High-frequency electromagnetic field coupling to long loaded non-uniform lines: an asymptotic approach

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1 CHAPTER 5 High-frquncy lctromagntic fild coupling to long loadd non-uniform lins: an asymptotic approach S.V. Tkachnko, F. Rachidi 2 & J.B. Nitsch Otto-von-Gurick-Univrsity Magdburg, Magdburg, Grmany. 2 Ecol Polytchniqu Fdral d Lausann, Lausann, Switzrland. Abstract In this chaptr, w prsnt and validat an fficint hybrid mthod to comput high-frquncy lctromagntic fild coupling to long loadd lins, whn th transmission lin approximation is not applicabl. Th lin can contain additional discontinuitis (ithr a lumpd impdanc or a lumpd sourc) in th cntral rgion. In th proposd mthod, th inducd currnt along th lin can b xprssd using closd form analytical quations. Ths xprssions involv currnt wavs scattring cofficints at th lin non-uniformitis, which can b dtrmind using ithr approximat analytical solutions, numrical mthods (for th scattring in th lin nar-nd rgions), or xact analytical solutions (for th scattring at th lumpd impdanc in th cntral rgion). Th proposd approach is compard with numrical simulations and xcllnt agrmnt is found. Introduction Th prsnt study considrs th important lctromagntic compatibility (EMC) problm of high-frquncy lctromagntic fild coupling to long transmission lins (TLs). W assum that th frquncy spctrum of th xciting fild and th transvrs dimnsions of th lin ar such that th TL approximation is not applicabl. To solv such a problm, on gnrally rsorts to th us of numrical mthods (.g. mthod of momnts) basd on th antnna thory. Howvr, a pur numrical mthod allows to hav a gnral physical pictur of th phnomna, only aftr larg sris of calculations. Morovr, a systmatic us of such mthods usually nds prohibitiv computr tim and storag rquirmnts, spcially whn analyzing long transmission lins. doi:.2495/ /5

2 6 Elctromagntic Fild Intraction with Transmission Lins Exact analytical xprssions of inducd currnt hav bn dvlopd for th cas of infinit ovrhad lins [, 2]. Using thos xprssions, it has bn shown that corrctions to TL approximation can b considrabl undr crtain circumstancs. In [3, 4] (s also Chaptr 4), a systm of quations is drivd undr th thinwir approximation dscribing th lctromagntic fild coupling to a horizontal wir of fi nit arbitrary lngth abov th ground plan. Th drivd quations ar in th form of tlgraphr s quations in which th lctrodynamics corrctions to th TL approximation appar as additional voltag and currnt sourc trms. Basd on prturbation thory, an fficint itrativ procdur has bn proposd to solv th drivd coupling quations, whr th zroth itration trm is dtrmind by using th TL approximation (s Chaptr 4). Howvr, this mthod (as formulatd in [3, 4]) dos not tak into account th vrtical lmnts and loads of th wir. Anothr disadvantag of th mthod is th divrgnc of th prturbation sris for high-quality-factor systms (.g. a horizontal opn-circuitd wir) nar th rsonant frquncis. In this chaptr, w considr th cas of a long lin, xcitd by a plan lctromagntic wav. Th lin can b trminatd at ach nd by arbitrary impdancs (with vrtical lmnts), a configuration that rquirs taking into account th intractions btwn wir sctions with diffrnt dirctions. Th lin can contain an additional discontinuity (in th form of a lumpd impdanc or a lumpd sourc) in th cntral rgion. Th proposd hybrid mthod to comput high-frquncy lctromagntic fild coupling with a long lin will b basd on th spcific faturs of wav propagation along long wirs, as dscribd nxt. An xciting plan wav gnrats fast currnt wavs (i.. with phas vlocitis largr than th spd of light) along an infinit straight wir paralll to a prfctly conducting ground. This currnt wav radiats uniformly along th lin [5 ]. Whn th homognity of th lin is disturbd by a discontinuity (lumpd impdanc or lumpd sourc, vrtical lmnts of th lin, bnd, tc.), th currnt distribution nar th discontinuity bcoms mor complx, involving diffrnt propagation mods, namly transvrs lctromagntic (TEM) mods, laky mods (which ar attnuatd xponntially with th distanc), and radiation mods (which ar attnuatd as /r n, r bing th distanc). Th xact analytical solution of this problm is known only for th cas of a lumpd sourc (lumpd impdanc) in an infinit wir (s,.g. [ 3]), or for th cas of a smi-infinit opn-circuitd wir [4]. At distancs larg nough from th discontinuity, in th so-calld asymptotic rgion, only TEM mods surviv, which propagat along th lin without producing any radiation [8, 5, 6]. Th amplitud associatd with th TEM mods can b xprssd in trms of scattring cofficints. Ths TEM mods, in turn, will scattr whn raching lin discontinuitis, nar which, again diffrnt typ of mods will b prsnt; howvr, nough distant from th discontinuitis in th lin asymptotic rgion, th only surviving mod is TEM, and th scattring procss can b dscribd by th rflction and transmission cofficints. To obtain th global solution, w hav to considr th scattring associatd with ach lin

3 High-Frquncy Elctromagntic Fild Coupling 6 discontinuity and join th solutions in th asymptotic rgion(s). As a rsult, w will obtain a closd form analytical xprssions for th inducd currnt along th lin [5, 6, 7, 8]. Ths xprssions involv scattring cofficints for diffrnt typs of currnt wavs on th lin discontinuitis, which can b dtrmind using ithr approximat analytical solutions, numrical mthods (for th scattring in th lin nar nd rgions), or xact analytical solutions (for th scattring nar a lumpd impdanc in th cntral rgion [8]). Th proposd approach will b compard with numrical simulations. 2 High-frquncy lctromagntic fild coupling to a long loadd lin 2. Asymptotic approach Considr a long currnt filamnt of finit lngth abov a prfctly conducting ground, in prsnc of an xtrnal plan wav (s Fig. ). Th lin is loadd at its trminals by impdancs Z and Z Solution for th inducd currnt in th asymptotic rgion Th xact spatial dpndnc of th inducd currnt can b dtrmind by th solution of th on-dimnsional (thin wir approximation) Pocklington s lctric fild intgral quation [9]. Th xamination of this quation for long lins Figur : Gomtry of th long trminatd lin xcitd by an xtrnal plan wav.

