Quick Computation of [C] and [L] Matrices of Generalized Multiconductor Coplanar Waveguide Transmission Lines

2328 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES. VOL. 42, NO. 12. DECEMBER 1994 Quick Compu a ion o [C] and [L] Ma ices o Gene alized Mul iconduc o Coplana Wa eguide T ansmission Lines En ique D ake, F ancisco Medina, and Manuel Homo, Membe , ZEEE Abs ac -An enhanced spec al domain quasi-TEM analysis o gene alized coplana wa eguide ansmission lines (GCPWTL) is p esen ed. The analysis s a s om he o mula ion o a con olu ion- ype in eg al equa ion o he elec ic ield a he slo s. Chebyshe polynomials including Maxwell singula i ies a e used as basis unc ions o sol e he in eg al equa ion by he Gale kin me hod. Fas and accu a e quasi-analy ical o mulas a e used o calcula e he Gale kin’s ma ix en ies, he eby signi ican ly educing he in ol ed CPU ime and inc easing eliabili y and accu acy. These ea u es make his echnique use ul and compe i i e as CAD ool o coplana wa eguide designs. I. INTRODUCTION OPLANAR wa eguide (CPW) ansmission lines a e C becoming a compe i i e al e na i e o mic os ip in many applica ions (including bo h hyb id and monoli hic echnolo- gies). A numbe o a ac i e ea u es-loca ion o he signal g ounds on he same subs a e su ace as he signal line, low pa asi ic induc ances, easy shun and se ies connec ions, a oidance o he need o ia holes, good isola ion in di ec- ional couple s, e c. [ I]-[3]-makes his ansmission medium pa icula ly in e es ing. Due o his ac oge he wi h he ela i e lack o design da a ii ailable o CPW s uc u es (in compa ison wi h he mic os ip line), esea ch on many aspec s ela ed o he cha ac e iza ion o CPW s uc u es is s ill going on [4]-[7]. The compu a ion o he p opaga ion cha ac e is ics o CPWs has ecei ed some a en ion in old and ecen li e a u e (see, o example, [8]-[ 1 11, which include exhaus i e bibliog aphy sweeping a wide a ie y o analy ical and nume ical ech- niques). As i is well known, he e alua ion o dispe sion, adia ion, highe -o de modes o leakage phenomena equi es igo ous hyb id-mode app oaches. Howe e , he quasi-TEM app oxima ion can be expec ed o yield use ul esul s in he equency band whe eon MIC’s usually ope a e oday, a leas o hose s uc u es and componen s which a e no pa icula ly equency-sensi i e [8], and e en up o 40 GHz in he design o coplana MMIC’s [9]. Since he quasi-TEM analysis equi es Manusc ip ecei ed No embe 17, 1993; e ised Janua y 27, 1994. This wo k was suppo ed in pa by he DGICYT, Spain, unde con ac TIC91-1018. The au ho s a e wi h he Mic oy es G oup. Depa men o Elec onics and Elec omagne ism. Facul ad de Fisica A da. Reina Me cedes s/n, 41012 Se illa, Spain. IEEE Log Numbe 9405369. much less compu a ional e o , i is mo e adequa e o design pu poses. In addi ion, quasi-TEM da a could e en ually be used as ini ial guesses in ull-wa e algo i hms, hus imp o ing hei e iciency. The quasi-TEM analysis o ce ain pa icula CPW geome ies has been al eady ca ied ou in a e y e icien way (sui able o CAD applica ions). Fo ins ance, a quasi- analy ical me hod o deal wi h a single CPW embedded in a s a i ied medium is epo ed in [l 11. In ha pape he eade can ind a lis o e e ences epo ing o he quasi- analy ical me hods o analyze a a ie y o symme ical and asymme ical single CPW geome ies (mos o hem based on con o mal mapping app oaches). One o he mos ecen con ibu ions based on con o mal mapping can be ound in [6]. Howe e , mo e complex CPW s uc u es in ol ing mul iple dielec ic laye s and coupled conduc o s ha e ecei ed less a en ion in spi e o i s ob ious in e es in p ac ical applica ions ( il e , couple s, e c.). Some analy ical o app oxima e solu ions o symme ical coupled s uc u es ha e been epo ed in he li e a u e [l], [12]. Mo e sophis ica ed geome ies ha e been conside ed in [13], [14]. A ecen wo k [ 151 p oposes an e icien Wiene -Hopi solu ion o a mul iconduc o CPW sys em ( o applica ion as in e digi al ansduce ), bu i is es ic ed o geome ies symme ically placed be ween wo g ound planes wi h homogeneous medium. The a ailabili y o quick and e sa ile mul iconduc o sol e s is impo an om he designe ’s pe spec i e, since op imiza ion p ocesses in ol e he i e a i e e alua ion o he pa ame e s o a s uc u e o a wide ange o design a iables. In his sense, ex emely e icien algo i hms ha e been al eady de eloped o he quasi-TEM analysis o gene al mul is ip geome ies [ 161-[ 181. Howe e , as a as we know, a sys ema ic and quasi-analy ical ea men o a gene alized coplana wa eguide ansmission line (GCPWTL) sys em-including an a bi a y numbe o me allic s ips be ween wo g ound planes embedded in a mul ilaye medium-has no been explici ly gi en ye . The cu en pape con ibu es o he compu e -aided design o coplana - ype ci cui s by o e ing a quasi-analy ical p ocedu e o compu e he quasi-TEM p opaga ion pa ame e s o he GCPWTL sys em in Fig. 1. The mul islo geome y o he GCPWTL sys em makes sui able o s a e he analysis in e ms o he ape u e elec ic ield. The e o e, a con olu ion in eg al equa ion is p oposed o connec he elec ic ield a he slo s wi h he ee cha ge on he me alliza ions. The 0018-9480/94$04.00 0 1994 IEEE Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on July 17,2020 a 15:06:06 UTC om IEEE Xplo e. Res ic ions apply. DRAKE e ai.: QUICK COMPUTATION OF [C] AND [L] MATRICES OF GENERALIZED MULTICONDUCTOR COPLANAR WAVEGUIDE TRANSMISSION LINES 2329 e.w., m.w. o 0.b. -------I ,///////,///,////, e.w., m.w. o 0.b. x=a 2- X=O Fig. 1, ansmission line (GCPWTL). C oss-sec ion o he gene alized mul iconduc o coplana wa eguide me hod used o sol e he elec ic ield in eg al equa ion is an enhanced Gale kin spec al domain analysis (SDA). Quasi- analy ical o mulas a e p o ided o compu e he Gale kin’s ma ix en ies. A dual ea men o mul is ip geome ies in e ms o he ee cha ge densi y on he me alliza ions was success ully used and epo ed in [17]. High speed o compu a ion and ex eme accu acy could make he use o his me hod use ul in he CAD o coplana wa eguide ci cui s. 11. OUTLINE OF THE PROBLEM Fig. 1 shows he c oss-sec ion o he GCPWTL analyzed in his wo k. T ansla ional symme y in he p opaga ion di ec ion (z-axis) is assumed. The whole s uc u e is enclosed in o a ec angula box de ined by he planes s = 0, J; = a, y = 0, and y = b. The la e al planes .I‘ = 0 and z = a a e elec ic walls (e.w.). The planes y = 0 and y = b can be chosen o be elec ic walls, magne ic walls (m.w.) o open bounda ies (0.b.). The subs a e is a Ni-laye ed 1osslessAossy iso/aniso opic linea medium. An a bi a y numbe (N) o conduc o s ips-alloca ed be ween wo g ounded me al ins-a e p in ed on he M h in e ace. Le us cha ac e ize each o he N + 1 slo s be ween hese conduc o s ips by bo h i s wid h (3%; z = 1.. . . ,N + 1) and he posi ion (zcz; 1 = 1,. . . , N + 1) o i s middle poin (wi h espec o he le la e al wall). This s uc u e includes a la ge g oup o CPW geome ies as pa icula cases. Mo eo e , he con igu a ion in Fig. 1 accoun s o possible echnological cons ain s (uppe and la e al shielding, conduc o backing, and line- o-line coupling). As is well known, all he quasi-TEM p opaga ion pa ame e s o a N-conduc o ansmission line may be ob ained om i s capaci ance, [C], and induc ance, [L], pe uni leng h (p.u.1.) ma ices. The de e mina ion o [C] implies o sol e an elec os a ic p oblem. [L] may be compu ed om he capaci ance, [C’], o a ela ed s uc u e [19]. I he elec os a ic p oblem is s a ed in e ms o he su ace ee cha ge densi y on he conduc o s, each coe icien , C2,, o [C] o [C’] is iden i ied as he ee cha ge on he z- h conduc o when he j- h conduc o is se o ol age uni y and he es o he conduc o s a e g ounded (canonical ol age exci a ion) [ 171. Howe e , he mul islo geome y is mo e e icien ly cha