Quick quasi-TEM analysis of multiconductor transmission lines with rectangular cross section

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 45, NO. 9, SEPTEMBER 1997 1619 Quick Quasi-TEM Analysis o Mul iconduc o T ansmission Lines wi h Rec angula C oss Sec ion Joaqu´ın Be nal, F ancisco Medina, and Manuel Ho no, Membe , IEEE Abs ac —This pape p esen s an e icien and accu a e p oce- du e o compu ing he quasi-s a ic ma ix pa ame e s ( [C] , [L] , [G] , and [R] ) o ec angula -shaped conduc o s embedded in a mul ilaye ed dielec ic medium o e an in ini e g ound plane. An addi ional op g ound plane can also be conside ed. The p oblem is o mula ed in e ms o he space-domain in eg al equa ion o he ee-cha ge dis ibu ion on he slab conduc o su aces. The spa ial G een’s unc ion is compu ed om i s spec al coun e - pa using sys em iden i ica ion echniques [P ony’s me hod o ma ix pencil me hod (MPM)]. The in eg al equa ion is sol ed by means o a Gale kin scheme employing en i e domain basis unc ions. This esul s in a small ma ix size. In addi ion, he quasi-analy ical e alua ion o he en ies o he Gale kin ma ix leads o a e y e icien and accu a e compu e code. A de ailed s udy on he con e gence and accu acy o he me hod has been included. Index Te ms—Losses, mic os ip, mul iconduc o ansmission lines, quasi-TEM analysis, hick conduc o s. I. INTRODUCTION ALARGE amoun o pape s ha e been de o ed o he analysis o plana ansmission lines h oughou he las h ee decades. Mos o he published wo k assumes negligible me alliza ion hickness. This app oxima ion is good enough o many p ac ical si ua ions, and pe mi s he simpli ica ion and e icien use o he analysis ad hoc ma hema ical ech- niques. Howe e , du ing he las ew yea s many au ho s ha e paid a en ion o he p oblem o accoun ing o he nonze o me alliza ion hickness. Apa om he aim o in- c easing accu acy, his in e es comes om he necessi y o analyzing he elec ical beha io o he ela i ely hick s ips employed in monoli hic mic owa e ci cui s and high-speed digi al ci cui s. In hese cases, me alliza ion hicknesses and s ip wid hs a e in he same o de o magni ude. The e o e, he o me can no longe be neglec ed. The s ip hickness mus also be conside ed whene e igh edge coupling is p esen , e en hough ela i ely wide s ips a e unde conside a ion. Mo eo e , he compu a ion o ohmic conduc o losses equi es explici accoun abili y o he s ip hickness. A numbe o au ho s ha e analyzed he e ec s o he me alliza ion hickness by using a ull-wa e analysis, bu in Manusc ip ecei ed Oc obe 4, 1996; e ised Ma ch 25, 1997. This wo k was suppo ed by he DGICYT, Spain, unde P ojec TIC95-0447. The wo k o J. Be nal was suppo ed by a g an om Jun a de Andalucia (Spain). The au ho s a e wi h he G upo de Mic oondas, Depa amen o de Elec ´ onica y Elec omagne ismo, Facul ad de F´ ısica, Uni e sidad de Se illa, 41012 Se ille, Spain. Publishe I em Iden i ie S 0018-9480(97)06066-3. his pape we a e only in e es ed in quasi-s a ic app oaches. This is because o i s compa a i e simplici y, which makes i use ul o de elop quick mic os ip sol e s. Full-wa e me h- ods, excep when applied o ze o- hickness p in ed lines, usually demand a lo o compu e ime. E en when a quasi- TEM app oach is used, accoun ing o s ip- hickness e ec s p ecludes, in p inciple, he use o some e y e icien analy - ical echniques de eloped o ze o- hickness plana s uc u es [1]–[3]. In p inciple, one migh use some ype o pu ely nume ical app oach, such as he ini e-di e ence [4] o ini e- elemen echniques [5]–[7]. Howe e , e en hough signi i- can imp o emen s ha e been in oduced in he o mula ion o hose me hods, hey a e mo e app op ia e when dealing wi h complica ed geome ies which canno be analyzed wi h lesse compu e esou ce-demanding