Spectral and variational analysis of generalized cylindrical and elliptical strip and microstrip lines

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 38, NO. 9, SEPTEMBER 1990 1287 Spec al and Va ia ional Analysis o Gene alized Cylind ical and Ellip ical S ip and Mic os ip Lines Abs mc -In his pape , he a ia ional echnique in he spec al domain (VTSD) is shown o be an e icien me hod o compu ing he quasi-TEM pa ame e s o a bi a y mul iconduc o and mul idielec ic cylind ical o ellip ical s ip con igu a ions. Simple con o mal mappings educe he cylind ical o ellip ical geome ies o an equi alen ec angu- la one wi h pe iodic bounda y condi ions. Mino modi ica ions o p e ious wo k on plana s uc u es allow us o analyze any cylind ical o ellip ical geome y, compu ing he capaci ance [Cl and induc ance E1 ma ices, om which he e ec i e dielec ic cons an s and mode impedances a e ob ained. I. INTRODUCTION N THE RECENT mic owa e li e a u e, se e al pape s I ha e been de o ed o he analysis o s iplike and mic os iplike ansmission lines p in ed on lexible di- elec ics w apped a ound cylind ical o ellip ical su aces. These sys ems can be used o exci e con o mal a ays moun ed on cylind ical o ellip ical objec s. Coaxial- o- plana line ansi ions, slo ed lines, and wa ping due o se e e en i onmen al changes can also be con enien ly modeled wi h cylind ical s ips. A numbe o au ho s ha e s udied cylind ical and ellip- ical con igu a ions using he quasi-TEM model. Wang uses a dual se ies ep esen a ion o analyze he homoge- neous cylind ical s ipline and he inhomogeneous cylin- d ical mic os ip [l]. The modi ied esidue calculus ech- nique is used by Joshi e al. o sol e cylind ical and ellip ical s iplines [2]. Con o mal mappings ha e been applied o se e al simple con igu a ions [31-[6] and, e y ecen ly, he mo e in ol ed mul iconduc o s iplinelike p oblem [7]. The G een’s unc ion o mula ion wi h a ia- ional exp essions o he capaci ance is used o analyze ellip ic single [81 and coupled [91 a c s ips and a nonsym- me ical pai o coupled cylind ical s ips [lo]. Single [ll] Manusc ip ecei ed Augus 29, 1989; e ised Ap il 6, 1990. This wo k was suppo ed by DGICYT (P ojec PB87-0798-C03-01, 1988-91) and by he Jun a de Andalucia (P ojec P5355-2, 1988-90). The au ho s a e wi h he Depa men o de Elec 6nica y Elec omag- ne ismo, Facul ad de Fisica, Uni e sidad de Se illa, 41012, Se illa, Spain. IEEE Log Numbe 9036755. and b oad-side-coupled [ 121 cylind ical s ips ha e been analyzed sol ing he Laplace equa ion subjec o he ap- p op ia e bounda y condi ions in a h ee-dielec ic medium. The spec al-domain echnique in cylind ical coo dina es has been used o analyze single and symme - ically coupled a c s ips [13]. A signi ican s ep in he analysis o sys ems o his kind has been gi en in [141, whe e a gene al class o mul iconduc o cylind ical lines was s udied by aking ad an age o he pe iodici y o hese s uc u es and using he FFT algo i hm in conjunc- ion wi h an i e a i e scheme. In addi ion, a ull-wa e analysis o single and coupled cylind ical s ips p in ed on one-laye subs a e has been epo ed in [151 and [161. The aim o he p esen pape is o show ha gene al- ized cylind ical o con ocal ellip ical s ip geome ies (Fig. l(a)) can be easily s udied by using he heo y and com- pu e p og ams p e iously de eloped o plana s uc u es wi h ec angula bounda y condi ions. No e ha mos o he ansmission lines men ioned in he p e ious pa a- g aph can