ac e ized in e ms o he ape u e elec ic ields han in e ms o he cha ge dis ibu ion. Owing o his, i is mo e di ec o calcula e i s coe icien s o po en ial ma ix [PI, i.e., he se o coe icien s, Pi,, which linea ly ela es he po en ial V, o any conduc o o he ee cha ges Q, on all o he conduc o s, including i sel : N , = Pi&), (i = 1,. . . ,N). J=l No e ha [PI = [GI-’. Each coe icien P;j may be de ined as he ol age o he i h conduc ing s ip when he j- h s ip is cha ged wi h cha ge uni y, and he es o he s ips a e discha ged (canonical cha ge exci a ion). The ol age o each s ip is compu ed in eg a ing he elec ic ield 2-componen along he slo s exis ing be ween one o he g ounded pla es and ha s ip. The e o e, [PI (and i s in e se, [C]) will be ob ained i we compu e he ape u e ields o N independen canonical cha ge exci a ions. 111. THE SLOT ELECTRIC FIELD EQUATION F om he G een’s heo em, he elec os a ic po en ial @(z) on he me allized in e ace o he s uc u e in Fig. 1 is ela ed o he ee cha ge densi y o(2) by whe e G(z, z’) is he po en ial G een’s unc ion associa ed o he hl h in e ace o he s uc u e. This is he ee cha ge den- si y in eg al equa ion usually sol ed when mul is ip geome- ies a e analyzed. Howe e , we a e now in e es ed in using an in eg al equa ion o he slo elec ic ield 2-componen . Owing o he exis ence o elec ic walls in bo h z = 0 and z = a, he Fou ie se ies expansion o G(z,z’) yields (3) whe e- om Fou ie ans o m heo y [20]--G,(a) is he spec al domain G een’s unc ion (SDGF) associa ed o he la e ally open s uc u e, i.e., a + 30. De ining he ollowing pai s o sine(cosine)-Fou ie ans o ms: and using (3), i is s aigh o wa d o con e (2) in o (4) by sine-Fou ie ans o m. Simple manipula ions le us o ew i e (5) in he ollowing way: Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on July 17,2020 a 15:06:06 UTC om IEEE Xplo e. Res ic ions apply. 2330 lEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 42, NO. I?, DECEMBER 1994 whe e (7) No e ha (6) may be iden i ied as he cosine-Fou ie ans o m o q(x) = .I” L(J,L’)E,(T’) ds’ (8) whe e Ez( ’) is he elec ic ield 2-componen along he me allized in e ace. q(~) in (8) is he amoun o ee cha ge alloca ed om d = 0 o T’ = L, i.e., x q(x) = 4, O(d) dxl and L(2.z’) is (9) a e used as basis unc ions in he expansion o he elec ic ield x-componen 2 -7 E+, (L’) = { 2 [I - (*) ] Tq( *); . ‘ E S, (15) el sew he e. The choice o his se o unc ions is sugges ed by he na u e o he in eg al ke nel. The applica ion o he Gale kin me hod con e s (12) and (13) in o a sys em o algeb aic linea equa ions o he coe icien s aq,,, A7+1 I IS ob ious ha q(x) s ands cons an along each slo (s E U0,J = 0 S, =. [L,, - s,/2 5 . 5 . ea s,/2]), and i s alue is J=1 i-1 q(: :€Si)=CQj; i=1 ...., N+l (11) j=O whe e Qj (.j = 1,. . . , N) is he o al ee cha ge on he j- h s ip, and Qo, he o al ee cha ge on he le coplana g ounded in. Since QO is no known, he o al alue o q(x) along a slo can no be compu ed when a cha ge exci a ion is imposed. Howe e , when z skips om a gi en slo o he nex one, he inc ease in q(x) is equal o he amoun o ee cha ge suppo ed by he s ip alloca ed be ween he wo slo s. The e o e, he elec ic ield 2-componen o he k h (k = 1: . . . , N) canonical cha ge exci a ion (cha ge uni y on he k h s ip keeping he es o he s ips discha ged) ul ils he ollowing condi ion 1“ L(z E ,Si+l,x’)Ez(x’) dd whe e he en ies A;:$ O , = 0, . . . , n i - 1; y = (I, . . . , i, ; - 1; i,j = 1,. . . , N + 1) o he sys em a e: A;;{ = J’ dz Exp,,(x) J’ d.c’ zc2 +s, /2 x,,+. ,!2 s,,-s,/2 a,,-s,/2 x EZq,, (d)L(X, d). (17) Howe e L(x,z’) in (17) is no known (excep o special cases) in closed o m. Fo una ely, (7) shows ha he spec al ans o m_, L(c ), o he ke nel o (17) is ela ed o he SDGF, Go(a), associa ed o he la e ally open e sion o he mul ilaye ed con igu a ion. The e o e, i is mo e sui able o ob ain a spec al domain exp ession o he en ies A;;:, and, hen, ake ad an age om he e icien algo i hm epo ed in [19], [21] o compu ing he SDGF o an a bi a y laye ed con igu a ion. The spec al e sion o (17) may be deduced om Pa se al and con olu ion heo ems (12) whe e kz0 a e he cosine-Fou ie ans o ms o II Ezq,, (d), Le., whe e b,k is he K onecke del a (1 i i = k, 0 i z # k). In J~(?)