echniques. A hyb id nume ical/analy ical me hod— he me hod o lines—has been ecen ly applied o he analysis o coupled mic os ips wi h ini e hickness [8]. Ne e heless, solu ions based on in eg al- equa ion o mula ions seem o be well sui ed o mos p ac ical cases i he goal is o ge high accu acy wi h low compu a ional cos . Thus, a bi a y-sec ion coupled conduc o s embedded in a laye ed medium ha e been analyzed in [9] by means o an in eg al-equa ion echnique based on he ee-space G een’s unc ion. This me hod was gene alized in [10] o accoun o nonlaye ed dielec ics. Impo an nume ical imp o emen s on his echnique ha e ecen ly been epo ed in [11]. O he au ho s p e e o use a dielec ic G een’s unc ion when dealing wi h laye ed dielec ic subs a es, since he numbe o unknowns is d as ically educed in his way [12]–[15]. The las i e pape s s essed he analy ical p ep ocessing o he compu a ions so as o enhance bo h accu acy and compu a ional speed. Some o he au ho s ha e epo ed di - e en echniques o de elop quick compu a ional ools o he quasi-TEM analysis o pa icula mic os ip s uc u es wi h nonnegligible me alliza ion hickness [16]–[19]. This pape i s in o his esea ch line. In his pape , we p opose a new me hod ha combines he ad an ages o di e en o mula ions in o de o build up a quick quasi-s a ic compu e sol e o ec angula -shaped conduc o s embedded in a mul ilaye ed dielec ic medium. The me hod is based on sol ing he space-domain in eg al equa ion o he ee-cha ge dis ibu ion on he conduc ing slabs. The app op ia e space-domain G een’s unc ion ( he ke nel) is con enien ly ob ained om he spec al one by means o sys em iden i ica ion echniques (using he complex 0018–9480/97$10.00 1997 IEEE Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on June 08,2020 a 15:49:53 UTC om IEEE Xplo e. Res ic ions apply. 1620 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 45, NO. 9, SEPTEMBER 1997 Fig. 1. The mul iconduc o ansmission line s udied in his pape . I consis s o N c ec angula conduc ing slabs embedded in he M h laye o a N -laye s dielec ic medium. images concep [20]). En i e domain basis unc ions a e used in a Gale kin scheme wi h he aim o keeping he size o he inal Gale kin ma ix small. In addi ion, he compu a ion o he elemen s o his ma ix is ca ied ou in a e y e icien way by using sui able nume ical quad a u es and closed- o m in eg a ion. Pu ing oge he all hese elemen s leads o an e icien and accu a e compu e code which is sui able o quick compu a ions e en on a PC pla o m. II. STATEMENT OF THE PROBLEM Conside a mul iconduc o ansmission line such as he one shown in Fig. 1. An a bi a y numbe o ec angula - shaped conduc o s a e embedded inside he h dielec ic laye o a mul ilaye ed ( laye s) dielec ic medium. A bo om g ound plane is always p esen , while he op g ound plane is op ional. Ou main pu pose is o compu e, in an e icien and accu a e way, he pe uni leng h complex capaci ance ma ix o he mul iconduc o sys em (i.e., he capaci ance and conduc ance ma ices). As is well known, he compu a ion o he pe uni leng h induc ance ma ix educes o he e alua ion o o he same s uc u e wi hou dielec ics. Finally, by using he Wheele ’s inc emen al induc ance ule, we can ob ain he esis ance ma ix o he mul iconduc o sys em om , as shown in [21]. As s a ed in he Sec ion I, he e exis s a wide a ie y o ech- niques o compu e he capaci ance ma ices o mul iconduc o ansmission sys ems. We ha e chosen o sol e he in eg al equa ion o he su ace cha ge dis ibu ion on he ec angula conduc o s. This means ha an app op ia e G een’s unc ion accoun ing o he mul iple bounda ies and he bo om (and op) g ound pla es has o be compu ed. The spec al-domain e sion o such a G een’s unc ion can be easily ob ained owing o he laye ed