RL-871 Complementary Reciprocity Theorems For Two-Port Networks and Transmission Lines C.T. Tai Radiation Laboratory Department of Electrical Engineering and Computer Science The University of Michigan Ann Arbor, Michigan, 48109 Abstract 'Two complllementary reciprocity tlleorenmis have been formulated in this work, one for two-port passive networks and another for transmission lines. The theorelms involve two networks or two identical sections of lines with different load imnpedances which must satisfy a conmplemlentary impedance condition relating to the cllharacteristic impedances of the network or thle characteristic impedance of a line. 1 Introduction In electromagnetic theory there are two well-known reciprocity theorems, one due to Rayleigh and Carson and another due to Lorentz. Recently, we encountered a field problem which requires a new reciprocity theorem in order to provide for tle answer. In this paper, we will present two simpler versions of that theorem; one applies to a pair of two-port networks and another to two identical sections of lines with different terminations. RL-871 = RL-871

2 Two-Port Networks A two-port passive network can always be represented by a T-network as shown ill Fig. 1 (a) with a shunt impedance Z., and two series impedances Z1 and Z,. The terminals at tile left side of the network will be identified as the left terminals and the others as the right terminals. Now if the left terminals are driven by a voltage kV. and the right terminals are connected to a load impedance ZR as shown in Fig. 1 (b), it is well known that the transfer function i/Va of that circuit is equal to the transfer function ib/1V of the same circuit with the roles of IVl and i,, replaced by Vb and ib as shown in Fig. 1 (c). This is the famous reciprocity theorem due to Rayleigh. It was later extended to field theory by Carson in the form I J ~ Ja bdV = j Jb EadV (1) J a Vb whlere Ja anld Jb denote two current sources in volume Va and Vb, respectively, whicl are responsive for producing the electric fields Ea and Eb in an identical environment containing isotropic media including the presence of electrically perfect conducting bodies in one of the media or in all media. When this theorem is applied to the networks shown in Fig. 1 (b) and (c), one can readily derive the reciprocity relationship ia/Va ib/Vb. (2) For the network under consideration, it is known that if a current source Ia is connected to the left terminals of that network and the right terminals are 2

still connected to a load impedance ZR with a load current ia, then the current transfer function ia/la is not equal to the transfer function ib/Ib when a current source lb is connected to tile right terminals in parallel with the same load impedance ZrI, and tile left terminals are short-circuited resulting in a current ib. In otlier words, for a single network, la/la # ib/Ib (3) or the current transfer fiunction is non-reciprocal. A reciprocity theorem for the current transfer function, however, can be formulated if we invoke two colmplementary networks as shown in Fig. 2 (a) and (b) where in network (b) tile locations of Z1 and Z2 have been interchanged and another load imlledance Zb is connected to tlle right terminals which is, in general, different from Za, the load impedance in network (a). By a straightforward linear network analysis it can readily be shown that the currents in these two networks satisfy tlle reciprocity relationship ia/Ia = ib/Ib (4) under the condition Za Z = Zc Zc2 (5) where Zc1 and Zc ( denote, respectively, the characteristic impedances of the T-network looking from the opposite terminals. They are given by 3

Zl =2 (Z -{ Z2) )+[(Zi + Z2)(Z1 + Z2+4Zm)]} (6) Zc2 2 {(Z - Z1) + [(Zi + Z-) (Z+ Z + 2 4Zm)]} (7) hence ZclZc2 = Z1Z2 + (Z1 + Z2) Zm (8) Equation (1) is designated as the complementary reciprocity theorem for the current transfer function, or (I-i)c theorem for short, in contrast to the V-i reciprocity llheoremn of Rayleigh-Carsonl for a single network. Equation (5) is designated as the comll)lenlentary inlpeclance condition. It. can be shlown\ thllat the product of the two characteristic impedances is also equal to ZS1ZQo or Z.sZ01, where Zsl,Zs2, and Z01,Z02 denote, respectively, the impedances looking into one pair of terminals of the T- network when the opposite terllillnals are either short-circuited or open-circuited. For a slymmetrical T-network, Z1 = Za, Z,1 = Z,2 = Z, = [Z1 (Z1 + 2Z2,,)], (9) and the physical configuration of the two complementary networks becomes identical, meanwllile, the impedance condition reduces to ZZb = Z2 (10) Only under the very special case corresponding to Za = Zb = Zc, that the two networks, includilg the load impedance, merge to one single entity. In 4

