Use of the Transmission-Line Super Theory to Determine Generalized Reflection and Transmission Coefficients of an Inhomogeneous Line with Risers

Transcription

1 Intaction Nots Not 66 Mach 5 Us of th Tansmission-in Sup Thoy to Dtmin Gnaizd Rfction and Tansmission officints of an Inhomognous in with Riss Sgy Tkachnko*, Ronad Rambousky**, and Jügn Nitsch* * Otto-von-Guick Univsity Magdbug, Gmany ** Bundswh Rsach Institut fo Potctiv Tchnoogis and NB Potction WIS, Munst, Gmany Abstact Th anaysis conductd within th famwok of Tansmission-in Sup Thoy TST has d to nw fomuations fo th fction and tansmission cofficints of inhomognous tansmission ins at high fquncis. In this pap, ths suts a divd and appid fo th fist tim on pactica non-unifom conducto configuations. Th suts obtaind ag xcnty with th xact TST soutions, athough th nw xpssions contain a ctain appoximation: th adiation phnomna a ony considd within th individua inhomognous pats of th ins. This wok was sponsod by th Bundswh Rsach Institut fo Potctiv Tchnoogis and NB Potction - WIS - Munst und th contact numb E/E59/DZ5/AF9

2 ontnt I. INTRODUTION... 3 II. ESSENTIAS OF THE TST AND STATEMENT OF THE PROBEM.. 4 II.. Dfinition of th goba paamts of TST 4 II.. Matizant and its appication fo soution of TST quations.. 6 II.3 3. Appoximat quations fo th goba paamts of TST ptubation thoy 8 III. DETERMINATION OF THE REFETION- AND TRANSMISSION- OEFFIIENTS.. 9 III..assica tansmission-in thoy. 9 III.. Th Rfction officints at th nds of th in in TST III.. 3. acuation of th fction cofficint at th nd of th in III.. 4. Th Rfction cofficint at th bginning of th in... III Th cunt ampitud function fo th stating outgoing wav..3 III Intmdiat Rsut fo th cunt of a hoizonta in with two iss.4 IV. TRANSMISSION AND REFETION OEFFIIENTS OF THE SATTERER..5 V. NUMERIA EXAMPES AND DISUSSION V.. Hoizonta in with on bnd V.. Simp on in simuato..3 VI. ONUSION 6 REFERENES..6

3 I. INTRODUTION In a numb of appications in ctica ngining and ctonics it is ncssay to cacuat cunts and votags popagating in wiing stuctus, ik,.g., antnnas and tansmission ins. A w-known, convnint too fo such cacuations is th cassica tansmission in appoximation T. assica T ads to xpicit anaytica xpssions and aows on to cay out quaitativ anaysis and ngining cacuations fo such systms, pacticay instantanousy. Howv, th T is vaid ony up to fquncis fo which th wavngth is compaab with th tansvsa dimnsion of th in. In modn ctonics, howv, th woking ang of th signa fquncis is continuay to incasing. Moov, th wiing stuctus can wok as a civ of intntiona and unintntiona intfncs of diffnt kind, and at th sam tim, th snsitivity of th modn smiconducto mnts is aso incasing. A ths cicumstancs qui th dvopmnt of cosponding cacuation mthods. Th pu numica mthods, ik MoM, TT, tc., hav ong cacuation tims; thy aso can ony consid th spcific cas and don t aow invstigating th pobm in gna. Th atnativs a diffnt anaytica and anaytica numica mthods. On of th most pomising mthods is th Tansmission in Sup Thoy, TST, [-8] which ducs th systm of xact intgo-diffntia quations Mixd Potntia Intga Equations MPIE to a systm of th fist od odinay diffntia quations. This systm ooks ik th systm of inhomognous Tgaph s quations, but with compx-vaud paamts and non-zo diagona mnts s Sction. Th paamts of th systm can b obtaind using an itation pocdu, which, in tun, quis quit ong cacuations. Howv, in aity, th ins consist of som non-unifom gions na tmina gions, diffnt bnds, umpd soucs and/o oads, tc., which a connctd with ach oth by gions having a unifom stuctu of cassica T wi is paa to conducting gound. Th soution fo th cunt na th non-unifomitis has a compx stuctu incuding diffnt typ of mods: aky mods, adiation mods and TEM mods. Usuay, th non-unifom gions a ssntiay shot than th unifom gions. This mans that th cunt in th ong, cnta gions can b sovd fo in a simp mann. Thi soutions ook ik soutions of inhomognous xact intgo diffntia quations fo th infinity ong in fo th cas of xcitation by an xtna fid pus two soutions of homognous intgo diffntia quations fo th infinit tansmission in fowad and backwad popagating TEM mods. Th cofficints fo th TEM mods a dfind by th so cad fction tansmission and ampitud cofficints fo cunt wavs [9,, 7]. Such a soution fo th inducd cunt th so-cad asymptotica soution fomay ooks ik a soution in T appoximation, but with oth vaus of fction and tansmission cofficints. Of cous, fo ow fquncis, whn on can us simp fomuas fo ths cofficints, th asymptotic soution bcoms a soution of th cassica thoy of tansmission ins. To obtain ths cofficints on can us an itation mthod fo th smi infinit in s [], [], [3] with th sam ft o ight tmina, but it quis som anaytica cacuations. Anoth mthod is a pocssing sut of th TST fo shot ins. It aows th dvopd mthods and softwa of TST to b usd. Moov, it instas a dp physica connction btwn th asymptotic mthod, TST and SEM [3, 8]. Th aim of th psnt sach is to obtain th fction tansmission and ampitud cofficints fo cunt wavs using th mthod of Tansmission in Sup Thoy. Gna fomuas a obtaind which Such psntation is convnint fo th anaysis of th Singuaity Expansion Mthod SEM of th pos in th fist ay fo such a tansmission in systm s,.g., []. nowdg of ths pos, in tun, aows on to obtain anaytica suts fo th spons in th tim domain. 3

