Improved quasi-TEM spectral domain analysis of boxed coplanar multiconductor microstrip lines

260 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 41. NO. 2, FEBRUARY 1993 Imp o ed Quasi-TEM Spec al Domain Analysis O Boxed Coplana Mul iconduc o Mic os ip Lines En ique D ake, F ancisco Medina, Membe .. IEEE, and Manuel Homo, Membe -, IEEE Abs ac - This pape p esen s a e y e icien quasi-TEM analysis o mul is ip ansmission sys ems embedded in a laye ed medium. The numbe o conduc o s and subs a es is a bi a y, and he whole s uc u e is assumed o be enclosed in a ec an- gula se o bounda y condi ions. The analysis makes use o he Gale kin me hod in he spec al domain. Chebyshe polynomials wi h edge condi ions a e used as basis and es unc ions o he s ips ee cha ge dis ibu ion. This s anda d echnique is conside ably enhanced by means o wo al e na i e p ocedu es o accele a e he compu a ion o he en ies o he Gale kin ma ix. Ex emely accu a e esul s o a mul is ip sys em, including he cha ge dis ibu ion, can hen be ob ained on a PC compu e in a sho CPU ime. I. INTRODUCTION ULTICONDUCTOR TRANSMISSION LINES (MTL) M a e widely used in (monoli hic) mic owa e in eg a ed ci cui s, high speed in e connec ing buses and o he applica- ions. Once he p opaga ion cha ac e is ics o a MTL sys em a e known, i s equency domain o ime domain elec ical esponses can be ob ained by means o well known me hods. The p opaga ion pa ame e s ha e been compu ed by means o bo h quasi-TEM and ull-wa e app oaches. In many p ac ical si ua ions he quasi-TEM analysis p o ides esul s which a e accu a e enough, and in hese cases i is p e e ed o he much mo e compu a ionally in ol ed ull-wa e app oach. In addi ion, quasi-TEM da a can be used as an ini ial guess in ull-wa e algo i hms, hus imp o ing hei e iciency. I quasi-TEM ope a ion is assumed. he p opaga ion pa am- e e s a e compu ed om he capaci ance, IC], and induc ance, [L], pe uni leng h (p.u.1.) ma ices o he MTL. Powe ul me hods ha e been epo ed in he li e a u e o compu e [C] and [L] o mul iconduc o sys ems ha ing a bi a y geome y [1]-[3] o plana geome y [4], [SI. Speci ic echniques ha e also been de eloped o mic os ip geome ies, which esul in pa icula ly e icien compu e algo i hms. Fo ins ance, some mul iconduc o s uc u es can be exac ly sol ed by using con o mal mapping [6], [7] o e y e icien ly handled by means o he in eg al equa ion echnique [8]. Fo he gene al mic os ip-like geome y embedded in a laye ed linea medium (see Fig.1) he spec al domain app oach (SDA) - combined wi h he Gale kin me hod [9, IO], a ia ional o mula ion [I I]. [I21 o i e a i e echniques 1131, [I41 - is p obably he mos Manusc ip ecei ed Feb ua y 24. 1992: e ised May 26. 1992. This wo k was suppo ed by he DGICYT. Spain (P ojec No. TIC91-1018). The au ho qa e wi h he Depa amen o de Elec onica y Elec omagne ismo. Facul ad de Fisica, Uni e sidad de Se illa. A da. Reinci Me ccdcc s/n, 41 01 2 Se illa, Spain. IEEE Log Numbe 9204482. Y Elec ic wall, magne ic wall 1 o open bounda y I =o 1- x=O Elec ic wall, magne ic wall x=a o open bounda y Fig. 1, C o $ sec ion o he gene alized boxed coplana mul is ip line unde s udy. simple and widely used ool. Al hough he di ec applica ion o hese echniques gi es place o accu a e, eliable and quick compu e codes, p ope analy ical p ep ocessing d as ically imp o es hei pe o mance. A a ie y o echniques in ol ing hea y analy ical wo k has been applied o he solu ion o he single mic os ip p oblem (see [ 151 and he e e ences he ein). The p esen pape is a meaning ul ex ension o he wo k in [ 151 which deals wi h mul is ip geome ies ha ing a bi a y s ip wid hs. The echnique is essen ially an enhanced spec al domain analysis. Two e icien schemes a e p o ided o ac- cele a e he compu a ion o he spec al se ies in ol ed in he Gale kin ma ix in a d as ical way. The applica ion o hese echniques makes i possible o compu e he cha ac e is ic pa ame e s o he mic os ip MTL in Fig. 1 wi h high accu acy in a sho CPU