Elements of Radio Waves

arXiv:0712.4029v1 [physics.gen-ph] 24 Dec 2007 Elements of Radio Waves Frank Borg Ismo Hakala Jukka Määttälä ∗ Contents 1 2 2.1 2.2 3 3.1 3.2 3.3 3.4 3.5 3.6 4 4.1 4.2 4.3 4.4 5 5.1 5.2 5.3 5.4 ∗ IH, Introdu tion . . . . . . . . . . . . . . . . . . . . . . . . . . Maxwell equations . . . . . . . . . . . . . . . . . . . . . . Quasi-stationary elds . . . . . . . . . . . . . . . . . . . . General ase  time dependent elds . . . . . . . . . . . . Diele tri s and ondu tors . . . . . . . . . . . . . . . . . . Ele tri sus eptibility . . . . . . . . . . . . . . . . . . . . . Magneti sus eptibility . . . . . . . . . . . . . . . . . . . . Ohm's law . . . . . . . . . . . . . . . . . . . . . . . . . . . Shielding . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Skin ee t . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Leakage through slots . . . . . . . . . . . . . . . . . Reexion and refra tion . . . . . . . . . . . . . . . . . . . Image harges . . . . . . . . . . . . . . . . . . . . . . . . . Interferen e and dira tion . . . . . . . . . . . . . . . . . . Geometri opti s . . . . . . . . . . . . . . . . . . . . . . . Fraunhofer and Fresnel dira tions . . . . . . . . . . . . . Fresnel zones . . . . . . . . . . . . . . . . . . . . . . . . . Kir ho equation . . . . . . . . . . . . . . . . . . . . . . . Radiation from antennas . . . . . . . . . . . . . . . . . . . The Hertz dipole . . . . . . . . . . . . . . . . . . . . . . . Dipole antennas . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Antenna ex itation . . . . . . . . . . . . . . . . . . 5.2.2 Re eiving dipole antenna . . . . . . . . . . . . . . . 5.2.3 Dipole eld . . . . . . . . . . . . . . . . . . . . . . Mutual impedan e . . . . . . . . . . . . . . . . . . . . . . Antenna above ground . . . . . . . . . . . . . . . . . . . . 5.4.1 Sommerfeld's analysis of the antenna above ground JM, . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 3 3 5 11 11 12 13 14 14 17 19 25 27 27 30 33 34 36 36 39 39 46 47 50 53 55 borgbrosnetti.fi; jukka.maattala hydenius.fi. Jyväskylä University, Chydenius Institute, Finland. Emails: FB, ismo.hakala hydenius.fi; . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 Introdu tion 2 5.5 Breaking point, non-smooth surfa e . . . . . . . . . . . . . . . 59 5.6 Ground wave . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 A Some denitions and results from ve tor analysis . . . . . . . . 65 B Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 C Bessel fun tions . . . . . . . . . . . . . . . . . . . . . . . . . . 69 C.1 Bessel C.2 Hankel fun tions D The . . . . . . . . . . . . . . . 74 E Chip on WCR2400 . . . . . . . . . . . . . . . . . . . . . . . . 78 F Re ipro ity theorem 80 G Measurements of the reexions from the ground . . . . . . . . 84 H C- ode for extra ting RSSI values . . . . . . . . . . . . . . . . 88 J -fun tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . able equation and impedan e . . . . . . . . . . . . . . . . . . . . . . . 69 71 Abstra t We present a summary of the basi properties of the radio wave generation, prop- agation and re eption, with a spe ial attention to the gigahertz bandwidth region whi h is of interest for wireless sensor networks. 1 Introdu tion Over a period of several months we have made measurements with a set of trans eivers with the purpose of investigating how the re eived power varies with the surrounding and pla ement of the devi es. The RF-devi es automati ally measure a parameter Indi ator, and thus provide a alled RSSI for Re eived Signal Strength onvenient means to tra k the power level of the signal. Sin e our measurements raised many issues about the behaviour of ele tromagneti elds, it was de ided to review some of the basi ele tromagnetism in the style of a handbook on the appli ation of the Maxwell equations to physi s of hapter. The emphasis here is on rete problems, not on the development of the theoreti al stru ture (in terms of dierential forms, gauge theory, et ). Of physi s books on EM theory we may mention [19, 26, 36℄, and the engineering style books [42, 47, 17℄. A good general physi s referen e in luding material on EM is [21℄. For appli ations to antenna theory see [28℄. General reviews of EM with wireless networks in mind an be found in [1, 6℄. Arnold Sommerfeld was a an eminent mathemati al physi ist who made, among other things, some signi ant ontributions to the propagation of EM elds; these  lassi al methods are des ribed in [14, 44, 45, 46℄. Of re ent texts on antenna theory we may mention [3, 16℄. Balanis has also written a 2 Maxwell equations 3 ni e minireview [2℄. For an interesting online olle tion of le ture notes and a sele tion of lassi papers on EM see [29℄. 2 Maxwell equations 2.1 Quasi-stationary elds Ele tromagnetism stands for one of the four fundamental for es in physi s. A stati point like harge q1 at the point r1 in an isotropi homogeneous medium exerts a for e on an other harge q2 at r2 given by (Coulomb intera tion), F1→2 = 1 q1 q2 (r2 − r1 ) . 4πǫǫ0 |r2 − r1 |3 (1) The quantity ǫ, relative permittivity, is a quantity hara terizing the medium, while ǫ0 (the permittivity of the va uum) is a universal onstant. Eq.(1) an be written as q1 r2 − r1 (2) F1→2 = q2 E(r2 ), where E(r2)✰ E(r2 ) = 1 q1 (r2 − r1 ) 4πǫǫ0 |r2 − r1 |3 (3) ✰ ❨ ✍ r1 r2 is dened as the ele tri eld at the point r2 generated by the harge q1 lo ated at r1 . The ele tri eld an also be expressed in terms of a potential fun tion φ, or vi e versa, E = −∇φ, φ(r) = − Z r r0 E · dr, (4) where the line integral is along a path onne ting the referen e point r0 and the point r. In the ase of Eq.(3) we have φ(r) = q1 1 . 4πǫǫ0 |r2 − r1 | When harges are in a relative motion with respe t to ea h other then we have to in lude in Eq.(2) a term depending on the velo ity, 2 Maxwell equations 4 F = qE + qv × B, B where denes the magneti eld strength. (Lorentz for e) a ting on a hara terized by E and B. BIl harge Thus, Eq.(5) gives the for e moving with velo ity v in an EM eld From this follows the familiar fa t, that a straight urrent I in a dire tion perpendi ular to Conversely, a magneti q harge ondu tor of length l , with the for e (5) q1 at r1 in a magneti B and the eld will sense a ondu tor. moving with the velo ity eld strength at the point B, r2 given by (in an isotropi v1 generates a homogeneous medium) B(r2 ) = µ0 µ q1 v1 × (r2 − r1 ) . 4π |r2 − r1 |3 The eld strength is thus ae ted by the magneti permeability µ µ ≈ 1, a terizing the medium. For most non-metalli s onstant. From the above it follows that two v1 and v2 will intera t via a magneti (6) while µ0 har- is universal harges moving with velo ities for e given by 1 q1 q2 µ0 µ v2 × (v1 × (r2 − r1 )) · = 4π |r2 − r1 |3 q1 q2 µ0 µ (v2 · (r2 − r1 ))v1 − (v1 · v2 )(r2 − r1 ) · , 4π |r2 − r1 |3 F1→2 = q2 v2 × B(r2 ) = (7) where we have used the rule that A × (B × C) = (A · C)B − (A · B)C. A urrent I in a ondu tor onsists of many moving ontributing to the total magneti onsider a small segment over the dl eld strength a of the harges, ea h one ording to (7). If we ondu tor, then the sum of all terms harges in this segment is equal to I dl. Therefore the magneti qv eld strength generated by this segment is given by (Biot-Savart law) 1 An interesting observation is that the for e F1→2 whi h the parti le 1 exerts on the parti le 2 is no longer, in general, the opposite of the for e that the parti le 2 exerts on 1, as would be demanded by the prin iple of is, we no longer have the losed system of F1→2 + F2→1 = 0. harge 1 + a tio est rea tio From between two ontribution of the urrent loops, with Newton's third law. one might on lude that harge 2 may start to move without an external However, the momentum of the total system is the momentum of Newtonian me hani s; that F1→2 + F2→1 6= 0 onserved if we also take into a ele tromagneti eld. on the other hand, we get Cal ulating the magneti F1→2 + F2→1 = 0, in a ause. ount for es ordan e 2 Maxwell equations 5 µ0 µ I dl × (r2 − r1 ) ; 4π |r2 − r1 |3 that is, moving harges forming a urrent I in a small ondu ting element dl at r1 generates a magneti eld dB(r2 ) given by Eq.(8) at the point r2 . In order to obtain the ee t of the dB(r2) = whole (8) I dl ✶ ondu tor one has to sum (integrate) (8) ▼ over all the segments. The permittivity into a ǫ and permeability µ take ount how the medium ae ts the ele - tromagneti eld. The harges in the medium are ae ted by the eld and may be ome dis- r2 − r1 dB(r2 ) ✌ pla ed, whi h leads to a modi ation of the r1 ▼ ② r2 eld (ba krea tion). This explains su h phenomena as polarization ( harge displa ements) and magnetization of a medium. From Eq.(3) and Eq.(8) we infer that by dening D = ǫǫ0 E, 1 B, H= µµ0 H (magneti eld), whi h are apparently independent of the material fa tors (ǫ, µ). From the denitions one an show that the integral of D over a boundary ∂V en losing a volume V is equal to the total harge Q ontained in V , while integrating H along a loop (boundary) ∂S en losing a surfa e S one obtains the total urrent I owing through that surfa e, we obtain the quantities D (ele (9) I ∂V I tri displa ement) and D · dS = Q, ∂S H · ds = I. 2.2 General ase  time dependent elds If we integrate B over a surfa e S we obtain a quantity Φ= Z S B · dS (10) 2 Maxwell equations 6 termed the magneti ux through the surfa e S . It is experimentally observed that when the ux en losed by a ondu ting loop hanges, this indu es a potential dieren e along the loop and auses a urrent to ow. More pre isely (Faraday's law if indu tion), ∂Φ = ∆φ ∂t ⇒ Z S ∂B · dS = − ∂t I ∂S E · ds. Using the mathemati al identity (Stokes' theorem) I ∂S A · ds = I S ∇ × A · dS, we obtain the indu tion law on the form, ∇×E=− ∂B . ∂t This links the time hange of the magneti eld strength to the spatial variation of the ele tri eld. The nal ru ial step is to nd an equation for the time hange of the ele tri eld. From the se ond equation in (10) one may infer that ∇ × H = J (for stati elds) but Maxwell realized that the right hand side of this equation must be omplemented with the term ∂D/∂t (Maxwell's displa ement term), whi h ontains the link to the time hange of the ele tri eld. This addition is needed for maintaining harge onservation (see Eq.(29)). Also, without this term no ele tromagneti waves would exist in the theory. Thus, J C Maxwell was able in 1864 to synthesize the known properties of ele tromagnetism in his now famous equations whi h give, as far as we know, a omplete des ription of the ele tromagneti phenomena in the lassi al regime, ∇·D=̺ ∂D ∂t ∂B ∇×E=− ∂t ∇ · B = 0. ∇×H=J+ Maxwell equations (11) Here ̺ denotes the harge density and J the urrent density. We note that there is an asymmetry between ele tri and magneti elds in the equations 2 Maxwell equations 7 in that there appear no magneti harges (no magneti monopoles) and no magneti urrents. No magneti harge has been ever dis overed, when e all magneti elds are assumed to be generated by moving ele tri harges (ele tri urrents) as des ribed above. Using the rule ∇ × (∇ × A) = −∇2 A + ∇(∇ · A), and the relations (9) one an show that Maxwell equations give the equations, 1 1 ∂2E ∂J + ∇̺, = µµ 0 c2 ∂t2 ∂t ǫǫ0 1 ∂2H ∇2 H − 2 2 = −∇ × J, c ∂t ∇2 E − (12) (13) where we have set (identied with the velo ity of light) c= √ 1 . ǫǫ0 µµ0 (14) Espe ially in the ase of the empty spa e (J = 0, ̺ = 0) we obtain the wave equation ∇2 E − 1 ∂2E = 0, c2 ∂t2 (15) whose plane wave solutions are of the form E = E0 cos (k · r ± ωt) . (16) (Here E0 is a onstant ve tor.) The magnitude of the wave-ve tor k is 2π/λ, where λ denotes the wave length, while ω ( ir ular frequen y) is related to the frequen y f by ω = 2πf . Inserting (16) into (15) we infer that |k|c = ω , whi h is the same as λf = c. Eq.(16) des ribes an os illating eld whose frequen y is f . The solution an be interpreted as a wave moving in the dire tion of the wave ve tor ∓k and with the velo ity c (light velo ity in empty spa e). For empty spa e we have ∇ · E = 0 whi h implies, that for the plane wave (16) we must have k · E0 = 0; that is, the ele tri al eld os illates in a dire tion orthogonal to the dire tion of propagation. From Maxwell equations we nd that the orresponding plane wave solution for the magneti eld is then given by 2 Maxwell equations 8 (17) H = H0 cos (k · r ± ωt) , 1 1k H0 = ∓ k × E0 = ∓ × E0 , ωµµ0 η k p where η = µµ0 /ǫǫ0 is alled the wave impedan e (≈ 377 Ω for va uum). This means that the magneti eld H0 is orthogonal to both k and the ele tri eld E0 . When both the ele tri and magneti elds are orthogonal to the wave-ve tor k the EM-wave is said to be transversal and of the type TEM. The dire tion of the ele tri eld E0 denes the polarization of the wave. For instan e, if E0 = (Ex , 0, 0) then the wave is polarized in the x-dire tion. ✿ E H A harge moving with velo ity v in an ele tromagneti eld feels a for e F given by (5). This involves a work per unit time (power P ) dened as P = F · v. Be ause of the identity v · (v × B) = 0 it follows from (5) that P = qE · v. If we have a urrent density J = ̺v this result is generalized to P = Z V E · JdV, (18) dening a power density P = E · J. Using the ve tor identity ∇ · (E × H) = H · (∇ × E) − E · (∇ × H), one an derive from Maxwell equations the following relation, ∂ E·J+ ∂t  1 1 µµ0H 2 + ǫǫ0 E 2 2 2  + ∇ · (E × H) = 0. (19) This an be interpreted as an equation of energy onservation where 1 1 E = µµ0 H 2 + ǫǫ0 E 2 2 2 2 Maxwell equations 9 represents the energy per unit volume asso iated with the ele tromagneti eld, and S= E×H (the Poynting ve tor) represents the ow of energy arried away by the ele tromagneti tion [39℄. That is, given an area element passing through If we dA (20) dA then S · dA represents per unit time due to the ele tromagneti ompute the ve tor S in radia- the energy radiation. ase of the plane wave (16), (17), we obtain, 1 k k S = ∓ E02 cos(k · r ± ωt)2 = ∓ηH02 cos(k · r ± ωt)2 . η k k (21) Thus we have the important result that the radiated power is proportional to the square of the ele tri If we and magneti cos(k · r − ωt) whi h varies as eld amplitudes. Also, for a wave the power is propagated in the dire tion of al ulate the time average of (21) over a period T = 2π/ω k. we obtain the fa tor 1 1 = 2 T Z T 0 cos(k · r ± ωt)2 dt. 2 Thus, if the average radiation power for a plane wave is 100 mW/m , then we obtain the orresponding ele tri al eld amplitude 100 mW/m2 = whi h yields E0 = √ Sin e ele tromagneti tion, whi h of 2 · 377 · 0.1 waves V/m E0 by setting 1 E02 , 377 Ω 2 ≈ 8.7 V/m (Volt per meter). arry energy, they an also arry informa- ourse make them useful in te hnology. The motion of harges at one pla e (transmitter) will thus intera t with harges at another pla e (the re eiver). This intera tion is des ribed in terms of the ele tromagneti (EM) elds. The transmitter generates EM-waves whi h are inter epted by the re eiver. As seen from the se ond equation in (12) the magneti by the urrent J; 1 H(r, t) = 4π Here one Z an write a solution of the ∇q × J(rq , t̄) 3 d rq = ∇ × |rq − r| t̄ = t − |rq − r|/c it takes time for the eld is the  retarded time eld is determined H-equation 1 4π Z as  J(rq , t̄) 3 d rq . |rq − r| whi h takes into a ontribution generated at the point rq (22) ount that to rea h the 2 Maxwell equations 10 r. This form (22) suggests introdu ing ve tor potential related to E and B by point the B = ∇ × A, E = −∇φ − an auxillary quantity A alled (23) ∂A . ∂t ∇ · B = 0 holds identi