geome y o he dielec ic egion (see, o ins ance, he ans e se ansmission line (TTL) me hod epo ed in [22] o he me hod in [23]). I sou ce and ield poin s a e inside he h laye , he spec al G een’s unc ion can be w i en in he ollowing o m: (1) in (1) s and o he e lec ion coe icien s seen om he lowe and uppe su aces bounding he h laye , p o ided we a e using he equi alen ansmission-line model o compu e he spec al G een’s unc ion [22]. O cou se, a e known closed- o m unc ions o he Fou ie a iable and o he pe mi i i ies and hicknesses o he dielec ic laye s below o abo e he h one. In o de o eco e he -dependence o he space-domain G een’s unc ion, a Fou ie ans o m in e sion has o be ca ied ou : (2) Apa om some pa icula cases, he in e se Fou ie ans- o m (2) canno be pe o med in closed o m. Howe e , he -dependen unc ions appea ing in (1) as mul iplica i e ac o s o he exponen ial e ms could be w i en as a ini e sum o complex exponen ials. In his way, we could apply he ideas epo ed in [20] so as o ge a e y close app oxima ion o he space-domain G een’s unc ion. This poin will be discussed in Sec ion III. Once he G een’s unc ion is known, we can sol e he in eg al equa ion o he ee-cha ge dis ibu ion by using, o ins ance, he Gale kin me hod. In Sec ion IV, we will gi e de ails on he ype o basis unc ions used in his pape and on he echniques applied in o de o speed up compu a ions. F om he cha ge on he conduc ing slabs, we ob ain he capaci ance ma ix. I he elec ical pe mi i i ies o he dielec ic laye s a e complex (dielec ic losses), he elemen s o his ma ix will be complex. Thei imagina y pa s gi e he elemen s o . On he o he hand, is he in e se o he capaci ance ma ix o he s uc u e wi hou dielec ics o e ( is he speed o ligh in acuo). Finally, unde s ong skin-e ec ope a ion, we can compu e he pe -uni -leng h esis ance ma ix , om by using he ex ension o a mul iconduc o ansmission- line sys em o he Wheele ’s ule [21]. The e o e, wi h an accu a e me hod o compu e complex capaci ance ma ices, we will be able o ob ain all he quasi-s a ic ma ix pa ame e s cha ac e izing ou lossy mul iconduc o sys em (including he ma ix, p o ided he skin e ec on me als is s ong). III. COMPUTATION OF THE SPACE-DOMAIN GREEN’SFUNCTION Following he guidelines in [20], he space-domain G een’s unc ion can be usually exp essed as a sho summa ion o e ms which can be easily compu ed om i s spec al-domain e sion. The spec al G een’s unc ion is known in closed o m o a laye ed s uc u e. The basic equa ion unde lying Chow’s p ocedu e is he ollowing ela ionship be ween spec al and spa ial unc ions: (3) Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on June 08,2020 a 15:49:53 UTC om IEEE Xplo e. Res ic ions apply. BERNAL e al.: QUICK QUASI-TEM ANALYSIS OF MULTICONDUCTOR TRANSMISSION LINES 1621 whe e s ands o Fou ie ans o m and is a gene alized complex dis ance. I we can w i e (1) as a sum o e ms such as he ones on he igh -hand side (RHS) o (3), i is clea ha we will ha e he space-domain G een’s unc ion o ou p oblem. This can be done almos in a s aigh o wa d way by expanding he e ms by mul iplying he exponen ials in (1) as a sum o complex exponen ials. The a gumen s o he exponen ials and he coe icien s o he expansion can be compu ed by using spec al es ima ion me hods. In his pape , we ha e used and compa ed he P ony’s me hod employed in [20] and he MPM, as epo ed in [24]. A cau ion should be exe ed when we a e conside ing s uc u es ha ing a op g ound plane in addi ion o he bo om one. In such a case, he spec al unc ions o be app oxima ed a e singula a he poin . This beha io comes om he ollowing ac o : (4) whe e and .A homogeneous s uc u e wi h ela i e pe mi i i y ha ing wo g ound planes sepa a ed by a dis ance has a spec al G een’s