be iewed as pa icula cases o his gene ic mul iconduc o sys em. The me hod is based on he appli- ca ion o con o mal mappings which ans o m he cylind ical o ellip ical geome y in o a plana one wi h ec angula pe iodic bounda y condi ions. This equi alen s uc u e is analyzed by using he VTSD app oach de- sc ibed in [ 171, [ 181 me ely in oducing sligh modi ica- ions in he analy ical ea men and he compu e p o- g ams. This analysis allows one o compu e he [,!,] and [ C] ma ices cha ac e izing he sys em unde quasi-TEM ope a ion and, om hese, he modal impedances and e ec i e dielec ic cons an s. The o iginal p og ams based on [17] and [18] ha e been imp o ed by using be e ial unc ions and a con enien asymp o ic analysis in o de o accele a e he con e gence o he se ies appea ing in he nume ical compu a ion. In his way, we ha e a e y e i- cien algo i hm o cha ac e ize, unde he quasi-TEM assump ion, e y gene al cylind ical o con ocal ellip ical ansmission sys ems in ol ing an a bi a y numbe o dielec ic laye s and conduc o s dis ibu ed be ween hem, e en i hey a e p in ed in di e en in e aces. 0018-9480/90/0900-1287$01.00 01990 IEEE Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on July 21,2020 a 15:01:28 UTC om IEEE Xplo e. Res ic ions apply. 1288 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 38, NO. 9, SEPTEMBER 1990 ELLIPTIC YULTXSTR PS SYSTEM w - plane I' I.' 2- - '1:. I N Y I I I I I 1,!2 1 I lh . 1 I E2 I I I I U (b) - I CL i y1 0 I 'I 2n I uR Fig. 1. (a) C oss sec ion o gene alized ellip ical mul is ip and mul i- dielec ic s uc u e. The no a ion o geome ical pa ame e s de ining a c s ips and in e aces is included. When c = 0, ai = bi = ,, we ha e he cylind ical case. (b) Image s uc u e in w plane o he con igu a- ion in (a). This s uc u e is pe iodic in he U di ec ion wi h pe iod 2iT. 11. GENERAL CONFIGURATIONS AND CONFORMAL MAPPINGS The c oss sec ion o a gene alized con ocal ellip ical sys em has been d awn in Fig. l(a). Ou pu pose is o compu e he [L] and [C] ma ices o his mul iconduc o ansmission line. Ins ead o wo king in an ellip ical ame, we will exploi he heo y and he p og ams de eloped by he au ho s o ea ing gene alized plana con igu a ions [17], [NI. In o de o do his, we ans o m he o iginal ellip ical (Fig. l(a)) geome y in o an equi alen one wi h pe iodic ec angula bounda y condi ions (Fig. l(b)) ia he ollowing con o mal mapping: whe e he as e isk deno es complex conjuga e; 2c = ocal dis ance; z = x + jy; w = U + ju; and b, and a, a e he mino and majo semiaxes o inne g ound ellipse. The o iginal ellip ical con igu a ion in he z plane is mapped in o he ec angula one in he w plane. Wi h he mapping gi en by (l), geome ical pa ame e s (a) and (b) o Fig. 1 a e ela ed in he ollowing way: Hi = anh-' ( bi+' /ai+') - anh-' ( bi /ai) u = an-' [(ai/bi) an&] uk= an-' [(ai/bi) ancpk] ( 2) (using he no a ion in Fig. l(a) and (b)). Fo a cylind ical con igu a ion in he z plane ( ha is, c = 0, ai = b, = i in Fig. l(a)) a sui able mapping unc ion is w(z) = jln(z*/a). (3) In his way he cylind ical sys em is ans o med in o he ec angula one shown in Fig. l(b): H~ = In( ip' / i) u = 'pi U$ = &. ' (4) The equi alen con igu a ion (Fig. l(b)) consis s o an a bi a y numbe (N,) o conduc ing s ips lying in a ce ain numbe (M) o plana in e aces be ween Nd lossless iso opic dielec ic laye s, he whole being en- closed by a se o ec angula bounda y condi ions. The cylind ical o ellip ical con igu a ion and hei