4 62 Elctromagntic Fild Intraction with Transmission Lins (L >> 2h) has shown that th currnt distribution along th lin may b dividd into thr rgions as illustratd in Fig. [5]. Rgions I and III ar locatd nar th trminal loads. Th main rgion II is constitutd by portions of th wir sufficintly far from th trminations, i.. l bound << z << L-l bound. In this cntral rgion, calld hraftr th asymptotic rgion, th influnc of lctromagntic filds arising from th load currnts may b nglctd in comparison with th filds gnratd by th currnts along th wir [5]. Th valu l bound dpnds rigorously upon th mods gnratd nar th lin discontinuitis, i.. lumpd loads and vrtical lmnts. Howvr, for most cass of practical intrst whn kh <, a valu l bound qual to about 2h can b adoptd. Thrfor, w postulat that th gnral solution for th currnt along th asymptotic rgion can xprssd as a sum of thr distinct trms I( z) = I xp( jk z) + I xp( jkz) + I xp( jkz) () 2 whr k = w/c, and k = k cos q, whr q is th lvation angl of th incidnt fild (th azimuth angl j = ). Th first trm is a forcd rspons wav, which corrsponds to th inducd currnt on an infinitly long wir. Th scond and th third trms ar positiv and ngativ travlling wavs and th cofficints I and I 2 dpnd upon th rspctiv gomtric wir configuration and loads at th two lin trminals. Th cofficint I of th forcd rspons wav is dtrmind from th solution of th Pocklilngton s quation for th cas of an infinitly long wir [9] w z (d / d z + k ) g( z z) I( z )d z = 4 π j E ( h, z), ka << (2) in which I(z) is th inducd currnt, and g(z) is th scalar Grn s function givn by gz ()= jk z + a jk z +4h (3) z + a z +4h Th trm E z ( h,z) is th tangntial xciting E-fild at th lin hight, which, for th cas of a vrtically polarizd plan wav is givn by i jk z 2jkhsinq z q z jk z E ( h, z)= E ( )sin = E ( h, jw) (4) Th analytical solution for th cofficint of th forcd wav for th cas of a vrtically polarizd xciting fild is givn by [2] (s also Sction 4.2.3) I 4 π cez ( h, jw) 2 (2) (2/ ) π π =, ka<< jwh sin q 2 ln g kasin q j + j H 2khsin q ( ) In ordr to dtrmin th cofficints I and I 2 of th positiv and ngativ travlling wavs for an arbitrary frquncy of th xciting fild, it is ncssary to know th xact solutions for th inducd currnt in rgions I and III, which may b (5)

5 High-Frquncy Elctromagntic Fild Coupling 63 obtaind by solving Pocklington s quations in ths rgions using a numrical mthod (.g. mthod of momnts). It is worth nothing that Pocklington s quation uss as sourc trm th tangntial componnt of th xciting lctric fild along th wir and along th conductors of th load impdancs. To obtain th cofficints from th numrical solutions nar th lin nds, it is ncssary to considr an intrmdiary stp, which consists of dfining two smiinfinit lins as shown in Fig. 2. (Th smi-infinit lin configurations will also Figur 2: Gomtry for th original lin (a), th right smi-infinit lin (b), and th lft smi-infinit lin (c).

6 64 Elctromagntic Fild Intraction with Transmission Lins allow us to obtain analytical xprssions for th inducd currnt nar and at th two xtrmitis of th long lin, as w shall s in Sction 2..2.) Th right smiinfinit lin xtnds from th lin lft-nd to + (Fig. 2b), and th lft smi-infinit lin xtnds from to th lin right-nd (Fig. 2c). Th gnral solution to this problm can b xprssd as a linar combination of th solutions for non-homognous (with xtrnal fild xcitation) and homognous (no xtrnal xcitation) problms. Th non-homognous solution for th right smi-infinit lin ( z < ) and for lft smi-infinit lin ( z ) can b xprssd rspctivly as in which and + + I ()= z I Ψ () z (6) I ()= z I Ψ () z (7) xact solution in th rgion, z l Ψ+ ()= z xp( jk z ) + C xp( jkz ), z >> l Ψ bound + bound xact solution in th rgion, lbound z ()= z xp( jkz ) + C xp( jkz ), z << lbound In qns (8) and (9), C + and C ar scattring cofficints, which dpnd on th frquncy and th angl of incidnc of th xciting lctromagntic fild C + = C + (k,q), C = C (k,q). Th homognous solution (no xtrnal xcitation) is givn by in which and + + (8) (9) I ()= z Ψ () z () I ()= z Ψ () z () xact solution in th rgion, z l Ψ+ ()= z xp( jk z ) + R+ xp( jkz ), z >> l Ψ bound bound xact solution in th rgion, lbound z ()= z xp( jk z ) + R xp( jkz ), z << lbound (2) (3) In qns (2) and (3), R + and R ar th rflction cofficints, which dpnd on th frquncy (R + = R + (k), R = R (k)). It is important to not that both rflction cofficints and scattring cofficints ar indpndnt of th lin lngth. Cofficints I and I 2 can b xprssd as a function of th cofficints C +, C, R +, and R by considring that th inducd currnt in th asymptotic rgion of th