4 a thn chckd by cacuating th simp, but not tivia xamps of a staight and a bnt hoizonta in with iss. II. ESSENTIAS OF THE TST AND STATEMENT OF THE PROBEM. Dfinition of th goba paamts of TST onsid a thin wi of abitay gomtic fom, wh is th coodinat aong th wi axis abov a pfcty conducting gound s, fo xamp, th non-unifom wi with vtica mnts in Fig. which may b oadd and xcitd by an xtna fid. E i Fig. : Gomty of th tansmission in stuctu. Using th zo bounday conditions fo th tota scatting pus xciting tangntia ctic fid on th sufac of th wi and th continuity quation fo th inducd cunt, w obtain a systm of intgo-diffntia quations fo th cunt and potntia pai Mixd Potntia Intga Equations MPIE: I I j d d I d g E d I g j d d 4,, 4 a, b H is an xciting tota tangntia ctic fid incidnt pus fctd, E is th scaa potntia aong th wi in th onz gaug, th adius of th wi, and a is th tota ngth of th wi. Th functions and a th Gn s functions aong th cuvd in fo th vcto potntia and scaa potntia, spctivy, which tak into account th fction of th gound pan:, g I, g I ~ ~, a a g a jk a jk I ~ ~ ~, a a g a jk a jk I 3 4

5 Th unit tangntia vcto d d of th cuv is takn aong th wi axis, ~ is th adius vcto fctd by th gound pan, and ~ d ~ d is th cosponding unit tangntia vcto. Now, in od to dfin th goba gnaizd tansmission in paamts, consid an xcitation of th tansmission in by a umpd votag souc U ocatd at th bginning of th in. Th in is aso assumd to b oadd by a umpd impdanc Z at th fa nd. Th a two possibiitis to account fo th souc and th oad: ith to tat both of thm as bounday conditions o tat both of thm as soucs with unknown ampitud fo th oad souc. Fo th scond possibiity th xciting fid E can b wittn as [5]: E U U 4 U Z I with 5 Of cous, in th foma viw 4 on can aso psnt a pobm with xcitations at two tminas, with xcitation at th ight tmina and oading at ft tmina, tc. Th admittanc functions, hav dimnsions of conductanc whi th tansf functions fo th potntia, a dimnsionss. Now t th spons functions, and, b soutions of th systm a,b fo th cunt and th potntia with soucs of ampitud V, ocatd in th points Δ and, spctivy. Du to th inaity of th considd pobm w can wit th soution fo th tota inducd cunt as I U U 6 Fo th potntia aong th wi on finds a simia quation: U U 7 Now on is ady to ook fo th systm of diffntia quations fo th potntia and cunt in T-ik fom: d jp d di jp d I jp jp I 8a, b U U Bcaus th vaus and a inay indpndnt, th quations 8 hav to b T T satisfid fo ach coumn, and, in matix fom: d P P j 9 d P P Using quation 9 on can vauat th quation fo th paamt matix P ˆ : Pˆ d j d It is asy to show that th choic of basic soutions dos not infunc th vau of th paamts. In fact, considing a nw systm of basic functions as a non-dgnat ina combination of th pvious on suts in: 5

6 : ~ ~ ~ ~ ˆ dt Thn, cacuating th paamts w hav: ~ ~ ~ ~ ~ ~ ~ ~ ˆ~ d d j P d d j a ˆ P d d j Thus, it is shown that th systm of intgo-diffntia quations, with a soution which is dfind by two indpndnt constants, can b xpicity ducd to th diffntia quations 8 with paamts 9. Ths paamts a ith goba paamts in th fuwav tansmission in thoy o th paamts of Maxwian cicuits, and thy a compx vaud. Moov, thy aso dscib th adiation of th systm [4, 5, 6]. Thy dpnd on th gomty of th systm, and, thfo, on th oca gomtic paamt aong th in. This fact was aady stabishd in [,, 3, 6] with th mthod of th poduct intga, and in [4] by pocssing th numica soutions fo th cunt and potntia with th Mthod of Momnts. Th paamt matix dpnds on th gaug of th potntia ˆ P. Fo xamp, it has a diffnt fom fo th ouomb gaug than it has fo th onz gaug.. Matizant and its appication fo soution of TST quations. As mntiond in th pvious sub-sction, th paamt matix can b dfind on any pai of ina indpndnt soutions of th intgo diffntia MPIE systm a homognous systm, and this systm is aso a soution of th cosponding ODE systm with paamt matix and som bounday conditions which hav not yt bn dfind. Futh, it is convnint to consid th matix of soutions, which satisfy th foowing bounday conditions: ˆ P ˆ ˆ : M j P I I ; a, b I I Du to th inaity of th pobm, if w hav abitay bounday conditions, th soution fo th coodinat ooks ik: T I, 6