(-I): cos(a,.~,~) Jq( )(-l)w sin(a,s,,]) i y is e en i q is odd (19) addi ion, he g ounding o he la e al ins makes he elec ic - ield x-componen ul ils E%,,(-) = 1‘ Ez(2-’)dT’ = 0. (13) wi h Jq(-) being he i s kind Bessel unc ion o o de q. Once he sys em (16) has been sol ed, he coe icien s o po en ial a e di ec ly compu ed om he expansion ze o h- IV. METHOD OF ANALYSIS o de coe icien s P obably, one o he bes known echniques o sol e in eg al equa ions as (12) and (1 3) is he Gale kin me hod. In his wo k, a ,=1 pzk = - i = 1, . . . , L k h exci a ion his me hod has been also chosen. The i s kind Chebyshe polynomials, Tq(.), weighed by he Maxwell edge singula i y (20) Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on July 17,2020 a 15:06:06 UTC om IEEE Xplo e. Res ic ions apply. DRAKE e a/ : QUICK COMPUTATION OF [CI AND [L] MATRICES OF GENERALIZED MULTICONDUCTOR COPLANAR WAVEGUIDE TRANSMISSION LINES 2331 0.1 0.0 1 I o-~ 2 3 4 5 6 7 8 9 1011 a /I Fig. 2. Rela i e di e ences be ween he capaci ance coe icien s o a la - e ally closed h ee-conduc o CPW- ype ansmission line (Ccl) and i s co esponding open e sion (C‘,,p), Da a: h = 0.635 mm, 1 = 2.4 mm, .sl = s2 = sa = sq = 0.2 mm, . l = u/2-1.1 mm, .cc2 = a/2-0.5 mm, . p:< = a/S + 0.5 mm, z,.4 = ~/2 + 1.1 mm, cZ*. = 13 o, Fyy = loco. since only he ze o h-o de basis unc ions in (15) ha e non- anishing in eg als along hei de ini ion in e als. The compu a ional s ep o he de e mina ion o [PI in ol - ing mos o he CPU ime is p ecisely he sum o he spec al se ies in (18). I s di ec sum is no ad isable because o i s e y slow con e gence. In his wo k, he Kumme ’s me hod (ex ac ion o an asymp o ic ail) is used o imp o e he se ies con e gence. The se ies o (18) a e hen spli as ollows: whe e <k (k = M, + 1) is he pe mi i i y (o he equi - alen pe mi i i y [E], [21] in he aniso opic case) o he k- h kye . Since La, has he same asymp o ic beha io han C o la ge an, he emainde se ies ( i s e m a he igh hand o (21)) con e ges e y quickly. The asymp o ic ails S;;: a e ex emely slow con e gen se ies, bu quasi- analy ical exp essions o hem a e p o ided in Appendix. The me hods employed o ob ain he o mulas in Appen- dix ha e been al eady desc ibed in [17]; hus, we ha e jus included in his Appendix he inal o mulas which ap- plies o he se ies in ol ed in he analysis o GCPWTL s uc u es. V. NUMERICAL RESULTS A FORTRAN p og am (MULTISLOT) has been de el- oped implemen ing he heo y in his pape . The compu e code uns on a PC/486/33 MHz. In o de o alida e ou me hod, we ha e ep oduced analy ical da a ob ained by means o exac con o mal mappings (which a e a ailable o some pa icula geome ies). Good ag eemen has also been ound wi h da a epo ed o mo e complica ed geome ies which we e ob ained by nume ical p ocedu es. An in e es ing compa ison has been ca ied ou wi h he esul s epo ed in 1151. The me hod used in ha pape is inhe en ly e y accu a e (al hough, in p inciple, i is limi ed o homogeneous o symme ical geome ies). The esul s epo ed in 11.51 a e he Fou ie ans o ms o he su ace cha ge dis ibu ions a he “ac i e” me alliza ion plane o se e al SAW (su - ace acous ic wa e) s uc u es o se e al elec ode exci a- ion condi ions. We ha e ep oduced hei esul s wi h e y good ag eemen and e y ew basis unc ions and sho CPU ime ( ypically less han one second on he a o e-men ioned compu e pla o m), Mino disc epancies we e de ec ed o la ge alues o he Fou ie a iable, due o he di e en na u e o he basis unc ions used in he expansions o he unknown unc