7 High-Frquncy Elctromagntic Fild Coupling 65 initial lin is idntical to th currnt in th asymptotic rgion for th smi-infinit lins (s Fig. 2). Th gnral solution for th currnt in th right smi-infinit lin is givn by th sum of homognous and non-homognous solution I ()= z I Ψ ()+ z I Ψ () z (4) in which I and Ĩ ar constant cofficints. In th asymptotic rgion of th lin, using qns (8) and (2), th abov solution can b writtn as I+ ( z >> lbound ) = I xp( jkz) + IC+ xp( jkz) + I xp( jkz) + I R+ xp( jkz) = I xp( jk z) + I xp( jkz) +[ I C + I R ]xp( jkz) (5) + + Similarly, th gnral solution for th currnt in th lft smi-infinit lin can b writtn as th sum of non-homognous and homognous solutions, but considring appropriat argumnt shifts causd by th nw coordinat origin which is shiftd by a lngth L (s Fig. 2c) I ( z) = I xp( jk L) Ψ ( zl) + I Ψ ( zl) (6) 2 Also in th asymptotic rgion, w hav I( z << lbound ) = I xp( jkz) + IC xp( jkz jkl jkl) + I xp( jkz+ jkl) + I R xp( jkz jkl) 2 2 = Ixp( jkz )+ IC xp( jkl jkl ) + I 2R xp( jkl) xp( jk z) + I xp( jk z + jkl) (7) 2 As it can b sn by qns (5) and (7), th solution in th asymptotic rgion is givn in th form of th proposd thr-trm approximation (). By imposing that th cofficints for th trms xp(jkz) and xp( jkz) ar idntical in qns (5) and (7), w obtain a linar systm for th unknown cofficints Ĩ and Ĩ 2, which yilds th following solutions: I = I I = I 2 C xp( jkl jkl) + C+ Rxp( 2 jkl) RR xp( 2 jkl) + C+ xp( jkl) + C R+ xp( 2 jkl jkl) RR xp( 2 jkl) + (8) (9) And, thrfor, th cofficints I and I 2 in qn () ar givn by I = I = I C xp( jkl jkl) + C+ Rxp( 2 jkl) RR xp( 2 jkl) + (2)

8 66 Elctromagntic Fild Intraction with Transmission Lins I = I xp( jkl) = I 2 2 C+ + C R+ xp( jkl jkl) RR xp( 2 jkl) + (2) In this way, th inducd currnt in th asymptotic rgion II is xprssd analytically by qn (), in which th cofficints, I, I, and I 2 ar givn by qns (5), (2), and (2). Th cofficints I and I 2 ar xprssd in trms of th asymptotic cofficints C +, C, R +, and R, which ar indpndnt of th lin lngth (for a lin lngth largr than a fw tims its hight) and ar charactrizd only by th currnt scattring nar th loads. For simpl lin trminal configurations such as an opn-circuit without vrtical lmnts, ths cofficints may b obtaind analytically using th itration mthod prsntd in Chaptr 4. For th gnral cas of arbitrary trminal loads, ths cofficints hav to b dtrmind numrically (using th mthod of momnts, for instanc). Sinc th asymptotic cofficints ar indpndnt of th lin lngth, thy can b valuatd using th numrical solutions for significantly shortr lins. In this way, th proposd mthod maks it possibl to comput th rspons of TL to xciting lctromagntic fild with a rasonabl computation tim, rgardlss of th lin lngth. In ordr to dtrmin th scattring cofficints, it is indispnsabl to considr two lins bcaus w hav to dtrmin four unknowns (C +, C, R +, and R ), and for ach lin w hav a st of two qns (2) and (2). Starting from qns (2) and (2), writtn for two lins with similar configurations (by which w man th sam wir radius, hight abov ground, trminal impdancs, and xciting lctromagntic fild), but with significantly shortr lngths L and L 2, th following xprssions for th scattring cofficints C +, C, R +, and R and can b drivd (s Appndix ) C R C + R + I2( L2) I2( L) = I ( L ) I ( L ) 2 I2( L) I( L2) I2( L2) I( L) = I I ( L ) I ( L ) 2 I( L2)xp[ j( k+ k) L2] I( L)xp[ j( k+ k) L] = I ( L )xp[ j( k k) L ] I ( L )xp[ j( k k) L ] I( L) I2( L2)xp(2 jkl) I( L2) I2( L)xp(2 jkl2) = I I ( L )xp[ j( kk ) L ] I ( L )xp[ j( kk ) L ] (22) (23) (24) (25) 2..2 Exprssion for th inducd currnt at th lin trminals (rgions I and III) Starting from numrical solutions for th two short lin configurations I(z,L ) and I(z,L 2 ), it is also possibl to driv analytical xprssions for th currnt at th trminals of th original lin. Th solution in th lft-nd rgion (rgion I in Fig. ) for th two short lins can b xprssd as (from qn (4)) I (, z L )= I Ψ () z + I ( L ) Ψ () z (26) + + +

9 High-Frquncy Elctromagntic Fild Coupling 67 I (, z L )= I ()+ z I ( L ) () z (27) + 2 Ψ+ 2 Ψ+ Not that in th abov two quations, th lngth dpndnc is containd only in th cofficints Ĩ (L ) and Ĩ (L 2 ), which can b calculatd by qn (8). From ths two quations, it is possibl to infr th functions Ψ + (z) and Ψ + (z). Aftr som straightforward mathmatical manipulations, w gt Ψ Ψ IzL (, ) IzL (, ) 2 + ()= z I ( L2) I ( L) IzL (, ) I ( L) IzL (, ) I ( L) ()= z I I ( L2) I ( L) (28) (29) Insrting th rlations (28) and (29) into qn (4) and considring that I = Ĩ, w gt th solution for th inducd currnt in th lft-nd rgion IzL (, ) I( L) IzL (, ) I( L) IzL (, ) IzL (, ) I (, z L) = + I ( L) lft nd I( L2) I( L) I( L2) I( L) (3) Similarly, in th right-nd rgion (rgion III in Fig. ), th solution for th two short configurations can b xprssd as (from qn (6)) I ( z, L ) = I xp( jk L ) ( z L ) + I ( L ) ( z L ) (3) Ψ 2 Ψ I ( z, L ) = I xp( jk L ) ( z L ) + I ( L ) ( z L ) (32) 2 2 Ψ Ψ 2 Again in th abov two quations, th lngth-dpndnc is containd only in cofficints Ĩ 2 (L ) and Ĩ 2 (L 2 ), which can b calculatd by qn (9). Aftr som mathmatical manipulations, it is possibl to intr from ths two quations th functions Ψ ( z) and Ψ (z) Ψ xp( jk L ) I( z + L, L ) xp( jk L ) I( z + L, L ) ()= z xp( jkl) I 2( L) xp( jkl2) I 2( L2) Ψ Iz ( + L, L) I ( L) Iz ( + L, L) I ( L) xp( ) ( ) xp( ) ( ) ()= z I jkl I 2 L2 jkl2 I 2 L (33) (34) Insrting th xprssions (33) and (34) into qn (6) and taking into account that I 2 = Ĩ 2 xp(jkl), w gt th solution for th inducd currnt in th right-nd rgion I (, z L) right nd xp( jkl ) I ( L ) I( z L + L, L ) xp( jkl ) I ( L ) I( z L + L, L ) = xp( jk L) I ( L ) xp( jk L jkl ) I ( L ) xp( jk L jkl ) xp( jk L ) I( z L + L, L ) xp( jk L ) I( z L + L, L ) I ( L) xp( jkl) 2 xp ( jk ( ) L ) I ( L ) xp( j( k k) L ) I ( L ) k (35)