7 ˆ ˆ M jp I I 3 Th matix ˆ M jpˆ is cad th matizant, poduct intga o popagato [5]. It tansats th soution fom th point to th point. Mo gnay wittn: Mˆ I jpˆ I 4 Th matizant can b asiy cacuatd by th division of th intva, into sma imitd, infinitsima subintvas i N and th cacuation of th matix poduct poduct intga is as foows: Mˆ N N I ˆ jpˆ i i im i i j Pˆ im xp j Pˆ 5 i i Fo th homognous paamt in, whn th paamt matix is constant, th matizant is just a matix xponnt: ˆ j Pˆ xp jpˆ 6 M Onc th matizant is known, it bcoms possib to find a soution with th givn bounday pobms. Fo xamp, fo th simp bounday pobm, dscibd in Fig., on has fo any point : i i Mˆ I U jpˆ I 7 But, h th cunt in th point condition: is unknown. To find it on uss th scond bounday Z I Mˆ I I U jpˆ I 8 This yids two quations fo th unknown cunts: ZM jpˆ M jpˆ I U M jpˆ ZM jpˆ M jpˆ M jpˆ M jpˆ M jpˆ I U 9a, b M jpˆ Z M jpˆ Anoth way to cacuat th matizant is to us an infinit sis of ptubation thoy Vota sis. This and many anoth poptis of matizant a dscibd in [5]. 7

8 Eq. 3 with 9a yids th soution in an abitay point. 3. Appoximat quations fo th goba paamts of TST ptubation thoy. Th soution of systm 8a, b with paamt matix P of and usua bounday conditions fo th cunts and votags diffncs of potntias in th points and obtaind by th matizant mthod dscibd in th pvious subsction o by any oth mthod yids th cunt and votag distibution aong th in fo abitaiy givn vaus of th tmina soucs and oads. Th pocdu is convnint, whn th xact vaus of th functions, and, a known, fom ith known anaytica soutions [4, 5] o numica soutions [5]. Anoth way is to oganiz som itation pocdu. Gnay, th appoximat soution of th systm is dfind in th fist stp. Thn this soution is usd to find th cosponding paamts, tc. In [-3] and [7, 8] th static distibutions fo th cunt and potntia w usd as th owst itation stp. Th fist itation fo th paamts was obtaind aft som numica pocdus. Anoth pocdu, which is basd on th assumption of a thin wi, is poposd in [4, 5, 6]. In th zoth stp in this pocdu th systm within th ogaithmica accuacy was ducd to a cassica T systm with constant paamts. Th soution of this systm with its soucs 4 yids th functions fo th cunt of th fist itation, and, of which th ina combinations can b psntd up to a constant facto as fowad and backwad popagating cunt wavs: jk jk I I a, b Howv, xact quations a, b a usd fo th scaa potntia in th fist itation and fo its divativ. Aft staightfowad cacuations on obtains th paamt matix in th fist od appoximation: c ˆ P c In Eq. on has usd th foowing xpssions fo th appoximat inductanc and capacitanc, spctivy: 4 g I, jk d ; g I 4, jk d 3a, b Futh, as mntiond in th Intoduction, assuming that th matix of paamts is known, on dfins th fction and tansmission cofficints using th componnts of th matizant. In th nxt sction, it wi b shown how this knowdg can b usd to cacuat th spons of a ong in with ativy sma non-unifom pics. 8

9 III. DETERMINATION OF THE REFETION- AND TRANSMISSION-OEFFIIENTS. assica tansmission-in thoy Fom cassica tansmission-in thoy fo a thin and ossss wi abov pfcty conducting gound on knows th cosponding physica quations fo votag and cunt du z di z di ji z and ju z, Uz 4a,b,c dz dz j dz Fom ths quations on can div fowad and backwad unning votag o cunt wavs and, with thi aid, dfin th fction cofficints at both nds of a tminatd in. In th foowing sction and thoughout th st of th pap, th us of cunt wavs is pfd. Fist, th fction cofficint at th bginning of th in sha b dtmind. Fo this pupos it is assumd that an incoming wav coms fom th ight, is fctd at th bginning of th in and uns back to th ight hand sid of th in which is tminatd by its chaactistic impdanc. This avoids fctions fom th nd of th in. Expssd in fomua, th cunt and votag thn ad using 4b, c : jkz jkz Iz I R jkz jkz and U z IZ R 5,a,b incoming wav outgoing wav Equations 5 a and b a now sovd with spct to R and sut in: jkz U z ZI z R UzZ Iz Insting th known soutions of cassica tansmission-in thoy fo whby th matizant mnts a sin and cosin functions, suts in M z, cos Z kz jz sin kz j sin kz coskz Uz 6 and I z, 7a and kz jz sin kz cos Uz U Z j I M z, Iz sin kz coskz I Z 7b on obtains fo z, T th constant fction cofficint: R Z Z In od to obtain th ight hand sid fction cofficint R on has instad of 5a: and instad of 6 on gts: Z Z jkz jk zt T I z I R incoming wav outgoing wav 8 9 9

10 R U z ZI z Uz ZIz U z and jk z T Now, again using th pop matizant xpssion fo I z in 3 3 U z U T Z M z, T T I T M z, T I z I 3 T and aivs at th known sut: Z Z R 3 Z Z Evntuay, it is aso known fom cassica tansmission-in thoy that th cunt aong th in can b xpssd with th aid of th fction cofficints as jkz jk T z I z U R jk 33 T Z Z RR H, th quantity T dnots th hoizonta ngth of th in. Th iss a not takn into account in ctt du to th condition kh<<. Th tm in 33 jkz U Z Z dnots th fist outgoing cunt wav bfo it is fctd fo th fist tim at th nd. Thus it can b wittn as jkz U z ZI z I z: z U Z Z Z fowad unning wav jkz 33a Th numato of th midd quotint in 33a is a fowad unning cunt wav which is nomaizd with Z. Th function z can b xpssd by: jkz U z ZI z z, 33b Z an xpssion that on wi ncount again at on. At this point th qustion aiss whth th abov suts of cassica T thoy can b appid in a gnaizd fom in TST, and if so, und which stictions.. Th Rfction officints at th nds of th in in TST Th fist obsvation in a dsciption of a cassica tansmission-in configuation in th famwok of TST is th incusion of th iss at both nds of th in. Moov, as a Maxwian thoy, in th TST th in paamts bcom compx vaud, dpnd on fquncy and oca coodinats, and th conducto adiats. Obviousy, most of th adiation of ctomagntic ngy is mittd aound th bnds conncting th hoizonta pat of th in with th iss. Sinc, howv, adiation is a ong-anging intaction vy pat of th in is pincipay affctd. Nvthss, th is an xtndd pat of th hoizonta pic of th in, in which th adiation can b ngctd and th TEM mod dominats. A oth mods mainy occu aound th bnds. Thus, aong th TEM mod sctions of th in th potntia psnts votag and can b masud.