ime. The cha ge dis ibu ion is simul aneously ob ained wi h ex eme accu acy. The analysis o a ypical mul is ip sys em can be ca ied ou on a PC/AT compu e wi h ma h cop ocesso in no mo e han one o wo seconds. The de eloped p og ams can be used o CAD applica ions on a wo ks a ion. This so wa e could be use ul o enginee s dealing wi h mul is ip geome ies. 11. STATEMENT OF THE PROBLEM The c oss sec ion o he mic os ip-like sys em conside ed in his wo k is shown in Fig. 1. T ansla ional symme y in he di ec ion is assumed. An a bi a y numbe , N. o ze o- hickness pe ec ly conduc ing s ips a e placed on he 31 h in e ace o a N,-laye ed medium. The i h s ip is 0~J18-9480/93$03.00 0 1993 IEEE Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on July 17,2020 a 15:45:33 UTC om IEEE Xplo e. Res ic ions apply. DRAKE c ul.: IMPROVED QUASI-TEM SPECTRAL DOMAIN ANALYSIS 26 1 cha ac e ized by i s wid h, w;. and he posi ion, si. o i s middle poin . The laye ed subs a e is composed o NE slabs o lossless/lossy/iso/aniso opic linea ma e ials. The j h laye is cha ac e ized by i s complex dielec ic pe mi i i y enso , Ai:: (I c;,,(an) . G(a,) . aq,j(an) (4) ij, (o equi alen pe mi i i y enso [lo]). The s uc u e is enclosed in o a ec angula ame bounded by he planes = b (see Fig. 1). A wide amily o coplana mic os ip-like ansmission lines can be conside ed o be a pa icula case o his gene ic s uc u e. As i is well known, all he quasi-TEM pa ame e s o he MTL sys em can be ob ained om i s capaci ance, [C]. and induc ance, [L], p.u.1. ma ices. Usually [L] is compu ed om 0q3J(”71)={ J4 ( y) (-l)(q-1)/2 cos(cy,,sj) i q is odd he capaci ance p.u.1. ma ix, [C’], o a p ope ela ed s uc u e a e he ollowing spec al se ies: 2m n=l = 0, = a, = 0, whe e (1, = nn/a is he Fou ie a iable, G is he SDGF, and cq,, (Q71) a e he sine-Fou ie ans o ms o he basis unc ions in (3): Ly W’ - J~ (y) (--1)q/2 sin(a,,sj) i q is e en (< [IO]. Then, he quasi-TEM analysis educes o sol ing wo elec os a ic- ype bidimensional p oblems. Each coe icien C,, (i.J = 1,. . . . N) o [C] o [C’] can be de ined as he ee cha ge on he i h s ip when he j h s ip is se o ol age uni y and he es o he s ips a e g ounded (canonical exci a ion). The e o e he compu a ion o [C] (o [C’]) equi es o sol e he ee cha ge densi y in eg al equa ion N imes ( o N canonical exci a ions): N nn whe e G(z.a:’) is he s a ic G een’s unc ion, @(x) is he ol age (one o ze o on each s ip depending on he pa icula exci a ion), and j(z’) is he ee cha ge densi y on he j h s ip. No gene al closed o m exp essions a e known o he spa- ial domain G een’s unc ion o ou p oblem, bu he spec al domain G een’s unc ion (SDGF) can be easily compu ed (see, o ins ance, [ 1 11, [IO]) o an a bi a y laye ed con igu a ion. Acco ding o his, i is mo e con enien o wo k wi h (1) in he spec al domain. A use ul echnique o sol e he spec al domain e sion o (1) is he Gale kin me hod. As i has been es ablished in he li e a u e on his subjec , Chebyshe polynomials weighed by he Maxwell edge singula i y a e pa icula ly sui able es and basis unc ions o he s ips ee cha ge densi y. When hese unc ions a e used aj(: ’) can be w i en as ollows: JI whe e .Iq is he i s kind o Bessel unc ion o o de q. The sum o he se ies in (4) is he compu a ional s ep in- ol ing meaning ul CPU ime cos . The e o e, he cons uc ion o a highly e icien compu e code equi es he analy ical p ep ocessing o hose se ies. Kumme ’s me hod (ex ac ion o an asymp o ic ail) is used in o de o accele a e he con e gence o he se ies. Acco ding o his me hod, he se ies (4) a e spli as ollows: 71=1 (7) k (k = h . h + 1) being he pe mi i i y (o he equi alen pe mi i i y [ 111, [ 101 in he aniso opic case) o he k h laye . Since Gas is chosen o be he asymp o ic beha io o G o la ge an. he emainde se ies ( i s e m o (6)) con e ges e y quickly. The asymp o ic ails Si:: a e ex emely slow con e gen se ies, bu hey can be educed o quasi-analy ical exp essions by means o he wo p ocedu es desc ibed in he wo