10 68 Elctromagntic Fild Intraction with Transmission Lins 2..3 Summary of th proposd procdur to dtrmin th inducd currnt along th lin and at th lin trminals Th procdur for th dtrmination of th cofficints of th analytical xprssions for th inducd currnt along a long lin (Fig. 2a) can b summarizd as follows:. Apply a numrical mthod (.g. mthod of momnts) to comput th rspons of two quivalnt lins having th sam configuration as th initial lin, but with shortr lngths L << L and L 2 << L. Typically, it is nough to considr L qual to about 5h and in ordr to avoid numrical instability, it is dsirabl to tak L 2 frquncy dpndnt, for xampl L 2 = L + λ/2. Starting from th numrical solutions for th inducd currnt on th two abov-mntiond lins, w dtrmin th cofficints I (L ), I 2 (L ), I (L 2 ), I 2 (L 2 ) using th last-squar mthod. 2. Th scattring cofficints C +, C, R +, R ar thn computd using qns (22) (25). 3. Th cofficints C +, C, R +, R, which ar indpndnt of th lin lngth, ar usd to comput th cofficints I (L), I 2 (L), for any lngth L using (2) and (2). Th analytical xprssions for th inducd currnt along th asymptotic rgion of th lin () and at th lin nds (3), (35) can b applid for th any lin lngth. Not that only th first stp of th abov procdur rquirs numrical computations, which is to b prformd not for th whol lin structur but on two significantly shortr lins. Thrfor, th computation tim can b drastically rducd for long lins. Additionally, onc th numrical solutions for th two short lin configurations ar known, it is possibl to comput analytically th solution for any similar lin configuration, but with any diffrnt lin lngth. 2.2 Accuracy of th proposd thr-trm xprssion for th inducd currnt along th asymptotic rgion of th lin In ordr to validat our assumption on th analytical form of th inducd currnt in th asymptotic cntral rgion II, w hav dvlopd a cod for th dtrmination of th cofficints I, I, I 2 in xprssion (), starting from numrical solutions obtaind using Numrical Elctromagntics Cod (NEC) [2]. Th ral and imaginary parts of cofficints I, I, I 2 wr dtrmind sparatly using th last-squar mthod. It has bn shown, considring svral load conditions, that th proposd xprssion () approximats vry wll th spatial dpndnc of th inducd currnt [6]. An xampl of comparison btwn th numrical solutions obtaind using NEC with th proposd approximat xprssion () is prsntd in Fig. 3. Th lin is charactrizd by a lngth L = 6 m, a conductor radius a = mm, and a hight abov ground h =.5 m, and is short-circuitd at both nds (Z =, Z 2 = ). Th xciting fild is a plan wav with f = 358 MHz (λ =.84 m), q = 45, E = 5 kv/m. Not that, th TL approximation is not applicabl to this cas bcaus th wavlngth of th xciting fild is practically qual to th hight of th conductor abov ground. In this figur, th abscissa l rprsnts th coordinat

11 High-Frquncy Elctromagntic Fild Coupling 69 Figur 3: Comparison of th inducd currnt flowing along th lin using th NEC solution and th proposd approximat formula (), with cofficints dtrmind using th last-squar mthod. Angl of incidnc θ = 45. (a) Ral part and (b) imaginary part of th currnt. along th whol wir lngth including th vrtical risrs, l = corrsponding to th point whr th vrtical conductor touchs th ground and l = h to th bginning of th horizontal part of th lin. Th total numbr of sgmnts along th lin considrd in th NEC cod was N sg = 245. It can b sn that an xcllnt approximation is found in th asymptotic rgion of th lin. According to our hypothsis, th cofficint I in th thr-trm spatial dpndnc () should b qual to th xprssion (5), corrsponding to th currnt inducd by an xtrnal plan wav for th cas of an infinitly long lin. Comparisons btwn th ral and imaginary parts of I obtaind from a last-squar approximation from th NEC solution and thos obtaind using rlation (5) hav also shown an xcllnt agrmnt [6]. Th rsults of comparison for th sam cas of a horizontal wir short-circuitd at its both nds ar prsntd in Fig. 4 as a function of th frquncy and angl of incidnc of th xciting fild. Not that th rsults ar practically idntical within th rsolution accuracy of th drawings. Othr succssful tsts of th proposd thory wr prsntd in [6] by comparing th rsults to analytical solutions for th cas of simpl lin configurations, such as an opn-circuit smi-infinit lin. 2.3 Application: rspons of a long trminatd lin to an xtrnal plan wav Th solution of th coupling quations for long trminatd lins using th proposd asymptotic thory is illustratd hr by two xampls. First, lt us considr an opn-circuitd wir of finit lngth abov a prfctly conducting ground (s Fig. in Chaptr 4). In this cas th xact xprssions for th asymptotic cofficints can b obtaind analytically using th Winr Hopf solution [4] (for th cofficints R + = R = R). Thy can also b dtrmind using