11 Fo th foowing anaysis it is assumd that aong th considd non-homognous in sctions xist that a dominatd by TEM mods which spaat thos pats of th conducto which a non-homognous. D facto, this mans that such non-unifom in pats which a spaatd by unifom in pics do not intact by adiation. Fo a cassica in configuation it foows that th adiation intaction btwn th two iss is ngctd. To fufi this quimnt, th ngth of th in must b much ag than its hight. In th TST th votag U z of ctt is pacd by th potntia and th cunt I z by i. Th paamt dnots th natua paamt ac ngth aong th in. Th soutions fo th potntia and th cunt a assumd to b known by th matizant. Thn in anaogy to th abov considations th gnaizd fction cofficints a dfind by th quotints of incoming and outgoing cunt wavs: jk i Z : i Z and R Z i jk : 34a,b Zi R Th invstigatd conducto configuation is dpictd in Figu. Fig.: Schmatic in configuation. dnots th tota ngth, bgins and wh th TEM gion nds. hight of th in abov gound. U is th fding souc, Z and is th coodinat wh th TEM gion Z a th tminations, h is th As can b sn fom Fig. th conducto is dividd into th pats: Two inhomognous pats which un fom to and fom to. Th thid pat of th in concns th cnta TEM gion, in which th cassica soutions appy. As assumd th soution fo th tota in is givn by th matizant M,. Du to th goup popty of th matizant th tota soution can b bokn down into patia soutions fo ctain sctions of th in. Fo th cas shown in Fig. on can wit:,,,, M M M M 35 III II I H M II, coincids with th soution in 7a. At th connction points and, th in paamts hav to b adjustd accodingy, so that th poduct soution givs th tota soution s Rf. [7]. 3. acuation of th fction cofficint at th nd of th in To cacuat R consid a cunt wav coming fom minus infinity and unning to th ight nd of th conducto. Th it is fctd by R and uns back to infinity. An incoming wav fom infinity was chosn to civ no fctions fom th ft nd of th in.

12 This can aso b achivd by tminating th in on th ft sid with th chaactistic impdanc s Figu 3. Fig.3: Fo th cacuation of R. Th quotint in 34a is now xpssd by th mnts of th matizant. Taking into account U Z i th bounday condition at th ight hand sid of th wi on gts via th matizant ation Z M, im, i i, 36 th dsid fina sut fo R : R, ZZ,,, ZZ,,, R jk M M M Z M Z, M M M Z M Z In a simia mann, 4. Th Rfction cofficint at th bginning of th in In this cas, it is assumd that a cunt wav coming fom th ight is fctd at th bginning of th in by R. This situation is iustatd in Figu Fig.4: onducto configuation to cacuat R. Now th quotint 34b is xpssd by th matizant mnts taking into account th Z i. On has: bounday condition at th ft sid of th conducto: Z M, i M, i i, 38 Thn, instion of and i fom 38 into 34b yids th foowing fo R :,,,,,,,, M Z Z M M Z M Z jk M ZZ M M Z M Z 39 R

13 5. Th cunt ampitud function fo th stating outgoing wav Th divation of is basd on Figus 5 and 6. Fig. 5: Iustation fo th divation of. Fig.6: Fig.6: Infinity fomay is pacd by Z An infinit wi with a ft-hand is is fd by a votag souc. This mans that ony an outgoing wav xists, and fo th cunt and fo th potntia on can mak th foowing quations: jk i jk and Z 4a,b Th assumption of an infinit in is mad to iminat fctions fom th ight hand-sid. Howv, fctions aso do not occu at th nd, vn if th conducto is tminatd by Z s Fig. 6. Thfo, th chaactistic impdanc occus in th xpssion fo th potntia. Taking th sum of 4a, b on obtains fo, th anaogu sut to 33b, namy th ampitud function of th outgoing wav. i Z jk 4 Z Nxt, th sut of 4 is psntd with th aid of th matizant and its mnts. It appis: U U Z i M, i i o in tms of th matizant mnts:, 4 i Z M,U Zi M,i i M,U Zi M,i 43 Division of 43 though 44 yids Z : 44 Z,,,,,, M U ZM M i M U Z M M i 45 Equation 45 now can b sovd with spct to i i U M, Z M,,,,, Z Z M M Z M Z M 46 3