ollowing sec ions. qn nx, 111. TRANSFORMATION OF THE TAILS INTO POWER SERIES dz’) = %.,%(J’) (2) I can be seen om (5) and (7) ha he compu a ion o S;:: in ol es he addi ion o slowly con e gen igonome ical se ies o he ollowing kind: q=O whe e w 71 71x1 mq , ( .E’) = (3) The applica ion o he Gale kin me hod leads o a sys em o algeb aic linea equa ions o uq ,. The en ies 24; : (JJ = 0. . . . . p,,,,,, : q = 0. . . . . qlIlaXl : 1. j = 1. . . . . N) o his sys em whe e d, = nw,/2u and : = (7 /a)(sJ S~). The esidues calculus echnique makes i possible o ans- o m (8) in o much mo e quickly con e gen powe se ies. The i s s ep is o iden i y (8) as he addi ion o he in ini e esidues --T T- Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on July 17,2020 a 15:45:33 UTC om IEEE Xplo e. Res ic ions apply. 262 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 41, NO. 2. FEBRUARY 1993 o he ollowing p ope ly chosen complex-plane unc ion: When (9) is in eg a ed along he closed pa h shown in Fig. 2, he esidues, Cauchy heo em p o ides an al e na i e exp ession o (S,;:),’: cos11 [(T - c* )y]: p + q e en x (10) . { si ih [(T - cL3)y]: p + y odd Now, he p oduc o modi ied Bessel unc ions IJ, is ex- panded as a se ies o powe s [[16], p. 9601 shown in (11) below. F being he hype geome ic unc ion, and being he gamma unc ion. The hype geome ic unc ion F[-k. -p - k; y+l; (d3/d,)2] is a k-deg ee polynomial in (d,/dO2 shown in (12) below. In addi ion, he in eg als appea ing in (1 1) a e known in closed o m. Two al ema i e exp essions [ [ 161, pp. 349-3501 o hem a e: whe e p = p + q + 2k, [j = T - cG1 and C is he Riemann’s ze a unc ion. The i s exp ession in (13) is used o he i s ew e ms o he k-se ies. This su ices o mos cases, bu i 1 jy z-plane Fig. 2. In eg a ion pa h in he complex plane o he compu a ion o he spec al se ies by means o he esidues calculus echnique. la ge alues o k a e needed, he second exp ession in (13) p o ides an al e na i e quick solu ion. This p ocedu e is s ill alid in he case p+q = 1 because he new pole o (z) in z = 0 p esen s a pu ely imagina y esidue. Howe e , he case p = q = 0 equi es a sepa a e ea men because o he double pole o (z) in z = 0. Le us conside : This auxilia y se ies can be compu ed by means o he echnique explained abo e when (z) in (9) is eplaced by g(z) = z . (z). I he powe se ies esul ing om his p ocedu e is in eg a ed wi h espec o c: exp ession (13, cos11 [(n - c*)yl: siiih [(T - c,,)y]: p + y e en p + q odd 2 !Jp+q+2k-l ’ /I’ d‘y sin11 (ny) Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on July 17,2020 a 15:45:33 UTC om IEEE Xplo e. Res ic ions apply. DRAKE e a/.: IMPROVED QUASI-TEM SPECTRAL DOMAIN ANALYSIS 263 which is shown a he bo om o his page, is ob ained. In his exp ession, h(d,, d,) is he in eg a ion cons an , which does no need o be calcula ed because i cancels ou when S,”:: is compu ed. I should be no iced ha he gene al exp essions p esen ed in his sec ion o he compu a ion o 5’;;: educes o he mo e simple exp essions appea ing in [ 151 when he case I = j is conside ed (basis and es ing unc ions on he same s ip). The powe se ies in his sec ion p o ide ex eme accu acy when jus a ew e ms a e e ained. In some pa icula cases (which co espond o a he heo e ical han p ac ical si ua- ions, such as ex emely high coupling le els o e y close p oximi y o he s ips o he side walls) mo e e ms need o be conside ed. Anyway, he Shanks ans o ma ion [ 17, pp. 369-3741 p o ides a use ul ool o accele a e he powe se ies in such a way ha he echnique is e en use ul in hese c i ical cases. I . SPATIAL DOMAIN COMPUTATION OF THE TAILS The second echnique conside ed o he compu a ion o (7) is based on he quasi-analy ical in eg a ion o he spa ial coun e pa o he se ies 5’;;;. By applying Pa se al and con olu ion heo ems, we can w i e (1 6), which is shown a he bo om o his page, wi h Gas(x. d) being he spa ial domain G een’s unc ion whose Fou ie ans o m is Gas(n), ha is: No e ha Gas may be physically in e p e ed