ions (su ace cha ge densi y o slo elec ic ield). We belie e ou esul s a e e en mo e accu a e since ou basis unc ions inco po a e he singula beha iou a he me allic edges and he esul s do no modi y when he numbe o unc ions inc ease abo e a ce ain alue (nume ical s abili y). In addi ion, exhaus i e con e gence es s ha e been pe - o med o iden i y he geome ical dimensionless a ios go - eming he con e gence o ou codes. This kind o s udy inc eases ou con idence in ou esul s. The conclusions om hese es s a e analogous o he ones epo ed in [I71 o he mul is ip case. We can summa ize he e he main poin s highligh ed by his s udy: The esidual spec al se ies ( i s e m a he igh hand in (21)) show exponen ial con e gence. The main ge- ome ical pa ame e go e ning his con e gence is he a io h/a (h being he hickness o he hinnes laye adjoining he me allized in e ace and a being he wid h o he enclosu e). The e o e, he pa ame e a. should no be chosen unnecessa ily la ge when la e ally open s uc u es ha e o be simula ed, since his could inc ease he numbe o Fou ie e ms o be added. Fo judicious alues o a jus a ew spec al e ms a e ypically equi ed. An example o he in luence o he box wid h on he capac- i ance coe icien s is shown in Fig. 2. In his igu e, he ela i e di e ence be ween he capaci ance coe icien s o a closed s uc u e and i s la e ally open e sion is plo ed. No e ha om a p ac ical poin o iew he box wid h does no need o be e y la ge o simula e he open s uc u e. I is impo an o highligh ha he wid h o he slo s/s ips egion (dis ance be ween he wo coplana g ound planes) has no in luence on he con e gence o he esidual spec- al se ies, hus a oiding he ypical con e gence p oblems a ising when o ce b u e summa ion is used o analyze na ow slo s/s ips egions. These s a emen s a e illus a ed wi h he example in Table I. No e he impo an CPU ime sa ings o all o he geome ies. The ob aining o he asymp o ic ails, Si.:, in ol es a neg- ligible compu a ional cos in compa ison wi h s aigh o - wa d Fou ie se ies summa ion. Ne e heless, i is use ul o know which ac o (s) may a ec o he con e gence o he powe se ies o Gauss-Chebyshe quad a u es shown Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on July 17,2020 a 15:06:06 UTC om IEEE Xplo e. Res ic ions apply. 2332 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES. VOL. 42, YO. I?, DECRMHER 1994 TABLE 1 NLHBEK OF TPRMS OF TH~ SPECTRAL SERIES WITH ()I, ) AND WITHOUT OBTAIV TIE CAPACITAYCE OF A CPW WITH 0.1% ACCURACY (ns) ASYMFTO~IC EXTRACTlOlV WHICH SHOULD BE ADDED TO (hi = >? = , = 9 9 o) RATIO OF CPU TIMES ( / a) [T/TqT- 0.01 0.2 0 28000 0.05 0.2 1 0.10 0.2 3 10.25 1 0.2 1 3 E 0.25 0.10 0.01 3 0 3 7 15 3 7 __ __ 5500 2900 1500 700 1500 1500 1500 1500 1500 0.025 0.046 0.088 0.086 0.086 0.086 0.100 in he Appendix. The el iciency o he compu a ion o he ails is essen ially con olled by he ela i e p oximi y be ween he slo s, in such a way ha when e y igh ly coupled slo s ( e y na ow s ips) a e p esen mo e e ms mus be e ained o add up he powe se ies (24) and mo e quad a u e poin s a e necessa y in Gauss-Chebyshe in eg a ions (28). Ne e heless, e y ew powe senes e ms a e equi ed excep o imp ac icable s ip wid h, and he numbe o quad a u e poin s do no need o be la ge han he numbe o basis unc ions (al hough i can e en ually be inc eased o accoun o ex emely na ow s ips). In any case, asymp o ic ex ac ion is al~ nys ad isable, since di ec summa ion has always much wo se nume ical pe o mance. 