unc ion ha can be exp essed as ollows: (5) The G een’s unc ion in (5) and he o iginal one ha e he same beha io a ound he poin . In addi ion, he space-domain G een’s unc ion o he homogeneous s uc u e is known in closed o m: (6) Exp ession (6) can be conside ed as he i s con ibu ion o he comple e G een’s unc ion o he laye ed s uc u e. The addi ional e ms a e exp essed in he spec al domain as he di e ence be ween (1) and (5). The e ms o his spec al unc ion ha e no singula i ies a and can now be ea ed wi hou p oblems. In b ie and a e some algeb a, he spec al-domain G een’s unc ion o he laye ed s uc u e can be w i en in he ollow- ing use ul o m: (7) whe e (8a) (8b) (8c) being In all he abo e exp essions in he p esence o a op g ound plane, and i he e is no op g ound plane. The i s wo e ms a he RHS in (7) co espond o he sou ce poin and he i s eal image. The spec al unc ions can be expanded as sums o complex exponen ials. These e ms can be iewed as he con ibu ions o ce ain complex images, ollowing he e minology in [20]. Usually jus a ew o hese complex images a e enough o ge a e y accu a e ep esen a ion o he G een’s unc ion. The e o e, a e app oxima ing he coe icien s in (8) as sho se ies o complex exponen ial unc ions wi h a gumen s and ampli udes , we ob ain om (3) and (7) he ollowing space-domain G een’s unc ion: (9) whe e is he o al numbe o employed complex images. Once again, he e m a ec ed by he ac o in (9) is p esen only i we ha e a op g ound plane. This e m does no p esen a loga i hmic singula i y in he de ini ion domain and, he e o e, will no lead la e o in eg a ion p oblems. Ne e heless, in he compu e implemen a ion o he code, we ha e ex ac ed ou no only he singula e m, bu also he i s wo eal images ( e lec ions on he op and bo om g ound planes) in o de o a oid leng hy in eg a ions when he sou ce and ield poin s a e e y close o he g ound pla es. In o de o ge some insigh abou he ea u es o he app oxima ion we a e using, we include in his sec ion some nume ical examples. Thus, Fig. 2 shows he ela i e e o when a ypical spec al unc ion—such as he ones ound in ou p oblem—is app oxima ed by an inc easing numbe o complex images. In his example, we ha e app oxima ed Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on June 08,2020 a 15:49:53 UTC om IEEE Xplo e. Res ic ions apply. 1622 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 45, NO. 9, SEPTEMBER 1997 (a) (b) Fig. 2. Rela i e e o in he app oxima ion o a ypical spec al unc ion in ou p oblem by using (a) P ony’s me hod, and (b) MPM. We p esen cu es o 2–5 complex images. 0 L (  ) co esponds o a single dielec ic laye ( " =2 ;h 1 =1 mm) on a pe ec g ound plane. , which is he sole unc ion o be app oxima ed when dealing wi h he s anda d mic os ip s uc u e. The complex images ha e been ound by applying P ony’s me hod [Fig. 2(a)] and he MPM [Fig. 2(b)]. F om hese igu es we conclude ha no mo e han ou o i e images a e necessa y o ge a e y good ep esen a ion o he o iginal G een’s unc ion in he spec al domain and, he e o e, in he space domain. Fig. 2(b) also includes he unc ion o be app oxima ed. No ice ha al hough he P ony app oxima ion is mo e accu a e when inc eases, he use o he MPM seems o be ad isable because his app oach yields be e esul s in he egion whe e he app oxima ed unc ion is meaning ully di e en om ze o. This ea u e has been con i med o a la ge numbe o nume ical examples. In b ie , we conclude a e a lo o nume ical es s ha jus a ew images compu ed wi h he MPM will ensu e a high-quali y app oxima ion o he equi ed space-domain G een’s unc ion. IV. APPLICATION OF THE GALERKIN METHOD In Sec ion III, we ha e desc ibed a simple me hod o ob ain an app oxima e closed- o m exp ession o he space-domain G een’s unc ion. Now we