co e- sponding ans o med s uc u es ha e he same [ C] and [ L] ma ices (nonmagne ic ma e ials a e assumed). A gene ic con igu a ion simila o he one in Fig. l(b) has been e icien ly analyzed by he au ho s using he a ia- ional echnique in he spec al domain in [171 and [181. Since he analysis is pa allel o he one epo ed in [17] and [18], we e e he eade o hese pape s and o he Appendix he ein o de ails. A b ie accoun will be gi en in he ollowing sec ion. 111. NUMERICAL RESULTS The [C] ma ix is compu ed om he elec ical ene gy s o ed pe uni leng h in he s uc u e o an app op ia e se o exci a ions o he conduc o s ips [18]. The un- known su ace cha ge dis ibu ion on he s ips o each exci a ion is expanded in o a se o basis unc ions and he expansion coe icien s a e compu ed o minimize he elec- ical ene gy (Rayleigh-Ri z me hod). In o de o do his, we wo k in he Fou ie ans o m domain, since he spec al G een's unc ion ma ix (SGFM) needed in he analysis can be easily ob ained using he me hod epo ed in [17]. As is well known, [L] is ob ained om he [C] ma ix o he same s uc u e wi hou dielec ics. The nume ical e iciency o he Rayleigh-Ri z algo- i hm s ongly depends on he choice o ial unc ions o app oxima ing he unknown su ace cha ge densi y on he s ips. In he u esen wo k we use a se o basis unc ions w( z) = cos-l( z*/c) - j an-' (b, /a,) (1) which has p b ed o be e y good o bo h quasi-TEM Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on July 21,2020 a 15:01:28 UTC om IEEE Xplo e. Res ic ions apply. MEDINA AND HORNO: SPECTRAL AND VARIATIONAL ANALYSIS 1289 and ull-wa e analyses because hey inco po a e he sin- gula beha io o he su ace cha ge densi y (p) a he s ip edges and he p ope ies o Chebyshe polynomi- nals: n p’( U) = pi( U) + aipi( U) (5) m=l S’ = (U:. + 4)/2 W’= uk - U:. whe e TJ ) is a Chebyshe polynominal o he i s kind o o de n. A his poin , some a en ion mus be gi en o he nume ical con e gence o Fou ie se ies appea ing in he analysis (see [18, eqs. (8)-(10)1). The se ies a ising om in e ac ions be ween noncoplana s ips a e quickly con- e gen , because he o -diagonal elemen s o he SGFM ela ing he Fou ie ans o m o he po en ial and he su ace cha ge in di e en in e aces dec ease exponen- ially. Howe e , in e ac ions be ween s ips lying a he same in e ace yield poo ly con e gen se ies (i he basis unc ions in (5) a e used, he con e gence is as l/n2). A signi ican imp o emen is achie ed i he asymp o ic ail ex ac ion echnique is applied (see he Appendix). In Fig. 2, we show a ypical con e gence pa e n wi h and wi hou asymp o ic ex ac ion. Clea ly, asymp o ic ex ac- ion will esul in an impo an educ ion o CPU ime. This ac is e en mo e p onounced o mul iple-s ip con igu a ions, and i has been e i ied o a wide a ie y o simple and coupled s uc u es. On he basis o his heo y we ha e w i en a compu e p og am whose eliabili y and nume ical e iciency ha e been con enien ly es ed. Table I shows he con e gence o he no malized cha ac e is ic impedance (E,“*z,) wi h he numbe o ial unc ions o a cylind ical homoge- neous s ipline. We ha e included some esul s abula ed in [14, able I, N = 20481 o compa ison. The pa icula case a = 180, which can be exac ly sol ed by con o mal mapping (exac alue = 5.361 O), has also been included. The ag eemen wi h da a epo ed in [14] is excellen , and he exac alue is also ob ained. We can see om his able ha bigge angles equi e mo e basis unc ions. Ne e heless, CPU ime does no inc ease as