3) The numbe o basis unc ions, n , ; i = 1,. . . , N+ 1. o be e ained o e each slo is ela ed o i s wid h. Thanks o he app op ia e ea u es o he basis unc ions, no mo e han wo o h ee o hem ha e o be used on each slo in mos cases. A ypical con e gence pa e n is shown in Table 11. An in e es ing poin o be emphasized he e is ha when he numbe o basis unc ions is inc eased, no nume ical p oblems a ise. On he con a y we ha e ound nume ical ins abili ies when di ec summa ion o Fou ie se ies is applied. As an addi ional ad an age o ou p ocedu e, we can say ha he expansion coe icien s in (14) a e compu ed wi h ex eme accu acy. The slo elec ic ield is hen ob ained in addi ion o he elec ical pa am- e e s. This is e y di icul wi h o ce b u e summa ion unless a p ohibi i ely la ge numbe o Fou ie e ms is e ained. In o de o check he esul s o ou compu e p og am when applied o a bi a y mul islo geome ies- o which we ha e no ound published da a-we ha e compa ed wi h esul s gene a ed wi h a p og am (MULTISTRIP) w i en o e icien ly analyze mul is ip s uc u es, [ 171. When his code is used, TABLE 11 CONVERGENCE OF THE CAPACITANCE COEFFICIENTS (NORMALIZED TO el)) WITH THE NUMBER OF BASIS FUNCTIONS AT EACH SLOT. DATA: ?’HE STKUCTURE IN FIG. 2 WITH a = 40 mm, 11 = 0.635 mm, 1 = 5.2 mm, .~i, = .s. = 1 .0 mm, .s2 = .s3 = 0.1 mm, : <.I = 17.9 mm, . C2 = 18.95 min, . ,3 = 21.05; = 1360, cy?, = 1060, n :i = j ~, n 4 = ) i mm, . 4 = 22.1 mm, TABLE 111 CAPACITANCE COEFFICIENTS FOR THE EQUIVALENT MI I.TIS TRIP TRANSMISSION LINE (MSTL) AND COPLANAR WAVEGUIDE TRANSMISSION LI~E (CPWTL) GEOMETRIES SHOWN IN (A) AND (B). DATA (I = 20 mm, 11’1 = w? = 0 5 mm, i 2 = 1.0; mm, 5 = 0 2 mm, h = 0 635 mm. = 9 G o EO MSTL (A) a Canaci ance coe icien s o s uc u A ! Capaci ance coe icien s o s uc u e B DC, I 16.6210 1-7.7423 1-0.2990 I 18.8537 he GCPW geome y is simula ed by using wide g ounded s ips o app oxima ely accoun o la e al g ound planes. Table I11 shows he capaci ance coe icien s o a h ee s ips CPW s uc u e when compu ed wi h MULTISLOT and when compu ed wi h MULTISTRIP. In he las case, he o iginal s uc u e is simula ed wi h a i e s ip con igu a ion wi h g ounded ex eme s ips. The esul s o his simula ion o se e al alues o he wid hs o he ex eme s ips a e shown in Table 111. Consis en esul s o he capaci ance pa ame e s ha e been ound wi h bo h compu e codes. Howe e , he CPU ime used by MULTISTRIP is en imes he CPU ime used by MULTISLOT (0.12 seconds in a PC/486/33 MHz compu e , including he compu a ions o he s uc u e in acuum). This di e ence is due o he ollowing ac : o his ype o geome ies he numbe o basis unc ions e- Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on July 17,2020 a 15:06:06 UTC om IEEE Xplo e. Res ic ions apply. DRAKE e a/.: QUICK COMPUTATION OF [C] AND [L] MATRICES OF GENERALIZED MULTICONDUCTOR COPLANAR WAVEGUIDE TRANSMISSION LINES 2333 TABLE IV CAPACII ANC~ COE~FICIENTS OF THI: S x STRIPS COPLANAR WAVEGUIDE TYPE GEOMETRY SHOWN ui THE FIGURE. THE PLANE AA' Is w:! = 0.6 mm, w3 = 0.8 mm, 51 = "4 = 0.3 mm, s2 = 0.2 mm, s1 = 0.1 mm, hl = 0.2 mm, hl = 0.2 mm, FI = 9.6~0, FZ = 4.0~0 A SYMMETRY PLANE. DATA: u = 20 mm, u'l = 0.4 mm, .-.- A -.- A' qui ed o app oxima e he su ace cha ge densi y on he s ips is much highe han he numbe o basis unc ions equi ed o accu a ely app oxima e he 2-componen o he slo elec ic ield. As a inal nume ical example, we ha e compu ed he pa ame e s o he six conduc o s geome y shown in Table IV. The ho izon al symme y plane (AA') is conside ed i s an elec ic wall and hen a magne ic wall in o de o exploi he symme y. Accu acy is se o ou decimal igu es o he no malized capaci ance coe icien s. This accu acy is achie ed by using ou basis unc ions a he 0.3 mm slo s and h ee basis unc ions a he 0.1 mm and 0.2 mm slo s. To al CPU ime was 0.7 seconds on a PC/486/33 MHz. VI. CONCLUSION In his pape we ha e p esen ed a echnique o deal wi h he quasi-s a ic analysis o mul iconduc o plana s uc u es belonging o he amily o coplana wa eguides. The me hod is based on he e icien solu ion o an in eg al equa ion o he elec ic ield exis ing a he slo s. E iciency is achie ed by means o analy ical p ep ocessing o nume ical se ies. The compu e p og ams de eloped on he basis o his me hod a e ex emely accu a e and nume ically