general, whllen network (a) and its load impedance Za are known or given-l we cani construct the miriror image of that network to form network (b) andl(l its load impedal-ice is delermlllilned by (5). As an illustration, we let Z - () 0, 1ill ia - ZinIal/ (Z — + Z,,, ). According to (5), we mlust have Zb 0 c, IIceC(, t}li' (I-i)0 theorem yields ib - iaIb/Ia = ZnIb/(Z2 + Z,,,) which is certainly true by inspecting the circuitry of network (b). The (I-i) theorem can be extended to cascad(le networks wXitllout llcllh difficulty. The above results call also be obtained by using a I-network to represenWt a twvo-port network. The twxo complementary 11-networks tllell have tlhe collfi,urations a.s sllown ill Fig. 3 (a) and (b). Inl terms of the admnittallce fu ltictes Yl, Y and yMI, hlichll are iiot equal to the reciprocals of Z1, Z.), and Z,,, we fi ild that the characteristic admittallce of thle network are given b)y I1 I /Z, l - {Y1 - Y - + [(y + Y2) (YI + Y- + 4y-,,,)]} (l l),; I = - '5 '- {Y2 - Yi + [(Y' + Y2) (YI + Y2 + 4ym)]} (1 2) then ~cl52 2- 1/ZZclZc2 = YIY2- + (Yl + Y2) Ym (13) Equation (5) can then be changed to anll equivalent form in terms of the admittance functions, niamely, YaY6- 1/ZaZb =l-cl2 (14) These results, of course, canll be obtained (lirectly by applying the duality prillncipl)e in network theory. 5

By taking a variation of (14) one finds 6Ya = (Ycl Y2) 6Zb or 6Zb (Zc Zc2) 6Ya =Z ZZc,26 (1/Za) that means when the load admittance of circuit (a) is increased by 6Ya, the load impedance of circuit (b) must be increased by an amount sucll that the two increments also satisfy the complementary impedance condition. It may be of some interest to remark that the two networks shown in Fig. 2 (a) and (b) within t.he boxes were used by Guillemin [1] t.o define two iterative impedances wheni they are connected in cascade. It call be shown tllat the product of the tw.o iterative ilmpedances is equal to the product of the two characteristic imrpedances of the individual network given by (8). Simlilar relation holds for the product of the iterative admlittances and that, of the characteristic admittances. For a. symmnetrical network (Z1 = Z,), we have only one characteristic impedance function and one iterative impedance function which are equal to each other. 3 Transmission Lines The complementary reciprocity theorem for two identical sections of transmission lines of length 'd' extending from z = 0 to d and with line constants L and C is based on two models slhown in Fig 4 (a) and (b). Line (a) is terminated at its t.wo ends by Za and Z' and Line (b) by Zb anld Zb at the corresponding ends. The lines are excited by two distributed current sources KIa(x) and I7\b(x) 6

are solutions of the equiationls (i.~.L) -, ~L 7,a(x) (15) d.V (Il~(r)- w~va(x) + K'a(x) (6 dx and two sim-ilar equations for tVb(XI), 'b(X), and Ab(X ). The boundary condition are ByN uisiul" tel( equa~tionis foi[ the line volta~ges anid currents itcaii be shown that [1'a(xI)16b(X) -?I(,X)Vb(X)/J I\JJl)*)+ KAt(X)i,(X) (19) Where Z, (LC d,(eiotilig the, characteristic impedance of the linies. An in-tegration of' (19) froii x: U- to d vil(,ds [Ka(X-) Ib(x ) ~ K14(x)i0,(x) ]dx [(3'(x'it(X) - V~a ( X )Vb(X)Z 0 (20) Ini view of the boundarv conditioi s stated by (17), and (18), we can put (20) in the fori n / [I- (XI( x) + f\I, (x ) 2a(xi (Ix 7