quickly wi h he a c wid h, since bigge angles equi e adding ewe Fou ie e ms. Typically, ou signi ican digi s a e ob- ained in 50-200 ms on a VAX-11/785 compu e . No e ha ou me hod does no impose es ic ions on he s ip wid h, in con as o he me hod in [14], based in he applica ion o he FFT. The mul iconduc o case is also checked by compa - isons wi h da a epo ed in [14, able 1111. In Table I1 we p esen esul s o he mode e ec i e dielec ic cons an s o six coupled s ips. Sligh di e ences could be due o he ela i ely small numbe o samples used in I141 o compu e hese da a. In gene al, e y good ag eemen wi h da a published in [14] has been ound. Disc epancies wi h ‘0 70 20 30 Numbos o Founes o- Fig. 2. Pe cen de ia ion om he exac alue o he no malized capaci ance o a cylind ical s ip as a unc ion o he numbe o e ained Fou ie e ms wi h (i) and wi hou (ii) asymp o ic ex ac ion. g aphic da a published in [l] and [ll] poin ed ou in [141 ha e also been de ec ed. Since his disag eemen has been ound using wo comple ely di e en me hods, we can also conclude ha he accu acy o he esul s in 111 and [ll] is ques ionable. Edge-coupled symme ical a c s ips ha e been consid- e ed o compa e wi h he esul s in [5] and 1131. The me hod in [5] is alid when he geome ical pa ame e s a e wi hin a ce ain ange o alues. Fo hese, e y good ag eemen is ound. Howe e , he au ho s in [13] use as ial unc ion o he cha ge densi y jus he ze o h-o de e m in (5). This choice esul s in se ious nume ical e o s when s ips a e s ongly coupled. This is because he cha ge dis ibu ion is no symme ical a ound he cen e o he s ip, as is assumed in [13]. When wide a c s ips a e in ol ed, he cha ge dis ibu ion ends o be almos uni o m. Thus, signi ican disc epancies wi h esul s in [13] also a ise in his case. Table I11 illus a es hese ac s. Resul s in he i s column (n = 0) co espond o he me hod in [13]. In o de o ob ain accu a e esul s, a ew basis unc ions mus be e ained. No e ha odd ial unc ions a e closely ela ed o he coupling e ec , since hese ake in o accoun he asymme y o he cha ge dis ibu ion on coupled s ips. E en unc ions mus be added when wide s ips a e in ol ed. Simila accu acy p oblems a e also expec ed o occu in [8]-[101, whe e poo ial unc ions a e used. Noncoplana s ips can be also ea ed wi h ou p o- g am. We ha e also es ed his case. Fo ins ance, we ha e conside ed he symme ical b oadside-coupled s ips analyzed in [6] and [12]. In hese pape s, he a io be- ween adii is chosen in such a way ha e en and odd Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on July 21,2020 a 15:01:28 UTC om IEEE Xplo e. Res ic ions apply. 1290 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 38, NO. 9, SEPTEMBER 1990 a Resul s numbe o ial unc ions (n ) 1.00 19.95 39.99 60.03 80.07 99.93 119.97 140.01 in [141 N=2048 38.14 21.27 14.75 j 11.29 0 12 34 5 6 189.0 39.18 38.11 22.97 21.27 21.26 16.61 14.76 14.75 13.16 11.31 11.29 10.97 9. 180 9.162 9.437 7.718 7.698 7.6 8.303 6.659 6.638 6.6 180.00 TABLE I1 EFFECTIVE DIELECTRIC CONSTANTS OF SIX COUPLED STRIPLINES AS A FUNCTION OF SEPARATION ANGLE (a) Resul s epo ed in 114, able 1111. (b) Ou esul s. 8.453 5.452 5.379 5.366 5.363 5.362 5.361 modes can be de ined, and he analysis is applied o he single conduc o geome y co esponding o each mode. We ha e analyzed hese s uc u es bo h by conside ing hem as pa icula cases o he mo e gene al asymme ic con igu a ion and by using he heo y o odd and e en modes, in oducing he