s able. In addi ion, CPU imes a e sho enough o conside hese p og ams use ul in he ame o a compu e aided design sys em. The elec ic ield and su ace cha ge densi y can be also compu ed wi h ex eme accu acy. APPENDIX Two al e na i e o mula ion-which we ha e called spec- al and spa ial domain o mula ions espec i ely-ha e been used o he quasi-analy ical de e mina ion o he asymp o ic ails S,"::. A. Spec al Domain Compu a ion The compu a ion o Si;: in (22) implies he addi ion o slowly con e gen igonome ical se ies o he ollowing ype: whe e di = % and e$ = I(xcj 2,i). These se ies ha e al eady appea ed in he analysis o a mul is ip con igu a ion [ 17, (S)]. The esidui calculus ech- nique may be used o con e (23) in o a much mo e quickly con e gen powe se ies (see [17] o mo e de ails) 23 F[-k, -p - IC; q +'l; (dj/ &)2] (iE + i) (p + IC + 1) X 00 yp+4+2k-1 1 dy sinh(7 y) . (24) whe e F is he hype geome ic unc ion, and , he gamma unc ion. The hype geome ic unc ion F[-k, -y - k; q + 1; (dj/d;)2] is a k-deg ee polynomial in (dj/di)2 cosh[(n - &)y] ; p + q e en ; p+q odd sinh[(n - cij)y] x F[-k, -p - k; q + 1; (dj/di)2] - k (k + i) (p + IC + i) (q + i)(dj/di)2m - n=O (25) Two al e na i e closed o m exp essions a e known o he in eg als appea ing in (24) whe e p = p + q + 2k, p = 7 - e$, and < is he Riemann's ze a unc ion. The i s exp ession in (26) is used o he i s ew e ms o he k-se ies. This su ices o mos cases, bu i la ge alues o k a e needed, he second exp ession in (26) p o ides an al e na i e quick solu ion. The case p = q = 0 equi es a sligh ly di e en ea men (2k + 1)F[-k, -k; 1; (d3/d2)'] di 03 (S,":,j)' = k=l ~IC~~(IC + 1) ( Jk x {+:$I +<[2k,L$]} - In [sin ($)I - 111(2) Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on July 17,2020 a 15:06:06 UTC om IEEE Xplo e. Res ic ions apply. 2334 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 42, NO. 12, DECEMBER 1994 No e he necessa y p esence o he in eg a ion cons an , ln(2), which was no calcula ed in [17] because in ha case i canceled ou when he de ini i e S,”:! was compu ed. B. Spa ial Domain Compu a ion Pa se al and con olu ion heo ems p o ide a second e - icien al e na i e o compu e S;:: om he quasi-analy ical in eg a ion o i s spa ial coun e pa s x,,+s,/z Tp(-) :Si;: =/ dx-- xLL-s*12 X (28) 27 7l = --In 4 sin -15 - .’I] sin [-(x + x0]}. 7 { 1:2: 2n (29) The squa e oo in he denomina o o he in eg ands in (28) makes he Gauss-Chebyshe quad a u e o mula o be spe- cially sui able o he compu a ion o hese in eg als. Howe e , di ec Gauss-Chebyshe summa ion o he con olu ions is no ad isable because o he loga i hmic singula i y o &(x, x’) in 5 = 2’. The e o e, a p e ious ex ac ion and sepa a e in eg a ion o his singula i y is e y use ul o inc ease he e iciency o he quad a u es. When mul is ip con igu a ions a e analyzed [17], he e en ual p oximi y be ween he s ips and he la e al walls in oduces quasi-singula beha io o he in eg and o z + x’ -+ 0 o x + x’ + 2n. In coplana wa eguide- ype con igu a ions he e is no such possibili y as a consequence o he p esence o he wo la e al g ounded ins. Consequen ly, he “singula pa ,” S(x, XI), o Lus(x, 2’) should be de ined as: (30) 26 S(z,x’) = --In Ix - J?I. 7T I he ke nel o (28) is spli in o he wo ollowing pa s: &s(x 2.’) = [&(z, x’) - S(X, d)] + S(Z, z’) (31) he con ibu ion o he i s e m ( e y smoo h unc ion) o he con olu ion in eg als is compu ed wi h a low o de Gauss- Chebyshe quad a u e, and he con ibu ion o he second e m (singula pa ) can be analy ically e alua ed. Le [q; x] be he con olu ion in eg al o he “singula pa ” excep a cons an ac o hen, i can be demons a ed ha o i # j (33) whe e sgn(-) is he sign unc ion, and o i = j The las s ep o he compu a ion o (28) is o ca y ou he inne p oduc s. Closed o m exp essions ha e been ound only o he case i = j The es o he inne p oduc s ha e been nume ically e alua ed by low