12 7 Elctromagntic Fild Intraction with Transmission Lins Figur 4: Comparison btwn th ral and imaginary parts of I obtaind by a last-squar approximation from th NEC solution and thos obtaind using rlation (5), as a function of (a) frquncy and (b) th angl of incidnc. th itrativ approach of Chaptr 4 (s Sction 2.4 in Chaptr 4 for th cofficints R + = R = R, and Appndix 2 for cofficints C + and C ). In Fig. 5, w prsnt a comparison btwn calculation rsults obtaind by th dvlopd asymptotic approach and thos obtaind numrically using th mthod of momnt (MoM). Th hight of th wir is.5 m, its lngth L = 6 m, its radius a = mm. Th wir is illuminatd by a vrtically polarizd plan wav with amplitud E = 5 kv/m, angl of incidnc q = 45º (j = ), and a frquncy f = MHz (kh ). Figur 5 shows a vry good agrmnt btwn th proposd analytical solution and numrical simulation obtaind using th MoM cod CONCEPT [22]. On th sam figur, w hav also plottd th rsults obtaind using TL thory, which dos not provid accurat rsults. For th gnral cas of arbitrary trminal loads and gomtrical configurations, th asymptotic cofficints R and C ar dtrmind using a procdur basd on th xact solutions of th intgral quations for two similar wir configurations, but having a significantly shortr lngth (as dscribd in Sction 2.). Th knowldg of th asymptotic cofficints R and C prmits th computation of th currnt cofficints I and I 2 for any trminal lin having th sam trminal loads and gomtry. In othr words, th numrical solution obtaind for a rlativly short lin prmits th analytical dtrmination of th solution for any longr lin having th sam configuration. Th aim of th scond numrical xampl is to illustrat th us of th proposd procdur to comput th inducd currnt by an xtrnal fild on a long loadd lin. Considr a 5 m long lin with a radius a = mm, and a hight abov ground h =.5 m. Th lin is short-circuitd at both nds (Z =, Z 2 = ). Th xciting fild is a plan wav with E = 5 kv/m, q = 45. Starting from th numrical solutions for th inducd currnt on two similar lin configurations with L = 6 m and L 2 = L + l/2, w dtrmin th cofficints

13 High-Frquncy Elctromagntic Fild Coupling 7 Figur 5: Currnt inducd along an opn-circuitd lin. I (L ), I 2 (L ), I (L 2 ), I 2 (L 2 ) using th last squar mthod. Th scattring cofficints C + (jw), C (jw), R + (jw), R (jw), ar thn computd using qns (22) (25) as a function frquncy. Th rsults ar shown in Fig. 6. It can b sn that for low frquncis (kh << ), th rflction cofficints R + and R tnd to, which is th TL currnt rflction cofficint associatd with a short-circuit trmination. Additionally th scattring cofficints C + and C tnd to zro at low frquncis. Lt us now dfin th currnt distribution, for xampl, for th frquncy f 358 MHz (l =.84 m, kh = 3.75). Not that at th considrd frquncy, th TL approximation is not valid sinc th wavlngth has th sam ordr of magnitud as th lin hight. Th scattring cofficints C +, C, R +, R for this frquncy ar as follows: R+ = R = j (36a) C+ =.25.29j (36b) C = j (36c) Using qns (2) and (2) th cofficints I and I 2 for a 5 m long lin can b dtrmind as I ( L = 5m) = j (37a) I ( L = 5m) = j (37b) 2 On th othr hand, th cofficint I calculatd using qn (5) is givn by I = j (38)

14 72 Elctromagntic Fild Intraction with Transmission Lins.5 R(R+) Im(R+).5 R(R-) Im(R-) Cofficint R+ -.5 Cofficint R kh (a) kh (b).5 R (C+) Im (C+).5 R (C-) Im (C-) Cofficint C Cofficint C kh (c) kh (d) Figur 6: Variation of th cofficints R +, R, C +, C, for th horizontal wir shortcircuitd by vrtical risrs, as function of frquncy. Th currnt along th 5 m long lin is simply givn by qn () with numrical valus for th cofficints givn by qns (37) and (38). Th rsults for th inducd currnt in th nar nd and in th cntral rgions of th lin ar prsntd in Fig. 7. It can b sn that th agrmnt btwn th proposd approach and th xact numrical solutions obtaind using NEC is vry satisfactory. Not that in this figur, as in Figs 3 and 4, th abscissa l corrsponds to th coordinat along th whol wir systms, including th vrtical risrs. 3 Asymptotic approach for a non-uniform transmission lin Th asymptotic approach prsntd in Sction 2.4 can b gnralizd to calculat th currnt inducd by an incidnt plan wav along a long lin, which has a local discontinuity rprsntd by a sris lumpd impdanc Z L at som intrmdiat point (s Fig. 8). This problm can dscrib, for xampl, a cabl with a damagd shild or a shild discontinuity [23].

15 High-Frquncy Elctromagntic Fild Coupling 73 5 NEC Proposd 3 2 NEC Proposd R(I) (Amps) -5 Im(I) (Amps) - (a) l (m) (b) l (m) 5 NEC Proposd 3 2 NEC Proposd R(I) (Amps) 5 Im(I) (Amps) -5 (c) l (m) - (d) l (m) 5 NEC Proposd 3 2 NEC Proposd R(I) (Amps) 5 Im(I) (Amps) -5 - () l (m) (f) l (m) Figur 7: Currnt spatial distribution along a short-circuitd horizontal wir (L = 5 m). Comparison btwn NEC solutions (solid lin) and th proposd asymptotic approach (dashd lin): (a and b) currnt in th lft-nd rgion; (c and d) currnt in th cntral rgion; ( and f) currnt in th rightnd rgion. Lft columns: ral part; right columns: imaginary part. Lt us first considr an auxiliary problm rprsntd by Fig. 9, whr th lin is infinitly long. Th Pocklington s intgral quation, assuming th lumpd impdanc is placd at th coordinat origin z =, is givn by 2 d 2 + k g( z z) I( z )d z = E (, )+ () ( ) 2 z h z ZLI d z π jw (39) z 4 d

16 74 Elctromagntic Fild Intraction with Transmission Lins Figur 8: Gomtry of th long trminatd lin with an addition lumpd impdanc. Figur 9: Infinitly long lin with additional lumpd impdanc. whr g(z) is th scalar Grn s function givn by qn (3). Th lumpd impdanc, which is usually considrd through a boundary condition, is takn into account in qn (39) as an additional trm in th total xiting tangntial fild E t z, accounting for th discontinuity at z = [24] t z z L E = E Z I() d ( z) (4) whr I() is th currnt in th impdanc. Considring that th intgration in qn (39) is carrid out ovr an infinit intrval and that th krnl of th intgral-diffrntial quation (39) dpnds on th diffrnc of argumnts z z', it is possibl to find a solution using th spatial Fourir transform. Th gnral solution for an incidnt plan wav can b writtn in th form I m ( z) = I y m ( z). Th subscript m indicats that th scattrr is placd in th main