as he G een’s unc ion o he “asymp o ic s uc u e” which esul s by p o- longing he M h laye o y = -m and he (A4 + 1) h laye o y = +w. The squa e oo in he denomina o o he in eg ands in (16) makes hese in eg als specially sui able o be compu ed by means o he Gauss-Chebyshe quad a u e o mula. Howe e , he di ec applica ion o his quad a u e o he compu a ion o he con olu ion in eg als is no e icien enough because o he loga i hmic singula i y o GaS(x..d) in .c = .E’. In addi ion, i he s ips #1 and/o #N a e e y close o he la e al elec ic walls, Gas (.I:, s’) exhibi s a quasi-singula beha io when LC + IC’ + 0 o x + :c‘ -+ 2a. In o de o o e come hese compu a ional d awbacks, he singula and quasi-singula con ibu ions o Gas (x, d) mus be ex ac ed ou and con enien ly ea ed. The h ee e ms causing he nume ical p oblems can be joined o gi e he ollowing “singula pa ” o Gas(x, d): I should be no iced ha om a physical poin o iew, S(a, 2’) accoun s o he con ibu ions o he eal cha ge line and he i s wo image lines e lec ed by he conduc ing walls. The con olu ion in eg als shown in (16) a he bo om o he page, can be e y e icien ly e alua ed by spli ing he kemel in o wo pa s: Gas(z.z’) = S(:E.:I;’) + [G,,(z.s’) - S(x, d)]. The second e m in his exp ession is a e y smoo h unc ion. The e o e, i s con ibu ion o he con olu ion is ob ained wi h a low o de Chebyshe quad a u e. On he o he hand, he con olu ion in ol ing S(: . d) has been analy ically e alua ed. Le us de ine I[; [q; z]: 111 lx - 2’1 i =O In (x + d) i =-1 (19) In (2a - :E - d) i = $1 s, - WL/2 5 s 5 s, + u1,/2 I complex plane in eg a ion echniques a e used, he closed o m exp essions shown in (20) a he bo om o his page a e ob ained o (19), whe e ’WZ ‘W 2- 2 s, - - < :I: 5 s, + 2: I1 Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on July 17,2020 a 15:45:33 UTC om IEEE Xplo e. Res ic ions apply. 264 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES. VOL 41. NO 2. FEBKLARY 1993 A 2.00 1.00 0.50 0.10 0.05 0.01 and sgn(.) is he sign unc ion. Fo he pa icula case i = j: u = 0: L2 S:;: S ,'," S,",': ($,':)* 0.43 0 0 1 O.GO 1 14 3 0.75 1 2 8 5 0.94 2 2 35 16 0.!)7 3 3 79 3F 0.99 3 0 229 94 The nex s ep o he compu a ion o (16) is o ca y ou he inne p oduc s. Only he pa in ol ing I:!/ has been ound o ha e a closed o m: -l/2p ln(w /4) i p = y # 0 i p = q = 0 ={() i p # q (22) The es o he inne p oduc s ha e been nume ically e al- ua ed by low o de Gauss-Chebyshe quad a u es. In mos cases, he numbe o poin s employed in bo h con olu ion and inne p oduc quad a u es has been wo o h ee mo e han he numbe o basis unc ions used on each s ip. This is su icien o ensu e mo e han eigh signi ican igu es in he calcula ions. Ne e heless, he compu e p og am which implemen s his echnique makes i possible o in oduce a la ge numbe o quad a u e poin s o he inne p oduc in eg als in ol ing I<:. 12:,11 . and (1 = 1:'. . N - 1). This possibili y can be use ul because hese inne p oduc s equi e a ew mo e quad a u e poin s in he c i ic,a/ cases men ioned a he end o Sec ion 111 (i simila accu acy is equi ed o all he inne p oduc s). V. NUMERICAL RESULTS Two double p ecision FORTRAN codes ha e been w i en o implemen he echniques discussed abo e. Exhaus i e nume ical wo k has been ca ied ou o alida e he compu e codes on bo h a PC/386 compu e (wi h ma h cop ocesso ) and a VAX/6410. This wo k has been use ul o es ablish he pa ame e s which ha e an in luence on he accu acy and e iciency o he wo al e na i e me hods p oposed in his pape . Fi s ly, we ha e checked he con e gcnce o he se ies (1 I) and (IS). The con e gence o (S;::)' has been ound o be mainly a ec ed by he a ios z = 111, + ,wJ/'2(.