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    The Volterra Series and Its Application
    The Volterra Series and Its Application
    The Volterra Series and Its Application
    Ebook130 pages53 minutes

    The Volterra Series and Its Application

    By Mark Dunn

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    Modeling of weakly nonlinear systems by means of Volterra series analysis is presented. Necessary conditions for representing nonlinearities by a Volterra series are developed analytically as well as heuristically. A two-condition convergence criterion for Volterra series and a method for determining Volterra transfer functions are established. For systems with multiple nodes, an extension of Volterra series analysis; method of nonlinear currents is developed and applied to a MESFET amplifier. Finally, methods of quantifying nonlinear behavior are discussed.
    LanguageEnglish
    PublisherLulu.com
    Release dateFeb 1, 2015
    ISBN9781312884083
    The Volterra Series and Its Application
    Author

    Mark Dunn

    Mark Dunn is a lifelong word lover and the author of Ella Minnow Pea (nominee for Book Sense Adult Fiction Book of the Year, and winner of the Borders Original Voices Awards for fiction), Welcome to Higby (short listed by Publishers Weekly as one of the best novels of the year), and Ibid: A Life. He makes his home in Albuquerque, New Mexico.

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      Book preview

      The Volterra Series and Its Application - Mark Dunn

      The Volterra Series and Its Application

      THE VOLTERRA SERIES AND ITS APPLICATION

      BY

      MARK ROBERT DUNN

      B.E.E (Georgia Institute of Technology) 1984

      THESIS

      Submitted in partial satisfaction of the requirements for the degree of

      MASTER OF SCIENCE

      in

      ELECTRICAL AND COMPUTER ENGINEERING

      in the

      GRADUATE DIVISION

      of the

      UNIVERSITY OF CALIFORNIA

      DAVIS

      Approved:

      Committee in Charge

      1992

      Copyright © 2013 Mark Dunn

      All rights reserved.  No part of this book may be reproduced in any form without the permission of the author.

      ISBN: 978-1-312-88408-3

      ABSTRACT

      Modeling of weakly nonlinear systems by means of Volterra series analysis is presented.  Necessary conditions for representing nonlinearities by a Volterra series are developed analytically as well as heuristically.  A two-condition convergence criterion for Volterra series and a method for determining Volterra transfer functions are established.

      For systems with multiple nodes, an extension of Volterra series analysis; method of nonlinear currents is developed and applied to a MESFET amplifier.  Finally, methods of quantifying nonlinear behavior are discussed.

      ACKNOWLEDGMENT

      I would like to thank Hewlett Packard for providing the financial support through its Fellowship program.  I am especially grateful to Kent Stalsberg.  I owe my acceptance in the Fellowship program to his perseverance, tenacity and ultimately to his belief in me.  I also thank Richard Spencer for his flexibility, patience and advice. 

      Finally, and most importantly, I thank my wife Sue without whom none of this would have been possible.