Now if xve impose the col(litiolls Z.Zb= Z- (22) anld ZZ> = Z: (23) simultaneously, (21) reduces to d (24) (/ [A'(x )i1,(x) + I\b(x )ia(X)] Cdx = 0 The above formula is a statelle'lit of the complementary reciprocity theorem for thle two lines, or (l - i),. thlleoreil for short, under the condition that their telrminlal iilll)tedaces sat isl'y tle coiln )lemlentary inmpedance conlitions stated by (22) and( (23). If'. Z,, all '1 ( tare givcen or known, these two equations can be used t.o (leterllille Z1, /alld XZ. Solme sl)ecial cases slholtl1( be pioilited out,. W hen d oo, the two sections of line become semli-inlfillitely loIlg. Thle case is also equivalent to z; = Z = Z, (25) \When Line (a) is short-circuited at both ends (Za = Z, = 0) then Line (b) must be open-circuited at b)otl eiids (Z, - o, Z -- oo). Other combinations can be easily visualized. These condlitions demonstrated very clearly the significance of the complemlentary stattus of the two lines. The two complementary lines under discussion are quite cliflerelt fromi the two circuits considered by Van Bladel [2] in his analysis of tlhe svillnetrical property of some scattering matrices in 8

t e theory of wave i(les. Il colntrast-, our complementary reciprocity theorem can l)e lused tlo ilvestigatl( (Il(e syliletrical property of tile scalar Green functionlls relatillg to the c iirret on) a transmission line without finding the explicit exp)lressiois of the (ree in l o l tioIs. \Whel(l thle c(irren'll sOlrce's onl the lines are localized we can write A'x() -, (a,) (x - Xa) (26) N1,(.') I, (.,) 6 (x - Xb) (27) w'llere (a' (. - a.) (eo'lltotes l ( delta ifunlction (lefined at x = x, and similarly for ((.- -'I,)..)Stiistlit ii iit" tI '.e two expressions into (24) we obtain the circuit -,l (,-), (a. ) = -I (Xb)ia (b) (28) 'I lie circit lrelaltioll for sillgle section of line terminated by any two load inpeCdalces at, the eilds, (derinva)le from an application of the Rayleigh-Carson recip)rocitv theorei l, \vwold be (a ( 1), ),b (,1e) lb (Xb) Va (X,) (29) where t( ('a,) alnd tj, (x./,) (lelote two line voltages measured across the lines. Thllis shows tlie difference of thle two reci)procity theorems both in the context and in the form Ilatiolls. As an exalil)le for t.le application of tlie complementary reciprocity theorem stated by (28), letl its coisider t.wo semi-infinite lines or with Z' - Zb = Zc and a 9

Line (a) is shor-t- cr(ireitedi at, x — 0 (Za, 0) that merians Line (b) must be opencircuitei at x — 0 (/b - -,cc). A localized current source with unit ami-plitude is How,;:tppiied to Line (a) ait r'. The solution for ia(x) obtainable from the theory of' transnii-~ssoij lilies [81,is i1sin kxlecx X > xi - (30) <ACI~x' cos kx, x <x By, nea(,Iis of (2hj, we find tha1-~t If' i a 1 uit. cenrrent source is applied to Line (b) at x, tile C1t rill;II at w1 1( 1~ 10.sI Isin kxlekx x > ii - ~(31) CA c os kx, x~ x lb\ ill ereha iilin".11 a 1(1 V in (81), we ohta~in the solution for i'b(x) when a unit ~l rnt Sllic i a) lidto thlit. Ii 1at~'lamely, L c. kx,(32) Ill othler wor-ds, it. I ~ eesayt ov ix) as a separa~te problem once tue1 soILution1 for1 tije eOlllj)ieilienta-ry problem- is kniown or vice versa. Both (30) and (32) a iso shlow ce~iea tlivIa. thei current, transfer function for a single line is Thie corn jiplel iwewtl eipoct theor-ems presented inthis paper represent thle cir-cuit. ver'Si1n aIld til 0iie-dliinensiona~l version of a more complex theoremr for a t.1h ree-di niellsiolil;1 elect roina-gnet ic field wvith multiple layers of isotropic mi-edia backed 1Ly a coiidnct~iig b~ody. In appearance, the theorem has a rather to