~,, + .s,) and 7.; = 'iu; + 1,/2(.s, - s ). In mos p ac ical si ua ions. he con e gence is eached in ew e ms. Ne e heless. he con- e gence becomes slowe when z and/o , ,j a e e y close o one (uni y). This occu s i some s ip is e y close o he la e al me allic walls (7.; E 1) and/o i e y igh ly coupled s ips a e p esen ( 6 E 1). In hese cases, mo e e ms mus be e ained o add up he co esponding se ies. In Table I we show a ypical con e gence pa e n o he se ies 5':::; associa ed o a pai o asymme ical s ips o di e en coupling le els. The numbe o e ms equi ed in he compu a ion o S::,': inc eases when y2 app oaches o uni . Fo his s udy i e signi ican co ec igu es a e always imposed in he compu a ion o ('l,l. TABLE I TO BE RETAINED FOR FIVE SK~NIFICAN~ Flm RES Acci RACY lh THE MAXIMUM NUMBER OF TERMS A,,,,,, OFl-H SERIES .$:,: I.,/ = 1.2 C4PAClTANCE. THE As1 FRISKED COLL IlN INCLUDES I HE RESLLTS OBTAINED BY USING THE TWICE ITERATED SHANKS TRANSFORMATION. (DATA: (I = 10.111 = 1.h' = 2. (I', = 1. 2 = 2.il = 3). €0 4WlA w2 I.- No e ha he use o he wice i e a ed Shanks ans o ma ion (as e isked column) in oduces a signi ican accele a ion (o e SO%) in he con e gence o his se ies. I mus be emphasized ha he cases in ol ing a e y la ge A:,,,,, a e no ealis ic. Anyway, he di ec summa ion o he o iginal Fou ie se ies would equi e much mo e compu a ional e o (p ohibi i e i e y high accu acy is desi ed). The e o e he applica ion o his echnique is always p e e able o di ec summa ion. The spa ial domain echnique p esen s simila di icul ies in he same cases, bu i has been ound o be less sensi i e o hose p oblems. In gene al, he con olu ion in eg al and he inne p oduc s a e e y accu a ely compu ed by using a numbe o quad a u e poin s equal o o sligh ly la ge han he numbe o basis unc ions employed on each s ip. Typical compu a ions a e so accu a e ha mo e han eigh meaning ul digi s can be ob ained o he coe icien s o he expansion o he cha ge dis ibu ion. Howe e , when I.: and/o I,; ake ( heo e ical a he han p ac ical) alues e y close o 1. he numbe o ' quad a u e poin s IV~T used o he compu a ion o he c i ical inne p oduc s in ol ing 1,;. I.:,'., and/o (i = I.. . . . .V - 1) mus be inc eased o keep he accu acy pa e n. In Table 11. we display he alues o lV,y needed o ge ho h i e and en signi ican igu es Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on July 17,2020 a 15:45:33 UTC om IEEE Xplo e. Res ic ions apply. DRAKE e[ a/ : IMPROVED QUASI-TEM SPECTRAL DOMAIN ANALYSIS Ihn I 2 3 1 5 7 8 9 Ill II I 2 3 1 5 G 7 S 9 6 265 A = 0. I mi CII Cl2 25.7199 -7.73533 30.8053 -9.00472 31.3117 -9.01270 31.1235 -9.1175G 31.1139 -9.13304 31.4455 -9.13GG9 31.1190 -9.13687 31.1191 -9.l3602 31.1191 -9.13604 31.4191 -9.13694 5.70529 -4.73082 11.9197 -7.53461 13.5927 -9.20794 13.9190 -9.19782 13.9535 -9.56750 13.9914 -9.57426 13.9020 -9.57489 l3.0920 -0.57191 1:1.9920 -9.57491 31.1179 -0.13600 TABLE I1 NUMBER OF GAUSS- CHEBYSHEV QUADRATURE POINTS . -,: USED IN THE CRITICAL INNER PRODUCT INVOLVING TO GET BOTH FIVE AND TEN SIGYIFICANT FIGURES CAPACITANCES. (DATA: EQUAL TO THOSE IN TABLE I). A 2.00 1 .oo 0.50 0.10 0.05 0.01 - 1'1 2 0.43 0.60 0.75 0.94 0.97 0.99 - iV; (5 digi s) 2 2 3 7 11 Nd (10 digi s) 13 20 35 I o CI,~ o di e en coupling le els. In ha example, he con olu ion in eg als and he non-c i ical inne p oduc s ha e been compu ed wi h 4 Gauss-Chebyshe quad a u e poin s. The aspec s discussed in he p e ious pa ag aphs deal wi h he quali y o he compu a ions o he Gale kin ma ix en ies. Howe e , he accu acy o he capaci ance and induc ance coe icien s also depends on he numbe o basis unc ions. The ial unc ions in (3) a e e y sui able o he mul is ip p oblem, since ex eme accu acy o bo h capaci ance coe i- cien s and cha ge dis ibu ion can be achie ed in all p ac ical si ua ions. Typical esul s ob ained wi h ou p og ams (bo h p og ams p o ide exac ly he same esul s wi hin he compu e accu acy) can be seen in Tables 111 and IV. F om Table 111, i can be seen ha mo e ial unc ions a e equi ed when he s ips a e e y close o when hey a e adjacen o hin dielec ic laye s ( he s ips a e in his case ela i ely wide in compa ison wi h he hicknesses o he laye s). Anyway, om a p ac ical poin o iew, no mo e han h ee o ou basis unc ions a e necessa y o ob ain use ul highly accu a e esul