission line should be terminated in its characteristic impedance. When RL* Z c the terminal conditions at the load end cannot be satisfied without introducing a reflected wave. The incident wave at z = Ms given by where V + is the amplitude of ?• j relative to 7/g. In order for a reflected wave to combine with the incident wave so as to satisfy the terminal conditions (3.9), the reflected wave must have the same time dependence as that of the incident wave. Hence the form of the reflected wave will be / z -v-^(< + 21 --- The argument must contain the factor t + z/v plus additional delay factors, so that at z = I the reflected wave has the form Tg(t - l/v). The reflected current wave is given by 1 At the load end the total current on the transmission line must equal the current JrL flowing through RL; thus 1 / I = S, -vr-TyrJt-- The total voltage on the transmission line must equal the load voltage; so (V~+V-)Vglt--\=VL=.fLRl, v. When we divide this equation by the first one, we obtain V* + VV* - V~ RL == ~Z~ 80 FOUNDATIONS FOR MICROWAVE ENGINEERING which gives V RL (3.1 + Z. The parameter T L is called the load voltage reflection coefficient. amplitude V~ is that of the reflected voltage wave and V~ is the ampljt* of the incident voltage wave. The ratio is determined by the conditions the load end only. Once a reflected wave has been launched from the load terminatio the total voltage on the transmission line will consist of the incident volt wave plus the reflected voltage wave until the time at which the refle wave reaches the generator end. If the generator internal impedance R equals the characteristic impedance Zc, then the reflected wave is absor" at the generator end. If R g # Z c then the reflected wave is reflected at generator end to produce another forward propagating wave. For the fleeted wave at the generator end, the terminal conditions are obtained short-circuiting the voltage generator. Thus the reflected wave sees a termi nation R and will be reflected with a reflection coefficient T given by ! Re-Zc (3.11 Ra + Z.. As long as the generator continues to produce a voltage 2^(0, it continu to launch a first forward propagating wave with voltage ^-(0, t) ZcTg(t)/(Rg + Zc) and with current J^(0, t) = Sg. Thus the superposi of a reflected wave at the generator end requires the launching of a seco forward propagating wave with a voltage amplitude that cancels that of reflected wave at the generator terminals, i.e., the generator is treated being short-circuited. The second forward propagating wave will also dergo reflection at the load termination, so that as time proceeds we will e up with a multitude of forward propagating and reflected waves on transmission line. This collection of waves can be described as follows: *-i v\s,t) = v+vt(t - ^ + rjr^lt + ^ — ) 2/ + rgrLv• » ; t + + rgilv T-e 2 t + + r*riv+yt t - v l -41 V Z + 4/ 2/ — v U\t- 31 U\t U t- 4/ + (3.1 where V+= Zc/(Rg + Zc) and U(t - a) is the unit step function wb» equals zero for t < a and equals unity for t SL a. The unit step function & TRANSMISSION LINES AND WAVEGUIDES 81 F I G U R E 3.4 Distance-time plot of a pulse undergoing multiple reflection on a transmission line when rL = -Te - 0.5. Reflection at the generator end causes a reversal in the polarity of the pulse. convenient function to use to specify when a waveform begins. In the case of multiple reflected waves on a transmission line, each reflected wave begins after time delays of l/u, 2l/v, 3l/v, etc., corresponding to the time delay to propagate a certain number of times back and forth between the generator end and load end.t The current wave can be obtained by multiplying the forward propagating waves by Y c and the backward propagating waves by —Y„. When the generator voltage 2 ^ ( 0 exists for only a finite time interval, the total voltage wave on the line will decay toward zero since each successive reflected wave is multiplied by a reflection coefficient, either Tt or r^, which is less than one in magnitude, and hence successive waves are of diminishing amplitude. The sequence of multiple reflected waves can be illustrated in a distance-time plot. In Fig. 3.4 we show this type of plot for a generator producing a rectangular pulse. We have chosen R L = 3Z C and R g = Z c / 3 so that Y L = 0.5, Y g = - 0.5. When the reflection coefficient is negative, the t T h e unit step functions were introduced for clarity in describing the physical process but are actually not required in (3.12) since 2^(* - T) = 0 for t < T. 82 FOUNDATIONS FOR MICROWAVE ENGINEERING sign of the reflected voltage wave is reversed and this is illustrated in p3.4. In high-speed digital circuits using interconnecting transmission U n multiple reflected pulses are undesirable and can be avoided by terminal' each transmission line in a resistance equal to its characteristic impedan 3.3 CAPACITTVE T E R M I N A T I O N When the transmission line is terminated in a reactive element such as capacitor as shown in Fig. 3.36, the reflected wave will have a waveform different from that of the incident wave. The solution for the reflected wave is readily found from the condition that the sum of the incident plus reflected voltage wave at z = / must satisfy the terminal conditions. The incident wave is again chosen to be ¥',(,1, t) = V+7SU - l/v). The reflected wave is initiated at time 1I = l/v and will propagate from the point z = i toward the generator. Therefore it is of the form For a capacitor we require d'ni,t) SL = c = Cl '-~dT dWr{t-l/v) dt dt where 7/'r{t - l/v) is the reflected voltage wave at z = I. In addition, we the condition SL = t-~ YC From these two equations we obtain d^t-l/v) 1 dt = -v+ / / cLzc • dVJt-l/v) dt V* + cLzc "K\t (3.13) r For a specific example we will consider the case when the generat° produces a rectangular pulse given by yg(t) = 1 0<t<T TRANSMISSION LINES AND WAVEGUIDES 83 The right-hand side of (3.13) will now become the source function V+ 1 cL z c t- -\-ult T t - -J +slt T where 8(t - a) is the Dirac delta function or impulse function t h a t arises from the derivative of the rectangular pulse. We can integrate (3.13) by introducing the integrating factor e ' / r where T = CLZC. We note that d K r dt ' I dVr 1 \ dt T so consequently (' J —7/-re,/Tdt = •TU - - ) e ' / r - 7^(0)e'/L' l/v dt \ vj = -f T H/v -vf t- -) - ult S\t T \-S\t T •l/rdt >"Tdt 'l/v V+(el/T I I - < t < - 2e'/UT) - V + / r V (-2e' " + 2e t,+vT)/ T ") V t> - + T Since we have included the impulse functions as derivatives of the applied rectangular pulse, the lower limit of integration is regarded to be just before t = l/v and thus 2 ^ 0 ) is equal to zero since it corresponds to 2^(0 - ). Hence we obtain "H*" r ( 1 V+(l-2e-""-')/"T) V+(-2e'/VT + l m 2e"+vT,/VT)e-'/r - < t < - + T v v I t>-+T v (3.14) At t = l/v the reflected wave has an amplitude equal to - V + which cancels that of the incident pulse. This is consistent with the requirement that initially the capacitor C L is uncharged and must have a zero voltage across it. The capacitor charges to a final voltage level K(T) = 2V + (1 -e~T/T) 84 FOUNDATIONS FOR MICROWAVE ENGINEERING f, -• v V 1I V L *• r 1 TT-r V -V y< ~ 1 +r F I G U R E 3.5 Incident and reflected voltage waveforms at z = I. at t = l/v + T and then discharges toward zero. The apparent discontinuous change that occurs in the reflected voltage wave at t = l/v + T is caused by the sudden drop to zero volts for the incident pulse, and in order to match the voltage across the capacitor, the reflected voltage wave must have a positive jump of value V4. The incident and reflected voltage waveforms at z = I are shown in Fig. 3.5. The reflected voltage waveform will propagate toward the generator and will begin to initiate a new forward propagating wave at time t = 2l/v. Clearly the capacitor has made a significant change in the waveform of the reflected wave. The analysis for the case of a capacitor-resistor termination as shown in Fig. 3.3c is similar. The terminal conditions are ^ = ^ G d?y'r t- L - t-~ so in place of (3.13) we have dK dt i 1 I dy\ 7/-= - - r, dt (3.15) cLzc The solution is similar to that for (3.13) except that the charging time constant is now TX = CLRLZC/(RL + Zc). Initially, Vr has a value equal - V * as before. In this case the capacitor charges toward a final voltag equal to V+\l C,RL RL-ZA + +Z RL c 2RL RL -F* determined by the steady-state voltage across R L if C L was absent. A' TRANSMISSION LINES AND WAVEGUIDES 85 t = l/u + T the capacitor voltage will be The reflected-wave voltage at this time will be V c ~ V*. When t becomes greater than l/v + T the incident voltage wave pulse drops to zero volts so the reflected-wave voitage jumps to a value equal to V c and will then decay toward zero with a time constant T ( . STEADY-STATE SINUSOIDAL WAVES When the generator produces a sinusoidal voltage 7/'g(t) = V cos wi, the steady-state voltage waves on the transmission line will be of the form cos a)(t - z/v) and cos <o(t + z/v). The steady state is achieved, for all practical purposes, after a few multiple reflections have occurred, since the amplitude of the successive reflected wave decreases quite rapidly because it is multiplied by V g or Y L upon each reflection. The solution for steady-state sinusoidal waves is most conveniently obtained using phasor analysis. The generator voltage is represented by VgeJwt. The voltage and current waves on the transmission line will then also have an eJ°" time dependence. The differential equations (3.1a) and (3.16) now become (the common time factor eJUI' is dropped) 3V(z) dz dl(z) = -jwLI(z) (3.16a) = -jioCV(z) (3.166) dz where Viz) and I(z) are complex phasor amplitudes. By eliminating the current we find that Viz) satisfies the equation d2V(z) - ^ - + u>2 -,V(2)=0 (3.16c) The solution for Viz) is of the form V(z) = V+e~J^ + V-eJf>z (3.17a) with a corresponding solution I(z) = I + e - * * - r > * (3.176) for the current waves. The constant /3 = io/v is the propagation phase constant. As before the current amplitudes are related to the voitage amplitudes through the characteristic impedance of the line, i.e., r=Ycv+ r=Ycv- When the time factor is restored, it is readily seen that e' 86 FOUNDATIONS FOR MICROWAVE ENGINEERING corresponds to a wave propagating in the +z direction, while eJliz+Ju" j s wave propagating in the -z direction. In the next section we will show that for a transmission line with finite conducting wires and possibly als surrounded with lossy dielectric materials, the waves attenuate in arnpli tude as they propagate. For this case the wave solutions are of the form V = V+e-Jf*-°« + Ve*'*" (3.I80) / = r* B -i*— - f > # » * - (3186) where a is the attenuation constant. 3.5 WAVES ON A LOSSY T R A N S M I S S I O N LINE Conductors used in a transmission line will always have a finite conductivity and will therefore exhibit some series resistance. Furthermore, because of the skin effect the current flows in a thin layer at the surface of the conductor, the effective thickness of the layer being equal to the skin depth 8S given by (2/w/iu)i/2 [see (2.104)]. Consequently, the series resistance increases with an increase in the frequency of operation. In order to account for this resistance, a distributed series resistance R per meter must be included in the distributed circuit used to model the transmission line. The two conductors in a transmission line are usually maintained parallel to each other by supporting them in a dielectric structure. For example, a coaxial transmission line is filled with a dielectric medium in order to keep the center conductor coaxial with the outer shield. Dielectric materials usually have a negligible conductance but do have a small amount of dielectric loss due to polarization loss in the dielectric. Consequently, a shunt conductance G per meter is added to the distributed circuit to account for this loss. Thus, for a lossy transmission line, the equivalent circuit of a differentia] length dz is chosen to be that shown in Fig. 3.6. If the voltage and current at the input are T(z, t), ,Hz, t) and if the voltage and current at the output are y+ — dz dz Ij) ^ 'I ) 3 • J + — dz dz Ldt Rd; o—'060o>—wv 8 + $d* i-Gd* <T F I G U R E 3.6 Equivalent circuit of a differential length of transmission line. \Vt'S7d/ TRANSMISSION LINES AND WAVEGUIDES 87 then KirchhofFs laws give / yr-\<zr+ \ 37' \ dS — dz = JR dz + Ldz — dz J dt -—=-JrR-L—dz dt or (3.19a) Similarly, / dS \ 37" S - \S + -—dz] = 7'Gdz + Cdz — \ dz ) dt bj d7' — ~-rG-C— (3.196) oz ol The first equation states that the potential difference between the input and output is equal to the potential drop across R and L. The second equation states that the output current is less than the input current by an amount equal to the shunt current flowing through C and G. Differentiating (3.19a) with respect to z and (3.196) with respect to time t gives or 2 _d 7' __ i = **" " _ fp l dS d2^ L r — - ^77Z dz dtdz (3.20a) d lT C~5(3.206) dtdz dt dt.2 ' ' Using (3.196) and (3.206) in (3.20a) now gives the following equation for the line voltage 7': d\y d27" dz2 or = -G I d7; r R\G% +C—\ "{" " at 37^ 37' d27 +L G — + C 2 dt 3t I d2T dT d2T —=- - (RC + LG)— -LC—5- - RG7'= 0 (3.21) The current .7 satisfies this one-dimensional wave equation also. If a solution in the form of a propagating wave ^=Re(Ve_vz+-'0") is assumed, substitution into (3.21) shows that the propagation constant y must be a solution of y2 -j(u( RC + LG) + w2LC - RG = 0 (3.22) If only the steady-state sinusoidally time-varying solution is desired, phasor notation may be used. If we let V and / represent the voltage and current without the time dependence ej°", the basic equations (3.19) may be 88 FOUNDATIONS FOR MICROWAVE ENGINEERING written as dV = -(R+jo,L)I az dl - - - =( u +-(G+jcoC)V j ^ ) v (3.23a) (3236) The wave equation (3.21) becomes -^-(RG-a>2LC)V-jo>(RC + LG)V=0 {3M) The general solution to (3.24) is V= V+e-v* + V-ey* (3.25) where y = a + jfi is given by y = [-«>2LC + RG +ja>(RC + LG)]1/2 (3.26) from (3.22). The constants V* and V~ are arbitrary amplitude constants for waves propagating in the +z and -2 directions, respectively. The solution for the current I may be found from (3.23a), that is 7 = f* < r y a - / - e + > 2 = „ y T ( V - g - ' " - V"e v z ) (3.27) The parameter Z = R+jcoL = {R+j(oL\l/2 (3.28) is the characteristic impedance of the line since it is equal to the ratio V+/I+ and V-/r. Note t h a t y = [(R +ju>L)(G +ja>C)]1/2. Loss-Free Transmission Line For a line without loss, i.e., for which R = G = 0, the propagation constant is y=,)3=,WLC (3-29) and the characteristic impedance is pure real and given by z,-/| <3-3W According to the field analysis, /3 is also equal to w(/xe) 1/2 , and hence LC = Me (331) for a transmission line. This result may also be verified from the solutio for L and C, as shown later in the section on transmission-line parameters TRANSMISSION LINES AND WAVEGUIDES 89 Using (3.31) in (3.30) shows that the characteristic impedance is also given by C " V C2 " CV c (3.32) C where Z is the intrinsic impedance of the medium. The characteristic impedance differs from the intrinsic impedance Z by a factor e / C , which is a function of the line configuration only. Low-Loss Transmission Line For most microwave transmission lines t h e losses are very small; t h a t is, R « coL and G « toC. When this is the case, the term RG in the expression (3.26) for y may be neglected. A binomial expansion then gives , y~j*JW + 1 . [R - { W \ - + G\ -\=a+jp (3.33) To first order the characteristic impedance is still given by (3.30) or (3.32). Thus the phase constant for a low-loss line is fi = wvXC (3.34a) and the attenuation constant a is 1 . where Y c = Z c sion line. ! (R G\ 1 (3.346) = \C/L is the characteristic admittance of the transmis- 3.6 T E R M I N A T E D T R A N S M I S S I O N L I N E : SINUSOIDAL WAVES In this section the properties of a transmission line terminated in an arbitrary load impedance Z L are examined. This will serve to illustrate how the forward and backward propagating waves can be combined to satisfy the boundary conditions at a termination. Figure 3.7 illustrates schematically a transmission line terminated in a load impedance ZL. The line is assumed lossless and with a characteristic impedance Z c and a propagation constant /•-/" F I G U R E 3.7 Terminated transmission line. 90 FOUNDATIONS FOR MICROWAVE ENGINEERING y =7/3. It should be noted that at microwave frequencies conventio low-frequency resistors, inductors, or capacitors, when connected across th two conductors of a transmission line, may behave as impedance eletne ? with quite different characteristics from the low-frequency behavior. If a voltage wave V+e~JP* with an associated current I'e'J"' • incident on the termination, a reflected voltage wave V'eJIS' with a curre '* -I~ejp* will, in general, be created. The ratio of the reflected and incident wave amplitudes is determined by the load impedance only. At the load th total line voltage must equal the impressed voltage across the load and thp line current must be continuous through the load. Hence, if Z L is located at z = 0, V=V++V-=VL (3.35a) i = r-r=h (3.356) T But r= FCV , / " = YCV', and VL/IL = ZL by definition of load impedance. Therefore V++ V~= VL (3.36a) V+-V-=~VL (3.366) The ratio of V to V * is usually described by a voltage reflection coefficient T defined as < 3 - 37 > r, = ~ In place of (3.36) we may write Dividing one equation by the other yields 1 +r L = i - rL ^ (3.38) zc The quantity ZL/ZC is called the normalized load impedance (load imped 3 " measured in units of Zc), and (1 + TL)/(l - VL) is then the n 0 " " * ^ input impedance seen looking toward_the load at 2 = 0. The norm. a load impedance will be expressed as ZL, with the bar on top signiv 1 ^_ normalized impedance in general. Solving for the voltage reflection c cient T gives ZL-ZC lL ZL + ZC ZL/ZC - 1 _ ZL - 1 ZL/ZC + \ ZL + 1 {3 .39) TRANSMISSION LINES AND WAVEGUIDES 91 Analogous to a voltage reflection coefficient, a current reflection coefficient r, could also be introduced. In the present case -r r '' 7+ = Yrv~ - + = -rL YV In this text, however, only the voltage reflection coefficient will be used; so the adjective "voltage" can be dropped without confusion. The incident voltage wave can be considered as transmitting a voltage V,_ across the load, and a voltage transmission coefficient T can be defined as giving V L in terms of V + ; thus V L = T V + = ( 1 + rL)V^ So P- 1 + T x (3.40) A corresponding current transmission coefficient is not used in this book. Returning to (3.39), it is seen that if ZL = Zc, the reflection coefficient is zero. In this case all the power in the incident wave is transmitted to the load and none of it is reflected back toward the generator. The power delivered to the load in this case is P = i R e ( V 7 * ) = i|V+|2Fc = i | y j 2 y L (3.41) The load is said to be matched to the transmission line when Y L = 0. If Z L does not equal Zc, the load is mismatched to the line and a reflected wave is produced. The power delivered to the load is now given by P=|Re(Vt7?)=|Re[(V^+V')(7 + -7-r] = iRe[y c (V + +V-)(V + -V-)*] = |Re[Y c |V i ] 2 (l+r,)(l-r / .)*] = |yciv+l2(i-irj2) (3.42) The final result states the physically obvious result that the power delivered to the load is the incident power minus that reflected from the load. In the absence of reflection, the magnitude of the voltage along the line is a constant equal to |V + |. When a reflected wave also exists, the incident and reflected waves interfere to produce a standing-wave pattern along the line. The voltage at any point on the line (z < 0) is given by v = v+e-rf, +YLV+ejllz and has a magnitude given by Ivi = IV + I u + rLe*J**\ = i v * M I + r,, e - 2 ^'i where / = -z is the positive distance measured from the load toward the 92 FOUNDATIONS FOR MICROWAVE ENGINEERING generator, as in Fig. 3.7. Let Y L be equal to pej0, where p = | r j ; thent IVI = | V + | |1 + pe""-^l>\ = \V*\[[1 + pcos{d - 2pl)]2 + p2sin2{e - 2pl)}1/2 = |V + |{(1 + pf - 2 p [ l - cos(0 - 2 / 3 / ) ] } V 2 Tl/2 = |V + (1 + p ) 2 - 4 p s i n 2 | / 3 / - - (3.43) This result shows that 1VI oscillates back and forth between maximum values of |V*|(1 + p) when fil - 6/2 = mr and minimum values ty*\ (1 - p) when pi - 6/2 = mr + TT/2, where n is an integer. These results also agree with physical intuition since they state that voltage maxima occur when the incident and reflected waves add in phase and that voltage minima occur when they add 180° out of phase. Successive maxima and minima are spaced a distance d = ir/fi = XTT/2TT = A/2 apart, where A is the wavelength for TEM waves in the medium surrounding the conductors. The distance between a maximum and the nearest minimum is A/4. Since the current reflection coefficient is equal to - r L the current waves subtract whenever the voltage waves add up in phase. Hence current maxima and minima are displaced A/4 from the corresponding voltage maxima and minima. Figure 3.8 illustrates the voltage and current standing-wave patterns that result when Z L is a pure resistance equal to 3ZC. The ratio of the maximum line voltage to the minimum line voltage is called the voltage standing-wave ratio S; thus |V+|(l+p) 1 + p IV-l(l-p) 1-p (3.44) This is a parameter of considerable importance in practice for the following reasons: At microwave frequencies instruments for the direct absolute measurement of voltage or current are difficult to construct and use. On the other hand, devices to measure relative voltage or current (or electric or magnetic field) amplitudes are easy to construct. A typical device is a small probe inserted into the region of the electric field around a line. The outp of the probe is connected to a crystal rectifier, and produces an outpu1 current which is a measure of the relative electric field or voltage at ti probe position. By moving the probe along the line, the standing-wave ra^i can be measured directly in terms of the maximum and minimum pr° t T h e symbol p denotes both charge density and the modulus of the reflection coefficient context makes it clear which quantity is under discussion; so confusion should not occur. TRANSMISSION LTNES AND WAVEGUIDES 93 F I G U R E 3.8 Voltage and current standing-wave patterns on a line terminated in a load impedance equal to 3Z,.. currents. The location of a voltage minimum can also be measured, and this permits the phase angle 6 of V L to be calculated. Since p is known from the measured value of S, \'L is specified, and the normalized load impedance may be calculated from (3.38). Although the reflection coefficient was introduced as a measure of the ratio of reflected- to incident-wave amplitudes at the load, the definition may be extended to give the corresponding voltage ratio at any point on the line. Thus, at z = -I, the reflection coefficient is V-e-jt>' !'(/) = y = + p-2jei = r p-W V e (3.45) where rL = V /V" denotes the reflection coefficient of the load. The normalized impedance, seen looking toward the load, at z ~ -I, is V Z„ = V*eipl + V~e~Jli' ~iz,. i + i'(0 1 + VLe-2jP' i-r(0 i-r,e- a -«" (3.46) By replacing YL by (ZL - Ze)/(ZL + Zc) and e±Jpl by cos pi ±j sin pi, this result may be expressed as Z, = Zs* ZL+jZe tan pi Zr Ze +jZL tan pi (3.47) A similar result holds for the normalized input admittance; so Y.= 7;„ YL+jYc tan pi YL + j tan pi Yc + jYL tan pi 1 + jYL tan pi (3.48) Of particular interest are two special cases, namely, pi = TT or I = A / 2 and 94 FOUNDATIONS FOR MICROWAVE ENGINEERING pi = -rr/2 or I = A/4, for which / = (3.49 «) 2> zji = 4 = Z, (3.496, The first is equivalent to an ideal one-to-one impedance transform^ whereas in the second case the impedance has been inverted with resDerf to Zc. The terminal conditions at the generator end are readily established bv using (3.47) to evaluate the input impedance Z in seen looking toward the load at the generator end. If the generator with open-circuit voltage V has an internal impedance Zg, then by the usual voltage division formula the total transmission-line voltage V at z = — / will be given by ft. Z;„ + Z„ But V is the sum of the forward-propagating-wave and refiected-wave voltages at z = -I, that is, V = V V " + V~e-jel = V+e""(l + TLe-2jf") By using this expression we can solve for V^ which is found to be (Zin + Zg)(e^ Zin(ZL + + YLe~M) Zc)Vg (3.50) 2(Z i n + Z8)(ZL cos pi +jZc sin pi) Terminated Lossy Line In the case of a lossy line with propagation constant y ~JP + a previous equations hold except that jp must be replaced by jp + a < w n e a is usually so small that, for the short lengths of line used in Ta0 experimental setups, the neglect of a is justified. Nevertheless, it is of so interest to examine the behavior of a lossy transmission line terminated in load ZL. One simplifying assumption will be made, and this is that characteristic impedance Z c can still be considered real. This assumption certainly valid for low-loss lines of the type used at microwave frequent TRANSMISSION LINES AND WAVEGUIDES 95 A detailed calculation justifying this assumption for a typical case is called for in Prob. 3.18. Clearly, the presence of an attenuation constant a does not affect the definition of the voltage reflection coefficient T L for the load. However, at any other point a distance / toward the generator, the reflection coefficient is now given by f(l) =YLe-*W- 2„l (3.51) As / is increased, T decreases exponentially until, for large /, it essentially vanishes. Thus, whenever a load ZL is viewed through a long section of lossy line, it appears to be matched to the line since Y is negligible at the point considered. This effect may also be seen from the expression for the input impedance, namely, 1 + rLe-*&-*"' ZL + Zc tanh( jpl + al) + Z Ltznh( jpl + al) (3.52) which approaches Zc for I. large since tanh x approaches 1 for x large and not a pure imaginary quantity. The losses also have the effect of reducing the standing-wave ratio S toward unity as the point of observation is moved away from the load toward the generator. As the generator is approached, the incident-wave amplitude increases exponentially whereas the reflected-wave amplitude decreases exponentially. The result is a standing-wave pattern of the type illustrated in Fig. 3.9. For illustrative purposes a relatively large value of a has been assumed here. The power delivered to the load is given by P, = i Re(VJl) = \\VL\*GL = - | l V + l 2 ( l - i r , | 2 ) Zc (3.53) j j / i = 34 1=T"+ F I G U R E 3.9 Voltage standing-wave pattern on a lossy transmission line. (1) Envelope of incident-wave amplitude; (2) envelope of reflected-wave amplitude; (3) standing-wave pattern. 96 FOUNDATIONS FOR MICROWAVE ENGINEERING as before. At some point z = - 1 , the power directed toward the load i 8 P(l) = ± R e ( V 7 * ) = ^\V\2YC = ^ | v + e " ' | 2 [ l - | r ( / ) | 2 ] = ~W-\2(e2"l~\rL\2) (3.54) where i r k " ' has been replaced by ITJ. Of the power given by (3.54) onk, that portion corresponding to P, as given by (3.53) is delivered to the load The remainder is dissipated in the lossy line, this remainder being given bv ^O-^-yl^lV'-i) (3.55) PART 2 F I E L D A N A L Y S I S O F T R A N S M I S S I O N LINES The first section of this part will show that Maxwell's equations can be reformulated so as to describe three classes of waves, TEM, TE, and TM waves. The TEM wave is the principal wave on transmission lines. From the solution for the electric and magnetic fields for the TEM wave, we will be able to establish that there are unique voltage and current waves associated with the TEM wave. We will also be able to evaluate the distributed-circuit parameters R, L, C, and G for a transmission line. The field analysis thus provides a theoretical basis for treating the transmission line as a distributed circuit. After the basic equations for TEM, TE, and TM waves have been derived, we present the field analysis for transmission lines that support TEM waves. This is followed by several sections dealing with planar transmission lines. Many of the planar lines that we examine support only quasi-TEM waves but can be analyzed as transmission lines once their equivalent distributed-circuit parameters have been determined. 3.7 CLASSIFICATION OF WAVE S O L U T I O N S The transmission lines and waveguides analyzed in this chapter are characterized by having axial uniformity. Their cross-sectional shape &* electrical properties do not vary along the axis, which is chosen as t " e axis. Since sources are not considered, the electric and magnetic fields TRANSMISSION LINES AND WAVEGUIDES 97 solutions of the homogeneous vector Helmholtz equation, i.e., V»E + k%E = 0 V 2 H + k*H = 0 The type of solution sought is that corresponding to a wave that propagates along the z axis. Since the Helmholtz equation is separable, it is possible to find solutions of the form f(z)g(x,y), where f is a function of z only and g is a function of x and y or other suitable transverse coordinates only. The second derivative with respect to z enters into the wave equation in a manner similar to the second derivative with respect to time. By analogy with e-""' as the time dependence, the z dependence can be assumed to be e±jp*. This assumption will lead to wave solutions of the form cos(w< ± pz) and sin(w/ ± /3z), which are appropriate for describing wave propagation along the z axis. A wave propagating in the positive z direction is represented by e~jP', and ejPz corresponds to a wave propagating in the negative z direction. With an assumed z dependence e~jPz, the del operator becomes V = V, + Vz = V, ~jfiaz, since Vz = azd/dz. Note that V, is the transverse part and equals a x d/dx + a y d/dy in rectangular coordinates. The propagation phase constant fi will t u r n out to depend on the waveguide configuration. Considerable simplification of Maxwell's equations is obtained by decomposing all fields into transverse and axial components and separating out the z dependence. Thus let (the time dependence eJ'"' is suppressed) E(*,y,2) = E,(x,y,z) + Ez(x,y,z) = e(x,y)e~Jli* + ez(x,y)e~Jp! H(x,y,z) = H,(x,y,z) + (3.56) Uz(x,y,z) = h(x,y)e-#* + hz(x,y)e ^ (3.57) where E , , H , are the transverse (x and y) components, and E>, H z are the axial components. Note also that e(x,y), h{x,y) are transverse vector functions of the transverse coordinates only, and ez(x,y),h2(jc,y) are axial vector functions of the transverse coordinates. Consider the V X E equation, which may be expanded to give V x E = (V, - . / / 3 a J X (e + ez)e~^ = -jton0{h + h j e " ^ or V, x e -jpaz X e + V, x e2 ~j/3az Xez = - j w M 0 h - > M o h = The term a ; X e, = 0, and V, X e 2 = V, X aze, = - a 2 X V,^. Note also that V, X e is directed along the z axis only, since it involves factors such as a, X a^,, ax X ax, ay X ax, and av X a v , whereas az X e and V, X e, have transverse components only. Consequently, when the transverse and axial 98 FOUNDATIONS FOR MICROWAVE ENGINEERING components of the above equation are equated, there results V, x e = -jun0hz (3.58 Q ) V, x ez -jpat X e = - a , X V,ez - . / / 3 a , X e = -ju>nQh (3.586) In a similar manner the V X H equation yields \ X h =jcoe0ez (3.58c) a, X V,hz +jpaz X h = -jwe0e (3.58d) The divergence equation V • B = 0 becomes V • B = V • MoH = (V, -jpa2) • (h + h , ) M o C - - " " = (V,-h-^a,-hJ or M o e ^ = 0 %'h=MK (3.58e) V, • e = j0ez (3.58/-) Similarly, V • D = 0 gives This reduced form of Maxwell's equations will prove to be very useful in formulating the solutions for waveguiding systems. For a large variety of waveguides of practical interest it turns out that all the boundary conditions can be satisfied by fields that do not have all components present. Specifically, for transmission lines, the solution of interest is a transverse electromagnetic wave with transverse components only, that is, Ez = Hz = 0, whereas for waveguides, solutions with Ez = 0 or Hz = 0 are possible. Because of the widespread occurrence of such field solutions, the following classification of solutions is of particular interest. 1. Transverse electromagnetic (TEM) waves. For TEM waves, Ez = Hz = 0. The electric field may be found from the transverse gradient of a scalar function * ( x , y ) , which is a function of the transverse coordinates only and is a solution of the two-dimensional Laplace equation. 2. Transverse electric (TE), or H, modes. These solutions have E z = 0, but H z ¥= 0. All the field components may be derived from the axial component H z of magnetic field. 3. Transverse magnetic (TM), or E, modes. These solutions have H z = "» but E z ¥= 0. The field components may be derived from Ez. In some cases it will be found that a TE or TM mode by itself will not satisfy all the boundary conditions. However, in such cases linear combinations of TE and TM modes may be used, since such linear combinations always provide a complete and general solution. Although other possible types of wave solutions may be constructed, the above three types are the most useful in practice and by far the most commonly used ones. TRANSMISSION LINES AND WAVEGUIDES 99 The appropriate equations to be solved to obtain TEM, TE, or TM modes will be derived below by placing E, and Hz, Ez, and Hz. respectively, equal to zero in Maxwell's equations. For TEM waves e, = hz = 0; SO (3.3) reduces to V,xe = 0 (3.59a) (iaz X e = wM0h (3.596) V, X b = 0 (3.59c) / 3 a z X h = -we0e (3.59d) V,-h = 0 (3.59e) V,-e = 0 (3.59 f) The vanishing of the transverse curl of e means that the line integral of e around any closed path in the xy plane is zero. This must clearly be so since there is no axial magnetic flux passing through such a contour. Although V, X h = 0 when there are no volume currents present, the line integral of h will not vanish for a transmission line with conductors on which axial currents may exist. This point will be considered again later when transmission lines are analyzed. Equation (3.59a) is just the condition that permits e to be expressed as the gradient of a scalar potential. Hence let e(x,y) = ~V,<S>(x,y) (3.60) Using ( 3 . 5 9 / ) shows that <t> is a solution of the two-dimensional Laplace equation V?4>(x,y) = 0 (3.61) The electric field is thus given by E,{x,y,z) = -V,<i>(x,y)e'^ But this field must also satisfy the Helmholtz equation V-'E, + k*E, = 0 2 Since V = V, -j/3az, V = V,2 - (32, that is, the second derivative with respect to z gives a factor - / J a , this reduces to V?E, + (kl- 0 2 ) E , = 0 or V,[V,24>+ {k2-fi2)<p\ = 0 This shows that /3 = ±k0 for TEM waves, a result to be anticipated from the wave solutions discussed in Chap. 2. The magnetic field may be found 100 FOUNDATIONS FOR MICROWAVE ENGINEERING from the V X E equation, i.e., from (3.596); thus WMo *o h = a 2 X e = Z0h (3.62) In summary, for TEM waves, first find a scalar potential <I> which is a solution of V,24>(*,;y) = 0 (3.63a) and satisfies the proper boundary conditions. The fields are then given by E = E, = ***** = - V/De*'*' 2 (3.636) H = H, = ±he*•'*«'* = ±Y0a, X •**-*•• (3.63c) where k0 = w(/z o e 0 ) I / 2 , Y0 = (e0//j.0)1/2, and e~Jk°* represents a wave propagating in the + z direction and e J °* corresponds to wave propagation in the — z direction. For TEM waves, Z„ is the wave impedance, and from (3.63c) it is seen that, for wave propagation in the + z direction, V F" 7T = - W = z o (364a) whereas for propagation in the —z direction, For transverse electric (TE) waves, h z plays the role of a potential function from which the rest of the field components may be obtained. The magnetic field H is a solution of v"2H + ft2H = 0 Separating the above equation into transverse and axial parts and replacing V2 by V2 - B2 yield V2k1(x,y)+k2h;(x,y) = 0 2 2 Vh + kh = 0 (3.65a) (3.656) where k2 = k\, - B2 and a z dependence e~iliz is assumed. Unlike the case of TEM waves, B 2 will not equal k% for TE waves. Instead, B is determined by the parameter h 2 in (3.65a). When this equation is solved, subject to appropriate boundary conditions, the eigenvalue k 2 will be found to be a function of the waveguide configuration. TRANSMISSION LINES AND WAVEGUIDES 101 The Maxwell equations (3.58) with es = 0 become V, X e = -7'w/i 0 h 2 /3a, X e = co^h V, X h = 0 (3.66a) (3.666) (3.66c) a, X V,/iz + jpaz x h = -jwe0e (3.66d) V,-h=jlih2 (3.66c) V, • e = 0 (3.66/") The transverse curl of (3.66c) gives V, x (V, x h ) = V, V, • h - V,2h = 0 Replacing V, • h by jph2 from (3.58e) and V,2h by -k'f.h from (3.656) leads to the solution for h in terms of hz; namely, t$ h =--^V>; (3.67) To find e in terms of h, take the vector product of (3.666) with a;, to obtain /3a 2 X ( a , X e) = p[(az • e ) a , - ( a , • a j e ] = -jSe = ^ 0 a . . X h or e= a ? X h = -— Z,.a, x h (3.68) The factor k0Z0/p has the dimensions of an impedance, and is called the wave impedance of TE, or H, modes. It will be designated by the symbol Zh, so that Z, - ^-Z0 (3.69) Thus, in component form, (3.68) gives T T - - ? - - ^ (3-70) for a wave with z dependence e~jliz. The remaining equations in the set (3.66) do not yield any new results; so the solution for TE waves may be summarized as follows: First find a solution for hz, where V?hz + k2chz = Q (3.71a) Then iP h= - S V » A « (3.716) and e = -Zhaz X h (3.71c) 102 where FOUNDATIONS FOR MICROWAVE ENGINEERING P = (k2 - k2)1/2 and Z h = -^ Complete expressions for the fields are H = ± h e ' • » • + hjt*** (3.71d) E = E, = ee*-** (3.71e) Note that in (3.71c?) the sign in front of h is reversed for a wave propagating in the —z direction since h will be defined by (3.716), with 3 positive regardless of whether propagation is in the +2 or -z direction. The sign in front of e does not change since it involves the factor /3 twice, once in the expression for h and again in Zh. Only the sign of one of e or h can change if a reversal in the direction of energy flow is to occur. That is, the solution for a wave propagating in the -z direction can be chosen as E = -eejPz, H = (h - hz)eJpz or as E = eeJpx, H = ( - h + hz)eJliz. One solution is the negative of the other. The latter solution is arbitrarily chosen as the standard in this text. "M Waves The TM, or E, waves have h. = 0, but the axial electric field e 2 is not zero. These modes may be considered the dual of the TE modes in that the roles of electric and magnetic fields are interchanged. The derivation of the equations to be solved parallels that for TE waves, and hence only the final results will be given. First obtain a solution for es, where V*et + k2ez = 0 (3.72a) subject to the boundary conditions imposed. This will serve to determine the eigenvalue k2. The transverse fields are then given by iB E, = e e * * * = - Ta V . 8 * * " (3.726) H, = ±he + J0z = ±Yeaz X ee*J0t (3.72c) 2 l/2 where f3 = (k'l - k ) given by and the wave admittance Y e for TM waves is k Ye = Z;> = J-Y0 (3.72a-) The dual nature of TE and TM waves is exhibited by the relation ZeZh - Z0 (3.73) TRANSMISSION LINES AND WAVEGUIDES 103 which holds when both types of waves have the same value of p and is derivable from (3.69) and (3.72d). The complete expression for t h e electric field is E = E, + E, = e e T ^ ± e;ef* = (-^V,e2±e le'^ (3.74) It is convenient to keep the sign of e the same for propagation in either the +z or -z direction. Since V • E = 0, that is, V, • E, + dEjclz = 0, this requires that the z component of electric field be -ezejliz for a wave propagating in the —z direction, because V, • E, does not change sign, whereas dEz/^z does, in view of the change in sign in front of ji. The transverse magnetic field must also change sign upon reversal of the direction of propagation in order to obtain a change in the direction of energy flow. For reference, this sign convention is summarized below. The transverse variations of the fields are represented by the functions e, h, ez, and hz, independent of the direction of propagation. Waves propagating in the +2 direction are then given by E = E + = (e + « , ) * - • * ' (3.75a) H = r= (h + h j e ' ^ (3.756) For propagation in the — z direction the fields are E - B T = (e - • , ) * * • (3.76a) H = H = ( - h + hje-** (3.766) Additional superscripts ( + ) or ( - ) will be used when it is necessary to indicate the direction of propagation. The previously derived equations for TEM, TE, and TM modes are valid in a medium with electrical constants e, ii, provided these are used to replace e0, n0. A finite conductivity can also be taken into account by making e complex, i.e., replacing e by e - ja/co. The wave impedance introduced in the solutions is an extremely useful concept in practice. The wave impedance is always chosen to relate the transverse components of the electric and magnetic fields. The sign is always such that if i,j,k is a cyclic labeling of the coordinates and propagation is along the positive direction of coordinate k, the ratio ^,/rlj = (Zw)k is positive. Here (Zu,)k is the wave impedance referred to the k axis as the direction of propagation. If i, j, k form an odd permutation of the coordinates, then EJH) is negative. The usefulness of the wave-impedance concept stems from the fact that the power is given in terms of the 104 FOUNDATIONS FOR MICROWAVE ENGINEERING transverse fields alone. For example, for TE waves, 1 t P=-RejExK*-azdxdy 1 , = — Re / e X h* • a z dx dy 2 1 Re f ZJa, X h) X h* • a, dxdy = —— / h • h* dx dy = —~fe-e*dxdy 2 Js 2 Js upon expanding the integrand. Thus the wave impedance enables the power transmitted to be expressed in terms of one of the transverse fields alone. A further property of the wave impedance, which will be dealt with later, is that it provides a basis for an analogy between conventional multiconductor transmission lines and waveguides. 3.8 TRANSMISSION LINES (FIELD ANALYSIS) Lossless Transmission Line A transmission line consists of two or more parallel conductors. Typical examples are the two-conductor hne, shielded two-conductor line, and coaxial line with cross sections, as illustrated in Fig. 3.10. Initially, it will be assumed that the conductors are perfectly conducting and that the medium surrounding the conductors is air, with e ~ e0, \i ~ fi0. The effect of small losses will be considered later. When the conductors are completely surrounded by a uniform dielectric medium, the principal wave that can exist on the transmission line is a TEM wave. The electric field for this wave can be found from the scalar potential which is a solution of Laplace's equation in the transverse plane. Microstrip lines and other planar transmission lines do not have the dielectric medium completely surrounding the conductors and therefore do not support a pure TEM wave. In this case it is found t h a t only in the low-frequency limit does the dominant mode of propagation approach that of a TEM wave. We refer to the principal wave on these lines as a O O (a) I O O 1 (*) FIGURE 3.10 (c) Cross sections of typical transmission lines, (a) Two-conductor line; (6) shielded two-co"" ductor line: (c) coaxial line- TRANSMISSION LINES AND WAVEGUIDES 105 s, «-/ F I G U R E 3.11 Cross section of a general twoconductor line showing transverse field patterns. quasi-TEM wave. The solution for the electric and magnetic fields of the quasi-TEM wave requires a separate solution for both the electric and magnetic fields in order to determine the distributed-circuit line parameters R, L, C, and G. This is because the electric and magnetic fields are no longer related in the simple way that they are for the TEM wave. The solution for the magnetic field can be found by solving for the vector potential function as will be shown later. In this section and the following one, we consider only transmission lines that support a TEM wave. With reference to Fig. 3.11, let the one conductor be at a potential V 0 /2 and the other conductor at - V 0 / 2 . To determine the field of a TEM wave, a suitable potential <I>(A:,y) must be found first. It is necessary that <t> be a solution of v;2^ = o and satisfy the boundary conditions V„ on S 2 4> = on S, Since <P is unique only to within an additive constant, we could equally well choose <J> = V0 on S2 and <J> = 0 on S1. If a solution for <l> is possible, a TEM mode or field solution is also possible. When two or more conductors are present, this is always the case. The solution for <t> is an electrostatic problem that can be solved when the line configuration is simple enough, as exemplified in Fig. 3.10. The fields are given by (3.63), and for propagation in the +z direction are E = E, = ee~jko* = -V,4>e_J'*<>* (3.77a) H = H, = y ( l a ! X e r ^ 2 (3.776) 106 FOUNDATIONS FOR MICROWAVE ENGINEERING The line integral of e between the two conductors is [% • dl = f J s, rS,2 - V.cl> • dl •'s, = ~fs!^dTdl = " [ $ ( S 2 ) - *(S'>J = - y ° I Associated with the electric field is a unique voltage wave V=V0e~^ (3.78) since the line integral of e between Sj and S 2 is independent of the path chosen because e is the gradient of a scalar potential. The line integral of h around one conductor, say S 2 , gives <f) h • dl = <f) Js dl = I0 by application of Ampere's law, V X H =j<uT> + J, and noting that there is no axial displacement flux D z for a TEM mode. On the conductors the boundary conditions require n X e = 0 and n X h = J s , where n is a unit outward normal and J s is the surface current density. Since n and h he in a transverse plane, the current J, is in the axial direction. In the region remote from the conductors, V, X h = 0, but the line integral around a conductor is not zero because of the current that exists. The current on the two conductors is oppositely directed, as may be verified from the expression n X h = J s . Associated with the magnetic field there is a unique current wave I = I0e-JI">* (3.79) Since the potential <P is independent of frequency, it follows that the transverse fields e and h are also independent of frequency and are, in actual fact, the static field distributions which exist between the conductors if the potential difference is V0 and currents /„, —7„ exist on S 2 , Sv respectively. The magnetic lines of flux coincide with the equipotential lines, since e and h are orthogonal, as seen from (3.776). Example 3-1 Coaxial line. Figure 3.12 illustrates a coaxial transmission Vine for which the solution for a TEM mode will be constructed. In cylindrical coordinates r, <t>, z, the two-dimensional Laplace equation is 1 d2* I A I <Tt>\ r Hr\ 2 r d<t>2 dr J or for a potential function independent of the angular coordinate i}>, r dr \ dr TRANSMISSION LINES AND WAVEGUIDES 107 FIGURE 3.12 Coaxial transmission line. *=0 Integrating this equation twice gives $ = C, In r - C, Imposing the boundary conditions 4> = V„ at r = a, <l> = 0 at r = ft, gives V0 = C, In a + C2 0 = C, in 6 + C, and hence C 2 = - C , In ft, C, = V 0 /[ln(o/ft)], )n{r/b) * = V0- (3.80) 'ln(a/ft) The electric and magnetic fields of a TEM mode propagating in the +z direction are given by (3.77) and are E = -ar— r e - > * < * =• - r - r ^ T T — « " • " " <?r fn(o/6) r _ ! l p .Mo* (3.81a) In ( f t / a ) r H = y 0 a . X ee J M - y V « » ' * ,. '.-...ln(6/a) r (3.81ft) The potential difference between the two conductors is obviously V0; so the voltage wave associated with the electric field is V = V „ e ;*•-• (3.82) The current density on the inner conductor is n X H = ar X H Wo a, ln(ft/a) a e-y*o z The total current, apart from the factor e "•'*"*, is Y V tor g 0 (3.83) I a d(b = 0 a\n(b/a)Jn ln(ft/a) The current on the inner surface of the outer conductor is readily shown to be equal to I 0 also, but directed in the -z direction. The current wave associated with the magnetic field is therefore In = I = V-'*"2 (3.84) 108 FOUNDATIONS FOR MICROWAVE ENGINEERING The power, or rate of energy flow, along the line is given by 1 rhr7.Tr P = - Re ("J 22 LJaino 1 YaVrf rb ,-2~dd>dr E X H* • asdrdt = 2-^—j J f J a o4 22 fln(6/a)] Urr( h/n\Y L TrY0Vi (3.85) ln(o/a) The power transmitted is seen to be also given, as anticipated, by the expression 1 1 1 „ 2rr, 2 v u ( 1 = -V 0 u -Re(V7*) = -V / 0 2 ' 2 " 2 \n(b/a) The characteristic impedance of the line is defined by the ratio — = Z t. (3.86) 'o in terms of which the power may be expressed as P = jZcI% = %YCVQ, where Yc is the characteristic admittance of the line and equal to Zc 1. The characteristic impedance is a function of the cross-sectional shape of the transmission line. Pransmission Line w i t h Small Losses Practical transmission lines always have some loss caused by the finite conductivity of the conductors and also loss that may be present in the dielectric material surrounding the conductors. Consider first the case when the conductors are surrounded by a dielectric with permittivity e = e' — je" but the conductors are still considered to be perfect. The presence of a lossy dielectric does not affect the solution for the scalar potential <P. Consequently, the field solution is formally the same as for the ideal line, except that ku and Y0 are replaced by k = k0{e'r -je"r)1/2 and Y = Y0(e'r -je"r)1/2, where the dielectric constant c r = e'r -je"r = e / e 0 . For small losses such that e" « e'r, the propagation constant is e"\l/2 / jk=a+j{3 = y K > Thus 1/2 *0 1 -y-f a = 6r ° 2(€'r)1/2 8 = {e'r)l/2k0 e"k */(4)1/2*o + -^-?72 (3.87o) (3.876) where a is the attenuation constant and B is the phase constant. The wave consequently attenuates according to e~"z as it propagates in the + z direction. It will be instructive to derive the above expression for a by means of a perturbation method that is widely used in the evaluation of the attenua- TRANSMISSION LINES AND WAVEGUIDES 109 tion, or damping, factor for a low-loss physicaJ system. This method is based on the assumption that the introduction of a small loss does not substantially perturb the field from its loss-free value. The known field distribution for the loss-free case is then used to evaluate the Joss in the system, and from this the attenuation constant can be calculated. In the present case, if e" = 0, the loss-free solution is E= -V/berm 1/z H- Ya2XE xn where k = (e'r) k0 and Y = (er) Y0. When e" is small but not zero, the imaginary part of e, that is, e", is equivalent to a conductivity a = (oe" = we 0 e" A conductivity a results in a shunt current J ~ ixE between the two conductors. The power loss per unit length of line is 1 , W<E" . P,= — - / J- J*dS = —- / E -E*dS (3.88) 2o Js 2 Js where the integration is over the cross section of the line, and the loss-free solution for E is used to carry out the evaluation of P,. Since loss is present, the power propagated along the line must decrease according to a factor e~2"'. The rate of decrease of power propagated along the line equals the power loss. If the power at z = 0 is PQ, then at z it is P = P0e~2a*. Consequently, dP -— - Pi = 2aP0e-2"-' = 2aP (3.89) which states that the power loss at any plane z is directly proportional to the total power P present at this plane. The power propagated along the line is given by 1 , P = - Re / E x H* • a , dS *• = J s Y Y - R e J/ " E x ( a 2 x E * ) • a,dS = -(E • E * d S zs 2 •'s Hence the attenuation a is given by P, a IP 2Y we" = A, 2Y0(e'r)l/2 ' °2(0'/2 which is the same as the expression (3.87a). For this example the perturbation method does not offer any advantage. However, often the field solution for the lossy case is very difficult to find, in which case the perturbation method is extremely useful and simple to carry out by comparison with other methods. The case of transmission lines with conductors having finite conductivity is an important example of this, and is discussed below. 110 FOUNDATIONS FOR MICROWAVE ENGINEERING If the conductors of a transmission line have a finite conductivity, t Q e v exhibit a surface impedance Z„,= 111 (3.90) . where 8 S = (2/w/icr) l / 2 is the skin depth (Sec. 2.9). At the surface the electric field must have a tangential component equal to ZmJs, where J ;,, the surface current density. Therefore it is apparent that an axial com'po. nent of electric field must be present, and consequently the field is no longer that of a TEM wave. The axial component of electric field gives rise to a component of the Poynting vector directed into the conductor, and this accounts for the power loss in the conductor. Generally, it is very difficult to find the exact solution for the fields when the conductors have finite conductivity. However, since \Z„,\ is very small compared with Z 0 , the axial component of electric field is also very small relative to the transverse components. Thus the field is very nearly that of the TEM mode in the loss-free case. The perturbation method outlined earlier may be used to evaluate the attenuation caused by finite conductivity. The current density J s is taken equal to n X H, where n is the unit outward normal to the conductor surface and H is the loss-free magnetic field. The power loss in the surface impedance per unit length of line is p, = - R e Z n6 R J,-J?* s2 (n X H ) ' ( n X f f )dl "> J. -s 2 R -*, ' -ri, H •H*dl (3.91) 'S 2 where R„, = l/trSs is the high-frequency surface resistance, and ( n x H) - ( n x H*) = n • H X ( n X H*) = n • [ ( H • H * ) n - ( H • n ) H * ] = H • H* since n • H = 0 for the infinite-conductivity case. The integration is taken around the periphery S { + S 2 of the two conductors. The attenuation constant arising from conductor loss is thus RmYS,+S. H • H* dl a = IP (3.92) 2Zj H • H* dS where the power propagated along the line is given by Re - JE X H* • a2 dS = -ZJH • H* dS and Z is the intrinsic impedance of the medium; that is, Z = (iJ./e)l/2- TRANSMISSION LINES AND WAVEGUIDES 111 W h e n b o t h dielectric a n d c o n d u c t o r losses a r e p r e s e n t , t h e a t t e n u a t i o n c o n s t a n t is t h e s u m of t h e a t t e n u a t i o n c o n s t a n t s a r i s i n g from each cause, provided b o t h a t t e n u a t i o n c o n s t a n t s are s m a l l . Example 3.2 Lossy coaxial line. Let the coaxial line in Fig. 3.12 be filled with a lossy dielectric (e = e' — je"), and let the conductors have finite conductivity a. For the loss-free case U" = 0, a = «) the fields are given by (3.81), with k0 and Y0 replaced by k = {e'/e0)1/2k0, Thus E = — ln(6/a) r YVU H Jk e a. = r7rrr— -- (3.93a) ' eJ <3-936) ' ln(b/a) r The power propagated along the line is 1 ' r2W rb TTYVZ The power loss P n from the lossy dielectric is, from (3.88), eve" rh <ye"VU7r *—£••*"**•-£(*£) (395a) The power loss from finite conductivity is given by (3.91), and is -^Cli+»/'* Y2VJ R,n ,2-/1 2 [ln(6/a)f RmirY*V* b + a [ln(6/a)]2 (3.95b) ob Hence the attenuation constant a for the coaxial line is given by °= we" RmY b + a Pn + Pn 2P • 2Y ' 21n(6/o) ab Rm b + a 2Zln(6/a) ab ~. -i- -= 2(e'r) ,/2 For the lossy case the propagation constant is consequently taken as a + j/3 = a +jk with a given by (3.96). (3.96) 112 FOUNDATIONS FOR MICROWAVE ENGINEERING TRANSMISSION-LINE PARAMETERS In this section the field analysis to determine the circuit parameters £ o C, and G for a transmission line is examined in greater detail. This will serve further to correlate the field analysis and circuit analysis of transmission lines. Consider first the case of a loss-free line such as that illustrated in Fig 3.11. When the scalar potential 4> has been determined, the charge density on the conductors may be found from the normal component of electric field at the surface; that is, p s = en * e = - e n • VO = -ed$>/dn, where e is th e permittivity of the medium surrounding the conductors. The total charge Q per unit length on conductor S 2 is Q = & en • e dl The total charge on the conductor S s is -Q per meter. The potential of is V0, and hence the capacitance C per unit length is Q efo n • e dl C= /!? (3.97 dl The total current on S 2 is l0 = <f)h-dl=(f) Yn-edl YQ '$•> since lh| = Y\e\ = Yn • e at the surface of S 2 because the normal component of h and the tangential component of e are zero at the perfectly conducting surface S 2 . The characteristic impedance of the Une is given by V, V0e eZ (3.98) A knowledge of the capacitance per unit length suffices to determine the characteristic impedance. To determine the inductance L per unit length, refer to Fig. 3.13, which illustrates the magnetic flux lines around the conductors. Since h & / ijj=0 line F I G U R E 3.13 Magnetic flux lines in a transmission line. TRANSMISSION LINES AND WAVEGUIDES 113 orthogonal to e, these coincide with the equipotential lines. All the flux lines from the $ = 0 to the 4> = V 0 / 2 line link the current on S 2 . The flux linkage is the total flux cutting any path joining the <t> = 0 line to the surface S 2 . If a path such as PiS2 or P2S2 is chosen, which is orthogonal to the flux lines, this path coincides with a line of electric force. The flux cutting such a path is <P J = f\hdl = fiY[Sa -ed\ = tiY— 2 J P, P, since I h | = Y\ e I for a TEM wave. The inductance of one conductor of the line is vV° * T The inductance of both conductors per unit length is twice this value; so L = ,MY-^ = liYZe (3.99) *o From this relation and (3.98) it is seen t h a t Z = \iZc/L = CZc/e, and hence £ _ M y.Zc CZC — — e L e which gives Z c = y - (3.100) Equations (3.98) and (3.99) also show t h a t fie = LC (3.101) for a transmission line. The above expressions for C and L can also be obtained from the definitions based on stored energy. The derivation is left as a problem. If the dielectric has a complex permittivity e = e' -j(", where e" includes the conductivity of the dielectric if it is not zero, the total shunt current consists of a displacement current I D and a conduction current Is. The current leaving conductor S 2 per unit length is I = ID + Is =jioe(p e • n dl =joje'(p e • n dl + a>e"(p e • n dl S-i Sz Sz where the first integral on the right gives the displacement current and the second integral gives the conduction current. The total shunt admittance is given by Y = jcoC + G = (Is + I/})/V0, and hence it is seen that 7, G = 7 S ID v = coe" fnvn = —C (3102) 114 FOUNDATIONS FOR MICROWAVE ENGINEERING since jwC = ID/V0 and jcoC/jcoe' = C/e'. This relation shows that Q differs from C by the factor toe" /e only. The transmission-line loss from finite conductivity may be accounted for by a series resistance R per unit length provided R is chosen so that The right-hand side gives the total power loss per unit length arising from the high-frequency resistance of the conductors. In terms of this quantity the resistance R is thus defined as R = R »> ,. ,. , JIX2 (3.104) (<psJh\dl) where R m = l/(r8s and 8 S is the skin depth. A further effect of the finite conductivity is to increase the series inductance of the line by a small amount because of the penetration of the magnetic field into the conductor. This skin-effect inductance L, is readily evaluated on an energy basis. The surface impedance Z m has an inductive part jXm = j/o-8s equal in magnitude to Rm. The magnetic energy stored in Xm is (note that Xm is equivalent to a surface inductance Xm/io = Lm) wm = ^6 ufdi 4w JSt+S2 4w JSt + Si 4w Rm 4ci> by using (3.103) to replace the integral. Defining L s by the relation \LJ20 = W,„ gives o>Ls - R (3.105) The series inductive reactance of the line is increased by an amount equal to the series resistance. However, for low-loss lines, R •« OJL, so that L s -^ *" and the correction is not significant for most practical lines. The inductance L s is called the internal inductance since it arises from flux linkage internal to the conductor surfaces. It should not come as a surprise to find that wLs = R since both the inductive reactance and resistance arise from the penetration of the current and fields into the conductor. The effect of this penetration into the conductor by an effective distance equal to the skin depth 5 S is correctly accounted for in a simplified manner by introduction of the surfa** impedance Zm = (1 +j)/a8s. TRANSMISSION LINES AND WAVEGUIDES 115 Example 3.3 Coaxial-line parameters. For the coaxial line of Fig. 3.12 the potential <I> is given by * v l n ( r / 6 ) The charge on the inner conductor is Q = e / " a r • ead<b = e I J J o ad<j> n dr m _=•* r^=J^L ln(a/6)'o ln(6/a) Hence the capacitance per unit length is e Q fer«* C=-—= (3.106) e V0 \n(b/a) since the capacitance arises only from that part of the charge associated with e' whereas e" gives rise to the shunt conductance. The magnetic field is given by (3.936) as YV a ln(6/a) r The current I 0 is rz*u , 2irYV ^ Q I„= I h • a.ad4> = •——, 0 J0 * ln(6/a) Thus the characteristic impedance is VQ Z c = -^ = Zln(6/a) 5-f-i (3.107) The flux linking the center conductor is •'o m( o / a ) -'n r Consequently, the inductance per unit length is <b ixYV0 6 y. b L = — = — = 7 - In - = — In /„ 2TTYV0 a 2TT (3.108) a from which it is seen that LC = ye' and Z(. = ( L / C ) 1 / 2 . The shunt conductance G is given by we"C/e', and is toe" G = 2-rre' 2-nwe" » (3.109) 6' l n ( 6 / a ) ln(6/a) To find the series resistance, the power loss in the inner and outer conductors must be evaluated. This was done in Example 3.2, with the result 116 FOUNDATIONS FOR MICROWAVE ENGINEERING [Eq. (3.956)] 1 „ RmTrY2V? b + a -RIS = Pl2 % 2 [ln(b/a)f ab Solving for R gives R = R„, b + a 'ITT ab (3l »0) The internal inductance L s is equal to R/io; so the total series line inductam ce per unit length is L + L = = h 'n a + 2,0,'ab^r ( ^ l ) The distributed-circuit parameters R, L, C. and G for a transmission line can also be determined from an evaluation of the stored electric and magnetic field energy and the power loss per unit length. Energy storage in the magnetic field is accounted for by the series inductance L per unit length, whereas energy storage in the electric field is accounted for by the distributed shunt capacitance C per unit length. Power loss in the conductors is taken into account by a series resistance R per unit length. Finally, the power loss in the dielectric may be included by introducing a shunt conductance G per unit length. Suitable definitions for the parameters L, C, R, and G based on the above concepts are L= ~rWH-H*dS (3.112a) *o'o s C= - ^ — [ E-E* dS W Js (3.1126) R = -^-(f) HH'dl (3.112c) G=-^-[E-E*dS (3.112d) where /„ is the total current on the line, and V 0 the potential difference. These expressions are obtained, for example, by equating the magnetic energy \l0lo L = W m stored in the equivalent series inductance L to the expression for W m in terms of the field. The above definitions are readily shown to be equivalent to the other commonly used definitions such as magnetic flux linkage ll3flj total current total charge per unit length voltage difference between conductors G total shunt current voltage difference between conductors 113ft) .O.M (3.U3C' Parameters of some common transmission lines are given in Table 3.*-j TRANSMISSION LINES AND WAVEGUIDES 117 TABLE 3.1 P a r a m e t e r s of c o m m o n transmission l i n e s t R 1/2 S> 1 2TT Ho) v e ; . D cosh ' — " , o Ina 1/2 In 2p 11 d [(D/df-1 2-n- \ a 1-?* 1+g* 1 + 4p 2 P= D/d 2Rr 1/2 bJ 2ff„ I + 2p2 +—V ( 1 + <?*)- ! • 4p- D 9 = tFor all TEM transmission lines e"C _ (Mop"* C= = L = ( Mo e ) ' Zt GZ, RK. <*</ = G= /?, _1 <r^, \ L./5\/ "M 2<r I 5, I/U U 310 INHOMOGENEOUSLY FILLED PARALLEL-PLATE TRANSMISSION LINE In Fig. 3.14a we show a parallel-plate transmission line (waveguide) partially filled with dielectric material having a permittivity e = ere0, where er is the dielectric constant. The plates are infinitely wide and spaced a distance b apart. The dielectric sheet has a thickness a and rests on the bottom plate. The purpose for studying this particular waveguide is that it exhibits a number of characteristics that are similar to those of the microstrip transmission lines examined in the following section. We will show that the dominant mode of propagation in the waveguide under consideration is an E mode and that as the frequency approaches zero this mode becomes a TEM mode. Furthermore, in the low-frequency limit, the propagation con- 118 FOUNDATIONS FOB MICROWAVE ENGINEERING y i i i i -IV f «S i i i i w F I G U R E 3.14 (a) Partially loaded parallel-plate waveguide; (6) parallel-plate waveguide with magnetic walk at x = + W. stant can be found in terms of the distributed capacitance; and inductance per meter by the usual transmission-line formula /? = covLC. As the frequency increases \i increases faster than to, in which case we say that the transmission line exhibits dispersion. Another feature that can be easily described for this waveguide is the existence of a surface-wave mode of propagation that consists of a field concentrated near the air-dielectric interface. Since the analytic solution for the partially filled parallel-plate waveguide is readily constructed, this waveguide serves as a useful example to provide some physical insight into the properties of microstrip transmission lines. An electric wall is a surface on which the tangential electric field must be zero. A good conductor such as copper provides a surface with a very small skin-effect surface impedance (see Sec. 2.9). When we let the conductivity a become infinite, we obtain an electric wall on which the boundary condition n X E = 0 holds. The dual of an electric wall is a magnetic wall on which the tangential magnetic field is zero, i.e., the boundary condition n X H = 0 holds. The magnetic wall does not have a physical realization but is, nevertheless, a useful theoretical concept. In practice, a magnetic wall can be inserted into a field region, without disturbing the field, along any surface on which the tangential magnetic field is zero. Such surfaces usually correspond to certain symmetry planes in a given problem. I" addition to the above boundary conditions, Maxwell's equations show that on an electric wall the normal component of H is zero, that is, n • H = 0. | The dual boundary condition n • E = 0 holds on a magnetic wall. For the E mode that we will consider in the partially filled parallel-pl a t e waveguide, we will assume that the fields do not depend on the x coordinsie but are functions of y and z only. A consequence of this assumption is that only the field components Ey, E z and H x are present. Thus we can place 8 magnetic wall along any x = constant surface without disturbing the fieldWe will now assume that magnetic walls are inserted at x = ± f f a s shown TRANSMISSION LINES AND WAVEGUIDES 119 in Fig. 3.146. By means of this artifice, we are able to talk about a closed waveguide structure, closed by electric walls at y = 0, b and by magnetic walls at x = ±W. In order to find the solutions for E modes having the z dependence e~Jpz, we must find solutions for the axial electric field component ez{y) first. In an ideal transmission line the propagation constant equals that for plane TEM waves in the surrounding medium. In the structure under investigation we have a nonuniform medium, namely dielectric in the region 0 < y < a and air in the region a < y < b. Consequently, we can anticipate that the propagation phase constant /3 for the dominant mode will take on an intermediate value, i.e., k0 = toj(i0e0 < fi < yfe^k0 = k The equation satisfied by eSy) is (3.72a) which is repeated below V2ez + k2ez = 0 Since we assume no variation with x, the transverse laplacian operator becomes simply d2/dy2. In this equation k 2 = k2, - P 2 in the air region and equals k 2 - [32 in the dielectric region. The propagation constant (3 must be the same in both regions because the tangential electric and magnetic fields must match at the air-dielectric interface for all values of z. For convenience, we will let kc = I in the dielectric region and let it equal p in the air region. We thus require that p2 — k\ = I'2 - k2 or l2-p2~(er-l)k2 (3.114) In the two regions the axial electric field is thus a solution of d2e, =0 * * * ' ' « • 0 £y < a (3.115a) a sy <b (3.1156) along with the boundary conditions y = 0,6 eAy) = 0 ez(y) continuous at y = a er de2 2 I dy 1 de2 = 2 « 7 9y (3.116a) (3.1166) (3.116c) a The third boundary condition comes from the requirement that H x be continuous across the air-dielectric interface. The transverse fields are given 120 FOUNDATIONS FOR MICROWAVE ENGINEERING by (3.726) and (3.72c). The generic form of the equations is P Jfiae "* *'k*dye ** -J/5* upon using a. X a y = - a , . In the equation for ff, the wave admittance y is given by kY/fi in the dielectric and by (k0Y0)/p in the air region where kY = o>yJ/x0€ i/e/fxo = fr^o^o- Thus we have - —5- —- dielectric region (3.117a) ev(:y) = 2 air region By jerk0Y0 de2 I2 K(y) = dielectric region 3y (3.1176) jk0Y0 dez P2 air region dy An examination of the expression for hx(y) shows that continuity of h x at y = a gives the boundary condition specified by (3.116c). We also find that in an inhomogeneously filled waveguide, the wave impedance, defined by the ratio -Ey/Hx, is not constant since it has a different value in the air region from that in the dielectric region. The reader can readily verify that the solutions of (3.115a) and (3.1156) that satisfy the boundary conditions at y = 0 and b are Cj(y) = Cl sin ly 0 <y <a e z(y) =C.2sinp(b - y ) a<y<6 where C 1 and C 2 are unknown amplitude constants. The boundary condition (3.1166) requires that C, sin la = C2 sin pc where c = b - a. The last boundary condition (3.116c) requires *r * — C, cos la — C29 cos pc I l p When we divide the first equation by the second one, we obtain / tan la = -erp tan pc (3.118) TRANSMISSION LINES AND WAVEGUIDES 121 This transcendental equation must be solved simultaneously with (3.114) to determine the allowed values of / and p. There will be an infinite number of solutions; consequently an infinite number of E modes are possible. Since /3 is given by (i = y/k2~p2 = y/k2 - I2 (3.119) most of the modes will be nonpropagating since increasing values of p and / give p > k0 which makes /3 imaginary. When /3 is imaginary the z dependence is of the form e~il3>z and the field decays exponentially from the point at which it is excited. These nonpropagating modes are called evanescent modes. We note from (3.119) that a value of /3 between k0 and k can occur only if p is imaginary. Thus we must consider the possibility that an imaginary p, say p =jp0, is a possible solution to (3.118). If we let /„ be the corresponding value of I, then our relevant equations become Z0 tan l0a = erp0 tanh p0c l2+p2 = (er~ l)kl (3.120a) (3.1206) We consider solutions of these equations and the corresponding fields in the low- and high-frequency limits in the next two subsections. Solution When the frequency is very low, k'l is a very small number (at 1 MHz, k 0 equals 0.02094 r a d / m ) ; hence / 0 and p 0 are then also small. We will assume that 6 is at most a few centimeters, then l0a and p0c are also small and we can replace the tangent function and the hyperbolic tangent function by their arguments. Thus (3.120a) becomes Upon using (3.1206) we readily find that (er- l)k2-p2= a a + erc The solution for /3 in the low-frequency limit is thus P = l/*o + Pi = \ —~ *o = \ / e > o (3-12D y a + erc where ee, given by this equation, is called the effective dielectric constant. 122 FOUNDATIONS FOR MICROWAVE ENGINEERING We will now show that this equals co</LC, where L and C are the stat" distributed inductance and capacitance per meter for the given structure If we have a uniform current density J z on the inner surface of n. upper plate and —Jt on the inner surface of the lower plate, the magneti field between the plates will be given by Hx = Jz. The time-average stored magnetic energy per unit length is given by Jn '-w J-w 4 2 We equate this to ^Llf where the total current 7. = 2WJZ and then find that L (3122 -w > The distributed capacitance C per meter is found by considering the capacitance of the dielectric and air regions as represented by two equivalent parallel-plate capacitances Crf and C a in series where e r e 0 2W O, = -^ C„ = e02W The capacitance Cd is that of a parallel-plate capacitor of width 2W, unit lit length, plate spacing a, and filled with dielectric. C a is the capacitance of the air-filled section which has a spacing c. The series capacitance is given by C nC d 2Were0 Ca + C d erc + a (3.123) The product LC = ere0ii0b/(erc + a) which gives the solution for /3 = io\/LC equal to that in (3.121). The expressions for the fields can be written down in simplified form using the small argument approximations and the relationship C 2 * C\ sin l0a/j sinh p0c = -jCj^/p^c obtained from the boundary condition requiring continuity of e z at y = a. We readily find that in the region 0 <y & a, ez = C^y e y = (3.124a) -v^-^i^s Jerk0Y0 *0 Ur(erc + a) V (er-l)c TRANSMISSION LINES AND WAVEGUIDES 123 and in the air region es = c Cl^-(b-y) JPUa (3.124rf) jfi jk0Y0 K= 1Q / * / (er-l)c ler(erc + a) c,=,y„c,y/ (6p _ 1)c (3.124e) (3.124/) We note that in the low-frequency limit e z vanishes as k(l, and hence / 0 , approach zero, while e v and h x remain constant. If we define the voltage V between the upper and lower plates by the line integral of ey, then The total 2-directed current on the upper plate is t z = 2WJ: = 2WHX, and hence the characteristic impedance is given by (3.125) Thus we find that in the low-frequency limit the dominant mode of propagation in the partially filled parallel-plate waveguide becomes a TEM mode and the waveguide may be analyzed as a transmission line. The propagation constant and characteristic impedance are determined by the static distributed inductance and capacitance. In general, at low frequencies the mode of propagation would be called a quasi-TEM mode since the axial electric field e2, even though it is small, is not zero. At high frequencies the mode of propagation is an E mode and departs significantly from a TEM mode in its field distribution. Solution At high frequencies k0, and hence l 0 and p 0 , are large. In this case p0c is large so we can replace tanh p0c by unity and (3.120a) gives Z0tanJ0a = erp0 = £ r V ( e r - l ) * o - ' o (3-126) upon using (3.1206) to ehminate p0. This equation is independent of the 124 FOUNDATIONS FOR MICROWAVK ENGINEERING plate separation 6. The solution for e z can be approximated as follows: et(y) = C, sin l0y e,(y) 0<y<a =C,jsinhp0(6 = C, sin lua (3.l27 a > -y) sinh p0(b - y) sinh p0(b - a) ePll<b-y) e* = C I sin(/ 0 o)e-'"><- 1 '- a , a<y<b (3.1276) This is a field that decays exponentially away from the air-dielectric surface and does not depend on 6 as long as pnc = pQ(b - a) is large. This field is guided by the dielectric sheet on the ground plane (lower conductor) even if the upper plate is removed to infinity. This type of mode is called a surface-wave mode because its field is confined close to the guiding surface. The axial electric field for this surface-wave mode is illustrated in Fig. 3.15. The first root for /„ in the eigenvalue equation (3.126) occurs for l0a < 7r/2 or /„ < 7r/2a. Thus, as k 0 approaches infinity, / 0 remains bounded but p0 will become large because pi = (er - Dk2, - /%. Consequently, for large enough kQ we will have 1% •« k2 and then /3 = k. As we go from zero frequency to very high frequencies, the propagation constant varies from a low value of ]/e^kQ given by (3.121) to an asymptotic value of i/erk0. We see that /3 is not a linear function of u> or k0 and for this reason is said to exhibit dispersion. The term dispersion arises from a consideration of signal propagation. In our discussion of waveguides later on in this chapter, we will show that a signal consisting of a band of frequencies will have its frequency components dispersed whenever j3 is not a linear function of (o. This is caused by the phase velocity vp, given by the relation /3 = co/vp and thus vp = <o/p = 1/ yjienlieQ, being a function of to. The ratio /32/&o gives the effective dielectric constant at any frequency. In Fig3.16 we show a plot of ee versus frequency for the case when er = ln> a = 0.4 cm, and 6 = 1 cm. This curve is derived by solving the pair ol equations (3.120). Microstrip transmission lines exhibit similar dispersion characteristics. ' F I G U R E 3.15 Axial electric field for surface-wave mode. TRANSMISSION LINES AND WAVEGUIDES j 40 GHz 125 F I G U R E 3.16 Effective dielectric constant as a function of frequency for cr = 10, Q =0.4 cm, and b = 1 cm. A second surface-wave mode solution can be found from (3.120a ) with l0a in the range TT < lQa < 3 - / 2 provided y e r - 1& 0 is larger than v so that (3.1206) can also be satisfied. For large kCla many surface-wave modes can propagate. In addition to the surface-wave modes, there are also an infinite number of solutions to (3.118) for real values of p. The higher-order solutions have values of p on the order of ntr/b in value, where n is an integer. Provided rr/b is greater than k0, these values of p will give imaginary values of fi and hence nonpropagating modes. The cutoff occurs when p = k0 giving /3 = 0. Thus at cutoff / tan la = —erk0 tan k0c = \/erk0 tan ^erk0a since / = k for (i = 0. This equation reduces to tan yj7~rk0a = - y/tTr tan k0c which can be solved for the values of k 0 at which the various modes cease to propagate. We will not consider the partially filled parallel-plate waveguide any further even though a good deal more could be said about its mode spectrum. The purpose of our discussion is to highlight those features that will be displayed by microstrip transmission lines, which is the next topic taken up. PLANAR TRANSMISSION LINES A planar transmission hne is a transmission line with conducting metal strips that lie entirely in parallel planes. The most common structure is one or more parallel metal strips placed on a dielectric substrate material adjacent to a conducting ground plane. A planar transmission hne that is widely used is the microstrip hne shown in Fig. 3.17. It consists of a single conducting strip of width W placed on a dielectric substrate of thickness H and located on a ground plane. By image theory this transmission line is equivalent to a line consisting of two parallel conducting strips placed opposite each other on a dielectric sheet of thickness 2H as also shown in 126 FOUNDATIONS FOR MICROWAVE ENGINEERING Ground plane |a) FIGURE 3.17 U>) (a) The microstrip transmission line; (6) equivalent parallel strip line obtained by using image theory. Fig. 3.17. Typical dimensions for a microstrip line are substrate thickness of 0.25 to 1 mm and strip widths of 0.1 to 5 mm. The microstrip transmission line can be fabricated using conventional printed-circuit-board techniques which result in good mechanical tolerances and a low cost. In addition to the microstrip line, there are many other planar-transmission-line structures that are used for various purposes. A number of these other transmission-line configurations are shown in Figs. 3.18 to 3.20. The coupled microstrip line shown in Fig. 3.18a is used in directional couplers. The coupled microstrip line supports two modes of propagation. The even mode of propagation has the same voltage and current on the two strips, while the odd mode of propagation has opposite voltages and currents on the two strips. The coplanar transmission line shown in Fig. 3.186 consists of a single strip mounted between two ground planes on the same side of the dielectric substrate. The coplanar line has an advantage over the microstrip fine in that shunt connection of components to the ground plane can be made on the same side of the substrate. In addition, it allows the series connection of components to be made with equal facility to that for microstrip lines. The coplanar strip line shown in Fig. 3.18c is similar to the coplanar line in tha all conducting strips are in the same plane (coplanar). It is less desirable than the coplanar fine because it is not balanced relative to a ground pla116 and t h u s wave propagation on this line is more strongly influenced DV nearby conductors such as a shielding enclosure. In practice, a shielding TRANSMISSION LINES AND WAVEGUIDES 127 (b) FIGURE 3.18 ( a ) Coupled microstrip lines; (6) coplanar transmission line; (c) coplanar strip transmission line. wmy/////A mmmm lb) F I G U R E 3.19 Suspended and crostrip line. )d suspended mi- 128 FOUNDATIONS FOR MICROWAVE ENGINEERING enclosure for a microwave circuit is needed to reduce spurious radiatin from the circuit, eliminate electromagnetic coupling with nearby circuit. and for environmental protection. In Fig. 3.19 are illustrated the suspended and inverted suspends microstrip lines which are quite similar to the conventional microstrip n n but involve less dielectric substrate material. Figure 3.20a shows a slot lin The open slot line is not as widely used as the microstrip and coplanar h n e are. The shielded slot line, which is typically a slot line placed inside a rectangular waveguide as shown in Fig. 3.206, is called a fin line and has found to be useful for many circuit applications. Only shunt-connected components can be used with a slot line. The final transmission-line structure illustrated here is the strip line shown in Fig. 3.20c. This line consists of a strip placed between two parallel plates that function as ground planes. The strip may be rigid enough to be suspended in air or it may be sandwiched between two dielectric sheets as shown in the figure. The strip line was often used in microwave filters and couplers before the other forms of planar transmission lines became popular. The strip line is somewhat more difficult to fabricate but has certain advantages for special applications to filters and couplers. Many directional coupler, power divider, and filter designs using strip lines were developed in the period from 1955 to 1975. An excellent reference source for design data for strip-line circuits is the book by Howe listed in the references at the end of this chapter. The methods used to fabricate planar-transmission-line structures and related circuit elements are compatible with integrated circuit fabrication and have allowed the development of microwave integrated circuits (MIC circuits). In integrated microwave circuits the active devices and all interconnecting transmission lines, impedance-matching elements, needed capacitors and resistors, etc., are fabricated on the same chip. In these applications the microstrip and coplanar transmission lines are the ones most easily adapted for on-chip fabrication. In MIC circuits the substrate thickness and line widths are generally much smaller than in hybrid circuits. The term hybrid is used to describe integrated microwave circuits where the discrete components such as transistors, capacitors, and resistors are soldered in place. The dielectric substrate material used in a planar transmission line must have low loss, i.e., a small loss tangent. A large dielectric constant results in a shorter propagation wavelength and hence a more compa c circuit. The substrate material should have good mechanical strength, be easy to machine, and have good thermal conduction. When active device are mounted into a planar-transmission-line circuit, the heat generated by the active device is in part conducted away to the ground plane through the substrate material. It is difficult to use metal heat sinks in a microwave circuit because these large metal structures would interact with the electrO' TRANSMISSION LINES AND WAVEGUIDES 129 magnetic field in an undesirable, and often unpredictable, manner. Consequently, in power amplifier circuits substrate materials with good thermal conductivity are required. Dielectric materials used in low-frequency circuit boards are generally too lossy to be used for microwave transmission lines. The dielectric constant and thickness must be maintained to a high level of uniformity in the manufacturing of substrates because otherwise the fabricated transmission lines will not perform according to the specified design since the propagation phase constant and characteristic impedance both depend on these parameters. Uniform thickness and dielectric constant is particularly important in the design of filters and impedance-matching elements whose dimensions are critical. Once a printed microwave structure such as a filter has been constructed, it is not very easy to add external tuning elements to bring the constructed filter performance into specifications. A commonly used substrate material is polytetrafiuoroethylene (PTFE)t which has a dielectric constant of 2.1 and a loss tangent of 0.0002 at 1 MHz and around 0.0005 at microwave frequencies. This material has excellent resistance to chemicals used in the photoetching process. In order to increase the mechanical strength, it can be loaded with woven fiberglass mat or glass microparticles. This increases the dielectric constant to the range 2.2 to 3. The use of glass fiber results in some anisotropy in the dielectric constant. In the manufacturing process the glass fibers are generally aligned parallel with the substrate so the dielectric constant along the substrate is typically 5 to 10 percent larger than that norma] to the substrate. By using ceramic powders as fillers, notably titanium oxide, much larger dielectric constants can be obtained. Typical values are in the range 5 to 15. Ceramic materials such as aluminum oxide (alumina) and boron nitride, as well as the glasslike material sapphire, are also used (or substrates. These materials are very difficult to machine. Alumina is perhaps the most commonly used material. It has excellent thermal conductivity. For integrated microwave circuits the usual semiconductor materials germanium, silicon, and gallium arsenide are used. These substrate materials have a high dielectric constant and may exhibit some conductivity depending on the doping level. In Table 3.2 we have summarized the important properties of a number of substrate materials. In this table e r is the dielectric constant along the substrate and e v is the dielectric constant normal to the substrate.$ tThis material is commonly known as Teflon, which is a registered trade name of Du Pont. t i n order to keep the notation as simple as possible, we use e y instead of try for the relative permittivity (dielectric constant) in the y direction. 130 FOUNDATIONS FOR MICROWAVE ENGINEERING TABLE 3.2 P r o p e r t i e s of substrate materials Material PTFE/woven glass PTFE/microfiberglass •CuFion *RT/Duroid 5880 *RT/Duroid 6006 •Epsilam 10 Boron nitride Silicon Germanium Gallium arsenide Alumina Sapphire Beryllium oxide Loss tangent 2.84 2.26 2.1 2.26 6.36 13 5.12 11.7-12.9 16 12.9 9.6-10.1 9.4 6.7 2.45 2.2 2.1 2.2 6 10.3 3.4 11.7-12.9 16 12.9 9.6-10.1 11.6 6.7 0.001-0.002 0.0005-0.001 0.0004 0.001 0.001-0.003 0.0005-0.001 0.0005-0.002 0.0002 0.001-0.002 Thermal conductivity Fair Fair Fair Fair Medium Medium Good Medium Medium Medium Good Good Good Ma. hioaba^ Good Good Good Good Good Good Poor Poor Poor Poor Poor Poor Poor 'CuFlon is a registered trademark of Polyflon Company. It is a Teflon material electroplated with copn» RT/Duroid is a registered trademark of Rogers Corporation. Rogers Corporation also manufactum substrates with dielectric constants around 10. Epsilam 10 is a registered trademark of the 3M Company It is a ceramic-filled Teflon material. The data in Table 3.2 have been compiled from a variety of sources.t Since the dielectric constant and loss tangent are frequency dependent and also influenced by the material processing, the listed data should be viewed as representative values at microwave frequencies. Substrate materials are usually plated with copper in 0.5-, 1-, or 2-oz weights (amount of copper per square foot). The use of 1-oz copper weight gives a plating thickness of 0.0014 in. Gold plating is sometimes used on top of the copper to prevent oxidation of the metal. In integrated microwave circuit construction a metalization thickness of a few microns is typical. One-half oz copper-clad board has a metalization thickness of 18 /xm. 3.12 MICROSTRIP TRANSMISSION LINE In a microstrip transmission line the dielectric material does not completely surround the conducting strip and consequently the fundamental mode of propagation is not a pure TEM mode. At low frequencies, typically below a t H . Howe, "Stripline Circuit Design," Artech. House Books, Dedham, Mass., 1974. 163. T. Laverghetta, Microwave Materials: The Choice is Critical, Microwave J., vol. 28, p1985. M. N. Afsar and K. J. Button, Precise Millimeter-Wave Measurements of Complex Refractive BeO. Index, Complex Dielectric Permittivity and Loss Tangent of GaAs, Si. SiO,2 Al 2 O.,, ^3 Macor, and Glass, IEEE Trans., vol. MTT-31, pp. 217-223, 1983. Some data were also obtained from manufacturers' literature. TRANSMISSION LINES AND WAVEGUIDES 131 few gigahertz for practical microstrip lines, the mode is a quasi-TEM mode. In the frequency range up to a gigahertz or somewhat higher, the microstrip transmission line can be characterized in terms of its distributed capacitance and inductance per meter in a manner similar to what was found for the partially loaded parallel-plate transmission line in the previous section. Unfortunately, there are no simple closed-form analytic expressions that can be derived for describing the field distribution or the characteristics of planar transmission lines. Formal solutions can be derived and evaluated on a computer and have been used to compile data on the characteristics of these transmission-line structures. Static field analysis has also been extensively used to obtain the low-frequency characteristics. However, even the static field analysis is quite complex. The analysis of planar transmission lines can be based directly on a solution for the electric and magnetic fields in the structure. An alternative approach is to first solve for the scalar and vector potential functions and from these find the corresponding electromagnetic field. In actual fact the propagation constant and characteristic impedance can be found from the potentials without a detailed consideration of the fields. The advantage of using the scalar and vector potentials in the analysis is that this approach provides a direct link to the quasistatic solutions in terms of more familiar low-frequency concepts. In this section we will develop the essential equations to be satisfied by the scalar and vector potentials for a microstrip transmission line. From these equations we then obtain simplified ones that will describe the quasi-TEM mode of propagation at low frequencies. The term low frequency is a relative one. It is the ratio of fine dimensions to wavelength that determines whether a microstrip line can be adequately described in terms of the quasi-TEM mode of propagation. In MIC circuits with line widths as small as 100 fj.m, the low-frequency region can extend as high as 20 to 30 GHz. Even though space does not permit a full development of analytic methods suitable for solving planar-transmission-line problems, some insight into the properties of these structures is obtained from the basic formulation of the relevant equations. After we have presented the theoretical foundations, typical dispersion curves and graphical results for characteristic impedances are given for a number of important substrate materials and a range of microstrip conductor widths. The vector and scalar potential functions are solutions of Helmholtz equations as described in Sec. 2.10 when the sources are located in vacuum (air). For the microstrip line shown in Fig. 3.17, two added complications enter due to the presence of the dielectric in only part of the region of interest and the anisotropic nature of some substrate materials. For this reason we need to derive new equations to be satisfied by the potential functions. The substrate material will be characterized by a dielectric constant e u in the y direction which is normal to the interface and by a 132 FOUNDATIONS KOR MICROWAVE ENGINEERING dielectric constant e r in the x and z directions. The unknown charge anj current densities on the conducting microstrip will be denoted by p an^ ~ These source densities are concentrated along y since they exist only on tk microstrip which is assumed to have negligible thickness. The source co * centration can be described by introducing the delta function 8(y - #, ' localize the sources at y = H. Thus we can write p(x,y,z) = ps(x,z)S{y - H) (3.128 a ) J(x,y,z) = Js(x,z)8(y-H) (3.1286) where J s and p s now describe surface densities rather than volume densities. We will assume that the dielectric constants ey(y) and e r (y) are functions of y that are constant in the substrate and undergo a rapid change in value to unity as the interface is crossed into the air region. The reason for doing this is that the equations we then obtain for the potentials will automatically give us the boundary conditions needed to properly join the solutions for the potentials in the substrate region to those in the air region. We begin the derivation by letting B = Vx A From Maxwell's equation V x E = -jwB = -jtoV x A we get V X (E + jo>A) = 0 which has the general solution E = -jwA - V<t> where <P is a scalar potential function. Up to this point we have followed the same steps as in Sec. 2.10. Maxwell's curl equation for the magnetic field is V X H =jcoD + J For an anisotropic dielectric we have D = e0er(Exax + E:az) + e0eyEyay We can replace H by P-Q ' V X A to obtain V x V X A = V V - A - V2A =j<op.0D + Mo J and express D in terms of the potentials as follows: D= - e 0 € r ju>(Axax + A 2 a J + a -eQey\jA,ay + &y ~dy~ x - + a,— TRANSMISSION LINES AND WAVEGUIDES 133 By adding and subtracting a term to the y component that includes the factor e r , we can reexpress D in the form D = - e r e 0 ( > A + V<D) - e 0 (e v - er)\ja>Ayay + a — We wish to eliminate the VV • A term in the equation for A by setting it equal to the gradient of another function. For this purpose we now express e V4> in the form V(er<P) - 4>vVr, where Ver has only a y component since er is a function of y only. We can set VV • A equal to -jioe0n0V(er<P) which gives the Lorentz condition (3.129) V • A = -jwe0ern0<t> The equation for the vector potential now becomes -V'A =j(onQ - y W 0 e r A + e 0 *Ve r -e0(ey -€r)\jmayAy+ av H0J The current J does not have a y component so the x and z components of this equation are V*Ax + V'A: + er(y)klAx= -»0JX (3.130a) er(y)klA2= -^Jz (3.1306) while the y component becomes V% + € y (y)AgA v = -jo)n0e0 *—-W-er)— oy *y d<P = Ju/J-oe0 ( 6 v - e r ) — + <t>(H)(er-l)S(y-H) tly (3.131) where — (e r - l)5(y - H) expresses the derivative of the step change that occurs in e r as y crosses the interface at H, that is, er(y) changes from c r to unity. The equations for A x and A 2 are of the same form as derived in Sec. 2.10, but the equation for A v is new. The equation for A v is coupled to the scalar potential <i>(//) at the boundary even if we have an isotropic substrate. Thus boundary conditions require the presence of an A v component even though there is no y component of current. 134 FOUNDATIONS FOR MICROWAVE ENGINEERING A separate equation for the scalar potential is obtained by usi Gauss' law V • D = p and the Lorentz condition (3.129). Thus we find th ^ V D= V -ere0(jojA + V4>) - e0(ev - e r ) \ j < o a v A y + av — ' dy I = - « o j<oV-(erA) fl + jcoj-(ey - er) Ay + V -(erV*) fl d<& fl = - e o M*J • A + A • V Cr ) + V - ( e r V * ) +J<o — (ey - er)Ay d d® By replacing V • A with -joj€0/j.0er<i> from the Lorentz condition, we obtain ()<P fl = d C ->^ ^fly i f - > W ^ - r ) ^ - P_ The last step is to simplify this equation using dA feder ( e v - e r ) A , . = ( e v - e r ) — - + Avk ~dy~ dy - \ fly dy fl flerd<t> V • e r V<t> = e r V 2 < P + —fly fly and fl d2<P fl<t> <9cp / fley *eT By using these expressions a number of terms cancel and we obtain the final form fl2<p fl2<S> fl fl<i>. + flz' dx' - = fly • - — +6?fc 2 0 * Cy— +>(f, - fly ' l)Ay(H)5(y - H) -jo>(€y - er) where we have also used A > ^ T = -(ey-l)Ay(H)8(y-H) (3.132) dy TRANSMISSION LINES AND WAVEGUIDES 135 This equation also displays a coupling between the scalar potential and Ay(H) at the interface as well as coupling within the substrate whenever e y does not equal e r , that is, for anisotropic substrates. In the air region y > H, both e r and e v are replaced by unity in (3.130) to (3.132). After the above lengthy derivation we can now obtain simpler equations to be solved in each separate region along with boundary conditions to use in joining the solutions at the interface y = H. The source terms p and J when expressed in the form (3.128) contain the 8(y - H) factor. In order for the left-hand sides of (3.130) to (3.132) to equal the corresponding right-hand sides, we must obtain a delta function Sly - H) from the derivative of the potentials with respect to y. In (3.130a) and (3.1306) this requires that dAx/dy and dAz/dy have a step change at the interface so that the second derivative with respect to y will produce a delta function. The required step change can be found by integrating both sides of the equation over a vanishingly small interval centered on y = H. The integral of terms not involving a derivative with respect to y will vanish since these terms must be continuous at y = H and the interval length vanishes. For example, if A s were not continuous at y = H, the second derivative with respect to y would generate a singular term corresponding to the derivative of the delta function and no such term exists on the right-hand side of the equation. Thus from (3.130a) we obtain rff+rrtt, A dA lim / —-=- ay = dy r - O - ' Ht-r -r ' V H- v lim H~ rM+r / - n0Jsx(x,z)8(y - H)dy= -IM0JS w or = -p&J* (3.133a) = (3.1336) H- In a similar way we obtain dy -Mo^s n The notation H^ and H~ means evaluating the derivative on adjacent sides of the interface at y = H. These two equations state that the tangential components of the magnetic field must be discontinuous across the current sheet J s since from the equation B = V X A: 17 = *°Hx = -Mo-ff, In a similar way we obtain the following boundary conditions by integrating 136 FOUNDATIONS FOR MICROWAVE ENGINEERING (3.131) and (3.132) about a small interval centered on y = H: dA"' = y<o/x 0 e 0 (e r - l)<t>(H) (3.133 C ) -fo+Mey-l)Ay(H) (3.md) «dy v ir ~dy~ H~ A term such as (e v - er)d<P/dy that occurs in (3.131), and a similar term occurring in (3.132), does not contribute because lim/ (e - e , ) — dy= l i m ( e - er ) / — dy = 0 since d<J>/dy is continuous in the interval H - - < y < H and for y > /f we have e„ =« €p = 1. The boundary conditions on <t> reflect the fact that the total y-directed electric field has a contribution from A y so that the discontinuity in D y across the charge layer is given by + € y€0 • JdiJ -e< — + J iioA. = P* ~dy~ ir dy * which is (3.133d). By using the above boundary conditions, we can solve (3.130) to (3.132) in each respective region away from the interface. Thus we need only to solve the following homogeneous equations, subject to the specified boundary conditions, in the substrate region: (v2 2 + £rk )Ax = o (3.134a) (3.1346) d$> V2Ay + eyk2Ay =j(oiu.0€0(ey - er) — <?2<1> d2<t> e„ el 2 * dA, + +f € 2 772 + T J ^ ^ 2 r*0* = -X«v r)^~ dx 9y dz *r <>y (3.134c) (3.134a") In the air region the equations to be solved are obtained by setting e r = fy = 1. There is no volume coupling between <J> and Ay in the air region or m an isotropic substrate region. Since we are interested in wave solutions representing waves propagating in the z direction, we can assume that the dependence is e~jt>z. The second derivative with respect to z can then be replaced by - / 3 2 . The common factor e~ip* can be deleted from the equations just as e-""' was dropped for convenience. Low-Frequency Solutions We can obtain the equations to be solved in the zero-frequency limit by assuming that the potentials and the source terms can be expanded a 5 TRANSMISSION LINES AND WAVEGUIDES 137 power series in to. Thus we let A = A° + <uA1 + « 2 A 2 + ••• (3.135a) <J> = <I>° + &><*>' + to2®2 + • • • (3.1356) J = J° + w J l + a r J 2 + ••• (3.135c) P =p° + top1 + w'Y + ••• (3.135d) The parameter k 0 = io2p.0e0 is of second order in to. The propagation constant (5 can be expressed in the form fi = ^ k 0 , where e e is a frequency-dependent effective dielectric constant. Consequently, p2 is also of second order in to. We now substitute these power-series expansions into (3.130) to (3.132) and equate all zero-order terms to obtain the following lowest-order equations: ( a ? + ^ 5 A°x - - / * 0 J , ° ^Z5 + -^\All= -n0J? dx* dy (3.1366) a2 , P d2 (3.136a) a a \ p° e r ^ r + — e v — $° = -— (3.136d) r 2 dx ay "ay J e0 Further information is obtained from the Lorentz condition (3.129) which gives 3A° M° —- + — y - = 0 (3.137a) dx dy -MAI = - > * o * r * ° (3.1376) In the air region e r and e„ are set equal to unity. From the continuity equation relating current and charge, namely, V • J = —jcop we obtain = 0 (3.138a) -jpj?= -jwp0 (3.1386) 138 FOUNDATIONS FOR MICROWAVE ENGINEERING In the above equations the e~jtiz factor is not included but any derivative with respect to z was replaced by -j/5. Since J° must be zero at the edges x = ± W/2 of the microstrip w p conclude that J® is zero because the integral of dj^/dx is at most constant. Hence, to lowest order, there is no x-directed current on thp microstrip and A" is zero. The Lorentz condition then requires that dA°/a v = 0 and hence A° = 0 aiso since a constant A\ is a trivial solution and would not produce any magnetic field contribution. Thus, to lowest order we only have to solve for an A^ and a scalar potential <t>". If we assume the microstrip to be at a potential V, then the boundary condition on 3>° is that it equals V on the microstrip and equals zero on the ground plane. We can integrate the continuity equation (3.1386) across the microstrip line to get 0 / / = "Q° (3.139) where 7Z° is the total 2-directed current on the microstrip and Q° is the total charge. On the microstrip the axial electric field must be zero. To lowest order this boundary condition is El = <?<p° - > A ° - — = -ju>A° +jp<b° = 0 az or wA° = pV (3.140) Hence A° is also constant on the microstrip. We will show shortly that the inductance L per unit length of the microstrip line is given by the equation I?L = A°z (3.141) The capacitance per unit length is given by C = — (3.142) By using these expressions to eliminate Q° in (3.139) and A" in (3.140), we obtain the pair of equations pi? = coCV »U„° = PV (3.143c) (3.1436) from which we find that p2 = u>2LC ^~ h° (3.1440) (3.1446) T TRANSMISSION LINES AND WAVEGUTDES 139 A F I G U R E 3.21 Surface used to find the magnetic flux linkage in a microstrip line. We have thus been able to show in a rigorous way that in the low-frequency limit the microstrip line can be analyzed as a static field problem and that its propagation constant and characteristic impedance are determined by the low-frequency distributed capacitance and inductance. The analysis leading up to (3.144) is quite general and applies to other planar transmission lines as well. At this point we return to the promised derivation of (3.141) giving the line inductance. With reference to Fig. 3.21 we note that the magnetic flux ip Unking the microstrip per unit length is given by the integral of B x over the area extending from the microstrip to infinity. Thus ip = f I B • axdydz By using B = v* x A and Stokes' law, we can write ip = f1 f V X A • a , dy dz J J 0 H 6\-d\ = c i where C, is the boundary of the area. Since A" is zero and A° is zero at infinity and is constant on the microstrip, we obtain i/< = A°z for the flux linkage. The inductance is given by i/»//£° and this gives (3.141). We can also derive equations for the next level of approximation. However, the solution of these equations is not much easier than the solution of the original equations; so it is not worthwhile to develop the power-series solutions beyond the lowest order. Thus the equations to be 140 FOUNDATIONS FOR MICROWAVE ENGINEERING solved are ;>2 a*5 dx2 d2 + dp A* = 0 y <H,y> H er dy2 9x* *2 \ ay2 „ (3.145aj y <H (3.1456) y>H (3.145c) with the boundary conditions (3.145d) = ~Mo^« !i a*0 M>° dy <t>° = V — €» P« (3.145e) »y on microstrip A° = constant on microstrip Along the interface and away from the microstrip, the right-hand sides of (3.145(f) and (3.145e) are zero. In addition, 4>° and A°z must be zero on the ground plane in order to make the tangential electric field vanish on this surface. The equations for A" do not depend on the dielectric constants of the substrate material. Hence the line inductance is the same as for an air-filled line. But for an air-filled transmission line with distributed capacitance C„, we have \/LC a = y>oe^ and hence L = so we can find L by finding the distributed capacitance of an air-filled microstrip line. By introducing Ca in place of L, the solutions for /3 and &c can be expressed in the form (3.146a) / ^ w i c ^ / — k 0 = jrek0 =l/ c: L C* •"cO (3.1466) where Zc0 is the characteristic impedance of the air-filled fine and the ratio C/C„ gives the low-frequency equivalent (effective) dielectric constant ee- TRANSMISSION LINES AND WAVEGUIDES 141 The effect of having an anisotropic dielectric substrate does not add any additional complication. If we introduce a new variable u = (er/ey)1/2y> then upon using d du S I e, d du dy we find that (3.1456) reduces to d2 dx* (3.147) du1 When y = H the corresponding value of u is Ur/ey)i/2H; so the solution of (3.147) is that for a microstrip with an equivalent substrate thickness H v given by (€r/evY/2H = He. The boundary condition (3.145e) becomes d<t>° h d<p° H, 9y a d<S>° - y ^ S du = - — (3.148) H; which shows that the equivalent dielectric constant of the substrate should be taken as the geometric mean e g = -Jerev. Thus, by modifying the substrate thickness and introducing the equivalent dielectric constant, the solution for the distributed capacitance C for the case of an anisotropic substrate can be reduced to that for an isotropic substrate. The distributed capacitance C a is that for the unsealed microstrip line. The unit of length does not enter directly into the differential equations for the potentials. Thus x and y can be in units of meters, centimeters, inches, or any other convenient unit. What this means is that the distributed capacitance and inductance per unit length is dependent only on the ratio of strip width to substrate thickness, i.e., on W/H. If we have found C and L for a given set of values for W and H on a per-meter basis, then if we change W to sW and H to sH, where s is a scaling factor, both C and L on a per-meter basis do not change. Hence the characteristic impedance, effective dielectric constant, and propagation constant /3 for any planar transmission line is invariant to a scaling of the cross-sectional dimensions. However, the attenuation caused by conductor loss does not scale since the series resistance is inversely proportional to the conductor widths. The attenuation due to conductor losses will double if the conductor size is reduced by a factor of 2. The scaling law is clearly illustrated for an ideal parallel-plate capacitor with a plate dimension of W? and separation H and having a capacitance e0Wl/H. Clearly keeping the ratio W/H fixed keeps the capacitance unchanged. A variety of methods exist for solving the two-dimensional Laplace equation (3.147). For planar transmission lines the conformal mapping method is widely used, generally along with some approximations that are necessary because of not having a dielectric medium filling all of the space 142 FOUNDATIONS FOR MICROWAVE ENGINEERING y W >,-;;u;ffi F I G U R E 3.22 A microstrip line with perfectly conductim, side walls inserted at x = ±a with ct »yy ?.m -a around the conductors. A number of useful solutions obtained by conforms mapping methods are described in App. III. We will refer to some of these solutions as needed. In order to illustrate the general method of solution, we will develop a Fourier series solution to (3.145) which will turn out to provide an efficient method to obtain the parameters of a microstrip transmission line. In order to use the Fourier series method, we place perfectly conducting (electric) walls at x = ±a as shown in Fig. 3.22. Provided a is chosen equal to 10VV or 10H, whichever is larger, the sidewalls have a negligible effect on the field which is concentrated near the microstrip. We can expand the unknown charge density p, into a Fourier series of the form Ps(x) = E Pncos n-1,3,... nvx 2a (3.149a) The charge coefficients pn axe given by Pn I fW/2 p s (x')cos nvx a '-W/2 '- dx' (3.1496) ~2a~ The charge density is an even function of x because of the symmetry involved; so only a cosine series is needed. The functions are chosen so that they vanish at x = ± a, a required boundary condition for the potential; so only odd integers n are used. The potential <i>(x,y) can also be expanded into Fourier series; so we let *(*.y) = E n=1.3,... E 11=1,3,... f,,(y)cos TITTX 2a mrx §n(y)cos 2a y>H 0 < y < He (3.150) where fn(y) and g„(y) are to be found. We are using an effective substra thickness He, so t h a t an anisotropic substrate can be accommodated. W have dropped the superscript 0 since it is understood from the context th» we are solving for a static potential field. TRANSMISSION LINES AND WAVEGUIDES 143 Each Fourier term in the expansion of <l> must be a solution of Laplace's equation. Hence we require that + *2 *2WfB(y)\ ^ ^)U(,)) C Ortirx S =0 ^ which gives d>f„(y) - ^ - -<fn(y)=0 - ^gn(y) = 0 where u>,2 = (nv/2a)2. In the region y < He, a suitable solution for g„(y) that vanishes on the ground plane is 8*(y) = Cn Sinn »>„y where C„ is an unknown constant. In the region y > / / . we need a solution that will vanish as >• approaches infinity; so we choose fn(y) = Dne-»»> where D„ is another unknown constant. At y = He. H the two potential functions must match; so we have CnsmhwnHe = Dne-»»» The Fourier series expansion of the charge density p s represents this charge density as sheets of charge p„ c o s n i r j ; / 2 a that extend from x = -a to x ~ a. By superimposing an infinite number of such charge sheets, we obtain a charge density ps that is nonzero only on the microstrip - W/2 < x < W/2. The boundary condition (3.148) is applied to each Fourier term to obtain dfn_ dg» H or — w„D„e *'*> w a * - egw„C„ cosh w„He ~ —- We now have two equations which we can solve to find C„ and Dn. The solutions are CL- Dm = where eg = J*res. e 0 u; n (sinh wnHe + eg cosh wnHe) w H Pne - sinh wnHe e0u>„(sinh wnHe + eg cosh waHe) 144 FOUNDATIONS FOR MICROWAVE ENGINEERING We now substitute our solutions for f n and g n into (3.150) and Usse (3.1496) for p„. Thus we obtain *(*,y)- E / »-l.3.... COS W„X COS U ) „ X ' W/2 J -wne0u/„a(siahu/nHt + egcoshwnHe) sinh /^.y X siimi« n i2 e e-^ ( } f - i r 3 Ps(x')dx- (3.151) where the upper term is for v < He and the lower one is for y > // -TJ. factor multiplying p s (x') under the integral sign represents the Green's function for this problem. It is designated by the symbol G(x,y;x', y')- so in abbreviated form we express (3.151) as r W/2 *(*,?)-/ J G{x,y;x',He)p,(x')dx' (3.152) -W/2 The last boundary condition to be imposed is the requirement that <f> = V on the microstrip; thus V=f J -w/2 G(x,Ht;x;He)p,(x')dx' - — W* W < x < — 2 2 (3.153) This is an integral equation whose solution would determine the unknown charge density p s (x'). Once we know the charge density, we can calculate the total charge on the microstrip using J -W/2 and find C = Q/V. Integral equations are most often not solvable by analytic means. However, various numerical schemes exist for obtaining good approximate solutions. The most popular method is the Method of Moments.t In "ris method the first step is to choose a finite number of basis functions and to expand ps(x') in terms of these in the form N W W where Q n are unknown coefficients. The basis functions could be the u height pulse functions shown in Fig. 3.23, the cosine functions cos 2nirx/ > or any other reasonable set that would give a good approximation to pM tR. F. Harrington, "Field Computation by Moment Methods," Krieger Publishing ComP"™' Inc., Malabar. Fla., 1968. TRANSMISSION LINES AND WAVEGUIDES tfi 145 *N F I G U R E 3.23 Unit height pulse functions for expanding the charge density. W 2 w 2 When this substitution is made in (3.153), we obtain W W - - < * < - N V- where L Q„Gn{x) Gn(x) = 2 f J (3.154) G(x,H,;x',Ht)*n(*')dx' -W/2 The next step is to convert (3.154) into a matrix equation for the unknowns either by making both sides of the equation equal at N points in x along the microstrip, or by using weighting functions to make N weighted averages of both sides equal. We can choose the <l>m{x) as weighting functions, in which case the method is called Galerkin's method. Other choices for the weighting functions can also be made. If we use Galerkin's method, we obtain £ GnmQa = Vm m = 1,2, ...,N (3.155) where the matrix elements are given by jW/2Gn{x)4,m(x)dx Gnm~ 1 -W/2 = / rW/2 / G(x,x-)Ux)<l>m(*')dxdx- -W/2 and the components V„, of the source vector are given by Vm= rW/2 J (•~V^m{x)dx m = 1,2,...,AT -W/2 The system of linear equations given in (3.155) can be solved for the unknown charge amplitude coefficients Qri. If N is chosen sufficiently large, we will obtain a good approximation for the charge density. If we know a priori how p,(x') is distributed on the strip, or a close approximation to it, we could get an excellent approximate solution using very few basis functions. From conformal mapping solutions we know that on an isolated infinitely long conducting strip of width W and with total 146 FOUNDATIONS FOR MICROWAVE ENGINEERING F I G U R E 3.24 Charge density on an idea) isolated strip of width W. charge Q per meter that the charge density is given by 2Q Ps(x) = TrWy7! -X2/{W/2)' (3.156) This charge distribution is illustrated in Fig. 3.24. At sharp corners and edges the charge density always exhibits an infinite behavior. However, the density is never so singular that it cannot be integrated. The reader can readily verify that if the substitution x = (W/2)sin 6, dx = (W/2)cos 0d6 is made, then iJ W/2P^)dx'^r/2deJ = Q -w/2 I ir -w/3 In a microstrip line the charge density is influenced by the dielectric substrate and the ground plane but surprisingly (3.156) remains a good approximation. By using a two- or three-term polynomial along with (3.156), i.e., choosing Q0 + Q,x2 + Q2x* Ps(x) = 2 }/l-x /(W/2) (3.157) 2 an excellent approximate solution for ps(x) will be obtained. We do not plan to solve the integral equation (3.153) numerically using the Method of Moments. It does require considerable numerical compu^' tion to evaluate the matrix elements because we only have the Green function expressed as an infinite Fourier series. What we are going to do to express the integral equation in a form that we can interpret as repr senting the capacitances between a conducting strip in air above a groun plane with spacings He, 2He, 3He, etc We can then make use of ^ conformal mapping solution for a pair of strips in air to evaluate t distributed capacitance C for the microstrip line. TRANSMISSION LINES AND WAVEGUIDES 147 Let us choose e = 1 but still retain the strip spacing above the ground plane as He. By using (3.151) in the integral equation (3.153) for this case, we obtain (note that sinh wnHr + cosh w„He = ew"H') rW/2 Q ' * C °S W„X COS W.x' L n 7 ! —(l - e-*"» H <) -^QJ-dx' = V (3.158) where we have set y = H e and multiplied and divided by the total charge Q. The part multiplied by - c - 2 " ' " " - represents the effect of the ground plane which is equivalent to that of the image strip at y = -He and having a charge density -ps. A solution of (3.158) provides the solution for the problem of a strip above a ground plane, as well as the solution to the problem of two parallel strips in air and separated by a distance 2HP. The capacitance per meter for a strip above a ground plane is given by Q Ca(He) = ~ while that between two strips separated by a distance 2He is We see that the integral, involving the normalized charge density, repre- sents 1/CJHJ. Consider now the factor sinh wn Hv sinh w„ He + eg cosh wn Hc that occurs in the Green's function in (3.151) at y = He. We can express this term in the form B"«ft _ e-*nHr (1 + 1 _ e~2w„Hr w e )e"'» . - (1 - e j c - " ' " " ' " I I - C- _ „ + C ( 1 _e-2-„^)(l_77e-2-n",)-1 1 + e g where 77 = (1 - e „ ) / d + e„). The last factor is now expanded into a power 148 FOUNDATIONS FOR MICROWAVE ENGINEERING series to obtain 1 1 + e (1-e -2u'„H, ) £ •n'"e-'imw-H' m=0 Y\ •nme~2mw"H'- - Y 7 i"' e - 2 < n ' + I)"',H„ m=0 We now add -ij"' to both series which has no net effect because of thp minus sign in front of the second series. This gives m*-0 I f " v'^l-e-2"""""') 1 +<r„ - m E =0 + " £ 1 rim(l ~ e-*"'*""'""') m=0 The m = 0 term is zero in the first series so we can change rn to m + 1 and still sum this for m = 0 , 1 , 2 , 3 , . . . without changing its value; thus we get i r 1 +e„ - ^ £ ij m "* 1 (l - e -2("> + !)«-„",) + y rim(l - e - 2 * 1 "+ »«'»«,\ m-0 m=0 •(n - 1) L *7 m (l -e" 2 ""*»«•<.»«) m-0 2e„ £ (1 + O «-o Vm(l ~ e~*m*v>w°'i') (3.159) Upon using this expansion in the Green's function, the integral equation (3.153) can be expressed in the form 4e, • = 0 ( 1 + 6,) rW/2 y -«72»-i,3,... cos wn x cos «;„ A: 4e 0 w„a (1 - ,-*»+*-»,*) M i l (&' = V (3.160) We note that the m th term considered by itself is an integral equation of the same form as that in (3.158) apart from the multiplying factor 4e^77 m /(l + eg)2. This integral equation would provide a solution for the capacitance of a strip in air spaced a distance (m + l)He above the ground plane. If we assumed that the normalized charge density ps(x')/Q was the same for any strip, independent of the spacing above the ground plane, eac integral would produce a constant voltage Vm but different from V. Wit increased strip spacing and constant total charge on each strip, the integr has to give a larger voltage since Q = Ca[(m + l)He\Vm and Ca[(.m + 1 ^ ' j | the capacitance between the strip and the ground plane, decreases wit increasing m. The approximation that the charge density is the sain independent of strip-ground-plane spacing is a necessary one to make sin c there is only one charge density expression in the integral equation. Tn TRANSMISSION LINES AND WAVEGUfOBS 149 approximation is a good one and by using it we can express (3.160) in the form since by our assumption W/2 £ c o s i e s cos «,„*' •'-w/2„ = 1 . 3 . . . . Q £.a)W+lltf 4COH;„C dx = V,„ = '" Ca[(m + l)ff e ] Prom (3.161) we now obtain the following solution for the distributed capacitance C = Q/V: C = ! a -J j (3.162) 1 + CJLH.) E v : ' CD[(m + l)tf c ] m x where M is the number of terms that are kept. Since -q is negative the series is an alternating one. Typically, from 10 to 40 terms are needed for good accuracy. The evaluation can be done on a computer very quickly and requires only a simple program to implement. However, we do require an expression for the capacitance between a strip in air as a function of the spacing above the ground plane, which is given below. The exact solution for the capacitance between a strip of width 2W and a distance H above a ground plane is given in App. Ill, along with tabulated values as a function of 2W/H. For practical applications it is desirable to have simple formulas that will enable the capacitance to be evaluated with an accuracy of 1 percent or better. A number of investigators have proposed such formulas which are based on approximate analytic solutions, along with empirical adjustment of various numerical constants so as to achieve the desired accuracy.! The following formulas give excellent results for the t H . A. Wheeler, Transmission Line Properties of a Strip on a Dielectric Sheet on a Plane, IEEE Trans., vol. MTT-25. pp. 631-647. August, 1977. E. 0. Hammerstad, Accurate Models for Microstrip Computer-Aided Design, IEEE MTT-S Int. Microwave Symp. Dig., pp. 407-409. 1980. E. 0. Hammerstad, Equations for Microstrip Circuit Design, Proc. European Micro. Conf.. Hamburg, W. Germany, pp. 268-272, September, 1975. Equations (3.163) and (3.166) come from this publication. S. Y. Poh, W. C. Chew, and J. A. Kong, Approximate Formulas for Line Capacitance and Characteristic Impedance of Microstrip Line, IEEE Trans., vol. MTT-29, pp. 135-142, February, 1981. 150 FOUNDATIONS FOR MICROWAVE ENGINEERING capacitance per meter of a strip of width W at a height H above a gfouiuj plane and with air dielectric: W 2776,, C = TBH In — W W H * 1 (3.163 a) W (W w — + 1.393 + 0.667 In — + 1.444 (3.1636) H > 1 H \ H These formulas give results that agree with those tabulated in App. m *. within 1/4 percent. The effect of finite thickness T for the microstrip on the distributed capacitance is normally negligible. If necessary the effect of finite thickness can be included by using an effective width W(, where We is given by the following expressions due to Gunston and Weale:t C=e, W 4TTW\ We = W+0.398T 1 + ln T 2H = W+ 0 . 3 9 8 : r | l + In I H 1 < 2^ W 1 — > 2^ H The above expressions can be used in (3.162) to evaluate the capacitance C for a microstrip line with an isotropic or anisotropic dielectric substrate. The effective dielectric constant e e for a microstrip line is given by c_ el (3.164) icteristic where C a is the capacitance of the unsealed air-filled line. The characteristic impedance is given by .1/5- i/^V Mo«o (3.165) e„ C„ Even though computations based on (3.162) are straightforward, there is an easier way to find ee. Schneider has presented a remarkably simp' formula for the effective dielectric constant of a microstrip line with an isotropic substrate.^ This formula was modified by Hammerstad to improve tM. A. R. Gunston and J. R. Weale, Variation of Microstrip Impedance with Strip Thickness. Electron. Lett., vol. 5. p. 697, 1969. tM. V. Schneider, Microstrip Lines for Microwave Integrated Circuits, Bell System Tech. •• vol. 48, pp. 1422-1444, 1969. Schneider's formula has a numerical coefficient of 10 instead ° 12 multiplying the H/W term. TRANSMISSION LINES AND WAVEGUIDES 151 the accuracy.t The modified formula, which we will refer to as the S-H formula, is + 1 e - \ ( 12H -1/2 where F(er,H) = 0.02(e r - 1X1 - W/H)2 for W/H < 1 and equals zero for W/H > 1. The last term accounts for the reduction in e e caused by the finite thickness of the microstrip. We have checked the accuracy of this formula against the results obtained by solving the integral equation and found the agreement to be better than 1 percent for 0.25 < W/H < 6 and 1 < e r < 16. We can also adapt the S-H formula to treat the case of an anisotropic substrate as follows: For an anisotropic substrate we replace the spacing parameter H by the effective spacing H e given by H -and use the geometric mean e g = ye^Cy f ° r *-ne dielectric constant e r . The S-H formula gives the capacitance of a microstrip line with spacing H e and dielectric constant e g relative to an air-filled line with the same spacing He; thus C{eg,He) e + 1 e„-l/ H„\'l/2 , tT v The effective dielectric constant is, however, given by the ratio of C(eg, He) to the capacitance of the air-filled unsealed line according to (3.164). Hence, for the case of an anisotropic substrate, we have U e„ = + l e g -l, H.A-^ 1 + 12-^ +F(c,,ff.) m **» The capacitances for the air-filled lines are readily computed using (3.163). For comparison purposes, typical results obtained for e e using (3.162), formulas (3.166) or (3.167), and those obtained by solving the integral equation are given in Table 3.3. Also listed are the number of terms needed in the formula (3.162) to give a numerical convergence of 0.3 percent. Overall, all three methods give values for e e that are in close agreement. Equation (3.162) gives values that are on the high side for wide strips and substrates with large dielectric constants. This is caused by the variation in charge density with strip spacing above the ground plane, which is more t E . O. Hammerstad. loc. cit. (1975 paper). 152 FOUNDATIONS FOB MICROWAVE ENGINEERING TABLE 3.3 Comparison of values for effective dielectric constant using different formulas Integral equation W/H 0.25 0.5 1 2 • 6 Eq. (3.162) E q . (3.166) Integral equation 1.583 1.605 1.639 1.689 1.75 1.789 1.589 an 1.612(7) 1.649 (7) 1.699(7) 1.761 (8) 1.799 (8) 2.671 2.694 2.731 2.797 2.906 2.979 -> 2 4 6 3.896 4.003 4.173 4.428 4.763 4.966 2.675(15) 2.698(15) 2.734(15) 2.802 (16) 2.929 (16) 2.944 (16) 2.69 2.721 2.731 2.794 2.890 2.963 *f=10 *r = 6 0.25 0.5 Eq. (3.166) er = 5.12, e v = 3.4t 6r=2 1.588 1.61 1.645 1.696 1.762 1.801 Eq. (3.162) 3.896 (22) 4.004 (22) 4.169(22) 4.447 (23) 4.837 (23) 5.012(23) 3.913 4.025 4.193 4.444 4.75 4.943 6.195 6.387 6.69 7.15 7.757 8.127 6.192(36) 6.40 (37) 6.70 (37) 7.16(38) 7.946 (39) 8.303 (40) 6.244 6.445 6.748 7.201 7.750 8.098 t B o r o n nitride. j T h e n u m b e r s in p a r e n t h e s e s a r e t h e n u m b e r of t e r m s used in t h e numerical solution. pronounced for wide strips. From the tabulated results it can be inferred that the modified Schneider's formula will be acceptable for most applications since even the dielectric constant of the substrate is often not known to an accuracy much better than 1 percent. The following two examples will illustrate the application of the above formulas to microstrip lines. Example 3.4. A microstrip line uses a substrate with dielectric constant cr = 9.7 (alumina) and thickness 0.5 mm. The strip width is also 0.5 mm. We wish to find the effective dielectric constant, the characteristic impedance, and the microstrip wavelength at a frequency of 2 GHz. Since the substrate is isotropic, we use (3.166) to find ee. Thus since W/H = 1, 10.7 8.7 -1/2 _ = 6.556 r ^ - ( 1 + 12) In order to evaluate Zc using (3.165), we first find Ca using (3.163). Thus e0 m(8 + | ) = 2.978 From (3.165) we get 120TT 2.978V^7 2.978^6.556 49.44 n TRANSMISSION LINES AND WAVEGUIDES 153 At 2 GHz the propagation constant is /3 = -^2TT/\0 = 1.0725 rad/cm. Hence A = 2ir//3 = 5.858 cm. The wavelength can also be found using A = A 0 / j/e^. Example 3.5. A microstrip line uses a sapphire substrate 1 mm thick and having «r = 9.4, ey = 11.6. We want to find the effective dielectric constant and characteristic impedance for the case when the strip is 0.5 mm wide. Since this substrate is anisotropic, we first find eg = Jerey = 10.44 and H e = yjer/*y H " °- 9 get the ratio mm - We n o w u s e (3-167) to find ee. By using (3.163a) we Ca(He) 2.347 ca(B) 2.26 = 1.0385 From (3.167) e„ = 1.0385 11.44 9.44 o,9 r 1 + 12-— 2 0.5) v + 0.02 x 9.44 x 4§ = 7.02 From (3.163a) we get CJ.H) = 2.26«0 and using (3.165) gives 120TT Z = 2.26/02 = 62.96 n Microstrip Attenuation Dielectric losses and conductor losses will introduce attenuation. The attenuation caused by the finite conductivity of the conductors is accounted for by the series resistance R, while attenuation caused by dielectric loss is modeled by the shunt conductance G in the distributed circuit model of the microstrip line. The separate attenuation constants are given by a „ = — 2ZC G «d - (3.168a) (3.168b) and the total attenuation is given by a = ac + ad (3.168c) The attenuation in decibels per unit length is obtained by multiplying a by 8.686. We will first examine the attenuation caused by dielectric loss. The dielectric loss arises when the permittivity e is complex, that is, e = e' -je". The loss tangent 6" tan 8, = — = 81 154 FOUNDATIONS FOR MICROWAVE ENGINEERING is the usual given parameter for a dielectric material. Maxwell's equation V X H = jweE + o-E = y W E + (<oe" + a)E shows t h a t cue" can be viewed as the effective conductivity of a In dielectric when the ohmic conductivity a is zero. Normally we can assu that cr = 0 except for a semiconductor substrate, in which case o- vrin depend on the doping level. The electric energy stored in the substrate region of the microstrip lin is given by We. = — ( E • E*tfV 4 -V, where V l is the volume of the substrate region per unit length of line. The power loss due to dielectric loss is given by P, = toe" r / E • E* dV 2 Jv. ' Thus we see that If the dielectric filled all of the space around the microstrip, we could equate Wel to C V 2 / 4 and P, to GV'2/2 and thereby obtain we G= — C € However, for a partially filled line some of the electric energy is located in the air region that occupies a volume we will call V2. Consequently, we have Wtl + We2 e' e0 , CV2 = - ^ E • E*dV+ -fvE • E*dV = — If we had an air-filled line, we could write e Wei + We2 = -? /" 4 -'v, E e C V2 * E*dV + 4 f E • E*dV= ~^r4 4 Jyz with the understanding that the electric field in the two cases will not be t same. There is no simple exact way to determine how the electric energy split between the two regions. There is, however, an approximate method find the division of the total energy between the two regions and ^">s based on the assumption that the volume integrals of E • E* in the t cases are approximately the same. By making this assumption we can vW eh + e0I2 = CV2 e0I, + e0I2 = CaV2 TRANSMISSION LINES AND WAVEGUIDES 155 where Ix and I2 represent the values of the integrals over Vl and V2, respectively. The above two equations can be solved for /, and / 2 to give e - e0 e ~ e0 where e'r = e'/e0. The fraction of the total energy in the dielectric region is <./, e'r(C-C„) e're,-l e'r where we have used C = e c C„. The parameter o: is called the filling factor. If q was found independently, then we could solve for ee to get ee = e'rq + (1 ~ q). The parameter q is the ratio of the integral of E • E* over the volume V y to the integral over the total volume V, + V2, that is, q = / [ / ( / j + I2) and clearly represents a filling factor. With the above assumption regarding no change in the volume integrals for the two cases, we see that G, as given earlier, should be reduced by the same fraction by which the total electric energy was split since there is essentially no loss in the air region. Hence an estimate for G is G=~J7—r — C (3.170) By using Z e = y e J y ^ t 0 e 0 / C we obtain ad = GZC — = ,7 e'r —-!=-- e, - 1 -tanS, (3.171) for the attenuation constant due to dielectric loss. In the derivation we used a>ijn0e0 = k0 = 27r/A 0 . As an example if e'r = 9.7, ee ~ 6.55, and tan 8, = 2 X 1CT4 we get ad = 1.52 X 10" 3 Np/wavelength. In decibel units this equals 0.013 dB/wavelength, which is a relatively small value. The attenuation caused by conductor losses will be significantly larger. Equation (3.171) is valid for isotropic substrates only. We now turn our attention to evaluating the attenuation caused by finite conductivity of the microstrip and the ground plane. The continuity equation (3.138b) shows that the current density J. along the conductors varies the same way as the charge density. Thus on a conducting strip of width W the current density will be similar to the charge density given by (3.156). At the edge of an infinitely thin strip, the current density will increase inversely proportional to the square root of the distance from the edge and becomes infinite at the edge. However, the density can be integrated to give a finite value for the total current. But since power-loss calculations require integrating the square of the current density, we would find that for the current density on an infinitely thin strip we would obtain infinite power loss. In practice, the conductors have a finite thickness and the current density is less singular at the edge and the power loss is finite. 156 FOUNDATIONS FOB MICROWAVE ENGINEERING Consequently, it is necessary to take into account the finite thickness of tL conductors. In addition, it is necessary to determine how the total cu r divides between the two faces of the microstrip since the presence /" ground plane results in an unequal division of the current for strip win! greater than one-half of the spacing above the ground plane. The current distribution, current division, and power loss can evaluated using conformal mapping techniques. In order to obtain usef formulas, some approximations are necessary. The analysis for a microsfJ; line is carried out in App. Ill and the results obtained are repeated hp^ (note that in App. Ill the strip width is 2w and the thickness is 2r, where here we use W for the width and T for the thickness). The normalized series distributed resistance for the microstrip is R l where R,W I 1 1 4irW = LR - + - ^ m — — (3.172) The loss ratio LR is given by IV LR = 1 — < 0.5 w /wy w (3.173) LR = 0.94 + 0.132— - 0.0062 — 0.5 < — < 10 H \ H} H The loss ratio gives the increase in resistance that results from an unequal division of the current. The normalized series resistance R 2 of the ground plane is given by R2 W/H W W— = 0 1 < — < 10 (3.174 v Rm W/H + 5.8 + 0.03H/W ~ H ~ This formula states that the effective width of the ground plane is W + 5.8H and having uniform current density. The skin-effect resistance R m is given by R m = ( w / i / a ) 1 / 2 . For copper with a conductivity of 5.8 X 10 7 S/m. w 6 have R m = 8.22 x 1 0 _ 3 / 7 ft where f is in gigahertz. The total series resistance is R x + R* and thus upon using (3.346) we get a„ = x?] + R2 (3.1751 2ZC for the attenuation caused by conductor losses. For the quasi-TEM m the magnetic field, and hence the conductor losses, do not depend on substrate material. . i The equations presented above predict somewhat higher theoreti attenuation than that obtained from a formula developed by Pucel, M» and Hartwig using Wheeler's incremental inductance rule.t Our form 1 "* tR. A. Pucel, D. J. Masse, and C. P. Hartwig, Losses in Microstrip, IEEE Trans., vol. *4T»" pp. 342-350, June, 1966. TRANSMISSION LINES AND WAVEGUIDES 157 a p p e a r t o b e i n b e t t e r a g r e e m e n t w i t h e x p e r i m e n t a l r e s u l t s . F o r practical m i c r o s t r i p lines surface r o u g h n e s s can increase t h e a t t e n u a t i o n b y a s m u c h as 50 p e r c e n t or m o r e d e p e n d i n g on t h e scale of surface r o u g h n e s s relative to t h e s k i n d e p t h . T h e etching process does n o t p r o d u c e a perfectly flat end face at t h e sides of t h e strip- S o m e u n d e r c u t t i n g of t h e edge occurs a l o n g with s o m e r o u g h n e s s , which will also r e s u l t in an increase of t h e a t t e n u a tion above t h e t h e o r e t i c a l values. E x a m p l e 3.6. We wish to find the attenuation for a microstrip line using a copper strip of width 0.5 mm, a spacing of 0.5 mm above the ground plane, an alumina substrate with er = 9.7, and a loss tangent of 2 x 10 4. The strip thickness T is 0.02 mm. The frequency of operation is 4 GHz. The effective dielectric constant and characteristic impedance were found in Example 3.4 and are e e = 6.556, Z c = 49.44 il. The wavelength of operation is 7.5 cm. The attenuation caused by dielectric loss was calculated after (3.171) was presented and is 1.52 X 10~ 3 Np/wavelength or 2.02 x 10~ 4 N p / c m . For this microstrip line W/H = 1 and W/T = 25. From (3.173) we find that the loss ratio is 1.0658 and (3.172) gives 1.0658 X 8.22 x 10 R. = 3 vf / 1 —— 1 - 0.05 \ir + — ? In IOOTT ir2 = 3.157 x 10~ 5 n / c m By using (3.174) we obtain 8.22 X 1 0 - 3 / 4 R * ~ 0.05 5.8 + 0.03 = 4.81 X 1 0 ' 2 f l / c m For this case the loss in the microstrip is a factor of 6.5 greater than that in the ground plane. The reason for this is the high current density near the edges in the microstrip as compared with a more uniform current distribution over a wider area on the ground plane. By using (3.175) we find that the attenuation due to conductor loss is 3.157 x 1 0 _ l + 4.81 X 10~ z "• : 2 X 49.44 = 3 "68 * 10 N p / C m This attenuation is 18.4 times greater than that caused by dielectric loss. The total attenuation is a = ac + ad = 3.88 X 1 0 " 3 N p / c m which equals 0.0337 d B / c m . Surface roughness could result in a real attenuation 15 to 25 percent higher than this at 4 GHz. In this example we neglected the correction to e c and Z c due to the finite thickness T since it is small. 158 FOUNDATIONS FOR MICROWAVE ENGINEERING High-Frequency Properties of Microstrip Lines ' The equations describing the quasi-TEM mode can be used with accent v accuracy for frequencies up to 2 to 4 GHz for a substrate thickness of l For a substrate 0.5 mm thick, the upper frequency limit would be 4 Dl' GHz. When these limits are exceeded, it is necessary to take into acco * the frequency dispersion of the effective dielectric constant and the cha in characteristic impedance with frequency. At the higher frequencies ti electric field becomes more confined to the region between the microst*' and ground plane. The greater concentration of the field in this rem ' results in an increase in the effective dielectric constant as well as increase if. I ________—- 6.0 __- 4.0 ^ 2.0 - ^ "~~ "w^*^ ——-""* _-——•"' 20 1.9 ^______ ^^" ___ 1.0 ———" 1.8 , o.s —_____I___!__^----- ' w7~H = 0.25 1.7 I 1R I I I I I I 12 I 16 I I I 20 I 24 I I ! FH in GHz mm (a) I HO Characteristic impedance _-_ 0.25 150 . 0.5 120 1.0 90 2.0 60 i OA "~Q 4 8 1—i : i 12 16 _ i — i_ 20 i 4.0 W/H - L__? J—Li 24 W i n GHz mm lb) FIGURE 3.25 ( a ) Effective dielectric constant for a PTFE/microfiber glass substrate with e r = 2.26, ty (6) characteristic impedance. %% TRANSMISSION LINES AND WAVEGUIDES Characteristic impedance ^^,^-0.25 120 100 : ^ _ _ _ - - 0.5 '- 80 _____ tso _________ I 0 -1.0 2.0 40 9n 159 • 4 I I 8 ! " 1 12 4.0 1 16 20 W/H = 6.0 24 FH in GHz mm M FIGURE 3.26 ( a ) Effective dielectric constant for an RT/Duroid 6006 substrate having e r = 6.36, e y = 6; (b) characteristic impedance. attenuation because of a greater concentration of the electric field in the substrate which has some loss. The conductor loss also increases because the skin-effect resistance R m increases and more of the current flows on the inner face of the microstrip. In order to determine the effective dielectric constant at high frequencies, it is necessary to carry out a full wave analysis, i.e., the complete set of equations given earlier for the potentials must be solved. In Figs. 3.25 to 3.28 we show the dispersive properties for four common substrate materials, a PTFE/microfiber glass substrate with e r = 2.26, e v = 2.2, RT/Duroid 6006 with er = 6.36, ey = 6, alumina with er = 9.7, and gallium arsenide 160 FOUNDATIONS FOR MICROWAVE ENGINEERING 0 4 8 12 J i i 16 20 24 FH in GHz mm 16 20 24 FH in GHz mm [ Characteristic impedance 100 r 8 12 (b) FIGURE 3.27 B ^ ^ (a) Effective dielectric constant for an alumina substrate with c r = 9.7; (6) characterise impedance. with er = 12.9. An examination of these figures shows that dispersion effects are more pronounced for wider strips and large dielectric constantsAlong with each figure giving the effective dielectric constant a s 8 function of normalized frequency is a figure showing the characterise impedance as a function of normalized frequency. The normalized frequency is the actual frequency in gigahertz multiplied by the substrate thickness^ millimeters. Thus, for substrates 0.5 mm thick, the frequency range covert is 0 to 48 GHz. When the propagating mode is not a TEM or quasi-TEM mode, thf** 1 is no unique value for the characteristic impedance because it is not P o S 9 '\J to define a unique value for the voltage as given by the line integral of** electric field between the ground plane and the microstrip. The characte1"18" TRANSMISSION LINES AND WAVEGUIDES Characteristic impedance 161 ^ ^ - — 0.25 100 05 ^ — 80 ________ 1.0 60 — 2.0 40 4.0 ?n • W / H = 6.0 — 0 i > 4 i— i 8 • • i 12 < 16 i 20 I . I i i 24 FH in GHz mm (to) FIGURE 3.28 ( a ) Effective dielectric constant for gallium-arsenide substrate with e r = 12.9; (6) characteristic impedance. tic impedances given in Figs. 3.25 to 3.28 are based on using the following definition for the equivalent voltage: A +d<P\ dy v=-/V* = / > > r 'o •'o \ dy, where the path of integration is along a straight line from the ground plane to the center of the microstrip. The current / is chosen as the total 2-directed current on the microstrip and Z c was calculated from the ratio V/I. In general, the power flow along the microstrip transmission line will not equal |V7. The characteristic impedance can be defined in terms of the power flow P by choosing either the current / or voltage V according to the 162 FOUNDATIONS FOR MICROWAVE ENGINEERING above definitions and using 2 Zl 2ZC to find Zc. These two equations will give different values for the characterk. tic impedance with neither one being equal to V/I. The lack of a unique value for the characteristic: impedance is not a great disadvantage since microstrip junctions and discontinuities can be described by equivalent circuits using any convenient definition for the characteristic impedance. In Chap. 4 we will find that an equivalent transmission-line circuit theory can be formulated for any waveguiding system and does not require that there be a unique characteristic impedance associated with the propagating modes. For computer-aided design (CAD) of microstrip circuits, it is important to have simple formulas that can be used to find the effective dielectric constant. Many different formulas have been proposed. The most accurate one that covers the full range of parameter values like!}1 to be encountered was developed empirically by Kobayshi and is given below:t (3 6) <">-'-itowu" '" where K- 0.75 + (0.75 ~ 0.332c; 1 73 )W/H 47.746 h Hfer - ee{0) m = mQmc < 2.32 mn = 1 + 1 + m, = ===== + 0.32(1 + JW/H) YTWTH(015 " °-235e~a45f/r°) -3 H w °"7 — > 0.7 H inCA» tM. Kobayashi. A Dispersion Formula Satisfying Recent Requirements in Microstnp IEEE Trans., vol. MTT-36, pp. 1246-1250, August, 1988. I TRANSMISSION LINES AND WAVEGUIDES 163 In these formulas H is in millimeters, the frequency f is in gigahertz, and whenever the product m0mc is greater than 2.32 the parameter m is chosen equal to 2.32. The effective dielectric constant at the frequency f is ee( f) and ec(0) is the quasistatic value which can be found using (3.166). It requires only a simple computer program to evaluate ee{f) using Kobayshi's formula. The accuracy is estimated to be within 0.6 percent for 0.1 < W/H < 10, 1 < er < 128 and for any value of H/A0.t In microstrip circuit design where junctions of microstrip lines with different widths are involved, it is necessary to characterize the junction in terms of an equivalent circuit. The parameters of this equivalent circuit will depend on frequency. The equivalent characteristic impedances that are assigned to the microstrip lines are arbitrary and often are simply chosen to have normalized values of unity. Any impedance level change that occurs at the junction is incorporated as part of the equivalent circuit of the junction. For these reasons we will not quote any of the formulas that have been proposed for evaluating characteristic impedance as a function of frequency because they are of limited use in practice. Simple formulas giving the attenuation of microstrip lines at high frequencies do not exist. Equation (3.171) can be expected to give a good estimate for the attenuation due to dielectric loss provided e e is replaced by the effective dielectric constant that applies at the frequency of interest. A realistic evaluation of the attenuation caused by the finite conductivity of the conductors requires evaluation of the current density on the microstrip and the ground plane at the frequency of interest. In general, the current tends to be more uniform across the microstrip at high frequencies, particularly for wide strips. In Fig. 3.29 we show several computed current distributions at frequencies of 10 and 20 GHz for an alumina substrate 1 mm thick. The quasistatic current distribution is also shown. The current density has been normalized so that the total current on the microstrip equals W. In view of the tendency for the current density to become nearly uniform at high frequencies for wide strips, the attenuation constant can be estimated with fair accuracy by assuming uniform current density over a width W on both the microstrip and the ground plane. In this limit the attenuation caused by the conductor loss for wide strips that are not too thin is given by ac=^£c WZC (3.177) v t T h e author has verified the accuracy of Kobayshi's formula by comparison with calculated numerical results for 0.25 <, W/H < 6 and 2 s er < 12. 164 FOUNDATIONS FOR MICROWAVE ENGINEERING I 0.64 10 GHz W = 2 mm 2 i\ i 2 u i It 1 if \\ i ij 1 •• ^ -~~? ^ 1 <«. 064 — 0.64 w. - 2 mm 20 GHz (b) ~" _ IV =6 mm —-' i 20 GHz [d) F I G U R E 3.29 Current distribution on the microstrip for an alumina substrate with H = 1 mm and two different widths. The broken curves give the quasistatic distribution, ( a ) W = 2 mm, f = M GHz; (6) W= 2 mm, f = 20 GH?,; ( c ) W = 6 mm, f= 10 GHz; id) W = 6 mm. f= 20 GHr For narrow strips and high frequencies, no simple formulas for atten ation appear to be available. For narrow strips, say W/H < 1, the qu» sistatic formula (3.175) is probably a reasonably good estimate since M* current density does not depart significantly from the quasistatic distn tion for narrow strips. 3.13 COUPLED MICROSTRIP LINES When two conducting strips of width W are placed side by side dielectric substrate above a ground plane as shown in Fig. 3.30a, we o a coupled microstrip line. Since this is a three-conductor transmission there are two fundamental quasi-TEM modes of propagation. The y mode is the mode corresponding to both strips being at the same potent* TRANSMISSION LINES AND WAVEGUIDES Odd mode 165 (c) FIGURE 3.30 ( a ) Coupled microstrip line; (6) the electric field distribution for the even and odd odes; (c) the current distribution for the even and odd modes. and on which the same currents exist. The odd mode corresponds to the strips being at opposite potentials, - V and V, relative to the ground plane. For the odd mode the currents on the two strips are also equal in ampUtude but of opposite sign. A sketch of the electric field lines for the two modes is shown in Fig. 3.306. For isolated strips in air, i.e., with no ground plane and substrate present, the theoretical current distributions for the two modes are:t J'(x) = </*(*) = V(* 2 -*?)(*£-* 2 ) ^(*2-*?)(*!-*2) (3.178a) (3.1786) tR. E. Collin, "Field Theory of Guided Waves," 2nd ed., chap. 4, IEEE Press, Piscataway, N.J., 1991. 166 FOUNDATIONS FOR MICROWAVE ENGINEERING For the even mode the factor x in the numerator reduces the arnpUturt the singular behavior near the inner edges at x = ±xv When x t equals * °^ the current singularity at ±.r, vanishes and J"(x) becomes the exDe^/ 0 current density on a single strip 2x2 units wide. For the odd tnodp current singularity at the inner edges ± x 1 is more like a 1/x sinmil • when *! is very small. This is caused by the strong electric field acrn very narrow slit with the adjacent conductors at opposite potential TO current distribution for the two modes is shown in Fig. 3.30c. The coupled microstrip line is used in various directional counl designs and these applications will be discussed in Chap. 6. The imports parameters describing the quasi-TEM mode properties of the coupled mi crostrip line are the even- and odd-mode effective dielectric constants e' e ° that determine the two propagation constants, and the even- and odd-mode characteristic impedances Zec, Z". An important parameter in directional coupler design is the coupling coefficient, which is given in terms of the characteristic impedances of the two modes by ye C = j ^ yo (3.179, The coupling coefficient is commonly expressed in decibel units, that is, 20 log C. In a coupled microstrip line it is not practical to achieve much more than a 2.5: 1 impedance ratio; so strong coupling cannot be realized in a simple coupled microstrip directional coupler. However, other designs are available, so this is not a problem for the microwave circuit engineer. There are no simple formulas giving the quasi-TEM properties of coupled microstrip lines that have an accuracy comparable to that for microstrip lines. Bahl and Bhartia list formulas characterizing coupled microstrip lines that give results which are acceptable for noncritical applications.! The computer program CMST implements these formulas. We have checked the accuracy against numerical results obtained from a full wave solution for an alumina substrate. The effective dielectric constants were found to be accurate to within 3 percent. The characteristic impedance for the even mode was also found to be accurate to within 3 percent. For tW odd-mode characteristic impedance, the error was as large as 8 percent w W/H = 1, S/H = 0.25, but considerably less for larger values of W/H ana S/H. of The dispersion properties for coupled microstrip lines on a variety different substrates have been computed by Morich.? In Figs. 3.31 to o- 9 Son5, t j , Bahl and P. Bhartia. "Microwave Solid State Circuit Design," p. 28, John Wiley * """ Inc., New York, 1988. JM. Morich, Broadband Dispersion Analysis of Coupled Microstrip on Anisotropic S u b s l . ^ , J , by Perturbation-Iteration Theory, M.S. Thesis, Case Western Reserve University, d e v ^ 8 ' * Ohio. May, 1987. TRANSMISSION LINES AND WAVEGUIDES 0 i • ' 2 4 6 I 1 1 1 8 10 12 14 FH in GHz mm 1 1 16 18 167 20 (a) ' 0 2 i—_i 4 6 8 1 1 1 1 ' 10 12 14 16 18 0.1 20 FH in GHz m m (ft) F I G U R E 3.31 Dispersion characteristics of a coupled microstrip line on an alumina substrate. S/H = 0.25, e r = 9.7. ( a ) Even- and odd-mode effective dielectric constant; (b) even- and odd-mode characteristic impedance and coupling coefficient C. 168 FOUNDATIONS FOR MICROWAVE ENGINEERING 12 W/H — 11 I 9 = 2 ^ ^ ^ - - ^ ' 1 ^ ^_____ 10 -Even mode f^-*"""^ Odd mote\^-^ 9 3 1 ; 1 1 1 " i i 8 10 12 14 FH in GHz mm i i i 16 18 20 8 10 12 14 FH in GHz mm 16 18 20 (6) F I G U R E 3.32 Dispersion characteristics of coupled microstrip line on a gallium-arsenide substrate. S/" 0.25, «r =* 12.9, (a) Even- and odd-mode effective dielectric constant; (6) even- and odd-m characteristic impedance and coupling coefficient C. rises to a value of 0.524 at FH = 20 GHz mm. A coupling coefficient of 0-' is needed for a 6-dB directional coupler. The effective dielectric constant the odd mode is smaller than that for the even mode because a '0^\ percentage of the electric field energy is located in the air regioncapacitance between closely spaced parallel strips at opposite potential large so the characteristic impedance of the odd mode is smaller than for the even mode for the normal range of parameters involved. For w strips \vith large spacing, there is very little coupling and the two n* impedances will be almost the same. TRANSMISSION LINES AND WAVEOUIDES 169 W/H=0.5 0 2 4 6 8 10 12 14 FH in GHz mm 16 18 20 (a) FH in GHz mm (ft) F I G U R E 3.33 Dispersion characteristics of coupled microstrip line on an RT/Duroid 6006 substrate. S/H = 0.25, e r = 6.36. ( a ) Even- and odd-mode effective dielectric constant; (b) even- and odd-mode characteristic impedance and coupling coefficient C. 1 7 0 FOUN DATtONS FOR MICROWAVE ENGINEERING STRIP TRANSMISSION LINES The basic strip transmission line consists of a conducting strip embedded • a dielectric medium between two ground planes as shown in Fig 3 o, l n Figures 3.346 and c illustrate coupled strip lines. For broadside CQU , a strips as shown in Fig. 3.34c, the coupling coefficient is significantly great* than for coplanar strips. Thus the strip line with broadside coupled string suitable for directional couplers where a coupling of 3 dB is required Si ' the dielectric completely surrounds the strips, the strip line and the COUDIBH strip line support pure TEM modes of propagation. There is no frequen dispersion or change in effective dielectric constant with frequency. ConsZ quently, for the coupled strip line the even and odd modes of propagation have the same phase velocity, which is also a desirable feature for directional coupler design. For the symmetrical strip line and coupled strip line, the TEM-mode characteristic impedance is readily found from the distributed capacitance with the latter determined by conformal mapping techniques. The distributed capacitance for the symmetrical strip line is found in App. in. W W 1 S 1 W ZH TT Even mode Even mode F I G U R E 3.34 . oU pKs<l ( a ) The basic strip-line configuration; (6> coupled strip line using coplanar strips: ic) L ^ strip line using broadside coupled strips. The electric field lines for the TEM modes are shown. TRANSMISSION LINES AND WAVEGUIDES 171 From the given expression (III.27) and using (III. 13c), we have Z0 Zr = K(k) Z0K *& K(k') (3.180a) Aj7rK' where k= K cosh -rt W/4H 1 = W ^ K k' = tanh iv / 1 + \[k ln2 0.7 < k < 1 TWT 1 / 1 - I n 21 - /*' ~K' W 0 < k < 0.7 and e r is the dielectric constant of the dielectric material that completely surrounds the center strip. For W > 2H the formula for Z c reduces to the simple form Z = — = ' — W>2H (3.1806) 8y/Zr(ln2 + TTW/4H) For a very narrow strip Z,.= 16H W<0AH In 2nyfc. TTW (3.180c) Since the dielectric material completely fills the strip line, the attenuation due to dielectric loss is given by (3.87a) and is aJ= ire, iTi/e'r tan o, (3.181) Aov^ Formulas for the attenuation in a strip line due to conductor losses and assuming an inner conductor of elliptical cross section have been derived using conformal mapping techniques.t In App. Ill it is shown that the series resistance of an isolated conductor of elliptical cross section is greater than that for a conductor with a rectangular cross section. By using an equivalent center thickness T e for the conductor with elliptical cross section, the two series resistances will be equal provided T e is chosen to be T. = e~^2 4WT ~ tR. E. Collin, loc. cit„ chap. 4, eqs. (73), (74). and (76). (3.182) 172 FOUNDATIONS FOR MICROWAVE ENGINEERING We will assume that this equivalence is also a good approximation f conductor placed between two ground planes. On this basis the a t t e n u a / a by the center conductor is '°n TTi?„, 16Hk' -lna,., = 16ZcHK' k' kirTe 2 ( 3 -183 Q ) and that due to the ground planes is -2R,nW a <2 64ZcH*K'2k' (3 18 - 36) For wide and narrow strips these formulas reduce to the following simplified forms: 8H a l = < ln Rj7r ~Wir TTW + ^fe JH ^W~ + ° W 2H * (3184a) »* !S rrR^W «c-2 = / ^ff\ 8Z0H2 l n 2 + —\ ,_ 2Rjer\n W>2H (3.1846) 4/7 4W — W<0AH ar, = «„, ^ = W < OAH 16H 2Z0H\n TTW Suitable formulas for evaluating K and K' are given by (III. 13) in App The total attenuation for a strip line is given by « = «d + «cl+«c2 (3-1^ Example 3.7. A strip line has a ground-plane spacing 2H = 1 cm and use= centered copper conducting strip of width W = 1 cm and thickness T - • cm. The dielectric filling material has a dielectric constant e'r = 2.2 and a , tangent equal to 10~3. We want to find the characteristic impedance attenuation at a frequency of 10 GHz. For this line W/H = 2, so (3.1806' c be used. Thus 120TT2 Z < = 8 v ^ 2 ( l n 2 + 7r/2) = 44 09 ' " TRANSMISSION LINES AND WAVECIIIDES 173 The wavelength of operation is 3 cm; so by using (3.181) we find the attenuation due to dielectric loss to be s ir&2 X 10 «*3 = 1.55 X 10 :i Np/cm 1.35 x 1 0 ' 2 dB/cm or From (3.182) we obtain T e = 0.00742 cm for the equivalent thickness. We can use (3.184a) to find 8.22 x 10- 3 y'10v f 2.2 m ' "- - x 240,7X0.5 4 0.00724 \n2 = 3.034 x 10" 4 N p / c m or + 2 2 2.635 x 10 :) dB/cm From (3.184b) we obtain 8.22 X 10 a 2 3 vTO J22v 2 '' ~ 8 x 12077 x 0.5 (ln2 + - / 2 ) 5 = 7.096 x 10 Np/cm or 6.163 x 10~ 4 d B / c m The total attenuation is 0.0167 dB/cm. For this transmission line the dielectric loss is greater than the conductor loss. According to a formula for strip-line conductor attenuation developed by Cohn, we would get a c = 3.236 x 10 ~ 3 d B / c m , which agrees very closely with 3.25 x 10 3 d B / c m obtained with our formulas.t COUPLED STRIP LINES F o r c o p i a n a r s t r i p s of width W a n d spacing S, t h e even- a n d odd-mode c h a r a c t e r i s t i c i m p e d a n c e s a r e given by$ c r 4 / e r K(kt) J^KiKl l ( 3 . 1 8 6 6) i{TrK(kn) tSee H. Howe. "Strip Line Circuit Design," eq. (1.5), Artech House Books, Dedham, Mass. 1974. iS. B. Cohn, Shielded Coupled Strip Transmission Line. IRE Trans., vol. MTT-!J. pp. 29-37, October, 1955. S. B- Cohn, Characteristic Impedances of Broadside-Coupled Strip Transmission Lines, IRE Trans., vol. MTT-8, pp. 633-637, November, 1960. S. B. Cohn, Thickness Corrections for Capacitive Obstacles and Strip Conductors, IRE Trans., vol. MTT-8, pp. 638-644, November, 1960. 174 FOUNDATIONS FOR MICROWAVE ENGINEERING where (w W\ [TT W+ S and A'o = l / l - * „ 2 . K = Vl - *? Note that KXA') = J?'(A). For the evaluation of K and #' see (111.13). For coplanar strips with a thickness T, Z' = Z0(2H-T) ln[l A=l + + W > Q.m i - e,o 4j7~r{W+(H/TT)CfA,) (3.187) tanh(7rS/4H)] In 2 ln[l + coth(irS/4/f)] A_ = l + In 2 (4H T ( 4 H - 7) -T :ln 2tf ( 2 H - T)2 For broadside coupled strips as shown in Fig. 3.34c, the even- and odd-mode characteristic impedances are given by [valid for W > 0.35S, W> 0.7/7(1 - S/2H)] 2 ^ _f W + ST» °- +1 4413 ; In S + 2T 2H 2H - S - 2T 2H -InS + 2T 2H-S-2T (3.188a) _f 2 ^ W W 2 f / - S - 2 T + S + C 2 ° V+ - T\ 1+ sr T T ln -s)-s -s (3.1886) where P" = 2 / / - 2T rS 2tf - 2T In 2H-S-2T + 8 2H-S-2T 2H-2T In All of the characteristic impedances given above refer to the impedance one strip to the ground planes. frofl1 TRANSMISSION LINES AND WAVEGUIDES 175 F I G U R E 3.35 (a) Basic coplanar transmission line; (6) a shielded coplanar transmission line; (c) electric field distribution. COPLANAR TRANSMISSION LINES An illustration of a coplanar transmission line is shown in Fig. 3.35a. A shielded coplanar line is shown in Fig. 3.356. The strip width is S and the strip to ground-plane spacing is W. The coplanar line is often called a coplanar waveguide (CPW), The most significant advantage that a coplanar line has over a microstrip line is the ability to connect active and passive circuit components in shunt from the conducting strip to the ground plane on the same side of the substrate. In a microstrip line a connection to the ground plane requires drilling a hole through the substrate which is somewhat difficult for ceramic materials such as alumina. Figure 3.35c shows the electric field distribution in a coplanar line. The characteristics of a coplanar line at low frequencies can be determined by conformal mapping techniques. A solution for the coplanar line is given in App. III. For the ideal case when the ground planes are very wide relative to the slot width W and the dielectric substrate is very thick, the electric field has the same distribution as for a coplanar fine in air. The reason for this is that the mapping of the air-dielectric boundary, shown as the intervals BC and EF in Fig. III.4, is the two parallel sides in the ideal parallel-plate capacitor shown in Fig. III.3. Thus the solution for the potential is not affected by the presence of the dielectric. The distributed 176 FOUNDATIONS FOH MICROWAVE ENGINEERING capacitance of the coplanar line is thus the capacitance of one-half 0 f air-filled line plus one-half of that for a line completely surrounded ^ dielectric. This capacitance is given by III. 14 and for the line with di e | e ^ on one side only we have K K ( e r + 1) K C ~ 2e0— + 2e,.e0 — - 4 , 0 — — (3 ^ where K(k) and K' = K(k') are again the complete elliptical integrals the first kind. The modulus k is given by the ratio 1 k = Y/k S/2 = S W + S/2 = S + 2W < 31 90) since in Fig. III.4 the center conductor extends from -1 to 1 and the ground planes begin at u = ±(l/k). From (3.189) it is clear that the effective dielectric constant for the coplanar line is given by e.+ 1 2 (3.191) Consequently, the characteristic impedance is given by ZQK' z < = TT^ <3192) When the substrate material is anisotropic with a dielectric constant e y in the direction perpendicular to the air-dielectric interface and e r in the direction parallel to the conductors, then e r should be replaced by e g = yjzrev in (3.189) and (3.191). In most applications it is necessary to provide shielding of a microwave circuit. If the shield dimensions are large, the shield will not produce a significant effect on the line characteristics. In monolithic microwave integrated circuits, the substrate is very thin and fragile; so it is desirable to use another ground plane below the substrate to mechanically strengthen tj overall circuit and to also provide a better heat sink to help dissipate tne power generated by active devices. Ghione and Naldi have considered number of different shield arrangements as well as coupled coplanar line* We will present results only for the shielded coplanar line shown in £'| 3.356. For this structure the effect of the widewalls, spaced by amount • is negligible provided 2A > 10(S + 2W). We will assume that this is tn ^ case. Ghione and Naldi make the assumption that the air-dielectric inter in the slot regions can be replaced by magnetic walls in order to simplify tG. Ghione and C. U. Naldi, Coplanar Waveguides for MMIC Applications: Effect of 'P $ Shielding, Conductor Backing, Finite Extent Ground Planes, and Line-to-Line Coupling. Trans., vol. MTT-35. pp. 260-267, March, 1987. TRANSMISSION LINES AND WAVEGUIDES 177 conformal mapping solution. The assumption is a correct one only for the case when the upper and lower shields are spaced the same distance from the coplanar line, that is, H = H Y in Fig. 3.356. For the line dimensions encountered in most practical applications of coplanar lines, the assumption does not introduce significant error. The advantage of this assumption is that it decouples the air and dielectric regions; so it is only necessary to find the capacitance of the air- and dielectric-filled sections separately. It was found that the effective dielectric constant is given by « e = 1 + q{er- 1) (3.193a) where the filling factor q is given by KjkJ/Kjk'J Q K(k2)/K(k2)+K(k)/K(k') (3-193*; K is the complete elliptic integral of the first kind, k\ and k'2 are the complementary moduli given by y 1 - k\ and yl - k%, and tanh(irS/4H) *1= .„-Lf-,o , n „ n „ „ t a n h [ i r ( S + 2W)/H \ (3.193c) tanh(TrS/4/7,) * 3 = tanh[7r(S+2W)/4ff1] (3.193c/) The characteristic impedance is given by Zn 2]/Te[K(kl)/K(k\) + K{k2)/K(k2)\ (3.194) The ratios K/K' are easily evaluated using III. 13c. For an anisotropic substrate e r should be replaced by e g = JereY and H should br replaced by the effective thickness He = Jer/ev H. When H and / / , are very large, &, and k,2 become equal to k given by (3.190) and the filling factor o equals 0.5. If the lower shield plate is replaced by a magnetic wall, we obtain an approximate model of a coplanar fine with a substrate of finite thickness H and with an upper shield at a spacing H, above the coplanar conductors. For this case Ghione and Naldi give q = (3.195a) K[k2)/K[k'a) +K(k)/K(k') Z. = • ,— , „ , " 2jre[K(k2)/K(k'2)+K(k)/K{k-)} r (3.1956) 178 FOUNDATIONS FOR MICROWAVE ENGINEERING where sinh(7rS/4H) 3 s i n h [ i r ( S + 2W)/4H] a n d k 2 is given by ( 3 . 1 9 3 d ) , k is given by (3.190), a n d e c i s g j V e , ( 3 . 1 9 3 a ) . W h e n H, is m a d e infinite k 2 becomes e q u a l to k. T h e result s t r u c t u r e is an u n s h i e l d e d c o p l a n a r line w i t h a s u b s t r a t e h a v i n g a fi n t h i c k n e s s H. E x a m p l e 3.8. A coplanar line with upper and lower shielding and used in *MMIC circuit has the following dimensions: S = 50 /j.m = 0.05 mm, IV = jr' nm, H = 250 jim, / / , = 800 Mm. The substrate material is gallium arsenide with er «= 12.9. We want to find the effective dielectric constant and characteristic impedance. The first step is to find k x and k 2 using(3.193 c )and (3.194c). A straightforward evaluation gives <fes = 0.3547, k2 = 0.33547. We now use (III. 13c) to find K{kx)/K(k\) = 0.6573, K{k2)/K(k2) = 0.6414. By using (3.193a) and (3.1936), we get q = 0.506 and ec = 7.0228. The last calculation is for Zc using (3.194) and gives Z,. = 54.77 12. This example shows that for the shield spacings used very little effect on the coplanar-line characteristics occurred as is apparent from the fact that k-, = k2 and q = 0.5. relationships that hold exactly when no shields are used. Attenuation T h e a t t e n u a t i o n in a c o p l a n a r line caused by dielectric loss is given by the s a m e f o r m u l a as for a m i c r o s t r i p line, i.e., by (3.171). By u s i n g the filling factor q this can be expressed as q t a n 8, (3-196) K T h i s f o r m u l a can be used for shielded a n d u n s h i e l d e d lines as long as tft a p p r o p r i a t e filling factor q is used. F o r t h i n c o n d u c t o r s w i t h t h i c k n e s s T less t h a n 0 . 0 5 S a n d with a sl» width W > 0 . 3 S , t h e formula for t h e unshielded coplanar-line attenuation derived in App. I l l can be used. T h e c e n t e r c o n d u c t o r h a s a series resistan p e r u n i t l e n g t h given by 4TTS «t = 4 S ( 1 - 2 1 +k (3.197" } 2 k )K {k) T h e d i s t r i b u t e d series r e s i s t a n c e of t h e g r o u n d planes is given by 4tr(S + 2W) *„ = 4S(1 2 2 -k )K (k) 77 + 1,1 f 1 ln 1+k £ TTT (3. 19?*) TRANSMISSION LINES AND WAVEGUIDES 179 w h e r e k = S/(S + 2W). T h e a t t e n u a t i o n d u e to c o n d u c t o r loss is given by R, + R2 (3 197c) * - sr - In normal s i t u a t i o n s t h e additional a t t e n u a t i o n i n t r o d u c e d by u p p e r a n d lower shields is small if H a n d / / , a r e g r e a t e r t h a n AW. Example 3.9. A coplanar line has a copper strip of width 0.6 mm, slot width W = 0.6 mm, metal thickness T = 0.005S = 3 ^m. The dielectric used is alumina with a dielectric constant of 9.7 and a loss tangent equal to 2 x 10 4 . We want to find the attenuation caused by dielectric loss and conductor loss at a frequency of 4 GHz. For this line k = S/{S + 2W) = 0.333. The effective dielectric constant, from (3.191), is 5.35. From (3.192) and (III.13c) we obtain Zr = 120TT 1 1 + v'fe7 • -In 2 -?= = 63.7 il 4j5M ~ 1 - y/F where we used k' = \Jl - k2 = 0.9428. From (3.196) we obtain 9.7TT a rf = ^ ^ - 0 . 5 X 2 x 10~ 4 x 8.686 = 1.526 x 10 7.5^5.35 3 dB/cm where q =* (ee~ l)/(e' r - 1) = 0.5 was used. To evaluate K(k) = K(Q.333), we use (111.13c/) to obtain 1 -k' K(0.333) = — - K — = : r ^ K ( 0 .v 0 2 9 4 ) = 1 + k' Xl + k'} 1.9428 1.9428 = 1.617 From (3.197) we get i?, = 19.25i? m /cm and R 2 - 5.97i?,„/cm. We now use R m = 8.22 X 10~ 3 V? and (3.197c) to get % = (2.158 x 10 z + 6.69 X 1 0 ~ 3 ) = 2.827 X 10 2 dB/cm The ground planes contribute 23.7 percent of the conductor loss. The total attenuation is 2.98 X 10~ 2 dB/cm. Jackson has made some loss calculations for coplanar and microstrip transmission lines at a frequency of 60 GHz and found that for typical line dimensions and characteristic impedances greater than 50 il, the coplanar line has a smaller attenuation.t The substrate considered had a dielectric constant of 12.8 and a loss tangent of 6 x 10 3 . The maximum strip width considered by Jackson was 0.3 mm. For these line dimensions the quasistatic formulas given above can be used. We have verified that indeed the quasistatic formulas give essentially the same attenuation. The lower loss in coplanar lines appears to be due to being able to use a wider center conductor for a given impedance as compared with that for a microstrip line. However, this is not always tR. W. Jackson, Considerations in the Use of Coplanar Waveguide for MjJIimeter-Wave Integrated Circuits IEEE Trans., vol. MTT-34, pp. 1450-1456, December, 1986. 180 FOUNDATIONS FOR MICROWAVE ENGINEERING necessarily true since the relative attenuation of the two types of transiv lines will depend on other factors such as substrate thickness and diel^ constant also. ^ic High-Frequency Dispersion Coplanar transmission lines exhibit dispersion effects similar to that f microstrip lines. There are less available computed results for the effect- ' dielectric constant and characteristic impedance for coplanar lines th what is available for the microstrip line. The coplanar line can be viewed two coupled slot lines and from this point of view it is clear that there ar two modes of propagation that are quasi-TEM in character. The reader i R referred to the paper by Nakatani and Alexopoulos for typical dispersive properties of a coplanar line.t In many integrated microwave circuit applies. tions, the line dimensions are so small relative to the wavelength that the quasi-TEM formulas can be used even though the frequency may be as high as 50 GHz. PART 3 R E C T A N G U L A R A N D C I R C U L A R WAVEGUHDES Hollow-pipe waveguides do not support a TEM wave. In hollow-pipe waveguides the waves are of the TE and TM variety. The waveguide with a rectangular cross section is the most widely used one. It is available in sizes for use at frequencies from 320 MHz up to 333 GHz. The WR-2300 waveguide for use at 320 MHz has internal dimensions of 58.42 in by 29.1 in and is a very large duct. By contrast, the WR-3 waveguide for use at 333 GHz has internal dimensions of 0.034 in by 0.017 in and is a very miniature structure. The standard WR-90 X-band waveguide has internal dimensions of 0.9 in by 0.4 in and is used in the frequency range of 8.2 to 12.5 GHz. The rectangular waveguide is widely used to couple transmitters and receivers the antenna. For high-power applications the waveguide is filled with j inert gas such as nitrogen and pressurized in order to increase the voltage breakdown rating. Circular waveguides are not as widely used as rectangular wavegu1 but are available in diameters of 25.18 in down to 0.239 in to cover tn frequency range 800 MHz up to 116 GHz. tA. Nakatani and N. G. Alexopolous, Toward a Generalized Algorithm for the Modeling ° ^ Dispersive Properties oflntegrated Circuit Structures on Anisotropic Substrates, 1E&E ' ^ vol. MTT-33, pp. 1436-1441, December, 1985. TRANSMISSION USES ANll WAVEGUIDES 181 RECTANGULAR WAVEGUIDE The rectangular waveguide with a cross section as illustrated in Fig. 3.36 is an example of a waveguiding device that will not support a TEM wave. Consequently, it t u r n s out that unique voltage and current waves do not exist, and the analysis of the waveguide properties has to be carried out as a field problem rather than as a distributed-parameter-circuit problem. In a hollow cylindrical waveguide a transverse electric field can exist only if a time-varying axial magnetic held is present. Similarly, a transverse magnetic field can exist only if either an axial displacement current or an axial conduction current is present, as Maxwell's equations show. Since a TEM wave does not have any axial field components and there is no center conductor on which a conduction current can exist, a TEM wave cannot be propagated in a cylindrical waveguide. The types of waves that can be supported (propagated) in a hollow empty waveguide are the TE and TM modes discussed in Sec. 3.7. The essential properties of all hollow cylindrical waveguides are the same, so that an understanding of the rectangular guide provides insight into the behavior of other types as well. As for the case of the transmission line, the effect of losses is initially neglected. The attenuation is computed later by using the perturbation method given earlier, together with the loss-free solution for the currents on the walls. The essential properties of empty loss-free waveguides, which the detailed analysis to follow will establish, are that there is a double infinity of possible solutions for both TE and TM waves. These waves, or modes, may be labeled by two identifying integer subscripts n and m, for example, TE„„,. The integers n and in pertain to the number of standing-wave interference maxima occurring in the field solutions that describe the variation of the fields along the two transverse coordinates. It will be found that each mode has associated with it a characteristic cutoff frequency f c „ m below which the mode does not propagate and above which the mode does propagate. The cutofT frequency is a geometrical parameter dependent on the waveguide cross-sectional configuration. When f c has been determined, it is found that the propagation factor [i is given by P~{H-k*y* (3.198) 182 FOUNDATIONS FOB MICROWAVE ENGINEERING where kn = w / / t o c 0 and k,. = 2Trfc)/p.0e0 . The guide wavelength is readji seen to be given by A0 2TT A s = T " "(1-A|/A») ^ ,/8 == A0 yJl-C/f* (3 199 - ) where A 0 is the free-space wavelength of plane waves at the frequen /*= W/2TT. Since kc differs for different modes, there is always a lower band of frequencies for which only one mode propagates (except when kc may ho the same for two or more modes). In practice, waveguides are almost universally restricted to operation over this lower-frequency band for which only the dominant mode propagates, because of the difficulties associated with coupling signal energy into and out of a waveguide when more than one mode propagates. This latter difficulty arises because of the different values of the propagation phase constant /3 for different modes, since this means that the signal carried by the two or more modes does not remain in phase as the modes propagate along the guide. This necessitates the use of separate coupling probes for each mode at both the input and output and thus leads to increased system complexity and cost. Another feature common to all empty uniform waveguides is that the phase velocity v p is greater than the velocity of light c by the factor Ag/A0. On the other hand, the velocity at which energy and a signal are propagated is the group velocity v. and is smaller than c by the factor A 0 /A # . Also, since /3, and hence As, u , and vg, are functions of frequency, any signal consisting of several frequencies is dispersed, or spread out, in both time and space as it propagates along the guide. This dispersion results from the different velocities at which the different frequency components propagate. If the guide is very long, considerable signal distortion may take place. Group and signal velocities are discussed in detail in Sec. 3.19. With some of the general properties of waveguides considered, it is now necessary to consider the detailed analysis that will establish the above properties and that, in addition, will provide the relation between k c and the guide configuration, the expressions for power and attenuation, etc. The case of TE modes in a loss-free empty rectangular guide is considered first. TE Waves For TE, or H, modes, ez = 0 and all the remaining field components can J> determined from the axial magnetic field h z by means of (3.71). The a» field h, is a solution of *?K» + *?*, = o 2 d h, or d 2k z -- o (3.200) TRANSMISSION LINES AND WAVEGUIDES 183 If a product solution hz = f(x)g(y) is assumed, (3.200) becomes 1 d2f 1 d2g fdx2^ g dy* + *? « 0 after substituting fg for h z and dividing the equation by fg. The term \/fd2f/dx2 is a function of x only, l/gd2g/dy2 is a function of y only, and k'l is a constant, and hence this equation can hold for all values of x and y only if each term is constant. Thus we may write d%f 111 _H fdx2 * 1 d2 g °r dz2 d2g g dy2 v dy'1 where k x + k'2 = k2m order that the sum of the three terms may vanish. The use of the separation-of-variables technique has reduced the partial differential equation (3.200) to two ordinary simple-harmonic second-order equations. The solutions for f and g are easily found to be f = A, cos kxx + A 2 sin kxx g = S 2 cos kyy + B2 sin kyy where A,, A 2 , B v B 2 are arbitrary constants. These constants, as well as the separation constants kx,ky, can be further specified by considering the boundary conditions that h, must satisfy. Since the normal component of the transverse magnetic field h must vanish at the perfectly conducting waveguide wall, (3.716) shows that n • V,h2 = 0 at the walls, where n is a unit normal vector at the walls. When this condition holds, tangential e will also vanish on the guide walls, as (3.71c) shows. The requirements on h z are thus —— = 0 dx at x = 0, o =0 at y = 0, b 3y where the guide cross section is taken to be that in Fig. 3.36. In the solution for f, the boundary conditions give — kxA1 sin kzx + A! x A 2 cos kxx = 0 a t x = 0,a Hence, from the condition at x = 0, it is found that A2 = 0. At x = a, it is 184 FOUNDATIONS FOR MICROWAVF. ENGINEERING necessary for sin kxa = 0, and this specifies k x to have the values rnr 1 = 0,1,2,... In a similar manner it is found that B 2 = 0 and rrnr * *•= m = 0,1,2,... Both n and m equal to zero yields a constant for the solution for h and n other field components; so this trivial solution is of no interest. If we use the above relations and put A , B , = Anm, the solutions for h, are seen to be nirx miry coscos- h=A. (3.201) a for n = 0,1,2,...; m = 0 , 1 , 2 , . . . ; n = m * 0. The constant Anm is an arbitrary amplitude constant associated with the n m t h mode. For the nmth mode the cutoff wave number is designated k c „ m , given by niry -,1/2 I mir (3.202) c.