app op ia e elec ic o magne ic wall [12]. The nume ical esul s ob ained wi h bo h p oce- du es a e indis inguishable. Ve y good ag eemen has been ound wi h he con o mal mapping esul s in [61 o a wide ange o alues o a. Howe e , sligh disc epancies wi h he a ia ional esul s in [12] we e ound o high alues o a. This is due again o a lack o p ecision in he es ima ion o cha ge dis ibu ion, as is e iden om Fig. 3. In his igu e, we show he a io o phase eloci ies o b oadside-coupled mic os ip lines using n = 0 and n = 6 in (5). The compu ed esul s coincide wi h g aphic da a epo ed in [12] when n = 0, bu signi ican de ia ion is ound (10% o wide s ips) when mo e e ms a e used o app oxima e he cha ge dis ibu ion. In he limi ing case a = 180, which can be exac ly sol ed, we ge i e signi i- can digi s using ou basis unc ions in (5). In o de o check he alidi y o he p og am o ellip ic a c s ips, we ha e compa ed ou da a wi h he e y accu a e con o mal mapping esul s epo ed in [3]. These TABLE I11 EVEN- AND ODD-MODE CHARACTERISTIC IMPEDANCE OF EDGE-COUPLED CYLINDRICAL STRIPS numbe o ial unc ions (n ) 50.92 49.89 48.80 48.77 48.77 48.77 37.07 35.59 35.55 35.47 35.45 35.45 16.79 16.63 14.29 14.27 14.26 14.25 14.25 13.67 13.45 12.87 12.82 12.81 12.79 12.79 45.52 45.47 45.02 45.02 42.47 42.41 42.23 42.23 20 80 20 15.88 15.88 14.02 14.02 14.01 14.01 14.58 14.58 13.46 13.46 13.46 13.46 b/a=1.4; c/a=1.8; c =2 n = 0 co esponds o he echnique in [131 1.S . 1 1-4 b- -P- - - - 1.11 ow’ 1 1 1 50 50 100 Hol --ongl <d g > Fig. 3. Va ia ion o mode eloci y a ios o b oadside-coupled mi- c os ip lines as a unc ion o hal -angle s ip a c. No a ion is he same as ha in [12, ig. 61: - - - - - esul s in [12]; 0 ou esul s wi h n = 0 in (5); - ou esul s wi h n = 6. (i) p/a = 0.1, (ii) p/a = 0.3, (iii) p/a = 0.6, (i ) p/a = 1.0. b/a = 1.4, E, = 2.32. esul s a e limi ed o single ellip ical s ipline in homoge- neous medium, bu hey a e good enough o ou compa - ison pu pose. We ha e made compa isons wi h all g aphic da a epo ed in [3], and he ag eemen is excellen in all cases. In Fig. 4, o example, we show he cha ac e is ic impedance o an ellip ical s ipline as a unc ion o he a c wid h o di e en loca ions wi h espec o he majo axis (0,) and wo di e en dis ances o he inne ellip ical g ound (h /a). As expec ed, he cha ac e is ic impedance depends on Bo, which en e s as a new a iable no exis ing in he cylind ical case. In conclusion, he compu e p og ams based on he heo y in his pape can be used wi h con idence o compu e he cha ac e is ic pa ame e s o a e y wide a ie y o cylind ical o ellip ical ansmission sys ems. Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on July 21,2020 a 15:01:28 UTC om IEEE Xplo e. Res ic ions apply. MEDINA AND HORNO: SPECTRAL AND VARIATIONAL ANALYSIS Q bo i. 1 0 1291 c: ___-----__ _--- - -1 ..' , , _.... ~---- ................. _._._._____.___.___._._._._. ................ .:::;:: i 0 0.02 0.04 0.06 a, .........'