ha e o sol e he in eg al equa ion o he su ace ee-cha ge dis ibu ion on he ec angula conduc o s by using he Gale kin me hod. I is well known ha he e iciency o his echnique depends o a la ge ex en on he sui abili y o he chosen basis unc ions. In ou pa icula case we should use unc ions accoun ing o he singula beha io a he me allic co ne s. Ne e heless, we ha e chosen a di e en c i e ion. We a e mo e in e es ed in using unc ions leading o closed- o m o mulas o he elemen s o he Gale kin ma ix in o de o speed up he illing o such a ma ix. Bu , in addi ion, he numbe o unc ions needed o ge a gi en accu acy should be kep as low as possible. These wo goals can be achie ed by using each o he aces o he ec angula conduc o s o he basis unc ions usually employed o app oxima e he cha ge dis ibu ion on ze o hickness s ips: i s -kind Chebyshe polynomials weighed by he Maxwell dis ibu ion. In his way, o a gi en conduc ing ace o wid h we w i e (10) whe e s ands o he o a iable ( o ho izon al and e ical conduc ing aces, espec i ely), is he middle poin o he in e al whe e he cha ge densi y is app oxima ed, and is he numbe o e ained basis unc ions on ha ace. An impo an poin is ha he unc ions in (10) only pa ially accoun o he singula i ies a he co ne s. Howe e , hey allow us a quick illing o he Gale kin ma ix. This is because some o he equi ed in eg a ions can be pe o med in closed o m. When we use he G een’s unc ion in he o m gi en in (9) and he basis unc ions in (10), we ha e o ca y ou he ollowing in eg a ions: (11) whe e s ands o o . The in eg als in (11) can be w i en in e ms o he ollowing one: (12) This in eg al has a closed o m and can be exp essed as ollows: Case a) : (13a) Case b) : (13b) Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on June 08,2020 a 15:49:53 UTC om IEEE Xplo e. Res ic ions apply. BERNAL e al.: QUICK QUASI-TEM ANALYSIS OF MULTICONDUCTOR TRANSMISSION LINES 1623 whe e (The sign be o e he squa e oo is chosen in such a way ha ). We mus men ion he e ha he eal pa o he in eg al in (12) was epo ed by Fikio is e al. in [1], and ha we ha e also essen ially used he same echnique o ge he imagina y pa o (12). The emaining in eg a ions (inne p oduc s wi h he es unc ions) can now be ca ied ou by using low-o de Gauss–Chebyshe quad a u es. I a op g ound plane is p esen we ha e o s ill accoun o he con ibu ion o he e m in (9). This is easily done pe o ming a double low-o de Gauss–Chebyshe quad a u e, owing o he ma hema ical na u e o he in eg and. The inal esul is ha he Gale kin ma ix is illed wi h low compu a ional e o . This could also be done o simple subsec ional pulse o iangula basis unc ions. The ad an age o using he unc ions in (10) is ha we do no need oo many o hem o ge e y accu a e esul s, as will be shown in he Sec ion V. The e o e, he Gale kin- ma ix size will be small and he o e all compu a ion ime will be low. V. NUMERICAL RESULTS As a i s s ep in he analysis o he nume ical beha io o he p oposed echnique, we ha e iden i ied he di e en ac o s a ec ing he accu acy o he inal esul s. In Sec ion III, we said some wo ds abou he compu a ion o he space- domain G een’s unc ion. Since his unc ion is nume ically compu ed by means o an app oxima e me hod, i s alues will be a ec ed by a ce ain e o . Howe e , his e o can be sys ema ically educed by inc easing he numbe o complex images. In ac , jus a ew o hese images ensu e a ela i e e o well below one pa in 10 in he whole ange o in e es . Ano he sou ce o e o o nume ical ype can be ound in he e alua ion o he nume ical quad a u es needed o compu e some in eg als. A e a lo o nume ical expe imen s, we concluded ha e y good esul s will be ob ained by using a numbe o quad a u e poin s exceeding in wo he o de o he highe o de Chebyshe polynomial