« m and is clearly a function of the guide dimensions only. The propagation constant for the nmth mode is given by 1/2 ynm =JPnm=J{kl-K.nm) 27r f V ^7/ m mr T- \2 1/2 (3.203) a When k0>kc,nm , fi nm is pure real and the mode propagates; when k0< „,, then ynm is real but (lnm is imaginary and the propagation factor is m e-yn,„U\t which shows that the mode decays rapidly with distance \z\ fr° the point at which it is excited. This decay is not associated with energy ^>ss, but is a characteristic feature of the solution. Such decaying, or evanescent, modes mav be used to represent the local diffraction, or fringing, fields tha exist in the vicinity of coupling probes and obstacles in waveguides. frequency separating the propagation and no-propagation bands is desig nated the cutoff frequency f c „ m . This is given by the solution of that is, C Ic.nm 'c.nm c riv- 2^ et + rrnr -,1/2 (3.204) TRANSMISSION LINES AND WAVEGUIDES 185 where c is the velocity of light. The cutoff wavelength is given by 2ab Ac „„, = (nV+inV) wo (3.205) A typical guide may have a ~ 26, in which case 2a Ac,nm 0.1/2 „ (n 2 + 4m 2 ) and A Cjl0 = 2a, A c 01 = a, A c ,, = 2 o / \/5, etc. In this example there is a band of wavelengths from a to 2a, that is, a frequency band c c 2c ' a for which only the H10 mode propagates. This is the dominant mode in a rectangular guide and the one most commonly used in practice. Above the frequency c/a, other modes may propagate; so the useful frequency band in the present case is a one-octave band from c / 2 a to c/a. The remainder of the field components for the TE, i m , or Hnm, mode are readily found from (3.201) by using (3.71). The results for the complete nmth. solution are mrx mny (3.206a) Hz = A„„, cos cos—:—g +-"*""•a b miry _ (3.2066) Hx = ± j — 2 Anm — sin cos k b cnm a a Pnm rmr mrx mny _ ._ (3.206c) Hy = i j p An,„—cos—-sin kcnm b a v B,,m rrnr nirx miry _ .„ Ex = Z , . n m A „ m j - ^ — c o s s i n - — »*•*«« (3.206a-) «c.nm b a b Ey~ -ZhiHmAamj-f Bnm nv kcnm a sin mrx a miry _ _ c o s — — e *-"W b (3.206c) where the wave impedance for the nmth H mode is given by Zh.„m = -^~Z0 (3.207) When the mode does not propagate, Z h nm is imaginary, indicating that there is no net energy flow associated with the evanescent mode. A general field with E2 = 0 can be described in a complete manner by a linear superposition of all the Hn m modes. 186 FOUNDATIONS FOR MICROWAVE ENGINEERING For a propagating Hnm mode the power, or rate of energy flow, Ln th positive z direction is given by J Z 2 -'0n Jn 1 -Ref(\E Re xH;-EvH:)dxdy 2 -o-'o = ^Re Z ; , nm / o 7 V , H ; + HxH:)dxdy (3.2o8) If we substitute from (3.2066) and (3.206c) and note that ^nvx .7rrnry rnrx miry ra fh i fa ,/, a •'o •'0 — 6 ""ujr UO -» 0 'O a6 4 = 1 a6 T dxdy Sill a 6 n * 0, m *• 0 n or m = 0 we find that l«& 2 P nm € fi*m ()ne0m „ R f,nm 2 / rmr \ 2 [I—) / n u \2] MTI \Zz \AnJ abi pnm (3.209) ,nm where e 0 m is the Neumann factor and equal to 1 for m = 0 and equal to 2 for m > 0. If two modes, say the H n m and Hrs modes, were present simultaneously, it would be found that the power is the sum of that contributed by each individual mode, that is, Pnm + Prs. This is a general property of loss-free waveguides. This power orthogonality arises because of the orthogonality of the functions (eigenfunctions) that describe the transverse variation of the fields when integrated over the guide cross section; e.g., £o sin nirx a sin rvx a dx n 3= r Even when small losses are present the energy flow may be taken to be to* contributed by each individual mode, with negligible error in all cases excep when two or more degenerate modes are present. Degenerate modes modes which have the same propagation constant y, and for these § presence of even small losses may result in strong coupling between * modes. TRANSMISSION L1NFS AND WAVEGUIDES 187 If the waveguide walls have finite conductivity, there will be a continuous loss of power to the walls as the modes propagate along the guide. Consequently, the phase constant jfi is perturbed and becomes y = a + j/3, where a is an attenuation constant that gives the rate at which the mode amplitude must decay as the mode progresses along the guide. For practical waveguides the losses caused by finite conductivity are so small that the attenuation constant may be calculated using the perturbation method outlined in Sec. 3.8 in connection with lossy transmission lines. The method will be illustrated for the dominant H10 mode only. For the Hnm and also the Enm modes, the calculation differs only in that somewhat greater algebraic manipulation is required. For the H 1 0 mode, the fields are given by (apart from the factor e-Jfov*) TTX /iz = A i 0 c o s — TTX TT h A = J • rP™ m -sin'kK ~A a c,K - z a .7010 » . TTX —sin — h, 10 •A 1 0 , a a as reference to (3.206) shows. From (3.209) the rate of energy flow along the guide is . ab P\o - l-^io I 4• '10 'h. 10 '<:. 10 The currents on the lossy walls are assumed to be the same as the loss-free currents, and hence are given by J, = n x H where n is a unit inward directed normal at the guide wall. Thus, on the walls at x = 0, a, the surface currents are J s a x X H = -avA10 x = 0 | - a r X H = -a,,AJ0 x = a whereas on the upper and lower walls the currents are a JPio * . X H = -a ,—s—A 10 —sin a «cio JPlQ -a v . 7T 10~sin x H = a ^2 l^, — o A a « a TTX a TTX + a^A to cosa y = 0 TTX axAlocos~ a y = 188 FOUNDATIONS FOR MICROWAVE ENGINEERING With a finite conductivity <r, the waveguide walls may be characterized exhibiting a surface impedance given by 1+./ tr8a = ( ! + . / ' ) # , Z„. - where S t is the skin depth. The power loss in the resistive part R per unit length of guide is Qe ?• * J*dl £ guide walls RJA in' 2 Since A t . -I0 = 7TX dy + 2 J J 0 0 ..a 7TX r ^ - ^ r s i inn- 2 — dx + 2 / c o s 2 — <& "c,io a •'o a " " / o . the above gives a Pw a + b + - c c. 10 2 2 a If P 0 is the power at z = 0, then P 1 0 = P 0 e . - 22Qi: is the power in the guide at any z. The rate of decrease of power propagated is P, = RJA 101 dPw dz = 2aP10 = P, and equals the power loss, as indicated in the above equation. The attenuation constant a for the Hl() mode is thus seen to be a R. b + a = 2P 10 ab 2 (110 0io ' c , 10 J ~2 R. I f\ ! abpl0k0Zo '7 h.W v.io 2 2 -(26& p /—m I 0 + —a- £ l " - " * " M ? III 1 1 1) -N **•*/ (3.210) •" ^ ^ ^ ^ ^ The attenuation for other T E n m modes is given by the formula m Table 3.4, which summarizes the solutions for TE„,„ and also TMn« modes. In Fig. 3.37 the attenuation for the TE 1 0 mode in a copper rectangular guide is given as a function of frequency. To convert attenuation given nepers to decibels, multiply by 8.686. The theoretical formulas for attenuation give results in good agf ment with experimental values for frequencies below about 5,000 MHz. higher frequencies, measured values of « may be considerably hig" ' depending on the smoothness of the waveguide surface. If surface im P er ^jjy tions of the order of magnitude of the skin depth 8 S are present, it is read"- TRANSMISSION LINES AND WAVEGU IDES 1 8 9 T A B L E 3.4 Properties of modes in a rectangular guidet TM m o d e s TE modes mrx tf.- cos cos a 0 E, Zh.nmHy y H, £ -JO M * Z-h.nm mrx a miry sin—;—c -"'n™-1 6 -#««,«* »« ~ COS 5 -Zn,nmHx miry . mrx JPnmn~ Cuo JPnmmTT mrx C S bkt„m ° a -j», m~y c.nm e •", »™- cos a * «, mz b sin . " " f . v _,„ Sin ; e Jf,*m4- sin 6* a H, 0 sin E, E 6 Z,,nm £, -Jt „ m ? z,.™ /3nmz° %e,nm [( = T • ( b "c.nm /mr\2l ~6~) tt#-' * * n m > ' Pnm 1/2 /2 2o6 *c,nm 2 2 ( n 6 + rr V ) 2«, 2/?„ - klnm/kl) 1/2 bZn{\ ,/s n2b3 + m 2 a 3 f c Z o ( l - A ? . , . ™ / * S ) , / 2 | , a * ^+m V 2 *£*„ \ " ^ + «V a 2 t f l „ • ( < U M O / 2 I T ) I / 2 , e 0 ,„ = 1 for in = 0 and 2 for m > 0. The expression for <J is not valid for degenerate modes. 190 FOUNDATIONS FOR MICROWAVE ENGINEERING 0.2 k - D 1 F I G U R E 3.37 Attenuation of TE 1 0 mode in a corm*. rectangular waveguide, a = 2 28R cn^ b= 1.143 cm. •o 8 » • 10 I 7 9 11 GHz appreciated that the effective surface area is much greater, resulting ir greater loss. By suitably polishing the surface, the experimental values of attenuation are found to be in substantial agreement with the theoretical values.t Dominant TE10 Mode Since the T E , 0 mode is the dominant mode in a rectangular guide, and also the most commonly used mode, it seems appropriate to examine this mode in more detail. From the results given earlier, the field components for this mode are described by the following (propagation in the +z direction assumed): TTX H=Acos—e"-"*2 a jB (3.211a) TTX H=i-Asm—e~Jli* k„ a 8 Ey (3.2116) TTX -jAZh~sin—< ,-m = (3.211c) where the subscript 10 has been dropped for convenience since this discussion pertains only to the T E ] 0 mode. The parameters 8, kc, and Z h & e given by *c = a (3.212a) 8 = k0 — £„ (z)1 v 1/2 zh = - -Hf = -^z 8 (3.212*) (3.212c) tSee J. Allison and F. A. Benson, Surface Roughness and Attenuation of Precision Dr Chemically Polished, Electropolished, Electroplated and Electroformed Waveguides, Proc(London), vol. 102, pt. B, pp. 251-259, 1955. ^ TRANSMISSION LINES AND WAVEGUIDES 191 The guide wavelength A„ is 2-rr A * = * il/2 [l - (A0/2O)2] 2 (3.212d) since the cutoff wavelength A c = 2a. The phase and group velocities are A„ op = —c (3.212c) "> = T; (3.212/*) and are discussed in detail in Sec. 3.19. In Fig. 3.38 the magnetic and electric field lines associated with the TEj 0 mode are illustrated. Note that the magnetic flux lines encircle the electric field lines; so these can be considered to be the source (displacement current) for the magnetic field. On the other hand, the electric field lines terminate in an electric charge distribution on the inner surface of the upper and lower waveguide walls. This charge oscillates back and forth in the axial and transverse directions and thus constitutes an axial and transverse conduction current that forms the continuation of the displacement current. The total current, displacement plus conduction, forms a i—r~n—r •I-EI3-W J| o o loi lei lb) F I G U R E 3.38 Magnetic and electric field lines for the T E 1 0 mode, ( a ) Transverse plane; (6) top view; (c) mutual total current and magnetic field linkages. 192 FOUNDATIONS FOR MICROWAVP ENGINEERING •' FIGURE 3.39 Decomposition of TE, 0 mode into two plane w •i closed linkage of the magnetic field lines, and as such may be regarded being generated by the changing magnetic flux these enclose. This com pletes the required mutual-support action between the electric and magneti* fields which is required for wave propagation. The fields for a TE 1 0 mode may be decomposed into the sum of two plane TEM waves propagating along zigzag paths between the two waveguide walls at x = 0 and x = a, as in Fig. 3.39. For the electric field we have 'LL—( JT /«-JP* h.t x E = e — e-Jnx/<>-JPz\ 2 kr If 17/a and (i are expressed as 7T the relation (TT/O)2 — = k0 sin 9 B = k0 cos 9 a + Bl = k$ still holds. The electric field is now given by Zh 8 ** 2 A/ ' which is clearly two plane waves propagating at angles ± 9 with respect to the z axis, as illustrated. Alternatively, the field may be pictured as a plane wave reflecting back and forth between the two guide walls. As shown in Sec. 2.7, the constant phase planes associated with these obliquely propagating plane waves move in the z direction at the phase velocity c/cos 6 | Bc/k0, and this is the reason why the phase velocity of the TE 1 0 mode exceeds the velocity of light. Since the energy in a TEM wave propagates with the velocity c in the direction in which the plane wave propagates, tnii energy will propagate down the guide at a velocity equal to the componen of c along the z axis. This component is vg = c cos 9 = (k0/B)c and is * group velocity for the T E l 0 mode. When 9 = TT/2, the plane waves reflec back and forth, but do not progress down the guide; so the mode is cuto The above decomposition of the TE 1 0 mode into two plane waves niaj be extended to the TE„„, modes also. When n and m are both differe from zero, four plane waves result. Although such superpositions of P 1 ^ waves may be used to construct the field solutions for rectangular gu1 ' this is a rather cumbersome approach. However, it does lend insight i° why the phase velocity exceeds that of light, as well as other properties the modes. TRANSMISSION LINES AND WAVEGUIDES 193 For TM modes, h z equals zero and e z plays the role of a potential function from which the remaining field components may be derived. This axial electric field satisfies the reduced Helmholtz equation V,2e, + fcf*a = 0 (3.213) of the same type encountered earlier for hz, that is, (3.200). The solution may be found by using the separation-of-variables method. In the present case the boundary conditions require that e z vanish at x = 0, a and y = 0, b. This condition requires that the solution for e z be n nx m vy e z = A„„, s i n — - s i n — — (3.214) a b instead of a product of cosine functions which was suitable for describing hz. Again, there are a doubly infinite number of solutions corresponding to various integers n and m. However, unlike the situation for TE modes, n = 0 and m = 0 are not solutions. The cutoff wave number is given by the same expression as for TE modes; i.e., nv c,n m a / T O 7 T \ 2• 2 1 / £. (3.215) + and the propagation factor /5nm by /? n m = ( ^ - * ; U ) , / 2 (3-216) The lowest-order propagating mode is the n = m = 1 mode, and this has a cutoff wavelength equal to 2ab/{a2 + 6 2 ) 1 / 2 . Note that the T E I 0 mode can propagate at a lower frequency (longer wavelength), thus verifying that this is the dominant mode.t It should also be noted that for the same values of n and TO, the TE„„, and TM„„, modes are degenerate since they have the same propagation factor. Another degeneracy occurs when a = b, for in this case the four modes T E „ m , TEWfflJ TM„ m > and TM„,„. all have the same propagation constant. Still further degeneracies exist if a is an integer multiple of 6, or vice versa. The rest of the solution for TM modes is readily constructed using the general equations (3.72). A summary of this solution is given in Table 3.4. The TM modes are the dual of the TE modes and apart from minor differences have essentially the same properties. For this reason it does not seem necessary to repeat the preceding discussion. tin any hollow waveguide the dominant mode is a TE mode because the boundary conditions z — 0 for TM modes always require e. to have a greater spatial variation in the transverse plane than that for h. for the lowest-order TE mode, and hence the smallest value of k c occurs for TE modes. Hence a TE mode has the lowest cutoff frequency, i.e., is the dominant mode. e 194 3.18 FOUNDATIONS FOR MICROWAVE ENGINEERING CIRCULAR WAVEGUIDES Figure 3.40 illustrates a cylindrical waveguide with a circular cross s of radius a. In view of the cylindrical geometry involved, cylindrical a<*ti0n nates are most appropriate for the analysis to be carried out. Since k general properties of the modes that may exist are similar to those for « rectangular guide, this section is not as detailed. TM Modes For the TM modes a solution of V,% + fe|e? = 0 is required such that e 2 will vanish at r = a. When we express the transverse laplacian V,2 in cylindrical coordinates (App. I), this equation becomes d2e, 1 de, 1 d2ez Sr' r 9r dtf (3.217) The separation-of-variables method may be used to reduce the above to two ordinary differential equations. Consequently, it is assumed that a product solution f(r)g(<j>) exists for ez. Substituting for e, into (3.217) and dividing the equation by fg yield 1 d2f ~fdV2 1 df + ^fd~r 1 + d2g ~^g~dl? k2 = 0 Multiplying this result by r 2 gives r* d2f 7^ I d2g r df + +rk 7d-r * = gd<},2 The left-hand side is a function of r only, whereas the right-hand side depends on <t> only. Therefore this equation can hold for all values of the variables only if both sides are equal to some constant v2. As a result, (3.217) is seen to separate into the following two equations: d2 f dT2 1 df + ~r~dr ,2 2 k- d<l>2 f=0 (3.218a) « ^=0 (3.218*) F I G U R E 3.40 The circular cylindrical waveguide. TRANSMISSION LINES AND WAVEGUIDES 195 TABLE 3.5 Values of p n m for TM m o d e s n 0 1 2 Pnl 2.405 3.832 5.135 P»2 P«3 5.520 7.016 8.417 8.654 10.174 11.620 In this case the field inside the waveguide must be periodic in rf> with period 2-TT, that is, single-valued. It is therefore necessary to choose v equal to an integer n, in which case the genera] solution to (3.218b) is g{4>) = Al cos n<t> + A2 sin n4> where A l and A 2 are arbitrary constants. Equation (3.218a) is Bessel's differential equation and has two solutions (a second-order differential equation always has two independent solutions) J,,(kcr) and Y,.(kcr), called Bessel functions of the first and second kind, respectively, and of order v.f For the problem under investigation here, only Jn(kcr) is a physically acceptable solution since Yn(kcr) becomes infinite at r = 0. The final solution for e, may thus be expressed as ez(r,<j>) = (Aj_cos n<f> + A2sinn<f>)Jn(ker) (3.219) Reference to App. II shows that Jn(x) behaves like a damped sinusoidal function and passes through zero in a quasiperiodic fashion. Since e 2 must vanish when r = a, it is necessary to choose kca in such a manner that Jn(kca) = 0. If the m t h root of the equation Jn(x) = 0 is designated pnm, the allowed values (eigenvalues) of k c are kc.nm = ~ (3.220) The values of p,im for the first three modes for n = 0,1,2 are given in Table 3.5. As in the case of the rectangular guide, there are a doubly infinite number of solutions. Each choice of n and m specifies a particular TM„ m mode (eigenfunction). The integer n is related to the number of circumferential variations in the field, whereas m relates to the number of radial variations. The propagation constant for the nmth mode is given by / 0 - - *S-^f tY,. is also catted a Neumann function. 2 ') 1 / 2 ( 3 - 221 > 196 FOUNDATIONS FOR MICROWAVE ENGINEERING the cutoff wavelength by 2TTO P„ m (3.222, and the wave impedance by „ Pixm y (3.223, «0 A cutoff phenomenon similar to that for the rectangular guide exists. p 0 the dominant TM mode, Ar = 2rra/p0i = 2.6'la, a value 30 percent greater than the waveguide diameter. Expressions for the remaining field components may be derived bv using the general equations (3.72). Energy flow and attenuation may be determined by methods similar to those used for the rectangular guide. A summary of the results is given in Table 3.6. TE Modes The solution for TE modes parallels that for the TM modes with the exception that the boundary conditions require that dh,/dr vanish at r = a. An appropriate solution for A, is hz(r,<j>) = (Bl<soBit6 + B2 sin n<f>)Jn(kcr) (3.224) with the requirement that dJn(kcr) dr =0 at r = a (3.225) The roots of (3.225) will be designated by p'nm; so the eigenvalues kcnm are given by h = ^2H a (3.226) Table 3.7 lists the values of the roots for the first few modes. Note that Pom =Pim s i n c e dJQ(x)/dx= -J^x), and hence the TE 0 m and TMln, modes are degenerate. ff The first TE mode to propagate is the T E n mode, having a cutofl wavelength A c n = 3.41a. This mode is seen to be the dominant mode\\o the circular waveguide, and is normally the one used. A sketch of the n lines in the transverse plane for this mode is given in Fig. 3.41attenuation constant for the dominant T E U mode is given by R„{ 1.8412rV2/l.8412 aZA kla* f [ kla* TRANSMISSION LINES AND WAVEGUIDES 197 TABLE 3.6 Properties of modes in circular waveguides TM m o d e s TE modes H, £. H "\ a 1 \sinn<j 0 " J^mPnm n JPnmPnm . , / Pl «PMn _,„ COS m r \ ,l / C 0 Srt (6 J oft* '[ a )e \sinn ' J"P„ H„. Pnm? -sin n<b cosnrf. ,-!»«„ rkc.nm Er kl- / \ sin mb £,., Kn, Er %... ( Pnmr \ '.(¥ J"P„ rkt Zh.nn,Hr Pnr n JPnmPnm Z h .m H t £.,. \ ,-j(in si 1 '' 2 /',,. *=r A?.- •'h.nm Pnr Zo Pnn Pnn a a 2-rra 2wa Pnm Pnm Z Power OkoPnm" , JUn * c.nmfOn 2 2 .r2, —{Pnn, ~ " . . )Jnlpnm> -1/2 aZ, »0 aZ„ Ml Ag (p;m)2-n2 1 - t2 f^JtMnf sin n (/» — sin n ib cos rc</> 198 FOUNDATIONS FOR MICROWAVE ENGINEERING TABLE 3.7 V a l u e s of p'nm f o r T E m o d e s « Pnl 0 1 2 3.832 1.841 3.054 Pn2 Pn3 7.016 5.331 6.706 10.174 8.536 9.970 F I G U R E 3-41 Field lines for the T E n mode in a circular waveguide. ffi •o 0.5 - 11.5 GHz F I G U R E 3.42 Attenuation of dominant T E U mode in a circular waveguide. Diameter = * cm. Figure 3.42 shows the attenuation in decibels per meter for a copper waveguide with an internal diameter of 2 cm. For this guide the cuto* frequency is 8.79 GHz. In the normal operating range from 9 to 11 GHz, the attenuation drops from 0.36 d B / m at 9 GHz to 0.11 d B / m at 11 GHz. 19 WAVE V E L O C I T I E S In any system capable of supporting propagating waves, a number of * 8 velocities occur that pertain to signal propagation, energy propaga £ '° wavefront propagation, etc. These various velocities are examined below' the context of propagation in waveguides. TRANSMISSION LINGS AND WA VKGUIDES 1 9 9 Velocity The phase velocity of a wave in a waveguide was introduced earlier and shown to be equal to vp = -f-C (3.228) for an air-filled guide. Here k g is the guide wavelength, A 0 the free-space wavelength, and c the velocity of light. The phase velocity is the velocity an observer must move with in order to see a constant phase for the wave propagating along the guide. It is noted that the phase velocity is greater than the velocity of light, and since the principle of relativity states that no signal or energy can be propagated at a velocity exceeding that of light, the physical significance of the phase velocity might very well be questioned. The clue to the significance of the phase velocity comes from the recognition that this velocity entered into wave solutions that had a steadystate time dependence of eJI"'. A pure monochromatic (single-frequency) wave of this type exists only if the source was turned on at t = — <» and is kept on for all future time as well. This is clearly not a physically realizable situation. In actual fact the source must be turned on at some finite time, which can be chosen as / = 0. The generated signal is then of the form illustrated in Fig. 3.43. Associated with the sudden steplike beginning of the signal is a broad frequency band, as a Fourier analysis shows. If this signal is injected into the guide at z = 0, an observer a distance I farther along the guide will, in actual fact, see no signal until a time l/c has elapsed. In other words, the wavefront will propagate along the guide with a velocity c. At the time l/c, the observer wi)) begirt to see the arrival of the transient associated with the switching on of the signal. After a suitable period of time has elapsed, the transient will have died out, and the observer will then see the steady-state sinusoidally varying wave. Once steady-state conditions prevail, the phase velocity can be introduced to describe the velocity at which a constant phase point appears to move along the guide. Note, however, that there is no information being transmitted along the guide once steady-state conditions have been established. Thus the phase velocity is not associated with any physical entity such as a signal, wavefront, or energy flow velocity. The term signal is used here to denote a time function that can convey Sit) <W F I G U R E 3.43 Sinusoidal signal applied at time t = 0. 200 FOUNDATIONS FOR MICROWAVE ENGINEERING information to the observer. Thus the step change at t = 0 is a signal K once steady-state conditions are achieved, the observer does not receive' more information. A better understanding of the above features will vZ obtained after the group velocity, discussed below, has been examined Group Velocity The physical definition of group velocity is the velocity with which a sign l consisting of a very narrow band of frequency components propagates. Th appropriate tool for the analysis of this situation is the Fourier transform rr a time function is denoted by fit), this function of time has associated with it a frequency spectrum Fiat) given by the Fourier transform of /"(*), F(u,) =f'j(t)e--""dt (3.229o) Conversely, if the spectrum F(io) is known, the time function may be found from the inverse Fourier transform relation 1 f(t) = —- .F{a))eJU" da> (3.2296) From Eq. (3.2296) it is seen that the Fourier transform represents fit) as a superposition of steady-state sinusoidal functions of infinite duration. These relations are a generalization of the Fourier series relations. If the time function is passed through a device having a response Ziw) that is a function of frequency, e.g., filter, the output time function f0(t) will have a frequency spectrum Fia»)Z(a>), and hence, by (3.2296), is given by f0(t) = — f F(o>)Z(w)e>»1 do> In general, Z(u>) = |Z(a»)[e~-'"Ww»; so W) = ^ / " j Z ( « ) | F ( » ) e * — * > d « (3-230) If the output f 0 is to be an exact reproduction of the input, then in ( 3 . ^ it is necessary for \Z\ to equal a constant A, and ip must be a Un function of w, say aw + b. In this case fo{t) A , ±-e-J<>r F(o>)e^<-°>da> X'TT Ytr (3-231°) J -^ *r If we put t ~ a = t', the above becomes fo(t< + b a)=^LLrF(a>)e^du>=Ae-; nt') Zv J - C O (3.23l*> TRANSMISSION LINES AND WAVEGUIDES 201 as comparison with (3.2296) shows. Thus the output time function is U* + «) - W) = Ae-'»an = Ae-»f(t -a) (3.232) i.e., an exact duplicate of the input, apart from a constant multiplier and a time delay a. Thus the conditions given on \Z\ and i// are those sufficient for a distortion-free system. Now, in a waveguide, the transverse variations of the field are independent of frequency. The only essential frequency-dependent part of the field solution is the propagation factor e jPx since 0 = (fci-*f)1/a=(^-ft?) is a function of frequency. Thus a waveguide of length /, in which the field has a time dependence eJ'0', w > 0, can be considered as a frequency filter with a response e~J^1. Since /3 is not a linear function of to, it may be anticipated that some signal distortion will occur for propagation in a waveguide. For an ideal TEM-wave transmission fine, )3 = k0 = io/c and distortion-free transmission is possible. However, practical lines have an attenuation which depends on frequency (Rm a -ff), and this will produce distortion. Fortunately, for narrowband signals neither waveguides nor transmission lines produce significant distortion unless very long lines are used. Consider now a time function fit) having a band of frequencies between —fm and /",„. This signal is used to modulate a carrier of frequency fc *• fm • T n e resultant is S(t) = f(t)cos coct = Re[ f(t)eJm<t) (3.233) If Fico) is the spectrum of fit), the spectrum of Sit) is Fs(w) = f_J-J""f(t) ~ da, = -[F(ai - ajc) + F(a> + wc)] (3.234) These spectra are illustrated in Fig. 3.44. For positive co the waveguide response is e ~&*"*, For negative w the response must be chosen as e^S("'" since, if the time variation is e~iQ", the sign in front of (i must be positive for propagation in the positive z direction. In other words, all physical systems will have a response such that |Z(w)| is an even function of co and iliiio) is an odd function of co. Since P is an even function, the sign must be changed for co < 0. These even and odd symmetry properties are required simply so that the output time function is real, a physical requirement. The output spectrum for the 202 FOUNDATIONS FOR MICROWAVE ENGINEERING F{u) Fsi(u) L\. -olm. -0- Um —A— FIGURE 3.44 Spectrum of /"(/land Sit). waveguide is thus F„(u,) = | [ F ( w - toje'"3'^' + F{a> -r c)eJPM'\ and the output signal is S„(t)~—f F0(u>)e;«'du> (3.235) The analysis that follows is simplified if the signal is represented in complex form as f(t)eJ"'c' with a spectrum F ( u — coc). In this case only the positive half of the spectrum needs to be considered, and the output signal is given by S „ ( 0 = R e — C' *"F(v - » J e * * - * w ' da> (3.236) since F(,a> - wc) is zero outside the band wc - wm < w < we + w m . If the band is very narrow, w,„ « w c , then /3(w) may be approximated by the first few terms in a Taylor series expansion about ae. Thus P(a) =/3(» e ) + d0 1 d2p (w - w c ) + 2 d!«r -2 ( w -o, c ) 2 +-- (3-237) Retaining the first two terms only and letting /3(wc) = 0 O and d(5/du>c = Po at wt„ (3.236) gives S0(t) = Re— r+WmF(w - aJeJt'-Me-M+JfV*da If this is compared with (3.231) and (3.232), it is seen that S0(t) = Refe^o'^'o'VU - P'0l)eJw'('-p'oh] - fit - M*»t*ji - M (3.238) To the order of approximation used here, the input modulating signal / is reproduced without distortion but with a time delay p'0L This is l ° t anticipated since /3 was approximated by a linear function of w in the TRANSMISSION LINES AND WAVEGUIDES 203 o)c - (om to <oc + (om (distortion-free condition). The signal delay defines the group velocity v which is equal to the distance / divided by the delay time; thus / V *=W* My1 (3.239) da ) This is also the signal velocity. Note, however, that this velocity has significance only if the band, or "group," of frequencies making up the signal is so narrow that fi may be approximated by a linear function throughout the frequency band of interest. If this is not the case, more terms in the expansion (3.237) must be retained and signal distortion will occur. In this case the group velocity as given by (3.239) is no longer the signal velocity. In fact, because of signal distortion, no unique signal velocity exists any longer. Different portions of the signal will travel with different velocities, and the resultant signal becomes dispersed in both time and space. In the case of a waveguide, d(co2/c2 v s = C dto J - k2) 1 '2 1 - 1 dji (3.240) CO «0 A g It is seen that vg < c and that vgvp = c2 for a waveguide. A typical plot of k0 versus /3 for a waveguide is given in Fig. 3.45. From this plot it can be seen that for a narrow band of frequencies a linear approximation for /3 is good. Also note that for high frequencies (large k0) (3 becomes equal to k0. Thus frequencies well above the cutoff frequency f e suffer very little dispersion and propagate essentially with the velocity of light c. No frequency components below the cutoff frequency f e can propagate along the guide. F I G U R E 3.45 Plot of k0 versus 0 for a waveguide. 204 FOUNDATIONS FOR MICROWAVE ENGINEERING The equality of the wavefront velocity and the velocity of light can k_ readily explained by means of Fig. 3.45. The switching on of a signal r e s 1 in an initial transient that has a spectrum of frequencies extending 0i »• infinity. Any small group of frequencies at the high end of the spectrum 7* have a group velocity equal to c since dk0/dp equals unity for k n infinity. Thus the high-frequency part of the transient will propagate al the guide with a velocity c. The lower-frequency components will propae t with smaller group velocities and arrive later. Energy-Flow Velocity Power is a flow of energy, and consequently there is a velocity of energy g 0w such that the average energy density in the guide multiplied by this velocity is equal to the power. In a waveguide it turns out that this velocity of energy flow is equal to the group velocity. A proof for the case of E modes will be given below, that for H modes being very similar. For E modes the field is given by [see (3.72)] JP "0*0 H, ~P~ a, X E, The average rate of energy flow, or power, is given by 1 /• IE, ? 1 k 0Y 0 dS = - P 7.E,fdS (3.241) where the integration is over the guide cross section. The energy density in the magnetic field per unit length of guide is CL 4 Js 4 p~ Js 'dS (3.242) The energy density in the electric field per unit length of guide is equal to that in the magnetic field. This is readily shown to be the case by using the complex Poynting vector theorem, which states that (Sec. 2.5) - J /' E X H* • a , dS = P + 2jw( Wm - We) s where the integration is over the guide cross section, and the term on right gives the power transmitted past the plane SS plus plus 2jco 2joi tiroes tn th reactive energy stored in the guide beyond the plane S. Since the integr ;egr the complex Poynting vector over a cross section 5 for a propagating nl in a loss-free guide is real, it follows that Wm = We. In addition, since location of the transverse plane S is arbitrary, it also follows that energy densities U m and U e per unit length of guide are equal. the TRANSMISSION LINKS AND WAVEGUIDES 205 The velocity of energy flow may now be found from the relation P v = U„ + U„ P k 0Y 0 H2 2t/„ P MoW 2v2 (3.243) and comes out equal to the group velocity as stated earlier. BIDGE WAVEGUIDE For a rectangular waveguide with a width a equal to twice the height ft, the maximum bandwidth of operation over which only the dominant TE 1 0 mode propagates is a 2 : 1 band. For some system applications it is necessary to have a waveguide that operates with only a single mode of propagation over much larger bandwidths. A transmission line supporting only a TEM mode can fulfill this requirement but must then have cross-sectional dimensions that are small relative to the shortest wavelength of interest. A coaxial transmission line will support higher-order TE and TM modes in addition to the TEM mode. Thus, to avoid excitation of a higher-order mode of propagation, the outer radius must be kept small relative to the wavelength. The small cross section implies a relatively large attenuation; so some other form of waveguide is needed. The ridge waveguide illustrated in Fig. 3.46 was developed to fulfill this need for a single-mode waveguide capable of operating over a very broad band. Physically, it is easy to understand why the ridge waveguide has a very large frequency band of operation. The center section of width W and spacing S functions very much like a parallel-plate transmission line and consequently the ridge waveguide has a much lower cutoff frequency for the same width and height as does the x FIGURE 3.46 Ridge waveguide. 206 FOUNDATIONS FOB MICROWAVE ENGINEERING conventional rectangular waveguide. Operation over bandwidths of 5 [_ . 1 more is possible. The ridge waveguide, when uniformly filled with dielectric, which be air, h a s the same general properties as the rectangular and circuT waveguides discussed earlier. If we can determine the cutoff wavelength for the dominant mode, then at any frequency above the cutoff freqUe * c the propagation constant ji is given by P = yjkl- (2tr/Xcf At the cutoff frequency (i = 0 and the electromagnetic field has no variatin with the axial coordinate z. The cutoff wave number kc = 2ir/\c can L found using the transverse-resonance method as described below. The transverse-resonance method is based on finding the resonant frequency for the transmission-line circuit that provides a model for the waveguide cross section. At cutoff we can view the electromagnetic field as a uniform plane wave with components Ey and H, that propagates in the x direction and is incident onto a second parallel-plate transmission line of reduced height. The equivalent transmission-line circuit is that of two parallel-plate transmission lines of height b, length (o - W)/2, and shortcircuited at x = 0 and a. These two transmission lines are connected to another parallel-plate line of height S and length W and placed between the first two as shown in Fig. 3.47. At the junction where the height changes, a local fringing electric field occurs and stores electric field energy in the vicinity of the step. The effect of this local fringing electric field is taken into account by a shunt capacitive susceptance jB at each junction as shown in Fig. 3.47. The standing-wave field pattern along the x direction can exist only at the resonant frequency for the transmission-line circuit shown in Fig. 3.47. For the dominant mode the voltage is a maximum and the current is zero at the midsection. Thus, at x = a / 2 , the impedance looking in the x direction toward the short circuit must be infinite. The corresponding input admittance will be zero. At the step the admittance looking toward the shofl circuit is a-W Y= -jYci cot kc- + JB By using the formula for admittance transformation along a transnu _ F I G U R E 3.47 Equivalent transmission-line circuit of section of ridge waveguide, TRANSMISSION LINES AND WAVEGUIDES 207 a 1 FIGURE 3.48 Normalized cutoff wavelength Kc/a for a ridge waveguide. W a line, we obtain Y-.„ = Y. c2 In order for Y Y+jYc3 tan keW/2 Yc2+jY tan kcW/2 to vanish, we must have ajB -jYcl cot kt W + jYe2tanke— W = 0 (3.244) which is the transverse-resonance condition. The two characteristic admittances are inversely proportional to the parallel-plate spacing; thus Yc2 = (b/S)Yci. An approximate expression for the normalized shunt capacitive susceptance can be found using quasistatic conformal mapping and ist B 26 1 a 2 ] „ l ^ 1 - l n 4 u + -u + - ( 1 - u2) ^ u = - < 0.5 (3.245) The eigenvalue equation (3.244) is a transcendental equation. The computer program RIDGEWG solves (3.244) for the normalized cutoff wavelength A c /a. Figure 3.48 shows typical results that are obtained. The numerical results obtained from (3.244) agree within 1 percent of the values given by Hopfer and Hoefer and Burton over the commonly used range of parameters.? t N . Marcuvitz, "Waveguide Handbook," MIT Radiation Lab Series, vol, 10, reprinted by Boston Technical Publishers. Inc.. 1964. t S . Hopfer, The Design of Ridged Waveguides IRE Trans., vol. MTT-3, pp. 20-29, October, 1955. W. J. R. Hoefer and M. N. Burton, Closed Form Expression for the Parameters of Finned and Ridged Waveguides, IEEE Trans., vol. MTT-30, pp. 2190-2194, December 1982. 208 3.21 FOUNDATIONS FOR MICROWAVE ENGINEERING FIN LINE If the width W of the ridges in the ridge waveguide is very small, we oht a fin line as shown in Fig. 3.49a. Usually the fins are metal foils on a th? dielectric substrate mounted in the E plane of a rectangular waveguide ft the dominant mode the current flows in the axial direction; so good elert^ cal contact between the fins and the waveguide is not essential. The fin I is a shielded slot line. The fin line can be matched to the rectaneui * waveguide by means of a tapered section or by using one or more quarto wave impedance transformers as shown in Figs. 3.496 and c. The fin line is suitable for use in microwave circuits that incorporate two-terminal devices such as diodes. Transistors cannot be connected to a fin line since they are three-terminal devices. The cutoff wavelength for a fin line may also be found by using the transverse-resonance method. The fins produce a shunt normalized capacitive susceptance across the center of the waveguide given byt B 26 (3.246) _}_ i XT T tt-i\ »° «2 n) "(£-*• where A=«i P2 = 2a\ + a* - 1 P3 = 4a? + %axal - 3Ul P4 = 8a? + 3a^ + 24ajfa| - 8a? - 4 a | + 1 "1 = C0S 27rS Jb a 2 = sm — 1/2 r„ = WHxf] The equivalent circuit of the fin-line cross section consists of two short cuited transmission lines of length a/2 with jB connected at the center tR. E. Collin, "Field Theory of Guided Waves," 2nd ed., chap. 8, IEEE Press, Piscataway. 1990. TRANSMISSION LINES AND WAVEGUIDES 209 ;^ s \ \ s s s s s s yj Dielectric -Fins bsgc^ssss^^sss 5 s 5 v 5 •• 5 5 ! (a) F I G U R E 3.49 ( a ) Fin line; (6) tapered matching section; (c) quarter-wave matching section. (c) (to) r JB - _— ZjB 2 Yc h-fH -}H F I G U R E 3.50 Equivalent circuit of fin-line cross section. shown in Fig. 3.50. The resonance condition is Y. jB _ = K 2K The computer program FINLINE the normalized cutoff wavelength merical results are shown in Fig. numerical results agree within 1 Hoefer and Burton.t The propagation constant fi conventional waveguide, i.e., . ira /cot— = 0 J A,. (3.247) solves this transcendental equation for A,./a for S/b = 0.1 to 0.9. Typical nu3.51 for the case where a = 26. These percent or better with those given by is given by the same formula as for a e = (H - k*c),/2 When the fins are mounted on a dielectric substrate, a correction is needed for the propagation constant if the dielectric has an appreciable thickness tW, J. R. Hoefer and M. N. Burton. loc. cie. 210 FOUNDATIONS FOR MICROWAVE ENGINEERING "0 0.6 0.8 1 S b FIGURE 3.51 Normalized cutoff wavelength \ / e/a fin line. for 5 a n d a large dielectric c o n s t a n t . E m p i r i c a l f o r m u l a s for t h i s case are avail able.t PROBLEMS 3 . 1 . For the ideal transmission line shown in Fig. P3.1. the switch is closed at t = 0 and opened 1 ixs later. The characteristic impedance of the line is 50 ft. The load resistance is also 50 fi. The battery has an internal resistance of 10 ft, ( a ) Sketch the voltage across R7_ as a function of time for a line 300 m long. The wave velocity « = 3x 10 8 m / s . (6) Sketch the voltage waveform across RL when RL = 25 fl and the line is 900 m long. (c) Sketch the voltage waveform across RL when RL = 100 fl and the lineis 900 m long. (.d) Repeat (6) and (c) for a line 75 m long. Z c = 50 !) V B «i T_ K FIGURE P3.1 3.2. Let a generator with internal resistance R g be connected to a transmission of length / and having a characteristic impedance Z,.. The line is terminatea a load resistance R,. Let r = l/v be the one-way propagation time d e ^ ' ~ , generator produces a pulsed waveform Pit), 0 < t < T. Show that the vo across RL is given by V '-Y. J—(i + rL)[p(t - r) + \LveP(t - 3T) + rlr;p(t - 5T) tK. Chang (ed.), "Handbook of Microwave and Optical Components." vol. 1, PP- 38-3" Wiley & Sons, Inc., New York, 1989. job" TRANSMISSION LINES AND WAVEGUIDES 211 Hint: See (3.12) and consider the total line voltage at z = I. 3.3. A pulse generator produces a sawtooth pulse P(t) = LOt/T, 0<1<T, where T= 1 ( T 8 s. The generator has an internal resistance R g = 200 O and is connected to a transmission line with Z c = 50 fi. The line is / meters long and terminated in a load resistance RL. The wave velocity equals 3 X 10 m / s . ( a ) Find and sketch the load voltage as a function of time when / = 3 m and RL = 200 O. (6) Repeat (a) when RL = 12.5 Q. (c) Find an analytic expression for t h e voltage across RL when RL = 200 $2 and / = 12 m. (d) Make a distance-time plot of the line voltage for (c). 3.4. A battery with voltage of 10 V is connected in series with a 50-JJ resistor to the input of a 50-fl transmission line at time / = 0. The transmission line, of length 12 m, is terminated in a 1-jtF capacitor. Find and sketch the voltage across the capacitor as a function of time. Hint: Apply Thevenin's theorem. Answer: V. = 10[1 - g-c- 0 - 0 4 )/ 5 0 ] / in microseconds 3.5. In the circuit illustrated in Fig. P3.5, the battery is connected at / = 0. Find and sketch the voltage across R L as a function of time. Assume that R L = R g = Zc = 50 ih C = 1 /iF, I = 300 m, and v = 3 X 10 8 m / s . R SW rAAAr/"*- I Z C = 50!J C = 1 ^F I = 300 m ;>fl L =50Il H FIGURE P3.5 3.6. The resistor R, is replaced by a capacitor CL *= 1 /xF in Fig. P3.5. Find the voltage across CL during the time interval 1 /xs < t < 3 its. Answer: VL = 5[l+e"~^25] - lOe -(«-n/5o where t is in microseconds. 3.7. Consider an ideal loss-free transmission line of length /. as shown in Fig. P3.7. The far end is short-circuited. At the input end a battery of voltage V 0 is switched across t h e Vine at time t = 0. Sketch the voltage wave on the line at the middle over the time interval 0 < / < ll/c. -2 FIGURE P3.7 212 FOUNDATIONS FOR MICROWAVE ENGINEERING ,r "o4=1 Zc = Ra —VW—o*0 3.8, Consider the transmission-line circuit illustrated in Fig, P3.8. At time / battery of voltage V 0 is switched across the line at the input. DetermiiT **' * output load voltage V as a function of time. Hint: This transient problem may be solved in a manner similar to tK used in low-frequency circuit theory. The governing equations for the transnv sion line are >?V BJF 3 — _£ Hz j= U Hz dV __ Q 8t ' The time derivative may be eliminated by taking the Laplace transform. The transformed solutions for V and S are y J e - w + V ~ e " '" V e " " ' + I~esz/V By transforming the circuit equations for the load termination and the input voltage, the resultant equations may be solved for the Laplace transform of the load voltage. The inverse transform then gives the load voltage as a function of time. The foregoing procedure may be simplified by first replacing the battery by a source V0e-""' and obtaining the transfer function V/V0 = Z,(ju) as a function of u> for this steady-state problem. Replacing ju> by s then gives the transfer function in the s domain. The Laplace transform of the output voltage is then V(s) = — Z , ( > = S ) s since the Laplace transform of the input step voltage is V0/s. The output voltage is obtained from the inverse Laplace transform of Vis). 3.9. Obtain expressions for the voltage and current standing-wave patterns on * lossless open-circuited transmission fine. Sketch these patterns. Assume an r time dependence. 3.10. A transmission line with Z c = 50 ft is terminated in an impedance 25 + . P 11. Find the reflection coefficient, standing-wave ratio, and fraction of tl incident power delivered to the load. Answer: VSWR = 2.618, 80 percent. 3.11. Verify (3.47) and compute Zin at a distance A 0 /4 from the termination gi in Prob. 3.10. 3.12. On a transmission line with Z c = 50 ft, the voltage at a distance 0.4A0 "°, the load is 4 +j2. The corresponding current is 0.1 A. Determine the nor ized load impedance. Answer: 0.145 +J0.397. 3.13. A 50-ft transmission line is terminated by a 75-ft load resistor. Fi n 0 t distance / from the load at which Y,„ = 0.02 ~jB. By connecting a ^ TRANSMISSION LINES AND WAVEGUIDES 213 susceptance -jB across the line at this point, the load will be matched to the transmission line. Explain why this is so. The distance / can be expressed in terms of the wavelength A0. 3.14. Figure P3.14 illustrates a three-conductor transmission line. Since potential is arbitrary to within an additive constant, the shield S„ can be chosen to be at zero potential. Show that there are two linearly independent solutions for TEM waves in this transmission line. If S„ encloses N conductors, how many TEM-wave solutions are possible? Hint: Note that the potential can be arbitrarily specified on S, and S2. V=0 FIGURE P3.14 3.15. Show that power transmitted along a transmission line is given by P = y/|V(<t>|2d*dy For Prob. 3.14 show that this equals jfV,/, + V21,) by using Green's first identity (App. I) to convert the surface integral to a contour integral around the conductor boundaries. I x and /., are the total currents on S, and S 2 . 3.16. Consider a three-conductor transmission fine as shown in Fig. P3.14 but assume that the cross sections of S, and S 2 are not the same. Let <$>„ and <Ph be two different solutions for the potential field. For <J>„ let Val, /„, and ^o2> A»2 De t n e voltage and currents on S, and S 2 . Similarly, for <t>b let Vbl, /;,, and Vb2, / 6 2 be the voltages and currents on S t and S2. For the TEM modes derived from 4>0 + <t>b, show that the power flow is given by HKi + vln)(iai + / M ) . |(V Q 2 + v, 2 )(/ a 2 + z62) It is convenient to choose the potentials so that the two TEM modes obtained from <t>„ and <1>6 have independent power flow. Show that this will be the case if the interaction term (ValIbl + VhiIal) + (Vo2Ib2 + Vb2Ia2) equals zero. Furthermore, show that the interaction term will vanish if the potentials are chosen so that c. K.2 " vb2 1 1/2 c„ + c, where C u is the capacitance between S 1 and S0, C22 is the capacitance between S 2 and S 0 , and C 12 is the capacitance between S, and S 2 . For a 214 FOUNDATIONS FOR MICROWAVE ENGINEERING symmetrical line, C u = C 22 and the two modes correspond to the ev a odd modes. *<4 Hint: Use the relations <?„, = C M V el + C12(VBl - Vo2), Q^ = c C 12 (V o2 - Vaj) and similar ones for the total charge Qhl and Q,,,, on <j 22 ,? s * terms of V6, and Vfc2. Also note that /„, = ( y 0 / e 0 ) Q a „ etc. "" '' »>n 3.17. Show that, for an air-filled coaxial line, minimum attenuation occurs x In x = 1 + x, x = b/a. What is the corresponding characteristic i m p e d a " Hint: Hold the outer radius b constant and find da /da. 3.18. Evaluate Zc for a lossy coaxial line using (3.28) and computed values of R n L, and C. Assume 6 = 3a = 1 cm, f= 10 9 Hz, a = 5.8 x l o 7 S / r a ' e = (2.56 -jO.OODe,,. Verify that ' and Im Z, <r_ Re Z,. and Z /M 1 / 2 -(c) See Table 3.1 for coaxiaf-iine parameters. 3.19. Use the energy definitions of L and C [Eqs. (3.112)1 to derive the results given by (3.106) and (3.108) for a coaxial transmission line. 3.20. A microstrip line has a substrate 1 mm thick and with a dielectric constant e r = 8. The strip width W = 2.5 mm. Find the low-frequency effective dielectric constant and characteristic impedance. Answer: ee = 5.953, Z,. = 32.13 il. 3.21. A microstrip line uses an anisotropic dielectric substrate with er = 10 and «v = 8. The substrate is 0.5 mm thick and the strip width W = 0.75 mm. Find the low-frequency effective dielectric constant and characteristic impedance. Answer: et = 5.895, Z,, = 42.97 fl. 3.22. A microstrip line has a 1-mm-thick dielectric substrate with a dielectnc constant of 6. Use the computer program MSTP to generate data giving the effective dielectric constant and characteristic impedance as a function of strip width W. Use these data to design a microstrip system with an input line having Z r = 50 ft, an output line having Z c = 75 il, and an intermediate quarter-wave transformer section with Z r = ^50 x 75 i l . Specify the three strip widths W,, VV2, W:i and the length of the quarter-wave transformer at 8 frequency f = 2 GHz (see Fig. P3.22). Answer: Widths are 1.505 mm, 1.0125 mm. 0.649 mm. length = l** cm. \ L | 1 \w2 w, ~~1 r~ i \— A 4 -A \% i FIGURE P3.22 3.23. Find the effective dielectric constant, characteristic impedance, and atte tion at 2 GHz for a microstrip line with the following parameters: f r "^ jg Joss tangent = 2 x 10" 3 , substrate thickness H = I mm. strip widthi * mm, and strip thickness T = 0.01 mm. Use the computer program MS 1 TRANSMISSION LINES AND WAVEGUIDES 215 3.24. Use the computer program MSTPD to generate dispersion data for the effective dielectric constant for a microstrip line having the following parameters: dielectric constant = 6, substrate thickness H = 0.5 mm, and strip width W = 2 mm. At what frequency has the effective dielectric constant increased by 5 percent more than the quasistatic value? 3.25. In a monolithic microwave integrated circuit, gallium arsenide with e r = 12.9 is used as a substrate material. Find the effective dielectric constant, characteristic impedance, and attenuation at 10 GHz for a microstrip line with the following parameters: substrate thickness = 0.1 mm, strip width = 0.05 mm, strip thickness = 0.002 mm, and loss tangent = 6 x LO -8 . For these dimensions the quasistatic parameters are accurate. The computer program MSTP can be used for the evaluation. What is the attenuation in decibels per wavelength (microstrip) for this microstrip line? 3.26. A microstrip line has the following parameters: strip width W = 1 mm, substrate thickness = 1 mm, and anisotropic dielectric with e r = 6.5, e v = 6. Find the following: distributed capacitance C and inductance L per centimeter. characteristic impedance, effective dielectric constant, and the quasiTEM-mode wavelength at 2 GHz. 3.27. Use the computer program CMSTP to find the even- and odd-mode characteristic impedances and the voltage coupling coefficient C for a coupled microstrip line having the following parameters: strip width W = 1 mm, strip spacing S = 0.1 mm, substrate thickness = 1 mm, and substrate dielectricconstant = 9.7. 3.28. A strip line has a ground-plane spacing of 2 mm, a strip width of 1 mm, and is filled with a dielectric medium with dielectric constant 2.3. Find the characteristic impedance. 3.29. Use the computer program STPL to evaluate the characteristic impedance and attenuation of a strip line with the following parameters: ground-plane spacing = 2 mm, strip width W = 0.5 mm, strip thickness T = 0.01 mm, dielectric constant = 6, loss tangent = 0.006, and frequency of operation = 5 GHz. What is the ratio of the attenuation due to dielectric loss relative to that of conductor loss? 3.30. A broadside coupled strip line is required for a 3-dB directional coupler. The even-mode characteristic impedance is to be 50 SI. The voltage coupling coefficient is 0.707. From this information determine the required odd-mode characteristic impedance. Find the required strip width W and spacing S for this coupled strip line. The ground-plane spacing is 4 mm and the dielectric constant of the dielectric filling is 5. The strip thickness T = 0.05 mm. Use the computer program CSTPL. You will need to follow an iterative procedure to arrive at the required parameters. An accuracy of ± 0.5 percent is adequate. Hint: Begin with W = 3.5 mm, S = 0.5 mm. 3.31. In a monolithic microwave integrated circuit, a coplanar transmission line with the following parameters is used: strip width S = 0.1 mm, slot width W = 0 . 1 mm, strip thickness = 0.002 mm, substrate thickness = 0.5 mm, dielectric constant = 12.9, loss tangent = 0.0008, and frequency = 10 GHz. Use the computer program CPW to determine the characteristic impedance 216 FOUNDATIONS FOR MICROWAVE ENGINEERING and attenuation. If the strip thickness is increased to 0.005 mm wjii significantly reduce the attenuation? ' 'h^ 3.32. Figure P3.32 shows a coplanar-transmission-line circuit for use in a v. amplifier circuit. The required input- and output-fine chara f r ^ impedances are 50 ft and 72 ft. The impedance of the quarter-wave sect- c v/50 x 72 = 60 ft. The ground-plane spacing 2W + S is kept c o n s t a n T ' ^ "* nun. The strip thickness is 0.002 mm. The substrate thickness is 0 4 nl * 0.3 the dielectric constant is 12.9 with a loss tangent of 0.001. Use the com" ^ program CPW to determine the required strip widths S 1 , S 2 , and S rj mine the length / of the quarter-wave matching section at a frequency f GHz. How much attenuation occurs in the quarter-wave section? An accu ^ of ±0.25 percent is sufficient. 0.3 mm s. no FIGURE P3.32 3.33. In a planar transmission line, the attenuation is 0.25 dB/cm. By what fraction is the wave amplitude reduced in propagating a distance of 1 cm on this transmission line? 3.34. Derive the equations (3.72) for TM waves. 3.35. Find the cutoff frequency for the TE 1 0 mode in a rectangular waveguide with dimensions 4 cm by 2 cm. Find the guide wavelength Ag and phase velocity at a frequency 25 percent higher than the cutoff frequency. 3.36. Derive the solution for a TE 1 0 mode in a rectangular guide of wide dimension a and height b when the guide is filled with dielectric of permittivity «• Show that the cutoff frequency is given by f, = c / 2 a e j / 2 , where c is the free-spi velocity of light and e r is the dielectric constant. Show that the gu> wavelength is smaller for a dielectric-filled guide than for an air-filled g«< 3.37. Obtain an expression for the attenuation of a TE 1 0 mode in a dielectric-fil guide when e = e l - je2 but the walls are perfectly conducting. Obtain ^ exact expression and compare it with the results deduced by an apphca the perturbation method. i* a witn 3.38. Obtain a solution for an H wave in the parallel-plate transmission tin ^ centered dielectric slab as illustrated in Fig. P3.38. Assume that the P 1 3 ^ ^ perfectly conducting and infinitely wide. Can a TEM wave propagate structure? Why? f0r Hint: Assume h, = cos kdx for |x| < a / 2 and h = Ae' a / 2 . Verify that k\ + p* = (e r - D*o- Match the tangential fields at x " to obtain an equation for A and one relating the parameters p and «rf- ' iltf TRANSMISSION LINES AND WAVEGUIDES 217 «b *0 FIGURE P3.38 3.39. Obtain solutions for T E „ 0 modes in the partially filled waveguide illustrated in Fig. P3.39. Hint: Assume that cos L i ft.= 1 A cos p(a - x) 0 <x < t t <x < a and match the tangential fields at x = t. Thus show that 02 = kl - p2 = er*g - kl and that p tan kdt = -kd tan pd Note that there are an infinite number of solutions for p and k d corresponding to various T E „ 0 modes. Obtain numerical values for /3, p, and k d when k0 = 2, t = I cm, d = 1.5 cm, and er = 4. Note that there is a lowest-order solution for p pure imaginary. 6o FIGURE P3.39 3.40. Obtain expressions for the surface currents of a TE 1 0 mode in a rectangular guide. A narrow slot may be cut in a waveguide along a current flow line without appreciably disturbing the field. Show that, for the TE 1 0 mode, narrow centered axial slots may be cut in the broad face of a rectangular guide. This principle is used in standing-wave detectors to provide suitable points of entry for a probe used to sample the interior waveguide field. 3.41. Use the computer program RECTWG to evaluate the parameters of a rectangular waveguide with width a = 1 cm, height 6 = 0.4 cm at a frequency of 20 GHz. How much attenuation will occur in a waveguide 5 m long? 3.42. For TE modes in a waveguide, write H, = -I(z)V,hz, E, = V(z)az X V,hz. Use Maxwell's equations to show that Viz) and I(z) satisfy the transmission- 218 FOUNDATIONS FOR MICROWAVE ENGINEERING line equations clV dl — = -i«wt ~ <- - | > , „ + 777 1 - | V — > M 0n2' -T az az \ ./<°M(i Construct an equivalent distributed-parameter circuit for these modes p TM modes put E, = - V(2)V,et., H, J U ) a 2 X V / e il and show that rfV / A,2 \ d/ dz ^ ./<o«o / "z Construct an equivalent distributed-parameter circuit for these modes.t 3.43. Consider an infinitely long rectangular guide. The guide is filled with dielectric for z > 0, having a dielectric constant er. An Hw mode is incident from z < 0. At z = 0, a reflected r/ 1(l mode and a transmitted H10 mode are produced because of the discontinuity. Show that the reflection coefficient is given by (Z 2 - Z , ) / ( Z 2 + Z,), where Z, is the wave impedance in the empty guide and Z 2 is the wave impedance in the dielectric-filled guide. Show that the ratio of the wave impedances equals the ratio of the guide wavelengths. 3.44. Find the surface currents for the Hnl mode in a circular guide. 3.45. Obtain an expression for power in a T E n mode in a circular guide. (See App. II for Bessel-function integrals.) 3.46. Derive an expression for attenuation for TE(1,„ modes in a circular waveguide. Answer: a = Rm fc20m/[aZ0 f( f1 - / ^ o , , , ) " 2 ] . 3.47. Find the attenuation in decibels per mile for an Hin mode in a circular copper guide of 1 in diameter when operated at a frequency of 10 times the cutoff frequency. 3.48. Show that, in a coaxial line with inner radius a and outer radius b, there are solutions for TE„„, and T M „ m modes. A suitable solution for e z and k z is [AJ„(kcr) +y n O,.r)]cosnrf> Obtain equations (transcendental in nature) for determining the cutoff wave number k v by imposing proper boundary conditions at r = a, b. 3.49. Use the computer program RIDGEWG to find the cutoff wavelength and frequency for a ridge waveguide with dimensions a = 1 cm, b = 0.4 cm, naj?J width = 0.5 cm, and ridge spacing S = 0.1 cm. Compare this with the cuto frequency of a standard waveguide with a = 1 cm and b = 0.4 cm. 3.50. Use the transverse-resonance technique to derive the eigenvalue equation T E „ 0 modes in the partially filled rectangular guide of Prob. 3.39. Verify ^ the wave impedances in the x direction in the two regions are k0Z0/P kz/kd = k0z0/kd. 4jn 3.51. A rectangular waveguide with internal dimensions a = 0.9 in and b = ^-^ (standard X-band waveguide) has a centered fin with a slot spacing * ~~. mm. Find the cutoff wavelength and compare it with that for the wavegu* without fin loading. Use the computer program FINLINE. tS. A. Schelkunoff, Bell System Tech. J-. vol. 34, p. 995. September, 1955. TRANSMISSION LINES AND WAVEGUIDES 219 3.52. The permittivity e is generally a function of e(u>) of to. Obtain an expression for the group velocity of a coaxial line filled with dielectric. Neglect the frequency dependence of the attenuation due to conductor loss. 3.53. A rectangular guide of dimensions a = 26 = 2.5 cm is operated at a frequency of 10 l 0 Hz. A pulse-modulated carrier of the above frequency is transmitted through the guide. How much pulse delay time is introduced by a guide 100 m long? REFERENCES 1. Ramo, S., J. R. Whinnery, and T. Van Duzer: "Fields and Waves in Communication Electronics," 2nd ed., John Wiley & Sons, Inc., New York, 1984. 2. Liboff, R. L.. and G. C. Dalman: "Transmission Lines. Waveguides, and Smith Charts." MacMillan Publishing Company, New York. 1985. 3. Pozar, D. M.: "Microwave Engineering," Addison-Wesley Publishing Company, Reading, Mass. 1990. 4. Rizzi, P. A.: "Microwave Engineering—Passive Circuits," Prentice-Hall, Inc., Englewood Cliff's, N.J., 1988. 5. Edwards, T. C: "Foundations for Microstrip Circuit Design," John Wiley & Sons, Inc., New York, 1987. 6. Bahl, I., and P. Bhartia: "Microwave Solid State Circuit Design." John Wiley & Sons, Inc., New York, 1988. 7. Chang, K. led.): "Handbook of Microwave and Optical Components," vol. 1. John Wiley & Sons. Inc., New York, 19S9. 8. Ishii, T. K.: "Microwave Engineering," 2nd ed., Harcourt Brace Jovanovich, New York, 1989. 9. Wolff, E. A, and R. Kaul: "Microwave Engineering and Systems," John Wiley & Sons, Inc., New York, 1988. 10. Baden Fuller, A. J.: "Microwaves," 3rd ed., Pergamon Press, New York, 1990. CHAPTER 4 CIRCUIT THEORY FOR WAVEGUIDING SYSTEMSt At low frequencies the interconnection of resistors, capacitors, and inductors results in a circuit. Such circuits are normally linear, so that the superposition principle may be used to find the response when more than one exciting source is present. Kirchhoff s laws form the basis for the analysis, whether in terms of loop currents or node voltages. In these low-frequency circuits the various elements are connected by conducting wires, and generally the length of these connecting wires is not critical or important. At microwave frequencies equivalent reactive and resistive elements may also be connected to form a microwave circuit. In place of connectin] wires, transmission lines and waveguides are used. The length or to connecting link is often several wavelengths, and hence propagation efleci become very important. The analysis of microwave circuits is therefore necessity somewhat more involved than that for the low-frequency ca~^ The circuit theory of transmission-line circuits has been well developed many decades, and, as will be shown the circuit theory for wavegu systems is formally the same. t T h e basic theory of microwave circuits is developed in C. G. Montgomery, B- H. Di y , , ^ E. M, Purcell, "Principles of" Microwave Circuits." McGraw-Hill Book Company, f J . e *..J U 1948. Much of the material presented here in Sees. 4.1 to 4.9 must of necessity be sirou view o! its basic nature. 220 CIRCUIT THEORY FOR WAVEGUIDINCi SYSTEMS 221 Many of the circuit-analysis techniques and circuit properties that are valid at low frequencies are also valid for microwave circuits. In actual fact, low-frequency circuit analysis is a special case of microwave circuit analysis. As a consequence, a study of microwave circuits provides a deeper physical insight into conventional circuit theory. In this chapter the physical basis for a circuit theory for waveguiding systems is developed. In later chapters we shall utilize this foundation in the study of impedance matching, waveguide devices, resonators, filters, etc. EQUIVALENT VOLTAGES AND CURRENTS At microwave frequencies voltmeters and ammeters for the direct measurement of voltages and currents do not exist. For this reason voltage and current, as a measure of the level of electrical excitation of a circuit, do not play a primary role at microwave frequencies. On the other hand, it is useful to be able to describe the operation of a microwave circuit in terms of voltages, currents, and impedances in order to make optimum use of low-frequency circuit concepts. For the most part this can be done. There is, however, a notable difference, namely, the nonuniqueness of the voltages and currents in most instances. It was noted in the preceding chapter that for the TEM wave on a transmission line there existed a voltage and a current wave uniquely related to the transverse electric and magnetic fields, respectively. In the case of TE and TM modes in a waveguide, no unique voltage or current waves exist that have the same physical significance as those associated with the TEM wave on a transmission line. This result might have been anticipated since the guide boundary is a closed conducting boundary, and one is at a loss as to the two points on the boundary between which the voltage should be measured. Furthermore, if voltage is defined as the line integral of the transverse electric field between two chosen points on the boundary it is found that for TM waves the line integral is zero (Probs. 4.1 and 4.2), whereas for TE waves the value of the line integral depends on the path of integration that is chosen. For these reasons the introduction of voltage and current waves, to be associated with waveguide modes, is done on an equivalent basis and has formal significance only. The basis for the introduction of equivalent voltages and currents is discussed below. In the previous chapter it was shown that propagating waveguide modes have the following properties; 1. Power transmitted is given by an integral involving the transverse electric and transverse magnetic fields only. 2. In a loss-free guide supporting several modes of propagation, the power transmitted is the sum of that contributed by each mode individually. 222 FOUNDATIONS FOR MICROWAVE ENGINEERING 3. The transverse fields vary with distance along the guide according t propagation factor e±Jl1' only. ie transverse magnetic field is related to the transverse electric field L 4. The constant, the wave impedance of the mode; i.e., a simple si Z„h = a . X e for a mode propagating in the + z direction. These properties suggest letting equivalent voltage and current waves be introduced proportional to the transverse electric and magnetic fields respectively, since the transverse fields have properties similar to those of the voltage and current waves on a transmission line. That is, in actual fact what is done. A propagating waveguide mode may be expressed in general as E = C*ee'jPz + C+eze'J0z (4.IQ) H = {The--*2 + C ^ e - ^ 2 (4.16) for propagation in the + z direction, and E = C~eejp' - C~eteJP* (4.2a) H = ~ < M w * » + C \izejtiz (4.26) for propagation in the —z direction. In (4.1) and (4.2), C + and C~ are arbitrary amplitude constants. Note also that if the mode is a TE or a TM mode, then e z and h.. is zero accordingly. Let the following equivalent voltage and current waves be introduced: V = f*-e-Jfi* + V~ejp' (4.3a) I^I+e~jPz ~I~eJPl (4.36) where V*= Kfi+, V~= KXC~, and /' = K2C^, I = K,C~. K} and K2 are constants of proportionality that will establish the relationship between voltages and the transverse electric field and currents and the transverse magnetic field. In order to conserve power, it is necessary that 1 * \C ^z. + ( / + ) * = — -Jsf e x h * - * z d S l or K^KZ = JeXh* • azdS (4-4) he By proper normalization of the functions e and h, the product K , K | c a P g made equal to unity. Although (4.4) provides one relationship D e t w e e I L h : B and K2, a second relationship is required before they are determined. * second relationship can be chosen in a variety of ways. For example- CIRCUIT THEORY TOR WAVIXIUSOING SYSTEMS 223 voltage and current waves given by (4.3) may be thought of as existing on a fictitious transmission line that is equivalent Lo the waveguide. As such, it may be desirable to choose the characteristic impedance of this transmission line equal to unity, in which case V+ V~ K, As an alternative, it might be desirable to choose the characteristic impedance equal to the wave impedance, in which case Z #1 . = 1T =Z, (4-6) Other possibilities are obvious and equally valid. In this text either the definition (4.5) or (4.6) is used. The one that is used will be stated, or else it will be clear from the discussion which definition is being utilized. When the equivalent voltages and currents are chosen so that the equivalent transmission-line characteristic impedance is unity, we shall refer to them as normalized voltages and currents. Note that even though equivalent transmission lines may be used to represent a waveguide, the propagation constant of this line must be taken as that for the waveguide. A waveguide supporting N propagating modes may now be formally represented as N fictitious transmission lines supporting equivalent voltage and current waves (from property 2 listed above for waveguide modes). Thus we have N V= £ (V;e-^> + V~e^-') (4.1a) a i '- t (Ke-^-I-e^) n=l - E n (V:Yne-^ - V-Yne'^) (4.1b) 1 where the Y n are arbitrarily chosen characteristic admittances for the equivalent transmission lines. When an obstacle is inserted into a waveguide supporting N modes of propagation, these modes are in general coupled together by the obstacle. This coupling can be described in terms of an equivalent circuit made up of impedance elements. This impedance description of obstacles in waveguides is developed in the next section. Once the equivalent voltage and current amplitudes have been determined, the wave- 224 FOUNDATIONS FOB MICROWAVE ENGINEERING guide fields are known from the relations n=l N n = l and the specified proportionality constants K]n, K2n for each mode. The axial field components may be found from (4.8) by the use of Maxwell's equations. Note that the equivalent current wave amplitude for propagation in the ~z direction is expressed by -I~, and hence the corresponding transverse magnetic field is proportional to -K:ir}I~. When a waveguide supports several modes of propagation simultaneously at the same frequency, the number of electrical ports will exceed the number of physical ports. That is, power can be fed to a given load by means of any of the propagating modes, and all these modes may be common to a single physical waveguide input port.f 4.2 IMPEDANCE DESCRIPTION OF WAVEGUIDE ELEMENTS AND CIRCUITS One-Port Circuits A one-port circuit (equivalent to a two-terminal network) is a circuit for which power can enter or leave through a single waveguide or transmission line. A short-circuited transmission line and a short-circuited waveguide containing a metallic post as illustrated in Fig. 4.1 are examples of one-port circuits. For one-port devices of the above type, a knowledge of any two of the four quantities V*, V~, V = V* + V~, I = 7 + - 7~ will serve to describe the effect of the one-port device on an incident wave (it is assumed that the waveguide supports only one propagating mode). These quantities must, ox course, be referred to a terminal plane such as t in Fig. 4.1 in order to be unambiguously specified. A terminal plane, or reference plane, is the equivalent of a terminal pair in a low-frequency network. In the present instance an impedance description is desired. If the total voltage and current at the t T h e microwave equivalent-circuit theory presented in this chapter may be extended to iuclu nonpropagating modes. However, when nonpropagating modes are included, the imped 8 0 and scattering matrices do not have the same properties as when only propagating modes 81* present at the terminal planes. See H. Haskal, Matrix Description of Waveguide Discontinuity in the Presence of Evanescent Modes, IEEE Trans., vol. MTT-12, pp. 184-188, March, 19 6 4 ' CIRCUIT THEORY FOB WAVEGU1DING SYSTEMS 225 f% t /=0 (a) F I G U R E 4.1 One-port circuits, ( a ) Short-circuited coaxial line; (b) short-circuited waveguide with post. terminal plane are y = v++ V- i~r-r=Yc(v+-v~) where Y c is the equivalent characteristic admittance (actual characteristic admittance for the transmission line), the input impedance is given by 7 m = V _ = 1 V+ v- 7 V* - V~ c (4.9) The complex Poynting vector may be used to establish the physical nature of one-port impedance functions. From (2.59) and (2.60) we have - 0 E x H * - n d S = P, + 2jw(Wm - Wt) (4.10) where n is a unit inward normal to the closed surface S, P t is the power dissipated in the volume bounded by S, and W m - W r is the net reactive energy stored within S. For the surface S choose the terminal plane, the guide walls, and the short-circuiting plane. For perfectly conducting walls and short circuit, n X E = 0; so the integral reduces to that over the terminal plane only. Thus - 0 E x H* • a, dS = P, + 2j<o(Wm - We) (4.11) Now at the terminal plane t h e transverse fields are [see (4.8)] E, - KJ1(V~+ V " ) e = K^Ve H, - / ^ ( / - - / - J h ^ Jq7h (4.12a) (4.12b) Hence (4.11) becomes -(K^Kty^Vl*je x h* • a2dS = i v 7 * = P, + 2j»(Wm - We) (4.13) 226 FOUNDATIONS FOR MICROWAVE ENGINEERING If now V is replaced by IZ^, we find that P, + 2jw(W„ - W.) Z;„ = {IP = R +JX (4. l 4 ) This relates the input impedance to the power loss and net reactive ene stored in the volume beyond the terminal plane. Inasmuch as the current may be an equivalent current, the corresponding impedance Z j s equivalent one also. Since P,, Wm, and W e are all proportional to |/"| 2 _ hence also proportional to |7| 2 in view of the linearity of the field equations the equivalent resistance R and reactance X in (4.14) are independent of the amplitude of the incident wave. By replacing 7* by Y*V* in (4.13), we obtain, after taking the complex conjugate, (4.15) for the input admittance of the one-port device. The susceptance B is positive (capacitive in nature) only if We > Wm. The evaluation of an input impedance by means of the general definition (4.14) will be carried out for the simplest case, that of a short-circuited coaxial line. In the short-circuited coaxial line of Fig. 4.1, the fields in the one-port device are given by V+ E = H = a. (e - > * o * ln(6/a) r \n{b/a) _ k PJ u' ) v r ) since the electric field must vanish at the short-circuited position z - O.u the terminal plane is located at z = -I, then 4 ^0 J a 27re 0 |V + | 2 ln(6/a) J -l - / J sin 2 k0zdz -l TT60|V-|2 l- sin 2k0l 2kn Similarly, it is found that ln(6/a) 1+ Wm = ln(6/a) sin2A: 0 / 2A„ CIRCUIT THEORY FOR WAVEGUIDINO SYSTEMS 227 The total current at the terminal plane at z = - / is / = YcV+(ejk°' + *-**»<) = 2YCV~ cos k0l Using (4.14) now gives 4ja>ire0\V*\2 Zm = ln(6/a) = j(OTre0 sm2k0l 2 A0(4Yr |V+|2)cos2V sin2/;0/ i5wiSwv' A l m W (416) on using the relations Yc = 2vY0/[]n{b/a)], sm2k0l = 2 sin kj cos k0l, and k 0 = w ( / i 0 e 0 ) ! / 2 . This result for the input impedance of a short-circuited coaxial line, as obtained from the general definition (4.14), agrees with the simple computation based directly on expressions for the total voltage and current at the terminal plane. However, the purpose of introducing (4.14) was not as a computational tool, but rather for the physical insight it provides into the nature of the impedance function for a one-port circuit. The second example of a one-port circuit as illustrated in Fig. 4.1 cannot be evaluated in as straightforward a manner as for the coaxial line because it does not consist of a uniform unperturbed waveguide. The presence of a conducting post within the termination results in induced currents on the post that will excite a multitude of waveguide modes. However, since it is assumed that only one mode propagates (the T E I 0 mode), all the other modes decay exponentially in both directions away from the post. By choosing the terminal plane sufficiently far away from the post, the fields at this plane are essentially just those of the incident and reflected dominant modes. The evanescent modes excited by the post will store reactive energy, and this will contribute to the input reactance as viewed from the terminal plane, as reference to (4.14) shows. The presence of the post within the termination modifies the input reactance by changing the amplitude of the reflected dominant wave in just the right amount to account for the additional reactive energy stored. As seen from the preceding discussion, it is important when dealing with waveguide structures to choose terminal planes sufficiently far away from obstacles that excite evanescent modes, so that only dominant-mode fields have significant amplitudes at these reference planes. This will ensure that all the reactive energy associated with the nonpropagating modes that make up the fringing field around the obstacle is taken into account in the expression for the input reactance. This precaution is particularly important in any experimental setup used to measure the impedance function for a particular obstacle. Once the impedance has been properly determined at a given terminal plane, it may be referred to any other terminal plane by 228 FOUNDATIONS FOR MICROWAVE ENGINEERING using the impedance-transformation formula Z(l)=Z 2 ' ZZ( (/ ,/ )' ) + ^+jZ t aetan nff(/2-/1) +jZ{ll)tanp{l ' ZZ,. + iZ(L)tanB(lo - L)2-ll) e (4. where lx is the location of terminal plane 1 and /., specifies the locatin the new terminal plane. In particular, shift in the terminal-plane positm*1)! a multiple of A g /2 leaves the impedance invariant. Thus an impedance be referred to terminal planes located in the near vicinity of an obst ? where now it is understood that this impedance describes the effect of th* obstacle on the dominant mode only, and does not imply that the total fi e t' at this particular terminal plane is that of the dominant mode only. In oth words, the impedance description of a waveguide element or obstacle give, information on the effect this element has on the dominant propagating mode but does not give any information on the detailed field structure near the obstacle. Fortunately, the latter information is rarely required. Lossless One-Port Termination If there are no losses present in a one-port circuit, the input impedance is a pure reactance given by jX^J^f31 (,18) The assumption of a lossless structure is often a very good approximation for microwave circuits. If Wm = We, the input reactance vanishes and a condition of resonance exists. There are actually two possibilities, namely, Wm = We b u t I * 0, and W„, = Wc but V * 0. The first corresponds to a zero in the input reactance (series resonance), whereas the second corresponds to a zero in the input susceptance (parallel resonance) as given by _ *MW,-Wm) 19) vv* When the input reactance is zero, the input susceptance must be infinite, which implies that V = 0 at the terminal plane. This latter condition u possible since, for a pure reactive termination, all the incident power reflected, so that the total voltage along the waveguide is a standing w a v e , the form sin /3/. In the case of a zero for the susceptance function #. reactance X must be infinite (have a pole) and / must vanish at terminal plane. It may be anticipated, then, that the reactance and sus tance functions will have a number of zeros and poles, i.e., frequencies^ which they vanish or become infinite. This behavior is clearly evident in expression for the reactance of a short-circuited coaxial line, which is jX = jZc tan k0l =jZc t a n — oil c , A on) (*•' CIRCUIT THEORY FOR WAVEGUIOING SYSTEMS 229 F I G U R E 4.2 Input reactance of a short-circuited coaxial line. A plot of X against frequency is given in Fig. 4.2. In particular, note that the slope of the reactance is always positive; that is, dX/dcu > 0. This positive-slope condition means that the poles and zeros of X must alternate as to is increased from zero to infinity. We shall show below that this is a general property of any reactive one-port circuit, a result known as Foster's reactance theorem. First, however, it will be instructive to rewrite (4.20) by using the infinite-product representation and also the partial-fraction expansion of the tangent function^ 0)1 « L X = 2(. tan k0l = n n=0 = Z„ 2wc ' L B-i.3.... 1 n= 1 ,„/ T17TC (ay 1 - (* + i)Ver (4.21) (nirc/2iy The first form contains the product of an infinite number of factors in both the numerator and denominator and clearly exhibits both the zeros and poles and their alternating occurrence. The second form exhibits the poles very clearly, but information on the zero locations is lost. In the vicinity of a pole, say that at w = to,, = nvc/2l, all terms in the partial-fraction expan- tE. A. Guillemin. " T h e Mathematics of Circuit Analysis," chap. 6, John Wiley & Sons. Inc.. New York, 1949. E. T. Copson, "Theory of Functions of a Complex Variable," Oxford University Press. Fair Lawn. N.J., 1935. J. Pierpont, "Functions of a Complex Variable," Dover Publications, Inc., New York. 195S. 230 FOUNDATIONS FOR MICROWAVE ENGINEERING sion are small except the nth term, so that cZ„ 2wc Z.. I (co„ - u>)(ion + a>) X « (4.22) l(w„ - w) since w ~ u>„. This behavior near a pole is similar to that for a s simple iC parallel network for which — coL X = l LC- (4.23, 1 where w0 = (LC)'1/2. However, the microwave network is a good deal m^t complicated, for it has an infinite number of poles and zeros, and not just" double zero and a single pole as a simple parallel LC circuit has [the zero occur at w = 0, where wL vanishes, and at infinity, where (wC) ' vanishes! These similarities and differences are important to note since they are characteristic of microwave networks in general, even though we have demonstrated some of these properties for a short-circuited coaxial line onlv. *4.3 F O S T E R S REACTANCE THEOREM The theorem that will now be proved is that the rate of change of the reactance X and susceptance B with o> is positive. Once this result is established, it follows that the poles and zeros of a reactance function must alternate in position along the o> axis. Figure 4.3 illustrates a general one-port reactive termination. The field within the termination satisfies Maxwell's equations V X E = -jujiH V X H =jaieE The derivative with respect to w of the complex conjugate of these equations gives dE* 9a€ rlH* 3H* dwfi r»E« -jwe yE*— VX +;H V X = ja>ix~ tS ti> diss Consider next the quantity Ex (9H* dE* do) Ho) XH = V XE + VX do) r?E* do) <9H* - E - V X —— 00) dE* H do) v", -r F I G U R E 4.3 A one-port reactive termination, ., VXH CIRCUIT THEORY FOR WAVEGUIDING SYSTEMS 231 Substituting from above gives ( <?H* V - E X —+ \ dto <?E* \ I due \ a <?ft>M = j H-H*-— + E-E* — I \ "to ''to J I SB* dH* dE* jeo\fiH dto• ^H • —— + eE • dto dot XH dto + 3>E* eE dtx) The second term on the right-hand side vanishes; so we have dH* EX dE* \ / dun Otoe \ + — X H =j H • H * ^ — + E • E * dto dto J \ dto do) j If we integrate throughout the volume of the termination and use the divergence theorem on the left-hand side, we obtain , / dH* dE* \ (6 E X + X H \-dS= ?S\ do) do) I Own dtoe \ e ( H H * + E - E * — \ dV J v\ dto doj ) = -4j(Wm + We) (4.24a) -jf where W„, + W c is the total time-average energy stored in the lossless termination, as reference to (2.53) in Sec. 2.5 shows, and dS is chosen directed into the volume. Since both n X E and n X dE/dto, where n is a unit inward normal, vanish on the perfectly conducting waveguide walls, the surface integral reduces to an integral over the terminal plane t only. On the terminal plane we have ./ dU* dE* \ dl* dV* HEX + xH - n d S = V + —~1 -' ( \ dto dto j du> (4.246) dto where V and / are the equivalent terminal voltage and current. Now V = jIX for a lossless reactive termination; so dV* dl* = -jX dl* Thus V d<i) jl* dto dto dV* dl* + / =jXI jXIdo) dtti dX — dto dl* dto dX jB*-rout and hence we find that (4.246) yields dX d(o ax MW„ + wf) to JF (425) The right-hand side is proportional to the total energy stored in the termination and can never be negative. Consequently, the slope of the reactance function must always be positive. If / is replaced by jBV in 232 FOUNDATIONS FOR MICROWAVE ENGINEERING (4.246), it is readily found that dB 4(Wm+We) dco ' W* (4.26) and hence the susceptance is also an increasing function of frequency m. above relations also show that the frequency sensitivity of the reactance susceptance is proportional to the total average energy stored. These rel^ tions are readily verified in the case of simple LC reactive networks and problem calling for this verification is given at the end of this chapter. *4.4 EVEN AND ODD P R O P E R T I E S OF Z, Before terminating the discussion dealing with one-port impedance func. tions, one further general property should be pointed out. This property jg that the real part of Zm = R + jX is an even function of w, whereas the imaginary part is an odd function. The physical necessity of this was pointed out in the previous chapter in the section dealing with group velocity. The property stems from the requirement t h a t the response of a circuit to a real-time-dependent driving function must also be real. That is, if W(t) is the applied voltage at the terminal plane, the frequency spectrum is given by the Fourier transform: V(a>) = f e~>"r(0 dt (4.27) The frequency spectrum of the current that flows is «-> -m - ^ The current as a function of time is v 2irJ-*R((o) + jX(u>) and must be a real function. This will be the case if V(-a>) R(-w)+jX(-<») V*(<o) ~ [«(*») +jX(a>)]* for then (4.29) becomes J_ ^ ~ f0 J_i ,= V{io)eJ"' 2^LXR(OJ) +jX(w) ^ + V(o>)eja" 2vJ 2irJ0 0 R(o>) + jX{u>) which we can show is a real function. In the first integral on the rig1 CIRCUIT THEORY KQR WAVEGUIDINU SYSTEMS 233 replace w by ~u> to obtain 6~ JQ V(»)e JU,( —Jiul V(-oj)e R( - a , ) + ;X( -at) -f R(u>) +jX(w) dot (4.30) The two terms in the integrand are complex conjugates of each other, and hence the sum is real; i.e., V(a)ejat S(t) = —Re/"' * aim) +M«Jdl* (4 31) ' The condition specified on V/{R + jX) is satisfied by V alone and by R + jX alone. Clearly, from (4.27), V(~a) - V * U ) . If R(-to) then + jX(-co) S(-»)-£(«) = [R(<o) +jX(<o)}* X(-w) => -X(o>) and R is an even function of w, whereas X is an odd function of to, as was to be proved. These even and odd properties are useful to know when approximate expressions for impedance functions are constructed from experimental data. For example, a series such as a-Aco' + a5u> + • a,to could be used to represent X, but not to represent R, since the series is an odd function of o». AT-PORT C I R C U I T S Figure 4.4 illustrates the junction of JV waveguides or transmission lines (or a combination of the two) that terminate in a common region or junction. The region between the N chosen terminal planes may contain any arbitrary collection of passive elements. If each guide can support only one mode of propagation, this circuit constitutes an iV-port microwave circuit. If one or more of the guides can support several independent modes of propaga- •v.-vOV/, v F I G U R E 4.4 An A/-port microwave circuit. 234 FOUNDATIONS FOR MICROWAVE ENGINEERING tion, the number of electrical ports exceeds the number of mechanical Each mode, since it carries power independently of all other modes P°*t*sponds to an electrical port through which power may enter or lea **** junction. To simplify the discussion, it will be assumed that each ' e supports only a single propagating mode. The extension of the theory t case when some or all of the guides may support several propagating J* is more or less obvious. ' ,:' Let the terminal planes be chosen sufficiently far from the juncti that the fields on the terminal planes are essentially just those of tl" incident and reflected dominant propagating modes. These fields may J uniquely defined at the terminal planes in terms of suitably defined equiv lent voltages and currents. Clearly, the amplitudes of all the incident wav<* may be arbitrarily specified, i.e., chosen independently. The amplitudes of all the reflected waves are then determined by the physical properties of the junction; i.e.. all the V~ are linear functions of the V,,'. When the Vn* and V" are known, the corresponding currents I*, I„ are known from the relations 7-= y y - *nrn 'n * n * r, ' n Since Maxwell's equations are linear and the junction is assumed to be linear in its behavior, any N linearly independent combinations of the 4iV quantities V,*, V„~> J„, and In may be chosen as the independent variables to describe the electrical behavior of the junction. For an impedance description the total currents /„ = /*— J~ at the terminal planes are chosen as independent variables. The N total terminal-plane voltages V„ = V*+ V„~ are then the dependent variables, and are linearly related to the currents as follows: Zn V, V, N Z» J N1 '22 zm %2S J NN h h (4.32) 1N The matrix of elements Z, y is the impedance matrix and provides a complete description of the electrical properties of the N-port circuit. Some 0 the properties of this impedance matrix are discussed below. If the junction contains a nonreciprocal medium such as a plasm (ionized gas) or a ferrite with an applied dc magnetic biasing field (fern are discussed in Chap. 6), then, in general. Z , ; * Z ; 1 ; that is, t h e impedance matrix [Z] is not symmetrical. The junction then requires 2N2 parame to describe it completely since each Z{j is complex and has two indepen terms. If the junction does not contain any nonreciprocal media, Z,j ** /j" and the impedance matrix is symmetrical. In this instance a total 01 ° • 2N2 - (N2 - N) = N{N + 1) independent parameters are required to scribe the junction since N 2 - N of the parameters are equal. Finally. » l . junction is lossless—and many microwave junctions may be approxiifl3 as such with negligible error—then all the Z,j must be pure imag" 1 • CIRCUIT THEORY FOR WAVF.GUIDING SYSTEMS 235 since there can be no power loss within the junction. In this case there are only ^N(N + 1) independent parameters required for a complete description. Any network containing the required number of resistive and reactive elements may be used as a representation for the junction at a given frequency. However, it must be kept in mind that when the frequency is changed, the values of the network elements (resistance, capacitance, and inductance) must also be changed. Rarely would any one particular network representation provide a complete description of the junction over a band of frequencies unless the network parameters are changed in value when the frequency is changed. The foregoing discussion applies also to the admittance matrix [Y], which relates the currents to the total voltages at the terminal planes; i.e., h Ml '.VI M2 M N v, Y,2 •2N V, y,N2 y wNN ,v (4.33) The impedance and admittance matrices are reciprocals of each other; so [Y] = [Z] ' (4.34) of of Symmetry for the Impedance Matrix The symmetry of the impedance matrix is readily proved when the junction contains media characterized by scalar parameters u and e. Let incidentwave amplitudes V^ be so chosen t h a t the total voltage V„ equals zero at all terminal planes except the i th plane. Let the corresponding field solution be E j , H ( . Similarly, let a second solution E J ( H correspond to the case when incident-wave amplitudes are chosen so that all V„ equal zero except V y The Lorentz reciprocity theorem [Eq. (2.135)] gives j>(E, x Hj - Ej x H , ) • n dS = 0 when there are no sources within the closed surface S. Let S consist of the conducting walls bounding the junction and the N terminal planes. The integral over the walls vanishes if they are perfectly conducting or if they exhibit a surface impedance Z m (Sec. 2.12). Therefore we obtain an integral over the terminal planes only, i.e., N £ j (E, X H, - Ej X H , ) • n dS = 0 n - 1 (4.35) 'n However, for the particular solutions considered here, n X E, and n X E ; , that is, Eti,E,j are zero on all terminal planes except t f and t,, respectively, since all Vn except V^ and V, have been chosen equal to zero. Thus (4.35) 236 FOUNDATIONS FOR MICROWAVE ENGINEERING becomes /"E, X H , • n dS = [E. X H, • n dS J J '. 'J where (I,-), is the current at the terminal plane i arising from an annl' voltage at plane ./', and similarly for (/,),. From the admittance descrinf (4.33) of the junction, we have £ « ( / , ) , - W forK = e,»#i Hence (4.36) gives VVY = VVY Y or = Y (4.37) which proves the symmetry of the admittance matrix. Since the reciprocal of a symmetrical matrix is a symmetrical matrix also, it follows that the impedance matrix is also symmetrical. The symmetry of the impedance and admittance matrices is a consequence of reciprocity. For nonreciprocal media, /x or e (or both) are nonsymmetrical matrices, and (2.135) no longer applies. In this case the impedance matrix is no longer symmetrical. Nonreciprocal microwave devices are discussed in Chap. 6; so no further comments on these are made in this section. *Proof of Imaginary Nature of [ Z] for a Lossless J u n c t i o n For a lossless junction all the elements in the impedance and admittance matrices are pure imaginary. Let [V]] and [/] be column matrices representing the terminal voltages and currents, respectively. The transposed matrices [V ],,[/], are row matrices of the form IN] The total complex power into the junction is 1 £ £ i:znmim = P, + 2>( wm - we) n *» 1 m — 1 1 aginaryor a lossless junction P, =* 0 and the double sum must be pure imag" * -^ For ince the /„ can be chosen as independent variables, they may all be chose Since CIRCUIT THEORY FOR WAVEGUIDING SYSTEMS 237 as zero except for the n t h one. In this case Re(/ n *Z n „/„) = 0 or ReZ„„ = 0 If all but /„ and l m are chosen equal to zero, we obtain •&z[{i*im + 1 * 4 ) 2 , . + ini*znn + imi%zmm\ = o But I„ I*, / „ , / * , and I*Im + I J* are all real quantities and Znn, Zmm are imaginary; so this equation can hold only if ReZ„„, = 0 Therefore all Znm are pure imaginary for a lossless junction. alized Impedance and A d m i t t a n c e Matrices Let us assume that we have chosen equivalent voltages and currents Vn, /„ so that the n th equivalent transmission line has a characteristic impedance Zn given by V+ V n ~ ~F ~ T For an N-port circuit the impedance description is given by (4.32). The impedance-matrix elements are Znm. We now wish to redefine the equivalent voltages and currents so that each transmission line has unity characteristic impedance and to find the new impedance-matrix elements which will be designated by Znm and called normalized elements. Let the new voltage and current amplitudes be V,f, V„~, /*, and 7„. In order to have the same power flow, we require •lvn l n 2vn1n We can express the power flow in the following two equivalent ways also: since for the normalized voltage and current amplitudes V n */7*= 1. From the last two relations for power flow, we see that the normalized voltage and current amplitudes are given by i:=yfz;i: From these relations it readily follows that K = fcvn /„ = {z~j„ and V„ = JZ~nVn In = JTJn FOUNDATIONS FOR MICROWAVE ENGINEERING If we substitute from these expressions into (4.32), we obtain {Yy \V1 % 0 0 0 0 — % 0 0 IT, o . x V, )/YN •• Vf "J Q ^11 0 z021 "2.V '.Yl 'AW 2,2 'W J fY~i 0 0 0 fc 0 4 ffi Jl>J I <Y,Y2ZX2 YxZn 21 Y%Z22 }/YtYNZ1N y/Y^Z, }/Y^Y~1Zl .VI 2N Yf,-ZNN '.v From this expression, we see that the elements of the normalized impedance matrix are given by nm ~ y[Z~ZB (4.38a) A similar analysis shows that the elements of the normalized admittance matrix are given by Y 4.6 (4.386) TWO-PORT JUNCTIONS At this point it seems advisable to examine the special case of the two-P0*1 junction rather than continue with the general theory of /V-port circuit* The derivation of some further properties of the TV-port junction is calle° for in some of the problems at the end of this chapter. Three examples ° f CIRCUIT THEORY FOR WAVEGUIDING SYSTEMS 239 two-port junctions are shown in Fig. 4.5. The first is the junction of two rectangular guides of unequal height (called an .E-plane step since the E vector of the T E I 0 mode lies in the plane containing the step geometry). The second is a symmetrical junction consisting of two similar rectangular guides joined by an intermediate guide of greater width. The third two-port junction consists of a typical coaxial-line-waveguide junction, where the center conductor of the coaxial line extends into the rectangular guide to provide an antenna radiating energy into or coupling energy out of the rectangular guide. A discussion of these particular junctions will serve to develop the general impedance description of two-port junctions or circuits. Since evanescent modes are excited at each discontinuity, the terminal planes are chosen far enough away so that these decaying waves have negligible amplitudes at the terminal planes. Equivalent voltages and currents are introduced proportional to the total transverse electric and magnetic fields, respectively, at each terminal plane. For example, for the junction in Fig. 4.5a, let the incident and reflected transverse fields of the TE 1 0 mode at each terminal plane be (coordinates x,y, z refer to the left-hand-side guide, and the primed coordinates x',y', z' refer to the righthand-side guide, as in Fig. 4.6) E, = (CjV*"' + C f e - ^ ' J a , s i n — at t t H, = -Yw(C{ejl3'> - C f e - ^ ' ^ a , s i n — at ( % E, = (C, V 3 ' 2 + Cje-Jli'')ay sin at t 2 H, = YjCieP"* - C 2 e ~ ^ ) a x sin at t 2 These expressions are obtained by choosing hs = (JTrYw/lia)cos(Trx/a) in order to simplify the expressions for the transverse fields. Let the equivalent voltages and currents be chosen as V,+ = Kfif eJ,sl' V{ = V2' = K2C;eJli'* V2~ = K2C2 e **** I2 = YwK2C;e*1* Kfiie ~m' /, = YwK2C2e ~** In order to simplify the notation, we have expressed K n as K x imd K2l as K 2 and eliminated the second constants Kl2 and K22 by introducing the wave admittance using (4.6). Thus the characteristic impedance of the equivalent transmission line is equal to the wave impedance Z w = Y~J of the TE 1 0 mode. To conserve power, it is necessary to choose K x and K 2 240 FOUNDATIONS FOR MICROWAVE ENGINEERING I- A h — — f c i _ i _i 4— « 5, ** j t * i « fa) T *- M "1 r f^2~ L (4) ki it* F I G U R E 4.5 Examples of two-port junctions. [el 0'" '* F I G U R E 4.6 Coordinates used to describe th junction in Fig. 4.5a. SO that vx Vtm)* = YwK?\Ctf = r j C j f f Pr*i_.* s i n 2 — dxdy J o Jo a or K, = y[abj2 and * ? ( # ) " = r ^ l l C ^ I 2 = y j C 2 + | 2 / " ° / " 6 W — dx'dy' J or K2 = jab2/2 . o Jo a CIRCUIT THEORY FOR WAVEGUID1NG SYSTEMS 241 FIGURE 4.7 Equivalent circuits for a lossless two-port circuit. If we use the above equivalent voltages and currents, we have fvll V 2. = 2j i Z12 V Z\-i ^22 h as a suitable description of the E-plane step. An equivalent circuit consisting of a T network joining two transmission lines as in Fig. 4.7a provides a convenient equivalent circuit for the junction. Other equivalent circuits are also possible. In particular, if the junction is lossless, any circuit consisting of three independent parameters may be used. Figure 4.76 illustrates a circuit consisting of two lengths of transmission lines of length d 1 and d 2 connected by an ideal transformer of turns ratio ml. Figure 4.7c is a variation of this circuit, where the transmission lines are replaced by reactive elements jXy and jX2. The parameters of any one of these circuits may be expressed in terms of the parameters of any of the others. The required derivations may be carried out in the usual manner. The foregoing discussion applies equally well to the other two-port junctions of Fig. 4.5 provided suitably defined voltages and currents are introduced. Likewise, the equivalent circuits of Fig. 4.7 may be used to describe the behavior of these other junctions. Although general forms for the equivalent networks can be readily specified, the values of the parameters are not so easily found. In some cases the network parameters can be evaluated analytically, whereas in many other cases they must be determined by experimental measurements.! +For typical analytical solutions and the methods employed, see R. E. Collin, " Field Theory of Guided Waves," 2nd ed., IEEE Press, Piscataway, N.J.. 1991. 242 FOUNDATIONS FOR MICROWAVE ENGINEERING -i Z* J* (a) V.a «•"! Z* h {b\ FIGURE 4.8 Equivalent circuits for a symmetrical two-port junction. [el F o r t h e j u n c t i o n of Fig. 4.56, perfect s y m m e t r y exists about the m i d p l a n e , a n d h e n c e t h e equivalent i m p e d a n c e m a t r i x h a s Z u = Z 2 2 . For a lossless j u n c t i o n of this type, t h e equivalent circuit can be any circuit c o n t a i n i n g two i n d e p e n d e n t p a r a m e t e r s in a s y m m e t r i c a l connection. Some typical c i r c u i t s t h a t m a y b e used a r e i l l u s t r a t e d i n Fig. 4 . 8 . T h e foregoing discussion could be r e p h r a s e d so as to apply to the a d m i t t a n c e - m a t r i x r e p r e s e n t a t i o n as well; i.e., h (4.40) 12 Y12 Y.22 V-2 T h e basic e q u i v a l e n t circuit described by (4.40) is t h e II n e t w o r k illustrated in Fig. 4.9. E x a m p l e 4 . 1 , To illustrate the use of equivalent circuits (assuming that tnei parameters are known at each frequency of interest), consider the coaxial line-waveguide junction of Fig. 4.5c. Let a generator of internal impedance e be connected to the coaxial line a distance / from the terminal plane t v L e t output guide be connected to a load that is matched to the guide, i.e., ^ presents an impedance 1 W at the terminal plane t2. The overall circuit is illustrated in Fig. 4.10a. We wish to evaluate the power transmitted to FIGURE 4.9 Equivalent circuit for admittance matrix of* two-port junction. CIRCUIT THEORY FOR WAVEGUIDING SYSTEMS Zn Zu-Az o- 243 \*zt-*t Az *c ie) i\ (A) FIGURE 4.10 Equivalent circuit for generator connected to a coaxial-line-fed waveguide load and the standing-wave ratio on the input line. The equivalent transformed load impedance at the plane t x is found by conventional circuit analysis to be Zf2 Z-l. - Z\\ r, ^22 + (4.41) Zu This impedance is transformed by the length / of coaxial line into an impedance Z'L ZL +jZc tan pi = Z, Z + jZ tan fil c L (4.42) at the generator terminals as in Fig. 4.106. This reduced circuit is easily solved. The current supplied by the generator is (4.43) ZS + Z'L and the power delivered to Z'L is P = i2 |"/*J' 2 Re Z-L (4.44) If the coaxial line and junction have negligible loss, this is also the power that is delivered to the load. To compute the standing-wave ratio, note that the effective impedance terminating the coaxial line at t x is ZL. Thus a reflection coefficient I' given by zL - z c r = zL + z c (4.45a) is produced. This reflection results in a standing-wave ratio S given by i + iri s= i - in (4.456) The reflection coefficient and standing-wave ratio depend only on the effective terminating impedance ZL. 244 FOUNDATIONS FOR MICROWAVE ENGINEERING Maximum power will be delivered to the load if Zg is made equal t complex conjugate of Z\, just as in the case of a low-frequency circuit n J low-frequency network theorems may also be applied. For example, Thev theorem may be applied at the terminal plane t 2 to reduce the circuit t n " S equivalent generator with a new voltage V g and a new internal impedance 7? The new impedance is readily found by transforming Z g to an equjVai * nt impedance Zg + jZ, tan (3/ Zgt = Zc Zc + jZg tan pi (4.46) at the plane /,. When viewed through the junction, this impedance appears as an impedance z; ~z'22" zz^e (4 47 -> at the terminal plane t2The evaluation of the open-circuit voltage at t 2 is not quite as straightforward. However, by using Thevenin's theorem twice in succession, the desired result can be deduced with a minimum of labor. We first construct a Thevenin equivalent circuit at plane /,. The equivalent internal generator impedance to use here is Zge given above. Let the voltage waves produced by the generator when the coaxial line is open-circuited at I, be ¥*«-*• + ?-#*' where z is measured from the generator. At z = / the coaxial line is opencircuited; so the total current must vanish at z = 1. Thus V = V+e~2jl". Hence, at the generator end where z = 0, V(0) = V ' ( l + e - 2 ^ ' ) /(0) = Y c V J ( l - e - 2 ^ ' ) But 7(0) = Ig and Vg = IgZg + V(0); so V(0) = V + ( l + e"*"") - Vg - lgZg =Vg- I(0)Zg = Vli-YiV'Ze(l-e -™>) When we solve for V' we obtain V* = (4.48) (1 + YcZg) + (1 - YcZg)e-W The open-circuit voltage at t 1 is now readily found to be V^ = v+e-Jf" + Ve-"" - 2V*e-m 2V e JP g ' ' " (l + YcZg) +(l-YcZg)e /4.4»). (*•' W The Thevenin equivalent circuit at t l is that illustrated in Fig. 4.H- CIRCUIT THEORY FOR WAVEOUIDING SYSTEMS 245 FIGURE 4.11 Thevenin equivalent circuit at terminal plane /,. FIGURE 4.12 Thevenin equivalent circuit at terminal plane t2- Application of Thevenin's theorem once more in the usual manner leads readily to the circuit of Fig. 4.12, where Z'g is given by (4.47) and V^ is given by zv/v,„. V" (4.50) Zii The reader may readily verify that the power delivered to Z w as computed from the circuit of Fig. 4.12 is the same as that given by (4.44) (assuming no circuit losses), i.e.. 1 P = 2 v i Z'e + Z„ (4.51) and is equal to (4.44). When there are circuit losses, not all the power delivered to the equivalent load impedance Z'L of (4.44) is absorbed in the load Zw. However, (4.51) does give the correct power delivered to the load even if other circuit losses are present. An alternative way of solving the problem is to replace the coaxial line by an equivalent T network also (Prob. 4.9), in which case the circuit is reduced to a conventional lumped-parameter network. e Equivalent Two-Port Circuits! F i g u r e 4.13 i l l u s t r a t e s a n u m b e r of useful e q u i v a l e n t circuits a n d s o m e of t h e i r d u a l s for r e p r e s e n t i n g lossless t w o - p o r t j u n c t i o n s . T h e i m p e d a n c e p a r a m e t e r s Zn, Z22, a n d ZVi a r e given below in t e r m s of t h e n e t w o r k p a r a m e t e r s , a n d vice versa. T h e s a m e e q u a t i o n s apply for t h e a d m i t t a n c e p a r a m e t e r s for t h e d u a l n e t w o r k (replace Z n by V",,, Z„ by Y0, etc.). N o t e t h a t i n t h e d u a l n e t w o r k s , t h e t u r n s r a t i o o f t h e ideal t r a n s f o r m e r s a r e reversed. I n addition, note t h a t Z 0 i n t h e s e circuits i s a n i n d e p e n d e n t tThe material in this section has been reproduced in modified form from C. G. Montgomery, R. H. Dickc, and E. M. Purcell, "Principles of Microwave Circuits." pp. 105-108. McGraw-Hill Book Company, New York, 1948. 246 FOUNDATIONS FOR MICROWAVE ENGINEERING parameter or characteristic impedance and does not equal ( M o / e ) i / 2 (a) Zn = -jZ0 cot 01 Z22=Z-jZ0 cot Z — Z22 pi Zu cos pi = - i i *t8 7 Z (b) 12 = ±jZ0 CSC pi Zu=Z{-j Z 22 Z cot = Z2 Z 0 =./2 1: 1 - Z\% pi Zx = Zu + yfiTz*. 12 -jCOtpl Z2 = Z 2 2 + / l + Zfj, I2 = ± 7 CSC pi sin Pl= ± — z- 1 2 If pi = T T / 2 . choose Zu = Z, = Z-Z2 Zl2 (c) z Z[ = Z,, Z2 z2 = z 22 ~jZ0 Z0 = - j Z 1 2 * 1 \ i = -./Z 0 cot /3/ cos / » = \ZV1Z22 '12 72 Z„ Z22 = - i cot pi Zl2=j—esc n (rf) Zu = Z„ pi = -jZu\ ^12 ZuZ22 'u n = '22 -j cot pi Z22 = Z cot pi cot pl=jZu n = '12 ZX2 = — esc pi (e) Zn =Z1+ n% z = z. 22 z u z ?12 1-Zfc zf2 Zi - Z u - — •^22 Z22 — Z2 Z o = Zoo Z 1 2 = ±nZ2 \ 2 1/2 rc = •"22 CIRCUIT THEORY FOR WAVEGUIDINC, SYSTEMS t > • 247 1 Y * \ Ifll(duoi) O 1 7 1—O A_ ' '—" I °~ 1 Y* ^0=' /o=' - 1 D 0 J o (/>)(duol) (6) I _ n:\ (fMdual! F I G U R E 4.13 Some equivalent circuits for lossless two-port junctions. An equivalent circuit incorporating a length of transmission line is particularly convenient to use since a shift in one (or both) of the terminal planes will reduce the circuit to a very simple form. For example, let the equivalent circuit of Fig. 4.13c be used to represent the junction in Fig. 4.14 between the terminal planes t x and t2. If the terminal plane <, is shifted a distance /' to the left so that pU + I') = w, that is, I + I' = A g /2, then the equivalent circuit has a section of transmission line one-half guide wavelength long. But since impedance is invariant to a transformation through a half-wavelength-long section of transmission line, this being equivalent to a 1:1 turns-ratio transformer, the section may be removed and the equivalent circuit reduces to a single ideal transformer. This new circuit represents the junction between the new terminal planes t\ and t2. 248 FOUNDATIONS FOR MICROWAVE ENGINEERING «:i /o FIGURE 4.14 t7 A junction and its equivaW circuit. SCATTERING-MATRIX FORMULATION The preceding section dealing with the impedance description of microwave circuits is in many respects an abstraction since the voltages, currents, and impedances cannot be measured in a direct manner at microwave frequencies. The quantities must therefore be regarded as secondary, or derived, quantities. The quantities that are directly measurable, by means of a small probe used to sample the relative field strength, are the standing-wave ratio, location of a field minimum position, and power. The first two quantities lead directly to a knowledge of the reflection coefficient. The measurement of power is needed only if the absolute value of the field in the device needs to be known. Another parameter that is directly measurable is the transmission coefficient through a circuit or junction, this being a relative measurement of the amplitude and phase of the transmitted wave as compared with those of the incident wave. In other words, the directly measurable quantities are the amplitudes and phase angles of the waves reflected, or scattered, from a junction relative to the incident-wave amplitudes and phase angles. Again, in view of the linearity of the field equations and most microwave devices, the scattered-wave amplitudes are linearly related to the inciden wave amplitudes. The matrix describing this linear relationship is called t scattering matrix. , Consider the N-port junction of Fig. 4.15. If a wave with an associate* equivalent voltage V,+ is incident on the junction at terminal plane t\> reflected wave S n V, + = V,~ .will be produced in line 1, where Sx\ & reflection coefficient, or scattering coefficient, for line 1, with a wave dent on, line 1. Waves will also be transmitted, or scattered, out of the o junctions and will have amplitudes proportional to V,+. These aropl l t u can be expressed as V~ = SnlVf, n = 2,2,...,N, where Snl is a t r a n s ^ sion coefficient on line n from line 1. When waves are incident in all ' the scattered wave in each fine has contributions arising from all tb« CIRCUIT THEORY FOR WAVEGUIDING SYSTEMS 249 F I G U R E 4.15 An N-port junction illustrating scattered waves. incident waves. Thus, in general, we can write = ^11 S12 ^13 S2l S22 ^23 SNl SN2 &N3 vr &2N S NN (4.52a) v; + (4.526) [v-] = [S][V ] or where [ S ] is called the scattering matrix. When dealing with the scattering-matrix description of a junction, it is convenient to choose all the equivalent voltages (and currents, which, however, do not enter the picture explicitly) so that the power transmitted is given by 5|V„+|2 for all values of n. This corresponds to choosing the equivalent characteristic impedances equal to unity.t The main reason for doing this is to obtain a symmetrical scattering matrix for reciprocal structures. If this normalization is not used, then because of different impedance levels in different lines, the scattering matrix cannot be symmetrical. Note that, with the assumed normalization, V =V*+V~ and I = T- I'= V + V , and hence V + = \{V + I) and V~ = | ( V - / ) . Thus the new variables V + and V are linear combinations of the variables V and / used in the impedance description. For this reason the currents do not enter into the scattering-matrix formulation. If desired, they may be calculated from the relation / = V - V~. y v .. es aJue different from unity would also be suitable, the only requirement being that all "aye the same characteristic impedance, so that power will always be equal to a constant 250 FOUNDATIONS FOR MICROWAVE ENGINEERING When it is necessary to distinguish between normalized and unn ized voltage and current amplitudes, then we use an overbar on the n ^^* ized variables. Throughout this section we are using only norm p variables so the overbar is not included, in order to keep the notaf ^ simple as possible. At any particular frequency and for a given location of the termplanes, the scattering-matrix elements S„„, have definite values. If !!/ frequency is changed, these elements change values also, in a manner readily deduced analytically in general. However, at a fixed frequency *u*1 change in the scattering-matrix elements arising from a shift in the termnal-plane location is readily found. For example, let terminal plane t \, shifted outward an amount /„ corresponding to an electrical phase shift n l tin = PJn> where /3„ is the propagation phase constant for the nth line If the incident-wave voltage is still denoted by V„" at this new terminal plane all the transmission coefficients S,„ „, m *• n. for transmission into line m from line n must be multiplied by e "-'"" to account for the additional path length over which the waves must travel. The reflected wave in line n has traveled a distance 2/ „ more relative to the incident wave at the new terminal plane. Thus the new value of S„„ is e~ 2 " / "S„„. Likewise, waves traveling from line m to line n must travel a distance / „ farther, and thus Snm is changed to e~jBnS„m. These results are readily expressed in the general case by the following transformation of the [S] matrix into the new [S'] matrix: e-ji, Sn .-;«! [S'] = o12 1M '22 &2N .-Jin . -S,vi SJV2 s NN- ,-J»i ,-J»2 (4.53) • • < ,~J°N where 0„ = Pnln is the outward electrical phase shift of the rath ten plane. inal Symmetry of Scattering Matrix For a reciprocal junction the scattering matrix is symmetrical, t 1 t Snm = Smn, provided the equivalent voltages have been chosen so ^ power is given by j\V„"\2 for all modes. The latter condition is eQ u l . jjpes choosing the characteristic impedance of all equivalent transmission ^ used to represent the waveguides equal to unity. If the voltages a1* CIRCUIT THEORY FOR WAVEGUIDINB SYSTEMS 251 hosen in this fashion, [S] will, in general, not be symmetrical. Problem 4 19 gives an example of a nonsymmetrical scattering matrix. The proof of the symmetry property of the scattering matrix is readily obtained by utilizing the known symmetry property of the impedance matrix. Thus the symmetry of the scattering matrix is basically a consequence of reciprocity. For the normalization used in the present section, i = i+- r= v + - v v. = v:+ v- Thus, since [V] = [V+] + [V] = [Z][I] = [ZJV+] - [Z][V~l we have ([Z] + [U])[V-] = ([Z}-[U])[V-] l [V-] = i[Z) + [U)y {[Z] - [U})[V<] (4.54) where [U] is the unit matrix. Comparing this result with (4.526) shows that the scattering matrix is related to the impedance matrix in the following manner: [S] = ([Z] + [ i / ] ) - ' ( [ Z ] - [ [ / ] ) (4.55a, Alternatively, we have [v+] = k[v] + in) = k[z] + WW] and [V-] = | ( [ V ] - [I]) = | ( [ Z ] - [U])[I] and this gives [V-] = ([Z]-[U])([Z] + [U])-1[V+] or [S] = ([Z]-[U])([Z] + [U])-< The transpose of (4.55a) is (4.556) [s], = ([z]-[[/]),([z] + [t/]);' But since the matrices in parentheses are symmetrical, they are equal to their transpose; e.g., az]-[u]), = [z]-[u] Hence [S]t - ([Z] - [V\){[Z\ + [U]} ' and using (4.556) now gives [S)r = [S] (4.56) a result that can hold only if [S] is a symmetrica] matrix. Scatte ringMat ^ for a r a los Lossless Junction ° sless junction the total power leaving the N ports must equal the total incident power. The mathematical statement of this power-c -conserva- 252 FOUNDATIONS FOR MICROWAVE ENGINEERING tion condition is N N n=l n=l Z \v-\2 = Z w;\2 •5?) This condition will impose a number of restrictions on the scatterin parameters such as to reduce the total number of independent par ^^ to \N(N + 1), the same number of independent parameters as ' * impedance matrix for a lossless junction. Replacing V~ by N K- L sniv; i = l the power-conservation condition may be expressed as A? B-l N Z sn,v; = Z w;\ i=i 2 (4.58i The Vn' are all independent incident voltages; so if we choose all V~=Q except V,+, we obtain 2 |T/+|2 Z \snyr?+ l =_ iv, (4.591 n=l N N Lis n ,i 2 = Es n ,s* =i or (4.60) The index i is arbitrary; so (4.60) must hold for all values of i. Equation (4.60) states that for a lossless junction the product of any column of tl scattering matrix with the conjugate of this same column equals unity. In addition to the above constraint on the Snm, a number of ad+d,t",j?7j constraints may be derived. If we choose all V* = 0 except Vs+ and V r , (4 gives N £ isnsv/+s„rv;i2= z z (snsv;+snrv;)(snsv;+snrv;f n=l n=l - w;\2 + IV;I2 Expanding the left-hand side gives W Z \B„av:f + Z \snrv;\2 z+ Z 8»s;xW n = l n=l N + Z S nr S n * s V r + (V s + )* = |V3+|2 + |VrH+ |2 n = l CIRCUIT THEORY FOR WAVEGUID1NG SYSTEMS 253 TTsine (4-59) results in a number of terms canceling, and we are left with N Z [snss:rv;(v;f + sn*ssnrv;(vs+)*] = o ra = l In view of the independent nature of V/ and V*, choose, first of all, V+ = V* We then obtain *s '' W;\2l ( S n s S n * r + S n * s S n r ) = 0 (4.61a) n = l If, instead, we choose V*=jV*, with V; real, we obtain JIV?? I (SnaS*r - S:sSnr) = 0 n (4.616) l Since neither V* nor V* is zero, both (4.61a) and (4.616) can hold only if £ S X = 0 **r (4.62) n = l This equation states that the product of any column of the scattering matrix with the complex conjugate of any other different column is zero. The conditions (4.60) and (4.62) are sufficient to restrict the number of independent parameters in the scattering matrix to | M A T + 1). A matrix with elements that satisfy these conditions is called a unitary matrix. To illuminate this unitary property further, it will be instructive to rederive the above results by means of matrix algebra. The power-conservation condition (4.57) can be expressed as [v-],[v-]* = [v+],[v+r = ([S][V+]),([S][V+])* = [v-],[s]f[snv+]* Upon factoring this equation, we obtain nH,U£7]-[sL[sr)[v + r = o This equation can hold only if [S]t[Sf = [U] °r IBf-lW (4.63a) (4-636) mat 6 ^•v i S n 0 t Z e r ° " T h e r e s u l t ( 4 - 6 3 6 ) i s t h e d e f i m t i o n o f a unitary rix. The conditions (4.60) and (4.62) are obtained by carrying out the tnx multiplication called for in (4.63a). 254 FOUNDATIONS FOR MICROWAVE ENGINEERING *«•* FIGURE 4.16 A two-port junction. 4.8 SCATTERING MATRIX F O R A TWO-PORT JUNCTION Since many common microwave circuits are two-port junctions, the scatter ing-matrix description of these is examined in greater detail. With reference to Fig. 4.16. let the scattering-matrix parameters of the junction be S S 2 1 , S 1 2 , and S22. The incident- and scattered-wave amplitudes are related by [v-] Vf = or = [s][v+] (4.64a) SUV?+ S12V? (4.646) v;=s21v?+s22vi (4.64c) If the output guide is terminated in a matched load, V2 = 0. From (4.646) it is seen that S £ I is the reflection coefficient in the input guide 1, with guide 2 terminated in a matched load. Also, S2l is the transmission coefficient into guide 2 from guide 1. Similar remarks, of course, apply to the parameters S22 and S 1 2 . If guide 2 is terminated in a normalized impedance Z2 at_the terminal plane t2, then V2~ may be regarded as the incident wave on Z2, and V2 >s the wave reflected from Z 2 . The ratio must be equal to the reflection coefficient of the load; hence YL z2-i v- ' z2 = r, (4.65) Substituting into (4.64), we obtain K-suv~=sl2vi=sl2rLv2 -s2lv^s22rLv2-v2 Solving for Vf /V* gives V\ _ c sl2s2lrL (4J661 which shows how the input reflection coefficient in guide 1 is modified the output guide is not terminated in a matched load. CIRCUIT THEORY FOR WAVEGUIDING SYSTEMS 255 POT a reciprocal junction, S i 2 = S 2 1 , and the scattering matrix conns at most, six independent parameters, which are the magnitudes and n a s e angles of Su, Sl2, and S 2 2 . If the junction is lossless, the scattering atrix contains only three parameters, since the S„ „, are related by conditions (4-60) and (4.62), which in the present case become Si.Sf, + Sl2Sf2 = 1 (4.67a) S22S*2 + Sl2St2 = 1 (4.676) SnS*2 + Sl2S$2 = 0 (4.67c) The first two equations show that IS„| = |S22| (4.68) and hence the reflection coefficients in the input and output guides are equal in magnitude for a lossless junction. In addition, (4.67a) shows that IS12I = Vl - I S n | 2 (4.69) If we let S u = ISule-"'1, S22 = \Su\ejB\ and S 1 2 = (1 - | S n | 2 ) l / 2 e ^ , then (4.67c) gives IS n l(l - ! S I 1 | 2 ) I / 1 V t ~ y ' * + e^->"0 = 0 or, equivalently, eJt" l+»2> = - V* - e + e2 = 20 - n ± 2n — Thus 01 and 04>= " 1 01 + e2 2 "* + -j- + nn (4.70) The two results (4.69) and (4.70) completely specify the transmission coefficient S I 2 in terms of the reflection coefficients S u and S 2 2 . Since S„ and S 2 2 are readily measured and a knowledge of these suffices for the complete description of a lossless junction, the scattering matrix is a particularly convenient way of describing a lossless microwave two-port circuit. The direct evaluation of the scattering-matrix parameters is illustrated by considering two simple examples. In Fig. 4.17a a shunt susceptance jB is connected across a transmission line with characteristic impedance Zc. To hn d Su, we assume the output line to be matched, so that V 2 = 0. The reflection coefficient on the input side is S - yc ~ ^ n " ~ Yc+Yin- Yc - Yc -jB -jB 2Yc+jB " 2Yc+jB rom symmetry considerations it is clear that S 2 2 = Sn. The third parameCan b evaluat ter e d using (4.69) and (4.70) or by finding the transmitage ^2 with the output line matched. For a pure shunt element, we 256 FOUNDATIONS FOR MICROWAVE ENGINEERING SB Zt (a) A h J* *-2i *-*! F I G U R E 4.17 Shunt and series elements on a transmission l i n e \b\ must have V,+ + V f = V2~ = V+(l + S „ ) . Since V^ = S21V+ also, we obtain 2K S 2 ] - 1 + S u - Sv, - 2Ye+jB For the second example we consider a series reactance jX connecting lines with characteristic impedances Zx and Z2, as in Fig. 4.176. In this example the characteristic impedances of the two lines are different; so we must first choose normalized voltages. Let V,+, V{, V.J, V2 be the actual transmission-line voltages for the waves that can exist on the input and output sides. Power flow for a single propagating wave is given by j ^ i y f l 2 and |y 2 |V 2 + | 2 . If we choose normalized voltages Vf = yil/2V1"h and V!j= Y2l/2V2+, then power flow is given by IIVJ'I2 and ±\V£\2 and is directly proportional to the voltage wave amplitude squared. If the output line is matched, we have V{ Vf + V, V* Zm-Zx " u Zto + Zt Z2~Zx+jX Z 2 + Z,+jX With the input line matched, we find % V2 + ?Z „ f* 22 z1-z2+jx Z2 + Z1+jX To find S 2 l , again consider the output line matched. On the input un1 have V, = V?+ Vf =» V~(l + Su) and I> = Y1(V?-Vn = YlV?(l-Sll) The current is continuous through a series element, and hence -/ 2 = 4-=A = W ( i - s u ) CIRCUIT THEORY FOR WAVEGU1DING SYSTEMS 257 But / 2 -= W; so Y2V2 = Yy;a - Su). We now obtain S 2 1 - Sl2 - h 5. 1/2 Y \ 1 / 2 VJ 2 I v Z, 1/2 (i-su) 2 V, 2^z7z 2Z, Zi+Z2+jX Zx + Z2+jX The equality of S I 2 and S 2 1 occurs because of the symmetrical manner in which Z, and Z 2 enter into this expression. If unnormalized voltages were used, the same expressions would be obtained for S u and S 2 2 , but for S 2 1 and Sl2 we would obtain instead S2i - 2Z, Z1 + Z 2 + y X 2Z, S{2 - Z , + Z + j X 2 TRANSMISSION-MATRIX R E P R E S E N T A T I O N When a number of microwave circuits are connected together in cascade, it is more convenient to represent each junction or circuit by a transmission matrix that gives the output quantities in terms of the input quantities. The reason for this is that, with such a representation, the matrix which describes the complete cascade connection may be obtained simply by multiplying the matrices describing each junction together. The independent variables may be chosen as the input voltages V„ and currents / „ , the incident- and reflected-wave amplitudes V„+, V~ on the input side, or any other convenient linearly independent quantities. When voltages and currents are chosen, we shall call the corresponding matrix the voltage-current transmission matrix. If incident- and reflected-wave amplitudes are chosen, we shall refer to the matrix as a wave-amplitude transmission matrix. To simplify the discussion, we shall consider the cascade connection of two-port circuits only. However, the general formulation is readily extended to cover the cascade connection of JV-port circuits. The transmission-matrix formulation is of great value in analyzing nitely l ° n g periodic structures such as those used in slow-wave circuits r traveling-wave tubes and linear accelerators. Since examples of these are •alyzed in Chap. 8, we shall consider only the basic formulation in this section. '"'^e-CW ent Transmission Matrix sure 4.18a illustrates a two-port junction with input total voltage and as th • ^' a n ( ^ o u t P u t quantities V2,12. Since V 2 an<^ h m a v De chosen e independent variables and the junction is linear, the dependent 258 FOUNDATIONS FOR MICROWAVE ENGINEERING a e Kf I* 35 (a) /. Kj a, A 4 ffi, fa =-*- a 2 CB? pj I*) FIGURE 4.18 (a) A two-port junction; (fc), l cade connection of two-noM v l ~~ ' June. variables Vj, I, are linearly related to V2,I2. Consequently, we may write Vx=s?V2+M2 (4.71a) Jj = WV2 + &I2 (4.716, where ^, 33, &, and ^ are suitable constants that characterize the junction. Note that we have chosen the positive direction of current to be to the right at all terminals. This is done so that the output current J 2 becomes the input current to the next junction, etc., in a cascade connection as illustrated in Fig. 4.186. In matrix form (4.71) becomes V, (4.72) <& The relationship of the voltage-current transmission matrix to the impedance matrix is readily found by rewriting the following equations in the form (4.71): Vi = I\ZU - I2Zi2 These equations may be solved to give Z\\/Zl2 [ZUZ22 h 1/Zis12 Zl2)/Zl2 v» (4-1 z^/z VI The s?£S'%Q> parameters of the junction are readily identified in ter the Znm from this relation. The determinant of the voltage-current t' mission matrix is = 1 for a reciprocal junction, as is readily verified from (4.73). of i* ,7*) CIRCUIT THEORY FOR WAVEGUIDING SYSTEMS 259 For a cascade connection as illustrated in Fig. 4.186, we may write \V1\ h. w A. fv2] V *i * i . sis 4 U v3] sr2 ®i V ki •A] and therefore hi [j/ 2 ^ 2 1 V . 1 = = r 2 ^2 \sfxs/2 + &,% J 3. s/i&t + MxBa 1 fv3] (4.75) Thus the input quantities are readily found in terms of the output variables simply by multiplying the transmission matrices together. The ratio of the output voltage to current is determined by the load impedance. Wave-Amplitude T r a n s m i s s i o n M a t r i x The wave-amplitude transmission matrix relates the incident- and reflected-wave amplitudes on the input side of the junction to those on the output side. It bears the same relationship to the scattering matrix as the voltage-current transmission matrix does to the impedance matrix. J u s t as in the case of the voltage-current transmission-matrix representation, it is convenient to choose the variables in such a fashion that the output variables from one junction become the input variables for the next junction. With reference to Fig. 4.19a, we thus choose (4.76a) e T - FT (4.766) c;= v- (4.76c) c2-= v; (4.70a") cs = v3- (4.76e) (4.76/-) he superscript + refers to the amplitude of the wave propagating to the 'gnt, and the superscript - refers to the amplitude of the wave propagatS to the left. The input and output quantities are linearly related; so we m * y write r "12 (4.77) c, are suitable constants that describe the junction. For the "i\ the Anm A22 260 FOUNDATIONS FOR MICROWAVE ENGINEERING ^21 ^22 [4! Au A,2 lb) F I G U R E 4.19 Wave-amplitude transmission-matrix representation of a junction. cascade connection of Fig. 4.196, we have An l 2J A12 A22 Au AI2 A 22 A22 c| (4.78) In terms of the scattering matrix for the single junction, we have vr v *. cf e s u s12 V1+ [ l s 1 2 s 2 2 [v-t\ t. [Sn S12 <••{ "12 s22 c2" (4.79) These equations may be solved for c,+ and c, to give 1/S]2 Sn/S12 _ -S22/S12 c, (4.80) (S]2 - SuS.^j/Sja from which the Anm are readily identified in terms of the S n the determinant of the [A] matrix equals unity, i.e., AnA-22 - A | 2 A 2 1 = 1 Note that (4.8D as is readily verified from (4.80). However, if the wave amplitudes ha been normalized, so that power was given by ^ie,71 ~, etc., the determin the [A] matrix would be different from unity in general (this follows^ ^ of the the nonsymmetry of the scattering matrix in this case). Some problems given at the end of this chapter illustrate these points. *4.10 SIGNAL FLOW GRAPHS ship-' A signal flow graph is a graphical representation of the r e l a t l \ t 0 » between a set of independent input variables that are linearly rela {if set of dependent output variables. For example, the impedance- CIRCUIT THEORY FOR WAVEGUIDING SYSTEMS Zn-Zn 261 Z22 ~ Z-,2 F I G U R E 4.20 ( a ) A generator connected to a load through an impedance network; (b) signal-flow-graph representation of t h e linear system in (a). description of a two-port network V, = Z u 7 , + Z 1 2 / 2 V2 = Z 2 I 7 t + Z 2 2 7 2 may be represented by a signal flow graph by drawing four nodes to represent the variables V v V2, 7 : , and 7 2 and connecting lines having transmission factors that show how the inputs 7, and I 2 feed signals to the output nodes labeled V t and V2. The graph is shown in Fig. 4.20. The node labeled V, has an input Z n 7 , from 7, and an input Z 1 2 7 2 from 72. These inputs are represented by the directed line segments having transmission factors Z n and Z 1 2 as shown in Fig. 4.20. Similar connections go from node V2 to nodes 7j and 7 2 . If the output is connected to a load ZL, we have a further condition, namely, V 2 = = —i - I , Z , which can be represented as a 2 2 L directed line segment from the 7 2 node to the V node and having 2 a transmission factor -ZL. If we connect a generator with voltage Vg and impedance Z g to the input, then one additional equation V l = V g — IxZg is imposed on the system. To represent this equation on the graph, a node labeled V g is added along with a directed line segment from this node to V, and with unity transmission factor. Also an additional line segment from node 7, to node V, with transmission factor -Zg must be added. From this example it should be clear how a signal flow graph is constructed to represent a linear system. Once the signal flow graph has been constructed, the solution giving any one variable such as V,, V2, 7„ or 7 2 in terms of the source variable V g an be determined from the topology of the graph and the application of a *« of formal rules known as Mason's rules.f For complicated graphs the use Mason's rules can be quite intricate and the chances of making a mistake - Mason, Feedback Theory—Some Properties of Signal Flow Graphs, Proc IRE., vol. 41, O r , ' 4 4 - H 5 6 , 1953; also J. Mason, Feedback Theory—Further Properties of Signal-Flow ra Phs, Proc. IRE., vol. 44, pp. 920-926, 1956. 262 FOUNDATIONS FOR MICROWAVE ENGINEERING are quite high. For this reason we will not give Mason's rules but will instead how the graph can be systematically reduced to a simple form gives the desired solution by inspection. The reduction is carried applying five basic straightforward rules which we describe below °* ^ RULE 1. A pair of linear relations such as x2 = C 21 x,, x3 = C. x h graphical representation shown in Fig. 4.21a. When there are no .u inputs to node 2, we have X = 3 ^2V~'Z2X\. Thus rule 1 states that two series paths are equivalent to a single path ftnode 1 to node 3 with a transmission factor equal to the product of th from node 1 to node 2 with that from node 2 to node 3 as shown in Fj 4.21a. RULE 2. If there are two or more parallel paths connecting node 2 to node 1, we have x2 = A2lxl + B2ix1 + C21xl + • • • = ( A 2 1 +B2l +C21 + ••• )*i Rule 2 states the obvious result that several parallel connecting paths are equivalent to a single path with a transmission factor equal to the sum of those of the individual paths. This rule is illustrated in Fig. 4.216. RULE 3. Consider the linear relationships X x 2 = ^ 2 1 xl 3 = ^32*2 + Q>3-r3 which have the graph shown in Fig. 4.21c. With no other inputs to nodes j and 3, we can eliminate x 3 to obtain X 2 = ^21*1 + ^'23^32'I2 *2(1 ~ C23C32) = C21X1 ^21*1 Xo = 1 — ^23^32 C32C22^J UooXo " C23C32 Thus rule 3 states that a feedback loop may be eliminated by d i v i W ^ input transmission factor by 1 minus the transmission factor aro ^ loop which is the product C23C32 by rule 1. If there are several mpu ^ outputs from node 2, each input transmission factor is divided by 1 pg. but all output transmission factors remain unchanged as shown ^j, 4.21c. If node 3 is isolated, then the feedback loop becomes a self-'°° p loop gain C23C32 = u 22- n * o5 "* < x * s? a z I o o x x ci J1 a: o 2 -c — 2 u V •a u a> i! O « OT3 -f C ~ «.2<* g Bjl E3^ . I 264 FOUNDATIONS FOR MICROWAVF. ENGINEERING RULE 4. Let node 2 have a single output and two or more inputs. Th have the relationships — C2lxx + C23x3 we '24*4 *„ * Cntxt = ( C n 2 C 2 l ) x , + (Cn2C23)x3 + ••• This shows that x 2 can be spht into a number of nodes x'2, x2, x"2 etc with separate inputs C21x1 for node x'2, C23x3 for node x2, etc., and'th all feed node n with a transmission factor C„ 2 . This rule is illustrated Fig. 4.21rf. If node 2 has a self-loop, it should be eliminated using mlet before the node is split. RULE 5. Rule 5 is similar to rule 4 except there is only one input but several outputs from node 2. In this case each output can be considered as coming from a single node such as x2, x2, etc., with each of these split nodes having the same input C 21 x,. This rule is shown in Fig. 4.21e. A self-loop at node 2 should be eliminated using rule 3 before the node is spht. The above rules are essentially those described by Kuhn.t We will now illustrate the application of the above rules to solve the circuit problem illustrated in Fig. 4.20. We will choose 72 as the variable to be found in terms of Vg. As a first step we combine the parallel paths between nodes V,, I x and V2,12 using rule 2. The new graph is redrawn in Fig. 4.22a. None of the rules we have given are of any help in reducing this graph any further and it seems as though we have come to an impasse. We can get around the dilemma by writing our load terminal condition in the form I 2 = - V2/ZL, which then provides an input to 7 2 , the variable we are interested in. Hence we undo the use of rule 2 between nodes V 2 and I 2 and redraw the graph as in Fig. 4.226. We now split node 2 using rule 4 ti obtain the graph shown in Fig. 4.22c. This graph contains a self-loop which we eliminate by using rule 3, which requires dividing the input transmission factors by 1/(1 + Z22/ZL) = ZL/(Z22 + Z,). The result is the graph shown in Fig. 4.22e. Note that we also combined the transmission factors from node It to V2 and from Vjj to l2 using rule 1. We again run into a problem in reducing the graph in Fig. 4.22e because there is no input to node r This is because of the way we chose to state the terminal conditions a generator end. Instead of using V, = V^ - Zglv we can use Ij = "gl K^ V1/Zg which shifts the generator input to an input to node I x **" ^e provides an input to I x from V v The new graph is shown in Fig. 4.5* 1^ can reduce this graph by splitting node V, into two nodes and appty1 » 4 again. This leads to the graph shown in Fig. 4.22g which has a se ' ' i l Pt N . Kuhn, Simplified Signal Flow Graph Analysis, Microwave J., vol. 6, no. » ' November. 1963. 59 CIRCUIT THEORY FOR WAVEGUIOING SYSTEMS 265 The self-loop is eliminated by applying rule 3 and results in the graph shown in Fig. 4.22A. This graph has a feedback loop so we apply rule 3 again to obtain h Zg + Z u 1 - (Z11 + Z12Z2i Zg)(Z22 Zr + ZL ) -221Vg (Zn + Zg)(Z22 + Z,J Zl2Z2l Zq + Z za (4.82) This example is simple enough so that we can easily solve for I 2 analytically and verify that (4.82) is the correct answer. We initially chose to express the terminal conditions at the load and generator ends so as to cause difficulty in reducing the signal flow graph in order to illustrate the importance of choosing the correct way to express the terminal conditions. The terminal conditions should be expressed in a form that will result in all nodes in the signal flow graph having both input and output signals. As a second example we will solve the same problem using a scattering-matrix representation of the impedance network; thus we use v 2 -=s 21 vr+s 22 v 2 + At the load end we have V 2 = FLV2~ where the load reflection coefficient is given by VL = (ZL - ZC)/(ZL + Zc) and Zc is the characteristic impedance of the transmission line assumed to be connected between the network and the load impedance. This line is considered to have a negligible length. At the generator end we have the terminal conditions (V.+ - Vf) which can be expressed as where the generator reflection coefficient is given by r s _z*~z< z, + z.. char 8 5 * 1 " a s s u m e t h a t a transmission line of negligible length and with net a ( T r i s t i c i m Pedance Z c is connected between the generator and the ? ork. The signal flow graph for the above system of linear equations is 266 FOUNDATIONS FOR MICROWAVE ENGINEERING Vg 1 Z V, " - Z,2 l2 z„-z Z 9\ •&.' Vg 1 V, Vg 1 V, Z, 2 f <-\- z m v2 /, (a) 1 V, Z,2 /, V, _L Z, 2 /2 'i (e) Zxz k r 1 / / -l2 % ZL+Z22 9 1 zn /, FIGURE 4.22 Illustration of steps used to reduce the signal flow graph in Fig. 4.206 to a simple form- shown in Fig. 4.23a. Note that we have chosen to express the ter conditions in such a form as to provide an input to the node V 2 *? vj output from node V-f so as to avoid the problems encountered earU reducing the graph to a simple form. The procedure that we will iable W parallels that used in the first example. We will choose V2~ as the varia . be solved since this is the amplitude of the wave that is incident on the CIRCUIT THEORY FOR WAVEGUIDINO SYSTEMS Vg Zc + Zg Vt S2, Vi v; s,2 v} v„ v\Ti-s22rL vi vf i - s ^ r ; v 2 V P s12rL 267 Zc {Zc+ZgW- V _ ^ 1-S„r g (e) F I G U R E 4.23 (o) Signal flow graph for the circuit in Fig. 4.20a but using a scattering-matrix formulation; (6)-(e) steps followed in reducing the signal flow graph to a simple form. The first step is to split node V,J using rule 5 to obtain the graph shown in Fig. 4.236. Rule 3 is now applied to eliminate the self-loop and produces the graph in Fig. 4.23c. A similar treatment of node Vf but using rule 4 followed by application of rules 2 and 3 results in the graphs shown in Figs. 4.23d and e. The final graph gives the desired solution upon applying rule 3 once more. Thus we find that (zc vKzcs2l + zg)[(i - snvg)(i - s22\\) - sl2s2lvLrt (4.83) The load voltage is given by VL = V2(l + TL). For simple two-port networks, the use of signal flow graphs does not offer a great advantage over the algebraic method of obtaining a solution. 268 FOUNDATIONS FOR MICROWAVE ENGINEERING However, for three- and four-port networks, signal flow graph techrii provide a useful tool with significant savings in the effort required to uS<*° a desired solution. ^n *4.11 GENERALIZED SCATTERING MATRIX F O R POWER WAVES If we have a load impedance equal to the characteristic impedance of transmission line to which it is connected, then all of the power in incident wave is delivered to the load. The reflected power will be zero Th'' would seem to represent an optimum situation. However, if the generat impedance does not equal the characteristic impedance of the interconner/ ing transmission line, we do not have an impedance match for maximi power transfer from the generator to the load. A partial standing wave oi the transmission line can result in a larger voltage being apphed to the load impedance and a greater amount of power delivered to the load. In general the generator should be terminated in a load impedance equal to the complex conjugate of the generator impedance for maximum power transfer. This conjugate impedance-matching criterion generally means that there will be a partial standing wave on the transmission line. It is possible to introduce new voltage and current variables and corresponding wave amplitudes that will result in the conjugate impedancematching condition being equivalent to having a zero reflection coefficient for the load. The new waves are called power waves. The scattering matrix that describes a microwave network in terms of incident, reflected, and transmitted power waves will be called the generalized scattering matrix. We will use script letters to represent the elements of the generalized scattering matrix and also to represent the voltages and currents associated with the power waves. The theory of the generalized scattering matrix and power waves is developed below. This theory provides a useful extension o the conventional theory for analyzing systems in which both the source an load impedances are complex. The theory can be developed by analogy w that of the conventional theory.! We present the key relations from the conventional theory first and then use these to obtain similar relations the generalized case. For the circuit shown in Fig. 4.24a, let the voltage and current *» amplitudes on the transmission line be V", V" and I*, I~ at the load. current amplitudes are related to the voltage amplitudes as follows: V* V- tK. Kurokawa. Power Waves and t h e Scattering Matrix, IEEE Trans., vol. MTT194-202, March, 1965. CIRCUIT THEORY FOR WAVEGUIDING SYSTEMS 269 i=i*-r zg-zt *t v = v*+v-\ zL vgQ w (a) iOI F I G U R E 4.24 ( a ) A generator connected to a load impedance by means of a transmission line; (6) a circuit with complex source and load impedances; (c) a two-port network. When Zt = Zc there is no reflected wave since TL = 0. Thus V~= 0 and the voltage on the line at the generator end will be V*eJ" and equals Ve/2 when Z= Zc. The incident power is given by IV* I2 I V / (4.84) p. = p = = —£— ,nc ava 2Z„ 8Z,. and is equal to the available power P a v a from the source. If Z, * Z c then the power delivered to the load is given by p,.= ( i - i r j 2 ) p i n c = ( i - i r , i 2 ) p a v a [4.85) The total voltage and current at the load are V = V+ + V" i = r-r= We can express the wave amplitudes V i and V relations V + ZJ 2 V-ZJ in terms of V and / by the (4.86a) (4.866) 270 FOUNDATIONS FOR MICROWAVE ENGINEERING The normalized voltage wave amplitudes are given by V y ^ _ y-= + Z ^ 2y/Z~ (4.87QJ V - ZCI — 2y[Z~e (4.876, Consider now a generator with internal impedance Z connected complex load impedance Z t as shown in Fig. 4.246. By analogy with (do we will choose the power wave amplitudes 7/'" and 2s-*- to be _ V+Z r+ ' ~ = ^~ f!1 5~~ (4-88Q) V - Z*I 2 < 4 - 886 ) where V and J are the actual voltage and current in the circuit. When the load impedance ZL is equal to the complex conjugate Z* of the generator impedance, then v= KZL s -g + zL = V8Z* * * /= zg + z* vg vg zg+zL zg + z* Upon using these expressions in (4.88), we obtain M vg z* - z* vg 2 - 2 Z # + Z;- Thus the definitions chosen for T" and '~Y have the desired property that when the load impedance is conjugate matched to the generatm impedance, for maximum power transfer, the amplitude of the reflect© power wave is zero. The power delivered to the load is given by 1 IV/ % |V/ z +z t s where Zg = Z£ = Rg + jXg. This also represents the available power hot the source and can be expressed in the following way: |r+P _ \¥'*\2 Fava " 2Rg ~ ~2~ j o b& where the normalized voltage wave amplitude has been chosen y+Zy/Rg tor sner?^e When the load impedance is not conjugate matched to the gen" impedance, we will have a reflected power wave with amplitude r\ CIRCUIT THEORY FOR WAVEGUIDINC SYSTEMS 271 ralized j o a ( j reflection coefficient, which we designate as V'L, is given by 1 L V~ V-IZ* V/I-Z* ZL-Z* <y+ V + IZg V/I + Zg ZL + Zg y ' ' When Z is real, we obtain the usual load reflection coefficient. For the nonmatched case the power delivered to the load is given by 1 l P ^ R e W * = -Re + Zg\2 SRg\ZL V R 2 / _; V ** ' ZL = MPava (4.90) where the impedance mismatch factor M = AR, Rg/\Z, + Zg\2. The impedance mismatch factor M is always less than or equal to one. For Z, = Z* we have M = 1. The mismatch factor determines the fraction of the available power that is delivered to the load. By using the definition for TJ and some algebra, we can readily show that M = 1 - \VL\2 (4.91) Consequently, we can write ^ = (l-HII2)Pava (4.92a) which is analogous to (4.85). Upon using V = ZLVg/(Zg + ZL) and / = Vg/(Z + ZL) in (4.88a), we find that ^"*"= V./2; so we can also write <y+ 1 (i - \rL\2) = (I - ir[( 2 )p i n c "••=2 (4.926) where P i n c is the power incident from the power wave launched by the generator. The new set of variables 7/'* and T~, which are linearly related to V and /, form a convenient pair of new variables for analyzing a circuit having complex generator and load impedances. If we have a two-port network as shown in Fig. 4.24c, we can introduce incident and scattered power waves on both the input and output sides. The power wave normalized amplitudes are linearly related to the normal voltages and currents on the input and output sides as follows: _ ^7= y + = V, + IXZX _ 2 ^' ^7= 2 2jR2 - e 2 ' V, - LZ* 2/i? 2 " 2JR2 Z\ and Z 2 , with real parts i?, and R2, are the complex generator 272 FOUNDATIONS FOR MICROWAVE ENGINEERING impedances on the input and output sides. The two-port network will <• two of the above variables to be linearly related to the remaining two i° rCe choose the incident power wave amplitudes as the independent vat- k e then the scattered power wave amplitudes are given in terms of the f • nnps hv the thf crfnoraliypd fir ones by generalized srat.tprincr-maf.rix scattering-matrix rp.nrpspntat.irm representation (4.93 0 ) ^ = •^21^ + ^22^7 (4.936) The generalized scattering-matrix parameters S?-^ cannot be measi «J directly. However, they can be expressed in terms of the normal two-no scattering-matrix parameters S,j. In practice, we would embed the two-no network into a transmission line with characteristic impedance Z. (usual! • 50 fi) and measure the S^. We can linearly relate Vt+, VL~, V2+, V^ to V f and V2, / 2 . The power wave amplitudes are linearly related to the tote! voltages and currents. By setting up these linear relationships and expressing V, and V.J in terms of V? and V.J, we can, after a number of matrix manipulations, show that \s*\ = [D*y \[S] - [ r * ] ) ( [ t / ] - [ H f S ] ) - 1 ^ ] (4.94) where [D] is a diagonal matrix with elements BH - ] i - T?r\l - Tt)Jl - \Tf * = 1,2 Z,-Z. r, = 2 , + Z, and [T] is also a diagonal matrix with elements fM = I",. The unit matrix is[U]. We will now examine the network shown in Fig. 4.25a. The two-port network can be described in terms of the normal scattering-matrix parameters S,j or in terms of the generalized scattering-matrix parameters. In tr network shown in Fig. 4.25a, we do not have a generator on the outpu side. The termination is the load impedance Z,. Also we have labeled the source impedance on the input side as Z s and the source voltage as ^ These changes correspond to setting V,2 = 0, Vgl = Vg, Zl = Zs, Z2 = t , the previous analysis. We can visualize the two-port network as conn to the source and load by transmission lines of negligible length. <Vg2 = 0 t h e output current is J 2 = - V2/ZL. Consequently, + hz = 0 n= v 2JTT7. 2 L so there is no power wave reflected from the load. The power derive the load is CIRCUIT THEORY FOR WAVEGUIDING SYSTEMS 273 I, ^11 '12 •^21 ^22 ^ Transmission lines of negligible length (a) j !/.=-/, ^e v,| F I G U R E 4.25 ( a ) a two-port network terminated in a complex load impedance; (6) Thevenin equivalent circuit. (6) The transducer gain of a two-port network, which could represent a microwave amplifier, is defined by the relation power delivered to load G = (4.96) available power from source Since the available power is simply \T?\2/2, we find that G = \y21\2 (4.97) From (4.94) we obtain the following expressions for the generalized scattering-matrix parameters: •*u- i i-r, wi-r* 5* = i [(«u - C X I - TLS22) + S 12 S 2i r L ] (i-rs)(i- (4.98a) 2\l'/2 s lr |2 1 |r li - rjii - rj ^ ~ » )( ~ ^ ) l So21 (4 986) " (4.98c) •^12 •^22 = ^ j ^ [ ( S 2 2 - rL*)(i - r s s u ) + s12s21r3] (4.98d) where w= (i - r s s u )(i - rLs22) - S12S21TJL °te that for a reciprocal two-port network with Sl2 = S2l t h a t .'Sx2 = y 2 274 FOUNDATIONS FOR MICR0WAVF. ENGINEERING also. We can now express the transducer gain in the form (i-ir/)(i-irj2)is21i2 2 G = i^2Ii = ;i - r.s M )(i - r t s 2 2 ) - s 1 2 s 2 1 r,rj 8 In microwave amplifier design a more useful expression for power gain Gp, which is defined by power delivered to load — amplifier input power to GP » (4.99, • gaj S<m ,s the (4.100) The input power is given by PiD = ( i - l^nl i ) p (4.101) so we obtain W2 G GP l - l ^ n I2 1 (4.102) " I^nl 2 Another gain expression is the available power gain, which is given by maximum available —— available input The maximum power that can be delivered when the load impedance is made equal to amplifier output impedance Zaut. We will delivered to the load is given by Ga = load power (4.103) power to the load is the load power the complex conjugate of the show that the actual power PL = ( i - L * 2 2 ! 2 ) P L . a v a ( 4 - 104) where PL<ava is the available load power. This relationship is of the saffl form as (4. 101) which relates the input power to the power available the source. When we use this expression in (4.103), we get Pin °ava 1 ~ l^Ml2 P l-|^ 2 2 | 2 = °£ G 1-|^22| = 2 W ^ (4.106» 1 - l ^ f liner** The above expressions for the various gains associated with an arn; ^ ^ t r i * all expressed in terms of the parameters of the generalized scatte and are the main reason for introducing the concept of power w a . s0tf& We now return to the derivation of (4.104). We can repla 0 6 . 0f t and two-port network by a Thevenin equivalent network cons CIRCUIT THEORY FOR WAVEGUIDING SYSTEMS 275 ltage generator with open-circuit voltage V T and a series impedance Z T as >nwn m pig. 4.256. The voltage V T is the voltage across the amplifier ' ltout terminals when Z L is removed. The impedance Z T is the impedance PPQ looking into the output terminals with the generator short-circuited. It j 3 equal to the amplifier output impedance Z o u t . From the Thevenin equivalent circuit, it is readily seen that the power delivered to the load is given by Pi. = o I'/JX = WT\2RL 2\ZT + ZL\ The maximum available load power equals P L when we choose Z L = Z£ and is iv r l 2 L. ava 8R T Thus, in general, we can express P L in the form |V T | 2 ARLRT 8R r \ZT + ZL\ 8R. -M where M = ARLRT/\ZT + ZL\2 is the impedance mismatch factor. If we can show that M = 1 - |,y 2 2 | 2 , then we can express G a as G„ = L.ava PL G MP,, M G \-\ST. 22> which is the expression we are trying to estabhsh. When V g = 0 we have ^i + I\Zg = 0; so %?= 0. The scattering-matrix relations now give ^7=^12^2 *Z -^ 12' 2 y2 - hzt IJRTL which gives us -L = ~ ^ ^ ^ ^ f? 22 v 2 - hzt V2 + /2ZL v2/is-zt V 2 /J 2 + Z L The Thevenin impedance Z T is thus found to be Z£ + Sfi22ZL \-s>. : ' • ; . ZT Z£ ZT + ZL 276 FOUNDATIONS FOR MICROWAVE ENGINEERING We now expand the expression 1 — \S"22\2 to obtain \ZT l A2 '- ' - " ZL\2-{ZT-Zt)(Z$-ZL) + \zr + zj* ( Z r + Z L ) ( Z ? + Z£) - ( Z r - Z £ ) ( Z * - z ^ \ZT + Z J ZriZt + iStt+ZftZt + ZZ) \zT + zL\2 4RLRT = M IZr + Zrl'' which completes the proof. As a final observation we note that when Z l = Z£ then M = l a n j GQ = G. Also if Zg = Z*, then ,/" u = 0 and Gp = G„ = G. Thus, with conjugate impedance matching at both the input and output, all three gains become equal. *4.12 EXCITATION OF WAVEGUIDES The preceding sections have dealt with the circuit aspects of passive microwave junctions. In order to complete the picture, it is necessary to consider also the equivalent-circuit representations for typical sources that are used to excite waves in a waveguide or transmission line. This particular aspect of waveguide theory is somewhat specialized, and it is not possible to give a complete analysis in this text without departing too far from the main theme. However, we shall present certain aspects of the excitation problem that provide a basis for choosing appropriate equivalent circuits and generators for representing typical sources and, in addition, make it possible t solve a number of coupling problems of engineering importance. The theor is, for the most part, developed by considering specific examples. P r o b e Coupling in a R e c t a n g u l a r Waveguide Figure 4.26 illustrates a typical coaxial-line-waveguide probe couplingshort-circuit position / and probe length d can be adjusted to acn J?, g maximum power transfer from the coaxial line into the waveguidecenter conductor of the coaxial line extends into the waveguide to tor electric probe. Any waveguide mode that has a nonzero electric field the probe will excite currents on the probe. By reciprocity, when the p current is produced by a TEM wave incident from the coaxial line, the waveguide modes will be excited.t It is thus easy to see that, for maxi tThis reciprocity principle is very useful for determining what modes a given probe can e»* r CIRCUIT THEORY FOR WAVEClttDrNG SYSTEMS 7,Id a 277 ——> End view Side view <S> Bottom view F I G U R E 4.26 Coaxial-line probe coupling to a waveguide. coupling to the dominant T E l n mode in a rectangular guide, the probe should extend into the guide through the center of the broad face so as to coincide with the position of maximum electric field for the TE 10 mode. The evanescent modes that are also excited are localized fields that store reactive energy. These give the junction its reactive properties. The section of short-circuited waveguide provides an adjustable reactance that may be used to tune out the probe reactance. The probe reactance can be evaluated by determining the amplitudes of the evanescent modes that are excited and computing the net reactive energy stored in these nonpropagating modes.t Since the details are rather lengthy, we shall evaluate only the amplitude of the radiated TE 1 0 mode. The current on the probe must be zero at the end of the probe. For a thin probe a sinusoidal standing-wave current distribution is a reasonable approximation to make for the probe current. Thus let the probe current be considered as an infinitely thin filamentary current of the form I ~ / 0 sin k0( d - y) 0 <y < d x = - 2 = o (4.106) We wish to determine the amplitude of the TE 1 0 mode excited by this current. A general technique for accomplishing this is a mathematical ormulation of the reciprocity principle invoked earlier to determine which waveguide modes a given source will excite. The required results are derived below. Figure 4.27 illustrates an infinitely long waveguide in which a current ' urce J is located in the region between z l and z2- The field radiated by s source may be expressed as an infinite sum of waveguide modes as p' f o r example, R. E. Collin. " Field Theory of Guided Waves," 2nd. ed., chap. 7, IEEE Press, p iscataw ;ay. N.J., 1991. 278 FOUNDATIONS FOR MICROWAVE ENGINEERING E*. H* E". H ' F I G U R E 4.27 A current source in a wavemjin follows: z > zn (4.107a) z>z2 (4.1076) z < z, (4.107c) z < z, (4.107d) n H-=Ec„-(-h n + h,j< Jti i n In (4.107) n is a general summation index and implies a summation over all possible TE and TM modes. The unknown amplitudes C„ may be determined by an application of the Lorentz reciprocity formula (2.135). For the volume V. choose that bounded by the waveguide walls and cross-sectional planes located at 2, and z 2 in Fig. 4.27. Let the field E p H j , to be used in the Lorentz reciprocity formula, be the field radiated by the current source. This field is given by (4.107). For the field E 2 , H 2 , choose the nth waveguide mode E ; , H ~ ; that is, E2 = E;=(e n + e,„)e^ I H2 = H ; = ( - h „ + h , n ) e ^ 2 Equation (2.135) gives <ft(E, r S xH--En-xH,) -ndS= J v JE;-JdV since the field E 2 , H 2 is a source-free solution ( J 2 = 0) within V. The surfed integral is zero over the waveguide walls by virtue of the boundary con n X E, = n X E , ; = 0. Since the modes are orthogonal, i.e., j E*XH„±-ndS = 0 n* m sh when all the terms except the nth in the expansion of E „ H , vani CIRCUIT THEORY FOR WAVEGUIDING SYSTEMS 279 integrated over the waveguide cross section S 0 . Thus we have f C n + [(e„ + e z n ) X ( - h „ + h „ ) - (e„ - e „ ) X ( h „ + h „ ) ] • a, dS -fC;[(e„-e M )X(-h n + l i J - ( e „ - e J X ( - h „ = -2C:j enXhn-a,dS + h j ] = -azdS JE-n-JdV since the integral over the cross section at z x vanishes identically. Hence C* is given by C > -•5-/E-'JdVr= -^-/(e„-e„)-Je*»'rfV (4.108o) If E+, H^ is chosen for the field E 2 , H.,, we obtain C~= -yfyK- *<W= ~yjv(en + e„) • Je~^ dV (4.1086) where P n = 2/" e„ X h„ • a, dS (4.108c) s0 and S 0 is a cross-sectional surface of the waveguide. The normalization constant P„ depends on the choice of expressions used for e„ and h „ , the latter being arbitrary. The above results are now applied to the probe problem introduced earlier. For the T E I 0 mode with fields given by "** E y = eye ~jf)z = sin —e ~** (4.109a) o TTX H x = hxe-Jti* = -Yw sin — e ~ j P ' a (4.1096) we have ea rb „ TX Pio = 2/J J[ Y w sin 2 — dxdy = abYw o o a (4.110) *nere Y w is the wave admittance for the TE 1 0 mode and /? is the propaga>ri constant. The probe in the short-circuited guide is equivalent to the original Probe plus its image at z = -21 placed in an infinite guide, as in Fig. 4.28. u we assume that the field radiated into the z > 0 region is tlc E;= C + sin —e--"" (4.111) 280 FOUNDATIONS FOR MICROWAVE ENGINEERING r •!• tmoqe probe Probe J. t F I G U R E 4.28 Probe and its image. /=0 z=-Zl then application of (4.108a) gives C+= - £ I0 sin k0(d abY, / o abk0 ( « " * * " - 1 ) ( 1 -y)dy- f I0 sin k0(d - y)e~j'w dy\ -cosk0d) (4.112) since ^•10 rrx . sin — c ^ 2 == 1 a - a at z == 0 , * - 2 ,-./20/ at * = —, z = -21 4t We have assumed that the current on the probe can be replaced by a line current along the probe axis and with a density given by (4.106). The volume integrals in (4.108) are consequently replaced by line integrals taken along the probe axis. Note also that the direction of the current in the image probe is reversed. This is necessary so that the fields radiated by the probe and its image will give a zero tangential electric field at the short-circuit position. The total transverse field of the T E l 0 mode radiated by the probe is thus, for z > 0, uz 7TX E y = ^rr-ie-2*' - 1 ) ( 1 - cosfe 0 d)sin — e ~ * * abk a (4.113a) Hx= (4.H36) -YwEy The total radiated power is given by P -TJDLV<M, 2 'o'o I27 Aabk20 \e-2Jf" - l\\l-cosk0d)' (4.U*) At the base of the probe antenna (y = 0), the total coaxial-line curr« is, from (4.106), / s I0 sin k0d CIRCUIT THEORY FOK WAVEGUIDING SYSTEMS » 7 t 281 the input impedance seen from the coaxial line, referred to the base, be = i?o + i%- The complex Poynting vector theorem then gives [Eq. (4.14)] zin = R0 F jx = + 2jmlWm-WM) ur where P is the power radiated into the guide and W„, - W e is the reactive energy stored in the vicinity of the probe owing to the excitation of nonpropagating (evanescent) modes. Since P has been evaluated and is given by (4.114), we can compute the input resistance. We obtain 2P R ° 2 I*sm k0d Z„ xll-e-^'^tan2-^2abk% (4.115) upon using the identities 1 - cos 2ff = 2 sin'2 6 and sin 20 = 2 sin 8 cos 0. This input resistance is called the radiation resistance of the probe. Note that its value can be varied by varying the parameters / and d, that is, the short-circuit position and probe length. Varying these parameters thus enables an optimum amount of power transfer to be achieved by adjusting fi 0 to equal the characteristic impedance of the coaxial line and introducing a suitable reactance to tune out the reactance jX. Suitable techniques for reactance cancellation are discussed in the next chapter. Radiation f r o m L i n e a r C u r r e n t E l e m e n t s Figures 2.29a and 6 illustrate linear current elements in a waveguide. For the case of the transverse current element, (4.108) shows that c;=C;=--/t J,dl (4.116) where J, is the line current density in amperes. This result may be interpreted to mean that a transverse-current element is equivalent to a shunt voltage source connected across an equivalent transmission line representing the waveguide when only a single mode, say the n = 1 mode, propagates. The reason for this is that the transverse current radiates a field with transverse electric field components that are equal on adjacent U s=0 (a) FIGURE 4.29 Linear "^rrent elements in a waveguide. J, 7=0 282 FOUNDATIONS FOK MICROWAVE ENGINEERING JX I , 0000 r— n:\ "9 (a) (A) F I G U R E 4.30 Equivalent circuits for current sources in a waveguide, ( a ) Transverse current source; (&) current source. - sides of the current source, and this is equivalent to continuity of the equivalent voltage across the source region. The transverse magnetic field is discontinuous across the source region, and thus the equivalent current is also discontinuous across the equivalent voltage generator. Figure 4.30 illustrates the equivalent circuit for this type of source for the dominant propagating mode. The ideal transformer provides a means of adjusting the coupling between the voltage generator and the transmission line so that the same amount of power is radiated as in the waveguide. The shunt susceptance jB represents the net reactive energy stored in the field of the evanescent modes that are excited. For an axial current located at z = 0, (4.108) gives C:=^jJre2ne^dl *n C~= -—fjrezne-^dl n If the current is a symmetrical function of z between -I < z <l> then, is not a function of z, we have C> -C n -= yf*i • e„ cos pnzdz (4.117) For this case the radiated transverse magnetic field is continuous across source but the transverse electric field is discontinuous. The source is equivalent to a voltage generator connected in series with an equrva transmission line, as illustrated in Fig. 4.306. . „ A linear current element may be viewed as an equivalent osciua electric dipole. From Maxwell's equation we have V X H =j(oeE + J =j<»e0E +ja>P + J and hence J enters into the field equations in the same manner as CIRCUIT THEORY FOR WAVECUIDING SYSTEMS 283 ff FIGURE 4.31 A current loop in a waveguide. polarization current jwP. Thus J may be considered equivalent to an electric dipole P given by J J«> Radiation f r o m C u r r e n t L o o p s Figure 4.31 illustrates a linear current loop in a waveguide. The amplitude of the rath radiated mode is given by C,:= -^-6E-n-rIdl where T/ is the vector current flowing around the contour C. r is a unit vector along C. By Stokes' law we obtain C:= -_0EB-.rfI= -—fvxE;-ndS But V X E ~ = ~ja>B~= -ju>n0H.~, and hence in) I , — C- (B--ndS (4.118a) Similarly, C-=—fB:-ndS (4.1186) It is seen that the excitation amplitude of the ra th mode is proportional to the total magnetic flux of this mode passing through the loop. If the current loop is so small that the field B„ of the n t h mode may be considered constant over the area of the loop, we obtain 70)/ C:=—B;-ndS r = J jo)I -—- B , t - S 0 ow / S 0 is the magnetic dipole moment M of the loop, where S 0 is the vector area of the loop; so we obtain c;=—B,T-M (4.119a) 284 FOUNDATIONS FOR MICROWAVE ENGINEERING and similarly /at (4.1 Radiation from a small current loop may be considered to be mam-. dipole radiation, as these equations show. For an axial magnetic J dipole (transverse current loop), the equivalent source is a shunt-connected vo source, whereas a transverse magnetic dipole is equivalent to a seri nected voltage source. *4.13 WAVEGUIDE COUPLING BY A P E R T U R E S t The foregoing formulation of the radiation from currents in a waveguide i terms of radiation from equivalent electric and magnetic dipoles is applicable to the coupling of waveguides by small apertures, or holes, in a common wall. To a first approximation a small aperture in a conducting wall is equivalent to an electric dipole normal to the aperture and having strength proportional to the normal component of the exciting electric field, plus a magnetic dipole in the plane of the aperture and having a streni proportional to the exciting tangential magnetic field. The constants of proportionality are parameters that depend on the aperture size and shape. These constants are called the electric and magnetic polarizabilities of t' aperture and characterize the coupling or radiating properties of the aperture. t A qualitative argument to demonstrate the physical reasonableness of these properties of an aperture is given below. Figure 4.32a illustrates the normal electric field of strength E at conducting surface without an aperture. When an aperture is cut in the screen, the electric field lines fringe through the aperture in the manner indicated in Fig. 4.326. But this field distribution is essentially that produced by an equivalent electric dipole as shown in Fig. 4.32c. Note that the dipole is oriented normal to the aperture. In a similar manner the tangential magnetic field lines shown in ^'"j 4.32d will fringe through the aperture as in Fig. 4.32e. These fringing neK lines are equivalent to those produced by a magnetic dipole located in plane of the aperture. . In Bethe's original theory the dipole moments are determined by ^ field in the waveguide in the absence of the aperture. Thus, for a cir aperture of radius r 0 •« A0, the dipole moments are related to the u t T h e theory was originally developed by H. A. Bethe, Theory of Diffraction by Sma" Phys. Rev., vol. 66, pp. 163-182, 1944. ±For a derivation of these results, see Collin, loc- cit. CIRCUIT THEORY FOR WAVEGU1D1NG SYSTEMS 285 „<S) (a) If) (£) F I G U R E 4.32 Aperture in a conducting wall. fields as follows: -e0af(n • E)n M= -amH, (4.120a) (4.1206) where n • E is the normal electric field and H, is the tangential magnetic field at the center of the aperture. The electric polarizability a e is given by ae=-|r03 (4.121a) and the magnetic polarizability a m is given by a, 3'0 (4.1216) The presence of an aperture also perturbs the field on the incident side of the screen. This perturbed field is t h a t radiated by equivalent dipoles which are the negative of those given by (4.120) and located on the input side of the screen. It is important to note that when the aperture is replaced by equivalent electric and magnetic dipoles, the field radiated by these is computed by assuming that the aperture is now closed by a conducting wall. The equivalent dipoles correctly account for the field coupled through the aperture in the conducting screen. Bethe's theory does not lead to an equivalent circuit for the aperture that includes a conductance to represent power coupled through the aperture. The reason for this is that the field assumed to excite the dipoles is chosen as the unperturbed incident field in the waveguide. In actual fact one should use the sum of the incident field and excited field as the polarizing eld. Since the excited field is small, the correction to the dipole moments is so small. However, by including the excited dominant modes (the propagating modes) as part of the polarizing field, we will obtain the needed Th" 1 " 6 ? i 0 " t l l a t r e s u l t s i n a conductance element in the equivalent circuit. e dominant-mode fields react back on the dipoles to account for the 286 FOUNDATIONS FOR MICROWAVE ENGINEERING radiation of power by the dipoles. Thus, in place of (4.120), the fonn . ft expressions are used for the dipole strengths: t g For radiation into the output waveguide, P = - 6 0 « e [ n • E^ + n • E S r - n • E 2 r ] n M= -am[Hgl Hllr, H" 2, r J , H-1226) For the radiation into the input waveguide, P = e0ae[n • E g l + n • E l r - n • E 2 r ] n M = a m [H g I + H l r - H 8» (4.122c) (4.122d) where the generator fields E R l , H g l are the dominant-mode fields in the input waveguide in the absence of the aperture, E l r , H l r are the dominantmode fields radiated by the dipoles in the input waveguide, and E 2 r , H 2 r are the dominant-mode fields radiated by the dipoles in the output waveguide. The unit vector n is normal to the aperture and directed from the input waveguide to the output waveguide. The subscript t denotes the tangential component of the magnetic fields. The field resulting from the aperture is determined by closing the aperture by an electric perfectly conducting surface and calculating the fields radiated by the dipoles given above and located at the center of the circular aperture. The theory is readily extended to include noncircular apertures. However, the procedure outlined above is restricted to circular apertures in a very thin common wall between two waveguides. There is considerable attenuation in the coupling through an aperture in a thick wall and in many practical applications this attenuation must be taken into account.^ The examples discussed next will illustrate the application of smallaperture coupling theory to an aperture in a transverse wall and an aperture in the broad wall between two identical rectangular waveguides. Aperture in a Transverse Wall Figure 4.33a illustrates a small circular aperture in a transverse wall& rectangular waveguide. To determine the exciting generator field, as»u t h a t the aperture is closed. A TE 1 0 mode incident from z < 0 is r e f l e c t ®L i o n the conducting wall at z = 0 to produce a standing-wave field in the reg tR. E. Collin, he. cit. tSee, for example, N. A. McDonald, Electric and Magnetic Coupling Through Small Ap^ in Shield Walls of Any Thickness, IEEE Trans., vol. MTT-20. pp. 689-695, 1972. ^ CIRCUIT THEORY FOR WAVBGUIDING SYSTEMS 287 (a) tM CD/M X ZH FIGURE 4.33 Aperture in a transverse waveguide wall. le) z < 0. This field is TTX Ev='C(e--""~e-"")an (4.123a) — a TTX Hx = -CYw(e'jez + e-"»*)sin — (4.1236) plus a z component of magnetic field which is not required to be known for the present problem. The normal electric field at the aperture is zero; so no induced electric dipole is produced. The tangential magnetic field at the center of the aperture is, from (4.1236), Hx = -2CYW and hence an induced x-directed magnetic dipole M is produced. In order to determine the total polarizing field using (4.1226) and (4.122d), we must find the fields H l r and H 2 r radiated into guides 1 and 2 by a magnetic dipole Max. The field radiated into the region z > 0 is that radiated by the magnetic dipole M, as illustrated in Fig. 4.336. This dipole js equivalent to a half circular current loop in the yz plane as illustrated. To nnd the field radiated by this dipole in the presence of the conducting transverse wall, image theory may be used. Since the image of the half 288 FOUNDATIONS FOR MICROWAVE ENGINEERING circular current loop in the transverse wall is the other half of the r loop, the image of M is another magnetic dipole of moment M. The eff ^ the transverse wall is equivalent to removing the wall and doubli strength of the dipole, as depicted in Fig. 4.33c. If the field radiated int region z > 0 is '"S irx B v + = Ae -** sin — = Ae v e"-"" a //;= TTX -AY^e-^sin — a = Ahxe~^z application of formula (4.119a) gives A = —^H;(2M) since the field B~ is -ti^h^ = ^ 0 Y U , sin(7rx/a) in the present case. The constant P 1 0 is given by ?io= -2/ / ra cb eyhxdxdy „ TX = 2 YJj J / s i n 2 — dxdy = abYw o o a Hence we obtain A = ^ 2 M = ^ 2 M (4.124) ao ab The presence of the aperture causes a field to be scattered into the region z < 0 also. For radiation into this region, the effective magnetic dipole moment is the negative of that used to obtain (4.124). Application of (4.119) now gives Ev = A sin —e J f i * z <0 a for the radiated field in the input waveguide and where A is given > (4.124). As expected, the magnetic dipole M a , acts as a shunt source. The * component of the radiation reaction fields H l r and H 2 r , at the center ot aperture, are and the generator field is HglI= -2CYW CIRCUIT THEORY FOR WAVEGUIDING SYSTEMS 289 Since M represents the dipole strength for radiation into guide 2, we use these fields in (4.1226) to obtain J4k0Z0 •M M = - a , -2CY„ ; abZ„ which can be solved for M to give 2a„YjC j4kQZQ M (4.125) abZ„ - « . We can now complete the evaluation of the constant A by using this expression for M in (4.124); thus Jk0Z0 A = (4.126) 4jk0Z0 i + abZ ' ul The total electric field in the input waveguide is TTX E y = [ O r * * + (A - C ) e ^ ] s i n — so the input reflection coefficient T is given by A-C The input normalized admittance is When we substitute for A from (4.126) into the expression for Y, we find that IL = 2 -A/C A/C 3ab = 1 ~jJ 8r 3/3 (4.12?) 0 l h e equivalent circuit of t h e aperture, as seen from the input waveguide, is a normalized shunt conductance of unit value plus a shunt inductive susceptance. The conductance term is called the radiation conductance and accounts for the power coupled, i.e., radiated, into the output waveguide. * he amplitude of the transmitted electric field is A which is given by (4.126). The transmission coefficient is A/C. From the equivalent circuit the transmission coefficient is 1 + T = 2 / ( 1 + F i n ) which gives the same result. The aperture is equivalent to an inductive susceptance connected across the transmission line. The conductance term represents the output transmission line terminated in a matched load. 290 FOUNDATIONS FOR MICROWAVE ENGINEERING 0 ©• -(D FIGURE 4.34 Aperture in a broad wall separating two waveguides. A p e r t u r e in B r o a d Wall of a Waveguide Figure 4.34 illustrates a circular aperture of radius r 0 placed in the broad wall separating two rectangular waveguides. The incident field is a TE U mode in the lower guide, and is given by z E=Csin—e-^ 3 a (4.128a) Hx= -CK.sin—e-JP* a Hz=j —-Ccos—e-"* lia a TTY„, (4.1286) TTX (4.128c) At the center of the aperture located at z = 0, x = d, the exciting field is vd (4.129a) Ev = C sin a ird H ' C Y J - a . s i n — +j^-a.cos — \ (4.1296) \ a " fia a J These expressions show that there will be a y-directed electric dipole and | and 2-directed magnetic dipoles excited in the aperture. Let the fields E ^ , Ef0, Bf0, and Bf0 be chosen as in E,' 1+„0 = a v sin — e ~ * * a E 1 0 = a , sin TTX ,jfc TTX TTX M 0 Hio= -»oYw\axsm~-j — azcos — \ TTX Mo TT TTX\ H r 0 - Br0= ^ o ^ l a ^ i n — + j ~ a z c o s — 1« -m CIRCUIT THEORY FOR WAVEGUIDING SYSTEMS 29 1 Also let the dominant-mode field radiated by the electric dipole in the upper guide be " •A&n \A2En z>0 z<0 l A ^ z > 0 \A2Hr0 <0 2 whereas t h a t radiated by the magnetic dipoles is ;A3E;0 2 AtE{0 >O z < 0 M3H;0 Z>0 \A4Hr0 2<0 The electric dipole P is equivalent to an electric current given by j w P . Since the dipole is oriented in the transverse plane, (4.108) gives 1 Trd A r = A2= - -^-JOJP • a v sin jo>Py = "l0 vd —- sin a a0 ^us (4.130) a Note that no integration is necessary since P is an infinitesimal dipole of total strength Py. The constant Pl0 is equal to abYw. The field radiated by the magnetic dipoles may be found by using (4.119). Thus . JaUfXw I . T?d ja»li0 I vd = — — M_sin ab \ x a , 77 vd \ vMz Trd \ +j——cos J /3a a ) (4.131) v Similarly, it is found that jo)u.a I rrd vM, vd \ = -—^-\-Mxsm—+j -cos— (4.132) ab \ a pa a ) With the above expressions for the amplitudes, the total field radiated tne upper waveguide is readily evaluated. It is given by ( A 1 + A 3 )E 1 - 0 (4.133a) 2 > 0 ( A 2 + A 4 )Er 0 2<0 (4.1336) A4 0 (A1+A3)Hr0 ( A 2 + A 4 )Hf 0 z>0 z<0 (4.133c) (4.133d) that the electric dipole and the magnetic dipole Af.. radiate the same in 292 FOUNDATIONS FOB MICROWAVE ENGINEERING both directions but the magnetic dipole M x does not. By correctly ch the aperture position d, it is possible to obtain zero radiation • Sl direction; that is, A 2 + A 4 can be made to vanish. We will return * 0 , , e property later. For radiation into the lower waveguide, the sign of the dipolp reversed. Since the mode functions are the same, this has the eff ^ changing the sign of the amplitude constants. By using the expression t the mode functions given earlier and the derived expressions for the am 1 tudes, we find that the radiation reaction fields are given by E\rv _ ^2ry ~ - ( ^ K v a , 10 vd abYw vd 2W/J.0VM 5111' cos 2 lia b vd (4.134a) sin a a » i « - H2rx = ~2H2rx = - ( A , + A 3 ) H 10 a t - ( A 2 + A 4 ) H f 0 - a x 2JI3MX ab -d sin' a (4.1346) ff. M - H2r! = -2H2r! = - ( A , + A 3 )Hf 0 a , - ( A 2 + A 4 ) H r 0 - a , 2a>vPy vd sm pa2b + cos a 2jojp.0Ywv%M: vd a vd cos 2 3 Pab a (4.134c) Note that in evaluating the reaction fields at the center of the aperture, we take one-half of the field at z = 0" plus one-half of the field at z = 0 + , that is, on adjacent sides of the center of the aperture. The above equations show that there is interaction between the two dipoles P y and M,. The above expressions for the reaction fields along with the generator fields given by (4.129) can now be used in (4.122a) and (4.1226) to write the following set of linear equations which will determine the dipole strengt vd vd 2jcoP vd 2(o(j.0vM; 2 sin cos — — ~ sin + - a „ee 0 pa2b a a a abY a vd 2jpMx vd -CY„. sin s m ' ab vd p y M, = c M . = -a C sin jvYw^ vd C cos Pa a a . vd vd — cos 2 ^ sm pa 2b a a Ba b a a 2OJVPV 2jo>fi0Yu,v2M 2 3 pab cos We can solve these equations for the dipole strengths which are found 2?q CIRCUIT THEORY FOR WAVEGUID1NCJ SYSTEMS 293 given by e 0 a e sin r-^~C Py= «m^,sin M * = Pam (4.135a) rrd — 2 C irrf <41356> •rrd ja„wY0 cos — ° C «0aA Mz= (4.135c) where I 2 • 2 " ^ 2 2 *** An sm z 77- cos^ — A = 1 + 2jac T~— + 2J<*, J m 3 I3ab "" pa b We can use these expressions to find the amplitudes A , , . . . , A., for the field in the upper waveguide. Thus for the electric field we have ( A , + A 3 )E^ 0 z>0 ' ( A 2 + A.,)E ! n 2<0 where i IT \2 rrd jaekZsm'~jam\-\ cos 2 \a „_ L A "a Cc ++ ^ -! ^ - C /3a6A 0a&A rrd A1+A, = jampwa2 •rrd zr-2 + —; aoJl + 2j«,„—sin TTTC 2 (4.136) — The amplitude A 2 + A 4 is given by this expression with the sign of the last term changed from positive to negative. If we now set A 2 + A 4 equal to zero, we find that this can occur for sin vd a An =-7=2\/6a (4.137) t h u s an aperture position exists such that there is no radiation through Port 4. Power entering port 1 in Fig. 4.34 is coupled into ports 2 and 3 only. If the incident field were through port 2, we would find that there is no Power coupled into port 3. A four-port network with these properties is a ectional cou pler> about which more is said in Chap. 6. 