ally; also it ∂A implies in onjun tion with Maxwell equations that ∇ × E + = 0 whi h ∂t suggests the se ond equation in (23). There is some freedom in hoosing A and φ; indeed, if we use the tilded versions  Ã = A + ∇ξ  (24) ∂ξ  Gauge transformation φ̃ = φ − ∂t (for some fun tion ξ ) then the elds E and B remain un hanged in (23). If we use as the supplementary ondition for A that (xing the gauge to the The rst equation in (23) implies that so 2 alled Lorenz gauge ) 1 ∂φ = 0, c2 ∂t (25) 1 ∂2A = −µµ0 J. c2 ∂t2 (26) ∇·A+ we get from Maxwell equations ∇2 A − This has a solution of the form ( onsistent with (22)) µµ0 A(r, t) = 4π In prin iple, if we know the ting antenna we an Z J (rq , t̄) 3 d rq . |rq − r| urrent distribution J (rq , t̄) (27) in the transmit- al ulate the radiated eld using (27) and (23). problem thus redu es to determining the urrent J (rq , t̄) very hard problem to solve analyti ally. However, in many The whi h often is a ases simple ap- proximations will do quite well. Finally we observe two important onsequen es of the above equations. If we dierentiate (25) with respe t to the time and use the se ond equation in 2 This was indeed introdu ed by the Danish physi ist Ludvig Lorenz (1829-1891) and not by the more famous Dut h physi ist Henrik Lorentz (1853-1928) to whom it is often attributed. 3 Diele tri s and ondu tors 11 (23) together with the rst equation in (11), then we obtain the relativisti form of Poisson equation 1 ∂2φ ̺ ∇ φ− 2 2 =− . (28) c ∂t ǫǫ0 This equation determines the potential φ when the harge distribution ̺ is known. A se ond observation is that the identity ∇ · (∇ × H) = 0 applied 2 to the se ond of the Maxwell equations (11) leads to the ontinuity equation, ∇·J+ ∂̺ = 0, ∂t (29) whi h expresses the law of the onservation of ele tri harge. This is of importan e when e.g. determining the urrent/ harge distributions in antennas. 3 Diele tri s and 3.1 Ele tri ondu tors sus eptibility Common ele tromagneti phenomena are due to the intera tion of ele tron and protons, the basi elementary parti les of ordinary matter. Ele tromagneti for es hold atoms and mole ules together. Sin e matter is thus made up of ele trons and protons (and neutrons) we expe t matter to ae t ele tromagneti elds and vi e versa. Although atoms and mole ules may be ele tri ally neutral ( ontain an equal number of ele trons and protons), the harges may be shifted so that one region is dominantly negative while another region is dominantly positive. The matter is then said to be polarized. The polarization an be understood in terms of ele tri dipoles. Suppose we have a positive harge q at the point r1 + l, and a negative harge −q at the point r1 , then the potential of the system measured at the point r2 be omes φ(r2 ) = 1 q q 1 − . 4πǫ0 |r1 + l − r2 | 4πǫ0 |r1 − r2 | When l approa hes zero su h that ql remains a nite ve tor p (the ele tri dipole moment), the potential be omes φ(r2 ) = 1 p·r 4πǫ0 r 3 (r = r2 − r1 ). The orresponding ele tri eld is given by 3 Diele tri s and ondu tors 12 1 E(r2 ) = −∇φ(r2 ) = 4πǫ0  3r(r · p) p − 3 5 r r  (r = r2 − r1 ). (30) The point is that even though the dipole is ele tri ally neutral it generates a non-zero ele tri eld whi h depends the orientation of the dipole. dipole is pla ed in an (homogeneous) external ele tri qE a ts on the positive end and a for e N = p×E a torque −qE a eld E, If a then a for e ts on the negative end reating trying to line up the dipole along the dire tion of the eld. This in turn ae ts the eld generated by the dipole. The harges asso iated with dipoles are one Thus, the total harge density we obtain whi h suggests that ǫ0 ∇ · E = ∇ · D − ∇ · P ǫ0 E = D − P. (31) Experimentally it is found, under not too extreme is a linear relation between the external eld E = ǫ0 χe P. sin e they While the displa ement eld an dene the ǫ0 ∇ · E = ̺, bound harges an be written as ̺ = D is dened su h that ∇·D = ̺free , polarization density P su h that ∇ · P = −̺bound . Sin e annot move freely. ̺bound +̺free . alled onditions, that there E and the indu ed polarization, This nally gives using (31) D = ǫ0 (1 + χe )E = ǫǫ0 E, (32) whi h is equation (9) with the relative permittivity given by where 3.2 χe is the ele tri sus eptibility. Magneti In the magneti ǫ = 1 + χe , sus eptibility ase we do not have magneti dipoles formed by magneti harges, be ause, as pointed out earlier, there appears not to exist any magneti harges in the nature. Instead the magneti by ele tri urrents. On the atomi elds are generated entirely and mole ular s ales we have ele tri ur- rents due the ele trons  ir ling around the atoms. Also the spin of the ele trons ontribute to magnetism. An external magneti the atomi urrents and thus hange the urrents in an analogy with the ele tri urrents Jbound generate a eld may dee t orresponding eld generated by the polarization. The atomi magnetization M dened by ∇ × M = Jbound . (bound) 3 Diele tri s and ondu tors Besides the bounded 13 Jbound urrents average to zero, we might in whi h ondu tors have a Jfree related to the −1 magneti eld by ∇ × H = Jfree . Sin e µ0 ∇ × B = J = Jbound + Jfree we on lude that free (ma ros opi ) urrent a ✻ µ−1 0 B = H + M. For para- and diamagneti magnetization and the magneti der normal M = χm H, I substan es the ✲ eld are, un- ir umstan es, linearly related, χm is the magneti sus eptibility. Thus, we get B = µ0 (1 + χm )H = µ0 µH, ability given by µ = 1 + χm , whi h is the se ond the ele tri where with the magneti equation in (9). Whereas polarization was analyzed in terms of ele tri dipoles, the mag- netization may be analyzed in terms of small urrent loops. If one su h a small hara terized by urrent loop in a magneti eld perme- B, onsider then the total for e a ting on it is zero whereas the torque be omes (applying (5)), N= with the magneti I Ir × (dr × B) = m × B, moment m dened by a= 1 2 I 3.3 I form a its urrent. The magnetization M moments. Ohm's law A free move. where r × dr is the area en losed by the loop and orresponds to the density of magneti m = Ia, harge q in an ele tri Thus, ele trons in the eld E feels a for e qE whi h auses it to ondu tion band in metals ( ondu tors) an urrent when a potential dieren e is applied over a pie e of a metal. The ele trons though meet resistan e aused e.g. by the thermal motion of the atoms. This is manifested in the well known law of Ohm a ording to U = RI in order to drive a urrent I R. (It is onventional to use U for uits.) In terms of the urrent density J and whi h one needs a potential dieren e through a ondu tor with the resistan e the potential in the theory of the ele tri al eld E ir driving the urrent, the law of Ohm J = σE an be written as (33) 3 Diele tri s and ondu tors 14 ondu tivity σ where the pre isely, for a is inversely related to the resistan e. ondu tor of length L and ross se tion A More we have for the resistan e R= where ρ = 1/σ for metals). denes the L L =ρ . σA A resistivity −7 (typi ally of the order of 10 When treating ele tromagneti waves in Ωm ondu tors we thus have to use the relation (33) in Maxwell equation (11). 3.4 Shielding 3.4.1 Skin ee t It is well known that metalli external ele tromagneti harge elds. This shielding is −z -dire z -dire tion on a metalli aused by the fa t that the surfa e with the normal dire tion in tion. The equation (12) be omes ∇2 E − 3 ∂E 1 ∂2E − σµµ0 = 0. 2 2 c ∂t ∂t Inserting a solution of the form (in the metalli we ages) prote t against arriers generate an opposing eld. Suppose we have an in ident plane wave along the the en losures (Faraday onsider only the transmitted omponents in (34) medium z > 0; here omponent, not the in ident and ree ted z < 0) Ex = E0x ei(kz−ωt) we obtain for k the equation k2 = ω2 + iσµ0 µω. c2 The imaginary part leads to exponentially de aying fa tor in (35) exp(ikz). For instan e, if the se ond term in (35) dominates then we have 1 + i√ k≈ √ σµ0 µω, 2 3 The ̺-term vanishes in this spe ial ase. Combining the ontinuity equation (29) with J = σE we obtain for an harmoni plane eld parallel with surfa e, E = (Ex , 0, 0), σ ̺ = −i ∇ · E, ω whi h is 0 sin e Ex depends only on z as the plane eld travels in the z -dire tion. 3 Diele tri s and ondu tors 15 whi h leads to ikz e     1 √ 1 √ ≈ exp i √ σµ0 µωz · exp − √ σµ0 µωz . 2 2 The se ond de ay fa tor shows that we have hara teristi penetration depth of r 2 . (Penetration depth, for high σ ) (36) σµµ0 ω As an example, for opper we have σ = 6 · 107 (Ωm)−1 , µ ≈ 1, whi h give at ω = 2 πf = 2 π 2.4 GHz a penetration depth of δ = 1.3 · 10−6 m; that is, about 1 µm. From this it follows that a 1 mm Cu-sheet will pra ti ally stop the eld ompletely. For an aluminum (δ = 3.8 · 10−6 m) foil of thi kness 0.01 mm we would get a suppression fa tor around exp(−0.01 mm/δ) ≈ 2.5 · 10−3, δ= or -52 dB. The shielding (absorption of radiation) an also be interpreted in terms of a omplex permittivity. From the se ond of Maxwell equations (11) we see that the urrent density J and the eld D o ur in form of the ombination J+ ∂D . ∂t Assuming harmoni elds depending on time as exp(−iωt) the time derivative above an be repla ed by the fa tor −iω , and if we repla e J by σE, the above expression be omes  σ  ∂D = −iω i + ǫǫ0 E. σE + ∂t ω This means that for ondu tors the ee t of the urrent on the elds an be taken into a ount by using a omplex relative permittivity given by σ ≡ ǫ′ + iǫ′′ . (37) ωǫ0 It is onventional to denote the real part by ǫ′ and the imaginary part by ǫ′′ . In many texts they write the omplex permittivity as ǫ′ − iǫ′′ , whi h follows from assuming a time dependen e exp(iωt) instead of exp(−iωt); ǫr = ǫ + i thus, it is purely a matter of onvention. The ee t of the ondu tor is also to make the wave impedan e η dened in (17) imaginary. Indeed, using Maxwell equations for the plane waves we nd that the E- and H-amplitudes are related by H= E , η η= k , ǫǫ0 ω + iσ (38) 3 Diele tri s and ondu tors where k 16 is given by (35). The above relation redu es to the one in (17) σ = 0. A material is alled a good ondu tor if the diele tri ′′ ′ part dominates, ǫ ≫ ǫ , whi h translates into when As an example, σ ≫ 1. ǫǫ0 ω for ω = 2 πf (Good = 2 π imaginary ondu tor riterion) 2.4 GHz we get ǫ0 ω ≈ 0.133 (Ωm)−1 (f = 2.4 GHz). 7 This an be ompared with the ondu tivity of opper, σ = 5.8 ·10 −1 (Ωm) , whi h thus, as all metals, qualies as a good ondu tor by a safe −1 margin. The human body has a ondu tivity around 0.2 (Ωm) and is thus a poor ondu tor at this frequen y. Sea water is a borderline ase at −1 this frequen y having a ondu tivity around 4 (Ωm) and ǫ ≈ 80. The penetration depth is an important mi rowave ovens; δ hara teristi must be of the order of for food that is heated in entimeters for the radiation to heat the food thoroughly. Be ause of the small penetration depth for good assumed that the ele tri that it is, like the eld is zero in the main part of the urrent, ondu tor, and onned to a thin layer of thi kness about the surfa e. This has important the ondu tors it is typi ally xy -plane, and that the ele δ near onsequen es. Suppose the surfa e lies along tri eld is tangential in the x-dire tion. Then we have from the third equation in (11), |∆Ex | = |∆z| If take the dieren e ∆Ex ∂By . ∂t to be over the interfa e, and let we obtain that (II) (I) = Et , Et that is, the tangential |∆z| → 0, then omponent Et of the ele tri (39) eld hanges ontin- uously a ross an interfa e between two mediums (I) and (II). Thus, if the (I) ele tri eld is zero inside the ondu tor (Et = 0) it must also be zero at the (II) outside surfa e (Et = 0). It follows that when an os illating ele tri eld Ei impinges on a ondu ting surfa e, it generates a surfa e urrent J ausing magneti H Er , eld ree ted eld whi h in turn, a ording to Maxwell equations, su h that the tangent omponent of the total eld is zero at the surfa e (this is a simpli ation valid only on a s ale large with the skin depth) auses a ompared 3 Diele tri s and ondu tors 17 i r Etot t = Et + Et = 0. (At the surfa e of a ondu tor.) (40) This is used as a boundary ondition when treating radiation in avities and antenna radiation. Thus, the urrent J aused by an impinging eld in an antenna will generate a eld outside the antenna from whi h one may, for example, al ulate the gapeld and orresponding potential dieren e in ase of a dipole antenna (see se . 5.2.1). Another onsequen e of the penetration layer is that ele tromagneti waves will loose energy due to ohmi losses; the waves indu e surfa e urrents whi h en ounter a resistan e given by 1 ρ Rs = = δ σ r σµµ0 ω = 2 r µµ0 ω . (Surfa e resistan e.) 2σ (41) Using previous data on opper we get for its surfa e resistan e Rs ≈ 1/78 Ω (at 2.4 GHz). 3.4.2 Leakage through slots If there are holes in the shield then there will be no ountera ting urrent at that pla e and radiation an leak through. This ee t an be used to an advantage when onstru ting slot antennas, but for shielding purposes the leakage is of ourse a nuisan e. In order to get an estimate of the leakage through a slot one may onsider a re tangular hole in a ondu ting sheet. We suppose the sheet is in the xy -plane with the normal along the −z -axis and has thi kness d. Further we take the hole to have the orners (0,0), (0,b), (a,0), (a,b) in the xy -plane. We will treat the hole as a wave guide on whi h impinges a planar EMwave along the z -axis. Intuitively it seems lear that waves with a wavelength λ ≫ a, b will have di ulty in passing through the hole; that is, the hole a ts as a high-pass lter damping waves with wavelengths ex eeding the dimension of the hole, but letting smaller wavelengths through (the high frequen y part). We will onsider a TE-wave passing the wave guide; thus, the ele tri eld is transversal while the magneti eld may also have a longitudinal omponent along the z -axis4 . We will therefore assume that 4 The TEM ase leads to a trivial solution of zero elds inside an empty wave guide. Indeed, in this spe ial ase one obtains from Maxwell equation that (T means here that the operators are restri ted to the transversal xy -plane) ∇T × E = 0, from whi h one may posit that there is a fun tion φ su h that E = −∇T φ. 3 Diele tri s and ondu tors 18 E(x, y, z, t) = (Ex (x, y), Ey (x, y), 0) · ei(βz−ωt) , H(x, y, z, t) = (Hx (x, y), Hy (x, y), Hz (x, y)) · ei(βz−ωt) . The dependen e on z is thus fa tored out as exp(iβz) sin e we are interested in wave solutions progressing in the z -dire tion. If we insert the above ansatz into the wave-equation (12) for the E-eld we obtain (J = 0 and ̺ = 0 in empty spa e) (Helmholz equation) ∇2T Ex = (β 2 − k 2 )Ex , (42) with ∇2T ≡ ∂2 ∂2 + ∂x2 ∂y 2 and k = with a similar equation for the y omponent. Here ∇2T refers to the Laplaian operator restri ted to the transversal plane. Sin e the tangential omponents of the ele tri elds vanish at the surfa e of the walls, they will be of the form Ex (x, y) = g(x) sin Ey (x, y) = h(y) sin  nπy  , b  mπx  a ω 2π = , c λ b x ✻ z ✲ ❘y ✲ a . Using the Maxwell equation ∂x Ex + ∂y Ey = 0 we an determine the fun tions h and g , obtaining nally (m, n 6= 0),  nπy   mπx  a sin , cos m a b  nπy   mπx  b Ey (x, y) = −Emn sin cos . n a b (43) Ex (x, y) = Emn Combining this with the equation ∇T × E = 0 gives the equation ∇2T φ = 0. Lapla e equation in two dimensions, and for a simple region (su h as the the wave guide) this has the trivial solution that φ is φ onstant along the boundary of the has the trivial solution for a magneti φ onstant. Indeed, Etangent = 0 implies ross-se tion, and therefore that ∇2 Tφ = 0 = = onstant in the ross-se tion. In the TE longitudinal omponent, ∇T × E = 0 and non-trivial solutions be ome possible. This is the ross-se tion of is repla ed by ase, where we allow ∇T × E = iωµ0 µHz , 3 Diele tri s and ondu tors The ase m = 0 19 orresponds to a solution of the form Ex (x, y) = an sin Ey (x, y) = bn sin  nπy  b   nπx a , . The general solutions may be onstru ted as superpositions of these (m, n)mode solutions. Inserting an (m, n)-mode solution into (42) we obtain the relation 2 2 β =k −  nπ 2 a − From this we see that if  mπ 2 b = a, b < λ a = 1 mm in a d = 2 mm thi k 2π λ 2 −  nπ 2 a −  mπ 2 b . (44) β must be ne essarily imaginary exp(iβz). Suppose we have a narrow then leading to an exponential de ay fa tor slot  ondu ting plate, then we may estimate the radiation through the slot to be damped by fa tor of the order   s   π 2  2π 2  − · d ≈ exp(−3141 · 0.002) ≈ 0.0019, exp −   a λ whi h m). For dB) orresponds to a -54 dB damping (power) at 2.4 GHz (λ = 0.125 a≪λ the above formula for damping damping [dB℄  − πd a = 20 · log e 3.5 Reexion and refra tion  an be approximated by (in d ≈ −27.3 · . a Reexion and refra tion of waves is a familiar phenomenon from our daily experien es with light, sound and water. Reexion is a basi ee t when a wave hits an inhomogeneity in the medium, typially the interfa e between two dierent mediums su h as air and water. We will dis uss this fundamental feature in terms of a very simple model. Consider a wave traveling along the x-axis in a medium (I, (I) ✲ ✛ (II) ✲ 3 Diele tri s and ondu tors 20 x < 0) whose propagation velo ity is u1 . At x = 0 starts another medium 0) with a dierent propagation velo ity u2 . (In ase of EM-elds (II, x > √ u = c/ ǫǫ0 where c is the light velo ity in va uum.) We use φ for a propa- gating eld satisfying the following equations: ∂2φ 1 ∂2φ − = 0, (I) (x < 0) ∂x2 u21 ∂t2 ∂2φ 1 ∂2φ − = 0. (II) (x > 0) ∂x2 u22 ∂t2 (45) We write the basi harmoni solutions for regions (I) and (II) as: (I) (II) φ1 (x, t) = ei(k1 x−ωt) + Rei(−k1 x−ωt) , φ2 (x, t) = T ei(k2 x−ωt) . Here the wavenumbers ki are given by ki = ω/ui . The interpretation of the solution (46) is the R-term represent the ree ted part propagating in the −x-dire tion, and the T -term the transmitted part. The oe ients R, T an be determined from the requirement that φ and ∂φ/∂x be ontinuous at the boundary x = 0; that is, φ1 (0−) = φ2 (0+), ∂φ1 (0−)/∂x = ∂φ2 (0+)/∂x. This yields the equations 1 + R = T, k1 (1 − R) = k2 T, (I) ✒ (46) ✻z ■ θr θ (II) y ✲ θt ✶ (47) (48) from whi h obtain R and T , u2 − u1 k1 − k2 = , k1 + k2 u1 + u2 2k1 2u2 T = = . k1 + k2 u1 + u2 R= (49) (50) The above model may e.g. be used to des ribe the ee t of onne ting two ables with dierent impedan es; the dis ontinuity at the onne tion gives rise to reexions. We may also note that the sign of the reexion oe ient 3 Diele tri s and ondu tors R 21 depends on whether the wave travels faster or slower in region II than in region I. Next we will on a diele tri onsider an EM plane wave in open spa e (z > 0) impinging surfa e in the side I (z > 0) will xy -plane at z = 0. The total ele tri eld on the onsist of the in oming and the ree ted part (R), while on the side II (z < 0) we will have the transmitted (refra ted) part (T ), Eei(k·r−ωt) + ER ei(k i(kT ·r−ωt) ET e R ·r−ωt) , (I) . (51) (II) Here the magnitudes of the waveve tors are given by √ k1 = |k| = |kR | = ω µ1 µ0 ǫ1 ǫ0 , √ k2 = |kT | = ω µ2 µ0 ǫ2 ǫ0 . As demonstrated in the ele tri onne tion with (39) the tangential eld does not omponent of hange a ross the interfa e. Hen e, oordinate system so that the tangent obtain at the I-II interfa e, z hoosing the omponent is along the y -axis we = 0, Ey ei(k·r−ωt) + EyR ei(k R ·r−ωt) This equality is only possible if (at z = EyT ei(k T ·r−ωt) . (52) = 0) k · r = kR · r = kT · r from whi h one dedu es (set the angle of reexion dent angle θr r = ŷ ) that is equal to the in i- θ (angles are here measured as those ✻ Dz made by the dire tions of propagation with the interfa e), while the angle of refra tion θt on the other hand is related by Snellius' law n1 cos θ = n2 cos θt where ni (Snellius) (53) are the indexes of refra tion of the mediums given by ni = √ ǫi µi . From (52) we also obtain that Ey + EyR = EyT . (54) 3 Diele tri s and ondu tors If we onsider the polarization the elds 22 E -eld to be polarized in the ase; H-eld will be along the y -axis) (verti al, V- omponents of an be expressed in terms of the amplitudes, Ey = E sin θ, Hy = 0, 1 Hx = E, η1 R Hy = 0, 1 HxR = E R , η1 T Hy = 0, 1 HxT = E T . η2 Ez = E cos θ, EyR = −E R sin θ, EzR = E R cos θ, EyT = E T sin θt , EzT = E T cos θt , In order to determine the amplitudes tion besides (52). This whi h yz -plane then the ER, ET , (55) we need one further equa- an be found by applying (10) to a very thin pill-box ontains the I-II interfa e, then one obtains that I where