17 High-Frquncy Elctromagntic Fild Coupling 75 cntral rgion of th lin. y m (z) is th solution of th non-homognous scattring problm in th main cntral rgion of th lin. Lt us also dfin: y m,+ ( z) as th solution in th cntral rgion of th homognous (no xciting fild) scattring problm for a currnt (TEM) wav xp(jkz) incidnt to th load from ; and y m, ( z) as th solution in th cntral rgion of th homognous scattring problm for a currnt (TEM) wav xp(jkz) incidnt to th load from. Using th spatial Fourir transform and aftr mathmatical manipulations, it can b shown that in which, th function F 2 (z) is m 2 y ()=xp( z jk z)+ F () z (4) m, 2 y + ()=xp( z jkz)+ F () z (42) m, 2 y ()=xp( z jkz)+ F () z (43) jk z L C L Z [ + F( a/2 h,2 kh, z/2 h)] F2 ()= z (44) 2 Z + Z [+ F( a/2 h,2 kh,)] and Z c = m / (/2π)ln(2h/a) is th charactristic impdanc of th lin, and th function F is and max 2y 2 ln(/ a) cos( yx) F (, y, )= dy (45) 2k h a x πj 2 2 G y y y y 2 2 ( a, ) ( a ) ( ) pj H y y H y y y< y 2 2 G( a, y y ) = 2 ln(/ a), y = y (2) 2 2 (2) K ( a y y) K ( y y ), y> y (2) 2 2 (2) 2 2, (46) It is important to not that th dlta-function in qns (39) and (4) is a mathmatical idalization. In rality, th rsistiv rgion along th wir has a finit lngth ( a). As far as w ar not intrstd hr in th dtaild structur of this rgion, it is possibl to limit th intgration at th corrsponding wav numbr k max π/2. In this way, th intgration convrgnc is nsurd.

18 76 Elctromagntic Fild Intraction with Transmission Lins Th z-dpndnc of th scond trm in qns (4) (43) is similar to that of th currnt inducd by a point voltag sourc in an infinit wir abov a prfctly conducting ground. Th corrsponding solution can also b asily obtaind by th spatial Fourir transform. This problm is a spcial cas of th wll-known problm of an infinit wir abov a ground of finit conductivity. In [], a numbr of paprs on this topic wr rviwd. Using mthods of complx variabl functions, it was shown that F 2 (z) can b rprsntd as th sum of thr trms: a main TEM mod, th sum of ign mods (or laky mods), and th so-calld radiation mod (th anti-symmtrical currnt mods in th two wir systm ar prsntd and invstigatd in [2, 3]). An invstigation of th z-dpndnc for th function F 2 (z) shows that for distancs from th load largr than about 2h(for kh <~ ), th transmission lin TEM mod xp( jk z ) prdominats. Othr mods dcay with th distanc from th scattring load as an invrs powr of z (radiation mod) or xponntially (ign mods), i.. F (a/2h, 2kh, z/2h). z As a consqunc, w can writ m m y ( z) = xp( jk z) + R xp( jk z ) (47) m, m y + ( z) = xp( jkz) + R xp( jk z ) (48) m, m y ( z) = xp( jkz) + R xp( jk z ) (49) whr R m is th rflction cofficint givn by R m = ZL (5) 2 Z + Z [+ F ( a /2 h,2 kh,)] C L In th low-frquncy limit, whn kh <<, F (a/2h, 2kh, ) Æ (s Fig. ) and th rflction cofficint rducs to th wll-known TL approximation, that is R m = Z L /(2Z C + Z L ). Figur shows th variation of th rflction cofficint as a function of kh. It can b sn that at low frquncis (kh << ), it rducs to th TL valu. Coming back to th finit systm of Fig. 8, and according to th asymptotic approach, w will sarch for a solution for th inducd currnt in th following form. In rgions I and II: Iz ()= I ()+ z I () z (5) y+ y+ In rgions II IV: Iz ()= I ( z L)+ I ( z L)+ I ( z L) (52) jkl ym 2ym, 3ym, + whr L is th distanc from th lin lft-nd to th impdanc Z L. Also, in rgions IV and V: Iz ()= I jk L y ( zl)+ I y ( zl) (53) 4

19 High-Frquncy Elctromagntic Fild Coupling 77 Figur : Frquncy dpndnc of th function F (a/2h, 2kh, ) for h =.5 m, a =. m, =.4 m. Figur : Frquncy dpndnc of th rflction cofficint R m (kh), qn (5), for a/2h =., = 4a, Z L = Z C. Cofficints Ĩ, Ĩ 2, Ĩ 3, Ĩ 4 in th abov quations can b dtrmind considring an asymptotic viw of ths solutions in rgions II, IV by formulas (8), (9), (2), (3), and (47) (49) and taking into account that in ths rgions, th solution can b writtn using th thr-trm form (qn ()). In this way, approximat analytical solutions for th problm of Fig. 8 can b obtaind (s Appndix 3).

20 78 Elctromagntic Fild Intraction with Transmission Lins R(I(l)), A CONCEPT Asympt. mthod for lin with non-uniformity l, m 6 Im(I(l)), A CONCEPT Asympt. mthod for lin with non-uniformity l, m Figur 2: Ral and imaginary parts of th inducd currnt along th lin. Comparison btwn th proposd approach and numrical rsults obtaind using MoM. To illustrat th proposd asymptotic mthod, lt us considr a simpl modl of an opn-nd straight wir abov a prfctly conducting ground. Th hight of th wir is h =.5, its lngth L = 6 m, its radius a = mm. Th wir is illuminatd by a vrtically polarizd plan wav with an amplitud E = 5 kv/m, angl of incidnc q = 45 (j = ), and a frquncy f = 358 MHz. Th wir contains an impdanc at its cntr qual to th charactristic impdanc of th lin, Z L = Z C. Th lngth of th impdanc rgion is = 4 mm. In Fig. 2, w show a comparison btwn th proposd asymptotic mthod (qns (5) (53) with asymptotic formulas (8), (9), (2), (3), and (47) (49)) and numrical rsults obtaind using th MoM cod CONCEPT [22], for th ral and imaginary parts of th inducd currnt. Th rflction cofficints for th currnt wav at th nds of th lin R + = R = R, C +, and C_ ar obtaind using an itrativ approach (s Chaptr 4 and Appndix 2). It can b sn that th rsults obtaind using th proposd approach ar in xcllnt agrmnt with xact numrical rsults. 4 Conclusion W prsntd in this chaptr an fficint hybrid mthod to comput lctromagntic fild coupling to a long trminatd lin. Th lin can also hav a discontinuity in th form of a lumpd impdanc in its cntral rgion. Th mthod is applicabl for high-frquncy xcitation for which th TL approximation is not valid. In th proposd mthod, th inducd currnt along th lin can b xprssd using closd form analytical xprssions. Ths xprssions involv scattring cofficints at th lin non-uniformitis, which can b dtrmind using ithr approximat analytical solutions, numrical mthods (for th scattring in th lin nar nd rgions), or xact analytical solutions (for th scattring at th lumpd