294 FOUNDATIONS FOR MICROWAVE ENGINEERING In the lower guide the aperture dipoles will radiate dominant fields with amplitudes that are the negative of those for the modes "^^ upper guide since the sign of the dipoles is reversed. Thus the reflected ^' transmitted electric fields are ^d E = (C-A1-A3)EI0 z>0 (4l3gQ E=~(A2+A4)Ef0 z<0 (4l38 Q 6j This shows that when the aperture is positioned so that there is no on coupled into port 4, then there is also no reflected power in port 1. We will define the aperture susceptanee jB and aperture reactance "f by the expressions 2aekl 2a np ird . _2am(7r/af vd rrd In terms of these parameters, the expressions for the amplitudes A, + A, and A 2 + A A can be written as jB/2 A +A jX/2 '-rr7i mfc <4140°> jB/2 jX/2 A „ + A4. = - ~ : C - ^ ^ = C 1+jB 1+jX (4.1406) ' c+ The condition A., + A4 = 0 is met_by setting X = B. Since ae is negative, we can obtain a negative value of_S. Thus the amplitude A, + A 3 can also be made small by setting X = - B . However, in this case Aj + A 3 does not vanish exactly. In order for A, + A 3 to equal zero, we would require X + B = 0 and XB = 0 which has only the trivial solution. By using X = - B we obtain an imperfect directional coupler. PROBLEMS 4.1. For TM modes in a waveguide, show that the line integral of the transve electric field between any two points on the boundary is zero. Hint: Note that V,et • d\ = (dejdl)dl = directional derivative of e, &°J the path. Integrate this and use the boundary conditions for ez- As an al tive, note that there is no axial magnetic flux, so that the line integral arou closed path in the transverse plane must vanish. .. 4.2. For TE modes show that the line integral of the transverse electric ^ between two points located on the guide boundary depends on the p integration chosen. . -ral Hint: Note that, because there is an axia] magnetic field, the line m « around a closed path does not vanish. 4.3. An obstacle located at z = 0 excites evanescent H modes that decay expo tially away from the obstacle in the positive z direction. Integrate the cor CIRCUIT THEORY FOR WAVEGUIDING SYSTEMS 295 Poynting vector over cross-sectional planes at z = 0 and z = w and the guide walls for the ramth evanescent H mode, and show that there is no power transmitted into the region z > 0. Show also that the reactive energy stored in the nonpropagating H mode in the region z > 0 is predominantly magnetic. Hint: Note from (2.59) that the total inward flux of the complex Poynting vector equals the power loss (which is to be taken equal to zero in this problem.) plus2jw(Wm -We). 4.4. Repeat Prob. 4.3 for the case of an E mode and show that nonpropagating E modes store predominantly electric energy. 4.5. For the circuits illustrated in Fig. P4.5, verify that the slope of the reactance function is given by (4.25). C o TiW- FIGURE P4.5 (a) *4.6. For the N-port junction choose an excitation such t h a t all /„ = 0 except I/, thus Vj = ZtJlj for all i. Show that all ZtJ must have real parts t h a t are even functions of ID and imaginary parts that are odd functions of u>. • 4 . 7 . Generalize the result (4.25) to show that for a lossless iV-port junction [i*L 8Z N A? u)=z z n n •»1 m = 1 HZ. So} -/ m = 4 j ( W ; + W m ) 4.8. Verify that (4.51) and (4.44) are equal. 4.9. Show that a length I of transmission line of characteristic impedance Z c is equivalent to a T network with parameters z n = 2 2 2 = ~jZc cot pi Zn = -JZC esc pi 4.10. Let Z^., Zfc, Z^, Z%. be the input impedance of a T network when terminals 2 are short-circuited, when terminals 1 are short-circuited, when terminals 2 are open-circuited, and when terminals 1 are open-circuited, respectively. In terms of these impedances show that the parameters of the T network are given by Zu = Zl ^22 zh = (zl-zl)zl = {zl-zl)zi - Zoc Use these relations to verify the equations for the circuits of Figs. 4.13d and e. •«1- For the mircowave circuit shown in Fig. P 4 . l l , evaluate the power transmitted to the load ZL. Find the standing-wave ratio in the two transmission-line f* Zt=Z< /W=% FIGURE P 4 . l i zc = z, Z fil***A Zi 296 FOUNDATIONS FOR MICROWAVE ENGINEERING sections. Assume Z L = 2ZU X x = X 2 = Z„ V, = 5V (peak). Check your swers using TLINE. ' ""' an4.12. FOT the microwave junction shown in Fig. P4.12, t h e equivalent-T parameters are Z n =j2, Zl2 =j/ y/2, Z22 = -jO.25. Find the p a r a m ^ ' * 0 ' * the alternative equivalent circuit illustrated. ** for &=1 01 n:\ FIGURE P4.12 4.13. For the three-port junction illustrated in Fig. P4.13, compute the power delivered to the loads Z, = 50 ft and Z 2 = 100 ft. Assume that V = IQ y peak. z, ~ so a FIGURE P4.13 4.14. For the transmission-line circuit shown in Fig. P4.14, find ( a ) the load reflection coefficient, (6) the impedance seen by the generator, (c) the VSWR on Z, = 40 il j fil'J FIGURE P4.14 transmission line, id) the fraction of the input power delivered to the following parameters apply: 50 ft Zg = 50 ft fil = IT/4 ZL = 40 ft liP* 4.15. For the circuit shown in Fig. P4.15, find the VSWR on both transmissij* ^ and the load impedance at the generator terminals. The following P& CIRCUIT THEORY FOR WAVEGUIDING SYSTEMS 297 )X zL fl=T '2=4 FIGURE P4.15 apply: I _ - l2 = - Z c = 50 n 2 g = 50 ft 2 L = 25 +>25 /X = / 2 5 4.16. For the transmission-line circuit shown in Fig. P4.16, find the required value of ' Z that will match the 20-ft load resistance to the generator. The generator internal resistance Rg = 60 (i. Find the VSWR on the transmission line. Is RL matched to the transmission line? RL = 20 Q Vg(~ £>=§ FIGURE P4.16 4.17. For the transmission-line circuit shown in Fig. P4.17, find the VSWR on each FIGURE P4.17 transmission line and the relative powers delivered to /?, and R2. The following parameters apply: 0i = IT 18 e2 - - R1 = 25 n R2 = 75 n zc = 50 n Rg = so n - f o r the transmission-Une circuit shown in Fig. P4.18, find the open-circuit voltage V x and the Thevenin equivalent impedance. Use these results to find the power delivered to RL. Assume that pi = TT/4, R g = 75 ft, Z c = 50 ft, « L = 30 n . 298 FOUNDATIONS FOR MICROWAVE ENGINEERfNG Rg |—WV & Pl=E FIGURE P4.18 4.19. Consider the junction of two transmission lines with characteristic imped Z, and Z 2 as illustrated in Fig. P4.19. When the usual transmission 1' -?12 s 2, 5,,^ ^ 5 2 FIGURE P4.19 voltages and currents are used, show that the scattering-matrix parameters are given by 2 1 z2 + z, 2Z, 821~ - ^ ~ S]., — Z, + Z, The normalized voltages V/ and V2+ are given by Vf = /F^VY, V2+ = / £ # , where the unprimed quantities are the usual transmission-line voltages. When normalized voltages are used, show that the scattering-matrix parameters are Z 0 ^11 _ ^22 2-2, „ "19 _ Z2 + Z , „ — "SI — 2 V /Z^ 2 " Z2+Zj 4.20. For the circuit shown in Fig. P4.20. find the scattering-matrix parameters. jX2 Zc Zc = 50 £2 FIGURE P4.20 When jX, =j25, jX2 = jlOO verify that ISu|a + I S , / = 1 Sn"i2 4.21. Find Z* T Z„ = 1 n:1 J •^12^22 12°22 - 0 Z £ , ZL and Z?. for the circuit shown in Fig. P4.21. "* FIGURE P4.21 CIRCUIT THEORY FOR WAVEGUIDING SYSTEMS 299 FIGURE P4.22 4 22. For the circuit shown in Fig. P4.22, find the input reflection coefficient !*,„. 4.23- For the circuit shown in Fig. P4.23, find the expressions for Z i n , r i n , and the input VSWR. Find the value of JX that will minimize the VSWR. Hint: The VSWR will be minimum when ir j n | 2 is a minimum. Why? e FIGURE P4.23 4.24. The field of a T E U mode in a rectangular guide of width a and height 6 is derived from irx vy h, = C cos — cos —ra b Determine the expressions for the equivalent-transmission-line voltage V* and current I' for the two cases (1) when Z c - Zu. = wave impedance of the T E U mode, (2) when Zc = 1. 4.25. Apply the complex Poynting vector theorem to show that, for a one-port microwave termination, the reflection coefficient V satisfies the relation when the wave amplitudes are normalized, so that W* = II*, that is, the equivalent characteristic impedance is unity. 4.26. Show that the j / ^ 4 f 3 parameters for a section of transmission line of length / and characteristic impedance Zt are given by .*' = 's = cos pi, .'£ = jZ,. sin fil, &= jYc sin j8/. 4.27. For a section of transmission line of length /, show that the wave-amplitude transmission matrix is a diagonal matrix with elements i n - * * 8- A22-e-'»l Al2 = A2i=0 Consider the junction of two transmission lines as in Prob. 4.19. Using conventional transmission-line voltages, show t h a t the [ A ]-matrix parameters describing the junction are A u = A & = (Zx + Z 2 ) / 2 Z 2 , A 12 = A 2 1 = (Z 2 - Z , ) / 2 Z 2 . When normalized wave amplitudes (voltages) are used, show that A,, = A22 = (Zj + Z 2 ) / ( 2 ^ Z , Z 2 ) , A 1 2 = A 2 I = (Z 2 - Z,)/(2 > /z7Z 2 "). • r i n d the [Aj-matrix parameters for a shunt susceptance jB connected across a transmission line of unit characteristic impedance. Repeat for a reactance jX connected in series with the line. • Show that when normalized voltages are used the scattering-matrix parameters ot a two-port junction are given in terms of the equivalent-T-network parame- 300 FOUNDATIONS FOR MICROWAVE ENGINEERING ters by A — 1 + Z u — Z22 11 ~ A + 1 + Z u + Z22 S\2 — S 2 1 — 2Z 1 2 A + 1 + Zu + Z22 A — 1 + Z 2 2 — Zj j •^22 - A + 1 + Zn + Z<22 where A — ZllZ22 Z12 4 . 3 1 . Show that the T-network parameters are related to the scattering-matri parameters as follows: (1 + S n ) ( l - S 2 2 ) + Sf2 Z„ ^ao o — » w _2S12 7 *12 W where W = (1 - S „ ) ( l - S 2 2 ) - Sfg 4.32. For a discontinuity in a waveguide, the following scattering-matrix parameters were measured: ^11 = 5 +Ja "12 = Ja ^22 = a ~J~S Find the parameters of an equivalent T network that will represent the discontinuity (Prob. 4.31). 4.33. For an £-plane step (Fig. 4.6), the following were measured: 611 3 +j 22 ~ 3 + j An equivalent circuit of t h e form illustrated in Fig. P4.33 is to be J represent the junction. Determine the susceptance jB and the ide former turns ratio n:l from the above given data. JB v.n FIGURE P4.33 1 ^. CIRCUIT THEORY FOR WAVEGUIDING SYSTEMS M f"l Z-c 301 M ix • FIGURE P4.34 *4.34. For the circuit illustrated in Fig. P4.34 construct a signal flow graph relating the variables Vg, Ig, V„ /„ V2, and I2. Use signal flow graph analysis to find the voltage V2 across ZL. Hint: See Prob. 4.9 for the T-network parameters of a transmission line. *4.35. For the circuit illustrated in Fig. P4.35, construct a signal flow graph. Use signal flow graph reduction to derive an expression for the load voltage V3. Use t h c w S ^ S * chain matrix to write relationships between the variables. Bint: See Prob. 4.26. JX k 'i — <2^ v2\ *t %—' zc v 3 | FIGURE P4.35 *4.36. For the circuit illustrated in Fig. P4.36, construct a signal flow graph using the scattering-matrix relationships between the variables. From a reduction of the signal flow graph find the load voltage VL. JX ^ P V* . i%+— Zc V{' w—- 0 FIG URE P4.36 • '• For the transmission-line circuit shown in Fig. P4.37, find the generalized scattering-matrix parameters for the series reactance jX. Use these results to derive an expression for the power delivered to ZL. Verify your answer using a more conventional method of analysis. 302 FOUNDATIONS FOR MICROWAVE ENGINEERING Rg+jXg v fx 9Q ZL=RL + jXL FIGURE P4.37 *4.38. Repeat Prob. 4.37 with jX replaced by a shunt element jXs. *4.39. Find the TE 1 0 field radiated by the current loop illustrated in Fj g p. Consider the loop area to be so small that (4.119) is applicable. The area nf i l0o equals S 0 . P a/2 /=0 FIGURE P4.39 *4.40. Find the TE„, field radiated by the current loop of Prob. 4.39 if a short circuit is placed at z = —/. *4.41. A linear constant current / extends across the center of a rectangular waveguide at J: = a/2, z = 0. Show that the total radiated electric field is -jwfigI Ev = * 1 nv nir: — sin — sin 2- a „ „ , 7„ 2 ,-y»lj a 1/2 where yn "J0n = •2 " 0 REFERENCES f Rtt- 1. Kerns, D. M.: Basis of Application of Network Equations to Waveguide Problems, J Nail. Bur. Std., vol. 42, pp. 515-540, 1949. l95j 2. Marcuvitz. N. (ed.): "Waveguide Handbook," McGraw-Hill Book Company, New Y ° r k " rt 3. Montgomery, C. G., R. H. Dicke, and E. M. Purcell teds.): "Principles of M** Circuits," McGraw-Hill Book Company, New York, 1948. Philip* 4. Pannenborg, A. E.: On the Scattering Matrix of Symmetrical Waveguide Junctions, Res. Rept., vol. 7. pp. 131-157, 1952. CHAPTER 5 IMPEDANCE TRANSFORMATION AND MATCHING In this chapter we are concerned with the important problem of impedance matching, such as the matching of an arbitrary load impedance to a given transmission line or the matching of two lines with different characteristic impedances. Methods of impedance matching to obtain maximum power transfer are presented, along with broadband design methods for quarterwave transformers and tapered transmission-line impedance transformers. To facilitate the development of the theory, the Smith chart, a graphical aid for the solution of many transmission-line and waveguide impedance problems, is described first. In a computer-oriented age the reader may very well question why one should be interested in a graphical aid, such as the Smith chart, to solve an impedance-matching problem. There are two basic reasons why the microwave engineer needs to be familial- with the Smith chart. One reason is at Using the Smith chart to solve an impedance-matching problem shows m a very vivid way how adding reactive elements moves the impedance >mt around and this provides considerable insight into the impedancenatching problem. The second reason is that the Smith chart is widely used 1 he industry to display the performance of a microwave circuit in terms of nput impedance versus frequency, VSWR or reflection coefficient versus ^ u ^ n c y , the frequency variation of scattering-matrix parameters, etc. in microwave amplifier design the Smith chart is indispensible as a aid to show how gain, noise figure, stability, and input and output cn i n g are interrelated and how these operating characteristics depend 303 304 FOUNDATIONS FOR MICROWAVE ENGINEERING on the load and source impedances. Without the aid of the Smith ch intuitive understanding of microwave amplifier design would be much ' ^ m difficult to acquire. °*e 5.1 SMITH CHART In Sec. 3.6 it was shown that a load impedance Z L was transformed in* impedance ZL + jZe tan pi Z ™ = Zc Zc+jZLtanpl (5.1) when viewed through a length / of transmission line with characterise impedance Zr. This formula is valid for any waveguiding system with phase constant /3, provided the impedances are properly interpreted in terms of suitably defined equivalent voltages and currents. Alternatively, the reflection coefficient T(Z), a distance / from the termination, is uniquely given by r/n r(/) Z i"(f>-2c <5-2> - zjiiTz; with Z-mil) given by (5.1). The reflection coemcient is a physical quantity that can be measured, and the normalized impedances Z-n/Zc and ZL/ZC may therefore be appropriately defined in terms of the reflection coefficient r at any point / on the line and the reflection coefficient YL of the load, as follows: z. . h.. I l l . iiiif^ m 1 zc i - r(/) Z, = ljjt* z< i - r £ i - y,e- 2jPl (J b) The Smith chart is a graphical representation of the impedance-trans formation property of a length of transmission line as given by i Clearly, it would be impractical to plot all values of Z, and Zm ° rectangular coordinate impedance plane, with one coordinate represei the real part, or resistance, and the other coordinate representing reactance, since this would require a semiinfinite sheet of paper, y other hand, all values of the reflection coefficient lie within a unit cir ^ the reflection-coefficient plane since |fl < 1. Furthermore, each value specifies a value of normalized input impedance by means of (5.3a). s . there is a one-to-one correspondence between reflection coefficient an *: »:O0 impedance. Instead of plotting contours of constant values of the ren coefficient, contours of constant values of input resistance and inp" 1 ^e tance are plotted on the reflection-coefficient plane. For a given value o ^ reflection coefficient, t h e corresponding input impedance can be rea IMPEDANCE TRANSFORMATION AND MATCHING 305 rectly from the plot. In addition, a movement a distance d along the line corresponds to a change in the reflection coefficient by a factor e~2Jpd only. This is represented by a simple rotation through an angle 2/3d; so the corresponding impedance point moves on a constant radius circle through this angle to its new value. The chart thus enables the transformation of impedance along a transmission line to be evaluated graphically in an efficient and straightforward manner. A more detailed description of the chart and its use is given below. In addition, a number of matching problems are solved with the aid of the Smith chart in later sections of this chapter. Let the reflection coefficient T be expressed in polar form as Y = peje where p = \Y\ and impedance be (5.4) 0 = Z T = LTU— 2/3/. Zm _ _ Let the normalized input 1 + peJ° 1 + T From (5.5) it is readily found that in the reflection-coefficient plane (p, 0 plane), the contours of constant R and constant X are given by (Prob. 5.1) R u — -== \2 K + l] +v2 = 1 5- (S+l)2 («-l)»+(*-jLj - i j (5.6a) (5.66) where u = p cos 8 and v = p sin 6 and are rectangular coordinates in the P, 0 plane. The above constant R and constant X contours are circles and plot as illustrated in Fig. 5.1. For convenience in using the chart, a scale giving the angular rotation 2/31 = 4TTI/X in terms of wavelength A is attached along the circumference of the chart. Note that moving away from the load (toward the generator) corresponds to going around the chart in a clockwise direction, as illustrated ui Fig. 5.2. A complete revolution around the chart is made in going a distance I = A/2 along the transmission line. At these intervals the input impedance repeats itself. The origin for the angular scale is arbitrarily chosen at the left side of the circle. To illustrate the use of the chart, let a line be terminated in a load impedance ft, + J X , = 0.5 +70.5. This point is located in Fig. 5.2 and labeled P v At a distance / = 0.2A away, the corresponding input impedance may be found as follows: A constant-radius circle through P, is constructed nr s t . The new impedance point P 2 lies on this circle at an angle 2/3/ = 0.8-n- 306 FOUNDATIONS FOH MICROWAVE ENGINEERING FIGURE 5.1 Constant R and X circle in the reflection-coefficiem plane. FIGURE 5.2 The Smith chart. rad in a clockwise direction from P-y. This angular rotation is re ^gctio" out by adding 0.2A to the wavelength reading obtained from the int e j.greIjce of the radius vector through P x and the angular scale at the circuit jzP j of the chart. From the chart it is found t h a t the new value of n° impedance is R2+jX2 = 2-jl.04 L IMPEDANCE TRANSFORMATION AND MATCHING 307 If we begin at a point Py, where the impedance is R x + jXlt and move n a constant-radius circle an amount A/4 to arrive at a point diametrically opposite, \\ changes into - I \ (2/3/ changes by TT), and we obtain an impedance «_ R22 +jX 7 22 = i - r, i — = •= =- = G, •JBt i + r, R1+JX1 Thus the input normalized admittance G, + jBt corresponding to a given input impedance Rx + jXl may be found from the value of impedance at a point diametrically across from the first impedance point, provided R 2 and jX2 are interpreted as^the input conductance and susceptance. To clarify this, note that R 2 + jX2 at P 2 is the normalized input impedance at point P 2 and equals the normalized input admittance at point P y at a distance / = A/4 away. The Smith chart may be used to find the transformation of admittances equally well. All that is required is_to interpret the constant resistance and reactance contours (constant R and jX contours) as constant conductance G and susceptance jB contours. Note J h a t a positive X corresponds to an inductive reactance but a positive B corresponds to a capacitive susceptance. In order to facilitate the use of the Smith chart in situations where it is necessary to convert back and forth between impedances and admittances, the impedance-admittance chart is used. This chart has a second Smith chart, rotated by 180°, superimposed on the regular Smith chart. Thus one set of circles gives impedance values and the second set of rotated circles, usually shown in a different color, give the corresponding admittances values directly. For the conventional Smith chart, the admittance F I G U R E 5.3 Inward spiraling of the impedance point on a Smith chart for a lossy transmission line. 308 FOUNDATIONS FOR MICROWAVE ENGINEERING values are obtained by rotating the impedance points by 180° impedance-admittance chart, the second set of rotated circles rn t ^ e unnecessary to rotate the impedance point in order to determ" e corresponding value of the admittance. 'he On a lossy line the reflection coefficient at any point is given h PL (5.7) As we move from the load toward the generator, p = />Le"2ul conti decreases, and hence we move along a spiral that eventually terminal the center, as in Fig. 5.3. In practice, we move on a constant p circle fi * through the angle 2/3/ and then move in radially until we are a dista p L e - 2 " ' from the center. Many practical charts have convenient seal attached to them, so that the amount of inward spiraling is readily obtained Note that the center of the chart represents a matched condition (p = 0) 5.2 I M P E D A N C E MATCHING WITH REACTIVE ELEMENTS When a given load is to be connected to a generator by means of a transmission line or waveguide many wavelengths long, it is preferable to match the load and generator to the transmission line or waveguide at each end of the line. There are several reasons for doing this, perhaps the most important one being the great reduction in frequency sensitivity of the match. Although the transformed load impedance as seen from the generator end of the transmission line can be matched to the generator for maximum power transfer, a small change in the operating frequency will change the electrical length fil of a long line by an appreciable fraction of v rad, and hence greatly modify the effective load impedance seen at the generator end and thus modify the matching requirements as well. To avo« this frequency sensitivity of the matching requirements, the load an generator should be individually matched to the transmission line or waveguide. . . Another disadvantage of not matching the load to the transmission line is that when a matching network is used at the generator end o . there may be a large standing-wave field along the transmission line I original load is badly mismatched. This reduces the power-handling ^ P ^ o ity of the system since, for a given power transfer, the ^iaximum,^ntL strength before dielectric breakdown occurs is reached sooner. In ao greater transmission losses are also incurred when there is a standing current along the line. t0 a The techniques that may be used to match a given load impedanc ^ transmission line or waveguide may equally well be used to mate ^ generator to the line. Hence it suffices to limit the following discussio ^ t h a t of matching an arbitrary load impedance to the transmission line: ^ convenience, normalized impedances are used. The first matching tecB IMPEDANCE TRANSFORMATION AND MATCHING 309 Y,=C V'V* FIGURE 5.4 Single-shunt-stub matching network. discussed employs s h o r t - c i r c u i t e d (or open-circuited) sections of t r a n s m i s sion lines as reactive e l e m e n t s , a n d is referred to as stub m a t c h i n g . However, t h e principles involved a r e g e n e r a l i n n a t u r e a n d m a y b e applied t o a n y waveguiding s y s t e m by s u b s t i t u t i n g s u i t a b l e s h u n t or series reactive elem e n t s for t h e t r a n s m i s s i o n - l i n e s t u b s . A description of s o m e typical reactive e l e m e n t s t h a t m a y be used is given in a l a t e r section of this c h a p t e r . Single-Stub M a t c h i n g Case 1 S h u n t s t u b . Consider a line terminated in a pure conductive load of normalized admittance YL = G, as in Fig. 5.4. At some point a distance d from the load, the normalized input admittance will be Y m = 1 + jB. At this point we can connect a stub with normalized input susceptance —jB across the line to yield a resultant that is, to arrive at a matched condition. The stub should be connected at the smallest value of d that will give Ym = 1 + jB in order to keep the frequency sensitivity as small as possible. The stub may be either an open-circuited or a short-circuited section of line, the latter being the most commonly used version for two-wire lines, coaxial lines, and waveguides because of ease in adjustment and better mechanical rigidity. In a microstrip circuit an opencircuited stub would be preferred since it does not require a connection to the ground plane. To find the position d, we must solve the equation YL+jt ^in = 1 +JB = t = tan pd 1 +JYL' If we assume that Y L = G is pure real, we require (l+jB)(l+JGt) =&+jt 310 FOUNDATIONS FOR MICKOWAVE ENGINEERING or by equating real and imaginary parts, we obtain 1 - BGt = G _ j(B (5-8a} + Gt)=jt (5.86) Equation (5.86) gives B = (1 - G)t, and substitution of this into (^, J s > ° a ) yield, 1-G 1-G EG ~ ( i - G)Ut il = tan 2 Bd = — G Replacing tan 2 fid by (1 - cos 2 Bd)/cos2 ,6c? finally gives or A d = / ^C0S G VTTg (5-9, where B = 2TT/A. Note that two principal values of d are possible, depending on which sign is chosen for the square root. An alternative relation is obtained if we replace 2 cos 2 Bd by 1 + cos 2Bd; thus 1 1 2G 1+G rncfftri G - 1 and (..(Jo A D U — ^~ G+1 which gives A u — G- 1 COS 4TT — (5.10) G+ 1 If dL is a solution of (5.10), then A/2 - d{ is another principal solution, sine* ±dx ± n A / 2 are all solutions of (5.10). The value of the input susceptance jB is given by 3 -{i-B)t-~ (5U) VG since tan Bd = 1/G. The required length l 0 of a short-circuited stub to { an input susceptance -jB is found from the relation 2 Yln= -JB= -j cot BlQ and (5.11); thus cot/3/ 0 = or 1 -G ._ VG A , /G / 0 = 2TT —tan"1 l - G where the sign of \ / G must be chosen to give the correct sign for B in (5.12' :: IMPEDANCE TRANSFORMAT1 OS AND MATCHING 311 •;o r^\+jB r^=s Z\»*°\*jX FIGURE 5.5 Location of stub relative to a voltage minimum. Z;»=Vs FIGURE 5.6 The series stub. 0 < d < A/4, the positive square root should be used, whereas if the other solution, A / 4 < d < A/2, is chosen, the negative square root must be used. A similar analysis may be carried out when Y, is complex, but it becomes more involved. The following procedure is usually followed instead. First locate a position of a voltage minimum from the load. At this point the reflection coefficient is a negative real quantity and the input admittance is pure real and given by Ym = 1-T i+r 1 +p = I -P =s (5.13) where S is the standing-wave ratio on the line. Let d 0 be the distance from this voltage-minimum point to the point where F in = 1 +jB, as in Fig, 5.5. The equations to be solved for the stub position d 0 and stub length /,, are the same as given earlier, but with S replacing G. Hence A S - 1 *•-*"• 'sTT A , (5.14a) v'S (5.146) The position of the stub from the load is readily computed by finding the distance from the load to the V^„ position and adding this to d 0 . Note that a stub position d 0 ± A/2 is also a suitable one. Thus, if d 0 - A/2 is still on the generator side of the load, the stub should be placed at this point instead of at "o in order to reduce the frequency sensitivity of the match. Case 2 S e r i e s stub. At a position of a voltage minimum, Zin = S'1. At some position d0 From this point, Z^ = I + jX. By connecting a stub with a normalized input reactance of -jX in series with the line at this point, the resultant input impedance is reduced to unity and a matched condition is obtained. This series stub-matching network is illustrated in Fig. 5.6. 312 FOUNDATIONS FOR MICROWAVE ENGINEERING To find d0, we must solve the equation _ S~l +j tan Bdr, 1 +>S tan/3rf0 This is the same equation as considered earlier, with X, S~l replac' and thus the solutions are g °- S, (5.15Q) (5.156) where the sign of JS must be_chosen to yield the correct sign for tan (3d • that is, for 0 < d0 < A/4, use + JiS, and for A/4 < d0 < A/2, use - js. The required stub length l0 is determined from the relation j tan /3/ 0 = -jX and hence, from (5.156), we obtain A / --2? ,1-S ton ~7T (516) The shunt stub is most commonly used for coaxial lines because it is easy to construct a shunt stub for a coaxial line, whereas a series stub is diflicult to build. A disadvantage with a single-stub-matching system is that every I08" requires a new stub position. The use of two stubs spaced by a fixed amount and located a fixed distance from the load may be used to overcome this disadvantage. However, a double-stub-matching system of this type will n« match all possible values of load admittance. The theory of double-stub matching is presented in the next section. DOUBLE-STUB MATCHING NETWORK n o t w n r l r is I'C illustrated illiic+ratoH schematically arhRTO&tiCSK The double-stub tuner, or matchingr network, Fig. 5.7. We may transform_the £iormahzed load admittance I t " 1 an^ equivalent load admittance Y L = G L +jBL at the plane aa and treat problem illustrated in Fig. 5.76 without loss in generality. _ -^ Let the point P K on the Smith chart in Fig. 5.8 represent YL- The ^ stub adds a susceptance jBl which moves P, along a constant-conduc . circle to point P 2 in Fig. 5.8. At the plane_66 just on^he right-hand si ^ the second stub, the input admittance is Y b = G b +jBb, and is obtain ,g moving along a constant-radius circle from P2 to P3 through an ^ #_ = 2/3rf = 47rd/A rad in a clockwise sense. The point P 3 must lie o B j G = 1 circle if the addition of a susceptance jB» contributed by the sec IMPEDANCE TRANSFORMATION AND MATCHING 313 F I G U R E 5.7 The double-stub tuner. stub is to move point P3 into the center of the chart (matched condition) along the G = 1 circle. From the description just given, it is clear that the first stub must add a susceptance of just the right amount, so that after the admittance at plane cm is transformed through a length of line d, we end up at a point on the G = 1 circle. The required value of susceptance jBl to be contributed by the first stub may be obtained by rotating the G = 1 circle through an angle - <f>. The intersection of the rotated G = 1 circle and the G L circle determines the point P2, and hence jBv as illustrated in Fig. 5.9. A point P'2 would also be suitable; the location P'3 then corresponds to the admittance just to the right of the second stub. _ From Fig. 5.9 it is clear that for all values of Y L that lie_within the G = G0 circle, a match cannot be obtained since all values G > G0 will not intersect the rotated G = 1 circle. The conductance circle G = G 0 is tangent F I G U R E 5.8 Graphical representation of the operation of a double-stub tuner. 314 FOUNDATIONS FOR MICROWAVE ENCilNEEKING F I G U R E 5.9 Graphical determination of ^ quired susceptance for the first stub in a double-stub tuner. to the rotated G = 1 circle at the point Q- It is easy to see that the smaller the distance d, the larger the range of load admittances that may be matched (see Fig. 5.10 for the case of d = A/8, 4> = TT/2). Also note that G0 will always be greater than unity; so all loads with_G L < 1 can be matched. At plane aa in FigJ5.7 we_have_y,, = G, +jBL. J u s t to the left of the first stub we have Ya = GL +jBL +jBy Just to the right of the second stub F I G U R E 5.10 of^ loa* Illustration of range be impedance which can" matched when d = < v 8 ' IMPEDANCE TRANSFORMATION AND MATCHING 315 we have GL +JBL + jBl +jt F*~ t = tan pd 1 +j*{GL +jBL +JBl) (5.17) Since Y b must equal 1 + jB, (5.17) gives, upon equating the real part to unity, = 0 _ 1 + t2 1± GL = 2«2 or U\\-Bht-Bxt) 1/1- (5.18a) (5.186) (1 + ' 2 ) 2 In (5.186) we note that the term under the radical sign equals one minus a positive quantity. Since G L must be real, the term under the radical sign must be positive, or zero. Hence the value of the_square-root term lies between zero and one. The corresponding limits on G L are _ 1 + t2 0<GL< 1 2 sin lid For any_ given choice of d, the whole range of load admittances outside the circle G0 = esc 2 fid may be matched. As an example, for d = A/8,__/3d = TT/4, <f> = 2(Zd = IT/2, and all values of load admittance outside the G0 =_2 circlejnay be matched, as shown in Fig. 5.10. For d = A/4, all values of Y L with G L < 1 may be matched. Although the theory predicts that virtually all load impedances may be matched by choosing d near zero, or A/2, so that esc 2 [id becomes infinite, this is not true in practice. The maximum value of stub susceptance that can be obtained is limited by the finite attenuation of the transmission line used. If jp were replaced by jp + a, it would be found that, even with A / 2 spacing, all values of load admittance could not be matched. In addition, stub spacings near A/2 lead to very frequency-sensitive matching networks, so that in practice spacings of A/8 or 3 A / 8 are preferred. The larger spacing is used at the higher frequencies, where the wavelength is too small to permit use of A/8 spacing. A complete analytical solution to the double-stub matching network is readily obtained. Solution of (5.18a) for the susceptance B, of the first stub gives n ts1 ere - nL + l±\/(l + t*)GL^GTt* (5.20) BL, GL, and t = tan pd are all known. Equating the imaginary part 316 FOUNDATIONS FOR MICROWAVE ENGINEERING r V=0 FIGURE 5.11 Illustration of the design of a double-stub matching network. of (5.17) to jB gives s = l^zIiLiIit)(B^—+ B, 2i + _t) GU 2*2 (l-Bj-Sjty+Gfc S u b s t i t u t i n g for B x into t h i s e q u a t i o n yields _ +M(l + ^ 2 )'^ 5 -G t B (5.2D G,t T h e u p p e r a n d lower signs i n (5.20) a n d (5.21) g o t o g e t h e r . T h e s u s c e p t a " * of t h e second s t u b m u s t be c h o s e n as — jB in order to provide a m a t e condition. E x a m p l e 5 . 1 . We want to design a double-stub-matching system to m a normalized load admittance i j , = 0.4 +jl. The stubs are spaced A / ° a P . _, We first construct the G = 1 circle rotated by an angle 1$d = 4 7 r ^g i r / 2 in the_counterclockwise direction as_shown in Fig. 5.11. Next wei ^ the point Y L at the intersection of the G = 0.4 and B = 1 contours. ^ { h e the point labeled Y L in Fig. 5.11. We can move Y L so that it interisects' IMPEDANCE TRANSFORMATION AND MATCHING 317 rotated G = 1 circle by moving it_to P, or P\ along the G = 0.4 contour. At p ( > jB =jl.8, so we need to_add jBx =y'(1.8 - 1) = j'0.8 to ?L to get to P±. At p-_ jB =j0.2, so to move YL to P[, we need a stub with susceptance jBl = # 0 , 2 - 1) = - J 0 . 8 . The next step is to rotate P, and_P', by TT/2 rad in the clockwise direction. The rotated points lie on the G_= 1 circle at P 2 and P2. At P 2 l jB = —j3, so we need a second stub with jB2 = -jB — j3 in order to move P2 into the origin along the G = 1 contour. Similarly, in order to move^P z in* 0 the origin, we need a second stub with jB2 =• -jB = -jl since jB at P/, equals jl. We can also use the Smith chart to find the stub lengths. For the first solution where we needed a stub with jB2 = j 3 , we will assume that we use an open-circuited stub. We draw a radius vector from the origin through the point where the jB = jZ contour cuts the outer boundary of the Smith chart as shown in Fig. 5.11. When we move along an open-circuited or short-circuited stub, we will be moving on the p = 1 circle^ or outer boundary of the Smith chart. At the open-circuit end of the stub, Y = 0 and this is the point on the left-hand side of the Smith chart as shown. We begin at this point and move on the G = 0 or p = 1 circle toward the generator (clockwise) until we get to the j3 point. From the normalized distance scale on the outer boundary of the Smith chart (not shown), we find that the required stub length is / 2 /A = 0.199. For the second solution where we need jB2 = —jl, we will assume that we use a short-circuited stub. At the short-circuit position Y =jx and this is the point on the right-hand side of the chart. We begin at this point and move on the G = 0 circle until we get to —jl as shown. This gives /'2/A = 0.125 for the stub length. The reader can verify that for stub 1 the required length is, for an open-circuited stub, h = 0.393 — =0.107 A while for a short-circuited1 stub, stub, — = 0.143 A A = 0.357 If series stubs are used, then we work with the normalized load impedance and the normalized reactances of the stubs. A point to keep in mind when designing t h e j e n g t h of a stub to give a specified reactance is that at a short-circuit Z = 0, while at an open-circuit Z = jto. Thus, for an opencircuited stub, we begin at the right-hand side of the Smith chart and move clockwise on the p = 1 circle until we arrive at the desired jX point. For a shunt stub that was open-circuited, we start at the left-hand-side edge of the Smith chart. ^^LE-STUB TUNER e d i s a d v a n t a g e of n o t being able to m a t c h all load a d m i t t a n c e s w i t h a o u b l e - s t u b t u n e r m a y be overcome by u s i n g a t r i p l e - s t u b t u n e r , j i s illusated in Fig. 5.12. S t u b 1 provides a s u s c e p t a n c e jBl s u c h t h a t Y L +jBi 318 FOUNDATIONS FOR MICROWAVE ENGINEERING No. 2 No FIGURE 5.12 Triple-stub tuner. 6" 0 =csc* Bd FIGURE 5.13 Transformation of Y^ into Y£. transforms to some new admittance Y'L just to the right of stub 2. Stubs 2 and 3 provide a conventional double-stub tuner for matching Y'L to the line. These two stubs will match all values of Y[ for which G'^< esc 2 pd. Thus the function of stub 1 is to ensure that a susceptance jBx is added 1 such that the transformed admittance Y'L has a G'L less than esc fid. To find a suitable value_of jBy, we note that, after moving a distance d "° YL, the_admittance Y, +jB1 must transform into a pc-int Pi outside circle G 0 = csc 2 /3d, as in Fig. 5.13. If we rotate the G 0 = c s c 2 0 d ci:irele through an angle -<£ = — 2(3d, we can_ readily see at once the_ range of susceptances jBx that may be added to Y^ to keep the resulting Y'L on s the G 0 = esc2 (3d circle. The procedure _is illustrated in Fig. 5.14. In example Y L falls within the rotated G 0 = esc 2 (3d circle, and b e ^ ^ susceptance jBx must be added to move the resultant load Y L +J&} Q r some point beyond P, or P2, say to P[ or Pg, which provides for a marg 1 " IMPEDANCE TRANSFORMATION AND MATCHING 319 FIGURE 5.14 Graphical solution for jBt for a triple-stub tuner. safety. The resultant load when transformed to the position of stub 2 will lie outside the circle G 0 = esc 2 /3d, and hence can be matched by stubs 2 and 3. A triple-stub tuner can match all values of load admittances. It may be considered to be two double-stub tuners in series, i.e., jB2=jB'2+jB2 so part of jB2 is associated with each end stub of the two double-stub tuners even though only one physical stub is present in the center. The triple-stub-matching system has more free variables so it can be optimized to increase the bandwidth. E L E M ^ ^ A N 0 1 3 MATCHING WITH L U M P E D Lumped-parameter elements such as conventional inductors and capacitors are not compatible with coaxial transmission lines and waveguides and consequently are generally not used with these structures for impedance matching. However, the widespread use of microstrip transmission lines and the minaturization of inductors and capacitors have brought lumped-parameter elements into prominence for impedance-matching purposes. In a nucrostrip-line, series connections of capacitors and inductors are easily made. The shunt connection of a capacitor or inductor is somewhat more difficult since a connection through the substrate to the ground plane must made. In the frequency range up to several gigahertz, the use of lumped capacitors and inductors for impedance-matching results in a more compact circuit. 320 FOUNDATIONS FOR MICROWAVE ENGINEERING Air bridge Microstrip line (b) (a) F I G U R E 5.15 ( a ) A spiral inductor connected in shunt across a microstrip line; (6) a series-connected spiral inductor; (c) a series-connected single loop inductor. For a microstrip circuit the most common form of inductor is the spiral inductor shown in Fig. 5.15a. For a shunt connection the center of the spiral can be connected to the ground plane. For a series connection an air bridge has to be used as shown in Fig. 5.156. If only a small inductive reactance is required, a one-turn loop as shown in Fig. 5.15c may be used. 11 a spiral inductor is to behave as a lumped inductor, its total length must a a small fraction of a wavelength. An estimate of the inductive reactance can be obtained by considering the inductor to be a length / of a transmission line. By using the formulas for the characteristic impedance and propagation constant, namely, T = y- p = cojw we obtain aiL = fiZc = k0Zc0 Sl/m As can where e0 is where Z 6c0 is the the characteristic characteristic impedance impeaance with wicn air air as as the tne dielectric aieiw-w'"- -7 ^j be seen from this expression, a high-impedance line (narrow width) s be used in an inductor. As an example, if we have a 100-fi l m < v i 1-mm-thick substrate with an effective dielectric constant of 4 at 2 Gnz> IMPEDANCE TRANSFORMATION AND MATCHING 321 get wL = — y ^ X 100 = —- X 200 = 83.8 ft/cm A0 ' 15 The inductance L is 6.67 n H / c m . A three-quarter-turn loop with a mean diameter of 1 cm would have an inductance of about 16 nH. Unfortunately, a single-turn loop this long would not function as an ideal inductor because every printed-circuit inductor has distributed capacitance associated with it. We can gain some insight into the length restriction by treating the singleturn loop as a short length of a high-impedance transmission line. With reference to Fig. 5.15c, let Z 0 be the characteristic impedance of the microstrip lines, let Z c be the characteristic impedance of the transmission line making up the inductor, and let Bl be the electrical length of the loop. Transmission-line theory gives Z0 + jZet in c Zc+jZ0t = Z0(Ze +JZ0t) + jt(Z? - Zj) ' Zc+jZ0t jt(Z?-Z*)(Ze-JZ0t) 0+ z* + z$ts where t = tan Bl. Since we normally would use a large value of Z r and keep Bl small, we will assume that Z%t2 « Z,2 in which case we get Zin = Z 0 + Z0t> Zl 1 - a- | \+jZct 72 ZJ For a short-circuited line, jXL ~jZc tan Bl=jZcBl Hence 72 Zin=Z0+Z0t* l~-%\+jX,\l- Thus the effective inductance is reduced by the factor 1 - Z%/Zf. A more serious departure from an ideal inductor behavior is the change in the real part of Zm. For an ideal series-connected inductor, we would have Z^ = o +J^L- ^ n order that the actual Z m should approximate this ideal result, we need to make the factor t \ \ - Z%/Z*) of the order of 0.05. In a typical situation where Z 0 = 50 ft and Z c = 100 ft, this requires that Bl be no arger than 0.26. The corresponding maximum value for the length I is '_ " For the earlier example where e c = 4 and f= 2 GHz, we have = 1 5 / 2 = 7.5 cm. Thus I should be no greater than 0.3 cm. This leads to roaximum usable inductance of only 2 nH. We can, of course, use a larger u e for / b u t the change in the resistive part of Z in away from the ideal 322 FOUNDATIONS FOR MICROWAVE ENGINEERING Open circuit stub (a) (£>) Dielectric / FIGURE 5.16 ( a ) A short open-circuited stub; (b) an interdigital capacitor; (c) a metal-insulator-metal (MLMl capacitor; (rf) a chip capacitor soldered across a microstrip-line gap. value Z 0 must then be taken into account in the design of a circuit requiring a series inductance. The spiral inductor can provide larger values of inductance. For example, a five-turn spiral inductor approximately 1.4 mm in diameter, with a conductor width of 0.06 mm and spacing 0.038 mm, has an inductance of 25 nH at 2 GHz.t A short open-circuited stub as shown in Fig. 5.16a will function as a lumped capacitor connected in shunt across a microstrip line. The distributed capacitance of a microstrip line is typically in the range 0.2y^«. to ^/f7 p F / c m ; so a short stub is suitable for providing a shunt capacitance up to about 1 pF. At 4 GHz a 1-pF capacitor has a reactance of about 40 ft. The interdigital capacitor shown in Fig. 5.166 can provide a series capacitance up to several picofarads depending on the number of fingers used and then length. For monolithic microwave integrated circuits, the metal-insulator metal (MTM) capacitor shown in Fig. 5.16c is generally used. A capacitance up to 20 pF or more can be obtained since the insulator thickness can very small. For example, an insulator 1 mm by 1 mm and 10 /xm thick an having a dielectric constant of 10 provides a capacitor with a capacitance about 9 pF. For hybrid microstrip circuits, the chip capacitor illustrated ^ Fig. 5.16a1 is used. It is soldered in place and can provide a capacitance up 100 pF or more. t D . A. Daly, S. P. Knight, M. Caulton, and R. Ekholdt, Lumped Elements in fJiiCT0' Integrated Circuits. IEEE Trans., vol. MTT-15. pp. 713-721, December, 1967. IMPEDANCE TRANSFORMATION AND MATCHING i*r IX: ;S; Ze=* (a) Z,= 1 :B? (b) L-i '(. I 323 F I G U R E 5.17 Two basic lumped-elemetit matching circuits, ( a ) Circuit used when GL < 1; (b) circuit used when RL < 1. The basic lumped-element impedance-matching circuit is a circuit consisting of a parallel-connected and a series-connected reactive element as shown in Fig. 5.17. The topology of the circuit is such that it is commonly referred to as an L matching network (compare with the designation of T and FI for the impedance and admittance networks for a two-port network). For the circuit shown in Fig. 5.17a, the shunt element jBl is connected in parallel with the load and the series element jX2 is connected in series. This configuration can be_used to match any load admittance Y, having a normalized conductance G, < 1. The circuit shown in Fig. 5.176 can be used to match any load impedance having a normalized load resistance R t less than 1. For some load impedances both G L and R L are less than 1 and either circuit may be used. When the normalized admittance lies inside the G = 1 circle on the Smith chart, the corresponding normalized load impedance, which is the reflection of the normalized load admittance through the origin, will lie outside the R = 1 circle and hence can be matched using the circuit shown in Fig. 5.176. When Z L lies inside the R = 1 circle, YL will lie outside the G = 1 circle and can be matched using the circuit shown in Fig. 5.17a. When both Z, and YL lie outside the R = 1 and G = 1 circles, respectively, either matching circuit can be used. The required values of the matching elements are easily found using the Smith chart. The procedure to be followed is described below. From this the reader will easily understand the rationale that underlies the use of the Smith chart to solve the matching problem using lumped reactance elements. Since the Smith chart procedure uses normalized immittance parameters, the first preliminary step is to determine the normalized values of Y L and Z L by dividing by the characteristic admittance Y e or characteristic impedance Zc, respectively, of the input transmission line.t Case 1. The circuit in Fig. 5.17a is used. A match can be obtained only if GL s 1. If GL > 1 use the circuit in Fig. 5.176 (see Case 2). With reference to Fig. 5.18. 1. Construct the G = 1 circle rotated by 180". 2. From the point 7t add jB1 to move along a constant-conductance circle until the rotated G = 1 circle is intersected. There are two possible solutions. term immittance is used to designate either an impedance or admittance. 324 FOUNDATIONS FOR MICROWAVE ENGfNEERlNG FIGURE 5.18 Illustration of steps followed in designing the matching circuit in Fig. 5.1 la (Case 1). The new values of Y, are Y'L and Y[. Note that if Y, is inside the G = 1 circle, it cannot be moved to intersect the rotated G = 1 circle because adding jB only moves YL on a constant-conductance circle. 3. Reflect Y'h and Y[ through the origin to get the corresponding impedance values Z\ and Z"L. These lie on the ft = 1 circle since we made Y[ and Y£ lie on the rotated G = 1 circle. 4. Since Z', and Z"L lie on the ft = 1 circle, these impedance_points can be moved into the origin by subtracting a reactance jX' or jX"._The origin represents a matched condition. Hence the required value of jX2 is_either -jX' or -jX". The required value of jB^ is ./'(B' - BL) or j(B" - BL)The greatest bandwidth is obtained when the reactive elements are a small as possible so that the circuit Q is as low as possible. We will return to this point after we provide the steps to be followed using the other matchin circuit. C a s e 2. The circuit in Fig. 5.176 is used. A match can be obtained only U RL < 1. If RL > 1 the circuit in Fig. 5.17a must be used (Case 1). 1. Construct the ft = 1 circle rotated by 180° as shown in Fig. 5.19The 2. Add jX1 to Z L to move Z u to intersect with the rotated ft = 1 circlemr\Hnn alnno- a constant cnncfant, resistance r e s i s t a n c e circlp h f r o aare r e Iwn o l u t i o n s &L motion ia is along circle. TThere two ssolutions ZZ3. Reflect Z'L and Z"L through the origin to obtain Y'L and 7/,'. IMPEDANCE TRANSFORMATION AND MATCHING 325 FIGURE 5.19 Illustration of steps followed in designing the matching circuit in Fig. 5Mb (Case 2), 4. The admittance points Y{ and Y[ can be moved into the origin along the G = 1 circle by subtracting jB' and jB", respectively. Hence the required value of jB2 is -jB' or ~jB". The required value of jX, is j( X' - XL) or j(X"-XL). Analytic solutions for the required values of the matching elements are readily derived. The analytic solutions are given in Probs. 5.17 and 5.18 but the reader has to supply the derivations. Circuit Q a n d B a n d w i d t h When a complex load impedance has been matched to a transmission line with characteristic impedance Zc, the input impedance looking toward the load equals Zc. Thus the reactive elements present in Z L and the matching network make up a resonant circuit that is loaded by R L and Zc. A resonant circuit has a quality factor, or Q, that can be evaluated from the general definition <o( average stored electric and magnetic energy) Q power loss (5.22) At resonance the average stored electric energy in the capacitors equals the average stored magnetic energy in the inductors. Hence Q can be expressed as 2ioWc _ 2toWm (5.23) <? = 326 FOUNDATIONS FOR MICROWAVE ENGINEERING <3 C > RL (a) •'-I « B = flt: C >R, *L %L 4 = c | f f t (&) 0.707V„, F I G U R E 5.20 ( a ) Parallel LCR resonant circuit; (6) loaded resonant circuit; (c) frequency-response curve showing half-power bandwidth. The bandwidth of the circuit is the frequency band over which one-half or more of the maximum power is delivered to the load. This bandwidth is called the half-power or 3-dB bandwidth and is inversely proportional to the loaded Q of the circuit. In order to clarify the above concepts, we will analyze the paralle resonant circuit shown in Fig. 5.20. For the circuit in Fig. 5.20a, the voltage across R, is given by V= (5.24) Y-m GL+ja>C-j/a>L where G L = 1/RL. The resonant frequency of the circuit is given by IMPEDANCE TRANSFORMATION AND MATCHING 327 The input admittance can be expressed in the form ( OJ 2 - ft) yfa = GL + > C ^r When o> = « 0 then V = 7 , / G i = IgRL, which is the maximum load voltage hat can be obtained. When jwC{u>2 - o»g)/w2 =y"Gz,, we see from (5.24) that |V| = / / v^Gx,, and hence the power in i ? L is only one-half that at resonance. In terms of the circuit elements, the Q is given by RL u>0C (5.26) io0L O7. The reader can verify t h a t this is the same Q as would be found using the definition (5.23) (see Prob. 5.19). When Q > 10 the frequency w is close to &)„ over the useful band of operation; so we have Q ^ — T ^ R L O O C - — 2 o) and 'I- CO'Q = (OJ - o»„)(w + «o) » 2w(w - w„] Yi„ * GL +jwC 2w(w — w 0 ) 2(w — &>0) —2 = G ; , +7'o>0CIO UJi, = Gjl + 2JQ^~\ (5.27) where A&> = w — w 0 . From this expression it is readily seen that the 3-dB fractional bandwidth is given by Aw BW = 2Q—- = 1 <o0 2 2Q or BW = 1/Q since the half-power points occur when 2jQ Ato/(o0 = ±j. When the circuit is connected to a matched source as shown in Fig. 5.206, it is clear that V=— S (5.28) 2GL(l + 2jQLlu>/w0) where Q L = Q / 2 is the loaded Q of the circuit. Consequently, for the loaded circuit, the bandwidth is twice as great. A typical response curve showing the bandwidth is given in Fig. 5.20c. In an impedance-matching problem, there are generally two solutions available. If we want a narrowband design, we should choose the solution that gives the largest loaded Q. On the other hand, when we want a broadband match, we should choose the circuit with the lowest loaded Q. load impedance plus matching network will contain either one capacitor and two inductors or one inductor and two capacitors. Thus, when using —3) to evaluate the circuit Q, the energy stored in the single reactive e ment or that stored in the two opposite reactive elements must be 328 FOUNDATIONS FOR MICROWAVE ENGINEERING V\2r C ifRL • « i (a) FIGURE 5.21 Two matching networks. evaluated. We will illustrate the evaluation of circuit Q by means of s. examples that show the steps involved. By making use of the facts th ' resonance Z in = Z c and that P i n = PL, the evaluation of the circuit Q J' be reduced to a few simple steps. For the circuit shown in Fig. 5.21a, we will base the evaluation of the circuit Q on the energy stored in the capacitor CL. The power dissipated in RL is given by IIVJ PL-T: 2 RL The average electric energy stored in C L is Hence from (5.23) we get 2u>0We Q = —5— = UOCLRL The loaded Q is QL = Q/2 since the system is matched at w0. The operating bandwidth is determined by the loaded Q. For the circuit in Fig. 5.2lb, we note that 1 |V|S P = — » 2 Z, IIVJ = PL= TT 2 RL since Z^ = Z c at w = w 0 . The energy stored in C 2 is we2 = {ivi 2 c 2 while that stored in C L is wei = \\vJcL - \cL^\vf of IVIwhere we used the equality of Pin and P L to express |V£I in t^ IMPEDANCE TRANSFORMATION AND MATCHING 329 The circuit Q is given by 2« 0 (W e 2 + WeL) Q = PL o>0(C2 + (RL/ZC)CL)\V\ WL\A/RL = to 0 C 2 Z f + co0CLRL This circuit Q is larger than that for the circuit in Fig. 5.21a. In general, the smallest circuit Q is obtained when the matching circuit contains two similar reactive elements that are opposite to the reactive element in the load- i.e., if the load contains an inductive element, the matching circuit should be made up of capacitive elements only, and vice versa. For circuits with a loaded Q of 5 or more, the frequency response is very nearly the same as that of the parallel resonant circuit shown in Fig. 5.20, over the useful operating frequency band. Thus the 3-dB fractional bandwidth is !/<?/,- For circuits with a low Q, the frequency behavior is different but similar. The discussion above has been based on the assumption that idea) lossless inductors and capacitors are used. As a general rule, lumped capacitors have very little loss and tbe assumption of negligible loss does not introduce significant error. However, lumped inductors do not have negligible loss, and in high-Q circuits inductor loss should be taken into account. The Q of an inductor is the ratio of the reactance o>L to the series resistance R of the inductor. Lumped inductors may have Q values in the range of 25 to several hundred. If the loaded circuit Q is less than about one-fifth of the inductor Q, then neglecting the loss in the inductor will not produce serious error. When several inductors are used in a matching circuit, the losses in the inductors should be taken into account since these losses can account for a significant fraction of the total power delivered to a circuit. For microwave circuits the useful bandwidth is often much smaller than the 3-dB bandwidth. It is not uncommon to require an impedance match providing a VSWR of no more than 2, or even less than 1.5 in critical applications. A VSWR of 2 corresponds to a reflection coefficient of 0.333. With this mismatch a fraction of 0.889 of the incident power is delivered to the load. The degree of mismatch is usually described in terms of the input VSWR or in terms of the return loss. The return loss is the ratio of the reflected power to the incident power, expressed in decibels; thus VSWR- 1 Return loss = RL = - 20 log p = - 20 log — S W R + 1 A well-matched system will have a return loss of 15 dB or more. A VSWR equal to 2 gives a return loss of 9.54 dB, while a VSWR of 1.5 gives a return loss of 13.98 dB. Thus a return loss of 15 dB, corresponding to a VSWR of • 4, J> does indeed represent a well-matched system. 330 FOUNDATIONS FOR MICROWAVE ENGINEERING In order to get a feeling for how narrow the bandwidth of using this criterion, let us assume that the loaded Q equals ' i a « y S t e n i i* VSWR = 1.43, we have p = 0.178. We can use ' ^ e n tL the A co 1 + 2jQ— a>0 Yin m Ye Y - Y P = Y, + Y.. 2QAco/co0 Aw where Q = 2QL. Hence the fractional bandwidth is Aw 2 to Q 2p p = — = — - 0.0356 Q QL This is much smaller than the 3-dB bandwidth which equals 1/Q or 0 9 5.6 DESIGN OF COMPLEX IMPEDANCE TERMINATIONS In microwave amplifier design, we often require a complex load impedance for the output and a complex source impedance for the input even though the final input and output connections are to 50-fl lines. In a typical amplifier circuit as shown in Fig. 5.22a, the function of the input and output matching networks is to transform the 50-fi line impedances into the required complex source impedance Zs and load impedance ZL. In a broadband amplifier design, the design of the matching networks can be quite complex. In narrowband amplifier design, elementary matching networks can be used. The matching problem for an amplifier is the reverse of that f matching a complex load impedance ZL to a transmission line. The t step is to choose a network topology to be used. In Figs. 5-226 to d we ithree possible networks that could be used. Many other choices caI} made. We will illustrate the use of the Smith chart to design m a t c H networks of the type shown in Fig. 5.22. The methods used can be exta to more complex networks. Design procedure for the circuit in Fig. 5.226. The matching netw^ ^ Fig. 5.226 consists of a transmission-line stub placed at a distance ^ git input. The stub may be either an open- circuited or a s h o r t - c i r c u r ^ ^ c e lumped susceptance since its function is to provide a normaliz j„pUt. ^e ra jBi connected across the transmission line at a distance l_ fr° following procedure should be used to determine / and B\1. Locate the_ required value of ZL on the Smith chart ani reflecting ZL through the origin. This procedure is illustra for the case where ZL = 0.4 -J0.2. find ? ^^ 5 IMPEDANCE TRANSFORMATION AND MATCHING Amplifier Zc 331 Zc 1 — i Output matching network Input matching network ,5. ZL G= 1 (o) jXs A. ,8. ZL (c) <3 = t G~ t 7»1 m FIGURE 5.22 (a) Microwave amplifier circuit; (6) transmission-line matching network; (c) and (d> alternative matching networks. 2. Rotate the point Ytj on a constant-radius circle until it intersects the G = 1 circle. There are two possible solutions. From the angle of rotation the normalized length l/\ is found. Note that the rotation is in a counterclockwise direction since we are mooing toward the load. 3. When we move to the right of the point where jBx is connected, Y m decreases by jBl to become Yjn ~ I + jB - jBv But to the right of jBv we require Y^ = G = 1 so Bx = B. Thus the values of jB on the G = 1 circle, where the rotated Y L intersected the circle, gives the required values of jBx for the two possible solutions. Note that if we begin at Ym = 1 and move left across JBU then Yin changes to 1 +jBx = 1 +jB. A rotation in the clockwise direction through an angle 2/3/ then makes J^ = YL. Thus using the reverse of the usual matching procedure results in the desired value of YL. For our specific example, / = 0.0626A or 0.4375A and the corresponding values required for jBx are jl or -_/l, respectively, as shown in Fig. 5.23. 332 FOUNDATIONS FOR MICROWAVE ENGINEERING FIGURE 5.23 Illustration of design procedure for the matching network in Fig. 5.226. The analytic solution for the above matching network is Bl + Gl + 1- 2GL B, = ± 1/2 G, tan pi = = !«/* R'i+Xl+l- 2RL (5.29a) R, GL-1 BL + BfiL Rl+Xt-RL (5.296) XL-B1RL D e s i g n procedure for t h £ circuit in Fig. 5.22c. The circuit shown in 5.22c can be used only if R L < 1. The following steps should be folio" design this circuit using a Smith chart: Fig^ 1. Construct the R = 1 circle rotated by 180°. 2. Locate the desired impedance Z L on the Smith chart. g * 3. Move Z L in a counterclockwise direction (toward the load) on t ^ ^g. constant circle until it intersects the rotated R = 1 circle as sho ^ . ^ a 5.24. There are two possible solutions. Therequired value for th e ' jXl is given by jXt =jXL -jX, where jX equals jX' or j* IMPEDANCE TRANSFORMATION AND MATCHING 333 FIGURE 5.24 Illustration of design procedure for the matching network in Fig. 5.22c. value obtained where Z L intersects the rotated R = 1 circle. The two possible values of jX are always equal in magnitude but opposite in sign. 4. Reflect the two new values of Z^ through the jprigin to obtain the corresponding values for F i n . This gives Vln_= 1 +jB. As we move to the right of the element jB2, we obtain Yin - jB2, which must be equal to 1. Hence the required value of jB2 is equal to jB. The reactance jXx and susceptance jB2 can be realized using lumped elements or transmission-line stubs. Design p r o c e d u r e for t h e circuit in Fig. 5.22<f. The matching network shown in Fig. 5.22d can be used only when GL < 1 (can you explain why?). The network can be designed using the Smith chart by following the steps given below. 1. Construct the G = 1 circle rotated by 180°. 2. Locate the desired load admittance point Y L (reflection of Z L through the origin). 3. Move Y L on the G = G L circle in a counterclockwise direction until it intersects the rotated G = I circle. There are two possible intersections. The required value of jBl is given by jBx = j(BL - B), where jB equals jB' or jB", as shown in Fig. 5.25. 4. Reflect the admittances 7{n and Y"n through the origin_ to get the corresponding impedance points Z'm and Z"„ that lie on the R = 1 circle. 5. When we subtract jX2 we must get Zm = 1, so jX2 =jX, where jX equals JX' or jX". 334 FOUNDATIONS FOR MICROWAVK ENGINEERING F I G U R E 5.25 Illustration of design procedure for the matching network in Fig. 5.22d. 5.7 INVARIANT PROPERTY OF IMPEDANCE MISMATCH FACTOR Consider t h e network shown in Fig. 5.26 a. The T network between the load impedance Z L and the source will be assumed to be lossless. The impedance looking into the network toward the load is Z m . For this network the input current is V. 1= z , + zin and the input power is P- = — 2 zs + zm Rm = gUlX. If Zi0 = Z* then Rm = Rs, Xio = ~XS, and we get maximum power t fer from the source to the load resistance RL. This power is P = IJKTfita 1.1V/ 2 2 4R< \Rin + Rs (5-30) = P- hav« 7 * Z* w e and is called the available power from the source. When Lxn * ^oTta an impedance mismatch and P m < P a v a . We can express P m m (5-31) 1 IVJ l WfR, *R*R-m — * avaM P„ = 2 \ZS + Z-\° 2 AR„ mismatch j 2 where M = 4RinRj\Zin + ZJ is called the impedance IMPEDANCE TRANSFORMATION AND MATCHING 335 ?-•• M (b) ML ava. L ava. 1 (,c) F I G U R E 5.26 (a) A T matching network; (6) Thevenin equivalent network; (c) a cascade connection of lossless networks for which P o v o , s = PKV» i = P a v a 2 = • • • P a v o ./. a n ^ W = M, = Af2 = • • = M,.. We can use the Thevenin equivalent circuit shown in Fig. 5.266 to calculate the power delivered to RL, which is P L and must equal Pin since the network with elements Z u , Z 1 2 , Z 2 2 is lossless. From circuit theory we find that V = 00 if.* Z;.= zn+z, A + Z22ZS Zll + Zs where A = Z , , Z 2 2 - Z 2 2 = Xfz - XUX22, upon using Z„ = . / X u , JX22, Z12 =jXl2. The available power from the network is UVJ 2 4fl, where ZT = RT + jXT The power delivered to R L is i IVJ 2 4RTRL P, = 2 4i?r |Zr + Z J 2 i iv; 2 4i2 -M, = P.. Z22 = 336 FOUNDATIONS FOR MICROWAVE ENGINEERING We will now show that the power available from the netw 0rlc that available from the source, i.e., ^u^ WJ 8fl, 8fl, We must then have iv/ M 8RS - = p in IK, 12 = 8R1 ir^ML~pL ^ ^ ^ ^ ^ ^ ^ which gives M = ML, that is, the impedance mismatch at the output I equals that at the input plane. The Thevenin resistance is mvfn given k„ by ] Z.„Z if — xvtr £j-r- 2 U + Zs We can express Z T in the form = T (A + Z ^ Z j t Z f . + Z * ) (A + Z 2 2 Z J ( Z * +Z S *) {Zn + ZMZ?i + z?) \zu+zf The numerator equals (X? 2 - XnX22 - X^Xs+jX22Rs)( -jXu -jXs + S,) and has a real part which is readily found to be simply Xf2Ra. Hence RT = X*2Rs/\Zn + Zf. We now get \yj 8R j. x&vf \Zn+Z/8Rr 8RS =p Since the available power at the output plane is the same as that from « source, the impedance mismatch M L = M. If we have several lossless networks (reciprocal) between the s and Z L as shown in Fig. 5.26c, we can repeat the analysis using Thevenin equivalent circuits at each stage to find t h a t at any plane the ava power and impedance mismatch are invariant quantities. ^en If the networks are not reciprocal, i.e., are linear active networ , ^ ^ the impedance mismatch is not invariant, e.g., an amplifier supp output power than is fed in at the input. etw°r'c The invariance of the impedance mismatch factor in a lossless jtj. places an important constraint on interstage matching networks i ^e stage amplifier. The design of the interstage matching network n this constraint into account. tra118'5 In Fig. 5.27 we show a multistage microwave amplifier. oJ» ggnet^ut tors at microwave frequencies are often not absolutely stable, ' 0 utp m u e not possible to use conjugate impedance matching at the P neinto ^ of each stage. For the first stage let Z jn . x be the impedance looW IMPEDANCE TRANSFORMATION AND MATCHING Input matching network Matching network Amplifier 1 337 Amplifier 2 , i Zc 1 PZ> Zsy Zlr , 1 ^oul. 1 Z\_\ Zs2 Z-m ? FIGURE 5.27 A multistage microwave amplifier. first amplifier and let Z s l be the source impedance presented to this stage by the input matching network (the networks are called matching networks even though they do not produce conjugate impedance matching). The input impedance mismatch is M, This same impedance mismatch must exist between the input transmission fine and the input to the matching network. Hence on the input line we will have a reflection coefficient p that satisfies the constraint 1 - p2 = M, Thus p = (1 - M x ) 1 / 2 and the input VSWR will be 1 + ^1 - M , VSWR = 1 - ^ 1 - Af" (5.32) Consequently, an input VSWR greater than unity is unavoidable when conjugate impedance matching cannot be used. A resultant problem in microwave amplifier design is to find an optimum load impedance that will result in an input impedance ZmA that is close to the complex conjugate of the optimum source impedance Z s l so as to minimize the input VSWR. The optimum source impedance is constrained by the requirement for a low noise figure for the input stage. For the interstage network the matching network must transform the >nput impedance Z j n 2 into the required load impedance Z £ 1 for stage 1 and simultaneously transform the output impedance Z o u l , of stage 1 into the required source impedance Z, 2 for stage 2. The design of the matching network is constrained by the condition t h a t the impedance mismatch be 338 FOUNDATIONS FOR MICROWAVE ENGINEERING 'out. I Z,= 1 ye, J zc=i ZC=A \ JB2 jB, I m F I G U R E 5.28 ( a ) A II matching network; (6) and (c) transmission-line matching networks. constant; thus 4fli„. 8 -B.2 4 * z . i * out, 1 I Z ^ + ZOUMI 2 12*,,, (5.33) 's2* The design of microwave amplifiers is described in detail in Chap, l As part of the design procedure, the four impedances ZLl, Zmli\. Z^.* a Zs2 are determined so that the impedance mismatch constraint given j (5.33) is satisfied. Hence we can assume that we know the admittances Y y out,i- y in.2. and Ys2. In terms of these self-consistent quantities, we c^ determine the parameters of the U matching network illustrated 5.28a. By conventional circuit analysis, we readily find that y yr 2 '.i - ? u " • '22 !2 + *in,2 2 y,12 * v9 ~ Y9.9 =*11 + *out.l 9 Yn=JBz2: After For a lossless network Yu =jBu, Yl2 = jBl2, and equate the real and imaginary parts of the above two e< number of algebraic steps, the following pair of equations IMPEDANCE TRANSFORMATION AND MATCHING 339 can be derived: {Gs2 + S L . » ) B n - (G o u t ., + GIA)B22 = GLlBin,2 + BLlGm2 - Bs2GoutA (Bs2 + Bin_2)Bn - (BIA - Gs2BuuiA (5.34a) + BimlA)B22 = G s 2 G o o t ,, - GLlGm,2 + BL,Bm.2 - B s 2 B 0 l l l | 1 (5.346) After B n and B 2 2 have been found, we use B\2 = BJK*.i + (Bn + BuulA)(B22 - Bs2) (5.35) to find B 1 2 . The expressions for Y u and Ys2 generate four equations which would generally not allow a FI network to be determined unless (5.33) was also satisfied. Many matching networks can now be designed by relating their parameters to those of the Tl network. Two possible transmission-line networks are shown in Figs. 5.286 and c. The parameters of the network in Fig. 5.28 are given by csc/» = |S12| (5.36a) jBx = j( B , , + cot pi) (5.366) jB2 = j( B 2 2 + cot pi) (5.36c) where jBl and jB., are the stub susceptances. This network can only be used if |Bi 2 | > 1 since esc pi > 1. When | B 1 2 | < 1 the network shown in Fig. 5.28c can be realized. Its parameters are given by jB, =j(Bu+Bi2) (5.37a) jB2=j(B22 + B12) (5.376) T*. = Bl2 (5.37c) where Y^ is the characteristic admittance of the center one-quarter-wavelength-long section. WAVEGUIDE R E A C T I V E E L E M E N T S In the place of transmission-line stubs, any other element that acts as a shunt susceptance may be used for the purpose of matching an arbitrary load impedance to a waveguide or transmission line. A number of such reactive elements for use in rectangular waveguides supporting the dominant TE 1 0 propagating mode are described in this section.! The formulas given for the normalized susceptance of these elements are approximate tDetailed information on susceptance values and equivalent circuits are given in N. Marcuvitz (ed.), "Waveguide Handbook," McGraw-Hill Book Company. New York, 1951. 340 KOI INDATIONS FOR MICROWAVE ENGINEERING a (4) -*i <r*t b la/2, \ mm a U) • d) FIGURE 5.29 Shunt inductive elements, ( a ) Symmetrical d (/>) asymmetrical diaphragm; (c) thin cireular'aP small circular aperture. ones, with accuracies of the order of 10 percent or better. The derivatio these formulas requires the detailed solution of boundary-value probk and is outside the scope of this text.f S h u n t Inductive E l e m e n t s Figure 5.29 illustrates a number of rectangular waveguide elements that act as shunt inductive susceptances for the TE, 0 mode. These consist of thin metallic windows extending across the narrow dimension of the guide as in Figs. 5.29a and b, a very thin cylindrical post as in Fig. 5.29c, and a small circular aperture as in Fig, 5,29rf. When a TE U , mode is incident on any of these discontinuities, evanescent T E „ 0 modes are excited in order to provide a total field that will satisfy the required boundary condition of a vanishing tangential electric field on the obstacle. These nonpropagating modes stoi predominantly magnetic energy and give the obstacle its inductive chart teristics. Approximate values for the normalized inductive susceptance of the: obstacles are: For Fig. 5.29a. _ 2TT -d I ay-, - 377 rrd B = — cot 2 — 1 + -^ sin 2 pa where p = [&2 - (TT/O)2]"* 2a I 4TT (5.38) a and y 3 = [(3ir/a) 2 - A 2 ,] 1 ' 2 - For Fig. 5.29b, _ 2TT . ird I „ ted B = — cot 2 -— 1 + esc' — /3a 2a \ 2a tN. Marcimte, he. eft. piartl taway R. E. Collin. "Field Theory of Guided Waves," 2nd. ed., IEEE Press, n s c L. Lewin, "Theory of Waveguides." Newnes-Butterworth, London. 19 75 " 5.39> (S IMPEDANCE TRANSFORMATION AND MATCHING tfSVMt / •WW/,'//?. 341 w ; -.' a rwi F I G U R E 5.30 Shunt capacitive elements, (u) Asymmetrical capacitive diaphragm: (6) symmetrical diaphragm; (c) capacitive nid; (d) capacitive post. WW a id) For the thin inductive post of Fig. 5.29c, 4- 15.40) where y„ = [(n~/a)'2 - A5]1'- and t is the post radius. For the small centered circular aperture of Fig. 5.29d, B = Sab (5.41! 8fR Shunt C a p a c i t i v e E l e m e n t s Typical shunt capacitive elements that may be used for matching purposes are illustrated in Fig. 5.30. Those consist of thin metal septa extending across the broad dimension of the guide to form capacitive diaphragms as in Figs. 5.30a and 6, a thin circular rod extending across the guide as in Fig. 5.30c, and a short thin circular post extending into the guide as in Fig. 5.30d. The post illustrated in Fig. 5.30a" behaves more like an LC series network connected across a transmission line. When the depth of penetration is between 0.76 and 0.96, it becomes resonant and acts almost like an ideal short circuit. For lengths greater than this resonant length, the post is equivalent to a shunt inductive susceptance. Actually, for a post of finite thickness, the equivalent circuit is a T network, but for small-diameter posts, the series elements in this T network are negligible (for post diameters less than about 0 .05a). Approximate expressions for the normalized susceptance of the obstacles illustrated in Fig. 5.30 are: For the asymmetrical diaphragm of Fig. 5.30a, 77 -a d 1 s = i ^ In esc •—• + 26 where 0 = [k\ - (TT/O)2]' 2 and y, - \byl -d \ 1 cos' ) 26 = [{v/bf - 0"]' 2 . (5.42) 342 FOUNDATIONS FOR MICROWAVE ENGINEERING For the symmetrical diaphragm of Fig. 5.306, S = ^ 2 2v vd I I 2ir In esc —— ++ 2b \b^' \ -ffd 1 cos 4 T 2b (5, 2 1/2 where y2 = [{2ir/b) - fi } . For the capacitive rod of Fig. 5.30c and the post illustrated • 5.30d, no simple approximate formulas are available. Analytical e • ln -^ for the T-network parameters for the capacitive rod are given by L e ^ 8 ' 0 " 8 n are not reproduced here.t ' "Jut The inductive and capacitive properties of the waveguide oh described above are determined by the electric and magnetic energy! th evanescent modes excited by the obstacle. For example, the inductive oh t cles discussed only excite TE evanescent modes and these modes store m magnetic energy than electric energy. The evanescent modes are nonpropa*. gating modes so they represent the local field in the vicinity of the obstacles The evanescent modes are excited with amplitudes such that the combination of all of the evanescent modes with the incident, reflected, and transmitted dominant mode, produces a total field with a zero tangential electric field at the surface of the obstacle. There is no way to assign a unique inductance or capacitance to these obstacles since there is no unique way to define the voltage, current, or characteristic impedance for a waveguide. Nevertheless, these obstacles do produce a reflected wave with a reflection coefficient that is correctly determined in terms of the equivalent normalized reactance and susceptance of the obstacle as connected into the equivalent-transmission-line model of the waveguide. Waveguide Stub Tuners An approximate equivalent of a single-stub matching network is the s screw tuner illustrated in Fig. 5.31a. This consists of a variable-depth sc mounted on a sliding carriage free to move longitudinally along the g u ^ over a distance of at least a half guide wavelength. The screw P® .^ into the guide through a centered narrow slot in the broad wall ( This slot is cut along the current flow lines so that it has a n ^ ^ perturbing effect on the internal field. Since the position of the ^ ^ adjustable over at least a half guide wavelength, its penetration ^ ^^ need to be so great that it will behave as an inductive element; ' " ^ ^vie* can be obtained with a shunt capacitive susceptance in all cases, of single-stub-matching theory will verify. - tOp. cit., chap. 2. This text contains many excellent and instructive deri circuit parameters for a variety of waveguide structures. <>(<*>tfi** IMPEDANCE TRANSFORMATION AND MATCHING Sliding corrioge 343 .2\ •> \ B 9 6 9 Variabledepth screw or post id) Adjustable short circuits, f-plone orm Input I Output F I G U R E 5.31 Waveguide tuners, ( a ) Slidingscrew If) tuner; (6) triple-screw tuner; (e) E-H tuner. Three variable-depth screws spaced a fixed distance of about 3 A g / 8 apart as in Fig. 5.316 are essentially equivalent to a triple-stub tuner and can match a large variety of loads even though the range of susceptance values obtainable from a single screw is limited. Circuits that are physically more like the actual short-circuited transmission-line stub are also possible. Figure 5.31c illustrates combined Eplane and /f-plane stubs, referred to as an E-H tuner. The positions of the sliding short circuits in the E- and H-plane arms are variable, so that a wide range of load impedances may be matched. The equivalent circuit of either an E-plane or H-plane junction is, however, much more elaborate than a simple shunt- or series-connected transmission line because the junctions are of the order of a wavelength in size and hence produce a very complicated field structure in their vicinity. Nevertheless, since no power now is possible through the arms terminated in the short circuits, these do still provide adjustable reactance elements t h a t may be used for matching purposes. WAVE T R A N S F O R M E R S quarter-wave transformers are primarily used as intermediate matching sections when it is desired to connect two waveguiding systems of different aracteristic impedance. Examples are the connection of two transmission e s with different characteristic impedances, connection of an empty wave- 344 FOUNDATIONS FOR MICROWAVE ENGINEERING Zc = Z> Zr = Z, ZL V« ~? H FIGURE 5.32 A quarter-wave transformer. guide to a waveguide partially or completely filled with dielectric of two guides of different width, height, or both, and the match11**1'011 dielectric medium such as a microwave lens to free space. If a match"8 ° f a narrow band of frequencies suffices, a single-section transformer <^*x* used. To obtain a good match over a broad band of frequencies, two three or even more intermediate quarter-wave sections are commonly used Th optimum design of such multisection quarter-wave transformers is * sented in this section. The essential principle involved in a quarter-wave transformer is readily explained by considering the problem of matching a transmission line of characteristic impedance Z, to a pure resistive load impedance 2,, as illustrated in Fig. 5.32. If an intermediate section of transmission line with a characteristic impedance Z., and a quarter wavelength long is connected between the main line and the load, the effective load impedance presented to the main line is ZL+jZ2tan(PA/4) 2 _Zj Z2+jZLtan(0A/4) ZL If Z 2 is chosen equal to }/Z^ZL~, then Z = Z, and the load is matched to the main line. In other words, the intermediate section of transmission line i length A/4 transforms the load impedance Z L into an impedance Z\ a hence acts as an ideal transformer of turns ratio y}Zx/ZL . A perfect i is obtained only at that frequency for which the transformer is a qua1 wavelength (or n A / 2 + A/4) long. / Let 8 be the electrical length of the transformer at the frequency ^ that is, p(f)l = 0, where the phase constant p has been ^ " t t e ? / r # function of frequency. For a TEM wave in an air-filled line, pi - 27r> any frequency the input impedance presented to the main line is ZL+jZ2t Z (545) '»-Z2Z2+jZLt where t = tan 0 = tan pi. Consequently, the reflection coefficient Z.n ~ ' zm + z, ZL ZL + Zx Zi(ZL'Zl)+jt(Zj_-LZM, z2(zL + zx) + jt(zi + zxzL) ~ Zx Zl+jt2yJZ^l (5.46) IMPEDANCE TRANSFORMATION AND MATCHING fll=6 8m'"h 345 FIGURE 5.33 Bandwidth characteristic For a single-section quarterwave transformer. The latter form is obtained by using the relation Z\ = ZXZ,. The magnitude of F, denoted by p, is readily evaluated and is given by \ZL - Zx\ P = [(ZL + ZX)Z + U*ZXZL\ 1/2 1/2 1+ (5.47) 2v^zT sec0 For 0 near TT/2, this equation is well approximated by \ZL -Z,\ ICOS0 P = 2/Z^ (5.48) A plot of p versus 0 is given in Fig. 5.33, and this is essentially a plot of p versus frequency. The variation of p with frequency, or 0, is periodic because of the periodic variation of the input impedance with frequency; i.e., the impedance repeats its value every time the electrical length of the transformer changes by ir. If p m is the maximum value of reflection coefficient that can be tolerated, the useful bandwidth provided by the transformer is t h a t corresponding to the range A0 in Fig. 5.33. Because of the rapidly increasing values of p on either side of 0 = rr/2, the useful bandwidth is small. The value of 0 at the edge of the useful passband may be found from (5.47) by equating p to pm; thus 0„ = cos - l 2PmJZ\ZL (5.49) (2L-2I>I/WI !n the case of a TEM wave, 0 = /3/ = (f/f 0 Y.ir/2), where f0 is the frequency for which 0 = TT/2. In this case the bandwidth is given by A/-=2(f 0 -f m ) = 2 ( / - 0 - ^ X ) 346 FOUNDATIONS FOR MICROWAVE ENGINEERING and the fractional bandwidth is given by A/" — = 2/o 4 -cos'1 "" ( Z / . - Z i ) ^ ! ^ (5.50, where that solution of (5.49) that gives 0 m < TT/2 is to be chosen Although there are a number of instances when the bandudrk vided by a single-section transformer may be adequate, there a" number of situations in which much greater bandwidths must be ^ for. The required increase in bandwidth can be obtained by using rjT tion quarter-wave transformers. The approximate theory of these mul tion transformers is discussed first, in order to develop a theory that has application in the design of other microwave devices, such as directir couplers and antenna arrays. This is followed by a discussion and present; tion of results obtainable from a more exact analysis. It should be noted that in the previous discussion it was assumed that the characteristic impedances Z, and Z 2 were independent of frequency. For transmission lines this is a good approximation, but for waveguides the wave impedance varies with frequency, and this complicates the analysis considerably. In addition, for both transmission lines and waveguides, there are reactive fields excited at the junctions of the different sections, brought about because of the change in geometrical cross section necessary to achieve the required characteristic impedances. These junction effects can often be represented by a pure shunt susceptance at each junctions The susceptive elements will also vary the performance of any practical trans former from the predicted performance based on an ideal model when junction effects are neglected. In spite of all these limitations, only theory for ideal transformers is developed here; i.e., junction effects an frequency dependency of the equivalent characteristic impedances glected. Thus the theory given will only be indicative of the perfor that can be obtained in the nonideal case.J IRE Tro"*-to1 t S . B. Cohn. Optimum Design of Stepped Transmission Line Transformers. ^ Q MTT-3. pp. 16-21, April. 1955. This paper presents an approximate theory^ ^ ^ transformers, together with a method of accounting for the reactances introdu tFor typical application to waveguide transformers, see: for firs R. E. Collin and J, Brown, The Design of Quarter-Wave Matching Lay 47 Surfaces, Proc. IEE, vol. 103. pt. C, pp. 153-158, March, 1956. *T M V r ^ pp L. Young. Optimum Quarter-Wave Transformers, IRE Trans., vol. M* ' j o n g . iWSeptember. 1960; also Inhomogeneous Quarter-Wave Transformers of Two 645-649. November. 1960. noubl*' f M E. S. Hensperger. Broad-Band Stepped Transformers from R e c t a n g u l a r Waveguide, IRE Trans., vol. MTT-6, pp. 311-314, July, 1958. IMPEOANCK TRANSFORMATION AND MATCHING o R Y 347 O F SMALL REFLECTIONS As a preliminary to the approximate analysis of multisection quarter-wave transformers, some results pertaining to the overall reflection coefficient arising from several small reflecting obstacles are required. Consider the case of a load impedance ZL connected to a transmission line of characteristic impedance Z, through an intermediate section of line of electrical length 81 = 0 and characteristic impedance Z 2 , as illustrated in Fig. 5.34. For each junction the reflection and transmission coefficients are Z, ~ Z, ?i z, + z, T2l = i + r, = r2= - r , 2Z, TV, = i + r2 = z, + za 2Z, z2 ZL - Z, r 8 = z L + z2 A wave of unit amplitude is incident, and the total reflected wave has a complex amplitude V equal to the total reflection coefficient. When the incident wave strikes the first junction, a partial reflected wave of amplitude T[ is produced. A transmitted wave of amplitude T2l is then incident on the second junction. A portion of this is reflected to give a wave of amplitude r 3 r 2 1 e - 2 - ' 8 incident from the right on the first junction. A portion Tv2T2ir3e~2J,> is transmitted, and a portion I'^/r^T^e'2'" is reflected back toward ZL. Figure 5.35 illustrates the first few of the infinite number of multiply reflected waves that occur. The total reflected wave of amplitude 1 is the sum of all the partial waves transmitted past the first junction toward the left. This sum is given by r = r, + T 12 T 21 r 3e - 2 ^ + r12r21r;?r2e-4-"' + • • • = F, + Tl2T2ir3e-W £ n r 2 'T 3 "e^"" 0 01 = 6 2J r, -5 C r , . •' F I G U R E 5.34 A microwave circuit with two reflecting junctions. 348 FOUNDATIONS FOR MICROWAVE ENGINEERING l^r,*-**r*r„rtrl*-*- '21 ' 3 ' r2,r2r5f-^ • /•ar|r|*-w F I G U R E 5.35 Multiple reflection of waves f a r . curt with two reflecting junction* This geometric series is readily summed to give [note that F x « n (1-r)-1] '°r Tl2T.nr3e-W r = i\ i - r 2 r 3 e- 2 ^ Replacing T 12 by 1 + f2 = 1 - I^ and T2l by 1 + I\ gives r, + r3e - 2 7 0 r= 1 + r^e" 1 ^ (5.51) If | r j and |T 3 | are both small compared with unity, an excellent approximation to T is r = r, r3e-2-"' (5.52) This result states that, for small reflections, the resultant reflection c cient is just that obtained by taking only first-order reflections into accoun^ This is the result that will be used to obtain a first-order t h e o r y ^ multisection quarter-wave transformers. As an indication of the acci docs the approximate formula, note t h a t if If,I = |T 3 | = 0.2, the error in not exceed 4 percent. 5.11 APPROXIMATE THEORY FOR MULTISECTION QUARTER-WAVE TRANSFORMERS Figure 5.36 illustrates an AT-section quarter-wave transformer junction the reflection coefficient is ^1 Z1+Z0 . At the fir?' (D.530) ZQ = Po IMPEDANCE TRANSFORMATION AND MATCHING 349 F I G U R E 5.36 A multisection quarter-wave transformer. Similarly, at the nth junction, the reflection coefficient is (5.536) rs- = P„ Z«+l+Zn The last reflection coefficient is z,.-z N " zL + z, N r = = (5.53c) PN Note that Z () is a characteristic impedance, and not necessarily equal to (fig/e0)i/2 here. Each section has the same electrical length fil = 6, and will be a quarter wave long at the matching frequency fQ. The load Z L is assumed to be a pure resistance, and may be greater or smaller than Z 0 . In this analysis it is chosen greater, so that all V n = p„, where p„ is the magnitude of f„. If Z, is smaller than Z„, all P„ are negative real numbers and the only modification required in the theory is replacing all p„ by — pa, For a first approximation the total reflection coefficient is the sum of the first-order reflected waves only. This is given by r = Po + Pi« -2jtt 4jll P 2e + +pNe -2J:V« (5.54) where e~2jn" accounts for the phase retardation introduced because of the different distances the various partial waves must travel. At this point it is expedient to assume that the transformer is symmetrical, so that p 0 = pN, p, = pN_l3 p 2 = p # _ 2 , etc. In this case (5.54) becomes T = e-' A W [ P o (e>"« + e~JN*) + Pi(e«N 2>" + e * * - « » ) + • • • ] (5.55) where the last term is p ( N _ 1 ) / a ( e ; 0 + e~->") for N odd and pN/;2 for N even. is thus seen that for a symmetrical transformer the reflection coefficient !s given by a Fourier cosine series: T = 2e'JNB[p0 cos NO + Pl cos( N ~ 2)6 + • • • + p„cos(N - 2n)0 + ••• ] 8h0 (5 (5.56) ^ P(N-i)/2 cos 0 for N odd and \pNn for N even. It be apparent that by a proper choice of the reflection coefficients «> and hence the Z„, a variety of passband characteristics can be obtained. ce t , l e series is a cosine series, the periodic function that it defines is 1 j5 6) t h e l a s t t e r m n0w 350 FOUNDATIONS KOR MICROWAVE ENGINEERING periodic over the interval - corresponding to the frequency r a n which the length of each transformer section changes by a half wavel °V The specification of />„ to obtain a maximally flat and an eon 1 ^^ rp Df passband characteristic is given in the following two sections. '- ' 5.12 BINOMIAL TRANSFORMER A maximally flat passband aand characteristic is is obtained obtained ifif pp =^ ||f] r | jand th N - 1 derivatives with respect to frequency (or ft) vanish at the mat w ^ match frequency /"„, where ft ~ TT/2. Such a characteristic is obtained if we ch " for which p = |ri=|A2;V(cos«)v| (6(WJ When 0 = 0 or ~, we have V = (ZL- Z0)/{ZL + Z„). and from (5.57a) we obtain T = A2 . Thus the constant A is given by , £"**• A = 2 .v ^ o ^ t + Zo (fi 58) However, if we use the theory of small reflections, then the constant A should be chosen in a different way, which we will explain shortly. Expanding (5.57a) by the binomial expansion gives r = 2-" ~ ~ d - e-W)S = 2"* | ^ £ C > — (5.59, _ ^ (5-60) where the binomial coefficients are given by _ N[N-l){N-2)---(N-n+l) 2)---(iV-. _ Note t h a t C* = C£_„, C„v = 1, C,A =N = C^_u etc. Comparing with (5.54) shows that we must choose since C,;V = C ^ „ . To obtain a simple solution for the characteristic impedances 6n> ", convenient to make a further approximation. Since we have already I fied that all p n are to be small, we can use the result z„ z „ + 1 + zn zpa IMPEDANCE TRANSFORMATION AND MATCHING 351 Thus we have l n % i = 2P„ = 2 - ^ v l n | t (5.62) where we have also used the approximation z0 zh + zn zL + z„ s\zL + z,j When we use the theory of small reflections (5.55) gives, for 0 = 0, HO) = P 0 + Pi + P 2 + ••• +/',v When we use (5.62) to evaluate the characteristic impedances, and also use A = 2'<N+l)MZL/Z0), we obtain 2NA = P(0) = - In — + In —- + •• • + In —Z0 Z, ZN \ tzlz 2 \Zn Zx Zj\ \ z^ ZN J 2 Z0 1 Thus the approximations introduced above for finding the characteristic impedances should be used along with (5.63) to evaluate HO). This will make the quarter-wave transformer designs using the theory of small reflections self-consistent.t In place of (5.58) we use the expression given above for A. Equation (5.62) gives the solution for the logarithm of the impedances, and since these are proportional to the binomial coefficients, the transformer is called a binomial transformer. Since the theory is approximate, the range of Z L is restricted to about 0.5Z 0 < Z, < 2Z„ for accurate results. As an example, consider a two-section transformer. From (5.62) obtain 2, ln T 1 , = 4 ln ZL T or z > = z ""z°/4 and Z Z In Z, -^ = -2 ln -± Z0 or Z„ ' 2 _="LZ¥*Ztf* "0 since C 0 — l and Cf = 2. Although the approximate theory was used, it urns out that the above values for Z, and Z 2 for the special case of the « author is indebted to Dr. E. E. Altschuler for pointing this out. 352 FOUNDATIONS FOR MICROWAVE ENGINEERING -8 #m "/t F I G U R E 5.37 Passband characteristic for a maximally flat transformer. two-section transformer are the correct nonapproximate solutions a res 1 that gives an indication of the accuracy of the approximate theory. The type of passband characteristic obtained with a maximally fiat transformer is illustrated in Fig. 5.37. Let pm be the maximum value of « that can be tolerated. The angle 0 m that gives p = pni is given by = cos ' 2/>„, m(Z,/Z0) l/N (5.64) as obtained from (5.576). In the case of transmission-line sections, 6 irf/2f0, and hence the fractional bandwidth is given by Af /o A, =2- 4 It cos 2p m \n(ZL/Z0) l/AT (5.65) since 6 m = irfm/2f0. Note that in (5.65) the solution to the inverse cosine function is chosen so that 6m < ~/2. By comparing Figs. 5.33 and 5.37 it» clear that a multisection maximally flat transformer can provide a muc greater useful bandwidth than a single-section transformer. 5.13 CHEBYSHEV TRANSFORMER Instead of a maximally flat passband characteristic, an equally uset ^ acteristic is one that may permit p to vary between 0 and pm, , ^ oscillatory manner over the passband. A transformer designed to y ^ equal-ripple characteristic as illustrated in Fig. 5.38 is of this y^ n5. provides a considerable increase in bandwidth over the binom 1 ' ^ p former design. The equal-ripple characteristic is obtained by m ^. behave according to a Chebyshev polynomial, and hence the nam ^ i fl- e ren t shev transformer. It is possible to have p vanish at as many ^ frequencies in the passband as there are transformer sections. ™°B^y W Chebyshev polynomials may be used in the design, it is nece consider the basic properties of these polynomials first. IMPEDANCE TRANSFORMATION AND MATCHING 353 F I G U R E 5.38 Equal-ripple characteristic obtained from a Chebyshev transformer. The Chebyshev polynomial of degree n, denoted by Tn(x), is an rath-degree polynomial in x. The first four polynomials and the recurrence relation are Tl(x)=x T2(x) = 2x°- - 1 T3(x) = 4x3 - 3x T4(x) = 8x 4 - 8x2 + 1 Tn(x) = 2xTn^ -T„_2 The polynomials T„ oscillate between ± 1 for x in the range 1*1 < 1 and increase in magnitude indefinitely for x outside this range. Figure 5.39 gives a sketch of the first four polynomials. If x is replaced by cos 6, we have r„(cos0) =cosrafl F I G U R E 5.39 Chebyshev polynomials. (5.66) 354 FOUNDATIONS FOR MICROWAVE ENGINEERING which clearly shows that |T„I < 1 for -1 < x < 1. As 0 varies fr 0 r n n the corresponding range of x is from +1 to - 1. Since we wish tn have the equal-ripple characteristic only over the range 0 to — _ ^ p cannot use T^teos 0) directly. If we consider instead »••• w * cos 0 \ / = cos n cos cos 0m J \ cos 0 cos 0„, J we see that the argument will become equal to unity when 0 = Q be less than unity for 0m < 0 < - - 0„,. This function will therefore co the equal-ripple oscillations of T n to the desired passband. The function given in (5.67) is an nth-degree polynomial in variable cos 0/cos 0 m . Since (cos 0)" can be expanded into a series of cn terms such as cos 0, cos 2 0 , . . . , cos nti, it follows that (5.67) is a series of th form (5.56). Hence we may choose p = 2e-JNB[Po cos NO + p, cos( N - 2)0 + • • • +p„ cos{N - 2n)B + •••] = Ae - • * " % ( s e c 0„, cos 0) (5.68) where A is a constant to be determined. When 0 = 0, we have Zr — Zr, 1 Z, r = ^ — / = ArN(Sec0m) ~ - I n - ^ and so A = ln(Z,./20) 2T A ,(sec0„,) Consequently, we have r = L - / « i n ^ 7'y(seC''"COs9) 2 Z0 (5.69) T i V (sec0 m ) In the passband the maximum value of !TjV(sec 0„, cos 0) is unity, and hen = Pm \n(ZL/Z0) 2Tw(secem) (5l0a) [B If the passband, and hence 0„„ is specified, the passband tolerance Pm fixed, and vice versa. From (5.70a) we obtain 1 Z , L T w ( s e c 0 m ) = -p>~^In — or by using (5.67) for cos 0 = 1, (1 sec Bm = c o s | - cos ln( ZL/Z0)\ 2pm {5.70b) ' ^ which gives 0„, in terms of the passband tolerance on p, that is, Pm- ™ IMPEDANCE TRANSFORMATION AND MATCHING 355 In o r d e r to solve (5.68) for t h e u n k n o w n p„, we n e e d t h e following results: n (cos0)" = 2-"e'-""'(l + e2j")" = 2" + l = 2 "e^"6 £ m I' C£eJ2mt [ C o " c o s ? i 0 + C , " c o s ( / i - 2 ) 0 + ••• + C^co8(n - 2m)0 + •••] (5.71) T h e last term in (5.71) is kC"n for n even a n d C , " _ 1 ) / 2 cosfl for « odd. U s i n g (5.71) a n d t h e earlier expression for T„(x), we can obtain t h e following: T x ( sec 0 m cos 0) = sec B m cos 0 (5.72a) r 2 ( s e c &m cos 8) = 2(sec 6m cos fff-l = sec2 8J1 + cos28) - 1 T 3 ( s e c 6 m cos 0) = s e c 3 0„,(cos 30 + 3 c o s 8) - 3 sec 0„, cos 0 (5.72b) (5.72c) T 4 (sec 0„, cos 0) = s e c 4 0 m ( c o s 4 0 + 4 c o s 2 0 + 3) - 4 s e c 2 0„,(cos20 + 1) (5.72d) T h e s e r e s u l t s a r e sufficient for d e s i g n i n g t r a n s f o r m e r s up to four sections in length. A g r e a t e r n u m b e r of sections would rarely be r e q u i r e d in practice. E x a m p l e 5.2 D e s i g n of a t w o - s e c t i o n C h e b y s h e v t r a n s f o r m e r . As an example, consider the design of a two-section transformer to match a line with Z 0 = 1 to a line or load with Z L = 2. Let the maximum tolerable value of p be P m = 0.05. Using (5.70a), we obtain T2(sec0m) = 2 s e c 2 0 m - 1 = — = 6.67 2 m " 3(0.05) and hence sec 0„, = 1.96, and 0„, = 1.04. Thus the fractional bandwidth that is obtained is A/ 1 4/w \ = — = 1.04 = 0.675 TT/2 f0 7T\2 A0 From (5.68), (5.69), and (5.726), we obtain [refer to the remarks following (5.56) as regards the last term in the cosine series for p] 2p o cos20 + p, = , , „ , r 2 ( s e c 0 m c o s 9 ) = p„, sec 2 em cos 20 + i>m{sec2 0m - l) and hence Po = {pm sec 2 0 m = p 2 = 0.099 P , = P m ( s e c 2 0 m - 1) = 0.148 The impedances Z, and Z% are given by Z, - e2""Z0 = 1.219 A Z2 = c 2 "'Z, = 1.639 plot of the passband characteristic is given in Fig. 5.40. As a check on the 356 FOUNDATIONS FOR MICROWAVE ENGINEERING i 0.33 r**-0.05 FIGURE 5.40 —H*—i— 1.04 »/2 1—•• Passband characteristic for a two «»-. * she, transformer with ,,„, » 0.05, P ^ T ^ accuracy, we calculated Zin for H = 0 using Z Z2 Z, i n = ip^i = 1-1063 From Zm we obtain Z - 1 P = - ^ — r = 0.05047 which is within 1 percent of the design value. *5.14 CHEBYSHEV TRANSFORMER (EXACT RESULTS) An exact theory for a multisection transformer having an equal-ripple passband characteristic has also been developed (see the references at t end of this chapter). Since the analysis is rather long, only the final res for the two- and three-section transformers are given here. In the exact theory of multisection ideal transformers, it is cotivei to introduce the power loss ratio P, K , which is defined as the < power (incident, power) divided by the actual power delivered to the ^j the incident power is P,, the reflected power is p2P, and the power to the load is (1 - p2)Pt. Hence P. Pr.R = and 1 2 (1~P )P, P = P ,I.R R (5.730' 1 ,5-73*1 - 1 LR 2 If T is the overall transmission coefficient, then \T\~ = 1 ~~ p u ta ined. ' For any transformer an expression for Z„, is readily o ^ ,g toUn< from this p, and hence P, R , can be computed. When this is IMPEDANCE TRANSFORMATION AND MATCHING 9 8 zrJs\~ 357 T1 * 1*1 Pm - 1 "T F I G U R E 5.41 Passband characteristic for a two-section Chebyshev transformer. I fl» & T/2 that P L R can be expressed in the form P L R = l + Q 2 *(CO S fl) (5.74) where Q2N(cos, fl) is an even polynomial of degree 2 AT in cos 6, with coefficients that are functions of the various impedances Z„. To obtain a" equal-ripple characteristic, P L R is now specified to be P L R = l + /e 2 7^(sec0,„cos0) (5.75) where k 2 is the passband tolerance on P L R ; that is, the maximum value of P L R in the passband is 1 + k2, since T'£ has a maximum value of unity. By equating (5.74) and (5.75), algebraic equations that can be solved for the various characteristic impedances are obtained. Figure 5.41 is a plot of p versus 6 for a two-section transformer. For this transformer P<LR» = 1 + \ZL-Z0f (sec 2 9t cos 2 0 - 1) tan40z *ZLZa (5.76) where 0 Z is the value of 0 at the lower zero where p vanishes. The maximum value of P L R in the passband is (ZL~Z0)A and hence 1/2 1 + k2 (5.77) iere k 2 = C ot 4 0 2 (Z^ - ZQ)2/4ZLZ0. The required values of Z, and Z 2 358 FOUNDATIONS FOR MICROWAVE ENGINEERING are given by 72 - 7a (zL-z0y AZl tan 4 Bt 1V2 + + ~~2~tan2_i z-—z (5.786) The value of B,„ is given by 9m = cos ' \/2 cos 0., and ±f 4 — = 2 /u W cos (5.79a, ' V2 cos ^ (5.796) provided 2 A « / - = A/•//'„ [if not, (5.796) gives 2 Afl/rr]. If the bandwidth is specified, then B2, and from (5.79a) B,„, are fixed. Equation (5.77) the specifies />,„. On the other hand, if />,„ is given, the bandwidth is determined. In the limit as B, approaches TT/2, the two zeros of p coalesce to give a maximally flat transformer. From (5.78) it is found that, for this case (compare with the approximate theory), zx = zrz^ (5.80a) Z2 = 2 * / % v < (5.806) For the maximally flat transformer, the value of 0„ at the point where P =Pm is given by (5.81) -i Bm = cos cote. where 0, is the previously defined quantity for the Chebyshev transfoi Equations (5.79) and (5.81) provide a comparison of the relative bandwidt obtainable from the Chebyshev transformer and the maximally flan former. This comparison is illustrated in Fig. 5.42 for N = 2 and j shows that the Chebyshev transformer can give bandwidths that art erably greater for the same maximum tolerable value pm- FIGURE 5.42 p. 0 0.4 0.6 1 ^ 1.6 2.0 FrocMonal bandwidth foe maximally Mot transformer Chebyshev and n"*" transformers. ,^ ft IMPEDANCE TRANSFORMATION AND MATCHING 359 FIGURE 5.43 *.g Passband characteristic for three-section Cbebyshev transformer. 9m Sr % Figure 5.43 illustrates the passband characteristic for a three-section Chebyshev transformer. The power loss ratio is given by ( Z L - Z0f (sec 2 0, cos 2 0 - l ) 2 cos 2 B ?LR=1 tan 4 fl *ZLZ0 (5.82) The passband tolerance A 2 is given by z fe = (ZL-Z0)2I 2cos0z (5.83) I 3 / 3 tan 2 6 Z 4ZLZ0 from which p m may be found by using (5.77). Again the general result that specifying k 2 determines the bandwidth, and vice versa, holds. The value of 6m is given by = cos -1 (5.84a) cos 6, 1/3 and for transmission lines for which A f/f^ = 2Aff/ir, Af — = fo 2(71/2 -tim) — 4 =2 T/2 TT cos 2 -7=- cos Bz V3 (5.84*) The characteristic impedance Z l is determined by solving z,. - z0 tan^ 0. zl ZL 2 ^ 1/2 Z,- Zi,Z0 1/2 2 z* ~ \fJ 0 zrzi (5 85a) ' and Z 2 and Z 3 are given by Z<i — ( Z h Z0) Za = z^z,, ^ 1/2 (5.856) (5.85c) 360 FOUNDATIONS FOR MICROWAVK ENGINEERING When 0 Z approaches TT/2, all three zeros coalesce at » , maximally flat transformer is obtained. The required value p " ^ ' ^ obtained from (5.85a) by equating the left-hand side to zero Tt° Z> "fy b! that Z, = ZlZl~", where i < a < \. With ZL/Z, near unity, „ ? > f ««3 to |, and for large values of ZL/ZI}, a will approach - By D i v °^ VarioUl values of a, a solution for Z, can be found quite readily by a tr ^ process (note that the equation for Z.x is a fourth-degree equati fnd"errii! maximallv fiat transformer the fl and O„J..^.... '°ruit maximally the value value of of 0,„ and rHo the Kbandwidth ** given by 1/2 cos (sinSJ 2 / 3 3\/3/ A/ 1 T = 2 cosfl (5.86a I 4 " v6 " (5.86b) /o ^ The Chebyshev transformer represents an optimum design in that no other design can give a greater bandwidth with a smaller passband tolerance. If it is assumed that some choice, other than (5.75), for the polynomial Q2iv in (5.74) can give a smaller passband tolerance for the same bandwidth. it will be found that a plot of the polynomial Q2N will intersect the polynomial T£ in at least N + 1 points. Since the polynomials are even in cos 0, they have at most N + 1 coefficients. Thus Q2N must be equal to TR since they have N + 1 points in common. But this equality contradicts the original assumption that Q2N could yield a better result, and hence proves that the Chebyshev transformer is an optimum one. 5.15 FILTER DESIGN B A S E D ON QUARTER-WAVETRANSFORMER PROTOTYPE CIRCUIT A very interesting filter design based on the theory of multisection q wave transformers was given by Young.t The quarter-wave trans'°" n e . yt bandpass filter but would normally not be used as a filter since t ^^ and output impedances are very different. In a multisection qu transformer, the impedances increase monotonically from &o ^8t most filter applications we desire equal input and output impedan ^ Young showed was that every other impedance step in a n, . m _ e danc* quarter-wave transformer could be replaced with an °PP° s ^, e a s t e p do** step. By alternating between a step up in impedance level an ^uflj # irn a in impedance level, we can end up with a final output * v jhed bel"* that at the input. The filter design based on this concept is desc tL. Young, The Quarter-Wave Transformer Prototype Circuit, IEEE Trans 483-489, September, 1960. vol MT TA ,«* IMPEDANCE TfUNSEOHMATION AND MATCHING 361 -2UL-22 2 T & " i . (a) 1 . (b) FIGURE 5.44 (0) An impedance step; (6) an equivalent junction when Z'.,/Z\ = Zl/Z.i. Consider the two junctions shown in Fig. 5.44. The first junction is a simple impedance step from Z, to Z 2 . The second junction is also an impedance step from Z\ to Z 2 , but, in addition, it has an idea) transmission-line section of electrical length TT/2 on either side. Furthermore, we assume that this electrical length does not vary with frequency. Obviously, we have introduced a nonphysical element and the reader may rightfully question whether anything useful can come from introducing such nonphysical elements. We will show that in the final filter configuration these nonphysical transmission lines can be eliminated. Thus their introduction is only to facilitate the development of the theory for the filter design. The two junctions will be fully equivalent if the scattering-matrix parameters S ' u and S 2 2 are the same as S u and S22. If the output is terminated in a matched impedance Z 2 , we get _Z " 2 - 2 , = (Z2/Zt) - 1 Z2 + Z, (Z2/Z,) + l for the first junction. Similarly, [or a matched termination on the input side, we get l-(Z2/Z,) i + (z2/Zl) - _ S j l ** For the second junction we use the quarter-wave-transformer formula to evaluate Z i n with the output matched. This gives 7 (*a* '2 from which we get _ (Z\)2/Z2~Z\ 11 (Z[f/Z2 + Z\ 362 FOUNDATIONS FOR MICROWAVE ENGINEERING F I G U R E 5.45 Microstrip filter design obtained from a multisection quarter-wave-transibrmer prototype circuit by replacing every other impedance step by the equivalent junction shown in Fig In a similar way we easily find that S22 = -S'n. By comparing the expressions for S u and S' n , we now conclude that the two junctions are equivalent if Z^ z, == Z^ z:2 (5.87) This is precisely the property we are interested in because if Z2/Zx >s a 8t up in impedance, then Z'2/Z\ must be a step down in impedance level. Our next step is now to replace every other impedance step »n multisection quarter-wave transformer by this new equivalent junction, vv» assume that the multisection quarter-wave transformer we use as type circuit has already been designed to give a desired power loss ra "* . on filter design procedure is illustrated in Fig. 5.45. In order that eac js. be one quarter-wave long at the center frequency, we make eac sion line have an electrical length 20 - ~/2. Thus, when ^. = . Y n r t h 20 TT/2 *= 77/2. In other words, we use a physical line of electnca simply use and a nonphysical line of electrical length - TT/2 for reasons^ fully explained later. For the first junction in the filter, we 5.88°' Z[ Zx which is the same as in the quarter-wave transformer . T h e second** IMPEDANCE TRANSFORMATION AND MATCHING 363 is replaced by the new equivalent junction; so we make ~ - ~ %~-—• = - + (5.886) This new junction incorporates the two nonphysical transmission lines of electrical length TT/2. The next junction is an impedance step like that in the quarter-wave transformer; so -£ = ^ Z'9=-~ = -=$ZZ (5.88c) The next junction is again the new equivalent junction which requires This procedure is continued until we have worked our way through all the impedance steps in the quarter-wave-transformer prototype circuit. The filter illustrated in Fig. 5.45 is a three-section filter. If this filter is examined, it will be seen that each section contains a transmission line of electrical length 20 — —/2 + —/2 = 20, so the nonphysical lines have been eliminated. We chose each transmission fine to have an electrical length of 20 — IT/2 specifically for the purpose of eliminating the nonphysical lines that are part of the new junctions that replace every other impedance step in the quarter-wave-transformer prototype circuit. However, this means that the power loss ratio of the filter is obtained by replacing 0 by 26 — IT/2 in the expression for the power loss ratio for the quarter-wave transformer. If the quarter-wave transformer is a Chebysbev transformer with power loss ratio given by (5.75), then the power loss ratio for the filter is given by P,. R = 1 + k2T* sec 0„, eos|2fl = 1 + & 2 T*[sec 0,„ sin 20] (5.89) i nus sin 20 replaces cos 0 as the frequency-dependent variable. At the center frequenc}' 28 = TT, SO each filter section is one half-wavelength long. For this reason the filter is called a half-wave filter. The band edges which occur at cos 0 = ±cos 8m for the quarter-wave transformer now occur when sec 0„, sin 2 0 = ± \ or sin 20 = ±cos0,„ = ± s i n | — ± 0m which gives pi = 20 = - ± e„ ± nv 364 FOUNDATIONS FOR MICROWAVE ENGINEERING The upper band edge will be at (3! = 3 w / 2 - 6 m and the lowe will be at /3/ = TT/2 + 0„,. Consequently, the fractional vJ "d <&» d l h MPD/IT = 1 - 29m/v, which is one-half that of the quarter former prototype circuit. Thus the prototype circuit should be'd^" 1 1 **have a bandwidth twice as large as that required for the filter t this change the frequency response of the filter is the same a<= u^ that prototype circuit. 'As a final point we note that, in a quarter-wave transform odd number of sections, the power loss ratio is unitv at thn ouencv. Since * ** <*"** fft. quency. Since zL z%^ ^m 72 zi 72 7'2 ^0 for an odd number of sections, the final impedance Z'L in the filter equal* Z<j. Hence the filter operates between impedance levels of ZQ and Z last nonphysical transmission line that appears in the output line (see Fk 5.45) for an odd number of sections can be deleted since this is a matched line of arbitrary length. For an even number of sections. Zin does not equal Z0 at the center frequency because the power loss ratio does not equal unity; it equals 1 + * 3 for this case. This means that the last impedance element in the filter. which is given by ~, _ y , _ Z I Z 3 Z N-l ? _ y will be different from Z0. By using (5.77) for p at the center frequew can solve for Z i n to get z =z - °f^ = ( * + / r T ^) 2 ^ (5.90) Since k is normally very small, the output line for the filter has i istic impedance not quite equal to that of the input line. ^ j - o ra In Table 5.1 we list the required values of the l m p e c i a ^ vg}aef, Chebyshev quarter-wave transformer with three sections for se o f the passband tolerance k2. W e will use this table t o design ^ ^ ; n t filter in the example that follows. More extensive tables are e literature.t - =— -— M. T. * tSee, for example. G. L. Matthaei, L. Young, o..Impedance Matching Networks, and Coupling Structures," Artec Mass. 1980. -"-5 IMPEDANCE TRANSFORMATION AND MATCHINO 365 Chebyshev quarter-wave-transformer data • — A/V/o-0.2 — Z*./2o 2 •1 10 20 100 z,/z0 k* 1.09247 I !<)474 1.349 1.48359 1.87411 7. . vZ 2o 1.19 X 5.35 x 1-92 x 4.29 x 2.33 x 10~7 10 7 10 ' 10 7 10 '' \f/f0Zx/Z0 0.4 fc2 A f / f „ = 0.6 Z,/Z„ 1.09908 7.89 > 10 " 1.1083 1.20746 3.55 X 1 0 - S 1 23087 1.37482 1.28 x 10 ' 1 42.12 1.52371 2.85 x 10 ' 1.60023 1.975 1.55 x 10 '•' 2.17928 fe* 9.57 x 10 6 4.31 x 10 •' 1.55 x 10 ;i 3.45 x 10 :' 1.87 x 10 ~ ^;i " ^fc^o/"] Example 5.3 Filter design. We want to design a bandpass filter with a fractional bandwidth of 0.2 and having a VSWR of 1.02 or less in the passband. From the given VSWR we find ,>,„ = (1.02 - 11/2.02 = 9.9 x 10 :>. By using (5.77) we get k'1 = t>f„/(l p%) = 9-8 X 10 r'. The quarter-wave-transformer bandwidth must be chosen as 2 x 0.2 = 0.4. From Table 5.1 we find that, for Z,JZ0 = 10 and Hf/f,, = 0.4. k* = 1.28 x 10 '. This value of k'~ would give a maximum value of VSWR equal to 1.023 in the passband. We will accept this value since an entry for k'1 = 9.8 X 10 5 is not given. From Table 5.1 we gel Z,/Z0 = 1.37482. We will use 50-S1 input and output lines. T h u s ^ Z ^ 1.3748 X 50 = 68.74 SI. The required values of Z 2 and Z :l are Z., = v .Wi 5 0 = 158.1 (1. Z, = 500 x 50/68,74 =» 363.69 (1. This completes the design of the quarter-wave-transformer prototype circuit. For the filter we need Z\ = Z, = 68.74 SI, and by using (5.886) and (5.88cI Z'2 = 29.89 il. Z':! = 68.74 SI. These impedance values are readily realized for a microstrip filter. The maximum out -of-band attenuation occurs at # = 0 and is the same as what is obtained when the input line is connected directly to a load ZL = 10Z0. The reflection coefficient will be 9 / 1 1 so 66.9 percent of the incident power is reflected. The transmission through the filter is reduced by —10 logfl - pz) = 4.8 dB. This is a small attenuation and shows that the particular filter configuration used here will generally not have a large out-ofband attenuation. If we use Z,_/Zu = 100. we would obtain an attenuation of 14 dB. But in this case the passband tolerance would be larger. The required passband tolerance could be achieved by using a filter with more sections. However, there are better filter configurations to use when large out-of-band attenuation is needed (see Chap. 8). C ^CpennSaCtroPnaCitanCe a n d L e n ^ h * an a b r u p t step in t h e width of a m i c r o s t r i p line, t h e r e will be an laditional fringing electric field from t h e open-circuited p o r t i o n s of t h e wider s t r i p as s h o w n pictorially in Fig. 5 . 4 6 a . T h e efFect of t h i s fringing e ' d can b e modeled a s a s h u n t capacitance a t t h e j u n c t i o n . T h e equivalent -ircuit of t h e s t e p is s h o w n in Fig. 5.466 a n d c o n s i s t s of an ideal i m p e d a n c e 366 FOUNDATIONS FOR MICROWAVE ENGINEERING -Of S„ 2 Z< au -? y' '* C * (b) V » C JR. 3 22 (C) F I G U R E 5.46 (a t Fringing electric field at a step change in width for a microstrip line; (6) equivalent i for a step-change-in-width junction: <c) an alternative equivalent circuit. step and a shunt capacitive susceptance. The susceptance is small but does modify the performance of the filter if not properly compensated. The junction capacitance can be compensated for by changing the length of each filter section. The equivalent circuit in Fig. 5.466 will be shown to be nearly equivalent to the circuit in Fig. 5.46c. The latter consists of the ideal impedance step plus two short lengths of transmission line. The electrical lengths of these transmission lines are denoted by 0, and 0-z. The two electrical lengths are of opposite sign. For the equivalent circuit in Fig. 5.466, we have jB S,,= 1- Yl-Y2-jB Yx-Yt Yt + Y2 +jB Yt + Y2l Y,-n + JB Yl + Y2 Since B/(YX + Y.,) is very small, we can use the binomial e x p a n d obtain B Y,~Y2 Yi + Y2 *i- Y2 r,+ Y-i r,- Y2 Y1 + Y2 B l-J B JB 1 - Yl _2jBY}_ 1 - Y? - Y£ Y\TY~2 + Y2 IMPEDANCE TRANSFORMATION AND MATCHING 36 7 »1 e, ~0/=;r-e2-e',- FIGURE 5.47 Length compensation of a filter section. For the circuit in Fig. 5.46c, we have Y,-Y2 Yr- Y, ,. (5.92) when 8, is small. In a similar way we obtain y2-y, '22 s:22 1 - F 2 + F, 2jBY2 n-y.^ r ( l - 2 ^ ) Y, + (5.93) (5.94) A comparison of S u and S 2 2 with S',, and S 2 2 shows that the two junctions are equivalent, to the order of approximations used, if we choose <?.= BY, 2 Y, - Y z r • » - - (5.95a) 2 BY, Y * - Y2* (5.956) In the half-wave filter the length of each section is now changed to compensate for the junction capacitance as shown in Fig. 5.47. The required section length is 2ft including 0., and 0',. Hence the physical length / is chosen to make /3/ = 20 - 0 2 - 0\ = 7T - 0 2 - 0\ (5.96) at the center frequency. Since 0, and 0 2 , etc., vary with frequency like B = coC, compensation is obtained at all frequencies. The parameters 0 2 and Q\ are small so the change in physical length of each section is small but important enough to take into account in the design of a filter. Similar junction capacitance effects also occur in some quarter-wavetransformer realizations and can be compensated for by changing the Physical length of each section by a small amount. The application of the above method of compensation will be illustrated in Example 5.4, but first we need data for the junction capacitance. A number of investigations of junction equivalent circuits for microstrip discontinuities have been carried out, but surprisingly very little data for design purposes are available. Gupta and Gopinath have evaluated the 368 FOUNDATIONS FOR MICROWAVE ENGINEERING 0.15 - 0.5 1 2 E J fc 0.1 $ ° " 0.05 W, 0 2 4 (a) 6 10 H FIGURE 5.48 (a) Junction capacitance Cf for a step change in width for a substrate wi PTlhlrO-taH n n r t i n r * of r . r curves . . . . « . r t ~ in I— (a). 'with enlarged portion • 0.6 _ 4 u. a. Ws T7=°1 «,= 9.6 E My 0.4 / . 0 , 1 1 / y . o" 0.2 --•^--—f^- '—• i . 4 H (b) FIGURE 5.49 (o) Junction capacitance C, for a step change in width for a substrate with e, enlarged portion of curves in (n). 4| shunt capacitance at a step in width for a microstrip line for subst dielectric constants of 2.3, 4, 9.6, and 15.1.t Their data give the i capacitance normalized by the distributed capacitance per meter c wider microstrip line. We have used these data to evaluate the s " capacitance for the two cases where the dielectric constant of the s is 2.3 and 9.6. The junction capacitance C s divided by the substrate ness H is shown in Figs. 5.48 and 5.49. Example 5.4 Microstrip half-wave filter. The filter d e s c r i b e ^ ^ 4 5.3 is to be built using microstrip construction. The substrate is a Wi**' fC. Gupta and A. Gopinath. Equivalent Circuit Capacitance of Microstrip Change m IEEE Trans., vol. MTT-25, pp. 819-822, October, 1977. IMPEDANCE TRANSFORMATION AND MATCHING 369 iM FIGURE 5.50 (a) Three-section half-wave filter using microstrip construction; CM scaled drawing of filter. a dielectric constant of 9.6 and is 1 mm thick. By using the computer program MSTP, we find after a few iterations the following required widths and the effective dielectric constants of each section: For Z, = 50 il W= 0.99 mm ee = 6.49 For Zc = 68.74 JI W = 0.466 mm ee = 6.18 For Zc = 29.89 (1 W= e, = 7.07 2.45 mm With reference to Fig. 5.49, we identify Wa/H and Wh/H for the junction between the 50- and 68.74-Sl lines to be H = 0.466 H = 0.99 Prom Fig. 5.496 we estimate CJH to be 0.03 and since H = 1 mm, C„ = 0.03 pF. The center frequency of the filter passband is to be 4 GHz. Hence B = ioCs = 7.5 X 10 4 S. The characteristic admittances y„ and Yt are (see Pig. 5.50) * • - » - * * *k- 1 68.74 = 0.01455 We now use (5.95) to obtain BY0 e, Y 2 - Vs = 0.08 >', 6.2 = --=-0, = - 0 . 0 5 8 For the junction between the 68.74-fl line and the 29.89-$! line, we have H 0.466 £-"• Prom Fig. 5.496 we estimate C, to be 0.18 pF which gives B = 4.52 x 10~ 3 S. 370 FOUNDATIONS FOR MICROWAVE ENGINEERING We now determine D\ and ff.z shown in Fig. 5.50. BY Y where Y3 = 1/29.89 = 0.03346 S. In the 68.74-fl section e wavelength is Atl/ JTe = 7.5/ \f6A8 = 3.017 cm. The reauini i ' this section is ^ ** le"e«» / 'l = A 2^{v ~ " 2 ~ ffi) = 3.27 "2V A = °- 5 2 A = L5 7cm For the 29.89-12 line, we have ec = 7.07 and a wavelength 7.5/ vT07 = 2.821 cm. The required length l.t is given by h = r - ( i r - 20'1) = 0.447A = 1.261 cm 2-ir Note that the length corrections are about 4 and 10 percent. If the filter were designed for 2-GHz operation, the length corrections would be half as large or about 2 and 5 percent. Below 1 GHz, compensation for the junction susceptances could be ignored. A proportionally scaled drawing of the foil pattern for the filter is shown in Fig. 5.506. 5.16 TAPERED TRANSMISSION LINES In a multisection quarter-wave transformer used to match two transmission lines with different characteristic impedances, the change in impedanc is obtained in a number of discrete steps. An alternative is to use a taper transition which has a characteristic impedance that varies continuous!} a smooth fashion from the impedance of one line to that of the other line. transition, or matching section, of this type is referred to as a P& transmission line. An approximate theory of tapered transmission analogous to the approximate theory presented earlier for mul transformers, is readily developed. This approximate theory is pf • below. A following section gives a derivation of the exact differenti ^ tion for the reflection coefficient on a tapered transmission line gives a brief evaluation of the validity of the approximate theory. ^ Figure 5.51a illustrates schematically a tapered transmission ^ ^ ^ to match a line with normalized impedance unity to a load wit n ^ . impedance Z L (assumed to be a pure resistive load). The 1*^ normalized impedance Z which is a function of the distance ^^ j , , taper. Figure 5.516 illustrates an approximation to the c o n t m u o ^ e r e n t i s considering it to be made up of a number of sections of one J olinis <+ length dz and for which the impedance changes by different1' from section to section. IMPEDANCE TRANSFORMATION AND MATCHING 371 FIGURE 5.51 Tapered-transmissinn-line matching section. The step change dZ in impedance at z produces a differential reflection coefficient dr0- Z + dZ - Z dZ Z + dZ + Z 2Z - ^d(\nZ) 1 d - -—(lnZ)ck 2 dz (5.97) At the input to the taper, the contribution to the input reflection coefficient From this step is dl , = e •+ml± (lnZ)rfz 2 dz If it is assumed that the total reflection coefficient can be computed by summing up ail the individual contributions, as was done in the approximate theory of the multisection quarter-wave transformer, the input reflection coefficient is given by *-ir*~so-*)* (5.98) where L is the total taper length. If the variation in Z with z is known, I', may be readily evaluated from the above. A problem of much greater practical importance is the synthesis problem, where Z(z) is to be determined to give I', the desired characteristics as a function of frequency. Before taking up the synthesis problem, two examples of practical taper designs are presented. 372 FOUNDATIONS FOR MICROWAVE ENGINEERING 2w 3ir 5ir 4JT 6TT 7V F I G U R E 5.52 Input reflection lor an e x p o n e n i k u j * " -—/3i.=2v-x Exponential T a p e r The exponential taper is one for which In Z varies linearly, and hence Z varies exponentially, from unity to In ZL\ t h a t is, In Z = — In ZL 2 = (5.99a j e{z/L)\n2,_ (5.996) Substituting (5.99) into (5.98) gives 1 , i , m Z, e dz r, = 2^^r 1 ~ r' sinSL JliL ta Zi -^r ,5100) where it has been assumed that we are dealing with a transmission line ' which (i = k = 2ir/A and is not a function of z. A plot of p, = (HI versus / is given in Fig. 5.52. For a fixed length of taper, this is a plot of P-. * function of frequency since k = 2TT f(, fj.e)> 2. Note that when L is grea than A/2, the reflection coefficient is quite small, the first minor lobe about 22 percent of the major-lobe maximum. Taper w i t h Triangular Distribution If dlln Z)/dz is chosen as a triangular function of the form. 42 d(lnZ) dz TllnZ, 1 4 i.2(L-z)lnZL 0<z<j — (5.10D <z <L Integ*ati^ a matching section with more desirable properties is obtained. IMPEDANCE TRANSFORMATION AND MATCHING H-t*i 373 FIGURE 5.53 Input reflection coefficient for a taper with a triangular distribution of reflections. (5.101) gives I e l»z/fcr*lbffc Z = e(4j/I,-2s 2 //-id-l)lnZ, L ~ < z < L 2 (5.102) Substituting (5.101) into (5.98) and performing the straightforward integration give I- = - e - ' « M n Z , sin(/3L/2) 0^/2 (5.103) A plot of p, versus /3L is given in Fig. 5.53. Note that, by comparison with the exponential taper, this taper has a first minor-lobe maximum which is less than 5 percent of the major-lobe peak. However, this small value of reflection coefficient occurs for a taper length of about 3A/2, or for a length twice that for the exponential taper. If Z L is considerably greater than unity, this latter taper will be preferable because of the much smaller values of p, obtained for all frequencies, such that the taper length is greater than 0.815A, which corresponds to the lower edge of the passband in Fig. 5.53. YNTHESIS OF TRANSMISSION LINE TAPERS Equation (5.98) is repeated here for convenience: 1 ,/ d([nZ) r,(2/3) = 2J- f' V * * - dz 0 (5.104) Ants equation may be interpreted as the Fourier transform of a function 374 FOUNDATIONS FOR MICROWAVE ENGINEERING diln Z)/dz, which is zero outside the range 0 < z < L.f As such inversion formula gives 1 d(\nZ) 2 dz e ^"HJIW 1 -« 2W-. This formula, in principle, solves the synthesis problem since "t required value of c/(ln Z)/dz to yield the .specified r,-(2/3). To '" ^^ d i s c u s s i o n l.n follow it will hp e n n v p n i e n f to i n t m r l n m 4 L - /• i> tne '">' the " u w , n e nor mal . ized variables: ^ - L/2 ' 5 106a, 0L 2L TT A (5.1066) In this case (5.104) becomes • 1 2e H, f* d(lnZ) dp dp J J (5.107) Now define g( p) to be d()n Z) (5.108a) and F ( « ) by F(M) Thus = /" e~J"ug(p)dp (5.1086) (5.109) F, = ±,e-jl,LF(u) The Fourier transform pair (5.104) and (5.105) now may be expressed as (5-Jl0o) Flu)-f_j-»"g(p)dp 1 /•» . , r T 1106) The synthesis problem may now be stated as follows: Specily a f o r . tion-coefncient characteristic F( u) that will give the desired ^ ^ j ^ a mance and yet be such that the g ( p ) computed by (5.1H .trjctio« function identically zero outside the range \p\> -._This iaitf* e n t from corresponds to the physical requirement that d(ln Z)/dz be di . Tranaf, t T h e Fourier transform relation was first pointed out by F. Bolinder. F o u r ' j g g 4 i Nov*"* the Theory of Inhomogeneous Transmission Lines, Proc. IRE, vol. 38, P1950. IMPEDANCE TRANSFORMATION AND MATCHING did pro only in the range 0 < z < L. Obviously, any arbitrary F(u) cannot be cified, for m general this would lead to a g(p) that exists over the whole 'nfinite range - * < p < x- For example, if F( u ) were chosen to be equal to nitv for — 1 < w S 1 and zero otherwise, then (5.1106) would give sin p g(p) = - x <p < » irp To realize such a g(p) would require an infinitely long taper, clearly an impractical solution. Before further progress with the synthesis problem can be made, restrictions to be imposed on F(u), to obtain a physically realizable solution, must be deduced. In order to derive suitable restrictions on Fin), let g(p) be expanded in a complex Fourier series as follows: g(P) Z aneJ,:p = ~rr < p < tr n * —M 0 \p\> (5.111) IT where the o„ are as yet unspecified coefficients subject to the restriction a„ = a* „ so that g will be a real function. Substitution into (5.110c) now gives sin F(u) = 2tr Z = 2TT Z <** — n) TT( U TT(U - n) sin 7r u a„(-l) v(u sin " TTU u Z a„(-l) "- - n) (5.112) The coefficients a„ can be related to F(u = n), for when u equals an integer «,, lim *n and lim tt—n sin TT( u — n) TT{U — n) = 1 sin TT(U — m) ~ —- = 0 ir(u — m) Thus F\n) = 2-an, or an = F(n)/2w; so F(«)= ™ sin-?r( « - n) £ *<»}— -i (5.113) his result is a statement of the well-known sampling theorem used in communication theory and states that F(u) is uniquely reconstructed from a knowledge of the sample values of F(u) at u = n, n = 0, ± 1, ± 2 , . . . 376 FOUNDATIONS FOR MICROWAVE ENGINEERING by means of the interpolation formula (5.113). One possible wav t F(u) is now seen to be a relaxation on the specification of F( ? FestHct specify F(u) at all integer values of u only. This, however, is not" ' t f l a t **» satisfactory solution, because we have no a priori knowledge tu e n i ' r e 'y specify F(u) at the integer values of u only, the resultant F(U) *' ' f We (5.113) will be an acceptable reflection-coefficient characteristic f ^^ ^ ues of u, even though it can be realized by a g(p) given bv f*vi ^ Val" y l0 11 an=F(n)/2ir. - D with We should like to obtain greater flexibility in the choice of F( \ see how this may be accomplished, let it be assumed that all a for i T^ ^ are zero. In this case •* " N L a„e jnp g(P)= sin vu (5 jj „ -ll*0 u F{u),^~zj-1)a.— (5,144) The series in (5.1146) can be recognized as the partial-fraction expansion of a function of the form Q(u) where Q(u ) is, apart from the restriction Q( -u) = Q*(u) so that a„ = a*., an arbitrary polynomial of degree 2N in u, and the denominator is the product of the N terms (u2 - 1),(u 2 - 4), etc. Using the partial-fractionexpansion formula, we have Q{u) n£.x(««-ii*) = = » E Q(m) _xn...nA? 77 "^(«-m)2 «Q(u) uWf.Au2 - n2) m = n^ * w mr_'iV 2 7772 2157 > -« ) + hm _ QOO -aw • ""^^ i^i^—^—r (" " / n ) 2 m 2 n n A ' = 1 . T T j ^ r Q(0) + (5.H5) 2 n^ 1 (-n ) where the prime means omission of the term m = 0. This is clear. same form as the series in (5.1146), with ,„ ( 1] °-= Q(m) 2m2n»_1.n*m(m2-n2) Q(Q) (5.U 60> [5.n6b} IMPEDANCE TRANSFORMATION AND MATCHING 377 e expression for F( u) now can be written as sin—u F{u) = 2v Q(u) nV , 2 ,• (5.117) •here Q(u) is an arbitrary polynomial of degree 2N in u, subject to the restriction Q(-u) = Q*(u). This result states that the first 2N zeros of in TTU which are canceled by the denominator in (5.117), can be replaced by 2N new arbitrarily located zeros by proper choice of Q(u). If g(p) were a constant (exponential taper), we should have F(u) proportional to (sin-u)/iru. But with 2N +• 1 coefficients available in the expansion of g(p), we are at liberty to rearrange 2N of t h e zeros of (sin TTU)/TTU to obtain a more desirable F( u). We have now reduced the synthesis problem to one of specifying an arbitrary polynomial Q( u ). To illustrate the theory, two examples are discussed below. A qualitative insight into how Q( u) should be specified may be obtained by imagining that F(u) is a rubber band stretched horizontally at some height above the fiL or u axis. The zeros of F(u) may then be thought of as points at which this rubber band is pinned down to the it axis. If the band is pinned down at a number of closely spaced points, it will nod rise much above the u axis in the regions between. The corresponding reflection coefficient will then also be small in this region. At a double zero the band is pinned down in such a fashion that its slope is zero at the point as well. This results in a less rapid increase in the height with distance away from the point. The number of zeros available (the points at which the band may be pinned down) is fixed and equal to those in the sin TTU function. The polynomial Q( u) permits only a relocation of these zeros. With reference to Fig. 5.52, which illustrates the characteristic for an exponential taper, let the zero at /3Z, = IT be moved to 2- to form a double zero at this point. Likewise, let the zero at 377 be moved to form a double zero at 4 - , etc. The function Q(u) that will provide this shift in every other zero so as to produce double zeros at u = ± 2 , + 4, ± 6, ± 8, etc., can be chosen as n = l From (5.117) we obtain ^ s i n ^ n ^ - 4 ^ e now wish to let N go to infinity. However, the products do not converge s e ; so we must modify the expression for F( u) to the following: m t n i s ca f(u) = 2 2 2 nsimrunZ=1(l-u /4n ) C , N . 2. 378 FOUNDATIONS FOR MICROWAVE ENGINEERING This modification is permissible since Q contains an arbitr multiplier. All constants can be incorporated in the one consb> following infinite-product representations for the sine functions sin TTU - / u2\ C I1st ° «iit —TTU = nn= l i i sin(7T«/2) iZ n -u/2 1 - n= 1 4u 2 it is readily seen that as N -* x we obtain W«)-C sin(-u/2) 12 TTU/2 (5.118, This is the reflection-coefficient characteristic for the taper with a tria lar d(ln Z)/dz function discussed earlier. In the present case we hav arrived at this solution for a taper with equally spaced double zeros bv a direct-synthesis procedure. As Fig. 5.53 shows, the specification of double zeros holds the values of Flu) (that is, p,) to much smaller values in the region between zeros. As shown earlier, the coefficients a „ in the Fourier series expansion of g(p) are given by 1 a„ = ^-F(n) = X 2TT' '' C rSin(mr/2)i: — 2TT rnr/2 (5.119) from (5.118). The reader may readily verify that the expansion (5.114a) g(p), with the above coefficients, is a triangular wave. To fix the constant ( we integrate (5.108a) to obtain ,,TT d(\n Z) _._ = [ g(P)dP= [ ; dp = In Z\U = In ZL •' rr f-ir dp But from (5.111) we have / and hence g(p)dp = 2 i r o 0 a = — In ZL 2TT (5.120) L From (5.119) we now find that 1 _ C L In Z, = — °° = 2 ^ " " 2TT so C = In Z,. With this value of C. the reflection coefficient c°"*=^en & to the F(u) in ((5.118) is easily verified to be the same as th fin' (5.103). As a second example, consider the synthesis of a t a P ^ r n)0vin*> having a triple zero at u = ± 2 . This can be accomplished by IMPEDANCE TRANSFORMATION AND MATCHING «. u 5 379 F I G U R E 5.54 Reflection-coefficient characteristic lor a taper with a triple zero at | « | = 2. zeros at ±1 and ±3 into the points u = ±2. The resultant rejection coefficient should remain very small for a considerable region around \u\ = 2. In the present case N = 3, and we choose Q(«) = C ( V - 4 ) ' : Thus F(u) («2-4r s i n 77 u 2TTC- = (z/2- l)(u2 - 4 ) ( u 2 - 9 ) 7T« A plot of |F(«)/F(0)l is given in Fig. 5.54. As anticipated, F(u) remains small in a considerable region around the point | « | = 2. Since the zeros at l « l = 3 have been removed, F(u) reaches a relatively large value at this point. However, for a range of frequencies around which L = A, this taper represents a very good design. The coefficients in the expansion for g( p) are given by 1 o , = a , = — F(l)= 0.316 ~—lnZL a.} = a_„ = 0 1 a , = a _o = -F(3) a „ = a_„ = 0 = 0.098 —2-rr^ - \nZL «>3 Hence d(lnZ) S(P) dp \nZL 2TT ( o 0 + 2 a , cos p + 2 a 3 cos3p) (1 + 0.632cos p - 0.196cos3p) 2TT 380 FOUNDATIONS FOR MICROWAVE ENGINEERING Integrating gives _ In Z, In Z = ~~( P + 0-632 sin p - 0.0653sin Zp) + c The^ constant of integration C is determined from the rea.ui In Z = 0 at p =,7, or In Z, at p = -; thus C = A | n Z L and r e m e W thai In Z, • n Z = - ^ — - ( p + - + 0.632 sin p - 0.0653sin3p) Replacing p by 2rr(z - L/2)/L from (5.106a) now specifies In Z function of g and completes the design of the taper. The foregoing synthesis procedure must be used with some cauti stemming from the approximations involved. The theory is valid as Inn </(ln Z)/dz is small; that is, In Z must be a slowly varying function of 2 order that the reflection coefficient everywhere along the taper be sma that is, \\'(z)\l « 1. This means that \g{p)\, and hence all |o„|, must not be permitted to assume excessively large values. Consequently, \F(n)\ must not be permitted to become too large. If too many zeros are closely grouped together around a particular vaiue of u, then outside this range F(u) mav become excessively large and the accuracy of the theory will suffer. Such ''supermatched" designs must be avoided. *5.18 CHEBYSHEV TAPER If the number of sections in a Chebyshev transformer is increased indefinitely, with the overall length L kept fixed, we obtain a Chebyshev taper This taper has equal-amplitude minor lobes and is an optimum design in tt sense that it gives the smallest minor-lobe amplitudes for a fixed tap< length, and conversely, for a specified minor-lobe amplitude it has t e shortest length. As such it is a good taper by which to judge how far tapers depart from an optimum design. It has been shown that, in t as the number of sections in a Chebyshev transformer goes to infim y reflection coefficient becomest 2 cosh li0L where 0„ is the value of 0 at the lower edge of the passband, as 1 s ^ , in Fig. 5.55. As ji increases from_ zero to /3 0 , the magnitude' ''^/a decreases to a final value of (In Z L )/(2cosh 0 O L), since in Proc « * " tR. E. Collin. The Optimum Tapered Transmission Line Matching Section. pp. 539-548, April. 1956. IMPEDANCE TRANSFORMATION AND MATCHING -0L 381 F I G U R E 5.55 Reflection-coefficient characteristic for a Chebyshev taper. cos L-\fp2 - Po = c o s n ^VP?> _ ^ 2 • Beyond this point the function in the numerator is the cosine function that oscillates between +1 and produces the equal-amplitude minor lobes. The major-lobe to minor-lobe amplitude ratio equals cosh jiQL. Hence, if this is specified so as to keep p, less than or equal to some maximum value p, n in the passband. the taper length L is fixed for a given choice of the frequency of the lower edge of the passband which determines /3„. We have cosh ji^L = InZ, (5.122) *P, Conversely, if /3„ and the taper length L are given, the passband tolerance pm is fixed. The theory given earlier may be used to determine the function g{ p) that will produce the reflection coefficient given by (5.121). Introducing the u variable again, we find that the function Fiu) is _ F(u) = (\nZL)- cos -Ju2 - u2 (5.123) C O s h 77 U i where -u = pL, -utl = fi,,L. The function cos TT-\Ju2 - u% pressed in infinite-product form as cos Try u2 - u\ = cosh TTU„ ]~I can be ex- 1 n • (n - | ) and this is the limiting value of the polynomial Q(u) in (5.117) a s N - " : . t h e sin — u term has been canceled by the infinite product rT^ =l Cl ~ u2/n2) 3S in the first example presented on taper synthesis. However, we do not need this product expansion since 2TTO„ = F(n) in any case. From (5.123) we have 1 «*-«—- 2~ In ZL cos ir-Jn2 - u% 2 77 Cosh 77 Un (5.124) 382 FOUNDATIONS FOR MICROWAVE BNGlNEKRINCi Thus InZ, #(p) = 9 ZT (cOshl7U„ + 2cOS7rv/l - ul COS fa + 2 cos Try7 4 - ul cos 2p + • • • ) [aZL 2TT cosh TT&(. (cosh iru 0 + 2 Z cos i r ^ " * 3 " ! c o s np n= l (5. - Integrating with respect to p gives InZ InZz. " cos Tr-Jn2 - ul I p C 0 8 h 2 , cosh ™ 0 ™» + 2 <|?1 n " ^ H + C (5.126) where C is a constant of integration. To render this result more suitable for computation, it is expedient to add and subtract a similar series: i.e., plnZL in Z = —z 2v + In Zh " cos nv . : 2sin np — cosh u j i r , . ] In Z L TT COSh U 0 7T x n= cos —y n 2 - u% — cos nir sin np + C , The second series converges rapidly because cos vyn2 - u% approach cos n IT as n becomes large. The first series may be recognized as Fourier sine series for the sawtooth function Sip), P &ZL S(p) = { 2TT cosh IT ut - T T <P < TT p = ±ir 5.56. and the periodic continuation of this function as illustrated in Sip) -ZlT FIGURE 5.56 Sawtooth funcWW- IMPEDANCE TRANSFORMATION AND MATCHING 383 Hence we obtain InZ P 1° Z/ + S( P) 2l7 Ir zL X L cos Txyn2 — K| - cos » TT sin np + C n At p = rr, we have In Z = In Z t , and since Sip) and sin np are zero at this point, C + | In Z L = In Z ; . or C = | in Z , . Our final result is + 77 CO lnZ = ih TTU„ p i p \ i- + - ; In Z ; 2JT 2 2T7 cosh — «„ | In Z, * cos ir-Jn2 - u'i - cos /? rr £ sinrap (5.128) 77 cosh 7ru 0 n j for -77 <p < TT. An interesting feature of the above result is that In Z changes in a stepwise fashion from 0 to (In Z ; )/(2cosh TT;/,,) as p changes from -IT — e to - T T + e, where e « 1. Likewise, at the other end of the taper, In Z changes abruptly from a value In ZL - (In ZL )/(2cosh -ua) to In Z ; as the point p = TT is reached. This means lhat the optimum taper has a step change in impedance at each end. The physical basis for this is readily understood by noting that when the frequency is very high, so that the taper is many wavelengths long, the reflection from the smooth part of the taper vanishes. Thus, in order still to maintain equal-amplitude minor lobes, the two-step changes in impedance must be provided to give a reflection coefficient In Z, cos jiL f>, = —zr— —; r for P *• Pn H H[> 2 cosh (i{lL As an indication of the superiority of the Chebyshev taper, computations show that it is 27 percent shorter than the taper with d(ln Z)/dz a triangular function, for the same passband tolerance and lower cutoff frequency. If the tapers are made the same length, the Chebyshev taper provides a major- to minor-lobe ratio of 84 as compared with 21 for the taper with a triangular distribution. C 0E9PFICnfT E Q U A T I O N F O R T H E REFLECTION The basic equation (5.98) for the input reflection coefficient T, was derived by neglecting all multiple reflections between individual differential sections. The exact equation, derived below, enables an estimate of the range of validity of the approximate theory to be made. First of all, the differential 384 FOUNDATIONS FOR MICROWAVE ENGINEERING equation describing the total reflection coefficient f at any point line, according to the approximate theory, is derived for later co With referenceto Fig. 5.57, let d\\, be the reflection coeffic- 11 ^ 50 ' 1 from the change dZ in characteristic impedance in the interv 1 ^'^"g This differential reflection was shown earlier to be given by [see te a 3t *• ld(lnZ) The total reflection coefficient at z is the sum of all differential tions rfr0 from z to L and is 1 ,/, .„ d(lnZ) where u is a dummy variable that measures the distance from the point 2 = 0 toward the load end. The phase angle of the reflected wave arising at u is 2(3(z - u) relative to the forward propagating wave at z. Differentiating H z ) with respect to z gives rfr 2jp ,L dz 2 /, M d(lnZ) 1 .„ du 2 d(lnZ) du = 2 ^ r - - \ (5-129) 2 dz This is the approximate differential equation for the total reflection coefficient at any point z along the taper. _ To find the exact differential equation for T, let Z in be the input impedance at z and Zm + dZin be the input impedance at z + dz. We have Zin + d Z i n + j Z t a n O d z ) Zj„ — L _ Z+./(Zin + dZm)tan(0dz) Zm + Z+j(Zin dZm+jZpdz + dZin)fSdz z ( z2 aZ-n + dZto+jfiZdz-j-fpdz H z upon replacing the tangent by its argument /? dz, neglecting orodu^ IMPEDANCE TRANSFORMATION AND MATCHING Z 385 FIGURE 5.57 Tapered transmission line. Z*dZ differential terms, and replacing the denominator by [Z+j(Zinpdz)\ ~JT** The above gives Z dz -JP ^-z Now we also have i + r_ Z„ = - — = z I - r and hence dZ„ dz r dz i - r dz 2Z dr + (i - n 2 ~dz~ Combining these two equations for dZm/dz and replacing Z,n by Z(l + T ) / ( l - r) finally give dV cfe 1 rf(lnZ) (5.130) 0*2 upon using the relation Z~x dZ/dz = d(ln Z)/dz. If we compare (5.130) with (5.129), we see that the approximate equation differs only by the factor 1 - T 2 that multiplies the dlln Z)/dz term in the exact equation. If IT2| <K 1 everywhere along the line, the approximate equation would be expected to yield good results. The exact equation (5.130) is called a Riccati equation. It is a nonlinear equation because of the term in f 2 and does not have a known general solution. This equation can be integrated only in certain special cases (one such case is for the exponential taper, as in Prob. 5.48). However, the practical difficulty in applying the exact theory does not stem so much from the lack of a general solution of (5.130), since numerical integration or iteration schemes can always be employed, as from being unable to specify what the characteristic impedance Z is along a general taper. If the taper is very gradual, then Z(z) can be taken as the characteristic impedance of a uniform line having the same cross-sectional dimensions as the taper has at the plane z. But for such gradual tapers the approximate theory is valid and 386 FOUNDATIONS FOR MICROWAVE ENGINEERING gives good results, so that the more complicated Riccati equaf required. For more rapidly varying tapers, the field structure T ' S n o t taper is perturbed to such an extent that no simple method of ° n ^ l ^ e Ziz) exists. In actual jact. the new boundary-value problem will i*'pec'Iying solved to determine Zlz), and this solution will also provide the the reflection coefficient along the taper. Thus one concludes tiT inability to specify Z(z), except for the case of gradual tapers m ^ exact equation of minor importance in practice. In the case of waveguide tapers, fi is a function of the cross-serf geometry and hence a function of z along the taper. In order to h the approximate equation (5.129), in this case, an auxiliary variable^ introduced as follows: 6 = (~2p(z)dz d0 = 2(idz d d dd d —- = — — = 26 — dz dO dz dd and hence We now have dY dY dd d(lnZ) J 2 dti This is readily integrated to give (multiply both sides by e'J" and note t dVe-'"/d8 = -JTe-* + e _ ' " r f r / d « ) ,« where 1 0Q = HI d(lnZ) "(In Z) (L2p dz If Z = ZL at z=L, then T10O) = 0, and since T(0) = T,, we have ° ^ essoin terms of the new variable 0. the problem is formally ^ ^ h e r d i ^ " ! However, the specification of Z(ti) as a function of 6 may be r syntt** to translate back into a function of z. Consequently, a gen ^ ^ u g h ' procedure applied to (5.132) may be difficult to carry o u ^ ^ p t principle it is formally the same problem as considered ear ^ jeter" the last step, i.e., expressing Z(H) as a function of * in or e the shape of the taper. IMPEDANCE TRANSFORMATION AND MATCHING 387 5 . 1 . Show that the R = constant and X = constant contours in the reflectioncoefficient plane are circles given by (5.6a ) and (5.66). 5.2. In the circuit shown in Fig. P5.2. what is the smallest value of d that will make the resistive part of Z m equal to 50 SI at the plane /,? Find the required value of jX to make Z m equal to 50 ii at the plane t2- Use a Smith chart to find d and jX. j* f j / ; =ton / f = 5on tz t, FIGURE P5.2 5.3. On a certain line terminated in a normalized load impedance ZL> it was found that the standing-wave ratio S was equal to 2 and that a voltage minimum occurred at A/4 from the load. What is Z t ? Find the position and length of a single shunt stub that will match the load to the line. Use a Smith chart. 5.4. What are the length and position required for a series stub to match the load specified in Prob. 5.3? Use a Smith chart. 5.5. A normalized load ZL = 2 terminates a transmission line. Does a voltage maximum or minimum occur at the load? Find the position and length of a shunt stub that will match the load to the line at a frequency /",, where the wavelength is A,. With the stub parameters and the load fixed, let the wavelength be increased to 1.1 A,. What is the new value of stub susceptance and the standing-wave ratio on the line? If at the wavelength A, the stub is placed A , / 2 farther toward the generator from its original position, what is the standing-wave ratio on the line when the wavelength is increased to 1.1A,? This illustrates the greater frequency sensitivity of the match when the stub is placed farther from the load. 5.6. A double stub with spacing 0.25A is used to match a normalized load admittance Y L = 0.5 + j \ . Find the required stub susceptances by using a Smith chart. Answer: Solution 1: Solution 2: fi, = -0.5 B, = - 1.5 L, = 0.176A Lx = 0.0936A B.z =1 B2 = - 1 L> = 0.375A L2 = 0.125A 5.7. Can a normalized load admittance Y L = 2.5 + j\ be matched with a doublestub tuner with stub spacing A/10? 5.8. What is the minimum stub spacing in a double-stub tuner that will permit a normalized load admittance YL = 1.5 + jl to be matched if the spacing is restricted to be greater than A/4? If the spacing must be less than A/4, what is the maximum stub spacing that can be used? 388 FOUNDATIONS FOK MICROWAVE ENClN'KERINc 5.9. Show that, for a double-stub tuner with d = A/4, the required val susceptances are given by -BL±[GL(l-GL)]l/'- B>= 1 B.,= -G, 1/2 GL Hint Take the limit of (5.20) and (5.21) as tan fid becon* es infinite. 5.10. A normalized load admittance Y L = 0.8 +./0.5 is to be matched to mission line using a double-stub tuner. Use a Smith chart to a find required stub susceptances and lengths. Assume that short-circuited spaced by a distance 0.4A are used. Answer: Solution I j Bx - - 3 . 1 7 1 i, = 0.0486A Solution 2: B, = - 0 . 5 8 2 L,=0.166A B 2 = -2.994 B.z = 0.2416 L 2 = 0.0513* L 2 = 0.288A 5.11. A line with an attenuation constant « = 0.01 N p / m is used as a short-circuited stub. Using the formula Y,„ = Yc coth(« + jB)l, find the maximum value of normalized susceptance this stub can give. If a = 0.02, what is the maximum value of stub susceptance that can be obtained? Use I = 1 m. 5.12. Consider the series-shunt matching circuit illustrated in Fig. P5.12. What values of series stub reactance and shunt stub susceptance will match an arbitrary load YL = G, + jBL a distance d away? Hint: Note that Y m at the position of the shunt stub equals (j.Y Z'mY '. where Z'm is the input impedance just to the right of the series stub. Next impose the condition r'm = I + jB to find the required value of jX. FIGURE P5.12 5.13. Develop a graphical method, using the Smith chart, to solve Pro • ^oVltig "-'-'- vV... must lie on the G = 1 circle. But Ym is obtained ^ Hint: 2Ln point. -jX+_Z[ point Hence, " ^ ^ toe mak r o t a 180° around the chart from the Z , on the G = 2 circie, choose jX to make Zin he on the n through 180°. TE o &(& 5.14. A horn antenna is fed from a rectangular waveguide in which l ^ ^ praduCe£ propagates. At the junction a reflection coefficient ' = ° * , t n e n What is the normalized input admittance to the horn? Fin _ ^fQltl t normalized susceptance and the spacing in guide wave l e n ^ e g U ide « junction of an inductive diaphragm that will match the horn. IMPEDANCE TRANSFORMATION AND MATCHING 389 -JBt FIGURE P5.14 5.15. A load impedance 40 I j l O is to be matched to a 50-ii line using lumped elements. Use a Smith chart to find the parameters of a suitable matching circuit. Evaluate the loaded Q of the circuit. Two solutions are possible; choose the one with the lowest Q. Check your answers using LCMATCH. 5.16. Use a Smith chart to design a lumped-parameter matching network that will match a load admittance Yt = 0.04 ; _/0.08 1.0 a 50-11 transmission line. Find both solutions and evaluate the loaded Q for each. Check your answers using LCMATCH. 5.17. For the matching circuit shown in Fig. 5.17«, shown that the required values of jBy and JX., are given by 0x -j[-Bt ±)/Gjl -GL)) Mi - ±J Hint: Note that Zm = JX., + (JBX + ¥h) ' must equal I. 5.18. For the matching circuit shown in Fig. 5.176. show that the required values of jX: and JB-, are given by ^ - / ( - . X i i / B ^ l - R i ) ) JBX =±j] I-Ml. RL 5.19. Show that the definition (5.23) for Q gives the expressions in (5.26). 5.20. A load admittance G, + ; ' w C t = GL +,jBL = 0.02 + ./0.05 is to be matched to a 50-11 transmission line using a double-stub tuner with short-circuited shunt stubs spaced by 0.18A. The stubs have a characteristic impedance of 40 11. Use the computer program STUBMACH to find the required stub lengths for the two possible solutions. Use the frequency scan option with a normalized frequency step of 0.02 and five steps below and above the matching frequency to evaluate the frequency sensitivity of the two solutions. Plot the return loss as a function of normalized frequency and find the bandwidth o( each circuit for which the return loss is less than - 20 dB. 5.21. Repeat Prob. 5.20 but use open-circuited stubs. Compare the bandwidths obtained with those when short-circuited stubs are used. 390 FOUNDATIONS FOR MICROWAVE ENGINEERING 5.22. A load impedance ZL = R,, + jioL = RL + jXL = 30 +j\00is tn to a 50-Ii transmission line using open-circuited series stubs apart- The stubs have a characteristic impedance of 75 Si. Use f ^ ^ °-l5>. program STUBMACH to find the required stub reactances and Carry out a frequency scan to determine the bandwidth ove^ which VSWR remains below 1.5. Answer: The two bandwidths are 0.045 and 0.087. 5.23 (a ) For Prob. 5.22 replace the first stub by a short-circuited shunt this improve the bandwidth? (6) Replace the second stub by a sh rt^ shunt stub but keep the original first stub. How does this a r W a m the b width? andAnswer: (a) BWs are 0.01 and 0.017. (6) BWs are 0.03 and 0 07~ 5.24 A load impedance Z,_ = 5 + .;50 is to be matched to a 50-!l transmission r Design a lumped-element matching network that uses a capacitor "in with the load as the first element plus a second shunt element Use computer program LCMATCH. Evaluate the return Joss as a functtor normalized frequency using normalized frequency steps of 0.01. For the ti solutions plot the return loss as a function of the normalized frequency Find the 3-dB bandwidth for the two circuits. Evaluate the loaded QL for the two circuits and compare the bandwidth with that given by \/QL. Answer: QL = 5 and 6.5. The bandwidths are 0.21 and 0.16 and are close to the respective 1/QL values. 5.25. Repeat Prob. 5.24 but use a shunt element for the first element in the matching network (option b in the computer program LCMATCH). 5.26. Aload admittance YL = 0.002 t j'0.1 is to be matched to a 50-Jl transmiffl line. Use a matching network with the first element being a shunt elemeni Evaluate the return loss as a function of (he normalized frequency and find the 3-dB bandwidth of the two circuits. Evaluate the loaded Q L and compar the bandwidth with that given by l/QL. Use LCMATCH. 5.27. Repeat Prob. 5.26 but use a series element as the first element. 5.28. A load admittance G, = 0.002 - ./'0.05 is to be matched to a 50-H tra sion line. Find solutions for four different matching networks t h a t c l " > t l t t , used. Evaluate the loaded Q L of each and identify the network gJ« greatest bandwidth. Use LCMATCH. ^ A load impedance Z = 80 ~J10 is to be matched to a 40-ii ^ " ^ r r * 5.29. L line. Find the elements for two possible matching networks an sponding bandwidths for which the VSWR remains below LCMATCH. .gj, tjse * 5.30. A microwave amplifier requires a load termination Smith chart to design a matching network of the form s h o * " o n , i n e s ha** The transmission line is an open-circuited one. The transr/us* characteristic impedance of 50 Jl. . e 5.31. Use a Smith chart to design a lumped-element network of t Fig. 5.22c t h a t will provide a load Z, = 30 + ./50 for an amp ^^ .0^ 5.32. What are the required length and impedance of a quarterthat will match a 100-fi load to a 5 0 - H l i n e a t T ~ " * » * £ , * line)? What is the frequency band of operation over whic coefficient remains less than 0.1? Sm*. IMI'KDANCE TRANSFORMATION AND MATCHING 391 5.33. Design a two-section binomial transformer to match the load given in Prob. 5.32. What bandwidth is obtained for pm = 0.1? 5.34. Design a three-section binomial transformer to match a 100-12 load to a oO-il line. The maximum VSWR that can be tolerated is 1.1. What bandwidth can be obtained? Plot /> versus f. 5.35. Design a two-section Chebyshev transformer to match a 100-ii load to a 50-11 line. The maximum voltage standing-wave ratio (VSWR) that can be tolerated is 1.2. What bandwidth is obtained? Plot p versus f. Use the approximate theory. *g.36. Use t h e exact theory to design a two-section Chebyshev transformer to match a normalized load Z, = 5 to a transmission line. The required fractional bandwidth is 0.6. What is the resultant value of p„,? *g.37. Design a three-section Chebyshev transformer (exact theory) to match the load specified in Prob. 5.36. The same bandwidth is required. Compute />„, and note the improvement obtained. *g.38. Let x„ be a zero of Ts(x); that is. T^ixJ = 0. Let the corresponding value of cos 0 be cos 02. Note that TlV(sec Hm cos 0) = 0 when sec fl„, cos " = sec 0 m cos H!. Use the result Tv(cos</>) = cos Ntl> to compute the zeros of '/'.,(.v). Note that the zeros occur when cos2<ft = 0, or <i> = TT/4. 3 - / 4 , and that .v„ = cos <!>.. Using these results together with the relation sec 0 m = x s sec «,. show that, for a two-section transformer, P LR = 1 + k2Ti(sect)mcos(t) reduces to (5.76). Note that when H = 0, p'2 = (ZL - Z0)-/{ZL + Z,,) 2 , and hence k2= (zL-zKtf 4ZLZ0Ti(sect),„) 5.39. For a particular application it is desired to obtain a reflection-coefficient characteristic The normalized load impedance equals 2. What must the value of C be? Use the approximate theory for an JV-section transformer to design a four-section transformer that will approximate the above specified /»,. Expand p t in a Fourier cosine series to determine the coefficients i>„. Plot the resultant p, versus 0 and compare with the specified characteristic. Show that the approximation to the specified />, is a least-mean-square-error approximation. If the number of sections in the transformer were increased indefinitely, could the specified p, be realized? 5.40. An empty rectangular waveguide is to be matched to a dielectric-filled rectangular guide by means of an intermediate quarter-wave transformer. Find the length and dielectric constant of the matching section. Use a = 2.5 cm, f= 10,000 MHz, er2 = 2.56. Plot /;, versus frequency f. Note that the appropriate impedances to use here are the wave impedances for the TE 1( , mode. t r 2 is the dielectric constant in the output guide. 392 FOUNDATIONS FOR MICROWAVE ENGINEERING FIGURE P5.40 5.41. Design a two-section binomial transformer to match the empty dielectric-filled guide in Prob. 5.40. Use intermediate sections with' constants en and er2. Plot the input reflection coefficient as a f. frequency. ' Unct ">n of 5.42. Find the required impedances for a three-section half-wave filter ha fractional bandwidth of 0.3 and a passband tolerance k2 = 1.87 x i n - ' " ? 3 input and output transmission lines have a characteristic impedance of "in n 5.43. Find the required impedances for a two-section half-wave filter havin • fractional bandwidth of 0.2 and a passband tolerance k2 = 10~ 2 . The in transmission line has a characteristic impedance of 50 il. What is the required characteristic impedance of the output transmission line? You will need to design the quarter-wave-transformer prototype circuit first. 5.44. The filter described in Prob. 5.42 is to be built using microstrip construction on an alumina substrate (e, = 9.6) 1 mm thick. The center frequency is 2 GHz. Determine the line widths and lengths including compensation for junction capacitance. Use the MSTP program. 5.45. Obtain an expression for p, for a taper which has d(\n Z)/dz = C sin THZ/L). Determine C so that In Z = In Z, at z = L and equals zero at z = 0. Plot p, versus pL. *5.46. Design a taper that has double zeros at 0L = ±2TT and ±3ir. Plot Fiu versus u for this taper. Determine In Z as a function of z. Achieve the design by moving the zeros at ±v and ±4TT into the points at ±2ir, ± 3~*5.47. Design a taper with single zeros at j8L = ±TT, ± 1.25tr, ± l.5ir, ± LW moving the zeros at ± 2 - , ± 3ir, ± 4TT into the specified points. Plot tt resultant Fiu). Determine the expansion for In Z as a function of z fo^ taper. Note that the close spacing of zeros keeps />, small m the .^ - <pi < 2TT, but that removal of the zeros at ± 3 i r and ±4rr lets p, large value in this region. „ | 6 r the *5.48. Show that, for the exponential taper, the exact solution ot IO.I input reflection coefficient is , l , = A 3in( BL/2) __ B cos( BL/2) + 2jp s i n ( B L / T ) where A = (In ZL)/L, B = v / 4/3 2 - A2. ^ *5.49. Show that the approximate differential equation for the inp along a slowly varying tapered line is (Z i n = Z at all pom - iinpe dao<* ^L=-2MZ-Zin) ential line « * * Integrate this to obtain the input impedance to an expon nated in a load impedance ZL. IMPEDANCE TRANSFORMATION AND MATCHING 393 Quarter-wave transformers 1. Collin, R. E.: Theory and Design of Wide Band Multisection Quarter-Wave Transformers. Proc. IRE, vol. 43. pp. 179-185. February. 1955. 2. Cohn. S. B.: Optimum Design of Stepped Transmission Line Transformers. IRE Trans.. vol. MTT-3, pp. 16-21, April, 1955. 3. Riblet, H. J.: General Synthesis of Quarter-Wave Impedance Transformers, IRE Trans., vol. MTT-5, pp. 3 6 - 4 8 , January, 1957. 4. Young. L.: Tables for Cascaded Homogeneous Quarter-Wave Transformers, IRE Trans., vol. MTT-7, pp. 233-237, April, 1959. See also IRE Trans., vol. MTT-8. pp. 243-244, for corrections. 5. Solymar, L.: Some Motes on the Optimum Design of Stepped Transmission Line Transformers, IRE Trans., vo). MTT-6, pp. 374-378. October. 1958. Tapered lines 6. Bolinder, F.: Fourier Transforms in the Theory of Inhoroogeneous Transmission Lines. Proc. IRE, vol. 38, p. 1354, November. 1950. 7. Klopfenstein, R. W.: A Transmission Line Taper of Improved Design, Proc. IRE, vol. 44, pp. 31-35, January. 1956. 8. Collin. R. E.: The Optimum Tapered Transmission Line Matching Section, Proc. IRE, vol. 44. pp. 539-548, April, 1956. 9. Bolinder. F.: Fourier Transforms in the Theory of Inhomogeneous Transmission Lines, Trans. Roy. Inst. Techno!., S/ockho/m, no. 48, 1951. 10. Matsumaru, K.: Reflection Coefficient of B-Plane Tapered Waveguides, IRE Trans., vol. MTT-6, pp. 143-149, April, 1958. 11. Johnson, R. C: Design of Linear Double Tapers in Rectangular Waveguides, IRE Trans.. vol. MTT-7, pp. 374-378. July, 1959. CHAPTER 6 PASSIVE MICROWAVE DEVICES A large variety of passive microwave circuit components or devices have been developed for use in both the laboratory and in microwave communication and radar systems. In this chapter we will describe the basic operating principles for a number of the most commonly used devices, such as attenuators, phase shifters, directional couplers, power dividers, hybr junctions, and ferrite devices such as isolators or gyrators and circulator The physical form that these devices take depends on whether waveguio coaxial transmission line, strip line, microstrip, etc., is used as the transn sion medium. The development of transistors and switching diodes 01 applications in the microwave frequency band has resulted in the c ment of a number of electronically controlled devices such as atteriu and phase shifters. Some of these electronically controlled circuit e are also examined in this chapter. 6.1 TERMINATIONS Two types of waveguide and transmission-line terminations are ^ use. One is the matched load, and the other is a variable short c^ produces an adjustable reactive load. These terminations a -^g paf" used in the laboratory when measuring the impedance or SC roVides a ameters of a microwave circuit element. The matched load P jvaJefll mination that absorbs all the incident power, and hence is ^jg sk° terminating the line in its characteristic impedance. The circuit is a termination that reflects all the incident power. 1 <> PASSIVE MICROWAVE DEVICES 395 la: F I G U R E 6.1 Matched loads for a waveguide, ( a ) Lossy wedge: (6) tapered resistive card. reflected wave is varied by changing the position of the short circuit, and this is equivalent to changing the reactance of the termination. The usual matched load for a waveguide is a tapered wedge or slab of lossy material inserted into the guide, as illustrated in Fig. 6.1. Since the material is lossy, the incident power is absorbed. Reflections are avoided by tapering the lossy material into a wedge. Thus the termination may be viewed as a lossy tapered transmission line. An overall length of one or more wavelengths is usually sufficient to provide a matched load with an input standing-wave ratio of 1.01 or smaller. Circuit The simplest form of adjustable short circuit for use in a waveguide is a sliding block of copper or some other good conductor that makes a snug fit in the guide, as illustrated in Fig. 6.2. The position of the block is varied by means of a micrometer drive. This simple form of adjustable short circuit. however, is not very satisfactory in its electrical performance. The erraticcontact between the sliding block and the waveguide walls causes the equivalent electrical short-circuit position to deviate in a random fashion from the physical short-circuit position which is the front face of the sliding block. In addition, some power leakage past the block may occur, and this results in a reflection coefficient less than unity. These problems may be overcome by using a choke-type plunger, as discussed below. The choke-type plunger is an example of the use of impedance transformation properties of a quarter-wave transformer. Consider, for example, a load impedance Z, that is approximately zero. If this impedance is Wtweguide Sliding plunger 1111111 MicromeTer dr F I G U R E 6.2 A simple short circuit for a waveguide. 396 FOUNDATIONS FOR MICHOWAVK KNUINEKR1NO X »/l x,/« F I G U R E 6.3 Two-section quarter-wave transformer viewed through a two-section quarter-wave transformer, as in Pi l 6 impedance seen at the input is e- -3,tr,e Z' = z2 '6.1, If Z 2 is chosen much greater than Z,, the new impedance Z\ will a n D mate a short circuit by a factor (Zi/Z2)2 better than Z, does. Thi essentially the principle used in choke-type plungers. The improveme factor of course deteriorates when the frequency is changed, so that th transformer sections are no longer a quarter wave long. However, by proper design, a bandwidth of 10 percent or more can be achieved. In very critical applications more than two sections may be used. The foregoing theory is now applied to a choke-type plunger for use in a rectangular guide. For the TE, 0 dominant mode in a rectangular guide, the surface currents on the interior wall flow up and down along the side walls and bath across and in the axial direction on the broad walls. In the simple type of short circuit illustrated in Fig. 6.2. the axial current must flow across the gap between the upper and lower waveguide walls and vertically across u front face of the sliding block. The currents that flow along the side wall flow in the vertical direction and do not need to cross the gap between tr waveguide walls and the front face of the plunger. Consequently, thee performance arises only from the axial currents flowing on the uppei lower walls. To avoid this erratic behavior, the plunger may be madeint ' form illustrated in Fig. 6.4. The width of the plunger is uniform and slig less than the interior guide width. However, the height of the P ' " " ^ ^ made nonuniform. The front section is a quarter guide wavelengt less than the guide height b by an amount 26,. The gap 6 t is ma d e ^J t p 0 t as possible consistent with the requirement that the front seen touch the upper and lower waveguide walls. The second sec ^ g. quarter guide wavelength long, b u t with the gap b% mao-e A ir i V4 , V 4 b2 • // z, F I G U R E 6.4 Choke-type adjust-" cuit (side view) s h o rt PASSIVE M1CIMWAVK DEVICES £t_. _ »,/4 * i>» * a -j 3 FIGURE 6.5 Alternative choke-type plunger design possible consistent with maintaining the mechanical strength of the plunger. The final back section makes a sliding fit in the guide. The quarter-wave sections have equivalent characteristic impedances proportional to 2 6 ( / 6 and 26. 2 /6 relative to that of the input guide. Thus application of (6.11 gives K for the normalized input impedance. By good mechanical design 6 2 can be made about 10 times greater than bx, and hence an improvement in performance by a factor of 100 over the non-choke-type short circuit may be achieved. A somewhat different design, as illustrated in Fig. 6.5, is also frequently used. In this plunger a two-section folded quarter-wave transformer is used. The inner line transforms the short-circuit impedance to an ideal open circuit at the plane aa. At this point, i.e., at an open-circuit or infinite impedance point, the axial current is zero. Hence there is no current present to flow across the gap between the waveguide wall and the plunger at the contact point aa. The next, or outer, quarter-wave transformer transforms the open-circuit impedance at aa into a short-circuit impedance at the front end of the plunger, i.e., at plane bb in Fig. 6.5. Short-circuit plungers of this type give very satisfactory performance. The above application of quarter-wave transformers is also used in the construction of choke joints for joining two waveguide sections together, in rotary joints, for plungers used to tune cavity resonators, etc.t ATTENUATORS Attenuators may be of the fixed or the variable type, The first is used only if a fixed amount of attenuation is to be provided. For bridge setups used to measure transmission coefficients, the variable attenuator is used. There are many ways of constructing a variable attenuator; only one type, the rSee, for example. G. L. Kagan (ed.). "Microwave Transmission Circuits." McGraw-Hill Book Company, New York. 1948. 398 FOUNDATIONS FOR MICROWAVE 15NI UNKKHIM ! rotary attenuator, is considered in detail. A simple form 0 f consists of a thin tapered resistive card, of the type used for mat t whose depth of penetration into the waveguide is adjustable T h ^ '°at*s inserted into the guide through a longitudinal slot cut in the ce f* C a r d •» broad wall of a rectangular guide. An attenuator of this form ha< complicated attenuation variation with depth of insertion and fr A better precision type of attenuator utilizes an adjustable I e n C y ' waveguide operated below its cutoff frequency. The disadvantae ^^ °^ type of attenuator is that the output is attenuated by reducing the between the input and output guides, and not by absorption of the ' power. As a result, a high degree of attenuation corresponds to a refl ' coefficient near unity in the input guide, and this is often undesirable Perhaps the most satisfactory precision attenuator developed fo AI • the rotary attenuator, which we now examine in some detail. The h components of this instrument consist of two rectangular-to-circular wa\ guide tapered transitions, together with an intermediate section of circular waveguide that is free to rotate, as in Fig. 6.6. A thin tapered resistive card is placed at the output end of each transition section and oriented parallel to the broad walls of the rectangular guide. A similar resistive card is located in the intermediate circular-guide section. Tlie incoming TE;o mode in ike rectangular guide is transformed into the TEn mode in the circular guide with negligible reflection by means of the tapered transition. The polarization of the T E n mode is such that the electric field is perpendicular to the thin res card in the transition section. As such, this resistive card has a negligible effect on the T E n mode. Since the resistive card in the center section can be rotated, its orientation relative to the electric field of the incoming TEn mode can be varied so that the amount by which this mode is attenuated: adjustable. , With reference to Fig. 6.7. let the center resistive card be orienti'd at an angle 0 relative to the direction of the electric field polarization of_ T E n mode. The T E , , mode polarized in the x direction may be deconip-* into the sum of two T E , , modes polarized along the u and v direct Resistive cards- •Rectangular to circular waveguide Transition Rotating section of circular guide F I G U R E 6.6 Basic construction of a rotary attenuator. PASSIVE MICROWAVE DEVICES 399 FICURK 6.7 Decomposition of T E , , mode into two o r t h o g o n a l l y polarized modes. illustrated in Fig. 6.7. That portion which is parallel to the resistive slab will be absorbed, whereas the portion which is polarized perpendicular to the slab will be transmitted. However, upon entering the transition section, the transmitted mode is again not perpendicular to the resistive card in this section; so some additional attenuation will occur. To derive the expression 1'or the dependence of the attenuation on the rotation angle 9, consider the analytic expression for the TE, ,-mode electric field. For polarization in the x direction the electric field is given by (Table 3.6) _ E = Ji(p\ir/a) r arcos<b . P'u ., Pur J, a a,,, sin <!> (6.2) apart from the propagation factor esliz. Since sin <b = sin( 4> - 0 + 8) = cos 8 sin( <b - 0) + sin 0 cos( <t> - 0) and similarly cos <b = cos( 4> - 0 + 0) = cos 0 cos( 6 - 0) - sin 0 sin( d> - 0) the above expression may be written as ,7, E = cosfll — a , . c o s ( * - 0) r p'u J[n. sin(4 - 8) a — sin 0 — ar sin(</> - 0) + — JJa,,, cos(</. - 0) which is equivalent to referring the angle variable <]> to a new origin at 8. The first term in brackets in (6.3) is a T E , , mode polarized with the electric field along the u axis as in Fig. 6.7, and the second term in brackets is a T E , , mode polarized along the v axis. Since the first part is completely absorbed, only the portion multiplied by sin 0 is transmitted into the output transition section. If we assume that the resistive card in this section is parallel to the y axis, only the component of the transmitted field which is polarized along the * axis is transmitted. We have, at the input to the 400 FOUNDATIONS FOR MICROWAVE ENGINEERING transition section, J, p'xl E = - s i n f l — a sin(<* - d) + —A** cos(0 - v r 0 A = sin 2 0 — a r cos 6 - sin P: --^a,., sin c6 cos «l — a r sin<j> + 1/,3^ cos 4> of which the first part is a T E U mode polarized along the x directio transmitted. Comparing (6.4) with (6.2) shows that the transmittea^tn' reduced by a factor sin 2 H from the amplitude of the incident field H the attenuation produced is given in decibels by « = - 2 O l o g ( s i n 2 0 ) = -401og(sinfl) /g5j A notable feature is that the attenuation depends only on the angle of rotation 0, a result that is verified in practice to a high degree of approximation. Electronically Controlled A t t e n u a t o r s For applications in various microwave systems, it is desirable to have an attenuator whose attenuation can be controlled by the application of a suitable signal, such as a dc voltage or a bias current. Two devices thai are suitable for use in an electronically controlled attenuator are the PIN dioc and a field-effect transistor. These devices can be used as variable resi=toi whose resistance is controlled by the applied signal. The basic attenuator network is a symmetrical resistive T or^ network as shown in Figs. 6.8a and 6. The resistor values fl, and / chosen so that when the attenuator is terminated in a resistance equ ^ the transmission-line characteristic impedance Zc, the input is that is, Z m = Z,, and to provide an output voltage reduction by *1 • ^ M «—Wv- — Zc (a) F I G U R E 6.8 The two basic attenuator networks. PASSIVE MICROWAVE DEVICES 401 factor K. For the T network we have Bm = fl R2{R1 + Zl.) . + R,+R + Z, 2 For Rin = Z r we get the equation Rl(Rl + 2R2) = Z* When R m = Z c the Thevenin impedance seen by the load equals Z, also. The Thevenin open-circuited voltage across R 2 is V ™ fi2 'Ri+Ri ~ ZcV' + The power delivered to the load is n-= y'TH 2 v 2Z, Z „ fl 2 ,tnrl2 \'iv,, P., + R.t + Z,. / 8Z, 2 The available power is IV^C/SZ,., so that the power attenuation K 2 is given by i 2 2 I «•-• = "2 * R} + R2 + Z, J When K has been specified, and also requiring that R m = Z,., we have two equations that are readily solved for the required values of ft, and R2. Thus we find that 1 -K R, = — — Z r (6.6a) i T A R 2K *=-, \^K2 ^ 2 Z , (6-66] For 10-dB attenuation in a 50-(2 system, we require K= /O.l which results in Rx = 25.97 fl and R2 = 35.14 ft. For a 3-dB attenuator, we require Rx = 8.58 ft and R2 = 141.4 ft. Thus, in order to produce a wide range of attenuation values, R x and P 2 must be capable of being varied over a substantial range of values. For the II network, P, and R 2 arc given by 1 + K 1-K2 R , - _ ^ - Z i (6.76) The P/iV diode differs from a conventional PN junction diode by having a thin layer of intrinsic semiconductor material between the usual Positive and Negative doped regions. The addition of the intrinsic region 402 FOUNDATIONS FOR MICROWAVE ENGINEERING reduces the junction capacitance, since the P and N regions apart. It also makes the forward conductivity of the diode a linear function of the diode bias current. In the forward bias "^ m 0 f e charge carriers are injected into the intrinsic layer and thus the c J 0 ^ 1 ' 0 " . is proportional to the injected charge, which in turn is proporti i bias current. The equivalent circuit of the PIN diode is shown in F The resistance R, and capacitance C, represent the intrinsic 1 ' g resistance R,, the junction capacitance C r and the diffusion capacit represent the PN diode characteristics. The resistance i? is th bulk semiconductor regions. The capacitance C f is the stray can between the terminals of the chip and can usually be neglected F packaged diode the equivalent circuit also includes the lead inducta and the package capacitance Cp. Under the reverse bias conditions, the diffusion capacitance is neel" ' ble and the junction resistance i? ; is very large. Also the parallel impedanct of R, and C, is negligible relative to the impedance of C; and ]?