D · dS = DzI − DzII S = Q, S is the top (bottom) surfa e area of the pill-box, and Q the surfa e harge ontained by it. Going to the innitesimal thin pill-box limit we obtain the general result on the normal omponent of the displa ement ve tor, DIn = DII n + ̺s , where ̺s is the surfa e (56) harge density (Q/S ). In our parti ular an assume that there are no extra surfa e harges (̺s = 0), ase we when e, using D = ǫǫ0 E, ǫ1 (Ez + EzR ) = ǫ2 EzT . (57) Another alternative is to use the boundary ondition that the tangent I II omponent of the H-eld is ontinuous a ross the interfa e, Ht = Ht , whi h an be derived in a similar way as in the ase of the E-eld. Note that the wave impedan es ηi in (55) are given by (38) whi h over the ondu tive media too. Combining (57), (54) and (53), we an obtain after some algebra, ase of 3 Diele tri s and ondu tors 23 p ǫr sin θ − ǫr − (cos θ)2 ER p ρv ≡ , = E ǫr sin θ + ǫr − (cos θ)2 √ 2 sin θ ǫr ET p τv ≡ , = E ǫr sin θ + ǫr − (cos θ)2 (58) (V-polarization ase.) where we have used the notation ǫr = ǫ2 /ǫ1 . Similar onsiderations an be applied in the horizontal (H) polarization ase, with the end result, p sin θ − ǫr − (cos θ)2 ER p ρh ≡ , = E sin θ + ǫr − (cos θ)2 ET 2 sin θ p τh ≡ . = E sin θ + ǫr − (cos θ)2 (59) (H-polarization ase.) The equations (58), (59), are known as Fresnel [FRA-nel℄ equations. These equations also apply when interfa ing a ondu ting material by repla ing ǫ by a omplex number as explained in onne tion with (37). Thus, if the medium II is a perfe t ondu tor this orresponds to letting |ǫr | → ∞, and we have then ρv = 1 for V-polarization and ρh = −1 for H-polarization. For diele tri s there is a spe ial angle, the Brewster angle θB , sin θB = √ 1 1 + ǫr (60) at whi h the ree ted verti al omponent goes to zero, ρv = 0. Thus, if the in oming eld is verti ally polarized, none of it will be ree ted at the Brewster angle (for a planar interfa e). This means that if the transmitter (using verti al antenna) and the re eiver are pla ed su h that the Brewster angle ondition is satised then the ree ting omponent of the radiation is eliminated from the transmission, and only the dire t eld is re eived. This onguration may be used to measure how the in lination of the re eiver antenna ae ts the re eption; that is, to measure the fun tion G(θ) for varying θ. The above analysis an be generalized to the ase where we have two mediums I, III, with a se ond medium II of thi kness d sli ed between them. One example might be air (I), and i e sheet (II) with water (III) below (whi h we have investigated experimentally). The ree ted eld in I is thus ree ted both from the interfa e I-II and from the interfa e II-III. We may 3 Diele tri s and ondu tors 24 treat the problem with the methods used above, pasting together plane wave solutions in the regions at the interfa es. We may also use the methods of geometri al opti s and sum the ontributions from all the additional reexions from the intermediary layer  e II-III. We denote by (and similarily  interfa ǫ2 for transmission oe ient) ρ ǫ1 , θ the reexion oe ient (subindexes will indi ate the polarization states) for an EM-wave in a medium I impinging on a surfa e of a medium II at the angle θ. The total reexion will be a sum of the primary ree tion at point A (see gure), the next ontribution omes from the transmitted part whi h ree ts from point B and then exits the ✒ ✒ surfa e at point C , and so on. It is important (I) θ to note that the parts that boun e through the ✒ ✻ intermediary layer II pi k up additional phase d (II) θt dieren es due to the fa tor ❄ θt′ (III) exp(ikII · r). ◆ ◆ The phase ontribution due to a a given opti al path is ks where s is the length of the path. Thus, after some trigonometri al exer ises, the phase dieren e between the paths ABC and AD will turn out to be, n2 2π2d n1 2π 2n2 2πd ∆= − 2d cot θt cos θ = λ sin θt λ λ  1 cos θt2 − sin θt sin θt  = 2kn2 d sin θt , where we have used the law of Snellius, and k = 2π/λ for the waveve tor magnitude in va uum, and ni for the indexes of refra tion. Summing all the reexion ontributions we get,         ǫ2 ǫ1 ǫ2 ǫ3 i∆ ρtotal = ρ (61) ,θ + e τ ,θ ρ , θt τ , θt + ǫ1 ǫ1 ǫ2 ǫ2           ǫ3 ǫ1 ǫ3 ǫ1 ǫ2 i2∆ ,θ ρ , θt ρ , θt ρ , θt τ , θt + · · · = e τ ǫ1 ǫ2 ǫ2 ǫ2 ǫ2       ǫ2 ǫ1 ǫ3   , θ τ , θ τ , θ ρ t t ǫ2 ǫ1 ǫ2 ǫ2    .  ρ , θ + ei∆ ǫ1 1 − ei∆ ρ ǫ3 , θ ρ ǫ1 , θ ǫ2 t ǫ2 t 3 Diele tri s and ondu tors 25 Here we have used the geometri summation rule 1 + x + x2 + · · · = 1/(1 − x). By a similar al ulation we obtain for the transmission oe ient τ for the radiation that enters into the medium III,     e τ τ ǫǫ23 , θt′    . τ= ǫ3 ǫ1 i∆ 1 − e ρ ǫ2 , θt ρ ǫ2 , θt i∆/2 ǫ2 , θt ǫ1 D ✒ θ A ✻ ❄ ✒ C d (62) (I) (II) θt B As an example we an al ulate the transmission oe ient for radiation impinging normally on a bri k wall of thi kness d = 10 m assuming ǫ1 = ǫ3 = 1 and ǫ2 = 4, leading to (setting ǫr = ǫ2 /ǫ1 ) τ= √ √ 4 ǫr eik ǫr d 1+
2 = −0.74 + i0.43, √ √
2 √ ǫr + ei2k ǫr d ǫr − 1 (63) whose absolute value is 0.86 ( 2.4 GHz) orresponding to a redu tion of power by the fa tor 0.862 = 0.74 (-1.3 dB). Sin e ǫr is assumed to be real there is no absorption in the wall. If we use ǫ′′ = 0.07 for the bri k wall then we have for a 1 m wall |τ | = 0.36 thus showing already signi ant absorption (-8.9 dB). In reality there would be a further loss of power due to s attering aused by the inhomogeneities in the wall. Bri k walls are seldom 1 m thi k, instead the radiation may have to pass several bri k walls whi h are say 10 m thi k. Then a qui k estimate would be that the power de reases with a fa tor about 0.74 per wall. For a more detailed treatment one an extend the above methods to an arbitrary numbers of diele tri layers [40℄. One an also apply the transmission theory for ables using for impedan e the wave p impedan es Z = µµ0/ǫǫ0 (see Appendix D). 3.6 Image harges We onsider a harge q above a plane perfe t ondu tor whi h we take to be in the xy -plane at z = 0. The plane will now ae t the ele tri eld. As explained earlier the ele tri eld must have a zero tangential omponent on the surfa e of the ondu tor. An equivalent formulation is that the potential ϕ is onstant along the surfa e. Thus, given this boundary ondition, one has to solve the Lapla e equation ∇2 ϕ = 0 whi h is valid for z > 0, ex ept at the pla e of the harge whi h we may suppose is at the point r0 = (0, 0, h). One an onvin e oneself that the solution must be 3 Diele tri s and ondu tors ϕ(r) = 26 q r − r0 r + r0 q − . 2 4πǫǫ0 |r − r0 | 4πǫǫ0 |r + r0 |2 (64) It satises the Lapla e equation for z > 0 ex ept at the point r0 , and it vanishes for z = 0; that is, z = 0 is an equipotential surfa e. Furthermore, if we integrate E = −∇ϕ over a small sphere ontaining q we obtain q/ǫǫ0 proving that it is indeed the potential of the harge q . The solution (64) means that the ee t of the ondu ting plane is the same as if we had an additional extra harge of the opposite sign at the pla e of its mirror image, r0 − 2(n · r0 ) = −r0 (n is the normal of the surfa e), in an empty spa e. One onsequen e of the mirror ee t is that the harge is attra ted toward the ondu ting plane by the apparent opposite image harge. Physi ally the ee t of the ondu ting plane is that the harge q polarizes the free harges in the plane by attra ting them if of opposite sign, and repelling them otherwise. In fa t, the indu ed surfa e harge at z = 0 an be al ulated from ̺s = −ǫǫ0 ∂ϕ/∂z by inserting the solution (64). If we integrate ̺s over the surfa e we get in fa t for the total indu ed harge the result −q . This mirroring method an be generalized to other surfa e that an be onstrued as equipotential surfa es for some distribution of harges. Consider the ase where we have two ondu ting planes meeting along the z -axis. We may take the ondu ting planes to be the xz -plane and the yz -plane (see part (b) in the gure). The potential in the open spa e region is obtained adding three image harges as shown in the gure. (I) q q ′′ q −q q −q (II) q′ (a) (b) Indeed, one sees that this arrangement makes the total potential zero along the planes. This example has appli ation in the ase we use 90◦ -degree bent sheet as an antenna ree tor. A somewhat more involved ase is that of pla ing a harge between to parallel ondu ting planes whi h requires an 4 Interferen e and dira tion innite number of image of pla ing a 27 harges (as one an see from the analogous ase andle between two parallel mirrors). The imaging prin iple an also be applied to the ase of a diele tri instead of a ondu ting plane. Thus diele tri z < 0. onsider the ase where we have a homogeneous z > 0, and a dierent diele tri II in the region harge q at the point r0 = (0, 0, h) in I, and the problem is I in the region We pla e a to determine the resulting potential in I and II. We make the ansatz that the potential in I is the sum of the potential generated by q and an imaginary ′′ ′′ harge q in II, and that the potential in II is generated by a harge q at r0 possible dierent from q due to s reening. q q′ 1 1 + , 4πǫ1 ǫ0 |r − r0 | 4πǫ1 ǫ0 |r + r0 | 1 q ′′ . φII (r) = 4πǫ2 ǫ0 |r − r0 | φI (r) = We an determine the unknown harges q′ (In I.) (65) (In II.) and q ′′ from the boundary onditions at z = 0 where we have the ontinuity of the tangent ele tri eld, I II EIt = EII t , and the normal ele tri displa ement, Dn = Dn (sin e no free surfa e harges are expe ted for the diele tri s). Expressing these onditions in terms of the potential (65) we obtain the equations q q′ q ′′ + = , ǫ1 ǫ1 ǫ2 ′ −q + q = −q ′′ . (66) These equations have the solution ǫ1 − ǫ2 , ǫ1 + ǫ2 2ǫ2 . q ′′ = q ǫ1 + ǫ2 q′ = q The in this (67) orresponds to the limit ǫ2 ′ ase we indeed re over the solution q = −q . ondu ting plane →∞ and we see that 4 Interferen e and dira tion 4.1 Geometri opti s As is well known, light, whi h is EM-wave of very short wavelengths (around 0.11 µm), an in many problems be treated as onsisting of rays. This is 4 Interferen e and dira tion 28 the method of geometri opti s. The plane waves dis ussed above do not exist in reality as they would be of innite extent. But lo ally EM-waves may often quite well be approximated by plane waves. The geometri opti al methods give good approximations far away from the sour e (transmitter) and when we onsider spatial dimension large in omparison with the wavelength λ. If we write ψ to represent some omponent of the EM-eld, then one may set ψ(r, t) = A · eiS(r,t) , ✯ (68) where A (amplitude) and S (phase) are real fun tions. In diele tri medium ψ satises the wave equation n2 ∂ 2 ψ ∇2 ψ − 2 2 = 0 (69) c ∂t √ where n = ǫµ is the index of refra tion. ✲ S ❥ A plane wave orresponds to the ase where A = onstant and S(r, t) = k · r − ωt. For a xed time t the sets of points r satisfying S(r, t) = onstant form surfa es (wavefronts) of onstant phase. If we draw the lines that are everywhere orthogonal to the wavefronts we obtain the rays. The on ept of rays is useful only when the hara teristi dimensions of the regions onsidered are large in omparison with the wavelength λ. Then one make the assumption that the amplitude A hanges only a little over regions of the dimension λ. Mathemati ally this ondition may be expressed as |∇A|λ ≪ |A|. If the phase lo ally is lose to that of the plane wave we may assume that |∇S|λ ∼ 2π . Using these assumptions one obtain from (68) the approximate equation n2 (∇S) = 2 c 2  ∂S ∂t 2 = n2 k 2 . (Eikonal equation.) (70) The last equality follows if we assume an harmoni wave for whi h S(r, t) is of the form S0 (r)−ωt. In (70) k = 2π/λ is the magnitude of the wave ve tor in va uum. The eikonal equation is equal to Fermat's prin iple5 a ording to whi h the rays are paths Γ whi h minimizes the traveling time dened by 5 This follows from the fa t the eikonal equation is analogous to the so alled HamiltonJa obi equation of lassi al me hani s for a parti le moving in a potential proportional to −n(r)2 . This in turn is related to the least a tion prin iple (Maupertuis) whi h nally leads to the Fermat's prin iple. 4 Interferen e and dira tion 29 tΓ = Z Γ nds . c (71) From this it follows that rays are straight lines in homogeneous regions (n independent of position). If we are far away from an antenna (r ≫ λ) then we may think of the EM radia- tion as arriving from the antenna along rays. If the antenna is r T R ✲ lose (in relation to the distan e to the observation point) to the ground, buildings et , then, besides the ontribution h oming ❃ θ along the straight line between the antenna and the observation point, we may also have rays ree ted from the ground, buildings et , arriving at the observation point. Thus, the eld at a point r may be a sum of ontributions due to many paths, ψ(r) = A1 eiS1 (r) + A2 eiS2 (r) + · · · . (72) For a at ground this sum redu es to just two parts (two-ray model): the dire t ontribution and the ree tion from the ground. In the terms in (72) will be of the simple form A exp(iks) ase of reexions where s is the total path length. Be ause the dierent paths may have dierent lengths may lead to either si the sum (72) onstru tive or destru tive interferen e. We have maxi- mum destru tive interferen e if the path dieren e is λ/2 + nλ (phase dif◦ feren e 180 ) and maximum onstru tive interferen e if the path dieren e ◦ is nλ (phase dieren e 0 ) in terms of the wave length λ. As a simple example we onsider the interferen e between the dire t and ree ted ray from h above a perfe tly r between R and T is onsiderably A1 ∼ A2 and the magnitude of the in- a transmitter (T) to a re eiver (R), both at the height ondu ting ground. When the distan e h larger than we may assume that terferen e be omes proportional to (this orresponds to the ase of verti ally polarized EM waves) ikr e +e This leads to a r √ ik2 h2 +r 2 /4 2 ≈ 1+e hara teristi with a separation ∆r proximately given by ikh2 /r 2   2 2 πh . = 4 cos λr interferen e pattern with (73) hanging distan e between the positions of maximum amplitude ap- 4 Interferen e and dira tion 30 2λr 2 . h2 ∆r = For r > 2h2 /λ the osines term approa hes 1. With referen e to the Fresnel equations (58), (59), we make the observation that, for a reexion from a perfe tly reexion oe ient ρh izontal polarization ondu ting surfa e, the is negative for the hor- ase. This is also true for the verti al polarization reexion in r ase of a diele tri oe ient ρv ground when the distan e (see Eq. (158)) is large enough to make smaller than the Brewster angle. In these θ ases S1 the  +-sign in (73) must be repla ed by a  −- sign, and the S2 osines term be omes instead a sinus-term,   2 2 πh . 4 sin λr This has the interesting property of approa hing 4  πh2 λr 2 as r → ∞, meaning that the interferen e redu es the power (whi h is propor−2 tional to the square of the amplitude) with an additional r -fa tor. Sin e −1 2 −2 the amplitude A falls of as r the power will fall o as A r ∝ r −4 in this ase. Thus the long-range behaviour is vastly dierent depending on whether we have a summation or a subtra tion in (73). One way to understand the propagation of the wavefronts is to imagine that every point on the wavefront is the sour e of an expanding spheri al wavefront (a Huygens wavelet), whi h together with the other spheri al wavefronts form the new wavefront. This prin iple was advan ed by Huy- gens (1678), and it gives a ni e explanation of why ree tion angle is equal the in iden e angle, and for the law of Snellius (noting that the velo ity of propagation of the wavefronts is c/n where n is the index of refra tion). It also provides a pi ture of the dira tion of EM waves e.g. through a hole in a s reen. 4.2 Fraunhofer and Fresnel dira tions Spe i ally x-dire onsider a re tangular hole extending from tion and from y = −b/2 to y = b/2 in the y -dire −a/2 to a/2 in the tion of a an opaque 4 Interferen e and dira tion 31 s reen. We suppose that a plane wave ψ propagates in the z -dire tion and impinges on the hole at z = 0. The eld at the observation point r0 = (z0 , y0 , x0 ), will be obtained by summing the phase fa tors exp(iks) over rays from the surfa e of the hole to the observation point ψ(r0 ) = A · Z a/2 −a/2 Z b/2 (74) eiks(x,y) dxdy. −b/2 If we take r to be the distan e from the enter of the hole to the observation r0 point then the path length s(x, y) an be evaluated as s=r s  1+ x − x0 r 2 +  y − y0 r 2 ≈r+ (x − x0 )2 (y − y0 )2 + . 2r 2r (75) Here we have assumed that r ≫ |a|, |b|. If x0 ≫ a and y0 ≫ b (Fraunhofer ase) one an retain only the linear term in (x − x0 )2 = x2 − 2xx0 + x20 (and similarly for y ) whi h simplies the integrals (74) to the form Z a/2 −a/2 Z b/2 −ikx0 x/r−iky0 y/r e −b/2 2r sin dxdy = kx0  kx0 a 2r  2r sin ky0  ky0 a 2r  . This means that the intensity at the point of observation will be proportional to the fa tor  sin(kax0 /2r) kx0 /r 2  sin(kay0 /2r) ky0 /r Closer to the hole we may no longer have 2 (76) . x0 ≫ a and y0 ≫ b. In this ase (Fresnel dira tion) we have to retain the full quadrati expression, whi h leads to integrals of the form Z a/2 −a/2 eik(x−x0 )2 /2r dx. This integral annot be solved analyti ally but is easily evaluated numeri ally using omputers. Traditionally the integral has been analyzed in terms of the Fresnel integrals ✲ ✒ ✲ ✲ ❘ 4 Interferen e and dira tion 32 C(σ) = Z σ cos(πs2 /2)ds, 0 S(σ) = Z σ sin(πs2 /2)ds, 0 and the so alled Cornu spiral dened by x = C(σ), y = S(σ). We may onsider the example of dira tion by an edge at y = 0 in the xy -plane with the open spa e for y > 0, and the s reen in the half y < 0. Suppose the observation point is at r0 = (0, d, Dp ); that is, Dp is the distan e from the xy -plane to the observation point. We have the geometri shadow for d < 0, and the illuminated region for d > 0. Repeating the same arguments as above the al ulation of the eld at r0 leads to an expression involving the integral r Z Z ∞ 2Dp ∞ iη2 ik(y−d)2 /2Dp e dy = (77) e dη, k −w 0 where the dira tion parameter w is dened by w=d s k . 2Dp (78) We onsider the shadow region d < 0. Then it is possible to estimate the integral (77) by partial integration [26, p.152℄  doing it twi e yields,   Z Z ∞ 3 ∞ −4 iη2 1 1 iη2 iw 2 − η e dη + − e dη = e (79) 2i|w| 4|w|3 4 |w| |w| For large |w| the rst term will dominate and the intensity of the eld |ψ|2 will therefore be proportional to 1/4w 2 . Comparing with the orresponding al ulation for d → ∞ (far into the illuminated region)6 , it follows that the intensity P in the shadow region varies as P0 (80) , 4πw 2 where P0 is the intensity in the illuminated region. We may apply this to an example dis ussed in [6, p.132℄ and treated by other methods there. P = 6 For this it is useful to know that Z ∞ −∞ exp(iη 2 )dη = p π/2 · (1 + i). 