14 This i is instd in quations 43 and 44, and thn th suting tms a usd in 4. If on sti taks into account that th dtminant of th matizant is, thn on finay aivs at th dsid sut fo : U jk 47. ZZM, M, Z M, Z M, appoachs th cassica vau if on insts th matizant mnts of 7a into 47. With this ampitud function fo th cunt on obtains an impotant intmdiat sut. 6. Intmdiat Rsut fo th cunt of a hoizonta in with two iss,, and At this stag th dvopmnt of th fomua R R it is possib to dscib th cunt distibution aong th in dpictd in Fig.. This is a conducto with two iss and a ativy ong hoizonta sction. In a at stp, such a in bcoms mo compicatd by adding an additiona scatt in th cnta pat of th in. jk To div th cunt on th in, on stats at th ft nd with th outgoing wav. This wav jk jk is fctd at th ight nd and uns back to th ft: R. At th bginning it is fctd th fist tim and again uns to th oth sid: jk jk R R jk. This pocss pats itsf infinity oftn and can b summaizd in two sums s Fig. 7. Fig.7: Schmatic psntation of summing up th fist th summands. o shot: n n jk jk jk jk jk jk jk i R R R R R n n i R RR jk jk jk jk This is an intsting intmdiat sut. Fist, it has its pndant in ctt with Equation 33a. Howv, 49 has a much wid vaidity gion: It is not ony vaid if kh cassica thoy but aso if kh. Thus, at ths high fquncis of -4GHz, th iss at th nds can aady b cognizd sovd, unik in th ctt wh th hight of th conducto abov gound 4

15 dos not occu in 33a. Anoth mak to b mad h concns th dfinition gims of th functions in 49. Th tota matizant soution is dividd into th spaat pats, which a aso sovd spaaty and thn at popy matchd up again to div th tota soution. Thus, th quantitis a dfind in th foowing intvas: in, R in, and R in. It shoud b notd that th is a common intva in whi ch a th functions a dfind and bcom constant:.in this intva on obsvs a dominating popagating TEM mod. Thfo, it can b concudd that 49 is ony dfind in. Howv, th is anoth point of viw fom which th pobm can b considd. Sinc th matizant is known fo th who soution aong th in, th oca coodinat is not stictd to ctain intvas in th abov fomua containing th matix mnts of th matizant. A quantitis which nt 49 a dfind ov th tota intva. Both pspctivs must, howv, guaant that th TEM mods dominat in th vicinity of th nds of th in. Btwn ths two TEM-pics th in may bcom again inhomognous. This wi bcom th subjct of th nxt chapts. IV. TRANSMISSION AND REFETION OEFFIIENTS OF THE SATTERER A t this stag of th wok th wiing is changd so that a scatt is instd in th cnta aa of th in. Th situation is shown in Fig. 8. Fig. 8: Schmatic psntation of a conducto with two iss, two paa sctions, and a scatt in th cnta pat. In Fig. 8 th in is composd of fiv pats: Th scatt in th midd pat pat III is mbddd in two asymptotic gions pats II and IV, ach of which mak connctions to th iss pats I and V. Th two paa pats of th in may hav diffnt hights abov gound. At an appopiat point insid th scatt a fnc point f. is sctd. If th scatt psnts a hoizonta bnd, thn th fnc point coud b xacty at th bnd point. Now, th a again two ways to cacuat th cunt distibution aong such a conducto. Fist, on can cacuat a fiv pats of th in individuay. Whn assmbing th individua soutions on must, howv, nsu that th in paamts a matchd against ach oth at th joints. O, on cacuats th matizant fo th nti in and thn spits it into sva pats accoding to M, M, M, M, 4 M 4, 3 3 M, 5 In th scond cas on aady has a fiv patia soutions av aiab and thy can b usd to cacuat a fction- and tansmission cofficints of th in. This taks pac in sva stps. 5

16 In stp on it is assumd that a wav coming fom th ft is appoaching th scatt whby on potion D of th wav is passing th scatt and anoth potion R is fctd by th scatt. Th configuation is dpictd in Fig.9. Fig. 9: Rpsntation fo th cacuation of R and D. hoosing a II, th TEM gion, thn on obtains th usua quations: jkf jkf jkf jk f i I R I Z R and Soving fo R givs i Z f i Z jk R 5a,b Not that th zo phass aways occu at th ocations of th fnc points of th scatts. Fo th tansmission cofficint, th foowing quations a vaid in zon IV. i I D On th oth hand, on has th ation: jk f and 5 jk f I Z D 53a,b M, i 54 i Equations 5a, b a instd into th ft hand sid of 54, whi on uss 5a, b togth with th known Matizant in th ight hand sid of 54. Thn, on aivs at two quations fo and, and aftwads can sov thm, which suts in: R and D Z jk M, Z M, M, Z, Z M Z 55 D R jk,,,, f,,,, M Z M M Z Z M Z M Z M M Z Z M Z 56 In stp two a wav coming fom th ight is considd which is patiay fctd R and patiay tansmittd by th scatt. Figu sktchs this situation. D 6

17 Fig. : Sktch fo th stimation of D and Anaogous to quations 5 and 53 on now stats with two simia quations: jk f jk I D Z i I D f 57a,b and. R and jk f jk f IZ R jk f jkf, i I R 58a,b Th ight hand sids of 57 and 58 a instd in th matizant ation 59 M, i 59 i and D. This again ads to two quations fo th two unknowns R jk Z D M, Z M, M, Z Z M, Z 6 R jk,,,, f,,,, M Z M M Z Z M Z M Z M M Z Z M Z 6 Basd on th suts of 55, 56, and 6, 6 th thid stp can b caid out: namy, th stimation of th cunt in th intva, f. By intoducing a futh fction cofficint R th subsqunt cacuation can b simpifid. Its ffct is shown in Figu. Fig.: Sktch of th ffct of R. 7