„ in series so the equivalent circuit reduces to that shown in Fig. 6.96. A typical value : ffs Cn± C„=k R, 4= C, R,: ± c, w =Ma cp4= R.+ R, (a) (c) F I G U R E 6.9 . (a) Equivalent microwave circuit of a PIN diode; (6) equivalent. conditions; (c) equivalent circuit for forward bias conditions. ircuit for re PASSIVE MICROWAVE DEVICES 403 for the impedance of the diode in the off state is predominantly a capacitive reactance of - J 4 0 to -j'400 fl. For forward bias conditions, the diffusion capacitance C d is large and provides a low-reactance shunt across the junction. Also C, is small, so that R, + R s becomes the controlling element for the diode impedance. The equivalent circuit under forward bias conditions is shown in Fig. 6.9c. In the fully on state, the resistance R, of the intrinsic layer is less than 1 Q in a typical diode. The resistance /?, varies inversely with the bias current. PIN diodes in the forward biased state can be used for the resistors R, and R 2 in the attenuator networks shown in Fig. 6.S. The presence of reactive elements in the equivalent PIN diode circuit will result in some inevitable input and output mismatch. The dynamic range of an attenuator using three PIN diodes as variable resistors is also limited to usually 10 dB or less. A more versatile attenuator can be built by using chip resistors to construct a cascade connection of Fl networks and using PIN diodes as switches to short-circuit the series arms for low attenuation or to be in the open state for one or more sections for various amounts of attenuation. An illustration of this type of attenuator is shown in Fig. 6.10. A basic section producing 2 dB of attenuation in a 50-fi system would have Rx = 436.2 iJ and #2 = 11.61 CI, In the fully forward biased on state, the PIN diode short-circuits R., and, since the parallel combination of /?, with a second /?, is a resistance of 218.1 $2, the loading effect on a 50-H line is relatively small, so that attenuation is also quite small. A field-effect transistor with zero voltage applied between the drain and source has a channel whose resistance can be controlled by the gate voltage. Thus FETs can be used as the variable resistors in one of the basic attenuator circuits shown in Fig. 6.8. In Fig. 6.11 we show a photograph of a commercial electronically controlled attenuator built as a MMIC circuit. This attenuator wil! operate over the frequency range from dc to 26 GHz and can produce a maximum attenuation greater than 30 dB. The return loss is greater than 10 dB and the insertion loss is 1.5 to 2.5 dB. This particular device is suitable for use in a microstrip circuit. The attenuator uses six MESFET devices as variable resistors in a basic T attenuation circuit. —WV^R2 fl2 fl, fl2 R, fl, fl, FIGURE 6.10 A swilchable attenuator network. PIN diodes are used to switch R.t in and out of the circuit. 404 FOUNDATIONS FOR MICROWAVE ENGINEERING F I G U R E 6. J1 A photograph of a MMIC electronically controlled attenuator. (Photograph courtesy of fo» Moskaluk. Hvwlett-Packard Company.) 6.3 PHASE SHIFTERS A phase shifter is an instrument that produces an adjustable change in the phase angle of the wave transmitted through it. Ideally, it should perfectly matched to the input and output lines and should produce zero attenuation. These requirements can be met to within a reasonable deg of approximation. There are a variety of designs for phase shifters mechanically adjustable type. The rotary phase shifter is the best in class and is the only one we will discuss. Electronically controlled phase shifters using PIN diodes as swwj have become popular for use in phased-array antennas. In a pnas .^ antenna, there are many radiating elements, such as printed-circui ^ and the radiated beam can be scanned in direction by varying i e J ^ excitation of each element in the array. In a large array this requ , ^ b e phase shifters; so a design that is small electronically controlled, t e c h n 0 Jeconomically produced in large quantities using integrated cir nts. ogy is desirable. The PIN diode phase shifters meet these requir Rotary Phase Shifter J in idelv use°. . -Molv . The rotary phase shifter is a precision instrument that is v - t oll microwave measurements. Its basic construction is similar PASSIVE MICROWAVE DEVICES 405 Rotating section */» plote x / 2 plate X M plote IJ; FIGURE 6.12 ( a ) Rotary phase shifter; l/>) quarter-wave plate. rotary attenuator, except that the center resistive card is replaced by a half-wave plate and the two outer resistive cards are replaced by quarterwave plates. The quarter-wave plates convert a linearly polarized T E , , mode into a circularly polarized mode, and vice versa. The half-wave plate produces a phase shift equal to twice the angle 0 through which it is rotated. The analysis of the principles of operation is given below. A circularly polarized field is a field with x and y components of electric field that are equal in magnitude but 90° apart in time phase.t A quarter-wave plate is a device that will produce a circularly polarized wave when a linearly polarized wave is incident upon it. Figure 6.12 illustrates the basic components of the rotary phase shifter. The quarter-wave plate may be constructed from a slab of dielectric material, as illustrated in Fig. 6.126. When the T E n mode is polarized parallel to the slab, the propagation constant /3j is greater than for the case when the mode is polarized perpendicular to the slab; that is, /3, > fi.,, where /3.2 is the propagation constant for perpendicular polarization. The length / of a quarter-wave plate is chosen to obtain a differential phase change ( 0 , - (i.Jl equal to 90°. The ends of the dielectric slab are tapered to reduce reflections to a negligible value. The half-wave plate is similar in construction, except that its length is increased to produce a differential phase change of 180°. fCircularly polarized fields are discussed in greater detail in Sec. 6.7. 406 FOUNDATIONS FOR MICROWAVE ENGINEERING FIGURE 6.13 l a ) Decomposition of incidi-nt T E , , mode; (6) oriental;' half-wave plate. In the rotary phase shifter, the quarter-wave plates are oriented at an angle of 45° relative to the broad wall of tiie rectangular guide. The incoming T E n mode may be decomposed into two modes polarized parallel and perpendicular to the quarter-wave plate, as illustrated in Fig. 6.13a. The incident mode is assumed given by (6.2) as J i E = — a r cos <& p'u «^i a * sm * If we replace cos $ by IT cos|(/.-- + -) = — cos\4> - — I - sin|<£ - and sin 6 by v2 in| <!> - cos| <t> ~ — + sin 2 the above expression for the incident field may be written as (6.8o> E = E , + E« where E, 2? 2 E2 = 2 J l I T\ P11 • ( . — a r cos^d. - - j - — J;a, ft s i n ^ 3\ — a r sin| <t> - 5 4 ?)*^M*-% (6-8 o) PASSIVK MICROWAVE DEVICES 407 The field E, is polarized parallel to the slab, and E 2 is polarized perpendicular to the slab. After propagation through the quarter-wave plate, these fields become E', = E i e -JJM E' 2 = E 2 e - ' (6.9c) J w = E2e -jis\te-juh-P\)t = JE e~ "'' (6.96) since (fi2- (ix)l = -IT/I. The resultant field consists of two orthogonally polarized T E U modes of equal amplitude and 90° apart in time phase, and hence constitutes a circularly polarized field. Consider next the action of the half-wave plate on the above circularly polarized field. Let the half-wave plate be rotated by an angle 0 past the quarter-wave plate, as in Fig. 6.136. The field E', + E'2 may be expressed in terms of T E , , modes polarized parallel and perpendicular to the half-wave plate by changing the origin of the angle variable <!> to 77/4 + 0; that is, we use the relations cos) <f> — — = cos rf> = cos ft cosUfr - 0 - — j - sin 8 sinj d> — 9 and 77 sin|<£ - - = sin 0 cos <{> - 0 — - + cos 0 sin </> - 0 — - 4 Thus we obtain J, Ei = — «-•"»•'< 1 77- — af cos\4> - 0 - — r \ 4 Pn / ~ -—--JJa,,, s i n U ~ 0 ~ ~ a \ - sin 0 — af sinlrf) - 0 - 4 Pn a E' -j& ,-jih' Pn „ a J, COS 0 / \ y a I ~ -</,'a,,, c o s U -Q- — / r (6.10a) 7T s i n ^ - 0 - - 774 + — - ^ a * cos <t> - 0 - — • + sin 0 Pn / , cos| <t> - 0 - — / ^ j;a, f t sin|</) - 0 - - J (6.106) The field polarized parallel to the half-wave plate has an r component varying as cos(<?> - 0 - TT/4), whereas the perpendicularly polarized mode has an r component of electric field varying as sin(<fc - 0 - TT/4). Hence, 408 FOUNDATIONS FOR MICROWAVE ENGINEERING from (6.10), we obtain E', + E'2 = Ej' + Eg (6. l i e , where -Jsintf) —arcos/<£ - 6 - /2 i) 8- ?)] - > S | ' -7« (6.116) -iV2 B«- e-jPii-jo — a r sin I <fi - 0 r \ + P u 4 / •"" j;acScosU - 6 - — a \ 4 (6.1k) After propagating through the half-wave plate of length 21, this field becomes E3 = E ' ; e - 2 ^ ' ' (6.12a) E„ = E £ e - W = Erge-****-**-***1 = - E J * - * * * (6.126] since 2 ( ^ 2 - pt)l = - T T . This new field may now be decomposed once again into two ' modes polarized parallel and perpendicular to the quarter-wave plate in » output guide. If we assume that this plate is parallel to the input quarte wave plate, we can obtain the required decomposition by referring the i variable <j> to i r / 4 as the origin. If we follow the procedure used earlier, obtain E 3 * «, - E'a t K ,613< " where E' V2 = - 2 e~3j0\i-r*< ,6-13*) Ji r a . cos " 4J a j\/2 J, a, s i n U - E'4 = : — i - W i ' - m — + —V,'a,» c o s U - 4 / J 2 r \ 4; a jate,t he - 3 Finally, after propagating through the second q u a r t e r * output field becomes ,g E 0 = E% + ET4 PASSIVE MICROWAVE DEVICES where 409 E"s = tfcr*** (6.146) E; = E'je--"**' =jE,4e--"i'1 (6.14c) When the fields E^ and E'j are combined, we obtain E n = e - 4 J W - W - l a f cos <b - — J,'a„, sin </>) I (6.15) which is again a linearly polarized T E U mode having the same direction of polarization as the incident field given by (6.2) and (6.8). Note, however, that the phase has been changed by an amount 4)3,/ + 2ft. Thus rotation of the half-wave plate through an angle 0 changes the phase of the transmitted wave by an amount 20. This simple dependence of the phase change on a mechanical rotation is the chief advantage of the rotary phase-shifter. Besides dielectric slabs, the circular guide may be loaded with metallic fins or rods to produce 90" and I80 c differential phase-shift sections. These methods are discussed in a paper by Fox.t Electronically Controlled Phase Shifters There are several basic designs that are used to build digital-type electronically controlled phase shifters. In all designs the PIN diodes are used to switch circuit elements in and out of the transmission path. Each switching operation adds or subtracts a finite phase-shift increment such as + 11.25°, ± 22.5°, + 45°, ± 90°, etc. By using a cascade connection of several phase shifters, the full range 0° to 180" can be covered with a resolution equal to the smallest phase increment that is made available. The simplest phase-shifter design uses PIN diode switches to switch one of two alternate transmission-line lengths into the transmission path as shown in Fig. 6.14. The bias currents can be supplied by connections at the midpoint of a half-wave open-circuited stub. The first quarter-wave section is a low-impedance stub and transforms the open-circuit impedance to a short circuit or low impedance at the point where the bias line is connected. The next quarter-wave section uses a high-impedance line and transforms the low impedance of the midpoint into a high impedance which produces negligible loading of the main transmission line. The dc return for the bias current can be obtained by connecting the input and output lines to the ground plane through short-circuited high-impedance quarter-wave-line sections as shown in Fig. 6.14. If the two transmission-line sections have lengths l x and l2, then the incremental phase change produced when line 2 is switched in to replace line 1 is A</> = /3(/ 2 - / , ) , where /3 is the propagation phase constant. This +A. G. Fox, An Adjustable Waveguide Phase Changer, Proc. IRE, vol. 35, pp. 1489-1498, December, 1947. 410 FOUNDATIONS FOR MICROWAVE ENGINEERING Bias circuit ground F I G U R E 6.14 fncremenlal-line-type phase shifter. type of phase shifter produces an incremental change in phase that depends on the frequency since p' is a function of frequency. If the dispersion of line is small, i.e., phase velocity is independent of frequency, then th increment in signal time delay is a constant and is given by A- =• lx)/vp = {li - / , ) # / « • When broadband signals are to be radiated phased array, the use of different phases for the excitation of the v s r l 0 ^ elements in the array will generally result in some signal distortion the phase shifters are true time-delay devices. ., oD fi Since the PIN diodes are not ideal switches and have a h n ^ j f t e r 5 resistance and some series lead inductance, the use of several phase in series will result in a significant overall insertion loss. A second type of phase shifter is shown in Fig. 6.15. In this ^ maln n s e t y n u type ui piiast- suttiei is suuwu ui z i g . W.J.^- ~~ rg susceptances of equal value jB are switched in shunt wi obtain . th? transmission line when the diodes are on. The stubs are used j if ai required susceptances and are usually short-circuited at ithe ' provide a dc return for the bias current. The stubs are sPa^ spa eing l apart; so the phase shift is jii = B when the diodes are off. T h * * p re5UlW chosen so that when the stubs are switched into the circuit ^ ^ t „ i network will still present an input and output impedance equa connecting transmission lines. I '.ASSJVK Mil -HOWAVE OEVII -JiS 411 •/}/=e- Itf- tf Bias currenl input 'S S3 •>• oc F I G U R E 6.15 A phase shifter using switched reactive elements. We can analyze this circuit in a straightforward way using wave-amplitude transmission matrices (Sec. 4.9). The wave amplitudes at various points in the circuit are designated by Vf, V, , V2*, V.J, etc., as shown in Fig. 6.15. The plus superscript designates the amplitudes of waves propagating to the right, while the minus superscript is used for the amplitudes of the waves propagating to the left. The wave-amplitude transmission matrices for a normalized shunt susceptance jB and a section of transmission line of electrical length 8 are B B >-% [A,] = B B J» [A,] = (I l J ~2 We can now write = = [A,][A2] AA Au Av, v:' "21 A2 2 v; = [A!][A2)[A3] vr If we choose V, = 0, then V~ = V{ /An; so the transmission coefficient from port 1 to port 4 is Tu = Aux. By multiplying out the matrix product shown above, it is readily found that 7\ 4 = B 2 B2 1+7-^-1 eJ" + - e--'" T-l 412 FOUNDATIONS KOK MIC KOWAV'E KNOINEERINC After some algebra we find that |T M ) = 1 provided we choose tan 9 = = B (6. 16, Wlien we use this relationship to replace B / 2 in the expression f obtain Tu = ~e*> = e -j(T?-0> (6.17, Thus the change in phase _when the diodes are switched on i (jr - 0) - e = 77 - 2«. For B = 2, 0 = TT/4 and A0 - - / 2 or 9(, B = 1, (V = 1.107 rad and A</> = 53.14°. Thus, by choosing B appropriat, a wide range of phase shifts can be obtained. To a first approximation the diode impedance in the on state is a sm n series inductive reactance. This should be included as part of the susce tance that is switched into the line. In the off state the diode impedance due to the shunt capacitance. Thus the shunt capacitance must be small if the phase shifter is to work properly. Since the off-state impedance is not always that large, an alternative design that allows some compensation for both the on-state and off-state diode impedance can be used. This alternate design is shown in Fig. 6.16. In this design two PIN diodes are used to open-circuit or short-circuit the tap points P and P' of two stubs that are connected in shunt to the main transmission line. The stubs are open-circuited at the remote end if shunt switches are used. Since the condition (6.16) for a matched circuit cannot be simultaneously met for two different susceptances, this last design requires that the two susceptances jB, and jB., that load the main line_when the diodes ai on and off be small. It is preferable to choose jB., = -jBt so as to maximiz the phase difference in the transmission coefficient for the two states. spacing the stubs A/4 apart, the reflections from the two stubs almost cancel. The stub lengths <7 and tap point distance d, can be chosen s< B 2 = - B , including the diode reactances in the on and off states. ^ When (I = TT/2 the use of wave transmission matrices shows ' / B\2 B2 B2 r 14 = - -/(1+/B)" F I G U R E 6.16 A phase shifter using >fe spaced A /4 apart. P and P' » the CTun when the diod*, respectively. The bias circuit *«!<*«• &. ff t ^v* '•>•'-. .., PASSIVE MICROWAVE DEVICES 413 when B « 1. Hence IT,,,!2 = d_+ B2)""' which is close to one. The phase angle of Tu is -tr/2 - t a n ' ' B = -jr/2 - B. The changes in_phase between the states when B = B, and - B , is thus 2 B , . A value of B, as large as 0.2 would not produce a significant mismatch and would result in a change in phase \<!> = 0.4 = 22.92° between the two states. This type of phase shifter is limited to relatively small phase shifts between states in order to keep the input VSWR small. A variety of other phase shifter circuits are also possible. A good summary of various circuits that have been analyzed and the performance that can be obtained can be found in the literature.! 6.4 DIRECTIONAL C O U P L E R S A directional coupler is a four-port microwave junction with the properties discussed below. With reference to Fig. 6.17, which is a schematic illustration of a directional coupler, the ideal directional coupler has the property that a wave incident in port 1 couples power into ports 2 and 3 but not into port 4. Similarly, power incident in port 4 couples into ports 2 and 3 but not into port 1. Thus ports 1 and 4 are uncoupled. For waves incident in port 2 or 3, the power is coupled into ports 1 and 4 only, 80 that ports 2 and 3 are also uncoupled. In addition, all four ports are matched. That is. if three ports are terminated in matched loads, the fourth port appears terminated in a matched load, and an incident wave in this port suffers no reflection. Directional couplers are widely used in impedance bridges for microwave measurements and for power monitoring. For example, if" a radar transmitter is connected to port 1, the antenna to port 2. a microwave crystal detector to port 3. and a matched load to port 4. the power received in port 3 is proportional to the power flowing from the transmitter to the antenna in the forward direction only. Since the reflected wave from the antenna, if it exists, is not coupled into port 3, the detector monitors the power output of the transmitter. If the coupler is designed for 3-dB coupling, then it splits the input power in port 2 into equal powers in ports 2 and 3. Thus a 3-dB directional coupler serves as a power divider. Directional couplers with 3-dB coupling are also called hybrid junctions and are widely used in microwave mixers and as input, and output couplers in balanced microwave amplifier circuits. There are many available designs and configurations for directional couplers, hybrid junctions, and power dividers. Space limitations will allow us to only examine some of these.t t l . Bah) and P. Bhartia, "Microwave Solid Slate Circuit Design," -John Wiley & Sons. Inc.. New York, 1988. K. Chang (ed.), "Handbook of Microwave and Optical Components, Microwave Solid State Components,'" vol. 2. John Wiley & Sons, Inc., New York, 1990. 414 FOUNDATIONS FOB MICROWAVE KNGINF.KRIM. F I G U R E 6.17 A directional coupler. Arrows jndi i«„eth, rection of power flow. © © r ±=t± •n p, © FIGURE 6 1 8 ^ rectional coupler. Directional couplers using waveguides usually consist of two rectanmi lar waveguides with coupling apertures located in a common wall as illus trated in Fig. 6.18. Since these devices are required to operate over a band of frequencies, it is not possible to obtain ideal performance over the whole frequency band. The performance of a directional coupler is measured by two parameters, the coupling and the directivity. Let P, be the incident power in port 1, and let Pf be the coupled power in the forward direction in arm 3. The coupling in decibels is then given by Pt C = 10log- (6.18) Ideally, the power Ph coupled in the backward direction in arm 4 should be zero. The extent to which this is achieved is measured by the directivity D, which is denned as D = 10 log - ^ (6.19) The directivity is a measure of how well the power can be coupled desired direction in the second waveguide. deduced A number of properties of the ideal directional coupler may w^ ^ from the symmetry and unitary properties of its scattering " i a * r * e d ln fig. The least stringent definition of a directional coupler as lllustr 6.17 is that it is a four-port junction with 5 1 4 - S->3 =0 = S 2 2 = 0, that is, ports 1 and 2 matched, and the coupling e^ ^ S u , S 1 3 , S 2 4 , and S3A are not equal to zero. The scattering ™< _ ^ « art" tA very good overall survey of types of directional couplers, hybrid ju M i c r o W ave dividers can be found in K. Chang. "'Handbook of Microwave ComP0"e",%9o. Components and Antennas," vol. 1. John Wiley & Sons. Inc.. New York, PASSIVE MICROWAVE DEVICES 415 the form 0 [S} = 0(2 SI2 0 s,, o 0 O [3 0 s2, s,, s,, 6'24 . , j o'-'3-1 o '-'4-1 If we form the product of row 1 with the complex conjugate of row 3, and also the product of row 2 with the conjugate of row 4, we obtain because of the unitary nature of the scattering matrix. Since S 1 3 and S.,4 are assumed to be nonzero, these equations show that S 3 3 = S 4 4 = 0; that is, all four ports are matched. Thus the scattering matrix becomes 0 [S] = O |2 ^13 0 SV2 0 0 S 24 ^13 0 S2A 0 0 "34 ^34 0 (6.20) If we take the product of row 1 with the conjugate of row 4, and similarly row 2 with the conjugate of row 3, we now find that Sl2S2i + Si 3 S.* 4 = 0 S 12 Sj* 3 + S2.1S3., = 0 If we note that l-S^S^I = | S 1 2 | IS24I, these equations are seen to give |S 12 I|S. 24 I = | S 1 3 | | S 3 4 | (6.21a) IS 1 2 I|S 1 3 | = | S 2 4 | | S 3 4 | (6.216) When we divide the first equation by the second equation, we obtain or '241 IS 131 ISI3| 10241 is,. |S 13 I = |S 24 I (6.22a) thus the coupling between ports 1 and 3 is the same as that between ports 2 and 4. Use of (6.22a) in (6.21a) also gives I S y = IS^I (6.226) so that the coupling between ports 1 and 2 equals that between ports 3 and 4 also. The product of the first row with its conjugate equals unity, so that | S , 2 | 2 + IS l 3 | 2 = 1 (6.23a) IS I2 I 2 + IS Z4 I 2 = 1 (6.236) similarly, By choosing the terminal plane in arm 1 properly, we can adjust the phase 416 FOUNDATIONS FOR MICROWAVE ENGINEERING angle of S 1 2 so that S , 2 is real [see (4.53)]. Thus let S l 2 be a a P st number C,. Similarly, by choosing the terminal plane in arm 3 n ° ' iv% 0per can make S 1 3 a positive imaginary quantity jC2, where C ^ 'y, w e z ;>positive. We now have real aiKj C« + C | = 1 (6.24) We can choose the reference plane in arm 4 so as to make S re-1 equal to C, by virtue of (6.226). It is now necessaiy for S2A\0 fe^ tl JC-z since Sl2SSA + SUiS^ = 0, as given earlier. Thus the simplest f o r " ^ the scattering matrix of an ideal directional coupler is 0 [S] = c, jc2 0 c,0 0 JC2 JC2 0 0 0 JC2 c, 0 (6.25) c, t/2 where C 2 = (1 - C f ) from (6.24). It may also be shown from the unitary properties of the scattering matrix of a lossless reciprocal four-port junction that if all four ports are matched, the device must be a directional coupler.t Directional-Coupler Designs There are a great variety of ways of constructing directional couplers. Some of the more common aperture-coupled types are described below. Their design is based on the small-aperture-coupling theory presented in S 4.13. This theory was originally developed by Bethe.l Bethe-Hole Coupler The Bethe-hole directional coupler consists of two rectangular wavegu)^ coupled by means of a small circular aperture located in the cen e ^ common broad wall. To achieve directional coupling, the axis o ,tnjj guides must be at an angle 6, as illustrated in Fig. 6.19a. A v a " a t J ° " rture design consists of a similar arrangement, with 0 = 0, but an o as in Fig. 6.196. 4 13- An The theory for the coupler in Fig. 6.196 was given in »ec-^ ^ ^ . j incident TE 1 0 mode in guide 1, with an amplitude A, P r o u C e ^ o p o I electric dipole in the aperture plus a tangential magnetic dipole p^ ^ ti,e and in the same direction as the magnetic field of the inciden t C . G. Montgomery. R. H. Dicke, and E. M. Pureed. "Principles of Micro*** 9.JO. McGraw-Hill Book Company. New York, 1948. 163 t H . A. Bethe, Theory of Diffraction by Small Holes, Phys. Rev., vol. 66- PP' - \9i* PASSIVE MICROWAVE DEVICES FIGURE 6.19 Bethe-hole directional pler. 417 cou- upper guide the normal electric dipole and the axial component of the magnetic dipole radiate symmetrically in both directions. The transverse component of the magnetic dipole radiates antisymmetrically. By varying the angle d or the aperture position d, the amplitude of the fields coupled into ports 3 and 4 can be controlled. For the directional coupler shown in Fig. 6.19 a, the optimum value for the angle 0 is given by cosfl = 2A* 0 (6.26) This choice for 8 will minimize the field coupled into port 4. Since the coupling is not zero, a perfect directional coupler is not obtained. A detailed analysis shows that the coupling and directivity that can be obtained are given by t 1 C = 20 log -= Xcos0 2cos0 D = C + 20 log 1 + cos 0 (6.27a) (6.276) where X = 16irr 0 l /3afcA /{ . The directional coupler shown in Fig. 6.19b was analyzed in Sec. 4.13. When the spacing d from the side wall is chosen to satisfy the following tR. E. Collin, "Field Theory of Guided Waves," 2nd ed., chap. 7, IEEE Press, Piscataway, N.J., 1991. 418 FOUNDATIONS FOR MICROWAVE ENGINEERING relation: sin /6 a (6. 28) there will be zero power coupled into port 4. The coupling into port ^ • (6.30a) with 1 + X 2 replaced by -J\ + Y ? and A x + 4, interchanged with 1 ^ ^ Directional-coupler c h a r a c t e r i s t i c s a r e also obtained by choo • Sln satisfy t h e e q u a t i o n t 8 d u} vd sin y/2{K% - a2) 9 (6.29) W h e n A 0 = / 2 a t h e a p e r t u r e will be located at t h e c e n t e r , t h a t is d = F o r t h e above condition t h e field coupled into portrt 33 is is minimized minimized and and t( that coupled i n t o p o r t 4 is a maximum. T h e ccoupling o u p l i n g aa nn dd directivity directivity that thai can . be achieved a r e given by [see (4.139) a n d (4.140) a n d let X <= - B] \C\ C = 20 log A2+A4, D = 20 log A.2 + A4 A, +Aa 1 +X2 = 20 log —g- (6.30a) = 2 0 log X - l (6.306) w h e r e X = ( 1 6 7 r r o / 3 a 6 A ^ ) s i n 2 ( i r t / / a ) . T h e s e formulas apply only at the design frequency. T h e above r e s u l t s a r e based on t h e a s s u m p t i o n t h a t t h e guide wall in which t h e a p e r t u r e is located is infinitely t h i n . F o r t h e normal thickne used in a waveguide wall, t h e coupling will be 1 to 2 dB smaller. Example 6.1. D e s i g n of a B e t h e - h o l e directional coupler. We design a directional coupler based on (6.28) and (6.30a). The wavegui dimensions are a = 0.9 in, b = 0.4 in. The center frequency is 9 < which 0 = [*§ - ( 7 r / a ) 2 l , / s = 1.29 r a d / c m . From (6.28) we obtain si = 3.333/2.286/6 which gives d <= 0.464 cm. Thus the center of the J is located at 0.464 cm from the side of the waveguide. We ^ - ^ f f j j f . coupler for_30-dB coupling. From (6.30a) this requires that (1 + x 31.623 or X = 0.03164. The required aperture radius is given by 3obAgX 16TT s i n 2 ( 7 r d / a ) •i 1/3 = 0.392 cm obtoi" This is already a large aperture; so clearly it would be difficu ^ funC,ion< coupling of 20 dB. The variation of the coupling and d i r e c t i w t y a s ^ frequency can be obtained by using the calculated values to tR. E. Collin, loc. cil. PASSIVE MICROWAVE W-.'VN IBS I GHz 419 F I G U R E 6.20 The coupling and directivity of the JSe! he-hole directional coupler as a function of frequency for the coupler in Example 6.1. expressions For A^ + A3 and A., + A4 given by (4.140) and using these in (6.30). In Fig. 6.20 we show a plot of coupling C and directivity D as a Function oF Frequency, The coupling remains almost constant. II increases From 28.46 dB at 8 GHz to 30.96 dB ai 10.5 GHz. The directivity, however, drops rapidly as the frequency changes From 9 GHz. From these results we can see that the Bethe-hole coupler is a narrowband device. By using a second aperture on the opposite side of the center line the amplitudes oF the coupled waves are doubled. This will increase the coupling by 6 dB without affecting the directivity. Note that increased coupling is measured by a decrease in the coupling C when expressed in decibels by (6.18). Hole Couplers Two-hole couplers consist of two r e c t a n g u l a r waveguides coupled by two identical a p e r t u r e s spaced a q u a r t e r guide wavelength A , / 4 a p a r t as in Fig. 6 . 2 1 . T h e a p e r t u r e m a y , in general, h a v e directive p r o p e r t i e s , i.e., r a d i a t e a field with different a m p l i t u d e s in t h e forward a n d reverse directions. With a wave of u n i t a m p l i t u d e incident at p o r t 1, let t h e field coupled into t h e second g u i d e have an a m p l i t u d e B f in t h e forward direction a n d B h in t h e backward direction. S i n c e B ( a n d B h are t h e a m p l i t u d e s of t h e coupled fields for an incident wave of u n i t a m p l i t u d e , t h e y a r e called t h e a p e r t u r e coupling coefficients. If only a small a m o u n t of t h e incident power is coupled by t h e first a p e r t u r e , t h e a m p l i t u d e of t h e incident wave is essentially u n i t y ® © **^ ~B, V^" e-j0d 1 f • ® F I G U R E 6.21 Two-hole directional coupler. 420 FOUNDATIONS FOR MICROWAVE ENGINEERING at the second aperture also. Thus this aperture couples the san power into the second guide. Note, however, that because of th ^ O u , l l 0 f eren in path length, the phase of the field coupled by the second apeM-6 te relative to that coupled by the first aperture. The total forward ««. 1 upper guide at the plane bb is 2Bfe •"'" a n d t "h e t o t a r b a c k w u a^ "^ wave i the plane aa is Bb(l •+ e -2jpd) Hence, since the forward-path I * in the two guides are always the same, same, the me iorwara forward waves Q i . , . „ v J e " ^ ys UC add phase. The backward waves will add out of phase whenever 2flrf n = 1,3,5 In particular, a value of d = A g / 4 will result in i cancelation of the backward wave. The coupling is given by C= -201og2|B,| (6.31 Q) and the directivity is given by D = 20 log = 20log 21B \Bf (' = 20 log BJIl + e-2'*'! Bh\ Icos /3d| B, 201og|sec/3d| Bb (6.316) The directivity is the sum of the inherent directivity of the single aperture plus a directivity associated with the array (in this case a two-element array only). Since Bf and Bh are the aperture-coupling parameters and are generally slowly varying functions of frequency, the coupling C is not particularly frequency-sensitive. However, the directivity is a sensitive function of frequency because of the sensitivity of the array factor sec fid. Sehwinger Reversed-Phase Coupler The Schwinger reversed-phase coupler is designed to interchange the frequency sensitivity of the coupling C and directivity D. This is accomp"^ by making one aperture radiate a field which is the negative of that racua by the other. With reference to Fig. 6.21, iet the first aperture r a d i a t e ^ d e Bf, B„ and the second aperture -Bn -Bh. At plane bb in the " P P ^ r n o j the total field is now B, ~Bf = 0 under all conditions. Hence port a ^ coupled to port 1. At the plane aa the total field is, after accounting phase change due to propagation, Bb - Bbe-2"'d = e'jlidBb2j sin (id Thus the coupling between ports 1 and 4 is C = -20\og2\BhsmPd\ . %%) p \s and is a maximum for d = A g /4. For this coupler the ^^ coUpling C theoretically infinite and independent of frequency, whereas t ^ fl5 t. is quite frequency-sensitive, although not as frequency-se° a u „d * directivity D given by (6.31*), since sin lid varies more s l o w V . ^ ^ e d* than does cos (3d. Actually, in practice, the directivity D is n0 PASSIVE MICROWAVE DEVICES 421 F I G U R E 6.22 ( a ) Schwinger reversed-phase coupler; (6) Moreno crossed-guidc coupler: (c) Riblet T-slot coupler. in the foregoing discussion, it was assumed t h a t the same incident field was present at each aperture, and each aperture radiated the same field into the upper guide. Because of interaction effects between the two apertures, the assumption of equal-amplitude fields coupled by both apertures is an approximation valid for a small amount of coupling only. Figure 6.22a illustrates a typical reversed-phase coupler. The TE,„ mode has a zero normal electric field and transverse magnetic field at the narrow wall, and hence the coupling to this mode in the upper guide is through the induced axial magnetic dipole moment of the aperture only. In the lower guide the axial magnetic field of the TE 1 0 mode is of opposite sign on the two sides of the center, so that induced dipoles M : and -Mz are produced. These dipoles radiate symmetrically in both directions, but are phase-reversed to obtain the desired reversed-phase directional coupler. The other double-aperture-coupled directional couplers in common use are the Moreno crossed-guide coupler and the Riblet T-slot coupler, illustrated in Fig. 6.22 also. Design nomograms for these couplers, as well as for the Schwinger reversed-phase coupler, are given in a paper by Anderson.t tT. N. Anderson. Directional Coupler Design Nomograms, Microwave J., vol. 2, pp. 34-38, May 1959. 422 FOUNDATIONS FOR MICROWAVE ENGINEERING Multielement Couplers To achieve good directivity over a band of frequencies, cour>le apertures may be used. The theory and design of such counl those given for multisection quarter-wave transformers in Ch ^ ' 3ara "el 6.2.3 illustrates an N + 1 element coupler with all aperture to d. If we assume that the total power coupled is relatively sm- l l " 1 ^ ^ " ^ with the incident power, the incident wave can be conside ,C°mi>ared. essentially the same amplitude A at each coupling aperture ana r additional phase change. Let the apertures having coupling coeffic"» n ~ 0,1,2,..., N, in the forward direction, and Dn, n = o, 1 2 1 the reverse direction. At the position of the Nth aperture the total V ' wave in the upper guide is N Br = Ae-*NdZ C„ (6.33) 71=0 At the plane of the first aperture, the total backward wave has an amplitude Bb = AZ Dne-><*"« (6.34) n =0 The coupling and directivity are given by iV C = - 2 0 log (6.35a) L c„ 71 = 0 D = - 20 log iI^0D„e-"J2«"| N = - C - 20log L Dn ,-jfiind (6.356) 71 = 0 In a multiaperture directional coupler the required coupling from ef aperture is small so the aperture radii are then also small. The correspoi ing aperture susceptance B and reactance X are then also small. T h u t _ . amplitudes of the waves coupled in the direction of ports 3 and 4 n 6.23, which are given by (4.140), can be approximated by Ax+A3=j IB + - A 2 + A 4 =j D0A C0A D,A C,A X\ - B X\ 2 2 DUA CNA _V_ F I G U R E 6.23 A multielement directional coupler. 2/ A A PASSIVE MICROWAVE DEVICES 423 In terms of the notation being used in this section, we can thus write B +X C„ =j (6.36a) 2 B - X (6.366) D„ =J By using these expressions we obtain the advantage that for fixed aperture offsets both C„ and Dti are constants multiplied by the radius r„ cubed for the n t h aperture. Hence we can express C„ and D„ in the form C„ = Tfr* = Tfd„ (6.37a) D„ = 7V.? hd b' n = Tb"„ (6.376) where d„ = r,f. Both Tf and Th depend on the frequency. We can now express the directivity D in the form N L dne - D = -C - 2 0 1 o g | T J - 20log j2ffnd (6.38a) 'i=0 and express C as C = - 2 0 1 o g l 7 > | - 20 log Z d„ r. (6.386) II In the expression for D the term 20 log \Tf/Tb\ gives the intrinsic directivity of a single aperture. This directivity is usually small except possibly at the design frequency. Thus, if we are to achieve a broadband design, we must design the array factor ;Y F = E drie j'ltJTIlJ (6.39) ra-0 to give a high directivity over the frequency band of interest. In order to obtain an equal-ripple characteristic in the passband, the array factor F is made proportional to a Chebyshev polynomial. If we choose a symmetrical arrav. with d v = d v , d, = d v _ , , etc., we obtain [Eq. (5.56)] M F = L n 2d„ cos(7V - 2n)pd (6.40) n where M = {N - l ) / 2 for N odd and N/2 for N even. Note that for N odd there are an even number of apertures, since the first aperture has been labeled the zeroth aperture. In (6.40) the Mth term is d M for N even and 2 d M cos(N - 2M)(id for N odd. To obtain a Chebyshev-type response, we 424 FOUNDATIONS FOB MICROWAVE ENGINEERING now choose M F= £ 2dncos{N - 71-0 2n)8 - K\TN(secdmcos8)\ (6.41) as in (5.68). In this equation 8 = (id and sec 8 m is the value of ser OJ at upper and lower edges of the passband. At the center of th ^ P Sbaad 8 = TT/2, corresponding to a spacing d = \/,/4. The constant K " l s ^ give the desired value of coupling C in the center of the ha J ° S e n u' d 8 = TT/2. Thus we obtain ' wh»e N C = -201og|7y| n=0 = -201ogK\Tf\\TN(sec8m)\ (6.42) since E *< «U = K\TN(sec 8„, )| from (6.41). J If we use (6.38a), (6.41), and (6.42), the expression for directivity mav be written as D = 20[log K\TfTN(sec 8m )\ - log \Tb\ - log tf|T„(sec 8m cos 8)\] = 20 log log T jV (sec0„ TN($ee8m cos 8) (6.43) Since Tf/Tb is a function of frequency, D will not have a Chebyshev-type behavior. However, the departure from a true Chebyshev behavior will usually be small since Tf/Th gives very little directivity, except perhaps near the center of the band. For a conservative design we choose the minimum value of D on the basis that Tf/Tb contributes negligible directivity. Certainly, for a broadband design, this will be the case at the edges of tl passband. The minimum value D m of'directivity in the passband as a tributed by the array factor F occurs when T v (secfl„,cos0) = 1 Hence let D,„ be defined as (6.44) D,„ = 2Ologir A r (sec0 m )l fixed, which in This equation snows shows that i n i s equation tnat if 11 we wespecify specnyDumm, , then tnen sec &e<-8„, u„,is .~ .\ sj) or turn fixes the bandwidth, and vice versa. Thus we must specify e i t o t * R sec 8m, and the other is fixed. We may then solve (6.42) for the con^ ^ in terms of the given value of the coupling C at the center hole Froml (6.41) coefficients a„ d„ are are iounu. found. \X3AU the tne coemcients three-'1" We will outline the procedure by considering the design ot a^ ^ ^ Chebyshev directional coupler utilizing offset circular apertu ^ ^ pig. common broad wall between two rectangular waveguides as s . fln wit 6.23. For the n t h aperture the field coupled in the forwardI air « feverSt A = 1, is C„ where C„ is given by (6.36a). The field coupled m PASSIVE MICROWAVE DEVICES 425 direction is given by Dn in (6.36/)). By using (4.139) we get 2/e2 sin 2 C„ = 3/3a6 a + . / — 0sin^ 3ao \ JTX 0 a 77 + 7 7 ^ cos 2 /3a" ~Xn (6.45a) 2/e2 o„ = 'J 3/3 a6 77' ~*r sin- 1TX, " • ' T3a6 Z I1P si " 2 ~~~ 7TI c o s 2 a ~ /3a~ (6.456) We have used x 0 for the aperture position so as not to confuse this with the aperture spacing d. The factors multiplying r,f are T f and Th, respectively, and x 0 is the aperture offset measured from the waveguide side wall. For a three-hole coupler, T N (sec e,„ cos 6) - T2( sec 6„, cos 6) = 2 sec 2 Q,„ cos 2 0 - 1 From a specification of D,„, we can solve for dm using D,„ = 20 log 17^ (sec 0 m ) | which for n = 2 gives sec H,„ = (10 < D,„ / 2 0 ) + 1) 1/2 /2 In general we can let sec 0ra = cosh </> and use ^ . ( c o s h <j>) = cosh Afy>; thus cosh N<f> = 10 (£, '" /20 » = y (6.46a) </> = — cosh^ 1 (10 < "•" / 2 0 , ) = T7ln(.V + \/y 2 - ! ) N (6.466) 1 Q„, = cos (6.46c) cosh <!> There will be two solutions for 6>„,, one less than 77/2 and one greater than 77/2. These two values determine the values of fid = 6 m at the lower and upper edges of the passband. When 8 m has been found, then if we specify the desired coupling C = C 0 at the center of the band we must have, from (6.42) with 0 = 77/2, C 0 = - 2 0 log K\Tf\ ITysec 0m)| or JQI-CO/20) K = 7^777^ 7—7 = \TA\TN(seC0m)\ 1Q(-C„/20. n ~n. \Tr\lO,D-/20> (6-47) where \Tf\ is found from (6.45a) at the center of the band where fid = TT/2. After we have found the constant K, we express TN(secOmcosO) as a 426 FOUNDATIONS FOR MICROWAVE ENGINEERING! F o u r i e r series. F o r N = 2 we have T,( sec e m cos 6) = s e c 2 0,„ cos 20 + s e c 2 B m - i F r o m (6.41) we can t h e n d e t e r m i n e t h e d,.. F o r o u r specific p case we 2 2 d 0 cos 2fld = 2d0 cos 20 = K s e c B m cos 20 d, = K ( s e c 2 0 m - 1) T h u s d Q = d 2 = ( X / 2 ) s e c 2 0 m . S i n c e d„ = r„3 we get 1/3 K r0 = *v = | — secz fl r, = ( i ? s e c 2 0 m - i f ) 1/3 T h i s c o m p l e t e s t h e design of t h e directional coupler. T h e coupling and directivity as a function of frequency can be e v a l u a t e d u s i n g (6.42) a (6.43) and requires t h e e v a l u a t i o n of T(, Tb, a n d 7 \ - ( s e c 0 m c o s 0 ) at each frequency of i n t e r e s t . T h e following n u m e r i c a l example illustrates the perf o r m a n c e t h a t can b e achieved. E x a m p l e 6.2 T h r e e - H o l e C h e b y s h e v d i r e c t i o n a l - c o u p l e r design. A three-hole directional coupler with a coupling of 20 dB and a minimum array factor directivity of 30 dB will be designed. The waveguide dimensions are a = 0.9 in, b = 0.4 in. The design frequency is 9 GHz. At 9 GHz the waveguide TE„, propagation constant is 0 = 0„ = •«-er 1/2 = 1.29 r a d / c m Thus the aperture spacing <i is ~/2(i = 1.218 cm. By using (6.46) we obta <b = ^assb~I(101') = 2.07338 and om = 1.3206 and TT - 1.3206 = 1.821. The values of /3 at the lower and upper. band edges are given (2fl„,/7r)0 o or 1.0845 and 1.4954. The corresponding values of *o (1.0845 2 + T T V O 2 ) 1 ' 2 = 1.7506 and (1.4954 2 + T r ' / a 2 ) " 2 - 2.03097. Fron^o we obtain the frequencies at the band edges and these are 8.358 GHz. The fractional bandwidth is \f'/fu = 0.149. The next step is the evaluation of" K using (6.47). The P a r a m 5 ! G . the magnitude of the coefficient of rf, in (6.45a) at the f r e ^ u e " ° | 3 p 7 a will choose an aperture offset given by (6.28), that is, x(l = • jjjy found then find that K = 6.0244 x 10 ~ 3 . The aperture radii are now r and are r0 = r2 = 0.3663 cm and r, = 0.4518 cm. . 3 g , in f The coupling and directivity at any frequency is given by -^ j ^ h e 6.24 we show the overall performance of the directional coupled ^ u tal p The factor t. ^ = curve shows the directivity contributed by the array that 7*! directivity is greater because we chose the aperture offset sue ^^. u ificant 9 GHz; thus the aperture directivity contributes in a sign PASSIVE MICROWAVE DEVICES /GHz 427 FIGURE 6.24 Performance of a three-hole Chebyshev directional coupler with C = 20 dB and Dm = 30 dI5. The dashed curve shows the directivity contributed by the array factor and is never less than 30 dB in the passband between 8.358 and 9.697 GHz. overall directivity. If we used a second set of apertures spaced a distance x0 from the opposite side wall, the coupling would be increased by 6 dB without changing the directivity. We could thus obtain a nominal coupling of 14 dB over the frequency band. Coupled-Line D i r e c t i o n a l C o u p l e r s Aperture-type directional couplers are not suitable for microstrip or strip-line construction. For planar-transmission-line structures, coupled transmission lines are frequently used for building directional couplers. In Fig. 6.25 we illustrate a microstrip directional coupler that involves two coupled microstrip lines. In practice, the printed circuit board would be housed in a shielded box and coaxial-transmission-line connectors would be bonded to each microstrip line. The analysis of the coupled-line directional coupler is readily carried out by taking advantage of the fourfold symmetry of the FIGURE 6.25 A microstrip coupled-line directional coupler. 428 FOUNDATIONS FOK MICROWAVE ENGINEERING structure. We can choose excitations so that the symmetry plan sponds to an electric wall (short circuit) or a magnetic wall (n QQ c and also so that the symmetry plane bb corresponds to an elect ° <"'rcu'U magnetic wall. When bb corresponds to an electric wall th W a " 0 r a propagation on the coupled line is the odd mode which has a ch ^^ of impedance Z u and propagation constant 0„. When 66 corresoo magnetic wall, the mode of propagation is the even mode wh' h ^ a : characteristic impedance Z,. and propagation constant /3.. The DP constants are different because the effective dielectric constants for t ^ ' modes are different. We will consider the following four different excitations: la) v+=v.;=v.;=v4>=v^ This case corresponds to both symmetry planes aa and 66 being magnet' walls. For this case we only need to analyze the equivalent circuit of one-quarter of the structure as shown in Fig. 6.26a. v:= v:= v* (b) v.t=v.:= -v 4 For this case the plane aa is an electric wall and the plane 66 is a magnetic wall. The equivalent circuit is shown in Fig. 6.266. v- = -y+= v* (c) --d v+= -v;=v+ -| oc " ^ ^ ^ r f l vO & zs (b) (a) OC (c) (rf) )lan** FIGURE 6.26 w h e n ,« • the P » Equivalent circuit for one-quarter of the coupled-line directional couple ^ ,,-r i. aa and 66 are magnetic walls, (61 aa is an electric wall and 66 is a magn magnetic wall and 66 is an electric wall, (d) aa and 66 are both electric PASSIVE MICROWAVE DEVICES 42S For this case the plane aa is a magnetic wail and the plane bb is an electric wall. The equivalent circuit for this case is shown in Fig. 6.26c. (d) v+~-Vf=V |£= ~ i £ - ~ y + For this case both symmetry planes correspond to electric walls and the equivalent circuit is that shown in Fig. 6.26d. For case (a.) it is readily seen from the equivalent circuit in Fig. 6.26c that Zm = -jZe cot j3ed Z = a -n ~ Z . Zm+Z(. -jZcCOtlied~Zc = " -JZecotM+Z, where Z,. is the characteristic impedance of the input microstrip line. Fron symmetry considerations we have Vf=r„v v2 = r 0 v v3 = r„v* v4-= r„v for this case. For case (6.) we have jZe tan Brf - Z,. * • - * - * - '•" = j Z . ^ A + z , and v,-= r,v + v2-= -r^ v3-= -vty v, = vy For case (c) we have -JZ„ Z,n= -JZ0 COt fij cot fi„d-Zc ~jZ„ cot p„d + Zc + r vr- = rev* v.; = rrv - y3- = - r v v, = - J ; v For the last case, namely case (d), we have Zm-jZ0 tan ftnd I r f -J Z o t a n M + Zr *T = C»r y,- = - rrfv* v3- = r,v • K, = - r„v' We now superimpose these four solutions. The superposition of th four incident waves at each port gives Vj* = 4V", V," = V£ = Y?=* 0. Tht only port 1 is excited. The superposition of the reflected waves gives + c a e v r - (>; + h + r + \;,)V = i( r + r„ + r + r,)v? (6.48a Va-fcr.-it + r.-r^vr (6.48* vr=7(r0-r6-r, . + rd)Vf (648( ^ - i O i + iv-r.-rjvz (6.48a 430 FOUNDATIONS FOR MICKOWAVE ENlilNKKRINO With some algebraic steps it is easy to show t h a t 2(ZX-ZeZ0to) r„ + r, jZ,.[Zt.~ZntJu)~Z2cte-ZeZ0tn where t„ = tan 0 o d and *,, = tan /3 f d. We also readily find that r +r = " 2(zc^-zez„f,) JZr(Zo-ZttJo)-ZX-Zj£ (6.496, For the ideal directional coupler, the two propagation constants sho irt equal, that is, (i„ = pr = 0, in which case te = /„. For a coupled-line cL 1 using strip-line construction, we have 0,. = £„. For a microstrip coupler). coupler, we can approach this ideal situation by placing a dielectric h" made from the substrate material on top of the conducting strips f Th surrounds the conductors with dielectric and will make the propagate constants equal to thai for TEM waves in the solid dielectric. The use of a dielectric overlay will change the even- and odd-mode characteristic impedances but they will remain different. When r„ = tn we see from (6.49) that, provided we choose the line dimensions to make zz, = zl (6.50) 1", + l'j =* 0 and [',, + l\. = 0. An examination of (6.48) now shows that V, = 0 and V3 = 0. Thus there is no reflection in port 1 and no power coupled into port 3 at any frequency. Thus the four-port junction is a directional coupler. When te = t„ = t and we make ZeZu = Z ; , then we also have ru + l'A - I',. - Yd = 2 ( | ; + Vh). The latter is given by 2(ZC2 - Zpt {Ta + Tb) ! ~ jzcze(i-t*)-(z? + z?)t This expression can be reduced to the following form: l'„ + W = - 2( Z(-Zn) sin 2fid j2Zt cos2 fid ~{Zt + Z„)wx2 fid The transmission coefficient into port 4 is £(r a + TA) and is the scat matrix parameter S. u = S,.,; thus Z S Z —sin2pd S4l = t^i 2Z„ cos 2 fid + j sin 2fid z„ + z, tM. Horno and F. Medina, Multilayer Planar Structures for High-D 1 "* Coupler Design, IEEE Trans., vol. MTT-34, pp. 1442-1449, 1986. (6-si) PASSIVE MICROWAVE DEVICES 43] FIGURE 6.27 A strip-liiie coupled-line directional couplei Coaxial-line connector The parameter (Ze - Z„)/(Ze + ZJ is the voltage-coupling parameter c For this coupler the coupling is given by (note that 2c? = I) C = 20 log = 20 log [(1 - c 2 )cos 2 /3/ + s i n - p / ] i '•> C sin fil [l -c2cos2/J?]1/2 c sin jil (6.52 The maximum coupling occurs when 2fid = pi = TT/2 which corresponds U a coupled-line one-quarter wavelength long. The maximum coupling i; 201og(l/c). In a microstrip coupled line it is not practical to obtain a rati< of Ze/Za greater than 2; so the maximum coupling is limited to a value o 9.5 dB or perhaps up to 8 dB with careful design. By using broadsidi coupled strips in a strip-line configuration as shown in Fig. 6.27, a couplinj of 3 dB is readily achieved. Figure 6.28 shows a plot of |S 4 1 | as a function o 2fid = fil for a coupled-line directional coupler. The directivity of this type of coupler is given by V^/V-J and is infinit since V3~ is zero at all frequencies. ' F I G U R E 6.28 The variation of |S 4 ,I as a function of pi for coupled-line directional coupler. 432 FOUNDATIONS FOR MICROWAVE ENGINEERING The transmission coefficient into port 2 is given by S2i - vT^T* 2 V1 - c cos /3/ +j sin pi (6.5gj The symmetry of the structure requires S u = S 2 2 = S 3;i = S "21 = "34 = "43> "14 = "41 = "23 = "32' anc » ^13 = = n o S 24 = S 3 , = g Thus all scattering-matrix parameters are known. If the propagatin' stants are not equal we can still use the coupled-line structmdirectional coupler, but it will not have infinite directivity. The ^i 3 performance can be improved for the case (ie =t (i0 by adding a small -h capacitor between the two coupled lines at the input and output t TK bandwidth can be increased by using several sections in cascade. Branch-Line Directional Coupler The branch-line directional coupler shown in Fig. 6.29 is readily fabricated using microstrip construction and can be designed for 3-dB coupling without any difficulty. The analysis of this coupler is also readily carried out using the fourfold symmetry that is inherent in the structure. With proper excitation the symmetry planes aa and 66 can be made to correspond to either electric or magnetic walls. If excitations are chosen the same as for cases (a) to ( d ) in the coupled-line coupler, then the equivalent circuits for the four excitations are those shown in Figs. 6.30a to d. From these equivalent circuits we readily obtain m Y r ~jYltl -jY2t2 ° h= r..= YA + Y J i -JY2tih (6.546> YJi -JYi +JY2tit2 Y,.t2-jY,tlt2+jY2 (654c) Yrt2+jY1t1t2-jY2 Yctlt2+jYlt2+jY2tl r„ = (654Q| Yr+jY,tl+jY2t2 (6 .54rfl Yct1t2-jY1t2-jY2tl A Y ]' where t x = tan 0, = t a n ^ j t / , and t 2 = tan 8 2 = tant p2d2 and ^ J t b e are the characteristic admittance of the input line, the through lme. f I. Bahl and P. Bhartia, "Microwave Solid State Circuit Design," John Wiley York, 1988. PASSIVE MICROWAVE DEVICES y ©C . d ' . ' ! i 1 ' 433 H ' — • — —7— , 1 i ©c= F I G U R E 6.29 A branch-line directional coupler. OC V, 0, OC F I G U R E 6.30 The equivalent circuit for one quarter section when ( a ) aa and bb are magnetic walls, (6) aa is an electric wall and 66 is a magnetic wall, (c) aa is a magnetic wall and bb is an electric wall, id) aa and 66 are both electric walls. branch line as shown in Fig.6.29. The relations (6.48) apply to the branchline coupler also; so we have s» = <i(ra + i; + r c + rj s 12 = s 2 I = icr. - rft + rc - r4) s 1 3 = s 31 = i ( r a - r 6 - r c + r(/) 5,, = s.u = H ra + r„ - re - r„) (6.55a) (6.556) (6.550 (6.55*0 If we choose «, = « 2 = 1 so that the through lines and branch lines are one-quarter wavelength lines and also choose K,2 - Kf = Y*, then we find that S„ = S 2 2 = S33 = S44 = 0, S u = 0, S 3 1 = i ( r o - i;.) and S z r = 434 FOUNDATIONS FOR MICROWAVE ENGINEERING i ( r a + T,.). These latter expressions give c °31 — 11 (6.56a) v S-zt = -Jxf Y i (8.566, for tx = 12 = 1. A 3-dB coupler is obtained if we choose Y - y 7, = \/2 Yc, a condition that is easily satisfied. A 3-dB directional with the two outputs 90° out of phase is also called a 90° hybrid i u C ° U ? ier The coupling and directivity at any frequency are given by C = 20 log—— = 20 log l«»l ^31 D = 20 log - — = 20 log «« i; + rrf - r 6 - g r„ + rd - rb - r, r„ + n, - r (6.57Q, (6.576) The branch-line directional coupler is a relatively narrowband device However, by cascading several sections a broadband coupler can be obtained by appropriate choices for Yl and Y2 for each section. Design formulas are available in the literature.t Lange D i r e c t i o n a l Coupler^: The last directional coupler that we will discuss is the Lange coupler shown in Fig. 6.31 This coupler uses several coupled lines in order to obtain larger coupling than what is possible in t h e simpler coupled-line coupler discuss earlier. The design of the coupler is such that wire connections betwee some of the lines are needed as shown in Fig. 6.31. This is the ma shortcoming of the Lange coupler since such wire connections are readily made in an MMIC circuit. The outstanding features of the coupler are its compact size and very broadband characteristics. coupler is often used as an input coupler in balanced microwave circuits. For this application it is designed as a 3-dB coupler and the o ^ ^ signals are in phase quadrature, so that it is a 90° hybrid junction^ ^ design formulas for a Lange coupler have been developed by " r e - , y tG. L. Matthaei, L. Young, and E. M. T. Jones. "Microwave Filters, I r o P ^ Networks and Coupling Structures," Artech House Books, Dedham, Mass., ±.J. Lange, Interdigitated Stripline Quadrature Hybrid, IEEE Trans., 1150-1151. December, 1969. r r r ' 6 PP §A. Presser, Interdigitated Microstrip Coupler Design, IEEE Trans., vol. N October, 1978. 80, ~ PASSIVE MICROWAVE DEVICES 435 1 (A)\ 6.5 Magic HYBRID 1!=J j FIGURE 6.31 Tlit' Lange directional coupler. JUNCTIONS T A waveguide hybrid junction, known as a magic T, is illustrated in Fig. 6.32. When a TE 1 0 mode is incident in port 1, the electric field within the junction is like that sketched in Fig. 6.326. This electric field has even symmetry about the midplane and hence cannot excite the TE 1U mode in arm 4 since this mode must have an electric field with odd symmetry (shown dashed in Fig. 6.326). Thus there is no coupling between ports 1 and 4. The coupling between ports 1 and 2, and 1 and 3, is clearly the same, as may be seen from the symmetry involved. For a TE 1 0 mode incident in arm 4. the electric field within the junction is sketched in Fig. 6.32c. Symmetry again shows that there is no coupling into port 1 (this is required by reciprocity as well). The coupling from port 4 into ports 2 and 3 is equal in magnitude but 180" out of phase. The scattering matrix of this hybrid T thus has the form [S] = 0 s 12 s** •S'24 £23 §33 -s24 -s24 ^24 s„ ^12 ^12 •^22 •^ 12 0 S44 since S, 3 = S 1 3 , So,, = -S34, from symmetry. Matching elements that do not destroy the symmetry of the junction may be placed in the E-plane and //-plane arms so as to make S , , = S44 = 0. For a lossless structure we may then show that the unitary properties of the scattering matrix require that S.J2 = S, ;i = 0, so that all ports are matched. In addition, S 2 3 = 0; so ports 2 and 3 as well as ports 1 and 4 are uncoupled. The hybrid T now becomes a directional coupler with 3-dB coupling, and is often called a magic T, even though there is nothing magic about its 436 FOUNDATIONS FOR MICKOWAVE ENGINEERING ^&n t -plane 3 2 arm •3C 5 U) lb) FIGURE 6.32 <a> Hybrid-T junction; ( M d f c trie field pattern for wave b d e n l in P° r ' 1; <c) electric field pattern for wave incident g port 4. operation. The magic T is commonly used in waveguide balanced mixers and in bridge networks. With S' n = S 4 4 = 0, the scattering matrix becomes [S] = 0 s12 ^12 SVz Sl2 S22 ^23 ^23 S33 0 s,24A 0 *^24 — *^24 -s.' 2 4 The product of the second row with its conjugate gives I S , / + I S , / + | S 2 / + \SMf = 1 (6.58a) and the similar expression for row 3 is \Sis s , / + i s 3 / + is'2-11 = 1 (6.586) If we subtract these two equations, we obtain (6.58c) is2/ - is3/ = 0 so |S 2 2 ! = |S 3 3 |. From rows 1 and 4 we have 21S 121 = 1 21S2/ = 1 or IS 1 2 I= or IS, J = 22 ft PASSIVE MICROWAVE DEVICES 437 and thus ft (6.59) |S 12 I - IS24I - -z~ Use of this relation in (6.58a) gives 1 + \S2.f + IS.J1 = I or ISwi + IS23I2 = 0. This sum can equal zero only if both STZ and S™ vanish. From the relation (6.58) it follows that .S*:(:t equals zero also. The reduced form of the scattering matrix becomes [*]- 0 8U S Ia 0 S 12 0 0 s 12 s 24 -*« 0 0 0 St 24 11 By proper choice of terminal planes in arms 1 and 4, we can make both SVi and S2., real. Thus the scattering matrix of a magic T can be exhibited in the form ft 0 I 1 0 1 0 0 I 1 0 0 -1 0 1 -1 0 16.60) upon using the relations (6.59). The branch-line coupler designed for 3-dB coupling is a 90° hybrid junction. The magic T is a 0 C or 180° hybrid junction since the two outputs are in phase if port 1 is the input port and are 180" out of phase if port 3 is the input port. A 180" hybrid junction that is readily made using microstrip construction is shown in Fig. 6.33. To understand its operation, consider a wave incident in port 1. This wave splits equally into two waves traveling around the ring circuit in opposite directions. The two waves will arrive in phase at ports 2 and 4 and out of phase at port 3. Thus ports 1 and 3 are uncoupled. Similarly, ports 2 and 4 are uncoupled since the two paths coupling these ports differ in length by A/2. A quantitative analysis of the hybrid ring is readily carried out. Each input line has a characteristic impedance Z, and the ring has a characteristic impedance Z,. We will let 0, be the electrical length of the ring between ports 1 and 2, 2 and 3 and 3 and 4, while 0., = 3ft, represents the electrical length of the ring between ports 1 and 4. We can choose excitations such that the symmetry plane aa is either an electric wall or a magnetic wall. This will allow us to characterize one-half of the structure in terms of 438 FOUNDATIONS POR MICROWAVE ENGINEERING FIGURE 6.33 The hybrid ring (•• rat-race"), scattering matrices for a two-port junction, ports 1 and 2. From the symmetry properties the complete four-port scattering matrix can be constructed. Let ports 1 and 2 have incident waves Vu~ and V6T. Also let ports 4 and 3 have incident waves V* and VA+, respectively. For this symmetrica) excitation the electric field must be a maximum on the symmetry plane aa and the magnetic field must be zero. Thus aa can be replaced by a magnetic wall, i.e., the ring is open circuited on the plane aa. The equivalent circuit of half of the structure is shown in Fig. 6.34a. It consists of two input lines with characteristic impedance Z,. and an interconnecting line of electric length (V, and characteristic impedance Zv The input port 1 is shunted bi susceptance jB, = jY, t a n 3 0 2 / 2 due to the open-circuited section of tl v: v; YC ye. JB2 Yc ©• FIGURE 6.34 one-b (a> v; IB, )B2 Equivalent^ J,* of hybrid ring for 0 P « n | B n e ditionsonthesvm"^-^ equivalent circuit J ^ ^ p ^ J J i w o - c i " ' =-— , for one-half of ters for one-hall structure. tprs PASSIVE MICROWAVE DEVICES 439 hybrid ring having an electrical length 30l/2. The output port 2 is shunted by a susceptance jB2 =jYl tan 0 , / 2 due to the open-circuited section of the ring of electrical length 0 , / 2 . We will let the two-port scattering-matrix parameters under open-circuit conditions on the plane aa be Sfj. Thus we can write sn vr Coc °2] ^ derations cocl v: QOC °22 CIK'1 ^4 (6.61a) ys °12 S-22 \v<; v? (6.616) °2I since port 4 is similar to port 1 and port 3 is similar to port 2. Let us now change the incident waves at ports 4 and 3 to - V£ and — V^. We now have an antisymmetrical excitation and the symmetry plane is an electric wall or short circuit. For this case the equivalent circuit is the same as that in Fig. 6.34a, except that the shunting susceptances are due to short-circuited transmission-line sections, so jBl is replaced by jB.d = -jYl c o t 3 0 ! / 2 and jB., is replaced by jB4 = -./T, cot 0 , / 2 . For this case we will designate the two-port scattering-matrix parameters under shortcircuit conditions on the plane aa by &f$. For this odd excitation we can write n CM S12sc v., [6.62a) •3 22 -v; -vs 12 and S sc 21 s.22 (6.626) Let us now superimpose the two solutions which then gives Vf= 2V„', V£ = 2V£, VQ — V£= 0. The superposition of the scattered waves gives K] .V [V4~1 .V 1 " 2 8£ + S% S S12 t>c 1 QOC ~ 2 °21 _ esc °21 (6.63a) BC i O B? '2:'. 22 C* Osc ~ °12 UC °22 Q Sc v; °22 (6.636) From these equations we can identify the following eight four-port scattering-matrix elements: &u - | ( S n + S,"J) s(S2] $21 = Q _ 1 / Q O C _ Q S C \ °ai 41 — _ + Sfj) S;2 _ a( ^ i 2 + S,s2) S 2 2 = a{Sa 2 + S ^ ) 2\ °2i °8iJ S32 2V ° 1 1 ^II ) 42 = - s( ^22 ~ Sat) 1 / Out 2 V 0 12 Csc \ °12) From symmetry considerations S M = S u , S 3 3 = S 2 a , So «3 = S™. The re- 440 FOUNDATIONS FOR MICROWAVE ENGINEERING maining five elements follow from reciprocity, namely, S, =5 T , four-port hybrid can be characterized in terms of two'"'sets "'of ' scattering-matrix parameters. °"Port In order to obtain the two-port scattering-matrix paramete the circuit shown in Fig. 6.346. We assume incident and scattered' COns "kr exist at ports 1 and 2. On the interconnecting line we assume the W 3 V e s **> of a forward and backward propagating voltage wave given by y+e-&* + y-em with associated current waves V+Yte~m - V~Yxe^z The continuity of the voltage at ports 1 and 2 gives VY + v r = v - + v v$+^=v+e-J*> + v-e* (664Q) (6646) The continuity of the current at each port gives (V,- - V-)YC = ( V + - V (V+e~^ - W - J Y , = ) 7 , + (V++ V{)jBl - ( V 2 + - V 2 )Yc + (V;+ V2)jB2 (6.64c) (6.64rf) We can solve the first two equations for V" and V". We then substitute these solutions into the last two equations and solve for Vf and V2~ in terms of Vf and V2. This will give us the two-port parameters S*. After carrying out the algebra, we obtain S u = j[Y* ~ Y i i 2 - Yi( Si + B2)cot B x + jYc(B2 - BJ] (6.65a) -,T'YlCSC^ (6-656) co, _ ™ . *JX(B,-B,) (6.650 «-*»«.» where 2 + B fi A = 7C2 - B , B 2 + Y,( Bj + B 2 )cot 0, + 7* + yTr(B, + B 2 - 2 7 i c o t 0 ! ) 30, B, = 7, t a n — - 0i B 2 = 7, tan — „, obtained W The short-circuit two-port scattering-matrix parameters are replacing B, by B 3 = - 7 , cot(30,/2) and B 2 by B 4 = - Y i * l ^ . * £ At the center of the frequency band of interest, we en°° h e s e con* for which B, = - Y „ B 2 = Y u B 3 = 7,, and B, = - 7 , . For^ ^ »* tions the expressions for the scattering-matrix parameters si PASSIVE MICROWAVE DEVH ES 441 readily find that »n - S oc _ 22 _ Y2 - 2Yj2 + 2jY,Yi = S22 2 + 2Y2 Y y;2 - 2Yf - 2jYrYl Y2 + 2Y 2 S' «CI IK: -_ 3o s| c» - = S? y£2 + 2Fj a From these expressions we get " a ~ Sjn S3:, S44 - y r 2 - 27, 2 YC2 + 2Y;2 (6.66a) (6.666) S:il = S42 = 0 -2iy,T, S i 2 ~ -S34 - - S 4 1 = S 3 2 = y 2 , 2y2 (6.66c) We see that port 3 is uncoupled to port 1 and ports 2 and 4 are uncoupled. Also ports 2 and 4 are coupled to port 2 but the port 2 output is 180° out of phase with the port 4 output since S 2 1 = — S 4 1 . All ports will be matched if we choose Y2 = 2Y,2 or When this latter condition is imposed, then | S 2 l | = | S 4 , | = / 2 / 2 and we obtain a 3-dB directional coupler or 180° hybrid junction. The four-port scattering matrix reduces to 1 J V2 0 1 0 -1 1 0 0 1 1 0 0 1 -1 0 1 0 (6.67) which is the same as that for the magic-T hybrid junction, apart from a different numbering of the ports and the choice o[ input terminal reference planes. At frequencies away from the frequency at which 0 1 = i r / 2 , the performance of the hybrid ring can be found by evaluating the various scattering-matrix parameters of interest as a function of the normalized frequency variable 2Bs/ir. In Fig. 6.35 we show a plot of the couplings )S i2 J, )SU), and IS 23 |, the isolation IS 13 |, and the input reflection coefficients \SU\ and |S 22 I as a function of 28x/v. 442 FOUNDATIONS FOR MICROWAVE F.NGINEKRISC IS 0.15 0.1 ?27lS, 3 l 0.8 L- // v\ Afarf - \\ 0.05 1 0.8 0.9 ^• J/ "to (a) 1 1 1.1 i 3 F I G U R E 6.35 Hybrid-ring performance as a fu of reflection coefficient and ISO coefficients. 6.6 port coupling coefficient; (6) magnitude of coupling POWER DIVIDERS Power dividers are used to divide the input power into a number of smaller amounts of power for exciting the radiating elements in an array antenna. They are also used in balanced power amplifiers both as power dividers and power combiners. A fundamental property of a lossless reciprocal three-port junction is that not all three ports can be simultaneously matched. If we assume that all three ports can be matched, then S n = S22 = S 3 3 and the scattering matrix has the form 0 [S] = I ••'. 5 12 S\3 0 &28 ^23 0 For a lossless junction the scattering matrix is a unitary matrix. ihM i that the sum of the products of the elements in any row with the i conjugate of the elements in another row is zero. For the Jun®~ discussion this would require S, 2 S|., = Sl:lS%3 = S12Sf3 = 0- * tions will hold only if two of the transmission coefficients S l 2 , 2(3'junction equals zero, in which case we do not have a functioning three-po .^£ If we want to use a lossless three-port junction to split or divi^ ^ (lthis power P, into fractions aPl = P 2 and (1 - a)P} = P a at ports 6.36. we is readily accomplished, accomplished. fFor the three-port three-portjunction junctions nshown is readily o r the o « » W J| ^ ^ire can choose Z 2 and Z 3 so that the input port 1 is matched an ,(2 sinc£ power split is obtained. If the input is matched, then Pi " f ' we b T V{ = V3 = V: because of the parallel connection of all three Pt- iY2mf+inn ~* PASSIVE MICROWAVE DEVICES f*o © ^ F f G U K E 6.36 A lossless three-port junction used us a power divider. For an impedance match we require Y x = Y., the desired power division, we require Ys and, in order to obtain Y., Y., 1 -« For example, if we want to split the input power so that P 2 = Pt/3 and Pa = 2Pl/3, then Y2/Y-A = | or Y3= 2Y.,. Consequently, in order to make port 1 matched, V","= Y2 +YZ = 3Y2; so" Z2 = 3 2 , and Z3 = 1.5Z,. This type of lossless power divider will not have matched output ports and since Si3 will not be zero it also does not have isolation between the output ports. If there is a shunt susceptance at the junction, such as would occur from excitation of evanescent modes in a waveguide T or Y junction, the input port can still be matched by placing a suitable shunt-compensating susceptance at an appropriate position in the input line. It is desirable to have S 2 3 = 0, so that reflected power at port 2 does not couple into port 3, and vice versa. Wilkinson developed an N-way power divider that would split the input power into output power at N ports and that would also provide isolation between the output powers.t A unique feature of the Wilkinson power divider is the use of resistors connected between the various output Ports. When the output ports are terminated in the correct load impedance, there is no current in the resistors; so they do not absorb any power. If one port is matched, then the reflected power from that port is partly absorbed by the resistor network and partly returned to the input, but no power is coupled into the other output ports as long as they remain properly terminated. Many different versions, including broadband designs, of the Wilkinson power divider have been developed. The multisection broadband f E Wilkinson. An A'-Way Hybrid Power Divider. IEEE Trans., vol. MTT-8. pp. 116-118 i960. 444 FOUNDATIONS FOR MICROWAVE ENGINEERING design for a two-way power divider was developed by Cohn a used for a number of practical designs.t ° hag be^,. The basic Wilkinson power divider is illustrated i n pi consists of two quarter-wave sections with characteristic impeda Z 3 connected in parallel with the input line, which has a ch impedance Z,.. A resistor R is connected between ports 2 and 3 r a -' ZL:i be the matched terminating loads for ports 2 and 3, respect" i t a a n < * want to split the input power P, into output powers P, and p 6 * ' " " * * P.j = K2P2 and also maintain zero current in R when ports 2 » T terminated, then the output voltage V.J at port 2 must equa] th* voltage V-j at port 3. In order to obtain the desired power ratio we U K2\V2 \2/ZL2 = |V3 \2/Zl3 for V J = y3". Hence we need K Z ' '-3 = Z" (6.68) For matched output terminations the resistor R has no effect on th operation of the circuit. In order to obtain a matched input at port 1 we require Ym = Yr. By using the transforming properties of the quarter-wave sections, we have *fc- zl + ^L3 -K (6.69o| From the above two equations we obtain (K Z3 + Z2jZL3 — Z72 Z'$Z$ (6.69b) upon eliminating ZL2 using (6.68). At port 1 the load impedance ZL2 is transformed into an admittance ym 2 = zi 2 / 2 ? and Z, 3 is transformed" into an admittance y w . 3 = z The'power delivered to ZL2 and ZL:i is the same as that delivered to we and y i n 3 , respectively. Thus, in order to get the desired power r require that i|V^!%.3 = KH\V^Y.m,2 which gives, upon using (6.68), Z2 - K Z3 fS. B. Cohn, A Class of Broadband 3-Port TEM Hybrids, IEEE Trans 110-118, 1968. .16.1* vol- MTT PASSIVE MICROWAVE DEVICES 445 G -VW Yx,*Y. 33 * Y„*Y '22 * X'23 '23 IMI (o) H SC Vc *.'• ^ F I G U R E 6.37 ( a ) The Wilkinson power divider; (6) the equivalent circuit with port I terminated and porta 2 and 3 excited, (c) equivalent circuit between ports 2 and 3 with port 3 short-circuited and R removed. Mf /a IC y3, B v3. (CI At any frequency YL2 transforms into *ta.2 YL2 + jY2 tan 6 2y2 +JYL2 tan e Y and y L 3 transforms into YL3+jY3Um0 n.,8 = *3 Y +jY tan 0 3 L:i m K.iy in, 2 by virtue of (6.68) and (6.70). Thus the input current on line 3 will be K' 446 FOUNDATIONS FOR MICROWAVE ENGINEERING larger than the input current on line 2. Since the line 2 and li are identical, apart from the relative impedance levels the I ^ netw ork> 2 curre: ' in Y, , Henp3 u U r r e n l it Y,_:i will also be K larger than the load current e the the load voltages will !be be ratio 0 f *L2 Vi.8 h,2^L2 IL2%L2 h*Zm KHL2ZL.A = 1 Since the load voltages are equal, there is noo current curre in R at anv f r Uenc as Jong 2 and 3 ' ai-e 5' —«" as" ports • — terminated "' —w" in •" their 'heir matched n,ai.v,iicu load ioaa im i ^ When there is no current in R, the input admittance at port 1 V',,,.:! = (1 + K2)Ym.,. The input reflection coefficient is given given bv by T. - Y-„, r,„In = Y,2-(l +K2)2Yi ' y/ + (1 + K'^Y2 + 2.7(1 + K*)YCY2 tan 0 This result is obtained by substituting the expression given earlier for Y at any frequency. Equation (6.71) gives the interesting result that input port 1 will be matched at a]] frequencies if we choose y, - T T F (6 72 - ' In order to analyze the coupling between ports 2 and 3, we will terminate port 1 in a load admittance Y.. We now have a two-port networi Apart from the resistor R, we can represent the network between pori and 3 by a 11 network as shown in Fig. 6.376. The resistor R is a si conductance G = 1/R in parallel with - Y.,3. From this network repres tation it is clear that ports 2 and 3 will be uncoupled if G = Y&, so «* ^ admittance between ports 2 and 3 vanishes. The circuit equations • removed are p z l T ^ Y&irvzi UJ ba rJW For V 3 = 0, that is, port 3 short-circuited, we have Yn ~v2 V 3=° Thus we can evaluate K>:) by finding the short-circuit c u r r f ? * g hoW n >r-F port 2 excited. The transmission-line circuit to be analyzed tf 6.37c. .1. PASSIVE MICROWAVE DEVICES 447 Let the voltage and current waves on line 2 be and those on line 3 be The terminal conditions at port 2 are v;+ v-= v2 (6.73a) (6.736) At port 1 the terminal conditions are v„v-"'+ va->"' = v;+v; (Vje-"' - Va-e»)Y2 - (V?+ V?)Yt + (V;- Vh)Y3 (6.73c) (6.73d) At port 3 we have V^e-J" + Vh'eje = 0 {V?e -"' - Vh'e>")Y3 = - 7 a (6.73e) (6.73/-) where 0 = f}2l2 = f}3!3 and h and 13 are the line lengths. From the last two equations we get I3 = -2V^e~JB. By adding arid subtracting (6.73c) multiplied by Y2 to (6.73d), we can solve for V,* in terms of either Vj or V~. We can combine the two solutions to get V,J in terms of V* + Va = V2 and then solve for Y13 = ~(2V+e-'»)/V2. The results are y,.. = ~ 2! ' 7,.( 1 - cos 26) - j( Y2 + y 3 )sin 20 (6.74) When 0 = 77-/2 we get Y23 - Y2Y3/Yf.. In order to uncouple ports 2 and 3, we thus require G = 7 or 2 3 = - ^ Z Z R = -^~ Y. (6.75a) (6.756) 448 FOUNDATIONS FOR MICROWAVK. ENGINEERING From (6.68), (6.696), (6.70), and (6.756), we can express all • irn'Pedant in terms of R in the following form: Z,^K{RZC <6.76Q) Z 3 = ~W< '6.766, K*R "X2 K2 + 1 z,., = K2 16. 76c, R + 1 (6.7W) The resistance R can be chosen arbitrarily. If we require port 1 matched hed at all frequencies, then R is determined by by the the condition condition given mw L by (6.72). We can also solve for 12 in terms of V., from (6.73). The ratio / /V for V3 = 0 gives the parameter Y, 2 . We find that Ym = Y, Y3 - Y2 + (Ya + Y2)cos 20 +jYe sin28 Yt.(cos20 - I) + j(Y2 + Y3,sin2<? (6.77) The parameter Y33 is obtained by replacing Y2 by Y3 and Y3 by y2 in the above equation. The parameters Y22 and Y33 are needed in order to determine the port 2 and port 3 reflection coefficients as a function of 8 or frequency. A 3-dB power divider is obtained by choosing K2 = 1. If we also specify that Z,,2 = Z / 3 = Z<:, we find from (6.76c) that R = 22,. From (6.76a) we get Z2 = Z3 = J2ZV for this case. A 3-dB power divider that has port 1 matched at all frequencies requires, from (6.72), Y2 = Ye/* ° Z., = 2ZC. From (6.76a) and (6.76c) we obtain R = 4Z f and Z w = A t 8 " 2Z,.. For this case the output lines have a characteristic impedance t\ that of the input line. We could choose R = 2Z,„ Z,, 2 = ZL3 = Zc and a* quarter-wave transformer with impedance ZJ v'2 to transform the p _ ^ input impedance, which now equals Z , / 2 , into an impedance A6.38a illustrates this type of 3-dB power divider. . j n 2 to For the unequal power division case if we choose Y-> a«^^ (6.72), the input port 1 is matched at all frequencies. From (6.7 that J? = (1 + K2)%/K2. By using (6.76c) we get Zl2 = <*•" ^ ]ofld and hence there is no discontinuity at the .junction of line 2 * of Z ; , 2 . Also Z w = Z 3 , so that there is no discontinuity at t h e j u n l lv/0 outP' and its load. The only disadvantage with this design is that ^ difli impedances Z,,, = (1 + K2)Zt and Z,..( = ( 1 + K')Z'fK'Afl\fl^ ent from Z(.. Output quarter-wave transformers can be adde ^ ZLZ and Z L 3 into the impedance Z.. The required transforms V^, J l + f f * and (1 + K2)/K2 and are different. Thus the m - PASSIVE MICROWAVE DEVICES 449 *Z2ZC z-i 4 Zc •-z,zr (b) FIGURE 6.38 ( a ) A 3-dB power divider with an inpuL quarter-wave transformer.' Id) a broadband unequal power divider. function of frequency will be different for the two output transformers. In order to equalize the transformation ratios and thereby obtain similar characteristics for ports 2 and 3, the output impedances should be chosen according to the relations ZL2 = KZt (6.78a) Z,j3=-^ K (6.786) We now require 1 +K2 R= K Z< (6.78c) 2 z2- -Zc)JK{\ + K ) (6.78d) V (6.78c) - Kz The required output quarter-wave transformers that will transform ZL.> and ZL3 into Z e will have characteristic impedances given by i/ZL2Zc = \[KZC and yjZLZZc = Zc/ \[K. A power divider designed on this basis will have a bandwidth approaching one octave. A modified design that incorporates an input quarter-wave transformer as shown in Fig. 6.386 will give a significant increase in performance.t The design equations are arrived at by noting that when ports 2 tL. I. Parad and R. L. Moynihan. Split-Tee Power Divider, IEEE Trans., vol. MTT-13, pp. 91-95, 1965. 450 FOUNDATIONS FOR MICROWAVE ENGINEERING and 3 are terminated in matched loads we can connect these location of the resistor R. The characteristic admittance of ]^ P ° l t s ^ th parallel is Y.> + Yz = (1 + K~)Y,. The terminating load admit']? 2 ^ 3 fa 7 i 3 = (1 + K2)YL2 = Y,. The structure shown in Fig. 6.386 c ^ t * Y U vie as a two-section quarter-wave transformer. Thus, for a maxim l). *«d reflection coefficient characteristic (see Sec. 5.12), we should ch * ^ ^ " t z; = ztf*z*s* where Z; is the characteristic impedance of the input transformer 1 7 4 2 = Zf/<Z>/ = (1 + K )Y., 1 + K2 With ZL2 = KZC and Z A;i = Z ( ./K, we have y, = Q + # 2 ) y c / # must choose 1 + K' 22 = ff3/4(l 23 = tf = 6.7 we 1/4 K z:.= H (6.79a, 1/4, +K2)1' Ze (H-JT')'" 1 + K2 K (6.796) (6.79c) (6.79a') MICROWAVE PROPAGATION IN FERRITES The development of ferrite materials suitable for use at microwave freqi cies has resulted in a large number of microwave devices. A number of tn have nonreciprocal electrical properties; i.e., the transmission coe through the device is not the same for different directions of propaga11 An understanding of the operation of ferrite devices may be ^ ^ T ^ t e the basic nature of microwave propagation in an infinite unbound ^ medium is understood. In this section we consider plane-wave pr p<* ^ in an infinite ferrite medium with a static biasing magne ic ^ _^ ^ present. It will be found that the natural modes of P r o p a g y ° h a t the* direction of B„ are left and right circularly polarized waves an ^ ^ modes have different propagation constants. In addition, we » ' . the permeability of the ferrite is not a single scalar quantity. tensor, which can be represented as a matrix. . . • s ^ a t fl Ferrites are ceramicljke materials with specific r e s j 5 l , l V ! t r j C coflsta> be as much as 10 H greater than that of metals and with dielet ^ ^^ around 10 to 15 or greater. Ferrites are made by sintering ^ Q fe, metallic oxides and have the general chemical c o m P O S J t " ? u , n , irofwhere M is a divalent metal such as manganese, magne PASSIVE MICROWAVE DEVICES 451 nickel, cadmium, etc., or a mixture of these. Relative permeabilities of several thousand are common. The magnetic properties of ferrites arise mainly from the magnetic dipole moment associated with the electron spin. By treating the spinning electron as a gyroscopic top, a classical picture of the magnetization process and, in particular, the anisotropic magnetic properties may be obtained. The electron has a number of intrinsic properties such as a charge of -e = - 1 . 6 0 2 x 1 0 - 1 9 C, a mass w = 9.107 X 1 0 - 3 1 kg, an angular momentum P equal in magnitude to hh, or 0.527 x iO - * 1 J • s (ft is Planck's constant divided by 2—), and a magnetic dipole moment m equal to one Bohr magneton, that is, m = eh/2w = 9.27 X 10 'M A • m 2 . For the electron, the angular momentum P and magnetic dipole moment m are antiparallel. The ratio of the magnetic moment to the angular momentum is called the gyromagnetic ratio y; that is, m *--= (6-80) If an electron is located in a uniform static magnetic field B 0 , a torque T given by T = m X Bn = - yP X B0 (6.81) will be exerted on the dipole moment. This torque will cause the dipole axis to precess about an axis parallel to B 0 , as illustrated in Fig. 6.39. The equation of motion is obtained from the condition that the rate of change of angular momentum is equal to the torque and hence is c/P —- = T = - y P x B 0 = wt XP (6.82a) y P B 0 sin <t> = w 0 P sin </> = mBu sin </> (6.826) or where w n is the vector-precession angular velocity directed along B„, and $ F I G U R E 6.39 Free precession of spinning electron. 452 FOUNDATIONS FOR MICROWAVE ENGINEERING is the angle between m and B„. For free precession the a is given by totl = yB0 8 3r V e l o <% <„ and is independent of the angle </>. The angular velocity at the Larmor frequency. • " * o f fen calk* If a small ac magnetic Held is superimposed on the static fieU o magnetic dipole moment will undergo a forced precession. Of !" t n e interest is the case where the ac magnetic field is circularly polari\^r' t U l a r plane perpendicular to B„. A circularly polarized field results wh and y components of the ac field are equal in magnitude and 90c time phase. Thus Jet the ac magnetic field be given by the phase sor Bl = Bx(ax+JBy) (6.83a i If we assume Bt to be real, the physical field is given by B, = £, R e ( a , vjay)eJU" = B , ( a , cos ,ot - a v sin wt) (6.836) The resultant field has a constant magnitude Bt, but the orientation of the field in space changes or rotates with time. At time t the resultant field vector makes an angle tan i = — tan tan tot = — tot with the axis and hence rotates at the rate -to, as in Fig. 6.40a. It is this rotation of the field vector in space that results in the field being called circularly polarized. If the above ac magnetic field is that of a wave pre gating in the z direction, it is said to be left circularly polarized. If I direction of rotation is clockwise, looking in the direction of propagation,! wave is called right circularly polarized. The latter type of wave would b an ac magnetic field given by (6-84) Br=Bi(a ->,) I the With a left circularly polarized ac magnetic field s u p e r i m p o s e f l " ^ static field B 0 = B „ a ; , the resultant total field B, is inclined at B, CQS Ult FIGURE 6.4°,. ^ l a r i y i la) \r-\ Magnetic f i e l d for J J Jarized waves-««• „vt, ocular P ^ c u l t f P right, or pos't'^larization- PASSIVE MICROWAVE DEVICES 453 (*) F I G U R E 6.41 Forced precession of spinning electron. 6 = t a n " 1 ( B , / B , | ) with the z axis and rotates at a rate -co about the z axis, as illustrated in Fig. 6.41a. Under steady-state conditions the magnetic dipole axis will be forced to precess about the z axis at the same rate. Thus the precession angle </> will have to be less than 6, as in Fig. 6.41a, in order to obtain a torque to cause precession in a counterclockwise direction. The equation of motion (6.82) gives dP T = m X B , = - y P x B , = -— = - w a , X P dt or - yPB, sin( fl - <£) = - wP sin 6 Expanding sin(0 -</>), replacing B, sin 0 by B , , B, cos 6 by B 0 , and solving for tan ifi gives tan <b = ?B, yB, yB0 + w tof> + to (6.85) The component of m which rotates in synchronism with B[ in the xy plane for the left (also called negative) circularly polarized ac field is m~ = m sin tf> = m 0 tan 4>, where m (l = m cos <b is the z-directed component of m. For B, very small compared with B 0 , the angle <b is small, so that m Q is approximately equal to m. Using (6.85) gives m = m u t a n <•/> = ym()B1 (6.86) Cl>„ + to If we have a right (or positive) circularly polarized ac field superimposed on the static field B 0 , the forced precession is in a clockwise sense about the z axis. A torque giving precession in this direction is obtained only if the angle il> is greater than the angle 6, as in Fig. 6.416. In this case 454 FOlfNDATIONS FOR MICROWAVE F.NGINKKRING the equation of motion (6.82) gives yB, sin( 4> 0) "= w sin <b from which we obtain t a n f(j = yB, w0 The component of magnetization in the xy plane rotating in svn -h with the positive circularly polarized ac field is m'= m„ tan <i> = v/«05, (6.88) The foregoing discussion has pointed out the essential features of th motion of a single spinning e)ectron in a magnetic field consisting of a stat field along the z axis and a small circularly polarized ac field in the x plane. A ferrite material may be regarded as a collection of IV effective spinning electrons per unit volume. Since the spacing between electrons is of atomic dimensions, we may regard the density of magnetic dipoles per unit volume as a smeared-out continuous distribution from a macroscopic viewpoint. The total magnetic dipoie moment per unit volume is M = Nm. When the static field B 0 is large enough to cause saturation of the magnetization in the ferrite, M = M s . In a saturated ferrite all the spins are very tightly coupled, so that the whole sample acts essentially as a large single magnetic dipoie. The magnetization M, produces a contribution to the total internal B field according to the relation B = M 0 ( H 0 + M s ). The torque acting on M„. is due only to the field B„ = ,u 0 H 0 , s i n c e t h e c r o s s P r o d u c t ' /j„M v with M s is zero and hence does not contribute to the torque. Thus, the equation of motion for the magnetization, the field producing the tore is M » H „ where H , is the total static plus ac magnetic field intensity U ferrite medium. That is, dt - = - y ( M x B ) = - 7 7 i „ M x (H + M) = - T M o M x H is* tt'jgB If the magnetic field intensity in the ferrite is H 0 + Hf, * ® ^ v g n by circularly polarized ac field, the 2-esuitant ac magnetization v "'' ^ expressions analogous to (6.S6) and (6.88), but with m0 replace j - p Nm0. The total ac magnetic field in the xy plane is the n e l d ) ^ fe\fcio* plus the contribution from the a c magnetization. Thus the tota ^ ^ ^ positive and negative circular polarization are [we are using B = Mo<H + Ml] u 0 y * M H ; (6.89"* B ' = M 0 M - + B r = M 0 ( ^ m ^ + H f ) = u 0 1 + io - to , 0 {6# MQVM0 B = M 0 M ~ + Bf=.^0 1 + te>n + t» Hf b) I>ASS1VE M1CROWAVK DEVICES 455 where M 0 = Mcosd>. B, = Mo#i> a n a H T = # i t a * ~.My) i n (6.89a), and Hj in (6.896) equals H,(a T +jay), as seen from (6.84) and (6.83a). The quantity M may be replaced by the saturation magnetization M, in the ferrite since the static field B„ is usually large enough to cause saturation. If we assume that B t « B 0 , so that M„ = Ms, the effective permeabilities for positive and negative circularly polarized ac fields are seen to be given by At.= M J l + M * (6.90a) =Mll 1 + — I w 0 + "> / (6.906) Plane circularly polarized TEM waves propagating in the direction of the static field B„ will have propagation constants /3.= ^7 (6.91a) p =<oyjipT (6.916) W v where e is the dielectric permittivity of the ferrite. The significance of the inequality of /3,. and (i_ is discussed later. The results expressed by (6.91) are also derived in an alternative way later. If small-signal conditions B, <£ B„ are not assumed, we cannot replace M0 by M g . In place of (6.85) and (6.87). which give solutions for tan <t>, we can solve for sin 4> to obtain, respectively, tan 4> sin <j> = -j= Vl+tan20 sin if> = yB, = (6.92a) r V ( r B , ) 2 + («« + » ) ' -yB, , V(yB,)2 + ( (6.926) W o - w ) 2 The magnetizations M* and M~ are given by M + =Mssin<b = M"= , •va„M v //, . ° 6 ' 2 l/(yti0H1) + (w0 + 2 2 (6.93a) (6.936) wf It is seen that the ac magnetization depends nonlinearly on the ac field strength / / , , and hence, under large-signal conditions, /x + and /x_ will be functions of the applied ac field strength. The nonlinear behavior of ferrites under large-signal conditions results in the generation of harmonics of the 456 FOUNDATIONS FOR MICROWAVE ENGINEERING fundamental frequency w. For this reason ferrites may be used generators.! ^ na rrr, 0E For Af it is clear that, if B x <K B 0 , that is, yB. « V D (6.936) is well approximated by ° * wo- then M ylx.0MiHx ton + CO Similarly, i[ to is not too close to the resonant frequency (6.93a) becomes AT = w •o. we see that yMoM,//, 0>n — (a These latter values of M ' and M~ lead directly to the expressions (6 901 for n 4. and ^ _. In any practical ferrite medium, damping effects are alwavi present, so that M' will remain finite and small compared with M eve" when a) = w 0 . Thus, for small-signal conditions, we can assume that*M = Ms in an actual ferrite medium. Damping effects are discussed in more detail later. It will be instructive to study the propagation of a plane wave in an unbounded ferrite medium by solving Maxwell's equations directly, together with the equation of motion for the magnetization. This analysis will illustrate the general technique of linearization to be applied in the smallsignal analysis of propagation through a medium such as a ferrite. However, it will not give as clear an insight into the physical reason why /x.. and >i. are different, as the analysis above did. That is, basically, M + sn^ M - differ because the precession angle 6 must be greater than the angle 0 in one case and less than 0 in the other case, and hence the projection of the magneti dipole moment onto the xy plane is different in the two cases. Consider an infinite unbounded ferrite medium with an applied magnetic field B 0 = ^ 0 H 0 = B0az. Let the magnetization in *® fe JJJJJ n8 M s per unit volume when no time-varying magnetic field is applied. gnt time-varying magnetic field /x.0H is also applied, a tinie-vaiying c o m p * ^ -*f of magnetization will be produced. The equation of motion lor ^ magnetization per unit volume is similar to that for a single e hence we have dt dt = -YMO(MS X H 0 + M , X ^ + / X H > fi$4i 0 +/ 5 tW. P. Ayres, P. H. Vartanian, and J. L. Melchor. Frequency "Doubling »»"""" in F « * * J Phys., vol. 27, p. 188, 1956; Microwave Frequency Doubling from 9 krne « Prize. IRE, vol. 45, pp. 643-646, May, 1957. PASSIVE MICROWAVE DEVICES 457 If small-signal conditions are assumed, i.e.. \jg\ <e | M J «Pol and the nonlinear term je x & in (6.94) may be dropped. We then obtain for the equation of motion the linearized equation = ~dt -y(M0M„ X^+Jt xB0) (6.95) since M s X B 0 = 0, because the saturation magnetization is in the same direction as the applied static field. Let the time dependence be e1"1', and let J! and & be represented by the phasors M and H. From (6.95) we obtain jcoM + yM X B 0 =./wM + w 0 M x a 2 = - y ^ M , , x H where w0 = yfi0H0 ~ yBa. In component form we have ,iioMx + w 0 M v = yMsix0Hy ju>My - u)„Mx = -yMsfj.uHx jioM, = 0 The solution of these equations for Mt, M v , and M, gives M = My = Wo Yd o Ma Hx + joj yfi 0 Ms R <D0yu0MxHy -jwyiJ.QMsHx. (6.96a) [6.96b) (6.96c) M, = 0 In the solution of Maxwell's equations it is convenient not to have to deal explicitly with the magnetization. The magnetization may be eliminated by introducing the magnetic susceptibility and permeability. In the scalar case this is done by means of the relations M = * „ , # , B = HQ(M + H) = n0{l + x,„)H = fiH. For a ferrite similar relations may be used, but \ m and n will not be scalar quantities. In matrix form (6.96) gives Mx My M, Xxx Xxy *vx Xyy 0 0 0" 0 HY 0 H: \K (6.97) 458 FOUNDATIONS F O R MICROWAVE ENGINEERING w{lyy.0Ms where = X vv ji»yn0Ms •%XV AVJ <#?, - at2 The matrix with the parameters Xxx, xxy, * „ . and * y v i n (6.97, ,.p ! the susceDtibilitv the. ferrite. ferritp The Thp relation rplafirm Koi„™. .. susceptibility tensor of the between the ftts ac B and H fields is B = M„(H + M) [ l + Xxx or = Mo . Xxy XyX 1 + Xyy 0 0 0 0 1 Hy (6.98) The matrix relating the components of H to B in (6.98) is the permeability tensor for the ferrite. It will be denoted by a boldface p. with an overbar i e 1 Xxy 0 + Xyy Xyy 0 + Xxx Xyx 1 0 0 1. In shorthand notation the matrix equation (6.98) will be written as B = H H (6.99) In the literature the minus sign in the equation of motion is often deleted, and this amounts to replacing y by -y in the equations used in thiLosses present in a ferrite may be accounted for in a phenomenologica way by introducing into the equation of motion a damping term that 1 produce a torque tending to reduce the precession angle 4>. The follow modified form of the equation has often been used in practice: djt dt a = -yMo(M, +*) X ( H 0 + ^ ) dJf M_X - j - (6.1001 where a is a dimensionless damping constant. With a small-sign the elements of the susceptibility matrix are now found to beT Xxx = Xyy = X -JX -X ,6 101*' Xxy--Xyx-J(K--jK")-jK Go*""* tR. F. Soohoo, "Theory and Application of Ferrites." chap. 5. Prentice-H Cliffs, N.J., 1960. PASSIVE MICROWAVE DEVICES 459 where 2 „,2 «j0a>m(o>o - to'2) + ^.wpfeTfl * [^-w2(l+«2)]2 + 4^>V " a)w„,ft[a)'f, + w 2 (l + a ' ) ] r= "MB[MJ - K' = + »2)j [w2-a>2(l +«*)]* +4w5»V 2w"W|,W„|« K" = and f2(l [tt§ ~a> 2 (l + o 2 ) ] +- 4fi>gwV w,„ = n0yMs As we derived the permeability tensor to use in the constitutive equation relating B and H, the only remaining task is to find solutions for Maxwell's equations in the form V X E = -y'wB = -y'wjl • H (6.102a) VxH=j«E (6.1026) \ " B = V-E = 0 (6.102c) For a TEM wave propagating in the 2 direction, i.e., along the direction of B 0 , solutions are readily found. Let the electric field be given by E = E 0 e ""** where E„ is a constant vector in the xy plane. Equation <6.102o ) gives V X E = - E 0 X Ve "u = ./'0E,, X a , e "i; = -jcop. • H Let the solution for H be H fl e -J * te > where H 0 is also a constant vector in the xy plane. From (6.1026) we obtain jpn„ X a 2 = y W E ( ) If this equation is substituted into the equation j p E , , X a, = -jojJL • H„ so as to eliminate E fl , we obtain M Jp-r—(Hn X a , ) X a, = -jwp. • H 0 Jioe. Expanding the left-hand side gives (note that a. • H„ = 0) p2His = co2eJL • H 0 (6.103a) 460 FOUNDATIONS FOR MICROWAVE ENGINE! This equation may be written in the following matrix formp2-u>2en0(l+x) -jo>2efi0K j(o2e^0K - u>2en0(l P* Ox +X ) " • . For a nontrivial solution for H,„ the determinant must condition yields the following eigenvalue equation for tho „ p constant 0 : >2-«V„(l fiz or = X )f w2en0(l (6. 1036) r o ' "^is ^ - <uVV0tf2 = 0 +X)±<»2en0K (6.104) If we substitute for x and K and assume a lossless ferrite so X" --= K" = 0, we readily find that the two solutions for /3 2 are li2 = p{ = w2eM + fi2 = fit = u>2efi _ (6.105Q) (6.1056) where /x, and n_ are given by (6.90). For each eigenvalue or solution for fi2, the ratio of H0t to U„ fa determined. For the solution /32, the first equation in the pair of equations (6.1036) gives [fit - « V o ( l +X)]H0t -jo>hn0KHQy = 0 'fi, or Ha ' = j But this condition means that H„ = H 0 ( a , - . / a v ) , or is a positive circula polarized wave. Similarly, the solution fi'i gives Hi0Y = -J (6.1066) which specifies a negative circularly polarized wave. Therefore it is s the natural modes of propagation along the direction of the static ferrite are circularly polarized TEM waves. If directions of Pr°^ vt% other than along B 0 were considered, it would be found that ^^ again, two modes of propagation, but these are no longer c i r c U "go_ the ized TEM waves. For a linearly polarized wave propagating a o ^ p h e . plane of polarization rotates, since 0. and ji are not e q U o t a l i o n . It « nomenon is a nonreciprocal one, and is called Faraday row discussed in the following section. FARADAY ROTATION apPlied Consider an infinite lossless ferrite medium with a static j ar j z ed °* along the z direction. Let a plane TEM wave that is linearly P 1'ASSIVK MICROWAVE DEVICES 461 FIGURE 6.42 Faraday rotation. the x axis at z ~ 0 be propagating in the z direction, as in Fig. 6.42. We shall show that the plane of polarization of this wave rotates as it propagates (Faraday rotation). The linearly polarized wave may be decomposed into the sum of a left and right circularly polarized wave as follows: E = axE0 - ( a , + . / a , ) y + (a., - j ' a j y 2 = 0 (6.107) The component waves propagate with different phase constants fi . and [1 , and hence the wave at z = I becomes E = = a, Bn eJ,il +ja,)~ (a.v E„ ,-JP i J1H '"" + e + -jit .i \ -fa (e (a * -J&y^Te ~"" B„ if + ./a v "" +JB.{e -'"' ' - ,."<•' H***'/* + « J W l< ) 2 -' i'->i*\ - <»•"" ^•"•/'a)l / = E„e " " '**•" 2 axcoS((i.- P )- (6.108) -avsin(p,-0 This resultant wave is a linearly polarized wave that has undergone a phase delay of {ft _ + p, ) / / 2 . The new plane of polarization makes an angle 0 = t a n ~ ' - f = tan ' - t a n ( / J . - / 3 / )- / - -(H^-fi )^ (6109> with respect to the ^ axis. When oi < m0, that is. below the ferrite resonant frequency, 0 + is greater than (i and the plane of polarization rotates counterclockwise, looking in the direction of propagation (Fig. 6.42). The rate of rotation is (p., - / 3 _ ) / 2 r a d / m . Rotation of 100° or more per centimeter is typical in ferrites at a frequency of 10,000 M H z / s . If the direction of propagation is reversed, the plane of polarization continues to rotate in the same direction. Thus, if we consider the propagation of the wave described by (6.108) back to the plane z = 0, the original 462 FOUNDATIONS FOR MICROWAVE ENGINEERING polarization direction is not restored; instead the wave will ar z = 0 polarized at an angle 26 relative to the x axis. This r e s " ^ ^ aclc at derived by noting that if the component circularly polarized w ' S e a s 'l v up the linearly polarized wave in (6.108) are propagated from z I ac 2 = 0, they undergo additional phase delays of amount p / a n H k to become, at z = 0, ^ -' and E-(..•;.,)£.-»-'•(.,-;.,,!«-«., (6ii0) By analog}' with (6.108) it is now clear that the new direction of nnls. i at 2 = 0 makes an angle carnation 26 = - ( 0 . - / 3 )l with respect to the x axis. Thus Faraday rotation is a nonreciprocal effect A practical ferrite medium has finite losses, and this will have significant influence on the propagation. The propagation constants y = JP + + «+ and y = j/i_ + u_ for circularly polarized waves will have unequal attenuation constants as well as unequal phase constants. When Josses are present, the propagation constants are given by (6.104) if p- is replaced by -y2; thus y + = > v f c o 7 ( l + X" -JX" + K' -jK")in (6.111a) •y_ = > v ^ 0 7 ( l + X' ~JX" ~ K' +jK")l/2 (6.1116) where x',X", K',K" are given after (6.101). The solutions of (6.111) are 1/2 V6 a + = a>Vo* 1 + *' ± K' + \ / ( l + x' ± K'f + (X" ± K"? V" ± g" 2/3 f = «»V± 2/3, <6 ' (6 .1126) The permeabilities for circularly polarized waves are M , = M W M W o ( 1 + * ' - • / * " ± K'+•/'*''> The values of u\ , u"_ as given by (6.101) and (6.113) and the prtjWBJ tion factors / 3 . and « + are plotted in Figs. 6.43 and 6.44 t0 0 ( „<, ferrite with parameters w,„ = 2TT X 5.6 x 10 9 . « = 0.05, as f/"" H z ' at a frequency of 10,000 MHz. Note that u>0 equals 2- X 2.8 ^ Tne applied field r/ 0 and that 4- x 10 :! Oe is a field strength oi Q(. 2 0 0 0 G. value of w,„ chosen corresponds to a saturation magnetization ^ (1 or n0Ms equal to 0.2 W b / m 2 . The curves in Fig. 6.44 sho ^ resoojj always very small but that «, is large in the vicinity ° Jon frequency «„ = co. For ion considerably above to, the a t t e n t l y i g0 th a l l comes small, but in this region (i . and ji _ do not differ grea j^ j s sma rate of Faraday rotation would be small. At low values of <•>,,. PASSIVE MICROWAVE DEVICES 463 n FIGURE S.43 Real and imaginary components of permeability for circularly polarized waves in a ferrite as a function of ui/m0 for w / 2 * «= 10 GHz, ">„,/2fr = 5.6 GHz. « = 0.05. 20 c uo/u 2.0 FIGURE 6.44 Propagation and attenuation cons t a n t s for circularly polarized waves in a ferrite. with parameters given in Fig. 6.43 U = 10e„). Note that 10a _ is plotted since a is very small. 464 FOUNDATIONS FOR MICROWAVE ENGINEERING 6.9 MICROWAVE DEVICES EMPLOYING FARADAY ROTATION Gyrator A gyrator is defined as a two-port device that has a relative diffe phase shift of 180° for transmission from port 1 to port 2 as compared ce in the phase shift for transmission from port 2 to Dort 1 A pvrat„. K . , : , . .. ,. . r , syrator may h e obtained by employing the nonreciprocal property of Faraday rotatii Figure 6.45 illustrates a typical microwave gyrator. It consists of a rectaneu lar guide with a 90° twist connected to a circular guide, which in turn connected to another rectangular guide at the other end. The two rectangular guides have the same orientation at the input ports. The circular guide contains a thin cylindrical rod of ferrite with the ends tapered to reduce reflections. A static axial magnetic field is applied so as to produce 90' Faraday rotation of the T E , , dominant mode in the circular guide. Consider a wave propagating from left to right. In passing through the twist the plane of polarization is rotated by 90° in a counterclockwise direction. If the ferrite produces an additional 90° of rotation, the total angle of rotation will be 180°, as indicated in Fig. 6.45. For a wave propagating from right to left. the Faraday rotation is still 90° in the same sense. However, in passing through the twist, the next 90° of rotation is in a direction to cancel Faraday rotation. Thus, for transmission from port 2 to port 1, there So 90° twist Ferrite rod ^ ^ F I G U R E 6.45 A microwave gyrator. ^ f PASSIVE MICKOWAVE DEVICES ki ^ 46 ^ ^ FIGURE 6.46 A gyrator without a twist section. net rotation of the plane of polarization. The 180° rotation for transmission from port 1 to port 2 is equivalent to an additional 180° of phase shift since it reverses the polarization of the field. It is apparent, then, that the device just described satisfies the definition of a gyrator. If the inconvenience of having the input and output rectangular guides oriented at 90° can be tolerated, a gyrator without a 90" twist section can be built. With reference to Fig. 6.46, it is seen that if the ferrite produces 90° of rotation and the output guide is rotated by 90" relative to the input guide, the emerging wave will have the right polarization to propagate in the output guide. When propagation is from port 2 to port 1, the wave arriving in guide 1 will have its polarization changed by 180°, as shown in Fig. 6.46. Hence a differential phase shift of 180° is again produced. The solution for wave propagation in a circular guide with a longitudinal magnetized cylinder placed in the center can be carried out exactly.t However, the solution requires a great deal of algebraic manipulation, and it is very laborious to compute numerical values from the resultant transcendental equations for the propagation constants. The solution does verify that Faraday rotation takes place as would be expected, by analogy with propagation in an infinite ferrite medium. tA. A. Th. M. van Trier. Guided Electromagnetic Waves in Anisotropic Media, Appl. S&. Ren.. vol. B3, p. 305, 1953. M. L. Kales, Modes in Waveguides That Contain Ferrites, J- Appl. Phys., vol. 24. p. 604, 1953. 466 FOUNDATIONS FOR MICROWAVE ENOINEEBING The isolator, or uniline, is a device that permits unattenuated trans • from port 1 to port 2 but provides very high attenuation for transmi* the reverse direction. The isolator is often used to couple a microwav S ' ° n *" Bignal generator to a load network. It has the great advantage that all th e a vaUable power can be delivered to the load and yet reflections from the load i get transmitted back to the generator output terminals. Consequently not generator sees a matched load, and effects such as power output variat and frequency pulling (change in frequency), with variations in the 1 impedance, are avoided. The isolator is similar to the gyrator in construction except that it employs a 45° twist section and 45° of Faraday rotation. In addition, thin resistive cards are inserted in the input and output guides to absorb the field that is polarized, with the electric vector parallel to the wide side of the guide, as shown in Fig. 6.47. The operation is as follows: A wave propagating from port 1 to port 2 has its polarization rotated 45° counterclockwise by the twist section and 45° clockwise by the Faraday rotator. It will emerge at port 2 with the correct polarization to propagate in the output guide. A wave propagating from port 2 to port 1 will have its plane of polarization rotated by 90 c and will enter the guide at port 1 with the electric field parallel to the resistance card, and hence be absorbed. Without the resistance card, the wave would be reflected from port 1 because of the incorrect FIGURE 6.47 A Faraday-rota""" |UIor . PASSIVE MICROWAVE HKVHIES 467 polarization, which cannot propagate in the guide constituting port 1. However, multiple reflections within the isolator will lead to transmission in both directions, and this makes it necessary to use resistance cards in both the input and output guides for satisfactory performance. Typical performance figures for an isolator are forward transmission loss of less than l dB, reverse attenuation of 20 to 30 dB, and bandwidth of operation approaching 10 percent- Isolator If the curves in Fig. 6.44 for the propagation constants of circularly polarized waves in an infinite ferrite medium are examined, it will be seen that the attenuation constant for negative circular polarization is always very small whereas that for positive circular polarization is very large in the vicinity of the resonance point w 0 = <u. This property may be used as the basis for a resonance isolator by using a negative circularly polarized wave for transmission in the low-loss direction and a positive circularly polarized wave for transmission in the reverse direction. In the latter case the wave is rapidly absorbed or attenuated. The condition for circular polarization is an inherent property of the dominant TE 1 0 mode in a rectangular guide at two positions within the guide. The TE u l -mode fields are K. = sin — e ' *• a > M | ) H V = ±70 s i n — e ' * • a >M0#- = - - c o s — e±Jt* a a Since Hx and H; differ in phase by 90", circular polarization occurs when \HX\ = \HJ, or when * = * j , where tan—L = ±— = ± - ^ (6.114) a Pa 2a For the solution in the range 0 < .*, < a / 2 , the ratio of H x to H 2 is ^=+j (6.115a) and the solution occurring for a / 2 < .v, < a gives — = +/' (6.115o) With respect to the y axis, the solution given by (6.115a) corresponds to a negative circularly polarized field for propagation in the +z direction and to 468 FOUNDATIONS KOU MICROWAVE ENGINEERING -*7» Dielectric S I loading £H— rwi'Fernre F I G U R E 6.48 Rectangular-waveguide resonance isolators. positive circular polarization for propagation in the -z direction If .v, solution given by (6.3 ISA) is considered, the direction of polarization reversed. The above property of the TE, 0 mode is utilized in the resonance isolator by locating a thin ferrite slab (or two slabs, as in Fig. 6.486) in a rectangular guide at a position where the RF magnetic field is circularly polarized. The ferrite is magnetized by a static field applied in the y direction, as in Fig. 6.48. Since the sense of the circular polarization depends on the direction of propagation, as (6.115) shows, it follows that. for propagation in one direction, the magnetic field is negative circularly polarized and suffers little attenuation, whereas in the reverse direction the field is positive circularly polarized and rapidly attenuated. By proper design the forward loss can be kept to under 0.5 d B / i n at A„ = 3 cm, and the reverse loss can be as high as 6 to 10 d B / i n or even more. Dielectric loading, as illustrated in Fig. 6.48c, gives an improved reverse-to-forward attenuation ratio. 6.10 CIRCULATORS A circulator is a multiport device that has the property (Fig- 6.4 ^ wave incident in port 1 is coupled into port 2 only, a wave incident i is coupled into port 3 only, and so on. The ideal circulator is also a_ device; i.e., with all ports except one terminated in matched loads, t Q{ impedance of the remaining port is equal to the characteristic impe« its input line, and hence presents a matched load. . q"S o A four-port circulator may be constructed from two product hybrid junctions and a gyrator as shown in Fig. 6.50. The gyr ^^ g u an additional phase shift of 180° for propagation in the directio^ ^ b in Fig. 6.50. For propagation from 6 to a, and also from c to t*o the electrical path lengths are equal Consider a wave incident in port 1. This wave is ^ h y b ' equal-amplitude in-phase waves propagating in the side arm PASSIVE M1CKOWAVK DEVICES 469 F I G U R E 6.49 Schematic diagram lor a four-port circulator. junction. The waves will arrive at points a and c in phase, and hence will emerge from port 2. A wave incident in port 2 will be split into two waves, one arriving at d with a phase d> and the other arriving at 6 with a phase <f> + TT because of the presence of the gyrator. These partial waves have the right phase relationship to combine and emerge from port 3 in the hybrid junction. A wave incident in port 3 is split into two equal-amplitude waves. differing in phase by 180°, and hence will arrive at the other hybrid junction with the correct phase to combine and emerge from port 4. In a similar manner a wave incident in port 4 will split into two equal waves 180° apart in phase. But now the gyrator will restore phase equality, so that the waves will combine and emerge from port 1. Consequently, the microwave device illustrated in Fig. 6.50 has the required circulating transmission property. A more compact form of four-port circulator may be constructed by employing 3-dB side-hole directional couplers and rectangular-waveguide nonreciprocai phase shifters. The nonreciprocal phase shifter will be described first. It consists of a thin slab of ferriie placed in a rectangular guide at a point where the ac magnetic field of the T E , 0 mode is circularly polarized, as in Fig. 6.51. A biasing field S„ is applied in the y direction. Gyrator F I G U R E 6.50 A four-port circulator. 470 FOUNDATIONS FOR MICROWAVE ENGINEERING -Ferriie slob -/ FIGURE 6.51 A nonreciprocal phas, Since the ac magnetic field is right circularly polarized at x f direction of propagation and left circularly polarized for the opposite" H 0 " 6 tion of propagation, the perturbing effect of the ferrite slab is different*! the two directions of propagation. Consequently, the propagation h constant /3_ for forward propagation is different from the propagati* constant p_ for reverse propagation. By choosing the length of slab so that ( £ . - # _ ) / = i r / 2 , a differential phase shift of 90 c for the two directions o| propagation can be achieved. A four-port circulator utilizing two 90 c nonreciprocal phase shifters is illustrated in Fig. 6.52. The phase shifters are biased, with oppositely directed static fields—an arrangement easily achieved in practice with permanent magnets, as shown in Fig. 6.52. One guide is loaded with a dielectric insert to provide an additional 90c of reciprocal phase shift. The coupling holes are arranged to provide 3 dB of coupling. The wave coupled Permonent magnets 90° 90° FIGURE 6.52 A compact form of four-port circulator. PASSIVE MICROWAVE DEVICES 47 1 through the apertures suffers a 90° change in phase, and this phase change is important in the operation of the circulator. Consider a wave in port 1. This wave is split into two waves by the first 3-dB coupler, the wave in the upper guide undergoing a 90° phase change because of the transmission properties of an aperture. The wave in the upper guide will arrive at the second coupler with a relative phase of 180*"'. and the wave in the lower guide with a relative phase of 90 c . The second coupler splits these waves in the manner illustrated in Fig. 6.52. It is seen that the resultant waves are out of phase in port 4 but in phase at port 2. Thus transmission is from port 1 to port 2. A similar analysis will verify that a wave incident in port 2 emerges at port 3, or, in general, that the sequence 1 -* 2 —> 3 -> 4 —»lis followed. -Port Circulator Carlin has shown that any lossless, matched, nonreciprocai three-port microwave junction is a perfect three-port circulator.t This theorem is readily proved from the properties of the scattering matrix. A perfectly matched three-port junction has a scattering matrix of the form " o sv, s v [S] = S2I S* (6.116) 0 S2:t S,2 0 For a nonreciprocai junction the scattering matrix is no longer symmetrical; that is, S u =f= Sj,. However, if the junction is lossless, conservation of power still requires that the [S] matrix be unitary. Thus (4.63a) will hold for any lossless microwave junction independently of whether or not the junction is reciprocal. Applying the unitary condition to (6.116) gives So2<S^8 + 5l3®13 0 2 ]02] + O23O23 **81®ffl + = * = 1 §32**32 = 1 S j 3 « 2 3 = S12S32 = S21S31 = u Let us assume that S21 * 0. The fourth of the above equations then gives S31 = 0. The third equation now requires | S 3 2 | = 1, and thus S , 2 = 0 from the fourth equation, IS13\ = 1 from the first equation, and S',23 = 0 from the fourth equation again. Thus we see that |S 21 I = 1 also from the second equation, so that \9m\ - \8J = \SJ = 1 S12 *" S 2 3 = S 3 1 = 0 *H. J. Carlin, Principles of Gyrator Networks, Polylech. Inst. Brooklyn. Microwave Res. Inst. Symp. Ser., vol. 4, p. 175, 1955. 472 FOUNDATIONS FOB MICROWAVE ENGINEERING Consequently, there is perfect transmission from port 1 into p o r ( . port 2 into port 3, and from port 3 into port 1. There is zero transm • *Tlitti any other direction. The resultant scattering matrix of any match '* °n in less, nonreciprocal three-port junction must then have the form [S]- 0 0 S.32 s« 0 0 (6-U7) If the locations of the terminal planes in the three input lines are n chosen, the phase angles of S 1 3 , S 2 1 , and S 3 2 can be made zero anrl fif e — Q _ c — i '13 - S2I - S32 Practical realizations for three-port circulators usually involve the symmetrical junction (Y junction) of three identical waveguides or "gtn line" type of transmission lines, together with an axially magnetized ferrite rod or disks placed at the center. Figure 6.53 illustrates both a waveguide version and a balanced strip-line version of the three-port circulator. The ferrite rod or disks are magnetized by a static B0 field applied along the axis and give the junction the required nonreciprocal property. By placing FIGURE 6.53 .:' Three-port « £ * , , . » Waveguide vertf* line circulator- PASSIVE MICROWAVE DEVICES 473 suitable tuning elements in each arm (these can be identical in each arm because of the threefold symmetry involved) the junction can be matched; that is, Su, Sn, and S 3 3 can be made zero. The analysis given above then shows that the junction must, of necessity, be a perfect circulator if all losses are negligible. Losses are, of course, always present, and this limits the performance that can be achieved. Typical characteristics that can be obtained are insertion loss of less than 1 dB. that is, |S 1 3 |, IS21I, |S 32 I greater than 0.89, isolation from 30 to 50 dB, and input reflection coefficients less than 0.2. The isolation that can be obtained corresponds to values of IS,,!, |S 1 2 |, and |S 23 I in the range 0.01 to 0.03. The junction circulator is an essential component used to separate the input and output ports in negative resistance amplifiers (see Sec. 11.5). Circulators are also used to couple a transmitter and receiver to a common antenna. Circulators ranging from miniature units that can handle a few watts of power up to units that can handle 100 or more kilowatts of average power are commercially available. Bandwidths of more than one octave have been obtained. of Three-Port Circulator The field analysis of a three-port circulator is based on the assumption that the electric field in the ferrite disk has only a single component £_. in the axial direction. The applied dc magnetic field is also in the z direction. The permeability tensor then has the same form as in rectangular coordinates; so we can write B., [6.118) Mo where \i.r = 1 + A'*.* a n d JK ~ Xxy ' n o r ^ e r lo obtain a good circulator, the losses in the ferrite must be very small. If we neglect losses W'Q Mr = ~ U? + (O0nU) "-". (On ~ (•> WW, K = 2 wn — w Equation (6.118) is easily derived from (6.98) by expressing Hx, Hy, Bx, and 6 V in terms of the r and <l> components of the fields. From Maxwell's equations we obtain SE, 1 AE. V x E = - a , X V£, = a - — - - a, •H7 r H& = -jcon • H = -j(on0ar(tLrHr +JKH.,,) -J«M»««J -jKHr + n.rH+) (6.119) 474 FOUNDATIONS IOR MICROWAVE ENGINEERING We can solve this equation for H r and H^ and use t b e m esult equation s in th e V X H =ywe£,a, to obtain 1 <52E 1 rV^ <V2 2 r <>r + r2 M > + * * £ , - 0 (6. 120) 2 where A = a V o < ^ r ~ K )e/fir. This equation has the same form as H o for the axial electric field for TM modes in a circular waveguide bat replacing kc. The general solution for Ez is, by analogy, of the'torm*1**1 *' *," M=o I (<*«« '"* + 6.«'"V.(V) (6.121, where o „ and b„ are amplitude constants and J„ is the Bessel function of order «. The corresponding solution for the <t> component component of of the trip magnetic ma.™.field is J'n(k*r)- K ,-jn4> Jn{k*r) e ;i = 0 nk (6.122) :«/-(*,r) J"* ,M r r where J'„{kt,r) ~ dJ „(kl,r)/d(ker). We can interpret this solution as waves that circulate around the ferrite disk in the ±<f> directions. The terms multiplied by a„e~'"'* are waves circulating in the +4> direction and the terms multiplied by b„ejn* are waves circulating in the ~4> direction. Since H„, is different for the two sets of waves, the ferrite disk clearly will exhibit nonreciprocal properties, an essential requirement for a circulator. Consider now the circulator configuration shown in Fig. 6.54. Each o the three microstrip input ports are 120° apart and extend over a coup angle i//. In the field analysis of the circulator, it is assumed that Ha « constant over each coupling region and is zero along the remainder boundary of the ferrite disk. Thus at r = a we let + 6, ^;(V) + 4> ff, 2 ~ 2v ", = i H2 «/' V ill T-?^T 4s- i4 2 + 2ir ip 4-rr <l> (6.123) 2 H3 The assumption of a constant magnetic field H,,, over ^ " J ^ mod* * is a first-order approximation to the magnetic field of the ig.122) '" ' exists in each port. At r = a the solution for H,,, given Y PASSIVE MICROWAVE DEVICES 475 Tapered transmission-line section F I G U R E 6.54 Three-port circulator showing coupling angle <li and tapered transmission lines for matching;. Fourier series; so the amplitude coefficients a „ and b„ are readily found by equating (6.122) to (6.123). When the coefficients a„ and 6„ have been found by Fourier analysis, the electric field given by (6.121) can be evaluated at r = a. In order to obtain a circulator, the condition E z - 0 at d> = 4 i r / 3 is imposed so that there will be no coupling into port 3. This condition establishes a constraint on the ferrite disk radius a, the magnetization M s in the ferrite, and the coupling angle ip. In practice, these parameters are optimized so as to obtain the best circulator performance possible, over as broad a band of frequencies as possible. It is found that the optimum coupling angle tp is given approximately byt 2-rr >!> = 1.84V3 K (6.124) M r « o Mr t H . Bosma, On Stripline Y-Circulators at UHK, IRE Trans., vol. MTT-12. pp. 61-72, 1964. V. S. Wu and F. J. Rosenbaum, Wide-Band Operation of Microstrip Circulators, IEEE Trans., vol MTT-22, pp. 849-856. 1974. 476 FOUNDATIONS FOR MICROWAVE ENGINEERING T h i s angle is typically a r o u n d 30° to 50" a n d r e s u l t s in a strin-l' t h a t c o r r e s p o n d s t o a low-impedance t r a n s m i s s i o n line. Conse " ^ ^ ^ h pered transmission-line sections are often used to t r a n s f o n n v , 1 ^ ' **" i m p e d a n c e into t h e 50-ii i m p e d a n c e of t h e input lines. T h e taper H '°* a r e i l l u s t r a t e d in Fig. 6.54. Satisfactory c i r c u l a t o r designs have beei hy t a k i n g as few as 3 to 6 m o d e s into account, t h a t is, n up to 6 'eved In t h e above account we have outlined t h e principal featur field a n a l y s i s of circulators. T h e references at t h e e n d of t h e chaDt.fr J c h a p t e r provfc m a n y of t h e details t h a t we h a v e o m i t t e d . 6.11 OTHER FERRITE DEVICES T h e devices utilizing ferrites for t h e i r o p e r a t i o n described in the preced" sections r e p r e s e n t only a small n u m b e r of t h e large variety of devices th-! have b e e n developed. In addition to t h e above, t h e r e are other forms of isolators, both reciprocal a n d nonreciprocal p h a s e shifters, electronically controlled (by varying t h e c u r r e n t in t h e e l e c t r o m a g n e t t h a t supplies the static b i a s i n g field) p h a s e shifters a n d m o d u l a t o r s , electronic switches and power limiters, etc. T h e n o n l i n e a r p r o p e r t y of ferrites for high signal levels h a s also been used in h a r m o n i c g e n e r a t o r s , frequency mixers, and parametric amplifiers. A discussion of t h e s e devices, t o g e t h e r w i t h design considerations, p e r f o r m a n c e d a t a , a n d references to t h e original literature, contained in t h e book by Lax a n d B u t t o n , listed in the references at the end of this c h a p t e r . T h e recent article by R o d r i q u e gives a good survey of the present s t a t u s of ferrite devices.t PROBLEMS 6 . 1 . Determine the values of /?[ and R., in the T and II attenuator n e t w l shown in Fig. 6.8 in order to obtain 6 dB of attenuation. 6.2. Derive the equations (6.7) for the II attenuator network. 6.3. For the electronically controlled attenuator shown in the photograpi in 6.10, the return loss is 10 dB. Calculate the input VSWR (see Sec.. 6.4. In the rotary phase .shifter, show that if the output quarter-wav^P^^ transition, and rectangular guide are rotated by an angle B v a phase change of 0, is produced in the transmitted wave. 6.5. For the phase shifter shown in Fig. 6.15. determine the length in incre menwavelength) for the two short-circuited stubs in order to obtain tal phase change of 4 5 \ , e s aCi l* e 6.6. For the phase shifter shown in Fig. 6.16. assume that the PW. g h u „ ( on t« idea) switches. When the diodes are on, the susceptance placed i tG. P. Rodrique. A Generation of Microwave Ferrite Devices, Proc IEEE, vc 1988. I'ASS1VK MICROWAVE DEVICES 477 main line is jBx = -j cot pdv When the diodes are off, the susceptance loading the main line is jB-2=j tan (3d. Show that by choosing d t = A„/4 I S and d = A„/2 - 8, If, = -B2. 6.7. For the phase shifter shown in_Fig. 6.15. the transmission coefficient 7'M = ~/'[l + jB - B2/2] ', where jB is the susceptance of the stub. By using 1 - / j - = \TjJr, find the magnitude p of the reflection coefficient and the input VSWR when B = 0.2 and when B = 0.4. Answer: VSWR = 2.04. 2.2 73 6.8. For a Bethe-hole directional coupler with the two guides aligned (0 = 0) and a centered aperture, why does not (6.28) give a useful solution for A0 as a function of a whereas (6.29) does? 6.9. Design a Bethe-hole directional coupler with a centered circular aperture. The waveguide size is a = 0.9 in. 6 = 0.4 in. The center frequency is 9.8 GHz. The required coupling is 30 dB. Find the aperture radius and the frequency band over which the directivity remains greater than 20 dB. 6.10. Design a Bethe-hole coupler based on (6.29) and (6.30). Assume a = 0.9 in, 6 = 0.4 in, f = 9 GHz. and C = 30 dB. Find the aperture position d and radius r. Evaluate C and D as a function of frequency over the band 8.5 to 10.5 GHz and compare the performance with that shown in Fig. 6.20. B.ll. Figure P 6 . l l illustrates two rectangular guides coupled by circular apertures in a common side wall. A T E „ , mode of unit amplitude radiates a field of amplitude j^r^-rr/a)Habli) ' in both directions in the other guide. Design a five-hole directional coupler of the binomial type. The coupling required is 30 dB at a frequency of 10 GHz. The guide width a is 2.5 cm. and the height b = 2.2 cm. Find the required aperture radii and the frequency band over which the directivity D remains greater than 40 dB. 6.12. For the coupler described in Prob. 6.11, find the aperture radii to give a Chebyshev coupler. The minimum value of directivity required is D m = 50 dB. Find the corresponding frequency band. How much greater bandwidth is obtained as compared with that of the binomial coupler of Prob. 6.11? 6.13. Design a three-hole Chebyshev directional coupler using rectangular waveguides with a = 0.9 in, * = 0.4 in. The center frequency is 9 GHz. The coupling and minimum directivity D m contributed by the array factor is 20 dB and 30 dB at 9 GHz. The apertures are located at x„ = a / 4 . Plot C and D as a function of frequency and compare the performance with that shown in Fig. 6.24. 6.14. Repeat the design problem given in Prob. 6.13 but instead of using Dm = 30 dB, the requirement is that the fractional bandwidth A 0 / 0 ( ) = 0.1, where B0 = 1.29 at 9 GHz and A# = # 8 - pl with /S2 = (S„ + 0.05/?u and /?, = /?„ 0.050,,. What is the resultant value of D,„ for this design? 478 FOUNDATIONS KOK MICROWAVE KN'MNKKRINCJ 6.15. Design a three-hole Chebyshev directional coupler using cent the common broad wall between two rectangular waveguides (f a p e r l u r e s i n center frequency is 9.5 GHz. The waveguide width equals Otf'- 6 ' 2 3 '' '"be height equals 0.4 in. The required coupling is 30 clB and th ' n ^" d l h e directivity is 30 dB. Find the aperture radii, spacing, and the band 6 ? ' n i n i u n i aw coupler. Why is this not a good design? Wth of the. 6.16. Design a broadside-coupled strip-line directional coupler with 3 HR See Figs, 6.27 and P6.16 for details. The ground-plane spacing " S 2 C Q U p h n g input and output lines have a characteristic impedance of 5n n p.1"' ^ e required strip width W and spacing S. Find W so as to obtain Z = -^ th * the input and output lines. For this calculation assume that th K "? centered between the ground planes. In the actual coupler the 1 C upper lines are not centered, but since the spacing S is small the chamT'' ^ will also be small. The computer program CSTPL should be used to obt '" i needed line parameters. Find the length of the coupled section when" frequency is 4 GHz. Assume that the strips have a negligible thickness t« 2 cm -IV- FIGURE P6.16 6.17. A branch-line coupler of the type shown in Fig. 6.29 and having 6-dB coupling at a frequency of 5 GHz is needed. The input and output transmission lines have Zc = 50 LI. Determine the characteristic impedances of the through lines and branch lines and their lengths in terms of wavelength. The coupler is tc be built using microstrip lines on a substrate with a dielectric constant of I and 1 mm thick. Determine the widths of all transmission lines and ike lengths of the through lines and branch lines. Use the computer prograi MSTP to determine the microstrip line parameters. Use the computer program MSTPD to determine the effective dielectric constants at 5 GHz a modify the transmission-line lengths so as to take into account the disper in the effective dielectric constant at 5 GHz. 6.18. The hybrid ring shown in Fig. 6.33 is constructed using microstrip lines on 1-mm-thick substrate with a dielectric constant of 2.3. The input a n d ^ ^ transmission lines have a characteristic impedance of 50 Si. ue ^ ,f,e width of the transmission lines and of the ring as well as the " " " ^ " ^ s l T ring. The frequency of operation is 2 GHz. Use the computer prog 6.19. Determine the impedances and the value of the resistor R °\e^%,eTei»" divider shown in Fig. 6.38. The power in port 2 is one-half of tna in 8 port 3. er ation 6.20. Use the equation of motion (6.94) to study second-harmonic ge . a l i o n at* ferrite. Assume that M = M ,*•""' + M , ? 2 " " for the ac « J ^ h B t is . the * that H has only an .r component with time dependence e • oCClix^' magnetic field is HseJ"''ax. Neglect the third-harmonic term show that 2j<oM2 = y»nHxa, X M, + yp0ti0*, * *** PASSIVE MICROWAVE DEVICES 479 Thus 2JMM2C = W u H , M l v . For M l v , take the small-signal solution * „ / / , and use the value of Xvx a * resonance for a lossy ferrite to show that Mas = — Note that, for good efficiency, a must be small (small damping), so that the precession angle will be large at resonance. 6.21. From the unitary properties of the scattering matrix for a lossless nonreciprocal two-port microwave junction, show that it is not possible to have S 2 1 zero while 5 1 2 is finite. Thus a lossless one-way transmission device cannot be built. 6.22. Continue the argument in the text to verify that the transmission sequence 1 > 2 -» 3 -> 4 — 1 is followed in the circulator illustrated in Fig. 6.52. 6.23. Show that the scattering matrix for an ideal lossless JV-port circulator can be put into the form [S] = 0 1 0 0 0 0 1 0 0 0 0 0 • • • • • • • • 0 0 0 1 1 0 0 0 by choosing proper terminal-plane locations in each port. 6.24. Show that, for TEM-wave propagation in a direction perpendicular to B„ in an infinite ferrite medium, the two solutions are linearly polarized waves with propagation constants ,11/2 >•+=/«•» 1 + .V Y- ! Hint: Consider propagation along x, and in one case assume E to have only a y component, and for the other case assume E to have a z component 1. Montgomery. C. G„ R. H. Dicke. and E. -M. Puree)): "'Principles of Microwave Circuits," McGraw-Hill Book Company, New York, 1948. 2. Ragan, G. L. led.): "Microwave Transmission Circuits." McGraw-Hill Book Company, New York. 1948. 3. Marcuvitz, N. (ed.l: "Waveguide Handbook." McGraw-Hill Book Company, New York. 1951. 4. Harvey, A. F.: "Microwave Engineering." Academic Press. Inc., New York, 1963. Ar outstanding handbook, offering a comprehensive survey of the whole microwave field together with an extensive bibliography covering the international literature. 5. Bahl, I., and P. Bhartia: "Microwave Circuit Design." John Wiley & Sons, Inc., New York 1988. 6. Chang; K. (ed.l: "Handbook of Microwave and Optical Components. Microwave Passiv and Antenna Components," vol. 1, John Wiley & Sons. Inc., New York, 1989. 7. D. M. Pozar, "Microwave Engineering." Addison-Wesley Publishing Company, Reading Mass.. (.990. 480 FOUNDATIONS FOR MICROWAVE ENGINEERING Microwave ferrites 8. Lax, B., and K. J. Button: "Microwave Ferrites and Ferrimagnetics " Mef" Company. New York, 1962. A very complete treatment of the physical"^* i1 ' 1! B°»k ferrites, electromagnetic propagation in ferrites. measurement techniques a of many ferrite devices. Includes a 24-page bibliography. ^ U s ^ , . 9. Soohoo, R. F.: "Theory and Application of Ferrites," Prentice-Hall Inr p J N.J.. 1960. "*' , , g l e w oodC!is i i 10. Clarricoats, P: "Microwave Ferrites." John Wiley & Sons, Inc., New York, IQRI A. G.: "Ferrites at Microwave Frequencies," transl. from the Russian bv A -p ?"**"'*. Consultants Bureau. New York, 1963. • 'vbulew^ 11. Roberts, J.: "High Frequency Application of Ferrites," D. Van Nostrand r Princeton, N.J.. 1960. <*»»Pwiy. Inc., Field theory of circulators 12. Bosma, H.: On Stripline Y-Circulators at UHF. IRE Trans., vol. MTT-12, p p . 61-72 l<*u 13. Fay, C. E., and R. L. Comstock: Operation of the Ferrite Junction Circulator /BPi Trans., vol. MTT-13, pp. 15-27, 1965. 14. Wu, Y. S., and F. J. Rosenbaum: Wide-Band Operation of Microstrip Circulators IEEF Trans., vol. MTT-22, pp. 849-856, 1974. 15. Schloemann, E., and R. E. Blight: Broad-Band Stripline Circulators Based on YIG and Li-Ferrite Single Crystals. IEEE Trans., vol. MTT-34, pp. 1394-1400, 1986. CHAPTER 7 ELECTROMAGNETIC RESONATORS RESONANT CIRCUITS Resonant circuits are of great importance for oscillator circuits, tuned amplifiers, frequency filter networks, wavemeters for measuring frequency, etc., at all frequencies from a few hertz up to and including light frequencies. Electric resonant circuits have many features in common, and it will be worthwhile to review some of these by using a conventional lumped-parameter RLC parallel network as an example. Figure 7.1 illustrates a typical low-frequency resonant circuit. The resistance R is usually only an equivalent resistance that accounts for the power loss in the inductor L and capacitor C and possibly the power extracted from the resonant system by some external load coupled to the resonant circuit. At resonance the input impedance is pure real and equal to R. This implies t h a t the average energies stored in the electric and magnetic fields are equal, since from (2.60) P, + 2ja,(Wm - We) Z;„ = (7.1) 211 This equation is valid for any one-port circuit provided a suitably defined equivalent terminal current / is used. Thus resonance always occurs when W^ = Wg, if we define resonance to be that condition which corresponds to a pure resistive input impedance. In the present case the time-average energy stored in the electric field in the capacitor is Wc = \VV*C 481 482 FOUNDATIONS FOR M1CKOWAVE ENGINEERING IL a ^ FIGURE 7.1 Lumped-parameter resonant circuit. and that stored in the magnetic field around the inductor 1 1 V Wm = -LI J? = —£ '" A 4 4w 2 L is W* The resonant frequency w 0 is now found by equating W and o,o = ( L 0 ) - v a (7.2) An important parameter specifying the frequency selectivity, and per formance in general, of a resonant circuit is the quality factor, or Q. A verv general definition of Q that is applicable to all resonant systems is w( time-average energy stored in system) Q= (7.3) energy loss per second in system At resonance Wm = W. and since the peak value of electric energy stored in the capacitor is 2W(, and occurs when the energy stored in the inductor is zero, and vice versa, the average energy W stored in the circuit is (7.4) W = W,„ + W= 2W„ = 2W = kCW* The power loss is ^GW* and is the energy loss per second. Hence, for the circuit of Fig. 7.1, tiiC R G coL (7.5) since u>2LC = 1 at resonance and G = R . ~wiance In the vicinity of resonance, say w = o>0 + Aw, the input tmpec can be expressed in a relatively simple form. We have 1 - AW/G>O R jwL J ~ I \R where the approximation l/'(w 0 + Aw) = (1 - A w / w 0 ) / ^ o Since ju0C + l/jco0L = 0, we obtain ZiD= (o'JRL R a>20L+j2RAa> 1 + j2Q{ba>/<*o) fc ; [ic«l n 8 iyp A plot of Zin as a function of Aw/w 0 is given in Fig. 7.2, and a n ^is v a j U e ; m a f "^espo^'1' resonancec curve. When \Zm\ has fallen to 0.707 of its phase is 45° if w < w0 and - 4 5 ° if w > w„. From (7.6) the ELECTROMAGNETIC RESONATORS 483 & -ft -\ • 90° 0.707-9 /l A \f~VLA \ BW XT' -9°" FIGURE 7.2 Z,„ for a parallel resonant circuit. value of Aw is found to be given by A to 2Q or A W = = 1 2Q The fractional bandwidth BW between the 0.707ft points is twice this; hence Q = 1 BW 2Au> [7.7) This relation provides an alternative definition of the Q; that is, the Q is equal to the fractional bandwidth between the points where |Z,„I is equal to 0.707 of its maximum value (for a series resonant circuit this definition applies to \Y:n\). If the resistor R in Fig. 7.1 represents the losses in the resonant circuit only, the Q given by (7.5) is called the unloaded Q. If the resonant circuit is coupled to an external load that absorbs a certain amount of power, this loading effect can be represented by an additional resistor R, in parallel with R. The total resistance is now less, and consequently the new Q is also smaller. The Q, called the loaded Q and denoted Q , , is RRL/{R QL- + RL) (oL The external Q, denoted Qe, is defined to be the Q that would result if the resonant circuit were loss-free and only the loading by the external load were present. Thus 484 FOUNDATIONS FOR IDCHOWAVE BMOI Use of these definitions shows that 1 J_ 1. Another parameter of importance in connection with a resonant circuit is the damping factor 6. This parameter measures the rate at which the oscillations would decay if the driving source were removed. For a high-Q circuit, 8 may be evaluated in terms ol the Q, using a perturbation technique. With losses present, the energy stored in the resonant circuit will decay at a rate proportional to the average energy present at any time (since P a W* and W a VV*, we have P,aW), so that dW — = -2SW at w=Wne^' or (7.9) (7.10) where Wu is the average energy present at / = 0. But the rate of decrease of W must equal the power loss, so that dW = 2 S W = P, dt Consequently, S = P, ^£i_ _ JL 2W ~~ 2 u>W 2Q (7.11) upon using (7.3). The damping factor is seen to be inversely proportional to the Q. In place of (7.10) we now have W = W0e'",,/Q I n (7.12) Q must b e replaced b y Q , i f a n ^ . d l ^ ^ ^ t S e ° circuit. The damping factor 8 i s also a measure o f h o w f a s ^ rf g ^g oscillations in the resonant circuit can budd up upon apphc S °™n microwave systems sections of transmission fines^J*^ sures called cavities are used as resonators m p l a c e o t h e £g* circuit. The reason for this i s that l u m p e ^ ^ J s ^ b e « * * J * losses from both conductor loss and radiation loss aV e ca ^ m " w a v e frequencies. In calculating the i m p e d a n * * a * s e n , T h e * J i t i s sometimes convenient t o assume there a e ^P imped*£ fa be evaluated separately, and in terms of ih^*™™fonant fire**** be modified to account for losses by replacing the ^ = *„ W» an equivalent complex resonant frequency w„U + . / / ELECTROMAGNETIC RBSI >NAT( >KS 485 that (7.6) can be written as w0R/2Q (7.13) j[w-u>0(l+j/2Q)} which shows that when losses are present this is equivalent to having a complex resonant frequency w 0 (l +j/2Q). Equation (7.13) neglects the small change in resonant frequency that occurs when small losses are present. 7.2 TRANSMISSION-LINE RESONANT CIRCUITS Series R e s o n a n c e ; S h o r t - C i r c u i t e d L i n e At high frequencies, usually in the range 100 to 1,000 MHz, short-circuited or open-circuited sections of transmission lines are commonly used to replace the usual lumped LC resonant circuit. It is therefore of interest to consider the order of magnitude of Q and impedance that can be obtained. It will be assumed that air-filled lines are xised, so that the only losses are those due to the series resistance R of the line. This is usually the case in practice since a dielectric-filled (ine has some shunt conductance loss and hence would result in a lower Q. Consider a short-circuited line of length /, parameters R,L,C per unit length, as in Fig. 7.3. Let / •= A 0 /2 at f = f0s that is, at to = <o„. For f near f0, say f = f0 + A f, fil = lirfl/c = TT(U/W0 = TT + 77 iw/01,,, since at w 0 , fil = v. The input impedance is given by Zin = Zv tanh( jliI + al)=Zc tanh al +,/' tan fit 1 + j tan j5l tanh «/ But tanh al = al since we are assuming small losses, so that al <s 1. Also tan fi! = tan(— + — Aa)/(o0) = tan IT i w / w , , = ~ Au>/<o(l since \u>/io0 is , 1 I «* < /i* Zcfl.K F I G U R E 7.3 Short-circuiled transmission-line resonator. if £3 / f -"p ^o F I G U R E 7.4 A series resonant circuit 486 FOUNDATIONS FOR MICKOWAVE ENGINEERING small. Hence al in = Z +JTT\W/O,Q <" 1 , • / A I ', =Z c\ a . l Aw +JTT (7. 14, since the second term in the denominator is very small Now a = \RYr = (R/2)JC/L, and 0/ = co0jLCl = TT; SO TT/ W O = / ^ c expression for Z m becomes Z,= \ fC - \-R\\-+ 2 f L J ' i W = 2 f i ' + i l l A a It is of interest to compare (7.15) with a series R0L0C illustrated in Fig. 7.4. For this circuit Z lr , = S 0 +J*>LQ[l - and the ( circuit tD~L0C0 If we let oig = 1/L 0 C„, then 2 in = ff o +7wZ. 0 - Now w2 - oij = (at - w 0 )(u + w 0 ) = AOJ 2w if a) - co0 = Aw is small. Z in = fi o +7'wL' 0 2(oAw 2~ to = ft0+j'2L0Ac (7.161 By comparison with (7.15), we see that in the vicinity of the frequency I which / = A 0 /2, the short-circuited line behaves as a series resonant arc with resistance RQ = ^Rl and inductance L0 = \Ll. We note that ««are the total resistance and inductance of the line; so we might wonde the factors \ arise. These enter because the current on the sbort-c ^ line is a half sinusoid, and hence the effective circuit parameters n o - ' only one-half of the total line quantities. vcuit °^ The Q of the short-circuited line may be denned as for the a Fig. 7.4: a>0L0 <oQL p (7. Q Rf R 2a W e ^ As an alternative, the general definition (7.3) may be used, ^ ^ s "* evaluate the Q of the short-circuited line from this definition j 2_ ELECTROMAGNETIC RESONATORS 487 an approximate method valid for high-Q (i.e., low-loss) systems in order to illustrate a method of great utility in connection with many microwave devices (see Sec. 3.8, where this perturbation method is discussed). For small losses the energy stored in the system is, to first order, the same as if no losses were present. For a loss-free short-circuited line, the current on the line is a pure standing wave / =/ 0 c o s / ^ <?-"•" where z measures distance along the line from the short toward the input end. In a length dz, the energy stored in the magnetic field is dWm = \H*Ldz = \l$L sin 2 iizdz The total time-average stored energy in the magnetic field is W„ -XLfA"/2cos^zdz=^L 4 •'o 16 The energy stored in the electric field, i.e., in the line capacitance, is equal to W m at resonance; so the time-average energy stored in the system is W= W,„ f f = -^LI 8 o To a first approximation the losses do not modify the current distribution along the line. Hence the power loss is given by p- ~rRn*dz = ~isr\os^zdz = ^m^ Thus, at <o = «„, Q = M<IW - p - = u>aAuLI*/8 K/»A 0 /8 = a>0L ~R~ (7 18) ' which checks with the earlier result. Typical values of Q range from several hundred up to about 10,000. As contrasted with low-frequency lumpedparameter circuits, the practical values of Q are very much higher for microwave resonators. It should be noted that in the above analysis the losses in the short circuit have been neglected. This does not introduce appreciable error if the length / is considerably greater than the conductor spacing. Line By means of an analysis similar to that used earlier, it is readily verified that an open-circuited transmission line is equivalent to a series resonant circuit in the vicinity of the frequency for which it is an odd multiple of a 488 FOUNDATIONS FOR MICROWAVE ENGINEERING ?t ]L° FIGURE 7.5 ? ^ ! n - d r C u i t e d "ansa,; missi,°<H.* ST C° Q resonator. quarter wavelength long. The equivalent relations are (Fig. 7.5) / - A° * 4" 7 ( , at(0 « Aw w (7.19a) 1 \„ (7.196) 2Rl (7.19c, LQ = -LI (7.19rf) <-(L0Co) (7.190 Ro = Antiresonance Short-circuited transmission lines behave as parallel resonant circuits in the frequency range where they are close to an odd multiple of a quarte wavelength long. The same property is true of open-circuited lines that are 1 multiple of a half wavelength long. When they behave as parallel resonant circuits, they are said to be antiresonant. ^^^* The case of parallel resonance is best analyzed on an admittance With reference to Fig. 7.6, let / equal A 0 /4 at w = w 0 . Then 111 = w)fLCl = w0\fl~Cl + Ato/LCl n. r<. oc.B C ° FIGURE 7.6 Short-circui& mission line. — — " • " ' ELECTROMAGNETIC RESONATORS 489 at co, and the input admittance is given by Yin ~Yccoth(a+jfi)l 1 +7 tan pi tanh al Yc tanh al + j tan /3/ 1 -jal/(±wljLC) al -j/(Acoly/LC) since 77 tan(w 0 \/LCZ + A W L C / ) = tan I— + Ao>\/LC/ = -(Awv/LC/) -l and tanh al ~ al. A further approximation yields j liojLCl + al Ym = Yc I + j boo al2/LC = Yc(jAa>yfLCl + al) = RC (7.20) — 1+jAcoCl after replacing Y c by ^ C 7 ? and a by i ; . R / 2 . For the parallel resonant circuit of Fig. 7.6, we have 1 K0 +JcoL0 jcoC0(R0 + jcoL0) + 1 Rn + jcoLn JCOCQRQ — co LQCQ + 1 since we assume Ra « wL 0 . If we define w0 by C 0 i? 0 <O'QL0C0 = 1, then conL0C0 - co L0C0 (a>0 - OJ)(<U0 + co) fi * o7 ^oT1 J L oC i + ./C02AW (7.21) 490 FOUNDATIONS KOR MtCROWAVK ENGINEERING Comparison with (7.20) shows that the short-circuited line in th a quarter wavelength long is equivalent to a parallel resonanr-o •VlclnUy 0 r CnRQ RC ~ l R, R CI = 2C 0 The Q of the circuit is given by a> Q 2i 0 • " • i r - -~R^ - 2o- (7.22) Although sections of transmission lines behave as simple Ii parameter resonant circuits in the vicinity of a particular resonim r quency, they are in reality a much more complicated network havin infinite number of resonance and anliresonance frequencies. The resona frequencies occur approximately when the short-circuited line is a multi of a half wavelength long, that is, /'„ = rac/2/, and the anliresonance frequencies (parallel resonance) when the line is an odd multiple of i quarter wavelength long, that is, f„ = (2n + l ) c / 4 / , where n is an integer. Thus the exact equivalent circuit would consist of an infinite number of resonant circuits coupled together. However, in practice, the frequency range of interest is normally such that a simple single-resonant-frequency circuit represents the transmission-line resonator with adequate accuracy. 7.3 MICROSTRIP RESONATORS In microwave circuits that use microstrip transmission lines, an open-cii cuited section of microstrip line may be used as a simple resonator. In Fig. 7.7a we show a microstrip resonator t h a t is capacitively coupled to the mpul microstrip line. In Fig. 7.76 we show a similar resonator that is capacitivel: coupled to the side of the microstrip line. An alternative resonator configu- FIGURE 7.7 \w consisting of C , ine ap P rov imately one-half *» IonK and c - p a ^ J & (a) W \ZZ (c) - * > « • t o a 55?*! ,6) the same re=o <a) bul coupled to <c)0*the microstnp ^ {) cular disk or reson-torw^p, tchcoupledtoam.** KI.EC I KOMAONET1C RESONATORS T 49 1 \—fih M FIGURE 7.8 The equivalent circuit of the resonators shown in Figs. 7.7o iind b. ration is the circular disk that is capacitively coupled to a microstrip line as shown in Fig. 7.7c. The equivalent circuit for the resonators shown in Figs. 7.7a and 6 is a transmission line of length / with a capacitor C„ at the open-circuit end and a capacitive II network that represents the coupling region as shown in Fig. 7.8. The fringing electric field at the open-circuit end results in additional charge on the microstrip line near the open-circuit end and this is modeled by the capacitor C„. The capacitor C„ makes the electrical length of the resonator appear to be longer than the physical length /, because the line length has to be increased beyond A/2 to provide the additional inductance that will compensate for the additional capacitance. Resonators of the type shown in Figs. 7.7a and b are commonly used in microstrip filters (see Chap. 8). In order to design a resonator of the type shown in Fig. 7.7a. the open-circuit capacitance C,, and the values of C„ C 2 , and C';, in the coupling network must be known. These parameters have been evaluated by Silvester and Benedek using the assumption of static potential fields.t The results are accurate for frequencies up to several gigahertz for microstrip lines with substrates 1 mm thick and even at higher frequencies for thinner substrates. In Fig. 7.9 we show typical values of the open-circuit capacitance Co/W, where W is the width of the microstrip line. In Fig. 7.10 we show typical results for C,, C2, and C:i for various microstrip parameters including the gap spacing s. t P . Silvester and P. Benedek, Equivalent Capacitances of Microstrip Open Circuit*:. IEEE Trans., vol. MTT-2G, pp. 511-516. 1972. P. Benedek and P. Silvester. Equivalent Capacitances for Microstrip Gaps and Steps, IEEE Trans., vol. MTT-20, pp. 729-733, 1972. 492 FOUNDATIONS FOR MICROWAVE ENtilNEERING 2 - L JTjty 1 N \S i — 9.6 0.2 -— 4.2 0.1 "*-• 2.5 «,= 1 1 < 1 1 IfH i 1.0 W/H i i i i 10 F I G U R E 7.9 Microstrip-line open-circuited capacitance. ( Based on Fig. o of Silvester and Benedek.) The resonator and coupling circuit shown in Fig. 7.8a can be represented by the equivalent lumped-parameter circuit shown in Fig. 7.11a, in the vicinity of the resonant frequency. In these circuits the capacitance C, and inductance L, are those contributed by the transmission line and are given by (7.230) c, = \ci (7.236) where C and L are the distributed capacitance and inductance per r for the microstrip line and I is the length of the resonator. The.factor ^ arises because the voltage and current standing waves on t n e ^ by sinusoidal, so that the stored electric and magnetic energies are r ^ one-half from what one would have for constant voltage and cu ^ ^ amplitudes, and this makes the effective capacitance and i n d u e t a r ^ ^ equivalent lumped-parameter model one-half of those tor resistance R represents the total losses in the resonator. , t j o n of ft For the circuit in Fig. 7.11a, the input admittance at the is given by j*>C0+jYetaxipl yiM - j « c 8 + YC ^ _ wCo tan +G pf wi H as reference to Fig. 7.8a clearly shows. For convenience we WCV OJC.,, and wC3 by B 0 , B „ B2, and S 3 . The input a denote »C^ i t t a n c e '- ELECTROMAGNETIC RESONATORS 493 u. u. Q. 0.1 _ u o 0.01 0.01 S/W S/W (a) (fe) •— - _^_ \ 1 1 1 1 LL a **~ 0.1 o 86 12. ~C2 96 3^ Ce - " 2.5 W/H = 2 6r=1 0.01 i 0.05 i ii i _1 L 0.1 0.5 S/W (O F I G U R E 7.10 The Il-network capacitances for a microstrip-line gap. 1 Based on Fig. 7 of Benedek and Silvester.) across C, is given by Y^-jB, j(B, JBJU JB2 + Y;n +B2)Y-;n-BlB2 JB.2 + Y.;n (7.24) In a gap coupled microstrip resonator B t «*: S 2 so it is operated as a series resonant circuit. Series resonance occurs when the imaginary part of the 494 FOUNDATIONS FOR MICROWAVE ENGINEERING (b) F I G U R E 7.11 ( a ) Equivalent lumped-parameter circuit for the resonator and coupling network 7.8a; (6) a simplified equivalent circuit incorporating an ideal transformer ?* denominator vanishes. The condition for resonance is thus ^in _ & = JB'm ™ ~JB2 (7.25) When we substitute for Y-n, we obtain an equation for tan pi which is tan pi = YC(B2 + B0 + B.3) B0(B2 + (7.261 Ba)~Yc2 Since B 2 , B :j , and B 0 are all very small, pi will be somewhat smaller than 77 since the expression for tan pi is small and negative. The resonant length is shorter than one-half wavelength because of the capacitive loading at the input and output ends. When we use (7.25) in the numerator in (7.24) and assume that we can approximate G + jB'm by -jB.2 for all values of <* h the vicinity of the resonant frequency, we are able to express (7.24) in t form Y, - B? G+JiB'n + Bz) C f Hi' When we compare this expression with that for the input admittance circuit in Fig. 7.116, we see that z = ju)L >» M '-^k + R. - j S l G + • / ( * . 1 Bt)\ so the ideal transformer turns ratio is n =B2 G where R c can be chosen arbitrarily. Thus we have -\ Fin = fif[/W(C2 + C3 + C„) +jY„ tan pi + G\ ELECTROMAGNETIC RESONATORS 495 In (7.28) we have used the approximation Y-n =jcoC:i + jwC0 + jYt. tan pi + G which is obtained by replacing the denominator factor Ye - wC 0 tan pi by Y c since wt?0 tan pi is a second-order small term near resonance because both ioC0 and tan ft I are small. The term jYc tan ft I can be represented by a parallel connection of C t and L, in the vicinity of resonance as discussed earlier for transmission-line resonators. Thus, in the equivalent series resonant circuit shown in Fig. 7.11b, the total inductance is the sum of C(, C 0 , C 3 , and C 2 , that is, n2 ^ = ^ l C 2 + C 3 + C 0 + t?,] (7.29) When the dispersion of the microstrip line can be neglected, the Q is given by (7.17) as P R Q = — = — - * «, 6 C,fl (7.30) where R is the equivalent parallel resistance that will account for conductor and dielectric losses in the microstrip line. From the examples considered in Chap. 3, we found that typical values of attenuation for a microstrip line was in the range 0.5 X 10 ~ 2 to 1.5 X 1 0 _ 2 Np/wavelength. Hence typical values of Q obtained from (7.30) would be 200 to 600. There is some additional loss caused by radiation of power from the open-circuited end. Measured values of Q for microstrip-line resonators are usually in the range 100 up to several hundred. When Kin = Yt. at resonance, all of the incident power will be coupled into the resonator. Since Y m = n2/Rr at resonance, the required value of n 2 for critical coupling is n2 = YCR,. The parameter 0tYe = ^-^Yc = nYc = ^ { U C ^ l = 2io0C, so R = 2Q/vYc, as may be seen by using (7.30) for Q. Thus for critical coupling. If Q = 157, then B.2 = y c / 1 0 . For a high-Q resonator, only a small coupling capacitance is required to obtain critical coupling. The loading of the resonator by the external transmission line is equivalent to a resistance n2Zt. connected in series with Rc. Thus the 496 FOUNDATIONS FOR MICROWAVE ENGINEERING external Qe is given by _ a>0Le OQCX n*Z,. Bf QYCG B-l The coupling parameter K is given by the ratio Q/Qe and K = Bl 18 2QBI Y,G (7.326) Q / T T YC C given given above W.31). found by using using RR == 22Q/TTY (7.31). Sin » irYe/2Q = G our earlier assumption that the numerator^ 7 i n in (7.24) could be approximated by Bf is justified. Prom Fig. 7 i 0 be found that for the typical values of C 2 that are required C and C negligible. A parallel resonant circuit coupled by a small series connect" capacitor functions as a series resonant circuit. For a transmission line without dispersion, that is, p is a linPa function of w, the Q of the transmission line is given by p/2a. U the propagation constant p is not a linear function of a>, the transmission line is said to be dispersive. For a microstrip line the effective dielectric constant et, is a function of oj; so ji = \Jee(oj) kn is not a linear function of w. In the frequency range where e c is changing quite rapidly with frequency, a different expression from that given above must be used for the Q. The power flow along the transmission line is given by Wvg, where W Is the average stored energy per unit length and vg is the group velocity which is given by (dp/dv)~\ Thus W = P/ug -= P(dp/d<o) and the Q is given by as YJTT/2Q IOW coP(dp/dw) 2aP w(dp/d<o) 2a When there is no dispersion, /3 = w/vp, where vp is a frequency-indeper dent phase velocity. For this case the general expression for Q re° u C e s P/2a. When there is significant dispersion, the capacitance C, in equivalent circuit of the resonator is also a function of o> because e, is. Circular Disk Resonator 7.12. An The circular disk or planar radial resonator is shown in Figr *ea£jngthe approximate analysis of this resonator is readily carried out by ^^ outer boundary at r = a as a perfect open-circuit boundary {ma^0n , gpi on which n x*H = 0. The field in the resonator will not Aepen ^^^ will have an axial electric field Er. Thus the modes in the reS°ut vv-ith d] like those of TM of E modes in a circular waveguide at cutott ^ sUju»bl* guide boundary being a magnetic wall instead of an electric expression for the axial electric field is Ez = CJ„(kr)cos n<i> ELECTROMAGNETIC RESONATORS H.=0 (a) 497 FIGURE 7.12 The circular disk resonator. (ft) where C is an amplitude constant, J„ is the Bessel function of order re, and k = JTrkfj with e r being the dielectric constant of the substrate material. Since this is the only electric field component present, the magnetic field is given by V x azE, = - a , X VE, = - j « ^ 0 H = -jk^YL or H = -J a , x VE,= r «0Z0 -^-^-a^ArJcos/J^ - fc y kuZnr - + a, x a(k—• r <ty> <ir _a r J„(Ar)sinn</> (7.34) where J'n(kr) = dJn(kr)/d(kr). In order for 72^ to vanish at the boundary r = a, we require J'n(ka) = 0. The roots of this equation are given in Table 3.7. The smallest root is 1.841 corresponding to the use of the junction Jx(kr). The resonant frequency is obtained by equating ka = ^erk0a to 1.841; thus 1.841c W'110 lln = 7= (7.35a) V^rO where c = 3 X 10 1 0 c m / s is the speed of light.t Since n = 1, 1.841 is the first root, and there is no dependence on z, this dominant mode is designated as the T M U 0 mode. As an alternative we can solve for the radius o which is given by 1.841 a = (7.356] \/e>o For a resonator at a frequency of 4 GHz and using alumina with e r = 9.7 as tMore accurate expressions for the resonant frequencies are given in I. Wolff and N. Knoppik. Rectangular and Circular Microstrip Disk Capacitors and Resonators, IEEE Trans., vol. MTT-22, pp. 857-864, 191 A. 498 FOUNDATIONS FOR MICROWAVE ENGINEERING the substrate materia], we require a = 0.7056 cm which resi ttively i v o l v compact c n m n f t r t resonator. recAnatfir rela. The Q of the resonator is readily evaluated. The stored oU etectr is given by »c e ner , We-\Cf- -JlJJjJtWcos^rdrdAdz „7re'e ( ,6 = | C ! 2 — ^ - a 2 J , ( k a ) + 11 - 1 k2ai )JAka) 1 (7.36) since J{(ka) = 0. In this expression f'r is the real part of the dielect i constant and b is the substrate thickness. The stored magnetic energy W equals V^, at resonance. The power loss in the dielectric is given by P n<=^T / \Efrdrd4dz = 2we: ~^Wt (7.37) 7, The power loss in the conducting disk and ground plane is given by 2 -"o •'o Rj2*fa(\H/ = J \H/)rdrd<l> + J •'n Q * '0n since the current density is given by a, X H on the ground plane and j - a , X H on the disk. After substituting from (7.34) for the magnetic fi we obtain P„ - icrjgr k2jf(kr) + ~^Jf(kr)\rdr after integrating over 4> which gave a factor of IT. We can integra term by parts to obtain ,adJ j-a d-Jx(kr) dJv(kr) 'o • i, dr dr . -rdr = ,, rJx{kr) dJ^kr) dr dJ^kr) -fj^kr) dr Since J't(ka) = 0 the integrated term is zero. We now differential equation d dJx — r—~ dr dr + I z \k r \ the first 1 J, = 0 r dr dr take note oft** ELECTROMAGNETIC RESONATORS 499 satisfied by Jx(kr). By using this equation to express the second derivative of Ji in terms of J , , the expression for Plr reduces to ,R„ir -**-sm*v-w™*f The Q is given by 2a>Wf Q k 0b ' pld + p h "" 2i?„,y 0 + 6';fe 0 6/ c ; ( 7 3 8 ) which is a surprisingly simple result in that there is no dependence on the Bessei function or the radius of the resonator. In (7.38) the substrate thickness is given by the parameter 6. As an example we will consider a disk resonator operating at 4 GHz. The substrate will be chosen as alumina with e'r = 9.7 and e". = 0.0002 and having a thickness 6 = 1 mm. For this resonator k0b = ( 2 - / 7 5 ) = 0.08377: 8.377 X 10~ Q ;, 2 X 16.44 x 1 0 - / 1 2 0 ~ + 2 x 8.377 X 10 */$.! 8.377 8.72 x 10~ 3 + 1,72 x 10 * = 942 The dominant loss mechanism is conductor loss. The Q of this resonator is higher than that of a transmission-fine type since the conductor losses are smaller. In practice, the Q can be expected to be less because of radiation loss and extra loss from surface roughness on the disk. There will also be a much larger azimuthal current density J,,, at the edge than is obtained using the approximate theory given above. For an infinitely thin disk, the current eL should exhibit the usual edge singularity and be proportional to ( a 2 - r 2 ) ~ 1 / 2 as the edge is approached. There will also be some current flowing on the top surface of the disk and this will also increase the conductor losses. The actual resonant frequency of the disk resonator is lower than that predicted by the simple approximate theory used above. More accurate formulas have been obtained that will give the resonant frequency to within about 1 percent. The reader is referred to the literature for a more comprehensive treatment.! f£ Woiff and N. Knoppik, Rectangular and Circular Microstrip Disk Capacitors and Resonators, IEEE Trans., vol. MTT-22. pp. 857-864, 1974. 500 FOUNDATIONS FOR MICROWAVE ENGINEERING r^ C^L Coaiioi line Coo»>o line (a) Aperture Waveguide tc) F I G U R E 7.13 Cavity-coupling methods, ( a ) Loop coupling. , . , probe coupling; (c) aperture coupling. In addition to the circular disk, other shapes such as ellipses, rines triangles, and squares, have been considered for use as resonators, as well as coupled sections of microstrip lines.t 7.4 MICROWAVE CAVITIES At frequencies above 1,000 MHz, transmission-line resonators haw relatively low values of Q, and so it becomes preferable to use metallic enclosures, or cavities, instead. A cavity can be considered as a volume enclosed by a conducting surface and within which an electromagnetic field can b excited. The electric and magnetic energies are stored in the volume o cavity. The finite conducting walls give rise to power loss and thus equivalent to some effective resistance. The fields in the cavity may excited, or coupled to an external circuit, by means of small coaxial- n probes or loops. Alternatively, the cavity may be coupled to a wave means of a small aperture in a common wall. These coupling met! illustrated in Fig. 7.13. Before considering the effects of coupling on_^ ^ performance, the field solutions in rectangular and cylindrical ca presented. Rectangular Cavity Figure 7.14 illustrates a rectangular cavity of height b, width a,^minsU* d. It may be considered to be a section of rectangular wavegui w a v e l e n gth in a short circuit at z = d. If d equals a multiple of a half gui \V.-~ tj. lielszajn and D. S. James, Planar Triangular Resonators with * Trans., vol. MTT-26. pp. 95-100. 1978. ,. I. Bahl and P. Bhartia. "Microwave Solid State Circuit Design,' J ° n New York. 1988. wi|ey 1E& & So"*- ELECTROMAGNETIC RESONATORS 501 F I G U R E 7.15 Standing-wave pattern in a short-circuited waveguide. F I G U R E 7.14 A rectangular cavity. at the frequency f, the resultant standing-wave pattern is such that the x and y components of1 electric field are zero at z = 0. Consequently, a short circuit can be placed at z = 0 as well, as in Fig. 7.15. The resultant structure is a rectangular cavity. This description of a cavity also shows that the field solution may be obtained directly from the corresponding waveguide solutions. For the nm th TE or TM mode, the propagation constant is given by <7-39a> *•-*«-(T) - I T ) where k() = 2-f0/c. We require p„,„d = lir, where I is an integer in order for the cavity to be a multiple of a half guide wavelength long. Thus, when d is specified, 0nm is given by fc„--? / = 1 , 2 , . . . (7.396) However, this relation is consistent with the earlier one only for certain discrete values of ktl. Only if k„ = k„m/, where k„ml is given by i/a rmr (7.40) kn ml ~b~ will (7.39a) and (7.39b) be satisfied. These particular values of k 0 give the resonant frequencies of the cavity; i.e., ck,' n I nm/ 2TT ml = c I m 2d 2b 2a) (7.41) where c is the velocity of light. Note that there is a triply infinite number of resonant frequencies corresponding to different field distributions. Also note that there is more than one field solution for a given resonant, frequency since (7.41) holds for both TE and TM modes. In addition, because of a lack of a preferential coordinate, in the case of a rectangular cavity, field solutions corresponding to TE and TM modes with respect to the x and y 502 FOUNDATIONS FOR MICROWAVE ENGINEERING axes could also be constructed, and these would have the sam frequency. However, these latter modes are just a linear comh' t ? S ° n a nt TE and a TM mode with respect to the z axis and therefore do not***011 0f a 0l r e r a new solution. P ese n t To illustrate the method of solution far the fields in a r cavity and the evaluation of the unloaded Q, the TE 1 0 J mode is ^^ detail. If b < a < d, this will be the mode with the lowest r e s o ^ ^ ' n quency and corresponds to the TE I(I mode in a rectangular wavem.^ 11 fre" mode subscripts nml pertain to the number of half-sinusoid variat* the standing-wave pattern along the x, y, and z axes, respectively 1 (3.206), the field solution for a T E ] 0 mode is B. = (A+e~Mu* + A V ^ - ' J c o s — a Hx = " — ^ - ( A ' e-JP* - A-e-*» z )sin — a 77- E = -j k(,Z,,a ,„ 77 x (A*e--"*">* + A - e r f » * ) s i n — 7T a where A + and A" are amphtude constants for the modes propagating in the +z and -z directions, respectively. To make Ev vanish at z = 0, d. we must choose A~= —A' so that A+e-rt™* + A-eJP">* = - 2 j A H sin pl0z and also choose liJ0 = ir/d. The corresponding value of k0 is thus TT K0 ^ 101 a f1 17 2 \ + — 11 \d t/a 1/2 = p L0 (7.42) and this determines the resonant frequency. The solution for the fiel 8 now be expressed as follows: ~2A^kmZtia TTX _ TT2 — sin — sin —TT a a 2jA *~a TTX TTZ Hx = — sin — cos —r d a d 43a I £,= TTX TTZ H; = ~2jA ' cos — sin — - 43*) (7.43c) 1 to the electa Note that the magnetic field is ±90° out of phase relative t^ ^ ^fgge field. This is always the case in a lossless cavity and correspo j ^ j g g s and current being ±90° out of phase with each other w circuit. KLECTROMAGNETIC RESONATORS 503 At resonance the time-average electric and magnetic energy stored in the cavity are equal. The average stored electric energy is given by En t" to tcl We = -±f / E,E*dxdydz 4 JQ -'O A) 6f 4-2 a 3 MA? 0 1 Z ( ?|AT (7.44) The reader may readily verify that Wm = ^ f f jd(HxH; + ff.Hf) dxdydz = Wt (7.45) 4 -"o •'o Jo In order to determine the cavity Q, the losses caused by the finite conductivity of the cavity walls must be evaluated. For small losses the surface currents are essentially those associated with the loss-free field solutions (7.43). Thus the surface current is given by J, = n X H where n is a unit normal to the surface and directed into the cavity. Hence the power loss in the walls is given by p =F i ¥f ^ J •J*dS = ^f! •'walls -^ IH -» |2 dS < 7-46> "walls where R m = 1/&SS is the resistive part of the surface impedance exhibited by the conducting wall having a conductivity a and for which the skin depth is 8 S = (2/wju.o-)1'2. In (7.46) H l a n is the tangential magnetic field at the surface of the cavity walls. Substituting from (7.43) into (7.46), a straightforward calculation gives Pl = \A-TRm 2a3b + 2d3b + ad:i + da3 -2 (7.47) With the use of (7.3), we find that the Q is given by Q = wW 2a>We <aJfef0,Z|o3d36e0 P, P, 2TT2R,„(2a3b + 2d*b + a3d + d3a) (ft l 0 1 acO J 6Z n 2i72R„,(2a3b + 2d3b + a3d + d3a) (7.48) upon replacing wZ D € 0 = w/^o^o by km. As a typical example, consider a copper cavity (a = 5.8 x 10' S / m ) with c = 6 = rf = 3 c m . The resonant frequency is found to be 7,070 MHz. The surface resistance R,„ = 0.022 fi, and the Q comes out equal to 12,700. The damping factor 8 = &/2Q equals 1.74 x 10 6 N p / s , or about 2.5 x 10 "4 Np/cycle of oscillation. Because of the high value of Q, 4,000 504 FOUNDATIONS FOB MICROWAVE ENGINEERING cycles of free oscillation can occur before the amplitude has rW aecre factor e->. a*Jd by a If the cavity is filled with a lossy dielectric material with oe e = e' -je", the time-average electric energy stored in the cavitv n ^ l t t ' v i t >' y Vol given by ume j s We=€-fm\2dV 4 Jv ('.49cj The lossy dielectric has an effective conductivity we", and hence th e 0 6 loss in the dielectric is P** P -=UJ-E^y=^/V'E'2^ (7.496) If Qd is the Q when a lossy dielectric is present but the walls are perfectly conducting, then 2 W W, Qd = e" Id (7.50) When wall losses are also present, the net Q is Q' and given by where Q is the quality factor when lossy walls are present and e" = 0. Q is given by (7.48), with e0 replaced by e'. Also note that, for a cavity filled with dielectric, the resonant frequency is given by fnml \l i e o nmt (7.52) ITT Cylindrical Cavity The cylindrical cavity is a section of circular waveguide of l e n g t j ! , . ^ radius a, with short circuiting plates at each end, as in Fig. 7.16. ^^ of cavity is very commonly used in practice for wavemeters to ^^ frequency, because of the high Q and wide frequency range of ope F I G U R E 7.16 Cylindrical cavity. ELECTROMAGNETIC RESONATORS 505 provides. A high Q is necessary in a frequency meter in order to obtain a high degree of resolution or accuracy in the measurement of an unknown frequency. When the cavity is tuned to the frequency of the unknown source, it absorbs a maximum of power from the input line. A crystal detector coupled to the input line can be used to indicate this dip in power at resonance. The fields in the cylindrical cavity may be determined from the corresponding waveguide solutions. The lowest resonant mode is the T E U 1 mode corresponding to the dominant T E l ] mode in the circular guide. This mode is examined in detail below. Using the field solutions tabulated in Table 3.6 and combining a forward and backward propagating T E U mode, we have Pnr a H, = J , (7.536) sinct>(A*e -"*"* - A"e-" , " z ) (7.53c) A.ZoO - J , ( — )sin d>{ A • e"•"'"* + A V * * * ) (P'u) (7.53d) J', P'u JPna2 H,= (Pn) E = iP\ir E* = a P'nr r Pn cos^{A*e-JP'" (7.53a) -A'eJ""z) H. £ cos<b(A+e-Jli»* + A~ eJ,i"z a (7.53e) v a I (7.53/-) =0 where p'u = 1.841. To make E r and E^ vanish at z = 0, d, we must choose A~= -A*. The factor A^e'-"1"' + A'e,is,': becomes -2jA' smfinz. /?,, must be chosen equal to -rr/d to make sin f$ud vanish. The resonant frequency is determined from the relation *o = *<l? ll/2 1/2 V 2^/-lu a (7.54) To find the Q of the T E U , mode, the time-average energy W stored in the cavity and the power loss in the wails must be calculated. The general expressions are I T - MIT. P< = 2 J walls $ f rf(\Ef + lEf)rd4> drdz mlan\2ds These integrals may be evaluated by substituting the expressions given 506 FOUNDATIONS FOR MICROWAVE ENGINEERING earlier for the fields. The final result obtained for the Q i s where I = n = m = 1 for the T E J H mode. For the T E „ m / m o c j e ,. also given by (7.55), with the appropriate values of n, m, / 'and • inserted. Note that all terms on the right are independent of frequen P"'n hence the Q varies as \0/S, for any given cavity and thus decreases*0 An analysis similar to the above may be carried out for the oth E „ , „ , and TM,,,,,, modes to obtain expressions for the fields. For the T E „ m , modes it is necessary only to replace cos <!> and sin tf> by cos mj, and sin n<b, J, by J „ , p'n by p'nm, and / 3 n by Irr/d in (7.53). Of particular interest is the T E 0 I I mode for wavemeters because its Q is two to three times that of the T E n , mode. Another advantage of the T E 0 U mode is that Hit = 0, and hence there are no axial currents. This means that the end plate of the cavity can be free to move to adjust the cavity length d for tuning purposes without introducing any significant loss since no currents flow across the gap; i.e., the gap between the circular end plate and the cylinder wall is parallel too the current flow lines. However, the TE 0 U mode is not the dominant mode; so care must be exercised to choose a coupling scheme that does not excite the other possible modes that could resonate within the frequency tuning range of the cavity. T To determine what modes can resonate for a given value of 2a/d an< frequency, it is convenient to construct a mode chart. For any given mcx we have knmI ' nml 21'/2 IT + C = 2l7 ~d ! or 2 (2a/,,,,,,) = cx„ /2c (7.56) + — _,. 7 i7 gives" where xllm = p„,„ for TE modes and p for TM modes. Figure i. ^ ^ plot of (2a/;,„,) 21 against (2a/d)2 for several modes and c o n s t l t u u ^ n c V and chart. Examination tion _of this chart shows over what range of lTeq e'^ t *o 2a/d only a single mode can resonate (in the case of degenera, betweeo 2 modes resonate at the same frequency). For example, f o r ( 2 a / u e nO' r8°t and 3, only the TE 0 1 , and T M , , , modes can resonate in the; tree.i - ^ corresponding to (2af)2 between 16.3 x 10 s and 20.4 x 10 J^^d, t dashed rectangle in Fig. 7.17). If the T M U 1 mode is n ° l ELECTHOMACNETJC HKSONATORS 0 2 , 4 507 6 F I G U R E 7.17 Mode chart for a circular cylindrical cavity. frequency range at least can be tuned without spurious resonances from other modes occurring. For the TM modes the Q can be evaluated to give «£- 2-(l + 2a/d) 2TT(1 (7.57) + a/d) Figure 7.18 gives a plot of Q8s/\0 against 2a/d for several modes. Note the considerably higher value oC Q obtained for the T E 0 U mode relative to that for the T E i n mode. Optimum Q occurs for d = 2a. At A 0 = 3 cm, S s /A 0 = 2.2 X 10 " 5 , and hence, from Fig. 7.18, it is apparent that typical values of Q range from 10,000 to 40,000 or more. At A 0 = 10 cm, the corresponding values of Q would be \ / 1 0 / 3 greater. 508 FOUNDATIONS FOR MICROWAVE ENGINEERING FIGURE 7.18 Q for circular-cylindrical-cavil v modes. 7.5 DIELECTRIC RESONATORS A dielectric resonator consists of a high dielectric constant cylinder, sphere. or parallelopiped as shown in Fig. 7.19. These structures will support resonant modes similar to those in metallic cavities but with the field extending beyond the boundaries into the surrounding air region. The linear dimension of the resonator is of order A 0 / )fer, where A„ is the free-space wavelength and e r is the dielectric constant. For e r = 100 the resonator is approximately one-tenth the size of a metallic cavity, and fc this reason it is a very attractive alternative to other types of resonators; use in microstrip circuits such as filters and oscillator circuits. Since required size of a dielectric resonator in order to resonate is very s relative to the free-space wavelength, the electromagnetic field outside t « resonator is quasistatic and very little radiation takes place. Thus d: resonators will exhibit a relatively high Q if a low-loss high d constant material is used. Typical values for the unloaded W o ^ electric resonator range from around 100 to several hundredonator is enclosed in a shielding box so that there is no energy ot>s ^ ^ radiation, an unloaded Q approaching the intrinsic Q of the ma obtained. W (c) FIGURE 7.19 , dieU Basic geometrical shapes usea ^ ^n onators. ( a ) Cylinder, ib) spher* ipecf. F.I KCTROMAONETICRBSOKATOBS 509 »! F I G U R E 7.20 A cylindrical dielectric resonator coupled to a mierostrip line. The resonant frequency of a dielectric resonator depends on the dimensions of the resonator and its dielectric constant. Since these parameters change with the temperature, it is necessary to use materials that have low coefficients of expansion and relatively temperature-independent dielectric constants. In recent years a number of ceramic compounds have been developed that are suitable for use in dielectric resonators and can provide a stability of a few parts per million per degree change in temperature. One such material is barium tetratitanate which has a dielectric constant of around 40 and a loss tangent of about 0.0005. The intrinsic Q of the material is the reciprocal of the loss tangent and equals 2.000 for barium tetratitanate. The dielectric cylinder with a height H approximately equal to the radius R and mounted on the substrate as shown in Fig, 7.20 is a popular choice for a dielectric resonator. The coupling between the resonator and a mierostrip line is adjusted by adjusting the spacing S. The dominant resonant mode is a TE mode having an azimuthal electric field E,,, that is independent of the angle <!