s. Table IV shows an example o he expansion coe icien s ob ained o he cha ge dis ibu ion when di e en numbe o basis unc ions a e used. The coe icien s o he expansion a e no sensi i e o he addi ion o new e ms when con e gence has been achie ed ( his ac sugges s ha (2, 3) is a quasi- o hogonal expansion). As i is expec ed om he s a iona y na u e o he i s coe icien , his is much mo e accu a ely compu ed han he cha ge dis ibu ion. In ou opinion, an impo an ea u e o he me hods epo ed in his pape is ha when hey a e used, no nume ical p oblems a ise i he numbe o basis unc ions is inc eased. In he pas , he au ho s ha e used o he asymp o ic ails-in ol ing la ge a gumen app oxima ion o he Bessel unc ions- o accele a e he con e gence o he spec al se ies [ 181. Fo mos p ac ical pu poses, ha app oach wo ks p ope ly (al hough mo e slowly han hose p esen ed in he p esen pape ), bu some p oblems can be obse ed when a ela i ely la ge numbe o basis unc ions a e used. This is a consequence o cumula i e nume ical e o s when se ies in ol ing la ge o de Bessel unc ions need o be compu ed. This d awback has no been de ec ed wi h he me hods de eloped in he p esen wo k. The nume ical da a gene a ed wi h ou p og ams ha e been compa ed wi h highly accu a e da a (o exac solu ions) e- ,' TABLE I11 TO 6") VERSUS THE NUMBER OF BASIS FUNCTIONS (33111.1). DATA: 6,-,.2 13. cyY2 = 10. ~,.~:3 = yy:i = 2.51. (1 = 20 mm, .s1 E 9.15 mm, u1 = 1 mm, (172 = 1.5 mm, CASE I (RI = 0.h2 = 0.635 mm, hg = O.h4 = 10 mm), CASE I1 (hl = 0.51 mm, 112 = 0.125 mm, h:j = 0.125 mm, hd = 9.875 mm). CONVERGENCE OF THE CAPACITANCE MATRIX (NORMALIZED Cases 1 ~ c22 - 37.1587 39.1877 39.4832 39.5505 39.5639 39.5663 39.5GG3 39.5671 39.5671 39.5671 39.5671 10.1251 13.OSG9 14.7678 15.0354 15.1115 15.1 170 15.1 1 77 15.1177 15.1 177 __ __ ~ C1 I ~ 27.1524 27.1841 27.2127 27.2132 27.2132 27.2132 ~ 6.57581 7.06826 7.18892 7. ISGO2 7.19835 7.19849 7.19850 7.19850 - 3 = lmn c12 -1.22612 - 1.22926 -1.19820 -1.19824 - 1.19829 - 1.19829 -1.74512 - 1.83526 -1.94558 -1.94596 -1.94900 -1.94901 -1.94901 - 1 .9ma 1 C12 35.1310 35.1583 35.2954 35.2955 35.2956 35.2956 - - 7.99884 8.14886 8.28897 8.29218 8.29925 8.29927 8.29928 8.29928 po ed in he li e a u e o pa icula s uc u es. The ag eemen has been always ound o be excellen (all he signi ican igu es epo ed ha e been in a iably ob ained). Fo example, nume ical da a o he capaci ance coe icien s o a i e s ips s iplike con igu a ion in homogeneous medium a e epo ed in [[8], Table VII] and [[7], Fig. 61. The o me uses an enhanced in eg al equa ion echnique and ex apola ion p ocedu es and he la e gi es a con o mal mapping solu ion (al hough nume - ical compu a ion o he hipe ellip ic unc ions is equi ed). Fi e igu es a e co ec ly gi en in [8] and six igu es a e gi en in [7] o he no malized capaci ance coe icien s. Ou p og ams ep oduce all he signi ican igu es epo ed in hose wo ks. To ob ain i e igu es accu acy, 4 basis unc ions ha e been e ained on each s ip, and he CPU ime was abou I .5 seconds on a PC/386 compu e (abou 0.15 seconds on a VAXJ6410). Six igu es we e ob ained wi h 5 basis unc ions (2.2 seconds on a PC/386 compu e , abou 0.2 seconds on a VAXJ6410). These CPU imes e e o he spa ial domain echnique. The same esul s we e ob ained by means o he o he echnique desc ibed in his pape wi h sligh ly highe CPU imes. As a inal example, in Table V we compa e ou esul s wi h 7 T Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on July 17,2020 a 15:45:33 UTC om IEEE Xplo e. Res ic ions apply. 