.... , 300 = .5 I A ,A, :: 1 10 1 Angle (deg ees) Fig. 4. Cha ac e is ic impedances o ellip ic s ipline in homogeneous medium (ai ) e sus s ip angle (A 4): - esul s epo ed in [31; A ou esul s. Wi h he aim o including an example no epo ed in p e ious li e a u e, we show in Fig. 5 he e ec i e dielec- ic cons an s and mode impedances o a pai o asymme - ical coupled cylind ical s ips p in ed on a wo-laye subs a e. In Fig. 5(a) we can see ha he c- and -mode e ec i e dielec ic cons an s can be made almos equal by a p ope choice o he hickness o he inne dielec ic (which has he lowe dielec ic cons an ). The ange o alues o (b - a)/a whe e he e ec i e dielec ic con- s an s a e e y close is a ound a c i ical poin de ined by he equaliza ion o he induc i e and capaci i e couplings. In his zone, he mode line impedances exhibi he singu- la beha io shown in Fig. 5(b). The c o T na u e o each mode changes o each side o he c i ical poin ; ha is, he co esponding mode numbe changes he sign. This beha io is simila o ha epo ed in [19] o coupled asymme ic mic os ips wi h o e lay. A de ailed analysis o hese phenomena and an analysis o o he pa icula s uc u es a e ou o he scope o his pape , which ocuses mo e on he compu a ional me hod o sol e his kind o s uc u e han on he analysis o he beha io o pa icula con igu a ions. IV. CONCLUSIONS In his pape we ha e s a ed ha e y gene al single- conduc o o mul iconduc o cylind ical/ellip ical s ip con igu a ions embedded in a laye ed dielec ic medium can be educed, ia con o mal mapping, o a ec angula geome y wi h he same [I,] and [C] ma ices. The analy- sis o his equi alen s uc u e is achie ed by aking ad- <b-a>/a (b) Fig. 5. (a) E ec i e dielec ic cons an s o c and n- modes o a pai o asymme ic cylind ical s ips on wo-laye subs a e. (b) Modal line impedances o he same s uc u e. No e he singula beha iou a ound ZzT. a1 = lo", az = 20", 0 = 5", c/a = 1.2, c1 = 2.32, c2 = 10. ._._ he c i ical poin . ---- Zlc; ...... ZIT; - Zzc; .- an age o p e ious wo k on boxed plana s uc u es. The nume ical con e gence o he p og ams is g ea ly accele - a ed inco po a ing he asymp o ic beha io o he se ies appea ing in he analysis, in such a way ha e icien p og ams ha e been w i en. I has also been poin ed ou ha excessi ely simple app oxima ions o he su ace cha ge dis ibu ion yield meaning ul nume ical e o s, mainly when s ong coupling o wide s ips a e in ol ed. As an applica ion example, he singula beha io o he cha ac e is ic pa ame e s o asymme ic coupled s uc- u es on mul ilaye subs a es has been shown. APPENDIX He e we include ce ain ema ks on h_e compu a ion o he spec al G een's unc ion ma ix, [Gij(n)], wi h i, j = 1,. e, M, used in he VTSD [17], [18] and on nume ical aspec s conce ning he addi ion o Fou ie se ies in ol ed in he me hod. Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on July 21,2020 a 15:01:28 UTC om IEEE Xplo e. Res ic ions apply. 1292 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 38, NO. 9, SEPTEMBER 1990 The s uc u e analyzed in his pape (Fig. l(b)) di e s om he con igu a ions s udied in [171 and [181 in he na u e o he la e al bounda y condi ions. The s uc u e in his wo k is pe iodic wi h pe iod 277 in he U di ec ion ins ead o ha ing elec ic o magne ic sidewalls. Then, all he quan i ies a e Fou ie ans o med using he de ini- ion: F(n) =/2T (u)e~nudu. 0 (AI) As a consequence, [GJn)]-' =[L,,(n)] can be com- pu ed om [17, eqs. (2)-(17)] making he ollowing subs i- u ions: k,-n E,*: = Ey*:-E, (A2) whe e- he subsc ip i e e s o he i h laye in Fig. l(b), and [L,,(n)] is a idiagonal symme ical ma ix wi h ele- men s de ined by whe e Ljk and pk a e he Fou ie ans o m o ee su ace cha ge and he po en ial dis ibu ions on he k h in e - ace (see Fig. l(b)). Tha is, he o iginal bounda y condi- ion p oblem is sepa a ed in o simple p oblems de ined by he bounda y condi ions in (A3). The solu ions o hese p oblems a e gi en in [17, eqs. (12)-(17)] in e ms o ce ain gij(n) unc ions de ined