>. The field has a dependence on the axial coordinate z but does not change with z rapidly enough to exhibit a standing-wave pattern along z. For this reason the mode is designated as the T E 0 U mode with the subscript 8 signifying less than one-half cyclic variation with z. Dielectric resonators in the shape of a cylinder or a parallelopiped do not have simple analytic solutions for the various modes of oscillation. On the other hand, there are simple solutions for the fields in a spherical resonator and also for a hemisphere mounted on a ground plane. The hemisphere on a ground plane shown in Fig. 7.21 is quite similar to the dielectric cylinder, so that an understanding of the field solution for the hemispherical resonator will provide good insight into the properties of a r„H -i • Oround plane F I G U R E 7.21 The hemispherical dielectric resonator on a ground r . plane. 510 FOUNDATIONS FOR MiCKOWAVE ENGINEERING cylindrical resonator as well. For this reason we will examin for TE modes in a hemispherical resonator. Solutions to Maxwell's equations in spherical coordinates rated into TE and TM modes with respect to the radial coordi ^ electric field for TE modes is given by Sepa - E = V X arknl> = M <'.0HQ) and the electric field for TM modes is given by E = - V x V X M r *=N (?5g6) where (//(r, 9, <j>) is a solution of the scalar Helmholtz equation in soh " coordinates, i.e., (V 2 + fe2).// = 0 (7.58c) The M and N functions have the useful property that V X M = kN (7.59a) V x N = kM (7.596) Thus, if the electric field is described by an M function, the magnetic field given by —jiofj.0H = ? X E will be described by an N function. For circularly symmetric fields, the solutions for ilt(r,6) are *„(r,0) =z„(kr)Pn(co$e) where z„(kr) is a spherical Bessel function and P„(cos0) is the t Legendre polynomial. The spherical Bessel functions are related to tl cylinder Bessel functions Zn + l/2{kr) as follows: *nikr) = IT ^ — Zn + l/2{kr) When the origin is included, the function 7T 2kr '••i/aW iherica1 is used. In order to represent outward propagating waves, Hankel function TT h*„(kr)~ 2kr HUU2(kr) is used. The Bessel functions of order n + \ and hence the functions can be expressed in terms of simple sine and cosin sp al**5 fo^cti0r ELECTROMAGNETIC RESONATORS 511 example, sin x Jo(x) (7.61a) = sin x cos x (7.616) M*) - (7.61c) / * - • * * - J J C (7.61c/) 2n + 1 (7.61*) The first few Legendre polynomials are P(,(cos8) = 1 (7.62a) (7.626) P,(cos0) = cosfl 2 P 2 (cos 6) = | cos 0 - \ = f cos 20 + j (7.62c) For circularly symmetric modes, we can use (7.60) in (7.58) to obtain M„ = -k-^zn(kr)*<, (7.63a) s(s-fl) 1 a7>„ d(krz') N „ - — — P..„(*r)., + 7 ^ - 5 ^ . . (MM We will use the above solutions to construct the solution for the dominant TE 0 1 2 mode for the hemispherical resonator. The electric field has an E,,, component only. The ground plane is located at 0 *= w/2, so that E,,, must vanish at 6 ~ TT/2. The Mj mode does not vanish at 6 = TT/2 but the M 2 mode does; so the first mode that can exist for the hemispherical resonator on a ground plane is the T E 0 ) 2 mode. A suitable solution for the electric field is dPo E= -A,/e—j2(fer)arf, dp 2 r<a » (7.64a) E = -A0kn-—h\(knr)n,„ r> a (7.646) dO where k = ^ ( l inside the sphere and A x and A 2 are amplitude constants. The solution for the magnetic field is given by "oA> «o^-o "Q^O 512 FOUNDATIONS FOR MICROWAVE ENGINEERING and hence is given by JBk H = Ak, -Z— -P2j2(kr)ar 0 0r jk A, K dP2d(krj,,(kr)) k0Zor de l d(kr) e r <a (7.65a) *« r>a (7.656) = A2~P2h\(kar)ar Z^r j + 2 dP2 Z0r dO ' d(k0rh\(kQr)) d(k0r) At r = a the tangential fields £,,, and He must be continuous across the boundary. Thus we must have =A2koh22(k0a) Axkj2(ka) d{krj,,(kr)) Alk ' = A2k0 d(kr) (7.66a) d(karh*(k0r)) (7.666) d(k0r) All the other factors in the expressions for the fields are the same and cancel. By using (7.61e) we find that j2(kr) = h\(kQr) = k3r3 - — sin kr - - J2 - J cos kr kr kr (7.67a) — (7.676) 37 klr3 k2Dr2 ft„r, \e-i*tf With the aid of these relations, the boundary conditions can be expressed as A xk ¥?-Ja)Sinka-kVC°Ska A2k„ 3 A xk ka 8j k%a3 k%a2 = 0 (7-680) k0a 6 a+ -^r l^"T o s i f l 6j - _ A- A2ku k*a3 + 6 Sj Ka2 k0 a -We ,-J*oa .68b) = 0 (7 ad In order for this homogeneous set of equations to have a solution^ ^ lgt0 A 2 ,the determinant must vanish. By equating the determin ^ kg c» we obtain the eigenvalue equation for the resonant wave n u ELECTROMAGNETIC RESONATORS 513 k = fco-v^- T h ' s equation is {r\2 +jr)x - l ) s i n x - [(ij 2 - I ) * +jvx2 - £x 3 ]cosx = 0 (7.69) 2 where we have let er = 17 and k0a = ka/-r\ = x/77. Since the roots of (7.69) are complex, the solution for x is not straightforward to carry out. An iterative method can be used to advantage. Since the dielectric constant is normally quite large, the approximate solution of (7.69) is obtained by keeping only the terms multiplied by r)z. Thus we have sin x - x cos x = 0 which has the solution x = 4.4934 ~ x0. If we denote the eigenvalue equation (7.69) as f(x), then a Taylor series expansion about the point x 0 gives df (*-*o) We can equate fix) to zero and then get the following improved solution A-t 0 ) df/dx, = *o + (jvx'o - l*o ~ *o)cos x0 - (jyx0 - l)sin xn (tjXo +JVXZ +j-q - | * | - jc 0 )sinx 0 - (717*0 _ *o)cosa: 0 (7.70) This second approximation can be called x 0 and used in (7.70) to obtain a third-order approximation. However, the second-order solution is usually quite accurate as the following example illustrates. If the dielectric constant e r = 49, then the second-order solution obtained using (7.70) is ka = x = x0 - 0.02619 4 J0.019497 « 4.4672 + jO.019497. The complex resonant wave number is given by kt) = x/r\a. Since we have chosen n as real, the l'esonant frequency is complex only because the oscillating resonator loses energy by radiation. The radiation Q r is given by Re ka * - iteS <7 71) ' and for our example equals 114.56. If the dielectric loss tangent were no greater than 0.001, the dielectric losses would reduce the Q by about 10 percent only; so the radiation loss is the major loss. By enclosing the resonator in a shielding box, a much higher unloaded Q can be obtained. However, when a shielding box is used, care must be exercised to avoid exciting cavity modes in the box. For a dielectric constant of 49 and a resonant frequency of 10 GHz, the r a d i u s of t h e h e m i s p h e r i c a l r e s o n a t o r is a = 4.4Q72/-qk0 = ( 4 . 4 6 7 2 / 2 T 7 T 7 ) A 0 = 0.1016A 0 or 0.3047 cm. Thus the resonator is very compact. The parameter k0a = 0.638 is quite small, so that the electromag- 514 FOUNDATIONS FOR MICROWAVE ENGINEERING netic field outside the resonator is approximately quasistatic Th A 2 can be expressed in terms of A, using (7.66a ) and approxiny t* °° nstai U by 4.4672, that is, neglecting the imaginary part. Outside the re "^ *° * * electric and magnetic fields are given by ^nator the E .k = - r A 2 £ o s i n 2 0 Hr =j-A2kl)Yl,(3cos20 *2r 3 + >~jkgr 2„2 k^r 1) Hll=j-A.,k0Yl)sm26 - a) k*0r< ,3„3 <,2„2 ,3-3 /,2 2 + • -JK . ~-/*qr (7.726, (7.72c) The terms in £,,, and H H that vary like l/knr represent the radiation field and account for the power loss due to radiation. If a cover plate at a height c above the ground plane is installed, the radiation field will be suppressed provided c < A 0 /2. Since E. must now vanish at z = 0 and z = c, the field outside the resonator can be expressed in terms of cylindrical waves. The lowest-order wave will have the form irz -H$\P c 17 where p is the radial coordinate T/X2 + y 2 and H% is the cylinder Hankel function. When p is large -,1/2 m (7 , 77P]/kf- jp\fH-i-/c)2 (TT/CY This is a nonpropagating wave when k0 < v/c or c < A 0 /2. The cover will not have much influence on the field close to the resonator s i n c e , terms that vary like \/k\rl and higher inverse powers of knr will be sir at the location of the cover plate when c is several times larger Thus when a cover plate is used, we can approximate the fields by • .g dropping the radiation terms. When the radiation from the res > ^ suppressed, the unloaded Q will be much larger since the l o s s e s on fa those arising from the loss in the dielectric and from the curr nat „r. ground plane that are associated with the resonating mode of t e r ( ) S t r ip The magnetic f i e l d from the resonator will link the nearbj ^ ^ ^ n line as illustrated in Fig. 7.22. As a result, an ostillating^reso^ excite propagating waves on the microstrip line. These wave. o r The flow of power and produce the external loading on the V . c0U pled'* resonator can be represented by a parallel resonant circuit t a ^Q n. the microstrip line by means of an ideal transformer with a jjP ^ „,ak as shown in Fig. 7.23. The resistor R must have a value ELECTROMAGNETIC RESONATORS R M I I L npprc | II 1 515 r 1 \JtM, ^nnp n:i zc z, ~. FIGURE 7.22 Magnetic coupling between a dielectric resonator and a microstrip line. FIGURE 7.23 Equivalent circuit for a dielectric resonator coupled !(• a microstrip line. 10CR equal to the unloaded Q, and L must have a value such that w2LC = 1, at the resonant frequency. These two conditions do not establish the impedance level or absolute value of R. We can choose R arbitrarily as long as the turns ratio n:l is adjusted to give the correct coupling between the resonator and the microstrip line. At resonance a series resistance n2R is coupled into the transmission line, and this makes the input reflection coefficient equal to (rc2i?+Zt)-Z, = (n2R + Ze) + Ze n2R = n2R + 2ZC when the output line is terminated in a matched load. By measuring T the parameter n2R ~z7 = 2T i -r is readily determined and this provides the additional information that is needed to specify the equivalent circuit. In some cases the turns ratio can also be evaluated analytically but in general it is preferable to measure it.t A number of papers have provided analysis of the cylindrical dielectric resonator. In Fig. 7.24 we show the dominant TE 012 -mode eigenvalue kR as a function of the ratio of the resonator height H to the radius R for a cylindrical resonator mounted on a ground plane. When the dielectric constant is greater than 20, the eigenvalue kR is essentially independent of the dielectric constant. This independence was also found for the hemi- +Y. Komatsu and Y. Murakami, Coupling Coefficient Between Microstrip Line and dielectric Resonator, IEEE Trans., vol. MTT-31, pp. 34-40, 1983. 516 FOUNDATIONS FOB MICROWAVE ENGINEERING 5.5 r^v* 5 h 4.5 kR 2R 4 /<r 3.5 =100 t , = 20 / 3 0.5 1 1.5 H/R FIGURE 7.24 The eigenvalue kR for the TE „ modcylindrical resonator on a groundpla,,. , Jf on Fig. 3 in R. E. Collin andD^ff Boundary Element Method for Dielectric R onators and Waveguides, Rod. Sci voi To pp. 1155-1167, 1987.) spherical resonator since, to a first approximation, (7.69) gave ka = 4.4934 independent of er. For a given cylindrical resonator, there exists an equivalent hemispherical resonator that will have the same resonant frequency. If we let kR = xc for the dielectric resonator and let ka = xs for the hemispherical resonator, then since k is the same we must have 1 = ^1 R xc From Fig. 7.24 we find that a dielectric resonator with radius R equal to a must have a height H equal to 0.7i? in order to have the same resonant frequency as a hemispherical resonator with the same radius. The inset in Fig. 7.24 shows the two equivalent resonator cross sections and clearly they are very similar. When a dielectric resonator is mounted on the substrate materia instead of on a ground plane, the resonant frequency will change by a srnai amount. The electromagnetic f i e l d around the resonator will also c h ^ but will remain similar to that for a resonator on a ground plane. quasistatic field outside the resonator is similar to that of a n»8 quadrapole. For a cylindrical or spherical resonator in free spai Cover plate Tuning disk Microstrip line Resonator FIGURE 7.25 . o l e C tric «***& A method for tuning « **£?&**» + The position of the metal du» le end face of the resonator 1- ELECTROMAGNETIC RESONATORS 517 dominant mode has a local field similar to that of an oscillating magnetic dipole. The image of a magnetic dipole in a ground plane, along with the original magnetic dipote, creates a magnetic quadrapole. Thus, when a resonator is mounted close to a ground plane, there is a significant change in the field around the resonator. The resonant frequency of a dielectric resonator in a shielded enclosure can be changed by a small amount by varying the position of a small metal disk relative to the end face of the resonator as shown in Fig. 7.25. When dielectric resonators are used as part of a filter, some form of tuning for adjusting the resonant frequency is normally required. 7.6 EQUIVALENT CIRCUITS FOR CAVITIES In this section the equivalent circuits of cavities coupled to transmission lines and waveguides are examined. A complete treatment is not given because of the complexity of the problem in general (see Sec. 7.9 for further details). Instead, a number of specific examples are examined in order to indicate the type of results that are obtained. Aperture-Coupled Cavity As an example of an aperture-coupled cavity, consider the rectangular cavity coupled to a rectangular guide by means of a small centered circular hole in the transverse wall at z = 0, as illustrated in Fig. 7.26. As indicated earlier, a small circular aperture in a transverse wall behaves as a shunt inductive susceptance with a normalized value given by (5.41) as 3ob (7.73) epr0 b is the guide height, r 0 is the aperture radius, the propagation constant for the T E I 0 mode. the aperture-coupled cavity is thus a shortof length d shunted by a normalized suscepBL=TRP where a is the guide width, and p = [kg - (Tr/af]l/2 is The equivalent circuit of circuited transmission line tance B,. To analyze this coupled cavity, we shall assume initially that there are no losses. The modifications required when small losses are present are given later. The cavity will exhibit an infinite number of resonances, and the input impedance Z m will have an infinite number of zeros interlaced by an infinite number of poles, this being the general behavior of a distributedparameter one-port microwave network. If we are interested in a resonance corresponding to a high value of Zin, infinite in the case of no loss, we should examine the nature of 2 i n in the vicinity of one of its poles. This case corresponds to a parallel resonant circuit, 518 FOUNDATIONS FOR MICROWAVE ENGINEERING Zc=\ ' J*L F I G U R E 7.26 ( a ) Aperture-coupied ' - cavity; ( W e q u S v ^ T S f mission-line circuit. ^=0 IA) The input impedance is given by the parallel impedance of iX J j tan pd and is '• ^ -XL tan pd 2;„ = — jXL + j tan pd (7.74) where jX, = (-jBL) '. The jmtiresonances occur when the denominator vanishes, i.e., at the poles of Zin, or when Xh = - tan pd = 8r20d (7.75) Zabd To solve this equation for the values of P that yield resonances, graphical methods are convenient. By plotting the two sides of (7.75) as functions of pd, the points of intersection yield the solutions for pd. When p is known, the resonant frequency may be found from the relation w 2^ =f= c /32 + I — 2^ a 11/2 (7.76) This graphical solution is illustrated in Fig. 7.27. Note that there are an infinite number of solutions. Normally, XL is very small, so that the valui of pd for the fundamental mode is approximately equal to TT. The higherorder modes occur for pd = P„d = (n - jfrr when n, an integer, is 1 FIGURE 7.27 frequenC °' Graphical solution for resonant n» ture-coupled cavity. F.l.KCTHOMAONETIC RESONATORS 519 The value of (i for the first mode will be denoted by /3,. and the corresponding value of w by w t, as determined by putting 0 = 0, in (7.76). An infinite number of equivalent lumped-parameter networks can be used to represent Zin in the vicinity of o>,. Usually, the simplest possible network is used. This equivalent network must be chosen so that its input impedance Z equals Z i n at w,. Likewise, for small variations AOJ about «>,, the two impedances must be equal. A general procedure for specifying this equivalence is obtained by expanding the impedance functions in a power series in w — w, = Aa> about w, and equating these series term by term. Since Z i n has a pole at w ~ u>v, the Taylor expansion cannot be applied directly to Z i n . However, it may be applied to (w — to,)Z m (w) to give (w - w,)Z i n (w) = lim (to - w,)Z i n (oi)+ — - ( « - w,)Z iri *>-»*>, l u to ; w - to,) d< 2 (u> ~ w,)Z, n + 2~ d-iio We now obtain + 3 ~ l w ~ "Mz„i Z jn (w) = to — w i + ••• (7.77) aio A similar expansion of Z gives (note that Z must have a pole at w ( also) Urn. Z(<o) = (to - w , ) Z ( w ) to — to, d Oto + ••• (7.78) Expansions of this type are called Laurent series expansions, and the coefficient of the <o> — to,)~l term is called the residue at the pole «>,. These two series must be made equal term for term up to the highest power in to — to, = Aw required to represent Z i n with sufficient accuracy in the frequency range of interest. For a microwave cavity, the Q is usually so high, and the frequency range d w / w , of interest is approximately the range between the two points where )Zm\ equals 0.707 of its maximum. This latter fractional frequency band is equal to 1/Q, and hence Aw/w is so small that normally only the first term in the expansion (7.77) would be required to represent Z in with sufficient accuracy in the vicinity of to,. In the present ease a simple parallel LC circuit would be sufficient to represent Z in (w) around «>,. In order to specify the values of L and C, we must evaluate the first terms in (7.77) and (7.78). For the LC circuit we have j(oL Z = 1 - o>2LC 520 FOUNDATIONS FOB MICROWAVE ENGINEERING We now choose wf LC = 1 to produce a pole at u>1 for Z. Henc ui'i - to2 i- , >v ~jao)\L hm (io - o>x)Z = <o + a), and -Jo>lL <a, Thus we have Z(co) = 2(cu — oij) for a> near Wj. To evaluate the behavior of Zin near w,, we can place w equal to u in the numerator in (7.74). The denominator is first expanded in a Taylor series in p about fi, to give XL((i) + tan pd =X,.(/3 X ) + tan/3jrf Pi dXL d tan pd dp dp iP~P>) ds<x2pld\{p-px) + since XL(pt) = X L 1 = - t a n / ^ d and dXjdp = (l/p)XL. Next we expand p in terms of o> about w, to give d£ P=Px + dw 'I If we denote dp/dot at « t by p\, we see t h a t Z-m can be expressed as - JXLl ten Pid i n _ _____ 2 [XL1+^,d(l + tan M)](/3'i/^.)(t^r^ upon replacing sec 2 pxd by 1 + t a n 2 /3,d. Replacing tan pxd by -A__ T gives « . NormaUy, X_._ « 1, and since /3,d ~_ir, we can make further app* tions to obtain (we shall verify that XLl <s 1 later) y2 in J p\d(o> (7.80) - ^ ELECTROMAGNETIC RESONATORS 521 Comparison with (7.79) shows that we must choose °r <o\L XfA 2 p\d L = 2Xf, ^l < 7 " 81 ) The capacitance C is determined by the condition u>\LC = 1 given earlier. Up to this point we have neglected the losses in the cavity. For a high-Q cavity these may be accounted for simply by replacing the resonant frequency w, by a complex resonant frequency w,(l +j/2Q), as indicated in Sec. 7.1. That is, the natural response of a lossy cavity is proportional to e -«+j«i* f aa^ n o t t 0 e > , / f w n e r e g = Wl/2Q. This is equivalent to having a complex resonant frequency Wjd + j/2Q). The field in the cavity is, apart from some local fringing because of the presence of the aperture, a TEi ()1 mode. Its Q was evaluated in Sec. 7.4, and is given by (7.48). For the lossy case we then have /3,d(w - OJ, - jw-i/2Q) At resonance (« = oi,), we now obtain a pure resistive impedance R m given by 2XhQ Rm-Zin = —^j (7.83) If we want the cavity to be matched to the waveguide at resonance, we must choose the aperture reactance X u so that Rin = 1; that is, This matched condition is referred to as critical coupling. If Rin is greater than the characteristic impedance of the input line (unity in the case_j3f normalized impedances), the cavity is said to be overcoupled, whereas if R,n is smaller, the cavity is undercoupled. If R m is the normalized input resistance at resonance for a parallel resonant cavity, then R m is defined as the coupling parameter K. In the case of series resonance, the coupling parameter equals the input normalized conductance at resonance. For the rectangular cavity discussed in Sec. 7.4, with a = b = d = 3 cm, we found f, = 7,070 MHz and Q = 12,700. For this cavity p\ = — ^ » 4.7 x 1 0 - l l s/cm and (7.84) gives XIA ~ 0.0157 for critical coupling. The corresponding 522 FOUNDATIONS FOR MICROWAVE ENGINEERING f&. y?=i ZZC FIGURE 7.28 Equivalent circuit f0r coupled cavity. ane, aperture radius r 0 from (7.73) is found to be 0.37 cm. Note that X so that our earlier approximation in neglecting XLI compared with u -ft justified. Also note that a solution of (7.84) for the required value of ? ^ 'S give critical coupling must, in general, be carried out s i m u l t a n e o u s l y ^ the solution of (7.75) for the resonant frequency a,. However, for a hiur, cavity, to, may be approximated with negligible error by the freque corresponding to [id = -rr in (7.84). This was done in the above calculation5' _ For the lossy cavity the equivalent circuit must include a resistance ftiri in parallel with L and C as illustrated in Fig. 7.28a. The reader mav readily verify that the input impedance Z now becomes, for w near <u z= -j iJ\L 2(w- w, [7.85) -JW./2Q) where Q = Rin/u>1L. Since the cavity is coupled to an input waveguide, the cavity terminals are loaded by an impedance equal to the impedance seen looking toward the generator from the aperture plane. If the generator is matched to the waveguide, a normalized resistance of unity is connected across the cavity terminals, as in Fig. 7.286. The external Q c is given by (Sec. 7.1) Qe = (7.86a) o>xL and the loaded Q L by -l 1 \Q + Air (7.866) (1+R^L Q. The loaded and unloaded quality factors are related as follows: 1 1 Q or l (1+ Rinhvl* _= W. jrL . + ^ 1 = 1 ( 1 x X ) Q R, *in Q = (7.87) (1+K)QL In general, the coupling parameter K may be defined as -I ( 7.8«) ELECTROMAGNETIC RESONATORS 523 For a parallel resonant circuit this is seen to give Bin oixL Vjl' 1 which agrees with the earlier definition, ^likewise, for a series resonant circuit with normalized jnput conductance G,„, the unloaded and loaded Q's are given by Q = IO^G^ Q(. = w,L. and hence K = Gm = Q/QL. again. The coupling parameter is a measure of the degree of coupling between the cavity and the input waveguide or transmission line. The external Q, Qe, is sometimes called the radiation Q. The reason for this is that the cavity may be considered to radiate power through the aperture into the input waveguide. This power loss by radiation through the aperture is equal to the power lost in the normalized unit resistance connected across the resonator terminals in the equivalent circuit illustrated in Fig. 7.286. For the rectangular cavity under discussion, the next-higher frequency at which a resonance occurs corresponds to a series-type resonance at which \Zm\ is a minimum. In the loss-free case, Zin = 0, and from (7.74) this is seen to correspond to tan jid = 0, or fid = jr. At a series resonance, F i n has a pole, and consequently an analysis similar to that presented for Z m is applied to Yin. It is readily found that in the vicinity of m = <u2> where u>2 is the value of w that makes )3d = ir, 1 ,n= J ~~ p.A<o-«,2-jco.z/2QJ In this case the aperture has no effect, as would be expected, because the standing-wave pattern in the cavity is now such that the transverse electric field is zero at the aperture plane. The input admittance near w.2 is just that of a short-circuited guide near a half guide wavelength long. This resonance is not of any practical interest since it corresponds to a very loosely coupled cavity. Cavity Figure 7.29 illustrates a cavity that is coupled to a coaxial line by means of a small loop. Since the loop is very small, the current in the loop can be considered to be constant. Any mode in the cavity that has a magnetic field FIGURE 7.29 Loop-coupled cavity. 524 FOUNDATIONS FOR MICROWAVE BNGINEEBOIC \M, >• .:., \M« „Cn '•R« F I G U R E 7.30 Equivalent circuit of loop-coupled cavity. with flux lines that thread through the loop will be coupled by the loop However, at any particular frequency to, only that mode which is resonant at this frequency will be excited with an appreciable amplitude. The fields excited in the cavity by the current / Mowing in the loop can be found by solving for a vector potential arising from the current I. From the vector potential the magnetic field, and hence the flux passing through the loop, may be found. For a unit current let the magnetic flux of the n t h mode that threads through the loop be i//„. This is then equal to the mutual inductance M„ between the coupling loop and the n th mode. Each mode presents an impedance equivalent to that of a series LCR circuit to the coupling loop. Thus a suitable equivalent circuit is an infinite number of series LCR circuits coupled by mutual inductance to the input coaxial line, as illustrated in Fig. 7.30. The input impedance is thus of the form w3M,?C„ Rq. ( 7 8 Zm = > L 0 +j £ iry^TJ-r Rn = 1 1 ~a> LnC„ -hja/L„2iB where L 0 is the self-inductance of the coupling loop. If we define < resonance frequencies w„ by io\LnC„ = 1 and the unloaded Q of the mode by Q„ = io„L„/Rnl we can rewrite (7.89) as * avX2 * (7.90) th term *TB If w = w„, then all terms in the series in (7.90) except the nm small. Thus, in the vicinity of the nth resonance, Z m =./<"-£ o + : L„(a>l-u2+ju>tan/Qn) ~J<->L0 -j ,7.91) o>'Ml 2L„(«.-wB -j<o„/2Qn) ELECTROMAGNETIC RESONATORS 525 F I G U R E 7.31 Illustration for Helmholtz's theorem. The equivalent circuit now reduces to a single LCR series circuit mutually coupled to the input line. For efficient excitation of a given mode, the loop should be located at a point where this mode provides a maximum flux linkage. The preceding results represent a formal solution to the loop-coupled cavity. In order actually to specify the circuit parameters, the boundary-value problem for the fields excited in a cavity by a given loop current must be solved. Also the Q's of the various cavity modes must be determined. For simple cavity shapes these calculations can be carried out with reasonable accuracy. However, they are too lengthy to include here. *7.7 F I E L D E X P A N S I O N I N A G E N E R A L CAVITY Considerable insight into the general properties of an electromagnetic cavity can be obtained by examining the problem of expanding an arbitrary electromagnetic field into a complete set of modes in a cavity of unspecified shape. Such general modal expansions are required in order to determine the fields excited in a cavity by an arbitrary source. These expansions are also required in the evaluation of the input impedance or admittance of cavities coupled to external transmission lines or waveguides. A fundamental theorem which is basic to general cavity theory is Helmholtz's theorem.t This theorem states that in a volume V bounded by a closed surface S as in Fig. 7.31 a general vector field P(x,y, z) is given by P(*,y,2) = V -V P(r') 4TTR V x dV + 6 , P(r') • n „ dS' 47r 4-R £ , V X P(r') , P(r') X n f — I dV + 6 ^ 7 ^ — dS' h 4-ff X 47ri? (7.92) where R = |r - r'|. Thus the volume sources for P are given by -V • P +R- '5. Collin, "Field Theory of Guided Waves," 2nd ed., Mathematical Appendix, IEEE Press, Pisc&taway, N. J.. 1991 526 FOITNDATIONS FOR MICROWAVE ENG1HEERING and V' X P, and the surface sources are given by P • n and P v 3nd only if, V • P = 0, n • P = 0, can P be derived entirely from the* suitable vector potential. Also, only if n X P = 0, V X P = Q °f a derived from the gradient of a scalar potential, as (7.92) shows IF V • P = 0 and V X P = 0 in V, then P is said to be a source-free fi il b ° t h In this case P can always be derived from the gradient of a scalar n t '"• ^' but this potential must be multivalued if n X P does not equal zero ' ° situations. This statement will be clarified later. In setting up a suitable set of modes in which to expand a vector fi P inside a given volume V, it is necessary to know the boundary condjf * that must be imposed on these modes in order that a unique set of m ri" S may be obtained. This uniqueness is needed so that when a solution for th field has been obtained we are assured of the uniqueness of that solutin Since the electromagnetic field satisfies Helmholtz's equation, we are concerned with obtaining a unique solution to this equation. Consider V2A„ + k'f,An = 0. We wish to determine the type of boundary conditions that must be imposed on A„ such that the solution to this equation is unique for a specific eigenvalue kn. Assume that a second solution B„ exists such that V 2 B„ + /? 2 B„ = 0 also. The difference solution C„ = A„ - B„ satisfies V 2 C„ + k'?,C„ = 0 also. We shall require of B„ the conditions V • B„ = V • A „ , V X B„ = V X A„ in V, so that both A„ and B„ have the same volume sources. Then V • C„ = V X C„ = 0 in V. Consider f c J* / | C J a W v = - J/ c „ - V 2 C „ d V v J j(Cn • VX V X C „ - C „ - W -C„)dV v Using the relations V • (C„ X V X C„) = IV X C „ | 2 - C„ • V X V X C„ V • (C„V • C „ ) = IV • C „ | 2 + C„ • VV • C„ and we obtain k2J\C„\2dV= Jy J (\VxC,fdV+ V [ |V-CJ 2 dV V ,c -^•(C,V-C, + C,XVXC,)* (7.93) The volume integrals on the right vanish, and to make ^ ' ' " j g va nishwhich implies C„ = 0, we must make the two surfaces " " ^ boU ndary This may be accomplished by making A„ and B„ satisly ELECTROMAGNETIC RESONATORS 527 conditions ?-A„ = n x A „ = 0 or on S n • A„ =n X V X A„ = 0 on S (7.94a) (7.946) and similarly for B„. These are the boundary conditions that will be used for the electric- and magnetic-type modes, respectively. Other possibilities are n • A„ = n X A„ = 0 on S or V • A„ = n X V X A„ = 0 on S. The imposition of these boundary conditions gives a unique solution. Note that two conditions must be specified for a vector function. Cavity Field Expansions in Terms of Short-Circuit Modes The short-circuit modes are those corresponding to the field solutions inside ideal (perfectly conducting walls), totally closed cavities. For these modes the electric field modes E„ satisfy the boundary condition n X E„ = 0 on S, and the magnetic field modes satisfy the boundary condition n • H„ = 0 on S. In addition, as (7.93) and (7.94) show, we must impose the additional boundary conditions V • E„ = 0 and n X V X H„ = 0 on S in order to obtain unique solutions. Sometimes it is convenient to consider a cavity as having a surface part of which is perfectly conducting and part of which acts as a perfect open circuit. In this case the boundary conditions for E„ and H„ are interchanged on the perfect open-circuit portion of the boundary. In practice, it is difficult to solve the cavity problem when the same boundary conditions do not apply on the whole surface. For this reason we restrict our analysis to the use of short-circuit modes, since these are readily found for the common types of cavities encountered. There are three basic types of cavities to be considered, as illustrated in Fig. 7.32. Type 1 is a simply connected volume with a single surface, whereas type 2 is a simply connected volume with a multiple (double) surface. Finally, type 3 is a multiply connected volume with a single surfa.ce. Examples of the latter type are toroids and short-circuited coaxial lines. Q ,@ S3) T ype I FIGURE 7.32 Basic types of cavities. Type 2 Type 3 528 FOUNDATIONS FOR MICROWAVE ENGINEERING Consideration of these basic cavity types is of importance in connect-the existence of zero frequency modes, as will be seen. Electric Field Expansion In general, both V X E and V • E are nonzero; so we require twn modes: a solenoida] set with zero divergence and nonzero curl irrotational set with zero curl but nonzero divergence. The solenoidal ^ mode are defined by ' s V Z E„ + fe*En = 0 inV on S n X E„ = 0 V - E„ - 0 (7.95, in Vand on S and the irrotational modes by / „ F„ = V<f>n, where V2«b„ + ll*„ = 0 inV <t>n = const on S V X F„ = 0 that is, n X V<S>„ = 0 on S (7.96) in V where k„,l„ are eigenvalues for the problem. The constant is chosen as zero, except for the n = 0 mode, which has /„ = 0. The above definition for F„ is chosen so that / ** dV = - 1/ *„ Vz<t>„ dV = - ~j (V • cb„ Vd>„ - V4>„ • V*„) dV V l „ V i, V d<S>„ Note that only one boundary condition is imposed on the F„ since scalar functions $„ are uniquely determined if either <t>„ or d<bn/dn is specified or S. The modes are assumed normalized, so that f <t$ dV - jWm • F„ dV - / E „ • E „ dV = 1 JV JV (7.97) JV ytiFor n = 0 the eigenvalues / 0 , k 0 are assumed to be the zero eig" ues. The corresponding eigenfunctions E 0 , F 0 are the zero-frequency n i ^ We shall show that E 0 cannot be distinguished from F 0 ; so only the ^ ^ will be retained. We have V 2 E 0 = 0 = VV • E0 - V X V X E 0 , or V E 0 = 0, since V • E 0 = 0 by hypothesis. Thus - /E0 - V x ? x E Jv o dV=0 {(V • E 0 X V X E 0 - V X E 0 • V X Eo) Jv ft ELECTROMAGNETIC RESONATORS 529 This gives jv\V x E 0 | 2 dV = &,-n • E0 X V X E0 dS = 0 since n X E0 = 0. Thus we must have V X E 0 = 0, which implies E 0 = V/\ where f is a scalar function which is constant on the surface S. This latter relation holds for F 0 as well, so that E 0 can be discarded as long as F 0 is retained. The zero-frequency mode F 0 will be denned by F0 = V<D0 V X F0 = 0 n X F() = 0 V2<i>0 = 0 on S that is, <t>0 = const on S (7.98) since lQ = 0 For cavity types 1 and 3, a solution for <t>0 other than a constant does not exist. Hence the F 0 mode is present only for the type 2 cavity, with tt>0 having different constant values on S t and S 2 . This mode is just the static electric field that may exist between two conducting bodies at different potentials. Orthogonality P r o p e r t i e s Nondegenerate E„ and F„ modes are orthogonal among themselves and with each other. Consider ^(EmV2E„-E„V*Em)</V = J/ ( E „ • V x V x E m - E „ • V x V x E„) dV v = / [V X E„ • V x E m - V X E m • V X E„ J v -V.(E„XVxE,„-Emx?xE„)|rfV = (£(nXEm -VxE„ - n x E „ - V x E m ) d S = 0 s Hence, if k2m * k\, fE„,-EndV=Snm (7.99) where the Kronecker delta Snm = 0 if n * m and equals unity for n = m. 530 have FOUNDATIONS FOR MICROWAVE ENGINEERING For the modes Fm we show first of all that the <l>n are orth °gonal. tye jy(% ?2*„, - *„, v^„) dV = (If, - ll )f^m ay so that f<Pn<P„,dV~0 when/,2*^ Next consider a<P JV • <Pn V<f„, dV = ^ „ ~ - dS = 0 = ^(V* B • V*„, + <!>„ v 2 * m ) dV = /„/„,/" F„-F„,dV-Z^/<J» n * m dV Since the latter integral is zero, we have (7.100) for n and m not both equal to zero. If m = 0, n ¥= 0, the proof of orthogonality still holds. However, the normalization for the F0 mode will be chosen as f F 0 - F 0 r f V = riVd> 0 | 2 dV=l J v v ( 7 - 102 ' J To show that the E„ and Fm modes are orthogonal, consider V • F„, X V X E„ = (V X F m ) • (V x E„) - F m • V X V x E , = -^F„,-E„ since V X F„, - 0, V X V X E„ = VV • E„ - V2E„ = k*nEn. We now obt k*JW • F„, x V X E„ dS m • E„ dV = - 6n J T v s c n = - fa. X F„, - V X E„ dS = 0 (7 102) For n = m = 0, the modes F0 and E„ are not orthogonal, J * " " ^ sets identical. Since the F,„ are orthogonal to the E„, it follows that_^ ^ are needed, including F0 in general, to expand an arbitrary elec ELECTROMAGNETIC RESONATORS 531 Magnetic F i e l d E x p a n s i o n To expand an arbitrary magnetic field (we consider the possibility of n • H ¥= 0 over a portion of the surface to exist), we shall set up a dual set of modes analogous to those used to expand the electric field. The solenoidal modes are defined by V*H„ + * * H „ = 0 9 • H = 0 n • H„ = 0 on S n X V x H„ = 0 on S (7.103) and the irrotational modes are defined by P,,G„ = v>„ #n (7-104) =0 on S (1/2 V X G„ = 0 When p„ = 0, we have (assuming this occurs for n = 0) V X G = 0, Y'Vo = v" • G0 = 0, n • G„ = 0. Helmholtz's theorem now states that G0 can be derived from the curl of a vector potential, say G0 = v* X A 0 , where V X V X A0 = 0. Since G0 has also been assumed to be given by G„ = Vi/#0, it follows that i//0 must be multivalued. The function G u corresponding to a static magnetic field can exist in the type 3 cavity only. For example, in a short-circuited coaxial line, H = /a f i 2-r Id a„ d le Ia„ la, = V— = = = - V X — - In r 2ir r <>6 2<rr 2-r 2v Note that the scalar potential I6/2TT is multivalued. If k„ = 0 for n = 0, then (7.103) gives V x v" X H„ = 0. A volume integral of H (1 • V X V X H„ similar to that employed earlier in connection with F 0 and E 0 shows that V x H 0 = 0. Hence H„ has the same properties as G„; that is, V X H„ = V • H 0 = 0, n • H 0 = 0 on S. Therefore we shall not retain the H„ mode, but keep the G„ mode for the zero-frequency mode. Orthogonality Properties We assume normalization according to the following: / H „ - H „ d V = (G„-GndV= J v v J fGl)-G0dV = j\V<l,l)\2dV= 1 {>\,ldV=\ J v (7.105a) (7.1056) 532 FOUNDATIONS FOR MICROWAVE ENGINEERING By methods paralleling that used for the electric-type modes th orthogonality properties may be derived: '°Uoy )Hn • H,„ dV = / v H „ • G m dV = )Gn • Gm dV = ^ dy = Q (7.106, for n * m. For n = m we also have Relationship B e t w e e n E„ and H„ Modes The eigenvalues for both the E „ , H „ modes were designated as k n because they are, in fact, equal. Furthermore, we can show that v x E„ = k„nn V x H„ = fe„E„ (7.108) The curl of the first relation gives V X V X E„ = VV • En - V 2 E„ = *„V X Hn = fc2E„ = - V 2 E 8 by using the second relation. Similarly, V x V x H n = - V 2 H „ = knV x E„ = *»H„ Hence (7.108) is consistent with the Helmholtz equation of which the E n , H „ are solutions. Furthermore, the boundary condition on E„ is n X En = 0 on S. This implies n X V X H„ = 0 on S, which is the boundary condition that has been imposed on the H„ functions. Also, we have n • H„ = 0 on S, which implies n • V X E„ = 0 on S. Now n • v" x E = n V -n dn X(E„ - n n - E J a +1 V - n — I X n n • E„ + n — X E„ on I on = n-V,xB, since only the term V, X E,„ is in the direction of n. Here : t' ^Q components tangent to the surface S. Since E,„ = 0 on S, n • f^ga on S and is consistent with the boundary condition n • W„ - v 0 ] u i n e (7.108) is the only possible relation between the E „ , H „ modes. A integral of v • E„ x r x E„ = - E „ • r x r x E„ + iv x EJ 8 = -M1*1"1 shows t h a t the normalization of H„ is consistent with the norm ^ ^odes the E„ also. It should now be apparent that the E „ , F„ and H « ^ ^gr&fc have the properties that enable them to represent the electric a ELECTROMAGNETIC RESONATORS 533 fields, respectively. An arbitrary field would require an infinite sum of these modes for its expansion. O S C I L L A T I O N S IN A S O U R C E - F R E E CAVITY Consider a type 1 cavity with perfectly conducting walls and free of all currents and charges. We wish to determine the possible modes of oscillation. Let the fields in the cavity be expressed in terms of infinite series of the form E= I>„(/)E„ + £/-„(0F„ n (7.109a) n H=£M*)H„+£&X*)Gfl n (7.1096) n where e„, fn, hn, and g n are amplitude factors that are functions of time. Since n x E = n - H = 0 o n S and the mode functions satisfy similar boundary conditions, t h e series are uniformly convergent and may be differentiated term by term. Maxwell's curl equations thus give V X E = £ e„V X E„ = E enknHn = -u E ^ H „ - u E ^ G „ n n n n V X H = E b$ X H„ = E h„k„En = eE ^ E „ + « E ^ F „ If the first equation is scalar multiplied by H„ and G„ in turn and integrated over the cavity volume, we obtain e„k„ = - 11 •it (7.110a) at (7.1106) = 0 by virtue of the orthogonality properties (7.106). Similarly, when the second equation is scalar multiplied by E„ and F„ in turn and integrated over the cavity volume, we obtain (7.111a (7.1116 at From (7.110a) and (7.111a) we obtain + k2 fie = 0 534 FOUNDATIONS FOR MICROWAVE ENGINEERING The solution for e„ is (7.1 12«) and from (7.111a) the solution for h„ is then , JK .TT . rr . "«M V M V M < '1126) where «„ = A > e ) '•' is the resonant frequency for the n t h mode In the absence of volume sources, the F„ and G„ for n * Q A exist. However, the zero-frequency modes F„ and G 0 may exist in ° "° types 2 and 3, respectively. These modes are independent of the E ^ ^ modes. The n th free oscillation in the cavity is given by E = C„E„ = E „ e - . ' H=jy-Hne^' (7Al3aj (7.lm) These results are valid if the material in the cavity is lossy as well, provided e and n are taken as complex quantities. In this case OJ„ is complex, with the imaginary part representing a damping of the mode. Cavity with Lossy Walls Consider a cavity with finite conducting walls on which n x E = Z , „ n X J s = Z„,H, (7.1H) where H, is the tangential magnetic field and n is a unit outward directed normal. The surface impedance Zm = (1 +j)/aS,. Let the fields be expanded as follows: Ee'"' = Z e„Ene^' + £ fmFH*** <7U5a) He'*" = £ n„H,y"'+ E g„G„e^" (7-115*) We where e„, /'„, h„, and §„ are amplitude constants independent of t i m e . ^ have assumed a time variation eJ'"' so that the concept of a ^ ^ impedance Z,„ can be applied to account for the finite conductivi. Walls - • u « «nd since the In the present case n X E and n • H do not vanish on o, « c o n ditio" s modes in which E and H are expanded satisfy the boundary ^ n X E„ = n X F„ = 0 on S and n • H„ = n • G„ = 0 on b' fyff & expansions for E and H will not be uniformly convergent at the t e r i n . T° Consequently, the curl of (7.115a) cannot be evaluated term ^ tia ]iy « overcome this difficulty, we use the divergence theorem ELECTROMAGNETIC RESONATORS 535 integration by parts) to obtain fv - E x H „ d V = f (VXE) J Jy -H„dV- fVxH J V = -EdV V tpnXE -HndS Replacing V x H„ by knE„ and V x E by ~ja>nH and using the expansions (7.115) and the orthogonal properties of the eigenfunctions now give -jtofxh,, - knen = djn X E • H„ dS = Zm(f)H • H„ dS (7.116a) s s Note that H„ • H = H„ • H, since n • H„ = 0. Similarly, we find ( V • E„ X H dV = f (V x E J •V v • H - V x H • E„ ) dV = ^ n x E / H d S = 0 8 Replacing V X E„ by knH„ and V x H by jweE and using the expansion (7.115) yield juee„=knh„ (7.1166) This result is the same as that obtained by taking the curl of (7.115b) term by term. However, (7.116a) cannot be obtained by taking the curl of (7.115a) term by term because of its nonuniform convergence. From the two relations (7.116) separate expressions for e n and h„ may be obtained. For h„ we find K = TTTuZm(bH • H„ dS R R n (7.117) S where k2 = w2ixe. Let us now assume that the field is essentially that of the n t h mode and that this mode is not degenerate, i.e., no other mode has the same eigenvalue k„. Then we have H = /*„H„, and the surface integral becomes hnZmj,H„ • HndS = hnZm<j)\Hn\- dS since H,, is real. The power loss in the walls for the n t h mode is ?~6lHnl2dS s and the average stored magnetic energy is P,= 2 W„, = - / | H J - d V = 4 Jy 7 4 536 FOUNDATIONS FOB MICROWAVE ENGINEERING The Q for the n t h mode is Qn 2wW.n: and thus 2P, SlH I2 dS 4coWm Ma > For (7.117) we now get = •/'<»€( 1 +J)R„ /*<•>*„ a relation that can hold only if k*-k*a Q„ 1 We thus find that, for the cavity with lossy walls, the resonant frequency differs from the no-loss resonant frequency <nn by a factor 1 — 1/2Q„. In addition, a damping constant 5 = ojn/2Qn is introduced. In terms of w and 8, (7.119) gives J J l '-'- '-{ -k)-k ( 20) " Degenerate Modes The volume orthogonality of the G„ and H m modes hold even if p» = *«* as an examination of the method used in the proof will show. Howeve , *n = * m . t n e n t n e proof of the volume orthogonality of the H„ » modes breaks down. In this case (Hn-HmdV v J may or may not vanish. If the integral does not vanish, the two ^ ^ coupled together, since the average magnetic energy stored i above modes will contain a nonzero interaction term arising t r o integral. it is also In addition to volume coupling between degenerate mo • ^ Qf $& possible to have coupling arising from finite wall losses. 1 " e surface coupling is examined below. ELECTROMAGNETIC RESONATORS 537 If two modes H„ and H,„ are degenerate, so that k n = k,„, and if in addition i H„-HmdS*0 (7.121) "s then the power loss associated with these two modes will contain a crossinteraction term arising from the above integral. In this case the two modes are said to be coupled together by the finite surface impedance of the cavity walls. It is not possible to have just one of these modes present since the presence of one mode automatically couples the other mode. However, for most practical cavities such coupling does not exist. Nevertheless, the possibility of mode coupling must be kept in mind since, if it exists, both modes must be included in any calculation of energy stored, power loss, andQ. In the case where k n and k m are not equal, the surface integral may be shown to vanish for rectangular and cylindrical cavities. Although a general proof is not available, we should anticipate that this is a general property of nondegenerate modes. The problem of coupled degenerate modes may be circumvented by introducing new modes that are linear combinations of the old degenerate modes in such a fashion that they are uncoupled. If H„ and H,„ are degenerate coupled modes, choose new modes H'„ = ClHn+CzH„, H'„, = d , H „ + d2Hm (7.122o) (7.1226) with c, and <f, chosen so that 0H'n H„, dS = 0 (7.123a) f H'„ • H'„, dV = 0 v (7.1236) s J [ | H ' „ | 2 d V = [\n'„fdV= 1 Jy (7.123c) J\r These new modes are uncoupled and can exist independently of each other in the lossy cavity. For these new uncoupled modes the Q may be evaluated for each mode individually, since the cross-coupling term in the expression for power loss has been made equal to zero and, similarly, the cross-coupling term in the expression for stored magnetic energy has been made equal to zero. If more than two modes are degenerate, a similar procedure may be applied to find a new set of uncoupled normalized modes. In a general 538 FOUNDATIONS FOB MICROWAVE ENGINEERING discussion we may therefore assume that all the degenerate been chosen so that they are uncoupled. *7.9 ^ hav e EXCITATION OF CAVITIES In this section we consider the application of the modal expand field in a cavity to the problem of finding the field excited by omen t° ^ electric dipoles, which may represent a current loop or probe resntr In addition, a small aperture may be described in terms of eo ' " electric and magnetic dipoles as well. Thus the theory to be d e v e l o p be sufficiently general to treat the three common methods of COUDI' *' cavity to an external waveguide or coaxial transmission line. Let a cavity contain infinitesimal electric and magnetic dipoles P«Poc'-'«(r-r0) M = M0^"'«(r - r0) (7 .124Q) (7.1246) at a point whose position is defined by the vector r 0 . The three-dimensional delta function 6(r - r n ) symbolizes that the dipoles are localized at the point r = r 0 . This delta function is defined in such a manner that, for an arbitrary vector A which is continuous at r 0 , we have fA(r)<5(r-r0)dV = A(r0) -V when the point r 0 is included in the volume V (Sec. 2.11). By analogy with the following equations governing polarization u material bodies, B = Mo(H + M) D = e0E + P it is seen that Maxwell's equations become VX E = -jwB = - j w ^ 0 H -jcoixaM08(r - r0) n.X2ob) V X H=jwD = > e 0 E + jo»P„5(r - r 0 ) where a time factor <?•"•" has been suppressed. We now use the general expansion (7.115) for the fields to .,e /7.126") E = L e„E„ + I fnV„ n.l26t» ELECTROMAGNETIC RESONATORS 539 To find the expansion coefficients e„, h „, we follow the derivation leading to (7.116) but note that V x E and V X H are replaced by the right-hand sides of (7.125). Thus ((V X E) • H dV - j r X H„ • E dV Jy Jy = Zm<f>H-H„dS S = [ [ ~jo,ft0B -jav0MQ8(r - r 0 )J - H„, dV - k„ JE,, • E dV Jy Jy = -j(o^0hn - k„e„ -J(OM-0M0 • H„(r 0 ) (7.127a) and similarly, k,,hn - > « o « » + > p o ' E «(*o) To obtain an equation for g„, consider (7.1276) f V • E x Gn dV = f (G„ • V X E - E • V x G„) dV Jy Jy = 6n X E • G„ dS = Z„, (f>H • G„ dS s s Using (7.125a) for V X E, the expansion (7.1266), and the orthogonal properties of the modes H„ and G„ now gives /wjiofti +>MoM 0 • G„(r 0 ) = - Z m 0 H • G„ dS (7.128) S since V X G„ is zero. In a similar fashion, use of the relation f V • F„ X H dV = ((H • V X F„ - F„ • V x H) dV Jy Jy = - [ F„ • V x H dV = 6n x F„ • H dV = 0 Jy I together with (7.1256) and (7.126a) yields j<oe0fn= -jcoP0-Fn(r0) (7.129) 540 FOUNDATIONS FOR MICROWAVE ENGINEERING We now have the following equations for the expansion m .o. ncoefRc Qts * K,fnlmdgn: e jwe0en = knhn - y ' w P 0 • E „ ( r ( l ) 7-13&a) -1 hl == " r Jwk Po W kl{ " E ' " + M H Z ^ ° ' " -J^ -4n • H„ d s | < 7-1306) > * / . - - * * • • * . _ ,7.130c, / • P d U - -Joifi0M0 • G„ - Zm(pU • G„ dS (7.i30rf> where /?„ = wzMoeo and (7.130a) has been used in order to eliminate t> obtain (7.1306). " and In many practical problems dealing with cavities, the above equations may be simplified. Usually, &> is very nearly equal to a particular resonant frequency w„. As (7.1306) shows, all coefficients hm, m * n, will then be small compared with h„. Thus all the coefficients em, in * n, will also be small compared with e„, and the field is predominantly that described by the H „ , E „ mode. In the surface integrals, which represent small perturbations from the loss-free solution, we may approximate H by A„H„ without appreciable error. In addition, in the equation for gn, we may neglect the surface-integral term to a first approximation. We may then use the relation , 2P, 0H„ • H„ dS - — «M„ Q„Rm derived earlier if the H„ modes have been chosen so that they are uncoupled. In place of (7.130) we now obtain the following simplified equations *SU or 1 -J -ft Q„ h = h„= -jwknP0-En -k20M0-H„ M , P « • E n + klUu • H„ k\-k% (7.131a) 1 + Qn jo>ti0{k„M0 • H„ + > P 0 • E„) 7.131*) e„ = kl- 4n(7.1 «o/„ = - P o - F » 8« = - M 0 - G „ Equations (7.131a) and (7.1316) may be used for all h m a n ^ have assumed u> to be equal or nearly equal to w„, the denorm Since may b t o r ELECTROMAGNETIC RESONATORS 541 replaced by k% - k2„, for m * n. For n we may factor the denominator to give I 2 i '„ - *o 1 + = -; !„ +k0{l H -2k. *„ - k. 1 - 1 I - ; 1/2 -j 2Q„ since Q„ is very large-. Usually, we are primarily interested in the strength of excitation of the resonant mode E „ , H „ . Its excitation coefficients are given by (7.13lo) and (7.1316). The coefficients /'„ and g„ describe the local field that exists around the dipole sources. This field is a quasistatic field in its configuration. The excitation of fields in cavities by volume distributions of currents may be solved by the same method outlined above. As an example, consider a cavity with a volume distribution of current J(rlt)e""••'. A differential element of current may be considered equivalent to an electric dipole P with a moment given by J/JOJ. Thus, from a current element J(r„)<5(r - r 0 ) located at the point r„, the amplitudes of the n th mode are given by (7.131) divided by jto. When the current varies with the frequency u>„ of the n t h resonant mode, only this mode is excited with a large amplitude. The amplitude of the electric field of the n th mode due to the volume distribution of current is found by superposition, i.e.. adding up the contributions from each current element. Thus we have e n = _ (vJ(r0) -Ejr^c/V,, Jw/x f 7.132) k'i-ki i + where the integration is taken over the volume of the current distribution. A similar expression holds for the amplitude constant /;„. The fields excited in cavities by volume distributions of currents may also be solved in terms of a vector potential function. A specific application of the use of the vector potential is given in Chap. 9 in connection with the klystron tube; so we do not consider this method here. CAVITY P E R T U R B A T I O N THEORY The resonant frequency of a cavity can be varied over a small range by inserting a small adjustable screw into the cavity as shown, for example, in Fig. 7.33a. A small obstacle, such as a dielectric sphere, placed in a cavity will also change the resonant frequency and the Q of the cavity if the dielectric is lossy. Small obstacles, when acted upon by electric and magnetic fields that are uniform over the volume occupied by the obstacle, can be 542 FOUNDATIONS KOR MICROWAVE ENGINEERING FIGURE 7.33 • (a) An adjustable screw used to tune a cavity; (b> a small dielectric sphere placed in a cavity characterized in terms of induced electric and magnetic dipole moments For example, a dielectric sphere of radius / and with a dielectric constant <f will have a total electric dipole moment P given by - 1 P =4n-/' e„ + 2 f-()*-<0 f i E n — # .t -. e£0n*E* i0 (7.133) where E 0 is the electric field at the location of the dielectric sphere when it is absent. The parameter a e is called the electric polarizability of the dielectric sphere. For other obstacles we can write, in a similar way, P - « U A (7134c) M=*„,H0 (7-1346) where M is the induced magnetic dipole moment and o m is the magneti polarizability. In Table 7.1 we Ust the polarizabilities for a number obstacles that have the shape of prolate or oblate spheroids or their degenei ate forms. The polarizabilities depend on whether the field acts along major axis or minor axis. .. When the dominant mode in the cavity is the n t h resonant m ° d e , T h ( fields E 0 and H () in (7.134) represent the fields of the n t h cavity m o d j t u d e 5 additional field radiated by the induced dipole moments have am given by (7.131). If We assume that the field in the cavity is essential the perturbed n t h mode, then the dipole moments are given by P w = e [aeu60>n, Enuau r n + , aeve0CE„v«aD]T / 7.1350' (7.1356) Let E„ • P = e„e 0 A and H„ • M = n„ *.„,, that is, ,7.l36o' ,7.1366) ELECTROMAGNETIC RESONATORS TABLE 7.1 Polarizabilities o f c o m m o n o b s t a c l e s Prolate spheroid (Mr-DV («,-l)V <>r, = ( e r - l ) L + l — U 2/, fr-( 2 1 ( M ,. - 1 ) L + 1 2(cr-l)V «... - r -+ Z,= ""' -e 2c3 , In f r 2(M,-1)V -l)L •"" 1+M, - ( M r J + e 2* 1 -e t/S 2/, « - ! - • V-<5»/,i3 = Oblate spheroid 1 ^ tan 2/,1 /"*" ^ v^ VI - e* 1/2 - u ,= 1 1 - - 2', Sphere 4TT/» «, + 2 2/ For conducting obstacles let tr -> * and set p.r = 0 Circular disk (metallic) 16 -l)L 543 544 FOUNDATIONS FOR MICROWAVE ENGINEERING Equations (131a) and (1316) give jk„Y0kn A e e„ kl±mhn h„ - = 0 k2 - k2\ 1 + -jkl)Zak„Amhn + kj^een e„ - = o ^~*S1 + Q„ This is a homogeneous set of equations and has a solution only if tk determinant vanishes. By equating the determinant to zero, we obi equation for k^ which determines the perturbed resonant frequency of th* n t h mode when the obstacle is placed in the cavity. When we neglect term' in A*, A2m, and A e A„,, it is readily found t h a t t »o — ^i A. + A r 1 - (7.137) where 8 = (1 ~j)/Q„. Normally Ae and A,„ are both very small; so the approximation is a very good one in many practical cases. As an example of the application of this formula, consider the cylindrical cavity with a tuning screw of length ll and radius I2 as shown in Fig. 7.33a. Let the mode in the cavity be the TM 0 , o mode with electric field given by E=e oio^oio — e ^o(Poi^/Q) ' ifirdaJiipoi) where d is the length of the cavity and a is the cavity radius. Weuv approximate the tuning screw as one-half of a prolate spheroid. \ li 3» l2 the induced electric dipole moment (see Table 7.1) is when the tuning screw is located along the axis of the cavity. The a ^ occurs because only one-half of the spheroid is present. The magnei along the axis is zero; hence (7.137) gives «... 1 k0 = k 1 - 2 Trda2Ji(Po\) assuff>e when we assume that the cavity Q is infinite so that 8 = 0. w r t 'urbed that a = 2 cm, d = 3 cm.. /, = 1 cm, and /2 = 0.1 cm. The un* +R. E. Collin, "Fieid Theory of Guided Waves." 2nd ed., chap. 5. IEEE Press, 1991. Piscai[ . , * : * • ELECTROMAGNETIC- RESONATORS 545 cavity r e s o n a n t wave n u m b e r is k010 = p0i/a = 2 . 4 0 5 / 2 = 1.2025 r a d / c m . T h e polarizability aeu is given by 212„3 fe V a eu = ^ hH 1 + e In 1 - e 2e w h e r e e = (1 - / f / Z 2 ) 1 / 2 . F o r o u r e x a m p l e aeu e q u a l s 2.065 X 10 p e r t u r b e d r e s o n a n t w a v e n u m b e r i s t h u s given b y 2 . The 2 . 0 6 5 X 10 - 2 kn = k 010 1 - 2ir X 3 X 4 X (0.5191)' = *OIO(1 - 0 . 0 0 1 ) T h e p e r t u r b e d r e s o n a n t frequency i s t h u s 0.1 p e r c e n t lower w i t h t h e t u n i n g screw p r e s e n t . By m e a s u r i n g t h e c h a n g e in t h e r e s o n a n t frequency a n d Q of a cavity w h e n a lossy dielectric s p h e r e is placed in t h e cavity, t h e complex p e r m i t t i v ity of t h e dielectric m a t e r i a l can be d e t e r m i n e d (see P r o b . 7.20). 7.1. Show that, on a short-circuited coaxial line one-half wavelength long, the time-average stored electric and magnetic energies are equal. Use the expressions for the fields given by (3.81). 7.2. For the folded coaxial line, show that b = /ad in order for the characteristic impedance of the inner and outer lines to be the same. Tins is a common form of line for use in high-frequency oscillators. The effective length / is about twice the physical length. At a frequency of 300 MHz, what must 1/1 be in order that / = A 0 /4? If 2c/ = 5 cm and 2a = 2 cm, what must lb equal for equal characteristic impedances? For a copper line (a = 5.8 x 107 S/m), find the Q and input impedance at resonance. For d fixed, what are the optimum values of a and b that will make Q a maximum? 1/2 -1 2d 24 J 1 Inner line<^ 2a\ 0 jier llne-< FIGURE P7.2 7.3. Verify that an open-circuited transmission line behaves as a series resonant circuit in the vicinity of the frequency for which it is a quarter wavelength long. Obtain an expression for the input impedance at resonance. 546 FOUNDATIONS FOR MICROWAVE ENGINEERING 7.4. A short-circuited two-wire line is made of copper. The condui 1 cm, the spacing is 3 cm. and the length is 40 cm. Find the "" '^^ rteterfrequency, the Q, and the input resistance at resonance. 7.5. Design a microstrip-line resonator like that shown in Fig 7 o crostrip input line and resonator are both of width W = l mm 1.°' T h e mi. thickness is 1 mm and made from alumina with e r = 9.7 a n j . e f u b s l rate equal to 2 x 10 '. The microstrip is made from copper 0.01 m m ' t v t a n g e n t the required capacitance C, for critical coupling and the gap spaci ^ F " l d is the length I and Q for the resonator? The frequency of operation^ *d W h a t How much shorter than one-half wavelength do you have to make nl!?** Z ' of capacitive loading at the two ends? The gap capacitance for e = Q i***" 1 * used. 7.6. Design a circular disk resonator operating in the TM, 10 mode at 6 GHz Th substrate to be used has a dielectric constant e'r -je" = 6 -J0.005 and * 1 mm thick. The disk is made from copper. Determine the required radiu of the disk and the Q of the resonator. 7.7. Find the resonant frequency and Q of a copper rectangular cavity of dimensions a = b = d = 10 cm for the TE,,,, mode. 7.8. A cylindrical cavity of radius a = 2 cm and a length of 6 cm is filled with a dielectric with permittivity e = (2.5 y0.0001)e 0 . The cavity is made of copper. Find the resonant frequency and Q for the T E I U mode. Note that, in the expression for resonant frequency, the velocity of light in free space, c, must be replaced by •7.9. Use the results given by (4.25) and (4.26) to show that the Q of a at given by 0B ax * 2R flu, 2(3 <?o) where Zm = R + jX and Ym = G + jB are the input impedance andl admittance at the terminals. Verify that these formulas give the usual results series RCL network and for a parallel RCL network. 7.10. A rectangular cavity of dimensions u,b,d is coupled to a r e c U " g ^ ^ a i n a n through a capacitive slit. The guide width is a, and the height is b. ^^ equation ror determining the first antiresonant frequency. Find ^ an(j a slit susceptance for critical coupling. For a = 26 = 2.5 cm, a f for copper cavity, compute the resonant_ frequency Q and su op ^ _j it critical coupling. Use the formula B, * ( 2 0 6 / i r ) t n c s c ( - r / 2 * ' susceptance. What is the loaded Q? tT. FIGURE P7.10 ELECTROMAGNETIC RESONATORS 547 7.11. For a capacitive diaphragm in a rectangular guide of the dimensions given in Prob. 7.10, obtain an equivalent circuit to represent the susceptance function Bc = (2/36/-)lncsc(777/26) correct to terms up to Aw = w - wl in the vicinity of the frequency Wj. Hint: Expand /? in a Taylor series about OJ, and choose a series LC circuit. 7.12. Design a rectangular cavity of length d. height b = 1.2 cm, width a = 2.5 cm that will resonate at 10,000 MHz. The cavity is critically coupled to a rectangular guide of dimensions a by b. Specify the cavity length d and the radius of the centered circular aperture. Determine the unloaded and loaded Q's if the cavity is made of copper. 7.13. For the aperture-coupled rectangular cavity discussed in the text, let the incident power at resonance be 100 mW. The cavity is critically coupledEvaluate the peak value of the electric field in the incident wave and in the cavity field. How does the peak amplitude of the cavity field depend on the cavity Q? (See Prob. 7.12 for cavity dimensions.) 7.14. A hemispherical resonator is to be used at 10 GHz. It is made from a dielectric with a dielectric constant of 100 and is mounted on a ground planeFind the required radius of the hemisphere and the radiation Qr. *7.15. A cavity is excited by an impressed electric field E„ tangent to an aperture surface Sa cut in the cavity wall S. Use the relation fvV - E X G„ dV to show that the amplitude g„ is given by -iu>nng„ = (bn X E • G„ dS = I n X E„ • G„ dS for a cavity with perfectly conducting walls. Using the general expansion (7.1156), show that an alternative expression is -y"wM,,g„ = -JW-of H ' G- dV = " j>n • Hl/ '« dS upon putting H • VI/I„ = V • Hi/>„ since V • H = 0 and using the divergence theorem. This last relation shows that the G„ modes are excited whenever n • H does not vanish over S. *7.16. Show that the two expressions for g„ in Prob. 7.15 are identical. Hint: Consider n • V x Ei/<„ = (/»„n • V X E - n • E X V</<„ = - jo»(X(j(&„n • H - n X E • Vi//„ and use Stokes' law to show that <^>n • V X Ei//„ dS = 0 s *7.17. Find the eigenfunctions E„, H „ , F„, and G„ for a rectangular cavity of dimensions a. b,c. *7.18. Obtain a modal expansion similar to (7.126a) and (7.130) for the electromagnetic field in a cavity excited by a volume distribution of electric current J ( r ) . 548 FOUNDATIONS FOR MICROWAVE ENGINEERING FIGURE P7.19 *7.19. ( a ) For the box cavity illustrated in Fig. P7.19, show that the normalized electric field mode function for the TE 1 0 1 mode is E101 abc vx TTZ sin — sin — a „ (b) Let a metal sphere of radius I be placed in the center of the cavity. Show that when a = 6 = c and / = a/20 that the TE l t l l -mode resonant frequency is lowered by T T / 2 0 percent. *7.20. A dielectric sphere of radius / and complex relative permittivity er = t'r - je" is placed in a cavity. Show that the perturbed resonant frequency for the nth cavity mode is given by k0 = k ^JL^n^u-wt'yjJ^ +l n 2Qn - o>+2)*+te:> where the cavity Q due to the lossy dielectric is given by to + 2) 2 + (e"rf Qd = 2 T H * E „ • E„3£-; REFERENCES 1. Montgomery, C. G-, R. H. Dieke, and E. M. Purcell: " Principles of Microw McGraw-Hill Book Company, New York, 1948. -^ 2. Ragan, G. L. (ed.); "Microwave Transmission Circuits," McGraw-HiU New York, 1948. biC 3. Slater, J. C: "Microwave Electronics," D. Van Nostrand CompoW, N.J.. 1950. ComP4"-''' ? ELECTROMAGNETIC RESONATORS 549 4. Goubau. G.: "Electromagnetic Waveguides and Cavities," chap. 2, Pregamon Press, New York, 1962. 5. Kurokawa. K.: The Expansions of Electromagnetic Fields in Cavities, IRE Trans., vol. MTT-6. pp. 178-187, April, 1958. 6. Kajfez, D., and P. Guillon: "Dielectric Resonators," Artech House Books, Dedham. Mass., 1986. 7. Van Bladel. J.: The Excitation of Dielectric Resonators of Very High Permittivity. IEEE Trans., vol. MTT-23, pp. 208-217, 1975. 8. Gastine, M., Courtois, L. and J. L. Dormann: Electromagnetic Resonances of Free Dielectric Spheres, IEEE Trans., vol. MTT-15, pp. 694-700, 1967. 9. Chow, K. K.: On the Solution and Field Pattern of Cylindrical Dielectric Resonators, IEEE Trans., vol. MTT-14, p. 439, 1966. CHAPTER 8 PERIODIC STRUCTURES AND FILTERS Waveguides and transmission lines loaded at periodic intervals with identical obstacles, e.g., a reactive element such as a diaphragm, are referred to as periodic structures. The interest in waveguiding structures of this type arises from two basic properties common to all periodic structures, namely, (1) passband-stopband characteristics and (2) support of waves with phase velocities much less than the velocity of light. The passband-stopband characteristic is the existence of frequency bands throughout which a wave propagates unattenuated (except for incidental conductor losses) along tl structure separated by frequency bands throughout which the wave is i off and does not propagate. The former is called a passband, and the latte referred to as a stopband. The passband-stopband property is of £ interest for its frequency filtering aspects. . a The ability of many periodic structures to support a wave havi phase velocity much less than that of light is of basic ' m P ° r t a n ^ t i o n traveling-wave-tube circuits. In a traveling-wave tube, efficient between the electron beam and the electromagnetic field is obtain "n0 the phase velocity is equal to the beam velocity. Since the latter is ^ greater than 10 to 20 percent of the velocity of light, considerable - ^ ^ down of the electromagnetic wave is required. Periodic structu aCt ual for use in traveling-wave tubes are discussed in this chapterprinciples of operation of the tube are covered in Chap. 9. micr0 w a v e The last part of the chapter is devoted to an introduction to ^ ^ filter theory. A complete treatment Df all aspects of filter theory sufl5l ., e nt would be much too lengthy to include in this text. However 550 PERIODIC S T R U C T U R E S AND KILTERS 551 material is covered to provide a background so that the technical literature can be read without difficulty. r APACITIVELY L O A D E D T R A N S M I S S I O N L m E - C I R C U I T ANALYSIS To introduce a number of basic concepts, methods of analysis, and typical properties of periodic structures, we shall consider a simple example of a capacitivity loaded transmission line. For a physically smooth transmission line, such as a coaxial line, the phase velocity is given by vp = ( L C ,-1/2 = (Mo^o) -1/2 [8.11 where e r is the dielectric constant of the medium surrounding the conductor. A significant reduction in phase velocity can be achieved in a smooth line only by increasing e r . This method has the great disadvantage that the cross-sectional dimensions of the line must also be reduced to avoid the propagation of higher-order modes. The phase velocity cannot be decreased by increasing the shunt capacity C per unit length because any change in the line configuration to increase C automatically decreases the series inductance L per unit length, since LC = /x0e. However, by removing the restriction that the line should by physically smooth, an effective increase in the shunt capacitance per unit length can be achieved without a corresponding decrease in the series inductance L. That is, lumped shunt capacitance may be added at periodic intervals without affecting the value of L. If the spacing between the added lumped capacitors is small compared with the wavelength, it may be anticipated that the line will appear to be electrically smooth, with a phase velocity U P = -f 1-1/2 (8.2) where C0/d is the amount of lumped capacitance added per unit length (a capacitor C 0 added at intervals d). The following analysis will verify this conclusion. One method of obtaining shunt capacitive loading of a coaxial transmission line is to introduce thin circular diaphragms at regular intervals, as in Fig. 8.1. The diaphragms may be machined as an integral part of the center conductor. The fringing electric field in the vicinity of the diaphragm increases the local storage of electric energy and hence may be accounted for, from a circuit viewpoint, by a shunt capacitance. The local field can be described in terms of the incident, reflected, and transmitted dominant TEM mode and a superposition of an infinite number of higher-order E modes. If the cylinder spacing b - a is small compared with the wavelength, the higher-order modes are evanescent and decay to a negligible value in a distance of the order of b — a away from the diaphragm in either 552 FOUNDATIONS FOR MICROWAVE ENGINEERING M-4 F I G U R E 8.1 Capacitive loading of a coaxial line by means of thin circular diaphragms. direction. An approximate expression for the shunt susceptance of th diaphragm ist B 8(6 -ef a = -— = V ln(6/a) In esc \2b-a (8.3) where Y c = [601n(6/a)J _ 1 is the characteristic admittance of an air-filled coaxial line. The expression for B is accurate for b - a < 0.1A0. In this low-frequency region, B has a_frequency dependence directly proportional to io. At higher frequencies B will have a more complicated frequency dependence, although the thin diaphragm can still be represented by a shunt susceptance. The circuit, or network, analysis of a periodic structure involves constructing an equivalent network for a single basic section or unit cell of the structure first. This is followed by an analysis to determine the voltage and current waves that may propagate along the network consisting of tl cascade connection of an infinite number of the basic networks. For the structure of Fig. 8.1, an_equivalent network of a basic section is a shunt normalized susceptance B with a length d/2 of transmission line on eitl side, as in Fig. 8.2a. Figure 8.26 illustrates the voltage-current relatior ships at the input and output of the « t h section in the infinitely cascade connection. . ul The relationships between the input variables Vn, I„ arid t h e °" ? n variables V„^„ J n M are readily found by using the .a/.W^J t r a ^ f " " ^ r e n t matrix discussed in Sec. 4.9. The V n and /„ are the total voltage a " d « ^ ^ amplitudes, i.e., the sum of the contributions from the inciden ^ fleeted TEM waves at the terminal piane. The circuit for a unit ^ ^ ^ n broken down into three circuits in cascade, namely, a section ° s hunt sion line of length d/2 (electrical length 0/2 = k0d/2\ followed y t N . Marcuvitz (ed.), "Waveguide Handbook," p. 229, McGraw-Hill Book 1951. PERIODIC STRUCTURES AND KILTERS *h> , .'/2 r— Fc = i 553 i -i r^ )B s=1 (a) (6) F I G U R E 8.2 (c> Equivalent circuit for unit cell of loaded coaxial line; (b) cascade connection of basic unit-cell networks. susceptance B, which in turn is followed by another length of transmission line. The &.'%'&5$ matrix for each of these individual networks is, respectively (Prob. 4.26), e cos- J sin- J sm- cos- JSin cos- 1 JB jsm- cos- The transmission matrix for the unit cell is obtained by the chain rule [see (4.75)], i.e., the product of the above three matrices, and hence we have e [V h cos 2 e jsm 1 2 e e j sin- cos- .0 0 1 8 cos- 7sin- 8 0 /sin- 8 cos- [K + I /„., B cos 8 - —- sin 8 ss 7 — cos0 + smfl - — 1 IB B' j — cos 8 + sin 8 + — B cos 8 - — sin 8 •lB • o S \ ] 'n* 1 (8.4) '„+, j Note that srf =9>, which is always true for a symmetrical network, i.e., a symmetrical unit cell. If the periodic structure is capable of supporting a propagating wave, it is necessary for the voltage and current at the (n + list terminal to be equal to the voltage and current at the rath terminal, apart from a phase delay due to a finite propagation time. Thus we assume that K*i~ *-**?* (8.5a) h ^ = e-T*ln (8.56) where y = jp + a is the propagation constant for the periodic structure. In 554 FOUNDATIONS FOR KtCROWSVE ENGINEERING terms of the transmission matrix for a unit cell, we now have \v„] \.'V m l [ v B + l l o-yrf [K + 1 l §>. . ' „ + , . — c f h '-> [ \se m\ \eyd 0 \ [V n + 1l t 9 J " 0 eyd 7„+i, = 0 1 or (8.6) This equation is a matrix eigenvalue equation for y. A nontrivial for V„ , i, /„ +, exists only if the determinant vanishes. Hence j / - ey" OS = tfB -&W+e2v>-e?d(sS +&) = Hon 0 (8.7) For a reciprocal network the determinant s?9 - <M<d? of the transmiss' matrix equals unity (Sec. 4.9); so we obtain sf +Qi COSh yd = — ^ — (8g) For the capacitively loaded coaxial line, (8.8), together with (8.4), yields B cosh yd = cos 6 - —sin 6 2 (8.9) When Icos 0 - (B/2)sin ()\ < 1, we must have y =jfi and a = 0; that is, B cos lid = cos 0 sin 6 (8.10a) 2 When the right-hand side of (8.9) is greater than unity, y = a and (i - 0; sc B cosh ad = cos 6 sin 8 > 1 (8.106) 2 Finally, when the right-hand side of (8.9) is less than - 1 , we must hav yd = jir + a, so that cosh yd = cosh(J7r + ad) = - c o s h ad B = cos0 - — sinfl < -1 (8.10O I t i s apparent, then, that there will b e frequency bands for whic u ^ i h e ated propagation can take place separated by frequency bands in sjble wave is attenuated. Note that propagation in both directions since ~y is also a solution. . j s ro ade in A detailed study of the passband-stopband characteristic^ f r e q u e n O Sec. 8.6. For the present we shall confine our attention to the W J ! * t h e n 8 i * limiting value of /?. When d « A0, 6 = k0d is small, and pa PERIODIC STIU CTi KKS AND FILTERS 555 be small. Replacing cos 6 by 1 - 0 2 / 2 and sin 9 by 0 in (8.10a) gives 02da fc^d2 cos£d - 1 ~ - g - - 1 - - | Using the relations k% = where wC 0 = B, we obtain W2M»«O 2 B*0d |- = a>2LC and B = B/Yc = »GJJL/CW*t 2 /3 = « LC + w2LC„ -u a and hence ,f = «tfZ|C7+-^J (8.11) Therefore we find that, at low frequencies where d <K A0, the loaded line behaves as an electrically smooth line with a shunt capacitance C + C^/d per unit length. The increase in fi results in a reduction of the phase velocity by a factor k0/fi. Another parameter of importance in connection^ with periodic structures is the normalized characteristic impedance Z B presented to the voltage and current waves at the reference terminal plane, i.e., input terminals of a unit cell. An expression for Zn may be obtained from (8.(5). which may be written as Hence ZH _ V,,,, -* = £B = — i = -M v' - e~"' - = (8.12) Replacing 2 e r " by .rf +0 ± [(.!/ +&)2 - 4 ] 1 / 2 from (8.7), we obtain %m Z§ = j= ~ •sf ± yO^ + ^ ) 2 - 4 (8.13a) where the upper and lower signs refer to propagation in the +z and -z directions, respectively. We are using the convention that the positive directions of V„ and /„ are those indicated in Fig. 8.2, independent of the direction of propagation. For a symmetrical network, si = .C*\ and since &&> -&<& = 1, we have .i/ 2 - 1 = m% • In this case (8.13a) reduces to = ±V< 813fe > 2 ±V4.W 2 - 4 In general, for a lossless structure, Z'a = ~(Z'b )* in the passband, since \st +S\ < 2, as (8.8) shows. zi- , 556 Fill INKS ' " i.vs FOR Mti PJ8 (WAVE BW1INKKBIW. Jf the unit cell is represented by a T network with paramet and Z22, then, by using the relations between the st/xi't. p- pg. the impedance parameters given in Sec. 4.9. we can also show th cosh yd = 2Z *^L1 — (8.1 4 ) 12 Z„ = "• —Iz, rS an ^ A1' ± ZVi sinh yd (8.15) The waves that may propagate along a periodic structure are oft* called Bloch waves by analogy with the quantum-mechanical electron wa that may propagate through a periodic crystal lattice in a solid. It is for t] reason that we have denoted the characteristic impedance as Z for th' Bloch wave. The voltage and current at the nth terminal plane will h denoted by V„;„ 7B*„ for the Bloch waves from now on instead of by the quantities V„. In. The + and - signs refer to Bloch waves propagating in the +2 and -z directions, respectively. We shall also adopt the convention that the positive direction_of current flowjbr Bloch waves is always in the + 2 direction; thus l'b = Y^V^ and Ilt = YjtVg. However, for a symmetrical structure such that -'/ = V, we shall have ?„ = - Y% = -(Z^)'l. If (8.13) is used, we find that, for the loaded coaxial line, 2 sin e + B cos 0 - B i (8.16) V 2 sin 6 + B cos 0 + B In the low-frequency limit, where we can replace sin 6 by 0 = k0d = iodjLC and cos 0 by 1, we obtain 20 2„ = 20 f 2B C + C0/d and thus Zp ~ ZffZ,. - C + C0/d ically Again we see that, in the low-frequency limit, the loaded h"l^ ^tidp'-1^ smooth and the characteristic impedance is modified in ^ ^ k>ngth manner by the effective increase in the shunt capacitance pe ^ ^ &&#& The characteristic impedance of a periodic structure is ^ ^ . ( (:e\\. quantity since it depends on the choice of terminal planes , 0 ^ t i o 0 i t he » the terminal planes are shifted a distance / in the - PBRfODIC STRUCTURES A.VD FILTERS 557 characteristic impedance becomes Z 8 +j tan A-n/ Z« = (8.18) 1 +./ZHtan/80/ WAVE A N A L Y S I S O F P E R I O D I C S T R U C T U R E S Periodic structures may be analyzed in terms of the forward- and backward-propagating waves that can exist in each unit cell with about the same facility as the network approach gives. In the wave approach the wave-amplitude transmission matrix \A\ discussed in Sec. 4.9 is used. With reference to Fig. 8.3, let the amplitudes of the forward- and backward-propagating waves at the n t h and (« + l)st terminal plane be c*, c,;, c* +1 , and c~ +1 . The e* + B e~ H , are related to the c*,C~ by the waveamplitude transmission matrix as follows: A 12 A22 "-AM A2, (8.19) •r, • ! The solution for a Bloch wave requires c~, , = e Hence (8.19) becomes A 21 A22-e>"' c„., y ''c* and c„ ., = e~**'en. = 0 (8.20) A nontrivial solution for c~rX,c~., is obtained only if the determinant vanishes. Consequently, the eigenvalue equation for y is A,lA22-A1,A2l + e 2 ^ - ^ " ( A , 1 cosh yd = or A u + A .r/, A22) = 0 (8.21) since the determinant of the transmission matrix, that is, AnA22 ~ Al2A2V equals 1 when normalized wave amplitudes are used. — T Cif' Fa %. Unit cell 1 «&1 Ca*t """" f n«l F I G U R E 8.3 Wave amplitudes in a periodic structure. 558 FOUNDATIONS FOB MICROWAVE KKG1NKKRINC The BJoch wave which can propagate in the periodic struct up from om forward- and backward-propagating propagating normal trancm;*-transmission.,. m a d e -«- - . i--o r. . -'""amission U 0r waveguide waves that exist between discontinuities. When v h determined from (8.21), the ratio of c~ to c' is fixed. This ratio • ***" the characteristic reflection coefficient Y . Thus the transverse c\ tL of the Bloch wave will have an amplitudeB ^ n c fielcj VBO = C O + CO = C O ( 1 + 1"B) at the zeroth terminal plane and an amplitude vft„ = <-•;+ c;= cr; (i + vB) = c; (i + vB)e~y"d (822a) at the n th terminal plane. The transverse magnetic field of the Bloch wa • will have an amplitude / / j „=c 0 *(l-r f i )e->""' (8.226) at the rcth terminal plane. The characteristic reflection coefficient may be found from the pair of equations (8.20) by eliminating e yJ by the use of (8.21). It is usually more convenient to express r B in terms of Z B by using the relation Z B « (1 + r B ) / ( l - YB). Thus we have zs-\ zi- (8.23) r = —1 R = where the + and - signs refer to Bloch waves propagating in the +z and —z directions, respectively. The above wave formulation is now applied to the capacitively loade transmission line discussed earlier. The unit cell is chosen as in Fig. 8 The wave-amplitude transmission matrices for the three sections of the unit cell are (Sec. 4.9 and Prob. 8.7) B 2+jB ,jk.Qd '•> 0 , -Jkud n 4+B2 B 2(2 +jB) J cell is and another matrix like the first one. The [A] matrix for the u » , obtained by multiplying the three component matrices together. 2 +jB [A] = ,>»/2 ,j«/2 0 ,-}»/2 B B 4 + B2 2(2 + jB) 0 PERIODIC STRUCTURES AND FILTERS 559 where 0 = kf)d. After multiplication we obtain 2+jB B _ e ;« [A] B -J- j l (8.24) 4 + B~ 2(2 J ii +jB)e Making use of (8.21), we find that (4 + B ' 2 ) e ' * + ( 2 + . / B ) V cosh yd " My^S) = cos e ' ^ S Sin ° which is the same as (8.9) obtained earlier. « 3 P E R I O D I C S T R U C T U R E S C O M P O S E D OF ^ S Y M M E T R I C A L TWO-PORT NETWORKS The capacitively loaded coaxial transmission line can be considered as made up of symmetrical sections by choosing terminal planes midway between each diaphragm. For other choices of terminal-plane positions the unit cell would be unsymmetrical, and its equivalent T network would then also be unsymmetrical. Other types of periodic structures are composed of intrinsically unsymmetrical unit cells such that there is no terminal-plane location that will reduce them to a symmetrical structure. Several unsymmetrical structures are illustrated in Fig. 8.4. For nonsymmetrical structures the Bloch-wave characteristic impedance is given by (8.15), which we rewrite as (8.25a) f W+z (8.256) nnnnnn /ii<< I* in I A. F I G U R E 8.4 Periodic structures with unsymmetrical unit cells, (a, 6) Rectangular waveguide loaded with thick unsymmetrical diaphragms; (c) coaxial line loaded with diaphragms and dielectric rings; (rf) equivalent T network of a unit cell. 560 FOUNDATIONS FOR MICROWAVE ENGINEERING where c= 2n Z2-, - ' • . : (8.26 a) Z = ±Zl2 sinh yd = ±JZl2 sin £d (8.266, and the sign is to be chosen so that Z has a positive real part Th constant $ is given by ' he Phase cos fid = Zu+Z, 2Z 12 22 (8.26c) in the propagation band. The physical length of a unit cell is d. Th quantities Z£ and [id are often called the iterative parameters of the T network. A consequence of the nonsymmetry of the unit ceil is that Z^ is B different from ZB. Let the voltage of the Bloch wave at the zeroth terminal plane be V where the signs + or - refer to Bloch waves propagating in the +z and -z direction, respectively. The corresponding Bloch-wave current is Z^0 = V&a/Z'B- At the « t h terminal plane the Bloch-wave voltages and current-; will be Vp* *e "Bn = VR BO* f y ;i ,1 ^.•2'c (8.27b) L B Recall that we are taking the positive direction of current flow to be in the + z direction, independent of the direction of propagation for the Bloc waves. If the restriction is made that the only points at which the voltages and currents will be specified are the terminal planes, the periodic stru< has properties similar to any uniform transmission line or wavegu • such, transmission-line theory can be applied to study the effects nating a periodic structure in an arbitrary load impedance, ' ^ matching sections for periodic structures, etc. These applications cussed in the following two sections. 8.4 TERMINATED PERIODIC STRUCTURES Figure 8.5 illustrates a periodic structure terminated in a k>a« ^ e „th 2,, at the JVth terminal plane. The total voltage and current a^ ^ ] o c h terminal plane will be a superposition of an incident and re PERIODIC STRUCTURES AND FILTERS 561 rx t "sol F I G U R E 8.5 Periodic structure terminated in a load Z,. wave; thus V = V p " : W + V~ pJP"<i v v + v Bn B~e B0e 1 1 I = I* p~.ll '" + J' u)V"<> T -Bn -B~e ~B0e (8.28a) (8.286) 1 where Y B = Zg . At the N t h terminal plane we must have 'L ~ *BN ~ "IABN ~ -'IJI. and hence V£N + vB-A. = zL( Y< v;N + ?B vHN) (8.29) The reflection coefficient f, of the load for Bloch waves is, from (8.29), ZLYS-1 r BN Z BN L^B~ Z-£ZL-Z- -'B Z;_ B 1 ZL Z,, - Z'R j (8.30) z + t zL + z- ( For a symmetrical structure f = 0 and the expression for I', reduces to the usual form. The Bloch-wave reflection coefficient at th$ n t h terminal plane is r„ = "Br, V BNC ,-j<N-n)-d = re-jxN-n)pd (8.31) v BNe The input impedance at the nth terminal plane is VL + v£„ z„ = Jin + I Bn vjWi + r j v s + „^+v Bn y B - 1 + r„ z B + Zfl(i + r„) *B + r„y8- z« + z : r (8.32) 562 FOUNDATIONS FOB MICROWAVE ENGINEERING From (8.32) we can also obtain the alternative expressions ZR+ZIV.B r„ = - (8.33a) ' n (z.-Q-z zB {zn-c)+z (8.336) If (8.31) is used to express F„ in terms of YL and (8.30> j,s ' used to express F, in terms of Z,, we find that (8.33a) gives « g ^ - £ + j Z " " tan( N-n)fid Z+./(Zt-f)tan(N-/,)/3d < 8 - 34 ) This equation gives the transformation of impedance along a periodi structure. It differs somewhat from the usual transmission-line formula when the unit cell is unsymmetrical, so that f * 0. For a Bloch wave propagating in the +z direction, the periodic structure must be terminated in a load Z, = ZB = £ + Z to avoid a reflected wave. Similarly, the matched-load_termination for a Bloch wave propagating in the —z direction is -ZB= Z - C- The two characteristic Bloch-wave impedances are the iterative impedances for the T network of the unit cell. With voltages and currents chosen as in Fig. 8.6, it is readily shown that an impedance Z B connected at terminals 2 is transformed into itself at terminals 1. Similarly, an impedance — Z B connected at terminals 1 is transformed into itself at terminals 2. It is for this reason that Zg is called an iterative (repeating) impedance. For a lossless T network, f is pure imagi; nary and Z is pure real in the propagation band. Ambiguity in the sign of Z as given by (8.15) or (8.266) may be avoided by noting that, in a passband, Z is real and positive, in order to be consistent with our choice of positn direction for current. We must have positive real power transmission, an hence P = RekV£(ltf = Rei|/«| 2 Z^= \\ltfZ > 0 (8.35) Another criterion that may be used is the one requiring the reactive part of *i.i=« (61 F I G U R E 8.6 Iterative impedance properties of a T network. PERIODIC STHUCTUBBS AND FILTERS n 1 1 I A, 563 1i r rX , 2 , 2 r c u J8 d _JU n ~f^~ir h [*] B_JLT ^ r ^ 1! ^ C T . FIGURE 8.7 (a) Tapered transition matching section for a diaphragm-loaded rectangular guide; (6( quarterwave transformer matching of a capacitively loadt-d coaxial line. Zg to have a positive derivative with respect to w (Sec. 4.3). There is also reactive power in a Bloch wave in a passband, and this is given by * reactive 2''«' & (8.36) Complex power for a Bloch wave propagating in the —z direction is given by - ^VH(IBY* because of our choice of direction for positive current. MATCHING OF PERIODIC S T R U C T U R E S If a periodic structure is connected to a smooth transmission line or waveguide, some means of matching the periodic structure to the input waveguide must be provided to avoid reflection of the incident power. A situation encountered quite frequently is the one where the periodic structure is identical with the input waveguide apart from the periodic loading. One way of providing a matched transition from the unloaded to the loaded waveguide is to use a tapered intermediate section. The matching taper section is similar to the loaded waveguide except that the periodic loading is gradually reduced to zero over a distance of about a wavelength. Figure 8.7a illustrates a tapered transition in a rectangular waveguide connected to a similar guide periodically loaded with diaphragms. Any of the matching networks discussed in Chap. 5 may also be used to match a periodically loaded guide to an unloaded guide. For example, at some distance d.'/2 in front of the first terminal plane for the periodic structure, the characteristic admittance Y B of the periodic structure is transformed into an admittance 1 - jB', so that placing a shunt susceptance jB' at this point provides a matched transition. Matching by means of a shunt susceptance may be viewed as an application of the quarter-wave transformer matching technique. The unit cell consisting of the shunt susceptance jB' plus a length d ' / 2 of transmission line (or waveguide) on either side, as in Fig. 8.76, may be considered as part of an infinite periodic structure with a propagation phase constant fi' and a normalized character- 564 FOUNDATIONS FOR MICROWAVE ENGINEERING istic impedance Z'a. If the parameters B' and d' are chc (5'd' = 7r/2, Z'B = Z^ / 2 , then at the input terminal to the matok- so t (8.34) gives aiciwng ^ . ^ 7 - ' ° = 1 (8.37) Note that the matching section is a symmetrical structure and £' = 0. We also require that Zu be real in order for (8.37) to h ° =^ solution for Z'B. When these conditions are met, we see that the ^ i ! ^ section behaves essentially as a quarter-wave transformer. For svmm structures the required values of B' and d' mav be found from (8 8?"°^ (8.136); thus, as cos B'd' = cos(~/2) = 0, we have = 9>' = 0 -. ,2 (z«) = r (8.38a) (8.386) For the capacitively loaded transmission line, we obtain, by using (8.4) and (8.16) applied to the matching section, 2 cot ftnd' = B' _ 2 {Z'B)I- (8.39a) 2 sin k„d' + B' cos k0d' - B' .*od' = ZJ (8.396) B — = tan z 2 sin £ n a" + B' cos 6 n a" + B' when (8.39a) is used to eliminate B ' . The above results are the equivalent of those derived in Chap. 5, i.e., given by (5.8) and (5.9). To obtain a match over a wide frequency band, more elaborate matching networks must be used since a single shunt susceptance usually does n< provide a match over a wide frequency band. Broadband matching i complicated by the fact that the characteristic impedance Z B of a periodic structure is a function of frequency. No general technique exists for design ing broadband matching networks because of the general nature of aEach periodic_structure must be considered by itself so that the treq variation in Z B can be incorporated into the design. For this reasi matching problem is not discussed any further. k0-B DIAGRAM We now t u r n to a detailed study of the passband-stopband c h a r a C ' f e earlier the capacitively loaded coaxial transmission line discussed m ^ tn sections. The information contained in the eigenvalue e t 5 u a t ' ° o n a M propagation constant B in a periodic structure is usually plot (or co-B) „. plane. . , curves of „ B _ versus k 0 show . -immediately _ -j-.ntrAv the " mThe bands for propagation and also the stopbands in which no prepay PERIODIC STRUCTURES AND KILTERS 565 F I G U R E 8.S k0d-jid diagram for a capacitively loaded coaxial line, B = 2k0d. place. The resultant plot is called the k0-fi diagram, or the Briilouin diagram, t For the capacitively loaded coaxial line, (8.9) gave B cos (id = cos k0d - —sin knd = cos k0d - Kknd sin k0d (8.40) where B/'l = wC0/2Yc has been expressed as Kk_ud. Curves of k0d versus fid are sketched in Fig. 8.8 for K = 1, that is, for B = 2k„d. A low-frequency passband exists for 0 < k0d < 0.4167T. This passband is followed by a stopband and further alternating passbands_ and stopbands. As knd becomes large, the loading is increased, since B increases with k0. This has the effect of decreasing the width of the passbands in terms of frequency. tNamed after Briilouin, who used diagrams of this sort to illustrate the energy-band structure in periodic crystalline media. 566 FobNDATIONS FOR MICROWAVE ENGINEBHIMG The edges of the bands occur when the magnitude of th side of (8.40) exceeds unity. The lower edge of the first na= t " g l u ~ when 0 < k0d < v and Passband cos k0d - Kk0d sin k0d = -1 This equation may be solved for k0d to give cot k0d = Kk0d kad = 0 cos- (8.416, The corresponding principal value of fid is -, and the values of h w obtained from (8.41) mark the edges of all the bands for this value of ft' The edges of the bands where fid = 0 are obtained by equating (8 40) t unity, in which case we obtain tan M = 2 -Kk0d i 8 42o (8.426) sin- One edge of the passband always occurs when the spacing between discontinuities equals one-half wavelength in the unloaded waveguide, in the present case, when k0d is a multiple of ir. When the spacing between discontinuities equals one-half wavelength, they may all be lumped together, with the result that the line becomes effectively loaded at a single point by an infinite susceptance (or reactance). Clearly, power transmission along the periodic structure must reduce to zero at this frequency. Only the principal value of fid is plotted in Fig. 8.8. In addition fid + 2n~, where n is an arbitrary integer, are solutions. These oth solutions are the propagation constants of the spatial harmonics into wn the Bloch wave may be expanded. The spatial harmonics are discuss SeC - 8'8h t feaThe ku-fi diagram for other types of periodic structures exhibit ^ tures similar to those in Fig. 8.8. For example, if the capacitive , 0 ^ £ ^ replaced by inductive shunt loading, the relative locations of the and stopbands are interchanged. The zero-frequency region will be '^ band since the shunt inductors will short-circuit the line at zero u *8.7 G R O U P VELOCITY A N D ENERGY FLOW The phase velocity for a Bloch wave in a periodic structure is gi v e oi feo M °P = With reference to Fig. 8.9, it is seen that k0d/fid is the slope < ( 8 - 43) PERIODIC STRUCTURES AhiD FILTERS 567 »0 d 0 50ff - 025TT- F1GURE 8.9 Enlarged drawing of first passbmd for a capacilively loaded transmission line, B = 2k0d. from the origin to a point P on the knd-fid diagram. Since fi is a function of w, the periodic structure has frequency dispersion. The group velocity v K as given by (Sec. 3.19) do) d(k0d) (8.44) is therefore different from the phase velocity. Again referring to Fig. 8.9, it is seen that the group velocity is equal to the slope of the tangent to the curve of k0 versus /3 multiplied by the velocity of light c. Thus we have vp = c t a n 4>,, vg = c t a n 4>g where <t>p and d>, are the angles given in Fig. 8.9. For the capacitively loaded coaxial line, use of the eigenvalue equation for 15, that is, (8.10a), enables us to obtain c sin fid dk, do (8.45) vg = c dft ~ c- d(lid] B \ B — — T + 1 sin kltd + —cos kX)d lk„d l I This expression shows that the group velocity becomes zero when /id = 0 or 77, except when kf) also equals zero. Thus, as the edges of the passbands are approached, the group velocity goes to zero. The group velocity is also the signal velocity for any signal consisting of a sufficiently narrow band of frequencies such that /? can be approximated by a linear function of w throughout the band. The signal delay r for propagation through a unit cell is given by d r= — (8.46) In Sec. 3.19 it was shown that for a waveguide, which is a dispersive medium, the velocity of energy flow in a propagating wave was equal to the group velocity. The same result will be shown to hold for a lossless periodic structure also. 568 FOUNDATIONS FOR MICROWAVE ENGINEERING Sc Periodic structure Unit cell S2 — F I G U R E 8.10 Se ^ ^ A unit cell of a periodic structure. Consider a unit cell of a lossless periodic structure as in Fie 8 10 & surface S is chosen to consist of surfaces S ; and S 2 at the input a rf output terminal planes plus a cylindrical surface S c surrounding the structure. If the periodic structure is enclosed by a perfectly conducting waveguide, the surface S c coincides with the waveguide wall. If the periodic structure is an open-boundary structure, the surface S c is that of a cylinder with infinite radius. In both cases n X E vanishes on S c , so that the Poynting vector is zero over this surface. For generality we shall let the unit cell contain regions with frequency-dispersive material, i.e., material with parameters n and e that are functions of to. In the derivation of Foster's reactance theorem in Sec. 4.3, it was shown that [see (4.24a)] dco ii dU* j) E X + dco dco x H • d S = -j( H • H = -4/(Wm dco <?<•)£ + E -E* (8.47a) We) since the latter integral is equal to four times the time-average energy stored in the volume bounded by S. Since the Poynting vector is zero on i>e and dS is directed inward, we have A / dU* dE? . E, X - = 2 - + -zrx H , |-a,dS dco dco IS_\E 2 X dn* XH 2 )-a.,dS=-4,(W m + We) (8.47« (1a> dco where E „ H ! are the fields at terminal plane 1 and E 2 , H 2 are terminal plane 2. For a Bloch wave, E 2 = E,e"-"", where pi » shift through a unit cell of length /. We thus find that dH* aE l da) m * - , ^ F dco "dot xH? ^ ^ ^ ^ + — - X H , + ^ du) Tdco E i x ^ PERIODIC STRUCTURES AND FILTERS 569 Consequently, (8.476) gives (note that the integral over S 2 can be evaluated as an integral over S,) dp f -2jt— Re [ E , X H t - a , dS = a co -'s. dp -4/7—P = a to - 4 / ( W,„ + W„) (8.476) where P = ^Re j s E, X Hf • a, dS is the power transmitted across a terminal plane. We now see that dco v g = dp (8.48) (Wm + Wt)/l But the energy density (Wm + WJ/l in a unit cell multiplied by the velocity of energy flow is equal to the power P, and therefore the group velocity is the velocity of energy flow. 8.8 F L O Q U E T ' S T H E O R E M A N D S P A T I A L HARMONICS It has been noted that in an infinite periodic structure the field of a Bloch wave repeats at every terminal plane except for a propagation factor e~yd, where d is the length of a unit cell. Since the choice of location of a terminal plane within a unit cell is arbitrary, we see that the field at any point in a unit cell will take on exactly the same value at a similar point in any other unit cell except for a propagation factor e~yd from one cell to the next. Thus, if the field in the unit cell between 0 < z < d is E(x, y, z),THx,y, z), the field in the unit cell located in the region d < z < 2d must be e-ydE(x,y,z - d),e"y,,H(x,y,z - d) Consequently, the field in a periodic structure is described by a solution of the form E(x,y,z) =e-y*El>(x,y,z) (8.49a) H ( x , y , 2 ) =e-**Hp(x,y,z) (8.496) where E^ and H p are periodic functions of z with period d; for example, E „ ( x , y , 2 + nd) = Ep(x,y,z) (8.49c) The possibility of expressing the field in a periodic structure in the form given by (8.49) is often referred to as Floquet's theorem.f From (8.49a) we tActually, Floquet's work dealt with differentia] equations with periodic coefficients. The case of periodic boundary conditions is an extension of thai work. 570 FOUNDATIONS FOR MICROWAVE ENGINEERING see that the electric field at z { + d is related to the field at ? E(x,y,zl +d)=e ^"WJx,?,^ + f0ll0W8: d) which has, indeed, the correct repetitive properties of a Bloch Any periodic function such as Ep(x,y, z) may be expan^fecTinfinite Fourier series; thus Ep(x,y>z) = Ep„(x,y)e->2"«/rf £ an 18.50) where Ep„ are vector functions of x and y. Multiplying both sid ^jimirz/d eJ— - g^fi anaintegrating integratingover over aa unit unit cell, ceil, i.e., i.e., from trom z2 == 00 to to dd!, eivp give *,J*<y) - yoX{x'y> *>*'*"""'* k (8.5i) since the exponential functions form a complete orthogonal set; i.e. fde j-lm,z/deJ2mTri/d fe = / 0 m * re ^0 ' \ rf m = re The field in a periodic structure can now be represented as x E(x,y,z) = Z - Epn{x,y)e--jl3*-j2n™/d E E p „(x,y)e-^ (8.52) where y = ,/"/3 and /3„ = /3 + 2mr/d. Each term in this expansion is called a spatial harmonic (or a Hartree harmonic) and has a propagation phase constant 0„. Some of the 0„ will be negative whenever the integer n sufficiently negative. The corresponding phase velocity of the nth spatis harmonic is w 10 (8.53) vnn = and will be negative whenever /?„ is negative. The group velocity of the harmonic is dp dm ven = dio dio -l (8.54) = v, t is see" tha* i'"** and is the same for all harmonics. From the above relations it ^g ^ some of the spatial harmonics (approximately one-half) have p e r t y is group velocities that are directed in opposite directions. This p ^ teT0 made use of in the backward-wave tz-aveling-wave-tube oscilla 0 - ^ ^ ^\ backward wave, or reverse wave, is often used to refer to PERIODIC STRUCTURES AND FILTERS 571 oppositely directed phase and group velocities. The voltage and current waves can, of course, also be expanded into an infinite set of spatial harmonics (Prob. 8.10). Although a Bloch wave can be expanded into an infinite set of spatial harmonics, all the spatial harmonics must be simultaneously present in order that the tota) field may satisfy all the boundary conditions. The eigenvalue equation for /3 for a periodic structure always yields solutions f}n = f} + 2n7r/d, in addition to the fundamental solution. These other possible solutions are clearly the propagation constants of the spatial harmonics. A complete kad-fid diagram thus exhibits k(ld as a periodic function of (id; that is, the fid curve is continued periodically outside the range — 77 s fid ^ TT. The slope of the line from the origin to any point on the curve still gives the phase velocity, and the slope of the tangent to the curve gives the group velocity, when multiplied by c. S T R U C T U R E S FOR TRAVELING-WAVE Traveling-wave tubes require a structure capable of supporting an electromagnetic wave with a phase velocity equal to the velocity of the electron beam. Since the latter is usually much smaller than the velocity of light, the required structure is commonly referred to as a slow-wave structure. A common type of slow-wave circuit used in traveling-wave tubes is the helix. The helix is treated in the following two sections, and hence this section is restricted to a discussion of some of the other types of slow-wave periodic structures suitable for use in traveling-wave tubes. A periodic slow-wave structure often used for the linear magnetron tube is the vane-type structure illustrated in Fig. 8.11. It consists essentially of a corrugated plane with thick teeth. It will be instructive to apply Floquet's theorem and carry out an analysis of this structure in order to illustrate the general techniques employed. Edge effects at * = ± o / 2 will be neglected for simplicity; i.e., we shall treat the structure as being infinitely wide. If a is large compared with the spacing b, and this in turn is small compared with A0, the edge effects will not produce a significant / / \b F I G U R E 8.11 Vane-type, or corrugated-plane, periodic structure. 572 FOUNDATIONS FOR MICKOWAVH ESdlSI-'.f.KiSU change in the characteristics of the ideal structure. For use in a mam a strong axial electric field is required, and hence we shall exam' ^ e possibility of having TM- or E-type modes. For TM modes having no variation with x, the field component be expressed in terms of the single magnetic field component H «.», present. We have V X H = ~ax X Vffx =jioe0E and so =0 (8.55 Q , bz E, = i-^ k0 9y (8.556) The field H, is a solution of a3 ;>2 (8.56) According to Floquet's theorem, the field / / , can be expressed in the form e~jPz\l>(y,z), where \}i{y,z) is periodic in z with a period d. Hence we shall assume that nx= L f„(y)i -JP„! where f}„ = fi + 2mr/d and the /"„(y) are functions of y to be determined. The substitution of this series into (8.56) shows that the f„(y) are solutions of d2f„(y) -02-*§)/"n(y) = o (8.57) Above the corrugations, i.e.. in the region c <y < b, we must choose the i so that E, will vanish on the perfectly conducting wall at y = b. Thus ^ require dfjdy = 0 at y = b. Since solutions to (8.57) are sinh h cosh hny, where hn = ( $ j - k'l)W2, we choose f„(y) =°n cosh k,,(b~y) • "For where a„ is a constant. At y = b, this function has a zero derivative^ the fields H x and E z in the region above the corrugations, we now n /« 58o I Hx= Z «., cosh hn(b-y)e~*«* *.58&' K--J-T upon using (8.556). £ anhnsmhk„(b-y)e-»'* PERIODIC STRUCTURES AND F1LTKRS 573 As a next step we must obtain a suitable expansion for H x in each corrugation, or slot. If H t (y, z) is the field in the slot extending from * = 0 to z = s and for 0 < y < c, then the field in the n th slot beginning at z = nd will be e~J'{"dHl{y, z — nd) according to Floquet's theorem. Therefore we need to concentrate on one slot only. We must determine Hx so that Ev will vanish at z = 0 and s and also so that E, will vanish at y = 0. A suitable expansion to use is * m 772 ,s m o since </[cos(mvz/s)l/dz vanishes at z = 0 and s. If this expansion is substituted into (8.56). we find that dsg>mf{y) ill mrr ~d? g,„(y) = o Normally, s <s: A0, so that mir/s > &„ for m * 0. Thus appropriate solutions that have a zero derivative at y = 0 are Smiy) = o ^ c o s h ^ y 2 where lm = [(m-/s) - &§]' "2 a n d °m 's a constant. For m = 0, the solution is gQ(y) = ft,, cos fej^y, and this part of the solution corresponds to a TEM standing wave in the slot. This mode has E v = 0. In the first slot we can thus write H, = * E b,„ cosh lmy cos m m -z (8.59a ) S II Z,, " K -j-;- E bjm sinh /,„ycos R 0 m=0 m —z (8.596) s The final step in the analysis is to determine the expansion coefficients a„ and b,„ by imposing boundary conditions at the plane y = c separating the two regions. We require the tangential electric and magnetic fields to be continuous across the gap y = c, 0 < z < s. In addition, we require the tangential electric field to vanish on the upper faces of the teeth, i.e., at y = c for s s z < d, or nd + s s z < (n +- l)d in general. Using (8.58) and (8.59), we see that the boundary conditions require * * mirz E a„e •"'"* cosh h„{ b - c) = E K, cosh lmc cos 0 < z <s n - - » m -0 (8.60a) E o^^-^sinhA^o-^) n= - = „ OTJT2 - E^A.sinh^ccos— >0 0 < ^ S s s z ^ d (g606) 574 FOUNDATIONS FOR MICROWAVE ENGINEERING If we multiply (8.60b) by «•**, we obtain «,,*« sinh h„{b - c)eJ2"Tr:,/'1 £ E K*mi, m-2 sinh /„,ccos- m=o f8.60c» S <Z <d Now the coefficients in a Fourier series are uniquely determined onlv T function which the series is to represent is specified for the co ' i interval over which the series is orthogonal. The functions e~' 2 "~- / d orthogonal over the range 0 to d, and thus, since the left-hand side*™? (8.60c) is specified for all z over one period, we can obtain unique exDre° sions for the a„ in terms of the b m from (8.60c). Note that this is not true for (8.60a), which holds only in the region 0 < z < s. If we multiply (8 60c) on both sides by e'-' rT7 -"' and integrate from 0 to d, we obtain (r is an integer) darhr sinh hr(b - c) m=0 = £ bjt rmrz sinh / „, (%M - a«r /d)z cosdz •'n sinh /,„c j{p m=0 2irr/d)[(-l)meJ"'* + (li + 2Trr/df - - l] (8.61) (imr/sf since / 0 V J2™/<'*'--'->dz = 0 for n * r and equals d for n = r. The above represents an infinite set of equations, i.e., one for each value of r. Although (8.60a) is not a unique equation for the a„, it does specify the o,„ uniquely in terms of the a „ . The a„ have already been expressed in terms of the b,„; so we may regard them as known. Multiplying (8.600) bv cos(r7T2/s) and integrating from 0 to s give s — 6, cosh lrc = £ rirz dz a„ cosh h„(b - c) I e -Jli„z cos- —3-|[l - (-1) e = -J E o „ c o s h h„(b - c) ^ -M+*"""\ + 2n1r/df - (rW») (8-62) 0. Th*8 lS where the Neumann factor e0r = 1 for r = 0 and equals 2 for r ^ ( g ' 6 l ) f l n d also an infinite set of equations since r = 0 , 1 , 2 , . . . , x - Equa ^j b„(8.62) constitute two linear systems of equations involving the PERIODIC STRIU T I R K S AND KILTERS 575 If the solutions for the a„ as given by (8.61) are substituted into (8.62), the result is a homogeneous set of equations for the bm. For a nontriviai solution, the determinant of this homogeneous set of equations must vanish. Setting the determinant equal to zero yields the eigenvalue equation for ji. However, the sets of equations are of infinite order, so that, in practice, an exact solution is not possible. Therefore we shall find only a first approximation to the exact eigenvalue equation. If the slot spacing s is small compared with A„, it seems reasonable to expect that the field in the slot can be approximated by the TEM standingwave field alone. Thus we shall take all b m except 6„ equal to zero. If we lump all the constants in (8.61) together and replace r by n. the equation is o[ the form > a„= L bmRma n = 0,±l,... (8.63a) m =0 Likewise, (8.62) is an equation of the form b,„= Z o„T„m m = 0,1.2,... (8.63b) Replacing a n by (8.63a) gives bm= L £ bmRmnTBm m =0.1,2,... (8.64) The determinant of this infinite set of homogeneous equations, when equated to zero, gives the exact eigenvalue equation for fi. When we take all 6 ni except 6„ equal to zero, we obtain instead 60 = £ n 1- £ b0Rl)„Tn0 -x /?o„T„«, = 0 (8.65) for a first approximation to the eigenvalue equation. Now i?„„ are all the constants in (8.61), multiplying b„ when the equation is solved for a„ and with r replaced by n. Likewise, Tn0 is the constant relating bu to the a, in (8.62). When these values for R 0 „ and T n „ are substituted into (8.65), we obtain 1 k0d tan k0c :-H s Uinfrt s / 2 1 d P„s/2 2 1 h,.d tanh h (b - c) (8.66) For slow waves /3 is much larger than kn. and hence /(„ can be replaced by /3„ in this equation with negiigible error. In this case the right-hand side is not dependent on k0. By evaluating the right-hand side for a range of assumed values for fi, the corresponding value of ka may be found by 576 FOUNDATIONS FOR MICROWAVE ENGINEERING J* A=A FIGURE 8.12 Simplified equivalent circuit for the vane structure of Fig. 8.11. solving (8.66). The numerical work is straightforward. Tynical r-^.u ts given by Hutter.t ^ A reasonably accurate description of the dispersion curve relating ft k 0 may also be obtained from a simple transmission-line analysis Th° region above the corrugations is essentially a parallel-plate transmission line (strip line) with a characteristic impedance Z, = Z 0 (6 - c) per unit width. The slots are short-circuited transmission-line stubs connected in series with the main line at periodic intervals d. The stubs present a reactance jX =jZ0s tan fc0c to the main line. The equivalent circuit of the structure is therefore of the form shown in Fig. 8.12. This periodic circuit may be analyzed in the same way that the capacitively loaded transmission line was. It is readily found that the eigenvalue equation for p is X cos fid = cos kad = cos kQd — -sin knd 2Z, (8.67) -tan A Q csin k0d This equation is quite accurate as long as s «d and also much small than A„. . For frequencies such that 0 < k0c < i r / 2 , the loading is inductive, ^ for 77-/2 < k0c < v, it is capacitive, etc. A typical dispersion curve lor case s = 2(6 - c) and d = 0.83c is given in Fig. 8.13. For these a"™*™ wr the phase velocity is reduced by a factor of about 3 only. M u c t l ^ & i a e reduction factors are obtained by making s/(b - c) larger so as t o . g o n the the normalized characteristic impedance of the stubs. For compar ^^ results from (8.66) for d = 2s are plotted also (broken curve), vervj^ ^ubs accuracy of (8.67). The first cutoff occurs approximately ^ h e n a J s 0 reduce become resonant, i.e., for kQc = ir/2. Increasing c will therelore tR. G. E. Hotter, "Beam and Wave Electronics in Microwave Tubes, Nostrand Company. Inc., Princeton, N.J., 1960. sec. 7.4. D V»J> -Zv -ir 0 rr PERIODIC STRUCTURES AND KILTERS 577 FIGURE 8.13 k0d-lid diagram for a loaded transmission line. stub- Zrr the phase velocity in the first passband, since (id will equal — for a smaller value of k0. The foregoing analysis is typical for periodic structures that cannot be represented by simple transmission-line circuits. The essential steps to be followed are summarized: 1. Obtain suitable field expansions in each region of the periodic structure. This involves solution of the Helmholtz equation and the use of Floquet's theorem. 2. Impose appropriate boundary conditions on the fields at all common boundaries separating the different regions. In general, it will be found that both E and H modes may be required in order to satisfy the boundary conditions. 3. By Fourier analysis convert the boundary conditions into algebraic equations for the amplitude constants. 4. The system of algebraic equations can be written in the form of a homogeneous set of equations. Equating the determinant to zero gives the eigenvalue equation for (3. Since the equations are usually infinite in order, some assumption must be made as regards the number of nonzero amplitude constants that will be chosen. Equating - the higher-order amplitude constants to zero results in an approximate eigenvalue equation. c Structures for Millimeter-Wave Traveling-Wave At millimeter wavelengths a helix has too small a diameter to be a useful slow-wave structure. Various forms of tape ladder lines, interdigital tape lines, and meander tape lines are preferred. Illustrations of these structures are given in Fig. 8.14. A discussion of these structures together with typical tA. F. Harvey, Periodic and Guiding Structures at Microwave Frequencies, IRE Trans., vol. MTT-8, pp. 3 0 - 6 1 . January. 1960. 578 FOUNDATIONS FOR MICROWAVE ENGINEERING Ladder .— Ladder Rectangular guide Ll (*l u 1_ Ridge guide (c) r £] (*) trf) F I G U R E 8.14 ( a - e ) Tape ladder lines; id) interdigital tape line; (e) meander tape line. k0-fi curves, is given in a paper by Harvey,t which also provides references to the original analysis of these structures. The two structures shown in Figs. 8.14 a 1 and e are complementary, or dual, structures. That is, the meander line is obtained by interchanging the open region and the conducting region in the interdigital line. For a complementary structure of this type, we can show t h a t the field is also a dual solution and hence both structures have exactly the same kQ-P dispersion curve. A detailed discussion of the dual properties is given below. Let the interdigital line be located in the xy plane as in Fig. 8.15. Let us consider a mode of propagation having an electric field for which t transverse U and v components) field E, is an even function of «r, that u>, the same on the upper and lower sides of the structure. Since V • E. - • • now have dEjdz = - V , • E„ and hence BEt/9z is an even function o . and E, must then be an odd function of z. From the curl « l u V x E = -j"oi/i 0 H, we can readily conclude that H, m u s t be an oaa ^ tion of 2 and Hz an even function of z. The field structure is llwstr_ ^ Fig. 8.15a. The field E, will vanish on the conducting surface, ana ^ ^ is an odd function of z, it must vanish on the open part of the *~v P 5 J l v jn the conductor surface, | H , | will equal one-half the total current ,- t h e the line since the total change in | H , | across the conductor musx total current density. A dual field E ' , H ' given by E'= ±Z0H H'= +70E PERIODIC STRl'CTDHKS AN!) FILTERS 579 F I G U R E 8.15 Illustration of dual properties of interdigital and meander lines. is easily shown to satisfy Maxwell's equations V X E' = -jto^H' V x H ' =./w€ tl E' if the field E , H does. The dual field is a solution to the meander-line problem (Fig. 8.156), provided we choose the dual solution E' = Z 0 H H' = - y 0 E (8.68a) above the meander-line plane and the solution E ' = ~Z0H H = 70E (8.686) below the structure. In both regions the primed fields satisfy Maxwell's equations. The field E', will vanish on the conducting portions of the meander line since the field H, was zero in the open regions of the interdigital line. Similarly, H'( vanishes over the open regions of the meander-line plane since the field E, was zero on the conducting surfaces of the interdigital line. All boundary conditions being satisfied, the solution is complete. It may now be concluded that both structures must have the same k0-fS dispersion curve. It should be noted, however, that duality applies only if the two structures are exact complements; i.e., superimposing the two structures must result in the whole xy plane being a single conducting sheet. The sides of the interdigital line must therefore extend to y = ± », and the line must be infinitely long. However, in practice, the field is 580 FOUNDATIONS FOR MICROWAVE ENGINEERING Ztra (a) -7 <j FIGURE 8.16 (a) A tape helix; (.6) sheath confined to the vicinity near the cuts, so that the sides do not have to extend much beyond the toothed region before they can be terminated with negligible disturbance of the field. The duality principle used above is often referred to as Babinet's principle. 8.10 SHEATH HELIX The sheath helix is an approximate model of a tape helix. The tape helix. illustrated in Fig. 8.16a, consists of a thin ribbon, or tape, wound into a helical structure. The pitch is denoted by p, and the pitch angle by i//. If the spacing between turns and the ribbon width are made to approach zero, the resultant structure becomes electrically smooth. At the boundary surface r = a, the boundary conditions for the electric field may be approximated by the conditions that the conductivity in the direction along the tape (air tion of current) is infinite, whereas that in the direction perpendicular to the tape is zero. The use of these boundary conditions permits a solution fc the electromagnetic field guided by the helix to be obtained with r ^ t " ' e ease. This anisotropic conducting cylinder model of a tape helix is canea sheath helix, illustrated in Fig. 8.166. The field solution, derived below, w show that the sheath helix supports a slow wave with a phase v p = c sin >li. The wave may be considered to propagate along the conductor with a velocity c, and hence progresses along the axial a with a phase velocity c sin i//. The sheath-helix model is valid at low e ^^ cies, where p is much smaller than A0. At higher frequencies a^ ^ realistic model must be used, and the existence of spatial harrnor becomes apparent, as shown in Sec. 8.11. .„ s sin<# The field solution for the helix consists of both E and H m ^ ] o n g the these are coupled together by the boundary conditions at r ~ «• ^ t he direction of the tape, the tangential electric field must vanish. ^m PERIODIC STKUCTURES AND KILTERS 581 conductivity in this direction is taken as infinite: thus fi^! cos ili + Ei{ sin tli = Etl,t cos tji + Ez2 sin I]I = 0 (8.69a ) where the subscripts 1 and 2 refer to the field components in the two regions r <a and r > a. The component of electric field on the cylindrical surface r = a and perpendicular to the tape must be continuous since the conductivity is taken as zero in this direction. Hence Ezl cos ili - £,,,, sin i]/ = Exi cos <// - EiU2 sin i/» (8.696) The component of H tangent to the tape must also be continuous since no current flows perpendicular to the tape; so a third boundary condition is Hzl sin 4i + H^j cos i// = H.2 sin i// + r/,, 2 cos i// (8.69r) Expansions for the E and H modes in the two regions r $ a may be obtained in terms of the axial field components E, and Hz, as shown in Sees. 3.7 and 3.18. The axial fields Ez = e,(/-,rf>><>L'^. H, = hAr,4>)e "" are solutions of Since we anticipate slow-wave solutions for which fi2 > kfly the solutions involve Bessel functions with imaginary arguments, that is, J„(ryftf, - iil) and Y^ryk2. - p2 ). In place of these functions, the modified Bessel functions In(ryP2 - k2, ), K„(ryp2 ~ /ef, ) are more convenient to use. These functions are related to the J„ and Y„ functions as follows: / „ ( * ) =./''•-/„(./*) (8.70a) K„( x) = jj""[ J„{ jx ) + jYJ jx) ] (8.706) For small values of x, the K„ functions approach infinity in a logarithmic fashion, and hence only the /„ functions are used in the region r < a. For r large, the asymptotic forms h(x) ^ihxeX K {x) " " (8 7l0) - ex fE ~ (8 716) - are valid. Since we require a field that decays for large r, only the functions Kn are employed in the region r > a. Suitable expansions for ez and hz in 582 FOUNDATIONS FOB MICROWAVE ENGINEERING the two regions are now seen to be Z a„e-J"AIn(hr) r<a * Z b„e-;"*Kn(hr) r>a c„e^In(hr) r<a dne-J"*Kn(hr) r>a n= — x Z h = « - i. Z where h = {fi* - t g ) " 2 and a„, 6„, c„, and d„ are unknown amplitude constants. For the sheath-helix model it is possible to find a solution for a field that satisfies the boundary conditions (8.69) for each integer n. We are primarily interested in the solution n = 0, which has circular symmetry. If we make use of (3.67), (3.68), and (3.72), together with the relations d h(hr) =/ (/ir) ^T ' dKn(hr) W = ~Ki(hr) we find that the field in the two regions can be expressed as follows: For r < a, Ez = o 0 / 0 ( A r ) e - - * * Er = J -^-a0Ix{hr)e^ Jfi IP H z = c0I0(hr)e-^ E„=- ~ ^ c 0 / , ( A r ) ^ e Hr = — c Z ^ r ) ^ !"> a , -w* H * = —h ^ ( f t r ) * (8.72o) For r> a, Et = b0K0(hr)e-*> Er Hz = Hr d0K0(hr)e~^ = = ft ~J±b0Kt(hr)e-^ -JAdKl(hr)e-^ ft E, = H, = J -^d0Ks(hr)e-jP' -J-^b0Kl(hr>t (8 .72ftl for the rc = 0 mode. . boun^* If the above expressions for the fields are substituted into ^^ f o r ary conditions (8.69), the result is four homogeneous ^ " ^ if the the constants a 0 , 6 0 , c 0 , and cf„. A nontrivial solution exists .^ tf,e determinant vanishes. Equating the determinant to zero res PERIODIC STRUCTURES AND FILTERS 583 1.0-1 0.5sin 10°- " 0 0.5 I.O , 1.5 »-*0a FIGURE 8.17 Phase-velocity reduction factor for a sheath helix with pitch angle i/> = 10°. eigenvalue equation for /3, which is ha) t a n 2 t/< K1(ha)Ii(ha {kQa)2 K0[ha)I„(ha) (8.73) For ha greater than 10, the ratio # , / , / / ? „ / „ rapidly approaches unity. In this region (8.73) gives h = k(l cot i//, from which we obtain ,1/2 n = (k20 + h2y/ =ft 0 csc// (8.74) The resultant phase velocity v p is V P= J = j c = c s i n * (8.75) and is reduced by the factor sin ill. A plot of vp/c as a function of k(la is given in Fig. 8.17 as determined by the solution of (8.73) with iji equal to 10°. For k0a greater than 0.25, the phase velocity is well approximated by (8.75). In the frequency range where vp = c sin i//, the group velocity vg is also equal to c sin i//, and there is no frequency dispersion. SOME GENERAL PROPERTIES OF A HELLX The tape helix consists of a thin ribbon of metal wound into a helical structure, as shown in Fig. 8.16a. A helix may also be constructed by the use of a round wire. The parameters describing the helix are the pitch p, or turn-to-turn spacing, the diameter 2a, and the pitch angle i//. These parameters are given in Fig. 8.16a, which shows a developed view of a tape helix. The helix is a periodic structure with respect to translation by a distance p along the axis and also with respect to rotation through an arbitrary angle 0, followed by a translation pti/2-rr along the axial direction. In other words, an infinitely long helix translated along the z axis by a distance p or rotated by an angle B and then shifted by a distance p8/2-nalong z will coincide with itself. 584 FOUNDATIONS FOR MICROWAVE ENGINEERING The above periodic, or symmetry, properties of the helix 1 restrictions on the nature of the field solutions. If E,(r, <ptZ) ^E a ° e cei are cylindrical coordinates, is a solution for the electric and E,(r,<ji + e,z + p0/27r) multiplied by a propagation factor e--»«M/2another solution, since the point r, <f> + 0, z + pd/2-n- is indisti from the point r, <£, z. The solution E ^ r . ^ . z ) must be periodic^lf 2 3 1 1 apart from a propagation factor e~jf>*, must also be periodic in 2 with* 3 n d " p. Hence E ( may be expanded in the double Fourier series E](r,<t>,z)= £ £ El„,n(r)c-^->2''«VJ/fe (8.76) where E 1 < m n ( r ) are vector functions of r corresponding to the usual amnl" tude constants in a Fourier series. The relationship between translation rotation noted above requires that eJfs'E^r, rf>, z) does not change when d> ^ are replaced by </> + 9, z + pB/2v. Thus, in (8.76), we require c -/m(rf. + H)-.;2;i7r(j \-pO/%ir\/p _ = g-jnUQ + ei-jnf)-j'ln-rrz/p g-Ji»4,—j2n?rz/p This condition will hold only if m = —n. Consequently, for a helix, the double Fourier series expansion for the electric field reduces to a single series of the form E,(r>,z)= £ ^ . . ( r j e ^ 2 " ^ ' ^ - * (8.77) The solution for a helix proceeds by expanding the field in the two regions r < a and r > a into an infinite series of E and H modes expressed in cylindrical coordinates. The boundary conditions at r = a will couple the E and H modes together, so that pure E or H modes cannot exist independently. For the rath term in (8.77), the radial dependence in t! region r > a will accord with the modified Bessel function of the second kind, t h a t is, Kn(hnr), of order n and with an argument 2nrr ft. r = -k\ The K„ functions are asymptotic to U/2h„r)>/2e h"r for r l a r ^ - f o r field will decay exponentially as long as all h n are real, i.e., g (0 + 2 n i r / p ) 8 greater than k%. When the field decays exponen corresponds to a surface-wave mode guided by the helix. g , are At a given frequency only certain discrete values of P, ^.'^'mode possible solutions. For each value of pm, corresponding to a par of propagation, the field is given by a Fourier series of the form (8.78) E m (r,*,z) = £ Em,„(r),->'2^-^'^ PERIODIC STRUCTURES AND FILTERS 585 Forbidden regions shaded FIGURE 8.18 Illustration of allowed and forbidden regions in the kn-p diagram for a helix. Each term in this expansion is called a spatial harmonic, and has a propagation phase constant /3,„ + 2mr/p. On a k^p-pp diagram the region above the lines kQ = ±/3 is a forbidden region, as shown in Fig. 8.18, since it corresponds to a situation where h0 = (fi'z - k\Yrz is imaginary and the n = 0 spatial harmonic does not decay in the radial direction. Since we also require |/3 + 2mr/p\ to be greater than k0, all possible allowed values of /3, corresponding to bound surface-wave modes, are further restricted to lie in the unshaded triangular regions in Fig. 8.18. The boundaries of these regions are marked by the lines 2nv * o = ± \P± where n is an integer. In the forbidden regions the propagation constant turns out to be complex rather than pure real, a feature which is different from that of a normal cutoff mode. A first approximation to the solution for a tape helix is obtained by assuming that the current is directed along the direction of the tape only, is uniform across the width of the tape, and has a propagation factor e -"*'. A typical k0-fi curve obtained on this basis and with </» = 10° is shown in Fig. 8.18. For further results the paper by Sensiper or the book by Watkins may be consulted.t INTRODUCTION TO MICROWAVE FILTERS The ideal filter network is a network that provides perfect transmission for all frequencies in certain passband regions and infinite attenuation in the t S . Sensiper, Electromagnetic Wave Propagation on Helical Structures, Proc. IRE. vol. 43, pp. 149-161, February, 1955. D. A. Watkins, "Topics in Electromagnetic Theory," John Wiley & Sons, I n c . New York, 1958. 586 FOUNDATIONS KOK MICROWAVE ENGINEERING stopband regions. Such ideal characteristics cannot be obt goal o[ filter design is to approximate the ideal requirements t ' ^^ c ^ e acceptable tolerance. Filters are used in all frequency ranges to* Wlt ^.' n an nearly perfect transmission as possible for signals falling with^' ^ ' ^ ^ passband frequency ranges, together with rejection of those H'"" d e s i r e d noise outside the desired frequency bands. Filters fall into t categories, namely, (1) low-pass filters that transmit all signals bet ^ m a " 1 frequency and some upper limit <oc and attenuate all frequencies ah* 0 ^ cutoff value « ( ., (2) high-pass filters that pass all frequencies above "\ the cutoff value w,. and reject all frequencies below W t , and (3) bandpass f j * er that pass all frequencies in a range w, to <o2 and reject frequencies out^H 3 this range. The complement to the bandpass filter, i.e., the band-reje< filter, which attenuates frequencies in the range m, to a>2, is also of int • \ in certain applications. At low frequencies the ''building blocks" for filters are ideal inductorand capacitors. These elements have very simple frequency characteristic* and a very general and complete synthesis procedure has been developed for the design of filters utilizing them. It is possible to synthesize directly filters with a wide variety of prescribed frequency characteristics. The filter design problem at microwave frequencies where distributed parameter elements must be used is much more complicated, and no complete theory or synthesis procedure exists. The complex frequency behavior of microwave circuit elements makes it virtually impossible to develop a general and complete synthesis procedure. However, in spite of these added complications at microwave frequencies, a number of useful techniques have been developed for the design of microwave filters. The case of narrowband filters is particularly straightforward since many microwave elements will have frequency characteristics essentially like those of an ideal inductive or capacitive reactance over a limited frequency range. In this case a low-frequenq prototype filter may be used as a model. The microwave filter is reahzec replacing all inductors and capacitors by suitable microwave circuit elements that have similar frequency characteristics over the frequency ran of interest. For this reason a good deal of the effort in microwave hie design has been based directly on the application of low-frequency synthesis techniques. . s jn There are essentially two low-frequency f i l t e r - s y n t h e s i s t e c h ^ ( a n d common use. These are referred to as the image-parameter me ^ ^ variations thereof, such as the constant-/? and in -derived f l l t e ^ t e r des ign insertion-loss method. The image-parameter method provides a^ ^ having the required passband and stopband characteristics, u inser tit>ne specify the exact frequency characteristics over each region. reaiizab^ loss method begins with a complete specification of a physica _^ s v n thefrequency characteristic, and from this a suitable filter netvyor -^^ 3 sized. The image-parameter method suffers from the s n o r * . C ° m i n order * good deal of cut-and-try procedures must often be resorted PERIODIC: STRUCTURES AND FILTERS 587 obtain an acceptable overall frequency characteristic. For this reason the insertion-loss method is preferable and is the only method considered in detail in the following sections. The image-parameter method is only briefly outlined, in order to show its relationship to the properties of periodicnetworks as already discussed. The labor involved in filter synthesis is largely obviated by the use of certain frequency transformations and element normalizations. These enable high-pass and band-pass niters operating over arbitrary frequency bands and between arbitrary resistive load terminations to be obtained from a basic low-pass filter design. The characteristics of any filter will, of course, be modified by the losses that are present in all physical network elements. To incorporate the effect of lossy elements into the synthesis procedure makes the synthesis theory a great deal more involved; so this is usually not done. At microwave frequencies losses can be kept reasonably small, to the extent that most filter designs based on the use of lossless elements do perform satisfactorily. The aim of the following sections is to present the essential features of low-frequency filter synthesis, frequency transformations, normalized filter design, and the applications of these techniques to microwave filter design. A number of typical microwave filters are also discussed. An extensive account of all aspects of microwave filter design is beyond the scope of this text. However, a number of selected references are given where further details may be found. 8.13 I M A G E - P A R A M E T E R M E T H O D O F F I L T E R DESIGN Filters designed by the image-parameter method have many features in common with those of periodic structures. As noted in the previous sections, a cascade connection of lossless two-port networks behaves similar to a transmission line. For unsymmetrical networks two characteristic impedances Z B ' = ±Z + £ occur, and each section has a propagation factor e±yd. A periodic structure of this form has passband and stopband characteristics and is therefore a bandpass filter. However, the proper load termination to prevent reflections is ZR and is complex when £ *• 0. Usually, a filter must operate between resistive load terminations, and it would not be possible to have matched input and output terminations in this case unless £ were zero, i.e., unless symmetrical networks were used or unless matching sections were used at the input and output. For this reason the imageparameter method of filter design is based on considerations somewhat different from those which have been discussed for periodic structures. Consider a single two-port network with parameters sf, *, %, and '/ Let the output be terminated in a load Z l 2 , and let the input be terminated in a load Zn, as in Fig. 8.19. For particular values of Z,, and Z, 2 , known as 588 FOUNDATIONS FOR MICROWAVE KNG1NEERING F I G U R E 8.19 Image parameters f0r _ . port network. the image impedances, the input impedance at port 1 equals Z and th port 2 equals ZlZ. These impedances then provide matched terminate !fe!! the two-port network, and if they are real, they also provide a mpower transfer when the generator has an internal impedance equal ti image impedance. The governing equations for the two-port network are Vl =.</K + .,• /, = Sf% +£>!, and hence V, _ .&V.Z +m2 _ sfZi2 +3g i, KZi2+& (8.79) If we solve for V2 and lz in terms of V, and /,, we obtain V 2 =£?V, - m T 2 = -&Vt +s^IJ t We thus have 7 A i«.2 _ - 9V} + Sfly wza +* (8.80) The requirement that Z,, = Z„, , and Z, 2 = Z-m2 gives Z,,(&Z iS + 0) = .a/Z, 2 + ^ Z l 2 ( STZ,, + st) = •SrZfi +« A simultaneous solution of these equations gives (8.81a) 2,,= 2l2 = v /, (8.816) Also we find that Z l 2 = l0/ss/)Zli. ^ ,I and I f a generator with internal impedance Z n i s connected a t ^ ^ ^ the output port 2 is terminated in a load Z, 2 , the voltage i transfer ratios are readily found from the relations I 2 = ~WV, +JTI, = ( - * f Z a + . » ' ) ' i PERIODIC STRUCTURES AND FILTERS 589 where V, is the voltage across the network terminals at port 1 (the generator voltage is 2V,). Thus we find that (8.82a) ~ = ^{{tfy-JIm) (8.82b) In a similar manner the transfer constants from port 2 to port 1 are found to be [or from (8.82)] V, /F" . v2 h y— (faa+ 499 (8.83a) fgT \j — (fas + JggW (8.8361 V .a/ The image propagation factor e~y is defined as e y = io/9> - 4&W (8.84a) whence it is found that e ' IB iw'9> + vfe«?if (8.846) cosh y = J$/@i (8.84c) s i n h y = J<M%' (8.84d) and The factor (y[a//@)2 is interpreted as an impedance transformation ratio and may be viewed as an ideal transformer of turns ratio fof/& . For a lossless network, sf and & are real and .£ and % are imaginary. In the passband of a filter, y is pure imaginary and equal to j/3, and this occurs for \jsf&\ < 1, as (8.84c) shows. Also, in the passband, the image impedances are pure real, whereas in a stopband they are pure imaginary, as the following considerations show. In a passband, SS and W must be of the same sign, so that .'M'i? = j\.<%\j\'*?\ = -\$g<&\ will make sinh y in (8,84a1) pure imaginary, that is, y =j(5. Thus, in (8.81), the quantity under the square root will be real and positive since sfSt must be positive to give a real solution for cosh y. Hence the image impedances are real in a passband. If N two-port networks are connected in cascade and these have propagation constants y„, n = 1 , 2 , . . . , N, and voltage transformation ratios r t-i/5 * t T.2,...,T„,...,TN 590 FOUNDATIONS FOR MICROWAVK ENOINF.EKINC, and the output section is terminated in an impedance equal to its image impedance, the overall voltage transfer ratio is YE \ 'N V,.V -TxTt TNe- •vi~vs~ -y.v 1 _ nr..-r. (8.•85) provided also that the output image impedance of any one section is eo the input image impedance of the adjacent section. With this filter net ' terminated in a load impedance Z, N equal to the image impedance of th output section, and with the generator at the input having an j n te i impedance Z,„ the overall network is matched for maximum power tran' fer. The filter operates between impedance levels of ZlN and Z wh"* provide an overall impedance-ratio change of amount Z ,A N Z,.\' Z.i,V - l Zrl nv »=! 18.86) If symmetrical two ports are used, </ = y and Z,x = Zi2, and both are equal to the Bloch-wave characteristic impedance Z'!f. For a symmetrical network no transformation or change in impedance level is obtained. The filter consisting of N symmetrical sections terminated in load impedances equal to the image impedance Z, behaves exactly like an infinite periodic structure, with its characteristic passband and stopband features. In the image-parameter method of filter design, the two-port parameters st, i&, &, & are chosen to provide for the required passbands and stopbands. In addition, the image parameters are also chosen equal to the terminating impedances at the center of the passband. The shortcomings o' the filter are now apparent, namely, the image impedances are functions oi frequency and do not remain equal to the terminating impedances over whole desired passband. This results in some loss in transmission (loss to mismatch) within the passband, an amount that cannot be prescribe determined before the filter has been designed. In addition, there i means available for controlling the rate at which the attenuation i n c r e a ^ with frequency beyond the edges of the passband, apart from increasu number of filter sections. Nevertheless, many useful microwave filter been designed on this basis, f , „u Freq" t S . B. Cohn. chaps. 26 and 27 in Radio Research Laboratory Staff, "Very « « " Techniques," vol. 2 McGraw-Hill Book Company, New York. 1947. Coille< For a discussion of image-parameter methods at low frequencies, see £•• "Communication Networks." vol. 2, John Wiley & Sons. Inc., New York, 1 9 3 ° PERIODIC STRUCTURES AND FILTERS 591 DESIGN BY INSERTION-LOSS METHOD The power loss ratio of a network was defined in Sec. 5.14 as the available, or incident, power divided by the actual power delivered to the load; thus 1 PLH = l _ r r 1 * - 737 (8.87) where T is the input reflection coefficient for a lossless network terminated in a resistive load impedance ZL = Rh. The insertion loss, measured in decibels, is L=101ogPLR (8.88) when the terminating resistive load impedance equals the internal impedance of the generator at the input end. In general, the insertion loss is defined as the ratio of the power delivered to the load when connected directly to the generator to the power delivered when the filter is inserted. The insertion-loss method of filter design begins by specifying the power loss ratio P L R or the magnitude of the reflection coefficient |T| ~ p as a function of w. A network that will give the desired power loss ratio is then synthesized. This procedure is seen to be essentially the same as was followed in the synthesis of quarter-wave transformers in Sees. 5.12 and 5.13. Indeed, the multisection quarter-wave transformer may be considered a particular type of bandpass filter. It must be kept in mind, however, that a completely arbitrary Hoi] as a function of w cannot be chosen since it may not correspond to a physical network. The restrictions to be imposed on f are known as the conditions for physical realizability, and some of these are discussed below. For a passive network it is clear that the reflected power cannot exceed the incident power, and hence one restriction on l'(w) is |r(w)|<l (8.89) If the normalized input impedance of the network is Z(w) = R(io) +jX(co) we have Zin-1 r ( w ) = •=Zin+ l R(w) - 1 +jX{») /?(«,) + 1 + JX(co) As shown in Sec. 4.4, ~R is an even function of 10 and X is an odd function of co. Hence g(a,)- W X(„) i (—01) = •= = R(w) + l-jX(M) — = r (10) ^ ' 592 FOUNDATIONS FOR MICROWAVE ENGINEERING and thus |T(«»)l*-**(•) =ir*-r(*>)r(-6») w and must therefore contain only even powers of at. Now anv WfctIonof impedance function (impedance of a network made up of resist" r e q u e r i c y tors, and inductors) can be expressed as the ratio of two poiynomi^31*' Consequently, T can also be expressed as the ratio of two polyn' S i" ( °' follows that />H a)) can then be expressed in the form °mial s . It 2 M( u>2) (•g-1) + X PZ M = (8.91) M(w*) + N(cu2) (R + if +X2 where M and N are real and nonnegative polynomials in loss ratio can now be expressed as M(u>2) _ Pi.« l.K = 1 2 w . The pow [fl(o) - l ] 2 + [Xja)]* N(<o2) ' 4R(a>) (8.92) The last result inj8.92) shows t h a t N(co2) must be an even polynomial in a> since it equals 4/?(«). Hence we write N(u>2) = Q 2 (w), which is clearly an even polynomial in w. If we denote M(ur) by the even polynomial P(ws) instead, we have P,. R = 1 + (8.93) The conditions specified on Put up to this point are necessary conditions i order that the network may be physically realizable. It may be shown th£ the condition that the power loss ratio P L R be expressible in the form (8.93 is also a sufficient condition for the network to be realizable.t In succe <j dl T| sections we consider suitable forms for the polynomials P and Q and types of networks required to yield the corresponding power loss rati 8.15 SPECIFICATION OF POWER LOSS RATIO could be There are virtually an unlimited number of different forms that ™woT^ specified [or the power ,ex power loss ratio and be realized as a physical compfe* However, many of these networks could be anticipated to b e v f 7 a v € and hence of little practical utility. The power loss ratios that ^ ^ ^ found most useful for microwave filter design are those that gi t G . L. Ragan fed.). "Microwave Transmission Circuits," sec. 9.13, McGrawpany. New York, 1948. PERIODIC STRUCTURES AND FILTERS 593 mally flat passband response and those that give an equal-ripple, or Chebyshev, response in the passband. Such passband-response characteristics correspond to those of the binomial and Chebyshev multisection quarter-wave transformers discussed in Chap. 5. The maximally flat filter (commonly called a Butterworth filter) and the Chebyshev filter are described below for the low-pass case only. In a following section it is shown that high-pass and bandpass filter characteristics may be obtained from the low-pass filter response by suitable frequency transformations, or mappings. ally Flat Filter Characteristic The power loss ratio for a maximally flat low-pass filter is obtained by choosing the polynomial Q equal to unity and choosing P(w 2 ) equal to k2(co/<oc)2N. Hence we have P L K = l + *2( — | (8.94) The passband is the region from 10 = 0 to the cutoff value <or. The maximum value of P IjR in the passband is 1 + ft 2 , and for this reason k 2 is called the passband tolerance. For w > wc, the power loss ratio increases indefinitely at a rate dependent on the exponent 2N, which in turn is related to the number of filter sections employed. A typical filter characteristic is illustrated in Fig. 8.20 for N = 2. ebyshev Filter The power loss ratio for the equal-ripple, or Chebyshev, filter is chosen as PLR=l + k 2 T 2 ^ (8.95) where TN(a)/(oc) is the Chebyshev polynomial of degree N discussed in Sec. Chebyshev Maximally Hat- F I G U R E 8.20 Low-pass-fiJter response for maximally flat and Chebyshev filters for N - 2. 594 FOUNDATIONS FOR MICROWAVE ENGINEERING 5.13. Recall that Tv| —) =COS|NCOS-1 — and thus TNi(o/w,,) oscillates between +1 for \w/coc\ < 1 an-i monotonically for w/w c greater than unity. The power loss" mcregsgg oscillate between 1 and 1 ++ kk2 inm the the passband, passband, equals equals 11 ++ k~ k2 at *u *^ ft _ - j -u ! .__ _ . _ • • 11 <. frequency, and will increase monotonically for w > u>c. A tvpica]' er ' CCUtntt ' e curve is shown in Fig. 8.20 for N = 22. Or.P One nartir-nWlv particularly <*„;u: striking featurer the Chebyshev response curve compared with the maximally flat curvp much greater rate of rise beyond the cutoff point. This means t h a t Y ^ corresponding filter has a much sharper cutoff" region separating the na band and stopband, which is usually a desired characteristic. For o>/<i)t large, the power loss ratio for the Chebyshev filter approaches al k2 /2(o LR l n 2JV (8.96) TUT Compared with the maximally flat response characteristic this is larger by a factor 22N'2. In fact, no other polynomial P(co2) yielding a passband tolerance of k 2 can yield a rate of increase of PLH greater than that given by (8.96). Conversely, for a specified rate of increase in the power loss ratio beyond the cutoff frequency, the Chebyshev polynomial gives the smallest passband tolerance. In this sense the Chebyshev filter represents an optimum design. The proof is similar to that used in Sec. 5.14 to prove the optimum properties of the Chebyshev quarter-wave transformer. When the power loss ratio is equal to 1 + ks, the magnitude of the reflection coefficient at the input is k P = (1 + A 2 ) The input voltage standing-wave ratio is given by 1 + p _ (1 +k2)U2 + k S = 1-P (1 + * 2 ) 1/2 -A If P is chosen as unity and Q is set equal to k2T$Uo/t»<X * btai** '^ 1 filter having Chebyshev behavior in the attenuation or stopband J^ h a v i o r » It is also possible to choose P and Q so as to give Chebyshev ^ow**8*' both the passband and the stopband. The required network is- ^^-av-e usually too complex to be realized in a satisfactory manner wit circuit elements. PERIODIC STRUCTURES AND FILTERS 8.16 595 SOME LOW-PASS-FILTER D E S I G N S The maximally flat and Chebyshev low-pass-filter power loss ratios discussed in the preceding section can be realized by means of a ladder network of capacitors and inductors in the form illustrated in Fig. 8.21. The load impedance is chosen equal to 1 12, and the generator impedance as R. The circuit in Fig. 8.21b is the dual of the circuit in Fig. 8.21a. Both circuits can be designed to give the same power loss ratio. For maximally flat or Chebyshev response in the passband, the ladder network is symmetric for an odd number of elements. This is also true for N even, in the case of the maximally flat filter. The element values are denoted by gk, and are the same in both circuits. However, the required generator impedance R' in the network of Fig. 8.216 is equal to 1/R. For N odd both R and R' equal unity. If we let Z m be the input impedance at the plane aa in Fig. 8.21a, the reflection coefficient will be r= z,n-R Z.„ + R In terms of Z i n and R, the power loss ratio is readily computed to be I2 in - J?|2 P,M = 1 + 2ff(Z i n + Z*) (8.97) At to = 0, all capacitors appear as infinite impedances and all inductors as zero impedances, and hence Z i n = 1. For a maximally flat filter or a -ntw<— 9z 9* 9B 9i 9% 9i (a) 97 ?5 ?i 9, 'WP—l—npSG^—f—^555^—t—^5W^?6 (A) F I G U R E 8.21 Low-pass ladder-prototype-filter networks. 9* 9z 596 FOUNDATIONS FOR MICROWAVE ENGINEER1NG Chebyshev filter with N odd, we must have unity power loss rat" 3 This requires that we choose the generator impedance equal to ' °' * 2 Un ty Chebyshev filter with N even, we have P I K = 1 + k at u> = ft ' - p or a u ' ^ d hence 1 +A2 = 1 + 4/? i? = 2k2 + 1 - v/ 4 * 2 ( 1 + £ 2 ) = — ' R' or (8.98) The required values of the elements g k in the ladder network w, obtained very readily by solving for Z-m and equating the power loss rat given by (8.97) to the desired power loss ratio for N up to 3 or 4 As° ^ example, consider the case of N = 2 for the circuit in Fig. 8.21a. We'readTfind that Zm =ja>L + 1 + jtoC where C = gx and L = g2. Using (8.97) gives (1 - R)2 + w 2 ( L 2 + C 2 fl 2 - 2 L C ) + *AL 2 C 2 J \ R = 1 4-ft To make PljR equal unity at to = 0 and to obtain maximally flat response, we m u s t choose R = 1 and L2 + C2 - 2LC = 0. If we specify cutoff to occur at w = 1 with a passband tolerance of k2, we also have 2 1 +& = 1 + L2C2 or LC = 2k. hence we find L = C = (2fc) 1/2 . For Chebyshev response we equate P , ^ to 2 1+ »2T2 w ^1 1 — = 1 + *>J fe - I "I in order to determine L and C. ^^ For large values of N the above procedure is very laborious t^ ^ n out. In place of this direct substitution scheme general solutions a^ ^ worked out.t In addition, tables of element values, i.e., values or have been prepared by a number of people.t . , 29 PP- l 0 * tV. Belevitch, Chebyshev filters and Amplifier Networks. Wireless Eng., volApril, 1952. T a b l e s . rW±L. Weinberg, Network Design by Use of Modern Synthesis Techniques an 195' Natl. Electron. Co/if., vol. 32. i956. ,196. February S. Cohn. Direct Coupled Resonator Filters, Proc. IRE, vol. 45, pp- 187 _1964See also " T h e Microwave Engineers' Handbook." Horizon House. I n c . '• PERIODIC STRUCTURES AND FILTERS 597 For the maximally flat network with a power loss ratio /> L R = 1 + " 2 N (8-99) t h e element values are given by R = l (8.100a) 2k - 1 gk = 2sm—-——7T k = l,2,...,N (8.1006) where g k is the value either of inductance in henries or of capacitance in farads. Each end of the filter is terminated in a 1-0 resistance. For a Chebyshev low-pass filter with w, = 1, the element values are given by gN+i (1 ~ \ 2k2 + 1 - 2*Vl + k2 Wodd N even (8.101a) (8.1016) When element gN is a capacitor, gN + 1 = R, but when gN is an inductor, gk = . where ak = sin ' (8-lOlc) 2k - 1 ——v ,. P , k-rr bk ~ sinlr —— + sin-2 -— * 2N N Vi + k2 + i p = In /T+ A2 -1 2a, gi sinh fi/2N Numerical values for the g k are given in Tables 8.1 and 8.2 for N up to 5, aic = 1, and a passband tolerance k2 equal to 0.0233 (a 0.1-dB ripple in TABLE 8.1 Values of g k for maximally flat filter N k 2 3 4 5 1 2 3 4 5 1.414 1.414 1.00 2.00 1.00 0.7654 1.848 1.848 0.7654 0.6180 1.618 2.000 1.618 0.6180 598 KOUNDATIONS FOR MICROWAVE ENGINEERING TABLE 8.2 Values of g k for Chebyshev filter with k 2 = 0.0233 N k 2 3 4 5 1 2 3 4 5 0.8430 0.6220 1.0315 1.1474 1.0315 1.1088 1.3061 1.7703 0.8)80 1.1468 1.3712 1.9750 1.3712 1.1468 the passband) for the Chebyshev filter. More extensive tables are givei the references cited. 8.17 FREQUENCY TRANSFORMATIONS The low-pass filter with cutoff at w,. = 1 and terminated in a 1-fl bad impedance may be used as a basis for the design of high-pass and bandpass filters with arbitrary resistive load termination. For this reason it is referred to as a prototype filter. For the purpose of this section, it is convenient to denote the frequency variable for the low-pass prototype filter by u>. The power loss ratio may be expressed in the form 1.K 1 + P(OJ'2) (8.102) for maximally flat and Chebyshev responses. If it is desirable to have a load termination RL different from 1 O, the required filter is obtained by multiplying all other reactances and the generator resistance by a factor R,. That is, the prototype-filter reactance can be viewed as normalized with respect to RL. The new values for t inductances and capacitances are (8.103a) (8.103b) (8.103c) (8 R=RLR ; where R' denotes the new value of R here, and not the generator " ^ f o r i n a . in the dual circuit of Fig. 8.216. In the discussion of frequency " . ^ ^ a n c e tions below, we assume a 1-il termination since the change i level of the filter can be made as a last step in the design P r ° ^ s ' w acCordIf we replace the frequency variable to' by a new varia ing to PERIODIC STRUCTURES AND FILTERS 599 the power loss ratio becomes PLR = 1 + P(a>'2) = 1+P[ f\m)] (8.104) As a function of a> or f, this power loss ratio is that of the low-pass prototype filter, but as a function of to, it has a different characteristic, depending on how /"(«) is chosen. A number of different frequency transformations, or mappings, are considered below. Frequency Expansion If it is desirable to change the cutoff frequency from unity to some other value o)c> we choose /H = and thus (8.1051 ^ L R = 1 Cutoff occurs when the argument w/w c equals unity or when m = <ou. The series reactances and shunt susceptances in the prototype filter must be replaced by new reactances and susceptances jX;=j{^-\Lk /Bi=/(—[i when a)' is changed to w/oi,. in order to yield the power loss ratio given by (8.105). Examination of the latter equations shows that the new values of the L k and C k must be (8.106a) L\-=± Q- c, (8.1066) The power loss ratios in terms of <o' and ai are compared in Pigs. 8.22a and b. * - P a s s tp High-Pass Transformation A high-pass filter is obtained by choosing f(w) equal to — <»c/o> to yield a power loss ratio LR 1 +P (8.107) a' 600 FOUNDATIONS FOR MICROWAVE ENGINEERING (4) FIGURE 8.22 •»-« ( a ) Low-pass prototype-niter response; (b) frequency exp a n s i o n ; (c) low-pass-tohigh-pass transformation, (d) low-pass-to-bandpass transformation. This frequency transformation maps the point w' = 0 into the points ID = ±*>, the points to = +1 into co = +w c , and the points u>' = ±<x. into co = 0. The effect is to interchange the passband and stopband regions, as illustrated by Figs. 8.22a and c. To obtain a power loss ratio of the form (6.107), the series reactances and shunt susceptances in the new filter must be chosen as follows: ior to. &Z = ~j—i* jB't = -j~ck CO CO From these equations it is seen that all inductances Lk must be replaced by capacitances C'k and the C k must be replaced by inductances L\ m following manner: P* = <»cLh L\ = (8.108a) 1 (8.108b) cZcl Low-Pass to B a n d p a s s Transformation To obtain a bandpass filter, consider a change of variable accordin co0 u> = f(to) co2 ~ (I) {8.lOS) PERIODIC STRUCTURES AND FILTERS 0 T^ i ^5^-H(_ y 601 FIGURE823 Series and parallel tuned circuits. This equation may be solved for to to give ( U ft) = 2- W I ft) ./ ,2|'W2-«V2 ± \l lt> If we also choose w'f, = o>|Co2, we obtain — 2 — lO] 1 / 2 2~~ to., a) <o = 0/ + —yio (a);, — o)y) + 4&),ft)2 (8.110) The point ft)' = 0 is seen to map into the points to = ±toQ, and to' = +1 maps into the four points ±(o>2 ~ tot)/2 + (w 2 + tot)/2 = ±ft)2 and +«>,. Thus the prototype-filter passband between ± 1 maps into passbands extending from ft), to o)2 and — ial to -ft>2, which represent bandpass filters with band centers at ±&0 equal to the geometric mean of <w, and a>2, as in Fig. 8.22c/. The required filter elements may be deduced by considering the frequency behavior of series and parallel connections of L and C, as in Fig. 8.23. For the series circuit we have ;X = 7ft)L + -^ = .Jn / ^ I WLC jwC V c \ to4W and for the parallel circuit we have JB' = jtoC + — mj-JT \W{LC jmL -'V L \ ' ftjyXC If we make vXC equal to (u 0 ', we obtain rn ((o ft)0\ .mi . rc~ (w «„ The required frequency transformation may now be seen to be obtained if we replace the reactance jXk = jtoLh and the susceptance jBk = j<oCk in the prototype filter by series and parallel tuned circuits such that 602 FOUNDATIONS FOR MICROWAVE ENGINEERING o TRHT'—|(—. TSUTT—| FIGURE 8.24 A bandpass-filter network. This requires that we choose 1 u kch 1 .2 0) l w 2 '-"n (8.111a) /t w0L 0> 2 - w, t or LA = (8.1116) for the series circuit. For the parallel circuit we m u s t choose 1 ^*CJk - 2 o>5 1 to 1 w 2 (8.112a) — Lk = o^C* = w2 - wx The resultant filter network is illustrated in Fig. 8.24. Period Bandpass Mapping -ariety of The general concept of frequency mapping may be applied in a v ,^ other ways as well. All that is necessary is to replace the r e a c t a n c e n c e and and susceptances jw'Ck in the prototype filter by other reac ^ ^ susceptance functions having <o' replaced by a new function /vw . rt-cirfurther example we shall consider the effect of replacing all w- ^ j a n C e 2. cuited transmission-line stubs of length / and characteristic '^L^cterist' 0 and the capacitors C k by open-circuited stubs of length ' a n d c PERIODIC STRUCTURES AND FILTERS 603 admittance Yk. The new reactance and susceptance functions become" jX'h = jZk tan | - / = jZk tan 0 = jgk tan 0 jB\ = jYk tan 0 = jgk tan 8 This filter has a power loss ratio given by PIM = 1 + P ( t a n 2 « ) (8.113) The frequency transformation w' = tan(wl/c) = tan 0 maps the whole <u' axis periodically into intervals of length -rr in the 0 domain or of length -c/l in the w domain. This filter would be satisfactory at low frequencies where junction effects at the points where the stubs are connected to the main line are negligible and where the separation between stubs could be kept very small in terms of wavelength.t At microwave frequencies the alternative occurrence of series and parallel tuned circuits in the filter derived from the low-pass prototype filter by a frequency transformation is an undesirable feature. It is difficult to construct a filter of this type using microwave elements. However, it is possible to convert the filter with series tuned elements into an equivalent filter containing only parallel tuned circuits in a cascade connection, or to convert the circuit into one containing only series tuned circuits. The desired transformation may be obtained by the use of impedance and admittance inverters or quarter-wave transformers, as discussed in the next section. IMPEDANCE AND ADMITTANCE INVERTERS An impedance inverter is an ideal quarter-wave transformer. A load impedance connected at one end is seen as an impedance that has been inverted with respect to the characteristic impedance squared at the input. Impedance inverters may be used to convert a bandpass-filter network of the type shown in Fig. 8.25 into a network containing only series tuned circuits. By using admittance inverters the bandpass filter can be converted into a network using only parallel tuned circuits. Furthermore, by choosing the inverters correctly all of the inductors and capacitors can be chosen to have the same values. Thus impedance and admittance inverters enable us tThis transformation is known as Richard/s theorem. i l f a filter is made up of transmission lines of commensurate length and resistors, the only frequency variable occurring is tan(w//c). In this case the frequency transformation iu = tan(<o//c) permits the use of conventional low-frequency-network synthesis techniques to be applied. 604 FOUNDATIONS FOR MtCROU'AVE ENGINEERING *«(») Yp(o» -« rf = 1 o— K=1 (a) Zs(<o) >/= 1 J=1 YP(<°) (b) FIGURE 8.25 la) Impedance inverter used lo convert a parallel admittance into an equivalent seriffi impedance: (6) admittance inverter used to convert a series impedance into an equivalent parallel admittance. to use identical resonators, either series or parallel tuned, throughout the network. In our initial discussion of inverters, we will assume that they are ideal frequency-independent elements. Later on we will consider the practical realization of inverters and how to compensate for their frequency-dependent characteristics. Consider the parallel admittance element Yp(a>) with an ideal impedance inverter with characteristic impedance K connected on D sides as shown in Fig. 8.25a. A short circuit at the output will be transformed to an open circuit in parallel with Yp. The input impedance given by K* Z in ~ li Thus the shunt admittance into parallel tuned converted into the inductance capacitance C j w C j d - wl/to'z) K Yp = Yp - Zs element with two impedance inverters converts t t e s ^ ^ an equivalent series impedance Zs(co) = YpUo>-^ 2 ' ^ ]g resonator with Y p = > C - j/coL =jotC(l ~ a%f%\ with a series tuned circuit with Zs =jioL(l " °*o/ ^ L in henries having the same numerical v a l u c e in farads. If we want to convert an admit j0ducinto a particular series tuned circuit with arbitra PERIODIC STRICTURES AND FILTERS 605 tance L, then we must choose K so that / K'VOJCAI\ «>n \ — | =ju>L GJ / 1 WQ - \ \ — <0 / Thus we can transform a shunt element into a series element having the same dependence on frequency and an arbitrary impedance level. Consider next a series element Zs(o)) with an admittance inverter with characteristic admittance J connected on both sides as shown in Fig. 8.256. An open circuit at the output is transformed into a short circuit in series with Zjo)) so the input admittance will be Ym =~ = J% = Y„ Thus the series element with the two admittance inverters will appear from the terminals to be an equivalent shunt admittance. If Z s is the impedance ju)L(\ - u>'o/w2) of a series tuned circuit, it is converted into a parallel tuned circuit with admittance Y= jwLil - wj/w 2 ) = jcoCil - to'i/w2), where C is numerically equal to L. For a series tuned circuit 7'a)L,(l - (O'Q/U>'2) to be transformed into an arbitrary parallel tuned circuit with Yp = ;'wC(l - ia\/m2), we must choose J2LX = C or From another point of view we see that a series element can be replaced by a shunt element with an impedance inverter at the input and output. Similarly, a shunt element can be replaced by a series element with an admittance inverter connected at each port. The admittance and impedance functions for parallel tuned and series tuned resonators can be expressed in the form The factor J(T/L is the admittance level of the resonator and y[L/C is the impedance level. The remaining factor is the frequency variable. 606 c FOUNDATIONS FOK MICROWAVE ENGINEERING 'TRRPK K C,L, L2 C,L, C2 L2 ,—/TJU^—1|— C,t, C2 —innp—|J— J J FIGURE 8.26 Use of inverters to convert a section of a filter into a network using only identical parallel series tuned resonators. We will illustrate the use of inverters to convert the circuit shown in Fig. 8.26 into one with two identical parallel tuned resonators or one with two identical series tuned resonators. For the first case we choose K so that K2jtoC^l - W'Q/OJ2) =./aiL 2 (l - a>oAo2) which requires that K = yjL2/Cl. For the second case we choose J so that J2jwL2(l - CO\/<D1) =jmCl(l co2)/to~) which requires J = yC]/L.2. Impedance and admittance level changing can be accomplished by using different input and output inverters as shown in Fig. 8.27. For example, in Fig. 8.27a the parallel admittance appears as a series element K'2Yp at the left-side port and as a serie element K2Yp at the right-side port. Similarly, in Fig. 8.276 the series element Z s is made to appear as a parallel admittance JfZs at the left-side port and as an admittance JjZs at the right-side port. We can modify t. impedance and admittance inverters to accommodate an arbitrary change I the impedance and admittance levels of a resonator. In Fig^ 8.2 ie impedance level y[Lx/Cx of the resonator is changed to ^Lu/Ca by c ing the impedance K' of the inverters to K, where K is chosen so tha K ,2 •JLJC-L or K2 /L0/C0 K- K" aioLy COQL.0 K = K V - mra- From the terminals the new circuit is equivalent to the o l d o n ^ . 8.270" we show a similar transformation of the admittance yj^i/ \ parallel tuned resonator to a value y/C0/L0 obtained by rep of t h e ^e PERIODICSTRVCTVRES AND FII.'IEII* e 607 —- Z^KfY. Y, K, K Y K2 Zin = j2 'm - J^Z. 2p (a) zs A &«</%. 1 0 (b) i L, C, —TRTtP—1|— • K |—iwr-—li- ° K- «.r^r fe) <-, J' 1 Jo c, -0 <--0 o J' J=J'. Co C, (tf) FIGURE 8.27 (o) Impedance level changing property of two impedance inverters; (6) admittance level changing properties of two admittance inverters; (c) equivalent basic sections with a series tuned resonator; id) equivalent basic section with a parallel tuned resonator. inverters with characteristic admittance J' by new inverters with characteristic admittance J = J'yJCQ/Cl . Impedance and admittance inverters are ideal quarter-wave transformers. There is no basic difference in their inverting properties. The only distinction that we make is to use the symbol K to denote the characteristic impedance of the impedance inverter and we use J to denote the character- 608 FOUNDATIONS FOR MICROWAVE ENGINEERING istic admittance of an admittance inverter. An impedance in F characteristic impedance K is equivalent to an admittance inv ^^ >rter characteristic admittance J = l/K. When K = J = l we \laJ ^th inverter and it does not make any difference whether we identify \w U n ' t j ' impedance inverter with K ~ 1 or as art admittance inverter with }*-?* ^ In Fig. 8.28a we show a bandpass filter terminated in a load r ~ l" RL and a source resistance R^. We will use impedance inverters ( this circuit into one using identical series tuned resonators with el mcn L 0 and C n and terminated at both ends in a resistance R. The transf C n t S tion process is easiest to understand by carrying it out as a successi ^ several intermediate transformations. We first introduce unity invertp ' convert all of the parallel tuned resonators into series tuned resonatn Thus, with all J, = K t equal to unity, we obtain the network shown in pit, 8.28o, where L\ = C, and Z/3 = C 3 . We now change the impedance level of resonators 1 and 3 to -JL0/C0 by changing the first and second inverters to inverters with K\ = K'2 = ^L0/L\ = yjLa/Cy and the third and fourth inverters to inverters with K'3 = K\ = ^L0/L':i = y]LQ/C%. The new network is shown in Fig. 8.28c. This impedance shifting is the same as that illustrated in Fig. 8.27c. As a next step we change the impedance level of resonator 2 from y/L 2 /C 2 = o>0L2 to u>0L0 by changing inverters 2 and 3 so that K? Kl o)0L0 <i>0L2 #3 0)QL0 ^3 ft) o^J and hence L0 , l Li The resultant network is shown in Fig. 8.28d. In order to change the load termination from R L to R, we modify first inverter so that K\ R Kf R and thus choose The last modification needed to complete the transforma change the fourth inverter to one with an impedance /Xo KA = K\ ' L4 ^ L\ y L4Cs PERIODIC STRUCTURES AND FILTERS (a) L4 L3 3 ' °' k^HH —-L^rSh C« K=1 K=1 K=1 K=1 (6) L4 C 4 <-o C 0 L2 C 2 | ^55!THt- -nm^-\\- 9 KS © _ t 0 Co, Kl K'2 *3 , ^ 0 0 0 ^-|p 1 T_ (O L4 L0 C 0 i-o C 0 C4 l-o C 0 | . -nm^-\\- kswMH K3 ^ —' 000 '—1|— Kt K', l-o Co -TSoTT,-||- l-oi Co : -^ooo^ii- } (d) / „ Cn l-o C„ ^6TSTH|- ^5WHh Ks K* K3 K2 [ K, 1 1 (e) S S S L 8 ? successive transforations used to convert the b a n d ^ s filter •" (^ ^ network with identical series tuned resonators and equal .nput and output term.na resistance R as shown in (e). 610 FOUNDATIONS FOR MICROWAVE ENGINEERING and to introduce a fifth inverter between R g and the fourth res place an inverter with impedance K'5 chosen so that r - If we *• -F-*we can replace R g by the new source resistance JR. After introrf f i f t h inverter t o accommodate the change i n source resistance w e " ^ t h e change it again when L 4 is replaced by L 0 . The new impedance K 5 must chosen so that *« and hence The final transformed network is the desired one and is shown in Fig. 8.28e. The required inverters that will transform the original filter network to the final form shown in Fig. 8.28e have impedances given by / LQR A/.C, K,2 = Jji-L (8-1W*) L 2 C, (8.114c) K3 = K4 = r2 lL0RRg K5 = (8.11*0 L4C3 L4 I L0R~ (8.H4e) V L-G* If instead of specifying identical resonators we choose L0v ^ Rg5 and L 0 4 for the new inductances, f l 0 0 for the new load resistanc^, ^ ^ ^ for the new source resistance (this is viewed as a parallel elemen PERIODIC STRUCTURES AND FILTERS 611 series of transformations would give Ki = = K-2 ^10 = ^21 = '-'01 " 0 0 L2Ci ^03^02 #3 K _ ^32 4 ~ #13 #5 (8.115a) Ct R L _ C 3L2 _ ~%U~ (8.115o) (8.115c) ^04^03 (8.115d) ^05^01 (8.115e) Ggi-4 These expressions exhibit a regular pattern and can, by induction, be extended to a network with a larger number of resonators or to one with fewer resonators. For example, if we have three resonators then in the expression for K43 we interpret L 0 4 as the new source resistance R M and interpret L 4 as the old source resistance. For a network with five resonators, we would obtain * M = "05 "04 ~C7L7 • " 06 ^ 0 5 and *65 = Rec5 The inverters shown in Fig. 8.28 can also be viewed as admittance inverters with J, = 1/K,. By using admittance inverters the bandpass filter shown in Fig. 8.28a can be converted to a network that uses only parallel tuned resonators as shown in Fig. 8.29. The analysis can be carried out in a manner similar to that used for impedance inverters as a series of elementary transformations of the type illustrated in Figs. 8.266 and 8.27d. The required expressions °»: Jn*\,N FIGURE 8.29 Equivalent filter network obtained by use of admittance inverters. 612 FOUNDATIONS FOB MICROWAVK ENGINEERING for the characteristic admittances of the inverters can be obt l expected, by interchanging the role of inductance and resistance win! of capacitance and conductance. It is readily found that Jl = / ^01^00 ^10 = 1 (8.116a) J2 = J 21 • 1 / (8.1166) L2Ct / ^03^-02 J, = J4 = ^43 = 1 -^32 = 1 (8.116c \j C3L2 / ^04^03 Js = (8.116d / & 05^04 ^54 = 1 \j (8.116e) C5Rg If we denote the load and source terminations by RL and Rg in the unmodified bandpass filter and by R0L and R0g for the transformed network, then the general expressions for Kkk.x and Jkik-\ f ° r a filter with N resonators are #io K*.*-! K,A . A - l ft N+l.N ftN _ "0t-^0*-l LkCk_1 fcodd "0*"0*-l A: even RogL'ON Nodd ^0g"«^,0A' + l.N (8.117a) " 0 1 **0L 'iV jV even (8.1176) (8.H7 C ) (8.H7 (8.n 7e? PERIODIC STRUCTURES AND FILTERS 613 and (8118fl) '"-VCKS; tf> '*-'sy t t * . ' feodd (8,1186) *•*-' = en V~L~C '-'ON '"--VmZfc Jw-UH-V p r V K og^.v Nodd N e v e n (8.118c) (8UM) (8.118e) The impedance and admittance inverters used in the preceding analysis were assumed to be ideal frequency-independent quarter-wave transformers of electrical length j r / 2 independent of frequency. Such ideal inverters do not, of course, exist. Nevertheless, the theory can be applied in practice. For very narrowband bandpass filters, say bandwidths of 1 or 2 percent, a quarter-wave length of transmission fine or waveguide does not depart appreciably from the ideal inverter having a i r / 2 electrical length. A filter designed on the basis of ideal inverters would have a response very close to the theoretical response in this case. For greater bandwidths the departure of a quarter-wave transformer from the ideal can be incorporated into the design by splitting the transformer into an ideal one with twt additional short lengths of transmission line on either side to account foi the excess or deficit in phase shift from the ideal phase shift of 90°. Fo: example, a transformer of length 0.3A0 at a wavelength of A0 can be treatec as a transformer of length 0.25A0 plus sections of line of length 0.025A0 oi both sides. The excess length of line may be incorporated into and mad part of the resonant circuit on either side of the inverter. Lumped-element circuits that act as impedance inverters and admi< tance inverters are illustrated in Fig. 8.30 along with their equivalen characteristic impedance or admittance. These circuits involve negativ elements and are frequency-dependent. However, the negative elemenl may be absorbed into the elements of the adjacent resonant circuits I eliminate them from the overall network. The resultant filter then consist of tuned circuits coupled by single reactive elements, as illustrated in Fi 8.31. Application of these techniques to microwave filter design is discuss* in the following sections. 614 FOUNDATIONS FOB MICROWAVE ENGINEERING FIGURE 8.30 I.umped-elernent inverters. Example 8.1. In this example we will design a two-resonator filter final form as shown in Fig. 8.32a. The filter is to be used in an intermediate^ frequency amplifier with center frequency of 10 MHz, a bandwidth of 0.5 MHz and having a load and source termination of 1,000 ft. The passband ripple is 2dB. (-^RRT I Li C * o I -c "1 fi i, -C Ll c-ct c-c, k o—omHh—If | l( j I H w o ^ X 1 ( T I i c I •~1 -HV-r. ' T-r> T 1 " 1 q ' ""/. AT i FIGURE 8.31 Reactance-coupled resonator-type circuits obtained by use of lumped-eleme inverted PERIODIC STRUCTURES AND FILTERS 615 R = 1,000 ft ? I L2 C 2 92 =r 9, $ 1 R- =rC,k-\ *; (*) L, C, C, L, 1 Ji -cc^= (O (e) FIGURE 8.32 (a) Bandpass filter; (6) low-pass prototype circuit; (c) bandpass prototype circuit; (d) transformed bandpass prototype circuit; (c) admittance inverter used to obtain the circuit shown in (a). We begin the design by first designing a low-pass prototype circuit with a cutoff radian frequency o>c = 1 as shown in Fig. 8.326. For a 2-dB ripple the maximum power loss ratio in the passband is 1 0 0 3 = 1.5849 = I + ft3. The parameter p is P = In \/TTki + 1 / T ^_i = 2.1661 From (8.101) we obtain R = 2fc2 + 1 - 2j%/l 4 k2 = 0.244175 for the lowpass filter input termination. We also obtain from (8.101) a, = sin-rr/4 = 0.7071, a 2 = sin 3TT/4 = 0.7071, 6, = 1.32306, 6 2 = 0.32305, and g x = 2.4881 and g2 = 0.6075. Hence, for the low-pass prototype circuit in Fig. 8.326, we have C, = gt = 2.4881 F and Z,2 = g2 = 0.6075 H, The values of a> at the band edges are <o, and to2. Since o>,<i>2 = u>% and u>2 - to, = 2ir x 0.5 x 10 B , we get <u,i = OJX + TT x 106 = ai%/oiy By solving this equation we obtain <ux = 6.1281 x 10 7 , w 2 = 6.4423 x 10 7 . The corresponding frequencies are 9.7532 and 10.2532 MHz. For the bandpass-h Iter prototype circuit, we use (8.1116) and (8.1126) to obtain C, = 2 . 4 8 8 1 / T T x 10 6 = 0.79199 MF and L 2 = 0 . 6 0 7 5 / C T x 10 6 = 0.19337 ,uB. The elements L, and C, are found using 616 FOUNDATIONS FOR MICROWAVE ENGINEERING <»%LiCl =ajlL2C2 = I and are Lx = 3.198 X W'10 H and C ~ i „ 2 10 9 F. " 1-3099 x By introducing two unity inverters, we can convert the resonator ator to a parallel tuned resonator with L'2 = C 2 = 1.3099 x in^9 t U n e d 3099 X 1 0 * H a n d CV, = L2 = 0.19337 M F . In order to shift the admittance level of thi from o)aC2 to OJQC|, we change lange the admittance inverters to ones w i » h e S ° n a t o r J, 7.9199 X 10 — = 2.0238 \ 1.9337 x 10 -«/,= For the admittance inverters we use the capacitive circuit shown in Fie 8 70 for which J x = J 2 = OJQCC; thus C c = ( 2 . 0 2 3 8 / 2 T T ) X 1 0 " 7 = 0.0322 M F Sin we want Jo use the same input and output terminations, we change J i so that R = 1 appears the same_to the network looking through the oriei' 1 inverter; thus J'2Z x 1 = J| x R so J 2 = 2.0238v'0.244175 = i. This l^st result shows that a 1-41 source resistance appears the same to the network ai the 0.244175-41 resistor seen through the inverter with J = J2. Thus we can discard the input inverter. We now scale the impedance level to 1,000 O by multiplying L, by 1,000 and dividing C x by 1,000. We also absorb the — C c in the remaining inverter as part of C,. Thus we obtain L = 0.3198 /xtl, C = (C, - C c )/1,000 = 760 pF. and C c = 32.2 pF, for the element values in the final filter shown in Fig. 8.32a. The double-tuned coupled circuit that we have designed is a classic one that has been widely used in low-frequency radio IF amplifiers. The power transmission coefficient is equal to 1/P L R and is shown in Fig. 8.33. If the circuit is designed for a maximally flat response, it is said to be critically coupled. When the response has a dip in the center, the circuit is overcoupled. The coupling coefficient for the circuit equals CC/(C + Cc). For critical coupling the coupling coefficient equals 1/Q = w0L/R = 0.02. For the circuit that we designed, the coupling coefficient equals 0.0406 so clearly the circuit is overcoupled. Two other well-known properties of this classic circuit are that 0 - PC/ co -10 •a c .9 1 1 -20 c a> < -30 h -40 \\ \\ Critically \ \ coupled \ \ filter 1 1 // \\ \ \\ X i \ \ i //Double . / tuned / / filter / / -.1 ' 0.8 1 0.9 1U W0 1.1 1.2 FIGURE 8.33 f r , the <*ou' Power transmission coefficient lor &&*** tuned filter shown in Fig. 83 , - ^ y coup^ curve shows the response for a en j fl o5/-„. circuit with a 3-dB bandwidth equaf PERIODIC STRUCTURES AND FILTERS 617 the 3-dB fractional bandwidth equals </2 /Q for critical coupling, and for an overcoupled circuit having a 3-dB dip at <o„, the coupling coefficient equals ( 1 + ^2J times critical coupling and the 3-dB fractional bandwidth equals 2i/2ll4/Q. l9 A MICROSTRIP HALF-WAVE FILTER In Chap. 7 we showed that open-circuited or short-circuited transmissionline sections one-quarter or one-half wavelength long were equivalent to either series or parallel tuned lumped-parameter LC circuits. Thus it is not surprising to find that sections of transmission lines are widely used as resonators in niters. Such structures are particularly appropriate for microstrip filters because of easy fabrication and low cost. In this section we will describe a microstrip filter that uses one-half wavelength open-circuited microstrip lines for the resonators. The resonators are coupled by means of the capacitance of the gap between the resonator sections. A typical half-wave filter consisting of three resonator sections is shown in Fig. 8.34. For an open-circuited transmission line the input admittance is given by Fjn=yy;tan^ We can expand Yin in a Taylor series about the frequency to0 where /3(w0)Z = TT; thus dp Yu, ~jYJ — [sec 2 /3(w 0 )Z](w - w 0 ) dco e rr<aQP'0 ft/ - ft>„ JW(«-»o) "JK 0n where /3'0 equals d/3/dio evaluated at circuit, 10L \ w<] For a parallel tuned LC w 1 = J«" 0 C (D — (Oc (w - w„)(w + w 0 ) = wc«0 2JOJ0C fc>o C 0) — a),, V L wn Thus, for frequencies within around ± 10 percent of ai 0 , the open-circuited transmission line is equivalent to a parallel tuned LC circuit if we choose C a> n 7r/3' n I - -ST* <8U9> 618 FOUNDATIONS FOR MICROWAVE ENGINEERING s, s2 s3 F I G U R E 8.34 A microstnp half-wave filter with three resonators. In the half-wave filter the open-circuited transmission lines can be used equivalent parallel tuned resonators. We can choose all of the resonato ' be identical provided we insert appropriate admittance inverters betw each resonator. The network shown in Fig. 8.35 functions as an admittance inverter It consists of two transmission lines with negative electrical length 6 and a series capacitor C g in the center. For this network a load admittance Y connected at one end is transformed into YL-jYrt YL = Y,c Yc-jYLt at the location of B = o>Cg and where t = tan 6. J u s t to the left of Cg, we have K> jBY-;n JB + Y{n and at the input Y,n yrK _ - m Y^YJi - 2BYt) +jY2[B(i -12) - m We now equate the coefficient of Y L in the numerator and the const term in the denominator to zero. Both conditions give t Y, -e 1 - = - tan 20 1 -t 2 (8.120a) 2 -0 F I G U R E 8.35 ^crostr-P A simple admittance inverter used in the filter. * * — PERIODIC STRUCT! KKr(AND KM ,TEES 619 W i t h t h i s condition i m p o s e d Y. Y'2t(2B-Yj) = — Y2tan-0 — = (Yc + 2Bt)YL (8.1206) Y,_ upon s u b s t i t u t i n g for B. When t h e r e l a t i o n s specified by (8.120) hold, we obtain a n a d m i t t a n c e i n v e r t e r w i t h J = yctan0 (8.121) T h e fact t h a t t h e n e t w o r k in (8.35) involves t r a n s m i s s i o n - l i n e sections w i t h negative electrical l e n g t h s does n o t c a u s e any difficult}' since we can a d d a length 6 at each e n d of t h e t r a n s m i s s i o n line a n d t h e n add l e n g t h s of electric l e n g t h ~ 0 t o s e r v e a s p a r t o f t h e i n v e r t e r n e t w o r k . Since b o t h B = <i)C„ a n d 8 a r e proportional to w t h e i n v e r t e r is n o t an ideal one. However, for n a r r o w b a n d filters J does n o t c h a n g e very m u c h over t h e passband t h a t the filter must operate. In t h e n e x t e x a m p l e we c a r r y o u t t h e design of a t h r e e - r e s o n a t o r filter of t h e type we j u s t described. E x a m p l e 8.2 D e s i g n of a t h r e e - s e c t i o n h a l f - w a v e b a n d p a s s filter. We will design a filter with a center frequency of 10 GHz, a bandwidth of 1 GHz. and having a passband tolerance k'1 = 0.1 which gives a maximum VSWR equal to 1.863. We first design the low-pass prototype circuit shown in Fig. 8.36a. From (8.101) we readily find that g, = C, = g:i = C;, = 1.5062 and g% = L2 = 1.1151. For the bandpass-filter prototype network shown in Fig. 8.366. we require C", = C,/( ( y 3 - w,) = C£ = ( 1 . 