4 Interferen e and dira tion 33 Thus onsider a transmitter at the height of 12 meter and a re eiver 37.3 meters away at the height 2 m. Between them is a house with 12 m to the not h, whi h in turn is 20 m from the transmitter. For a rough estimate of the redu tion of signal strength due to the building, whi h is assumed to a t as a s reen, we an use (80) and (78) with Dp ∼ 17 m and d = -10 m. In the ase of f = 900 MHz we nd the redu tion to be about -28 dB, whi h an be ompared with the value -25.8 dB given by Bertoni. From (78) we observe that the boundary of the shadow region w = onstant is given by the lower part (y < 0) of the parabola z= k 2 y . w2 (81) Above we assumed that the waves from the transmitter are plane waves (parallel rays), but if we take into a ount that it is a nite distan e Dq from the s reen we need only to modify the expression for dira tion parameter w (78) a ording to w =d· s kDq . 2Dp (Dp + Dq ) (82) The Fresnel and Fraunhofer ases above be roughly distinguished by that the Fresnel approa h deals with the situation where we have diverging rays while the Fraunhofer approximation treats the ase with parallel rays. Sin e the Fraunhofer approximation uses linear terms in the argument of the expfun tion it lends itself readily to Fourier methods, and this has lead to the development of Fourier opti s. 4.3 Fresnel zones Consider a transmitter (T ) and re eiver (R) whose line of sight (LOS) distan e is D = d1 + d2 . Let U be a point on the line of sight with T U = d1 , UR = d2 , and let UV be perpendi ular line to the line of sight with length s. Denoting s1 = T V and s2 = V R the dieren e between lengths of the paths T R and T V R be omes q q 2 2 F = s1 + s2 − d1 − d2 = d1 + d + d22 + d2 − d1 − d2 . (83) The n:th Fresnel zone is the region bounded by F = nλ/2 whi h forms an ellipsoid with the transmitter and re eiver at the fo uses. The rst Fresnel zone F1 thus orresponds to maximum phase shift of 180◦ due to a ree tion. As a rule an undisturbed path between the transmitter and re eiver requires 4 Interferen e and dira tion 34 that there be no obsta les in the Fresnel zone F1 . If D ≫ d then one an, for d1 = d2 = D/2, approximate (83) by 2d2 Fn = , D from whi h we obtain the width of the n:th Fresnel zone, r r DFn Dnλ = . rn = d = 2 4 Inserting this into (78) and (80) we nd that the redu tion of the intensity of the eld due to a shadow of depth rn would be 1/πn. Roughly, if the Fresnel zone Fn is free then the transmission power is disturbed at most by a fa tor of (1 − 1/πn). In de ibels 1/π orresponds to 10 log(1/π) ≈ -5 dB. The parameter d rn where d is shortest distan e from LOS to obstru ting obje ts is alled the Fresnel zone learan e and is used in investigating the path learan e. While obje ts protruding into the Fresnel zones ause s atterings, one draws a series of Fresnel zones also in order to see whether there are some surfa es lose to the tangent of the Fresnel zones; su h surfa es may ause reexions and thus interferen e effe ts. s1 d1 d s2 d2 4.4 Kir ho equation Above we presented the Fresnel-Huygens prin iple in an intuitive fashion. It an also be justied starting from the wave-equation (69) whi h in the harmoni ase ψ(r, t) = ψ(r) exp(−iωt) be omes, ∇2 ψ(r) + k 2 ψ(r) = 0. (k = ω/c.) (84) Suppose ψ(r) is known on some surfa e S1 , the idea is to try to express the value of ψ(r) at an observation point r0 in terms of its values at the surfa e S1 . First we note that the fun tion (a Green fun tion) 4 Interferen e and dira tion 35 G(r) = − satises 1 ikr e 4πr (85) (86) ∇2 G(r) + k 2 G(r) = δ(r). From that follows (using Green's theorem) I Z∂V ZV Z (ψ(r)∇G(r − r0 ) − G(r − r0 )∇ψ(r)) · dS = (87) (ψ(r)∇2 G(r − r0 ) − G(r − r0 )∇2 ψ(r))dV = V (ψ(r)(∇2 + k 2 )G(r − r0 ) − G(r − r0 )(∇2 + k 2 )ψ(r))dV = V ψ(r)δ(r − r0 )dV = ψ(r0 ). We an take V to be a volume en losed by a s reen S1 (and some hole in it) and an innitely far away surfa e S2 where the eld disappears. It may assumed that the eld is zero at the s reen and is approximately a plane wave at the hole area A, where we thus have ∇ψ = ik ψ . Hen e (87) be omes (Kir ho equation, 1882) in ψ(r0 ) = Z A Sin e ψ(r)(∇G(r − r0 ) − ikin G(r − r0 )) · dS. (88)   r − r0 r − r0 r − r0 ≈ G(r − r0 )k ∇G(r − r0 ) = G(r − r0 ) − +k , 2 |r − r0 | |r − r0 | |r − r0 | when k|r − r0 | ≫ 1 (that is, |r − r0 | ≫ λ), we get nally,   r − r0 ψ(r0 ) = −i ψ(r)G(r − r0 ) k − kin · dS. |r − r0 | A Z (89) Noting that the normal of the surfa e A is pointing away from the point r0 we infer that   r − r0 − kin · dS = k(cos θ + cos θin ) dS, k |r − r0 | 5 Radiation from antennas where θ 36 is the angle between r − r0 and the normal of the surfa e dS at the hole. The new thing obtained here beyond the Fresnel-Huygens prin iple used above is the geometri al fa tor cos θ+cos θin . The Kir ho method gives often good approximations though it is not entirely mathemati ally tent [19, se t. 9.8℄. Indeed, the assumption of the vanishing of ψ and onsis- ∂ψ/∂n at the s reen is no longer ne essarily valid for the Kir ho solution itself. Sommerfeld and Rayleigh have introdu ed slight modi ations (employing the mirror prin iple) that remove these in onsisten ies. There are also a few exa t solutions of the díra tion problem  the half plane s reen and the wedge shaped s reen  presented by Arnold Sommerfeld and others, whi h are of great theoreti al interest though approximate methods will typi ally su e in pra ti al problems. 5 Radiation from antennas 5.1 The Hertz dipole If we onsider the situation of an antenna in free spa e, then in the region of the antenna (and the RF- ir uit of magneti simplied elds satisfy the free spa e equations. ase where we have short pie e dire ted along the z -dire L J 6= 0 only ourse); elsewhere the It is useful to onsider a of a wire as an antenna at tion [18℄. We suppose that the r0 , urrent density in this antenna may be des ribed as using the Dira J(r, t) = δ(r − r0 )LI0 cos ωt, (generalized) delta-fun tion δ(. . . ) (90) in three dimensions, δ(r) = δ(x)δ(y)δ(z). I0 denotes the onstant urrent along the z -dire tion. Inserting (90) into (27) gives A(r, t) = from whi h we an al amplitude dire ted µµ0 LI0 cos ω(t − |r − r0 |/c), 4π |r − r0 | ulate the magneti eld H = ∇ × A/µµ0 , (91)   1 cos ω(t − |r − r0 |/c) × (LI0 ) = H(r, t) = (92) ∇ 4π |r − r0 |   (LI0 ) cos ω(t − |r − r0 |/c) ω sin ω(t − |r − r0 |/c) + (r − r0 ) × . − 3 2 |r − r0 | c |r − r0 | 4π R ≡ r − r0 , we an see that the cos-term in (92) on the right −2 −1 hand side falls o as R while the sin-term falls o only as R with growing R. Thus, for R−2 < (ω/c)R−1; that is, for (the radiation eld region) If we use 5 Radiation from antennas 37 R> λ 2π (93) the se ond term (radiation eld) will dominate giving7 H(r, t) ≈ LI0 ω sin ω(t − R/c) R × . c R2 4π (94) This an be interpreted as a planar eld propagating in the dire tion R̂ = R/R with an amplitude H0 = ω LI0 sin θ , c 4πR (95) where θ is the angle between R̂ and I0 (= the z -dire tion). As the antenna is along the θ = 0 dire tion we an see that there is no radiation in this dire tion. The interpretation of our results is that the os illating urrent in a short wire antenna generates a eld whi h, far away form the sour e, approximates a plane wave with an amplitude whi h falls o with distan e as 1/R. Sin e the ele tri and magneti elds are related by (17) (se ond equation), it follows that the ele tri eld amplitude E0 will be orthogonal to both H0 and R̂; in fa t, we have E0 = ηH0 × R̂.8 Inserting (95) into the expression for the Poynting ve tor S (21), and integrating over the surfa e of a sphere of radius R, entered at the antenna (r0 ), we obtain, Z η P = S · dS = 12π |r−r0 |=R  ωLI0 c 2 , (96) 7 The near-eld region is of interest when one onsiders ases of energy transfer not by radiation but through indu tive ouplings [25℄. 8 One an work out the ele tri al eld dire tly (see e.g. [20, se . II℄) from the relation ∂A ∂t by al ulating φ from the harge distribution ̺ whi h in turn is obtained from the E = −∇φ − ontinuity equation ∇·J+ ∂̺ = 0. ∂t Another pro edure is to rst determine the magneti eld from H = ∇ × A/µµ0 and then the ele tri eld from E = i∇ × H/ωǫǫ0 , whi h follows from the Maxwell equation ∇ × H = ǫǫ0 ∂E/∂t in the urrent-free region (J = 0). 5 Radiation from antennas 38 for the average (over time) power radiated by the antenna. If we ompare RI02 /2 dissipated by a resistor R through whi h ows an os illating urrent I0 cos(ωt), then (96) suggests that the antenna this to the average power an be asso iated with a 2η Rs = 12π radiation resistan e  ωL c 2 2π =η 3  2  2 L L ≈ 790 Ω. λ λ One impli ation of this analysis is that, if an AC- ir uit whose lengths approa hes the order of λ = c/f , of the AC- urrent, then one has to take into a where f (97) ontains wires is the frequen y ount that the wires may radiate a signi ant power and have to be treated as antennas. if the typi al dimensions D are mu h less than treated as an ordinary point-to-point λ, However, then the system ir uit where Kir ho 's laws an be an still be applied [55℄. The above model based on (90), and the assumption L ≪ λ, is referred to as the Hertz dipole. It already shows some general features. Thus, in the −1 far away region the magneti and ele tri eld amplitudes fall o as R , and sin e the radiated power per unit area is proportional to the square of −2 the eld amplitude, it falls o as R . This is logi al sin e the power owing 2 through the spheri al surfa e area 4πR must be onstant (in empty spa e) and independent of R sin e energy is onserved. We also found that the dire tion of the ele tri al eld (dire tion of polarization) lies in the plane of E = E θ̂. If the antenna is of the size of the wave length, L ∼ λ, then the variation of the antenna and the radius ve tor the R; more exa tly, urrent along the antenna be omes an important problem. Basi ally one would have to solve the eld equations for the propagation of the eld along the antenna. The the surfa e of the the ondu tors a t as waveguides dire ting the elds along ondu tors. For good ondu tors there is no eld inside ondu tor and thus no energy is transported in or out the there is slight resistan e in the ondu tor. If ondu tor then some of the energy transported into the ondu tor and will be dissipated as heat. In this sense the antenna does not as su h radiate, instead it guides the elds along its surfa e. This is vivildly illustrated by the diagrams omputed by Landstorfer by elds will ow and oworkers [27℄ whi h show how the ele tromagneti energy ows along the surfa e of antennas and is spread into the surrounding spa e. If an os illating voltage sour e is onne ted to two parallel wires it will generate a traveling wave between the wires whi h will deta h from the free ends of the wires and radiate into the surrounding spa e. By bending the wires at the free end forming a T-dipole the part of energy radiated into spa e may be in reased. The problem of determining the radiating eld thus 5 Radiation from antennas 39 be omes a boundary value problem, with ex itation voltage given at the feedpoint, and with the boundary vanishes at the surfa e of the ondition that the tangential ele tri eld ondu tors. However, linear wire antennas are often treated by the simpler method of making some more or less well justied assumptions about the elds are urrent distribution in the antenna, from whi h the al ulated as exemplied by the Hertz dipole ase. The problem of determining the antenna urrent given the ex iting voltage is alled the antenna ex itation problem. Mathemati ally it leads to integral equations that are numeri ally solved using various dis retization pro edures, su h as the moment method and the method of Galerkin [13℄. For a freely available software ( 4Ne 2, numeri al ele tromagneti ode) for omputing antenna elds and parameters see [49, 7℄. 5.2 5.2.1 Dipole antennas Antenna ex itation The dipole antenna onsists of two wires extending in opposite dire tions with the feeding point at the meeting ends. A simple dipole from a oaxial λ/4 able by exposing a sleeve of the outer ondu tor as shown in the adjoining gure. The added advantage of the sleeve is that it a ts as a so study the urrent distribution in a the wires are an be made end of the inner wire and make a ylinders of radius alled balun. In order to ommon dipole antenna we assume that a ≪ λ, oriented along the z -dire tion, and Ez drives the separated by a small gap where an os illating ele tri al eld antenna urrent. For a thin wire dipole-antenna it is typi ally assumed (based on the thin-wire approximation; see below) that the sinusoidal along the enter of the antenna (x = 0, J(z) = Im sin (Note that the feedpoint to the maximum antenna extends from z (98) I(z = 0) = I0 is related Im by I0 = Im · sin(kL/2).) Here the z = L/2, and (98) is onsistent with urrent amplitude = -L/2 to urrent must vanish at the endpoints. (As usual denotes the wave-number 2π/λ.) The approximation (98) for the however is in onsistent with the fa t that the the surfa e of the urrent distribution is = 0),  kL − k|z| δ(x)δ(y)ẑ. 2 urrent amplitude the requirement that the k  y ondu tor. Still, for antenna (in terms of the radius a urrent is urrent on entrated on al ulating elds far away from the of the wire) the dieren e will be small. 5 Radiation from antennas 40 The mathemati al form of the antenna exitation problem for wire antennas is usually developed as follows. Dierentiating the se ond equation in (23) with respe t to time, and using (25) to eliminate φ we obtain ∇(∇ · A) − 1 ∂2A 1 ∂E = . c2 ∂t2 c2 ∂t (99) Be ause of the ylindri al symmetry we may assume that A = (0, 0, Az ). Se ondly we express A as a fun tion of the urrent J using (27), leading to the equation ✻ λ 4 ❄ ✻ λ 4 ❄ ∂ 2 Az (z, t) 1 ∂ 2 Az (z, t) − = ∂z 2 c2 ∂t2   Z µµ0 L/2 ∂ 2 1 ∂2 I(u, t̄) du = − 2 2 p 2 4π −L/2 ∂z c ∂t (u − z)2 + a2 1 ∂Ez (z, t) c2 ∂t (100) on the ylinder (wire) surfa e (to repeat this is stri tly speaking in onsistent, see e.g. [20℄). Finally, using the time-harmoni form I(u, t̄) = I(u)e−iω(t−R/c) Ez (z, t) = Ez (z)e−iωt , (R = p (u − z)2 + a2 ), (101) (102) we end up with Po klington equation (whi h does not appear in this form in [38℄) Z L/2 I(u) −L/2  ∂2 + k2 ∂z 2  √ 2 2 eik (u−z) +a p du = 4π (u − z)2 + a2 −iǫǫ0 ωEz (z) (103) where we have used (14) and the fa t that ω/c = k. This is an integral equation in the unknown urrent I in terms of the ex itation eld Ez . Sin e it solves for the urrent I , given the input potential V , it gives the impedan e Z = V /I of the antenna. If the antenna is assumed to be a perfe t ondu tor then Ez = 0 at the surfa e of the antenna sin e a non-zero tangential eld 5 Radiation from antennas 41 would generate an innite urrent (more pre isely, in a perfe t ondu tor it generates a eld that an els it su h that the total tangential eld be omes zero  more on this in the se tion on shielding). Ez diers from zero only at the feedgap where it may be assumed to be −V /∆, where V is the voltage fed into the antenna, and ∆ is the gap size. The same equation applies (in) also to the re eiving antenna in ase whi h Ez = −Ez (in the right hand side of Eq.(103)) at the surfa e of the ondu tor; i.e., the s attered eld osets the in oming eld su h that the total tangential omponent vanishes. Po klington developed a somewhat more general theory of wire antennas in [38℄ as he onsidered wire antennas of arbitrary shapes, su h as ir ular rings and helixes. His basi idea was to onsider the wire antenna as being omposed of an array of innitesimal Hertz dipoles. The total ele tri eld an then be obtained by summing (or integrating) the ontributions from the elementary Hertz dipoles, ea h hara terized by a urrent I(s) and dire tion ds, E(r) = −∇ Z ∂G(r − s(u)) I(u)du + k 2 ∂u Z I(u)G(r − s(u)) ∂s du, ∂u (104) where eikr . (105) 4πr The integration in Eq.(104), whi h an be derived from Eq.(144) and Eq.(145), is along the wire antenna (u is the length parameter along the urve dened by du = |ds|). Finally Po klington imposed the boundary ondition that the tangential omponent of the ele tri eld (104) vanishes at the surfa e of the ondu tor. In the limit of a vanishing small radius this leads to the equation9 G(r) = µµ0 9 For losed antenna loops, or where urrent is assumed to be zero at the ends of the antenna, the rst term in (104) be omes by partial integration ∇ Z G(r − s(u)) ∂I(u) du. ∂u Po klington then evaluates (104) at surfa e of the ondu tor of a small radius δ . For small δ the leading ontribution to the eld at a point on the surfa e is from the nearby se tion of the ondu tor yielding a term proportional to ln(M/δ) (for some onstant M ). √ RC Indeed, note that 0 dz/ z 2 + δ 2 → ln(2C/δ) for small δ . The ele tri eld parallel to the ondu tor at its surfa e thus be omes proportional to   ∂ 2 I(u) 2 + k I(u) ln(M/δ) ∂u2 5 Radiation from antennas 42 ∂ 2 I(u) + k 2 I(u) = 0, ∂u2 (106) i.e., the urrent is a sinusoidal fun tion of the wire length parameter u. (This equation follows from the approximation of (27), that Az on the ondu tor is proportional to the urrent at that point, in ombination with Eq.