o de Gauss-Chebyshe quad a u es. REFERENCES [I] C. P. Wen, “Coplana -wa eguide di ec ional couple s,” IEEE T ans. Mic owa e Theo y Tech., ol. MTT-18, pp. 318-322, June 1970. [2] R. A. Pucel, “Design conside a ions o monoli hic mic owa e ci cui s,” IEEE T ans. Mic owa e Theo y Tech., ol. MR-29, pp. 51 3-534, June 1981. [3] R. W. Jackson, “Conside a ions in he use o coplana wa eguide o millime e -wa e in eg a ed ci cui s,” IEEE T ans. Mic owa e Theo y Tech., ol. MTT-34, pp. 1450-1456, Dec. 1986. [4] A. A. Oma and Y. L. Chow, “A solu ion o coplana wa eguide wi h ai -b idges using complex images,” IEEE T ans. Mic owa e Theo y Tech., ol. 40, pp. 2070-2077, No . 1992. [5] N. I. Dib, G. E. Ponchak, and L. P. B. Ka ehi, “A heo e ical and expe imen al s udy o coplana wa eguide shun dubs,” IEEE T ans. Mic owa e Theo y Tech., ol. 41, pp. 38-44, Jan. 1993. [6] M. Gillick, I. 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Mic owa e Theo y Tech., ol. 40, pp. 41-48, Jan. 1992. E. D ake, F. Medina, and M. Homo, “Quasi-analy ical s a ic solu ion o he gene alized boxed coplana wa eguide,” In . J. Micmwza e and Millime e -Wa e Compu e -Aided Enginee ing, ol. 4, pp. 163-174, Ap . 1994. J. S. McLean and T. I oh, “Analysis o a new con igu a ion o coplana s ipline,” IEEE T ans. Micmwa e Theo y Tech., ol. 40, pp. 772-774, Ap . 1992. Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on July 17,2020 a 15:06:06 UTC om IEEE Xplo e. Res ic ions apply. DRAKE e al.: QUICK COMPUTATION OF [C I AND [L] MATRICES OF GENERALIZED MULTlCONDUCTOR COPLANAR WAVEGUIDE TRANSMISSION LINES 2335 T. Ki azawa, Y. Hayashi, and R. Mi a, “Asymme ical coupled coplana - ype ansmission lines wi h aniso opic subs a es,” IEE P oc., Mic owa es, Op ics An ennas, ol. 133, p . H, pp. 265-270, Aug. 1986. T. Ki azawa and T. I oh, “P opaga ion cha ac e is ics o coplana - ype ansmission lines wi h lossy media,” IEEE T ans. Mic owa e Theo y Tech, ol. 39, pp. 1694-1700, Oc . 1991. A. F. Molisch, A. R. Baghai-Wadji, and C. 0. Schiebl, “On he applica ion o he Wiene -Hop echnique o elec os a ic ield p oblems in in e digi al ansduce s,” IEEE T ans. Mic owa e Theo y Tech., ol. 41, pp. 318-324, Feb. 1993. G. E. Howa d, J. J. Yang, and Y. L. Chow, “A mul ipipe model o gene al s ip ansmission lines o apid con e gence o in eg al equa ion singula i ies,” IEEE T ans. Mic owa e Theo y Tech., VOL. 40, pp. 628-636, Ap . 1992. E. D ake, F. Medina, and M. Ho no, “Imp o ed quasi-TEM spec al domain analysis o boxed coplana mul iconduc o mic os ip lines,” IEEE T ans. Mic owa e Theo y Tech., ol. 41, pp. 260-267, Feb. 1993. -, “Un anilisis e icien e de lineas mic o i as mul iconduc o as pa a PC’s,” P oc. o VII Symp. Nacional U.R.S.I., pp. 831-835, Milaga, Spain. M. Ho no, F. L. Mesa, F. Medina, and R. Ma quis, “Quasi-TEM analysis 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 T ans. Mic owa e Theo y Tech., ol. 38, pp. 1059-1068, Aug. 1990. S. Haykin, Communica ion Sys ems. New Yo k: Wiley, 1983. F. Medina and M. Homo, “Uppe and lowe hounds on mode ca- paci ances o a la ge class o aniso opic mul ilaye ed mic os ip- like ansmission lines,” P oc. Ins . Elec. Eng. (Mic owa es, Op ics An ennas), ol. 132, no. 3, pp. 157-163, June 1985. En ique D ake was bo n Sep embe 4, 1966, in Mon illa, Chdoba, Spain. He ecei ed he Licenci- ado 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 es. He is Assis an P o esso a he Depa men o Applied Physics a he Uni e si y o Se ille since 1992. His esea ch in e es ocus on he analysis o plana s uc u es and mul iconduc o lines. F ancisco Medina, o a pho og aph and biog aphy, see page 1631 o he Sep embe issue o his TRANSACTIONS. Manuel Ho no (M75), o a pho og aph and biog aphy, see page 432 o he Ma ch issue o his TRANSACTIONS. Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on July 17,2020 a 15:06:06 UTC om IEEE Xplo e. Res ic ions apply.