5 0 6 2 / 2 - ) x 10 9 = 239.72 pF. L'., = L 2 /(a> 2 - to,) = ( l . f l . 5 1 / 2 - ) x 10 9 = 177.47 pH. By introducing unityinverters we can convert the series tuned resonator to a parallel tuned resonator with C 2 = L'2 = 177.47 pF. We now change these inverters into inverters with J chosen so that J2/C\ = 1/C 2 or J = \JC\/Cl = ijC\/L'.z = 1.16221. The network in Fig. 8.36a* is the final one after we scale all impedances by a factor of 50 so that the input and output transmission lines will have a characteristic impedance of 50 i! instead of 1 SI. Thus we require C = (239.72/50) pF = 4.7944 pF, and J = 1.16221/50 = 0.023244. The ratio of the resonator admittance \jC/L = w 0 C to V", of the input and output lines is 2- X lO™ x 4.7944 x W~'2/0.Q2 = 15.062. If we use 50-fl transmission-line sections for our resonators, then the ratio of the resonator admittance given by (8.119) to the characteristic admittance of the input and output lines is a>„Trp'„/2lin. If we neglect dispersion, then 0 O = ^u>0/c and p0/pn = \/u>„, where c = 3 x lO" m / s is the free-space velocity of light. In this case the ratio equals - / 2 which is different from the required ratio of 15.062. This problem is overcome by inserting an admittance inverter between the input line and the filter and also one between the output line and the filter. These inverters are chosen so that the line admittance Y c will appear to be 15.062 times smaller than the 620 FOUNDATIONS TOR MICROWAVE ENH1SEKUISG La fi = 1 T=C, c, 4= ^fl=1 (a) Cj Ls j—TJTO"»—| LJ " 1* c; L\ II " Cj =4= =4= *1 (b) Co 1 a U, C'i =f= Lg 14 c, J' j' (c) 50 O 5012- (O FIGURE 8.36 <rf> final (o) Low-pass prototype circuit; (6) bandpass filter; (c) transformed bandpass niter, filter network. resonator admittance vtjt. Thus we require 4 TTV;. Yc. 2 x 15.062 y = 0.3229V, c/„ = "* V 2 x 15.062 '-f or If dispersion is important then we note t h a t dp ~d7o = d^.oj/c _ y/ej_ du> c a <^£ 2y/Tdc do) which now gives (8 2\ €, do 1'ERIODIC STRUCTURES AND KILTKKS 621 in place of (8.119) for the resonator admittance. This expression can be evaluated once the substrate material and microstrip width W have been chosen to give a 50-JI transmission line. The frequency dependence of e,, can be evaluated using Kobayshi's formula (3.176). For simplicity, we will neglect dispersion effects. Our last step is to design the required inverters. For the two middle inverters we require the ratio of J to the resonator admittance to be 0.023244/<,.„C = 0 . 0 23244/(2- x 0.047944) = 0.07716. Hence we must scale J from the value 0.023244 to 0.023244 x 7rF r /2w„C since we are using resonators with admittances equal to —Yc/2. Our new and final value for J is 0.07716-Y ( ./2 = 0.1212Y r . From (8.121) we find that we require 0 = tan ' 0.1212 = 0.1206. From (8.120a) we obtain B = O.Oltan20 = 0.00245 and hence a gap capacitance CA, of 0.00245/w o = 0.039 pF. For the input and output inverters, J0 = 0.323Yr = Yc tan l)n so 0O = 0.3124 and B 0 = 0.01 t a n 2 « 0 = 0.00721 which gives a gap capacitance Cg0 of 0.1147 pF. In this example we determined the inverter parameters from basic principles in order to illustrate again the theory involved. We could have made the evaluations more directly by using the general formulas (8.118). Thus we would have '/ c^Z =J ™ ~ J" Coi^m where C0l = C02 ~ TrYr/2t»0 is the chosen value for the resonator capacitances and C, = C\ and L 2 = L 2 are the values used in the prototype bandpass-filter circuit. The reader can readily verify that the above expressions give the same results that we found after a series of steps. The equivalent circuit of a symmetrical microstrip gap (equal width input and output lines) is a symmetrical capacitive n network shown in Fig. 8.37a. Therefore, when we introduce a gap capacitance C g for our inverters, we will also introduce a shunt capacitance C, at the ends of the two coupled fines. These capacitances are equivalent to a small effective lengthening of each line. For the two circuits in Fig. 8.376, we have Y,„ = > C , =jYc tan /3 M *>jYe0 M and hence the line will appear to be ,1 = g mm longer due to the capacitive loading. Thus we need to reduce the physical length of each resonator by an amount proportional to the capacitive loading at each end. For the two middle gaps we will let A/ be t h e length reduction required and for the two outer lengths we will let the length reduction be denoted by A/„. The designed filter is symmetrica) about the midsection. In Fig. 8.38e we show one-half of the filter with t h e various parameters that we have 622 FOUNDATIONS FOB MICROWAVE ENGINEERING (a) (o) FIGURE 8.37 (a) Equivalent circuit of a gap in a microstrip line; (6) equivalent transmission-line used to represent shunt capacitive loading. determined. At each end of the transmission-line sections, we add and subtract lines of length I) and - 0 or «0 and - 0o. The lines with negative electrical lengths are then part of the required inverter circuits. The additional lengths 6 and 0 O are absorbed as part of the resonators and require that the physical length of each resonator be shorter by the corresponding amounts. We must also account for the line lengthening due to capacitive end loading. The length of the first resonator is chosen so that 0l)ll + p0($-l0 + AO + 00 + 8 = ir; thus 'i = r-»t-$ - M0- M Similarly, we find that the required physical length of the second resonator IT - 26 -2M h= -p0(l2+2 Al)—~}g -B .. -0 6 • • II • • ' - li * (a) 2 mm * FIGURE 8.38 filter (a) Half-wave filter equivalent circuit; (6) metalization pattern tor nau microstrip construction on a substrate with thickness 0.5 mm and dielectric »*** ngtan t-1-2 PERIODIC STRUCT! IRES AND FILTERS 623 The third resonator has the same length as the first one. We can ignore the effects of A/ 0 and 0 0 at the input and output lines since the length of these lines are arbitrary. We will complete the design by assuming that the filter will be built using microstrip construction and a substrate 1 mm thick and with a dielectric constant e r = 4.2. This thickness of substrate at 10 GHz results in negligible dispersion of the characteristic impedance. The required strip width is 2 mm for a 50-fl line. The low-frequency effective dielectric constant is 3.2. By using Kobayshi's formula we find that e c = 3.37 at 10 GHz. The dielectric constant is about 5 percent larger at 10 GHz and since fi = \ft,.kn this will result in about a 2.5 percent shorter resonator length than what would have been specified if we used the low-frequency value of e... If we did not change the resonator lengths, the filter center frequency would have been about 2.5 percent below the design value. Since the filter bandwidth is 10 percent, this amount of detuning is significant so a satisfactory design would not have resulted if we had ignored the dispersion in tir For the two outer gaps we need a capacitance of 0.1147/2 = 0.0557 p F / m m . For the two inner gaps we need a capacitance of 0.039/2 = 0.0195 p F / m m . From the data given in Fig. 7.10c, we see that the required gap spacings will be very small, to the extent that the shunt capacitance C, is negligible; so we do not need to adjust the resonator lengths to compensate for C s . When C t in Fig. 7.10c can be neglected, t h e gap capacitance CA, = C 0 / 2 . For the inner gaps we thus require C 0 = 0.39 p F / c m , which is obtained using a gap spacing of about 0.22 W = 0.44 mm. The data in Fig. 7.10c do not give values for the capacitance for S/W less than 0.05 for which C„ = 0.6 p F / c m . For the outer gaps we require C 0 = 1.147 p F / c m . The data in Fig. 7.10 apply to infinitely thin strips. For very small gaps the strip thickness will be comparable to the spacing, so that the parallel plate capacitance of the end faces must be accounted for. If we use 1-oz copper-clad board, the strip thickness will be approximately 0.036 mm. If the gap spacing were 0.05 mm, the parallel plate capacitance between the two end faces would be C P P " eo 0.036 x 2 x 10 7T^ 0.05 = 0.0127 P F for a strip 2 mm wide. This amount of capacitance is about 10 percent of the required amount. Since the data for determining the outer gap spacing are not available, they need to be found either experimentally or numerically before the filter design can be completed. The resonator lengths can be specified since we can neglect A/ and A/„, Thus for the center resonator we require - - 20 77 - 2 x 0.1206 l2 = — ~ — = — p = = 0.754 cm /3 1/3^37 2 - / 3 The required length for the two outer resonators is /, = v - e - e0 = 7T - 0.1206 - 0.3124 = 0.704 cm The metalization pattern for the filter is shown in Fig. 8.386. 624 FOUNDATIONS FOR MICROWAVE ENGINEERING 0 - 1 • m v. - 1 0 § sa - -20 i g < -30 f it - a Ph* '-.id • ifilter 1 0.9 — i FIGURE 8.39 (a) Comparison of attenuation of half-wave filter and ideal Chebyshev filter in the passband (6) comparison of half-wave and ideal Chebyshev filter attenuations as a function of normalized frequency over an extended frequency band. The dotted curve shows the deterioration in the filter response when the two outer gap capacitances are reduced in value by 10 percent- Prom the attenuation of the microstrip line, which is approximately 0.0029 N p / c m , we obtain the intrinsic resonator Q = p/2a = 665. The e x t e r n a l Q of t h e first r e s o n a t o r is <o0C/G = <o0CYc/Jjj = (flT r 2 /2)/(0.323y c ) 2 = 15.06. Since this external Q is much smaller than the unloaded resonator Q, the losses in the resonators will have a negligible effect on the performance of the filter. The calculated insertion loss or attenuation for the filter is shown in Fig. 8.39 as a function of the normalized frequency o>/(o0. Also shown is the response or attenuation of the ideal Chebyshev filter which has the power loss ratio P I . „ = 1 + fe 2 r 3 2 (o»') = 1 + * 2 ( 4 » ' 8 - 3 a / ) 2 where o> = (a>/o>0Ka>/u>a - wQ/io) = U 2 - w 2 )/a> 0 . Since P J J , = l A 1 ~P^ where f> is the magnitude of the input reflection coefficient, the p< transmission coefficient 1 - p 2 equals 1/P L R . The filter insertion l o s s ^ attenuation in decibel units is given by 10 log P L R . In Fig. 8.39a we s*0Jf filter attenuation and that of the ideal Chebyshev filter in the P ^ s b ^ b y s h e v be seen that the filter response is very close to that of the ideal C e _ ^ ^ filter with the main difference being a somewhat smaller a t t e n u a j t s ov er OI/OIQ =» 1.025. In Fig. 8.396 the two responses are shown in decibel un ^^ the not-malized frequency range 0.8 to 1.25. The response of the ^ g( filter is remarkably close to that of the Chebyshev filter with the e * ^ £ u a t j 0 n somewhat greater attenuation below the center frequency and l e s s _ a 2 5 ) The above the center frequency (33.8 dB versus 40.9 dB at w/o>o " '"funCtion half-wave filter was designed on the basis that the a d r n i t t a n C ^ m a t e d W jYe tan pi for an open-circuited transmission line could be apP r(W PERIODIC STRUCTURES AND FILTERS 625 .'.• (a) cs Z\n *" Zc *-c \—p{l2*2M)—H 2 |- /?</,+A/+A/0) -| (b) FIGURE 8.40 (a) Illustration of point at which the impedance mismatch is evaluated; (6) equivalent circuit used to evaluate Z". jYcv(a) - w0)/io0 in the vicinity of the frequency at which the resonator was one-half wavelength long. Thus it is not surprising that there should be some difference in the attenuation at frequencies that are far from the center frequency. The results shown in Fig. 8.39 do, however, show t h a t the approximations made are acceptable and lead to useful filter designs. If we use w' = (a» — w0)/u>0 in place of the expression (8,109) in the Chebyshev polynomial, we will obtain a symmetrica) response about the point <o„ and a closer agreement with the response of the half-wave filter. The latter has a symmetrical response curve because tan 0 is symmetrical about H = 0. The impedance mismatch M at the input to the filter is equal to 1 - pl. In Sec. 5.7 we showed that in a lossless network the impedance mismatch is invariant throughout the network. Hence we can evaluate it at the center of the filter. If we let Z,J, and Z i n be the impedance at the center of the filter when looking toward the right and left as shown in Fig. 8.40a, then the impedance mismatch is given by A/ = 4^;/?- (R,:,)2 iz£ + z , j 2 " |z:i a since the filter is symmetrical so Z l n = Z'n. The evaluation <^f Z* is easily carried out using the circuit model shown in Fig. 8.40b. This method was used to evaluate the filter attenuation shown in Fig. 8.39 and is much simpler than evaluating the input impedance for the whole filter network. The construction of a filter requires good control on the electrical parameters if the theoretical performance is to be met. In Fig. 8.396 we show the response of the filter when the two outer gap capacitances are reduced in 626 FOUNDATIONS FOR MICROWAVE ENGINEERING H • b. # JT1 FIGURE 8.41 S A P a r a l l e l coupled microstrip filter value by 10 percent from the design specification of 0.1147 to 0.1032 DF F the figure it can be seen that the passband attenuation at u/a, = n°o increased to 1.43 dB which exceeds the passband tolerance of 0.414 dR I about 1 dB. Nevertheless, the overall filter response is still quite good 8.20 MICROSTRIP PARALLEL COUPLED FILTER The parallel coupled filter shown in Fig. 8.41 is more compact than the half-wave filter described in the previous section. Since the coupling between resonators occurs over a quarter-wave-long side of each resonator the slot width is larger and the tolerance on the slot width is not as critical. The ends of each resonator section may be open-circuited or short-circuited. The design of parallel coupled filters is readily carried out by using an equivalent circuit of the filter which is easily designed. This equivalent circuit for each pair of coupled resonators is derived below. Consider the pair of coupled microstrip lines shown in Fig. 8.42a. The strips will be considered to have unequal widths. When the voltage applied to each strip is the same (even mode) as shown in Fig. 8.426, the currents on the two strips will not be the same because of the different widths, We will let the currents be /, = YtaV and I 2 = YebV, where i^" and Ycb represent the characteristic admittance of strip a and strip b relative to the ground plane. For the odd mode with a voltage V applied to strip a and - V applied to strip 6, the currents are given by /j = Y°V and I 2 = ~ ".' where Yva and Ynb are the characteristic admittances of the two strips for the odd mode. We will show that the coupled-line circuit shown in Fig 8.42a is equivalent to the circuit shown in Fig. 8.42c, where Y x = y(Y0°-Ye°)(Y0b-Yj>) (8-l24a> r.-W + m-r, (8 1246) - The equivalence will be established by showing that the input admittan^ ^ port 1 with port 2 terminated in an open circuit or a short circui same for both circuits. _ ipa can The voltage and current waves on the circuit shown in Fig- • wjth be expressed in terms of a superposition of odd and even rno ^t voltages Va and Ve. Thus on strip a we express the voltage a*1 PERIODIC STRUCTURES AND FILTERS (D 627 >L (a) /T\ D y.-= — v.= '2 (ft) v,„ = v, (0 F I G U R E 8.42 ( a ) A pair of coupled niicrostrip lines; (6) illustration of even and odd modes; (c) equivalent circuit of coupled strips shown in (a), waves in the form Va(z) = V;e~Jlii + V~em + V*e-** + V„ e"5'' IA*) = Y?V?*~i0* ~ Y?V~eif>* + Y?V*e &* - Y°V„ eJIU It is assumed that both modes have the same propagation constants. On strip b the voltage and current waves are V*(z) = Ve-1** + IT**" - T*"*' - K «•** / 6 ( z ) = 1*V+e~** - YebV;e^ - Y0hV?e-J'<>* + y„*VnTe-"i2 When 2 = 0 we must have VA(0) = 0; thus (K.+ + V;) - ( V; + V-) - 0 (8.125a) since the end at 2 = 0 on strip 6 is short-circuited. If we place a short 628 FOUNDATIONS FOR MICROWAVE ENGINEERING circuit at port 2, we will require that at z = I, where /3Z = o which eivp.R gives ' *>'•) = (Vr+ - V:)e~j0 + {V; ~V,;)eJJB (8.1256) On strip a we must have VaU) = 0 since this is also a short-circuited end; thus ( v ; + v;)e ~J0 + (V; +\7)«-*«-o (8.125c) The last terminal condition is t h a t Va(0) must equal the applied vnlto , l a g e at port 1 which is Vx. Hence ^ = K , ( 0 ) = K+ + K - + V O + + VO- (8.125$) From the above four equations we can readily solve for V*, V~, V* and V~ The input admittance at port 1 is given by h V. Y.°{V; ~v;) + Ya°(v; -vo~) v: + v~ + v: + v- Vt = -j cot 6 (8.126) When we place an open circuit at port 2, the terminal condition Vb(l) = 0 given by (8.1256) is repJaced by 7A(/) = 0 or Y*{V?e'» - Ve~e^) - Y*(V?e-* - V;eJ") = 0 The other terminal conditions remain the same. After solving for voltage amplitudes for this case and using (8.126), we find that •*in.oc 2(YJ> + Y0h) J . •> ( yb ^ i + CTXt, (« yb For the network shown in Fig. 8.42c, a straightforward evaluation I that when port 2 is short-circuited 18 1280 Y>n.»:=-J(Yi + Y2)<:ot0 and when port 2 is open-circuited r,.=y^^tan^y^^^^cot, (»•*» Yi Yl + Ys PERIODIC STRUCTURES AMI FILTERS 629 From a comparison of (8.126) and (8.128a), we see that Y, + Y2 = y„" + Yf If we excite the networks at port 2 and place a short circuit or an open circuit at port 1, the expressions for Yin ^ and Y m (11, are the same as given by the above equations but with the superscripts a and 6 interchanged and the subscripts 1 and 3 interchanged. Thus, for the short-circuit condition at port 1, we will find that Fi + y» - Yh + Y* Hence we have n + Y," * i - • = = - 2= - = - - * i Ynh+ Y* y3=-^T^-y. When we use the expression for V, + Y-, in the coefficient for tan ti in (8.1286), we easily find that Thus we have shown that the circuit in Fig. 8.42a is equivalent to that in Fig. 8.42c. For the special case when the coupled strips have the same width, Yeb = Y° = Yc and Y* = Y° = Y0 and we obtain Y - Y Y, = — g - ± (8.129a) Yz = Y3 = Ye (8.1296) Consider now the filter structure shown in Fig. 8.43a. By replacing each coupled strip pair by its equivalent circuit in the form shown in Fig. 8.42c, we arrive at the circuit shown in Fig. 8.436. This circuit is readily reduced to the one shown in Fig. 8.43c by combining the admittances of the stubs that are connected in parallel. A Chebyshev filter based on the circuit shown in Fig. 8.43c is readily designed. For parallel coupled filters using microstrip construction, it is desirable to use open-circuited coupled microstrip sections in place of short-circuited coupled sections. A parallel coupled filter using open-circuited coupled transmission-line sections is shown in Fig. 8.44a. The equivalent circuit for this filter is shown in Fig. 8.446. A basic coupled section is shown in Fig. 8.44c and its equivalent circuit is shown in Fig. 8.44rf. 630 FOUNDATIONS FOR MICROWAVE ENGINEERING The equivalent circuit for the basic section is readily obtj set of equations (8.124) through (8.129). If we regard all voltage v "* representing currents, current variables as representing V n i * » - ^ l a ^ e s voltages, tances as representing impedances, and open circuits/short ' ' admit. • ' representing short circuits/open circuits, then all the equations and"' 1 termi nal conditions remain the same. Thus we obtain immediately Z^ = Z-i = \{(z°~Za0){Zhe-Z>) \{Zl+Zl)~Z, zz = \w.+z:)~zl (8.130a) (8.1306) (8.130c and when Z« = 2* Zft = Z\, 2,= z.. - z„ (8.13la) z., = z« = z„ (8.1316) The equivalent circuit shown in Fig. 8.44b for the filter is obtained by replacing each coupled section by its equivalent circuit shown in Fig. 8.44a" and combining the adjacent series-connected transmission-line stubs. This filter and its equivalent network is the dual of the one shown in Figs. 8.42 and 8.43. In extracting the square root of (8.130a) for the case of symmetrical strips, we choose the positive root which is the one given by (8.131a) since Z(, > Z„. For both of the parallel coupled filters described, each section is A/4 long at the center frequency and hence each transmission line is A/2 long. In the equivalent circuits of these filters, we have transmission-line stubs separated by A/4 transformers. These quarter-wave transformers function as nonideal admittance and impedance transformers. In the circuit shown m Fig. 8.43c, each short-circuited stub is the approximate equivalent of a parallel tuned LC resonator, while in Fig. 8.446 each open-circuited stub « the approximate equivalent of a series tuned LC resonator. Equations for designing Chebyshev bandpass filters, with bandwd up to one octave, using parallel coupled strips have been derived oy Matthaei.t The filter design equations were obtained by setting the i impedance and phase of each section of the network shown in rig. form equal to that for a bandpass-filter prototype circuit of the \e Fig. 8.29 at the center frequency « 0 and at the passband edge w ^ m = w,. The design equations for the filter circuit shown in Fig. 8. the dual of those for the filter circuit in Fig. 8.43c. We will give the equations below without derivation. t G . L. Matthaei, Design of Wide-Band (and Narrow-Band) Band-Pass Insertion Loss Basis, IRE Trans., vol. MTT-8, pp. 580-593, 1960. Microwave F i ! " « 0 0 PERIODIC STRUCTURES AND FILTERS 1 ya 63 1 \— ya *i 'e' 'o Y * G' Y . / * '< (3) y n a -yf v«-tf Yn-Y„ C K- v-'+y.? (0 F I G U R E 8.43 ( a ) A parallel coupled filter; (6) equivalent circuit; (c) reduced equivalent circuit. For a Chebyshev filter with N sections, there are N + 1 impedance inverters and N + 1 even- and odd-mode line impedances to specify. The filter is assumed to be terminated in input and output lines with characteristic impedance Zc. Each resonator has an electrical length of IT at the center frequency co0. The frequency at the lower edge of the passband is w, and file = 0j at this frequency, where l c is the effective length of each resonator after correcting for the capacitive end loading at each open-circuited end. The impedances of the input and output impedance inverters are 632 FOUNDATIONS FOB MICROWAVE ENGINEERING given by K in KN Z,. - 1, N (8.132,a) The following parameters are also required: TTto, »l-- 2w0 K10/Zc P sin 01 = [UanO.-h S = (8.1326) (8.132c) 21 V 2 (Kl0/Zcf] <8.132rf) Itanft. + CX^/Z,)2 Z* = Zf+I ^ ( 1 + P s i n f l J Z,l = Z » * l - £ c ( l - P sin 0,) (8.132e) I (8.132/) The remaining impedance inverters and even- and odd-mode impedances are given by Z,. K A--l,i K Ntt + i,ft -2,. k = 1,2,..., N- 1 (8.133o) + 5 tan 2 0X (8.1336) (8.133c) (8.l33rf) In the above equations the g k are the element values for a loss-pass prototype filter with cutoff frequency coc = 1, and are given by (8.101). For the dual network all Kk + Uk are replaced by </* + i.*» d\ ^ ' replaced by Y*, and all 2* are replaced by Yc*. In the next example we will illustrate the application of the t equations to the design of a two-section Chebyshev filter. Example 8.3 Parallel coupled Chebyshev bandpass filter. Tte ^. will be designed for a 30 percent bandwidth, a passband tolerance a and a center frequency of 6 GHz. The input and output lines characteristic impedance of 50 ft. The filter is illustrated in Fig. 8.4 5a- ,ated The element values for the low-pass prototype circuit are yg" using (8.101) and are g„ = 1, g} = 2.33754, g2 = 0.62634, and g3 The 3.7321. The frequency at the lower band edge is f0 - 0.15 f„ = °.%\?& value of <?, is 0.85n/2 and sin 0, = 0.97237, tan #, = 4.165. By uS,n_S04i2»and (8.133) we obtain Kl0 = 0.65406Zr, K21 = 0.826447Z,, P sin <»i = PERIODIC STRUCTURES AND FILTERS 633 -26- fa) zc + z" 7 a * 7° Z-z; zb-z" zc-zc zi-4 m \A Id) FIGURE 8.44 (a) Parallel coupled filter using open-circuited coupled-line sections; (6> equivalent circuit of filter: (c) basic coupled-line section: (d) equivalent circuit of basic section. s = 19.9165, and JV 12 = 2.8865. The following even- and odd-mode impedances are found Zj = 70.64 Z] = 29.36 Z 2 = 73.95 Z 2 = 41.03 In the equivalent transmission-line circuit shown in Fig. 8.456, the line impedances are found to be Z, = Zxa, Z 2 = (Z,f - Z„')/2, Z 3 = Z„' + Zf, Z 4 <* ( Z 2 - Z 2 ) / 2 , Z 5 = Z3, Z 6 = Z 2 , and" Z 7 = Z,. For this filter it is a straightforward transmission-line circuit analysis problem to evaluate the impedances Z m = Zt'n at the center of the filter and evaluate the impedance mismatch (R,n)2/\Z,J . In Fig. 8.46 we show the computed filter response with that of the Chebyshev filter with power loss ratio I + k2T%{o>') and using to' = (u) — u>0)/io0 as the frequency variable. The design equations are approximate ones only. Since we are specifying a 30 percent bandwidth for a filter using" only two sections, it is not surprising 634 FOUNDATIONS FOK MICROWAVE ENGINEERING (a) Z, = Z , z7=z, Ze = Z , (b) FIGURE 8.45 (a) A two-section parallel coupled filter; (6) equivalent trans. mission-line circuit. that the actual filter performance does not quite meet the specifications. The response curve shown in Fig. 8.46 indicates that the actual filter has a 25 percent bandwidth but a more rapid increase in attenuation than the ideal Chebyshev filter has. We have also designed a three-section filter with the same passband tolerance and bandwidth. The calculated parameters are g, = g 3 = 2.5547, K w = KM = 0.62565, K2l = K32 = 0.672, P s i n 0 , = 0.39776, s = 20.21, Z\ = 69.89, Zl„ = 30.11. Ze2 = Z'f = 57.807, and Zf, = Z03 = 30.645. In Fig. 8.47 we show the computed response of this filter. The agreement is now much closer to that of the corresponding three-section Chebyshev filter. The difference in the attenuation does not exceed 0.3 dB over the frequency range 0.7/„ to 1.3 /"„. At the band edges where f= 0.85 f„ and 1.15 f0, the parallel coupled filter has an attenuation of 2 dB instead of the theoretical design value of 1.761 dB, which is a difference of only 0.24 dB. 5 I g -10 -20 0.7 0.8 0.9 1 f/fn 1.1 1.2 1.3 FIGURE 8.46 „rthetwo-^°* A comparison of the response oi ^ lde0) parallel coupled filter with that ^ fa u« Chebyshev filter. The P a s s b a I T,h e a U als 0.3dB and the fractional bandwidth PERIODIC STRUCTURES AND FILTERS 635 FIGURE 8.47 Response of a three-section parallel coupled filter. The passband tolerance is 1.76 dB and the fractional bandwidth is 0.3. JZ^L/'.'^Z Coupled-line sections 'JJL S / / / ,CEZ FIGURE 8.48 Shielded suspended microstrip line used for constructing parallel coupled Biters. The design formulas for parallel coupled filters are based on the assumption that the even- and odd-mode phase velocities are equal. Thus, if conventional microstrip-line construction is used. it. would be necessary to use a dielectric overlay so as to equalize the two phase velocities. The parallel coupled filter can also be built using strip-line or shielded suspended microstrip line as shown in Fig. 8.48. The suspended microstrip line employs a thin low dielectric constant sheet on which the transmission-line sections are printed. Since the dielectric sheet is thin, it has about the same effect on the phase velocity of both the even and odd modes. For parallel coupled strips a ratio ZJZn less than 3 is needed in order to avoid very close spacing hetween the strips. For the filters discussed in Example 8.3, the largest ratio was 2.4, so that these filters can be built using edge-coupled strips. When tighter coupling is required, the adjacent strips can be overlapped by printing every other transmission-line section on the opposite sides of the supporting dielectric sheet. QUARTER-WAVE-COUPLED CAVITY FILTERS Quarter-wave-coupled cavity filters a r e s i m i l a r t o t h e f i l t e r discussed i n t h e preceding section except t h a t t h e t r a n s m i s s i o n - l i n e s t u b s a r e replaced b y cavities. T h e filter is realized in practice by placing d i a p h r a g m s in a waveguide. T o u n d e r s t a n d t h e basis for design i n t h e n a r r o w b a n d case, w e m u s t first consider t h e e q u i v a l e n t circuit of a section of waveguide loaded with two identical d i a p h r a g m s a distance l k a p a r t , as s h o w n in Fig. 8.49. 636 FOUNDATIONS FOR MICROWAVK KNGINF.KRINC; u e, =01, I \-jB» -jB, -> If) F I G U R E 8.49 I a ) Rectangular waveguide loaded with twr, indu tive diaphragms to form a cavity, (6) exact equivalent circuit. (<•) approximate equivalen! circuit. For a waveguide, the important frequency variable is not cu but rather ({3/k0)u> = fie, since waveguide diaphragms have susceptances that vary very nearly as /3 or jl~' and the electrical length of a section of guide is proportional to fi. The normalized frequency variable to/w0 = A0/A is therefore replaced by A g 0 /A g = /3//3 0 , where \g0 is the guide wavelength at w = o>0 and Afi is the corresponding value at any w. Consequently, in ail design formulas, we replace w by /3c, where c is the velocity of light. The exact equivalent circuit for a waveguide loaded with two identical inductive diaphragms with normalized susceptance -jBk is shown in Fig. 8.496. For filter design according to the methods developed in preceding sections, we must replace the exact equivalent circuit by an approximate shunt circuit. Mumford has shown that an equivalent circuit of the form illustrated in Fig. 8.49 c has the same frequency characteristics as the exact equivalent circuit has over a narrow band of frequencies around <wu-T *»e results obtained by Mumford are presented here without derivation. 1 derivation is straightforward, and may be found in Mumford's paper. The shunt susceptance B is expressed in the form B= where A/3 = 0 - /3 0 is small. When a resonant circuit of this 0T* ^ connected across a transmission line, it is loaded by a shunt condu 27. PP tW. W. Mumford, Maximally Flat Filters in Waveguides, Bell System Tech. J684-714, October 1948. v PERIODIC STRUCTURES AND FILTERS 637 normalized value unity on each side. The loaded Q of the circuit is thus 1 Q, = - ( ^ 0 1 fC c)C=-y- (8.134) since /30c = (LC)~' 2. Hence we may express B in terms of the loaded Q: thus B = 4Qk— (8.135) Po The value obtained for Q k by Mumford is jr-lan-1(2/Bt) Q* = 2 sin (^ + 4lf) 1/2 (B,! + 4B, 2 )' ~ 1 2 = ^— -+—tan" ^4 Bk (8.136) since B / ; is large compared with unity for a narrowband (high-Q) filter. The required diaphragm spacing / /; to give perfect transmission through the cavity at <u = w0 is given by tmfi0li = - w (8.137) The two sections of line with electrical length 0lk in the circuit of Fig. 8.49c are to be chosen so that / M A + 2 e u = § A + 20 l f t =TT (8.138) at the frequency » - , These additional lengths of line in the equivalent circuit of a single cavity are absorbed into and made part of the quarter-wave coupling lines in the filter. The design of maximally flat and Chebyshev filters with N odd is straightforward. If the prototype circuit of Fig. 8.29 is used, it is only necessary to make 1 Q k = ^\-r^ /C„, i<o* <8-139> and to choose Cok, L0k so that C0llL0k = ( 0 o c ) ~ 2 and all Jkk , equal unity. The section of waveguide between cavity k and k + 1 has an electrical length equal to - / 2 . Since this includes tt Ul . , and 0lk from the adjacent cavities, the physical length of the quarter-wave coupling line between 638 FOUNDATIONS FOR MICROWAVE ENGINEERING r^rr -JS, F I G U R E 8.50 Quarter-wave-coupled waveguide-cavity filter. cavities k and k + 1 will be 1 iir _ 0»H ij 1 *go I *k + h 2- ' 2 "21/ '*+'*+, 2 A 4'0 T4 (8.140) upon using (8.138). A schematic illustration of the filter is given in Fig. 8.50. Formulas for the required diaphragm dimensions to yield the specified value of B,. are given in Sec. 5.8. The power loss ratio for the filter is obtained by replacing io/(oa by / 3 / / V For a Chebyshev filter it is given by [see (8.104) and (8.109)] i.u = 1 +k2T2 Po P_ 00 .A -ft (8.141) where fi2 and /3, are the values of /3 at the edges of the passband. If /3, and 13., are specified, then It should also be noted that for the waveguide filter, w0, W], and a)> m tl design formulas (8.111) and (8.112) must be replaced by / V - Pic< a n d &'£' where c is the velocity of light; i.e.. replace k0 = co/c by /3. To illustrate 11 procedure, we shall evaluate the required susceptance —jB} for tne cavity in the filter shown in Fig. 8.50. . ^ Assume that a five-element filter will be needed. The response i s ^ of the Chebyshev type with a passband tolerance k2 = 0.0233. T ne ^ guide to be used has a width of a = 0.9 in. The passband is to exten ^ A = 10,000 MHz to f2 = 10,400 MHz. The corresponding values o to/c are 2.1 and 2.18 r a d / c m . The values of 0, and /3 2 are thus «.*«*'* fix = (2.1) - |a 0 a = [ ( 2 . 1 8 ) 2 - 1.89] i = (4.4-1.89)l/2=l-59 - = 1.7 PERIODIC STRUCTURES AND FILTERS 639 The center of the band occurs at p0 = (j6,/3 2 ) I/2 = 1.64, which gives /",, = 10,200 MHz. From Table 8.2 we find g x = 1.1468. Using (8.1126) gives ' 0 2 - ft /V 0 2 - 0, fi0C Since J , 0 is to equal unity and R L = i ? 0 / = 1, we have C„, = CJ from (8.118a). Using (8.139), we obtain For the feth resonator we should obtain Q-Jk-4L (8.143) For Q, we obtain 8.56. and from this result we can determine B, by using (8.136). For SA. large, we can replace tan~,(2/Bk) by 2/Bk, and we then find that B, = 2 ( Q f - l ) ' " (8.144) Thus we find that B, = 17. The required diaphragm dimensions can now be determined, and also the cavity length from (8.137). The above procedure has to be repeated for each cavity in the filter chain. D I R E C T - C O U P L E D CAVITY F I L T E R S Direct-coupled cavity filters have the advantage that the physical structure is more compact than the corresponding quarter-wave-coupled cavity filter. A design procedure for direct-coupled cavity filters that is accurate for bandwidths up to 20 percent has been developed by Cohn.t Cohn's design method is based on the use of the network in Fig. 8.28 as a prototype. The design formulas will be presented here without the detailed derivation. The waveguide cavity and its equivalent circuit shown in Figs. 8.49a and b may also be represented by a II network shunted with inductive susceptances at each end, as in Fig. 8.51. The two shunt susceptances B = -cot(0,./2) may be neglected^ compared with By. since Bk will be large, and 0^ is nearly equal to -r; so B is small compared with unity. The series arm X is thus used as the series resonant circuit in the prototype filter. For impedance inverters Cohn uses the shunt inductive reactance plus two short sections of waveguide (equivalent transmission lines), as shown in t S . B. Cohn, Direct Coupled Resonator Filters, Proc. IRE. vol, 45. pp. 187-196. February. 1957. 640 FOUNDATIONS FOR MICROWAVE ENGINEERING X = sin <?. la) F I G U R E 8.51 A waveguide cavity and its equivalent circuit. Fig. 8.52. For this circuit the impedance inverting properties are obtained "r 1 k i = --tan^ 2 B, 1 2 (8.145a) •=B„ (8.1456) K where K is the characteristic impedance of the quarter-wave impedance inverter. With 0 U and B k determined at a frequency w0, it is found that the inverter does not depart appreciably from its ideal characteristics over a 20 percent band. In the vicinity of to = to0, where 8,. = v, the series reactance X behaves as X = sin 9k = sin(fl, ~ tr + TT) = -(0,-77) = - ( p - 0 o ) / = ft,-/? rr( 0 ''""alA where /3 0 / = IT. This frequency behavior is similar (apart from the sign, which Js immaterial) to that for a series resonant circuit for which X = jL/C(a>/u>0 - « „ / « ) = 2{L/C(a> - to0)/to0 if « i / » 0 is replaced by H new frequency variable /3//3 0 . a h. When the negative line lengths of the impedance inverters are ^ sorbed as part of the cavity length, the physical length of the ktti i becomes i - A*° + ^ r « +e F I G U R E 8.52 An impedance inverter. ^ <8-l46) PERIODIC STRUCTURES AND FILTERS 641 -SM f^H h« -M -0i, F I G U R E 8.53 A direct-coupled waveguide-cavity filter. In the prototype circuit of Fig. 8.28, we must choose all ^Lok/C0/l equal to 7r/2 to obtain a correspondence with the type of series resonant circuit employed here. In addition, we choose ^Ok^Ok ($0C) ~ -2 (8.147) The impedance-inverter parameters as given by (8.115) thus become known in terms of the C k and Lk, which are related to the glt in the low-pass prototype. From the known values of the Zt^.l A.,the shunt susceptances B k may be found. The filter is illustrated schematically in Fig. 8.53. The design formulas obtained as outlined above are _ _ 1 - w/gl 1 (8.148a) yWtfi B2=- 2 \ l (8.1486) w * - » ' * - 1 - wR/gs-\ N = — F = = = = = ^ (8.147c) yJSkSk-i gkSk-\ R = 1 for N even (8.148rf) •/U'R/gN-l where •* P-i-Px W = 2 da and the g* are the element values from the low-pass prototype filter. Note that R = 1 for N even, and also for N odd in the case of maximally flat filters; otherwise R is given by (8.98). The length of the Ath cavity at /3 = 0o is -go * 2 "go i 4TT 2 _, I B + t a n - ' •=*+ i (8.149) 642 FOUNDATIONS FOR MICKOWAVK ENGSNZERiNC The power loss ratio is given by substituting P0 P, - Pi Po P for w' in the low-pass prototype filter response.! Note also that 8 a 0ifl2. 8.23 OTHER TYPES OF FILTERS In the preceding sections we have discussed only a few specific fib order to illustrate the insertion loss method of filter design. In add'r the maximally fiat and Chebyshev-type filters, there exist two other to ' that have the feature that, in addition to producing equal-ripple respon the passband, they produce a number of attenuation poles outside th* passband. These filters are the elliptic filters and the generalized Chebyshe* response filters. By producing attenuation poles, i.e., frequencies at which the attenuation is infinite, the skirts of the filter response curve are much steeper so the attenuation will increase much faster outside the passband It is much more difficult to find microwave circuit components that can be used to implement elliptic-type filters so these filters are not very commonly used. Another important filter topic that we have not addressed in the phase response of a filter. In order that the filter should not produce a distortion of the signal transmitted through it, the amplitude response should be independent of frequency and the phase response should be a linear function of u> so that the group delay will be a constant. If we express the transmission coefficient of the filter in the form A(w)e""'""", then the time delay experienced by a signal propagating through the filter is given by d<b/dw as shown in Sec. 3.19. Thus <!> should be a linear function of u> so as to give the same time delay for each frequency component of the signal. R filters only satisfy this criterion in an approximate way. For critical apphca tions, especially when broadband signals are involved, it may be necesss to insert a phase equalization circuit in series with the filter. The reader is referred to the references at the end of this chapter tor^ discussion of a number of filter topics that we could not include becai space limitations. PROBLEMS 8.1. Find the ../. rf^'S* matrix for_the following networks: (a I a s h u n ' , S y „ series jB. (b) a series reactance jX, (c) a shunt reactance jXx followed . reactance jX2. b:i" d*i tCohn uses a somewhat different frequency variable, which, however, (or the considered, is essentially the same as we have used. PERIODIC STRUCTURES AND FILTERS 643 8.2. Derive the relations (8.14) and (8.15). 8.3. Consider a T network terminated in a load Z. Evaluate the input impedance Z m and show that the condition that Z transforms into Z,„, that is. Z m = Z. leads to the characteristic values Z§ for the periodic structure. 8.4. Show that the eigenvalue equation for the propagation constant of a Bloch wave on a transmission line loaded at intervals d with a series reactance jX is cosh yd = cos kud - (X/2)sin kQd. 8.5. Show that (8.21) may be expressed in the form COS(/> cosh yd = —— l*»12l where <b is the phase angle of S 1 2 , and Sr, is the scattering-matrix off-diagonal element for the unit cell (Sec. 4.8). 8.6. Express \'B in terms of the A u by noting the similarity between (8.20) and (8.6) and that between I"e and ZB'. 8.7. Show that the wave-amplitude transmission matrix for a shunt susceptance jB is B 2+jB [A]' 4 + B2 B j -2 2(2 +jB) 8.8. A load ZL on an ordinary transmission line gives a reflection coefficient 1 = <Z, - \)/(ZL + 1). Show t h a t (8.30), giving VL fw a Bloch wave, may be expressed as r, - i •*- rj f'a-n i + J* i l - r» where rB" are the characteristic reflection coefficients of the component waves making up the Bloch wave. 8.9. For Bloch waves in the capacitively loaded coaxial line, show that the TEM voltage waves between any two consecutive diaphragms are given by V = y-e-./*(iU-»rf> + y ejk„iz~nd) _ y-t-g 7*n<* "d) + piy-gy*«l.- "<*> for the Block wave propagating in the +2 direction, and V = fy«-J**f »<*• + VLI V,; ejk"u"d' for the Bloch wave propagating in the — z direction. The zeroth terminal plane has been chosen as 2 = 0 and VB = V*(l + V,', ), VB = V„'(l + Tfi ). 8.10. Consider an infinite transmission line loaded with shunt capacitive susceptances jB at z = nd, n = —cc to *. Show that the current and voltage waves that make up a Bloch wave are given by VB = V¥e~jk"! + V-ejh°s 1B = I'e Jk "' + l^ejk"- = V*e~jk"* - V eJk°* 644 FOUNDATIONS FOJt MICROWAVE ENGINEERING where V = - V ( l - e J ^ " ) / ( l - e**-""*) and 0 = k0d. Let V Vp(z)e J<" and expand Vp(z) into an infinite series of spatial harm • * that the relation between V* and K~ may be obtained by using xh^^' ? con that V„(z = d) = e •»>dV„lO) and that /? is given by (8.10a). diti 0 n 8.11. For the sheath helix show that the eigenvalue equation for the nth mode is I ' i V + nfia cot * ) * (k„ka2cot K";,(/ia)/;(/ta) Kn{ha)l„(ha) i!/)" 8.12. Consider an N-section filter made up of a capacitively loaded coaxial li unit cells. The filter is terminated in a resistive load R equal to th ' impedance at zero frequency, i.e., equal to 1. The generator at the input 1 an internal resistance equal to R. Show that the power delivered to the load given by p = V2 "- - .ReZin I* + Z-J where V is the generator voltage and Z i n is given by _R + Z, tanh yNd in ' Z{ + R tanh yNd and Z,, yd are the image parameters at any frequency. In the passband where Z, is real and tanh yd =j tan pd, verity that V2(l+t2) V*ZfR{\ + t2) 4R%2 + (R2 i Zf)t~ _l t = tan Npd R2-Z2 Thus show that the power loss ratio becomes V2/4R R2 - Zf " = 1 Pun- 2RZ, sin Npd Plot P, R as a function of «. in the passband for the case where .3 R = 1, and N = 4. See (8.10a) and (8.16) for expressions giving Z, - ** lid. Verify that in the stopband the power loss ratio is given by R * L R = 1 + 7 R ^ sinh 2 N n d 8.13. ( a ) For the two circuits shown in Fig. PS.13a show that the mismatch is the same. . . .n£hes8Bie (6) Similarly, show that the two circuits shown in Fig. P8.136 h mismatch, Thus in a filter the insertion loss does not change when an K = Z,, or J = Yt is inserted at either end of the filter. PERIODIC STRUCTURES AND FILTERS fl- Zm - ^in + zm /*ir R' 645 (flm-yxj M M la) € Y,„ = G,n+jB,r 6 (Gin-/S|n) M M (*»> FIGURE P8.13 8.14. Design a two-section lumped-element bandpass filter similar to that in Example 8.1. Assume that k 2 = 1 and that the bandwidth equals 0.1 f„. The terminating resistances equal 1,000 il and f 0 = 1 MHz. Verify that the coupling coefficient defined in Example 8.1 equals (1 + </2 )/Q, where Q is the loaded resonator Q. 8.15. Design a three-section lumped-element filter of the form shown in Fig. P8.15. Assume a passband tolerance k'~ = 0.5, a bandwidth equal to 0.1 /*„, terminating resistances of 1,000 ft, and a center frequency of 455 kHz. Plot the filter response using P, R = 1 + k2T$(ut'), where u> is given by (8.109). FIGURE P8.15 8.16. Design a two-section half-wave filter with a 5 percent bandwidth and a passband tolerance k 2 = 0.2. The frequency of operation is 5 GHz. The filter uses 50-ft transmission lines. Specify the three required gap capacitances. 8.17. Design a three-section half-wave filter with the same specifications as in Prob. 8.16. Specify the required gap capacitances. 8.18. Design a two-section parallel coupled filter with a bandwidth of 0.1 f0 and a passband tolerance k'z = 0.2. Specify the required even- and odd-mode characteristic impedances for each section. Assume that Z r = 50 ft- 646 FOUNDATIONS FOR MICROWAVE ENGINEERING 8.19. Verily the parameter values given for the three-section parallel p c discussed in Example 8.3. °upled fill 8.20. Design a three-section parallel coupled filter with a bandwidth en i and a passband tolerance k'2 = 0.1. Specify all even- and odd-mode h° ° ' 2 c istic impedances. Assume Z c = 50 II "aract» 8.21. Design a three-cavity quarter-wave-coupled filter with the followm tions: waveguide width a = 0.9 in, band edges at f 1 = 10,000 M H ^ ' * " 10,400 MHz. passband tolerance It2 = 0.0233. Chebyshev response Ind % diaphragms with circular holes are to be used. Determine the hole rad""* diaphragm spacings. 8.22. Design a four-cavity direct-coupled cavity filter having Chebyshev response The passband tolerance is k 2 = 0.0233, band edges occur at f 1 = 9 500 MU. f 2 = 10,500 MHz, and the guide width is 0.9 in. Specify the diaphragin dimensions and spacing. Use any convenient inductive diaphragm. 8.23. Design a four-cavity direct-coupled maximally flat waveguide filter with the specifications given in Prob. 8.22. Note that the maximally flat filter designed from the low-pass prototype has a passband tolerance of 1. To obtain a passband tolerance of It'2 between 0, and 0 2 , the design must be carried out for a wider bandwidth, say 0', to 0'2. Thus we should have flo [ 0 0» 02 - 0'i \ 0o 0 for0=0'„0 2 = 1 Also 0',0 2 = li2. Determine 0 2 and 0j so that 00 = k2 02-0', 1.0 when 0 = 0, and 0 2 If the design is carried out using these values of 0", and 0'2, the requ passband tolerance of k 2 wjj) be maintained in the passband between 0, an 0 2 . Show that, in general, 0',0' 2 = 0 , 0 2 = 0« and 0'2 - 0\ = (0a _ 0 i , f t 8.24. For the circuits in Figs. 8.496 and c show that the normalized it impedances are given by -\,/2 (l-B2~2B„cotek) 1/2 and '2sm28u+cos2Bu^B\ 2sin20j* + cos20u + BI When w = ioQ, B = 0, show that B k must be related to 0 k by l ° - 1 ' Z, equal to unity. Show that the image phase constants <b for the are given by cos <t> = cos Qk Bk sin flk and and will be equal at w0 if 0A + 20u = IT. circU cos«i = c o S 2 ^ - Y S i n 2 " ' * PERIODIC STRUCTURES AN D FILTERS 6 4 7 ;FEB ENCES Periodic structures 1. Brillouin, L.: "Wave Propagation in Periodic Structures," 2d ed.. Dover Publications, Inc., New York, 1953. 2. Slater, J. C: "Microwave Electronics," D. Van Nostrand Company, Inc., Princeton, N.J., 1950. 3. Watkins, 1). A.: "Topics in Electromagnetic Theory," John Wiley & Sons, Inc., New York, 1958. 4. Bevensec, R. M.: "Electromagnetic Slow Wave Systems," John Wiley & Sons, Inc., New York, 1964. Microwave filters 5. Rhodes, J. D.: "Theory of Electrical Filters," John Wiley & Sons, Inc., New York, 1976. 6. Malherbe, J. A. G.: "Microwave Transmission Line Filters," Atech House Books, Dedham, Mass., 1979. 7. Matthews, EL (ed.): "Surface Wave Filters," John Wiley & Sons, Inc., New York, 1977. 8. Alseyab, S. A.: A Novel Class of Generalized Chebyshev Low-Pass Prototype for Suspended Stripline Filters," IEEE Trans., vol. MTT-30, pp. 1341-1347, 1982. 9. Mobbs, C. I., and J. D. Rhodes,: A Generalized Chebyshev Suspended Substrate Stripline Bandpass Filter, IEEE Trans., vol. MTT-31. pp. 397-402. 1983. 10. Williams. A. E.: A Four-Cavity Elliptic Waveguide Filter," IEEE Trans., vol. MTT-18. pp. 1109-1114, 1970. 11. Makimoti, M., and S. Yamashita: Bandpass Filters Using Parallel Coupled Stripline Stepped Impedance Resonators, IEEE Trans., vol. MTT-28, pp. 1413-1417, 1980. 12. Wenzel. R. J.: Exact Theory of Inter-Digital Bandpass Filters and Related Coupled Structures. IEEE Trans., vol. MTT-13. pp. 558-575. 1965. 13. Cohn, S. B.: Microwave Bandpass Filter Containing High-Q Dielectric Resonators. IEEE Trans., vol. MTT-16. pp. 218-227. 196S. CHAPTER 9 MICROWAVE TUBES 9.1 INTRODUCTION Microwave tubes are the prime signal sources in high-power radar systems. The magnetron is the tube most frequently used and can provide many kilowatts of continuous-wave (CW) output power and a megawatt or more of peak power with pulsed operation. Magnetrons are also used for industrial heating applications and in microwave ovens for consumer use. The traveling-wave-tube amplifier with power outputs up to 10 W or more is the workhorse in satellite communications. The klystron tube can function as an oscillator or as an amplifier. It can be designed for either low or high output power applications. In low-power applications the klystron was once widely used as the local oscillator in microwave receivers b u t has now been replaced by solid-state oscillators. Solid-state oscillators are replacing n crowave tubes in many low-power transmitter applications also. Even thoug many of the applications for microwave tubes have been taken over solid-state devices, the requirements for high power can only be me microwave tubes, so they are an essential device for many systems. Conventional low-frequency tubes, such as triodes, fail to °Pe[\ w microwave frequencies because the electron transit time from the cat the grid becomes an appreciable fraction of the period of the sinu ^ signal to be amplified. In other words, propagation times becomes S ' B " . ^ cant, and the same limitations that are inherent in low-frequency c ^ are present in low-frequency tubes also. Microwave tubes must be t o utilize the wave-propagation phenomena t o best advantage. .^ Broadly speaking, there are two basic types of microwave tubes,^ ^ that employ electromagnetic cavities (klystrons and some magnetr fi48 MICROWAVE TUBES 649 those that employ slow-wave circuits (traveiing-wave tubes). Both types of tubes utilize an electron beam on which space-charge waves and cyclotron waves can be excited. The space-charge waves are primarily longitudinal oscillations of the electrons and interact with the electromagnetic fields in cavities and slow-wave circuits to give amplification. The properties of cavities and slow-wave circuits have already been discussed. What remains to be done is to examine the propagation of space-charge waves on electron beams and then to consider the interactions that take place between electron beams and the fields in cavities and slow-wave circuits. The purpose of this chapter is to examine the nature of electron beams and the space-charge waves that they can support. In addition, the interaction of the beam with a microwave cavity or slow-wave circuit is to be studied in order to explain the operating principles of a number of different microwave tubes. Space does not permit a detailed treatment of the many different varieties of microwave tubes in existence. We shall concentrate on fundamentals that form, more or less, the basic operating principles of all microwave tubes. Two approaches may be used in analyzing the dynamic behavior of the electron beam. The earliest approach used was the ballistic, or lagrangian, approach. In this method the motion of an individual electron is studied in detail, and it is assumed that all other electrons behave in a similar way. The ballistic approach has the advantage of permitting certain nonlinear, or large-signal, effects to be treated fairly easily. The other approach is the field approach, sometimes called the eulerian, or hydrodynamical, approach. In this method the electron beam is essentially treated as a charged fluid. Field variables that describe the velocity, charge density, ac current, etc., at an arbitrary point as a function of time are introduced. However, no attempt is made to follow the motion of a single electron. The field approach, which loads to the space-charge waves, is more unifying and lends itself to the treatment of all different types of microwave tubes within the same general mathematical framework. Therefore only the field approach is used in this text. An exact analysis of a microwave tube would be very difficult and laborious to carry out. As in any other physical problem, it is necessary to introduce a number of simplifying assumptions in order to arrive at a mathematical model that can be analyzed without too many complications. The success of a simplified theory must then be judged by the extent to which it predicts and agrees with experimental results. The first few sections of this chapter discuss a number of models used for the electron beam and the propagation of space-charge waves on these beams. The governing equations are Maxwell's equations and Newton's laws, together with the Lorentz force equation. The equation of motion for a charge element is a nonlinear equation, but may be linearized by assuming small-signal conditions; i.e., all ac quantities are small compared with dc quantities. We shall consider only the small-signal situation since this will 650 FOUNDATIONS TOB MICROWAVE ENEINEBRISG suffice to develop the operating principles of microwave tubes I analysis is a great deal more difficult, and the theory, in general • ^ e " si ^ n 9l developed. ' s n o t fii After treating the dynamics of the electron beam, the klvst traveling-wave tube are examined in detail. A number of other t ^ also discussed, but in a more qualitative way. 9.2 ELECTRON BEAMS WITH dc CONDITIONS By means of a suitable electron gun consisting of a cathode, accele electrodes, and focusing electrodes, a beam of electrons with essenti U^ uniform velocity v 0 can be produced.! Figure 9.1 is a schematic illustratk) 3 of a cylindrical electron beam with a radius a. If the potential differen through which the electron is accelerated is V, the velocity u0 is given bv 2Ve\in = 5.93 x 1 0 5 V 1 / 2 m / s (9-1) where -e is the electron charge and m is the mass of the electron. For V = 1,000 V, vQ = 1.87 X 10 7 m / s = 0.0625c. The beam perveance is defined by the quantity TV~' 2 , where / is the total beam current. The coulomb repulsive force, or dc space-charge force, will tend to cause the electron beam to disperse, i.e., cause outward radial motion of the electrons. The space-charge force will be proportional to the density of the beam, i.e., to the number of electrons per unit volume. For the usual density of beams employed in microwave tubes (10 12 to 1 0 l o electrons per cubic meter), the dispersion of the beam due to space-charge forces is negligible if the drift space is short (d is small in Fig. 9.1). This condition exists in many klystrons, but in traveiing-wave tubes the beam must travel over distances which are so long that considerable dispersion may take place unless some means of keeping the beam together or focused is employed. The means by which the dc space-charge forces are counteracted leads to three common used beam models. These models are discussed below. Ion-Neutralized Beam Even with the high vacuum employed in a microwave tube, a gre£ >*>^ neutral gas particles are still present. Many of these gas molecules ^e ionized by means of collisions with the relatively high energy electro t T h e design of electron guns is not treated in this text. For a discussion of these ^ _ y i J. R. Pierce. "Theory and Design of Electron Beams," D. Van Nostrand ° Prince-ton, N.J., 3950. . tg4s. K. R. Spangenberg, -Vacuum Tubes," McGraw-Hill Book Company. New Yo MICROWAVE TUBES 6 5 1 Electron gun Beam ic J Anode or collector Ccthode 4HH— F I G U R E 9.1 A cylindrical electron beam. presence of positive ions will tend to neutralize the negative space charge of the electron beam. The positive ions, however, need not be considered in the interaction of a high-frequency electromagnetic field with the beam because their mass is at least 1,800 times greater than the electron's mass, and hence the ac motion of the ions is negligible by comparison with that of the electrons. Although all electron beams are ion-neutralized to some extent, complete electron space-charge neutralization is rarely achieved. However, for the purpose of mathematical analysis, a completely ion-neutralized electron beam is sometimes postulated as a model. Beam spreading due to spacecharge forces is discussed in Spangenberg's book. with Axially Confined Flow If a very large static magnetic field B 0 in the direction of the beam velocity is applied, the effect is to constrain the electrons from moving in the radial direction. The space-charge forces tend to impart a radial velocity to the electron. The magnetic field B„ produces a force - e v , X B 0 , which causes the electrons to execute circular motion about the magnetic field lines and thus prevents the beam from dispersing in the radial direction. In the magnetically focused beam the field B„ has its flux lines threading through the cathode surface, as in Fig. 9.2a. Some electron diffusion across the magnetic field lines will occur, but if B„ is made large enough, the amount of beam dispersion can be kept small. For the purpose of mathematical analysis, it is convenient to assume that B 0 is made infinite since in this case no electron motion in a transversedirection can take place. The analysis of the behavior of the beam under ac conditions is thereby greatly simplified since electron motion can now occur only in the axial direction (one-dimensional motion). The axially confined flow model is commonly used in the treatment of traveling-wave tubes. 652 FOUNDATIONS FOH MICROWAVE BM3MEERJNC Solenoid (mognct) Cothode Anode FIGURE 9.2 ( a ) Magnetic focusing for axially confined flow; (6) magnetic focusing for Brillouin Brillouin Flow In Brillouin flow (to be described), the axial magnetic field B 0 is not permitted to thread through the cathode surface. Since the field lines are continuous, they must move away from the beam region in the radial direction near the cathode, as shown in Fig. 9.26. When the beam enters the magnetic field region, it is given a uniform rotation at the Larmor frequency w, = eB0/2m by the magnetic field. In cyclindrical coordinates, r,4>,2 equation of motion for an electron, dv m dt = - e ( E + v X B) may be written in component form as I dd> \2 d2r 'dt e | _ dd> d26 "dF (9.2o) dt 1 dr dtf> + 2 e (9.2b) dr B * ~d7~~^ °~di d*2 lie =o dz ~dt (9.2c) = Un in the region where B = B0a, and is uniform. It is assumed t E = 0. The radial electric field may be found by using Gauss' law- ^^fc MICROWAVE TUBES 653 charge density of the beam is - p 0 , then 2—rDr = - 7 r r 2 p 0 , or E,. = - r p 0 / 2 e 0 . The radial space-charge force on an electron is thus -eEr = r e p 0 / 2 e 0 . If d2d>/dt* = 0, we find from (9.26) that (9 3) "*—'~te ' 2 If this solution is to satisfy (9.2o) and also make d' r/dt require rwf e [ m •f- 2 vanish, we r Po 2m e0 + rS a ••••',• 2 where «_ = ( e p 0 / / n e o ) 1 / 2 is called the plasma frequency. Typical values of o)p for beams used in microwave tubes range from 10 r to 10 9 . If the focusing field B 0 is chosen to satisfy (9.4), there will be no radial acceleration of the electrons. The equilibrium condition in the radial direction is actually a balance of outward radial forces — eE, due to space charge and mwjr due to centrifugal acceleration against the inward magnetic radial force ew/rB,,. Electron-beam flow under these conditions is referred to as Brillouin How. Although we have given the conditions for steady-state Briiiouin flow within the uniform B„ field region, we did not show that a beam leaving a cathode with a velocity v0a, will assume Brillouin-flow characteristics as it enters into the uniform B„ field region through the nonuniform field region in front of the cathode. To show this requires demonstrating that the change in angular momentum of the beam from an initial value of zero to its final value for Brillouin flow is equal to the time integral of the torque ev0Bar produced by the radial magnetic field component in the nonuniform region. The reader is referred to Brillouin's original paper for the derivation. t The conditions required for Brillouin flow can be achieved in practice. Even if the beam is partially ion-neutralized, as long as p 0 is not zero, a value for Bu such that (9.4) holds can be found. However, the behavior of a beam with Brillouin flow under ac conditions is more difficult to treat since transverse motion of the electrons is permitted. For this reason the ideal axially confined flow model is more commonly used. ff.. Brillouin, A Theorem of Larmor and Its Importance for Electrons in Magnetic Fields. Phys. Rev., vol. 67, p. 260. 1945. W. G. Dow, Nonuniform D.C. Electron Flow in Magnetically Focused Cylindrical Beams, Advan. Electron. Electron Phys., vol. 10. 1958. Pierce, op. ai. 654 FOUNDATIONS FOR MICROWAVE ENGINEERING In the magnetron-type (M-type) travehng-wave tube, a r beam is used. For this type of beam an analogous flow, referred t ^ S n e e l Brillouin flow, can take place. The properties of sheet beams are H S - P ' a n a r in Sec. 9.11, dealing with M-type traveling-wave tubes, and henc 1SCUSsed are covered in this section. not 9.3 SPACE-CHARGE WAVES ON BEAMS WITH C O N F I N E D FLOW This section is devoted to an analysis of space-charge waves on an a • confined electron beam inside a cylindrical waveguide of radius h TK^ radius of the beam is a, as in Fig. 9.3. Small-signal conditions are assumed8 The beam is considered to be uniform in density in a cross-sectional plane. The dc charge density is —p 0 , and the axial velocity is u 0 . The d current density in the z direction is J„ = -p0v0. The dc parameters p v and J a are independent of space and time coordinates. Under ac conditions' with time dependence e •""', there will be ac components of charge density velocity, and current that vary with time and the spatial coordinates. These ac components are denoted by p, v, and J. The ac fluctuation in electron density from the dc value N will be denoted by n, The electromagnetic field satisfies the equations V X E = ~jwfj.uH (9.5a) V X H =./we 0 E + J (9.56) V-E = ^ (9.5c) V -B = 0 (9.5c/) V • J = ~j(t>p (9.5c) A unit volume of the beam with charge density -p0+ P and charge-mass ratio n equal to that for electrons, that is, TJ = e/m, has a motion govern by the equation (N + n)m~ = (p - p 0 ) ( E + v, XB + v, X B„) (if. where N + n is the number of electrons per unit volume and v, = v o ^ the total velocity. For small-signal conditions, (Bl « | B 0 | ; so the ioi B0 24 * F I G U R E 9.3 Electron beam inside a cylindrical w a vecu' de - MICROWAVE TUBES 655 v( X B can be neglected compared with v, X B„. In addition, since B = /x0H = /j-0Yf,E = E/c, we see that |v, X B| is smaller than |E| by a factor v,/c. Hence ac magnetic forces are negligible. The total velocity v, is a function of x, y, 2, and t. In addition, the position x,y, z of a charge element is a [unction of time. Consequently, d\, dv, ^ - " dv, dx + ^dr^ dv, dv + dv, dz ^7d7 = ^ dv, + (v '-V)v' (9 6 - > since v, = ax dx/dt + aydy/dt + a.dz/dt. Thus expansion of d\,/dt leads to a nonlinear term <v, • V)v,, depending on of. However, for an ac velocity v that is small compared with the dc velocity v„, we have [(v + v 0 ) • V](v + v 0 ) = (v + v 0 ) • Vv = (v 0 • V)v (9.7) since v 0 is constant and the second-order term (v • V)v is negligible and may be dropped. Thus we obtain (AT + n)m T7 + ( v „ - V)v = ( p - , > 0 ) ( E + v X B 0 ) PC But the terms involving n and p are products of two ac quantities and may be dropped for small-signal conditions. Hence the first-order linearized equation of motion becomes dv — + (v0-V)v= - i j ( E + v x B „ ) (9.8) <</ since Ne = p0 and e/m = 77. For the cylindrical beam under consideration, we also have v 0 = a,v0 and a time dependence e•""'. If we let B n approach infinity, the transverse components of v must vanish, so that the term v X B„ in (9.8) will vanish. Thus v has a component in the z direction only, and (9.8) gives Hu jwv + u0— = -7)E, (9.9) dz When v has only a z component, the ac current density J has only a z component since the total current is J0 + J = ( -f,0 + p ) ( v 0 + v) = -p„V Q + (pV„ - />0v) + pv = -P0v„ + ( p v 0 - p 0 v ) (9.10) after dropping the second-order term pv, which is the product of two small ac quantities. The dc and ac currents are thus J o = -P0V0 J =pv0- P o (9-Ha) v (9.116) 656 FOUNDATIONS FOK MICROWAVE ENGINEERING From the continuity equation (9.5e) we obtain dj Hz = -Jwp Equations (9.9), (9.116), and (9.12) permit us to express J as a fun r Ez. Maxwell's equations (9.5a) and (9.56) may then be solved in th^ ' ° n manner to obtain wave solutions. Since we are looking for wave solutions, we may assume that li quantities have a z dependence e~jpz. In this case (9.9) and (9.12) ei VP (j« -JPv0)v = -TJE, jpj (9.13a) = jwp (9.136) For convenience, o>/v0 will be denoted by f30, which may be interpreted as the dc propagation constant for the beam. Using (9.116) and (9.13), we find that J- ~j Pl^Es (9.14) » (Po-P)' where iop = p0v/eo ' s the plasma frequency squared, and — p 0 is the electron charge density of the beam. To solve Maxwell's equations for the beam inside a cylindrical guide, it will be convenient to introduce the vector potential. For a mode havir azimuthal symmetry (no 4> dependence), all boundary conditions can be satisfied by a vector potential having only a z component Az(r, z) = i//(r)e ~Jfiz. The equation satisfied by A, is VzAe + k\Az = -n0J From A, we obtain VV - A , i\ + d2Az/dz2 Ez = -j<oAz + JV(t0€0 /a¥*o*o k20-p2 Ji»Hoeo A> Using (9.14) to express Elr in terms of J gives »*"-i-i) ®TtW* (9-1 The Helmholtz equation for A z now becomes V*At + p2Az = 0 0 <r £a V/A, - h2Az = 0 a <r sb (9.l6 f l (9-1^ MICROWAVE TUBES 657 where we have replaced V2 by V,2 - p2 and put 2 p 2 - k2 "0 p>_ _ £ • + * « + l -111 f Pi (0o-0)* --(**-*« 1 - 0«-l8 /i 2 = /3 2 - fe (9.17a) (9.176) after using (9.15) to express J in terms of A,. The analysis, when completed, will show that the space-charge waves are slow waves, with 0 ~ p0 ~» k0, and hence p and h will be real. With no 4> variation, (9.16) reduces to dH 1 d<l> dr' r ill = 0 dr (9.18) -h'< where Az = ili(r)e -"3j. The equation for i/< is Bessel's equation of order zero, and the solutions are J0(pr), Yu(pr), J0(jhr), and Y{)(jhr). Instead of using the Bessel functions with imaginary argument, we use the modified Bessel functions I0(hr), K0(hr). In the region 0 < r < a we cannot use Y0 since it becomes infinite. Therefore we let i/»(r) = C|J„(p/-) 0 <r <a i/»( r) = C.2I0( hr) + C3K0( hr) a s r < 6 where C v C2, C 3 are arbitrary constants. The axial electric field must vanish at r = 6 and must be continuous at r = a. These conditions hold for Az, and hence for i// also. Thus C{J0(pa) = C2I0(ha) + C3K0(ha) 0 = C2I0(hb) + C3K0(hb) (9.19a) (9.196) Besides E z , the only other field components present are E r and Htll. These are given in terms of ^4. by p Ma Pc M , k0 wii0e0 <>r 1 dA, Hj.= Mo dr dr k, •Y0Er Continuity of H^ at r = a requires ClpJ,n(pa) = C2hl'a(ha) + C3hK'0{ha) (9.20) where the prime denotes differentiation with respect to the argument pa or ha. In order for (9.19) and (9.20) to have a nontrivial solution for Cv C2, 658 FOUNDATIONS FOR MICROWAVE F.NCilNEERINO and C 3 , the determinant of the coefficients must vanish. Thus J»(pa) m J »{pa) K0{ M>)l0{ ha) - K'„l ha)I{)(kb) »d that # o( H>) 70( ha) - K0{ ha) I~JJb) (9.2l) This transcendental equation, together with the relations (9 17» 2J.2 p2 = We fi alh (p0 - //1 2 + ktf -k2 + to L- ' 'i which give (9.22) determines the propagation constant fi. Two special cases are now examined. First consider the case where 6 and a are made very large. Then, since /<,(*) - ] >2ie~X V^ - . V for large x, we find that (9.21) gives p tan(pa - TT/4) = h. But since we are letting a go to infinity, the only possible solution independent of a is p = h = 0. From (9.17) we then obtain the following solutions for (i: P = ±k0 0=/3 o (9.23a) (l±^j (9.236) But with p = h = 0, (i = ±kQ, all field components vanish as reference to the equations given earlier for E,, H,lt, and E r shows. Thus this is a trivial solution. The other solutions /3 = /30(1 + wp/to) correspond to the spacecharge waves. The wave velocities are 1 -. - w p 1 ± <ou/co since top <K to for conditions that are typical in microwave tubes. The wave velocities are slightly greater and slightly smaller than the dc beam velocity v0. The two waves are called the fast and slow space-charge waves. For p = 0, both E r and H* vanish but E, remains finite. The space-chai| waves may thus be viewed as a longitudinal oscillation of the electro the beam. When to = top, one solution corresponds to /3 = 0, that is, propagation. Thus it is seen that the plasma frequency is a nat eTse quency of oscillation for electrons in an infinite beam. Since the trap ^ fields are zero, the space-charge waves are not changed even if &a u as long as the beam has infinite radius. beam As a second special case consider the situation 6 = a so that ^ fills the waveguide. From (9.21) we now see that the T i ^ j ( p a ) ^ becomes infinite since the denominator vanishes. Thus we require " u ] a r 0. Hence pa takes on values typical ot those for TM 0 „, modes in a er3] guide. The lowest-order solution is (Sec. 3.18) pa = 2.405, or m &* MICROWAVE TUBES 659 pa = p0m, where the p0m are given in Table 3.5. Using (9.22), we now find that Po, Pi = -K' 1 - (ft, ~ yfh2 + kl) Pi = (*2"/32) 1 w I (9-24) (Po-P) For the field waves we expect j3 to be approximately equal to k0. Then, since wp « w and P0 » &0, an approximate solution of (9.24) is F = k% - Po, (9.25) This is the unperturbed propagation constant for TM„„, modes in a cylindrical guide. A correction to /3 may be obtained by using the solution given by (9.25) in the term multiplied by w 2 in (9.24). Of greater interest are the space-charge waves for which ji ~ p0. For these k'l « /3 2 ; so (9.24) may be approximated by Po,. A = -&' 1 <•> ' (Po-P) 2 which is a quadratic equation in /3 . To obtain an approximate solution, let P = p0(l + 8), where <5 will be small. Then we obtain Po,, = ~Pl CO which gives ±(w 5 = /M)PC /302 + (Po m /<*)T and hence P=Po •i 1 + rf 1/2 (9.26) PW (O Note that 5 is small, which justifies the approximations made. We may express (9.26) in the same form as (9.236) by introducing an effective plasma frequency u>q given by 2 », =«„ 1 2„2 Pi* \"l/2 = Ftor (9.27) 660 FOUNDATIONS FOB MICROWAVE ENGINEERING 1.0 08 §.0.6 •g «»• 3 II k 04 02 FIGURE 9.4 Space-charge reduction factor for a cylindrical electron beam of radius a, velocity v„. inside a circular guide of radius b. The data apply to the dominant TE„, space-charge mode. £na = : where F is called the space-charge reduction factor; for example, w* = F2p{-,r\/ea\ so the effective space charge is F 2 p 0 . Hence we can write /3 = /3 0 l i ft) (9.28) For the beam completely filling the guide, there are again a slow and a fast space-charge wave, with velocities slightly greater and slightly smaller than the beam velocity v0. However, the effective plasma frequency is reduced because of transverse variations in the field. Nevertheless, the space-charge waves have very small transverse field components Er, H,y Only the axis electric field Ez is large. - , In the general case, when a ¥= b, the solution for /3 is tedious. The results may be expressed in the form (9.28) for the space-charge w j ^ J , introducing the effective plasma frequency or the space-charge redu ^ factor. Some typical results computed from curves given by Branc Mihran are shown in Fig. 9.4.t , j ^ n The space-charge-wave theory was first developed by Hahn an ^ . ^ in 1939. Since that time space-charge waves under a variety o l , c o " r wiU have been studied. The references cited at the end of this chap tG. M. Branch and T. G. Mihran, Plasma Frequency Reduction Factors in Electn [HE Trans., vol. ED-2, pp, 3 - 1 1 , 1955. MICROWAVE TUBES 661 provide an introduction to the literature on this topic. For the analysis of the ordinary, or O-type, traveling-wave tube, the model discussed above is accurate enough to describe the main operating characteristics. 4 SPACE-CHARGE WAVES ON U N F O C U S E D BEAMS Many low-power klystrons employ electron beams without, magnetic field focusing when the distance (drift-space length) the beam must travel is short. The propagation of space-charge waves on this type of beam is therefore of interest. We shall consider a beam of radius a and with dc parameters -p0,v0az. For space-charge waves with axial symmetry, the only field components present are Er, Ee, and HA. The governing equations are Maxwell's equations (9.5) and the force equation (9.8), with B 0 equated to zero.t For space-charge waves we may assume a z dependence e "i~. In component form (9.5a) to (9.5c) and (9.8) are 9JL + ~=jwp.QHA JPEr JtiHi!,=jlo€nEr + (9.29a) Jr i a (9.29c) r or Id - r E r »r r (9.296) p - m ^ e0 JvEr w - pv0 v = JvEz (9.29d) (9.29e) (9.29/-) In addition, we have the relation J =pv0 - p0v which gives <f* - -Pn»r "" "Br J z = ~PoVz + Pv0 = <rEz + PV0 (9.30a) (9.306) where (9.29e) and (9.29 f) have been used, and the effective conductivity <r tStatic space-charge forces are assumed lo be negligible, which is a valid assumption for a low-density beam and a short drift space. Alternatively, the beam may be assumed to be ion-neutralized. 662 FOUNDATIONS FOR MICROWAVE ENGINEERING has been introduced as follows: a = -JVPo -Jeo*>l u> - /3u 0 w ~ fivQ (9.3i) When we make use of (9.30), the continuity equation V • J = found to give 1 <> I ~ j;rEr ~ JPEt | a - J&HP = J<op is -Je>P If (9.29d) is used to replace the term in parentheses, we obtain jpv0 + Jo\P = 0 (9.32) If the ac charge density p does not vanish, we must have oJPva +ju) = 0 This requires that ji be given by <i) ± <i> V„ \ ft) The corresponding wave solutions are the space-charge waves in an infinite beam for which E r = H^ = 0. For a beam with finite radius the bounc conditions at r = a cannot be satisfied with these values of 0. Consequently, the space-charge waves that we are looking for must have different value of p and, in addition, must have a zero ac space-charge density p, so that (9.32) will hold. For these waves, (9,30) now gives J = crE, or J = J w e o — e -E J<° o Maxwell's curl equation for H becomes V X H =jioe0E + J =jioe0 I + a E = ./WE jwe0 where the effective permittivity e of the beam is given by MICROWAVE TUBES 663 By using the effective permittivity e, the beam may be treated as a dielectric cylinder. The equation satisfied by E, is thus V 2 ^ + k2Et = V?Et + (k2 - p-)Ez - 0 where k* = io2p0e = kl t (9.34) S—1 <o(w - pu0) For space-charge waves we anticipate slow waves, for which /S » k(). Hence a suitable solution for Ez is E, = CJn(pr) 0<r<a (9.35a) Es = C2K0(hr) r>a (9.356) where p = (/32 - * 2 ) l / z , ft = (/32 - A 2 ) 1 ' 2 , and C „ C 2 are amplitude constants to be determined. The boundary conditions at r = a are different from those for a beam with confined flow, for the following reasons.t The electrons have a radial ac velocity, and hence the boundary of the beam does not remain at its dc position r = a since electrons will oscillate back and forth about the dc boundary. The resultant boundary thus becomes rippled, as shown in Fig. 9.5a. The positive charge shown in this figure represents a deficit of negative charge. The effect of the rippled boundary on the radial electric field under small-signal conditions may be accounted for by replacing the rippled boundary by a layer of surface charge. This is, of course, exactly what is done in the case of a dielectric boundary in which polarization charge oscillates back and forth about a mean boundary surface. The surface charge is given in terms of the dielectric polarization P by P • n = (D - e 0 E) • n. For the electron beam the corresponding surface-charge density p s arises from two causes, namely, charge flowing toward the boundary because of the radial current J r and charge carried to a given point z owing to surface charge moving with the beam. That is, the rate of increase of surface charge is (Fig. 9.5b) — =j(oPs = Jr- t ' ( ) — = Jr +JpvIJps The term -1>0 Hpx/dz arises as follows: Let the surface charge density at a point z be ps. Then at the point z - dz the charge density is approximately Ps ~ (<>ps/(iz)dz. The rate at which charge is carried away from the point z on the boundary is v0fis, and the rate at which charge from the adjacent tSee also W. C. Hahn, Small Signal Theory of Velocity Modulated Electron Beams, Gen. Elec. Rev., vol. 42, pp. 258-270. 1939. 664 FOUNDATIONS FOR MICROWAVE ENGINEERING r ^-Unperturbed boundary •"o Rippled boundary (a) -o<A-£*' \b\ c F I G U R E 9.5 Boundary conditions at the surface of a rippled beam. point z - dz flows toward z is v0[Ps - Opjdz) dz}. The net rate of accumulation of charge in an interval dz, due to the finite beam velocity u0, i s i -v0Vps/dz)dz =JI3voPsdz, since the z dependence is e• ' '. ' *e c ^ density is obtained by dividing by dz. Our final expression tor tne equ surface charge density is crE. Ps = j(to - pv0) ~ j(o> -&v0) -VP0Er (9.36) (<o-pv0Y The amount of charge which has crossed the unperturbed *°unjg -p0r, where r is the ac displacement of a unit volume of charg • -£ = v r = -^ + (v 0 • V ) r = j(a> ~ Pv0)r at at MICROWAVE TUBES 665 we see that = ~ P o r = j(co-l3v ) 0 j(<o - Pva) which is an alternative derivation of (9.36). The boundary condition to be applied to the radial electric field is E2r ~Elr=~ = — —3 = e 0 €„(w-/3U0) p ~—rA (to - flv0) (9.37) where the subscripts 1 and 2 refer to the fields in the regions r < a and r > a, respectively. The above boundary condition may also be expressed in the form a ^0E2r = jlv-fa) K ° This latter result is similar to that which holds at the boundary of a dielectric cylinder except that <u is replaced by w - fiv0 because of the uniform motion of the cylinder, in the z direction, with velocity i>(). Associated with the equivalent surface charge is an equivalent surface current of density J s = Jsa_„. To obtain an expression for J s , consider Fig. 9.5c. The total rate at which charge flows into a small region of length dz on the boundary is Jrdz W- dz dz and must equal the rate of increase Jaips dz of surface charge density in an interval dz. Consequently, - — =jNs =jfp, - Jr =JfioDp9 dz or J, = ftB, (9.38) after replacing Jr by j(io — fiv0)ps from (9.36). For w » wp, the total surface current is usually much larger than t h e total volume current, and it is therefore very important to include it. The boundary condition to be applied to H^ is # 2 * - #u = ^ = P s "o (9-39) When Hlt, satisfies this discontinuity relation, the boundary condition on the radial electric field is also satisfied. With the above boundary conditions we are now able to complete the solution to our beam problem. 666 FOUNDATIONS FOR MICROWAVE ENGINEERING When we combine (9.29a) and (9.296), we obtain ja>e0 + a H 2 2 * " p - k <)EZ jwe = 0 dE 2 2 + ja>p0cr 77 " fi -k 77 Referring to (9.35), we thus find that H * = Hh > = j(oe ~yCir»ipr) = jioe —CJi{pr) /we„ H„ = H2iil = -^C.K^hr) = ] r < a (9.40o) /we,, -^CK^hr) r > „ (9 m) We require Ez to be continuous at r = a and H,4 to satisfy the condition (9.39). Therefore CM Pa) = C,K0(ha) /we,, (9 Ala) /we -—CMka) - (TVn —CJl{Pa) =p s „ 0 = ^ - _ f L _ % From (9.296) we find E lr (y"we 0 + a-) = jpHVtU, and hence the second boundary condition becomes C.,Kx(ha) _ Cj^pa) h p wr (9.416) (to - pv0y Dividing (9.416) by (9.41a) gives Kx(ha) 7,(pa) />/„( pa) 1 - (w - pv0y (9.42) hK0(ha) The propagation constant p is determined by a solution of this equation, together with the relation 02 = p2 + k2 = p2 + k-Z 1 *S w(w - pv0) - t f + Aj (9-43) For space-charge waves, p » k0, so that h and p can be replaced by P' approximation that may be used to simplify (9.42) and (9.43). . -n The results obtained by solving (9.42) and (9.43) may be express the form (9.28): At microwave frequencies, w /w is usually in the range 0.01 to u- > ' MICROWAVE TUBES 667 0.8 0,63 II \4 F I G U R E 9.6 Plasma-frequency reduction factor F for an unfocused cylindrical electron beam as a function 0.2- ~&a of Ha. differs from p„ by only a few percent or less. In Fig. 9.6 the plasma-frequency reduction factor F = wq/u>p is plotted as a function of fia. If p and h are replaced by j3 in (9.42), that equation may be put into the form KJ0 /3a = /3na 1 + CO -1/2 K0Ix from which it is seen that -1/2 K0(130)1,(130) (9.44) This expression was evaluated to obtain F, as given in Fig. 9.6. The data in Fig. 9.6 may be used to solve for the corresponding values of /3 n a; that is, /30a = fia l±F(cop/co) (9.45) In a beam with confined flow only axial ac convection currents are permitted. For this reason an ac charge density p exists since the current J 2 is a function of z and must have associated with it space-charge-density fluctuations. For the unfocused beam, ac radial convection currents are also present, and this makes it possible for the total current J +jwenE to be solenoidal, i.e., to form continuous closed flow lines without terminating in ac space charge. ac POWER RELATIONS The ac power associated with the space-charge waves on an electron beam is of importance for understanding the gain mechanism of traveling-wave 668 FOUNDATIONS FOR MICROWAVE ENGINEERING Lubes and, of course, for power calculations. The small-signal DO rem for beams with confined flow was first derived by Chu f p" l extensions have been made by Haus and Bobroff? and Kluver.§ Th ? r t .^ t ' r tion is straightforward, but the interpretation of the various te ^ enter in is not always clear, depending on the type of beam irmri , ' \ t h a l 0deI considered. **«* Maxwell's equations for the fields associated with the beam are V X E = ->>MoH V XH =_/ W e 0 E + J where the small-signal ac current density is given by (the surface current" is included for generality) J = - n 0 v + pv„ + ps\Q In addition, we have the equation of motion jcov + (v 0 • V)v = - T J ( E + v x B 0 ) Expanding the following expression and using the above equations, we obtain V • ( E x H*) = H* • V x E - E • V X H* = -jioii0H • H* + j<oe0E • E* - E • J* (9.46) The continuity equation in the interior of the beam may be written as v„V • J = -jwpv0 = = -jai( pv 0 - p 0 v) - jwp0v -ja)J-jojp0v If we multiply the complex conjugate of this equation by v / i j , we obtain v - vn V • V* J* = — v J* + v fV] n ^ ^ ^ ^ ^ ^ Multiplying the equation of motion by J * / i j gives J* • E + J* • v X B 0 = > J v -J* * (v 0 • V)v V V the Since the transverse components of J and v are in the same direction, term J * • v X B„ = J * X v • B (l = 0 tL. J. Chu, A Kinetic Power Theorem, paper presented at the IRE-PGED Electrc Research Conference, Durham. N.H.. June. 1951. J. AppltH. A. Haus and D. Bobroff, Small Signal Power Theorem for Electron Beams, vol. 28, pp. 694-703, June, 1957. SJ. W. Kliiver, Small Signal Power Conservation Theorem for Irrotational Electro J. Appl. Phys.. vol. 29, pp. 618-622, April, 1958. Tub* phys " MICROWAVE TUBES 669 The addition of the above two equations thus gives j * . _ ^ v . v * = - ^ r . j * - * '7 *J 1) If we introduce the term V • (v - v 0 J * ) = v • v„V • J* + J* • V(v • v 0 ), we obtain for the right-hand side E -V - v • v„ 7)v - «r* - vv • v0] -r[J* 1 By expanding the bracketed term in rectangular coordinates, it is found to reduce to — a, X J s • V X v when v 0 = v0az. Hence we have v • v * = - v" v • v„ a. X J* • V X v (9.47) V The ac kinetic-power theorem is obtained by adding (9.47) to (9.46) to obtain J*-E v T-EXH* v • vn . Po = ~/OJ/J..,H • H* - jto—v • v:i -J* '7 / >7 + jwe0E - E* + — a . X J* • V X v (9.48) n The term (p(,/2n)v • V* is the ac kinetic-energy density in the beam since PO/TJ = ((>Q/e)m is the mass density per unit volume. Let us now specialize to the case of a beam with confined flow for which J = a, J, and the last term on the right-hand side of (9.48) vanishes. The real part of the above equation then gives ReV • EXH* - v • v., = 0 (9.49) The volume integral over a volume enclosing the beam between cross sections S, and S 2 as in Fig. 9.7 may be converted into a surface integral 5c -rfs S, Sc FIGURE 9.7 A section of an electron beam. 670 FOUNDATIONS FOR MICROWAVE ENGINEERING over the surface S, + S 2 + Sc. Thus the following power-co rem is obtained: 'i R e < 2 > - E x H * • dS = - Re J" s., a.rfS 2?7 + Re f ion 2T, a,dS '9.50 The term ( - v • V„)/TJ = -im/e)v • v„ = Vh has the dimensions of a volt age, and is called the kinetic voltage. It is the term t h a t gives the conversi' of kinetic energy of the beam into electromagnetic energy. In order that" 1 net amount of electromagnetic energy may flow out of the surface S quantity must be negative. A fuller appreciation of the above relations may be obtained by considering the application to space-charge waves in an axiaJly confined infinite cross-section electron beam. For the slow and fast space-charge waves, the ac kinetic power may be obtained by using (9.13a), (9.14), and (9.49). It is found that Be\-VkfJf\ 1 Re Re =/.*=- -vQvfJf 2n tow2pec 2uS(0,-/3o) wo)pe 0 2vl(f}f-p0) E m. :r l Since ps > p0 > fif, the slow space-charge wave has a negative ac kine power, whereas the fast space-charge wave has a positive ac kinetic power If the beam is excited, with only the slow space-charge wave, the significance of the negative ac kinetic power is that some of the dc flow e n e / ^ , the beam has been extracted and converted into negative ac energy > excitation process. The negative ac kinetic energy must in turn be conv into electromagnetic energy flow in order to maintain power consei In the discussion of the traveling-wave tube, it will be seen that the space-charge wave is the one that produces amplification. VELOCITY MODULATION The preceding sections have established the existence of space-charg^ ^ on electron beams. We must now examine the problem of exci , j^ __ i.e., ..»__• . •. modulation L.I.I; __ the ii.„ beam, u^om Inj «°y= waves, producing an ac velocity on tubes velocity modulation is commonly produced by passing the MICROWAVE Ti m-.s 671 Ac inpul Cylindrical Duncher cawi'> cross section F I G U R E 9.8 Velocity modulation of an electron beam. through two closely spaced grids located at the center of a cylindrical reentrant cavity, as in Fig. 9.8. The particular form of cavity used is chosen in order to satisfy the requirement of high ac electric field strength across the grids (which requires small grid spacing) and yet maintain a high cavity Q. The latter requires a large volume-surface area ratio. If we let Re( Ege•""') = Eg cos wt be the cavity electric field across the gap (often referred to as the buncher gap), those electrons entering the gap when Eg cos wt is directed in the negative z direction will be accelerated and will leave with a velocity greater than <.'„. Electrons entering the gap region when Eg cos i»t is directed in the positive z direction are slowed down and will leave with a velocity less than B 0 . It is apparent, then, that an applied ac electric field between two parallel grids will velocity-modulate an electron beam. The analytical details of the modulation process are discussed below.t We shall consider an unfocused electron beam of the type discussed in Sec. 9.4. Let the cavity field in the gap region be Efi cos wt. Electrons will traverse the gap with essentially the entrance velocity v„. If the time at which a particular electron passes the midplane z = -d/1 is /!,, then the field in the cavity at time / when this electron is at a position z = -d/1 + v0(t - <[)is (D E cos mt = E cos — 2 + — + f 0 /i The work done by the cavity field on the electron during its transit through TThis analysis is based on a ballistic formulation, and not on a field approach, since the former is more straightforward. 672 FOUNDATIONS FOR MICROWAVK ENGINEERING the gap is W= -C ~ eEg ° OS Po\Z + ~2 + °°'>) dz sin(/3„d/2) (9. where @ 0 = ai/v0 is the dc propagation constant for the beam Th hoo e bea coupling parameter M is denned to be m- sin(fl,rf/2) /30d/2 (9.52) For an electron passing the midplane at time t, the work done on th electron is clearly given by - eEg dM cos cot The work done on an electron results in an increase in its kinetic energy If the exit velocity from the buncher cavity is i>0 + v2, we have jm(v0 + vzf - \mvl = | m ( 2 u 0 i ; 2 + vf) = mv0ut = -eEgdMcoswt (9.53) since for small-signal conditions, vz « v0. In complex form (9.53) may be written as v,eJ"' = EgdMeJ"" Thus the axial ac beam velocity at the exit grid has a value v2 = ^-M{Egd) (9.54) The foregoing first-order analysis predicts that there will be zero average work done in bunching the electron beam since the average of (9.51 over one period from f, to C, + l/f is zero. This result is not correct, and in actual fact, a net amount of average work is required to velocity-modula the beam. To determine the average work done, a second-order analys must be performed.! The principal effect of requiring a finite amount o^ work to velocity-modulate the beam can be represented by an eqwv shunt conductance loading the buncher cavity (beam loading of the bun ^ cavity). The magnitude of this shunt conductance is typically such a New * tK. R. Spangenberg, "Vacuum Tubes," chap. 17, McGraw-Hill Book Company. 1948. ic-d-P-S.**M. Chodorow and C. Suskind, "Fundamentals of Microwave Electronics, Hill Book Company. New York, 1964. MICROWAVE TUBES 673 reduce the unloaded Q of the buncher cavity by a factor of 2 or so. However, even though the first-order analysis given above is not sufficiently accurate to give the beam-loading equivalent conductance, it does give a satisfactory answer for the velocity modulation of the beam, which was the information we were interested in obtaining from the analysis. Since we now know the velocity modulation of the beam at the exit grid of the buncher cavity, we are in a position to evaluate the amplitudes of the space-charge waves that will be excited on the beam in the drift space beyond the buncher cavity. We shall treat the case of an unfocused beam in detail. The case of a beam with confined flow is somewhat easier to analyze and is the model usually assumed in the analysis of the klystron, even though it is not the type of beam used in most klystrons. However, it turns out that the results for the unfocused beam and the beam with confined flow are essentially equivalent, the main difference being that, for the unfocused beam, the major contribution to the ac current comes from the equivalent surface current on the beam, whereas for the beam with confined flow, the ac current is a volume current distributed over the cross section of the beam. During the course of the analysis, the results for the case of the beam with confined flow will be given for comparative purposes. In the drift space z > 0, space-charge waves will be launched because of the ac velocity modulation of the entering beam. At the plane of the exit grid, the radial electric field of the space-charge waves is short-circuited and must be zero. This condition can be met by a suitable combination of the fast and slow space-charge waves. If we let £ r , and Erf be the radial electric field of the slow and fast space-charge waves, we require Ers = -Erf at 2 = 0. But Ers and Erf depend on r according to the first-order modified Bessel function Ix(pr), where p is different for the fast and slow waves. However, for typical beams, p ~ (5 = /?„; so the radial dependence can be taken as lx(fi0r) with negligible error, in which case the boundary conditions at z = 0 can be satisfied without introducing higher-order space-charge modes. The required boundary conditions on E r can be met if we choose the two space-charge waves to have the same amplitude and to combine in phase for the ac velocity v x at z = 0. Thus let ots and v,t be the amplitudes of the slow and fast space-charge-wave axial velocities, so that we may write Vg = vzse~ip'z + vzfe~^'z = uzf(e-J"r~' + « r * * ) where ps = /30il + o^/co) and pf = /30(1 - o>q/o>), and ioe/ is the effective plasma frequency equal to Fwp. Introducing these expressions gives vz = 2vzfcospqze J»»-' (9.55) where f3q = (30wg/oi = (oq/vn. Note that va, is a function of r according to I0(P„r). However, /30a is small, so that vzf is almost constant. Thus we may 674 FOUNDATIONS FOR MICROWAVE ENGINEERING equate 2vzf at r = 0 to v z as given by (9.54) to obtain v2 = —-MV g cos pqzl0( p0r)e-J*» (9.56) for the ac axial velocity at any point in the drift space z > 0. In (9 -c we have p u t V R for the exciting gap voltage Egd. For the unfocused beam the ac space charge density p i s 2 e r _ from (9.306) we find J, = -p0u2. Consequently, the ac axial current de'nsT in the drift space is «/.= VPo MVgco8pqzI0(P0r)e-J"9* (9.57) The surface current J s will be evaluated later, and will t u r n out to be more important than the volume current. To find Et, we use (9.29/") to obtain E,= * - ftPe <o, in Jl JV U) - PfU0 E *f = D = *r »*/ Hence the axial electric field is given by E2 = 2w -Av; *P - e ~ « * )= = v.Aef(e &C-<>-*»*) '^-vzfe-JPoZsm pqz V JT) Introducing the earlier expression for v-t* gives E t= (9.58) -pqMVgI0(p0r)sinpqze-^ Note that E z vanishes at the exit grid, where 2 = 0. From (9.29tO we have l a ~rBn-jfi.B„- -(P0 --rErr=JPfE!f= (ft, +Pq)j».f" - ' ^ ^ fiq)-±v„~—W.f w„ since /3 0 » /3„. The boundary conditions on £ r are consequently s a t , S Q f t h e a very good degree of approximation. If desired, a small adjustmen ^ ^ j , amplitudes of the two space-charge waves could make Ers + rf MICROWAVE TUBES 675 exactly at a particular value of r. However, since we have already approximated It(pr) by J £ (£ 0 /0 for both waves, the present approximation of dropping p q relative to 0 O is consistent with our earlier assumptions. If we examine the expressions for J z and vz, we see that, because of the beating or interference between the two space-charge waves, the ac current and velocity vary according to cos fiqz in the drift space. Maximum volume current density occurs when 2 = "Y (9.59) where X q = 2v/pq is the space-charge wavelength and n is an integer. If we had considered a beam with confined flow as discussed in Sect. 9.3, we should have J, = — • w - pv0 by using (9.116) and (9.12). In this case If v: varies according to cos/3,,2, then J. will vary as sin (iqz for a beam with confined flow. The different behavior of the two beam models arises because of the zero ac volume space charge density in the unfocused beam, a condition that can exist because radial ac motion of the electrons is permitted. The axial current for the two beam models is given by J z = -N°t + Pu0 J z = -Pffiz The space charge p changes the relationship between J. and v. from Jz = —pQvz to Jz = -p0v2to/((i> — pv0), which in turn produces the difference in z variation of the current amplitude. In some klystrons where a high power and long drift spaces are used, magnetic focusing is employed. In this case the confined-flow-beam model would be the appropriate one to uste. For the beam with confined flow, the axial current density at r = 0 would be found to be IVPn J, = to — MV, sin /J a ze-•"'"' (9.60) v o % " The peak current density is a factor co/co greater than the maximum current density for the unfocused beam as given by (9.57). This suggests that the confined-flow beam is superior. This is usually not the case because the surface current J s for the unfocused beam may contribute in a very substantial way to the total ac axial current, as the following analysis will show. 676 FOUNDATIONS FOR MICROWAVE ENGINEERING The surface current density is given by (9.36) and (9.38) p n - — o r t h e space^-charge waves, we obtain _ __W>o t w o JL--^St*L When we combine (9.29a) and (9.296), we find 7/3 3JS. 7/3 where CsI0(pr) = Ez. Consequently, / , ( j 8 0 a ) ( C I / c - > ^ * + C 1 J e-^.*) J. = - Now E ; s = -(oiq/jr,)vzf and E^ = (wq/jr})vzf; so we have C l s = -Uoq/jv)v!r, Clf=(<oq/J7])vzr, where 2u z/ -= -(ij/i/ 0 )Jlf\^. The final expression for Ja becomes J = ^/.(/Soojsin^ee-^ (9.61) We shall now compare the relative contributions to the total axial current. The total surface current is given by /5 2ira—MVgI1(l30a)Bmpqze-J** = (9.62a) and the total volume current flowing in the axial direction is 7, = Povo ira—-MVgI^0a)cosPqze-^ (9.62b) where we have used the result ,a I0{fi0r)27rrdr -'n = 2ira - ^ - I ^ a ) fit The ratio of the peak amplitudes is (4). 0) (9.63) (h), This gives the very interesting result that the total surface c u r r e ^ ' v 0 ] U [ n e case of an unfocused beam is a factor oj/a>q larger than the total ^e current. In fact, the total surface current given by (9.62a) is e q u a i ^ total volume current in a small-radius confined-flow beam, that is, ira MICROWAVE TUBES 677 F I G U R E 9.9 Electric field lines associated with ac space-charge bunching in (a) a beam with confined (low, (b) an unfocused beam. where J, as given by (9.60) for confined flow has a peak amplitude equal to that of 7 S if we replace 7,(£„a) by / 3 n a / 2 . We therefore conclude that both types of beams are about equally efficient, at least for short drift spaces, where beam spreading would not be important. It is apparent that ac space-charge bunching is an important mechanism in the production of high-density ac currents in the velocity-modulated beam. For confined flow the ac space-charge bunches are formed within the beam, and for the unfocused beam the ac space-charge bunches appear on the beam surface in the form of a rippled boundary. A sketch of the electric field associated with the two beam models is given in Fig. 9.9. The positive charge shown is only an equivalent charge that accounts for a net migration of electrons out of the region, leaving a net negative charge density less than p 0 , which can be viewed as a superposition of a small positive charge density on the constant dc background density ~p0. In judging the relative amplitudes of the volume current and surface current in an unfocused beam, the ratio to /u> must be known. Usually (oq does not differ by more than a factor of £ or so from the plasma radian frequency wp. In Fig. 9.10, fp = <n p /2ir is plotted as a function of the beam 678 FOUNDATIONS FOR MICROWAVE ENGINEERING 2.5xlOa - 2 x10° - 1.5 x 108 - F I G U R E 9.10 Plasma frequency as a function of beam current density. 5x10 50 100 150 / o . mo/cm 2 200 250 current density J 0 in milliamperes per square centimeter for several values of beam-accelerating voltage V. Note t h a t fp is proportional to JQ/2/V1/4. 7 TWO-CAVITY K L Y S T R O N A schematic illustration of a typical two-cavity klystron amplifier is shown in Fig. 9.11. The first cavity is excited by the input signal, which can be coupled to the cavity by a coaxial-line loop or a waveguide aperture. The first cavity acts as the buncher and velocity-modulates the beam. The Input siqnol , Output signol |l l„_, Buncher covity Catcher cavity Electron gun Cathode node --z--- • " Th • F I G U R E 9.11 The two-cavity klystron amplifier. t TTS•=-=-T MICROWAVE TUBES 679 second cavity is separated from the buncher by a drift space of length /, which should ideally be chosen so that the ac current at the second (sometimes called the catcher) cavity is a maximum. The second cavity is thus excited by the ac signal impressed on the beam in the form of a velocity modulation with a resultant production of an ac current. The ac current on the beam is such t h a t the level of excitation of the second cavity is much greater than that in the buncher cavity, and hence amplification takes place. The output signal is taken from the second cavity. If desired, a portion of the amplified output can be fed back to the buncher cavity in a regenerative manner to obtain self-sustained oscillations. One form of klystron analysis begins with an assumed lumped-parameter equivalent circuit for the output cavity and evaluates the current flowing in this equivalent circuit from an electron beam passing through the cavity by calculating the rate of change with time of the charge induced on the grids at the center of the cavity. This analysis gives a correct picture of klystron behavior, but it fails to illustrate the mechanism of the electromagnetic field interaction with the beam as it actually takes place within the cavity. A more satisfactory approach is to begin with a field analysis that will eventually lead to an equivalent Circuit and the basis for a circuit-type analysis of the problem. This is the approach presented below. We first evaluate the field set up in a cavity by the passage of an ac current through the cavity. This leads to an equivalent circuit for the cavity in the vicinity of one of the resonant frequencies. The next step is to evaluate the response of the cavity (or its equivalent circuit) to the passage of a velocity-modulated electron beam on which the ac current is in the form of a propagating current wave, This leads to a definition of the beam coupling coefficient, which is a measure of how effective the modulated electron beam is in exciting a response in the cavity. The third step, which we have presented in Sec. 9.6, is the evaluation of the ac current produced on an initially unmodulated electron beam when it passes through a cavity in which an oscillating electric field exists. These three phases of the analysis substantially provide the complete picture of the operation of a klystron. of a Cylindrical Cavity In a klystron cavity it is desirable to have a small grid spacing in order to make the beam coupling parameter M = [sin([i u d/2))/(p (l d/2) close to unity. The transit angle fiQd = wd/v0 should be kept small. In addition, a high cavity Q is desired, and this leads to the use of a reentrant cavity. The analysis of the modes in a cavity of this configuration is difficult, and therefore we shall consider instead an ordinary cylindrical cavity. The principal features involved in the excitation of the latter type of cavity by a modulated electron beam are the same as for the reentrant-type cavity. The cylindrical cavity to be studied is illustrated in Fig. 9.12. The cavity radius is b, and the cavity length is d. Two small cylindrical holes are 680 FOUNDATIONS FOR MICROWAVE ENGINEERING 8 F I G U R E 9.12 Cylindrical cavity excited by an axial current. cut in the center and replaced by grids to allow an electron beam to pass through. The beam radius is a, and is considered very small compared with the cavity radius ft. We shall first study the excitation of this cavity by an axial ac current of the form Jz = JeJml 0 < r s a (9.64) Later on we shall consider a traveling-wave current JeJU,'~-ip,>z of the type existing on a velocity-modulated beam and shall find that, for the latter, the cavity response is modified by the beam coupling parameter M. In view of the uniformity of the current in the z direction and the axial symmetry, only TM I l m 0 modes are excited. These have Er, E^, Hr, and H. equal to zero. It is convenient to introduce the vector potential A z , which is a solution of V2A. + kU, O^z = -H0J ,0 0 < r <a r> a (9.65) At /• = b we have A2 = 0, so that Ez will vanish on the boundary. From i we obtain (9.66) Et = -ju)Az 1 0A, tiQ dr (9.666) since there is no z or <£ variation. The natural modes of the cavity are solutions of the equation (9.67) where kQm0 = a.0„,o<Mo*o>1/2 and <o0m0 is the resonant frequency T M 0 m 0 mode. With no z or <t> variation, V z becomes of the MICROWAVE TUBES 681 and the solutions to (9.67) are Bessel functions. That is, A2.0mo = CmJo(^jf) (9.68) where p 0 „, is chosen so that =/ 0 (p 0 m ) = 0, C m is an arbitrary constant, and for no axial variation k0m0 = p0m/b (Sec. 7.4). For a solution to (9.65) we may choose] v /-. r t Pomr \ L CmJ0( — ) . A;= (9.69) since the Bessel functions are analogous to the sine and cosine functions and may be used as such in a Fourier series expansion of the vector potential. When we substitute (9.69) into (9.65), we obtain £ Cn,(k20m0 - * g ) j 0 ( ^ j = MoJ 0 < r < a (9.70) since (9.69) is a solution of (9.67). The following orthogonality property holds: n ¥= m (9.71) n = m Thus, if we multiply both sides of (9.70) by rJ0(p0mr/b) and integrate, we obtain 2 r ^o f° ,, IPnmr\ JM¥)-* 2uoJaJt(Poma/b)c2 (9.72) W Pom 0 m 0 ~^~ O) after replacing k0 by w/c, and similarly for kQm0. This equation holds for all values of m. We see immediately from this expression that only for those modes for which w = w 0 m 0 will the excitation amplitude be large. In addition, we see that, if co = w 0 m 0 for a particular value of m, C,n becomes infinite. An infinite response occurs because the cavity is ideal and is driven at one of its natural resonant frequencies. A practical cavity has a finite Q and will not respond with an infinite amplitude. As shown in Chap. 7, the effect of a finite Q is to replace the resonant frequency by 1+ 2Q0m0) where Q 0 m 0 is the Q of the T M 0 m 0 mode. The unloaded cavity Q is given 682 FOUNDATIONS FOR MICROWAVE ENGINEERING by (7.57) as Q = P nC ° ' (1 + b/d)w0m08, (9.73) Thus the excitation amplitudes for a cavity with finite Q are given h c„, = - 2Mo<fc/cV > (p 0m q/6) PamM?(p0m)(<o0m0 + w)(w - w 0 m 0 - > ( ^/2Q^)' If we choose to = to0i0, then C, will be large and all the other C small. In this case , nilP'nr\ A . = C,J0( — J - 2 >oO'^3Jr1(/J01a/6)Q /p0Irx ^o(-^-) 6Vf(poi)w3 (97 4) win k_ (9.75) where w and Q now refer to the TM 0 1 0 mode. Note that A z is proportional to the Q, and hence a high Q is desirable. If the cavity is coupled to an external load, we must replace Q by the loaded Q, Q,. The total ac current is 7 = wa2J. Also, since a « 6, we can replace a J\(Pa\ /b) by p01a/2b. In the vicinity of the resonant frequency w 010 , that is, for to = w 010 + A to, we find that the electric field E, is given by J ( P * - -J**. = o ., °ir/6) — (9.76) f / " 2 2 ' 27r6 60J, (pOI)(Ato-yto010/2Q) If we introduce an equivalent voltage V0 as the line integral of Ez across the cavity gap at r = 0, that is, Vn = -E,d, we may define an admittance Ye for the cavity as follows: y.-v.-^'^AJ^w) (9 ) " For a lumped-parameter LCG0 circuit with resonant frequency w010 = ( L C ) ~ 1 / 2 as in Fig. 9.13, we have / y„, = Gn + 2jC\co = G0\l+j 2Aa> Q\ where Q = o> u l 0 C/G 0 . Comparison with (9.77) shows that the equivalen cavity conductance G 0 is given by (note that e 0 = Y0/c) ,»„ G0 = «oio r2, Trb2Y0J{(p01)-^ F I G U R E 9.13 .j^, Equivalent circuit of excited cavity with no to < (9.78) MICROWAVE TUBES 683 The above equivalent circuit thus seems a possible circuit to represent the cavity in the vicinity of the resonant frequency w 0] „ of the TM 01 „ mode. However, we must show that it correctly accounts for the properties of the cavity even in the absence of the beam current /. For the TM 0U , mode, the energy- stored in the electric field is given by [we assume E, = Ju(p0ir/b)] Ai •'o -'M 4 -"n •'o -'o —— /• dr d<b dz = —7-b'2 dJ'f (p 0 1 ) The corresponding voltage across the cavity at r = 0 is V = Etd = d. Since Q = 2aiWt,/Pl, the power Joss in the cavity is given by 2»K Pi- Q If we define a conductance G so that Pt = \V'~G, we find that G = 2P, V2 Tre0bzo>mJ?{pas) 4^. = V'Q dQ = G, since <o = w 010 . Thus the two definitions for the cavity conductance lead to consistent results. If required, C and L are given by tonwC = GG„ and to'fIU)LC = 1. Thus L and C can be found from the known values of Q and G„. Cavity Excitation by a Velocity-Modulated Beam In a velocity-modulated unfocused beam the ac current is predominantly the ac beam surface current, Is = 2traJs, where Ja is given by (9.61) since co :» ioq for typical beams. The z dependence of the current is sin (i^ze '""J when 2 is measured from the output grid of the buncher cavity. The output cavity should be located a distance / = (n + ,jXA, ; /2) from the input cavity so that sin f}qz will equal unity and I s will have a maximum. Because fiq •« /3„, we have fi d <K 1 and the ac current variation with z can be taken as e^'^'e'"^ in the output cavity, where we have replaced z by / + z, so that the new origin is at the center of the output cavity. Thus the cavity is excited by a traveling-wave current. When the ac current has a z dependence, all the TM l ) m „ modes are excited. The solutions for A t are thus of the form G„,„t/,) riirz / Pi>, cos ~d~ We then have E E c„ Po„ tlTT ~d~ -k] Jo Pu. n-z cos—— = ftnJ2(r, z) (9.79) 684 FOUNDATIONS FOR MICROWAVE ENGINEERING in place of (9.70). However, if OJ = <"0I0, only the m = l , 7 , = Q mode is excited with a large amplitude. To find CU), we multiply (Q 7 rJ0(p0mr/b)cos(n7rz/d) with m = 2, n = 0 and integrate over r anY and obtain 2M 0 £-10 / (ki10-ki)by?(Poi)J0 Jt(r)rJ0 dr J -d/2d 0l ° ~e-jB, dz This latter result shows that the excitation amplitude is modified bv factor d/2 / : d/za ~e^dz~ sin(p0d/2) p0d/2 M which is the beam coupling factor. The integral over r may be replace jPi,z since J2 = 0 except at r = a by aJs without the factor e ~ J„(p{na/b) ~ 1 for a •«: 6. Hence, for w = <DOU \a>, we obtain for E„ r = 0, jM(jIx)e-^' E = i 2-6%j;- (p01)(Aw-./Woio/2Q) (9.80) where 7, is obtained from (9.62a) and is <0% 7, = 27ta — M / ^ / V * ) ^ TT-a^e, 'doMVe (9.81) since for /30a «1 we have /,(/3 0 a> = /3 0 a/2 and are assuming that an optimum length / of drift space is employed so that sin Pql equals unity. If we compare (9.80) with (9.76), we find that the only change in E, is replacing the current / by 7, and multiplying by the beam coupling coefficient M and an irrelevant phase factor e~JI>"'. The voltage developed across the cavity is modified by the factor MConsequently, the effective current that flows in the equivalent circuit I product the voltage V is I,. MI, <M»J from which it is seen that the term beam coupling coefficient is c l J ^ appropriate. For good coupling between the beam and the cavity « transit time [i0d is required. . can be When an external load is coupled to the output cavity., 1 The represented by an additional conductance G L in shunt ^t "where equivalent circuit of the output cavity is shown in Fig. 9er 7 f = M7, is the equivalent current that flows in the circuit. supplied to the external load GL is (9.83) \IXG, Po= o 2 (G0 + G,y MICROWAVE TUBUS 685 fe = MI, la) 1*1 F I G U R E 9.14 ( a ) Equivalent circuit of output cavity: (6) equivalent circuit of input cavity. A similar equivalent circuit may be assumed for the input cavity, as in Fig. 9.146. For identical cavities the cavity conductance is G 0 for both. If we assume that the beam produces negligible loading on the input cavity, the total conductance in the equivalent circuit of the input cavity is G 0 + G where G g is the equivalent conductance of the signal source.! The input power to the buncher cavity at resonance is 1 P i n " o2 H/G0 ' » .2 (9-84) (Gg + G„y which results in a voltage L 1/2 i IP developed across G0. This is also the velocity-modulating voltage. We can now evaluate the power gain, or amplification, of the klystron. Combining (9.81), (9.83), and (9.85), we obtain A= £o = \I/GL(G0 G^ + 2 Pin \IJ G0 \ G0 + GL Tra%p0*>l '.'. MAGL G0(G0 + GLf 2 a>p o>p V2 7ra Y0p0-^ -J- | n M4GL i n \ , C «J GQ(G0 + GL)' (9.86) t T h e conductance representing the beam loading on the input cavity is given by G* = l / 2 ( I0/V)M[M - cos(fS„d/2)] and is easily taken into account by adding it to G„ + G M when it is not negligible Note that /„ is the dc beam current and V the dc accelerating voltage. See Spangenberg, op. cit., chap. 17. 686 FOUNDATIONS FOR MICROWAVE ENGINEERING As an example consider G, = G„ and a beam radius a = ( also assume a beam current density of 100 mA/cm 2 and an ^ ^ *"* voltage of 1,000 V, we find, from Fig. 9.10. that w = i 0 2 ? w v P , t i f t 8 /•= 10'", a. = 6.28 x 10"> and /3() = w / c 0 = 33.6 r a d / c m . ' H e n c e * F ° r 6.72. Since u>p <e w, we have jia - /iaa, and Fig. 9.6 then sho ajq * 0.1OJP. Using these data, the power amplification A is foiinlT t h a t 0.094(y i ) /G„) a M 4 . To evaluate G 0 , we make use of (9.78) and '9 7 s l ° ^ have p 0 l = 2.405, and since p n i = & 010 6, we get ' ™e 2.405c 6.28X10'° = 1 1 5 c m To keep p0d small, we must choose d very small. If we take c/ = 0 05 find = 1.68 and M'1 = 0.62. It would be desirable to make d even small'" but then the Q, and hence G0, become small for the type of cavitv we a r ' considering. For a copper cavity we find from (9.73) that Q = 785, which is not very large. If a reentrant-type cavity were used, a Q about 10 times larger could be obtained. Using (9.78) gives G 0 = 0.06V o . The power amplification is thus 16.2, or 12 dB. Considerably higher gain would be obtained by using a reentrant-type cavity since d could then be made smaller and still a high unloaded Q maintained. However, even with the nonoptimum cavity that we have considered, the gain is quite good. For the particular example we have evaluated, the beam loading conductance G h is very small compared with the cavity conductance G 0 . For a more efficient cavity with a much higher unloaded Q, the cavity conductance G 0 would be much smaller and the beam loading conductance G h might not then be negligible. In order to obtain greater power gain than can be obtained from a two-cavity klystron, multicavity tubes are used. The gain increases exponentially with the number of cavities employed. In the multicavity klystron the first cavity is again used to provide the initial velocity modulation of the beam. The last cavity in the chain is used as the output cavity. ' intermediate cavities are kept unloaded by any external circuits and used t increase the modulation and hence the ac current on the beam. Power g of 50 to 60 dB can be achieved with multicavity klystrons. 6 = 9.8 REFLEX KLYSTRON The reflex klystron is an oscillator tube with a built-in feedback m e c h a £ " J ^ f It uses the same cavity for bunching and for the output cavity. A ske c ^ the reflex klystron is given in Fig. 9.15. The operation is as follo, *"V u , aie d assume an initial ac field in the cavity, the beam will be v e l o c l t y " / n < T e a m is as it passes through the cavity. Upon entering the drift space, the ^e decelerated and reversed (reflected) by the large dc field set up - j e t0 repeUer or reflector electrode at potential - Vr. Thus the beam is ^ pass through the. cavity again, but in the opposite direction. . v ^e choice of the reflector voltage Vr, the beam can be made to pass tnr MICROWAVE TUBES 687 Ac o u l p o l f' Electron gun .--Cavity Reflector s r Beam y,\ 1 FIGURE 9.15 The reflex klystron. /. cavity on its return flight when the ac current phase angle is such that the fieid excited in the cavity by the returning beam adds in phase with the initial modulating field. The feedback is then positive, and oscillations will build up in amplitude until the system losses and nonlinear effects prevent further buildup. When the velocity-modulated beam enters the drift space, it is subjected to a constant decelerating field V,./s, where s is the cavity output grid-reflector spacing. As a result, the beam propagation constant /3 0 = <o/v0 is gradually reduced to zero, and then increased back up to - / 3 0 . The total phase change undergone by the ac current on the beam will be given by B = 2f",p0(z)dz where z m is the maximum distance an electron can penetrate into the drift space. We can evaluate I) in terms of the transit time T for an electron to return to the cavity. We have, in the drift space, dva( z) dt = --q~ s which integrates to v0(z) = w0 — T\tVr/s. Hence vQ(zm) =v Vr T o-r> — ^ =0 which gives T / 2 = v0s/r)Vr. The return time is equal to T/2 also, so that 6 = wT = 2v0sw (9.87) If we let V g be the accelerating-gap ac voltage, the ac beam current reflected back through the cavity is given by (9.62a) when fi0z is replaced 688 FOUNDATIONS FOR MICROWAVE ENGINEERING by 0 and /3,2 by Pq0/Po = taHd/u. Thus JTra'^PoPoMV /, = mi sin wO -JL-e~J" to where we have approximated /,(/3 0 a) by (S0a/2. In a reflex klystro usually quite small, so that sin<w (/ 0/W may be replaced by u> 0/°° Tk 8 effective current for excitation of the cavity is /e = 7,Af, and is give ^K given by 'oV„ f>e^-=-fM^sine+jcose) Bo (9.88) , 22 where 7„ is the total dc beam current Tra p0u0 and V a is the acceleratin voltage from which v% = 2nV 0 . The ac electronic admittance of the beam is denned by Y. = — = -~6(siti0+jcos0) (9.89) The equivalent circuit of the reflex klystron consists of the electronic admittance Y c in shunt with the equivalent circuit of the loaded cavity, as in Fig. 9.16. Oscillations can take place when the net conductance is less than zero, or more specifically when / 2 Aw ye + ( G L + G 0 ) 1 + 7 QL\ \ w (9.90) =0 010 where QL is the loaded Q of the cavity. Since 0 is a function of the reflector voltage, as given by (9.87), oscillations depend on an appropriate choice of Oscillations -Y FIGURE 9.16 Equivalent circuit for a reflex klystron. FIGURE 9.17 4^mi».or,ro diagram Aitumm for a Admittance ^ re MICROWAVE TUBES 689 n=\ 'iA A #w i c ^ v r Vr F I G U R E 9.18 Reflex-klystron tuning curves. Vr. In Fig. 9.17 we have plotted the admittance Y in polar form as a function of 6. Note that \Ye\ increases with 6. On the same plane we have plotted the negative cavity admittance / - F = -(G0 + GL)\l+j 2 Aw QL which is a straight line parallel to the jB axis at G = - ( G 0 + GL). provided we assume Ga,GL,QL independent of w in the vicinity of the resonant frequency w 010 . The construction shows that oscillations are possible for 6 in the vicinity of 3 i r / 2 , 7ir/2, etc., since in this region Gc + G0 + GL < 0. In addition, oscillations will take place for a range of values of Vr about the points that make 6 = 3TT/2 + 2nir. Each value of n gives a mode of oscillation. In typical klystrons as many as seven or more modes of oscillation can be obtained. For stable oscillations Yc + Y = 0, and consequently the frequency of oscillation varies as V r is varied to tune across a given mode. Typical tuning curves giving power output and frequency as a function of reflector voltage are shown in Fig. 9.18. Physically, the various modes arise because of the increased transmit time for electrons into the drift space when V r is reduced. Oscillations occur when the transit time T equals (| + n)f ' or \ + n ac periods since the ac current has the proper phase under these conditions. Commercially available reflex klystrons range from small-size units producing 100 mW of power up to units capable of delivering several watts of power under continuous operation. Klystron amplifiers employing two or more cavities are available in a size range from units capable of handling a few hundred milliwatts up to several hundred kilowatts of amplified output power. 690 9.9 FOUNDATIONS FOR MICROWAVE ENGINEERING MAGNETRON This section is devoted to a qualitative description of the magnetror tor. The basic structure of a magnetron is a number of identical reson arranged in a cylindrical pattern around a cylindrical cathode, as show, °*' Fig. 9.19. A permanent magnet is used to produce a strong magnetic fi T normal to the cross section. The anode is kept at a high positive voltap v relative to the cathode. Electrons emitted from the cathode are accelerator toward the anode block, but the presence of the magnetic field B Q produ a force ~evrB„ in the azimuthal direction which causes the electr trajectory to be deflected in the same direction. If the cathode radius is a and the anode radius is 6, the potential at any radius r is V(r) = V0Un(r/a)]/lln(b/a)]. The velocity of an electron at this radius is given by v(r) = \2vV(r)) 1/2 The electron can execute circular motion, at the radius r, about the cathode if the outward centrifugal force mv%/r and the radial electric field force -eEr = eVa/[r ln(6/a)] are exactly balanced by the inward magnetic force ev(r)B0. For circular motion at radius r, we therefore have mv2 eVa + —-—; r = evBn r rln(b/a) or since v = (oer, where o e is the electron's angular velocity, vVa We ~ VBQUe + 2 r \n(b/a) =0 (9.91a) (9.916) For later reference, we solve (9.91) for the cathode-anode acceleratin Anode block ot potentiol Va Interaction space Coouo line output Cathode F I G U R E 9.19 A multicavicy magnetron. Heduced side view MICROWAVE TUBES 691 voltage VQ: va = // bM\ I/ .„ w,. w. S *o" kr*fa\ a/ \ »J (9.92) This value of V*a will permit an electron to execute circular motion at a radius r and with an angular frequency a»e. If now there is present an ac electromagnetic field that propagates in the azimuthal direction with a phase velocity equal to the electron velocity wt.r, strong interaction between the field and the circulating electron cloud can take place. The possibility of this type of electromagnetic field being present is discussed below. The multicavity magnetron is a periodic structure in the azimuthal, or <l>, direction. If there are N cavities, the period in </> is 2—/N. According to Floquet's theorem, each field component can be expanded in the form n = —x = $„Lr)e~JM'-Jnm L (9.93) n = —x where the period p ~ 2—/N. But since the structure closes on itself, (Mr, 2TT) = i/>(r, 0). The only possible values of (i that will make [i'2- equal a multiple of 2-rr are fr» = m m = 0, ± I , ± 2 , . . . (9.94) With the value of /3 specified, a corresponding frequency w becomes specified, say iom, which is the resonant frequency for the m t h mode. In other words, when w = mm, we obtain a value m for p"m. Thus a typical field component will have the form x <l>in(r,<t>)ej"-"'= L ^ ( r )«-.*«+<«*»+*.' n- « The phase velocity in the azimuthal direction 4> for the n t h spatial harmonic of the m t h resonant mode is o> ,„ r {3„in m + nN (9.95) at the radius r; that is, angular phase velocity is w m //3,„„. The usual mode employed in a magnetron oscillator is the TT mode, where the phase change between adjacent cavities is TT rad, or 180°. Each cavity with its input gap acts as a short-circuited transmission line a quarter wavelength long, and hence has a maximum electric field across the gap. For the TT mode the field is oppositely directed at adjacent cavities. A sketch of the electric field lines in two cavities is given in Fig. 9.19. For the TT mode, 692 FOUNDATIONS FOR MICROWAVE ENGINEERING Pm4> = mt{> must equal -n for a change in <}> equal to one 1 Hence m = N/2, and the phase velocity for the rcth spatial h 2~/Nhai becomes "'nonic p.nN/2 N(l + 2n) (9.96) In order to obtain interaction between the electron cloud and the spatial harmonics at a particular radius r, we must choose Vv - So that w„r = v(r) = v p.nN/2 or '.v 2 (9.97) AT(1 + 2n The required voltage V a to obtain synchronism between the electron cloud and the ac field may be found from (9.92). If we choose a value of r midway between the cathode and anode., that is, r = (b + a)/2, and note that in typical magnetrons b/a is small enough so that l n ( 6 / a ) = 2(6 - a)/ (6 + a), we obtain 6 2 - os V = m B0- nN m + nN (9.98a; in general, and K = 2w N/2 N(l + 2n) 62-a2 B0- 2(0 N/2 nN(l + 2ra) (9.98ft) for the - mode, where m = N/2. From a physical viewpoint the synchronism between the electron cloud and the rath spatial harmonic and the ac field implies that those electrons located in the field where £,,, acts to slow down the electrons will give up energy to the field. As the electrons slow down they move radially outward [see (9.91)], and eventually are intercepted by the anode. Electrons that are accelerated by the ac field move in toward the cathode until they get into a proper phase relationship such as to give up energy to the field. When latter happens, they begin to slow down and spiral out toward the m^°^ Thus the only electrons that are lost from the interaction space are too; that have given up a net amount of energy to the ac field. g The ac power may be coupled out from one of the cavities coaxial-line loop as shown in Fig. 9.19 or by means of a waveguide. 9.10 O-TYPE TRAVELING-WAVE T U B E The ordinary, or O-type, traveling-wave-tube employs a m a g " e ' cused electron beam and a slow-wave structure such as a helix, disc Chap. 8. The electron-beam velocity is adjusted to be approximately ssed ^^ MICROWAVE TUBES 693 the phase velocity for an electromagnetic wave propagating along the helix. Under these conditions a strong interaction between the beam and the field can take place. From another viewpoint we can consider the presence of the slow-wave circuit to modify the space-charge wave-propagation constant in such a manner that it becomes complex and represents a growing wave. We shall present a more satisfactory picture of the gain mechanism after we have analyzed a particular tube configuration in detail. A full appreciation of the physical principles involved is somewhat difficult to obtain without a detailed study. For simplicity we shall use the sheath-helix model discussed in Sec. 8.10 and an axialiy confined flow beam model ( S „ infinite) of the type treated in Sec. 9.3. In addition, we shall assume that the beam completely fills the region interior to the helix. This assumption is not true in practice, but we make it, nevertheless, in order to simplify the analysis. The basic principle of operation of the tube is not changed by this assumption. The traveling-wave tube is operated in an axialiy symmetric mode; so all field quantities will be independent of the angle <f>. Figure 9.20 illustrates the construction of a typical traveling-wave tube. The main components are an electron gun, a helix, a solenoid to produce the focusing field B„, and suitable input and output ac coupling to the helix. The helix is taken to have a radius a and a pitch angle \l>. It is approximated by a cylindrical sheath with infinite conductivity along the direction oi the winding and zero conductivity in the perpendicular direction. In Sec. 8.10 it was shown that both TM and TE modes were required in order to satisfy the boundary conditions at r = a. However, for a beam with axialiy confined Row, where only a z component of ac velocity is permitted, the TE modes are not affected by the beam since these have E z = 0. Hence the field components for the TE mode for n = 0 are those given by (8.72) in Sec. 8.10. Similarly, for r > a, that is, outside the helix, the field components for the TM mode are those given by (8.72) in Sec. 8.10. Cooxiol-line F I G U R E 9.20 O-type traveling-wave tube. 694 FOUNDATIONS FOR MICKOWAVE ENGINEERING Inside the helix region the TM field in the presence of the beam is that a v in Sec. 9.3. However, the pertinent equations are repeated here for C O n ^ n nience. The vector potential A, is a solution of (9.16a), Vt2A, + p2A, - 0 where 0o * » - ( « - * • ) 1 - to I \Po-P (9.99) For the present problem p 2 will t u r n out to be negative; so we shall replace p 2 by -g2. The solution for A z is then proportional to I0(gr). Since E is proportional to A., we can choose K = a0IQ(gr)e-J,i* where a„ is an amplitude constant. The field components E r and Hit, are readily found from Maxwell's equations; i.e., j/i E = -r ~2 !>E: U^~ p2 - K •»• k0 i.-~^YoEr H P Thus we can write the following expressions for the fields in the two regions for the n = 0, or axially symmetric, case: For TE modes, H2 = c 0 J 0 O ) e - ; * Mr=~cQUhr)e -jez -J-^c0ll(hr)e-^ E„,= Bz~dQKa(kr)e Hr= r <a -jfiz -j±d0Kx(hr)e-^ E.-J-^dnKl(hr)e->» r>a MICROWAVE TUBES 695 For the TM mode, Ez = a0I0(gr)e-^ JPg Er=J-^-a0I1(gr)e-^ h r <a E,_ = bnK0(hr)e ~&* Er= -^b0Ki(hr)e-^ H„= -J—±bliKi(hr)e^ j- > a where h2 = p2 - k2. The boundary conditions at r = a for the sheath helix are given by (8.69). For the present problem they yield -juv-u h c 0 /,(/ia)cos tli + a0Ia(ga)sin ip = 0 d0K1(ha)cos \\i + b0KQ( ha)sin tli = 0 a0h(§a )cos I/I + /"'Mo ——cu/,( ha )sin 0 = bnK0( ha)cos i/» : — d 0 / f [(/ia)sin i/< and JW£0g c0I0( ha.) sinib + 2 anI{(ga )cos i/> = d 0 K 0 ( Aa)sin iA —&„/?,(/m)cosi/r If we solve for c 0 and c/0 from the first two equations and substitute into the latter two equations, we obtain two homogeneous equations for a 0 and b0. For a nontrivial solution the determinant must vanish. Equating the determinant to zero gives Ii(ga) 8 I0(ga) /i a tan 2 <// k o Ip(ha) It(ha) K0(ha) + K,(ha) 696 FOUNDATIONS FOR MICROWAVE ENGINEERING For most traveling-wave tubes the parameters are such th ha are large. In this case the ratio of the Bessel functions *' 1 m approaches unity, and we obtain ^100) h3 g -2-2 "o tan 2 *-h (9-IOIQ) From (9.99) we have g2 = h2 1 - ^ in Po Po~P and hence 1 - Po -S *> I \Po-p 211/2 h2 = 2~tan2tlt- 1 (9.1016) where h2 = fi2 - k\. The above is a sixth-degree equation in p, and cannot be solved exactly. Since we are dealing with a slow-wave system, /3 2 will be large compared with k2, and h2 = fl2. In addition, we can equate k0cotili to 0„ since the phase velocity of the helix in the absence of a beam is chosen equal to the beam velocity v0. That is, k 0 esc 4i is the propagation constant for the helix, and for t// small, sin iji can be replaced by tan iji. We thus obtain to Po 2p2 Po-P ~PJ - 1 We now assume that /3 = y30(l + 8), where 8 is small. With this substit tion we get ^ = 82(1 + 48 + 282f = 4<56 + 1 6 5 s + 20S 4 + 8 5 3 + 8s (9.102) For 8 small, we can drop all but the term involving the lowest power This is the cubic term, and thus The three solutions for 8 are \(w„/<o)2/3 multiplied by the cube of 5- of MICROWAVE TUBES 697 - 1, which are - 1 and (1 ±7'v/3 ) / 2 . Hence %>2' (9.103a) *"I 1 / 01. \ 2 / 3 (9.1036) (9.103c) Since top/co is small, the assumption that 5 was small is justified. The corresponding propagation constants are 1 ">„ 2/31 JPt =JPo 1 - - 2 1 -* W ,2/3 n JP±=JPu 1 + - ^ £ . (9.104a) w ' w ) 4 —i W JP3=.)P0 1 + - 4 n P_ [9.1046] i + 7fVf) (9.104c] ,2/3 ) CO (W73) < < The first solution corresponds to a wave with a phase velocity slightly greater than the beam velocity. The other solutions have phase velocities slightly less than the beam velocity, and in addition jfS2 corresponds to a decaying wave, whereas ,//3 3 corresponds to a growing wave. The growth constant ag is V3 «,-*-T- wr 2/3 (9.105) — If all three waves are present at the input, only the latter wave will predominate at the output. There are additional solutions to the eigenvalue equation (9.1016). We should expect a wave propagating in the -z direction, with (i ~ — kQ csc >li ~ — 0 O , which is not significantly perturbed by the beam. We therefore assume that fi = -jG„(l + 5) and consider 8 small. Substituting into (9.1016) and retaining the smallest power term in d give 5= - 1 32 CO Hence a fourth solution is JPA = -JPo 1 - 1 / 32 = -JPo (9.106) The remaining two solutions of (9.1016) give values for /3 approximately 698 FOUNDATIONS FOB MICROWAVE ENGINEERING equal to ±k0. However, the eigenvalue equation (9.1016) is a tion to the true eigenvalue equation (9.100), obtained by assumi ° xini aand ha are large and that fi is large compared with k„. Therefore h Sa solutions p = ±k0 to the sixth-degree equation (9.1016) are not , t w ° olut of (9.100) and do not correspond to physical waves. ions The ac current and velocity are given by (9.14) and ( 9 . 1 3 Q ) a<? v = f9 2 u0(Po-P) J = -J- " 07) —snE (9108 » (fio-P)*° ' ) These equations show that v and J are negligible for the three waves for which p is significantly different from pn. Thus v and J arise from the first three slow waves discussed. The fourth wave can be excited by reflection at the output end of the tube. If it is reflected at the input end also, it will be amplified and, with continued reflection and amplification, will result in oscillations. To avoid this undesirable feature, an attenuating resistive vane or an integral ferrite isolator is built into the traveling-wave tube. At the input end we must have the total ac current and velocity associated with the three forward slow waves vanish. Thus the initial conditions at the input z = 0 are t/[ + J2 + J3 = 0 vl + v.2 + v3 = 0 When we assume that Ez = I0(p0r){Cie-'^ + Ctf-J*" + C 3 e - - " ^ ) and make use of (9.108) and the initial conditions, we find that 2 = g/Zir/3 _* _ e -j2-/3 Consequently, all three waves at the input have equal magnitudes; i.e., « find that Gj = C 2 = C 3 . The growing wave will have an amplitude equal ^ one-third that of the input signal. Therefore the amplitude gain c traveling-wave tube is E0- = 1- « ,' e e E, 3 where a e is given by (9.105) and / is the tube length. The power gau g decibels is A = 20logO.333 + 20a / l o g e UJ = - 9 . 5 4 + 3.75/V 2/3 9 l 0 9) MICROWAVE TUBES 699 With the aid of the preceding results we can now describe the physical mechanism of the gain. We note that the growing wave has a phase velocity slightly less than the beam velocity. This growing wave is the perturbed slow space-charge wave. The ac kinetic-power density of the fast and slow space-charge waves are [see (9.49), (9.13a), and (9.14)] The slow space-charge wave has fis > 0 O and hence has a negative ac kinetic-power density, whereas the fast space-charge wave has a positive ac kinetic-power density. Since the slow wave grows, it therefore loses energy, and the conservation theorem (9.49) then requires that the electromagnetic power increase. The ac current of the slow wave will have a phase angle relative to E, such that Re(E.,.J*) is negative and the current continually gives up energy to the field. This may be verified by substituting /3 3 for /3 in (9.14) to obtain -f) fel£,/ As a further aid to the understanding of the traveling-wave tube, it may be noted that it can be viewed as a large number of closely spaced cavity gaps operating as a multicavity klystron. The adjacent turns of the helix are then considered as constituting a gap. The main advantage of the traveling-wave tube over the klystron is its relatively broad frequency band of operation. Typical units provide gains of 30 to 50 dB over an octave or more in frequency. Power-handling capability ranges from milliwatts to megawatts. M-TYPE TRAVELING-WAVE T U B E The magnetron-type (M-type) traveling-wave tube is a linear version of the cylindrical magnetron. Figure 9.21 is a schematic illustration of an M-type tube using a corrugated, or comblike, slow-wave circuit. The electron beam is much wider than it is thick and approximates a sheet beam. A potential V a is applied between the sole and the anode block. A large static magnetic field is applied in a direction perpendicular to the beam velocity vna,, and the static electric field - £ „ a v arises from the anode to sole potential Va. The electrons moving upward from the cathode at potential V e are deflected by the magnetic field into a beam moving in the positive z direction. The desired type of flow is the one where there is only a z-directed velocity 700 FOUNDATIONS FOK MICROWAVE ENGINEERING Ac input Outpu' , Slow-wove circuit Collector Beam f - m Cathode 1 Vc FIGURE 9.21 M-type traveling-wave tube. va(y), which in general is a function of v. Electron flow takes place in a crossed E and B field, which is typical of magnetron-type tubes. For stable flow, v0(.y)a^ does not vary with z. If we denote by V(y) the potential at an arbitrary value of y between the sole and anode block, we must have a balance between the magnetic force -ev0(y)a, x a,B0 = -eB0u0(y)ay and the electric field force aye<)V/dy. Thus nv = (9.110) v0(y)Bc ay The velocity o 0 (y) may be found from the energy equation imv20(y)=e(V-Vr) The derivative with respect to y gives (9.11D The potential Viy) arises from the applied potential V„ and from ^^ space charge within the beam. Under equilibrium c o n d l t 1 0 " ' this of - e ( E + v X B 0 ) acting on an electron is zero. The divergence equation thus gives dvf (9.112) Pos VXB0=0= --— +B0— V E • since v = v0a, and B 0 = B o a , . In this equation -p0 & the space charge density and s is a factor giving the fraction o space charge which is not neutralized by positive ions. For no y negative negative - t i v e j„ n s ^ ^ B MM ROWAVB IVHKS 701 present, S = 1. If we assume thai S = 1, the set of relations (9.110) to (9.112) can hold only if 2 n S B 2 Or = T »(> or = I'"11 m e = w;, = 2 Up (9.113) as can be determined by eliminating 8V/9y and 9va/9y. When this condition holds, the flow is referred to as planar Brillouin flow. With the above model for the beam, it is possible to solve for spacecharge waves that can propagate on the beam. In the presence of a slow-wave structure, the propagation constants become perturbed and a growing wave is produced similar to that in the O-type tube. For a detailed analysis the reader is referred to the citations given at the end of this chapter. The principles involved are not sufficiently different from those already discussed to warrant inclusion in this text. 12 GYROTRONS Magnetrons and klystrons require resonant cavities to support the electromagnetic field that interacts with the electron beam. The traveling-wave tube requires a slow-wave structure. These structures have dimensions linearly proportional to the operating wavelength and become very small at millimeter wavelengths. The consequence of having to reduce the dimensions as the frequency increases is thai the available area for the electron beam decreases and the power output that can be achieved decreases rapidly, approximately proportional to l//~ 2 . Thus, at frequencies of 100 GHz and above, conventional microwave tubes are not capable of producing power outputs in the kilowatt range. A relatively new tube, the gyrotron, has been developed in more recent years and does not rely on the use of resonant cavities or slow-wave structures. In the gyrotron the electromagnetic field interacts with the cyclotron motion of the electrons in a strong static magnetic field. When an electron is acted upon by the force of a steady magnetic field, its motion in the plane perpendicular to the magnetic field is a circular- one. By using a sufficiently strong magnetic field, the frequency of rotation, called the cyclotron frequency, can be in the frequency range corresponding to millimeter waves. The interaction of a microwave field having the same frequency as the cyclotron frequency of the electrons results in growing waves. Thus the waveguide through which the electron beam passes and which supports the electromagnetic field is not restricted in diameter by the need to provide either a resonant structure or a slow-wave structure. As a result the fundamental size restrictions of conventional microwave tubes are not present in the gyrotron. There are three common forms of gyrotron tubes. These are the gyromonotron oscillator; the gyro-TWT, a traveling-wave amplifier tube: 702 FOUNDATIONS FOR MICROWAVE ENGINEERING Electron gun Output waveguide Cavity Ring cathode Solenoid (a) Electron gun Output waveguide Cathode Input waveguide First anode Second anode W Ring cathode Input waveguide Electron gun Gun solenoid Vacuum seal window Output waveguide Magnetic field solenoid (c) Input cavity F I G U R E 9.22 Jifier (a) The gyromonotron oscillator; (6) the gyro-TWT amplifier; (c) the gyroklystron ami MICROWAVE TUBES 703 and the gyroklystron, another amplifier tube. Simplified drawings of the three gyrotron tubes are shown in Fig. 9.22. Each tube has a magnetron-type electron gun which imparts a high radial velocity to the electrons before they enter the high magnetic field region. The large static magnetic field is provided by either a liquid cooled solenoid or a superconducting solenoid. In the gyromonotron shown in Fig. 9.22a. the interaction region is an enlarged circular waveguide that can support many different propagating modes. The output is taken from an output waveguide through a transparent window which is also used as a vacuum seal for the tube. In the gyro-TWT amplifier the input signal is coupled into the input of the interaction region through a waveguide as shown in Fig. 9.22/). The input microwave signal provides the initial bunching of the electrons in the beam. The input signal is amplified in the circular waveguide whose dimensions are large enough to support many possible propagating modes. The electron beam is in the form of a hollow beam with a radius such that it interacts strongly with only one, or at most only a few, of the circular waveguide modes. The gyroklystron uses an input and output cavity as shown in Fig. 9.22c. The signal to be amplified is coupled into the input cavity. The output signal is taken from the output cavity. Gyrotron amplifiers that provide power gains of as much as 24 dB and output powers as high as 50 kW at 5 GHz have been built. Ferguson, Valier, and Symons describe a 5.2-GHz gyrotron tube producing 128 kW of output power.t This tube uses an 8-A-65 kV electron beam. Pulsed power outputs from gyrotron oscillators have been produced at levels of several hundred megawatts. It has been reported that 28-GHz gyrotrons with 200 kW of continuous-wave output power are in operation at Oak Ridge National Laboratory.? The high power capability of gyrotrons has been amply demonstrated; so these tubes will become more important for millimeter-wave systems in the future. Particle Interaction in a Gyrotron In a gyrotron electron beams having a very large azimuthal velocity are used, so that the relativistic increase in the mass of the electron must be taken into account. The electron gun injects the beam into the high magnetic field region with the electrons initially having a large radial velocity component. The v(l X B„ force then causes the electrons to follow a helical path with a velocity v„ = vlhla,., + v0ja: with vlllt, being several times larger than u„,. If we treat the electrons as a charged fluid, then under the action t P . E. Ferguson. G. Valier. and R. S. Symons, Gyrotron-TWT Operating Characteristics, IEEE Trans., vol. MTT-29. pp. 794-799. 1981. +J T. Coleman, "Microwave Devices," Reston Publishing Company, Inc.. Reston. Va. 1982. 704 FOUNDATIONS FOR MICROWAVE ENGINEERING of a microwave field the velocity of a differential volume elemp charged fluid will have an ac component v in addition to the dc co '^ v 0 ; so the total velocity field will be v, = v 0 + v. The effective m a s ^ T 6 electron will be my = mil - vf/c2r*/2, where y is the relativists c ° tion. The momentum is myut. The particle density is N + n , w h e r ° the ac variation in the number density from the average value r ) particles per unit volume. The charge density is given by -e(AT +° ^ _ - p 0 •+ p. If E and B represent the microwave field and B 0 is the stai~ magnetic field, then the equation of motion for a volume element of th° electron fluid is [see (9.5/")] N + n) m = (p - P o ) ( E + dt V/ XB + v, X B 0 ) (9.114) When there is no ac field present, the steady-state motion is a drift with a constant velocity v0s along the direction of B 0 , which we take as the z axis, and rotation about B n at the cyclotron frequency il given by eB, PpBo n = Nmyf) (9.115) my, where y 0 = (1 - V'Q/C'Z) l / 2 . By using a very large magnetic field, fl can be in the microwave or millimeter-wave range of frequencies. For example, v0/c = 0.8 and B0 = 3 W / m 2 (30 kG), we get fl = 8.8 X 10 11 , wh corresponds to a frequency of 140 GHz. A magnetic field as large as 3 requires the use of superconducting solenoids. A gyrotron can operate harmonics of the cyclotron frequency. The advantage of operating at harmonic of the cyclotron frequency is that a smaller static magnetic field required, but this is accompanied by a lower efficiency for power generatio The possibility of field interaction with the beam at a harmonic of th cyclotron frequency is readily demonstrated by considering the curren associated with a circulating electron. Consider an electron with azimuthal velocity v04l and located at 4> = 4>u r = rn, at time t = 0. The current is in the form of an impulse J+ = -ev^Slr - r0)S(z)<5(</> - <£,), where the delta functions localize the current element at the position r0, z = 0, </> - d>\can make a Fourier series expansion of J^ in terms of the angle 4>; tnu J„ = S(r-r0)S(z) I I„e">< (9.11 where the /„ are given by Li. e " ' " * ( - e » # ) ( * - * i ) ^ J j 0 2TT -fit,-**, 2v [9.H MICROWAVE TIU1ES 705 Hence we have, at t = 0, J*= -^Hr-r^Siz) £ «#•«*-** (9.118) At a later time the position of the electron will be at <t> = <£, + ill. A Fourier series expansion of the current at time t may also be carried out and gives • W = -^r-8(r-r0)8{z) £ e'""-'^"iU (9.119) upon replacing rf>, by c6x + fit This expansion shows that the current associated with a single rotating electron is composed of an infinite number of equal-amplitude harmonics of the cyclotron frequency, a result due to the impulsive nature of the current. Consider now a very large number N of electrons spaced at random around the orbit. Each electron contributes a current given by (9.119) but with <6, replaced by <£, for the ; t h electron. The total current is obtained by averaging over all electrons and will involve the average of the following quantity over all phase angles 4>,: ev '*»-—7^S(r - r0)5(z) A £ e'—"I> For large N the average will be zero except for the n = 0 term which gives a factor N. Thus the current becomes **»=—^-S{r-r0)Hz) (9.120) LIT which is a dc current. These results show that, in order to obtain an ac current at the cyclotron frequency or its harmonics, the electron distribution around the orbit must be nonuniform. We require bunching of the electrons around the orbit. If electron bunching occurs, then we will obtain an ac current with which the microwave field can interact. We will show next that the dependence of the electron mass on the velocity provides a mechanism that will cause electron bunching to occur. If we take a scalar product of the equation of motion with v,, we obtain d (N+n)mVl -\— y v , | = ( p - p 0 ) v , - E We now note that d dy dv, v,--yv, = v , . v , - + v v , - — 2dy 1 duf (9.121) 706 FOUNDATIONS FOR MICROWAVK ENGINEERING since d(v, • \,)/dt = 2v, • dv,/dt. We also have dy 2. d in dt\ 1 - T [ dt i 2c 1 ,.** A By using these results we find that dy 2- •dt^ r - K ' dt so dy — c2\d y dvf /» _ + /'n 2 ^ = C"1 - dt B- — E 2 *' (9.122) mc This equation tells us that when v, • E is positive y will decrease and whe v, • E is negative y will increase. From (9.115) we see that when y decreases the cyclotron frequency will increase and when y increases the cyclotron frequency will decrease. Consequently, those electrons that have a phase angle greater than + 90° relative to the electric field will have thencyclotron frequency reduced, whereas those with phase angles less than ± 90° relative to that of the electric field will have an increased cyclotron frequency. This process results in bunching of the electrons in the azimuthaJ direction in a manner similar to the longitudinal bunching that occurs in a klystron. When the electric field adds energy to the electron, its azimuthal velocity increases. Paradoxically, this increases the value of y hut reduces the cyclotron frequency. What happens is that the radius of the orbit increases so that even though the azimuthal velocity has increased it takes longer for an electron to execute one circuit around the orbit so the cyclotron frequency is lower. dt m ( N + n ') c 2 "/ Gyrotrons generally use hollow cylindrical beams as shown in Fig. 9.23. In Fig. 9.23a we show a conventional beam in which the electrons revolve around individual magnetic field lines and do not have a common center of rotation (guiding center). This type of beam is best suited for field interaction at the cyclotron frequency. The beam shown in Fig. 9.23o is used in large-orbit gyrotrons. All of the electrons in this beam have common center of rotation. As a typical example a beam with a cyclotn frequency of 10 GHz and an azimuthal velocity of 0.8c will have an orbital radius equal to vM/i\ = 2.4 x 1 0 1 1 / ( 2 T T X 10 10 ) = 3.82 mm. This beam * large enough to provide good field interaction with microwave fields various harmonics of the cyclotron frequency, . * The small-signal analysis of a gyrotron can be based on the linear^ =*• equation of motion. If only terms linear in the ac quantities are retain (9.114), the equation of motion, we obtain v+4- yoNm\~ -pQE - p0(v0 xB + vXB0)+/>voxB(, (9 123) MICROWAVE TUBES 707 I I H I I \ \ \ S (a) lt» FIGURE 9.23 ( a ) Cross section of a solid electron beam with each electron having its own guiding center. (6) Cross section of a cylindrical sheath beam in which all electrons have a common guiding center. In addition, the current J is given by Po v + />vo (9.124) and the continuity equation dp V • J = - (9.125) m must hold. The above equations can be solved For J and p in terms of the ac fields E and B. Maxwell's equations must then be solved in the circular (or other) waveguide, both within the beam region and outside, including the source terms. The results will show that growing waves are produced. All ac quantities can be assumed to have the form •x where C„ is an expansion coefficient for the quantity of interest. Since the equations are linear, the solution can be carried out for the nth term by itself. The electromagnetic field in the circular waveguide can be described in terms of left and right circularly polarized waves. Only those fields that rotate in synchronism with the electrons will interact with the beam ac currents. For large power applications nonlinear effects must be included. The commonly used approach is to solve for the perturbed orbits of the electrons 708 FOUNDATIONS FOR MICROWAVE ENGINEERING and then evaluate the field interaction that take place. It is necessa numerical methods in order to solve the nonlinear equations. Typicaf ' obtained by this method are given in the papers cited at the end J "t SU ' ts of thi chapter. ~ 9.13 OTHER TYPES OF MICROWAVE T U B E S In addition to the main types of microwave tubes already discussed th are a variety of others as well. In one form of traveling-wave tube th' resistance-wall amplifier, the helix is replaced by a circular guide lined with a resistive material. The resistive lining enables a slow wave to propagate in the guide, a wave that is highly attenuated in the absence of a beam If an electron beam is present, amplification takes place with a growth constant a B large enough to offset the attenuation due to the resistive lining. Thus a net overall amplification is obtained, In another form of traveling-wave tube, the double-stream amplifier. two parallel electron beams are used. In this tube one of the beams provides the slow-wave structure, or circuit, for the other beam. It is also possible to amplify the space-charge waves directly by passing the beam through a succession of accelerating and decelerating regions. This type of tube is called a velocity-jump amplifier because the beam velocity v{l is periodically changed, or jumped, to new values. For both the O-type and M-type traveling-wave tubes, it is possible to adjust the beam velocity so that it is equal to the phase velocity of any one of the spatial harmonics making up the Bloch wave that can propagate along the periodic structure used for the slow-wave circuit. In particular, interaction between the beam and one of the backward-propagating spatial harmonics is possible. Consider a Bloch wave propagating in the —z direction. For this wave, E, has the expansion where p is the period of the periodic structure in the z direction. If we w interaction between the beam and the n = -1 spatial harmonic, it is necessary to choose u<> = v„ P = - (i / ft3 --2 i r / P ) 2ir/p~P If the period p is small enough, the n = - 1 spatial harmonic has a P velocity directed in the +z direction and its group velocity is in direction. If the » = -1 spatial harmonic is amplified, all the other s tiai MICROWAVE TUBES 709 h a r m o n i c s a r e also amplified, since t h e y m u s t all be p r e s e n t w i t h very definite a m p l i t u d e s in order t h a t t h e b o u n d a r y conditions m a y be satisfied. T h e amplification o f t h e n o n i n t e r a c t i n g spatial h a r m o n i c s comes a b o u t because of t h e i n c r e a s i n g surface c u r r e n t a n d c h a r g e induced on t h e metallic b o u n d a r i e s b y t h e amplified s p a t i a l h a r m o n i c t h a t i n t e r a c t s w i t h t h e b e a m . T u b e s e m p l o y i n g i n t e r a c t i o n with a backward spatial h a r m o n i c a r e usually used as oscillators a n d a r e called b a c k w a r d - w a v e oscillators, or c a r c i n o t r o n s . T h e y h a v e t h e i r o u t p u t coupling a t t h e cathode end. T h e r e a r e still o t h e r forms of m i c r o w a v e t u b e s , a n d no doubt more will be developed. F o r m o r e extensive discussion t h e cited references at t h e end of this c h a p t e r s h o u l d be consulted. 9 . 1 . Consider an electron beam of radius a, velocity u„, and space charge density fin. The dc current density is then JQ = p«O0. Show that a magnetic field rp.0v0 H.,. 0 < r < a a2p0v0 2r r > a is produced. Verily that the compression force - e v „ X B,,, is much smaller than the radial outward force due to the space-charge electric field and may therefore be neglected. 9.2. Show that an electron with velocity v perpendicular to B„ executes circular motion at the cyclotron frequency o>,. = e B 0 / m = TJZ?„ by equating the centrifugal force to the — ev X B u force. 9.3. An electron beam has a radius of 0.2 cm. The accelerating voltage is 1,000 V. The total beam current is 0.03 A. Calculate the beam perveance, the space charge density p* and velocity v„, the number of electrons per cubic meter, and the radial electric field due to space charge. Estimate the radial displacement of an electron located at the beam boundary during the time it takes the beam to move a distance d = 5 cm. Use the equation m d'lr/dtl = -eE,., and assume E,. to be constant and equal to its value at the beam boundary. Is the beam dispersion significant in this case if d is kept less than 5 cm? 9.4. Consider an electron beam with dc parameters />„, v 0 = v „ a , immersed in a field B0 = a;Dn. Assume a time dependence eJ'"' and a z dependence e •"'"' and solve the linearized equation of motion (9.8) for v, = vxax + L v a v + t'.a, to obtain = —n j{w-pu0)/^ co,/A 0 where A = co2 — (co — fivo)2- -av/A j(<o-pv0)/± 0 0 0 l/j(a, - livlt) E, Ey A', 710 FOUNDATIONS FOR MICROWAVE ENGINEERING From the continuity equation (9.5e) and (9.116), show that J = v„V • J Pov -jio u0V - J '•J,= " Po«. -ju> ~PoV, V • J = -jpj; - Pcft • v< ./ - JJ&t • v - Wp0Vt u> - fiun 9.5. Using the i-esults of Prob, 9.4, obtain solutions for p for waves in an infinite electron beam when all ac quantities are independent of x and y. Note that for space-charge waves, Ex = Ey = 0 but E. is finite. For the field waves' E: = 0. Hint: Note that V X JH = -y'/?a_, X H, T X E = - y , 8 a ; X E, which leads to the equation .Up2 - k'^E, = w/i„J, = -co/*0/>0v,. Answer: For field waves, p is a solution of (1)(0 <o..u>to~ («-/»«„>—r -(0*-4§)A = ± — where A is given in Prob. 9.4. Note that two solutions are given by w - f}va = ±01,.. These are the cyclotron waves. For p == k0, so that w » /3f0, four other approximate solutions are P - ±*o 1 - O)((0 ± ft),.) 9.6. Compute the gain of a klystron amplifier of the type considered in the text where the following data apply: Beam radius = 0.3 cm, beam current density = 100 niA/cm 2 . Accelerating voltage = 1,000 V, frequency = 3,000 MHz. GL = G0, cavity width d = 0.2 cm, cavity conductivity = 5.8 X 107 S / m . Compute the gain for d = 0.3 cm also, and compare with the earlier calculation. 9.7. Consider a reflex klystron employing a cylindrical cavity of the form shown in Fig. 9.12. The data of Prob. 9.6 apply, with d = 0.2 cm. The external loading GL = G0. The cavity grid-reflector spacing s is equal to 1 cm. Calculate an^ plot the electronic admittance spiral as a function of reflector voltage r a frequency of 3.000 MHz. Plot also the negative cavity admittance - Y on same susceptance plane. Determine the reflector-voltage variation o ^ across the n = 2 and n = 3 modes. Evaluate the change in oscillation quency a s the modes are tuned across. ., 9.8. Consider a cylindrical waveguide of radius a lined by a resistance shee so the boundary conditions at /• = a are Ez = -Z„,HA, where Z,„ = ' J"' a V e is the surface impedance of the wall. Analyze this structure as a trave MICROWAVE TUBES 7 1 1 tube when an electron beam (axially confined flow) with velocity i'„a. completely fills the guide. Determine an appropriate value of Z m in order to obtain amplification. Find the optimum value of Z,„ to give a maximum gain. Answer: fi is a solution of g I^ga) (P-1$ For ga large, so that i 0 / 7 , = 1, ji •= (1 - <S)£„ with 5 small. Sa is given by g2 = J .2kURm/Zlt)-(o,p/w) lil+.j2kl(Rm/Z{)y where Z,„ = ( l + . / ) / ? , „ . 9.9. Consider a cylindrical waveguide of radius a uniformly filled with a stationary plasma (an ionized gas with an equal number of electrons and ions per unit volume). At high frequencies the motion of the ions may be neglected because their mass is much greater than that of the electrons. Thus the guide will have the same properties as one filled with an electron beam with zero axial dc velocity. Use (9.33) to show that the guide may be considered as filled with a dielectric medium with permittivity e = e„(l atg/to ), where iu„ is the plasma frequency for the plasma. Find a solution for the lowest-order circularly symmetric E mode and show that for w < w0 such thai « is negative the wave impedance is inductive. 9.10. The results of Prob. 9.9 may be used to analyze the beam-plasma amplifier. Consider an electron beam passing through the plasma-filled guide. Use a confined-flow model to describe the beam and show that for a beam completely filling the guide the equations of Sec. 9.3 are valid provided e„ is replaced by £ = eu(l - WQ/IU 2 ) throughout, where <«,, is the plasma frequency for the plasma. In particular, (9.24) to (9.26) hold. Thus in (9.26), if u>;, = ( e p o / m e , , ) ' ' 2 is replaced by (ep^/me)1'2 = (epa/me0 )'-(<.)/< «* - <o'i)'/2h it is seen that /3 becomes complex for <o < w0 and a growing and decaying pair of waves are obtained. Show that the gain constant is given by Re 0O ph epe \ a-flf, (V-<4) and is very large when w is close to <u0. Note also that for a finite-radius beam with unconfined flow passing through an unbounded plasma medium the equations of Sec. 9.4 apply with e0 again replaced by e. For this model of the beam-plasma amplifier (9.45) may be used in place of (9.26) and will predict a gain constant of the same order of magnitude as does the confined-flow model. 9.11. Show that when both the plasma electrons and the beam electrons are subjected to the confined-flow condition the only change which occurs in the result given in Prob. 9.W is the replacement of w0 by Fu„. where F = (1 + Pom/Poa2) " / a ' s l n e plasma-frequency reduction factor. 712 FOUNDATIONS FOR MICROWAVE ENGINEERING REFERENCES 1. Slater, J. C.: "Microwave Electronics." D. Van Nostrand Company Inc Prinrot 1950. "' "*"• N -J.. 2. Pierce, J. R.: "Traveling Wave Tubes," D. Van Nostrand Company, Inc., Prin ceton, N.j 2950. 3. Kleen, W. J een, J.: "Electronics of Microwave Tubes," Academic Press, Inc., New York, lae 4. Beck, A. H. W.: "Space Charge Waves," Pergamon Press, New York, 1958, 5. Mutter, R. G.: "Beam and Wave Electronics in Microwave Tubes," D Van Niwt US., Princeton, N.J.. 1960. oatrand Company, Inc 6. Spangenberg, K. R.: "Vacuum Tubes," McGraw-Hill Book Company, New York 194R 7. Hamilton. D. R.. J. K. Knipp. and J. B. Horner Kuper: "Klystrons and Microwav Triodes," McGraw-Hill Book Company, New York, 1948. 8. Collins, G. B.: "Microwave Magnetrons," McGraw-Hill Book Company, New York 194R 9. Chodorow, M.. and C Susskind: "Fundamentals of Microwave Electronics," McGraw-Hill Book Company. New York, 1964. 10. Reich. H. J.. P. F. Ordung, H. L. Krauss, and J. K. Skalnik: "Microwave Theory and Techniques." D. Van Nostrand Company. Inc., Princeton, N.J., 1953. 11. Reich, H. J.. J. K. Skalnik, P. F. Ordung, and H. L. Krauss: "Microwave Principles," D. Van Noslrand Company, Inc., Princeton. N.J., 1957. Space-charge wave theory 12. Ramo, S.: The Electronic Wave Theory of Velocity Modulated Tubes, Proc. IRE, vol. 27, p. 757, 1939. 13. Ramo, S.: Space-Charge and Field Waves in an Electron Beam, Phys. Rev., vol. 56, p, 276, 1939. 14. Hahn, W. C: Small Signal Theory of Velocity Modulated Electron Beams, Gen. Elec. Rev., vol. 42, p. 258, 1939. 15. Chodorow, M., and L. Zitelli: The Radio Frequency Current Distribution in Brillouin Flow. IRE Trans., vol. ED-6, p. 352, 1959. 16. Rigrod, W., and J. Lewis: Wave Propagation along a Magnetically Focused Cylindrical Electron Beam, Bel! System Tech. J., vol. 33, p. 399, 1954. 17. Brewer. G. R.: Some Effects of Magnetic Field Strength on Space-Charge Wave Propagation. Proc. IRE, vol. 44, p. 896. 1956. Gyrotrons 18. Sprangle, P. and A. T. Drobot: The Linear and Self-Consistent Nonlinear Theory of the Electron Cyclotron Maser Instability, IEEE Trans., vol. MTT-25, pp. 528-544, 1977. 19. Li, Q. F., S. Y. Park, and J. L. Hirshfield: Theory of Gyrotron Traveling-Wave Amplifiers, IEEE Trans., vol. MTT-34. pp. 1044-1058, 1986. 20. Silverstein, J. D., M. E. Read. K. R. Chu. and A. T. Drobot: Practical C o n s ' ^ r ° 2 g in the Design of a High-Power 1 mm Gyromonotron, IEEE Trans., vol. Ml pp. 962-966. 1980. Compact 2 1 . Vitello. P.. W. Miner, and A. Drobot: Theory and Numerical Modeling of a ^ J ^ Low-Field High-Frequency Gyrotron. IEEE Trans., vol. MTT-32, pp. 373-38 CHAPTER 10 SOLID-STATE AMPLIFIERS The first solid-state amplifiers for microwave applications were negativeresistance diodes such as the tunnel diode. This was followed by the development of parametric amplifiers that used a variable-capacitance diode (varactor) and an oscillator (pump source) to vary the junction capacitance at the pump frequency. An outstanding feature of parametric amplifiers was the low noise that could be achieved by cooling the diode to liquid-nitrogen temperatures. The theory and design of parametric amplifiers is described in Chap. 11. Parametric amplifiers became the prominent and most widely used solid-state amplifiers during the period 1958 to about 1970. By 1970, improvements in materials preparation and processing technology had resulted in development of npn silicon bipolar transistors with a maximum frequency of oscillation greater than 10 GHz. During the next two decades further progress in the design and manufacture of high-frequency microwave bipolar transistors and field-effect transistors was dramatic. The key to successful microwave transistor design is miniaturization which is a necessity in order to reduce device and package parasitic capacitances and lead inductances and to overcome the finite transit time of the charge carriers. An appreciation for the need to reduce parasitic capacitance and inductance can be obtained by referring to Table 10.1 where representative values of reactances are given at several frequencies. For example, an inductance of 0.1 nH at 10 GHz represents a reactance of 6.28 (1 which is not a negligible value in a 50-11 system. A capacitance of 0.1 pP at 10 GHz has a reactance of 159 fl and would be a significant shunt reactance across a 50-il transmission line. Transit times are dependent on the electron mobility and saturation velocity in the semiconductor material. In this regard gallium arsenide 713 714 FOUNDATIONS KUK MICROWAVE ENGINEERING TABLE 10.1 R e a c t a n c e as a function of frequency Frequency K;HZI Reactance /. - 0.1 nH L = 1 nil C=0-lpF C = 1 pF I 10 100 0.628 6.28 1592 159 6.28 62.8 159 15.9 62.8 628 15.9 1.6 (GaAs) is significantly better than silicon for high-frequency devices Bv 1980, the design and fabrication of metal-semiconductor field-effect transis tors (MESFETs) were well established. In the frequency range above 5 GH7 MF.SFET devices are widely used. In order to achieve the high-frequency performance in transistors it was necessary to develop the technology that would enable key device dimensions to be less than 1 ,w.m, e.g., gate lengths with submicron dimensions. By means of molecular beam epitaxy (MBE), it has been possible to grow high-quality epitaxial layers and controlled doping profiles in highly localized regions. MBE techniques also led to the development of heterostructures which, in turn, led to the development of the high-electronmobility transistor (HEMT) which can operate at frequencies as high as 100 GHz. Microwave amplifiers are usually constructed either as hybrid microwave integrated circuits (MIC) or as monolithic microwave integrated circuits (MMIC). In hybrid construction the transmission lines and matching networks are usually realized as microstrip circuit elements on a suitable substrate material and then the discrete components such as chip capacitors, resistors, and transistors are connected in place by soldering or using wire-bonding techniques. Discrete devices are available with beam leads for easy insertion into the hybrid circuit. The word monolithic is derived from the two Greek words monos meaning single and lithos meaning stone. Thus a monolithic microwave integrated circuit is a circuit where all active devices, e.g., transistors, an passive circuit elements such as transmission lines, capacitors, resistors, and spiral inductors are fabricated on a single semiconductor crystal. substrate material used has typically been gallium arsenide because ot 1 s high resistivity in the undoped state and because of its superiority ° high-frequency field-effect device construction. A number of processes su as ion implantation for active devices, metal deposition and evaporation^ form ohmie contacts, electrode pads, and transmission lines, via hole e ing and plating, dielectric deposition, etc., is involved in monolithic c construction. The overall design and mask making is facilitated by t SOI.IIJ-ST.VI K AMPUFJBRS 715 of computer-aided design (CAD) programs. Electron-beam lithography and plasma-enhanced etching and deposition techniques are used for fabrication of submicron device elements. The cost of a monolithic microwave integrated circuit is related directly to how many circuits can be built on a single wafer since the processing of a single wafer is generally a fixed-cost operation. Consequently, in the frequency range below 10 GHz, where distributed circuit elements are relatively large, the hybrid form of construction is often less costly than monolithic construction. However, in the frequency range of 0.1 to 10 GHz, the ability to produce miniature inductors and capacitors has led to the development and production of many MMIC systems using lumped circuit elements instead of distributed circuit elements. In the millimeterwavelength band monolithic microwave integrated circuit construction promises to be more cost effective and to yield circuits with greater reliability and uniformity. In this chapter we are primarily concerned with the design of microwave amplifiers from the engineer's point of view. That is. starting from the measured or manufacturer's given two-port parameters of the device. we want to design an amplifier that meets a set of given system requirements such as gain, noise figure, bandwidth, and input and output VSWR. For this reason we only give a short discussion of the main characteristics of bipolar and field-effect transistors. Bias requirements and some typical bias circuits are also described. The design methodology7 that we will develop is based on the use of the scattering-matrix parameters for the device. At. high frequencies a transistor will have some intrinsic feedback from the output to the input, usually caused by a finite capacitance from collector to base and emitter lead inductance or in an FET the capacitance from drain to gate. Thus the device may be potentially unstable, and unless the amplifier is properly designed, it may oscillate, in which case it would not be useful as an amplifier. Thus, after the basic equations for amplifier gain have been derived, we examine the conditions under which the device is unconditionally stable or potentially unstable. If it is potentially unstable we will find that only for a certain range of values for the source and load impedances will the amplifier be stable. The available source and load impedances are easily displayed by constructing the source and load stability circles on a Smith chart. The equations for these stability circles will be derived and their interpretation and use will be examined. The Smith chart is an indispensable aid in the visualization of the different constraints that the engineer must take into account in the design of a microwave amplifier. In addition to the input and output port stability circles already mentioned, there are a number of other useful circles that aid the design process and can be plotted on the Smith chart. The most important of these are circles of constant gain, circles of constant noise 716 FOUNDATIONS KDU M1CROWAVS KN<;INI-KHIN<- figure, and circles of constant input and output mismatch. ' for these other circles will be derived and their interpretation ^uat'°ls Use amplifier design will be discussed. in The chapter will conclude with the description of a design strat ingleand two-stage amplifiers that are subjected to various systi <jv« egy f ° r si.» s .c- a. l u U«.U-,-M,«6C ampiiiio.o niaL me suujecieu quirements. A semi interactive computer program that implements th" reef.»lf>crv will w i l l Allan t a ^ r r i h o r i TThis V i i e nrtrr».-.%iffvi. sign strategy also hp be rdescribed. computer .-,„..._ program ren drudgery of carrying out all the computations that are required and wifl • the user valuable experience in a design process where a number of ^ V straints are imposed and tradeoffs among conflicting requirements musi'T made. Most of the relationships involved in amplifier design are of the for of a bilinear transformation from one complex variable to another complex variable. If Z and W are two complex variables, then an equation of the form W = AZ + B CZ + D where A, B, C, and D are complex constants, is a bilinear transformation of Z into W. This transformation has the property that circles in the Z plane will map into circles in the W plane, with straight lines as limiting forms of circles, with infinite radii, and some points as circles with zero radius. The relationship between impedance and reflection coefficient, namely. _ 1 + T is a bilinear transformation. For this transformation the straight 1 R = constant and X = constant in the Z plane map into circles in the plane. These mapped circles make up the Smith chart. Since the bilinea transformation occurs over and over again in amplifier design, we discuss its circle-mapping properties before we take up the design theory. This will provide results that enable us to readily identify circle mappings, in particu lar. the center and radii of mapped circles, from the particular bilm transformation involved. 10.1 BIPOLAR TRANSISTORS The basic principle of operation of a bipolar transistor designed for crowave applications is the same as that of low-frequency transistors. device must be biased to set the operating point. In a c o m r n o n . e n l j t a £ amplifier circuit, an input network must be designed so that a sign ^ can be applied to the base. A suitable load impedance must be c 0 " n e c T h e n the collector and the output signal is developed across this m i P , - c e r at main difference in the analysis of the operation of a transistor amp SOLID-STATE AMPUFtERS 7 1 7 microwave frequencies relative to low-frequency operation is due to the fact that the intrinsic device, along with package parasitic capacitances, resistances, and lead inductances, requires a much more complex equivalent-circuit model. In addition, there is often sufficient capacitive feedback from the collector to the base so that the device is potentially unstable and will be prone to oscillate unless the input and output circuits are designed to prevent oscillations from occurring. In the frequency range below 5 GHz, silicon bipolar transistors are generally preferred over GaAs FETs except for very low noise amplifier designs. Silicon bipolar technology is more mature and manufacturing costs are less. The gain obtained from a bipolar transistor is inherently greater than that from a field-effect transistor because of a much higher transconductance gm. Bipolar transistors are suitable for oscillator and power amplifier applications in addition to small-signal amplifiers. Power gains of 15 to 20 dB can be obtained at 2 GHz with noise figures of around 2 dB. At 10 GHz the power gain for many presently available bipolar transistors is around 5 dB and the GaAs F E T is then a better alternative. It is expected that, by perfecting the technology for making the critical dimensions of the emitter structure smaller, silicon bipolar transistors with a maximum frequency of oscillation approaching 100 GHz can be achieved. In recent years the heterojunction technology, originally applied to the construction of field-effect transistors, has also been used to improve the high-frequency performance of bipolar transistors. These transistors are called heterojunction bipolar transistors (HBTs) and exhibit very low base resistance, high current gain, and a speed increase by a factor of 2 to 3. An AlGaAs/GaAs HBT with a cutoff frequency of 105 GHz and maximum frequency of oscillation of 175 GHz has been reported.! Thus the future application of bipolar transistors can be expected to extend well into the millimeter-wavelength region. The basic construction used in a microwave bipolar transistor involves a multifinger interdigitated emitter-base construction. A simplified drawing of the cross section and top view of a bipolar transistor is shown in Fig. 10.1. The use of this particular design is for the purpose of overcoming transit-time limitations and yet maintain a sufficient emitter area. An equivalent-circuit model of the intrinsic bipolar transistor and the additional parasitic elements added by the package is shown in Fig. 10.2. A drawing of a packaged microwave transistor with beam leads is shown in Fig. 10.3. tN. H. Sheng, el al.. High Power GaAIAs/GaAs HBT's for Microwave Applications, 1987 IEEE Int. Electron Devices Meeting Digest, pp. 619-622, 1987. C. 11. Liechti, High Speed Transitors: Directions for the 1990's Microwave pp. 165-177, September, 1989. 718 FOUNDATIONS FOR MICROWAVE ENGINEERING B Oxide- % nitride insulator E B E H ^/^f-np ~ N~ e 7* /-\ Collector • e) (a) Bont ' «nB Pad (6) F I G U R E 10.1 ( a ) Cross section of a microwave silicon bipolar transistor; (b) top view showing interdigitated emitter-base construction. -VvV- 4=c, •AW c, 4= VW c, 4= AV»- -»c Oo/» -VW 1 «„ £ 4=c,9B 4= C, F I G U R E 10.2 Equivalent-circuit model of a silicon bipolar transistor. F I G U R E 10.3 A packaged transistor with beam leads. SOLID-STATE AMPLIFIERS 719 The equivalent-circuit model shown in Fig. 10.2 is based on that given by Vendelin, Pavio, and Rohde.t In this circuit model the various circuit elements are identified as follows: C bp —base bond pad capacitance C ep —emitter bond pad capacitance Rbl.—base contact resistance Rm—emitter contact resistance /?,, R2, R3—base distributed resistance Cv C2, C;j—collector-base distributed capacitance Re—dynamic emitter-base diode resistance Ce—emitter-base diode junction capacitance Rc—collector resistance Lh, Lt.