266 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 41, NO. 2. FEBRUARY 1993 3 1 2 4 1 2 4 5 1 2 4 5 G 3 3 3 TABLE IV CHARGE DISTRIBUTION COEFFICIENTS OF A PAIR OF ASYMMETRICAL COUPLED STRIPS FOR DIFFERENT VALUES OF THE NUMBER OF BASIS FUNCTIONS ~n ey. DATA: EQUAL TO THOSE IN TABLE 111, CASE I, 1 = 1 mm. -1.95733 27.2132 0.10152 0.01094 27.2132 0.10156 0.01094 0.04654 27.2132 0.10156 0.01094 0.04654 0.00033 -1.95706 -1.95705 -1.95705 Qmax 1 2 3 4 5 G Dics cl [5] 4.633 7.857 5.723 -2.547 -2.338 -0.oso -0.553 -0.064 -0.013 Exci . (1,O) This wo k 4.642 7.871 5.738 -2.555 -2.346 -0.080 -0.553 -0.064 -0.013 a92 -1.22612 -1.22926 1.70030 1.68933 -1.19820 -0.82322 - 1 A9824 1.68674 0.31610 1.68675 0.31610 -0.82325 - 1.19829 -0.82309 -0.09211 - 1.19829 1.68676 0.31609 -0.0921 1 0.02020 -0.82309 Exci , %,I - 1.226 12 -1.22926 -1.51918 - 1.19820 - 1.48469 -0.55950 -1.19824 -1.48439 -0.55952 -0.15337 -1.19829 -1.48446 -0.55955 -0.15338 -0.03038 -1 A9829 -1.48446 -0.55955 -0.15338 -0.03038 -0.00439 :oJ) a9,2 35.1310 35.1583 35.2954 -0.11748 -0.11830 -4.65142 35.2955 -0.11827 -4.65132 -0.02298 35.2956 -0.11828 -4.65161 -0.02298 0.16107 35.2956 -0.11829 -4.65161 -0.02298 0.16107 -0.00 150 hose ob ained by he me hod o lines (wi h nonequidis an disc e iza ion) in [5] o a i e conduc o mic os ip s uc u e wi h di e en wid hs and inhomogeneous subs a e. To ob ain 4 digi s accu acy we ha e used 5 basis unc ions, and he CPU ime was less han 1.5 seconds (PC/386). A ac ion o a second was necessa y o ge he same le el o accu acy o he example in [5]. In gene al, e y eliable and accu a e esul s can be ob ained o he capaci ance and induc ance ma ices o mic os ip s uc u es on a PC/386 compu e wi h CPU imes anging om a ac ion o a second o wo o h ee seconds (depending on he numbe o s ips and basis unc ions). The su ace cha ge dis ibu ions a e also p o ided wi h e y good accu acy. VI. CONCLUSIONS In he p esen wo k, a nume ically imp o ed spec al domain app oach is employed o he e icien and accu a e quasi- TEM analysis o a wide class o mul is ip ansmission lines. A p ope analy ical p ep ocessing is inco po a ed in he compu a ion o he en ies o he Gale kin equa ions sys em in o de o achie e ex eme accu acy in a sho CPU ime. To each his goal, wo di e en echniques ha e been p oposed and compa ed. The double p ecision FORTRAN p og ams implemen ing hese echniques a e able o analyze mul is ip con igu a ions embedded in mul ilaye ed subs a es on a PC/386 compu e in less han a ew seconds. To al ag eemen has been ound wi h exac esul s (up o he accu acy epo ed in he li e a u e) o simple pa icula con ig- . i -~ TABLE V COMPARISON BETWEEN OUR RESULTS AND THOSE REPORTED IN [S] FOR THE CAPACITANCE MATRIX (NORMALIZED TO F~, ) OF A FIVE- CONDUCTOR MICROSTRIP CONFIGURATION. DATA: 12 = 0.8 mm, = 2.5~0. (1’1 = 0.1C mm, = 0.47 mm A = 0.07 mm, I( = 3.7 mm. €0 I 4 w1 kkW24i w1 H--w2-l-l w1 k- I I €0 i w1 k-k-wzii w1 H--w244 w1 k- 4k- n € h Ik- - n € Tl I u a ions. Good ag eemen has been also ound wi h many o he esul s epo ed o mo e gene al s uc u es. The su ace cha ge dis ibu ion can be ob ained wi h accu acy and eliabili y i his quan i y is equi ed. REFERENCES [I] C. Wei, R. F. Ha ing on, J. R. Mau z, T. K. Sa ka , “Mul iconduc- o ansmission lines in mul ilaye ed dielec ic media,” /€E€ T ans. Mic owa e Theo y Tech.. ol. MTT-32, pp, 439450, Ap . 1984. [2] Z. Pan ic and R. Mi a, “Quasi-TEM analysis o mic owa e ansmis- sion lines by he ini e-elemen me hod,” IEEE T ans. Mic mu e Th o y Tech.. ol. MTT-34, pp. 1096-1103, No . 1986. [3] F. Olyslage , N. Fach6, and D. de Zu e , “New as and accu a e line pa ame e calcula ion o gene al mul iconduc o ansmission lines in mul ilaye ed media,” IEEE T ans. Mic owm Theo y Tech.. ol. 39, pp. 901-909, June 1991. [4] V. K. T ipa hi and R. J. Bucolo, “A simple ne wo k analog app oach o he quasi-s a ic cha ac e is ics o gene al lossy, aniso opic, laye ed s uc u es,” IEEE T ans. Mic owwe Theo y Tech.. ol. MTI-33, pp. 1458-1464, Dec. 1985. [5] H. Dies el, “Analysis o plana mul iconduc o ansmission-line sys ems wi h he me hod o lines,” AEU, ol. 41, pp. 169-175, 1987. [6] L. J. P. Linne , “A me hod o he compu a ion o he cha ac e is ic immi ance ma ix o mul iconduc o s iplines wi h a bi a y wid hs,” IEEE T ans. Mic, owa e Theo y Tech.. ol. MlT-22, pp. 930-937, No . 