      Table of Contents

      1.0 INTRODUCTION

      1.1 DEFINITION OF NONLINEAR SYSTEMS

      1.2 STRONGLY vs. WEAKLY NONLINEAR SYSTEMS

      1.3 METHODS OF ANALYZING NONLINEAR SYSTEMS

      1.3.1 PIECEWISE LINEAR

      1.3.2 NONLINEAR DIFFERENTIAL EQUATIONS

      1.3.3 HARMONIC BALANCE

      1.3.4 VOLTERRA SERIES

      2.0 MODELING SYSTEM NONLINEARITIES

      2.1 INTRODUCTION

      2.2 POLYNOMIAL REPRESENTATION

      2.3 CURVE-FITTING TECHNIQUES

      3.0 POWER SERIES ANALYSIS

      3.1 INTRODUCTION

      3.2 GENERAL APPROACH

      3.3 SINUSOIDAL RESPONSE OF

      MEMORYLESS NONLINEAR SYSTEMS

      3.4 LIMITATIONS

      4.0 VOLTERRA SERIES ANALYSIS

      4.1 CONVERGENCE

      4.2 DEVELOPMENT OF VOLTERRA SERIES

      4.3 FIRST-ORDER VOLTERRA SYSTEMS

      4.3.1 FIRST-ORDER IMPULSE RESPONSE

      4.3.2 CAUSALITY

      4.3.3 STABILITY

      4.3.4 FIRST-ORDER VOLTERRA TRANSFER FUNCTION

      4.3.5 SINUSOIDAL RESPONSE OF

      FIRST-ORDER VOLTERRA SYSTEMS

      4.4 SECOND-ORDER VOLTERRA SYSTEMS

      4.4.1 SECOND-ORDER IMPULSE RESPONSE

      4.4.2 KERNEL SYMMETRIZATION

      4.4.3 CAUSALITY

      4.4.4 STABILITY

      4.4.5 SECOND-ORDER VOLTERRA

      TRANSFER FUNCTION

      4.4.6 SINUSOIDAL RESPONSE OF SECOND-ORDER

      VOLTERRA SYSTEMS

      4.5 nth-ORDER VOLTERRA SYSTEMS

      4.5.1 nth-ORDER IMPULSE RESPONSE

      4.5.2 KERNEL SYMMETRIZATION

      4.5.3 CAUSALITY

      4.5.4 STABILITY

      4.5.5 nth-ORDER VOLTERRA TRANSFER FUNCTION

      4.5.6 SINUSOIDAL RESPONSE OF nth-ORDER

      VOLTERRA SYSTEMS

      5.0 DETERMINATION OF VOLTERRA TRANSFER FUNCTIONS

      5.1 INTRODUCTION

      5.2 THEORETICAL DEVELOPMENT

      5.3 APPLICATION

      6.0 METHOD OF NONLINEAR CURRENTS

      6.1 INTRODUCTION

      6.2 THEORETICAL DEVELOPMENT

      6.3 FREQUENCY-DOMAIN REPRESENTATION OF

      NONLINEAR CURRENTS

      6.4 APPLICATION TO NONLINEAR SYSTEM

      7.0 NETWORK ANALYSIS

      7.1 INTRODUCTION

      7.2 METHODOLOGY

      7.3 APPLICATION

      7.4 CONGERGENCE REVISITED

      8.0 QUANTIFICATION OF NONLINEAR BEHAVIOR

      8.1 INTRODUCTION

      8.2 HARMONIC DISTORTION

      8.3 INTERMODULATION DISTORTION

      8.4 GAIN COMPRESSION

      9.0 CONCLUSION

      REFERENCES

      Table of Figures

      Figure 1.1 Block Diagram of Typical Nonlinear System

      Figure 2.1 Interpolation vs. Approximation

      Figure 3.1 Power Series Analysis Model

      Figure 3.2 Complete MESFET Model

      Figure 3.3 Modified MESFET Model

      Figure 4.1 The basis function

      Figure 4.2 Input function & approximation

      Figure 4.3 Second-order impulse response

      Figure 4.4 Second-order transfer function

      Figure 5.1 Nonlinear Circuit

      Figure 6.1 First-order Model

      Figure 6.2 Second-order Model

      Figure 6.3 Third-order Model

      Figure 6.4 Fourth-order Model

      Figure 6.5 First-order Model

      Figure 6.6 Second-order Model

      Figure 6.7 Third-order Model

      Figure 5.1 Nonlinear Circuit

      Figure 7.1a Nonlinear Network

      Figure 7.1b Conceptualization of Nonlinear Network with

      M nonlinear elements pulled out

      Figure 7.1c Associated linear network driven by

      M nonlinear current sources

      Figure 7.1d First-order Solution

      Figure 7.1e Second-order Solution

      Figure 7.1f nth-order Solution

      Figure 7.2 Complete MESFET Model

      Figure 7.3 Simplified MESFET Model

      Figure 7.4 MESFET Model with Nonlinear Currents

      Figure 7.5 MESFET Model with Second-order

                               Nonlinear Currents

      Figure 8.1 Plot of 2nd- and 3rd-order Intercept Points.

      Figure 8.2 Plot of Gain Compression.

      1.0 INTRODUCTION

      Many papers and books have been devoted to the analysis and synthesis of linear systems, in spite of the fact that no system is entirely linear.  All systems, whether electrical, mechanical, biological, or whatever, exhibit varying degrees of nonlinear behavior.

      In medicine a patient's condition may improve with a single dose of medication, be cured with a double dose or made critical with a triple.  In electrical engineering one is constantly faced with nonlinearities due to the fact that the constitutive relationships of any electrical device are nonlinear if a large enough range is considered.  Diodes, for example, are nonlinear regardless of the input, whereas resistors will be nonlinear only at the extremes of their operating range, when the power dissipated causes thermal effects to vary their resistance.

      Generally when an engineer is faced with a nonlinear problem one of two approaches is taken.  The first approach is to set up a measurement system that quantifies the problem (i.e. measures the power of an undesired harmonic) and then proceed to tweak the design until the undesired response is minimized.  This empirical approach sometimes

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