—base and emitter bond wire inductances For an Avantek AT-60500 silicon bipolar transistor operated with a collector current of 2 mA and a collector-emitter voltage of 8 V, typical values for these parameters are: C b p + C 3 = 0.055 pF C, = 0.01 pF C^ « 0.026 pF C, = 0 . 7 5 p F R^ + Ra = 4 . 2 ( 2 . C 2 = 0.039 pF R„ = 0.66 1>, /?, = 7.5 12 R2 = 10.3 12 Re - 12.9 n Rr = 5 12 L6 = 0 . 5 n H Lt = 0 . 2 n H This transistor has a base cutoff frequency of 22.7 GHz. The common base current gain is given by 1 +jf/fh where f h is the base cutoff frequency and 7 d is the collector depletion region delay time (6.9 ps). The only device parameters that can be easily measured at microwave frequencies are the scattering-matrix parameters Su, Sl2, S.n, and S22. These can be measured by embedding the transistor in a 50-12 microstrip transmission line and using a network analyzer for the measurements. It is also possible to measure the StJ parameters with on-wafer probes. By measuring the S u parameters over a range of frequencies, the equivalent- fG. D. Vendeiin. A. M. Pavio. and U. L. Rohde, "Microwave Circuit Design Using Linear and Nonlinear Techniques." chap. 3, John Wiley and Sons. Inc.. New York. 1990. 720 FOUNDATIONS FOIt MICROWAVE KNC.INK.KRINO circuit parameters can be adjusted, by using a computer program they produce a circuit model with calculated scattering parameter tk correlate with the measured ones. The equivalent-circuit model Drov'H better physical understanding of the various circuit parameters that ^n affect the operation of the transistor. However, for amplifier design > ' easier to use the scattering-matrix parameters instead of the equivalent- "^ cuit model. Transistor Biasing There are two main considerations involved in the design of a bia circuit: (1) The biasing circuit must provide a stable operating point that is insensitive to variations in the device parameters and temperature changes and (2) the biasing circuit must be isolated from the high-frequency circuit so that high-frequency signal currents do not flow in the dc biasing circuit. The first requirement is met by incorporating dc feedback in the biasing circuit. The second requirement is met by inserting high-impedance highfrequency circuit elements in series with the dc components and by using low-impedance capacitive bypass circuits to shunt high-frequency currents around the dc circuit elements. The overall bias circuit and RF matching circuits must provide stable terminations for each active device outside the frequency band of interest in order to ensure that oscillations do not occur at any frequency. The bias circuit shown in Fig. 10.4a provides a stable operating point. It is commonly used in low-frequency electronic circuits and can also be used in microwave amplifier circuits but with more difficulty because of parasitic inductance associated with the capacitor leads. The bias circuit is shown isolated from the transistor by incorporating series inductors (RF chokes) between the device terminals and the bias-circuit resistors. The -> V,. R* r ' RFC Q Q • RFC W 0.- F I G U R E 10.4 , ( a ) A passive bias circuit, an active bias circuit. SOLID-STATE AMPLIFIERS 721 emitter terminal is maintained at RF ground by means of the bypass capacitor C. An active bias circuit is shown in Fig. 10.46. In this circuit the collector current in transistor Q t is established by means of the resistors Ry, r? 2 , and i? 3 . The base current in the microwave transistor Q., is the collector current of Q t . Since the bias circuit for Q, is a stable one, the collector current of Q u and hence the base current of Q.2, is maintained at a value that is essentially independent of the transistor parameters. This circuit has the advantages that it consumes less dc power and requires only two RF chokes for isolation as compared with the circuit shown in Fig. 10.4a which is shown with three RF chokes and a bypass capacitor for isolation purposes. In a microwave amplifier the RF chokes are often replaced by a quarter-wave high-impedance transmission line or a combination of transmission-line sections. FIELD-EFFECT TRANSISTORS There are two main characteristics of field-effect transistors that make them superior to bipolar transistors in microwave amplifiers. These are the lower noise characteristics and the higher frequency of operation. The higher operating frequency is due to the higher electron mobility in gallium arsenide, which is the material used in field-effect transistors, compared to that in silicon, the standard bipolar transistor material. The higher electron mobility and the absence of shot noise are important Features that result in low noise. The first, microwave field-effecl transistors were metal-semiconductor field-effect transistors (MESFETs). High-frequency operation required a very short gate length, typically less than 1 fim. Thus it was only after the processing technology had advanced to the stage that submicron device features could be reliably made that microwave solid-state device development advanced rapidly. Currently produced MESFETs have gate lengths of the order 0.3 to 0.5 /im. The frequency at which the short-circuit current gain becomes equal to unity is given approximately by the relationship fT = us/2-rrLg, where vs is the electron saturation velocity and L/: is the gate length. In gallium arsenide (GaAs) the maximum drift velocity is about 2 x 10 7 c m / s , so that for a gate length of 0.5 nm, fT = 60 GHz. Clearly, very short gate lengths are essential for high-frequency operation. Beginning around 1980, a new technology involving heterojunctions began to find applications in device construction. A heterojunction is a junction formed at the interface of say an aluminum-gallium-arsenide (AlGaAs) doped alloy and an undoped GaAs layer. The use of a heterojunction enables a channel with a very high electron mobility to be obtained. The field-effect transistor which is made using a heterojunction is called a high-electron-mobility transistor (HEMT). Since high mobility is achieved by doping only the large bandgap material, the name modulation-doped 722 FOUNDATIONS FOR MICROWAVE ENOJ.VEEH)S<; Source Drain VZZZZZZA, N'GaAs Gale zzzzzzz* _E22_ \ Undoped AIGaAs \ N' AIGaAs KS5^X^sX^X^^\\^\\SS\\y Undoped GaAs y Heterojunclion high-mobility channel Semiinsulating substrate Source Gate pad (c) F I G U R E 10.5 ^^, (a) Cross section of a high-electron-mobility transistor (HEMT); (6) source, drain, and gale structure for a low-power FET; (c) interdigitated construction used for a power FET. The source metalization passes above the gate structure and is insulated from it. field-effect transistor (MODFET) is also used. The HEMT device has a higher frequency of operation and a lower noise figure than the standard MESFET device. A good discussion of the technology and fabrication of HEMT devices and the theory of operation can be found in the books by Chang and Pengelly.t . Figure 10.5a shows the cross section of a typical HEMT device, while Fig. 10.56 shows a top view of the source, drain, and gate structure. I « gate width W g is much greater than the gate length Lg. In a power Fh ^ many as 10 source and drain fingers adjacent to gate fingers are use: obtain large drain currents and hence large output powers. The interc t K . Chang led.). "Handbook of Microwave and Optical Components. Microwave Sob Components." vol. 2. John Wiley & Sons Inc.. New York, 1990. •• .^Dd R. S. Pengelly. "Microwave Field-Effect Transistors—Theory, Design and Applications ed.. Research Studies Press, l.etchwarth, England. 1986. SOLID-STATE AMPLIFIERS 723 5W>—« D FIGURE 10.6 Simplified small-signal equivalent-circuit model of a microwave GaAs FET. tated construction is shown in Fig. 10.5c. The transconductance g m of the FET is increased by using a large gate effective width and this is needed for large power output. However, with a larger gate width the input capacitance is increased. A simplified small-signal equivalent-circuit model of a microwave GaAs FET is shown in Fig. 10.6. Typical values for the circuit parameters for a device with a l-/i.m gate length and a 300-fj.m gate width are listed below: Gate-to-source capacitance C„s = 0.4 pF Gate-to-drain capacitance CRd = 0.01 pF Channel resistance R ds = 500 ii Transconductance gm = 30 mS R, = 3il Cdl. = 0.015 pF The circuit parameters Rg, Rd, Rs, and C^ are extrinsic elements. The gate, drain, and source ohmic contact resistances Rg, Rtl, and R s are typically a few ohms. The drain-to-substrate capacitance C ds = 0.07 pF. The inductors Lg, L,, and L d have inductances in the range 0.05 to 0.3 nH. The equivalent current source is Vggm, where V^, is the signal voltage across C „ . The microwave FET can also be described in terms of measured scattering-matrix parameters. Gallium-arsenide MESFET devices can give a single-stage gain of 8 to 15 dB at 2 GHz with noise figures below 1 dB. For HEMT devices a 724 FOUNDATIONS FOR MICROWAVE ENGINEERING F I G U R E 10.7 Output characteristics for a GaAs MESI showing suitable operating points P, and p. for small- and large-signal inputs. single-stage gain of ] 5 dB al 8 GHz and 6 dB at 50 GHz can be achieved with corresponding noise figures of 0.4 and 1.8 dB, respectively.! For power applications output powers of several watts from a single device can be obtained. Several devices may be operated in parallel using power-combining techniques to achieve higher output powers. FET B i a s i n g The output drain current versus drain-to-source voltage Vds is shown in Fig. 10.7 as a function of gate-to-source voltage for a typical microwave GaAs MESFET. For small-signal application an operating point in the vicinity of the point P l would be suitable. For maximum dynamic range the operating point should be placed in the central region of the output characteristics which is depicted by point P2. In either case the dc voltage of the gate must be negative with respect to that of the source for a depletion mode device. This bias condition can be achieved by grounding the gate through an RF choke or a high-impedance quarter-wavelength transmission line and obtaining the desired bias voltage from the voltage drop across the source resistance R s as shown in Fig. 10.8. The required value of R s is given by R s = - V ^ / / ^ . The source resistance R„ should be bypassed to ground for RF signals by means of a capacitor C„. In small-signal low-noise applications, the best noise figure is obtained for a dc drain current equal to abou 20 percent of the drain saturation current at zero gate-to-source voltage. At low values of drain current, the transconductance is reduced, si that gain must be sacrificed to achieve a low noise figure. t K . Chang, toe. cit.. p. 465. SOLID.STATE AMPLIFIERS RFC or ft 725 ft= < 4= Cs FIGURE 10.8 FET bias circuit- 10.3 CIRCLE-MAPPING PROPERTIES OF BILINEAR TRANSFORMATIONS A circle with center at x0,y0 and with radius R is described by (x — xu)~ + (y-y0)2-R2 = 0oT x2+y2- 2*x u - 2yy 0 + (x$ 4 y2 - i? 2 ) = 0 Now let Z = x +jy, Z0 = ,r0 + jy(). The circle equation can be written as \Z - Zf - R2 = 0 or (Z - Z^Z* - Z*) - fl2 = 0 which gives ZZ* - ZZ% - Z*Z0 + (Z 0 Z c t - ft2) = 0 ;io.i) Consider now the bilinear transformation from the complex Z plane to the complex W plane given by W = AZ + B CZ + D This transformation will map circles in the Z plane into circles in the W plane (straight lines are limiting cases). Consider the circle \W\~ = p 2 or WW* - p2 = 0. Using the transformation, we get AZ + B A*Z* + B* CZ + D C*Z* + D* -p2 = 0 By expanding this equation we get ZZ*(AA* - p2CC*) - Z{p2CD* - AB*) - Z*(p2C*D - A*B) + BB* - p2DD* = 0 (10-2) By comparison with (10.1) we see that this is a circle with center at (coefficient of - Z * ) 2 Z„ = C*D - A*B p2C*D - A*B AA* - p*CC* " |A|2-p2|C|2 P (10.3) 726 FOUNDATIONS FOR MICROWAVE ENGINEERING The constant term equals [Z 0 | 2 - R2, so that we can identify the as given by R - Z0ZQ - \B\2 - P W IA12 - p 2 |C| 2 from which we get R \AD - BC\ |(A[2-p2|C|2| <10-4) If the circle in the W plane is \W - Wa\ = p, then we note that W-Wn AZ + B CZ-t- D Wn (A- CW0)Z + (B-DW0) CZ + D If we define A = A - CW0, B' = B - DW0, then the above formulas (10.3) and (10.4) apply with A, B replaced by A', B'. Many of the relationships that occur in amplifier design involve bilinear transformations and their circle-mapping properties can easily be identified by comparison with the above equations. 10.4 MICROWAVE AMPLIFIER D E S I G N U S I N G Su P A R A M E T E R S At microwave frequencies impedance and admittance parameters of a transistor cannot be directly measured. The scattering-matrix parameters can be measured and therefore a design methodology based on using the S,, parameters is widely used. The S parameters are measured by inserting the transistor into a test circuit with 50-9. input and output lines, applying appropriate bias voltages and currents, and measuring the S,-j parameters. In any design using StJ parameters, it should be kept in mind that these vary with bias conditions, temperature, and £rom transistor to transistor even if it is the same device number. Thus the design should leave some margin for S,j variations. The following are the usual microwave amplifier design goals: 1. Maximum power gain. 2. Minimum noise figure for the first stage. This requires a specific s ° u r ^ impedance Z s for the input stage. The optimum Z s giving the low noise figure is generally given by the manufacturer of the transistor3. Stable gain, i.e., no oscillations. 4. Input and output VSWR as close to unity as possible. 5. Adequate gain and uniformity of gain over a specified frequency " SOLID-STATE AMPLIFIERS 727 6. Phase response that is a linear function of to (no distortion, orly group delay). 7. Insensitivity to nominal changes or variations in the device S,, parameters. These objectives cannot all be realized at the same time; so the design procedure must trade off one objective against another one, e.g., gain must be sacrificed for stability. Input VSWR must be sacrificed for a low noise figure since the normalized source impedance Z s is fixed for minimum noise. Designing a microwave amplifier using potentially unstable transistors is like designing a bridge using interconnected beams whose lengths can be telescoped and with joints that are free to rotate with insufficient external constraints to define a rigid structure. It is a "loose-jointed" problem without a unique solution. The many design specifications are all interrelated which makes the problem almost unmanageable without some computer optimization strategy. In the next several sections we will examine in detail the constraints that are imposed by stability requirements, by the need for large power gain, low noise, and low input and output VSWRs. We can obtain a good physical understanding of these often conflicting requirements from a study of the stability circles, the constant power-gain circles, the constant impedance-mismatch circles, and the constant noise-figure circles plotted on a Smith chart. From the insight obtained from such a study, we will be able to formulate a design strategy using some computer optimization that will lead to satisfactory designs. The reader will need to bear with us as we work our way through the maze of details but will in the end be rewarded with the satisfaction of having obtained the necessary insight and understanding to be able to implement the theory for practical amplifier design. All the various low-frequency amplifier circuits such as balanced push-pull amplifiers, cascode amplifiers, and traveling-wave amplifiers can also be used at microwave frequencies. A device that is potentially unstable can be stabilized by resistive loading at the input, output, or both input and output, with a resultant reduction in power gain and increase in noise figure. Stabilization can also be achieved by using negative feedback. Space limitations do not allow a detailed discussion of the variety of amplifier circuits and configurations that are used in practice. Many of these circuits are described in the book by Pengelly and the one by Vendelin, Pavio, and Rohde already cited. The reader is referred to these texts for a detailed discussion. In this text we will limit our attention to amplifier design based on the use of scattering-matrix parameters and linear two-port design. For high-power amplifier design, it is necessary to consider the nonlinear characteristics of transistors and to pay more attention to device power 728 FOUNDATIONS FOR MICROWAVE ENGINEERING dissipation and the design of adequate heat sinks. Small-signal » r design can often provide a first approximation to the design of lar? • Sl ra amplifiers, particularly in connection with stability. Thus the linear t ** ^ design methodology developed in this chapter will also provide useful' ^^ into the large-signal design problem even though we do not cover the 1 t ter topic. 10.5 AMPLIFIER POWER GAIN We will begin our discussion by deriving expressions for the power gain of an amplifier. There are several definitions used for the gain of an amplifier and they are given below. power delivered to load Power gain Gp = input power to amplifier (10.5a) power delivered to load Transducer gain G = Available power gain Ga = available input power from source ' available load power available input power from source (10.5c) If the device is unconditionally stable, then conjugate impedance matching can be used at both the input and output. If this is done then G p = G Ga = Gmax - maximum gain. For a device that is only conditionally stable, conjugate impedance matching at both the input and output cannot be used when the stability parameter K < 1. The power gain achieved in this case is G . Power gain is the most useful definition in practice since it applies to any actual amplifier independent of whether conjugate impedance matching is used. For the two-stage amplifier shown in Fig. 10.9, the incident power u given by 50O F I G U R E 10.9 A two-stage amplifier with matching networks. SOLID-STATE AMPLIFIERS 72}) and the input power is pin = (i - inVta where Y c is the characteristic admittance of the input line and Y is the input reflection coefficient. The power P L delivered to the output 50-J1 line is given by GplG2P-,„• Hence the two-stage power gain is Gp = -^ = GplGp2 (10.6) and the corresponding transducer gain is G = -^ = (l-|lf)G„1G/,.2 (10.7) For lossless matching networks the impedance mismatch is the same on the input side as on the output side as shown in Sec. 5.7. If M x is the impedance mismatch between the first amplifier input and its source impedance Z„ as seen looking into the output side of the first matching network, then 4fl 5 fl,„ M, = — =—: iz, + zj2 where Z i n = R m + jXin is the amplifier input impedance. On the input line 4Z.fi 4R 2 |1 + Z\2 \ZC + Z) where Z = (1 - D / ( l + D. When we use the relations 1 + T 1 + V* ouR p and 7 i 7* i ~z ' " 11 i-r ' I-r i 7 '" 2 i+r we find that M= i - rr* = I -\n2 = Mx (io.8) upon substituting for R and 1 + Z in terms of 1* and simplifying the resultant expression. The input voltage standing-wave ratio is given by VSWR,1 = i + in — = 1 - HI i + i/i - M , = 1 - vT - M 11 i / i - j ^ t , 1 - -fi^Ms (10.9) Thus the degree of mismatch between Z s and the input to stage 1 determines the input VSWR. If we have a constraint on VSWRj in our design and we want to use an optimum source impedance Z s for minimum noise, then we must terminate stage 1 in a load that will produce an impedanct; mismatch M s that will keep the input VSWR, at the specified value (oi- 730 FOUNDATIONS FOR MICROWAVE ENGINEERING e C~s r,n^ F I G U R E 10.10 A basic amplifier circuit. Cr o u l r L j> lower). The required load termination must, of course, not lead to unstable (oscillating) amplifier and must also yield good power gain A power gain of 10 or more is desirable, since this will make the effect f following stages on the noise figure small. Sometimes the input VSWR constraint has to be relaxed because it is in conflict with other requirements. •Derivation of Expressions for Gain Consider the basic amplifier circuit shown in Fig. 10.10. The source and load are viewed as connected to the amplifier by means of transmission lines with characteristic impedance Z c and having negligible lengths. For this circuit the source and load reflection coefficients are given by Z, - 1 'L ?T and Z, + 1 r = ~— s z. + i For the amplifier we have vf = s „ v r + s12v2 V2 = S2lV{ + S22Vi But v2 = r ; y, so v2- = s2lv; + s22rLv2 or v2 = s2yt/a - s22rLi we use this in the first equation to get vf = vr s n + s, 2 s 2l r L i - s 22 r L and hence (10.10) = r*, 1 - S22TL where A = S U S 2 2 - Sl2S2l. In a similar way we find t h a t r„.„ = s 22 - AI; l - s„r. (lO.U) SOUDSTATK AMPLIFIERS 731 The input power to the amplifier is given by W/R, ^ 8*. * 2IZ. + Z J * where |VS| /8RS is the available power from the source. First, we will express M s in terms of l's and l' jn . To do this, we use i - 1 ; 2R„ = Zs + Zf and i + rs i-i; i + r* i-ir 2(1 - \rf) ii-i; since (1 - r„Xl -r*) = \l - l'f. Similarly, we get li - r,J 2 Hence we obtain M, - 4(i-irj2)(i-\rf) \i - \\f\i - if i + rs 4RmRs -= \zs + 2J i - r. 1 + r„ - r„ + I 4(i-ir,j2)(i-ir/) ii - r / l i - rinl 2 li - r/nl |i - rf |(i + j ; ) ( i - r , j + (l - r j ( i + v,jf (i - ir,„l2)(i - wf) li - r,r,j 2 (10.12a) Similarly, at the output M,= II - i'J'ouJ2 (10.12o) The load voltage VL = V2 + Vg = K f d + FL). The load current is IL = Yc( V2~ - V£), so that the power delivered to the load is given by pL = |Rev,./* = iy,.iv2i2(i + r j ( i - vt) Now (i + r t ) ( i - Tt) - 1 - r j T + rE - rr = 1 - |r,. 2 so ^•= W ( i - i r j ) y r 2Imr, 732 FOUNDATIONS FOR MICROWAVE ENGINEERING as expected. We now use the expression for V.2 given above (in ini °btain The input power is i|V, + | 2 (l - |f i n | 2 ) Yc. Hence G ," Pm (i-irj 2 )is 2 1 i (i-irmi2)u-s22rL (10.13) The transducer gain is given by PL G = Pava W//8R, The available power P ava is related to the input power by P MsPava and thus the expression for transducer gain becomes G=ELM pm 2\io C1 - irj')(i - ir.l')is, " u - s 22 r,i 2 |i - r i n r / = MP = |2 = (10.14) We now eliminate r jn from the gain expressions by using (10.10) for rlnI that is, 2 i - inj = i - ii - s 22 rj li-s22rj2-isu-Arj ii - s B iy" The elimination of I'in gives us (i - irj2)is21l2 G = " n-s22rj2-|sH-ArLi2 Next we use |1 - S22VL\2 = (1 - S ^ X l - S & I ? ) and similarly \Sn - &TL\2 to get for the denominator i + s*s&rtr£ - s22i\ - s;2n - s„sf, + s„ A*TL* = 1 - |S„| 2 + irj 2 (|S 22 l 2 - |A|2) - 2Rer t (S 2 2 - ASf.) Hence G„ = (i-ir,r)is2,i 2 2 i - is„i + irj (is 22 i 2 - IAI2) - 2 R e r a s 2 2 io.i5) ASf,) SOLID-STATE AMPLIFIERS 733 The same substitution for !"„, reduces the expression for G to G = (i - irj2)(i - ir/)is21i2 DL-««r i -s u r f + Ar,rI.r (i - irt,i2)(i - ir,ia)is2)ia |(i-s,2rj(i-snrj-s12s21rsrj (10.16) These are the final forms for the power gain and transducer gain. When the device (transistor) is absolutely stable, we can use conjugate impedance matching, i.e.. choose Zs = Zj*, ZL = Z* ul . In this case Vs = T^, VL = r*u[, and Ms = M, = 1. Clearly, when Ms = I, the power gain Q equals the transducer gain G. For conjugate impedance matching we require i* = r. = 1 o and r1 = r* = i, • in ' ..in S* J 22 -1*V* 1-SfiI? We can substitute the second equation into the first one and solve for Y s which gives r.a sM 2B, A, + (^ -4IB.I2)' (10.17a; A, ± ( A 2 2 - 4 | B a | 2 ) I / '10.1761 Similarly, ' L ~ 'LM ~ where 2B2 Ax = 1 + IS„I 2 - |S 22 I 2 - |A| A.2 = 1 + | S 2 2 | 2 - | S n | 2 - | A | 2 B, = S | , - AS22 B2 = S^v - AS,*", The minus sign is used when A, > 0 and the plus sign is used when A, < 0 in order to get |l' sM l < 1 and \VLM\ < 1. We will show in the next section that for an absolutely stable device A, > 0 and A2 > 0, so the minus sign is the appropriate one to use in (10.17). When A\ < 4 | B 2 | 2 the solution for VLM can be expressed as [A.2 ± y ( 4 | B 2 | 2 - A | ) 1 / 2 ] / 2 B 2 . For this case the magnitude of YLM is equal to unity which corresponds to a pure reactive load termination and zero power gain. Thus, for an absolutely stable transistor, we must have |A 2 | > 2|B 2 |. It is this condition that leads to the choice of sign to use in (10.17). 734 FOUNDATIONS FOR MICROWAVE ENGINEERING In the next section we will show that a device is absolutely <*t stability parameter K is greater than one, i.e., K _ l - )S„| 2 - }S22) 2 ee '«f the 2 + \A\ 2|S12S2,| (10.18) In terms of the parameter K, the power gain for an absolutely stabl H e V | ce, using conjugate impedance matching, is given by G p = G '21 p.max = G = Gmax = (#-V/r^T) '12 The parameter When K = 1 expression for can be derived can show that (10.19) IS 2 1 /S ! 2 I is called the "Figure of Merit" for the transistor it gives the maximum stable gain G M S G = \S.n/S I The Gmax for an amplifier with conjugate impedance matching following the steps outlined below. By direct expansion we A | - 4 i 5 / = 4|Sl2S21|2(/iry- 1) From the equations for A 2 and K, we can show that 1 - )SU\2 = A., + lAl2 - \S22\2 = 2K)S,2S2i\ ~ (\A\2 - \S22}2) so by addition and subtraction we get 1 - IS,,! 2 = K\Sl2S2l[ + -^ |A| 2 - | S 2 / = K\Sl2S21\ - — We also take note of the relationship 2 W S22 - S*XA) = A2±(A%- 4|5./)l/2 which is a real quantity. By using these expressions the denominator (10.15), which is the equation for Gp, can be written as K\Sl2S2l)+^ A., J A.. +\h\H~- - K\S]2S2I\) / r +2is 12s21iv ^ ^T = (i-ir,/) i - \rtf - A2+2\Sl2Sn\JK _lA1_K]Si2s2i ^ ^ ^ ^ The fast step is to use the solution F L M for T L to get i - irJ - , 8 " a » h f r I (4 - is.sa.l^1) \B.,r \ l SOLIDSTATK AMPLIFIERS 735 With this substitution and a few more algebraic steps, we get is„i' G P \SV2S2,\(K+ 2 i/K -l) s« s„ (ff± V^2 - 1 ) There are two possible solutions for G p which correspond to the two possible solutions for VLM. Since Gp should not keep increasing for K » 1. only the solution given by (10.19) corresponds to the gain obtained with a passive load. We will have a further comment on these two possible solutions at a later point. For now we note that the chosen solution corresponds to using the minus sign in (10.17/;) which implies that A2 > 0. 6 AMPLIFIER STABILITY CRITERIA When the transistor is potentially unstable, which occurs when the stability factor K given by (10.18) is less than one, a stable amplifier can still be designed but only for restricted values of source and load impedances. Furthermore, it will not be possible to use conjugate impedance matching at both the input and output ports. In this section we will derive the expressions for the allowed terminating impedances in terms of input and output reflection-coefficient stability circles. These stability circles can be plotted on a Smith chart and will show what values of source and load impedances can be used in order to achieve a stable (nonoscillating) amplifier. The conditions for amplifier stability are established by requiring that the reflected power from the amplifier ports be smaller than the incident power. This means that the reflection coefficients looking into the amplifier ports must have a magnitude less than one for all passive source and load impedances. If a reflection coefficient has a magnitude greater than unity, the amplifier input or output impedance would have a negative real part, e.g., if Z , „ = -R + jX then IU = -R +jX - Zc (R + Z,.f 4 l -R +jX + Zc (Z,.-R)2+X2 ! 1 2 > 1 If Zm *= -R„, +jXw then the input current is 7= V. Rs-R,n+j{Xm+Xj If Rx = Rm and Xm + Xs = 0 which can occur at some frequency, then / becomes infinite. We can set V s = 0 and thermal noise in the input can produce self-sustained oscillations at the frequency where the total loop impedance in the input equals zero. Oscillation at any frequency generally makes the amplifier unusable. 736 FOUNDATIONS FOR MICROWAVE ENGINEERING The conditions for stability are - AJ\ irj = s„ 1 - s22rL s22 - AI; i-sui; < i forall|rt|<l <I forallirj<l (10. 20a) (10.206, impedance Zs. In this circumstance the device is said to be condition T stable. If ZL = Ze, then \\ = 0 and | r j < 1 only if | S „ | < 1. Similarly ff Z, = Z C , i r ( ) U J < l only if | S 2 2 | < 1. Hence two necessary conditions for absolute stability are IS„|<1 and >22 < 1 Values of T, that result in \rm\ < 1 are called stable ones. The corresponding region of the Smith chart is the stable region. The boundary between stable values of T L and unstable ones is the circle in the V L plane that corresponds to the mapping of the circle jF^] = 1 in the Tjn plane. From the bilinear transformation r.„ Su - AfL i s22rL ArL - s u s^r,, I we find that the center of the mapped circle is at (A = A, -SU=B, S.22 = C, - 1 = D) SUA* - S22 r*.c = |A|2 - \Sj (10.21a) and the radius of the circle is ) 12 S 21 | l Rr.r LC ~ (10.216) |(|A|2-|S22|2)| This load stability circle may or may not include the origin r L = 0 as s ow in Figs. 10.11a and b. When Y L = 0, Tin = S„ and | S „ I < 1 for a «J circuit. Hence, if the stability circle encloses the origin, then all v a l u * ? es 0 f inside the circle will give values for Ym such that I T J < 1- These v r L are stable ones. If f L = 0 lies outside the circle, then all y f j j j f ^ u s outside the circle are stable ones. The origin is included only " t ^s situ tlon RLC is greater than the distance | r i C | to the center. If this * n g i d e the then we require RLC > 1 + \TLC\, so that all values of JfJ < 1 ue g^goBcircle and represent stable values for all possible passive load ternu that If the origin is not included, then we require that |FLCI > 1 + " ' • ' ' ' .so SOLID-STATE AMPLIFIERS 737 ir, n l = 1 circle r,nl = 1 circle ^ Smilh chart boundary (a) FIGURE 10.11 Load stability circles plotted on the Smith chart, (a) Origin is included within stable region; (6) origin is excluded so load impedances outside stability circle are stable values. t h e e n t i r e stability circle \Fin] = 1 m a p s into a circle o u t s i d e t h e S m i t h c h a r t b o u n d a r y \TL\ = 1. T h e n a g a i n all values of \VL\ < 1 will be stable o n e s . We need to find an expression t h a t will e n a b l e us to s t a t e , in t e r m s of t h e s c a t t e r i n g - m a t r i x p a r a m e t e r s Su, S 1 2 , S 2 1 , a n d S 2 2 of t h e device, w h e t h e r or not it is an absolutely s t a b l e device. If t h e amplifier will be stable for all passive s o u r c e a n d load impedances, t h e n t h e device is absolutely stable. If it is stable only for a limited s e t of source a n d load t e r m i n a t i o n s , t h e n t h e device is only conditionally stable, T h e r e are t w o cases to consider a n d they b o t h lead to t h e s a m e s t a t e m e n t or criterion for stability. C a s e 1. Origin (1*^ = 0 ) lies outside the circles of V L values that make ir, n l = 1. As noted earlier this case corresponds to all values of I", outside the load stability circle |I" m | = 1 being stable values. All passive values of Z/ will be acceptable if the \Tm\ ~ 1 circle lies outside the Smith chart boundary |I"J = 1. This requires that the distance \VLC\ to the center of the circle be greater than the radius plus one, that is, |ftc-| > 1 + RLC. In order to have r / r | > 1 + Rlr, we must have tic > ff£ c . From (10.21) we see that this requires Ii>t,A* — S ^ T > I S ^ S ^ J|2 . By direct expansion we can show that Ou-i &22I ~ l^ , 12^ l 2ll (l-|Sn|2)(IS,/-|A|2) (10.22) Hence our stated condition is equivalent to (1-IS„|2)(IS22I2-|A|2)>0 which is possible only if IS 2 2 | > IAi since IS n l < 1. Thus Case 1 can occur only when this condition is true. When the condition is true, then |(|S22I - |A| 2 )| = | S 2 , | 2 - |A| 2 . The stability condition 1 + R, r < \l'LC\ can be stated in the form (Su* - S}f >d + RLCf(\M2-\s2./y 738 FOUNDATIONS FOR MICROWAVE ENGINEERING upon using (10.2la). From (10.216) we obtain ««:- is 2 2 r - IAI2 + is, a s 2 ,i IS 22 I 2 ~ IAI2 and hence |S„A* - S | 2 | 2 > ( | S 2 2 | 2 - IAI2 + \Sl2S2i\f Upon using the expansion (10.22), we can restate the condition for stabilit IS1252ir2+(l-|5u|2)(|S22l2-|A|2) > ( I S , / - IAI2 + \Sl2S2,)f After multiplying out the term on the right, canceling the common factor | S l a S 2 i l on both sides, and also canceling a common factor IS 2 .,| 2 - |^i 2 we get ( | S 2 / - | A | 2 ) + (2|S 1 2 S 2 X |) < 1 - I S U | 2 We now introduce the stability parameter K and express the final result in the form 1 - I S , / - |S 2 2 I 2 + \M2 2IS| 2 S 2 1 I |SU|<1 > 1 (10.23a) (10.236) C a s e 2. It may happen that the circle of T L values giving If^l = 1 encloses the origin (F L = 0, Tin = S u ) . In this case all Y L values inside the circle are stable ones. For absolutely stable devices the circle must then be large enough to enclose the entire Smith chart \YL\ < 1, so that any passive load impedance Z L can be used. This requires that the radius RLC be greater than one. With reference to Fig. 10.12 it is seen that we now require d > 1 or RLC ~ \^LC> > l or equivalently |F £C I < Rlc - 1. This case can occur only if i r i n l = 1 circle FIGURE 10.12 , i t i o n when Illustration for deriving stability to the stability circle encloses the origi SOLID-STATE AMPLIFIERS 739 using (10.21) we find that we require |S 22 I < IA|. We now use (10.21a) and (10.21A) aJong with the expansion for \SnA* - S$J2 given by < 10.22) to state the stability criterion in the form [|(|A| 2 - | S 2 2 | 2 ) | - | S 1 2 S 2 1 f > | S 1 2 S 2 , i 2 + (1 - \Snf)(\Snf - |A| 2 ) or (|AI 2 - I S , . / ) * + I S 1 2 S 2 1 | 2 - 2 | S u S 2 1 ( ( | A r ' - [ S 2 / ) >|S]2S2J|2-(l-|Sn|2)(|A|2-|S22|2) When we cancel the term IS 1 2 S 2 il 2 on both sides and a common factor )AJ - IS-^/l in the remaining terms, the final result is the same as obtained earlier for Case 1. The requirement that RLC be greater than one places a further restriction on the condition for absolute stability. From (10.2.16) we readily see that, when IAI > |S 2a l in order for RLL > 1. we must have |A|2 - | S 2 2 | 2 < | S 1 2 S 2 1 | From the expression for K we gel 1 - |Sn|2 |A| 2 - | S 2 / |S]2S2|| lo|2S2|l = 2K Let (|A| 2 - I S 2 2 | 2 ) / | S J Z S 2 l | = 1 - 5 , where .5 is a positive quantity since this term is less than one. Thus we get -\Sn)2 1 IS12S21I = 2K- l + 5 > 1 since K > 1. Hence we require that IS, 2 S 2 J I < 1 — IS,jl in order that we can have Rlc > 1. The condition K > 1 is a necessary condition for absolute stability but may not always be sufficient since the condition that R,c > 1 when |A| > |S 22 J may be a more stringent condition as we will shortly see. T h e source stability circle is t h e circle of source reflection-coefficient \\ values t h a t m a k e \l'oull = I. By direct analogy with t h e derivation of (10.21), we find t h a t t h e c e n t e r for t h e s o u r c e stability circle, in t h e r„ p l a n e , is given by r s r =-^, £ (10.24a) —r (10.246) a n d its r a d i u s is given by RsC=-, sc |(|A|*-lS M f)| T h e s e e q u a t i o n s a r e t h e s a m e a s (10.21) w i t h S n a n d S 2 2 i n t e r c h a n g e d . 740 FOUNDATIONS FOR MICROWAVE ENGINEERING Since K is symmetrical in the variables S u and S 2 2 , it c inferred that the same condition K > 1 for absolute stability • " ^ J be from the requirement |r o u t | < 1, along with the two conditions lo* i° t a i n 6 d \Sr2S3}\< 1 - \ST/. ^ 2 2 ^ land The necessary and sufficient conditions for absolute stabilit y are thu s n 1 - \SU\2 - \S.,f + |A| 2 > 1 2)S 1 2 S 2 1 | (10.25 Q , |S„I< 1 (10.256) IS 22 | < 1 |S12S2,|< 1 - IS,,| (10.25c) (10.25d) 2 IS 1 2 S 2 1 | < 1 - | S 2 2 | By adding (10.25A?) (10.25e) and I10.25e) and using (10.25a), we get 2\S12S2i\ < 1 - | S „ | 2 - IS 2 2 | 2 + |A| 2 + (1 - |A| 2 ) <2|S12S21|K + (1-IAI2) Thus |A| 2 - 1 * > 1 + 2 1 S ^ < U J V> When |A| < 1 then clearly this condition holds whenever K > 1. However, when |A| > 1 the condition (10.25 f) is more stringent than (10.25a). For most devices |A| < 1, so that (10.25a) to (10.25c) are sufficient to guarantee absolute stability. Conditionally Stable Devices For GaAs MESFETs in the common source connection and bipolar transistors in the common emitter connection, one generally finds that l^u 1 *" | S 2 2 | are both less than unity. For bipolar transistors in the common base connection, ( S , , | and | S 2 2 | are usually greater than one. L l k e ™ s e ; a n d MESFETs in the common gate connection, one often finds that \SU\ I S ^ will be greater than one. The same is true for the common connection. h'l'tv circle For an unstable device there are four possible load s t a ^ ' ' y T h e s e configurations and a similar number for the source stability circ es.^ ^ g i n four cases are described below, along with the necessary conditions to occur. Case 1. The load stability circle may lie entirely outside the Smit ^ ^^ | S „ | > 1. In this case all values of fL on the Smith chart are uns^ a b g o , u t e ly since the origin is an unstable point. The device in this case SOLID-STATE AMPLIFIERS 741 unstable and would be of no interest or use for an amplifier. The necessary conditions for this case to occur are K < -1 | S 2 2 | < |A| Wnf>l C a s e 2. The load stability circle may enclose the Smith chart, and when |S,,I > 1 all values of l'L on the Smith chart will be unstable values. Again, the device will be an absolutely unstable one. The necessary conditions for this case to occur are K < -1 IS22I > IA| I S , , | - > 1 - (2K + 1)IS I 2 S 2 1 I C a s e 3. The load stability circle may lie entirely inside the Smith chart. In this case the device is conditionally stable since a region of the Smith chart will represent stable values of VL. In order for the stability circle to lie inside the Smith chart, we require that | r / f . | + RLC < 1 and RLC < 1. When IS 2 .,| < IA| these two conditions lead to the following necessary conditions for this case to occur: K> 1 \SW\ < |A| |S„r-> 1-(2K-1)|S12S21| The last condition is equivalent to RLC < l.When | S 2 , | > |A| the corresponding necessary conditions for the load stability circle to lie entirely inside the Smith chart are K < -1 |S22I > 141 |Sn|2< 1 -(2K+ 1)|S 1 2 S 2 1 I These conditions are obtained by requiring that \YLC\Z < (1 - RLVSZ and using (10.21) and (10.22). When K > 1, | S 2 2 | < |A|, and IS,,I 2 < 1 - (2/f - 1)IS 12 S 21 I, the device is absolutely stable. Devices for which K > I are almost always stable, and since devices with K < - 1 are not likely to occur, Case 3 is not likely to be encountered in practice. Case 4. The usual situation that occurs in practice when a device is only conditionally stable is the one for which the load stability circle intersects th€; boundary of the Smith chart. This case is the one on which we will focus our attention. The load stability circle |r j n | = 1 then maps into a circle of |I' L | values that intersects the Smith chart boundary I f J = 1 at two points. The load stability circle may or may not enclose the origin. In order to establish 742 FOUNDATIONS FOR MICROWAVE ENGINEERING whether values of \'L inside or outside the |I"ln| = 1 circle are stahl following rules apply: (a) ir.nl =1 circle encloses the origin is,,l<i all interior values are stable vaJues is„|> l all exterior values are stable values r. = l (6) Ues . the circle does not enclose the origin is„i< i all exterior values are stable values is,,i> i all interior values are stable values These rules are established by simply noting that the origin 1", = 0 wh h gives r,n = Sn is a stable point when | S , , | < 1 and is an unstable point when IS,,I > 1. A similar set of rules applies to the source stability circle which is a plot of all values of the source reflection coefficient J's that gives ir | = 1 For example, when | S 2 2 | < 1 the origin I"s = 0 giving r<ml = S 2 2 is a stable point. Thus, if the circle |l"n,J = 1 encloses the origin, all values of r„ inside the circle are stable values. They are unstable values when |S 22 I > 1. By plotting the load stability circle whose center and radius is given by (10.21), it is easy to determine what values of YL, and hence load impedances ZL, can be used to ensure that |f'jn| < 1. A similar plot of the source stability circle ir, ml | = 1 shows what values of source reflection coefficient Ys can be used and will ensure a stable circuit. It is useful to have an analytical test of when the load or source stability circle will enclose the origin. Such a test is easily established by examining the ratio WLC^/R'IC- F r o m (10.21) and using the expansion (10.22). we obtain ir, : . ( • sic =I+ (1-|SUI2)(IS2/-IAI2) \Sl2S 2V In order for the origin to be enclosed, we must have # / x > ' ^ " ' ' ^ requires that the last term on the right-hand side of the above equation negative, since the ratio must be less than one. Hence we can sta following rule: The load stability circle encloses the origin when |S U I < * |A| > | S 2 2 | or when \SU\ > 1 and |A| < |S 2 2 |. When these conditions do not hold, the origin is not enclosed. A applies to the source stability circle. The origin is enclosed c |S 22 I < 1 and |A| > \Sn\ or when |S 22 I > 1 and |A| < IS n l. After the derivation of the equations for the load and source coefficients needed for conjugate impedance matching at bo (10.17), we promised to show that for an absolutely stable device l sim ^^ reflection ^ .^ v ^^ ^ i SOLID-STATE AMPLIFIERS 743 a r e positive. T h e expression for A 2 is A2 = ( l - | S n | 2 ) - ( | A | 2 - | S 2 2 | 2 ) Since I S , , ! " < 1 for an absolutely s t a b l e device, t h e first t e r m is positive. F o r | S 2 2 | > |Al t h e second t e r m is also positive a n d A 2 is positive. W h e n |S 2 2 1 < \\\ t h e n we m u s t h a v e |A| 2 - I S 2 2 | 2 < | S 1 2 S 2 1 I in o r d e r to get RLC > 1. T h i s condition c o m e s directly from (10.216) which gives RLC. In o r d e r for A 2 t o b e positive, w e r e q u i r e 1 - | 5 „ l 2 > IA| 2 - | S 2 2 | 2 B u t o u r stability r e q u i r e m e n t s specify t h a t 1 - \SU\2 > I S ! 2 S 2 1 | > |A| 2 | S 2 2 | a n d h e n c e A.2 is positive. A similar proof s h o w s t h a t A, is positive. E x a m p l e 10.1 S t a b i l i t y c i r c l e s . A MESFET has the following scatteringmatrix parameters at 5 GHz: S , , = 0.75^ - 120= S , , = 0.08^50° S 2 I = 3.9^.90° S 2 2 = 0.4Z - 25° and at 10 GHz: S n = 0.72/170° S 2 , = 2.3/i45 0 S 1 2 = 0.1Z40° S 2 2 = QA/L - 55" We wish to find the stability parameter K, the maximum stable gain, and the load and source stability circles. At 5 GHz we find from (10.23a) that K = 0.64234: so the transistor is not absolutely stable at 5 GHz. By using (10.21) we obtain the load stability circle parameters which are YLC = 3.7678 +78.3712 KLC = 8.5057 From (10.24) we obtain the source stability circle parameters which are I' s c = - 0 . 9 9 7 1 +jl. 19296 Rsc = 0.7104 These circles are shown plotted in Fig. 10.13. Neither circle encloses the origin, and since I S , , | < 1 and |S 22 l < 1, all values of I", and r, outside their respective stability circles are stable values and may be used in an amplifier design. At 10 GHz we find that for the given parameters K = 1.0891; so at this frequency the transistor is absolutely stable. The stability circle parameters are l' /x . = - 2 . 1 6 7 7 -ylO.5364 Ru: = 11.8549 r s c = - 1.47156 - i O . 2 8 3 3 6 R^ = 0.47035 For the load stability circle ii',.,-1 = 10.757, and since RLC > \rLC\ + 1, the circle encloses the Smith chart. Since IS,,I < 1 all values of F L inside this circle are stable values. The circle is a very large one, so that only a small portion of it is shown in Fig. 10.13 (dashed line). The source stability circle is also shown in Fig. 10.13. This circle lies entirely outside the Smith chart, and since |S 22 I < 1 all values of H, outside this circle are stable values. 744 FOUNDATIONS FOH MICROWAVE ENGINEERING Imr Load stability circle Load stability circle, 10 GHz Source stability circle Rer Source stability circle. 10 GHz FIGURE 10.13 Load and source stability circles for a MESFET at 5 GHz (solid circles) and 10 GHz circles). The device is conditionally stable at 5 GHz and absolutely stable at 10 GHz. 10.7 CONSTANT POWER-GAIN CIRCLES It is convenient to introduce the normalized power gain g given by G„ Sp = (10.26) \S3l\2 From (10.15) we obtain i - Vt gp 2 2 1 - | S n | + VrJt(\S.Z2\ - \A\2) - TL(8n - AS*,) - I?(S& - * * * (10.27a) upon writing \TL\2 = YhXl and using the relationship 2 Re Z - ^ can rewrite the expression for normalized power gain in the form r,A* - g „ ( S 2 2 - A S * ) r t + g.,(S2*2 - A*sn)rL* (|S 22 | 2 - \A\2)gp + l =o - (i -^uQirjJ. 110.27*) SOLID-STATK AMPLIFIERS 745 by multiplying g p by the denominator on the right-hand side and bringing the term 1 - \YL\2 over to the left-hand side and then dividing by the coefficient of V^V*. When we compare the above expression with (10.1), we see that it describes a circle in the T L plane. The center of the circle is at \'L = VLg, where VLg is the coefficient multiplying T£. The radius RLg of the circle is obtained by noting that the constant term is equal to | I j _ | - R'f ,. Thus we readily find that the constant normalized power-gain circles have a center at r„- 'a--,r,f,"*r. IS22I'2 - |A| 2 )g„ + 1 and a radius given by 1 Ri.= - - 2Kgp\SV2S.2l\ + g2\S12S2i\2)' r^—•—~^r^ i (io.28ft) 2 |(WJ" - lAI )^ + l| where we have used the expansion (10.22) to replace \S22 - A*S,,| 2 and expressed 1 - | S n l ~ as \S2.2\2 - |A| 2 + 2K\Sl2S2i\ in order to simplify the expression for RLg. When g p approaches infinity the equations above become those for the load stability circle. Thus, on the load stability circle, the power gain is infinite. When gp = 0 we obtain I", g = 0, RLg = 1, which is the boundary of the Smith chart. On the boundary of the Smith chart, \Y,\ = 1. which corresponds to a pure reactive load impedance and consequently no power is delivered to the load and g p = 0, as the above equations verify. It is common practice to plot the normalized constant power-gain circles that correspond to gains 1 dB, 2 dB, 3 dB, etc., less than the maximum normalized power gain (K - iJK2 - 1 ) / | S l 2 S 2 1 | [see (10.19)] for an absolutely stable device. For a potentially unstable device, constant power-gain circles for normalized gains 1 dB, 2 dB, etc., less than the normalized "Figure of Merit" gain (|S 2 1 I/|S 1 2 |)/|S 2 1 | 2 = 1 / | S I 2 S 2 I | are usually plotted. The equation for the center of a constant power-gain circle lies on the same ray from the origin as the center of the load stability circle does, since both r i C and \"Lg have the same complex numerator except for the factor By plotting the constant power-gain circles and the load stability circle. we can easily determine those values of load reflection coefficients T L that will give the largest power gain and yet result in a stable amplifier design. Those values of Tt in a stable region and lying on a given gp = constant circle will give a power gain G p = \S2i\2gp. Later on we will find that there will be additional constraints imposed on TL because of low-noise and low input and output VSWR requirements. 746 FOUNDATIONS FOR MICROWAVE ENGINEERING Properties of t h e Constant G a i n Circles By examining the geometric properties of the constant gain circles obtain considerable insight into the operational characteristics ' ^ *'*' crowave amplifier. The characteristics of the constant power-gain ci ( 3 "^ quite different for absolutely stable devices and potentially unstable d ^ " ceS; so we examine each case separately. Stable D e v i c e s For a stable device one set of constant gain circles lies entirely inside th Smith chart. The g p = 0 circle coincides with the boundary of the Smirl chart. As g p increases from zero, the radius RLg decreases and becomes zero when K ± VK"' - 1 op |S 1 2 S 2 1 I For circles inside the Smith chart, RLg = 0 when \/K2 - 1 K- |SI2S21| When K = 1, RLg = 0 when gp - l / | S l 2 S 2 1 l , which gives the maximum gain. A second set of circles exists outside the Smith chart and gives values of gp for values of \VL\ > 1, which implies the use of an active load. For passive loads these circles are not relevant. In the region K+fK2-l K- JK'1 - 1 S„ = to \Sl2!i> S 0 2|l \S12S21\ there are no real solution for RLs, when g p > 0 as illustrated in Fig. 1014. Typical constant power-gain eireies for a stable device are shown in Fig. 10.15 for the case when the load stability circle lies outside the Smith chart. Inside the Smith chart boundary the maximum normalized power gam * ^ No real solution for RLQ in this region lor g p > 0 9p U FIGUHE J0.14 h e is " =°' " Illustration of region where t er ^ tion for the radius of the consttu gain circles. SOLID-STATE AMI'I.IKIKKS 747 Load stability circle FIGURE 10.15 Circles of constant normalized power gain for an absolutely stable transistor when the load stability circle lies outside the Smith chart. The circles of constant negative power gain are shown by the dashed circles. (K - y/K2 - 1 ) / | S 1 2 S 2 1 | is obtained at the single point where V, = VLM\ as given by (10.176). This is the gain for the conjugate-impedance-matching condition. Away from this point the constant gain circles have increasing radii and decreasing gain values. The limiting circle coincides with the Smith chart boundary on which g p = 0. A second set of similar constant power-gain circles exists inside the load stability circle. In this region the minimum normalized gain is (if + \JK2 - 1 )/\SViS2\\ and occurs for YL = VUMZ, where now VLM2 is the other solution given by (10.176) for conjugate impedance matching, i.e., using the positive sign in front of the square-root term. As we move away from this point, the constant gain circles have increasing radii and increasing gain values. The limiting circle is the load stability circle on which the gain is infinite. Inside the load stability circle, which is the unstable region 748 FOUNDATIONS FOB MICROWAVE ENGINEERING of F, values, each FL results in an input reflection coefficient I' magnitude greater than unity. Since \V,J > 1 also in this region it ^^ a that both Z„, and Z L have negative resistive parts. This type T i ^ impedance could arise by connecting an unstable microwave amplifi t '^ he output of the amplifier under discussion. When |Tin > 1 and IF I > e , power is reflected from the amplifier input and load termination th ^ incident on either one. The input power and load power are thu negative but the ratio is positive, so that the power gain is positive A boundary of the load stability circle is approached, which is the circle f r values t h a t give |F i n | = 1, the gain becomes infinite since, when IF I = \ there is zero input power to the amplifier. In the region between the Smith chart, and the load stability circle |F L | > 1, but since this is a stable region each F, produces a I' with |F j n | < 1. We now have a situation where there is a finite positive input power to the amplifier but a negative load power since, with |FJ > 1, the load termination reflects more power than is incident upon it. Thus the power gain is negative. Indeed, if we assume a negative power gain, we will get another system of constant negative power-gain circles that fills the region between the Smith chart boundary and the load stability circle. As the load stability circle is approached from this region, the gain approaches minus infinity. These negative power-gain circles are shown as dashi circles in Fig. 10.15. The circle with infinite radius occurs when SP = IAI2 - | S 2 . / 10.29) i.e., when the denominator in the expression (10.286) for RLg vanishes. Or this straight line the gain is constant and is given by (10.29). In order to obtain finite values for F, from (10.276) when gp is given by (10.29), the numerator of the second term in (10.276) must vanish. If we let V L = x +jy, we find that setting the numerator equal to zero gives the following equation for the straight line which is the constant power-gain circle having an infinite radius Re(S22-ASr,) y = * Im(S -AS* 22 | A | - - | S 2 . / + |Snl - 1 10.30) In Fig. 10.16o we show the gain profile along the line c o n n e ^ . i n t [ h e origin of the Smith chart to the center of the load stability circle. I> regions in which the power gain is negative. por cn When |A| > | S 2 2 | the load stability circle encloses the Smith ^ w n i n this case the constant power-gain circles have the configuration s ^ Fig. 10.17. For this case the circles again begin at the point It ^ inside the Smith chart and increase in radii and decrease in gain v • ^ circle is IFJ = 1 on which gp = 0. Beyond the Smith chart boun J .^ inside the load stability circle, IF, I > 1, |F i n | < 1, and the power g SOLIO-STATR AMPLIFIERS 749 F I G U R E 10.16 Gain profile along the line joining the Smith chart center and the load stability circle center. (o) Gain profile when the load stability circle lies outside the Smith chart; (6) gain profile when the load stability circle encloses the Smith chart. 750 FOUNDATIONS FOR MICROWAVE ENGINEERING K-VK2-I Load stability circle S„I2-IAI2 Smith chart boundary F I G U R E 10.17 Circles of constant normalized power gain for an absolutely stable transistor when the load stability circle encloses the Smith chart. The dashed circles are circles of constant negative power gain. creases in a negative direction and approaches minus infinity on the load stability circle. Beyond this circle the radii keep increasing until the straight line given by (10.30) is reached. On this line gp is given by (10.29) but is now positive since |A| > |S 22 I. The radii then begin to decrease until th< limit point at r L = YLM2, the second solution given by (10.176), is reachw and at which the normalized gain is (K + J~K2 - 1 )/)Sl2S21l A gain profile along the ray from the origin to the center of the load stability circle for ' case is shown in Fig. 10.16*. When WJ approaches infinity the g» approaches the value given by (10.29). Unstable D e v i c e s For an unstable device the load stability circle will usually i n t e r ^ f Smith chart and may enclose the origin. For unstable devices with I no value of g p that is real will make RLg = 0. Again the g p = ° c i r c e ^e SOLID-STATE AMPLIFIERS 751 boundary of the Smith chart. For most unstable devices the load stability circle will cut the Smith chart boundary at two points. We can show that these two points are invariant points t h a t all constant gain circles pass through. The constant gain circle on which the gain is given by (10.29) is the circle with infinite radius. This circle is the straight line given bv (10.30). We can prove that all constant gain circles intersect the Smith chart boundary at the same two points that the load stability circle does as follows: The load stability circle is given by fit - hd2 = R'tc = irj2 -r irLCl2 - r,j-,*. - iyru. We now let \'L = x +jy. The stability circle intersects the Smith chart when |T L | = 1. By setting \VL\ =1 we then find that the above equation is that of a straight line given by \ru:\2 ~ Ric + i Rer LC y = — % -(- I m l'LC . 2 Im r i C This line intersects the circle |T; | = 1 at two points. In a similar derivation we find that the constant gain circles will intersect the circle ll'J = 1 at the two points where the line Reru ir,,/-fl|,+ i ImT,./ 21mr,. f i intersects the WL\ = 1 circle. By substituting for r,_r, RLC from (10.21) into the first equation and for [\ , RLg from (10.28) into the second equation, we find that both lines have the same slope and intercept. We omit the detailed algebra involved but it is straightforward. As a consequence of the above property, the constant normalized power-gain circles for an unstable device have the configuration shown in Fig. 10.18. When Y L approaches either one of the two invariant points, both the numerator and denominator in the expression (10.27a) for the power gain approach zero. The ratio is thus indeterminate and must be evaluated as a limit with the result that the limit depends on the direction along which the invariant points are approached. Thus we can have many constant gain circles with different gains at the two invariant points, as is clearly evident from an examination of Fig. 10.18. We will now examine the constant gain circles for the four special cases that can occur in practice. The usual situation is the one where \Sn\ < 1. When this is true the origin, that is, F, = 0, is a stable point. The load stability circle may or may not enclose the origin. When the origin is not enclosed, the constant gain circles within the Smith chart but outside the load stability circle are in the stable region and \'L may be chosen to lie on one of these gain circles as shown in Fig. 10.19a. In the region common to both the Smith chart and the load stability circle, the power gain is 752 FOUNDATIONS FOR MICROWAVE ENGINEERING 9o = IAI2 - I S * / load stability cin 9A<93 FIGURE 10.18 Constant normalized power-gain circles for an unstable device. negative. In the region outside both circles, the power gain is also negative; in the region inside the stability circle but outside the Smith chart, the gain is again positive. The gain always changes sign when the circle IfJ = >- o which g p = 0 is crossed and also when the load stability circle on which the gain is infinite is crossed. On one side of the load stability circle boundary. gp = oo, and on the other side of the load stability circle boundary, g„ For this case the centers of the constant gain circles move from zero ^ YLC as g p increases from zero to infinity, as shown in Fig. 10.19a. for o locations of the centers on the ray joining the origin to the stability c center, the constant gain circles have negative gain values. When the stability circle encloses the origin, then the stable region ^ the region inside the load stability circle as shown in Fig. 10.19b .^ stable amplifier design Y L must be chosen to lie on a constant gain circ this region. For this case the centers of the positive constant gain cir move from zero to infinity in the direction away from YLC and return SOLID-STATE AMPLIFIERS 753 (a) F I G U R E 10.19 Constant normalized power-gain circles when IS U I < 1 so that the origin is a stable point, (a) Constant gain circles and stable region when the origin is not enclosed; (6) constant gain circles and stable region when the origin is enclosed. infinity toward YLC as the gain increases from zero to infinity, as shown in Fig. 10.196. The other two cases occur when | S U | > 1 for which the origin is an unstable point. The constant gain circles for these two cases are shown in Fig. 10.20. The stable and unstable regions are interchanged from those shown in Fig. 10.19. The values of YL must be chosen to lie on a constant gain circle in that part of the Smith chart in which the origin is not included. Hence, when the stability circle encloses the origin, Y, must be outside the stability circle for a stable design. When the stability circle does not enclose the origin, then Y L should be chosen as a point inside the stability circle. For an absolutely unstable device, K < — 1, and there are two values of g p that are negative and correspond to circles of constant gain that have 754 FOUNDATIONS FOR MICROWAVE ENGINEERING 1 Lg 'Bturr(S r r o m toward r, from 0 to ~.away from VLC Load siability circle Smith chart boundary F I G U R E 10.19 Continued. zero radii. These gain values are given by gp = ~(\K\ ± vK" - 1 )/IS l2 S 2 jl. The constant gain circles are similar to those shown in Figs. 10.15 ana 10.17 but g p is negative inside the Smith chart. For the case when the load stability circle lies inside the Smith chart, K may be greater than 1 or less than - 1. In both cases there are two points corresponding to constant gai circles with zero radii. The constant gain circles are again similar to shown in Figs. 10.15 and 10.17. On the load stability circle g p equals plus or minus infinity and on the Smith chart boundary g p = 0. If the interio the load stability circle is a stable region, g p will be positive in this| r ^f 1 ° ' Between the load stability circle and the boundary of the Smith cha^'t^ will be negative, and outside the Smith chart it will be positive. interior of the load stability circle is an unstable region, the sign ol gp be opposite to that described above. SOLID-STATE AMPLIFIERS 9p=' 755 \'Lg moves Irom co toward lLC Smith charl boundary Load stability circle fLg moves from 0 to ~ away from VLC (a) FIGURE 10.20 Constant normalized power-gain circles when |S' U | > 1 so that the origin is an unstable point. (a) Constant gain circles and stable region when the origin is not enclosed; (61 constant gain circles and stable region when the origin is enclosed. E x a m p l e 10.2 P r e l i m i n a r y amplifier design. A given microwave transistor has the following scattering-malrix parameters: S n = 0.9Z.60". S l 2 = 0.06/160°, S 2 I = 3^120°, S.i2 = 0.82^ - 30°. We will use this device in the design of a single-stage amplifier having 50-12 input and output transmission lines. For the given transistor we find that Stability parameter K = 1 - 0.9- - 0.82* + |A|8 2 x 3 x 0.06 = 0.902026 where \M = \SuS2-2 ~ S 12^2,1 = 0.898404. The transistor is potentially unstable since K < 1. The "Figure of Merit" gain is I S 2 1 / S 1 2 | = 50 or 17 dli. The normalized value is 5 0 / I S 2 , | 2 = 5.555. The parameters for the stability circles are obtained using (10.521) and (10.24) and are r t c = - 0 . 4 0 0 2 6 +J0.46312 r s c = 6.8262 - 7 6 3 . 2 4 3 Klc = 1.336 Rgc = 62.7068 The centers and radii of the constant normalized power-gain circles for g p = 5, 756 FOUNDATIONS FOR MICROWAVE ENGINEERING Stable region Load stability circle Smith chart boundary m FIGURE 10.20 Continued. 4, and 3 are 8,, 5 rLg = 0.8262 - J 0 . 9 5 5 9 6 RLg= gP 4 TLg = 0.4678 -.,-0.5413 RLg = 1.01606 VLg = 0.2715 - j O . 3 1 4 1 7 RL gp = 3 1.3227 = ^0.9456 ^ ^ ^ ^ ^ ^ The stability circles and constant gain circles are shown in Fig- ' The load stability circle encloses the origin. The source stability circle m t e r s e * ' ^ only a small portion of the Smith chart, so that almost all values of sou reflection coefficients T s will be stable values. Since ( S u l < 1 t n e °. . represents a stable point, so t h a t all values of f L inside the load stability are stable values. The source stability circle appears as a straight line i 10.21 since its radius is very large, that is, Rsc = 62.7. . a From the figure we see that I"t = 0 is a stable point and would g" ^ normalized gain somewhat greater than 5. From (10.27a) we see "f 1 ^ normalized gain, when T,. = 0, is 1/(1 - | S n l z ) which equals 5.2b. ^ advantage gained by using Y L = 0 is that no output matching net*" SOLID-STATE AMPLIFIERS 757 FIGURE 10.21 Load and source stability circles and constant normalized power-gain circles for the amplifier designed in Example 10.2. required. On the straight line joining the two invariant points, g p = 1/(|AI IS 22 |") = 12.75 which is considerably larger than the "Figure of Merit" gain. We could choose I~, to he on this constant gain circle and still have ir in l < 1 with a sufficient stability margin. However, as we will see, the choice of too large a gain will result in poor stability at the output of the amplifier or a very high input and output voltage standing-wave ratio. 758 Ft IUNDATIONS FOR MICROWAVE ENGINEERING If we choose I,. = 0, then r,„ = S „ . For conjugate impedance mo* ._at the input, r. = T* = Sf,. If we choose this for the source Zf coefficient, then the amplifier output reflection coefficient becomes ^ion S22 - Ai; r„ u t = . _ s r = S 2 2 - AS?, — 5 - = 0.2838 +70.3384 The magnitude of the output reflection coefficient is 0.4416 Since r n*t L the output VSWR is (1.4416)/(1 - 0.4416) = 2.58. The point r, = Sf, is shown in Fig. 10.21 and is seen to lie quite close the source stability boundary. A relatively small increase in T s could lead oscillations in the output circuit of the amplifier since i r ^ l > 1 when V li i inside the source stability circle. A more conservative choice for r would ho 0.85Z. - 60°. This point is also shown in Fig. 10.21. For this choice the input is not conjugate matched. The mismatch at the input is, from ( 1 0 . 1 2 B ) (1 - i r i n | 2 ) ( l - | r , | 2 ) M = n-r,r,ni' 0.19X0.2775 = 0.05523 =0-9547 The input VSWR is now VSWR,1 = 1 + v'l - M 1 - /l - M = 1.54 With the above choice for l\, we find that r„ u , = 0.3846 +J0.1615 and the output VSWR is 2.43. For this choice we improved the stability margin but at the expense of having to accept an input VSWR of 1.54. Before we leave this example, we will introduce a useful method that helps one to choose a value for l"s that gives a good stability margin, a low input VSWR, and a given normalized power gain. The input reflection coefficient is given by in Sn-Ary, i-sz2rL The complex conjugate of this equation is in A*l?-Sf 1 sf 2 r L *-i This is a bilinear transformation. Thus all values of V L t h a t lie on a constant gain circle will map into a circle of T* values. If we choose r„ to lie on t h e o ^ circle, then the input will be matched. We will discuss the FJ circles in m detail in Sec. 10.9. For now we note that the centers of these circles lie 0 ^ e s ray from the origin to the center of the source stability circle. The r, n c ^ d a r y also have the interesting property that all circles cut the Smith chart b 0 " ^ at the same two points that the source stability circle does. This p r ° ^stable similar to the corresponding property of constant gain circles for u ^ , devices. In Fig. 10.21 we show the F* circle for all values of T L on the ^ . ^ circle. Some of the T* values will correspond to Y L values outside U^ ^^ chart. The best stability margin is obtained by choosing F, as the pomi ^ as T s in Fig. 10.21, since this is farthest away from the source stabui . SOLID-STATE AMPLIFIERS 759 For this example the T* circle has a center at 0.11833 -j 1.09625 and a radius of 0.46922. The value of \\ at the illustrated point is 0.06797 - j'0.6297. With this choice for Ts = I,*, the input is matched and we get a normalized power gain of 5 or a real power gain of 5 x | S 2 ] | = 45 or 16.53 dB. However, we must check if the corresponding value of I"; is satisfactory. With the above choice of Ts, we find that r„ ul = 0.7819 - / 0 . 2 0 7 7 - For the chosen value of r*„ we can solve for V, using the equation r ^ = S - I A~" o r = ° - 7 0 9 7 +70.36165 The value of \'L for the chosen F,,, is well within the load stability circle. In Fig. 10.21 we show the points T L and r* ut . The output VSWR is easily calculated and is 2.52. Thus this design gives a matched input, a good power gain of 45. good stability, and an output VSWR equal to 2.52. However, we do need an output matching network that will transform the line impedance of 50 17 into a load impedance given by 1 ZL = 1 + '"/. —Z, = 85 * j168.2 " <i. A better output VSWR could be obtained by relaxing the input VSWR. By trial and error we find that, by using T, = 0 . 3 5 - . / 0 . 6 0 6 2 and l\ = 0.47237 — /0.31859, we can achieve a normalized power gain of 5. an input VSWR of 1.366, and an output VSWR of 1.8515. These final values of T, and Y L are shown in Fig. 10.21 as the points I" and Y\. This last design results in good gain, low input and output VSWR, and good stability margins. The last item that we will examine is what happens if we try to design the amplifier for a gain significantly larger than the "Figure of Merit" gain. A very large gain means that the amplifier is very close to oscillation. Hence it can be anticipated that either Ts, YL, or both will be close to their respective stability circle boundaries. The [",* circle for all values of I", on the gp = 12.75 gain circle is shown as the dashed circle in Fig. 10.21. This r,* circle lies very close to the source stability circle. For a matched input, I", would have to be chosen to lie on this I-*, circle and clearly this would not give an adequate stability margin. If we choose Ts sufficiently far away from this circle to obtain a good stability margin, then we will end up with a very poor impedance match at the amplifier input. The consequence of a poor impedance match al the input is that only a small fraction of the incident power on the input transmission line will be delivered to the amplifier input. Thus, even though the power gain will be high, the transducer gain will be low. In practice, microwave amplifiers are generally designed for power gains no greater than the "Figure of Merit" gain because of the poor stability and high input and output VSWR that results for larger gain designs. As is apparent there is no direct way to control all design objectives individually, since improvements in some areas generally are accompanied by a deterioration in some other characteristic. In Sec. 10.9 we will discuss low-noise amplifier design and will find that this will result in an additional constraint on Vs, since the best noise figure is obtained only for a particular value of I,. 760 FOUNDATIONS FOR MICROWAVE ENGlNF.F.KlNr; e n «) "0Noiseless • resistor'. R 'n(t) = (a) F I G U R E 10.22 • ( a ) Thevenin equivalent circuit which uses a noise voltage generator; (6) equivalent circuit for a noisy resistor in which a noise current source is used; (c) typical noise voltage waveform. 10.8 BASIC NOISE THEORY As a result of thermal agitation, the electrons in a resistor have an inherent random motion which results in a random voltage appearing across the resistor terminals. This random voltage is referred to as noise. There is no analytical way to describe the exact voltage waveform; so we must be content with a description of certain average characteristics of the noise. We can model a noisy resistor as a noise-free resistor in series with a noise voltage generator en{t) or in shunt with a noise current source i„(t) as shown in Pigs. 10.22a and b. In Fig. 10.22c we show a typical noise voltage waveform that might be produced. Noise is a random process and its effects in a linear system are analyzed using statistical methods. For this purpose we construct an ensemble of macroscopically identical systems, e.g., we consider an infinite number of resistors with each one producing its own noise voltage. Averages < various products of the noise voltages at different times, such I e„(f,), e„(t,)e„(r2), etc., are obtained by averaging over the ensemble of noise waveforms. . Thermal noise is generally regarded as a stationary ergodic nois^ process which is a random process for which ensemble averages c replaced by time averages. Thus, in our brief summary of basic n o l s e -1 we will use time averages. Since our objectives are only to obtain t .... r results needed to derive the equations required for low-noise amp ^ design, our discussion will be brief. Much more complete treatme noise are widely available in texts on statistical communication theory < SOLID-STATE AMPLIFIEKS 761 the reader should consult some of these for the details that are missing in our treatment. The time-average value of the noise voltage, given by < e„( t)) = Hm — / % „ ( 0 dt = 0 (10.31) is zero. The correlation function for the noise voltage is the average value of the product of the noise voltage at time t and that at a later time / + r, thus C(r) = Hmgyj ea{t)en(t + T)dt = (en(t)en(t + 7)) (10.32) where C(r) is the correlation function. If r = 0 we obtain the average power {e'l) associated with the noise (the noise voltage is thought of as being applied to a 1-0 resistor, so that the dimensions are those of power). The average power in noise is distributed over a broad band of frequencies because noise voltage waveforms contain a broad spectrum of frequencies. The power spectral density S„(w) of noise is given by the Fourier transform of the correlation function; thus S„(o>) = C C(r)e J r - dT (10.33a) The inverse transform relationship is C(r)= ,-* du> S„(W)(>.'-— (10.33ft) The power spectral density represents the noise power in the spectral domain; so S„(w) A f is the noise power in a frequency increment \f. At room temperatures the power spectral density of thermal noise is constant up to frequencies of the order of 1,000 GHz and decreases at higher frequencies. Thus, at microwave frequencies and below, we can assume that the spectral density is a constant or is flat. This is equivalent to having a correlation function that is a constant multiplying the delta function §(r), that is, C(r) = CQt(r) since the Fourier transform of 8(r) is a constant equal to unity. Noise with a constant power spectral density is called white noise and is uncorrected noise. The power spectral density is an even function of ft>; so we can choose the spectral density such that only positive values of w need to be considered. For thermal noise in a resistor, the power spectral density for the noise voltage is given by Nyquist's formula Sc(to) = 4kTR <o>0 (10.34) 762 FOUNDATIONS FOB MICROWAVE ENGINEERING Z L zs i(V) z, F I G U R E 10.23 A two-port network connected to (o) a voltage source; <b) a current source. where k = 1.38 X 10 '23 J / K is Boltzmann's constant and T is the absolut* temperature of the resistor R. Thus the amount of noise power P i n frequency interval A f is given by P„ = UTRAf (1035) Thermal noise in a resistor is also called Johnson noise, after one of the early investigators of noise. If we use the equivalent current source model shown in Fig. 10.226, then the average power, if the current in{t) flows in a 1-il resistor, is given by (if,(0) and has a power spectral density given by Sf(ft») = - 4kT S R w>0 (10.36) for thermal noise. Filtered Noise Consider a sinusoidal voltage generator with a complex r m s voltage V that is connected in series with a source impedance Z x and a two-port network as shown in Fig. 10.23o. The input current produced by V is V/(Z„ + Zm>. where Z m is the input impedance to the network. The input power produced by V is given by V P.i n . l z. + z, ivi_2 4R. -M 2 fife = 47?, IZS + Zia (10.37) where \V\2/4RS is the available power and M is the impedance-misma factor. The power transfer function is M / 4 i ? s and is a function of *» su Zs and Zin are functions of co. If the voltage generator is replaced by a n voltage source ejt) with a power spectral density Seia>), then the noise power in a frequency band A f centered on w is given by mu P the source power spectral density by the power transfer function SOLID-STATE AMPLIFIERS 763 factor A /'; thus M(w) (10.38) The total input noise power is obtained by integrating over all frequencies; thus ,* M(ui) dai .10.39) Consider next the circuit shown in Fig. 10.236. The input current from the current source is Z!II/(ZS + Z, n ) and the input power is P in .2 = izx zs + zin l/l 2 4& |/|2|Z/ fl.„ = 4fl„* i n 4/?. Z,. (10.40) -M where G s = R e ( l / Z s ) = RH/\ZS\ . If the sinusoidal current source is replaced by a noise current source i„(t) with power spectral density S,(«), then the input noise power in a frequency band A f is given by M(w) P„,,(W)=S,(W)^-A/- (10.41) The power spectral density of the noise power delivered to Z, would, by analogy, be given by the product of the power spectral density of the source and the power transfer function from the source to the output load impedance. Since we can describe the power delivered to Z L by the product of P m with the power gain Gp(co), we see that the noise current source will produce an output power in Z L with a spectral density given by G„(w)Af(w)S,(w)/4G s . A similar expression holds for the output power spectral density produced by the noise voltage source en(t) acting alone. When both sources V and I are acting, the input current will be V + /Z. and the input power will be given by (V+ IZS)(V* + I*Z*) R,» in, 3 ivr i/r vi*z*Rin 4fl<- M + T4G„ ^ - M + 2 R e -IZ. ' "2 (10.42) 764 FOUNDATIONS FOR MICROWAVE ENGINEERING Because of interaction between the two sources, the input simply the sum of that from each source acting independently When we have two noise sources ejl) and in(t) acting sirnulta there is no input power caused by the interaction between e (/) a ^ 0Us v> when e„U) and i„(t) are uncorrelated or statistically independent ' " ' " sources. For this case the input noise power in a frequency band A f - " ° ^ e sum of that given by (10.38) and (10.41). When there is a de * correlation between e„(t) and iri(t). there will be some input noise n due to the source interaction. The cross-correlation between the current source in(.t) and volta source e„(l) is given by Cx( T) - hm 1 (T wJ JMKit + r)dt ( io.43« , The Fourier transform of CJT) gives the cross-power spectra] density Sx(o>), t h a t is, S x (w) = S„(o>) +jSxl(u>) = J C Cx(r)e-JmTdr (10.436) — X If we replace to by -to, we see that Sx(—u>) = Sx(to) since Cx(ri is real. From this result we find that SXr(to) is an even function of to and Sr,(o>) is an odd function of to. For input noise power calculations, we replace |V| by the noise voltage source power spectral density Se(,to), replace |/|~ by the power spectral density S,(io) of i„(t), and replace VI* by the cross-power spectra) density SJto) in (10.42). Thus, for partially correlated noise sources, the total input noise in a frequency band A f is given by Pn = Af(Se( M 1~R~ + S,(to) 4[Sxr{to)Rs M 4a + + \z. + z„ Sxi(to)Xs] s -K. (10.44) The extra factor of 2 in the last term is due to the fact that we have combined the contribution from negative values of to with those nom positive values of to using the fact that SxrRs and SxlXs are even junctions of to. Also, we have defined the spectral densities Se(to) and Sjlf s ^ that only positive values of to are to be integrated over to get the total mpu noise power. H ffers The power spectral density of the noise produced in a network ^ from that of the noise source because the network response d e P e " l f S t h e frequency. The source noise spectrum is filtered by the network^ ^ source noise spectrum is flat (white noise), the noise spectrum P r o < ^ U C ^ w e r some point in the network is not flat. Noise with a nonconstant P spectral density is called colored noise. SOLID-STATE AMPLIFIERS 765 From (10.44) we find that the power spectral density of the noise power delivered to Z, is M M 4G. 4R. M)Gl,(co) " (10.45) after multiplying by the power gain G p of the two-port network. The total output noise power delivered to Z L is P„,,n, = f dc S(,0)- (10.46) Noise in Active Devices In an active device such as a transistor, there are three main contributions to the noise produced by the device: (1) thermal noise in the resistive elements that are present, (2) shot noise due to the discrete nature of the charge carriers that constitute the current flow across p-n junctions, and (3) flicker noise that has a power spectral density proportional to 1 / / ' in the frequency domain. The current that flows across a p-n junction is not a smooth stream of charge. It consists of discrete charges that cross the junction in a more or less random manner like raindrops falling on a (in roof. The noise associated with this current is called shot nu ; se and is directly proportional to the dc bias currents. Shot noise, like thermal noise, has a flat power spectral density. For low noise an active device is operated with low dc bias currents. However, since the transconductance, and hence the gain of the device, decreases with a reduction of the bias current, there is an optimum lower value of bias current that gives the best compromise between low noise and gain and produces the optimum noise figure for the device. In Fig. 10.24 we show the typical behavior of noise figure versus drain current for a MESFET. 2 - 1.5 10 20 30 mA 40 FIGURE 10.24 Variation of the noise figure F of u MESFET as a function of drain current. FOUNDATIONS FOR MICROWAVE ENGINEERING edt) : in 766 '„((} I eJLt) <$_ Noise-free two-port network lb) FIGURE 10.25 Equivalent input noise sources for a noisy linear two-port network. Flicker noise is usually negligible relative to thermal noise and shot noise at frequencies greater than a few kilohertz for bipolar transistors but may be important in MESFETs up to frequencies as high as 100 MHz. Noisy Two-Port Networks In analyzing the noise produced at the output of a linear two-port network due to the internal noise sources, we can replace all of the internal noise sources by a series noise voltage generator e„(t) and a shunt noise current generator in(t) at the input as shown in Fig. 10.25. The total noise power at the output can be found by evaluating the noise output produced by en(t), i„(t\ and the thermal noise in the resistive component R s of the source impedance. Two equivalent noise sources are needed at the input because if the input is short-circuited, that is, Z s = 0, the source in(t) does not produce any output noise, yet the noisy two-port does have a noise output under short-circuit conditions at the input so a noise voltage source e„(t) is required. Similarly, under open-circuit conditions e„(t) does not produce any output noise; so a noise source /„(/) is needed to represent the equivalent input noise source under open-circuit conditions. The two nois sources e„(t) and in(t) axe not completely independent since a part of e„\t and in(t) may arise from the same basic noise-producing mechanism within the two-port network. Thus, in general, there is some cross-correlation between e„{t) and i„(t) with a resultant nonzero cross-power spectr density. . j It is common practice to express the power spectral density asstx with the two noise sources e„(f) and in(t) in a form similar to that ^ v e " . J, (10.34) and (10.361 for thermal noise. When this is done, nicker noise, w ^ is low-frequency noise with a X/f spectrum, is not included, bine microwave amplifiers do not produce an output for low-frequency ^e signals, the neglect of flicker noise can be justified. We will thus specity SOLID-STATE AMPLIFIERS 767 spectral densities as follows: Fore„(t) S,.(w) = 4A77?,. For /„(/) and 2[Sxr(") S,(w) = AkTG, +JSx,(u>)} = 4kT(y, + jy,) where R c is an equivalent noise resistance, G, is an equivalent noise conductance, and yr -i-jy, is a complex equivalent noise impedance. A total of four parameters, Rt„ Gh yr, y,, are needed to describe the noise properties of a noisy two-port network. In terms of the above spectral densities, we can express the total noise input to the noise-free two-port network, in a frequency band A /'. as follows by using (10.44): p D Pnin = hTlfM + kTAf-^M R,y,. + X,y. + 2kTbf—^-——M + kTLf-^-M (10.47) In this equation the first term on the right, liTIfM, is the input thermal noise from the source resistance fi g . The output noise in Z ; in a frequency band A/'is obtained by multiplying by the power gain Gr(ui) of the network. The noise produced at the output termination by the equivalent sources enit) and i„(t) placed at the input of the network is fully equivalent to that produced by the internal noise mechanisms in the real noisy two-port network. 10.9 LOW-NOISE AMPLIFIER DESIGN In a typical microwave communication system, the information to be communicated is modulated onto a microwave carrier and radiated into space by means of an antenna. At the receiving site an antenna is also used to intercept a small portion of the radiated signal energy. The receiving antenna will also pick up a certain amount of noiselike radiation from atmospheric disturbances, radio stars, the sun, and other celestial bodies. The received signal, along with some noise, is very weak and must be amplified to a level where it can be used to produce the desired video, audio, or digital output information that was transmitted. The function of the first amplifier stage is to amplify the signal with the addition of a minimum amount of extra noise. Thus the first amplifier stage should be designed for minimum noise. If the power gain of the first stage is around 10 or more, the signal will be sufficiently large at the output of the first stage, so that additional noise contributed by the following amplifier stages will have a 1 768 FOUNDATIONS FOR MICROWAVE ENGINEERING negligible degrading effect on the overall signal-to-noise power rat vided that the noise contribution of the second stage is moderate I ^ F ° design of the first stage, the minimum noise requirement is more i m D " than maximum power gain or output VSWR, provided a power gain f ^If"1 more can be achieved. Hence we can relax the gain and output V^wr"" requirements in order to achieve the objective of a minimum noise cont "k tion from the first stage. In this section we will discuss noise figure and the design of amplifier for minimum noise. We will show that there is an optimum sot impedance Z, (or source reflection coefficient Vs) that will result in the lowest noise figure. We will also introduce constant noise-figure circles that can be plotted on the r s plane and which will show in a pictorial way the increase in noise figure that occurs when the optimum source reflection coefficient cannot be used. For transistors that are not absolutely stable, the use of the optimum source reflection coefficient for minimum noise could result in an unstable amplifier, in which case a noise figure somewhat larger than the minimum one will have to be accepted. Noise Figure With reference to the circuit shown in Fig. 10.25a, the definition of noise figure F (also called noise factor) is signal-to-noise ratio at input F = — — signal-to-noise ratio at output (10.48) The output noise is the amplified thermal noise from the source resistance plus the noise produced by the amplifier. The standard definition of F requires the source to be conjugate impedance matched to the network, that is, Z„ = Z * , and the source resistance R„ to be at the standard temperature T 0 = 290 K. Very often in practice the source resistance is at a different temperature and the source is not matched to the network. In this case t ' definition (10.48) gives the operating noise figure. If F is given at a single frequency and is based on the noise power in a small frequency band h then the noise figure is called the spot noise figure. When all of the n o i ^ sources are referred to the input as equivalent noise sources, then the sp noise figure can be defined as follows: total input noise power to network , ^n.49) thermal noise input power from source resistance where the noise powers are those in a narrow frequency band A/. i> this latter definition, the spot noise figure for the system shown m . SOLID-STATE AMPLIFIERS 769 10.25 may be obtained from (10.47) by dividing by kT AfM. Thus we get /?, F = ^ T , G, + Rsyr + X„n +2 -G, —B— | 1 0 5 0 ) This noise figure is seen to depend on the source impedance as well as on the noise parameters Re, G,, y r , and 7,. In (10.50), G s = RjiR'i + X*\ The noise figure does not depend on the frequency bandwidth Af. However, the input signal-to-noise ratio will deteriorate if the amplifier bandwidth is greater than that required to accommodate the signal. The optimum source impedance that will minimize the noise figure is obtained by setting l>F/dRs = 0, 9F/dXa = 0, and solving for RK and Xs. It is readily found that X, = X,„=-^- (10.51a) G, and R* + X* = Re/G„ which gives When these values for i?, and X a are used in (10.50), we obtain the minimum noise figure Fm. We now replace y, by -X„,G, in (10.50) and consider F - Fm which is given by F- F m = — [Re+ ( 8 * -f X*)G, - 2X m X,.G,l **« -^-[Re (R2m+Xi)G,-2XiG,} + - -jf[n. + (R« - -R,-,)2G, + ( * , - * » ) * $ + 2fl„*„ I G I -i?*,G,-A';?,G,] - ^ - [ « e + ^.G1-Zr2„G,] ft = ^-[(Rs-Rmf + 1 + (Xs-X„,)2] G } 1 \ {R RiG x2 \T<-R-j *- >2 /"!'~ . "> ' V2 1 upon using 2RsR,„Gi/Rs = 2R2mG,/Rm and putting this term with the 770 FOUNDATIONS FOR MICROWAVE ENGINF.KRING factor - 1/Rm. The factor Re ~ (R2m + X 2 ) G , is zero as ma (10.51). Thus we obtain *-rm+ G YI<R> - * - . ) * + ( * . - xm)2] Wn tr0l » (10.52) This is a very useful relationship in practice, since it determines the n " figure in terms of the minimum value F m obtained when the optimu source impedance R m +jXm is used along with only one additional parameter G,. By adjusting the source impedance, both the optimum source impedance and the minimum noise figure can be determined experimen tally. Transistor manufacturers will often give the minimum noise figure Fm, the optimum source impedance Rm +jX„, or source reflection coefficient r,„ = (Rm + jXm ~ ZC)/(R„, +jXm + Zc), and the noise conductance G, or noise resistance Re. If Re is given the equation i ? 2 + Xf = R ,/G which gave (10.516), can be used to find G,; thus G, = R, Rl+Xi (10.53) The given data can be used in (10.52) to find the noise figure F. Noise Figure for Cascaded Stages Figure 10.26 shows a two-stage microwave amplifier. The power gains of the two stages are GPl and G P 2 . The input impedance mismatch is M, for the first stage and M 2 for the second stage. The equivalent noise sources for stage 2 are designated as e'„(t) and i'n(t). The source impedance for stage 2 is the output impedance of stage 1. The noise figure for stage 1 will be called F, and that for stage 2 will be called F2. The noise figure for the system will be called F. Prom the definition (10.49) for noise figure, we see that the total input noise to the first stage is F^kTAfM,). At the output of stage 2, this noise 1 zs V en{t) •#») iC\ /O [ 9 h A- Gpi r GP2 Kh F I G U R E 10.26 A two-stage microwave amplifier showing equivalent noise sources. SOLID-STATE AMPLIFIERS T71 has been amplified by the factor GP]GP2 to give an output noise contribution Pm = Gpfi^F.kTlfM, A similar analysis shows that the noise sources e'„(t) and /'„(/) produce an input noise power equal to kT \[M2(F2 - 1) to stage 2. Note that we do not include the thermal noise from the source resistance for stage 2. since this comes from the resistive part of the output impedance of stage 1. The latter is an internal noise source for stage 1 and its effects have already been included in the equivalent noise sources for stage 1. The total output noise power in a frequency band Af is readily seen to be Pn,^ = GPlGP2FikTAfMl + GP2(F2 - Y)kTAfM2 where the last term is the amplified noise produced by e'„(t) and i'„(t). The output noise can be considered to be the amplified thermal noise in /?,. multiplied by the two-stage amplifier noise factor F; thus P*,m = GPfiP2FkT\fM, By equating these two expressions for output noise power, we get F r -> + ip>-»l£t <10 54 ' » This result shows that the second-stage noise does not produce a large change in the noise figure above that of stage 1 alone, since the contribution from the second stage is divided by GPl, provided GP[ is of order 10 or more, and the second-stage noise figure is on the order of that of the first stage. For a cascade of three or more amplifier stages, a similar analysis shows that the overall noise figure is given by Clearly, successive stages do not degrade the overall noise figure significantly. The noise in an amplifier can be accounted for by imagining that it comes from the thermal noise in the source resistance by assigning an equivalent noise temperature T„ to the source resistance. Thus we can write Pn,m = = GP]GP2 GPiGP2 •••MxkTL,\f ••• FM^kTXf and hence Te = FT. The excess temperature Tc - T = (F - l)T is called the noise temperature of the amplifier. A low-noise amplifier with a noise figure of 1.4 would have a noise temperature of 0.4 X 290 = 116 K. If all sources of noise in a system, which includes amplifier noise, thermal noise, and radiation noise picked up by the antenna, are regarded 772 FOUNDATIONS FOR MICROWAVE ENGINEERING as thermal noise in the source resistance Rs, then the temperat must be assigned to R s so as to give the same amount of total no" » and is called the system noise temperature. l ls Tt Constant Noise-Figure Circles For the purpose of low-noise amplifier design, it is useful to plot con t noise-figure circles on the source reflection-coefficient plane. Since Z„, ~ Z, Z, - Zc zs+zc •- zm+z/ the expression (10.52) for noise figure can be written as F # " --S*l i + rr i - r. +r i - r. I G, „, 4irs - r„ R, ' i i - r / i i - r Next we use 2ft, = z. + z: i + r. = 2 Re z, i - r,. (i-rs)(i-rs*) 2 Re ii - r„ 2(1-irj 2 ) ii - rj to get F~Fm = AG, K-U (10.56a) 2 li - r j ( i - i r / ) where G, = G,Zt, is the normalized value of G,. A similar expression derived involving RN = Re/Zc, namely, F _ F - 4 j?..~N l i ; :r"l'.- g - U + rj2(i-ir/) can be (10-56*) We now introduce a parameter N, defined as AT. = (fl-irju + r j 2 4ftN (F ; -F m )ii-r„ (10.57) 4G, where W, corresponds to a chosen value F", for F. From (10.56) " now SOLID-STATE AMPLIFIERS 773 obtain N = r -r i r r * - r r * - r r + r r* i-!rj i-r,r; 2 which c a n be w r i t t e n in a form such as (10.1) a n d identified as d e s c r i b i n g a circle in t h e r, plane. T h e c e n t e r of t h e circle is located at r.,= J ^ - „0.58a, a n d t h e r a d i u s of t h e circle is given by yV R,= + N,(i-ir m i 2 ) — (10.586) We can plot t h e circles for v a r i o u s values of N n d e t e r m i n e d by (10.57) for chosen values of F:. E a c h circle s h o w s t h e values of Y s t h a t can be u s e d in order to get a noise figure equal to F1. F o r N, = 0 t h e circle d e g e n e r a t e s to a single p o i n t at Y m giving t h e m i n i m u m noise figure Fm. If F, = Y m is a s t a b l e point, t h e n t h e amplifier can be designed to give a m i n i m u m noise figure Fm. Usually, we can choose t h e load Z L so t h a t F, = Y m gives a s t a b l e design. However, s o m e t i m e s t h e i n p u t VSWR is too high if we u s e F, = F„ ( . In s u c h a case a choice for T s on a c o n s t a n t noise-figure circle with F, > P m would be used in o r d e r to obtain a b e t t e r i n p u t VSWR. Example 10.3 Low-noise amplifier design. A silicon bipolar transistor has the following parameters at 4 GHz: Su = 0.36X148° S , , = 0.11X42° S 2 2 = 0.67Z. - 64° Tm = 0 . 3 8 ^ - 153° S2I = 1.57X27° RN = 0.4 Fm = 1.905 (2.8 dB) By using (10.25a) we find K = 1.2421 > 1; so the transistor is an absolutely stable device. From (10.19) we find that the maximum power gain is 7.2123 for which the maximum normalized gain is 7.2123/IS 2I I = 2.926. Since the device is absolutely stable, we can use conjugate impedance matching. From (10.17) we find that this requires F. = -0.53287 jO.40911 and T L = 0.34159 + y 0.74723. With conjugate impedance matching, the input and output VSWR equals unity. If we use the above value of T in (10.566). we find th;n the noise figure is 2.49 or 3.96 dB. This is 1.16 dB greater than the minimum value. It is desirable to have a lower noise figure and yet not sacrifice any gain which already is on the low side. We can accept an increase in VSWR at both the input and the output of the amplifier. A design using g p = 2.9, which is very close to the maximum value, will be attempted. For an aid in the design process, the two constant normalized power-gain circles g p = 2.9 and g p = 2,3 774 FOUNDATIONS FOR MICROWAVE ENGINEERING FIGURE 10.27 Constant normalized power-gain circles, constant noise-figure circles, and J"* the low-noise amplifier design example. m have been constructed and are shown in Fig. 10.27. The centers and radii of these circles were found using (10.28) and are rLg = 0.34026 + J 0 . 7 4 4 3 RLg = 0.03574 for gp = 2.9 ['Lg = 0.30532 + /0.6679 RLt, = 0.2052 for gp = 2.3 As explained in Example 10.2 the values of VL on a gp ~ constant circle generate a set of values for r i n and hence a circle of 1",* values. The 1*^ circles for gp = 2.9 and 2.3 are also shown in Fig. 10.27. The centers and radi) of these two circles [the equations for these circles are given in Sec. 10.10 a (10.59a) and (10.596)] are 1* , = - 0 . 5 3 1 9 8 - . / 0 . 4 0 8 4 3 lfn c = -0.51225 -jO.39328 tf, n Rin = 0.03029 for g„ = 2-9 = 0.148 for g p = 2.3 Also shown in Fig. 10.27 are the constant noise-figure circles for F = 2. (3.3 dB) and 2.4 (3.8 dB), which correspond to 0.5 dB and 1 dB greater t"^ Fm. The centers and radii for these circles are found using (10.57) and and are rxf= - 0 . 3 1 7 0 6 -y"0.16155 r g / = - 0 . 2 9 5 9 - J0.1508 R r = 0.23444 for F„, + 0.5 dB R f = 0.3317 for F m + 1 dB If we want an input VSWR of unity, then we must choose f s to ' * ° f* circle. The dot shown inside the smallest f* circle is t h e value of J in SOLID-STATE AMPLIFIERS 775 conjugate impedance matching is used. The figure clearly shows that if we choose T„ to lie on this point, the noise figure will be more than 1 dB greater than Fm. Our earlier calculation gave a value of 1.16 dB greater. If we are willing to relax the gain requirement to g p = 2.3, then the figure shows that we can obtain a unity input VSWR and a noise figure somewhat better than Fn, + 0.5 dB by choosing F, as the point Vsi shown in Fig. 10.27. If we choose rt = i'm and gp = 2.3, then the best input VSWR is obtained by choosing a l'L that will produce a Tfn that lies as close as possible to \\. = Ym. This is the point rj = - 0 . 3 3 8 6 - j'0.1725 shown in Fig. 10.27. The corresponding value of VL is l' £ 2 = 0.1818 + ./0.504 and is also shown in the figure. These choices result in a minimum noise figure, an input VSWR equal to 1.392, and an output VSWR equal to 2.069. The noise figure is a minimum value and the VSWRs are acceptable but the gain is only 2.3 X \S21f = 5.67. If we insist on having gp = 2.9 which gives a power gain of 7.15 and we also want a good input VSWR, then we have to accept some increase in noise figure. If we accept a noise figure 0.5 dB greater than F,„, then for the best input VSWR we choose rs to lie on the F = Fm + 0.5 dB constant noise-figure circle and as close as possible to the [** circle for g p = 2.9. This point is Ts = - 0 . 4 7 1 - /0.338 and is shown in Fig. 10.27 as the point !'„.. The required value of TL that will place I"* as close as possible to !"„. is I, = 0.3217 I- yO.7137 and is shown in Fig. 10.27. These choices result in an input VSWR equal to 1.22 and an output VSWR equal to 1.065. This last design can be considered to be acceptable. The input and output impedance-matching networks using parallel open-circuited transmission-line stubs can be designed using the method described in Sec. 5-6 (i.e., (5.29a ) and (5.296>i. Since we use parallel stubs each stub is required to produce only one-half of the susceptance given by (5.29a). From the chosen values of 1', and Y L given above, we readily find that the required normalized source and load admittances are Ys = 1.683 +71.716 YL = 0.1715 -./0.6327 The matching networks along with the stub positions and lengths are shown in Fig. 10.28. This figure does not show the dc bias circuit which is also required. FIGURE 10.28 Matching networks lor the low-noise amplifier 776 FOUNDATIONS FOB MICROWAVE ENGINEERING AJI of the calculations for the above amplifier design tlle determination of the values of TL and \\ that give the best input and Utput VSWR for the chosen gp and F were carried out using the computer ° MICROAMP. The matching networks were also designed using this PTO^^ 10.10 CONSTANT MISMATCH CIRCLES In Examples 10.2 and 10.3 we used the f£ circle as an aid in the design f microwave amplifier with a low input VSWR. The parameters that describe the Tj* circle will be derived here. For an amplifier terminated in a load having a load reflection coefficient rL, the input reflection coefficient is given by '"in = 1 - S22VL The complex conjugate of this equation is r* = 'in a*rL*-s* $22^1. _ 1 This is a bilinear transformation, so that all values of T t that lie on a constant g p circle will map into a circle of r i n values. The center rLlj and radius RLg for a constant normalized power-gain circle are given by (10.28). By using these circle parameters, we can obtain the corresponding parameters for the circle of Tjn and circle of Pj* values as described in Sec. 10.7. The center f£ c for the circle of f£ values is given by " 6 -* r* 1**, in. V ' v -ii2 s is given by Rir — l«a SJR^g s22\ - \s22rLg -- n 2 | (10.596) K* In the design of a microwave amplifier, the choice for the s o U reflection coefficient is constrained by the requirements that are " ^ f ^ . j u to obtain a stable amplifier with a low noise figure. Unity input VS ' be obtained if T s can be chosen to lie on the r£ circle for the cho=ei P circle. If the constraints do not allow this choice or, as sometimes h * P j ^ ^ the r£ values inside the Smith chart are unstable values for I s . . eS[ should be located as close as possible to the r£ circle to obtain the ^e input VSWR. When the choice for f, has been made, then, if f~s l i e S SOLID-STATE AMPLIFIERS 777 T,* circle, we will require Fin = F* and we can then find the required value of FL using If T s cannot be placed on the r*n circle, then the best value for T*n is the one that is closest to r,s but on the specified rz circle. The complex conjugate value of this should be used for r, n in (10.60) to find T/.. For an unstable device a part of the g p = constant circle will lie outside the Smith chart boundary. The values of \'L on the g p = constant circle outside the Smith chart produce values of \'*t that lie outside the Smith chart. As explained in Example 10.2 the F*, circles have two invariant points for an unstable device. These invariant points coincide with the two points at which the source stability circle intersects the Smith chart boundary as illustrated in Fig. 10.21. The proof that these points are invariant points is as follows: For an unstable device we have shown that the g p = constant circles have two invariant points that coincide with the two points at which the load stability circle intersects the Smith chart boundary (see Sec. 10.7). Since F,* is given by A* Ff - S f , in s2*2r* - 1 it is clear that these two particular values of F,, which we will call VLl and r / 2 , that are invariant points for the g p = constant circles will map into two fixed values for F£ that are common to all \'*n circles. Thus the r;* circles have two invariant points. Since F,, and l\2 also lie on the load stability circle that makes |r i n | = 1, the two points, which we will label as r in , and l m 2 , must lie on the Smith chart boundary. Thus the two invariant points for the F*; circles lie on the Smith chart boundary. It remains to be shown that these points coincide with the two points at which the source stability circle intersects the Smith chart boundary. The source stability circle is the circle of Fs values that make |F011t| = 1, where Ar; - s 2 2 I' = — " (10.61) We can rewrite (10.60) in the following form: 1 1 A— r„ '' s, - S22 (10.62) ' 1 which is a bilinear transformation of the same form as in (10.61). Let us choose r s l = F* ,. Since I V , 1 2 , = 1 we have r r i - 1/F in ,. From (10.62), 778 FOUNDATIONS FOR MICROWAVE ENGINEERING l / r i n , maps into the point l/\'Ll for which | r t l | = 1. The bilinear tr mation in (10.61) is the same, so it follows that F/Sl = 1/f point r out , for which )l't Thus r , is a point on t n e ' s o u r c e ^ a h T * 3 circle. For the same reasons r s 2 is a point on the source stabilit . Consequently, F*,, = Vsi and T* 2 = Ts2 must be on the source s t a S h l circle. Hence the invariant points for the ["£ circles coincide with th t points at which the source stability circle intersects the Smith chart h W ° u ary. Constant Input Mismatch Circle If we want to design an amplifier with a specified input VSWR, then if th load reflection coefficient VL has been chosen, there will be a circle of l values that can be used which will produce the specified input VSWR Let the required input VSWR be VSWR,. The reflection-coefficient magnitude is given by VSWR, - 1 P= VSWRj + 1 and the input impedance mismatch M x is given by M, (10.12a) the input mismatch is given by 1 - p2. From | 32\ / 1 1 1 — 1 1 :i .n 4R R :_ «t/t x. i liD (i-ir i ) ( i -i i i- r,2/ ) (10.63) Let \"t be the chosen value of VL. From (10.60) we can calculate the corresponding value of f,n, say r,'n. By using this particular value of Tin in (10.63), we can express (10.63) in the form of an equation for a circle in the 1*. plane. The center of this circle is located at sM (10.64a) l-o-iiWJ 2 and the radius of the circle is given by /r=nr(i-iiy*) l R. sM u = (10.646) i-(i-M,)ir;nr The following example will illustrate the application of the constant inp«' mismatch circle in amplifier design. Example 10.4 Application of constant input mismatch circ amplifier design. An FET has the following parameters: SM = 0.8Z. - 140° S, 2 = 0.2Z30° S 21 = 2.8^60° 5 2 2 = 0.2Z11500 T,„ = 0.7Z.100" RN = 0.4 F,„ = 1.5 (1.76 dB) We want to design an amplifier with low noise and an input VSW ill tO BOUD-STATE AMPLIFIERS 779 ssc F=F_+0.5 dB LSC FIGURE 10.29 Constant gain, constant noise-figure, and constant input impedance-mismatch circles used for the amplifier design in Example 10 4. 1.6. The required value of the input impedance mismatch is found to be M, = 0.94674. In Fig. 10.29 v>'e have plotted the £ ; , = 1.6 constant normalized powergain circle, the load stability circle, the source stability circle, and the optimum value r,„ for I's to obtain a noise figure equal to the minimum value /''„,. The F = 1.683 constant noise-figure circle is also plotted. This noise figure is 0.5 dB greater than Fm- For this example we have chosen \'L = 0.307 /0.55, which is a point on the g ; , = 1.6 constant gain circle and not too close to the load stability circle, f o r this choice of l\ = \"L, the input reflection coefficient r; n = - 0 . 3 3 0 2 -./0.3062, The input mismatch circle that will give VSWR, = 1.6 has a center and radius given by [•„# = 0.316 ( ./0.293 ft,,,, = 0.186 and is also plotted in Fig. 10.29. All values of VH on this circle will give an input VSWR equal tc> 1-6. In order to obtain the best possible noise figure, we should choose \\ to lie on this circle and on the line that joins the center of this circle to the optimum point I",,,. The best choice for I", is shown in Fig. 10.29 and lies inside the F = Fm + 0.5 dB constant noise-figure circle, so that the input VSWR requirement can be met with a noise figure somewhat less than 1.683. The power gain obtained is 1.61 S 2I \ 2 = 12.54, which is quite close to the "Figure of Merit" gain of 14. 780 FOUNDATIONS FOR MICROWAVE ENC.INEERING Output Impedance-Mismatch Circle If it is required to design an amplifier with a specified output V'SWR for each chosen value of I's = \"s, we can plot a circle of r ; values t h - ^ 6 " ' ensure t h a t the specified output VSWR is obtained. The equation f^ * ' " output impedance-mismatch circle are of the same form as (10 641* T^ center of the circle is located at 0* " MAKucf i-(i-Af2)ir^ (10.65a and the radius of the circle is given by CM where - /i~^2(i-ir;,j2) 1 - (i-M2)ir;uti2 M2- = 1 - = (10.656) ( VSWR2 - 1 ^2 ( VSWR2 + 1 ] AS - s.i2 snr; - I In the design of a two-stage amplifier, the design of stage 1 leads to a specified output impedance mismatch for stage 1. Consequently, the design of stage 2 is constrained by the requirement that the input mismatch to stage 2 be equal to the output mismatch of stage 1, since the impedance mismatch is conserved in the lossless matching network that is used to couple the first and second amplifier stages. For this reason the constant impedance-mismatch circles described above are useful aids in the design of a two-stage amplifier. The application of the constant impedance-mismatch circles in two-stage amplifier design is described more completely in the next section. 10.11 MICROWAVE A M P L I F I E R D E S I G N In this section we will present a design strategy for designing narrowb* one- and two-stage amplifiers, The first stage can be designed for a low noise figure. The method to be described can be used with both stable an^ potentially unstable transistors. It is assumed that the scattering-ma parameters S,j, the optimum source reflection coefficient Ym for mini noise, the minimum noise figure F„„ and the normalized noise resis R N or noise conductance G, are all known. The design specifications assumed to be a power gain greater than some minimum value, a figure no greater than a specified maximum value, and input and o VSWRs that do not exceed specified maximum values. SOLIDSTATK AMPLIFIERS 781 There is no unique design for an amplifier that meets the design specifications. Also, there is no unique method for carrying out the design. The method described in this section works quite well for achieving a satisfactory design, but many other systematic approaches could also be developed. In general, we have to examine a range of possible load and source reflection coefficients in order to obtain an optimum design. It would be very tedious to carry out the required optimization procedures using hand calculations. Consequently, in practice a suitable computer program is used. The design strategy that is described in this section has been implemented as the computer program MICROAMP. Stage Amplifier D e s i g n The first stage of a multistage amplifier or a single-stage amplifier is normally designed for a minimum noise figure, maximum power gain, and a chosen maximum input and output VSWR. In a multistage amplifier the output VSWR of the first stage is usually not a critical parameter. When the constraint on the output VSWR is relaxed, there is a greater degree of freedom available that makes it easier to achieve the other design requirements. The design of the second stage is also easier to carry out when the first-stage output VSWR is relatively large. We will describe the design of the first stage as a series of steps or procedures to be carried out. 1. The first step is to evaluate the stability parameter K given by (10.25a), i.e., %\SuSn\ and also to check if (10.256) to (10.25/') are satisfied. When these conditions hold and K > 1, the transistor is absolutely stable and steps 2 to 4 should be followed. If K < 1 the device is potentially unstable and steps 5 and 6 should be followed. 2. For K > 1 conjugate impedance matching can be used. The required values for the source and load reflection coefficients are given by (10.17). Let r*s, be the solution for I"s given by (10.17a). From this value of I", the noise figure F can be calculated by using (10.56). The power gain with conjugate impedance matching is given by Gp = Gl>,max=(K-^i-l) "12 If the noise figure is acceptable, then the design is finished except design of the input and output matching networks. In practice, it turns out that the noise figure obtained using conjugate impedance ing is not satisfactory. In order to obtain a better noise figure, it for the usually matchwill be 782 FOVNDATIONS FOK MICROWAVE ENGINEERING F I G U R E 10.30 Constant gain. I'*,, and constant noise-figure circles used for amplifier design. necessary to design for a lower gain and some mismatch at the input and output ports, as described in the following steps. 3. In order to visualize the design procedure, we have plotted three constant normalized power-gain circles, the corresponding three V*n circles, and two constant noise-figure circles for a hypothetical device in Fig. 10.30. We first try to obtain a satisfactory design using l"s = Tm for minimum noise and a chosen normalized power gain. For example, we can choose § P = £i/2) in Fig. 10.30. We now construct an objective function OF = WlMl + W2M2 where M, is the input mismatch and M.> is the output mismatch and W, and W 2 are weights that can be chosen to place different levels of impor tance on the input and output VSWRs. A value of VL, say r t „ is chosen on the g (2) constant gain circle. From this value of fL we can calculate ,„ using n„ = Mi,-S u ^ 2 2 ^ 1 1 We can also calculate foul using Ts = F„,; thus r 1 = nil! AT... - S,22 slxrm-t SOLID-STATE AMPLIFIERS 783 From these we calculate M, and M 2 using M,= M2 = (i-irini2)(i-irj2) li - rinr„,i2 (i-irou[i2)(i-irL10 li - i ; u ^1A The objective function for this value of P ; is now evaluated. The best input and output VSWRs are obtained by maximizing the objective function. Thus we search the gp(2) circle for the value of Y L that maximizes OF. This search is carried out by incrementing \\ in the direction that increases OF, that is, r L is set equal to r / 2 , P t 3 , . . . , as shown in Fig. 10.30. If satisfactory values for the input VSWRj given by VSWR, = 1 + fl - M , l-}fl-M1 and the output VSWR 2 given by VSWR, = 1 + / l - M2 are obtained, then the design process is terminated. If we obtain a good value for the output VSWR2 but an unacceptable value for the input VSWRj, then we can set the weight Wi = 0 and W\ = 1 so as to place all of the emphasis on achieving a good input VSWR!. With all of the emphasis placed on M\. the optimum value of T/. will be that value which produces a F"n that is as close as possible to r m . since this produces the best input impedance match. However. M] and M->_ have the same dependence on rL so the largest values of M\ and Mi occur for the same value of TL, so changing the weights will not improve the results. It will then be necessary to search for an optimum value of Ti on a lower constant gain curve, say gr(i). The corresponding r^ circle is larger: so clearly HJ, can be brought closer to r m , thereby improving the input VSWRj. The above process is repeated until the lowest acceptable constant gain circle has been searched. If this does not result in acceptable input and output VSWRs, the only alternative left is to accept a noise figure greater than Fm. In this case it is helpful to compile a table of best possible input VSWR, values for a given noise figure and normalized power gain, as described in step 4. 4. Figure 10.31 shows the F = F, constant noise-figure circle and the [^(2) circle for gp = gp(2). If we are designing the amplifier for this noise figure and power gain, then the optimum choice for Y^ and T* that will result in the best input VSWR, is the value of Ts on the F = Fx circle and the value of F£ on the V*a(2) circle that are as close together as possible. 784 FOUNDATIONS FOR MICROWAVE ENGINEERING F I G U R E 10.31 A constant noise-figure circle and a 1",* circle and the optimum choice for Vx and T*, that will give the lowest input VSWR. These points lie on the line that joins the center of the F = Fj circle to the center of the F*(2) circle as shown in Fig. 10.31. The center for the noise-figure circle is \'Sf given by (10.58a) and the radius R f of the circle is given by (10.586). The center V*n ( . and radius r?m for the Fj* circle are given by (10.59). The vector from the center of the F = Fl circle to the center of the F*n(,2) circle is given by r = r*.c-iv A unit vector pointing from Ts/ to f* c is r / | r | . The point Ts lies a distance R f from Vst and in the direction of r; hence r --w*'- r* - r ir* ^T~l r The optimum value of f* is similarly given by T* = 1 ;Ein = fi,_ in Thus we can calculate the optimum choice of T s and I'* and from these SOLID-STATE AMPLIFIERS 785 evaluate the input mismatch M t and VSWR,. The value of V L that produces rjjj is given by < 10.60). The above calculation can be repeated for various choices of gp and F and allows us to compile a table of best VSWR, values as a function of g and noise figure F. By consulting such a table we can easily see the tradeoffs between power gain, noise figure, and input VSWR,. We now choose fe for the lowest noise figure consistent with the lowest value of acceptable power gain and the largest acceptable value of input VSWR,. The next step is to repeat the optimization procedure described in step 3 by searching the identified g p = constant gain circle so as to optimize the objective function. The maximum value of the objective function might not correspond to the optimum choice for V* t h a t maximizes M|. Also, the output VSWR 2 might be higher than specified. If this is the case, then either the design requirements have to be relaxed or a different transistor must be used. 5. If the transistor is potentially unstable, then we cannot carry out a design with conjugate impedance matching. For this case a design for minimum noise using \\ = l'„, should be explored first. This requires that we determine the source stability circle and check that l',„ lies in a stable region of the Smith chart and not too close to the boundary of the source stability circle. When T m is a stable value, the design procedure is the same as described in step 3, i.e., a chosen gp = constant gain circle is searched for the best value of V, that will maximize the objective function OF given in step 3. The ''Figure of Merit" gain is | S 2 , / S , 2 | with a normalized value |S 1 2 S 2 1 j • It is good practice to design an amplifier for a normalized power gain that does not exceed this value. Thus constant power-gain circles with S P = | S a i S l 2 | ' and smaller are searched. If acceptable values of gp, VSWR,, and VSWR.,, are obtained, then the design process is terminated. The resultant amplifier will have a minimum noise figure Fm. If a satisfactory design cannot be obtained using T, = f"„, or if l'„, is an unstable value, then the procedure outlined in step 6 should be followed. If f„, is an unstable value, it would be advisable to use another transistor. 6. In order to minimize the amount of searching for the optimum value of V, subject to the constraints on gp, F, VSWR,, and VSWR 2 , it is again helpful to compile a table of best VSWR, values as a function of gp and F. By consulting such a table we can determine if the design objectives can be met a n d / o r the best power gain, noise figure, and input VSWR, that can be obtained using the chosen transistor. When the I",* values inside the Smith chart represent stable values of I",, then the input VSWR, for the optimum choice of T*n and fs on a chosen F = constant circle are calculated the same way as for a stable device. The procedure is described in step 4. The optimization of the input VSWR, and output VSWR., for the chosen normalized power gain and noise figure can be carried out in the same way as described in step 4. A bipolar transistor used in a common base configuration or an F E T used in a common gate configuration often have | S , , | > 1 and |S 22 I > 1. 786 FOUNDATIONS FOR MICROWAVE ENGINEERING When this latter set of conditions holds true, it is not possible to des' amplifier with a low input or output VSWR. The reason is that stable if*? ^ of T, produce values of V*n that lie in the unstable region of the r nl a we cannot choose Ts equal to 1',*. Similarly, stable values of r produce r*° values that lie in the unstable region of the rL plane so the output n ""' cannot be matched. Since the common base and common gate connectin usually have poorer noise performance as well, the common base or commo gate circuits are not used in low noise amplifiers. In Example 10 5 w illustrate the impossibility of matching the input port of a common base amplifier. Example 10.5 Common base amplifier. A bipolar transistor in the common base connection has the following parameters at 5 GHz: S u = 1.3^140° S 1 2 = 0.2^130° S2I = 2 ^ - 8 5 ° S 2 2 = I.lfiZ - 50° r,„ = 0.7^135° Fm = 2.5 RN = 0.4 For this transistor K = - 0.579 so it is potentially unstable. In Fig. 10.32 we have plotted the load stability circle, the source stability circle, the gp= 1.5 F I G U R E 10.32 , 6 ircle. base Load stability circle, source stability circle, g p = 1.5 circle and corresponding common i&i^ F = F„, + 1 dB constant noise-figure circle for the bipolar transistor, in a configuration as used in Example 10.5. SOLID-STATE AMPLIFIERS 787 circle and corresponding !',* circle, and the F = F„, + 1 dB = 3.147 circle. Since I S n l > 1 and | S 2 2 | > 1, the origin is an unstable point for both \\. and !'/_. The regions of the Smith chart where stable values of T v and V, are located are shown cross-hatched. The point r £ ] on the g p = 1.5 circle maps into the point l~* , which lies in the unstable part of the Tv plane. All values of l'L in the stable region map into T* values that lie in the unstable part of the Ys plane. Hence we cannot choose 1", = 1'* and consequently the input port cannot be matched, If we try to design a low-noise amplifier by choosing I.. = [*„, and with a normalized power gain of 2. the resultant input VSWR l = 15.42 and the output VSWR., = 19.03. If we reduce the gain requirement G p to 5. for which gp = 1.25, we obtain VSWR, = 12.76 and VSWR., =• 26.56 for a design with minimum noise. There is a small improvement in the input VSWR, but the output VSWR 2 is increased. From Fig. 10.32 it is quite clear that any value of r„ on the F = Fm + 1 dB noise-figure circle and in the stable region will still be far away from all I'* values inside the Smith chart. Thus a low-noise amplifier with acceptable gain and input and output VSWRs cannot be designed using the above transistor in a common base circuit. Example 10.6 Low-noise amplifier design. A GaAs FET has the following parameters at 10 GHz: S u = 0.73^42° S.,2 = 0.5^34° S 1 2 = 0.2A - 58° r,„ = 0.52Z. - 70° S.2I = 1.52i - 66" F„, => 1.25 A\v = 0.75 We want to use this device in a low-noise amplifier design which meets the following specifications: Noise figure /•' < 1.5 Input VSWR t < 1.5 Output VSWR 2 < 1.5 Power gain G(, as large as possible For this device K = 1.071 so that the FET is absolutely stable. For a conjugate-impedance-matched design, we get Gp = Gpmm = 5.222. VSWR, = VSWR 2 = 1. and F *= 1.7. The noise figure is too large; so we must consider a design that is not matched at the input and output. The power gain which can be obtained is no greater than 5.222; so we do not want to sacrifice much power gain for an improved noise figure. The best VSWR, that can be achieved using gp = 2.25, which gives Gp = 5.1984. for F = F„, -i 0.5 dB is 1.3644 and for F = Fn, + 1 dB it is 1.026. This shows that the design objectives can be met by allowing a noise figure equal to Fm + 0.5 dB = 1.4. By searching the gp = 2.25 circle for the optimum value of I", and using the optimum value of I",, which was determined so as to lie on the F = Fm + 0.5 dB noise circle and give VSWR, = 1.3644. we obtain Gp = 5.1984, VSWR, = 1.3643. VSWR;, = 1.1179, and F = 1.4. The required values of Tv and V, are I"v = 0.389 -./'0.534 and IV = - 0 . 0 0 2 8 - / 0 . 3 2 S . This design meets all of the specifications. 788 FOUNDATIONS FOR MICROWAVE ESCINEER1NC, M, My Zc MN 1 M, J pi M, M. M, G-pa MN2 MN3 CJ5 rs r,n ruit I r F I G U R E 10.33 A block diagram of a two-stage amplifier. D e s i g n of S e c o n d S t a g e for a Two-Stage Amplifier The design specifications for the second stage of a two-stage amplifier emphasizes power gain and output VSWR. The noise figure of stage 2 is not very critical since the noise contribution of the second stage is reduced by the power gain of the first stage as shown by (10.54). However, at the higher microwave frequencies, the power gain of the first stage is often not very large, so that some consideration of the noise figure of stage 2 is necessary. If we assume that the matching network used between the output of stage 1 and the input to stage 2 is a lossless network, then, since the impedance mismatch is constant throughout a chain of lossless networks, the impedance mismatch at the input to stage 2 is the same as that at the output of stage 1. This places a constraint on the design of stage 2, namely, that the input impedance mismatch must equal M.2, where M 2 is the output mismatch for stage 1 and was determined in the design of stage 1. In order that the interstage matching network be physically realizable, this constraint cannot be violated. In Fig. 10.33 we show a block diagram for a two-stage amplifier. The source reflection coefficient, input reflection coefficient, output reflection coefficient, and load reflection coefficient for stage 2 are identified by a superscript prime. The output impedance mismatch of stage 2 is M 3 and the corresponding output VSWR will be called VSWR 3 . It is assumed that the same type of transistor that was used in stage 1 is also used in stage. The design of stage 2 will be based on the optimization of the noise figure and output VSWR or mismatch M 3 for a chosen power gain gp- T h l £ optimization is carried out subject to the constraint that the input imped mismatch equals M2. The following objective function is used for the desi of stage 2: F OF = W,M 3 + W2 — nd where W, and W2 are suitable weights. We choose Fm/F for the se ^ ^ term, since this quantity is of the same order of magnitude as M 3 t becomes larger for smaller values of F. Thus our goal is to find t h e SOLID-STATE AMPLIFIERS 789 F I G U R E 10.34 A constant power-gain circle and a constant input impedance-mismatch circle used in the design of t h e second stage of a two-stage amplifier. value of rL on a chosen gp = constant gain circle so as to maximize OF. The design procedure is the same for both stable and unstable devices. A visualization of the optimization process can be obtained by referring to Fig. 10.34. In this figure we show the load stability circle, two constant power-gain circles, and a constant noise-figure circle for a hypothetical transistor. Let us assume that we will choose g p = gl. We then pick an initial value of V,, say T,',, on the gt circle. From this value of V'L we can calculate I7n using nrLl - su 1 in Q i- _ i The input mismatch M 2 is given by (i-irj')(i-ir/) |i - r;nr;i2 Since M2 is fixed by the stage 1 design, this equation determines a circle of 790 FOUNDATIONS FOR MICROWAVE ENGINEERING V s values that can be used. The center and radius of this input circle are given by (10.64) and are KM ~ R'SM - mis match M2d7nf i-(i-M2)ir;j2 Each value of T's such as 1^, r; 2 , etc., shown in Fig. 10.34 enables us to calculate a corresponding noise figure using (10.56). Also for each IT we can calculate an output reflection coefficient I"1,, ,. From r; u l , and the chosen value Y'Ll for V{, we can evaluate the output mismatch Af3. Thus we can evaluate the objective function. Our procedure is now to search the input mismatch circle for the value of l"s that maximizes the objective function. The maximum value is recorded. We now increment Y"L to a new value Y',.,. This results in a new input mismatch circle which is searched for the value of l"a that maximizes OF. This value of OF is compared with the previous one, and if it is greater then l"L is incremented to a new value T£3 in the same direction on the gp = gx circle. If the second value of OF is smaller than the first one, then i"L is incremented in the opposite direction. The search process is continued until the optimum values of V L and T'a are found. The optimization routine would be very time consuming to carry out with hand calculations but can be done very quickly on a computer. The optimization carried out on the input mismatch circle will result in a choice for H that lies as close as possible to the optimum point l'm if the weight W, is set equal to zero (see Fig. 10.34). When WL is not zero, TJ will generally deviate from this point in order to get a lower output VSWR 3 . If satisfactory values of noise figure and output VSWR 3 are not obtained from a value of Y'L on the chosen gp = constant circle, then the procedure outlined above must be repeated on a constant gain circle having a lower gain. If, after searching the lowest acceptable power gain circle, an acceptable design is not obtained, then the design specifications will have be relaxed, a different transistor used, or a third stage added. A third stag* can be designed using the same approach as used for the second stageSome of the V s values on an input mismatch circle may he '"* *? unstable region of the Smith chart. If Y^ is an unstable point, t h e " t e ' r produces an output reflection coefficient r„ u [ ,, with a magnitude 8 ^ ^ than unity. When i r ; u t J > 1 the output mismatch M 3 and output VbW ^j will be negative. Consequently, M:) will contribute a negative quanti } the objective function. Since the objective function is being maxim' ^ unstable values of f^ are not selected since they tend to mininuz objective function. SOLID-STATE AMPLIFIERS 791 W h e n t h e design of t h e second s t a g e h a s b e e n completed, t h e two-stage amplifier will have a power gain Gp = Gpfip2 a n d a noise figure F = FX + ( F a - 1) M2 Mfipl The interstage matching network m u s t transform the output admitt a n c e YtMl of s t a g e 1 i n t o t h e r e q u i r e d s o u r c e a d m i t t a n c e FJ for stage 2. A t t h e s a m e t i m e it m u s t t r a n s f o r m t h e i n p u t a d m i t t a n c e V,'n of stage 2 into t h e r e q u i r e d load a d m i t t a n c e Y L for s t a g e 1. T h i s m a t c h i n g n e t w o r k is physically realizable because of t h e c o n s t r a i n t t h a t was placed on t h e s t a g e 2 i n p u t i m p e d a n c e m i s m a t c h when it w a s designed. T h e design of an inters t a g e m a t c h i n g n e t w o r k is described in Sec. 5.7. Example 10.7 Two-stage amplifier design. At 6 GHz an PET has the following parameters: S n = 0 . 8 ^ - 130° Sts = 0.2,630" S 2 I = 3z60° S , 2 = 0.3Z.140" r„, = 0.6/Ll6(T PL, = 1.6 Rx = 0.6 The design specifications are: Two-stage power gain O > 120 Total noise f i g u r e F s 1.8 Input VSWR, < 1.5 Output VSWR, < 1.5 For this device K = 0.513. so that the transistor is not absolutely stable. The "'Figure of Merit" gain equals [£jj,/£, 2 l = 15. The normalized value is g p = 1.666. For the design of the first stage, the following table of minimum VSWR, values as a function of normalized power gain and noise figure was compiled: gp 1.6 1.4 1.2 1.6 1.4 1.2 From the above table normalized power gain normalized gain of 1.2, gp = 1.2 and a value of F *m Fa Fm 1.68 1.68 1.68 VSWR, 2.39 2.084 1.809 1.71 1.49 1.294 r. -0.564 +70.205 -0.564 +70.205 - 0.564 -+ 70.205 - 0 . 5 8 1 -70.304 - 0.576 7'0-306 - 0 . 5 6 9 t-7 0.307 we see that we can obtain a noise figure of 1.68, a of 1.4, and an input VSWR, = 1.49 or, with a reduced an input VSWR t of 1.294. We will choose a power gain I % midway between those giving VSWR, equal to 1.294 792 FOUNDATIONS FOR MICROWAVE ENGINEERING and 1.809, so as to obtain a noise figure somewhat smaller than 1.68. Thu choose r„ = - 0.568 + j'0.27. We now use this value and search the g = i gain circle for the value of ["L that will give the smallest values for VSWTl VSWR 2 . The result is T L = 0.458 + ./0.14 and an amplifier having a nT figure 1.572 and an input VSWR, = 1.484 and a power gain G , = 10 8 Th output VSWR 2 = 6.5015 which gives M 2 = 0.4621. The design of stage 2 is carried out using the procedure described earlier The objective function W^AT, + W.,F„,/F is optimized by searching for the value of l", on a chosen g p = constant gain circle and the value of rj on the associated input impedance-mismatch circle using M 2 determined above The search on the gp = 1.6 circle resulted in a design with VSWR., = 1.65 and F2 = 3.693 using W, = 1, W2 = 0. For this example a search on lower gain circles gave higher values of VSWR a . If we relax the design specifications to allow the somewhat larger output VSWR 3 value, then for the two-stage amplifier the following performance is obtained: Gp = 10.8 X 14.4 = 155.5 F0 - 1 F = F, + M., Mfi •• VSWR, = 1.484 VSWR 3 = 1.65 2.693 = 1.572 + 0.4621 x 0.962 x 10.8 ssc Input mismatch circle FIGURE 10.35 Illustration for the two-stage amplifier designed in Example 10.7. = 1.692 SOLID-STATE AMPLIFIERS 793 For all specifications except the output VSWR3, this design meets the stated criteria. For practical applications an output VSWRa equal to 1.65 instead of 1.5 is acceptable. In Fig. 10.35 we show the load and source stability circles, the g = 1.6 and 1.2 constant gain circles, the values of r, and f", for stage 1, the values of i"s and \"L for stage 2, and the point fs = !"„,. Also shown is the point I",;, for 1"^ and the input mismatch circle for V'L. Note that V's lies on this circle but is not as close as possible to the optimum point I m . The reason for this is that V was chosen to obtain the best output VSWR3, not the best noise figure. The figure clearly shows that t'v, \'L, P., and \"L are sufficiently far away from the stability circle boundaries so the design has an adequate stability margin. A second design was carried out for which the input stage was designed for a power gain of 7.2 and a minimum noise figure and a resultant input VSWR| = 1.358 was obtained. For the second stage we were then able to obtain GpS = 14.4, VSWR3 = 1.517, and F = 4.28. For the two-stage amplifier we obtained Gp = 103.7, VSWR, = 1.358, VSWR., = 1.517, and F = 1.685. This design has an input VSWR, lower than required, a noise figure essentially the same as for the first design, an output VSWR,, nearly equal to 1.5, but a significantly lower gain. The small increase in output VSWR., in the first design is only a small price to pay for the much larger power gain that was obtained, so the first design is a better one. 10.12 OTHER A S P E C T S OF MICROWAVE AMPLIFIER D E S I G N Microwave amplifier design as described in the preceding section is only a small part of the overall design problem. Once it has been verified that the specified performance can be obtained, a decision has to be made as regards to whether hybrid construction or monolithic integration will be used in the fabrication of the physical amplifier. A decision of whether to use mierostrip circuits or eoplanar-waveguide circuits must also be made, as well as a decision of whether to use lumped elements or distributed elements for the impedance-matching networks. The layout of the circuit must be designed and the circuit must incorporate both the bias circuit and the RF elements with suitable decoupling of the RF circuit from the dc bias circuit. The physical dimensions of ail transmission lines and other printed-circuit elements must be calculated and will depend on the substrate material used. Suitable masks must be prepared for use in the fabrication of the amplifier. After the amplifier has been built, it must be tested to determine if it meets the design specifications. After the RF circuit and dc bias circuit have been designed, a theoretical evaluation or computer simulation of the amplifier should be carried out to verify that it is stable at all freqxiencies. This check should be performed before the amplifier construction is undertaken. For a broadband amplifier the design of the matching networks is considerably more complex than that for narrowband amplifiers. The 794 FOUNDATIONS FOR M1CROWAVF. ENGINEERING matching networks must be designed so as to provide adequate stabilitgain throughout the passband, and constant group delay. The latt quires that the phase function <M«) in the overall amplifier transfer f ' tion H(<o) = \H(u)le-,M<"' be a linear [unction of w (see Sec. 3.19). For a power amplifier, dynamic range, nonlinear distortion, and i modulation characteristics must be taken into account. Also, suitable n sion must be made to remove the heat produced in the active devices Companies that manufacture microwave amplifiers use a number f computer-aided design (CAD) software packages to facilitate the overall design. The effort expended in carrying out a thorough design before construction is undertaken pays large dividends since there is very little that can be changed in either the circuit component values or the circuit layout once the amplifier has been built, particularly so for monolithic microwave integrated circuits (MMICs). Hewlett-Packard manufactures modular microcircuit packages that are very useful in the eaHy stages of microwave amplifier development. These packages provide a ready-made miniature box with input and output miniature 3-mm coaxial-line connectors and dc bias terminals that allow for easy mounting of a prototype circuit so that it can be tested. A photograph of these package modules is shown in Fig. 10.36. F I G U R E 10.36 Modular microcircuit package for prototype circuit design and testing. (Photograpn Ray Moskaluk, Hewlett-Packard Company.) ortes>. of SOLID-STATIC AMPLIFIERS 795 FIGURE 10.37 A broadband traveling-wave amplifier (MMIC) circuit. All components including matching networks are built on a single chip. (Photograph courtesy of Kay Moskaluk. Hewlett-Packard Company.) A typical M M I C circuit is s h o w n in Fig. 10.37. T h i s is a b r o a d b a n d g e n e r a l - p u r p o s e traveling-wave amplifier. It is a GaAs MMIC chip u s i n g seven M E S F E T gain stages a n d h a s a flat gain of 8.5 ± 1 dB over t h e frequency r a n g e 2 to 26.5 G H z . A large n u m b e r of gain stages is r e q u i r e d since for a n y amplifier t h e gain-bandwidth p r o d u c t t e n d s to r e m a i n cons t a n t , so t h a t t h e gain p e r stage is necessarily low in a very b r o a d b a n d amplifier. T h e noise figure r a n g e s from a r o u n d 5 dB at the low-frequency e n d to 7 to 8 dB at t h e high-frequency end. T h e input and o u t p u t V S W R s a r e less t h a n 1.5. PROBLEMS 10.1. A bipolar transistor has the following scattering-matrix parameters at 2 GHz: S,, = 0.56Z.170" S I 2 = QM/-75* S , , = 4.04 Z. 76° S.u = 0.41Z -23° Find the stability parameter K, and if K > 1 also find the maximum stable gain. Find the load and source stability circle parameters and plot these on a Smith chart. FOUNDATIONS FOR MICROWAVE KNGINF.KRING 10.2. A silicon bipolar transistor is used in a common base amplifi er circuit at 5 GHz. Its scattering-matrix parameters are: S u = 1.3^140° S ( z = 0.2Z130" S 2 1 = 2Z. - 85° S 2 2 = 1.15/. - 55° Find the stability parameter K, the maximum stable gain, and the load source stability circle parameters. Plot the stability circles on a Smith ch 10.3. A GaAs FET has the following scattering-matrix parameters at 2 GH? S„ = 0.91Z - 42° S l z = 0.05^33° S 2 , = 6Z.105° S22 = 0.62Z-95° Find the stability parameter K and the load and source stability circles Pint the stability circles and show the regions of the Smith chart where stable values of the load and source impedances lie. 10.4. A bipolar transistor is used as a common collector {source follower) amplifier at 5 GHz. Its scattering-matrix parameters are: S „ = 0.63A - 9 6 ° S 1 2 = 0.8^15° S21 = 2.3Z. - 5 3 ° 5 2 2 = 0.62Z98" Evaluate the stability parameter K. Find and plot the stability circles and show the regions of the Smith chart where stable values of Y L and i"8 occur. Answer: K = 0.3993, rLl. = - 1.298 +J4.0718. RLC = 4.5735, r s l = 2.347 - J 3 . 8 7 9 2 , Rsc = 4.8395. Stable values he inside the stability circles. 10.5. An engineer adds an external network to a bipolar transistor and finds that in a common base connection its scattering parameters at 5 GHz are S„ = 1.3^140° S 1 2 = 0.2^130° S 2 1 = 2zL85" S 2 2 = 1.1SZ - 5 5 ° Find the stability parameter K and the stability circles. Show in what regions of the Smith chart stable values of Y L and Ts occur. Answer: K = 1.439 but the device is only conditionally stable since | S „ ) > 1 and ISajJ > 1; Vu- = 0.174 + J0.6258, Ru: = 0.2172, TS( - 0 . 2 1 7 9 - jO.495," ft sc = 0.2714. Origin is unstable. 10.6. For the amplifier discussed in Example 10.2 and using the design that requires ZL = 85 +j 168.2. design a matching network consisting of an open-circuited stub located a distance d from the output that will transform the 50-11 line impedance into the required load impedance (see Sec. 5.6). 10.7. Redesign the amplifier discussed in Example 10.3 so as to get a noise figure of 2 and a normalized power gain of 2.9. Find the required values of I,, w and the resultant input and output VSWRs. . Hint: Construct the F = 2 noise-figure circle. The best choice for I, a point on this circle lying on the line joining the center of the F - & with the center of the P* circle for g p = 2.9. This I s can be found since^* centers and radii of the circles are known. The corresponding E can also be found. From P. calculate P,.,, and from F* calculate T,.- <"* input and output impedance mismatches can now be found using U and (10.126). From these the input and output VSWRs can be found. SOLID-STATE AMPLIFIERS 797 10.8. A MESFET has the following parameters at 8 GHz: S , , = 0.65Z - 1 5 0 " S 2 1 = 2.2Z.6T S i a = 0.12^32° S22 = 0.1/150 I"„, = 0.45^130° RK = 0.32 F„, = 1.3 Use the computer program MICROAMP to design a single-stage low-noise amplifier using the MESFET described above. The design requirements are: power gain of 9 or more, noise figure equal to 1 4 6 or less, input VSWR no greater than 1.5. What noise figure is obtained if conjugate impedance matching is used? Answer; G;, = 9.196, F = 1.4586, input VSWR = 1.346, output VSWR = 1.166, i; = - 0 . 5 0 4 3 -j'0.3715. \\ = - 0 . 2 1 8 8 + . / 0 . 1 1 5 7 . For conjugate impedance matching F = 1.87. 10.9. Show that for a microwave amplifier the available power gam G a can be expressed as G„ = Mfip/M.,, where Gp is the power gain and A/, and M., are the input and output impedance-mismatch factors. By using relations of this type, show that the noise figure for a cascade connection of amplifier stages can be expressed as /•', - 1 F3- 1 10.10. Verify that the matching networks shown in Fig. 10.28 will transform the line characteristic admittance 1' « 0.02 S into the required source and load admittances needed for the amplifier discussed in Example 10.3. 10.11. Design a single-stage low-noise amplifier using an FET having the following parameters: S , , = 0.74Z - 115° S 1 2 = 0.14Z40" S2l=2.7z87° S,2 = 0.13Z - 6 0 ° r„, = 0.5^100' ^,,,= 1-3 R x - 0.24 The design specifications are: G. > 15. VSWR, < 2, VSWR 2 < 2, and F < 1.5- If you cannot meet the design specifications, relax one or more of the requirements. Answer: A design with G. = 18.95, F= 1.33. V'SWRj = 2.11. and VSWR 2 = 2.265 can be achieved using l\ = - 0 . 1 7 +./0.61. 10.12. The transistor whose parameters are given in Prob. 10.11 is to be used in a two-stage amplifier. The design calls for Gp > 300, F < 1.5. VSWR, < 1.5. VSWR:i < 1.5. Plot the load and source stability circles, the gp = 2.6 and 2.4 constant gain circles, and the F = F,n + 0.5 dB constant noise-figure circle. On this figure show your final design values for V. and \'L for the first stage and I" and V't for the second stage. Does your final design have a good stability margin? 10.13. A transistor has the following parameters: S„ = 0.5Z 160'- S , , = 0.06^50° S 2 , = 3.6/.60 0 S,., = 0.5Z -45° l„, = 0.4/. 145° Rs = 0.4 Fm<*> 1.6 Design a single-stage amplifier with the best possible noise figure subject to the constraints Gp > 10, VSWR, < 2. VSWR 2 < 2. 798 FOUNDATIONS FOR MICROWAVE ENGINEERING 1 0 . 1 4 . A t r a n s i s t o r with t h e Following p a r a m e t e r s is to be u s e d in a t tw amplifier: °-stage Su = 0.85L159° S12 = 0 . 0 6 ^ - 7 4 ° S 2 „ = 0.66,1 - 162° I m = 0 . 8 A - 160° S 2 1 = 1.8z - 3 4 ° F,„ = 1.74 RN = Q 3 T h e design objectives a r e : G ; , = 250, V S W R j < 2, VSWR., < 2, F < 2 2 F y o u r f i n a l design c o n s t r u c t t h e load a n d s o u r c e s t a b i l i t y circles t h e ° 6, 7 , 8 , 9 gain circles, a n d t h e F = F,„ + 0.5 dB a n d F = F„ + 1 dB c o n S ' a m noise-figure circles. On t h i s figure s h o w t h e design v a l u e s of Ts, r, a n d I" i" a t w h i c h you a r r i v e d . REFERENCES 1. Vendelin, G. D.. A, M. Pavio, and U. L. Rohde: "'Microwave Circuit Design Using Linear and Nonlinear Techniques," John Wiley & Sons. Inc.. New York, 1990. This is a very comprehensive text covering small-signal and low noise amplifier design, power amplifier design as well as oscillator and mixer design. 2. Pengelly, R. S.: "Microwave Field-Effect Transistors—Theory, Design, and Applications," Research Studies Press, Hertfordshire. England. 1986, distributed by John Wiley & Sons. Inc., New York. 3. Gonzalez, G.: "Microwave Transistor Amplifiers. Analysis and Design," Prentice-Hall, Inc.. Englewood Cliffs, N.J.. 1984. 4. Vendelin, G. D.: "Design of .Amplifiers and Oscillators by the S-Parameter Method," John Wiley & Sons, Inc., New York, 1982. 5. Ha. T. 'I'.: "Solid-State Microwave Amplifier Design." John Wiley & Sons, Inc., New York, 1981. 6. Gentile, C: "Microwave Amplifiers and Oscillators," McGraw-Hill Book Company, New York, 1987. 7. Carson, R.: "High Frequency Amplifiers," 2nd ed., John Wiley & Sons, Inc., New York, 1982. 8. Chang, K. (ed.): "Handbook of Microwave and Optical Components. Microwave and SolidState Components," vol. 2, John Wiley & Sons, Inc., New York, 1990. This is a very good reference source on semiconductor theory, device design and fabrication, device modeling, and applications, for microwave transistors and diodes. CHAPTER 11 PARAMETRIC AMPLIFIERS A parametric amplifier is an amplifier utilizing a nonlinear reactance, or a reactance that can be varied as a function of time by applying a suitable pump signal. The time variation of a reactive parameter can be used to produce amplification. This is the origin of the term parametric amplifier. The possibility of parametric amplification of signals was shown theoretically, as long ago as 1831, by Lord Rayleigh. The first analysis of the nonlinear capacitance was given by van der Ziel in 1948.f He pointed out that this device could also be useful as a low-noise amplifier since it was essentially a reactive device in which no thermal noise is generated. The first realization of a microwave parametric amplifier was made by Weiss, following the earlier proposal (1957) by Suhl, suggesting the use of the nonlinear effect in ferrites (Sec. 6.7). In the following few years, the semiconductor-diode (sometimes called a varactor, for variable reactance) parametric amplifier was developed through the combined efforts of many workers. At the present time the semiconductor junction diode is the most widely used parametric amplifier. For this reason we limit the discussion in this text to this particular type of parametric amplifier. The p-n junction diode has a nonlinear capacitance. If a pumping signal at frequency wr, and a small-amplitude signal at frequency «„ are applied simultaneously, the device behaves like a time-varying linear capacitance at the signal frequency a>.. As we show in later sections, a time-varying capacitance or a nonlinear tA. van der Ziel, On the Mixing Properties of Nonlinear Capacitances. J. Appl. Phys., vol. 19. pp. 999-1006, November. 1948. 799 800 FOUNDATIONS FOB MICROWAVE ENGINEERING capacitance can be incorporated into a circuit to give linear amphficat' a small-amplitude signal. Before presenting this analysis a brief desc-'°r ° f of some of the properties of junction diodes is given, followed by a p r e ' P "^ tion of the Manley-Rowe relations. The latter are a set of power-cons tion relations, of considerable value in determining the maximum gain & A other performance features of parametric amplifiers. 11.1 p-n J U N C T I O N D I O D E S The diodes used for parametric amplifiers consist of a junction of rc-type and p-type semiconductor material. An n-type semiconductor has an excess supply of electrons, which is why it is called n-, or negative-, type. An example of an n-type material is a pure semiconductor such as germanium with a small amount (about I part in 10°) of impurity doping with an element such as arsenic or antimony. Germanium has a valence of 4 whereas arsenic or antimony has a valence of 5. Thus, at each site in the germanium or host crystal where an arsenic or antimony atom replaces a germanium atom, four of the valence electrons are used up to form the bond, and this leaves one excess valence electron. The valence electrons left over are relatively free to move around in the crystal under the influence of applied electric fields and make the material a donor of electrons, or re-type. In p-, or positive-, type material, the impurity atoms are chosen to have a valence less than that of the host atoms. For example, gallium, with a valence of 3, may be used in a germanium crystal. When a gallium atom replaces a germanium atom, there are only three available valence electrons to form the required bond. A stable bond requires four valence electrons, and consequently, at each site where a gallium atom is located, a hole is created which can be filled by an electron that may be passing by. If an electron from some other bond moves over to fill the hole, the result is the creation of a new hole at some other point. The overall effect is as though the holes were positive carriers of electricity, i.e., equivalent positive electrons, that can move through the crystal. The holes do, in fact, behave as equivalent positive carriers, and thus p-type material can be considered a essentially the same as n-type material except that the signs of the charg carriers are opposite. . , Consider now a linearly graded junction of n-type and p-type maten , as shown in Fig. 11.1a. In the linearly graded junction, the n-type materi ^ changes gradually and in a linear fashion over to p-type materiai i distance d. This variation is obtained by gradually reducing the doping, ^ concentration of donor atoms, down to zero in the region x = d/2 do _^ x ~ 0 and then linearly increasing the concentration of acceptor atoT"_g is the region x = 0 to x = -d/2. If the number density of acceptor ato ^ N 0 and the density of donor atoms is Nd, the difference will vary u° PARAMETRIC AMPLIFIERS 80 1 IVa-A/d d/2 /jtype -d/z n type -d/2 d/2 * /7tyoe - * n type * ' • FIGURE 11.1 The linearly graded junction. V : with x across the junction; thus N-N, = kx (11.1) where k is a suitable constant. When there is a gradient in the impurity-concent ration densities, electrons will diffuse from a region of high concentration to one of low concentration. Holes will diffuse in a similar manner. Thus the electrons will diffuse into the p-type side of the junction and holes will diffuse into the n-type side. This diffusion continues until a space-charge distribution, together with a resultant electric field, is set up of sufficient strength to produce a force that is equal and opposite to the effective force created by the concentration gradients. When equilibrium has been reached, a small region, called the depletion region, substantially free of charge carriers, is produced at the junction. The space charge built up on either side of the depletion region, together with the electric field existing across the depletion region, constitutes an equivalent capacitor. If a reversed-bias voltage is applied across the junction, the electron distribution and hole distribution will be forced farther apart. This widening of the depletion layer results in a decrease in the junction capacitance. It is now apparent that if an ac pumping voltage is superimposed on the bias voltage, the equivalent junction capacitance can be varied as a function of time. In the graded junction the space-charge density will vary linearly across the junction so that a depletion layer completely free of carriers is not 802 FOUNDATIONS FOR MICROWAVE ENGINEERING produced. The effect of having the space charge vary linearly aero junction instead of being concentrated at x = ±d is, however mn h same. If we consider a linearly varying space-charge density ( p e r cross-sectional area) p = qx, where q is a suitable constant, Poiss^"' 1 equation gives d2<& p q 2 e e dx for the potential 4>. Integrating and using the boundary conditions that thp electric field, and hence d<P/dx, is zero for \x\ > d/2 and that 4> = Q t x = 0 from symmetry considerations, we get * = <7* A- d2 2e 3 4 (11-2) The potential difference across the junction is qd3 -iM-i- 127 Under equilibrium conditions this potential difference must equal the contact potential <t>c plus the negative applied bias voltage - V; thus qd^ <t> - V = (11-3) 12e The total stored charge per unit area is given by Q= [ qxdx = qd2 8 Eliminating d by means of (11.3) gives ^ ( c p , . - V) Q = ,2/3 (11.4) 8 Since the capacitance is a function of the voltage, it must be denned as the ratio of an incremental change in Q to incremental change in & c - *• * " U t the capacitance per unit cross-sectional area is l'/3 dQ c = d(%-V) = € (11.5) 12e(<&c - V) As seen from this equation, the junction capacitance C is nonlinear sine depends on the voltage V. If C is a linear element, Q = CV. In an abrup junction diode C is proportional to (<J>C - V ) _ 1 / 2 . PARAMETRIC AMPLIFIERS 803 If we denote <t>. - V by V0 and apply in addition a pumping voltage vp = Vp cos u)pt, the capacitance becomes a function of time: * \ , / 2 1r +vD c o s cw e (1L6) - lu^J ( ^ ^l We now have a nonlinear capacitance that is also a function of time. The capacitance is a periodic function of time, and can be represented by a Fourier series expansion of the form C(t)= £ cvmsnapt fl (11.7) 0 The coefficients are given by • (i C n = 2ir — ^ 12fV0 C„ - e 1/3 I --^— I \2eVj / 11 + -?• cos A J-w\ V0 - 1/3 cos nttdtt where 0 = oi,,/. To evaluate the coefficients would require a numerical procedure. However, we do not need to know the values of the C„ in order to analyze the general properties of a parametric amplifier. The important feature brought out in the foregoing analysis is that C(t) is a function of time that can be represented by a Fourier series involving all harmonics of the pumping frequency f.. It is important to note that the coefficients are not, in general, linear functions of the ac voltage V. or the voltage V0. Thus, since the junction capacitance C(l.) is a nonlinear capacitance, the principle of superposition does not hold for arbitrary ac signal amplitudes. Under small-signal conditions. A Taylor series expansion of C(l) about an operating point may be used and only the linear term in the signal amplitude retained. In this case superposition does hold. The situation here is no different from that in any other amplifying device since all are nonlinear for sufficiently large applied signals. In addition to the capacitance associated with the diode junction, there is a shunt conductance arising from the bulk resistance of the material in the depletion layer. This shunt conductance is proportional to the area, and since C is also proportional to the area, the ratio is independent of the cross-sectional area of the diode. The shunt, conductance of the depletion layer is small, and can often be neglected. Of more importance is the series resistance of the n- and p-type semiconductor material outside the depletion layer. When the p-n junction is encapsulated and connecting leads are put on, an additional shunt capacitance C p due to the cartridge and a series inductance arising from the leads are also present. The overall equivalent circuit is thus of the form shown in Fig. 11.2. Typical values of C p and L 5 are somewhat less than 1 pF and 1 nH, respectively, at microwave frequen- 804 FOUNDATIONS FOR MICROWAVE ENGINEERING I1* Equivalent circuit Encopsulofed diode FIGURE 11.2 Equivalent circuit of a parametric diode. cies. The junction capacitance C is also about 1 pP, and typical values of R are a few ohms. 11.2 MANLEY-ROWE RELATIONS Manley and Rowe have derived a set of power-conservation relations that are extremely useful in evaluating the performance which can be achieved from a parametric device consisting of a nonlinear reactance.t These relations are also derived below. The circuit considered by Manley and Rowe is shown in Pig. 11.3. It consists of resistive loads in series with ideal bandpass filters connected in shunt with a lossless nonlinear capacitance. Two sinusoidal signals at frequencies /", and f2 are applied. The nonlinear capacitance causes (requencies at the harmonics of /", and f 2 to be generated. Each bandpass filter is considered to pass only one harmonic component n/\ + mf2. The overall circuit thus isolates all the harmonics and dissipates their power in separate resistive loads. The Manley-Rowe relations establish two constraints governing the conversion of input power at the frequencies /"j and f 2 into power at other frequencies. Let the charge Q on C be a single-valued function of the ac voltage v = L>J + v2 = Vl cos toxt + V2 cos w2t applied across it. Thus Q = Q(v). We may expand Q in a Taylor series in u to obtain dQ 1 d2Q (11.8) Q = Q(0) + ~v + -—2v2 + dv 2 ov where all derivatives are evaluated at v = 0. Since all powers of v occur, i is clear that, because v = <Vj/2Xe J '" 1 ' + e~J">') + (V2/2Xe-''"«' + e~JW1']' "JJ charge Q will have frequencies at all harmonics of /", and f2- If currents • t j . M. Manley and H. E. Rowe, Some General Properties of Nonlinear Elements, P General Energy Relations, Proc. IRE, vol. 44, pp. 904-913, July, 1956. See also Proc. / » * • 47, pp. 2115-2116. December, 1959. ^ PARAMETRIC AMPLIFIERS f i*fi\ \'\-fi\ K'*2'? 805 \nf,+mf; C(/) ~)fy (~ )h F I G U R E 11.3 Circuit for illustration of the Manley-Rowe relalions. the various harmonics are permitted to pass through C, the voltage developed across C will also contain all possible harmonics. In this case Q is a function of all voltages present. However, the expansion (11.8) will still be valid, except that now the coefficients will have different values. Consequently, the general expansion of Q has the form Q= E E Q„. >a /(tt*>4 -"nai.^l The charge is a real function of time; so we must have Q „_,„ = order that the n,m and —n,~m terms will combine to form function of time with frequency n to, + m w2. The total voltage V can be expressed as a function V(Q) of the A similar Taylor series expansion of V{Q) then shows t h a t V expressed in a form similar to (11.9); thus v- E E vni ?j(nui[ -t-mio^tt (11.9) Q*m in a real charge. can be (11.10) n ~ - ot rt j = — x For V to be real, we must have V_„ „, = V*m. The current through C is the total rate of change of Q with time, and is given by / = _ dQ ~di y = E E V(™ l + ma>2)Qn„y<'""'""•"*" n = — -x in --x Y j i 1 ej(> *\+™<»2) (ii.il) n = —Gt m — —x where lnm = j ( n w , + m<o 2 )Q„ m . Since C is a pure reactive element, there can be no net power into or out of C. If we assume that w, and io2 are incommensurable, there will be no time-average power due to interacting harmonics. The average power at 806 FOUNDATIONS FOR MICROWAVE ENGINEERING the frequencies ±\nujl + mw,| is p = (V I* + V* I \ nm \ ' It m * n m 'nm'nml = ( V I* + V T* \ = P (11.12) since the time average of I »™c VI + V 'nm'-n-m T 1 I = V -n-mK T* + V* I 'nm1nm ' -n-m'nm ) T = V / * + T / "nm'nm "nm'nm ^ 7* ' -n-m*-n- Conservation of power can therefore be expressed as X X E E Pnm = o (ii.i3) n = — x m= —x since />„„, = /»_„ .„, from (11.12). To obtain the Maniey-Rowe relations, we multiply each term by (nco1 + mco2)/(niol + mui2) and split the sum into two parts; thus x x "iE nP x x E -——-— + co2 z E ._ p — = o(ii.i4) „ = - « „ , = -* " w , + mw 2 „__«,„,_ -^ ww, + mco2 We can now show that each double sum must vanish separately. We can replace each Inm/(nwl + mw2) by jQnm, and then P n m / ( r a « , + ma>2) becomes -JVamQ*m -jV_„_mQ*n_m and does not depend explicitly on OJ1 or u>2. For any choice of w t and o 2 , we can always adjust the network external to C so that the currents which result keep all the voltage amplitudes V nm unchanged. The Q„m are then also unchanged since they depend only on the Vnm. When this is done we see from (11.14) that it is possible to change w 1 and w 2 arbitrarily but keep the two double sums involving P„mAna>l+ma>i)= -jVnmQ*m - JV.„ „,«*„_„ unchanged. Consequently, (11.14) can hold for arbitrary at, and w2 only if E E ni0 n=-*m=-x l * x E E nP ^ ^ mc= 0 + °2 mP ""' -o ,^-xm = -x"co1 +ma)2 That is, the coefficients of w, and <o2 must vanish separately. The above two relations are the Manley-Rowe relations. They are usually written m somewhat different form, however. We may write the first sum as p» np » x —nP E= om=-oc «">, *-> n "-'nm + mco2 \ - V n = 0m=-x -n-m -nwl - mo>2 + Eby E-n and -m in the second ter where n and m E have been ;replaced PARAMETRIC AMPLIFIERS Since P _ n m 807 = Pnm, the two parts are equal; so we obtain " nP E -o (11.15Q) E E mPnm = 0 nco. + into* (11.156) L n«0 m= — =e tlOJy + 77! W-2 Similarly, we can obtain m=0n=—* These are the standard forms for the Manley-Rowe relations. The Man ley Rowe relations are general power-conservation relations, and do not depend on any specific circuit such as that in Fig. 11.3. This is apparent since no reference to an external circuit was made in the derivation. For an example of the application of the Manley-Rowe relations, consider a circuit similar to that in Fig. 11.3 with generators at frequencies fx and f2. Let all harmonics be open-circuited except /", + f2. Thus currents at the three frequencies fx, f2, and /", + f 2 are the only ones that can exist. The n = + 1, m = 0 and n = 0, m = +1 and n = m = ±1 terms in (11.15) are the only ones present. Thus we get 10 0), + !1 Pn 01, = 0 (11.16a) = 0 (11.1661 &>., Since power is supplied at the frequencies w, and w 5 we P 0 , positive. Therefore P u is negative, and power is nonlinear capacitor C at the frequency w, + w 2 . If ai, frequency and m 2 is the pump frequency, then «J 3 = co frequency. The maximum signal gain is -Pll P\<\ M l + ^2 '"I = (0i = 1 must have P I 0 and delivered from the is the input-signal + 01., is the output (11.17) as obtained from (11.16a). A parametric amplifier of this type is called an up-converter. Because of losses that are always present in a practical amplifier, the gain will be less than w 3 /a),. The Manley-Rowe relations give the maximum gain possible and hence provide a criterion by which a practical up-converter can be judged. P-3 LINEARIZED EQUATIONS FOR PARAMETRIC AMPLIFIERS Consider a linear capacitance C for which the charge-voltage relationship is Q = Cv. The current flowing through C is given by 808 FOUNDATIONS FOR MICROWAVE ENGINEERING If C is made a function of time, for example, a parallel-plate capacitor • a plate separation that is varied with time, the current will be given b d dC dv i = — (Cv) = v- , dr ' dt + C ~dt (U.18) If instead of a time-varying linear capacitance we have a nonlinea capacitance, where the charge Q is a nonlinear function Q(v) of the volta the current is given by 9Q dv i = dt dv dt If the voltage v is the sum of a pump voltage vp at frequency &> signal voltage vs at frequency tos and \vs\ <tc Iw^l, we can expand and Q(v) = Q(vp + vs) in a Taylor series about the point v s = 0. Thus we obtain 1 d2Q a Q(vp + v.)=Q(vp)+ -£ + ",'0 2 ^ " vt + ••• o.-O For fs sufficiently small, we can obtain satisfactory accuracy by retaining the first two terms only. The current is then given by i = dQ{vp) dt d IclQ dt \ dv (11.19a) o.-0 Let the quantity dQ/dv for vs = 0 be denoted by C(t), in which case we can write . i = dQ{v.) d dt dt [C(t)vs (11.196) ^ ^ ^ ^ ^ ^ ^ If we compare this result with (11.18), we see that the nonlinear capacitance behaves like a time-varying linear capacitance for signals with amplitudes that are small compared with the pump signal amplitude. The first term, dQ(vp)/dt, in (11.19) gives a current at the pumping frequency, and is not related to the signal current. If the pumping voltage is also small compared with the dc bias voltage in a junction diode, we can assume C(t) to have trie form [see (11.6)] C{t) = C 0 (1 + 2Mco8o» p 0 (11.20) since <V0 + V p cos » p O _ 1 / ' * V / 3 d " ( V 3 V 0 ) c o s u>pt] for V p « V0. The linearized equations (11.19) and (11.20) are the ones we shall use in t analysis of parametric amplifiers. The equivalent circuit of a p-n junction diode is illustrated in Fig•&• 11.2. For the purpose of analysis it is more convenient to use an equiv series circuit of the form shown in Fig. 11.4. The two circuits are equivale PARAMETRIC AMPLIFIERS •TCs I 809 F I G U R E 11.4 Equivalent series circuit for a junction diode. if we choose R. RPc2 (u>CCpRpf+(C c= Cpf + (<oCCpRpf + (C + Cpf (a>CRpfcp + C + C„ which makes the input impedance the same for both circuits at the frequency w. In most diodes the resistance R p is small compared with the cartridge reactance 1/ioC ; so coC.R « 1. In this case we find that C R. c + c„ R. C. c„ When these approximations are valid, the two circuits are equivalent, independently of frequency. This is a necessary requirement if the series circuit is to be useful for analysis purposes, since in a parametric amplifier currents and voltages at several different frequencies are simultaneously present. When C is a function of the voltage, Rs will also be voltage-dependent. The effect of a voltage-dependent resistance Rs will, for simplicity, be neglected, since it is not too important. In other words, we shall consider R K to be a constant resistance. PARAMETRIC UP-CONVERTER In the up-converter, a pump voltage at frequency <u and a signal at frequency w s are applied to the diode, and the output signal is taken at the higher frequency to, + cop. Mixing effects take place that give rise to all possible harmonics of w p and tat. However, in the up-converter, the circuit external to the diode is chosen so as to permit currents only at the signal frequency w s , the pump frequency top, and the output frequency <o0, which is chosen as the sum of the pump and signal frequencies, that is, at w 0 = «, + w„. There will, consequently, be a voltage across the diode at the 810 FOUNDATIONS FOR MICROWAVE ENGINEERING three possible frequencies. If we let the diode voltage be represented vs = Re(V>'™>') = i(V 5 e""-' + y*e~**) vp = Re(V p e>p') = | ( V , , e " V + ?*«"*•**) u a = Re(V0e-""»') = | ( V 0 e ^ « ' + V J * - * * ) we may generalize (11.19) and (11.20) to give dQ(vp) i = 1 d ~dT~* 2d7C«(1 + 2Mcos V) x( V,e>-' + Vfe •'-•' + V 0 e ^ ° ' + V?e-**>) (U.21) Let the current at the frequencies cos and w 0 be expressed as i.~k(l.e*"'+l*e-^') (11.22) i* <*${!***+ !$*-**) (11.23) When the time derivative of (11.21) is taken and the terms at frequencies w and ± io0 only are retained, we obtain (a knowledge of the pump current at frequency w,, will not be required; so we do not need to evaluate it) *• Co + ***• -fU"?.*"* +jw0MVse'"'«' -JueV*e-JO'' + jto0V0e*<* -ja>sMV£e~-""*' -j<o0V*e-->»°' ~ju0MV*e_-"u°' + j^.MV^"'1) Using (11.22) and (11.23) gives h - j*&y, + MA ^ o /o=J^oC0Vo+>oCoMys (11-2*0) (11.246) These two equations show that, for the input signal current i s and output signal current iQ, the junction capacitance may be represented by an admittance matrix such that I: jwsC0 j(osC„M I ju>aC0M jo)0C0 (11.24c) V0 The parameter M is proportional to the pump voltage and gives the coupling between the voltages at the two frequencies u>a and o>0. Figure 11.5 illustrates an equivalent-circuit model of an up-converter. The series tuned circuits are chosen so that the three circuit loops hav resonant frequencies of <os, o>0, and wp and only currents with these respective frequencies can exist in each loop. Thus, in the input circuit 1 P< only /„ is present. The three circuit loops are coupled together through time-varying part C of C s only. Therefore, for the two frequencies w 5 an o)0, the equivalent circuit can be reduced to that illustrated in Fig. ll-obox labeled C(t) in this circuit is an equivalent impedance network maintains the relationship given by (11.24) between the terminal curre PARAMETRIC AMPLIFIERS 811 FIGURE 11.5 Equivalent-circuit model of an up-converter. R, R, LS'L, Cf+Ei Cg+fj, t2+tt R, Jus't Ke FIGURE 11.6 Reduced equivalent circuit for an up-converter. and voltages. The analysis of the parametric amplifier is a conventional network-analysis problem since the diode has been replaced by an equivalent linear two-port network with terminal relations described by (11.24 ).t Each loop in the circuit is assumed to provide a very high impedance to currents at all frequencies present except the resonant frequency for that loop. The resonant circuits in Fig. 11.5 have been assumed to have zero loss. Circuit losses can be considered included in R p and R,. At the end of the analysis, R g and R L can then be split into two parts so as to separate the circuit losses from the generator and load impedances. In practice, circuit losses are small compared with the loss arising from the diode resistance R y and the external loading represented by R e and RL. We may solve (11.24c) for Vv and V0 in terms of la and 70 to obtain V. -M/jco0C0 2 1 - M 11.25) tCare must be exercised in the analysis since currents and voltages at several different frequencies are simultaneously present. 812 F( lUNDAHONS FOR MICROWAVE ENGINEERING For the input circuit we may now write ^ = I, ft + /?, +j*Ji La + L,) + = & ft,+ / ? „ + > , ( L. + LJ + + v„ yWs(c„ + c,) i yw^Cp + Cj) + yWs(i-M2)c y«,(l(l-M2)C0 whereas for the output circuit 0 = /, ** + *. + M ( £ . + £*) + -jm {€ 6 + C2) p + Vn 1 = /, > o ( C ^ + C2) + jw0(l-M2)CD M7. jo.s(l-M2)Cu Let us now assume that the circuits are tuned so that the following conditions hold: 0>i = i oil0 = + C p + C, 1 1 L.+U (1-A/2)C0 i- (1-M2)C0 Cp+C2 We then obtain y^/o w0(l - M 2 ) C 0 0 = 70(ft,, + ft,) jM7 s «,(l-Af2)Co We may solve for 7 0 to obtain -JV.MwoC^l h= 2 - M2) M + (Rg + ftj( ft,. + ft,)» 0 *»,(l - ^ (11.26) 2 ) Co The maximum available input power from the generator is and the output power developed in ft,, is |!7 0 | 2 ft,.. The midband transdu PARAMETRIC AMPLIFIERS 813 power gain is thus given by G0 = 4|/0|2fit/? Vf ARLRgM2 a . 2 ( l - M T C 2 (Ra + Rt){Rh + Rs) + M'' <o0cos(l - M2VC* (11.27) when (11.26) is used to express I 0 in terms of V,. If desired, circuit losses can be included at this point by replacing R g and R L in the denominator in (11.27) by Rg + Ru and R, + R2/, where Ru and R2I represent the loss resistance in the input and output circuits. For simplicity we shall take To achieve maximum gain requires adjustment of R and RL. Since R K and R, occur symmetrically in the expression for Gn, the optimum values of R L and R a are equal. Hence we need to maximize ARIM' G0 = <o*,(l-M*)~C* M2 (RL + R,y + l2 Equating dG0/dRL to zero and solving for RL give 1/2 M2 RL = R* I + jW,J? 2 (l-M2)Cu2 (11.28) The effective Q of the diode may be defined as 1 i?.,^(l (11.29) -M*)Ca We then find that 1/2 RL = R, -{MQ)' (11.30) and the maximum gain is G0 = m (11.31) * (i + v T T T ) where 5 = (w s /o> 0 XMQ) . According to the Manley-Rowe relations discussed in Sec. 11.2, the maximum gain of an up-converter is w0/o>„. The quantity 8/(1 + fl + <5 )2 may therefore be regarded as a gain-degradation 814 FOUNDATIONS FOR MICROWAVE ENGINEERING L,*L, <V<T, CZ*CP F I G U R E 11.7 Equivalent circuit for a negative-resistance parametric amplifier. factor. As the diode Q approaches infinity, that is, as R s goes to zero, 8 approaches infinity, and the gain-degradation factor becomes equal to unity. Hence, for a lossless diode, the gain becomes equal to « 0 /a> s , as predicted by the Manley-Rowe relations. In a typical microwave diode, MQ could be equal to 10. If o>0/ws = 10 also, the maximum gain as given by (11.31) is 7.3 dB. To achieve high gain with an up-converter requires a large ratio <w0/to of output-to-input frequency. At the higher microwave frequencies this is not a very practical requirement, and for this reason up-converters are usually restricted to operation at signal frequencies f s below 1,000 MHz. Higher gain can be obtained from the negative-resistance parametric amplifier, which is discussed in the next section. 11.5 NEGATIVE-RESISTANCE PARAMETRIC AMPLIFIER In the negative-resistance parametric amplifier currents are permitted to exist at the signal frequency cos, the pump frequency cop, and the idler frequency w, = <op - w s . The equivalent-circuit model that will be analyzed is shown in Pig. 11.7. . J When we replace the voltage <;0 in (11.21) by v, = {(V^' + Vt*e J > and introduce the idler current i, = \(l,e^' + / , * e - " " ' ) , we may solve for / s and /, in terms of V s and V, in the same manner t h a t was used to treat the up-converter. It is readily found that, for w, = w p - w s , jcOiCoM ju.^o (11.32a) V* -Jf^o CnM and M V, V* h Jo>iCQ 1-M2 -M -1 j<osC,o 7* (11.32*) PARAMETRIC AMPLIFIERS 815 For the circuit of Fig. 11.7, we can write the following equations: 1 R, 1W, + Rs+jws(Ll +L,) 1 C,) + >x(C. + ju,,{l-M2)C0 M ! 11.33a) + (1-M2)>,C ' 0 V2 = 7, R2 + Rs+j<oi(L2 + Ls) >,(1-M2)C0 Jo>,(C„ + C2) M (1-M2)>SC0 (11.336) •I? If we impose the tuning conditions «! = + (l-Af2)C0 L9 + L. Cp + C2 + (1-M*)C0 C'. + C, 1 ' we obtain V, = ( * , + « „ ) / , V2 = (R2 + RS)I,- (11.34a) a>s(l-M2)C0 9 (11.346) Let us now assume that V 2 = 0. We can then determine /, from (11.34) and evaluate the gain G 0 = 4i? 1 /? 2 |/,| 2 /V, 2 . We readily find that 4Z?,fl 2 M 2 On = M2 (R2 R.+R.- + Rs)-<Q(l-Mi) (K2 + / * „ K « . ( i - J n cs (11.35) The term -M2/[(R2 + BJwta/£ - M 2 ) 2 C 0 2 ] may be interpreted as an equivalent negative resistance —/?. Introducing i?, we may express G„ as G0 = 4/?,rt2[tt,(l -M"2)C0/f]V Mz(fl, +RS (11.36) -R)' It is clear that a very large gain can be obtained if R is made almost equal to i ? ! + Rs. However, care must be taken not to make 7? too close to R l + R s because a small change in parameters will then cause large changes 816 FOUNDATIONS FOR MICROWAVE ENGINEERING in the gain and will cause oscillations to occur if R becomes qual to R + Rx. The parametric amplifier discussed above is called a negative tance converter. It is possible to take the output at the same frequen^'^ as the input. If we split R } into a generator internal resistance R D i to * 3 load resistance RL, the power delivered to R, is 5-RJJJ 2 . The transd power gain will be G0 = We may evaluate I. from (11.34) to obtain 4RRL G0 = (Rg + R^ + Rx-Ry (11.37) where, as before, R (R.z + Rs)<0,<os(l~M2)2C* 11.38) The effective negative resistance -R arises in the following manner: The application of signal plus pump power to the nonlinear capacitance causes frequency mixing to occur. When current is permitted to exist at the idler frequency u>p - w s , further frequency mixing of power at the pump and idler frequencies occurs. This latter mixing causes harmonics of o»p and <!§, = (op - OJS to be generated; in particular, power at the frequency <w5 is generated. When the power generated through frequency mixing exceeds that being supplied at the signal frequency ws, the diode appears to have a negative resistance. If idler current is not permitted to exist, the negative resistance vanishes, as reference to (11.38) shows when R% is made infinite (open circuit for the idler signal). The negative-resistance parametric amplifier with input and output a the same frequency is not very stable. The reason is that in a microwave system R g and R L are the impedances seen looking into the input an output transmission-line ports. If the loads connected to these transmission lines are not matched, reflected waves occur. Reflected waves in the outp fine return to the amplifier and are amplified and fed into both the inp and output lines. The result is t h a t the gain becomes a sensitive function the external generator and load impedances. The stability of the a m P t L g greatly improved by the use of a circulator, as illustrated in Fig- H-°use of a circulator makes the load termination R L for the amplifier eq the characteristic impedance of the transmission line independently o external generator or load impedances Z g andZL. The available power the generator is still given by ~V?