18 In od to xpss R in ation to a th oth fction and tansmission cofficints, R,, D R,, and D R, on assums a wav coming fom that is patiay passing though th scatt D and patiay bing fctd by th scatt R. At th nd of th in it is again fctd R and uns backwad, again patiay passing though th scatt D and patiay bing fctd R. A pats of th wavs passing though th scatt coming fom th ight vanish at. Howv, th scatting pocss btwn th inhomognous pat of th conducto and th nd of th in is patd an infinit numb of tims s Fig.. Fig.: Dfinition of th fction cofficint R Th distanc btwn f th coodinat oigin fo this pocss and th nd of th in is dnotd by :. Thn, on gts fo : f R o R R D R D D R R R D jk jk jk jk jk... piodica pat n jk jk jk jk jk DRD jk n R R R R D R R R D R 6 63 At this point, a sut 49 of th pvious chapt is usd and wittn on th psnt fction cofficint R. jk jkf jk R i,, jk f f R R 64 Rpacing R in 64 by xpssion 63 on obtains a ong quation fo i : jk jk jk jkf jk i R R R R D D R R R R R R R R D D R R jk jkf jkf 65 i fo f, mains to b stimatd. jk An incoming wav fom th ft I : f passs th scatt and is fctd at th nd by R, uns via back to th scatt and th is fctd again. This pocss is patd infinity oftn fo xpanations s Fig.3. Finay, in th ast stp th cunt 8

19 Fig.3: Scatting pocss btwn scatt and th nd of th in. Tansating th abov scatting pocss into a fomua yids: jk jk jk jk jk jk jk jk jk jk i I D D R D R R D R R R... jk jk jk jk D D R f jk jkf I R jk jk ID RR RR jk RR 66 In 66 a quantitis a known xcpt th ampitud function I. This function is takn fom th outgoing wav pat fom 64: Equation 67 is now usd in 66, R fo i : i jkf jkf R R I 67 is pacd by 63, and finay on obtains th sut jk jkf jk D R R R R R R R D D R R jk jkf jk f, f, 68 With 64 and 68, th cunt distibution aong th nti in is known, xpssd in tms of fction and tansmission cofficints. Th quantity is takn fom th pvious chapt, as w as th quantitis R R. and A spcia cas of th abov gna scatt woud b a hoizonta in conductd hoizontay abov gound with a bnd. In this cas, som fomua can b simpifid accoding to R R R, D D D, and Z Z. 9

20 V. NUMERIA EXAMPES AND DISUSSION Rcnty, a numica xamp was shown [7] fo a unifom T abov a conducting gound pan with iss on ach nd. Th hoizonta ngth of th in was cm and th hight ov gound was 5 cm ading to a tota ac ngth of cm fo th T. It coud b shown that th matizant of th who in can b composd of th matizants fo th is gions using TST and th cassica matizant fo th asymptotica unifom pat btwn th is gions. It was aso shown that th cunt in th asymptotica gion can b cacuatd using th concpt of advancd fction cofficints R and R which can b cacuatd using th paamt matix mnts of th TST anaysis of th is gions. In th foowing sction, non-unifomity is aso aowd in th midd gion of th T. This non-unifomity can b oca, as fo xamp a sing bnd in th in, o th can b a gion with distibutd nonunifomity.. Hoizonta in with on bnd Fist, a sing oca bnd in th midd of an othwis unifom T with iss on ach nd is invstigatd. Th T configuation is shown in Fig. 4. Fig. 4: Unifom T with iss at ach nd and a 9 bnd in th midd of th T. Th hight of th iss is h = 5 cm. Th ngth of th hoizonta pat btwn th iss is cm and th is a 9 bnd xacty in th midd of th in. In TST th natua paamt of th T is th ac ngth. Th top of th ft is is at = 5 cm, th hoizonta bnd at = 5 cm, th top of th ight is at = 5 cm and th nd of th in at = cm. This ads to a tota T ac ngth of = cm. Th position of th hoizonta bnd is dsignatd by f = 5 cm. On th ft sid th T is divn by votag souc with U = V and souc impdanc Z = 5. Th in is tminatd with oad impdanc Z = 5 at th ight nd. Fo th T configuation of Fig.4 a TST anaysis [8] was pfomd ading to th oca, fquncy dpndnt and compx vaud T paamt matix mnts shown in Fig. 5.

21 Fig. 5 : TST paamt matix mnts fo th T with iss and 9 bnd fom Fig. 4 at a fquncy of GHz cosponding to th p unit ngth inductanc of a cassica T. Rgading th a pat of th TST paamt matix mnt P * cosponding to th p unit ngth inductanc of a cassica T th dviations fom th cassica vau in th is gions and at th 9 bnd in th midd of th T a obvious Fig. 5 ft. But th a two gions btwn th iss and th midd bnd w th cassica vaus a achd and, thfo, th TEM mod is dominant. This is tu fo th ac ngth gion fom about = cm to = 88 cm and = cm to = 88 cm. In Fig. 5 ight th imaginay pat of th cospondnc is shown to b zo fo th cassica T, bcaus in cassica tansmission-in thoy adiation ffcts do not occu. In TST adiation ffcts a incudd and, thfo, th paamt matix mnts a compx in gna. Athough adiation is a coopativ pocss of th who T [7], th toughs at th is positions and at th hoizonta bnd indicat th most adiating pats of th T wh th is a distinct non-unifomity of th in. Th matizants intoducd in sction II a cacuatd using th mnts of th TST paamt matix P * [8]. Fo futh invstigation of th T mod Fig. 4 th ac ngths and a st in th spctiv asymptotic gions using = 55 cm and = 55 cm. Accoding to th fomuas in sction III. th fction cofficints R and R fo th in nds a cacuatd wh R R R bcaus of symmty. Fo th oca scatt hoizonta 9 bnd in th midd of th T at ac ngth = f = 5 cm fnc point th fction and tansmission cofficints w cacuatd accoding to th fomuas in sction III. Bcaus of th symmty of th T in ation to th fnc point, on can wit R R R and D D D. Fig.6 shows th fquncy dpndncy of th mntiond cofficints. Both tminating impdancs on th ft and ight nd of th T hav a vau of 5 suting in a constant cassica fction cofficint R cass =.73. Bcaus of adiation ffcts and th gnation of non TEM mods in th non-unifom, bnt T, th TST fction cofficint R dcass fom th cassica vau at high fquncis.