(107). For a pedagogi dis ussion of various approximation s hemes for the antenna urrent see [30℄.) If we return to the version (103) and let ∆ → 0 we arrive at the deltagap ase where the feedgap is assumed to be innitesimally small at z = 0. Speializing the earlier onsiderations of the dipole antenna to this ase (usually asso iated with the name Hallén [15℄) we get the equation ∂ 2 Az + k 2 Az = 0 (z 6= 0), 2 ∂z whi h suggests that Az is of the form Az = (107) 1 (B cos(kz) + C sin(k|z|)) , c (108) where B and C are onstants to be determined. This form ensures the symmetry Az (z) = Az (−z) and I(z) = I(−z). When dierentiating the |z|-argument this introdu es a dis ontinuity at z = 0. In fa t, using the equation iω ∂ 2 Az (109) E = + k 2 Az , z c2 ∂z 2 R in order to ompute −V = Ez dz we obtain the value 2Ci whi h determines C in terms of V . B nally is determined by the ondition that the urrent I , solved (numeri ally) from − 1 Az = (B cos(kz) + C sin(k|z|)) = µµ0 c Z I(z ′ ) eikR ′ dz , 4πR (110) should vanish at the ends of the antenna, I(±L/2) = 0. Having solved for I(z) we obtain the impedan e Z of the antenna from Z = V /I(0). (If we use the approximation that Az is proportional to the urrent I at the wire when negle ting terms small ompared with ln(M/δ) as δ → 0. The vanishing of the tangential eld omponent therefore leads in the limit of a thin wire to (106) ex epting the end points and feedpoints of the antenna. It is to be emphasized that the above argument does not demonstrate that sinusoidal urrents do produ e zero tangential elds for nonzero radius. In fa t, for sinusoidal urrent the tangent eld will in general look sinusoidal too. 5 Radiation from antennas 43 I(±L/2) = 0 would simply be surfa e, then the ondition C sin(kL/2) = 0, but su h an approximation is too ome B cos(kL/2)+ rude in order to yield a reliable estimate of the impedan e of the antenna.) urrent I is at the axis of /4πR used above ( alled the In the integral in Eq.(110) it is assumed that the ikR the antenna leading to the simplied kernel redu ed kernel approximation ), with R e denoting the distan e from a surfa e point to a point on the axis (see Eq.(101)). In the ylindri al antenna theory the redu ed kernel √ 2 2 eik (u−z) +a p 4π (u − z)2 + a2 in Eq.(103) be omes repla ed with 1 K(u − z) = 2π Z 0 2π √ 2 ϕ 2 2 eik (u−z) +4a sin 2 q dϕ. 4π (u − z)2 + 4a2 sin2 ϕ2 This kernel is based on the repla ing an axial urrent z with a surfa e the se tion z. urrent I at the I/2πa uniformly distributed on the (111) ross se tion ir umferen e at One sees that (110) is, in fa t, mathemati ally in onsistent if one uses the redu ed kernel, in the sense that the left hand side is ontinuously dierentiable at z=0 (supposing the right hand side is not [13℄. The urrent I is well behaved) while the ylinder kernel fares better on this point sin e R = 0 (for the redu ed kernel one has always R ≥ a). K(z) grows like − ln(z) when z goes to zero. However, R R integrals K(z − u)I(u)du still remain nite be ause ln(z)dz = z ln z − z stays nite when z approa hes 0. it has a singularity Indeed, the kernel Returning to Eq.(110) we N = 2Q + 1 an dis retize it by dividing the antenna into ∆ = L/N with the feedgap at z = 0 ( orren = Q). The midpoints of the sli es are given by sli es of length sponding to the index zn = (n − Q)∆. A dis retized version of (110) an thus be written as the matrix equation Vn = N −1 X k=0 where Mn,k Ik (112) 5 Radiation from antennas µµ0 = 4π 2 Mn,k Z π 0 44 Z ∆/2 −∆/2 q eik √ ((n−k)∆−z)2 +4a2 sin2 ((n − k)∆ − 1 (B cos(kzk ) + C sin(k|zk |)) , c C = iU/2 (U is the applied feedgap Vk = The matrix equation software su h as V = MI z)2 + 4a2 ϕ 2 sin2 ϕ2 dϕdz, (113) (114) voltage). (115) is easily implemented with mathemati s −1 , et . One solves for I = M V and Math ad Matlab , B su h that I0 = IN −1 = 0. Writing the ve tor V (1) in (113) as V = (B/c)V + (C/c)V (2) one obtains B from B(M −1 V (1) )0 + C(M −1 V (2) )0 = 0 (whi h is thus generally a omplex quantity). The adadjusts the oe ient joining gure (Fig.1) shows the results for imaginary and real parts of the urrent in the As ase an be seen the Q = 100, a = 0.001λ al ulated half-wave dipole to be L = 0.48λ (λ m). Z = U/I(z = 0) = 73.4 − i4.3 Ω Ω) for the ompared to the radiation resistan e (73 al ulated below assuming a sinusoidal result indi ates that by making the dipole a bit shorter than make the = 12.5 urrent deviates slightly from the sinusoidal urrent. The impedan e was obtained as whose real part may be and urrent. This λ/2 one an omplex part (rea tan e) of the input impedan e disappear, whi h simplies the impedan e mat hing of the antenna (su h an antenna is a resonant antenna). alled Current, (mA) 5 Radiation from antennas Fig. 1: 45 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 -1 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 -0.0 0.1 / (m) 0.2 0.3 0.4 0.5 0.6 The gure shows the distribution of the urrent along a dipole antenna, with an applied deltagap potential of 1 V at the enter. The urrent has been al ulated by solving the dis retized Hallén equation (112). The solid line represents the real part of the urrent, the dashed line the imaginary part, while the dotted line represents a pure sinusoidal urrent. Parameters: L = 0.48λ, antenna radius a = 0.001λ; the al ulated input impedan e be omes Z = 73.4 −i4.3 Ω. Note that in the literature it is often the absolute magnitude of the urrent that is display instead of the real and imaginary part separately. Another approa h to solving the Hallén equation (112) is to express the urrent I as trigonometri sum [32℄ I(z) = N X n=1 Dn sin  nπ L  L − |z| 2  . (116) The idea is thus to try to approximate the urrent using basis fun tions that are expe ted to be similar to the true urrent. We note that (116) automati ally satises the ondition I(±L/2) = 0. If the expression (116) is 5 Radiation from antennas 46 inserted into Eq.(110) we obtain an equation of the form Z L/2 N X Dn sin −L/2 n=1  nπ L  L − |u| 2  K(z − u)du = B cos(kz) + C sin(k|z|). (117) Choosing N + 1 points in the interval (−L/2, L/2) we an write (117) as a matrix equation N X n=1 where Km,n µµ0 c = 4π Z Km,n Dn + Km,N +1 B = C sin(k|zm |), L/2 −L/2 K(zm − u) sin Km,N +1 = − cos(kzm ).  nπ L  L − |u| 2  du (n ≤ N), (118) (119) (120) As before C is related to the applied feedgap voltage U by C = iU/2, and K(z) denotes the kernel (111). When sele ting the points zn one should take are not to make the (N + 1) × (N + 1) matrix K singular. (This may happen if the points are sele ted symmetri ally along the z -axis with respe t to z = 0.) Using N + 1 = 6 points we obtain a ve term trigonometri al approximation of the urrent whi h hardly improves mu h going to higher values of N  the omputational ost in reases rapidly due to the integration over the interval (−L/2, L/2). 5.2.2 Re eiving dipole antenna For simpli ity assume that the in oming eld E(in) is parallel with the dipole antenna, hosen again to be along the z -axis. For a perfe t ondu tor the in oming eld will indu e a urrent and a ree ting eld E su h that the tangential omponent of the total eld E + E(in) is zero. From this we obtain (in) (in) that Ez = −Ez . For an in oming plane wave we an assume that Ez is onstant along the antenna.10 We an therefore write the solution to Eq.(109) as, 10 In the more general ase the last term in (121) is repla ed by ik ω Z 0 z Ez(in) (ζ) sin(k(z − ζ))dζ. 5 Radiation from antennas Az = 47 i 1 (B cos(kz) + C sin(k|z|)) + Ez(in) c ω Making use of the Lorenz gauge (∆/2 ≤ |z| ≤ L/2). (121) ondition ∂Az iω 1 ∂φ = − 2 φ = 0, 2 c ∂t ∂z c expressed in terms of Az , ∇·A+ the potential φ an be φ= c2 ∂Az . iω ∂z For a thin wire dipole antenna we tion and assume that ir uit ase we have the The onsequent limit ∆=0 Az an use the Po klington approxima- is proportional to the ondition that the ondition Az (z) = 0 for to the solution, In the same limit urrent is zero at |z| = ∆/2, L/2, (in) icEz C= ω ic B = Ez(in) , ω urrent [31℄. In the open ·  then leads in the 1 − cos (kL/2) . sin (kL/2) ∆ → 0 we then get for the indu c2 U = φ(∆/2) − φ(−∆/2) = iω |z| = ∆/2, L/2. ed open (122) ir uit voltage, ∂Az (z = ∆/2) ∂Az (z = −∆/2) − ∂z ∂z (in) 2Ez = k · 1 − cos (kL/2) . sin (kL/2) Espe ially we obtain for the half-wave dipole (L  (123) (in) = λ/2): U = Ez λ/π . 5.2.3 Dipole eld In order to be able to derive some analyti al results for the dipole antenna of length L = 2l we will assume a sinusoidal urrent (98) (whi h, as pointed out above, is typi ally su iently a urate for estimating the far eld.). As in the an then ase of the Hertz dipole we For an antenna oriented in the radiation eld (in z -dire al ulate the radiation eld. tion the ele tri omponent of the omplex representation) in the far-eld region (r the distan e from the antenna and E(r) = θ̂2ηIm λ > λ, r is is the wavelength) thus be omes cos(kl cos θ) − cos(kl) eikr · . sin θ 4πr (124) 5 Radiation from antennas Here I0 48 is the amplitude of the sinusoidal antenna θ-rotation, k = 2π/λ ve tor in the dire tion of the η urrent, θ̂ is the unit is the wave-number, and is the wave impedan e. In Eq.(124) we have ignored the ee ts of the surrounding, su h as the ir uit board (PCB) of the transmitter. measurements it was found that the PCB In our ontributed to an anisotropi of the radiation eld in the horizontal plane whi h an be form ured en losing the PCB in a symmetri al ( ylindri al) shielding box. Eq.(124) is a spe ial ase of the more general form E(r) = of the far-eld ele tri eikr · G(θ, φ) r (125) omponent. Sin e the power ux of the radiation 2 eld is proportional to E , Eqs.(124, 125) predi t that the power depends on −2 the distan e r as r . Power P is usually measured in the units of de ibel (dB) relative to a standard power P0 (su h as 100 mW), PdB = 10 · log10  P P0  . (126) whi h is equivalent to the eld strength. −2 Thus, if we have a r -dependen e of the power then it is redu ed by 20 dB for every de ade of distan e and by distan e. From this relation one a 6 dB for every doubling of ould dedu e that, if the power at r = 4 m is -10 dB, and we measure a power -40 dB at an unknown distan e r , then (40−10)/20 ≈ 126.5 m. Besides the distan e r the re eived must be 4 m × 10 r signal strength depends also on the antenna orientations. A ommon dipole antenna is the half-wave (λ/2) T-dipole with In this l = λ/4 ase the eld (125) kl = π/2. omponent Eθ beand omes,
ηIm cos π2 cos θ ikr e . Eθ = 2πr sin θ θ (127) ✻ The total radiated power is obtained by in2 tegrating Eθ /2η over a spheri al surfa e r = onstant, 1 Ptot = 2η  ηIm 2π 2 4π Z 0 π/2 cos π 2 cos θ sin θ
!2 sin θdθ. l ❄ 5 Radiation from antennas 49 The integral annot be solved analyti ally but an be evaluated numerially on the omputer. The end result is that, 2 Ptot ≈ 0.097 · η · Im , (128) Rn = 30 Ω · Cin((4n − 2)π), (129) and identifying this with Rs I02 /2 (note that for an λ/2-dipole Im = I0 ) we obtain for the radiation resistan e Rs of the λ/2-dipole, Rs ≈ 2 · 0.097 · η ≈ 73 Ω. We an generalize these al ulations using (127) to antennas of lengths Ln = (2n − 1)λ/2 (the al ulations are simplied for these spe ial lengths) whi h yield for the orresponding antenna resistan es the expression where the osine integral is dened by Cin(x) = Z x 0 1 − cos y dy. y (130) A variant of the dipole antenna is the folded λ/2-dipole where the endpoints of the T-dipole are onne ted by a wire. Thus the urrent will ounted twi e when evaluating the radiation eld whi h therefore will be twi e ompared to the T-dipole. The radiated power will onsequently be four-fold. Hen e the radiation resistan e of the folded dipole is also four-fold, or about 4 × 73 Ω = 292 Ω. In a bit more areful treatment of the folded dipole ase [3, se tion 9.5℄ the onventional analysis is based on the tri k of viewing the system as onsisting of two parallel oupled dipoles separated by a small distan e d (see Is ✻ ✻ Ia ✻ ❄ gure). These are ex ited by voltages U/2 and U/2 (symmetri al ase (a)), and U/2 and U U U − U2 2 2 2 −U/2 (asymmetri al ase (b)). The urrent I of the original problem is then obtained as the sum of the urrents Ia and Ib (at the feed(a) (b) points) for the separate ases. In ase (a) we have the equation U/2 = Z11 Ia + Z12 Ia , and assuming that for small separation d the mutual impedan e Z12 is lose to the self-impedan e Z = Z11 of the single dipole we obtain, 5 Radiation from antennas 50 U . 4Z The asymmetri al ase (b) an be handled with the help of the transmission line theory (see Appendix D). The input impedan e of two-wire able of length L/2 short ir uited at the end is Zb = −iZc tan(kL/2) where Zc is able impedan e. Thus the urrent Ib be omes Ia = U ÷ −iZc tan(kL/2). 2 The impedan e of the Zf of the folded dipole is therefore given by Ib = Ia + Ib 1 i 1 = = + . Zf U 4Z 2Zc tan(kL/2) When L = λ/2 we obtain that the folded dipole impedan e Zf is four times the simple dipole impedan e, Zf = 4Z . In this ase only Ia ontributes to the total urrent sin e the short ir uited λ/4-length transmission line has innite impedan e whi h suppresses the asymmetri al mode (b). In the ases onsidered we have assumed that the feedpoint of the dipole antennas is entered at the midpoint ( orresponding to z = 0). If the feedpoint is o- enter by an amount h, then the urrent at the feedpoint  assuming a sinusoidal urrent Im sin(kL/2 − k|z|) for the antenna  will be Im sin(kL/2 − k|h|) whi h for a λ/2-dipole be omes Ih = Im cos(kh). 2 The radiated power an therefore be written P = 1/2 · ℜ(ZIm ) = 1/2 · 2 2 ℜ(Z/ cos(kh) )Ih implying that the o- enter input impedan e is Zh = Z/ cos(kh)2 . 5.3 Mutual impedan e A urrent in a ondu tor 1 generates an ele tri eld whi h indu es a urrent and its rea tion eld in any nearby ondu tor. This mutual inuen e is the basis of the use of antennas. The mutual inuen e is measured in terms of the mutual impedan e. If we onsider two (unloaded) dipole antennas and denote the voltages and the urrents at the feedgaps (antenna terminals) as Vi , Ii , then we have the relations, V1 = Z11 I1 + Z12 I2 , V2 = Z21 I1 + Z22 I2 , (131) (132) where Z11 , Z22 are the antenna impedan es, and Z12 = Z21 is the mutual impedan e (see Appendix and the Re ipro ity Theorem). Following 5 Radiation from antennas 51 I2 r ✾ z V1 Fig. 2: ✾ r0 θ Field by antenna 2 indu es an open- ir uit voltage V1 over the terminals of antenna 1. Be hmann [4℄ one an ompute the impedan e for a set of N ondu tors by starting from the Poynting's energy theorem (19). Taking the time average for harmoni elds the term ∂/∂t(· · · ) disappears, and we obtain using phasors, Pav 1 =− 2 Z ℜ (E · J⋆ ) dV. (133) Denote by Ei the ele tri eld generated by the urrent density Ji of the ith ondu tor. Then the equation (133) be omes Z
1X ℜ Ei · J⋆j dV. Pav = − (134) 2 i,j
R The term Ei · J⋆j + Ej · J⋆i dV measures the mutual inuen e between ondu tors i and j . Sin e Ei an be supposed to s ale with the amplitude of the urrent density Ji , due to the linearity of the Maxwell equations, one an dene a urrent-independent quantity alled (mutual) impedan e by 1 Zij = 2Ii Ij Z
Ei · J⋆j + Ej · J⋆i dV, (135) where Ii denotes the hara teristi urrent amplitude of the ith ondu tor (one may refer to the urrent amplitude at the antenna terminals, or the maximum urrent amplitude of the ondu tor). 5 Radiation from antennas 52 In the Appendix on the Re ipro ity relations we onsider the transmission between two antennas. If the antenna 2 is a transmitting dipole while antenna 1 is the re eiving dipole then it was demonstrated that the open ir uit voltage V1 indu ed by the eld generated by 2 is given by V1 = Z12 I2 (see Eq.(215) in the Appendix) Z 1 V1 = (136) E2 · J1 dV. I1 This is typi ally al ulated by inserting a sinusoidal urrent for J1 , letting E2 be the in ident eld and integrating over the antenna. In the situation onsidered by Be hmann E2 would refer to the in oming eld and J1 to the indu ed urrent, and the total eld (satisfying the BC) would be E1 + E2 . Then the integral in Eq.(136) over the antenna an be interpreted as an ele tromotive for e generated in the antenna.11 This voltage an furthermore be expressed as V1 = h1 (θ) · E2 where h1 (θ) dened by (for a linear antenna) 1 h1 (θ) = I1 Z L/2 eikz cos θ I1 (z)dz (137) −L/2 is alled the ee tive antenna length (θ is the angle between the wave dire tion and the linear antenna  see illustration). The exponential fa tor omes from the fa t that the far eld E2 (r) varies as exp(ik · r) = exp(ik · r0 ) exp(ikz cos θ) along the re eiving antenna. Again using sinusoidal urrents as an approximation for a λ/2-dipole we obtain 11 Carter [8℄ argues that the impressed eld Ez (z) at z of the antenna indu es a voltage dz , and a urrent I(z), and that Ez (z)dz I(z) by re ipro ity is equal to I(0)dV where dV is the orresponding indu ed voltage at the antenna terminal z = 0. Summing over all se tions gives the indu ed total voltage V (0). In [48℄ the authors adopt the approa h to let E2 be an arbitrarily pres ribed in oming eld whi h generates a urrent σE2 in the antenna whi h is assumed to have nite ondu tivity σ . This urrent is then added to the Maxwell equations for the rea tion elds E1 , H1 in the antenna ( ondu tor): Ez (z)dz at the point z over the se tion ∇ × E1 + iµµ0 ωH1 = 0, ∇ × H1 − σE1 = σE2 . (In the se ond equation the displa ement term has been dropped as negligible for a good ondu tor.) Combining this with the equations for the elds outside the ondu tor (free spa e or a diele tri ) leads to a system of equations des ribing waves traveling along ondu tors whi h was treated by Sommerfeld already in 1899 (a more re ent referen e is [45℄). 5 Radiation from antennas h(θ) = Z 53 L/2 ikz cos θ e −L/2 Hen e for a 2 cos(kz)dz = k Z π/2 cos (u cos θ) cos u du