in [17, eq. (611. The physi- cal meaning o hese unc ions is e iden om [17, eq. (5)l. They a e he ac o s ela ing he Fou ie ans o m o he ee cha ge densi y and he po en ial dis ibu ions in he con igu a ions shown in Fig. 6. On he o he hand, all he Fou ie se ies appea ing in [171 and [181 mus be added om n = --M o n = +-M ins ead o om n = 1 o n =W. In Sec ion I11 we e e ed o he need o aking in o accoun he asymp o ic beha - io o hese se ies in o de o achie e good con e gence. F om [17, eqs. (6) o (1711 we can see ha [Lij(n)l exponen ially con e ges o a diagonal ma ix whose ele- men s a e ik,k(n +w) =€o(El+E,+l)lnl=G~:(n +w), k=l,..* ,M (A41 whe e i and i + 1 e e o he dielec ic laye s adjacen o he k h in e ace Fo he basis unc ions in (5), he Fou ie se ies appea - ing when he me hod in [18] is applied has he ollowing o m: whe e Jp,q, Bessel unc ions o he i s kind o o de p,q, a e he Fou ie ans o ms o he basis unc ions in (5). The con e gence o S is no sa is ac o y. Howe e , i we conside he asymp o ic beha io o A(n) when n - CO, namely A,(n), we can ew i e (A9 as m m S= C [~(n)-~,(n)]+ C AAn) (~6) n= -m n= -m whe e cos (nu - p. /2- ~/4) cos( nb - q /2- ~/4) n2 A,( n) = C wi h C = 2/ [ . ( ab) 1'2~o( E: + E:' ')I . Now, a e some manipula ions, he second e m in (A6) is exp essed as a combina ion o igonome ic se ies. One o hese se ies can be exp essed in closed o m; he o he is educed o an equi alen one con e ging in a ew e ms by means o he esidue echnique. In ou p og ams we ha e used T2 x(27-x) , 0<X<2T 6 4 n=l l-log(x)+x2 ( -+- 7: 1:4i0)], 4 9 sinh[m. (. -x)/2] 7 m,odd m2sinh(m. 2/2) [l- anh(n)] - - sin(nx) c ------"( -(2. -x) l-log(2~-x)+(2. -x)2 n=l n2 { n=l x < 0.5 , (27-X) <OS 0.5 < x < (2~ -0.5) (eigh digi s being co ec in he wo s case using hese exp essions). Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on July 21,2020 a 15:01:28 UTC om IEEE Xplo e. Res ic ions apply. 1293 MEDINA AND HORNO: SPECTRAL AND VARIATIONAL ANALYSIS (b) S uc u es de ining he elemen a y spec al unc ions needed o comp e he global ma ix G een’s unc ion: (a) de ini ion o 2Jn) 7 c?~/Y; (b) de ini ion o ~ ,l~i(n)=C?l/~~i. The quan i ies C?* and V, a e he Fou ie ans o ms o he ee su ace cha ge densi y and he po en ial dis ibu ion. Fig. 6. [14] C. H. Chan and R. Mi a, “Analysis o a class o cylind ical mul iconduc o ansmission lines using an i e a i e app oach,” IEEE T ans. Mic owa e Theo y Tech., ol. MTT-35, pp. 415-424, Ap . 1987. [15] N. G. Alexopoulos and A. Naka ani, “Cylind ical subs a e mi- c os ip line cha ac e iza ion,” IEEE T ans. Mic owaue Theo y Tech., ol. MTT-35, pp. 843-849, Sep . 1988. [16] A. Naka ani and N. G. Alexopoulos, “Coupled mic os ip lines on a cylind ical subs a e,” IEEE T ans. Mic owa e Theo y Tech., ol. MTT-35, pp. 1392-1398, Dec. 1987. 1171 F. Medina and M. Ho no, “De e mina ion o G een’s unc ion ma ix o mul iconduc o and aniso opic mul idielec ic plana ansmission lines: A a ia ional app oach,” IEEE T ans. Mi- c owaue Theo y Tech., ol. MTT-33, pp. 933-940, Oc . 1985. [18] F. Medina and M. Ho no, “Capi ance and induc ance ma ices o mul is ip s uc u es in mul ilaye ed aniso opic dielec ics,” ZEEE T ans. Mic owa e Theo y Tech., ol. MTT-35, pp. 1002-1008, No . 1987. [ 191 K. Sachse, “Analysis o asymme ical inhomogeneous coupled-line di ec ional couple ,” in P oc. 18 h Eu opean Mic owa e Con ., S ockholm (Sweden), Sep . 1988, pp. 985-990. REFERENCES Y.