used in he basis- unc ion’s expansion. In he case o inne p oduc s co esponding o unc ions de ined on ouching s ip segmen s, his numbe should be inc eased o 10. Wi h his choice, we do no de ec e o s associa ed wi h e oneous compu a ion o de ini e in eg als. Howe e , he main ac o a ec ing he accu acy o he inal esul s is he numbe o basis unc ions e ained in he expansion o he ee-cha ge dis ibu ion. We a e in e es ed in ge ing good enough esul s wi h ew basis unc ions so as o keep he size o he Gale kin ma ix small. Ob i- ously, ou compu e code pe mi s us o conside he case o ze o- hickness s ips as a pa icula case. We ha e made compa isons o he esul s p o ided by ou code wi h o he p ac ically exac esul s epo ed in he li e a u e [1]–[3]. Ou TABLE I C 11 ( +) AND C 1 2 ( 0 ) (pF/m) FOR THE STRUCTURE IN THE FIGURE AGAINST THE NUMBER OF BASIS FUNCTIONS ON VERTICAL ( N ) AND HORIZONTAL ( N w ) CONDUCTOR FACES. w =1 , s = =0 : 2 , h 1 =3 , h 2 =2 (mm). " 1 =2 : 5 , " 2 =10 p og am also yields i ually exac esul s because o he ze o hickness case, i is a quasi-analy ical mic os ip sol e (such as he me hods epo ed in he ci ed pape s), wi h ex emely low cen al p ocessing uni (CPU) ime consump ion. This excellen pe o mance is ela ed o he na u e o he used basis unc ions—which a e especially sui able o ze o hickness s ips—and o he analy ical ea men o he compu a ion o he Gale kin ma ix en ies. The e o e, as a subp oduc o ou analysis, we ha e an ex emely e icien code o he analysis o ze o hickness coupled s ips a ailable. Ne e heless, in he con ex o his pape , we a e mo e in e es ed in he case o nonze o hickness s ips. The e o e, we show in Table I he con e gence o he capaci ance coe icien s o a pai o hick coupled mic os ips. F om Table I, we can see ha con e gence is no so good as he one we achie ed in he analysis o ze o- hickness coupled s ips [2]. The eason o his is ha he employed basis unc ions do no exac ly i he co ne beha io . Howe e , hese unc ions a e e y good om a p ac ical poin o iew: wo basis unc ions pe s ip side yield esul s wi h an accu acy be e han 0.3%. O cou se, he numbe o basis unc ions has o be inc eased when he geome y is mo e c i ical ( o example, when coupling be ween he s ips is ex emely s ong o in o he geome ically complica ed si ua ions). Ne e heless, we usually ob ain e y good esul s wi h jus a ew basis unc ions pe s ip side. Ge ing simila accu acy wi h subsec ional- ype basis unc ions would equi e many mo e basis unc ions, he e o e inc easing he CPU ime. A e analyzing he con e gence o he me hod, i is in- e es ing o check he ypical a ainable accu acy using some Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on June 08,2020 a 15:49:53 UTC om IEEE Xplo e. Res ic ions apply. 1624 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 45, NO. 9, SEPTEMBER 1997 TABLE II Z 0 OF A SYMMETRICAL SHIELDED SLAB LINE N w; IS THE NUMBER OF BASIS FUNCTIONS W = SLAB WIDTH. = SLAB THICKNESS b = SEPARATION BETWEEN GROUND PLATES app op ia e benchma k. We can use as a benchma k he ec angula slab symme ically placed be ween wo in ini e g ound planes, since o his s uc u e he e exis s an ana- ly ical solu ion. Table II shows he compa ison be ween exac solu ions aken om [25] (con o mal mapping app oach) and nume ical alues compu ed wi h ou me hod. I is clea om his able ha ou me hod can p o ide e y high accu acy i we use enough basis unc ions. Howe e , e en using jus one unc ion on each side o he slab, we ge esul s wi hin a 0.4% e o excep o he wo s case ( ), whe e he e o is a ound 1.5%. In his case, using jus one mo e unc ion on he long sides o he slab esul s in a d as ic imp o emen o he accu acy (less han 0.02% e o ). O he compa isons ha e been made wi h some exac esul s epo ed in [26] wi