22 Fig. 6: Advancd and fquncy dpndnt fction and tansmission cofficints fo th T with iss at th nds and a oca bnd in th midd of th in. Th TST tansmission cofficint D fo th oca scatt is fo ow fquncis and dcass aso fo high fquncis. Th cosponding fction cofficint R is fo ow fquncis and incass fo high fquncis bcaus of th mntiond adiating ngy osss and th gnation of non TEM mods, mainy in th vicinity of th nonunifom pats of th T. Th fction cofficint R is a kind of combination of th ight hand fction cofficint R fo th tmination of th in incuding th is ffcts and th tansmission and fction cofficints of th oca scatt D and R. Fig. 7: unt in th ft hand asymptotic gion at position of th T configuation accoding to Fig.4. Pat of th initia ctomagntic wav coming fom th votag souc on th ft sid of th T bouncs back and foth btwn th scatting cnt at th ight hand is and th oca scatt 9 bnd in th midd of th T. This is th ason fo th osciation in th cous of R.

23 Using fomua 64 fo th cunt in th ft hand asymptotic gion at it can b shown that th is an xcnt agmnt with th cunt suting fom a pu TST cacuation. This situation is shown abov in Fig. 7. Th cospondnc fo th cunt in th ight hand asymptotic gion at position accoding to fomua 68 with th pu TST cacuation is aso xcnt. Th fquncy dpndnt cous of I is shown in Fig. 8. Fig. 8: unt in th ight hand asymptotic gion at position of th T configuation, accoding to Fig.4.. Simp on in simuato Now, th oca non-unifomity in th midd of th T is pacd by a distibutd nonunifomity. Th cous of th T is dfind so that th is sti a ft and ight hand asymptotica gion btwn th ft and ight hand iss and th distibutd non-unifomity is in th midd pat. Th actua gomty is dispayd in Fig. 9. On can think of a pojction of an opn TEM wavguid stuctu onto th vtica axia pan yz-pan. Th tota dimnsion of th T in z-diction is again m. Bcaus of th iss and th distinct vtica xtnt, th tota ac ngth with = 5.5 cm is significanty ag. Stating with ac ngth = 6 cm th is a sop up to th top bnd which is aso dfind as th fnc point with f = 44 cm. Fom th top bnd th is a ativy stp dcin towads th ight hand asymptotica gion, which again is at a hight of 5 cm abov th conducting gound pan. Fig. shows th ik a pat of th P * paamt matix mnt of TST anaysis aong th T ac ngth fo a fquncy of f = GHz. It is obvious that th is significant dviation fom th cassica constant vau. Th bnds of two iss, i.., th top bnd and th bnd btwn th ight-hand nd of th distibutd non-unifom gion and th ight-hand asymptotica gion, ad to distinct toughs in th a pat of. In th ft sop gion th is itt osciation, whi in th stp ight hand sop gion osciation is obvious, which shows th psnc of aky mods [8]. It is aso obvious that actua T vaus a pat of in th asymptotica gions do not fit to th cassica vaus as w as in th pvious xamp s Fig.5. Th ason fo this is that th assumptions fo an asymptotica gion a not fufid vy w bcaus of th shotnss of th hoizonta T pats. But, at on, it is shown that vn with ths stictd conditions th cunt in th asymptotica gions can b cacuatd quit accuaty. 3

24 Fig. 9: onstuction dtais of th T with ft and ight hand iss and a vticay distibutd non-unifomity in th midd pat. Fig. : Ra pat of ik TST paamt matix mnt fo th T with iss and distibutd non-unifomity in th midd pat, accoding to Fig.9. Th appopiat fction cofficints fo th is pats and th fction and tansmission cofficints fo th distibutd non-unifom midd pat w cacuatd accoding to th fomuas in th pvious sctions using th TST matizants fo th actua T configuation fom Fig.9. Th suts a shown in Fig.. Of cous, now th vaus fo R and R diff bcaus th is no ong any symmty fo th T mod. Howv, it can b gnay shown fom 55 and 6 that D and D a aways qua to ach oth. If this dos not sut fom th numics thn th asymptotic gions aound th scatts w not chosn ag nough. In od to not ovoad Figu ony th + vaus a shown. 4

25 Fig. : Advancd and fquncy dpndnt fction and tansmission cofficints fo th T with iss and distibutd non-unifomity in th midd of th in simp on in simuato Bcaus of th stong non-unifomity of th midd pat in th actua T mod, th tansmission cofficint D is significanty sma and th fction cofficint R significanty ag than in th pvious xamp, whi fction cofficints fo th in nds main nay th sam. Th shown cofficints of Fig. w cacuatd fo ac ngth = 35 cm, which is in th ft hand asymptotic gion s Fig 9. Th - cofficints ndd fo th futh cunt cacuations w cacuatd fo ac ngth = 6 cm ying in th ight hand asymptotic gion. Fig. : unts magnitud in th asymptotica gions at ft and ight using advancd fction and tansmission cofficints in compaison with pu TST cacuations. Using th appopiat cacuatd cofficints fo th cunt dtmination accoding to 64 and 68 it can b shown that th agmnt with pu TST cacuations is again vy good. Th suts a shown in Fig. fo th cunt at ac ngths = ft and = ight. Athough th quimnts fo an asymptotic gion a not pfcty fufid, as discussd abov, th dvopd vauation pocdu sms to b quit toant of this fact. 5