1974. [7] D. Homen co schi, A. Manolescu, A. M. Manolescu, and L. K eindle , “An analy ical solu ion o he coupled s ipline-like mic os ip line p oblem,” IEEE T ans. Mic owa e Theo y Tech.. ol. MTT-36, pp. 1002-1007, June 1988. [8] D. W. Kammle , “Calcula ion o cha ac e is ic admi ances and coupling coe icien s o s ip ansmission lines,” IEEE T ans. Mic owai Theo y and Tec,h.. ol. MTT-16, pp. 925-937, No . 1968. Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on July 17,2020 a 15:45:33 UTC om IEEE Xplo e. Res ic ions apply. DRAKE e a/.: IMPROVED QUASI-TEM SPECTRAL DOMAIN ANALYSIS 261 [9] T Ki azawa and Y. Hayashi, “Asymme ical h ee-line coupled s iplines wi h aniso opic subs a es,” IEEE T,-ans M c m~ai Theo y Tech , ol. MTT-34, pp 767-772, July 1986 dnaly is o mul ilaye ed, mul iconduc o coplana s uc u es wi h dielec- ic and magne ic aniso opy including subs a e losses,” IEEE Tiuns M ciowaie Theo y Tech, ol. 38, pp. 1059-1068. Aug. 1990. [I I] F Medina and M Homo, “Uppe and lowe bounds on mode ca- paci ances o a la ge class o aniso opic mul ilaye ed mic os np- like ansmission lines,” P oc Ins Elec Eiig (M c ou~u es. Op i s & An ennuhi, ol. 132, no 3, pp 157-163, June 1985 lines. 1121 A Sawicki and K Sachse, “Lowe and uppe bound calcula ions on he capaci ance o mul iconduc o p in ed ansmission line using he spec al-domain app oach and a ia ional me hod,” IEEE Ti ans Mi ou~a Theo y Tech, ol. MTT-34, pp 236244, Feb. 1986. [I71 C H Chdn and R. Mi a, “Analysis o MMIC s uc u es using an Manuel Ho no (M’75) was bom in To e del e icien i e a i e app odch,” IEEE T ans Micinna e Theo b Tech . ol. Campo, Jain, Spain. He ecei ed he Licenciado 76, pp. 96-105, Janua y 1988. deg ee in Physics in June 1969, and he Doc o [ 141 E. D ake, F Medina, and M. Homo, “An imp o ed i e a i e echnique deg ee in physics in Janua y 1972, bo h om he o he quasi-TEM analysis o gene alized plana lines,” IEEE T ans Uni e si y o Se ille, Spain. Mi nwm3e Theo Tc h, ol. 40, Ap 1992 Since Oc obe 1969 he has been wi h he [IS] F. Medina and M. Homo, “Quasi-analy ical s a ic solu ion o he boxed Depa men o Elec onics and Elec omagne ism mic os ip line embedded in a laye ed medium,” IEEE T ans Mic I ouwe a he Uni e si y o Se ille, whe e he became an Theo Tech , ol 40, Sep . 1992. Assis an P o esso in 1970, Associa e P o esso [16] I. S. G adsh eyn and I. M Ry hik, Tuhle oj I eq als. Se ies and in 1975 and P o esso in 1986. He is a membe P ndicc s New Yo k. Academic P ess, 1980 o he Elec omagne ism Academy o M.1 T., [I71 c M Bende and s A O szag. Ad anced Mo hema cal Me hodc oi Camb idge. HIS main ields o in e es include bounda y alue p oblems S ie ilim and Enginee s New Yo k McG dw-Hill, 1978. in elec omagne ic heo y, wa e p opaga ion h ough dniso opic medid, and Medim and M. Homo, “SFc al and a ld lonal analYsl5 o gene - mic owa e in eg a ed ci cui s He is p esen ly engaged in he analysis o dized cYllnd lcal and elllP lcal ip and mlc o5 lP Ilnes,” [EEE T ans plana ansmission lines embedded in ani o opic ma e ials, mul iconduc o M ciouai Theo Te l . ol 38, pp 1287-1293. Sep 1990 ansmission lines, and plana slow-wa e 5 uc u es F ancisco Medina (M’91) was bom in Pue o Real, CBdiz, Spain, on No embe , 1960. He ecei ed he Licenciado deg ee in Sep embe 1983 and he [IO] M Homo, F L Mesa, F Medina, and R. Ma quis, “Quasi-TEM Doc o deg ee in 1987, bo h in physics, om he Uni e si y o Se ille, Spain. He is cu en ly Associa e P o esso o Elec ici y and Magne ism in he Depa men o Elec onics and Elec omagne ics, Uni e i y o Se ille. His esea ch deals mainly wi h analy ical and nume ical me hods o plana s uc u es dnd mul iconduc o En ique D ake was bom in Mon illa, Co doba, Spain, on Sep embe 4, 1966. He ecei ed he Licenciado deg ee in physics om he Uni e si y o Se ille, Spain, in 1990. He is cu en ly ollowing a Ph.D. p og am in Mic owa e5 wi h a schola ship o he Spanish Go e nmen . His esea ch in e es ocus on he analysis o plana s uc u es and mul- iconduc o line5 Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on July 17,2020 a 15:45:33 UTC om IEEE Xplo e. Res ic ions apply.