2 cos π2 cos θ = . k (sin θ)2 λ/2-dipole at θ = π/2 (138) 0 we obtain h = λ/π . This agrees with the result in se tion 5.2.2. As dis ussed in an earlier example, a near 2 planar EM-eld with power ux of 100 mW/m orresponds to an ele tri eld amplitude of E0 = 8.7 V/m. λ/2λ= h × 8.7 If this is aligned with a re eiving antenna (i.e. the antenna is parallel with the polarization), and given 0.125 m, the eld will indu e an open 0.125/π × 8.7 V/m = 5.4 ir uit voltage of amplitude V = 0.35 V between the antenna terminals. Antenna above ground We have dis ussed the antenna radiation eld in the open spa e. However, antennas are usually situated near the ground whi h will thus ae t the radiation eld. As shown in se tion 3.6 the stati diele tri or ondu ting plane eld of a harge in presen e of a an be handled using the image method; that is, the total eld above the ground is the sum of the dire t eld and the one generated by an opposite mirror also in the antenna T ase of moving is situated at r harge. An analogous situation prevails harges in antennas. Thus, if the transmitter- = 0, and the ground is on the level the radiation eld is the same as in the free spa e mitter TM situated at in the plane z = −h r = (0, 0, −2h). TM and is fed with the z = −h, ase with a se ond trans- is a mirror image of urrent then IM = −I . T ree ted For a verti- ally oriented dipole antenna above the ground the resulting far eld Etot (r) be omes, Etot (r) = E(r) + E(r − D), where (139) D = (0, 0, −2h) and E is the dipole eld of Eq.(124). for the total eld is justied in the it satisfy the ase of a perfe tly This equation ondu ting ground sin e ondition of a zero tangential total eld at the interfa e. 5 Radiation from antennas 54 T R θ Ground TM The eld Etot represents the sum of the dire t eld and the ree ted eld. One thus speaks of a two ray model. If we use Eq.(139) and Eq.(124) in 2 order to al ulate the power (whi h is set proportional to Ez ) as fun tion of the distan e from the transmitter we obtain Fig.3. The solid the urve represents ase when the re eiver and the transmitter are at the height above the ground. As h = 1 m an be seen the interferen e has a marked ee t and the power does not de rease monotoni ally with distan e as it would do in the free spa e. In ontrast, the dashed m shows a monotoni urve representing the Our measurements in an outdoor so 3. In this h = 0.1 er eld revealed that the ground (sand, gravel) was best des ribed as a diele tri ǫ≈ ase fading with distan e. with the relative permittivity ase we need to employ the Fresnel relations for the ree ted part of the horizontally (H) and verti ally (V) polarized omponents of the eld, (58), (59). Eq.(139) has then to be modied by adding the reexion fa tor, Etot (r) = E(r) + ρv (θ)Ev (r − D) + ρh (θ)Eh (r − D), for the horizontally and verti ally polarized antennas Ev = E .) Eq.(140) the interfa e when the eld omponents. (140) (For verti al an be justied by the boundary onditions at an be assumed to be approximately planar. A. Sommerfeld has obtained an exa t (for the idealized model) solution for the Hertz dipole above the ground [43, 44℄. An alternative (but mathemati ally equivalent) approa h has been presented by Weyl [53℄ where he de omposes the dipole eld into a sum (integral) of planar waves. Thus one may apply the Fresnel relations to ea h planar wave whi h yield the solution. omponent an then sum the omponents 5 Radiation from antennas 55 102 101 100 2 10-1 10-2 10-3 10-4 10-5 1 Fig. 3: 5.4.1 10 Distance(m) 100 Variation of power with distan e due to interferen e al ulated a ording to the model of Eq.(139). Antennas (transmitter and re eiver) are supposed to be verti ally oriented. Solid line orresponds to antennas at the height h = 1 m; dashed line to h = 0.1 m; dotted line to hsender = 0.5 m and hre eiver = 0.1 m. Sommerfeld's analysis of the antenna above ground We onsider a verti al Hertz dipole at a height h above a planar ground hara terized by parameters ε and σ . Above ground the radiation eld an be onsidered as a sum of the primary (open spa e solution) ex itation plus a ree ted part, while the eld in the ground onsists of the transmitted part. The task is to is to determine the ree ted and transmitted elds by invoking the boundary onditions at the interfa e. The dieren e here ompared to the earlier ree tion problems is that we no longer assume that the primary eld is a planar wave but is given by the full Hertz dipole eld (in omplex notion), 5 Radiation from antennas 56 eiω(r/c−t) , A(r, t) = µµ0 lI0 4πr where (141) r = |r − r0 | is the distan e from the antenna. E = −∇φ − ∂A/∂t. However, the φ-part then given by The ele tri eld is an be avoided if we use the Maxwell equations and the Lorenz gauge to derive the equation ∇(∇ · A) − Then for harmonious waves we introdu ing the Hertz fun tion ~ Π 1 ∂E 1 ∂2A = 2 . 2 2 c ∂t c ∂t an repla e ∂/∂t(· · · ) through A= (142) by −iω(· · · ), ~ 1 ∂Π , c2 ∂t and (143) we obtain ~ + k2Π ~ = E. ∇(∇ · Π) (As usual kc = ω .) (144) In open spa e we have the Hertz dipole solution whi h we write simply as (the important part here is only how it depends on r) ikr ~ = ẑ e Π r , (145) for a dipole oriented along the verti al dire tion ẑ. It satises the free spa e wave equation ~ + k2 Π ~ = 0. ∇2 Π z =h z > 0). It will be assumed that the dipole is at ground orresponds to primary eld ~ prim Π z = 0 (and air for (146) ex ited by the antenna. while the surfa e of the Eq.(145) represents the With no ground this would onstitute the total eld. However, with the ground present the total eld will be ome a sum ~ =Π ~ prim + Π ~ sec , Π where ~ sec Π (147) represent the se ondary rea tion eld generated by intera - tion with the ground. In found the solution for ase of the perfe tly ~ sec Π through the image ondu ting ground we already harge prin iple. Now we as- sume that the ground has nite ondu tan e σ and an index of refra tion nE 2 given by nE = ǫ + iσ/ωǫ0 . Earlier in se tion 3.5 we dis ussed reexion and transmission for planar elds and how the ree ted/transmitted part was 5 Radiation from antennas 57 found by onsidering the boundary onditions of the eld at the interfa es. The same applies here. For this end one may take advantage of the ylindri al symmetry of the situation and employ Bessel fun tions. In terms of ylindri al oordinates (ρ, ϕ, z) the distan e r from the antenna is given by r 2 = ρ2 + (z − h)2 . ~ prim and Π ~ sec have omFrom the ylindri al symmetry it follows that Π ponents only along the z -dire tion for whi h we also write Πprim and Πsec . Now the primary eld an be expressed in terms of the Bessel fun tion J0 as an integral [44, p. 243℄ (known as Sommerfeld's equation ) eikr Πprim (ρ, z) = = r for the region z > 0, and where γ= Z p ∞ J0 (ρζ) e−γ|z−h| 0 ζdζ , γ (148) ζ 2 − k2 . (The real part of γ is taken to be positive in order to ensure the onvergen e of the integral.) This suggests the ansatz Πsec (ρ, z) = Z 0 Πearth (ρ, z) = Z ∞ ∞ F (ζ)J0(ρζ) e−γ(z+h) dζ, (149) FE (ζ)J0(ρζ) eγE z−γh dζ, (150) 0 for the se ondary eld and the transmitted eld in the ground. Here γE = q ζ 2 − kE2 , with kE = nE k (the term γh in the exponents is added only for onvenien e in dealing with the boundary onditions). Πearth represents the eld transmitted in the ground. The fun tions F and FE will be determined by the boundary onditions for the ele tri and magneti eld at the interfa e z = 0. The onditions are that the tangential omponents Eρ and Hϕ shall be ontinuous a ross the boundary z = 0. These omponents are given in terms of Π by, ∂ ∂Π , ∂z ∂ρ −ik 2 ∂Π , Hϕ = ωµµ0 ∂ϕ Eρ = (151) (152) 5 Radiation from antennas 58 and similarly for the earth elds (z < 0). The boundary onditions at z = 0 an thus be written, ∂ ∂Πearth ∂ ∂Π = , ∂z ∂ρ ∂z ∂ρ ∂Πearth ∂Π = kE2 . k2 ∂ϕ ∂ϕ (153) (154) (We have assumed that the magneti permeability is µ = 1 also for earth.) The rst ondition an be integrated over ρ leading to ∂Π ∂Πearth = . ∂z ∂z Inserting the expressions for Π = Πprim + Πsec and Πearth one is nally led to the following equations for the fun tions F and FE ,  1− ζ F = γ FE = 2γE 2 nE γ + γE 2ζ . + γE  (155) , (156) n2E γ These fun tions are related to the Fresnel oe ients ρ and τ Eq.(58), although the relation is not that transparent due to the integral form in the present ase.12 However, the solution for z > 0 an be written [44, p.251℄, ⋆ eikr eikr + ⋆ −2 Π= r r Z ∞ J0 (ζρ)e−γ(z+h) 0 γE ζdζ . + γE γ n2E γ (157) The se ond term orresponds to the ree ted eld in the ase of a perfe tly ondu ting ground where r⋆ = p ρ2 + (z + h)2 . Indeed, we see that the integral in Eq.(157) vanishes in the limit of |nE | → ∞ whi h orresponds to the ase of a perfe tly ondu ting ground. Hen e the integral term des ribes the deviation from the ase of a perfe tly ondu ting ground. 12 As pointed out above there is an approa h by H Weyl [53℄ from 1919 (des ribed in [47, se exp(ikr)/r exp(ikr)/ikr = i∞ exp(ikrη)dη . 9.29℄) that may be more transparent in this regard, sin e it develops in terms of plane waves starting from the identity R1 5 Radiation from antennas 5.5 59 Breaking point, non-smooth surfa e In the ase of a ondu tive medium the permittivity be omes a omplex ′ ′′ quantity, ǫ = ǫ + iǫ . For a typi al ground the imaginary part is negligible at the frequen ies onsidered here (2.4 GHz). Assuming a real permittivity there is a spe ial grazing angle, the Brewster angle at whi h the ree ted verti al the re eiver are at the heights angle θB , as explained above, omponent disappears. If the transmitter and h1 and h2 above the ground, then the Brewster orresponds to a distan e √ h1 + h2 = (h1 + h2 ) ǫ. tan θB e is of interest sin e for r > DB the reexion DB = This distan verti al (V) in (158) oe ient for the omponent turns negative and approa hes the value -1. (This is ontrast to a perfe tly ondu ting ground where the V-reexion oe ient is always 1.) From this it follows that, for verti al antennas at large distan es, r ≫ DB and r ≫ 2πh1 h1 /λ, the power varies as P ∝ where h1 and (h1 h2 )2 , r4 (159) h2 are the heights of the antennas above the ground. The −4 breaking point where the r -dependen e starts to dominate is dened as rBP = 4h1 h2 . λ (160) Interesting it seems that the early investigators of radio ommuni ation, su h as Sommerfeld, did not pay mu h attention to the drasti interferen e −4 ee t of the diele tri ground resulting in the r power-distan e relation. The mirror (or two ray) model dis ussed above assumes a smooth ground. If we measure roughness s of the surfa e as the typi al height varia- tion of the surfa e, then the Rayleigh s< for the grazing angle θ. This riterion for smoothness requires that λ , 16 sin θ (161) orresponds to the idea that su h a height variation auses a phase dieren e less than 2s sin θ < λ/8; that is, less than ◦ 45 . Along similar lines of thought we an obtain an estimate how mu h a rough surfa e ae ts the ree tion in terms of a s attering loss fa tor Suppose height variation s around an average height tributed with standard deviation the grazing angle θ. δs . s fs . = 0 is Gaussian dis- Consider the parallel ree ted rays with Due to the height variation we get an extra phase fa tor 5 Radiation from antennas 60 exp(ik2s sin θ). Summing over all the rays in the θ dire tion the ontributions have to be weighed by the Gaussian distribution, whi h leads to the integral, 1 fs = √ 2πδs Z (   2 )  πδs sin θ s2 . exp ik2s sin θ − 2 ds = exp −8 2δs λ −∞ ∞ (162) The orre ted reexion oe ient is thus obtained as fs ρ. Thus in the ase the distan e between the transmitter and the re eiver is 10 m, the devi es are at the height 1 m ( orresponding to θ ≈ 11.3◦ ) and δs = 1 m we get fs = 0.98, and when δs = 5 m we get fs = 0.62 ( 2.4 GHz). 5.6 Ground wave The two ray model seems to a ount rather well for the ground ee ts when omparing to measurements. It is based on al ulating the ree tion of rays, or planar waves, from the ground whi h usually are good approximations in the far eld region. The mirror method gives the exa t solution in ase of a perfe tly ondu ting ground but for a general diele tri ground the situation is dierent. Near the ground the propagation may be altered sin e, due to the air-ground interfa e, the EM-eld might no longer be a pure transversal TEM eld. This is in analogy to the waveguide ase dis ussed earlier. The so alled ground or surfa e wave (Oberä henwelle) may be ome of interest in settings where the trans eivers are dire tly on the ground. Sommerfeld in fa t originally got interested in the problem of the eld of the dipole near the ground in order to nd out whether the dipole solution ontained the surfa e wave on eived earlier by Zenne k [54℄. It may be however in the interest of larity to treat the simpler ase of Zenne k. We onsider a plane wave that travels in the x-dire tion over a plane ground dened by z = 0. Thus we have a ground for z < 0 and e.g. air for z > 0 with the interfa e at z = 0. We assume that ele tri al eld is polarized along z . However from the dis ussion of the skin ee t we already learned that the eld be omes damped in a ondu ting media and that Ez therefore also is a fun tion of z (besides of x), Ez (z). Then from the ∇·E = ∂Ex /∂x+∂Ey /∂y +∂Ez /∂z = 0 it follows that Ex and Ey annot both vanish be ause then we would have ∂Ez /∂z = 0. We may thus assume that E is of the form (Ex , 0, Ez ) and H is of the form (0, Hy , 0). From Maxwell equations we obtain the wave equations in the harmoni ase, 5 Radiation from antennas 61 ∂ 2 Ez ∂ 2 Ex 2 + k E + iσµµ ωE = , z 0 z ∂x2 ∂z∂x ∂ 2 Ez ∂ 2 Ex 2 + k E + iσµµ ωE = , x 0 x ∂z 2 ∂z∂x (163) (164) k 2 = µµ0 ǫǫ0 ω 2 . For air we set ǫ = µ = 1 and σ = 0, and for the ground ǫ = ǫE , µ = µE = 1 and σ = σE . From Eq.(164) we an again infer that ∂Ez /∂z 6= 0 leads to a non vanishing Ex - omponent. For these linear equations we have solutions of the form (due to their ouplings Ex and Ez where must have the same exponential fa tor) Ex = A exp(ikx x + ikz z), Ez = B exp(ikx x + ikz z), whi h inserted into the equations (163, 164) yield the relations (and their dupli ates for ground) kx2 + kz2 = k̃ 2 = k 2 + iσµµ0 ω, Akx + Bkz = 0. From the boundary Ex , and the normal onditions ( ontinuity of the tangential omponent Dz ) at the air-ground interfa e (165) omponent z = 0 we further obtain, kxE = kx , A = AE , B = ǫ̃E BE , with ǫ̃E = ǫE + i The index E σE . ωǫ0 refers to earth/ground. Summing up we have, after setting A = 1, altogether 7 equations for the 7 unknowns (kx , ky , kxE , kyE , AE , B, BE ). From these relations we obtain, 5 Radiation from antennas 62 r ǫ̃E , 1 + ǫ̃ r E 1 kz = ±k · , 1 + ǫ̃E kzE = ǫ̃E · kz . kx = k · This is nothing but a solution of the Fresnel equations in reexion ( orresponding to the Brewster angle). enfor ed by the initial assumption Eq.(165). general to omplex. ase of zero Zero reexion was here The kx In order that the eld stay nite as and kz , kzE are in z → ±∞ we have hoose the negative sign of the square root in the above equation. ℑ(ǫ̃E ) ≪ exp(ikx x) results in  1 then  p as exp −x · k 1/2σ/(2ǫ0 ω) whi h If a damping fa tor that an be written x → +∞. These surfa e vanishes as waves were studied by Cohn (1900), Uller (1903) and Zenne k (1907) [47, se . 9.10℄. Sommerfeld also derived a surfa e wave from his verti al dipole solution dis ussed above in the limit (z ≥ 0) kx r → ∞. In this limit he obtains for the Hertz fun tion a term [44, p.256℄, A Π = √ · eikx r+kz z , r where kx and kz are as above (A is a of a two-dimensional wave in the instead of ongoing 1/r as in the z -plane onstant). (166) This des ribes a sort sin e it is proportional to √ 1/ r ase of spa e waves. Interestingly there has been an ontroversy about the signi an e and reality of the surfa e waves that has lasted for over a entury [54, 43, 53, 33, 34, 22, 51, 11℄. One of the famous legends promulgated as re ently as 1998 in a review of the eld by Wait [51℄ is that Sommerfeld made a sign error in his pioneering paper [43℄ with serious onsequen es for the interpretation of the results. Sommerfeld never admitted to any su h errors (see for instan e [44℄ se tion 32 and h. 23 in [14℄), and the re ent study [11℄ by Collins indeed rearms that the famous sign error is a myth. Collin tra es the myth ba k to a short paper [33℄ by K A Norton in 1935 whi h asserts that there is a sign error in Sommerfeld's 1909 paper  this allegation was later un riti ally repeated by numerous authors (for instan e in the book [47℄) although the exa t lo ation of the error was never revealed. More importantly Norton did endorse the on ept of the surfa e waves based on his  orre ted version of the theory. Nevertheless, the asymptoti evaluation of the Sommerfeld solution remains a somewhat tri ky business. In fa t, Kahan and E khart have presented a 5 Radiation from antennas 63 series of studies [22℄ where they argue that no surfa e waves are in the dipole solutions, others. They assert 13 ontained ontrary to the opinions of Sommerfeld, Norton, and that a areful evaluation of the Sommerfeld solutions shows that the surfa e term will be an eled by an equal term of opposite sign missed by Sommerfeld. Kahan and E khart thus agree with the 1919 analysis by Weyl. The interesting thing is that Sommerfeld's result agrees with Weyl's result [14, p.937℄. The starting point in Sommerfeld's approa h (whi h no one diagrees with) is to repla e J0 (ζρ) J0 (ζρ) = in Eq.(157) by  1  1  (1) (2) (1) (1) H0 (ζρ) + H0 (ζρ) = H0 (ζρ) − H0 (−ζρ) . 2 2 This makes it possible to rewrite the integral in Eq.(157) as an integral extending from innity to innity, Π= Z (1) W H0 (ζρ)e−γ|z| n2E ζdζ . n2E γ + γE (167) The path W extends from −∞+iδ to 0+iδ , and from 0−iδ to ∞−iδ where δ > 0 is an innitesimally small real quantity. This rule is enfor ed in order to ensure the orre t evaluations of the square roots involved in the integrand. The tri ky part is the asymptoti evaluation of the integral for The reason is that the pole of the integral may be very bran hing points whi h is given by k and whi r → ∞. lose to one of the h will ae t the steepest des ent evaluation of the integral. This was the point overlooked by Sommerfeld in his earlier works a ording to [11, 22℄ though Sommerfeld mentions the problem [44, p.258℄ later but refers for further details to a paper by H. Ott [35℄. In [14, p.932℄ he also a knowledges omments from F. Noether and V. Fo k on the perils of the approximation pro edure whi h in the end means that in pra ti e the surfa e wave annot be separated from the spa e wave (Es dürfte keine Bedienungen geben, unter denen si h der Oberä henwellentypus P rein ausbildet und den Hauptbestandteil des Wellenkomplexes darstellt). However if the ground is overed by a diele tri layer (su h as i e over the sea) then there may indeed arise trapped surfa e waves in the layer as pointed 13 They laim to have settled the issue in the present paper by proving in a quite general way that this surfa e wave annot be in luded in the said dipole radiation and by pointing out a thus far hidden error in Sommerfeld's the problemati s of the asymptoti by e.g. F Noether and V Fo k and omputation [22, p.807, abstra t℄. However, evaluation of the integral involved was early re ognized ommuni ated to Sommerfeld. 5 Radiation from antennas 64 out by Wait [50℄14 . The bran hing points mentioned above arise be ause of the square root expressions when the integral is evaluated using the residue al ulus. In this pro edure the path of integration is losed by an innite half ir le in the upper half plane inside whi h there will be one pole and two bran hing points (at k and kE ). Cutting up the plane along lines from k and kE to k + i∞ and kE + i∞ will make the integrand single-valued and the residue theorem be omes thus appli able. For further dis ussion of this topi see the referen es that have been listed above. Returning to the dis ussion of the Zenne k waves we may envisage that they an appear if a plane wave travelling along the x-axis hits a diele tri / ondu tor at x = 0, whi h extends to z < 0 and x > 0. Part of the wave s atters but part of it will ontinue along x > 0 in the half-spa e z > 0 and is expe ted to approa h the form of a Zenne k wave far from x = 0. 14 The question of the presen e of surfa e waves may be ompared to similar problems of identifying parti le states and resonan es in quantum me hani s; investigations whi h too involve s rutnizing the poles in evaluating the transfer fun tions of the ψ -wave. A Some denitions and results from ve tor analysis 65 A Some denitions and results from ve tor analysis Ve tors are denoted by A = (Ax , Ay , Az ) = Ax x̂ + Ay ŷ + Az ẑ in a Cartesian oordinate system; A = (Aρ , Aϕ , Az ) in a ylindri al oordinate system; A = (Ar , Aθ , Aϕ ) in a spheri al oordinate system. Note that the ϕ- omponent appears in dierent order in spheri al and ylindri al oordinate systems though it has the same geometri al meaning in both systems. A · B = Ax Bx + Ay By + Az Bz (dot produ t) A × B = (Ay Bz − Az By , Az Bx − Ax Bz , Ax By − Ay Bx ) ( ross produ t) A × (B × C) = (A · C)B − (A · B)C (Lagrange identity)   ∂φ ∂φ ∂φ ∇φ = (gradient) , , ∂x ∂y ∂z   ∂φ 1 ∂φ ∂φ , , ∇φ = ( yl. oord.) ∂ρ ρ ∂ϕ ∂z   1 ∂φ ∂φ 1 ∂φ , , ∇φ = (spher. oord.) ∂r r ∂θ r sin θ ∂ϕ ∂Ax ∂Ay ∂Az + + ∇·A= (divergen e) ∂x ∂y ∂z 1 ∂(ρAρ ) 1 ∂Aϕ ∂Az + + ∇·A= ( yl. oord.) ρ ∂ρ ρ ∂ϕ ∂z 1 ∂(sin θAθ ) 1 ∂Aϕ 1 ∂(r 2 Ar ) + + ∇·A= 2 (spher. oord.) r ∂r r sin θ ∂θ r sin θ ∂ϕ   ∂Az ∂Ay ∂Ax ∂Az ∂Ay ∂Ax ∇×A= (rotor) − , − , − ∂y ∂z ∂z ∂x ∂x ∂y   ∂Aϕ ∂Aρ ∂Az 1 ∂(ρAφ ) 1 ∂Aρ 1 ∂Az − , − , − ∇×A= ( yl. oord.) ρ ∂ϕ ∂z ∂z ∂ρ ρ ∂ρ ρ ∂ϕ    ∂(sin θAϕ ) ∂Aϕ 1 ∂Ar 1 ∂(rAϕ ) 1 , − − , ∇×A= r sin θ ∂θ ∂z r sin θ ∂ϕ r ∂r   1 ∂(rAϕ ) ∂Ar − (spher. oord.) r ∂r ∂θ ∂2 ∂2 ∂2 ∇2 = ∇ · ∇ = 2 + 2 + 2 (Lapla ian operator) ∂x ∂y ∂z 1 ∂2  = ∇2 − 2 2 (wave operator) c ∂t ∇ × (∇ × A) = −∇2 A + ∇(∇ · A) ∇ · (A × B) = B · (∇ × A) − A · (∇ × B) A Some denitions and results from ve tor analysis I A · ds = I∂S I∂V ∂V Z ZS A · dS = 66 ∇ × A · dS ∇ · A dV V (Stokes-Green theorem) (Gauss' theorem) Z (F ∇G − G∇F ) · dS = (F ∇2 G − G∇2 F ) dV V (Green's theorem)