21 High-Frquncy Elctromagntic Fild Coupling 79 impdanc in th cntral rgion). Th proposd approach has bn compard with numrical simulations and xcllnt agrmnt is found. Appndix : Dtrmination of cofficints R +, R, C +, C as a function of cofficints I and I 2 W start from qns (2) and (2) r-writtn blow I ( L)= I C xp( jkl jkl) + C+ Rxp( 2 jkl) RR xp( 2 jkl) I ( L)= I 2 + C+ + C R+ xp( jkl jkl) RR xp( 2 jkl) + (A.) (A.2) W will first dcoupl th abov quations to obtain th sparat quations for R +, C +, and for R, C. To do that, lts us considr th quantitis I 2 (L) R + I (L) and I (L) R I 2 (L)xp( 2jkL). Aftr straightforward mathmatics, it can asily b shown that I ( L) R I ( L)= I C (A.3) I ( L) R I ( L)xp( 2 jkl) = I C xp[ j( k+ k ) L] (A.4) 2 Now, lt us writ qn (A.3) for two diffrnt lin lngths I ( L ) R I ( L )= I C I ( L ) R I ( L )= I C (A.5) (A.6) It is asy now to xprss R + and C + in trms of I and I 2 C + R + I2( L2) I2( L) = I ( L ) I ( L ) 2 I2( L) I( L2) I2( L2) I( L) = I I ( L ) I ( L ) 2 (A.7) (A.8) Writing qn (A.4) for two diffrnt lin lngths yilds And consquntly I ( L ) R I ( L )xp( 2 jkl ) = I C xp[ j( k+ k ) L ] (A.9) 2 I ( L ) R I ( L )xp( 2 jkl ) = I C xp[ j( k+ k ) L ] (A.) R I( L2)xp[ j( k+ k) L2] I( L)xp[ j( k+ k) L] = I ( L )xp[ j( k k) L ] I ( L )xp[ j( k k) L ] (A.)

22 8 Elctromagntic Fild Intraction with Transmission Lins C I( L) I2( L2)xp(2 jkl) I( L2) I2( L)xp(2 jkl2) = I I ( L )xp[ j( kk ) L ] I ( L )xp[ j( kk ) L ] (A.2) Appndix 2: Drivation of analytical xprssions for th cofficints C + and C for a smi-infinit opn-circuitd lin, using th itrativ mthod prsntd in Chaptr 4 In this appndix, w us th itrativ procdur prsntd in Chaptr 4 to driv an approximat analytical xprssion of th zroth and th first itration trms of th asymptotic cofficints C + and C, for th cas of smi-infinit opn-circuitd lin (for a right smi-infinit lin (Fig. 2b), th gomtry is idntical to th on shown in Fig. of Chaptr 4, with L Æ ). Th systm is xcitd by a vrtically polarizd xtrnal lctromagntic wav with an lvation angl q. Th azimuth angl of incidnc is assumd to b zro, j =. For th right smi-infinit lin th solution in th asymptotic rgion z >> 2h can b xprssd as (s Sction 2..) + z>> 2h + I ( z) I (xp( jk z) + C xp( jkz)) (A2.) whr k = w/c, k = kcosq and I is th currnt inducd on an infinit lin, givn by th xprssion (A2.2), which can asily b drivd (s qn (53) in Chaptr 4) I 4 cez ( jw) ( j )= 2 (2) (2) w hw sin q ( H ( kasin q ) H (2khsin q )) (A2.2) whr E z (jw) is th total xciting (incidnt + ground-rflctd) tangntial lctric fild, H (2) (x) is th zro ordr Hankl function of th scond kind [25], h = m. Th zroth itration trm of th prturbation thory, which is dtrmind by th TL approximation and which satisfis th opn-circuit boundary condition for z =, is givn by,, I+ ( z) = I (xp( jk z) xp( jkz)) (A2.3) in which I, is th inducd currnt on an infinit lin calculatd using TL approximation [2] (qn (58) in Chaptr 4) I Ez (, h w), = 2 /2πln(2 h/ a) j sin m w q (A2.4) W will driv now an xprssion for th first itration trm I (z) using th gnral quation of th prturbation thory for th nth itration trm (qn (4) from Chaptr 4) which rducs, for th right smi-infinit lin ( z <, k Æ k jd, d Æ ), to th following xprssion: +, n n n I ( z) = F ( z) F ()xp( jkz) (A2.5)

23 High-Frquncy Elctromagntic Fild Coupling 8 Using th gnral quation for th function F n (z) (qn (4) from Chaptr 4) and making us of qn (A2.3), w can obtain th xprssion for th function F (z) for th first itration F z ()= I [ ], jk z jkz jk ( z z ) + a jk ( z z ) +4h jkz jkz I, dz 2 ln(2 h/ a) ( z z ) + a ( z z ) +4h For small argumnts, qn (A2.6) yilds F () = I D in which, 2 2 ( jk x h ) jk xp + 4 xp( jkx) x jkx D dx 2 ln(2 h/ a) x 2 2 x +4h And, for larg argumnts z Æ, w gt (A2.6) (A2.7) (A2.8) whr F ( z) = I xp( jk z) D, 2 z (A2.9) j 2 j (2) D2 = π ln( gkhsin q) H (2khsin q) 2 ln(2 h/ a) π (A2.) To obtain th larg argumnt xprssions (A2.9) and (A2.), w hav usd th intgral (qn (5)) from Chaptr 4 and th wll-known formula from th thory of Bssl functions [25] (2) 2 j H ( z) ln( gz/ 2), whr g =.78 z π (A2.) Th knowldg of function F for z = and z Æ maks it possibl to obtain a closd-form solution in th asymptotic rgion z >> 2h for th first itration trm of th inducd currnt I (z). Using qns (A2.5), (A2.7), and (A2.9), w gt +, z, 2 I ( z) I [ D xp( jk z) + D xp( jkz)] (A2.2) Th total inducd currnt in th asymptotic rgion is thn givn by + +, +, z, 2 I ( z) I ( z) + I ( z) I {(+ D )xp( jk z) + ( D )xp( jkz)}, 2 2 = I (+ D ){xp( jk z) + ( D )(+ D ) xp( jkz)} (A2.3)