/4Rg. However, the amplifier p°m £1 PARAMKTRIC AMPLIFIERS 817 f, ' - ? f-' t-s*Ly Rs Cp* Circulotor Matched load ffa=4 F I G U R E 11.8 A negative-resistance parametric amplifier using a circulator. now all delivered to the load RL = Zc, and none of it is dissipated in the internal generator impedance Rg. Consequently, the power gain is nearly four times greater since I s is nearly twice as large, because the series resistance in the input circuit is now RL + R„. instead of RL + Rg + Rs 2RL, when RL = Rg » Rs. The power gain is the square of the voltage reflection coefficient, and is given by G0 = Z,n - Z. (R„-R-RLy Z-m + Zr (R, + Rt~R? since Zm = Rs - R at resonance and we are taking RL = Zc. For high gain, R ~ RL + Rs and is large compared with Ra. Consequently, the gain can be expressed as 4*1 G0 = (11.39) (RL+RS-R)' The maximum value of negative resistance that can be obtained is determined by the diode that is used. If we make R2 = 0 in (11.38), we see that maximum R is R„, where Rm= -(MQVR. (11.40) to, (11.29) having been used to introduce the diode Q. If we regard R as fixed, we see that, for the amplifier without a circulator, we must make R g + RL + Rs = R, as (11.37) shows. But with Rg = RL = Z,. large compared with Rs, we get Rg = RL~ R/2, and (11.37) gives R' G0 = (2RL+R,-Ry (11.41a) 818 FOUNDATIONS FOR MICROWAVE ENGINEERING Pump c a v i t y / ^-Tuni Tuning scrj screws^ Coupling h o l e ^ Stgnal input 'Idler cavity Signal output FIGURE 11.9 A microwave negative-resistance parametric amplifier. whereas for the amplifier with a circulator, R, = R and 4R2 G " = <«,.,*,-«,* <11 -41i) Note that when a circulator is used, the optimum value of R L is twice what it is when a circulator is not used. Thus the denominators in (11.41a) and (11.416) are the same. These relations then show that the use of a circulator gives 6 dB more gain for the same amount of diode loading, and hence will have a gain-bandwidth product twice as great. If the pump frequency iop is chosen equal to twice the signal frequency w s , the idler frequency a>i = o>p - ws = tos is equal to the signal frequency. In this case the amplifier is called a degenerate negative-resistance amplifier. For the degenerate amplifier the signal and idler circuits would be a single resonant circuit. The analysis of the degenerate amplifier is similar to that already carried out, and so will not be presented here (some results on noise properties are given in Sec. 11.6). There are many different ways of building microwave parametric amplifiers. Transmission lines, waveguides, or a combination of the two may be used to construct suitable cavities to use as resonant circuits. A tyP'^ microwave negative-resistance amplifier is illustrated in Fig. 11.9.1 pump and idler cavities are formed in an X-band rectangular wavegu The signal cavity is a coaxial-transmission-line cavity. The varactor diode is mounted in the center of an inductive diaphragm located between the Pu™^ and idler cavities. Coupling to the signal cavity is achieved by h a v i n g T h e diode terminate in the center conductor of the coaxial-line signal cavity. pump is coupled to its cavity by an aperture. The pump frequency is c o ^ around 9,200 MHz, and the idler frequency is 7,900 MHz. The mpu 1961tW. O. Troetschel and H. J. Heuer. A Parametric Amplifier for 1296 Mc. QST. January, PARAMETRIC AMPLIFIERS 819 output signal frequencies are the same and equal to 1,300 MHz. This amplifier gives a gain of 25 dB or more with a bandwidth of about 5 MHz. The bandwidth over which high gain can be obtained in a negativeresistance amplifier of the type discussed above is small. The negative resistance has the effect of increasing the loaded Q, which results in a high-Q resonant circuit with a narrow bandwidth. To analyze the bandwidth properties, we shall assume that the circuit model given in Fig. 11.7 is valid. For high-Q circuits the impedance of the signal and idler circuits may be expressed in the form Z s = (Ri + R.M + 2 j ^ Q l \ (11.42a) Z, =(R2 + Rjll + 2j^Qi) (11.426) where <?!=» : Li + Lt R,+R, L2 + L, "* •R> + Rs The derivation of these expressions is as follows: From (11.33a) the impedance of the signal circuit at a frequency ws + Aw, is Zs = Rl+Rs+j(<os + Acos) ( * - i + £ . ) (a>s + A W s ) 2 l C „ = /?, +Rs+j(u>s + A<os)(Ll + LS + C, C0-M2C0 ml 1 - > * + A».) = fl, + R, +j- , ' (2w s Aw, + Awf) coc + AOJK Aw, = (#, +R„)\l+j2Ql A similar derivation holds for Z,. From (11.33) we obtain V, =IJR. +Rs)(i V2 = I,(R2 + RS) \l + + 2jQl^\ <os I 2jQ2—L + <o, / Mlf (1-M')j(w, + Aw,)C 0 MI! 2 ( l - M ) 7 ( w s + Aw s )C Q (11.43a) (11.436] 820 FOUNDATIONS FOB MICROWAVE ENGINEERING If i?j = R g + R L and the output is taken at the frequency «>,., the e i gain is given by G = 4ftgflJ/s V,2 When we solve (11.43) for / s , we find that (note that V 2 = 0 in this case) G = 2 AT Z. - (H.44) 7 * », + Aw,)(w s + A w s ) ( l - M 2 ) C022Z; Since the pump frequency is fixed, Aw, = - A w s . The midband gain G is given by (11.37). To determine the bandwidth, we equate G given by (11.44) to G 0 / 2 and solve for Aw s . When the gain G 0 is high, we find that, to a good approximation, Aw.. \ (Rt+R.-RY (J? I + fl,) 2 (Q 1 +«,Q 2 /«0 2 (11.45) Thus the gain-bandwidth product becomes Aw, . G ^yft^ft,, ° (R1+fts)(Q1 + Q 2 w s /w,.) (11.46) If we assume as a typical case Rg + RL = ft, + fts and Rg = RL and note that, for high gain, ft, + Rs = R, we get 2,/G;0 Aw, w.s Qi + ( w s / w , ) Q z The smallest possible value of Q, occurs if C + C, equals zero. In this case <?> = «.(^i + ^.) ft, + R„ w s ( l - M 2 ) C 0 ( f t , + ft J '" » , ( 1 - M 2 )C 0 ft Similarly, the smallest possible value of Q 2 is obtained if C 2 + ^P " and is ea = w , ( l - M 2 ) C 0 ( f t 2 + ft s ) If we refer to (11.38) for ft, we now find that M2Q}Q2 = 1- Thus 2/Go Aw„ 1 ws Q, + w s / M 2 w , Q , PAKAMKTBK" AMPLIFIERS 821 This expression has a maximum value for 1 Hence the maximum gain-bandwidth product is (U.47) V s / max For 20-dB gain we obtain a bandwidth of 2 Aa»s ws Af = rWj 2oy «T Usually M is no greater than about 0.2, and consequently the fractional bandwidth in percent is approximately equal to Jiat/ws. In a practical amplifier the gain-bandwidth product will be less since the capacitances C l + C p and C 2 + C p cannot be reduced to zero. The parametric amplifier may, of course, be broadbanded by using broadband circuits at the signal and idler frequencies. An alternative scheme for obtaining broadband operation is the traveling-wave parametric amplifier, where resonant circuits are avoided entirely. In the traveling-wave amplifier a waveguiding system is loaded periodically with varactor diodes. With the application of pump power and signal power, mixing occurs, with resultant power generation at the signal frequency. For a detailed analysis the references cited at the end of this chapter may be consulted. 16 NOISE PROPERTIES OF PARAMETRIC AMPLIFIERS The noise produced by parametric amplifiers is the thermal noise in the resistances in the equivalent circuit. For the up-converter illustrated in Fig. 11.6, the input thermal noise at frequency w, is that arising from the generator resistance R In analyzing the noise power in a circuit, it is useful to consider noise as consisting of a spectrum of noise waveforms with an effective root-mean-square (rms) voltage e(o>) equal to the square root of the noise power spectral density. Thus, for the generator resistance Rg, the mean-square noise voltage across Rg is e\ = 4k.TRg A f. If we neglect circuit losses compared with the diode resistance R s (this is a reasonably good approximation), the only other noise originating in the signal circuit is that generated in R s across which a mean-square noise voltage e\ = 4kTsRs A f at frequency toa exists. We denote the equivalent noise temperature of the diode at frequency <us byTs. The noise at frequency w s in the signal circuit is amplified and converted to noise at the output frequency o>0. 822 FOUNDATIONS FOR MICROWAVE ENGINEERING The amplified output noise at midband is given by (replace Vj by J72~ V in Fig. 11.6) ' + RL(e2 Pi = U i l X = + e22)MW0C*(l M2)' - [ M 2 + ( R g + RS)(RL + Rs)a>0cos{l - M 2 ) 2 C * | where the noise current /, is obtained from (11.26), with V, replaced h (e2 + e 2 ) 1 / z . If we introduce the midband gain G 0 , we obtain [see (11 27)1 e2 + e? kAf(RJ 'G0 = u **. ~ R8 + RsTs) There is also noise generated in the output circuit the resistance Rs. If the diode noise temperature at the an equivalent noise voltage ej = 4kT0RsAf appears output circuit. When we solve for the noise current compute P 2 = \I2\2RL, we obtain w*(l-M2fc*(Rs + n G, (11.48) at frequency u> in frequency <o0 is T across Rs in the I 2 that results and R„)2G0el 4:RgM2 co2(l - M 2)C2{RS EM + RjkT0Rs A fG0 2 (11.49) The total output noise power at frequency w 0 is P n = P, + P 2 - The available input noise power from R is kT A f. Hence the noise figure is given by (T = 290° in the definition for F) Pi+P-z " G0kTAf «>2{l-M2)C2(Rs+Rg)RsT0 R,T, + RgT + " } 2 RgM T Note that the gain G 0 was defined as output power divided by the availabli power from the input signal source, so that F as evaluated above conforms to the accepted definition of noise figure. If we take the diode noise temperature at the two frequencies w s and w 0 equal to Td, we obtain (Rs + Rs) R, Td F = \ + — — 1 + M2Q2R'i Rg T after introducing the diode Q from (11.29). For maximum gain, ,'/2 Z R = RL = Rs 1 + — M r2n2 Q (11.5D PARAMETRIC AMPLIFIERS 823 from (11.30). Thus the noise figure under maximum-gain conditions is given by F=l+ -£ = i+ T,! '<i OJ, . Q2 1 + — M2Q Y(1+<^M2Q2) \-V» l + (i [iMV^ff| + 2 M Q2 (11 52) - As a typical example consider Td = T, to0 = 10w„, and MQ = 10. We then find that F = 1.36, or 1.3 dB. This example clearly demonstrates the low-noise property of the parametric amplifier. The diode noise temperature T d is somewhat greater than the ambient temperature T because of shot noise that arises from random motion of the carriers across the junction. In addition, the resistance R s is usually not exactly the same at the two frequencies. However, this difference can be taken into account by choosing appropriate values of T0 and T,.. The thermal noise arising from circuit losses can be included in (11.50) very simply by replacing RSTS by RUT + RSTS and RST0 by R2/T + RBT0, where R u and R.2t represent the equivalent-circuit resistances. However, since the effective Q of most diodes, that is, [(1 - M'2)cosC,iRs]~\ would rarely exceed 50 whereas circuit Q's would normally be 1,000 or more, the resistances R u and R.,, are negligible compared with the diode resistance Rx. The noise properties of negative-resistance parametric amplifiers have been analyzed and measured by Uenoharat and others. The noise theory for a negative-resistance parametric amplifier employing a circulator, and with the output signal taken at the same frequency cos as the input, is presented below. With reference to Fig. 11.10, the following sources of noise are the main ones that need to be considered: (1) input noise from R g = Z c at temperature T x and frequency « , ; (2) input noise at the idler frequency w, = o>p- a»4 that arises in i? 2 ; (3) noise arising in the diode resistance Rs at frequency <MS (equivalent noise temperature T s ); and (4) noise arising in R s at the idler frequency w, (equivalent noise temperature T,). It is important to consider noise sources at the idler frequency because these noise signals are converted into noise at the frequency o>8 by frequency mixing that takes place in the diode. The equivalent circuit and noise voltage sources are shown in Fig. 11.10. tM. Uenohara, Noise Considerations of the Variable Capacitance Parametric Amplifier, Proc. IRE, vol. 48, pp. 169-179, February, i960. 824 FOUNDATIONS FOR MICROWAVE ENGINEERING r"v,An •-? H e\=HkTs hf R s J ffs LS*LX Cp+C> Ls,L2 CP*C2 R «^n .2 = 4*^- A / ff, e\ = Atrff2A/ /?,= /•: *f = 4 */",#„ A' FIGURE 11.10 Noise circuit for a negative-resistance parametric amplifier. The noise-power input from R g is kTx A f. This will appear at the output load R L as amplified noise of amount 4RjkTtAf Px = GakTxtS.f = (RL (11.53) + RS-R? where G 0 was obtained from (11.39) for the high-gain case. The noise contributed by the amplifier is represented by the voltage sources e2, e 3 , and e 4 . Since the three noise sources are uncorreiated, the noise powers add. Hence the effective noise voltage in the idler circuit is e = i ije3 + e l • The equations describing the noise currents /, and I 2 at midband are obtained from (11.34). Thus jMI* e2 = (RL + Rs)I1- e, = (R2 + Rs)I2- 0,,;(1 (11.54a) ~M2)C0 mt (11.54b) 2 Ws(l-M )C0 since the loading R x of the signal circuit as seen by the source e 2 is Z c = RL. When we solve (11.54) for I v we can evaluate the noise power P2 = \h?Ri delivered to the external load RL. Giving the final results, we find that _ II ~ e2 RL+RS-R JMe* + o>,(l 2 - M )C0(RL + Rs - R)(R2 + *•> We must find the noise power contributed by e 2 and e, separately, s PARAMETRIC AMPLIFIERS 825 noise voltages do not add. From e 2 we obtain P (Ke , )2 = ' (RL + RS-R)2 and from e, we obtain e?R,M2 P(e.) = [<„,(1 - M*)C0( R L + R, - fl)( H 2 + 8 , ) ] efMLMma *>,(/?*,+ fl.-AH K2 + «J Hence P2 = P ( e 2 ) + P ( e , ) = G0k±f (11.55) R.T. RL If we assume that Ta = T, = T = Td, we obtain the simplified expression P2 = RK ( G0k*fTd—\l + cos R -- 11.56) The corresponding noise figure F is given by R F = kT^fG0 = 1 TlR1 (11.57) 10, where T x must be taken to be 290 K for the standard definition of F. In (11.57) R may be replaced by R, + Rs since this is the requirement for high gain. "We then obtain F= 1 + T,(wpRL d 1 - + Ra 1 (11.58) RL As an example, assume Td = Tt, choose wp = 1.5&),-, and let / ? t be much greater than jRg. We then obtain a noise figure of 1.5, or 1.76 dB. The foregoing analysis is valid only when the two frequencies w s and w, are spaced by an amount greater than the passband of the signal and idler circuits. When this is not the case, a»8 ~ to,, and the amplifier is a degenerate negative-resistance amplifier. Noise current from the input source resistance R e at the frequency coi will now exist in the signal circuit. Similarly, noise currents at the frequencies w, and w s arising from R s will be present in both the signal and idler circuits. In the degenerate amplifier the signal and idler cavities are identical and the equivalent circuit is that shown in Fig. 11.11. If the amplifier passband is symmetrical about 0 ^ / 2 , 826 fXIUNOATIONS FOR MICRO WAVE ENGINEERING £,+£, &*G ffs Clt) e\s=Ai'raff, A/, ^e\i = ^*rd/ft Ar e*s=4*r,/tf&f, \^=Akr,ng Lft FIGURE 11.11 Noise circuit for a degenerate parametric amplifier. noise at the frequency w, < o> p /2 is amplified to give noise output at the frequency ots, and also converted and amplified to give noise-power output at the frequency o>, = o>p — u>s. The two frequencies u)s and iot are symmetrically located about (Dp/2 and fall within the amplifier passband. Therefore we must consider the noise from R g and R s at the two frequencies u s and a>r as separate uncorrelated noise in evaluating the total output noise power. The present situation is illustrated schematically in Fig. 11.12. The passband is split into two equal parts A /", and A f2, located symmetrically on either side of cop/2. Noise in the band A /', is amplified to give noise in the same band A f l at the output. In addition, noise in the band A /\ is amplified and converted into noise in the band A f,2 at the output. The lower sideband A /", will be regarded as the signal sideband. Since w s = «,, the noise power in the band A /", arising from R g and R s produces the same amount of noise power in the output bands A /\ and A f.2 as does the noise power arising from Rg and Rs in the band A/ 2 . Hence we need to consider only the noise in the lower band A /\ in detail. The noise from R g in the band A f\ is represented by an equivalent voltage source eu in Fig. 11.11. To evaluate the noise power delivered to Ri from e l s , we shall make the approximation of taking the series impedance o the circuit as R 5 + R L throughout the passband. We denote the noise currents at frequencies o>a and co, by /, and 7 2 . Input _ Output F I G U R E 11.12 Illustration of noise conversion. PARAMETRIC-AMPLIFIERS 827 At the signal frequency tts the available gain is G 0 = |r| 2 . Under high-gain conditions R L + R s = R\ so (11.39) may be used for G 0 . The available input noise power from eu is e\t/lRg = e2u/4RL, since we assume that R = RL = Zc. The noise power Pu delivered to R, in the band A /", is thus When there is no impressed voltage at the frequency u>t, the currents /, and / a are related by [see (11.34) with R 2 = RL] jMI? (R, +R )I.,= -j— The power P h delivered to i?, in the band A/"2 is given by I// R P„ « \I./RL = -fpPu = Pu (11.60) w Uil / . + Hs since fl = M 2 / [ w 2 ( l - M 2 ) 2 C 0 2 (ft,. + H g )] when w, = OJ,. The source e,, delivers an amount of power equal to Pls to i? L in the band A /'2 and an amount equal to P u into the band A fx. Hence the total noise power delivered to RL from R^ is the same in the two bands A /", and A f2, and is given by Pi = P u + Pu = G o ^ A A l l + R R L + R ) = 2G„AT, A/", = G , , ^ , A/" (11.61) since R =* RL + Rs and 2 A /", = A /". To evaluate the noise power delivered to R L from Rs, we consider the noise in the band A/', first. The equivalent voltage source is e 2 s , and the circuit equations are (we can put w i = w s ) fc -(Ji A +*,)/, - , t1 W,( jMIj „„,„ 1-M2)C0 (11.62o) jffl? 0 = (*A+*,)/,- (11-626) 2 «J.,(1 - M )C0 The output noise in the two bands A f x and A f 2 is given by p 2s = \ifRL p2l = uj»jrx When we solve for 7, and I2, we obtain p * = <*t+ *.-*>* - ^ - « ; ( 1 1 6 3 ) 828 FOUNDATIONS FOR MICROWAVE ENGINEERING The output noise from e 2 s in the band A f 2 is given by R ~Pg* = 2i The noise source e 2 , contributes a noise power P 2 s in the band \ r and P2l in the band A /",. Hence the total noise power delivered to 7? fr the internal amplifier noise sources is (in each band A f l and A f ) 2kTdAf\RsG0 Pz = P* + P2l = kT„Rfiatif RL RL ; n.65) The total output noise from both R g and R s is P„=Pi +P2 = kAfG0 T, + J?. -iT, (11.66) If the degenerate amplifier is used as a single-sideband amplifier (signal input ixi the lower band A /", only), the single-sideband noise figure F S S B is defined by the ratio of the total output noise power in the band A f = 2 A fi divided by the available input noise power in the signal band A /",. Thus ^SSB _ kAf1GaTl 1 + = 2 1 PL\ TyR•L5 - (H-67) It is seen that, for single-sideband operation, the noise figure cannot be less than 2, or 3 dB. The signal-to-noise power ratio in the signal band A f Y at the input is PJik A /", 7\). The signal-to-noise power ratio in the band A /", at the output, given by G0Ps/Pn, is worse by a factor equal to the singlesideband noise figure F S S B . ' ^ n e n ° i s e degradation is due to noise entering in the idler band A f2, in which no signal is present. For double-sideband operation input signal power is present in hot) bands A f x and A f2. In this case the available input noise power is taken to be that in the band A f. Hence the double-sideband noise figure F D S B 1S kAfGQTx DSB kAfGJ, 1 + T, R, + Td f^± R T, R (11.68) and is a factor of 2 (3 dB) better than for the single-sideband case. The double-sideband noise figure has been measured by Uenohara' a number of different diodes. It is found that the theory given ab°^g[Il reasonably accurate. Typical noise figures that were measured ranged 0.9 to 4.5 dB. For diodes with wsC0Rs less than 0.1, the noise fig ur *J w ^ ge dB or better. There was a strong correlation between the measured i* figure and the diode quality factor Q. This is predicted by the theory as » ilbid. PARAMETRIC AMPLIFIERS Since R L + R 829 = R for h i g h gain a n d R can be expressed by from (11.29) a n d (11.38), we see t h a t t h e o p t i m u m value of R L + R s is MQRX. T h e factor R^/RL in t h e expression for noise figure m a y n o w be replaced by (MQ - 1 ) " ' to give ^DSB - 1 + T, MQ - 1 s h o w i n g t h a t t h e noise figure i m p r o v e s w i t h diode quality factor Q. P a r a m e t r i c amplifier noise is p r i m a r i l y t h e r m a l noise in t h e diode r e s i s t a n c e Rs. T h e e q u i v a l e n t amplifier noise t e m p e r a t u r e is given by T A = (F — l)Tlt w h e r e T l = 290 K [for t h e d e g e n e r a t e amplifier w i t h single-sideband o p e r a t i o n T A = (F - 2 ) T , ] . By cooling t h e amplifier to liquidn i t r o g e n t e m p e r a t u r e , noise t e m p e r a t u r e s below 100 K have been o b t a i n e d . 1 1 . 1 . Consider a square-law mixer for which the output, current Ut) = k[v(t)]2, where v(t) is the applied voltage and k is a constant. Let a local-oscillator signal Vp cos iopt and a signal Vs cos coj be applied, with Vs « Vp. Show that the output current at the sum or difference frequencies to ± m.x is a linear function of Vs when Vs <K V . Thus the square-law mixer is a linear converter for small-signal amplitudes. 11.2. Consider a parallel-plate capacitor with capacitance Cn. Let a voltage V = Vs cos a> J be applied. At time / = 0 the plate separation is suddenly increased to change the capacitance from C0 to C = C„ - AC. Since the charge cannot change instantaneously, the voltage must increase. At time t = {4fs) '. when V = 0 and Q = 0, let the plate separation be brought back to its original value. There is no change in V produced since V = 0 at this time. When t = ( 2 / 9 > - 1 , let the capacitance be suddenly decreased to a value C again. When this process is continued, the resultant voltage across C is amplified and will have the waveform illustrated. This is an example of a linear capacitance varied at a rate twice that of the signal frequency. Evaluate the incremental change in voltage that occurs every half cycle and the power supplied by the pump. To evaluate the latter, determine the change in stored energy that occurs every time C is suddenly decreased. FIGURE PI 1.2 8 3 0 FOUNDATIONS FX'R MICROWAVE ENGINEERING 1 1 . 3 . A d o w n - c o n v e r t e r is a p a r a m e t r i c amplifier w i t h an i n p u t signal f r e q u e n c y w„ = a>„ + o>p a n d t h e o u t p u t signal t a k e n at frequency <„ "r,.' t h e c i r c u i t of Fig. 11.6. s h o w t h a t t h e down-con v e r i e r gain (actually » I ^ g i v e n by (11.27), with <o„ a n d », i n t e r c h a n g e d . 1 1 . 4 . D e r i v e (11.32). 1 1 . 5 . D e r i v e t h e expression (11.35) for t h e gain o f t h e n e g a t i v e - r e s i s t a n c e n a r ric. amplifier. 1 1 . 6 . D e r i v e ( 1 1 . 4 9 ) For t h e noise p o w e r P2. 1 1 . 7 . O b t a i n an expression lor t h e e q u i v a l e n t noise t e m p e r a t u r e of a p a r a n i e t i up-converter. 1 1 . 8 . D e r i v e a n e x p r e s s i o n for t h e gain o f t h e n e g a t i v e - r e s i s t a n c e degenerate p a r a m e t r i c amplifier i l l u s t r a t e d in Fig. 1 1 . 1 1 . 1 1 . 9 . C o n s i d e r t h e d e g e n e r a t e p a r a m e t r i c amplifier w i t h c i r c u l a t o r s h o w n i n Fig 1 1 . 1 1 . A s s u m e an i n p u t g e n e r a t o r voltage V, at frequency w, in place of e a n d eu. T h e g e n e r a t o r s e n d s a w a v e w i t h c u r r e n t /,* i n t o t h e amplifier' w h e r e /," m u s t be equal to V,/{ R g + Zc) = VH/2Z,., w h e n R g = Z,., since V" s e e s a m a t c h e d load. A reflected wave is set up w i t h a c u r r e n t I[ = -1 / •" T h e load current in R L is / , " , a p a r t from a phase angle. T h e total amplifier c u r r e n t a t frequency w , i s / , = / f + / , ' . W i t h t h i s i n f o r m a t i o n determine t h e a p p r o p r i a t e circuit e q u a t i o n s , a n a l o g o u s t o (11.34), for / , a n d I,,. 1 1 . 1 0 . A p a r a m e t r i c diode h a s t h e following p a r a m e t e r v a l u e s : C 0 = 2 p F , R, = 1 ft. T h e m o d u l a t i o n index M = 0.25. T h e frequency f s = 5,000 M H z and f = 12,000 M H z . E v a l u a t e t h e d i o d e effective Q. D e t e r m i n e t h e load resist a n c e R t to give a gain of 20 dB for a n e g a t i v e - r e s i s t a n c e amplifier of t h e form s h o w n in F i g . 11.8. A s s u m e R g = R L = R2. C a l c u l a t e R for 20-dB gain. REFERENCES 1. Blackwell. L. A., and K. L. Kotzebue: "Semiconductor-Diode Parametric Amplifiers," Prenlice-H»Jl. Inc., Englewood Cliffs. N.J.. 1961. 2. Penfield, P., and R. P. Rafuse: "Varactor Applications," The M.I.T. Press, Cambridge, Mass.. 1962. 3. Chang, K. K. N.: •'Parametric and Tunnel Diodes." Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. Traveling-wave parametric amplifiers 4. Cullen, A. L.: A Traveling Wave Parametric Amplifier, Nature, vol. 181. February, l 9 ° 8 5. Honey, R. C, and E. M. T. Jones: A Wide-Band UHF Traveling Wave Variable Reactan Amplifier, IRE Trans., vol. MTT-8, pp. 351-361, May, 1960. , ., 6. Heilmeier, G. H.: An Analysis of Parametric Amplification in Periodically Loaded 1 rana sion Lines. RCA Rev., vol. 20, pp. 442-454, September. 1959. Broadbanding techniques , 7. Matthaei, G. L.. A Study of the Optimum Design of Wideband Parametric Amplifiers Up-Converters, IRE Trans., vol. MTT-9, pp. 23-28. January, 1961. Wjlie 8. Gilden. M., and G. L. Matthaei: Practical Design and Performance of Nearly Optimu bg[ Band Degenerate Parametric Amplifiers, IRE Trans., vol. MTT-9, pp. 484-490, N° 1961. CHAPTER 12 OSCILLATORS AND MIXERS In this chapter we will examine the operating characteristics of two types of negative-resistance solid-state devices, namely, Gunn devices and IMPATT diodes. These devices are widely used for low-power oscillators in microwave and millimeter-wave transmitters. The Gunn device is useful as a local oscillator in receiver front ends. We will also discuss the use of bipolar and FET transistors in oscillators. The last part of the chapter provides an introductory treatment of mixers. A mixer is a nonlinear device, very often a diode or several diodes in a bridge arrangement, that will cause the microwave signal and local-oscillator (LO) signal to mix to produce a translation of the signal spectrum to a lower frequency called the intermediate frequency (IF). All superheterodyne receivers use a mixer for this purpose. The advantage gained is that the amplification of the signal, before demodulation, is more easily accomplished at the lower fixed IF frequency. The IF amplifier also establishes the bandwidth of the system since selectivity in the high-frequency RF amplifier is usually low. The main purpose of the RF amplifier is to increase the signal amplitude to a level such that the mixer noise will not produce a significant degradation of the signal-to-noise ratio. Apart from economic factors the requirements of an oscillator include: 1. Excellent frequency stability, i.e., negligible variation in the frequency of oscillations due to variations in temperature, power-supply voltage, and oscillator loading (load pulling). 2. Adequate power output for the intended use. 3. Low amplitude-, phase-, and frequency-modulation noise. 831 832 FOUNDATIONS TOR MICROWAVE ENGINEERING 4. Variable tuning, including mechanical tuning and voltage control 5. Capability to be modulated in amplitude (AM), frequency (FM) or (PM). ' 6. Simple circuit requirements. i, phase A variable-frequency oscillator (VFO) whose frequency is varied by means f an applied control voltage is called a voltage-controlled oscillator (VCO) Good frequency stability and low noise are obtained by employing high-Q resonator in the oscillator circuit. The resonator should have a resonant frequency that is insensitive to variations in the ambient tem perature, and this usually translates into low thermal expansion of the resonator. For critical applications the oscillator may be placed in a temperature-controlled oven. The frequency of an oscillator will vary with the dc bias voltages that are applied. This effect is called oscillator pushing and can sometimes be used to advantage to fine-tune an oscillator over a narrow band of frequencies. The frequency of oscillation is determined by the resonant frequency of the input and output networks. Consequently, any change in the impedance of the load connected to the oscillator will result in a change in the oscillator frequency. This effect, which is referred to as oscillator load pulling, is usually undesirable. The pulling effect can be minimized by using loose coupling between the oscillator and the load, i.e., the external Q should be large. A disadvantage of using a loosely coupled load impedance is that the output power will be smaller and the oscillator efficiency will be reduced. Load pulling can also be minimized by using a very high Q resonator as the main frequency-determining element in the oscillator circuit. Oscillator pulling can be determined by measuring the change in oscillator frequency as a function of the phase angle of the load reflection coefficient. 12.1 GUNN OSCILLATORS Some bulk semiconductor materials, such as gallium arsenide (GaAs), dium phosphide (InP), and cadmium telluride (CdTe), have two closely spaced energy bands in the conduction band. A typical energy vers momentum band structure is shown in Fig. 12.1. At low electric fie strengths in the material, most of the electrons will be located in ^ lower-energy band. At high electric field strengths, most of the electrons be transferred into the high-energy band. In the high-energy band ^ effective electron mass is larger and hence the electron mobility is than what it is in the low-energy band. Since the conductivity is d i r e f i e | j proportional to the mobility, there is an intermediate range of electric OSCILLATORS AND MIXERS 833 Low-energy High-energy high-mobility low-mobility band band I Band gap 1.4 eV ! Valence band Momentum F I G U R E 12.1 Typical double-energy conduction band for a Gunn material such as GaAs. strengths for which the fraction of electrons that are transferred into the high-energy low-mobility conduction band is such that the average mobility, and hence conductivity, decreases with an increase in electric field strength. Thus there is a range of applied voltages over which the current decreases with increasing voltage and a negative incremental resistance is displayed by the device. A typical current-voltage characteristic for a Gunn device is shown in Fig. 12.2. A Gunn device is also called a transferred-electron device since the negative resistance arises from the transfer of electrons from the low- to the high-energy band. The oscillations that occur in materials with the energy band structure noted above was discovered by J. B. Gunn. The possibility of obtaining negative differential resistance had been predicted earlier by Ridley and Watkins. There are two principal modes of operation that result in oscillations for a Gunn device. When the applied voltage exceeds the threshold value, a dipole domain (a region of electron concentration and depletion) forms near the cathode end with most of the voltage drop appearing across the highresistance part of the device. A short section of the input region is in the low-energy high-mobility state and electrons leave the cathode with a large High-mobility region Threshold voltage F I G U R E 12.2 Current-voltage characteristics for a Gunn device. Note the negative-resistance region- 834 FOUNDATIONS FOK MICROWAVE ENGINEERING High-mobility state \ =1+ J. Low-mobility state =1+ —i+ Cathode Dipole domain Anode Low-impedance RF circuit F I G U R E 12.3 A simple Gunn Oscillator circuit u the transit-time mode of oscillation. velocity. At the point in the material that separates the high-mobility anc low-mobility states, electrons accumulate on the left side and are depleted on the right side by virtue of the different mobilities. This dipole arrange ment of charge is shown pictorially in Fig. 12.3. This dipole domain swee*across the device, and when it arrives at the anode, the device is in a high-mobility state and a new dipole domain forms at the cathode end an J moves toward the anode. This mechanism is self-repeating and represen an oscillation with a period equal to the transit time. This mode of oscillation has a low efficiency (a few percent) of power generation and a frequency that cannot be controlled by the external circuit. This mode of oscillation " called the transit-time mode or Gunn mode. The second mode of oscillation is the limited-space-charge (LSA) mode. Operation of a Gunn oscillator in the LSA mode can produce several watts of power with efficiencies of around 20 percent or more. The power outputs that have been obtained decrease with frequency and are below 1 W at frequencies greater than 10 GHz. Output power of several milliwatts can obtained at 100 GHz. In the LSA mode the Gunn device is incorporated as part of a resonant circuit as shown in Fig. 12.4a. The frequency of the resonant circuit is adjusted so t h a t it is several times greater than that of the transit-time mode. As a consequence, dipole domains do not have sufficient time to form and the device operates essentially as a negative-resistance device. The dc bias is adjusted to a value somewhat greater than the threshold voltage. The RF voltage of the oscillations will build up to a peak-peak value approximately equal to the voltage increment over which the device resistance is negative as shown in Fig. 12.46. The resonator loading, represented by | resistor R, is adjusted to a value about 20 percent greater than the maximum negative resistance of the device. This will ensure that oscil ations will start. The amplitude of the oscillations will build up until the average negative resistance of the Gunn device becomes equal to the re* onator resistance R. f If the resonator frequency is adjusted to a value slightly a b o v e r i a t h e the transit-time mode, the Gunn device will operate very much hke basic Gunn mode, but the dipole domain will be quenched before it a r r . „ at the anode by the negative-going oscillation voltage. This type of ope « OSCILLATORS AND MIXERS 835 mode is called a quenched-domain mode. Oscillations can also occur by adjusting the resonator frequency, so that it is lower than the frequency of the Gunn mode. In this cas>e the dipoie domains have sufficient time to sweep across the device and arrive at the anode. However, the initiation of a new dipoie domain is delayed until the oscillation voltage rises above the threshold value. This mode of operation is called the inhibited or delayed mode. Circuits The equivalent circuit of a Gunn device operating in the LSA mode is a negative resistance -Rd in parallel with a capacitance Cd as shown in Fig. 12.5. The negative resistance has a value that typically lies in the range - 5 to —20 ft. The required resistive loading from the cavity and the external load should be around 20 percent higher than the Gunn device resistance so that the parallel combination -RRd/{R - Rd) will be negative. The cavity used for the resonator must generally have an impedance-transforming property in order to reduce the high impedance of the output waveguide to I (t>) F I G U R E 12.4 ( a ) A basic G u n n oscillator operating in t h e LSA mode; (6) RF oscillating voltage across a Gunn device operating in the LSA mode. 836 FOUNDATIONS FOR MICROWAVE ENGINEERING C„ £ - R , F I G U R E 12.5 M Equivalent circuit of a Gunn device operating in the LSA mode the appropriate Jow value required by the Gunn device. A simple cavitv structure is shown in Fig. 12.6. The Gunn device is located under a post in a rectangular waveguide. The cavity is resonated at the desired frequency bv adjusting the short-circuit position. The degree of coupling to the external waveguide is adjusted by changing the window opening in the inductive diaphragm located at the front of the cavity. The top of the post is insulated from the waveguide. The dc bias voltage (typically around 12 V) is applied to the post. There is sufficient capacity between the post and the surrounding waveguide to provide an adequate low-impedance RF bypass capacitance and thus RF currents do not flow through the bias voltage supply. Fine tuning of the cavity can be obtained by means of a tuning screw. Another simple cavity arrangement for a Gunn oscillator is shown in Fig. 12.7. In this cavity the high impedance of the waveguide is transformed into a low impedance at the location of the Gunn device by means of quarter-wave transformers. The cavity resonant frequency can be adjusted by changing the location of the short circuit. A tuning screw can be used for fine tuning of the cavity. The cavity shown in Fig. 12.6 is easily modified to have two posts, one for mounting the Gunn device and a second one for mounting a varactor diode. The capacitance of the varactor diode is a function of the control voltage Vc. By varying V c the resonant frequency of the cavity can be varied. V Tuning screw i" ; V / /•/ / / j / ; / / j ; ; / ' / 7^77-r-A / / W Output waveguide %.//// Post Z 2 3 Z Z Z Z Z Z Z Z Z 2 5 / /// Inductive diaphragm (a) S//-7-. Short circuit V /• / ^7 - Gunn device zzzzfczzz^ z z z z Inductive diaphragm (") F I G U R E 12.6 j - ^ t the ( a ) A simple waveguide cavity for a Gunn oscillator; (6) inductive diaphragm used to coupling between the output waveguide and cavity. OSCILLATORS A N D MIXERS 8 3 7 Tuning screw , Gunn device >, / / /Sf/fJ/SSS 7-77 Output waveguide S / J S ; / ; / / / ; / ; ; / / / / F I G U R E 12.7 A Gunn oscillator cavity which uses a two-section quarter-wave transformer to transform the high impedance of the waveguide to a low impedance at t h e Gunn device. Two-section quarter-wave transformer K • y/////// 'Mz. ^ZTzzzzzmzzzzzzzzzzzSzzzzzzzzzL Varaclor diode Gunn device F I G U R E 12.8 A Gunn oscillator cavity which has a post-mounted varactor diode for cavity tuning. A sawtooth sweep voltage applied to the varactor diode will produce frequency modulation of the oscillator. If a sawtooth sweep voltage is applied to the varactor diode, the Gunn oscillator will be frequency-modulated. The cross section of the cavity is shown in Fig. 12.8. The Gunn device can be operated as a pulsed oscillator by applying the dc bias voltage in the form of a pulse train of short rectangular pulses. If the duty cycle is low enough and very short bias pulses are used, the peak power output will be limited only by the peak current, since thermal heating of the device will be negligible during the short on time. For pulsed oscillator applications, the IMPATT diode, or variations of it, is preferred because of higher output power. The Gunn device can also be, and has been, used as a negative-resistance amplifier. DIODES The acronym IMPATT stands for IMPact Ionization Avalanche Transit Time and describes the phenomenon associated with reverse voltage breakdown in a p-n junction diode and the transport or transit of charge carriers 838 FOUNDATIONS FOR MICROWAVE ENGINEERING I I I Avalanche region k: Drift region F I G U R E 12.9 Structure of a Read (IMPATT) diode and the electric field profile across the diode. through a drift region. W. T. Read had proposed in 1958 that there should be a phase delay of more than 90° between an applied RF voltage and the avalanching current if the RF voltage caused the total voltage to exceed the reverse breakdown voltage in a diode. In 1965, R. L. Johnson verified Read's prediction. When the current lags the RF voltage by more than 90°, the diode will exhibit a negative resistance and can be used as a source of microwave power in an oscillator circuit. The Read diode, designated as a p*nin* type, consists of a heavily doped p* region, a normally doped n region, an undoped or intrinsic semiconductor section, and an n + region as shown in Fig. 12.9. The p~ n diode junction will break down when the applied reverse bias voltage exceeds a threshold value. The current-voltage characteristic is shown in Fig. 12.10 and is similar to that in the familiar zener diode. The rapid increase in current at the breakdown voltage is caused by avalanche multiplication of the density of the holes and electrons. If a Read diode is placed in a cavity and a reverse bias somewhat smaller than the breakdown voltage is applied, along with a small RF voltage, then breakdown will occur when the RF voltage becomes positive. When breakdown is initiated a large number of holes and electrons are created at the p*n junction. The electrons are swept across the n region into the intrinsic semiconductc drift region. After a transit-time delay, the electrons are collected at the n terminal. The current pulse moves through the diode from right to I' When the time Cor avalanche charge buildup plus that for charge transi through the drift region exceeds one-half RF period, the output current w lag the RF voltage by more than 90°. With these conditions the diode v exhibit a negative resistance for RF currents. In an oscillator circui initial RF voltage comes from the cavity resonant-frequency componen the noise that excites the cavity. Once the oscillations start they gr° , amplitude until the average negative resistance of the diode becomes eq to the total equivalent resistance of the cavity and the external load. OSCILLATORS AND MIXERS 839 Reverse voltage breakdown F I G U R E 12.10 Current-voltage characteristic of a diode. Note the reverse voltage breakdown re- Since the introduction of the Read diode as a generator of microwave power, a number of other diode structures have been developed that will also produce microwave oscillations. The two most common variations of the Read diode are the BARRITT (BARrier /njection Transit-Time) diode and the TRAPATT (Ti?A pped Plasma Avalanche Triggered Transit) diode. The BARRITT diode is a p * tip ' or back-to-back diode. The charge carriers that traverse the drift region in a BARRITT device consist of minority carriers that are injected from the forward-biased p+n junctions. Since the BARRITT diode does not involve an avalanche breakdown, it produces less noise than an IMPATT diode does. However, the power output and efficiency is less. The TRAP ATT diode is a p'nn* diode and is driven by a large repetitive pulse of current. Breakdown will occur at one of the p'n diode junctions, and since the current drive is very large, a large collection of electrons and holes (a plasma) is generated. The violent breakdown creates a high electric field shock front that moves across the n-type drift region. After passage of the shock front, the plasma is located in a low-field region and is said to be trapped because it takes a long time to clear the drift region of charge carriers. When the plasma has been cleared from the drift region, the cycle will repeat. In a TRAPATT diode, oscillations start with the diode operating as an IMPATT device. When the amplitude of oscillation becomes large enough, the TRAPATT mode of oscillation is established. The TRAPATT diode will not operate at as high a frequency as the IMPATT diode does. It is also noisier, so its use is decreasing. IMPATT diode oscillators are used with the same cavity structures that are used with Gunn devices. IMPATT diodes are much noisier than 840 M3UNDATIONS FOR MICROWAVE ENGINEERING Gunn devices and so are generally not used for local oscillators E crowave and millimeter-wave receivers. For power generation, IMPATT diodes are superior to Gunn devices. Output CW powers of as much as 10 W at a few gigahertz and around 1 W at 100 GHz can be obtained from a sin l device. At frequencies above 100 GHz, the output power from eurrentJ produced IMPATT diodes decreases approximately as l/f3. 12.3 TRANSISTOR OSCILLATORS Silicon bipolar transistors are a good choice for oscillators at frequencies UD to 5 GHz. From 5 GHz up to about 40-GHz, MESFET devices can be used in oscillator circuits. In the frequency range 30 GHz up to around 100 GHz the high-electron-mobility transistor (HEMT) would be used because of it' higher frequency of oscillation. In Chap. 10 we were concerned with the general problem of designing amplifiers t h a t would not oscillate. We noted t h a t many microwave transistors were only conditionally stable and only a restricted range of load and source impedances would ensure amplifier stability. The stable and unstable regions were shown graphically by plotting the input and output stability circles on the Smith chart. In the design of an oscillator, we choose the input and output port terminations in the unstable regions. In the unstable regions both the input and output impedances of the transistor circuit will have a negative resistance and oscillations will occur at a frequency at which the total reactance in the input and output circuits vanish (resonance condition). Transistor oscillators can thus be viewed as negative-resistance oscillators. If a transistor is absolutely stable, it can be made unstable by using feedback from the output to the input of the device. Common base or common gate circuit configurations tend to have the greatest amount of instability as Example 10.5 showed. One suitable feedback arrangement is the use of a reactance in series with the emitter or source lead. A series reactance can also be used in the common base or common gate circuits. The criteria for a transistor terminated in an impedance Z s at the input port and Z L at the output port to oscillate are readily_established. Initially, assume that a voltage source V g acts in_series with Z s + Z i n . The input current to the transistor will be Vg/iZ± + Z±n) = Iin. If we now reduce V g to zero but at the same time make Z s + Z i n tend to zero, we can maintain the current 7in and a finite output power. Thus, for oscillations occur when Vg = 0, we must have Zs or = Rx+jXs= flin=-«s «.--/*. -Rin-jXin (12. l a ) lb ^; l Thus the transistor must have a negative input resistance and the i P must be tuned to resonance. The frequency of oscillation is deterrn m . , (12.16). From the relations f s = (Z, - D / ( Z S + 1) and T m - (#in \ OSCILLATORS AND MIXERS 841 (Zin + 1), we readily find that the condition for oscillations can also be stated as rsrm = i (12.2) Since Rm is negative |r i n | > 1, which is in accordance with our requirement for instability as given in Chap. 10. We_will now show that when (12.1) or (12.2) hold, at the output ZL = -Z < m , or ^ou, = -%t (12.3a) Mmt" ~JXL (12.36) l'J'out = l (12.3c) Thus the conditions for oscillations are satisfied at both ports if they are satisfied at one port. The proof of this property is as follows: The following relations hold = "* _ s22 - yr i n = s22rin - A i - ' ; s n " i-su/r,n " r,n-su S22-M; upon using (12.2). We also have r. = s„ - &rf, which can be solved for T, to give h= 'in ~ S u ^22'm - A r out which is the relationship we wanted to prove. Since the impedance looking into both ports has a negative real part, the transistor will deliver power to the external circuit at both ports. Since it is operating in the unstable region, the power gain is negative as explained in Sec. 10.7. An absolutely stable transistor with a suitable external feedback network can be viewed as a new modified potentially unstable device for which the above relations will hold true. We normally view an oscillator as an amplifier with a feedback loop that feeds a fraction of the output power to the transistor input and the input circuit. Part of the power fed back is absorbed in the resistance associated with the input circuit. When an unstable transistor is used as an oscillator, the feedback takes place internally and is described in terms of the reverse transmission coefficient S, 2 . Thus the power dissipated by the input circuit as well as the input power that drives the oscillator is provided by internal feedback of power from the output to the input of the oscillator. The only significant difference in the two oscillator types is the feedback path which can be either external or internal, In order for oscillations to start, it is necessary to choose Rs < \Rm\ using the small-signal scattering-matrix parameters to evaluate Rm. Oscil- 842 FOUNDATIONS FOR MICROWAVE ENGINEERING lations will then build up until the nonlinear characteristics of the trai tor cause the power gain to saturate. Thus steady-state oscillations wll cause the transistor to operate under large-signal conditions. For la signals the nonlinear behavior means that the large-signal scattering-mat ^ parameters will be different from those that apply for small-signal cond" tions. Furthermore, the nonlinear behavior will cause harmonics of th fundamental frequency to be generated. Usually the presence of harmonic in the output is undesirable unless the oscillator is specifically designed for an output at one of the harmonics. The basic problem in oscillator design is choosing the port terminations so that the desired output power is obtained the harmonics are adequately suppressed, and the desired frequency of oscillation with good stability against variations due to load, temperature and bias conditions is obtained. Some general guidelines that should be followed in order to achieve the above objectives are: 1. The loaded Q of the output circuit should be at least 10 to give good harmonic suppression. 2. The input reactance jXa should have a large-frequency derivative or slope dXs/dw. This will mean that a small change in jXin can then be matched by a small change in to to bring jXs back to —jXm. Usually this requirement is met by incorporating a high-Q temperature-stable resonator in the input circuit or in the feedback path. 3. For minimum harmonic generation the oscillator should operate as a class A oscillator. 4. For best efficiency and largest amount of output power, the oscillator should be designed for class B or class C operation. This will require a self-bias circuit, so that initially the circuit operates as a class A oscillator in order for oscillations to start. In class A operation the collector (drain) current flows continuously over a complete RF cycle. In a class B oscillator the current flows for one-half of the RF cycle, while in class C operation the current flows for less than half of one period in each RF cycle. In class C operation the maximum collector (drain) current flows when the RF voltage at the collector (drain) is negative. Thus the power dissipation in the device is minimized and efficiency (RF output power/dc input power) can be high. At lower frequen cies class C efficiencies as high as 80 percent or more can be achieved, t>u microwave frequencies the efficiency is much lower because of circuit 1< relatively low power gain, and limited supply voltages to avoid damaging transistor. The efficiency of microwave oscillators and amplifiers is described in terms of the power-added efficiency. The power-added efficie is defined as follows: Power-added efficiency = RF output power - RF input power — dc input power , , 2 .4) * OSCILLATORS AND MIXERS 4 843 T H R E E - P O R T D E S C R I P T I O N OF A T R A N S I S T O R In order to facilitate the analysis of an oscillator circuit, when an impedance is connected between one of the transistor terminals and the ground plane, it is useful to have a scattering-matrix description of a transistor viewed as a three-port circuit. In Fig. 12.11 we show a transistor with a microstrip line connected to each terminal. For clarity we have labeled the base terminal as port 1, the collector terminal as port 2, and the emitter terminal as port 3. Any other labeling could be used equally well. The normalized amplitudes of the incident and reflected voltage waves will be denoted by at,a2,a5 and bt,bs,b3, respectively. The three-port circuit is described by the scattering-matrix relation 6, b, = b, s„ s12 s„ s,, s22 S-,3 a., 4. a:< S3I °SS (12.5) where S,, are the three-port scattering-matrix parameters. We will show that the three-port parameters have the property that the sum of all elements in any row or in any column equals unity; thus /= £ s,., = 1 (12.6a) 1,2,3 i = 1,2,3 (12.66) j= 1 Consequently, the three-port scattering-matrix elements are not all independent. In fact, they can be expressed in terms of the two-port scatteringmatrix parameters that describe the transistor when one of the terminals is grounded, e.g., common emitter parameters when the emitter is grounded. In order to prove (12.6a) we note that, provided there is negligible stray capacitance between each transistor terminal and the ground plane, the sum of all currents entering the three-transistor terminals must be zero. Thus we have £ (o,-6,)=0= £ L- £s!Ja]=0 i=l 1=1 \. j 1 / FIGURE 12.11 t7777777?77777777PV77777777/r A transistor viewed as a three-port network. 844 FOUNDATIONS FOR MICROWAVE KNG1NEERING We can choose the incident-wave amplitudes independently; so if we cho a2 = a 3 = 0, we obtain 3 «. - E £„«, = o or £ S„ = 1 i = i which is one of the relations in (12.6a). By choosing in t u r n a, and a as the nonzero amplitude, the other two relations are obtained. The relations given by (12.66) are obtained by noting that, when all port (terminal) voltages are equal, the input current at each port will be zero, provided again that there is negligible capacitance between each terminal and the ground plane. Thus we must have 6, = a, when a = a = a 2 -.i = a > a n ^ consequently, 6, = b 2 = 6 3 = a and b,= Es,,«,= EV = " .; i j \ which gives the desired result. The total normalized voltage at port 3 is V 3 = a 3 + 6 3 . Let us define new voltage-wave amplitudes at ports 1 and 2 as follows: V r - O i - - ^ - ^ ^3 V{ = 6X - -f = 6, - (12.7a) 2 a 3 + &8 \ a. 2 * 4 (12.76) 63 (12.7c) ^ - 6 , - ^ - 6 , - &3 ^ (12-7^) V?-« 2 - £ - « , - =2 ^ a3 + The total new voltages for ports 1 and 2 are now Vx = Vj* + V, = «i 61 - V, and V., = a 2 + 6 2 - V 3 and are thus referenced to the port A voltage. The above definitions for the new incident- and scattered-wa amplitudes were chosen so as to leave the port currents unchanged, tna Vf - Vf = a, - bv V2+ -V2-=a2- b2. The common emitter < c o m m ° " port 3 terminal) two-port scattering matrix is the scattering matrix relates V, and V2 to Vj" and V2+; thus (12.8a) Vf = S^? + S 12*2 v2 =s2y[+s22v; Upon expressing the V, and V, <l2-86) in terms of the a, and 6, using 1(\9 *•*•7) we oscrrxAToRS AND MIXERS 845 obtain 62 _ H l _ ^ ==S 2S.,.\ I | aa.i - _ L _ ^ | + S 2 2 j a ^ 2 Since the sum of all currents flowing into the three transistor terminals is zero (Kirchhoff's law), we must also have o 1 - 6, +Q2 - 62 + a 3 — 63 = 0 The above three equations are easily rearranged to give three equations expressing the bi in terms of the a,. The first two equations can be written in the form *i = -SnO, + 5,2.3:2 + ff V* \ % b, = S2)at + S22a2 + "T>-^ where <ru = 1 — § u — S , 2 and o"22 = 1 — S 2 2 — S 2 1 . The third equation can be solved for b3 to give b3 ~ a, + G2 + a3 — 6, - b2. When we substitute for 6, and 6 2 , we readily find that 2a.9 2<r?, a b8 = - ^ - a ! + - - ^ a 2 + - a3 (12.9a) 4 — <J 4 — (T 4 — cr where o-]2 = 1 — S u - S 21 , o"21 = 1 — S 2 2 — S 1 2 . ir = S n + S 1 2 + S 2 1 + S22 = 2 - cri2 - (Tz, = 2 - iru — IT.,.,. By using this expression in the equations for 6, and 6 2 , we obtain / (r,,.}ir,., \ + a I IT.,;<T.,, \ + s + + 2(To? "> = ( a - T^) > I - Trf h 5 ^ 0 a (12-9c) Equations (12.9a) to (12.9c) provide the three-port scattering-matrix description of the transistor in terms of the two-port scattering-matrix parameters. The S,j are given by c _Q •^21 - -321 + * ** - 2o 4 tTi2(Ta • "l2 - O- 4 _ a c °22 3 32 _o °22 2 <r , + «"21 4^7 ™(r*i 4 _ 2fr 6 a ff 23 4 ™ _ ,5 °" ^3 - 4 _ ff a 846 FOUNDATIONS FOR MICROWAVE ENGINEERING where a = S n +Sl2 + S2i + S22 <ru = l - S trl2 = 1 — Sn — S2l u - S l 2 <r22 = 1 — S22 — S21 a2i = 1 — S22 - Sl2 The reader can readily verify that the above scattering-matrix parameter1? satisfy the relations (12.6). If the three-port scattering-matrix parameters have been measured it is easy to obtain expressions for the two-port scattering-matrix parameters Let us assume that we want to find the common emitter two-port scattering parameters. This requires that we make V a = 0 or b a = - a 3 . By setting 6 3 = - a 3 in the last equation in the set given by (12.5), we can solve for a, in terms of a, and o 2 - By using this solution in the first two equations, we obtain the solutions for 6, and b 2 in terms of a, and a 2 from which the two-port scattering-matrix parameters are readily identified. It is found that £11 - ^11 °A <S Sl!> — S19 12 — 1+^33 3»1 — i + s33 i+$ 33 (12.11) ^23^32 •^23 ^ 3 1 "9.1 21 <5 °I3°32 &«9 '22 — " 22 99 1 + 333 Consider now the case when a series normalized impedance Z s is connected between the emitter and the ground_plane. For this situation, as shown in Fig. 12.12, a 3 = T6 3 where T = (Z, - 1 ) / ( Z , + 1). The same procedure used to obtain (12.11) can be used to find the two-port scattering matrix for the transistor with a series feedback impedance Z s connected in series with t h e common emitter lead. It is readily found that the new scattering-matrix parameters are given by (12.11) upon replacing S33 + by S 3 3 - r _ 1 . As a final note we point out that, if the relations (12.10) are used to express the § u in terms of the Sijt (12.11) is satisfied identically. i.e., it gives S 0 • Stj. F I G U R E 12.12 feedback A transistor with a series impedance in the commdn emitter lead. OSCILLATORS AND MIXERS 847 The scattering parameters S u and S.,.z are given by S " = ~$^~T (12 12fl " ) A2r - 3 2 2 where A1 = SnS33 - Sl3S3l and A2 = S22S33 - S23S32. The above equations can be solved for f to give r = c " ' °33°H ~ a ] r =* I _A °.S3'-,22 d2.13a) < 12-136) **S If we restrict the series impedance to be a pure reactive element, then |l"| = 1. The circle of |T| = 1 values maps into circles in the Sn and S22 planes in accordance with the bilinear transformations given by (12.13). The center and radius of the S u circle are Center = Radius = s„ -Mf ~-~ 3 i - iW \Sl3S31l ^ ^ II - IS33I I (12.14a) (12.146) while those for the S.,., circle are &. ~ A J? Center = — ^-~ 1 ~ I £«,! Radius = \§ § I ^V (12.15a) (12.156) n —1<» 1 By plotting these circles it is possible to visually see the range of values t h a t can be obtained for_S,, and S 2 2 by varying the series feedback reactance. For each value of jX the stability parameter K for the equivalent two-port network can also be evaluated [see (10.18)]. This parameter is a useful measure of the degree of instability a series feedback impedance can produce. Some impedances, particularly resistive ones, will actually improve the stability of the device. Such impedances would be useful in amplifier design where instability is undesirable. Some caution should be exercised in using the above two-port to three-port relations since they are based on the assumption that the stray capacitance from each transistor terminal to the ground plane is negligible. 848 FOUNDATIONS FOR MICROWAVE ENGINEERING T h e above r e s u l t s s h o w t h a t m a t r i x p a r a m e t e r s i s very useful scattering-matrix parameters with one c o m m o n lead. T h e following above r e s u l t s . a knowledge of t h e t h r e e - p o r t scatterin in determining the equivalent t w o - D J l a series feedback i m p e d a n c e connected ' e x a m p l e i l l u s t r a t e s an application of th" E x a m p l e 12.1 Transistor w i t h c o m m o n source f e e d b a c k impedance A MESFET device has the following common source two-port scattering-matrix parameters at 10 GHz: Sn = 0.73/1172° S 1 2 = 0.093,139° S 2 I = 2.31/144° S n = 0.09Z - 5 5 ° This device has a stability parameter K = 1.13 > 1 and is absolutely stable. By using (12.10) the following three-port scattering-matrix parameters were computed - . S , , » 0.735/1241.5° Sl2 = 0.52^27.8° S, :1 - 0.98^24.4° S 2 1 = 1.71/161.4° S22 = 0.517Z - 79.2" S23 = \ L - 85° S31 = U - 5 8 ° S32 = 0.517/130.9° S. w = 0.59,187.8° If a normalized series reactance jX is inserted into the common source lead, a potentially unstable equivalent transistor or two-port network can be obtained. The possible range of S n and S 2 2 values that can be obtained are shown by the circles plotted in Fig. 12.13. Also shown on these circles are the values of S n and S2i obtained using a normalized inductive reactance of j0.5 and jl and the values obtained using normalized series capacitive reactance of —jO.5 and —jl. The computed values of the equivalent two-port network and the stability parameter K for the four reactances considered are tabulated below. jX =j0.5 K - 0.921 Su = 0.244/1181.7° si2 = 0.33Z66.3° S 21 = 1.83/140.3° s 22 = jX=jl AT =0.915 Su = 0.126Z263.30 S 21 = 1.57/139.7° s l 2 = 0.463/. 65-1° s 22 - 0 . 4 7 5 ^ - 41.37° 0.342/1 -38.9° jX = - / 0 J K = - 0.49 Sn = 1.818/1188.2° S, 2 = 0.556^ -63.5° S 2 , = 3.23/158.7° jX= -jl S M = 0.545/1185.7° K = -0.453 S u = 3.12/1235.2° S 1 2 = L52Z - 19.7° S2l = 3.57/198.7° S 2 2 = 1.52Z.231.1" OSCILLATORS AND MIXERS 849 S,, circle FIGURE 12.13 Circles showing the values of the scattering-matrix parameters S,, and SB2 for a MESFET with a series reactance in the common source lead. For the above reactance values K < 1, so the MESFET with common source series feedback is unstable (will oscillate). Note that a series capacitive reactance produces a highly unstable device, in particular, for jX = -j\ the reverse transmission coefficient S 1 2 is very large relative to normal values and both IS,,I and | S 2 2 ! are greater than unity. The results given above were obtained using the computer program TRIPORT. J L 2.5 OSCILLATOR CIRCUITS A microwave oscillator can be designed using any of the standard lowfrequency oscillator circuits such as the Hartley, Colpitts, or Clapp circuits. Various variations of these circuits can also be used. The frequency stability of the oscillator is generally achieved by incorporating a resonator in either 850 FOUNDATIONS FOR MICROWAVE ENGINEERING r~i o 50 a DR RFC \G RFC D ft 50 a )X RFC F I G U R E 12.14 A 5-GHz FET oscillator using ! dielectric resonator DR in the input circuit for frequency stabilization. The feedback is obtained using a series capacitive reactance in the common source lead. the input or output circuits or as part of the feedback loop. A disk resonator can be used but its Q is relatively low; so the resultant frequency stability will not be very high. A high-Q metallic cavity can be used, but because of its large size the compact high-Q, temperature-stable, dielectric resonator is often the preferred choice. In Fig. 12.14 we show a 5-GHz FET oscillator that is stabilized by using a dielectric resonator in the input circuit. The magnitude of the source reflection coefficient is controlled by the coupling to the resonator which can be varied by changing the spacing d between the resonator and the microstrip line. The phase angle of the source reflection coefficient is controlled by the length /, of the input line. The output circuit is a standard stub-matched circuit that transforms the 50-fi load impedance to the required value for the oscillator. The FET is made to oscillate by using a series capacitive reactance in the common source lead. This feedback arrangement makes the equivalent transistor two-port circuit unstable as was shown in Example 12.1. The dc bias currents are applied through Kr chokes. The output load is isolated from the oscillator dc voltages by the low-impedance dc blocking capacitor C.t In Fig. 12.15 we show an FET oscillator using a dielectric resonator in the feedback path from the drain to the gate. The amount of feedback can be adjusted by the coupling to the dielectric resonator. The resonator I located a distance A/4 from the open-circuited ends of the coupling lin since the standing wave of current and magnetic field is greatest at location from the open ends and thus provides the strongest coupling to resonator. The correct phase of the feedback voltage is controlled by the tV. Rizzoli, A. Neri, and A. Costanzo. Analysis and Optimization of DROs Using a Purpose CAD Program, Alia Frequenza. vol. 57, pp. 389-398. 1988. Gen,eral OSCILLATORS AND MIXERS 851 h ^k OR C '. Is], 50 £} T J F I G U R E 12.15 An FET oscillator using a dielectric resonator in t h e feedback path from t h e drain to t h e gate. lengths /, and l2, as is the phase of the source reflection coefficient seen at the gate terminal. Electronic control of the oscillator frequency can be obtained using a varactor diode as part of the input circuit. Variable-frequency oscillators are also built using a yttrium garnet (YIG) ferrite sphere whose resonant frequency is controlled by the dc magnetic biasing field. OSCILLATOR DESIGN When maximum power output and efficiency are not of prime importance, a satisfactory oscillator design can be achieved using the small-signal scattering-matrix parameters. The amount of power generated can be varied by adjusting the dc bias voltages. The major shortcoming of small-signal oscillator design is that it does not provide any way of predicting the steady-state oscillating signal level. Oscillator design based on large-signal scattering-matrix parameters is much more difficult because of the difficulty of obtaining large-signal parameters. Two approaches are possible, namely, measuring the scattering-matrix parameters under large-signal conditions or obtaining these from computer simulations using a theoretical nonlinear model of the transistor. Neither method is easy to carry out so as to achieve a high accuracy. Space limitations will not allow us to discuss the large-signal approach. However, the references at the end of this chapter provide information on methods that have been developed and used successfully. We will only consider the small-signal design approach, and even this in a limited way, by means of two examples. 852 FOUNDATIONS FOR MICROWAVE ENGINEERING E x a m p l e 12.2 Oscillator design u s i n g an unstable transistor. A ail" bipolar transistor has the following scattering-matrix parameters at 6 GH ° n S„ = 0.65^130° S 12 = 0.2^80= S 2 1 = 2/142° S™ = 0.4Z. - 6 0 ° The stability parameter K = 0.646, which is less than 1, so the transistor i potentially unstable. In Fig. 12.16 we have plotted the input and output stability circles. One of the primary effects of large-signal operation is reduction of gain because of gain saturation. For simplicity, we will assume that under large-signal conditions S 2 1 changes to a value 1.5<L42° and all other scattering-matrix parameters stay the same. Thus, under large-signal conditions, we find that the stability parameter K has increased to 0.798. As a result the stability circles move. The large-signal stability circles are shown bv the dashed circles in Fig. 12.16. Let us choose a source reflection coefficient Ts = 1^210° which corresponds to a pure capacitive reactance load at the base. For this value of f ZL = 0.14 +/'1.774 ZL = 0.509 + /1.67 Output stability circles Input stability circles FIGURE 12.16 . Exam ple Stability circles for small- and large-signal conditions for the oscillator discussed in ^ 12.2. The source reflection coefficient and two values of the load impedance are also sn OSCILLATORS AND MIXERS 853 we find that the output normalized impedance of the transistor is 2, and = — 0.509 —j 1.67 Z o u t = - 0 . 1 3 9 - jl-774 for small signal conditions for large signal conditions In order to satisfy the conditions (12.3) for oscillations, we must choose a load termination Z L at the collector where Z L = 0.509 +J1.67 for low-level oscillations and Z L = 0.139 fj1.774 for large-signal oscillations. The reactive parts are almost of the same value, but the resistive part for large-signal conditions is only 0.27 of the required value for small-signal conditions. In the small-signal design approach, it is usual practice to choose the load resistance around a factor of 3 smaller than what is required by the condition (12.3a) for steady-state oscillations. This will allow the oscillations to build up in amplitude until gain saturation makes —Rout equal to R,_. There will_be some shift in the resonant frequency as the oscillations build up since X o u l changes and thus the frequency must change until X o u [ = —XL. When the input network produces a very rapid change in the phase of F.. with frequency, only a small frequency shift will occur. Some means of tuning the oscillator is normally used so as to establish the desired frequency of oscillation. In the example we are discussing, we will choose ZL = 0.14 + y 1.774. This impedance point is shown in Fig. 12.16, as is the point ZL = 0.509 + ./1.67. The former point lies on the stable side of the assumed large-signal stability circle and the oscillator would not oscillate under these conditions. Our choice for Z L will, in actual practice, limit the oscillation amplitude at a value for which IS 2 1 | is somewhat greater than 1.5 so as to keep the point Z L in the unstable region. Since we have used a pure reactive termination at the input, there is no power delivered to the input circuit. Thus the power gain of the circuit is negative and infinite, so that the stable point of oscillation occurs when the stability circle moves outwards so as to make Z t lie on the circle since the output stability circle coincides with the infinite gain circle. When the input termination has a resistive part, the power gain must be negative and finite, so that Z L must lie on the appropriate negative gain circle and inside the unstable region (see Chap. 10 for a discussion of negative gain circles ir the unstable region). The circuit for the oscillator is shown in Fig. 12.17. An open-circuitec transmission line is used to produce the input reactance. Since r, = e"2jl" tht minimum length I is equal to 5A/24 in order to make the phase angle of F equal to - 150", which is equivalent to 210". The frequency sensitivity of tht phase angle can be increased by using a transmission line A /2 longer, that is / = 17A/24. A 1 percent change in frequency or /3 will then change the phasi angle of F s by ( - 1 5 0 - 1801/100 = - 3.3°. The computed values of Z„u, for i ±2= change in_the phase angle of Ts are Z o u l = - 0 . 0 8 3 -j 1.697 for F. = 1/12180 and Z o u , = - 0 . 2 0 2 - j l . 8 6 3 for F s = 1^222° under large-signa conditions. It is apparent that Z 0UI is quite sensitive to small changes in thi phase angle of Ts. The output matching network consists of a 50-ft transmission line o length 0.1975A and an open-circuit 50-fi stub of length 0.22A. This circui transforms the 50-O load to a normalized impedance of 0.14 +_/1.774 at th collector. The bias voltages are applied through 150-ft quarter-wav 854 FOUNDATIONS FOR MICROWAVE ENGINEERING 50 Q FIGURE 12.17 The oscillator circuit designed in Example 12.2. transmission lines that are bypassed to the ground plane by capacitors C^p. These lines are connected at low-impedance points on the input and output circuits. The 50-11 output is isolated from the transistor by the dc blocking capacitor C. The design carried out above is a hypothetical one since we do not know how the scattering-matrix parameters change with signal level. The example does show the basic physical principles involved in determining the steady-^tate operation of the oscillator. The small-signal design approach that uses Z L = - ( R o u l / 3 + jXout) usually leads to a reasonably satisfactory design. E x a m p l e 12.3 Oscillator d e s i g n u s i n g a dielectric resonator. In this example we will use the MESFET described in Example 12.1 with a series reactance -JO.5 in the common source lead. The equivalent two-port scattering-matrix parameters are S u = 1.818Z.188.20, S 1 2 = 0.556A - 6 3 . 5 ° , S 2 1 = 3.23^58.7°, S 2 2 = 0.545^185.7°, and the stability parameter K = - 0 . 4 9 . The input and output stability circles for the equivalent two-port network are shown in Fig. 12.18. Since | S 2 2 | < 1 the origin is a stable point for r„, so that values of Y s inside the source or input stability circle are unstable ones. The values of Z L outside the load stability circle are unstable ones since ISnl > 1. We will choose T s = - 0 . 8 for the initial design of the oscillator. This value of T 5 is s h o w n i n Fig. 12.18 and is in the unstable region. For_this value of T, we find that ZoM = - 1 . 6 - J0.906. For Z, we will choose Z L = 0.6 jO.906 for which R L = - 0 . 3 7 5 R o u t . The oscillator circuit used is shown ^ Fig. 12.19. The resonator equivalent circuit is a parallel combination of R> ' and C which is series-coupled to the microstrip line by an ideal translo with turns ratio n : 1 as shown in Fig. 7.23. We can choose R equal to unity OSCILLATORS AND MIXERS 855 Load stability circle l l ' J = 1 Source stability circle | F _ FIGURE 12.18 The input and output stability circles for the oscillator in Example 12.3. The design values of T, and Zj are also shown. choosing an appropriate value for the turns ratio n:\. The resonator Q is given by R/w0L, which determines the inductance in terms of the resonator Q. The capacitance in the equivalent circuit is given by the resonance condition IOQLC = 1. At w = io0 the impedance coupled into the microstrip line is a series resistance equal to n2R = n 2 , so n 2 represents the coupling coefficient. The transmission line of length A/4 can be replaced by a short circuit at the Rg c BP 1 T • vvv , 1 T X 4 o 5012 DR 0.426X • 0.144X FIGURE 12.19 The oscillator circuit designed in Example 12.3. 856 FOUNDATIONS FOR MICROWAVE ENGINEERING location of the resonator. Thu.,, at the resonator location, the reflect" coefficient on the microstrip line is I" = (n 2 - l ) / ( « 2 + 1). In order to tn lc" T = - 0 . 8 , we must choose « 2 = (1 + D / ( l - D = 0.2/1.8 = 0.111; so th resonator is undercoupled. By using a transmission line A /2 long between H resonator and the gate, we obtain Ts = V = - 0.8. The output circuit consists of a 50-il transmission line 0.213A long and an open-circuited 50-ft stub 0.144A long. This network transforms the 50-O load impedance to the required normalized value 0.6 + jO.906 at the drain terminal. The capacitive feedback normalized reactance of —j0.5 is obtained by using a short-circuited 50-fl stub of length equal to 0.4262A in the common source lead. In general, the resonator impedance coupled into the input microstrip line is given by n* £ — F.I — *.. 1 + 2jQ ">o in the vicinity of the resonant frequency <o0. If we assume that the resonator Q equals 500, then a 0.1 percent change in frequency will change T from - 0.8 to n2Z - 1 n2 - 1 - / - 8 -ffl r = —= -= = — = 0.895 L186.38° n2Z+l n 2 + l+j 10+^9 The new value of Z ou , becomes - 1 . 3 7 - y l . 4 1 , where we have assumed, for a 0.1 percent change in frequency, that there is a negligible change in the S,y parameters and the transmission-line length. If we used stronger coupling to the resonator, we would obtain a larger change in I" and this would improve the frequency stability of the oscillator, since the resonator would have a stronger control of the oscillation frequency. A better choice for I"s would be - 0.4 which gives n 2 = 3 / 7 and Z o u l = - 0 . 4 9 4 - j0.089. A 0.1 percent change in frequency would now give T= 3-7-77 3 + 7 +j7 = -4-/7 10+jl = 0.6605^205.3° and Z o u l = - 0 . 5 2 3 -7 0.905. The change in the phase angle of I" has been increased by a factor of about 4 by using the larger coupling. The frequency stability is also increased by about the same amount. The design of the oscillator circuit using f s = - 0 . 4 is left as a problem to be solved (Prob. 12.5). 12.7 MIXERS In Fig. 12.20 we show a block d i a g r a m of a microwave s u p e r h e t e r o d y n e receiver. T h e signal from t h e a n t e n n a is first amplified by a low-noise amplifier. After amplification, t h e signal is mixed w i t h a local-oscillator signal to o b t a i n t h e original signal t r a n s l a t e d to a m u c h lower i r e q u called t h e i n t e r m e d i a t e ( I F ) frequency. T h e mixer is a n o n l i n e a r device sue as a diode or d u a l - g a t e F E T . If t h e m i c r o w a v e carrier frequency is o>c OSCILLATORS AND MIXERS 857 Antenna V Jit Mixer IF amplifier Demodulator Signal processor Local oscillator F I G U R E 12.20 A block diagram of a microwave receiver. the oscillator frequency is « 0 . the nonlinear mixer device will produce signals at the IF frequency w!F = &>„ - «,., at the frequency <«0 + to,., and, in general, at many harmonic frequencies no>0 ± mm,, where n and m are integers. The signal at the IF frequency is further amplified, then demodulated, and finally processed for the intended output application. A single-tone AM signal with modulation frequency w m has the form (1 + Mcos <omt) cos o)rt = cos u>ct -I- jM[cos(w,. + wm)t + cos( w,. - wm)t] After mixing, the sideband spectrum becomes cos[(w ( . - <tf 0 )i + »mt] + cos[(w r - w0)C - bjmt] when the carrier frequency w e is greater than the local-oscillator frequency run. When the carrier frequency is less than the local-oscillator frequency, the sideband spectrum, after mixing, remains unchanged because an AM signal has symmetrical sidebands. Thus a local-oscillator frequency greater or smaller than the carrier frequency can be used. For an FM or phase-modulated signal of the form cos[<o,.l + <f>(t)] the spectrum, after mixing, is of the form cos[(wr - w„)r + <t>(.t)] when w,. > w0 and cos[(w0 - uic)t — <b{t)] when w,. < a>0. For the case when a>0 is greater than the carrier frequency wv, the sideband spectrum is reversed with the high frequencies becoming low frequencies, and vice versa. This phenomenon occurs because FM and phase-modulated signals do not have symmetrical sidebands. The subtraction of the spectrum of <f>( t) from the IF frequency reverses the high- and low-frequency components. In order to avoid this inversion of the signal spectrum, the local-oscillator frequency must be less than the carrier frequency in FM and phase-modulated systems. In this section we will discuss those mixer characteristics that are important from a systems point of view. For the purpose of this discussion, 858 FOUNDATIONS FOR MICROWAVE ENGINEERING Vce"*<' FIGUBE 12.21 A single diode mixer circuit. we will use a simple single-diode-mixer circuit to help clarify some of the operational characteristics of diode mixers. In the following section we will examine some of the more complex mixer circuits that enhance the overall mixer performance. In Fig. 12.21 we show a simplified circuit for a single diode mixer. The RF signal with carrier frequency w r is connected to the diode through a filter network with impedance Z R F . Similarly, the local-oscillator signal at frequency w 0 is applied to the diode through the impedance Z0. The IF signal appears across Z 1 F which represents the input to the IF amplifier. If a point contact diode is used, a dc biasing circuit is not used but the diode must have a dc current return path to ground which is through the RF choke RFC. When a Schottky barrier diode is used, a small forward bias is normally applied to the diode. This serves to overcome the barrier potential and increases the sensitivity of the diode. In a conventional p-n diode, the junction capacitance is quite large and will shunt the RF and LO signals across the junction, thus making these diodes ineffective as mixer diodes at frequencies greater than 1 GHz. The minority carriers also limit the diode recovery time. The Schottky diode, consisting of a metal-semiconductor junction does not have a depletion layer and also has very little store charge at the junction. Hence it has a very small junction capacitance. It usable as a mixer diode for frequencies in the microwave and millimeter-wa range and beyond. The IF frequency is much lower than the LO and RF frequencies, to example, if the RF frequency f e = 10 GHz and a typical IF frequency ot MHz is assumed, then the local-oscillator frequency must be (10 ± • GHz. In this case the IF frequency is a factor of 200 smaller than the and LO frequencies. Hence the capacitor C h in the circuit shown in 12.21 can be chosen large enough to short-circuit the high-frequency rents at the IF amplifier input. Consequently, the IF port is isolated OSCILLATORS AND MIXERS Ls fl. i—'TnRP—vw 859 Ideal diode -*- c, C„ F I G U R E 12.22 Equivalent circuit for a microwave diode. the RF and LO ports. On the other hand, the RF and LO frequencies are almost equal, so it is not practical to use filter networks Z 0 and Z R F that provide good isolation between the RF and LO ports. We can, however, assume that Z n and Z R F are essentially zero-impedance elements at the IF frequency. We could, for example, insert a series resonant circuit across the RF and LO ports (shown by dashed hnes in Fig. 12.21) with L and C chosen so that LCUJ\V = 1, where ct>l(. is the IF frequency. This circuit provides a zero-impedance path across the RF and LO ports at the IF frequency. At the RF and LO frequencies, jwL represents a very high reactance that shunts the RF and LO ports and can be neglected. The equivalent circuit of the diode is shown in Fig. 12.22. In this circuit we have an ideal diode shunted by the diode junction capacitance C, along with a series resistance Rs, a series inductance Ls, and a package shunting capacitance C . The ideal diode is described by the equation <,i = /s(ev""-l) (12.16) where irt is the diode current, / s is the reverse saturation current, v d is the voltage across the diode; y = e/kTn, where e is the electron charge, k is Boltzmann's constant, T is the absolute temperature, and n is a diodedependent parameter having a value between 1 and 1.5. At room temperatures y has the approximate value of 40. For the purpose of the discussion in this section, we are going to neglect all of the parasitic elements in the equivalent circuit of the diode. This can be justified only if we assume that the RF and LO frequencies are quite low. We will carry out a more careful analysis of the diode mixer in a later section. With the assumptions we have made, the mixer equivalent circuit for RF, LO, and IF signals reduce to those shown in Fig. 12.23. The network that the diode is embedded in has been assumed to have a zero impedance at all frequencies except those in the vicinity of the RF, LO, and IF frequencies.t Thus the only voltages that can exist across the diode are those at the RF, LO, and IF frequencies since all other frequency components are short-circuited by the embedding network. iWhen diode biasing is used, the RF and LO circuits must have nonzero dc resistance so as not to short-circuit the dc voltage applied to the diode. 860 FOUNDATIONS FOR MICROWAVE ENGINEERING + *- Z,s -Vr -•0 :& r (a) ? »f >d lb) (c) F I G U R E 12.23 (a) Equivalent circuit for RF signals in the mixer; (6) equivalent circuit for LO signals; (c) equivalent circuit for IF signals. The current i d through the diode can be expanded in a Taylor series about the dc operating point. This is equivalent to assuming a power-series expansion of the form h = J o aiv + a2v2 + a 3 t ; 3 + ••• (12.17) where / 0 is the dc biasing current, v is the voltage excursion about the operating point, and av, o 2 , a 3 , . . . are suitable coefficients. We will assume that the voltage v consists of a local-oscillator signal v0 = V0 cos w 0 /, an RF signal Uj = V, cos to^t, and an IF signal — vlF at the IF frequency <n0 - " I The IF signal has the form wIF = \V[F\cos(wiFt + </>), where d> is the phase angle. The complex-phasor IF voltage is Viy = |V IF |e-'*. When we expand (12.17) we obtain id = /„ + as(v0 + Vl - u, F ) + a2{vl + v\ + vjF + 2v0v1 - 2v0vlF - 2W,«IF) + a3(vl + u? - ufp + Sv'lv^ - Svluw + 3y 0 u? + 3f 0 ff F - S i ^ j p - Ba0vtvw) (12-1 8 ) In a number of published analyses of mixers using a power series sue as that in (12.17), it is assumed that the voltage across the diode consists only of the RF and LO signals. This assumption would imply that the embedding network that the diode is connected to has a zero impedance a the IF frequency, which is an unrealistic assumption. If power is to delivered to one port of the embedding network at the IF frequency, then the network must have a nonzero impedance at the IF frequency at OSCILLATORS AND MIXERS 861 port. The impedance of the embedding network at the various harmonics of the applied signals determines the harmonic voltages across the diode. Operation In practice, the local-oscillator signal has an amplitude much larger than the RF signal and the IF signal. When the RF signal amplitude is small, we can neglect terms that are proportional to higher-order powers of the RF and IF signals. Thus, when we retain only those terms that are linear in u, and v[F, the diode current is given by id =I0+ c , ( j ' 0 + t>, - vw) + a2(4 + 2v0vl - 2£/0d/IF) + a3{u% + 3v$vt - 3u 2 i;, F ) (12.19) A term such as u^ equals V02 cos 2 a>0t = iV<2 + \v*)cos2ajiit. The product term 2U 0 L , 1 equals V0V,[cos(w0 - «,)/ + cos(w 0 + co})t] and has an IF frequency component. The product term -2v(lvlF equals -V0|V,K|{cos[(a»0 wlF)t - <!>] - cos[(w0 + wlF)l + </>]} and does not contain frequency components falling within the IF amplifier passband. The term —3a 3 i>|oj F contributes an IF current component equal to - fcig^fvjF. The total IF current is given by j I F = - a , i / 1 F + a.y0Vl cos w, F r - fa 3 V ( fi;, F We now let ilF = Re / I K e"" I F ' and use phasor analysis to obtain ht = -«!*» + »*W - hsV$Vw At the IF frequency the circuit equation or constraint imposed by the embedding network shown in Fig. 12.23c gives VlF = / | F 2 I F . Thus we find that a o V f . Z • r- VIF= ,F : .,v V. 1 + ( o , + 1.5a 3 Vo a )Z lF ' (12.20) The power-series expansion up to terms in a 3 is valid only for small values of V0. Consequently, the term 1.5a:)V,2 is usually small relative to a,. The above equation shows that under the assumptions made, the voltage at the IF amplifier input is linearly proportional to the RF voltage amplitude V,. In this operating range the mixer functions as a linear mixer. The conversion loss, in decibels, for the mixer is given by available RF power L = conversion loss = 10 log——— (12.21) IF input power Typical values for conversion loss for a single-diode mixer are 6 to 10 dB. 862 FOUNDATIONS FOR MICROWAVE ENGINEERING N o n l i n e a r Mixer Operation For larger values of the RF voltage, we must take additional terms ; m account. From the term multiplied by a3, we have a term - 3 a a w f & 1 P = -1.5a 3 V, 2 U| K - 1.5a 3 V 1 2 u I F cos2w 1 i and a term - a 3 u ? F = -a.jVlF\3 cos3(o]Ft + <j>) = - 0 . 7 5 a 3 | V I F | 3 c o s ( w , F i + <£) - 0.25a 3 |V I F | 3 cos(3w 1F / + 3$) The IF frequency component of this latter term can be expressed as — 0.75a 3 |V ir ,| 2 t/ 1F . We can use (12.20) to obtain an approximate solution for |V IF | 2 and then find that a 2 V n Z 1F V, V!y = l + [a1 + 1.5a3(V02 + V 2 ) + 0.75a 3 |V 1 F l 2 ]Z I F VIF + at + 1.5a3V02 , «M V2 a 2 V 2 |Z IF | 2 V02 211 + (a, + 1.5a3V02)ZIF|2 (12.22) where YlF = Z^F. This equation shows that the effect of the nonlinear terms - a 3 i ; 3 F and - 3 a 3 u 2 L ' I F is to reduce the IF voltage. Thus a mixer will exhibit nonlinear saturation and the range of allowed input RF voltage amplitudes (dynamic range) must be limited in order to avoid nonlinear distortion of the down-converted RF signal. The dynamic range of a mixer is specified by giving the RF power level at which a compression of 3 dB occurs in the IF power.t In Fig. 12.24 we show a plot of IF power versus RF input power and the 3-dB compression point. The units are dBm or decibels relative to 1 mW. The nonlinear behavior of a mixer also results in intermodulation distortion. Consider an input signal consisting of two closely spaced sinusoidal signals at the frequencies w, and w2, where |w, - io2\ « « I F - l e presence of two closely spaced RF signals results in the generation of a larger number of frequency components. For linear operation we obtain IF signals |VIFI|cos[(a«0 - uix)t + <£,] and |VIF2|cos[(o>0 - <o2)t + <b2i T h e ' £" bic term -a 3 i> 3 F will result in new additional frequency components come from the products VjFlulF2 and ulF1v'iF2. The new frequencies g e ^^ ated, which fall within the IF amplifier passband, are a>0 - 2<ol + <*2 tSometimes the 1-dB compression point is used. OSCILLATORS AND MIXERS 863 3-dB compression point -40 -30 -20 -10 RF input power, dBm FIGURE 12.24 Illustration of nonlinear mixer response and the 3-dB compression point. - 2w 2 + w,. We will let o>1F:i = (w n - w,) + (w 2 _ w0 a n ^ -• i r--i = (u>0 a>2) + (o)| — &>2). For a self-consistent solution for (12.18), we see that we must assume the presence of at least four IF signals, namely, w V \F = V lFl COS( W, KI < + ( / > , ) + V , F 2 C O S ( w , F 2 f + (j>2) + V, F 3 cos(«| K 3 * + d>3) + VIF., cos(w I F 4 / + d>4) (12.23) When v I F 3 and ulFA are included, then the term multiplied by o 3 will also have product terms of the form vjPlvlF3, v'iP3vn-4, etc. These product terms result in additional new frequency components that fall within the IF passband. Since vlFl and L'| F 2 are proportional to V, and V2> we s e e that 2 2 I>I F3 and uiV4 will be proportional to V| V*2 and V2 V,, respectively. A term such as i^f^ip;! is thus proportional to V*V2 and can be neglected since its amplitude will be much smaller. The production of new frequency terms that arise from the mixing of vSF1 and ulF2 is called two-tone intermoduiation distortion. If V, = V2 = V these terms are proportional to the third power of the RF signal amplitude. The IF power associated with the intermoduiation terms will increase proportional to the third power of the input RF power. The intermoduiation distortion of a mixer is specified by giving the RF power level at which the intermoduiation IF power becomes equal to the IF power associated with the desired IF signals t;)F, and v 1F2 . This is a theoretical value that is obtained by extrapolating the linear mixer respon.se or IF power output at the desired frequencies <0|F| and <0|F2 and that of the IF power output for the intermoduiation terms until they meet. On a decibel scale the slope of the curve giving the intermoduiation IF power is three times that of the 864 FOUNDATIONS FOR MICROWAVE ENGINEERING / / l . Third-order intercept Intermodulation IF power -40 -30 -20 -10 RF input power. dBm F I G U R E 12.25 Illustration of diode mixer response for the desired IF power and the IF power at the intermodulation frequencies. The intersection of the two extrapolated response curves gives the third-order intercept point. curve giving the desired IF signals. The intercept point is called the thirdorder intercept and is illustrated in Fig. 12.25. In practice, the RF input power to a mixer must be kept below the 3-dB saturation point and also below the third-order intercept point to avoid signal distortion. When the amplitude V 2 of the RF signal at the frequency co2 is held constant, but Vj is increased, then the slope of the line giving the intermodulation power in the IF passband will be 2. 12.8 MIXER N O I S E F I G U R E The noise figure of a mixer is another important parameter that specifies the performance of a mixer. Thermal input noise at frequencies around <o0 - w I F will be down-converted into noise in the IF passband. Thermal noise at the image frequencies around w 0 + « I F when mixed with the LU signal will also fall within the IF passband. The mixer will also introduce additional noise with frequency components in the IF passband. additional noise arises primarily from thermal noise in the resistive ' ments of the mixer and from shot noise. The noise figure of a mixture is defined by the relation F = (12.24) SJN0 where S,/N, is the input signal-to-noise power ratio and S0/Nn lS output signal-to-noise ratio. If the local-oscillator frequency is <o0, the signals at <o0 + <o]F and « 0 —w IF are converted to IF signals. quency u>0 + coie is called the upper sideband and the frequency w 0 ~~ the _ .fl a OSCILLATORS AND MIXERS 865 the lower sideband. Noise at both sidebands is converted into noise in the IF passband. In most systems the signal is present in one sideband only. The unused sideband is called the image frequency. When the signal is present in one sideband only, the noise figure is designated as the single-sideband (SSB) noise figure F S S B . If the signal is present in both sidebands, the corresponding noise figure is called the double-sideband (DSB) noise figure and denoted by FDSB. A double-sideband signal would be of the form A[l + m(t)][cos(<u0 + (Ojp)t + cos(<u0 — wlF)t], where m(t) is the modulation. When down-converted the IF signal voltage will be proportional to 2A[l + rn{t)]cosa)1Ft, and consequently, the IF power is increased by a factor of 4. Thus for double-sideband signals, the ratio Sl/Su is one-half of that for a single-sideband signal since the output IF power is increased by a factor of 4 but the input RF power is increased by a factor of 2 only. This assumes that the mixer conversion loss is the same for both sidebands. As a consequence of the above, the single-sideband noise figure is twice as large as the double-sideband noise figure, that is FSSB = 2 ^ D S B - The double-sideband noise figure is easiest to measure since noise sources are usually very broadband and would provide noise signal input in both sidebands. Schottky diode mixers have typical single-sideband noise figures in the range of 4 to 8 dB. The IF amplifier noise figure will also increase the noise figure of the mixer-IF amplifier combination because of the mixer conversion loss. The noise figure of the mixer-IF amplifier combination is given by F = Fm + Lr(FlF- 1) (12.25) where F m is the mixer noise figure, F w is the IF amplifier noise figure, and L(. is the numerical value of the mixer conversion loss. As an example assumes that the conversion loss is 4 (6 dB), the IF amplifier noise figure is 2 (3 dB), and the mixer noise figure is 4. The noise figure of the combination is thus 8 which is a 3-dB increase over that of the mixer by itself. Since the signal level at the output of a mixer needs additional amplification, the mixer conversion loss has a significant effect on the overall noise figure. For this reason it is important to have sufficient signal amplification ahead of the mixer. The noise figure of the preamplifier will then govern the system noise figure. 9 BALANCED MIXERS There are several disadvantages associated with a single-diode mixer. These are: no isolation between the RF and LO ports, poor isolation between the IF port and the RF and LO ports, a high level of oscillator noise input to the IF amplifier, and the generation of many spurious signals. Balanced mixers can improve the characteristics of a mixer and alleviate many of the listed shortcomings. The basic circuit for a single balanced mixer is shown in Fig. 12.26. The two diodes need to be well matched in their electrical characteristics in 866 FOUNDATIONS FOR MICROWAVE ENGINEERING ^Vtfi 1 -^^J,fl F I G U R E 12.26 Basic circuit for a single balanced diode mixer. order to achieve a well-balanced mixer. the circuit, we can see that the voltage while that acting across diode D 2 is u 0 a 2 in the expression (12.18) for the diode l d\ From the symmetry properties of across diode D x is u 0 + v - v u x + u I F . If we retain terms up to current, we find that for diode D = I0 + o^Uo + Vt - i/ IP ) + a 2 (ug + vf + v^ + 21;,,^ - 2u 0 u I F - 2u,ulF) while for diode D 2 , l d2 = h + Ol("o _ y I + U IF) + Q 2(t;0 + U l + 1>IF - 2 ^ 0 ^ , + 2y 0 y I F - 2v1vlF) The input current to the IF low-pass filter is •dl — i dl = 2 a 1 ( u , - o,p) + 4 o 8 ( u 0 o 1 - u 0 o 1 F ) The IF frequency component is given by / 1 F = - 2 a 1 V I F + 2a 2 y 0 V 1 (12.26) From the above equations we see that there is no local-oscillator voltage at the IF port so the LO and IF ports are isolated. Amplitude-modulation noise from the oscillator, as well as up-converted (or down-converted) thermal noise in the oscillator circuit, will consequently not be present at the I* input. From an examination of the circuit in Fig. 12.26, it is also apparent that the LO and RF ports are isolated. , For broadband mixers operating up to 1 GHz or so, the single balanced mixer can be built using transformers consisting of windings on a ' e r r toroidal core. At microwave frequencies the single balanced mixer can constructed using a 180° 3-dB hybrid junction such as t h e magic T or hybn ring described in Chap. 6. A single balanced mixer using a magic T is^sno in Fig. 12.27. A 90° hybrid junction can also be used, but the mixer will the = not be fully balanced with respect to LO and IF port interactions. In a hybrid junction reflected waves at the output port appear at both m P ports with a phase that makes the reflected waves add out of phase. OSCILLATORS AND MIXERS 0 867 Signol F I G U R E 12.27 Single balanced mixer using a waveguide magic T. the LO and RF port input VSWRs tend to be small. For a 180° hybrid junction, the reflected waves add in phase at each input port so the resultant input VSWRs are higher for the same degree of output impedance matching. A VSWR of 1.5 using a 90° hybrid junction can be readily achieved, whereas the VSWR for a 180° hybrid junction is around 2. It is relatively difficult to provide a good impedance match between a diode and a transmission line over a wide band of frequencies. By using the four diodes in a bridge network, a double balanced mixer can be built. The basic circuit for a double balanced mixer is shown in Fig. 12.28 and provides isolation between the RF and LO ports, as well as between the IF and the RF and LO ports. In addition, many more spurious signals cancel at the IF port. The double balanced mixer suppresses the even harmonics of both the RF and LO signals and hence has a lower level of intermodulation distortion than that of the single balanced mixer. At the lower frequencies double balanced mixers are readily constructed using transformer hybrids. At microwave frequencies transmission-line baluns are used and the overall mixer configuration can become quite complex. F I G U R E 12.28 Basic circuit for a double balanced mixer. 868 FOUNDATIONS FOR MICROWAVE ENGINEERING OTHER TYPES OF MIXERS When the local-oscillator frequency is higher than the RF signal frequen cies, the IF spectrum contains those signal components in the frequencv band o>0 - <uIF. The undesired frequencies in the image band GJ0 + w osuppressed by means of a filter ahead of the mixer. It is possible to arranep two mixers in such a manner that IF signals corresponding to signals in the upper and lower sidebands are down-converted and appear at separate output ports. The unwanted sideband signals can be dissipated in a resistive termination. A mixer of this type is called an image-rejection mixer and is particularly useful in wideband signals where it is difficult to eliminate signals in the undesired sideband by filtering. The up-converted signal proportional to V0V, cos(&>0 + w,)( represents a transfer of RF power to an undesired frequency. By reactively terminating this frequency component, it can be reflected back to the mixer and mixed with the second harmonic of the LO signal to produce a desired IF signal. With a proper adjustment of the circuit parameters, the conversion loss can be reduced by 1 to 2 dB. This type of mixer is called an imagerecovery or image-enhanced mixer. At millimeter wavelengths it is not always possible to build a suitable oscillator at the very high required frequency. In such circumstances a lower-frequency oscillator can be used and the desired IF frequency signals can be obtained by mixing the RF signal with one of the harmonics of the oscillator. These mixers are referred to as subharmonic mixers. By using two well-matched diodes connected in antiparallel as shown in Fig. 12.29, a second harmonic mixer is obtained. This particular diode arrangement results in an absence of all mixing products that involve odd harmonics of the local-oscillator signal. FET mixers can also be configured to operate as balanced mixers, although current practice is to use an FET in an unbalanced circuit configuration. Discussions of F E T mixers can be found in the references given at the end of this chapter. F I G U R E 12.29 Basic circuit of a subharmonic m where the RF signal is mixed with second harmonic of the oscillator sig" OSCILLATORS AND MIXERS 869 j2.11 MIXER ANALYSIS USING HARMONIC BALANCING A power-series expansion of the diode current, such as that given by (12.17), is valid only when the voltage across the diode is of the order of a few millivolts. In practice the local-oscillator-signal voltage across the diode is much larger in order to obtain a relatively low conversion loss. The harmonic-balance method of mixer analysis is based on first analyzing the nonlinear mixer circuit with only a local-oscillator signal applied. This leads to a description of the diode in terms of a time-varying conductance and capacitance. In this section we will develop the basic concepts associated with the harmonic-balancing method, which are similar to those used in describing the operation of parametric amplifiers in Chap. 11. The equivalent circuit for the diode is shown in Fig. 12.22. The junction capacitance Cj is a function of the voltage across the ideal diode junction. The other circuit elements, namely Rs, Ls, and Cp, can be considered to be part of the embedding network. We can replace the network as seen from the terminals of the ideal diode by a Thevenin equivalent circuit which includes an equivalent local-oscillator source as shown in Fig. 12.30a. The impedance Z(w) of this network has a different H Zlu.) LO (a) lb) F I G U R E 12.30 (a) Thevenin equivalent circuit of a diode mixer as seen from the terminals of the ideal diode; (6) Thevenin equivalent circuit of a diode mixer at m = 0. 870 FOUNDATIONS FOR MICROWAVE ENGINEERING value at each harmonic n ai n of the local-oscillator signal. The total current iT flowing through the diode and junction capacitor is iT = id -+- j -pi current through the capacitor is given by , dQ(vj) dQ^dv^ dt ' dVj dt where Q(vj) is the charge stored at the junction. The incremental capacitance dQ/dvj = CjiVj) is a voltage-dependent capacitance since, in general Q is not a linear function of Vj. The current i d through the ideal diode is given by id = Is(e^ - 1) When only the local-oscillator signal is applied to the circuit, the voltage Vj and the currents iT,id,ic will contain all harmonics naj0 of the fundamental frequency a>0. Hence we can write jj=» — x x «&- E v^"-"' n = — <x h- E ic„e>«»« Since i d and v t are real, /_„ = 7 * and V_n = V*. The voltage i>, is periodic, so CjiVj) is a periodic function and has a Fourier series representation of the form oo C,= E C„e-"""°' n = —w where the C„ will be dependent on Vj. C, is a time-varying circuit element. The presence of a dc bias voltage simply changes the values of J0, V0 of the n = 0 terms. Consider now the presence of an additional small-signal voltage across the junction, with \v\ •« \vj\. The current through the diode is now given by This equation can be expanded in a Taylor series about Vj, the dyn 3 1 " OSCILLATORS ANI> MIXERS 871 operating point; thus dQ(vj) ir + i-IAt*j-l) + d IdQ(Vj) dt dv, dt ) + d dt The current i is given by dC, \ dv i = \yl„e^ + -£\v + CJ(VJ)~ (12.28) This is a linear differential equation of first order with time-dependent coefficients. Under small RF signal conditions the diode behaves as a time-varying linear circuit element and superposition holds for the RF signal. If an RF signal at frequency w, is applied to the circuit, the time-varying diode-operating characteristics will cause currents and voltages at all the frequencies co1 + nio0, n = 0, ± 1, + 2 , . . . , to be generated. Hence ix and y, will have the expansions ;l " - * ,i = - X There are complex-conjugate terms at - w , . The term ylseyv' = y(id + Is) = yid since ls is very small. Thus yIsey"-> can be replaced by yid. Each term y/„ has the dimensions of a conductance and will be called g„, that is, yln °° Sn- From the basic equation describing the diode small-signal behavior x I — Y i + ei<«i **»«M n " —« + L Cs £ vJ(o}l + n<o0)e^W«'-"">' (12.29) s — - x n= — x The harmonic terms on each side of this equation must be equal (principle of harmonic balancing); so upon putting w, + (s + n)io0 = w, + mwp, where m = s + n, we can write q - -or = £ [*«-„ +./(«i + ™ « o ) C m _ J y n (12.30) 872 FOUNDATIONS FOR MICROWAVE ENGINEERING In matrix form we get [»»] = [Yam][vm] (12.31) where Ynm = gn m +j(<o1 + nu0)Cn_,n. The [Ynm] matrix describes the mixing action of the diode. It relates the harmonic amplitudes of the Rp signal current through the diode to the harmonic amplitudes of the Rp signal voltage across the diode junction. Each harmonic current i ej( has a constraint imposed on it by the external circuit (embedding network) in that the voltage around any closed loop, including the diode, must sum to zero(Kirchho£f s law) at each frequency OJ1 + nw 0 . These circuit constraints will be given later. In order to determine the elements in the mixer conversion matrix [Y„ m ], it is necessary to know the function that describes the junction capacitance CJ(VJ) in terms of the junction voltage. The solutions for iT, j ic, and Vj are constrained by circuit relations imposed by the embedding network. The Thevenin equivalent circuit of the mixer for dc currents is shown in Fig. 12.30b. At w = 0 we must have VB = I0(R(I + RS) + V0 (12.32a) At the local-oscillator frequency, we have •j =ITiZ(wu) + V, (12.326) ^ = IT_XZ( - o , 0 ) + V.x = IfcZ*(«»o) + v* (12.32c) At all other harmonics of the local-oscillator frequency 0 = ITnZ(nco0) + Vn (12.32d) These equations along with the known function C,(y,) and the ideal diode equation must be solved in order to find the current amplitudes / „ , the voltage amplitudes V„, and the capacitance elements C„. Numerical methods are required because of the nonlinear - behavior of the diode. After the elements of the mixer conversion matrix have been found, we can solve the small RF signal problem by solving a linear network problemThe embedding network can be replaced by a Norton equivalent network a each frequency w l + mcon as shown in Fig. 12.31. The RF signal source can be chosen as Ieoej"'>'. The circuit equation at the frequency w, is / V " • ,- Z(ft»!) ^ + Z(t»x) y v „ (12.33a) m=_x At the frequency <D1 + mo0 the circuit equation is v„ Z(iov + nw0) L Ynmvm = 0 (12.336) , + no>o) At the frequency w, + nw 0 the current through the impedance Z(o»i1 OSCILLATORS AND MIXERS 'o , 1 I** " (T k 873 — t zu,> v0 '1_» 1 Z((U, + <D„) 1' l KJ ' 2 _ Zio>, + 2ai0) V2 '*_•• Zito, + nio0) Vfi FIGURE 12.31 Equivalent circuit for a mixer for linear RF signal response. is given by t h e first t e r m on t h e left-hand side in (12.336). T h e c u r r e n t i n is given b y t h e second t e r m i n (12.336). T h e e q u a t i o n s t a t e s t h a t t h e s u m o f t h e s e two c u r r e n t s m u s t be zero. I n principle, t h e solution o f t h e mixer problem u s i n g h a r m o n i c b a l a n c ing is s t r a i g h t f o r w a r d . In practice, it is a complex problem t h a t r e q u i r e s n u m e r i c a l e v a l u a t i o n u s i n g a c o m p u t e r p r o g r a m . Space limitations do n o t allow us to develop t h e m e t h o d in g r e a t e r detail. However, s o m e of t h e references given at t h e end of t h i s c h a p t e r provide m o r e details. 12.1. The three-port scattering-matrix parameters are given by (12.10). Verify that the sum of the elements in any row or column equals one. 12.2. In a transistor a series impedance producing a reflection coefficient F is connected in the common lead. Show that the equivalent two-port network scattering-matrix parameters are given by (12.11) upon replacing S33 + 1 by •S 33 - T . 12.3. For the unstable bipolar transistor in Example 12.2, find the smallest series resistance to be added in the common emitter lead to obtain an absolutely stable two-port network. Use the computer program TRIPORT. 874 FOUNDATIONS FOK MICROWAVE ENGINEERING 12.4. Design an oscillator following the procedure described in Example fj assume that [*, = 0.9/1225°. For this design it will be necessary to desien Ut matching network that will transform a 50-il input termination into a sou a impedance at the base that corresponds to the chosen value of r . 12.5. Redesign the oscillator in Example 12.3 using a resonator coupling with n 2 = 2 / 7 . Specify the lengths of all transmission lines used in the oscillato circuit. 12.6. Design a 10-GHz common gate oscillator using a GaAs MESFET having the following scattering-matrix parameters: Su = 1.1/1160= S , , = 0.16^130° S 2 , = 2.7/1 - 62° S 2 2 = 1.4.1 - 57° Use a pure reactive termination for the input (source terminal) and a circuit topology similar to that in Fig. 12.17. The parameters of the stability circles can be found by using the MICROAMP program (arbitrary values for T and noise resistance can be entered. Arbitrary gain circles can also be chosen since this information is not used). The output impedance for the chosen value of r, can be found using the TWOPORT program but the scatteringmatrix parameters must be entered in rectangular coordinate form (real and imaginary parts). 12.7. Design an 8-GHz MESFET oscillator similar to that in Example 12.3 using a device with the following common source scattering-matrix parameters: 12.8. 12.9. 12.10. 12.11. 12.12. 12.13. S , , = 0.25^8° S 1 2 = 0.11^145° S«, = 3.5/1168 0 S 2 2 = 0.43/1 - 60° This device is absolutely stable so a series reactance must be connected into the common source lead. Choose this reactance to obtain a value for the stability parameter K that is less than 0.6. For the oscillator in Fig. 12.19 the dc gate voltage is applied through a high-impedance line A/4 long. The feed line is connected at a point where Z m = Rm. Find the distance from the load end at which the feed line should be connected. Show that a term such as V^cos 3 w,/ has signal components at the two frequencies o>, and 3o>lt while V 4 cos 4 u,t has components at the frequencies 0, 2<t>,, and 4<u,. Show that a term such as V,2V22 cos 2 w^ cos 2 ui.J has signal components at oi = 0, 2a),, 2<o2, a n d 2(<o, ± a>2). Find all of the frequencies generated from the term multiplied by aA H> (12.18). When a 3 |V, F | 3 « a 2 |V 1F |, show that, by using (12.20) to evaluate the term - fa 3 |V, F | 2 in the expression for the diode current, in place of (12.20) the input voltage is given by (12.22). 2 When (12.19) with terms up to a 2 is used to evaluate ily and v\fl and Vffv show that the term -3a.,((;?,I F l t I F 2 i- v2W2v, ) results hVthe following contra the If bution to the intermodulation component of the diode current in OSCILLATORS AND MIXERS 87i amplifier p a s s b a n d : ~3q3<^V'' [V?V 2 c a s t a ) - 2w, + wz)i + V, V22 c o s t a - 2 « , + 2|l+fl,ZiF|2' ' t o l )r} „^V°|Z|J 2(1 + n , Z i F Assume that Z,K is constant in t h e passband. 1 2 . 1 4 . A m i x e r h a s a noise figure of 8 dB a n d a conversion loss of 10 d B . T h e If amplifier noise figure is 3 d B . F i n d t h e noise figure of t h e c o m b i n a t i o n . 1 2 . 1 5 . Find t h e noise figure of a mixer a n d IF amplifier c o m b i n a t i o n if t h e c o n v e r sion loss is 5 d B . t h e m i x e r noise figure is 6 d B , a n d t h e IF amplifier noise figure is 4 d B . REFERENCES 1. Chang, K.: "Handbook of Microwave and Optical Components. Microwave Solid Stat* Components.'' vol. 2, John Wiley & Sons. Inc., New York, 1990, 2. Vendeiin. G. D., A. M. I'avio, and U. L. Rohde: "Microwave Circuit Design Using Linear and Non-Linear Techniques," John Wiley & Sons. Inc., New York, 1990. 3. Bahl I., and P. Bhartia: "Microwave Solid-State Circuit Design," John Wiley & Sons, Inc.. New York, 1988. 4. Maas, S. A.: "Microwave Mixers," Artech House Books, Norwood. Mass., 1986. 5. Maas. S. A.: Two-Tone Intermodulation in Diode Mixers. IEEE Trans., vol MTT-35. pp. 307-314. 1987. 6. Koilberg, E. L. ted.>: "Microwave and Millimeter Wave Mixers," IEEE Press, Piscataway. N.J,, 1984. 7. Kerr, A. R.: A Technique for Determining the Local Oscillator Waveforms in a Microwave Mixer, IEEE Trans., vol. MTT-23, pp. 828-831. 1975. 8. Held, D N.. and A. R. Kerr: Conversion Loss and Noise of Microwave and MillimeterWave Mixers. I. Theory. IEEE Trans-, vol. MTT-26. pp. 49-55, 1978. APPENDIX I USEFUL RELATIONS FROM VECTOR ANALYSIS 1.1 VECTOR ALGEBRA Let vectors A and B be expressed as components along unit vectors a,, a 2 , a 3 in a right-hand orthogonal coordinate system. Then A±B = (A1±Bl)a1 + (A2±B2)a2 + (A3±B3)a3 A • B = |A| IBicos e = A1B1 + A.2B2 + A3B3 (1.1) (1.2) where 0 is the angle between A and B. A X B = a,( A.,B3 - A3B2) + a3(A1B2 + a 2 ( A 3B t - A tB 3) -A2BX) | A x B i = |A||B|sinfl A-BXC=AXB-C=CXA-B A x B = - B XA A X ( B X C) = (A • C ) B - ( A • B ) C 876 (I.3o) (1.36) (1.4) (1.5) (1.6) USEFUL RELATIONS FROM VECTOR ANALYSIS 877 FIGURE 1.1 Rectangular coordinates. 1.2 VECTOR OPERATIONS IN COMMON COORDINATE SYSTEMS Rectangular Coordinates d<$> d<& V<t> = ax-— + a . , — dx - dy M, divA = V • A = —- + dx curl A = V X A == a, <?* (1.7) + a,-r— z dz 3A¥ AA. —- •+dy dz + a, (1.8) fa7 <?2 MY 3y ) (1-9) <?2<t> <?2* a2* dx- Sy dz' v 2 * = — s + —^ + 2 (1.10) V2A = a,V 2 A, + a v V 2 A v + a,V 2 A z (Ml) Cylindrical Coordinates d(t> 1 a* d$ V<l> = a r — r r d(t> *••»"£ dr l M. aA. l a V-A = - —(rAr) + + r a^> e>z r c'r 7 A * -^7« (1.12) (1.13) ld(rA.) + + " * 7 J M V " *r/ **[r" 0T 1 Mr ' r d,b (1.14) 1 a a* V** = —— r — 1 «92* V2A = r r - A - vx rx A a2<P (i.i6) Note that V2A # a,.V 2 A r + a,,, V2Arf, + a z V 2 A e since V 2 a r Ar * a r V 2 A r , etc., because the orientation of the unit vectors a r , a,,, varies with the coordinates r,(}>. 878 FOUNDATIONS FOR MICROWAVE ENGINEERING .•v.* FIGURE 1.2 Cylindrical coordinates. Spherical Coordinates dQ> 1 d<P V* = a , — + aflSr " r dO 1 d „ r* dr 5<t> 1 d — (sin 9 A„) r s i n 0 dd dA VXA r sin 0 + V2<1> = a ** — (A, sin0) 5 — (rA9) e> dr( (1.17) r s i n 6 d<t> 9A.< (1.18) r sin 0 d<f> 1 dAr 1 s i n 0 d({> d (rA \ drK *' M. (1.19) dB 1 ('J / a * \ 1 3 ( d<t> \ rl— + — — sin©— I + do r2 dr\ dr r 2 sin 0 38 V 2 A = VV • A - V X V X A FIGURE 1.3 Spherical coordinates. 1 2 r sin 2 0 dtf (1.20) (1.21) USEFUL RELATIONS FROM VECTOR ANALYSIS 3 8 79 VECTOR IDENTITIES V($(A) =<AV4> + <t>V</' (1.22) V-(^A) = A « V^ + tfrV-A V - ( A x B ) = (VX A) -B - (V X B ) • A V X (e/<A) = (Vtfi) X A + (AV X A V x (A x B) = AV • B - BV • A + (B • V)A - (A • V)B (1.23) (1.24) (1.25) (1.26) V(A • B) = (A • V)B + (B • V)A + A x (V x B) + B x (V x A) V • Y<S> = V 2 * (1.27) (1.28) V-VxA=0 (1.29) V X V* = 0 (1.30) V X V X A = VV • A - V2A (1.31) If A and 4> are continuous functions with at least piecewise continuous first derivatives in V and on S (or on S and the contour C bounding S), J( v J(V v V<P dV =T6<t> dS s • AdV = (f)AdS T s J[VxAdV v = (f>nXAdS T s (1.32) (divergence theorem) (1.33) dS = ndS (1.34) jnxVQdS = <f)Q>d\ fVxA-dS = (f)A-d\ (1.35) (Stokes' theorem) F I G U R E 1.4 Surface S bounded by contour C. (1.36) 880 FOUNDATIONS FOR MICROWAVE ENGINEERING GREEN'S IDENTITIES If A, B, <P, and i/> are continuous with piecewise continuous first derivatives J(V<D • V0 + <AV2<D)cfV = c£ l //VcD-dS J J v s which is Green's first identity. Green's second identity is J ( ^ V 2 < i > - cDV2</<)dV = (J>(iAV<D - <J>V./0 • dS (1.37) (1-38) In two dimensions (1.37) becomes /' (V,* • V,i/» + >/, V,24>) dS = r6il>Vt<t>-Tdl (1.39) s c where V, is the del operator in two dimensions and T is a unit vector normal to C and in the plane of S. The two-dimensional form of (1.38) is similar. The vector forms of Green's identities are J f v - ( A x V x B ) d V = f [ ( V X A ) -(V X B ) - A - V x V X B ] d V J •V v = $ A X (V X B ) - d S (1.40) /"(B-VxVxA-A-VxVxB)dV J v = (£[AX (VXB) - B x ( V x A ) ] -dS (1.41) APPENDIX IT BESSEL FUNCTIONS 1 ORDINARY BESSEL FUNCTIONS The wave equation and Helmholtz's and Laplace's equations are separable in cylindrical coordinates. The differential equation describing the radial dependence of the solution is BesseVs differential equation. BesseVs equation is 1 d df r dr dr / , \ n•2 r~ J When k2 is real and positive, the two independent solutions of Bessel's equation are called Bessei functions of the first and second kind, denoted by J„(kr) and Yjkr), respectively. These solutions may be expressed as power series as follows: JAkr). Yn(kr) = f '-""<^>;;„~0 2 - _\_ m\(n+m)\ kr\ 1*-* + In — \Jn(kr) - - E 2 } ^„fTn £ TT„, = 0 (-l)m(kr/2)"'2m (n-m-ni! : m\ 2 \» T" \kr) m\(n + m)\ 1 1 1 1 1 xi + - 2+ 3- + - - - + — m+ i + - 2+ 3-r + --- + n + m I (II 3) 881 882 FOUNDATIONS FOR MICROWAVE ENGINEERING where y = 0.5772 is Euler's constant. The subscript n denotes the order of the function and is usually an integer in physical problems. The Y n fum. tions become infinite at r = 0. For large values of kr, the Bessei functions approach damped sinusoids: z -rr nrr (II -4a) —— cos kr — irkr 4 2 I IT nn lim Yn{kr) = (11.46) —— sin kr — irkr , 4 2 A few of the lowest-order Bessei functions are plotted in Fig. II. 1. To represent radially propagating waves, linear combinations of the J and T„ are formed, called Hankel functions of the first and second kind. Thus the Hankel function of the first kind is lim«/„(Ar) = Wn(kr)=Jn(kr)+jYn(kr) (11.5a) and the Hankel function of the second kind is given by H^kr)=Jn(kr)-jYn(kr) 1.0 'Vo Y-\ 0.5 0 0.5 (11.56) z & V \ ^/~ - r 0.5 ° * ^ > & - Ys 0 \ / f, ~2x y* v»y \6 A 8/ /WV / -0.5 -1.0 I / // F I G U R E H.1 Ordinary Bessei functions. BESSEL FUNCTIONS 883 For large values of kr, the Hankel functions are given by the following expressions: (11.6a) -j(Ar-7r/4-mr/2) (11.66) Some useful relations that hold for any of the Bessel functions Jn, Yn, or H n are given below, where Z„ denotes J „ , Y„, or Hn. xZ;,(x) =nZn(x) -xZntl(x) = -nZn(x) + xZ„_,(x) (II.7) where the prime denotes differentiation with respect to x. f Zn(kx)Zn(lx)xdx = ^—^[kZa(lx)Zn+1(kx) - lZn(kx)Z„.1(lx)} (II.8) *-2 fzl(kx)xdx 2 Z'n<(kx) + \1- k22XV 2 \ZZ(kx) (II.9) MODIFIED BESSEL FUNCTIONS When A 2 is negative, k is pure imaginary. If we let k = jh, the solutions are given by JJjhr) and Yn(jhr). However, for convenience, new modified Bessel functions are introduced and denoted by I„(hr) and K„(hr). The modified Bessel function of the first kind is I„(hr), and is given by In(hr) =j'"Jn(jhr) =j"Jn( -jhr) (11.10) and the modified Bessel function of the second kind is given by Kj.hr) = \jnil[Jn(jhr) +jYn(jhr)\ = ^J^B&kr) (H.11) For large values of hr we have Uhr)~ Kn(hr) (II.12a) •J2irhr ~ 77 ,-hr 2hr The first few modified Bessel functions are plotted in Fig. II.2. (11.126) 884 FOUNDATIONS FOR MICROWAVE ENGINEERING 5 x FIGURE II.2 Modified Bessel functions. A number of useful relations that hold for the modified Bessel functions are given: xl'n(x) = nl„(x) + xln+l(x) = -nln(x) + * / „ _ , ( * ) (II.13a) Io(x)=I1(x) (11.136) fx-"In 1(x)dx=x-"In(x) + fx"In_1(x)dx = (II.14o) (11.146) x"In(x) When n > - 1, we have f\(kx)In(lx)xdx = —^[kln(lx)ln^(kx) - lln{kx)ln + l(lx)} (11.15) -X- rn\kx)-\\ ffZ(kx)xdx= xK'n(x) = nRn(x) K'0(x) = -K^x) fx-"Kr^l(x)dx= fx"Kn_l(x)dx= -xKm+l(x) -x-"Kn(x) -x"Kn(x) + = k22X„ 2 (II.16) n(kx) -nK„(x) - xKn.x(*) (II. 17a) (11.176) (II.18a) (11.186) BESSEL FUNCTIONS 885 When Re(k + 1) > 0, we have iy,Akx)K,,(,lx)xdx = -^-j-2[kKn(lx)Kn^kx) - lK„(kx)Kn^(lx)} (11.19) For Re k > 0 j^xK?,(kx)dx-- x2 K'n<{x)-\1 + kh K*(kx) (11.20) REFERENCES 1. Watson, G. N.: "Theory of Bessel Functions." Cambridge University Press, New York. 1922. 2. McLachlan, N. W.: "Bessel Functions for Engineers," 2d ed., Oxford University Press, Fair Lawn, N.J., 1948. 3. Bowman. F,: "Introduction to Bessel Functions," Dover Publications, Inc., New York, 1958. APPENDIX in CONFORMAL MAPPING TECHNIQUES The determination of the distributed capacitance and inductance of a transmission line requires a solution of Laplace's equation in two dimensions. For transmission-line structures like those used in many planar transmission lines, it is difficult to construct solutions for Laplace's equation if one used the system of coordinates that is the natural one for describing the transmission-line configuration. A powerful method for solving two-dimensional potential problems is to use conformal mapping to map the boundaries into a simpler configuration for which solutions to Laplace's equation are easily found. This conformal mapping technique is equivalent to a coordinate transformation and its application to planar transmission fines is described in this appendix. ni.l CONFORMAL MAPPING Let £ = cr + jri be a complex variable and consider the function sin I = sm( a + jr\) = sin cr cosh rj + / cos cr sinh r\ This function is periodic along the real a axis with a period of 2ir. T» e function takes on all its possible values in a strip extending from - i r / 2 ™ i r / 2 along cr as shown in Fig. III.l. Along the contour labeled A-B-C-" shown in the figure, the sin £ function goes from - °° to — 1 and then to and finally to + °=. If we let a new complex variable W be defined by W= u+jv = sin r 886 (Ifl.l) CONFORMAL MAPPING TECHNIQUES 887 FIGURE m . l The sin f function. then the chosen contour in the £ plane maps into the real axis in the W plane as shown in Fig. III.2. All values of £ in the cross-hatched region shown in Fig. III. 1 map into the upper half of the W plane. In the W plane, which we will regard as our real physical space, let a total charge Q per meter be placed on each side of the strip extending from u = - 1 to u = 1. Furthermore, let us regard the strip as a conducting strip at constant potential. By symmetry the two boundaries -» < u < -1 and 1 < u < o= are magnetic walls on which dfy/dv = 0. We can find the solution for Laplace's equation o + K =0 du2 dv2 /v -1 FIGURE III.2 The mapping W = sin f. -u 888 FOUNDATIONS FOR MICROWAVE ENGINEERING in the W plane by solving the potential problem in the £ plane. In the W plane the potential gradient is given by d<t> V<t> = — a , du If we regard u and v as rectangular coordinates, then a and represent new curvilinear coordinates. In the o-, 17 coordinate system Laplace's equation has the form (I 2 h „ <9<I> da ha da d h„ 3* d-q h d-q =0 where k v and hIT are the metric coefficients. For a coordinate transformation that is obtained through a conformal mapping, the metric coefficients are equal and given byt dW K = hn = dC Consequently, Laplace's equation reduces to the same form as it has in a rectangular coordinate system. Thus a and 77 can be treated as rectangular coordinates when solving for the potential field. However, the gradient of 4> must be evaluated using 1 d<t> V* = h„da l d<i> + — — a „ = V<1> hn dr, " a dC dW (HI.2) = Vd> w where V<t>|^ represents the gradient in the £ plane obtained by treating a and 77 as rectangular coordinates. The conformal mapping of the boundary of a polygon in the { plane into the real axis in the W plane is called a Schwarz-Christoffel transformation. For a differential element dW along a contour in the W plane, the corresponding differential element along the mapped contour in the £ plane is d£ = (dt/dW)dW. The angle that d( makes with the real axis is the sum of the angles of d£/dW and dW. When the contour in the W plane is the real axis, the_angle of dW is zero When d£/dW has the form i/W - W1 / JW^'Wz, then the angle of jW - Wx is TT/2 for W < W, and zero for W > Wv while that for 1/ JW - W2 is -TT/2 for W < W2 and zero for W > W2. Hence, as W moves along the real axis, there will be a change of - T T / 2 in the angle of df as W moves past the point W, and a change of TT/2 when W moves past the point W2. This causes the contour in the I plane to change direction in a step-like fashion by + 90°. This property & m. E. Collin, "Field Theory of Guided Waves," 2nd ed., Chap. 4. IEEE Press, Pisacataway. N.J.. 1991. CONFORMAL MAPPING TECHNIQUES 889 used to pick the appropriate mapping functions that are used in this appendix. We will use the above results to find the charge distribution along the conducting strip in the W plane. In the £ plane the presence of magnetic walls along a = ±TT/2, 17 > 0 requires that the electric field be uniform and in the r\ direction. Hence the charge is uniformly distributed on the lower boundary -IT/2 < a < i r / 2 with density Q/~. Thus the potential gradient is given by <9* Q £ =— ~ °J~ nri TT so V4>|, = -Q/Tre0. From (III.2) we find that in the W plane dt, Upon using dW/d£ = cos ( = (1 - sin 2 Ol/2 = (1 - W 2 ) 1 / 2 for W = u, we obtain ps = Q , (in-3) 2 TTVI - u2 for the charge density on one side of an isolated conducting strip two units wide and located in air or completely surrounded by dielectric. This example clearly shows the power of the conformal mapping method of solving potential problems. However, it does depend on our ability to find a mapping function that will transform the boundaries of the physical problem into a configuration for which the solution to Laplace's equation is easily obtained. In the transformed plane the new coordinates may be treated as rectangular coordinates provided we relate the gradients in the two planes by the relation (III.2). Furthermore, the capacitance between conductors remains invariant under a conformal mapping, so that it may be found for the transformed configuration. 2 ELLIPTIC SINE FUNCTION For the function W = sin <T, we have dW/d£ = cos [ = (1 - W2)W2, so dt —- = 1 . (III.4) dw /T^w^ which gives the inverse function Z = sm-'W=\ rw , dW (III.5) The period along the a axis is obtained from the integral from 0 to 1 which 890 FOUNDATIONS FOR MICROWAVE ENGINEERING gives one quarter of the period, i.e., = 2 dW L /I (III.6) W A function that is much more useful in the solution of a number of planar-transmission-line problems is a function that is periodic along both the <r and -q axis in the complex C plane because it would take on all of its possible values inside a rectangle. A doubly periodic function is the elliptic sine function, which is expressed as W=sn(i,k) (IH.7) The parameter k is called the modulus and determines the two periods. The elliptic sine function has a period of 4K along u and 2K' along 77. The inverse function is given by { = sn-1(W,k) = fw [ '0 dW 7 ( 1 - W z ) ( l -k2W2) (in.8) The two quantities K and K' are given by *-C-r ") yj(l-W )(l-k W 2 2 (III.9a) 2 J o VU i/k K + 0 or * " L'1 dW (111.96) / ( l - W 2 ) ( l - k2W2) i/k dW (111.9c) y/(W2-l)(l-k2W2) By using the substitution W = sin 8, the expression for K becomes 1 ao K = K(k)= P' IT . =F\k,l/l -k2sin20 \ 2 which is the complete elliptic integral of the first kind. In the expression for K' we can let k\W2 - 1) = (1 - &2)cos2 0, which reduces the expression to the form K> - K>(k) = f / 2 , •'o ^ 2 Vl-fc'2sin20 = Flk>, J) = K(k') (ffl.ll) * 2/ where the complementary modulus k' = Vl - k2. From (III.8) and (III.9) we also see t h a t sn(0,*) = 0 (111.120) sn(K,k) = l sn(K+jK',k) = ^ (ffl.126) (ffl.12^) CONFORMAL MAPPING TECHNIQUES 891 A further useful value is sn(jK',k) = ±00 (III.12d) The elliptic sine function is a generalization of the sine function and reduces to the latter when k = 0. Two useful approximations for the K and K' parameters for the cases k •« 1 and k' <K 1 are tf(A) = * ' ( * ' ) = - | l /T(*') = * ( * ) = / kl2 + T + k'2 1 + - 9 + k < 0.4 -*< ,\ (III.13a) 4 -Hln- 21 k > 0.65 (III.136) k* 4 168 Note that k and &' can be interchanged. The ratio of K(k)/K'(k) = K(k)/K(k') can be evaluated to an accuracy of one part in 10 5 using the following expression:! K 1 ( 1 + Jk\ —1 = - I n 2K IT l-ilk 0.7 < k s 1 -i — Ill - -S- 0<A<0.7 7= (III.13c) i - VF For intermediate values of k and k\ the following formula can be used: K1 = K(k') = 2 -if [1 - k (III. 13d) where k and k' can be interchanged. When k' is less than 0.707, then (1 - /e)/(l + k) is always less than 0.172 and (III.13a) can be applied. When k' is greater than 0.707, we can use (III.136). Consider now the rectangle shown in the t plane in Fig. III.3. Also shown are the values of sn(<T, k) at the points labeled A, B, C, D, E, F. The mapping W = sn(£, k) maps the rectangle into the real W axis and all interior values into the upper half of the W plane. If the segments AB, CDE, and FG are conducting metal boundaries, then the configuration in the W plane is a coplanar transmission line in air. Its distributed capacitance is twice that between the plates CDE and BAF for the ideal parallel- t T . S. Gradshteyn and T. M. Ryzhik, "Table of Integrals, Series, and Products," Academic Press, Inc.. New York, 1965, p. 925. formula 8.198 and using q = expf -trK'/K). 892 FOUNDATIONS FOR MICROWAVE ENGINEERING IV - i X )K 1 k B A G F -1 C D E —< 1 k FIGURE m . 3 1 The sn(f, k> function. K plate capacitor in the f plane and hence is given by 2K 2e0— = C 4c K 0 - (111.14) The relevant parameter for the coplanar line is the ratio of the width of the center strip to the spacing between the two ground planes which is given by 2/2ui = k, the modulus of the elliptic sine function. Note that in the f plane the boundaries BC and EF are magnetic walls so there is no fringing capacitance. If we choose the segments BC and EF to be conducting strips, we obtain a coplanar strip transmission line. Its distributed capacitance is that associated with the ideal parallel-plate capacitor with the boundary BC at potential V and EF at potential - V; thus K' K' (111.15) 2K = When we let k tend toward zero, the points ± u, move out to infinity and we obtain a slot line. However, we cannot let k = 0 because the capacitance between two semiinfinite planes separated by a slot becomes infinite. C - 2 eOr„ — m.3 C A P A C I T A N C E B E T W E E N TWO P A R A L L E L STRIPS We will now consider the problem of two parallel strips in air as shown in Fig. III.5. The plate separation is 1H and the plates have a width 2w. The following mapping will map the boundary shown by the dashed fine in f 'gIII.5 into the real axis in the W plane shown in Fig. III.4: = / J o (1 - k*W*) / ( l - W 2 ) ( l -k2W2) where k 0 = l / « 0 and k = l / u , . When Z = H on the interior side of the CONFORMAL MAPPING TECHNIQUES 892 /" —- U - 1 u0 y-O, FIGURE m . 4 The mapping W = sn(f,ft). right-hand strip, W = 1; so i-klw2 -w2)(i vTi (III. 17a) -kzw2) At the outer edge of this plate, Z = H + jw, and we make this correspond to the unknown point u „ lying between 1 and u,; thus -k2W2 1 H (1 - Wz)(l = dW -k2W2) (III.176) At the point Z <= H on the outer side of the strip, we set W = «,; thus H = / J o 1 - k2W2 _ /(l dW (III.17c) -W2)(l - k2W2) The distributed capacitance between the parallel strips is equal to that for iy /w- _ n c "J L -H F I G U R E III.5 Two parallel strips and the boundary to be mapped. 894 FOUNDATIONS FOR MICROWAVE ENGINEERING the coplanar strip line that results from the mapping and is given hv y (111,15). For the parallel strips the capacitance depends only on the ratio w/H When we compare (III.176) and (III. 17c), we see that we require jw = - f ' -k2W2 1 , y/{l Wz)(l - = dW -k2W2) A similar comparison of (IV.17a) and ("IV. 176) shows that ,«. 1 - k2W2 jw = f • dW h / ( i - W 2 ) ( l -k2W2) By combining these two expressions, we obtain l~k2W2 .8,' I*^ W2)(l -k2W2) dW = 0 (III.18) which is the equation that will determine k 0 and hence « 0 . The integral in (III.18) can be separated into two integi'als, and by using (111.9c) to replace the integral that goes from 0 to 1, we find that (111.18) can be expressed in the form K ' = k2fL/K , l/(W2- ^ d W l)(l-k2W2) We can express this integral in terms of complete elliptic integrals. However, in practice it is more expedient to evaluate the integral numerically. By using the substitution kHW2 - 1) = (1 - fc2)cos2 0 as before we obtain T - fejf*/Vl-*W*# = fMk', j) (IH.19) where E is the complete elliptic integral of the second kind. From the above we get k2K' 0 20) E(k',?r/2) The expression for H given by (III. 17a) can be reduced to the form (we use W = sinfl) sin 2 6 H = K-k%r-l de L2 . 2 % Vl-62sin20 /2 CONFORMAL MAPPING TECHNIQUES 895 which gives H = K + \ E \ k , - \ - F \ k , k^2rv"2i ' v" 2 K'[E(k,TT/2) - K] +KE(k',Tr/2) E(k',ir/2) For convenience, the complete elliptic integrals of the second kind will be written in the compact form EJ*,^) =E(k) =E E(*'.?) -*(**) =E\k) = £' We now use E(k', TT/2) = E'(k, TT/2) and the following identity: KE' + K'E - J 3 T = Z which allows us to express H in the simpler form * = 2 E ( * W 2 ) - 2 ^ (HI -21) The last reduction is the expression for u; to the form w = 1 TT / r„kl(l ~ k'2siD?6) - k2 / cifl l/l - fc' 2 sin 2 0 kl = ~E{k',8) -F(k',6) k (111.22) where k'Q = y 1 - k\ and ffc/c ft 0 = cos > - (111.23) k k0 This equation gives w in terms of incomplete elliptic integrals. Values for the complete and incomplete elliptic integrals are available in the book by Jahnke and Emde.t The easiest way to use the above formulas is to choose a value for k, solve (111.20) for k0, (111.21) for H, and (111.22) for w. The integrals are readily evaluated numerically. By this means values of w/H and the capacitance C given by (III.15) can be compiled as a function of k. The capacitance between a strip of width 2w and spaced a distance H above a ground plane is twice as large as that given by (III.15). Table III.l gives *E. Jahnke and F. Emde, "Tables of Functions," Dover Publications. Inc. New York, 1945. 896 FOUNDATIONS FOR MICROWAVF. ENGINEERING TABLE I I I . ] C a p a c i t a n c e b e t w e e n a s t r i p of w i d t h 2u> at a h e i g h t H a b o v e a g r o u n d p l a n e C 2w C H «o H «o 0.05 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.238 1.436 1.703 1.912 2.094 2.26 2.415 2.563 2.706 2.845 1 1.5 2 2.5 3 4 6 6 8 10 2.981 3.621 4.232 4.822 5.399 6.529 7.63 8.72 10.86 12.98 2ic computed values of C/eQ as a function of 2w/H. The integrals were evaluated numerically using Simpson's quadrature formula. HI.4 STRIP TRANSMISSION LINE The strip transmission line is shown in Fig. III.6. The boundary A-B-C-D-E is mapped into the real axis in the W plane by means of the mapping function (111.24) W = sin Z = sin x cosh y + j cos x sinh y In the Z plane H is IT units, so we make the point B be jirw/H - ir/2 so as to maintain the same width-to-spacing ratio w/H as in the physical strip line. The real axis in the W plane is mapped into a rectangle in the L, plane, shown in Fig. III.6d, by the mapping function W= 1 + 2 s n ( f , A ) The point B corresponds to W = -cosh IT w/H and sn(-K +jK', k) = —1/k. Consequently, we must have 2 1 J' cosh 7TIV to sn(£, k) = rcw H which gives k = (111.25) cosh TTW ,/2H In the C plane the capacitance is given by eQK'/K and represents onequarter of the capacitance of the strip-line configuration. Hence for the strip CONFORMAL MAPPING TECHNIQUES jy A e w B ! C B 1 ' \D 2H 1 1 A E C 0 — • A 2 2w ' (at 2 tfej > cosh -jj- -1 Icl -K F I G U R E III.6 The strip transmission line and its mapping 897 898 FOUNDATIONS FOR MICROWAVE ENGINEERING line K'(k) C = 4t (III .26) The corresponding characteristic impedance is mr-K(k) * = ^ = m.5 K'(k) (HI .27) CONDUCTOR LOSS Since the current density and charge density are related by the continuity equation [see (3.1386)], the current density has the same functional form as the charge density. If we know the current density J , , then the power loss is given by where R m is the skin-effect surface resistance and the integral is taken around all the contours along the conductor surfaces. If we have infinitely thin conductors with edges, then the current density becomes infinite at the edge inversely proportional to the square root of the distance from the edge. For example, for an isolated strip of width 2w the current density, according to (III.3), would be 2-rrwy/l - x2/w2 on each side of the strip. This current density is too singular to allow a finite value for the power loss to be evaluated. Consequently, it will be necessary to take into account the finite thickness of the conductors. The simplest case to consider is that of an isolated strip of width 2w and thickness 2t as shown in Fig. III.7. The labeled boundary in the Z plane can be mapped into the real u axis in the W plane using dZ d~W W2 =A 1 -k2W2 (111.28) which upon integration gives Z = A(W l - W1 - k2W2 dW+jt (III.29) The constant jt is added since W = 0 is made to correspond to Z = jt. When W = 1 we require Z = w + jt and when W = 1/k we want Z =* u>- These two conditions provide a solution for A and the modulus k. By following a CONFORMAL MAPPING TECHNIQUES 899 iy c E 2.1 F iv C L- G t _l 1 1 -1 FIGURE III.7 An isolated conducting strip with finite thickness and the mapping of one-half of the strip boundary into the W plane. procedure similar to that for the parallel strips, we find that wk2 A = -k,2K(k) E(k) E(k') - kzK(k') t w E(k) - k,2K(k) (III.30a) (III.306) In the W plane the conducting strip extends from - 1/k to 1/k along u. Since it is an isolated infinitely thin strip, the current density in the W plane is of the form A J " = A - k2u2 where the factor A has been inserted arbitrarily. In the Z plane the current density is proportional to dW V4>lz = V * | w dZ 900 FOUNDATIONS FOB MICROWAVE ENGINEERING In order to find Jt{x), we need to express u as a function of x, which is not easily accomplished. Fortunately, we will not need this functional relationship. Let us consider the integration of (.111.28) very close to the 90° corner where Z = w +jt, of the conducting strip. When x = w, u = 1; hence 1-u2 * v 1 - k2u2 -'i, du For u very close to one, we can use the approximations 1 — « 2 = (1 + a) (1 - u) => 211 - u) and 1 - k2u2 = 1 - k2. We then obtain w - x = r^Al!^du = 2\T^A(l-u)3/2 which gives (3A)V32"2/3 J,= v^/T^T £l~**)l/V-*)1/a (III .31) This fundamental result shows t h a t close to a 90° corner the current density has a weaker singularity, namely, it is inversely proportional to the cube root of the distance from the corner. This current density, when squared, can be integrated to give a finite result for the power loss. The edge behavior is a local field phenomenon, so that the result obtained is true near any 90° corner. At some distance away from the corner, the expression [(1 — u )/ (1 - k2u2)]1/2 can be approximated by unity since k is close to one for a thin strip, i.e., for t/w small. Thus we have / dx ~ A du o -'o J or x = Au. Hence away from the corner the current density is given by 1 J.= v'l -u2 1 y T - (x/A)2 1 \A ~ (x/wY since A ~ w then t/w is small. We can summarize the above result with the statement that for a thin conducting strip the current density over most of the strip is the same as for an infinitely thin strip, but as the corner is approached the one over the square root of the distance from the corn f^ behavior changes over to a less singular behavior going as one over the cu root of the distance from the corner. CONFORMAL MAPPING TECHNIQUES 901 For the isolated finite thickness strip, we can evaluate the power loss per meter by doing the integration in the W plane. Thus p dx wk du 2 A) II - u \ du <-4/„ du . _ \f(l - u2)(l - k2u :) = 2R,„A ; « = 2RmA[K(k)+K'(k)] (111.32) The integral gives the loss for one quarter section, so we multiplied by a factor of 4 to get the total loss. The total current on the strip is given by 7 = 4/ dx \/k Jl-u 2 du = 4Af du du i/k fiT~k2u2 2 IT A (III .33) The integral was evaluated by using the substitution ku = sin 0. The series-distributed resistance R per meter for the conductor is defined by the relationship m* = p, From the derived expressions we obtain k2Rm R = (111.34) K + K') A In practice, the ratio t/w is small. For example, a board plated with 1-oz copper has a metalization thickness of 0.036 mm; so a microstrip of width 1 mm will have t/w = 1/28. When t/w < 0.05 the following approximations are valid since k ~ 1: Z E(k')~-\: E(k) 4 TT *<*>- ¥ |1+ K(h)~ht- 4 By using these approximations in (III.30), we obtain it A = k2w and TTW We now find that the expression for H reduces to the simple form R = R. 2 2TT W ATTW In (111.35) 902 FOUNDATIONS FOR MICROWAVE ENGINEERING In this expression w is one-half of the strip width and t is one-half of the strip thickness. The part Rm/2vw is the contribution from the currents that flow on the two end faces of the strip. From the expression used for J we find that the current density at the center of the broad face where u = fj is unity. At the center of the end face where u = 1/k, the current density is k/k' = ^TTW/21 and hence is much larger. It is for this reason that there is a significant contribution to the power loss from the current on the end faces even though t is very small. If the conductor cross section was approximated by an ellipse with major axis equal to 2w and minor axis equal to 2t, it would be found that the series resistance is R = —y— In TT2W t when t/w is small. For very small values of t/w, this resistance is twice as large as that for a rectangular bar of width 2w and thickness 2t. The reason is that for the elliptical cross section the current density at the narrow ends is a factor w/t greater than that at the center of the broad face and thus exhibits a more singular behavior than that for the current at the center of the narrow face for a rectangular bar. A useful approximate way to find the series resistance of a thin conducting strip is to use the current density for an infinitely thin strip but to integrate the square of the current density up to a point a distance d from the edge. The distance d is chosen so that the same resistance as given by (III.35) is obtained. Thus for the isolated strip we use P '= 2fl , u ,-d dx w + x w d ~ 4 T TZPOT-*«•»• m l -xyw' w-x 2w * Rmw In — The total current on the strip is 2mv = /; so by equating I2R/2 to P, we obtain R 2w R = —mJ - In — 217*10 d We now equate this expression to (III.35) and solve for d, which gives 2t 2t d = Aire" 290.8 In the derivation we expressed v as In e~. This method of finding the loss resistance was proposed by Lewin.t This method is based on the fact tha tL. Lewin, A Method of Avoiding the Edge Current Divergence in Perturbation Loss Calcu ations, IEEE Trans., vol. MTT-32. pp. 717-719, July, 19B4. CONFOBMAL MAPi'INU TECHNIQUES 905 the effect of finite thickness is to change the local behavior of the current near the edge but leaving the current density on the major portion of tht conducting strip relatively unchanged. ni.6 CONDUCTOR LOSSES FOR A MICROSTRIP TRANSMISSION LINE Consider a microstrip fine with a strip of width 2u>, thickness 2t, and spacec a distance H above a ground plane. In order to apply the rule establishec above, we need to know the division of the total current on the upper anc lower faces of the strip, which is affected by the presence of the grount plane. Apart from an unequal division of the total current, the density o current is very nearly the same as on an isolated strip. From the mappin; function (III.16), we find that the current density is proportional to J * v lvv * dW - V4»|f dz ""*dWdw dZ \i-klw2\ At the center of the upper face u = u, = 1 /k and at the center of the lowe face u = 1. Hence the ratio of the currents on the two faces is II - k2/k2\ Jz2 k2 - k* The fraction of the total current on the upper face is p, where p is given b; k2(l-k2) Jzi JtX +J,2 ££(1 - k2) E'-k2K' " (1 - k2)Kf [III.36 where (III.20) has been used to eliminate fef. The power loss on the uppe face is proportional to (1 - p ) 2 . The total power loss is proportional b p 2 + (1 - p) 2 . For an isolated strip p = 1/2, so that the series resistanc fi, of the microstrip in the presence of the ground plane is given b, multiplying R by the ratio [ p 2 + (1 - p ) 2 l / 0 . 5 ; thus i?, = 2 ( 2 p 2 - 2p + 1)R (111.37 where p depends only on the ratio 2w/H and R is the same as in (III.35 When p = 0.25 we find that Rx = 1.25 R which shows that an unequf division of the total current increases the resistance. The ground plane corresponds to the .y axis in Fig. III.5, which map into the u axis in the VV plane. Thus the current density will be proportion? to the gradient along the v axis in the W plane which is given by I J, a l - ^ O ) 1 2,,2 1+*0W 904 FOUNDATIONS FOR MICROWAVE ENGINEERING Let J, be chosen as J,= i + k'y where / 0 is to be found so that the total current will equal I. The total current on the ground plane is given by = 2 / 0J / 2 2 o 1+ kv dZ dv dW dv o J(l + k v )(l + v2) = 2 / 0J/ * 2 2 The integral is easily done in the g plane by using dv = \dW\ dv = y ' a + A V ) ( i + v2) d-q dC and the limits 0 to K'(k) for -q. Thus we find that / = 21 0 K'(k). The power loss in the ground plane is given by -r—— 2 P, = R 4K R„I2 dZ dv dW dv i-f _ 4A" 2 Jo (1 + k2v2)y/{l + k2v2)(l + v2) The series resistance R 2 of the ground plane is obtained by equating the power loss to 7 2 R 2 / 2 . The series resistance will depend on the absolute dimensions of the microstrip line. Thus we will multiply P l by (2w/H) and divide by this same factor but use (111.21) for H. We thereby obtain R, R TV 2w 2lV 2w 4 m 1 \ f<* dv ~H ElK'2jh (l + k2v2)J(l +k2v2)(l + v2) (III.38) The normalized resistance 2wR2/Rm is a function only of the ratio 2w/H and thus only needs to be evaluated once. The relevant expressions for the loss ratio LR = 2 ( 2 p 2 - 2p + 1) (111.39) and the normalized resistance 2wR2/Rm have been evaluated numerically as a function of 2w/H. We have fitted simple polynomial functions to the resultant data, so that for application purposes the loss ratio and normalized resistance can be computed from the following expressions to an CONFORMAL MAPPING TECHNIQUES 905 accuracy of 2 percent or better: 2w LR = 1 3 LR = 0.94 + 0.134 2w 'if- 0.0062 2 JO/H 2w (III. 40a] < 0.5 m 2 "Xt2 — 2w/H + 5.8 + 2w 0.5 < — < 10 ~H (III.406) 2w 0.1 < —- < 10 (m.41) 0.03H/2LV This formula states that the resistance of the ground plane is that of a strip of width 2w + 5.8H and with a uniform current density. It is remarkable that this relationship holds for such a wide range of 2w/H ratios. In practice, surface roughness increases the resistance by 10 to 50 percent so an accuracy greater than a few percent in the theoretical formulas is not needed. III.7 ATTENUATION FOR A COPLANAR LINE We consider a coplanar line as shown in Fig. III.8a. The ground planes and center conductor have a thickness 2t. The strip width is 2a and the ground plane separation is 2b. The upper half of the Z plane will be mapped into the upper half of the W plane with the contour labeled 1-2—8 mapping into W u--\ V- 2aIV 3 -U, - -r -1 5 4 -Un 6 * "• ib) FIGURE m . 8 Conformal mapping of the cross section of a coplanar transmission line with finite thickness conductors. 906 FOUNDATIONS FOR MICROWAVE ENGINEERING the real axis in the W plane as shown in Fig. III.86. The required mapping function is (l-k2W2)(l rw -k2W2) (HI .42) where k 0 = l/u0 and k l = l / « , . The parameters A, k0, k, and k t are determined by requiring that the four points Z = a + jt, a, b, and b + jt map into the points W = « 0 , 1, 1/k, and u v These requirements lead to the following four equations: =A(U°F{W)dW a (III.43a) t=jApF(W)dW (III.436) b A("'F(W)dW (III.43c) -jA(U,F(W)dW h/k (HI.43d) = t= where F( W) is the integrand shown in (111.42). The real axis in the W plane can be mapped into the rectangle in the £ plane using (III.8). The boundary-value problem in the C plane is that of an ideal parallel-plate capacitor. Hence V<J>|{ is a constant which we set equal to one. The current density J z is t h u s given by J, = V<D|. u d; *= = dZ dW dZ dW dZ Consequently, J,dZ = dC dC dW dZ = dW dW dW dZ The total current on the center conductor is 7 d£ d w 4 dW 4 = //0; ddWw " = 'J/ 0; ^/(l - W 2 ) ( l -k2W*) = 4K(k) The power loss on the center conductor is given by d( Z 2 -R a ' - /J \Jt\'dZ = 0 2RmJ- o dW2 dZ dW dW du f1 mJ 2 2 2 2 (III.44o) 2 2 2 | ^ ( 1 - u ) ( l - k u )(l - k u )(l - k\u ) CONFORMAl. MAPPING TECHNIQUES t while that on the two ground planes is given by /2 2 -R„l a ru, du • , = = T (111.44 J i/k |^/(1 - « * ) ( ! - k2u2)(l - k2u2)(l - k2u2) I The integrals in (111.43) and (III.44) can be evaluated numerically but iterative technique is necessary in order to find the required values of k0, and kx. We will only consider the case when the thickness t is very sm; say t < 0.05a. For this important special case, a number of approximate can be made that will lead to relatively simple expressions for the pov loss. When t/a is small k n is approximately equal to one and k x approximately equal to k. Thus the intervals u 0 to 1 and l/k to u, i small. We also note that if t =* 0, then k0 = 1, kx = k and (111.42) gi\ Z = AW from which we obtain a = A and b = A/k or k = a/b. We will i these values for A and k for the case of small but finite values of I. (III.436) we can set u equal to 1 in all but the critical factors 1 — u a 1 - k0u, since the interval of integration is very small. Thus we find tr. t = aJ 1/ -1 2(l-**Xl-«) /&{,« - 1 1 - u du This integral can be evaluated in terms of elementary functions and give; TT k0 - ira '-riir--»-<*•-^ Hence we obtain 2' k0 = 1 + — (in.4.' which verifies that k 0 is close to one. In (III.43rf) we put u = l/k in all factors except the critical om 1 - ku and 1 — Ajtt, since the interval of integration in this equation is ah small. An integral of the same type as found above is obtained and readi leads to the result which shows that k x is close to k = a/b. 908 FOUNDATIONS FOR MICROWAVE ENGINEERING The power-loss terms involve the following four integrals: 7, = /., = f1G(u)du ~ Ju„ f""G(u)du J o I3= f G{u)du 74 = f"'G{u)du 'i/k where G(u) - l[(l - u 2 Xl - k2uHl 7 2 and 7., will give us the power loss considered first. For the evaluation [(1 - k2uHl - k'fu-)]'l/2 = (1 - k2) ' A, ka = l/k0 to obtain 72 = - k2u2)(l - k2u2)V1/2\. The integrals on the conductor edges and will be of I.2 we use the approximation and'make the substitution k0u = d\ M 1 - **} h fitf *"(*«) 2 2 * 0 (1 ~ k2) - 1)(1 - k A ) Since k a is close to one, we can use K'(fe a ) « T7/2 and fe 0 = 1 to get h= (III.47a) 2(1 - a 2 / 6 2 ) The same approach is used to evaluate 7 4 to obtain h~ — r \-k K d\ rh 2 26(1 2 k 2 V(A -l){l-felA ) 1 - *2 ^ ' ( * 6 ) —a -a*/b2) (III.476) since kh = kx/k is close to one. In order to evaluate 7, we first make the substitution knu = cos 8 to obtain 7, = i-f 7T/2 ^ 2 rffl y/{k% - cos 2 0 ) ( £ 2 - k 2 cos 2 0 ) ( £ 2 - *f cos 2 0) We now split the integral into one over a small interval 0 to 0, plus aa integral from 0, to v/2. Over the first interval the expression under the square-root sign is approximated by ( K + l ) ( * o - **)(*o - *i)( *o - cos B) = ( l - * 2 ) 2 [ ( f c o - l ) 2 + 02] CONFORMAL MAPPING TECHNIQUES 909 The resultant integral is an elementary one and easily integrated to give kl / l o == r»\ ^o 1 4 i - 1k' k2 ^ — ^ dO -l)+62 y/2(kQ ' 4t/ira 1 2 -k ln[0 1 j2(k0-l)+0* 20! A: 2 1 - k y/4t/va + In ij4t/TTO 2 Even if 4t/ira is of the same order as 0 , the error in the last approximation is small. For At/-a = ft2 it is In(1.207) = 0.207. which is small relative to other terms that occur for Pn. In the integral from 0 t to 7r/2, we can assume that t = 0 so k{) = 1, k x = k, and we then obtain j _ pr/8 d0_ lh sin0(l-/e2cos20) 16 By using A = cos 0 this integral becomes dK rcos0, '»-! (1-A2)(1-A2A2) and is readily evaluated. We obtain ',6 = In 2 2(1-A ) 2 4- ) 8? A In 2(1 + A) - Atf 2(1 - k) + /efljf upon using cos0, = 1 - 0 2 / 2 . If we choose 0, = 0.25, then a number of terms involving 0 2 can be dropped. When we combine Ila and Iih, we get 1 /.= Arra 2 2(1-A ) In 1 +k A In (111.48a; t \ - k which has the nice feature that it does not depend on 0,. For the final integral / 3 , we make the substitution kxu = cosh 0 and again split the integral into one over the interval 0 to 0, plus one over the interval 0, to ^. In the first interval cosh 0 is replaced by unity in the noncritical terms and by 1 + 0 2 / 2 in the 1 - k2u2 = 1 - i.k2/k2)cosh2 0 term. In the second integral we use k0= 1, k, = k. We then find that k 4TT6 2 2(1-A ) In 1 1 + k In (III.486) t k 1 - k The series resistance of the center conductor is i?, and is defined by the relationship 2 U2R, =Pn = -Rm(lx +I2) The series resistance R2 of the ground planes is defined by a similar 910 FOUNDATIONS FOR MICROWAVE ENGINEERING relationship, namely, W 2 f l 2 = P , 2 = r i ? m ( / 3 + /4) 2 a The total current / equals 4K(k), so we find that R. 2 8aK (k)(l R„,k - 2 k) [ 4ira 7r + In - + In t 4-n-b k In 1+k 1 -k 1 l —In l-k (III.49a) +k' (HI.496) where k = a/b. The attenuation is given by /?, + R-2 (111.50) where Z r is the characteristic impedance of the coplanar line. When t is very small and the ground planes are widely separated, the expression for R x reduces to that for an isolated strip, as one would expect. Apart from the factor 1/(1 - k2) the ground-plane losses are approximately what one would find for an isolated conductor of width 2b. The formulas derived above are estimated to be accurate to within 10 percent for t < 0.05a and k < 0.8 More accurate approximations can be made but the resultant formulas would be more complex. The expressions obtained above can also be derived with fewer steps by using Lewin's method. The alternative derivation was chosen so as to provide an example of a more complete derivation and to more fully show the approximations involved. APPENDDi IV PHYSICAL CONSTANTS AND OTHER DATA IV. 1 PHYSICAL CONSTANTS Permittivity of vacuum = e a = 8.854 x 10~ 1 2 = ( 1 / 3 6 T T ) X 10~ 9 F / m Permeability of vacuum = fi0 = 4v x 1 0 - ' H / m Impedance of free space = Z0 = 376.7 = 120TT $1 Velocity of light = c = 2.998 X 10 8 m / s Charge of electron = e = 1.602 x 1 0 " 1 9 C Mass of electron = m = 9.107 x ! 0 - 3 1 kg ,, =e/m = 1.76 x 1 0 " C / k g Mass of proton = M = 1.67 x 10 -7 kg Boltzmann's constant =• k = 1.380 x 10 ~ 23 J / K Planck's constant - ft *= 6.547 x 10 '3'f J • s 10 7 ergs = 1 J 1 J = 0.6285 x 10 19 eV 1 eV = energy gained by an electron in accelerating through a potential of I V Energy of 1 eV = equivalent electron temperature of 1.15 X 10*4 K e / N \' 2 Electron plasma frequency f. = — = 8.97N1 " Hz. where iV is 2TT \ me0 J the number of electrons per cubic meter Electron cyclotron frequency fc = eB/2vm = 28,000 B MHz for B in webers per square meter; fc = 2.8 B MHz for B in gauss 10" G = 1 W b / m 2 91 912 IV.2 FOUNDATIONS FOR MICROWAVE ENGINEERING CONDUCTIVITEES OF MATERIALS Material Conductivity, S/m 5.8 x 3.54 x 6.14 x 1.28 x Copper (annealed) Aluminum Silver Nickel IV.3 S/m 10 7 10 7 10 7 10 7 0.5-1.0 x 10 7 2 X10"4 3-5 < 2 X10~17 Steel Water (distilled) Sea water Quartz (fused) DEELECTRIC CONSTANTS OF MATERIALS Frequency, Material Polystyrene Polystyrene (foam) Lucite Teflon Fused quartz Ruby mica Titanium dioxide Mahogany wood IV.4 Conductivity, Material MHz 3,000 3,000 10,000 10.000 10,000 3,000 10.000 10,000 *'Ao Loss tangent e"/e' 2.54 1.05 2.56 2.08 3.78 5.4 90 1.7 0.00025-0.0016 0.00003 0.005 0.00037 0.0001 0.0003 0.002 0.021 SKIN DEPTH IN COPPER Frequency, Hz Skin d e p t h s , , cm in 2.08 60 0.85 toa 0.66 10 a 0.208 S, = ft/io/ur " 6.6 f- 1 / 2 cm for copper (a - 5.8 X 107 S/m). 10 4 6.6x10" 10s 6.6 x K T 4 AUTHOR INDEX Afsar. M. N.. 130 Alexopolous, N. G., 180 Allison, J., 190 Alseyab, S. A.. 647 Altschuler. E. E-, 351 Anderson, T. N., 421 Ayres, W. P., 456 Baden Fuller, A. J., 219 BabJ, J., 166, 219, 413, 432, 479. 500, 875 Bailey. A. E.. 16 Balanis, C. A., 16 Beck, A. H. W., 712 Belevitch, V.. 596 Benedek, P., 491 Benson, F. A.. 190 Bethe, H. A., 284, 416 Bevensee, R. M., 647 Bhartia, P., 166, 219, 413, 432, 479, 500, 875 Blackwell, L. A., 830 Blight. R. E., 480 Bobroff, D., 668 Bolinder, F., 374, 393 Bosma, H„ 475, 480 Bostian, C. W., 16 Bowman, F., 885 Branch, G. M., 660 Brewer, G. R., 712 Brillouin, L„ 647, 653 Brown, J., 346 Burton. M. N., 207, 209 Button, K. J„ 130, 480 Carlin, H. J.. 471 Carson. R„ 798 Caulton, M., 322 Chang, K„ 210, 219, 413-414. 479. 722, 724, 798. 875 Chang, K. K. N., 830 Chew. W. C. 149 Chodorow. M.. 672, 712 Chow, K. K.. 549 Chu, K. R., 712 Chu, L. J., 668 Clarricoals, P., 480 Cohn, S. B., 173, 393, 346, 444, 590, 596, 639. 647 Coleman, J. T., 703 Collin, R. E., 16, 43, 165, 171, 208, 241, 277, 284, 286, 340, 346, 380, 393, 417, 418, 525, 544, 888 Collins. G. B., 712 Comstock, R. L., 480 Copson. E, T., 229 Costanzo, A., 850 Courtoise, L., 549 Cullen, A. L., 830 Q I Q 914 AUTHOR INDEX Dalman, G. C, 219 Daly, D. A., 322 Dicke, R. H-, 220, 245, 302, 416, 479, Dormann, J. L., 549 Dow, W. G., 653 Drobot, A. T., 712 Edwards, T. C, 219 Ekholdt, R.. 322 Elliott, R. S., 16 Emde, F., 895 Fay, C. E., 480 Ferguson. P. E., 703 Fox, A. G., 409 Fung, A. K., 16 Gastine, M., 549 Gentile. C, 798 Ghione, G., 176 Gilden, M., 830 Ginzton, E. L., 16 Gonzalez, G., 798 Gopinath. A., 368 Goubau, G.. 549 Gradshteyn, T. S., 891 Guillemin, E. A., 229, 590 Guillon, P.. 549 Gunston, M. A. R., 150 Gupta, C, 368 Ha. T. T., 798 Hahn, W. C, 663, 712 Hamilton, D. R.. 712 Hammerstad, E. O., 149, 151 Harrington, R. F., 63, 144 Hartwig, C. P.. 156 Harvey, A F., 479, 577 Haskal, H., 224 Haus, H. A., 668 Hayt, W. H., Jr., 70 Heilmeier, G. H. 830 Held, D. N., 875 Helszajn, J., 500 Hensperger. E. S.. 346 Heuer, H. J., 818 Hirshfield, J. L., 712 Honey. R. C, 830 Hopfer, S., 207, 209 Horner, J. B., 712 Homo, M., 430 Howe, H., 130, 173 Hutter. R. G. E., 576, 712 Ishii, T. K., 219 Ivanek, F., 16 Jackson, R. W., 179 Jahnke, E., 895 James, D. S., 500 Johnk, C. T. A., 70 Johnson. R. C., 393 Jones, E. M. T., 364, 434, 830 Kajfez. D., 549 Kales, M. L., 465 Kaul, R., 219 Kerns, D. M., 302 Kerr, A. R., 875 Kleen, W. J.. 712 Klopenstein, R. W., 393 Kluver, J. W., 668 Knight, S. P., 322 Knipp, J. K, 712 Knoppik, N., 497, 499 Kobayshi, M., 162 Kollberg. E. L., 875 Komatsu, Y., 515 Kong, J. A., 70, 149 Kotzebue, K. L., 830 Kraus, J. D., 16, 70 Krauss, H. L., 712 Kuhn. N.. 264 Kurokawa, K, 268, 549 Lange, J., 434 Laverghetta, T.. 130 Lax. B., 480 Lewin, L., 340, 342, 902 Lewis, J., 712 Li, Q. F., 712 Liboff, R. L., 219 Liechti, C. H., 717 Makimoti, M., 647 Malherbe, J. A. G., 647 Manley, J. M., 804 Marcuvitz, N., 207. 302. 339-340, 479. 552 Mason. S. J., 261 Mass, S. A., 875 Masse, D. J., 156 Matsumaru. K, 393 Matthaei, G. L., 364. 434, 630, 830 Matthews, H., 647 McDonald, N. A., 286 McLachlan, N. W., 885 Medina, F., 430 AUTHOR INDEX Meixner. J., 43 Melchor, J. L., 456 Mihran, T. G., 660 Mobbs, C. I., 647 Montgomery, C. G., 16, 220, 245, 302, 416, 479, 548 Moore, R. K., 16 Morich, M., 166 Moynihan, R. L., 449 Mumford, W. W., 636 Murakami. Y., 515 Nakatani, A., 180 Naldi, C, U., 176 Neri, A., 850 Okress, E. C, 16 Ordung, P. F., 712 Pannenborg, A. E., 302 Parad, L. I., 449 Park, S. Y., 712 Pavio, A- M., 719, 798, 875 Penfield, P., 830 Pengelley, R. S., 722, 798 Pierce, J. R., 650, 653, 712 Pierpont, J., 229 Poh, S. Y., 149 Pozar. D. M.. 219, 479 Pratt, T., 16 Presser, A., 434 Pucel, R. A., 156 Purcell. E. M., 220, 245, 302. 416, 479. 548 Rafuse, R. P., 830 Ragan. G. L., 397, 479, 548, 592 Ramo, S., 219, 712 Read. M. E., 712 Reich, H. J.. 712 Rhode, U. L., 719, 798, 875 Rhodes, J. D., 647 Riblet. H. J., 393 Rigrod, W., 712 Rizzi, P. A., 219 Rizzoli, V.. 850 Roberts, J., 480 Rodrique, G. P., 476 Rosenbaum, F. J., 475, 480 Rowe, H. E., 804 Ryzhik, T. M., 891 Schelkunoff, S. A., 218 Schloemann, E., 480 Schneider, M. V., 150 Schwarz, S. E., 70 Sensiper. S., 585 Shen, L. C., 70 Sheng, N. H., 717 Silverstein, J. D., 712 Silvester. P., 491 Skalnik, J. K., 712 Skolnik, M. I., 16 Slater, J. C, 548. 647, 712 Solymar, L., 393 Soohoo. R. F., 458, 480 Spangenberg, K. R., 650, 672, 712 Sprangle, P., 712 Stratton, J. A., 70 Stutzman, W. L., 16 Suskind, C, 672, 712 Symons, R. S„ 703 Thiele, G. A., 16 Troetsehe), W. O., 318 Uenohara. M., 823 Ulaby, F. T., 16 Valier. G., 703 Van Bladel, J., 549 Van der Zeil, A., 799 Van Duzer, T., 219 Van Trier, A. A. Th. M.. 465 Vartanian, P. H.. 456 Vendelin. G. D., 719, 798, 875 Villeneuve, A. T.. 63 Wait, J. R., 70 Watkins, D. A., 585, 647 Watson, G. N.. 885 Weale. J. R„ 150 Weinberg, L., 596 Wenzel, R. J.. 647 Wheeler, H. A., 149 Whinnery, J. R., 219 Wilkinson, E., 443 Williams, A. E., 647 Wolff. E., 219 Wolff, I., 497, 499 Wu, Y. S., 475, 480 Yamashita, S., 647 Young, L„ 346, 360, 364, 393, 434 Zitelli, L., 712 915 SUBJECT INDEX Admittance characteristic, of transmission line, 76 electronic, in klystron, 688 input, for transmission line, 93 inverters, in niters, 603-614 Amplification of klystron, 685 of parametric amplifier, 813, 815-820 of traveling-wave tube, 698 Amplifier, design of. 755-759, 780-795 double stage, 788-793 low noise. 773-776, 787 single stage, 781-788 gain of. 274, 728-735 stability of, 736-744 Angular momentum. 451 Anisotropic media, 2 6 - 2 8 Antenna, pi-obe in waveguide, 276-281 Aperture coupling by, in waveguide, 284-294 polarizabifity of circular, 285 in rectangular cavity, 517-523 Attenuation for circular waveguide, 196-197 for coaxial transmission line, 111, 117 for coplanar line. 178-180 for microistrip line, 153-157, 163-164 for rectangular waveguide, 188-189 for strip line, 171-173 for transmission line, 108-111 Attenuator electronic, 400-404 rotary. 397-400 Babinet's principle, 580 Backward-wave oscillator, 709 Bandwidth of matching network. 325-330 of resonant circuit, 483 Beam, electron (see Electron beam) Beam coupling parameter. 672 Bessei functions, 195, 581-583. 881-885 spherical, 510-511 Bethe directional coupler, 416-419 Bilinear transformation, 716, 725-726 Binomial quarter-wave transformer, 350-352 Bloch wave, 556 impedance of, 556-557 (see also Periodic structures) Boundary conditions at conducting edge, 43-44 at conducting surface. 4 1 - 4 3 for electromagnetic field. 39-44 at infinity, 44 917 918 SUBJECT TNDEX Branch line directional coupler, 432-434 Brillouin flow, for electron beam, 653, 701 Capacitance distributed of coaxial line, 115 of coplanar line, 176 of microstrip line, 147-151 of strip line, 896-898 of transmission line, 7 2 - 7 3 of microstrip gap, 493 of microstrip open end, 492 of microstrip step, 368 Capacitors, for microstrip circuits, 322 Carcinotrons, 709 Cavity coupling parameter for. 496, 521-523 cylindrical, 504-508 mode chart for, 507 Q of, 507 resonant frequency of, 506 degenerate modes in, 536-538 excitation of, 538-541, 683-686 field expansion in, 525-533 filter, 635-641 loop-coupled, 523-525 oscillations in, 533-536 perturbation of, 541-545 rectangular, 500-504 aperture coupled, 517-523 Q of, 503-504 resonant frequency of, 501-502 Chebyshev filters, 593-598 Chebyshev polynomials, 353, 355 Chebyshev quarter-wave transformer, 352-360 Chebyshev tapered transmission line. 380-383 Choke joint, 397 in variable short circuits, 395-397 Circles, constant gain, 744-755 mismatch, 776-780 noise figure, 772-776 stability, 736-744 load, 736 source, 739 Circular polarized field, 405-407, 452 Circulator, four-port, 468-471 for parametric amplifier, 816-817 scattering matrix for, 471-472 three-port, 471-476 Coaxial transmission line attenuation in, 111, 117 characteristic impedance of, 115 distributed parameters for, 115-116 fields in. 106-108 ConformaJ mapping, 886-889 and conductor loss, 898-910 for coplanar line, 905-910 for microstrip line, 903-905 for coplanar line, 892 for microstrip line, 892-896 for slot line, 892 for strip line, 896-898 Constitutive relations, 23-28 Continuity equation for current, 20 Coplanar transmission line, 126-127, 175-180 attenuation in, 178-180 impedance of, 176-178 Corrugated plane as periodic structure, 571-577 Coupled microstrip line, 126-127 for directional coupler, 427-432 Coupled strip line, 173-174 Coupling in directional coupler, 414 of modes in lossy cavity, 536-538 Coupling coefficient, for coupled microstrip line, 166 Coupling parameter, for cavity, 496, 521-523 Current, equivalent, in waveguide, 221-223 linear, excitation of waveguide by, 281-283 loop, in waveguide, 283-284 normalized. 223 on transmission line, 106 Cutoff frequency (see Waveguide, circular; Waveguide, rectangular) Cyclotron frequency, 701, 704 Damping of cavity, 484 Delta function, 59-60 Diaphragm capacitive, in rectangular guide, 341-342 inductive, in rectangular guide, 340-341 Dielectric constant, 25 Dielectric resonator, 508-517 cylindrical, 515-516 hemispherical, 509-515 Q of, 513 Directional coupler Bethe type, 416-419 branch line, 432-434 Chebyshev. 422-427 coupled line, 427-432 coupling in, 414 directivity of, 414 Lange, 434-435 SUBJECT INDEX multielement, 422-427 scattering matrix for, 414-416 Schwinger reverse phase, 420 two-hole, 419-420 Moreno crossed guide, 421 Riblet T-slot, 421 Disk resonator, 496-500 Dispersion in microstrip line, 158-163 of signal in waveguide, 198-204 Double-stream amplifier. 708 Double-stub tuner, 312-317 for waveguide, 342-343 E mode. 102-104 in circular guide, 194-196 in rectangular guide. 193 E-H tuner, 342-343 Electron beam ac power relations for. 667-670 with axially confined flow, 651 beam coupling parameter for, 672 Brillouin flow for. 653, 701 dc conditions for, 650 ion-neutralized, 650-651 kinetic-power theorem for, 670 perveance of, 650 space-charge waves on. 654-667 velocity modulation of, 670-678 (see also Space-charge waves) Electron precession in ferrite, 451-460 Electronic admittance of reflex klystron, 688 Elliptic sine function, 889-891 Energy electric, 34-36 magnetic, 3 4 - 3 6 velocity of, in free space, 48 in periodic structures, 566-571 in waveguides, 204-205 Excitation of cavity, 538-541. 683-686 of waveguides, 281-294 Exponential taper for transmission line, 372 Faraday rotation in ferrites, 460-464 Faraday's law, 18 Ferrite electron precession in, 451-460 Faraday rotation in, 460-464 magnetic permeability of, 455, 457-459 in microwave devices, 464-476 plane-wave propagation in, 459-460 Filling factor, 155 919 Filters cavity direct-coupled, 639-642 quarter-wave-coupled, 635-639 frequency transformations in, expansion, 599 low-pass to bandpass. 600-602 low-pass to high-pass, 599-600 periodic, 602-603 half-wave. 360-370, 617-626 image-parameter design of. 587-590 impedance inverters in, 603-615 insertion-loss design of, 591-592 low-pass designs for, 595-598 parallel coupled, 626-635 power loss ratio in, 592-594 for Chebyshev, 593 for maximally flat, 593 Fin line, 208-210 Floquet's theorem, 569-571 Foster's reactance theorem, 230-232 Frequency bands, 2 - 3 Fresnel reflection coefficient, 51-52 Fresnel transmission coefficient, 51-52 Gain definitions of available, 274, 728 maximum, 274, 728 power, 274, 728-735 transducer, 273-274, 728 of klystron, 685 of parametric amplifier. 813, 815-820 of traveling-wave tube. 698 Gauss' law. 19 Group velocity in periodic structures, 566-571 in waveguide. 204-205 Gunn oscillator, 832-837 Gyrator. 464-465 Gyrotron, 701-708 H modes, 98, 100-102 in circular guide, 196-198 in rectangular guide, 182-192 Half-wave filter. 360-370, 617-626 Half-wave plate, 405 Hankel functions. 881-885 spherical. 510-511 Helix general properties of, 583-585 sheath. 580-583 dispersion equation for, 583 in traveling-wave tube, 693 920 SUBJECT INDEX Helmholtz's equation, 32, 97 Helmholtz's theorem, 19, 525 HEMT transistor, 722 Hybrid junction as balanced mixer, 865-866 branch line coupler as, 432-434 magic T as, 435-437 ring circuit as, 437-442 scattering matrix for, 436-437, 441 Image parameters of filters, 587-590 IMPATT oscillator, 837-840 Impedance characteristic of capacitively loaded transmission line, 556 of coaxial line, 115 of coplanar line, 176-178 of microstrip line. 150-153 of strip line, 171 of transmission line, 76 general definition of, 38 input, even and odd properties of, 232-233 input, on transmission line, 93 matching, with lumped elements, 319-330 with stubs, 309-319 (see also Quarter-wave transformers; Transmission line, tapered) matrbc imaginary property of, 236-237 symmetry of, 235-236 normalized, 90, 237-238 surface. 56 wave for circular guide, 196-197 for TE waves in rectangular guide. 185. 190 for TM waves in rectangular guide, 189 of waveguide elements, 339-342 Impedance inverters in filters, 603-615 Impedance mismatch factor, 334 invariance of, 334-339 Impedance termination, design of, 330-334 Inductance, distributed, for transmission line, 72-73 Inductor, for microstrip circuits, 320-322 Insertion loss in filters, 591-592 Interdigital line, 577-579 Isolator, 466-468 Johnson noise, 762 k0-p diagram, 564-566 Kinetic power theorem for electron beam, 670 Kinetic voltage, 670 Klystron reflex electronic admittance in, 688 oscillation conditions for, 688 tuning curves for, 688-689 two-cavity. 678-686 equivalent circuit for, 684-685 excitation of fields in, 683-686 gain of, 685 Laplace's equation, 29 Larmor frequency, 452 Lorentz condition, 57. 133 Lorentz force, 17-18 Lorentz reciprocity theorem, 62-64 Loss tangent, 26 Magic T, 435-437, 865-868 Magnetic permeability, 18, 27 for ferrite, 455. 457-459 Magnetic susceptibility, 27 Magnetron. 690-692 Manley-Rowe relations, 804-807 Matching network design of for amplifier, 330-334, 338-339 lumped element, 319-330 Q of, 325-330 with transmission line stubs, 309-319 Maxwell's equations, 21 Meander line, 577-579 MESFET, 721 MIC circuit, 714 Microstrip line, 125-128, 130-169 attenuation of, 153-157, 163-164 coupled, 164-170 dispersion in, 158-163 effective dielectric constant for, 149-152 impedance of, 150-153 inverted-suspended, 126-127 Microstrip resonator, 490-496 disk, 496-500 Q of, 499 Mixer, 856-868 balanced, 865-868 compression in, 862-863 harmonic balance method for, 869-873 image-enhanced, 868 image-rejection, 868 intermodulation in, 863-864 noise figure. 864-865 subharmonic, 868 SUBJECT INDEX MMIC circuits, 714 Mode chart for cylindrical cavity, 507 Negative-resistance amplifier, 814-821 Noise, conductance equivalent. 767 equivalent temperature of, 762 figure, 768-773 circles for, 772-776 of cascaded stages, 770-772 of mixer, 864-865 optimum source impedance for minimum. 769-770 of parametric amplifier, 821-829 Johnson or Nyquist, 762 temperature of amplifier, 771 of system, 771-772 resistance, equivalent, 767 theory of, 760-765 in two-ports, 766-767 Normalized current, 223 Normalized load impedance. 90 Normalized voltage, 223 TV-port circuits, 233-235 Oscillators, design of, 851-856 Gunn, 832-837 IMPATT diode, 837-840 three-port scattering matrix for. 843-849 transistor. 840-856 O-type traveling-wave tube, 692-699 Parallel plate transmission line, 117-125 Parametric amplifier linearized equations for, 807-809 Manley-Rowe relations for. 804-807 negative resistance, 814-821 gain of. 815-820 gain-bandwidth product for. 821 noise in, 823-825 noise figure, of degenerate negative resistance, 825-829 of negative resistance, 823-825 of up-converter, 821-823 p-n junction diodes for, 800-802 up-converter, 809-814 gain of, 813 Periodic structures Bloch-wave impedance for, 555-556 Bloch waves in, 556 921 energy flow velocity in, 566-571 and filters, 587-590 Floquet's theorem for, 569-571 group velocity in, 566-571 k0-p diagram for. 564-566 matching of, 563-564 spatial harmonics in, 569-571 terminated, 560-563 for traveling-wave tube, corrugated plane, 571-577 helix, general properties of, 583-585 interdigital line, 577-579 meander line, 577-579 sheath helix, 580-583 tape ladder line, 577-578 unsymmetrical two-ports in. 559-560 Permeability, 18, 27 for ferrite. 455. 457-459 Perveance of electron beam. 650 Phase shifter, electronic, 409-413 rotary, 404-409 Phase velocity, 47. 198-199 in waveguides. 182 Physical constants, 911-912 PIN diode, 401-403 Plane waves, 44-48 Plasma frequency, 653 effective, 659 Poisson's equation, 29 Polarization circular. 405-407. 452 of circular aperture, 285 in dielectric, 23-27 p-n junction diode, 800-802 Post capacitive, in waveguide, 342 inductive, in waveguide, 341 Potential scalar, dynamic, 57 static, 28 vector, dynamic, 57 static, 30 Power, in circular guide, 197 for TE waves in rectangular guide, 186-187 Power added efficiency. 842 Power divider, 442-450 Wilkinson, 443-450 Power gain, 274, 728-735 Power loss ratio in filter, 591-594 in quarter-wave transformer, 356-357 Power orthogonality, in waveguides, 186 Power waves, scattering matrix for, 268-276 Poynting vector, 3 8 - 3 9 complex, 37 922 SUBJECT INDEX Probe, radiation resistance of. in waveguide, 281 Pulse propagation, on transmission line, 78-85 Quality factor or Q, 325. 503-504 of cylindrical cavity, 507 of dielectric resonator, 513 of disk resonator, 499 external, 483 loaded, 483 of matching network, 325-330 of rectangular cavity. 503-504 unloaded, 483 Quarter-wave plate, 405 Quarter-wave transformers Chebyshev, exact results. 356-360 three-section, 359-360 two-section, 356-358 N-section, approximate theory for. 348-350 binomial. 350-352 Chebyshev, 352-356 prototype circuit for filter, 360-370 single-section, 343-346 Reactive elements in waveguide. 339-343 shunt capacitive, 341 -342 shunt inductive, 340-341 stubs as, 342-343 Reciprocity theorem, 62-64 Reflection from conducting plane. 53-56 from dielectric surface parallel polarization, 49-52 perpendicular polarization, 5 2 - 5 3 small, theory of. 348-350 Reflection coefficient, current, 91 for tapered transmission line, 371 and Riccati equation. 383-386 for terminated transmission line. 90-91 voltage, 90 Reflex klystron. 686-689 Resistance, radiation of probe in waveguide, 281 of transmission line. 114 Resistance-wall amplifier, 708 Resonant circuits bandwidth of. 482-483 damping of, 484 Q of. 482-484 transmission line antiresonant. 488-490 open circuited, 487-488 short-circuited. 485-487 Return loss, 329 Riccati equation for tapered transmission line 383-386 Ridge waveguide, 205-207 Ring circuit, 437-442 Scalar potential dynamic, 56-59 static, 28 Scattering matrix of circulator, 471-476 of directional coupler, 414-416 of hybrid junction, 436-437, 441 for lossless junction, 251-253 for power waves, 268-276 symmetry of, 250-251 and transformation of terminal planes, 249-250 for transistor, 843-849 for two-port junction, 254-257 unitary property of, 253 Schwinger directional coupler, 420 Separation constant, 45 Separation of variables method, 44, 183 Sheath helix, 580-583 in traveling-wave tube, 693 Short circuit choke-type, 397 variable, in waveguide, 395-397 Signal flow graphs, 260-268 Signal velocity, 200-204 Skin depth, 54 Slot line, 127 Smith chart. 304-308 Snell's law. 50 Space-charge reduction factor, 659-660 Space-charge waves ac power relations for, 667-670 and kinetic-power theorem, 670 and kinetic voltage. 670 on axially confined beam, 654-661 dc propagation constant for. 656 effective plasma frequency for, 659-660 fast and slow, 658 reduction factor for. 659-660 on unfocused beam, 661-667 Spatial harmonics in periodic structures, 569-571 Stability, of amplifier, 735-744 Stability circles. 736-744 load. 736 source, 739 Standing wave ratio, 92 Standing waves, on transmission line, 9 1 - 9 2 SUBJECT INDEX Static fields, 28-30 Strip line, i TO-174 attenuation on. 171-173 coupled, 173-174 impedance of, 171 Stub matching with, 309-319 double. 312-317 single, 309-312 triple, 317-319 in waveguide, 342-343 Substrate, properties of, 130 Surface impedance. 56 Surface wave, 124 Susceptibility electric, 25 magnetic, 27 TE waves, 98. 100-102 TEM waves. 98-100 Termination, waveguide, 394-397 TM waves, 98, 102-104 Transducer gain, 273-274. 728 Transmission coefficient, 51 Transmission line capacitively loaded. 551-557 Bloch waves in. 556 characteristic impedance of, 556 circuit analysis of, 551-557 eigenvalue equation lor, 554 kn-p diagram for. 564-566 wave analysis of, 557-559 distributed circuit analysis of. 86-89 field theory of. coaxial line. 106-108, 111 lossless iine, 106-108 lossy coaxial line. I l l with small loss, 111 parallel-plate, with dielectric, 117-125 parameters of capacitance, 112, 117 characteristic impedance. 113, 117 coaxial line, 115, 117 conductance, 115. 117 inductance, 115, 117 resistance, 116. 117 resonant circuit, 485-490 antiresonant. 488-490 open-circuited, 487-488 short-circuited, 485-487 tapered, Chebyshev, 380-383 exponential, 372 reflection coefficient on, approximate equation, 371 reflection coefficient on, Riccati equation for, 385 923 synthesis of, 373-380 triangular, 372-373 terminated, 8 9 - 9 6 Transmission matrix, for cascade network voltage-current, 257-259 wave-amplitude, 259-260 Transverse resonance method, 206-208 Traveling-wave tube M-type- 699-701 O-type, 692-699 gain of, 698 periodic structures for. 571-585 Two-port junctions. 238-248 equivalent circuits for. 245-248 Vector formulas. 876-880 Vector potential dynamic, 56-59 solution for. 59-62 static, 30 Velocity energy flow in periodic structures. 566-571 for plane waves, 48 in waveguides, 204-205 group in periodic structures. 566-571 in waveguides, 200-204 phase for plane waves, 47-48 in waveguides, 182 signal, in waveguides, 199-204 wavefront, in waveguides. 199 Velocity-jump amplifier. 708 Velocity modulation, of electron beam, 670-678 beam coupling parameter in. 672 Voltage, equivalent, in waveguides, 221-224 normalized, 223 Voltage standing wave ratio, 9 1 - 9 3 Wave classification of, 9 6 - 9 9 impedance of TE mode, 185, 190 of TM mode. 189 plane. 44-48 reflection of, from conducting plane, 53-56 reflection of, from dielectric surface. 49-53 TE. 98-102 TEM, 98-100 TM, 98. 102-104 924 SUBJECT INDEX Wave (Cont'd) transmission matrix, 259-260 (see also Periodic structures; Space-charge waves; Transmission line; Waveguide I Wave equation, 31 Wave number. 32 Waveguide capacitive diaphragm in, 341-342 capacitive post in, 342 capacitive rod in, 342 circular, attenuation in, 196-197 solutions for, 194-197 TE waves in, 196-197 TM waves in, 194-196 equivalent current and voltage for, 221-224 excitation of, by aperture, 284-294 by current loop, 283-284 by linear current element. 281-283 inductive diaphragm in, 340-341 inductive post in, 341 properties of, 180-182 rectangular, attenuation in, 188 cutoff frequency of, 184 dominant TE mode in, 190-194 power in, 186-187 solutions for, 189 TE waves in. 182-190 TM waves in, 193 wave impedance for, 185, 189, 197 ridge. 205-207 termination, 394-397 velocity in. energy, 204-205 group, 200-204 phase, 182 signal, 199-204 wavefront, 199