eikr ∇2 + k 2 = −δ(r) 4πr r Z ∞ π 2 e−ax dx = (ℜ(a) ≥ 0) a −∞ The s alar Lapla ian ∇2 φ an be 2 Lapla ian operator al ulated for ylindri al and spheri al ∇ φ = ∇ · (∇φ). For ve tor fun tions the 2 an be evaluated using the identity ∇ A = ∇ (∇ · A) − oordinates by using the identity ∇ × (∇ × A). (Gauss integral) The Dira 15 delta δ(x) is a so alled fun tional dened by the property Z R for any fun tion f (x)δ(x − x0 )dx = f (x0 ), f. (Dira The Fourier representation of the Dira delta.) delta an be expressed as 1 δ(x) = 2π We may extend the Dira ordinates) Z eikx dk. (168) R delta to ve tors by dening (in Cartesian o- δ(r) = δ(x)δ(y)δ(z). In the text we have often used an harmoni E-eld is written 16 omplex notation for the EM elds. Thus, E(r, t) = E(r)e−iωt . The physi al eld will then (169) orresponds to the real part of the phasor (169), 15 Ephys (r, t) = ℜ(E(r, t)). It be ame well known through Paul Dira 's but the 1925) from where Dira 16 Prin iples of Quantum Me hani s pi ked it up. We here adhere to the physi s tradition using the time fa tor exp(−iωt), while many exp(iωt) (also often engineering texts su h as [3℄ assume a time dependen e of the form denoting (1930) on ept had been introdu ed in ele tri al engineering by Oliver Heaviside (1850- √ −1 by j instead of i). B Tables 67 When omputing dot-produ ts and have to use ross-produ ts for omplex onjugation in order to obtain the quantities. For instan e, Ephys · Ephys omplex elds, we orresponding physi al is evaluated as 1 E · E⋆ . 2 The fa tor 1/2 omes from the fa t that a time average is implied (RMS- value), whi h for real elds amounts to the fa tor 1 T Z T 1 (cos(ωt))2 dt = . (T = 2π/ω.) 2 0 Similarly the Poynting ve tor is expressed as 1 ℜ (E × H⋆ ) , 2 P = E · J be omes S= and the power density quantities P= likewise in terms of phasor 1 ℜ (E · J⋆ ) . 2 B Tables Constant ǫ0 µ0 η0 c0 |e| Value 8.854187817 ·10 −7 4π · 10 −12 Unit Legend As/Vm Permittivity of va uum Vs/Am Permeability of va uum Ω Wave-impedan e of va uum m/s Velo ity of light in va uum 376.7303 ·108 ·10−19 2.99792458 1.60217733 Tab. 1: Here c0 and µ0 As Ele tron harge Constants are exa t values. These and 1 c0 = √ , ǫ0 µ0 η0 = r η0 and are µ0 , ǫ0 from whi h we obtain η0 = c0 µ0 ≈ π120 Ω ≈ 377 Ω. Sometimes tables list the loss tangent dened by onne ted by B Tables 68 Material Con rete Plasterboard Glass Dry bri k Limestone Wood Douglas r (plywood) Soil (sand) Water (sea) Water (lake) I e Snow Air Human tissue Iron Copper Aluminum ǫ 812 14 8 4 7.5 3 1.82 3.4 80 80 3.2 1.5 1 70 NA NA NA Tab. 2: σ (Ω m)−1 10−5 10−7 10−11 0.01  0.02 ( 4.3 GHz) 0.03 10−13 10−4 0.049 ( 3 GHz) 10−5 35 10−3 ( 3GHz) 5 ·10−4 ( 3GHz) 10−3 0 0.2 106 5.8 ·107 3.5 ·107 µ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 5000 1 1 Ele tri properties tan δ = σ ǫ′′ = . ǫ′ ǫ0 ω That is, ǫ′′ = ǫ′ tan δ . Ele tri properties depend generally on temperature and frequen y. The omplex permittivity of water, for instan e, an be quite well represented for f < 50 GHz and 20◦ C by the Debye model (quoted by [23℄) ǫ = ǫ∞ + σ ǫs − ǫ∞ , +i 1 − iωτ ωǫ0 where ǫ∞ = 5.27, ǫs = 80.0, τ = 10−11 s, and the nal term a ounts for the ee t of salt if present (salt water having a ondu tivity around 3  5 (Ωm)−1 ). About wood note that is usually anisotropi due to the bers. Fields polarized along the dire tion of the bers pass more easily through the material [37℄. For human tissue there has been reported [41℄ the following expression for (real) permittivity and ondu tivity, C Bessel fun tions 69 70 − V , 1 + (1.5/λ)2 70 − V (1.5/λ)2 σ = σ0 + , 60λ 1 + (1.5/λ)2 ǫ′ = 5 + V ∼ 5 (volume fra tion o upied by ma romole ular σ0 ≈ 1/70 ( m Ω)−1 (observe the unit) and λ is expressed where nents), These values are to be (170) (171) ompoin m. onsidered as averages sin e the properties vary with the type of organ and tissue (blood, bone, mus le, brain, et ). C Bessel fun tions C.1 While Bessel cos- plane (the and J -fun sin-fun tions tions are asso iated with rotational symmetry in the ir le), Bessel fun tions are asso iated with rotational ( ylindri al) symmetry around an axis in the 3-dimensional spa e. Bessel fun tions ome 2 typi ally into play when we have to solve the Lapla e equation ∇ u = 0 in spa e for a system with in ylindri al ylindri al symmetry. The Lapla e operator is given oordinates as, 1 ∂ ∇ u= ρ ∂ρ 2   ∂u 1 ∂2u ∂2u ρ + 2 2 + 2. ∂ρ ρ ∂ϕ ∂z (172) eigenvalue equation ∇2 u = αu in the ase α = -1. Apparently u = exp(iy) = exp(iρ sin ϕ) is a solution. We develop it as a series in exp(iϕ), X eiρ sin ϕ = Jn (ρ)einϕ (173) Consider the n Jn of integer order (n = 0, ±1, ±2, · · · ). ∇ u + u = 0 we nd that the Jn satises whi h denes the Bessel fun tions Inserting the expression (172) into the equation, 1 ∂ ρ ∂ρ 2     ∂Jn (ρ) n2 ρ + 1 − 2 Jn (ρ) = 0. ∂ρ ρ From Eq.(173) it follows also that 1 Jn (ρ) = 2π Z 0 Jn (174) an be dened via the integral, 2π eiρ sin ϕ e−inϕ dϕ, (175) C Bessel fun tions 70 and that we have a series expansion (for n ≥ 0), Jn (ρ) = ∞  ρ n X 2 k=0 1 k! Γ(n + k + 1)  ρ2 − 4 k (ρ > 0). (176) R We have in Eq.(176) employed the Gamma-fun tion17 Γ(t) = 0∞ st−1 e−s ds (ℜ(t) > 0) whi h for integer values m > 0 satises Γ(m) = (m−1)!. Eq.(176) an in fa t be extended by repla ing n by fra tional values ν to Bessel fun tions Jν of fra tional order whi h also satisfy the Bessel equation (174). As a histori al note we may mention that Daniel Bernoulli studied the series expansion of what is now alled J0 already in 1738 in onne tion with the problem of determining the shape of an os illating hain ( atena ) supported from both ends [52, se . 1.2℄. Using the method of steepest des ent or stationary phase [44, se . 16.E℄[12, se . VII.6℄ one an show that the asymptoti value of Jn (ρ) for ρ → ∞ and xed n is given by Jn (ρ) ≈ r  2 π . cos ρ − (2n + 1) πρ 4 (177) 1 Γ(ν + 1) (178) Thus for large ρ the fun tion Jn(ρ) behaves as the cos-fun tion. One also sees from this that Jn (ρ) has an innite number of zeros, as already pointed out by Daniel Bernoulli in ase of J0 . Below we will also allow Jn (z) to take omplex valued arguments z . From small ρ we have from Eq.(176) the asymptoti relation, Jν (ρ) ≈  ρ ν 2 Sin e exp(iζy) = exp(iζρ sin ϕ) = Jn (ζρ) satises the equation, 1 ∂ ρ ∂ρ  ∂Jn (ζρ) ρ ∂ρ  (ν 6= −1, −2, · · · ). P n Jn (ζρ) exp(inϕ) it follows that   n2 2 + ζ − 2 Jn (ζρ) = 0. ρ (179) Espe ially we have in the ase n = 0,  17  ∂2 ∂2 2 + + ζ J0 (ζρ) = 0. ∂x2 ∂y 2 (180) A more general denition is (L Euler 1729) nt n! n→∞ t(t + 1) · · · (t + n) Γ(t) = lim whi h is valid for all points t in the omplex plane ex ept for 0 and the negative integers (the poles). C Bessel fun tions 71 This implies that for any fun tion F (ζ) Π= Z ∞ ± F (ζ)J0(ζρ)e √ ζ 2 −k 2 z (181) dζ 0 satises the wave equation ∇2 Π + k2 Π = 0. This fa t was employed in se tion 5.4.1 in order to write exp(ikr)/r on the form (181). One makes the ansatz that there is a fun tion F su h that (181) is satised in ase of Π = exp(ikr)/r . One studies rst the instan e z = 0, eikρ = ρ Z ∞ (182) F (ζ)J0(ζρ)dζ. 0 Using the general orthogonality property Z ∞ 0 (183) Jn (ζρ)Jn (τ ρ)ρdρ = τ −1 δ(ζ − τ ), of the Bessel fun tions one an invert Eq.(182) obtaining [44, p.243℄, ζ −1 F (ζ) = Z ∞ 0 eikρ J0 (ζρ)dρ = p 1 ζ 2 − k2 . (184) On e we have the result for p z = 0 (182) we obtain the general solution by inserting the fa tor exp(± ζ 2 − k2 z) as in Eq.(148). The orthogonality relation Eq.(183) an be demonstrated by a lever use of the identity f (x, y) = ZZ f (ξ, η)δ(x − ξ)δ(y − η)dξdη, inserting the representation Eq.(168) for the Dira deltas and then going over to polar oordinates assuming the spe R ial form f (r, ϕ) = g(r) exp(inϕ) for the fun tion f [44, se . 21℄. That 0∞ Jn (τ ρ)Jn (ζρ)ρdρ = 0 for τ 6= ζ an be seen also dire tly by multiplying the Bessel dierential equation for Jn (τ ρ) by Jn (ζρ) and vi e versa and subtra ting the expressions. C.2 Hankel fun tions As we found above the Bessel fun tion Jn resembles the cos-fun tion. It is often more useful to employ the omplex exp-fun tion than the cos-fun tion, for instan e when dealing with time-harmoni elds. It likewise exists a Bessel ounterpart to the exp-fun tion that may be preferable to the J fun tions in some ir umstan es. These are the so alled Hankel fun tions (1) (2) Hν and Hν . In order to arrive at these one may start by generalizing the C Bessel fun tions 72 integral representation (176) to one that overs also the ase of non-integral orders (S häi 1871), 1 Jν (ρ) = 2π Z (185) eiρ cos z+iν(z−π/2) dz. W0 The integration is along an innite path W0 in the omplex z -plane, hosen su h as to make the integral onvergent for real positive ρ. The standard hoi e for W0 is the path − π2 + i∞ → − π2 + i0 → 3π + i0 → 3π + i∞. 2 2 In ase of an integer order ν = n the expression (185) redu es to the earlier one (176), sin e the integrals along the verti al se tions of the path W0 an el ea h other in this ase. In the adjoining gure we introdu ed two other innite paths W1 and W2 whi h also dene two onvergent integrals (introdu ed by Sommerfeld in 1896) and the orresponding fun tions alled the rst and se ond Hankel fun tions, W0 ❄ ❄ - π2 Z 1 eiρ cos z+iν(z−π/2) dz, = π W1 Z 1 (2) eiρ cos z+iν(z−π/2) dz. Hν (ρ) = π W2 Hν(1) (ρ) W1 ✻ ❄ W2 3π 2 (186) Now we have formally W0 = W1 +W2 sin e the integrals along the negative imaginary axis an el ea h other. We therefore obtain,
1 Hν(1) (ρ) + Hν(2) (ρ) . (187) 2 This an be ompared to cos ϕ = (exp(iϕ) + exp(−iϕ))/2. Indeed the asymptoti form as ρ → ∞ is given by Jν (ρ) = r 2 · ei(ρ−(ν+1/2)π/2) , ρπ r 2 (2) · e−i(ρ−(ν+1/2)π/2) . Hν (ρ) ≈ ρπ Hν(1) (ρ) ≈ For ρ → 0 we have instead (188) C Bessel fun tions 73 2 (2) H0 (ρ) ≈ −i ln ρ, π  ρ −ν Γ(ν) Hν(2) (ρ) ≈ i π 2 (189) (ℜν > 0). One property following from the denitions (188) is that for real ρ and ν (190) (2) Hν(1) (ρ) = Hν (ρ), where the bar denotes omplex onjugation. The Bessel fun tion orresponding to the sin-fun tion is alled the Neumann fun tion and is dened by Nν (ρ) =
1 Hν(1) (ρ) − Hν(2) (ρ) . 2i (191) The Hankel fun tions play an important role in Sommerfeld's theory of the dipole over ground and e.g. in the treatment of EM-elds along ir ular ondu tors. Using the de omposition (187) of the R∞ R ∞Bessel fun tion J0 in Eq.(157) one an transform the 0 -integral into an −∞ -integral. This makes it possible to apply the al ulus of residues and evaluate part of the integral in terms of a pole of the intef2 grand. An important step is the relation W (2) (1) (1) (2) H0 (−ρ) = −H0 (ρ), H0 (−ρ) = −H0 (ρ), (192) −3π −2π −π 0 2π π whi h is a spe ial ase of the Umfor the Hankel fun tions [44, p.314℄. Note that the integration paths used in order to dene Bessel fun tions need to stay in the limits of |z| → ∞ in the he ker board patterns shown in the adjoining gure in order to ensure onvergen e for ρ > 0; for ρ < 0 the he ker pattern is shifted by π along the real axis). The paths may though be deformed as long as they do not ross borders and still dene the f2 is an allowed version of W2 des ribed earlier. Now same fun tions. Thus W (1) H0 (−ρ) (ρ > 0) is dened using the path W1 translated by π along the real axis, whi h we denote W1 + π . Sin e −ρ cos z = ρ cos(z ± π) we get laufrelationen D The able equation and impedan e 74 R (1) H0 (−ρ) = (1/π) W exp(iρ cos(z))dz where W1 + π is shifted by π for z > 0 and by −π for z < 0. This gives a path equivalent to W2 ex ept that it is traversed in the opposite dire tion. D The able equation and impedan e Waves do not travel only in open spa e, but also along ables. Thus, for high frequen y urrents, ir uit hara teristi s an no longer be treated a ording to the usual point-to-point models. For instan e, the lengths of the onne tions may ae t the ir uit properties. One entral property of ir uit elements, ables and loads, is that of impedan e Z . Generally speaking, if we feed a voltage V = V0 · exp(−iωt) into a load and measure a resulting urrent I = I0 · exp(−iωt), then the impedan e (at the given frequen y) of the load is dened by the quotient Z= V . I We will onsider a dis rete model of a able made up of a series of indu tors L and apa itors C (in parallel). L In−1 Vn−1 Fig. 4: C In+1 In Vn Vn+1 Dis rete model of a transmission line as a series of indu tors L and apa itors C . These elements are dened by their impedan es ZL = −iωL and ZC = 1/iωC ; that is, the voltage over an indu tor is given by V = L ∂I/∂t and over a apa itor by V = Q/C . If one applies the laws of Kir ho and Ohm to the urrents and voltages over the n − 1, n, and n + 1:th elements one obtains the equations −C V̇n = In − In−1 , −LI˙n = Vn+1 − Vn . (193) (194) D The able equation and impedan e Letting the spa ing ∆x 75 between the units go to zero we may repla e the dieren e by derivatives, ∂I , ∂x ∂V , −LI˙ = ∂x −C V̇ = where C = C/∆x is the (195) (196) apa itan e per unit length, and indu tan e per unit length of the L = L/∆x the able. This leads to the  able equation ∂2V 1 ∂2V − =0 ∂x2 u2 ∂t2 1 u= √ . CL with (197) This is a wave equation for EM-waves propagating along the able with u. Inserting solutions of the form V = V0 · exp(i(kx − ωt)), I = I0 · exp(i(kx − ωt)) into (195), where k = ω/u, we obtain for the impedan e, r k L V = = . Z= (198) I Cω C q L gives the able impedan e for this model. Thus, the quantity Z0 = C the velo ity One an also add resistive elements to the model, but the general features are already apparent in this simple model. impedan es of 50 Ω or 75 Ω, Standard oaxial ables have but one may note that these values apply only in some restri ted frequen y range. In the above we have a ase we onsidered an innite long able with impedan e Z0 of length l able. Suppose instead whi h is ended by a load ZL . What will then be the impedan e of this system? Assume the feeding point is at x = 0, and the load is at the point x x = l. If we feed a voltage V0 · exp(−iωt) at = 0, then the solution for the system is of the form V (x, t) = aei(kx−ωt) + bei(−kx−ωt) , i(kx−ωt) I(x, t) = ce The terms with −ikx i(−kx−ωt) + de in the exponents . orrespond to the ree ted part of the wave. The impedan e will thus be given by the quotient Z= (199) a+b V (0, t) = . I(0, t) c+d D The able equation and impedan e 76 The oe ients a, b, c, d are determined by the ondition V (l, t) = ZL · I(l, t), and using (195) (whi h yields the relations a = Z0 c, b = −Z0 d). After some algebra one obtains nally Z = Z0 ZL − iZ0 tan(kl) . Z0 − iZL tan(kl) Z0 ZL (200) We have the interesting result that the ✛ ✲ l impedan e depends on the able length l and the wavelength λ through the term tan(2πl/λ). For instan e, for a quarter wave able, l = λ/4, we get Z = Z02 /ZL. In the short- ir uited ase (ZL = 0) we obtain Z = −iZ0 tan(kl). (Short ir uited ase.) In this ase we have innite impedan e for quarter wave ables. This property is used in mi rowave hokers whi h are quarter wave avities designed to trap unwanted leakage of EM-waves e.g. through the door slits mi rowave ovens. For an open ended able (ZL = ∞) we have instead Z = iZ0 cot(kl), (Open ended ase.) and this orresponds to an innite impedan e for a half wave able. We an al ulate the rms power dissipated by the load ZL from (assuming Z0 is real) 1 1 P = ℜ(I ⋆ (l)V (l)) = (|a|2 − |b|2 ) = 2 2Z0 V02 2Z0 1− 1+ ZL −Z0 ZL +Z0 2 ZL −Z0 i2kl e ZL +Z0 2. The quotient |b/a| des ribes the fra tion that is ree ted, and it is obtained from b ZL − Z0 i2kl = ·e . a ZL + Z0 (201) From this we infer that there is no reexion when we have impedan e mat hing, ZL = Z0 . In this ase the energy transferred to the load is simply V 2 /2Z0 . The term impedan e mat hing is also used for point-to-point D The able equation and impedan e 77 ir uits where reexion