simplle form, nainlk,-I hu m -1- HI3dV IiIJ B -fI dV' (33) wxhere JA aii(l JJ demmote two electric cii rr1emit djensi ty fu nctions iii two comnplen-entary iimodels like Ka mi1 Ik' iii the t ranusmission flne theor~, occupying, reslpectiVelV, voIl mme" V'4 and V13. H-, ~ an(l "1Li dlenote the mnagmietic fiel(ls prodUcedl by thlese two criirriits ill two complodillemit arv (ii1vironnlienits, Corresponiding to 1U, and it, ill the t rm I snils-siom Iile i lo(l( Is,. 'Ille p roof of (.33 ) lowever, Is m noi moore in vol vedl. 1wwork will be reportedl elsew here together wIt hi ai a pp 1cationl of that. theorem to a Lou ndarv-vahuie prohleim inl elect romnlagmiletics. A, part, of the presenlt. paper is ihase(l oil a tech ii cal rep~ort, onl the1 compilemenetary reciprocity thicorellis Ill i-lect roiiuigiiet-i thleory [1], availalde np)on requiest. Inl conlnsiw-onl It h101( e eipaieltihat thle compjlemneltary reciprocityV thle-orem-1 fom' the t wo-portI iet works nid(- thlit for thle.Nraisnili~siomi hiues are, quit~e distinct, fronm t. lie B ave ighi- (<arsoi thleorome. The li tter Involves only a single network Or' a siieSecou ofIi mOF 11 vl wile, the niew theorem-l-s apply to two comnpienme~nta~ry clircuits and( tw%\o id-entical sec tions of hune wvitl dlifferent. load impedances Whic~h are related t~o each othec-r iii termis of the characteristic impeclances of the struct~ur15. The auother i's inm(lelbt-e(l t~o Prof. A Ian B3. la-cnee for calling attention to the identi tv betweemi the prod mects of' thle iterative i impedances and that of the characteristic imipedanices of- a. tw\o- piort. un1svm-rirneitric netwvork. H~e is m-ost grateful t~o Ms. Boo iie lKidd for the lpreparation of this manuscript. 11I

References [1] E.A. G uillcmii, (Coin i nuiiiicatioii Networks, V~ol. II, John Wiley, Newv York, NY, 1935, j). 166. [2] J. vaii Bla~del, E.lect.romaiioiit.ic Fields, McGraw-Hill, New York, NY, 1964, p.436. [3].1. Li [va di c ( oci s liii F(tctio() [In i i lect roi 1agie. i0c Ilicorv, I ntext, Puhi lishers, scraiinoii, PA I 1971. ( lhiPvil 3. [4I] (2. 'I. ''li, Coiplieiin ecijpiocitv tlieor~eiis InI electroiiagciietic theorv,l 'l(ch~l incal Rep orit P H 69U81ad.io im Lihora~tory I1)ejpartmenet of 1Ele'-ct rl cal L"iigiiieeriiit nid (1 Coinpiier Science, The University of' Michigani, Ann Arhor, NI I, Seiiii1991 12

K -C K -z K N: -C 0 ~1 '- 0 ___ 0 C ______ t LN I I H I I - ____ - IK I -C C0 -C N 0 U - 0 0 K 0 0 K H 0 C H __ 0 ___ CN d K1 C

- ia L, ib la IY Ya L. Y2 (a) Fig. 3. Two complementary n-networks. (b)

Ka(x) ia(x) KaI I a I ZaI x = 0 ) Zc n~" x=d x = d (a) A section of transmission line terminated by two load impedances Za and Z'a. ib(x) Kb(x) r- w Zb4 Z ) Zb x=d x-=0 (b) An identical section of line terminated impedances Zb and Z'b. by two load Fig. 4. Two sections of line with the complementary impedance conditions ZZ = Z c2, Za Z = Zc Z7-Z =Z4, ZalZb=Zc -