-C. Wang, “Cylind ical and cylind ically wa ped s ip and mi- c os iplines,” IEEE T ans. Mic owaue Theo y Tech., ol. MTT-26, pp. 20-23, Jan. 1978. K. K. Joshi and B. N. Das, “Analysis o ellip ic and cylind ical s iplines using Laplace’s equa ion,” IEEE T ans. Mic owaue The- o y Tech., ol. MTT-28, pp. 381-386, Ap . 1980. K. K. Joshi, J. S. Rao, and B. N. Das, “Cha ac e is ic impedance o nonplana s iplines,” P oc. Ins . Elec. Eng., p . H, ol. 127, pp. 287-291, Oc . 1980. L.-R. Zeng and Y. Wang, “Accu a e solu ions o ellip ical and cylind ical s iplines and mic os iplines,” IEEE T ans. Mic owaue Theo y Tech., ol. MTT-34, pp. 259-265, Feb. 1986. c. J. Reddy and M. D. Deshpande, “Cha ac e is ics o inhomoge- neous coupled cylind ical s iplines,” Elec on. Le ., ol. 23, no. 16, pp. 821-822, July 1987. C. J. Reddy and M. D. Deshpande, “Cha ac e is ics o b oadside coupled cylind ical s iplines,” M c owaue and Op . Tech. Le ., ol. 1, no. 4, pp. 133-136, June 1988. D. Homen co schi, “A cylind ical mul iconduc o s ipline-like mic os ip ansmission line,” IEEE T ans. Mzc owaue Theo y Tech., ol. 37, pp. 497-503, Ma . 1989. B. N. Das, A. Chak abo y, and K. K. Joshi, “Cha ac e is ic impedance o ellip ic cylind ical s ip and mic os iplines illed wi h laye ed subs a e,” P oc. Ins . Elec. Eng., p . H, ol. 130, pp. 245-250, June 1983. B. N. Das and K. V. S. V. R. P asad, “E en- and odd-mode impedances o coupled ellip ic a c s ips,” IEEE T ans. Mic owaue Theo y Tech., ol. MlT-32, pp. 1475-1479, No . 1984. C. J. Reddy and M. D. Deshpande, “Analysis o coupled cylind i- cal s iplines illed wi h mul ilaye ed dielec ics,” IEEE T ans. Mic owaue Theo y Tech., ol. 36, pp. 1301-1310, Sep . 1988. C. J. Reddy and M. D. Desbpande, “Analysis o cylind ical s ipline wi h mul ilaye dielec ics,” IEEE T ans. Mic owa e Theo y Tech., ol. MlT-34, pp. 701-706, June 1986. C. J. Reddy and M. D. Deshpande, “Analysis o b oad side coupled inhomogeneous cylind ical s iplines,” P oc. Ins . Elec. Eng., p . H, ol. 134, pp. 545-549, Dec. 1987. M. D. Deshpande and C. J. Reddy, “Spec al domain analysis o single and coupled cylind ical s iplines,” IEEE T ans. Mic owaue Theo y Tech., ol. MTT-35, pp. 672-675, July 1987. F ancisco Medina was bo n in Pue o Real, CQdiz, Spain, on No embe 9, 1960. He ecei ed he Licenciado deg ee in Sep embe 1983 and he Doc o deg ee in Sep embe 1987, bo h in physics, om he Uni e si y o Se illa, Spain. He is cu en ly Assis an P o esso o Elec- ici y and Magne ism in he Depa men o Elec onics and Elec omagne ism, Uni e si y o Se illa. His esea ch ocuses mainly on nume i- cal me hods o plana s uc u es and mul icon- duc o lines. Manuel Ho no (M75) was bo n in To e del Campo, Jah, Spain. He ecei ed he deg ee o Licenciado in physics in June 1969 and he deg ee o Doc o en Ciencias in physics in Jan- ua y 1972, bo h om he Uni e si y o Se illa, Spain. Since Oc obe 1969 he has been wi h he Depa men o Elec ici y and Elec onics a he Uni e si y o Se illa, whe e he became an Assis- an P o esso in 1970, an Associa e P o esso in 1975, and P o esso in 1986. His main ields o in e es include bounda y alue p oblems in elec omagne ic heo y, wa e p opaga ion h ough aniso opic media, and mic owa e in eg a ed ci cui s. He is p esen ly engaged in he analysis o plana ansmission lines embedded in aniso opic ma e ials, mul iconduc o ansmission lines, and plana slow-wa e s uc u es. 8 Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on July 21,2020 a 15:01:28 UTC om IEEE Xplo e. Res ic ions apply.