plays no part. If we have resistan es R0 and RL in series and apply the voltage V , then the power dissipated by the resistan e RL be omes 2 PL = RL · I = RL ·  V R0 + RL 2 , whi h for xed V and R0 attains its maximum when RL = R0 . Thus, for this ase, in order to deliver maximum of power to the load, its impedan e must be mat hed to the ir uit impedan e R0 . If we repeat this exer ise with the omplex impedan es Z0 and ZL we obtain the mat hing ondition ZL = Z0⋆ . These results an be applied to the ase where we have an RFir uit with impedan e Z0 oupled to an antenna with the impedan e ZL . From (201) we get for the magnitude of the reexion oe ient, |ρ| = ZL − Z0 . ZL + Z0 (202) Two antenna design parameters are dened in terms of r: the Voltage Standing Wave Ratio, VSWR = and the Return Loss, 1 − |ρ| , 1 + |ρ| (203) (204) R = 20 log(|ρ|). Designing the antenna onditioning one has to make a hoi e between minimizing the ree tion (Z0 = ZL ) and maximizing the radiation power (ZL = Z0⋆ ). One wants to get rid of the reexions be ause they an interfere with RFir uit operations. Above we onsidered a disrete model of a transmission line. The straight oaxial able an be easily solved too be ause of the ylindri al symmetry. Thus assume the enter wire has the radius a and outer ondu tor an inner radius b. The transversal ele tri and magneti mode (TEM) orrespond to the ase where the ele tri al eld E = (Er , 0, 0) is radial while the magneti eld is azimutal, H = (0, Hϕ , 0). The Maxwell equations of interest in the region a < r < b be ome, ✻ ✛ H E E Chip on WCR2400 78 ∂Er = iωBϕ , ∂z ∂Hϕ = iωBϕ , ∂z 1 ∂(rHϕ ) = 0. r ∂r (205) (206) (207) From this follows that ∂ 2 Er + k 2 Er = 0, ∂z 2 √ where k = ω ǫǫ0 µµ0H. It follows from the third equation that Hϕ ∝ 1/r. More pre isely we have H · dr = I where the integral is along a ir le of radius a < r < b en ir ling the inner wire whi h arries a urrent I . Thus Hϕ (r) = I , 2πr whi h in ombination with the equation (205) gives Er = r I µµ0 I =η . ǫǫ0 2πr 2πr The potential V between between the inner and outer ondu tor be omes therefore Z a b ηI  a  . ln Er dr = 2π b Finally the hara teristi impedan e Z = V /I of the able be omes E a η Z= . ln 2π b Chip on WCR2400 The Chip on devi e CC2420 [10℄ operates in the 2.4 frequen y band (λ = 12.5 m). IEEE 802.15.4 denes 16 hannels in 5 MHz with the frequen ies fk = 2405 + 5 · (k − 11), with k = 11, 12, . . . , 26. The ee tive data rate is 250 kbps (2 MChips s−1 ) and the devi e uses a oding with 4 bit symbols in 32 hip spread sequen es. The RSSI values are determined from the average over 8 symbols orresponding to a time interval ∆t = 128 µs. The signal strength is determined from the automati gain ontrol fa tor in the signal E Chip on WCR2400 79 amplier part (variable gain amplier, VAG). From the bit rate we gather that every bit has a 4 µs time window. Sin e the velo ity of light c is about 3 × 108 ms−1 we get c × 4 µs ≈ 1200 m for the distan e that the radiation an travel during the time window whi h means that the ree ted and refra ted parts (from obje ts within 600 m whi h by far ex eeds the normal range of the devi es) an ontribute to the bit-signal, either by destru tive interferen e. The orresponding onstru tive or hip time window is 1 µs whi h translates into a 300 m distan e. In our measurements we used a standard oaxial λ/2-dipole antennas model WCR-2400-SMA. The dipole antennas ir ular radiation pattern in the horizontal plane [9℄. 5 31 22 5 270 approximate well a 0 90 13 45 5 180 Fig. 5: Variation of RSSI (radius orresponds to absolute RSSI value) with angular orientation of the re eiver with shielding ( ir le) and no shielding (lled ir le) in outdoor setting. The re eiver-transmitter separation was 10 m. Origin of the diagram orresponds to RSSI = -50, and the radial intervals to 10 RSSI units. However, the antenna + RF devi e results in orientation ee ts apparently be ause of the PCB ae ts the radiation pattern of the antenna in the horizontal plane. Shielding the RF devi e using an appropriate metalli en losure (Faraday shown by Fig.5. age) redu es the orientation ee t signi antly as The orientation ee t is similar in the transmitting and re eiving mode. One may also note that we get a better RSSI value in the shielded ase. Cylindri al en losures of diameter 90 mm and height 40 mm F Re ipro ity theorem 80 ✛ ✻ ✲ 13 mm 40 mm ✛ ✲ 82 mm ❄ Fig. 6: Shielding box. were ustom made in 4 mm aluminum. The en losure housed the PCB plus the battery, with the antenna tted to the enter of the top surfa e (top and bottom plates 1 mm thi k). The en losure + antenna naturally will be expe ted to have a dierent input impedan e than the antenna alone. However, there seems not be any readily available treatments of how su h a ylindri al en losure modies the impedan e. This problem as well as impedan e measurements using a network analyzer will be left to a separate study. F Re ipro ity theorem A basi property used in antenna measurement [24℄ is the so alled re ipro ity whi h says that an antenna has the same eld pattern both in the re eiving and transmitting mode. Therefore one needs to test it only as either re eiver (usually the simpler alternative) or as a transmitter. More generally, suppose an antenna T1 at r1 is transmitting radiation at the frequen y f and is inter epted at r2 by an antenna T2 , then eld strength measured with T2 is the same as would be measured T1 if T2 is transmitting with the same energy and at the same frequen y f . If we imagine the antennas as part of of ir uit [8℄ with voltages and urrents related by theorem V1 = Z11 I1 + Z12 I2 V2 = Z21 I1 + Z22 I2 (208) then re ipro ity boils down to the symmetry of the mutual impedan e, Z12 = Z21 : A given urrent in one antenna auses the same (open ir uit) F Re ipro ity theorem 81 voltage in the other antenna irrespe tive whi h of them a ts as the transmitter or re eiver. Z11 − Z12 Z22 − Z12 V1 V2 I1 I2 ✲ Fig. 7: ✛ Z12 Antenna system as a two-port network The ele tromagneti re ipro ity theorem goes ba k to the work by H A Lorentz in 1895-1896 and an a ount is given by Sommerfeld in [14, p.653663℄ (whi h however rests on the assumption that the antenna length L is insigni ant ompared to the wavelength λ). It is built on the fundamental equation ∇ · E 1 × H 2 − ∇ · E 2 × H 1 = E 2 · J1 − E 1 · J2 (209) ∇ × Ek = iωBk , ∇ × Hk = −iωDk + Jk . (210) whi h is obtained from Maxwell's equations for (E1 , H1, J1 ) and (E2 , H2 , J2 ) in the harmoni ase (time derivatives of the elds are repla ed by multipli ation by −iω ) assuming we have a linear and isotropi medium (Dk = ǫǫ0 Ek ; Bk = µµ0 Hk ; k = 1, 2), We may take the index 1 to refer to the ase when devi e 1 a ts as sender, and the index 2 to refer to the ase when devi e 2 a ts as sender, everything else remaining similar. The elds have to satisfy the boundary onditions on the ondu tor surfa es whi h is why we have in general Z12 6= 0.18 If one integrates Eq.(209) over a spheri al volume ontaining the antennas and the devi es we get for the left hand side, by using Stokes' theorem, the surfa e integral 18 If the indexes 1 and 2 refer to dierent instan es, then Ek will be the total ele tri eld and the boundary ondition has to be applied to it. In this ase Jk will dier from zero not just for the sending antenna but also for the re eiving antenna be ause of the indu ed urrent. If the indexes refer to the same instan e but where J1 is the urrent of antenna 1 alone (and vi e versa for J2 ) then the boundary ondition has to be applied to the total eld E1 + E2 . F Re ipro ity theorem 82 Z Sr E1 × H2 · dS − Z Sr (211) E2 × H1 · dS whi h goes to zero as the radius r of the sphere goes to innity. The reason for this is that the eld at a far away distan e approa hes lo ally a planar ele tromagneti radiation for whi h H = ±k × E/(kη). Inserting this in Eq.(211) we see that the dieren e vanishes. Thus we get from Eq.(211) Z E2 · J1 dV = Z (212) E1 · J2 dV. If the antenna 1 a ts as a transmitter it emits a power PT = 1/2·ℜ(V1I1⋆ ) = 1/2 · ℜ(Z11 |I1 |2 ) from whi h we dedu e that Z11 = R1s + iX1 where R1s is the radiation resistan e of the antenna while X1 is alled its rea tan e, and mutatis mutandis for Z22 . If the re eiving antenna 2 is furnished with a load ZL the urrent I2 will be19 −Z21 I1 /(ZL + Z22 ) and the re eived power hen e given by PR = 1/2 · |Z21 I1 /(ZL + Z22 )|2 ℜ(ZL ). This attains the maximum ⋆ for a mat hed load ZL = Z22 in ase whi h we obtain the ratio of re eived to transmitted power as, PR |Z12 |2 = , PT 4R1s R2s whi h is, as seen, symmetri al in the in the indexes 1 and 2 if Z12 = Z21 . We will onsider the re ipro ity relation in a bit more detail using as surfa es of integration the surfa e S1 whi h follows the surfa e of the antenna 1+devi e 1 ex ept where it rosses a se tion of the the oaxial able (see the right part of Fig.(8) and [5℄), and the surfa e S2 dened mutatis mutandis for the se ond antenna + devi e. One thus integrates over a volume V with the boundary S1 + S2 whi h ex ludes the interior of the antennas+devi es and where we therefore have J1 = J2 = 0. One obtains then the re ipro ity theorem in the form Z S1 +S2 E1 × H2 · dS − R Z S1 +S2 E2 × H1 · dS = 0. (213) R Here, for instan e, the integral S2 E1 × H2 · dS be omes A2 E1 × H2 · dS where A2 is a ross se tion of the oaxial able interfa ing the antenna 2, 19 Indeed, the antenna resistan e has to be in luded in the total impedan e be ause the antenna reradiates also in the re eiving mode. The present analysis assumes that the distan e between the antennas is large ompared with the wavelength λ and thus Z12 small in omparison with Z11 and Z22 ; that is, the oupling between the antennas is weak  see further [6, se . 4.2℄. F Re ipro ity theorem 83 be ause elsewhere the ele tri eld is orthogonal to the surfa e meaning that E1 ×H2 ·dS = −H2 ·E1 ×dS = 0 . In this integral H2 is the eld in the oaxial able generated by the impressed urrent I2 while E1 refers to the eld in the oaxial able indu ed by the impressed urrent I1 in the antenna 1 via the transmitted eld impinging on antenna 2. In [5℄ the integration is only over S2 and the volume R V thus ontains the antenna 1 and we will therefore have a urrent term V E2 · J1 dV on the right hand side of Eq.(213). In [5, Eq.(15) and the pre eding one℄ I1 apparently refers to the urrent indu ed by the eld E1 in antenna 2 (a point left somewhat un lear in [5℄). Fig. 8: Sket h of the dipole antenna link to the oaxial able. On the right the arrows indi ate the ow of eld energy in the re eiving mode. The dipole ondu tors steers the ow along the ondu tor surfa e into the oaxial able. In the transmitting mode the energy ow is outwards from the oaxial able and dire ted to the surrounding spa e by the antenna surfa e. On the left there is a s hemati outline of the surfa e S of integration. The surfa e S follows the dipole antenna and the re eiver/sender along the ondu ting surfa es where the tangential elds are zero, ex ept where it forms the ross se tion A. For a TEM wave in a oaxial able we have the solution G Measurements of the reexions from the ground 84 I , 2πr I , Er (r) = η 2πr (214) Hϕ (r) = ± where I is the urrent of the oaxial able. The ele tri eld is thus radial and orthogonal to the magneti eld whi h is along the angular dire tion. R Inserting this solution into A2 E1 × H2 · dS we obtain Z η I1→2 I2 E1 × H2 · dS = (2π)2 A2   b η = Zc I1→2 I2 . I1→2 I2 ln 2π a Z a b 1 2πrdr = r2 Here a is radius of the inner wire of the oaxial able and b the radius of the outer ondu tor, while Zc denes the impedan e of the able. We use I1→2 to denote the urrent indu ed in antenna 2 by the eld radiated by R antenna 1 with the impressed urrent I1 . The evaluation of A2 E2 × H1 · dS will give the same result ex ept being of the opposite sign (this has to do with the orientation of the elds and the fa t that the impressed and indu ed elds go in opposite dire tions in the oaxial able). This leads nally to the relation 2Zc I1→2 I2 = Z V E2 · J1 dV. If the re eiver ir uit is oupled to a mat hed load RL = Zc then 2Zc I1→2 will orrespond to the indu ed voltage V2 = Z21 I1 . From Eq.(212) and repeating the above onsiderations for antenna 1 we obtain Z12 = Z21 . The indu ed voltage in antenna 2 an, a ording to these results, be written as20 1 V2 = I2 Z V E1 · J2 dV. (215) G Measurements of the reexions from the ground The measurement pro edures were designed su h as to obtain knowledge how the RSSI depends on the distan e the re eiver and transmitter, and on 20 When E1 refers to the total eld then the result orresponds to losed ir uit voltage, when E1 refers only to the in oming eld then the result orresponds to the open ir uit voltage. G Measurements of the reexions from the ground 85 the environment. The devi es were typi ally atta hed to wooden supports (poles) of height 1 m. was Power sour es were 9V batteries. The transmitter ongured to send 20 pa kets per transmission. The transmissions were inter epted using a snier devi e (Chip on Pa ket Snier) laptop omputer. The data was saved on the of the RSSI values, pa ket information and other data. the tests the onne ted to a omputer for later extra tion For a majority of hannel nr 20 was used and the power set to level 11 (0xA0B) orresponding to -11 dBm (see Tab.(3)). When al ulating the average RSSI value the maximum and minimum values were dropped in order to eliminate possible outliers. Alternatively we used the median value. The RF- ir uit was shielded by putting the whole PCB plus battery into a metalli an of 90 mm diameter and 40 mm height. PA level Output power (dBm) Register 31 0 0xA0FF 27 -1 0xA0FB 23 -3 0xA0F7 19 -5 0xA0F3 15 -7 0xA0EF 11 -10 0xA0EB 7 -15 0xA0E7 3 -25 0xA0E3 Tab. 3: In Power level assignments ase of a smooth ground the two-ray (mirror) model quite well to the data, see Fig.9. In the gure we (RSSI values) with data (power in dB) des ribed above with the diele tri an be tted ompare measured data omputed using the two-ray model permittivity ǫ set to 3 providing a good t. Measurement points were sampled more densely (∆r = 10 m) where we expe ted the hanges to be largest. In this ase the Brewster angle θB will ◦ be 30 . The Brewster distan e is about 3.9 m, and the breaking point is at r≈ 42 m. G Measurements of the reexions from the ground 86 -50 -55 -60 RSSI, dB -65 -70 -75 -80 -85 -90 0 Fig. 9: 5 10 15 20 Displacement(m) 25 30 RSSI and power dB vs distan e, measured data ( ir le) and data omputed using the two ray model (+) with ǫ = 3. The tops of the antennas were about 1.14 m above the ground (sand). At the time of ◦ measurement the temperature was -5 C and the ground was overed with a a 5 h1 (m) m layer of fresh snow. Correlation Error (stdev) 0.64 0.99 0.95 49.4 1.14 0.99 1.15 48.3 1.64 0.98 1.30 49.3 Tab. 4: Oset (mean, Statisti s of model-data Similar measurements were made with h1 µ) omparison = 0.64 m, 1.64 m, while h2 remained 1.14 m, with equally good ts between model and data. Tab.(4) shows the statisti s of tting the measured RSSI values to where P is the power al ulated from the model, and µ 10 · log(P ) − µ, is the oset value between measured RSSI and model power (in dB). The dieren e (error) G Measurements of the reexions from the ground 87 between the model and the data are seen to orrespond to about 1 RSSI unit when measured in terms of the standard deviation. Note that the data points are not evenly distributed but were hosen su h as to best over the pla es where the hanges in the RSSI were expe ted to be largest. Measurements were also made on a frozen river with a smooth i e of thi kness around 10 m. The results were modeled using the diele tri layer model treated in se tion 3.5, see Fig.(10). A hara teristi dieren e when ompared with Fig.(9) are the mu h deeper troughs of the interferen e pattern in ase of sandwi hed stru ture air-i e-water. In the model this feature is sensitive to the thi kness of the i e. -50 RSSI, dB -60 -70 -80 -90 -100 0 Fig. 10: 10 20 30 Displacement (m) 40 50 RSSI and power dB vs distan e measured on a frozen river. Measured data (+) and data omputed using the two ray model ( ontinuous) with two ree ting interfa es, with ǫ = 3 for i e (10 m thi k) and ǫ = 80 for water. The tops of the antennas were about 1.14 m and 1.16 m above the i e. H C- ode for extra ting RSSI values H 88 C- ode for extra ting RSSI values Below is an example of the C-implementation of the fun tion extra t() whi h has been used to extra t the RSSI values from the psd-les obtained with the Pa ket Snier. The C- ode will depend on what sort of information the devi es are programmed to relay besides RSSI (e.g. battery power level, sensor data). #in lude <s t d i o . h> #in lude < s t d l i b . h> #in lude <ansi_ . h> / ∗ S t r u t u r e used ∗ / stru t s RS S Iva l { int int // s i z e o f RSSI−a r r a y // a r r a y o f RSSI− v a l u e s n; ∗ RSSI ; }; typedef stru t s RS S Iva l t RS S Iva l ; / ∗ e x t r a t d e f i n e s a f u n t i o n whi h opens a b i n a r y psd − f i l e and r e t u r n s t h e RSSI v a l u e s ∗ / har t RS S Iva l e x t r a t ( ∗ fname ) { i , j = 0; temp ; dummy [ 1 2 1 ℄ ; x=0 , y=0; temppu ; t RS S Iva l data , e r r ; int unsigned har unsigned har int int err .n = 0; e r r . RSSI = 0 ; f p = f o p e n ( fname , " r+b" ) ; data . n = 0 ; H C- ode for extra ting RSSI values if ( fp == NULL) err . n = return 89 { − 1; err ; } f r e a d (&temp , s i z e o f ( har ) , 1, fp ) ; s i z e o f ( har ) , 1, fp ) ; x=temp ; f r e a d (&temp , y=temp ; y=y <<8; // number o f p a k e t s y=y+x ; if (y < 1){ err . n = return − 2; err ; } d a t a . RSSI for ( i = = 0; ( i int ∗ ) < y; allo (y , s i z e o f ( int ) ) ; i ++) { f r e a d ( dummy , f r e a d (&temp , s i z e o f ( har ) , s i z e o f ( har ) , 26 , 1, fp ) ; fp ) ; temppu=temp ; i f ( temppu ! = 0 ) { i f ( ( temppu &0x 4 0 ) > 0 ) { temppu=temppu &0 x 7 f ; temppu=temppu ^0 x 7 f ; temppu=temppu + 1 ; −45−(temppu ) ; temppu = d a t a . RSSI [ j ℄= temppu ; j ++; // i n r e m e n t s t r u e // number o f RSSI v a l s } el se { temppu=temppu &0 x 7 f ; −45+(temppu ) ; temppu = d a t a . 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