26 VI. ONUSION In th TST, th divd gnaizd fction and tansmission cofficints in this pap a usd to cacuat th cunts aong pactica ayouts of non-homognous tansmission ins. Th obtaind suts w compad with thos of an xact TST cacuation, and an xcnt agmnt was obsvd in th invstigatd high fquncy gion. Not, that fo th divation of th R s and D s, paa conducto sctions with dominating TEM mods w ncssay btwn th cnta scatt and th non-homognous iss. Thus, adiation intaction was stictd ony within th individua inhomognous conducto pats. Evn an xtnsion in th high GHz am up to 4 GHz did not chang th xcnt agmnt of th suts fo th cunts. Th psntation of th cunts in tms of oca and fquncy dpndnt fction and tansmission cofficints ad to fomuas which smb thos of cassica T thoy. In paticua, th fomuas a quit pactica whn soving fo cunt pos in th compx pan, and thus faciitat an SEM anaysis [9]. REFERENES. H. Haas and J. Nitsch, Fu-wav tansmission in thoy FWTT fo th anaysis of th-dimnsiona wi-ik stuctus, in Pocdings of 4 th Intnationa Zuich Symposium on EM, Fb., pp H. Haas, J. Nitsch, and T. Stinmtz, Tansmission - in Sup Thoy: A Nw Appoach to an Effctiv acuation of Ectomagntic Intactions, Radio Scinc Butin, 37, 3, pp H. Haas, T. Stinmtz, and J. Nitsch, Nw Popagation Mods fo Ectomagntic Wavs aong Unifom and Non-unifom abs, IEEE Tansaction on Ectomagntic ompatibiity, EM-47, 3, 4, pp J. Nitsch and S. Tkachnko, Nwst Dvopmnts in Tansmission - in Thoy and Appications, Intaction Nots, Not J. Nitsch and S. Tkachnko, Goba and Moda Paamts in th Gnaizd Tansmission in Thoy and thi Physica Maning ; Radio Scinc Butin, 3,Mach 5, pp J. Nitsch, S. Tkachnko, High-fquncy Muticonducto Tansmission in Thoy, Foundation of Physics 4: 3-5, DOI.7/s J. Nitsch, G. Wonbg, and F. Gonwad, Radiating Nonunifom Tansmission-in Systms and th Patia Emnt Equivant icuit Mthod, Wiy R. Rambousky, J. B. Nitsch, and H. Gab, Appication of th Tansmission-in Sup Thoy to Muti-wi TEM-Wavguid Stuctus, IEEE Tansaction on Ectomagntic ompatibiity, EM-55, 6, 3, pp S. Tkachnko, F. Rachidi, and M. Ianoz, "High-fquncy ctomagntic fid couping to ong tminatd ins", IEEE Tansactions on Ectomagntic ompatibiity, Vo. 43, No., May, p F. Rachidi and S. Tkachnko, ds., Ectomagntic Fid Intaction with Tansmission ins: Fom assica Thoy to HF Radiation Effcts hapts 4 and 5, WIT Pss 8. S.V. Tkachnko, J. Nitsch, R. Vick, F. Rachidi, D. Pojak, Singuaity xpansion mthod SEM fo ong tminatd tansmission ins, 3 Intnationa onfnc on Ectomagntics in Advancd Appications IEAA, Toino, 9-3 Spt. 3, pp: S.Tkachnko, F. Rachidi, M. Ianoz, "Ectomagntic Fid ouping to a in of Finit ngth: Thoy and a Fast Itativ Soutions in Fquncy and Tim domains," IEEE Tansaction on Ectomagntic ompatibiity, vo. 37, No. 4, Novmb

27 3. S. Tkachnko, F. Middstadt; J. Nitsch, R. Vick, G. ugin, F. Rachidi, High-fquncy ctomagntic fid couping to a ong finit in with vtica iss, In th Tansaction, Pocdings of th Gna Assmby of th Intnationa Union of Radio Scincs URSI, Bijing, 4 DOI:.9/URSIGASS Mi, Thoy of Maxwian icuits, Rad. Sci. Bu., 35, 3, pp F. R. Gantmach, Th Thoy of Matics, vo., hsa, Nw ok H. Haas, J.Nitsch, S. Tkachnko, A Fu-Wav Tansmission in Thoy, 5 Intnationa onfnc on Ectomagntics in Advancd Appications IEAA, Toino, Itay, Spt R. Rambousky, J. Nitsch, and S. Tkachnko, Appication of Tansmission in Sup Thoy to assica Tansmission ins with Riss, submittd fo pubication in Adv. Rad. Sci., Spt J. Nitsch, S. Tkachnko, Physica Intptation of th Paamts in th Fu-Wav Tansmission-in Thoy, ISTET 9, XV Intnationa Symposium on Thotica Ectica Engining, übck, -4 Jun, Gmany. 9. F. Mittstädt, S. Tkachnko, R. Vick, and R. Rambousky, Anaytic Appoximation of Natua Fquncis of Bnt Wi Stuctus abov Gound, submittd fo pubication to th joind Intnationa IEEE Symposium and EM EUROPE 5, Dsdn, Gmany. 7