24 82 Elctromagntic Fild Intraction with Transmission Lins Now using qns (A2.2) and (A2.), and assuming D 2 << w can writ for th currnt amplitud and qn (A2.3) bcoms I I I ( + D ),, 2 D2 2 (A2.4) I+ ( z) I {xp( jk z) + ( D )(+ D ) xp( jkz)} (A2.5) Expanding in trms of [2ln(2h/a)] and taking th two first trms, th cofficint (D )( + D 2 ) in qn (A2.5) rducs to (D + D 2 ), and qn (A2.5) bcoms I+ ( z) I{xp( jkz) + ( D+ D2 )xp( jkz)} (A2.6) Th cofficint C + may b obtaind by idntification of qns (A2.) and (A2.6) C + D + D (A2.7) + 2 Th cofficint C can b dtrmind in a similar way considring a smi-infinit lin < z, for which th currnt in th asymptotic rgion z << 2h is givn by I ( z) I (xp( jk z) + C xp( jkz)) (A2.8) z<< 2h It can b asily shown, following similar mathmatical dvlopmnt, that th xprssion for C is th sam as for C +, qn (A2.7), but rplacing k by k in qns (A2.8) and (A2.). A comparison of th frquncy dpndnc of th asymptotic currnt cofficint C(jw) for an opn-circuit smi-infinit lin undr normal incidnc (C + = C = C ) obtaind by th proposd asymptotic mthod (Sction 2..) and th on drivd by itration mthod (qn (A2.7)) is prsntd in Fig. A2., and again, a vry good agrmnt is found. Appndix 3: Analytical xprssion for th inducd currnt along th asymptotic rgion of th lin containing a lumpd impdanc Lt us considr th solution for th currnt in th asymptotic rgions II and IV (s Fig. 7). In rgion II, starting from th lft nd of th lin, using th xprssion (5) and taking into account th asymptotic rprsntation (qns (8) and (2)), w will gt Iz ()= Iy ()+ z I + y+ () z I [xp( jk z) + C xp( jkz)] + I [xp( jk z) + R xp( jkz)] z>> 2h + + = I xp( jk z) + I xp( jkz) +[ C + I R ]xp( jkz) + + (A3.)

25 High-Frquncy Elctromagntic Fild Coupling 83 Figur A2.: Comparison of asymptotic cofficint for an opn-circuit smiinfinit lin obtaind by itration thory (curvs and 2) and th on drivd by th asymptotic thory (curvs 3 and 4). Normal incidnc. Now, starting from th cntr of th lin, using xprssion (52), and taking into account th asymptotic rprsntation (qns (47) (49)), w will gt Iz I z L I z L I z L ( )= y ( )+ y ( )+ y ( ) L z>> 2h jk L m 2 m, 3 m, + I R I R jkl jk( zl) jkz ( L) jkz ( L) jkz ( L) m 2 m [ + ]+ [ + ] jk( zl ) jk( zl ) 3 m + I [ + R ] I jk z L 2 jk z jk L jk ( zl ) m 2 m 3 m = I +[ I R + I R + I (+ R )] + ( ) (A3.2) As it can b sn from qns (A3.) and (A3.2), th solution in th asymptotic rgion is givn in th form of th proposd thr-trm approximation (). By imposing that th cofficints for th trms xp(jkz) and xp( jkz) ar idntical in qns (A3.) and (A3.2), w obtain two quations to dtrmin th unknown cofficints Ĩ, Ĩ 2, and Ĩ 3 jk ( k) L jkl jkl I + = I R + I R + I (+ R ) (A3.3) m 2 m 3 m (A3.4) IC + IR = I jkl + + 2

26 84 Elctromagntic Fild Intraction with Transmission Lins In a similar way, using th xprssions for rgion IV, w can obtain two othr quations for th unknown cofficints Ĩ, Ĩ 2, Ĩ 3, and Ĩ 4 jkl j( k k) L jkl I + = I C + I R (A3.5) 3 4 IR I R I R I (A3.6) jk ( k) L jkl jkl jkl m + 2 (+ m)+ 3 m = 4 Th final solutions for th systm of qns (A3.3) (A3.6) ar givn by I I R R C R R 2jkL jkl jk L 2jkL 2 m + + m 2 = {( )( + ) 2jkL jk LjkL jk L 2jkL + m m R (+ R ) ( C + R R )} 2jkL 2jkL 2 2jkL + m m + m 2 {( RR )( RR ) RR(+ R ) } I I R R C R R 2jkL jk LjkL jk L 2jkL 3 + m m 2 2 = {( )( + ) 2jkL jkl jk L 2jkL m + + m 2 + R (+ R ) ( C + R R )} 2jkL 2jkL 2 2jkL + m m + m 2 {( RR )( RR ) RR(+ R ) } (A3.7) (A3.8) jkl I =( I I C ) A (A3.9) jkl2 jkl I =( I I C ) A (A3.) whr L 2 = L L in qns (A3.7) and (A3.8). Using th abov cofficints, th inducd currnt in th asymptotic rgions will b givn by In rgion II: jk z jkz jkz 2 Iz I I I ()= + + (A3.) In rgion IV: jk z jkz jkz 3 4 Iz I I I ()= + + (A3.2) whr I = Ĩ (A3.3) I2 = IC + I + R+ (A3.4) I jk( k+ k ) L jkl = I C + I R (A3.5) 3 4 I4 = Ĩ 4 jkl (A3.6)

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