h simila conclusions. Con e gence o he co ec alue has hen been demons a ed. No e ha his poin is impo an since ou basis unc ions do no exhibi he igo ously co ec edge beha io . In [19], he au ho p esen s a echnique based on spec al- domain analysis (SDA) o compu e he quasi-s a ic pa ame e s o boxed coupled s ips o a bi a y hickness. We ha e made compa isons wi h he esul s epo ed in [19] and we ha e ound e y good ag eemen . In pa icula , we ha e compu ed he modal pa ame e s o wo e y igh ly coupled s ips, which canno be e icien ly ea ed by using he simple mul is ip model epo ed in [15] (as was claimed in [19]). Ou esul s a e e y accu a e o his case e en using only wo basis unc ions on he aced sides o he s ips and one on he emaining sides (0.2% e o ). This esul s in CPU imes much sho e han he ones epo ed in [19] (ou s is below 1 s on a 66-MHz pen ium-based PC). Ou p og am can be used o calcula e he esis ance ma ix o he mul iconduc o sys em jus applying he Wheele ’s ule, such as discussed in [21]: whe e is he su ace esis ance o he me al, is he induc ance ma ix, and is he coo dina e a iable no mal o he conduc o su ace. The de i a i e is nume ically compu ed, so we only need o accu a ely compu e . Table III shows he no malized esis ance o a ec angula slab on a g ound TABLE III NORMALIZED RESISTANCE OF RECTANGULAR SLAB OVER GROUND PLANE AS A FUNCTION OF THE NUMBER OF BASIS FUNCTIONS ( n ) ON EACH STRIP SIDE AND OF THE 1  INCREMENT USED IN THE NUMERICAL DERIVATION SLAB WIDTH =2 a ,SLAB THICKNESS = a ,DISTANCE TO GROUND PLANE = a Fig. 3. A enua ion ac o s o he undamen al quasi-TEM modes o he ou mic os ips sys em analyzed in [27]. S ip wid hs: w 1 = w 4 =0 : 6 mm, w 2 = w 3 =0 : 3 mm, sepa a ion be ween s ips: s 1 = s 3 =0 : 3 mm, s 2 =0 : 2 mm, s ip hickness: =0 : 01 mm, subs a e hickness: h =0 : 635 mm, heigh o co e : d =6 : 0 mm, " =9 : 8 " 0 ; =5 : 1 1 10 7 S/m plane as a unc ion o he numbe o basis unc ions used on each s ip side. The esul is close o he one epo ed in [13] . We ha e included se e al columns o show how he inc emen used o nume ically pe o m he de i a ion a ec s he inal esul . We can ypically choose a ound 10 imes he smalles dimension o he conduc o . The esul emains s able o lowe alues o . No e ha he de i a i e is e y accu a ely compu ed e en hough he o iginal unc ion is a ec ed by a ce ain e o , since his is a sys ema ic e o ha a ec s in he same magni ude he wo alues o he unc ion equi ed o nume ically pe o m he de i a ion. Recen ly, Gen ili e al. [27] ha e epo ed a echnique o compu e he modal a enua ion ac o s o a mul iconduc o line. The alues o he pa ame e s can be eadily ob ained om he cha ac e is ic ma- ices compu ed in his pape . This way, we ha e compa ed ou esul s wi h hose epo ed in [27], wi h e y good ag eemen . Fo ins ance, we show in Fig. 3 he ou modal a enua ion ac o s co esponding o he undamen al modes suppo ed by a ou -coupled-s ip lossy sys em. Ou esul s a e e y close o he ones compu ed by he me hod p oposed by Gen ili e al. (sligh di e ences could be a ibu ed o he side walls conside ed in ha pape ), bu some impo an disc epancies ha e been ound when compa ing he esul s compu ed by means o a ini e-elemen -based code. Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on June 08,2020 a 15:49:53 UTC om IEEE Xplo e. Res ic ions apply. BERNAL e al.: QUICK QUASI-TEM ANALYSIS OF MULTICONDUCTOR TRANSMISSION LINES 1625 TABLE IV COEFFICIENTS OF [C] , [L] ,AND [R] FOR THE FIVE CONDUCTORS’ STRUCTURE IN THE FIGURE. w =3 , s =2 , h =1 , =1 (mm), " 1 =2 , " 2 =1 . Finally, Table IV shows he elemen s o and o a i e-coupled-s ip sys em. All he igu es in Table IV a e co ec . We ha e used 15 basis unc ions on each s ip side, i.e., a o al o 300 basis unc ions, o ensu e he accu acy o hese esul s. This esul s in a compu a ion ime o 75 s on a Pen ium PC/66 MHz. Howe e , accu acy be e han 0.2% o all he ma ix elemen s is achie ed by using i e unc ions and 13 s o CPU ime. I an e o in he o de o 1% can be ole a ed, we only ha e o use wo unc ions pe s ip side. We ha e comple ed Table IV, including he no malized esis ance ma ix elemen s. VI. CONCLUSIONS This pape desc ibes an e icien and accu a e echnique o compu e he quasi-s a ic ma ix pa ame e s ( , , , ) o a sys em o ec angula c oss-sec ion coupled con- duc o s embedded in a laye ed dielec ic medium. Accu- acy and nume ical e iciency a e achie ed by means o wo main issues. Fi s , we use en i e domain basis unc- ions o app oxima e he ee su ace-cha ge dis ibu ion on he s ips. These unc ions allow us o keep he Gale kin ma ix size small when compa ed wi h ypical ma ix size associa ed wi h he use o subsec ional unc ions. Second, we ge quasi-analy ical e alua ion o he in eg als de ining he Gale kin-ma ix en ies. This has been done by aking ad an age o he use o he complex image concep and he ma hema ical p ope ies o he employed basis unc ions. Any spec al es ima ion echnique can be used o compu e he complex images, bu he MPM seems o pe o m be e han P ony’s me hod. The inal p oduc is an accu a e and quick compu e code ha pe mi s one o analyze unde quasi- TEM assump ion a a ie y o ansmission lines consis ing o coupled ec angula conduc ing slabs. The pe o mance o he code has been exhaus i ely checked by making con e gence es s and compa isons wi h o he echniques, some o which ha e been included in Sec ion V. No e ha al hough in his pape i has been assumed ha all he s ips we e embed- ded in he same dielec ic egion, a mo e gene al si ua ion can be ea ed by applying he same gene al ideas epo ed he e. REFERENCES [1] J. G. Fikio is, J. L. Tsalamengas, G. J. Fikio is, “Exac solu ions o shielded p in ed mic os ip lines by he Ca leman–Vekua me hod,” IEEE T ans. Mic owa e Theo y Tech., ol. 37, pp. 21–33, Jan. 1989. [2] E. D ake, F. Medina, 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. [3] D. Homen co schi, G. Ghione, C. Naldi, R. 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De Zu e , “Space-domain G een’s unc ion app oach o he capaci ance calcula ion o mul iconduc o lines in mul ilaye ed dielec ics wi h imp o ed su ace cha ge modeling,” IEEE T ans. Mi- c owa e Theo y Tech., ol. 37, pp. 1562–1568, Oc . 1989. [13] F. Olyslage , N. Fach´ e, 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 owa e Theo y Tech., ol. 39, pp. 901–909, June 1991. Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on June 08,2020 a 15:49:53 UTC om IEEE Xplo e. Res ic ions apply. 1626 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 45, NO. 9, SEPTEMBER 1997 [14] K. S. Oh, D. Kuzne so , and J. E. Schu -Aine, “Capaci ance compu- a ions in a mul ilaye ed dielec ic medium using closed- o m spa ial G een’s unc ions,” IEEE T ans. Mic owa e Theo y Tech., ol. 42, pp. 1443–1453, Aug. 1994. [15] G. Plaza, F. Mesa, and M. Ho no, “Quick compu a ion o [C] , [L] , [G] , and [R] ma ices o mul iconduc o and mul ilaye ed ansmission sys ems,” IEEE T ans. Mic owa e Theo y Tech., ol. 43, pp. 1623–1626, July 1995. [16] V. Rizzoli, “Highly e icien calcula ion o shielded mic os ip s uc u es in he p esence o unde cu ing,” IEEE T ans. Mic owa e Theo y Tech., ol. MTT-27, pp. 150–157, Feb. 1979. [17] E. D ake, F. Medina, and M. Ho no, “Quasi-TEM analysis o hick mul is ip lines using an e icien i e a i e me hod,” Mic owa e Op . Technol. Le ., ol. 5, no. 10, pp. 530–534, Sep . 1992. [18] G. G. Gen ili and G. Macchia ella, “Quasi-s a ic analysis o shielded plana ansmission lines wi h ini e me alliza ion hickness by a mixed spec al-space domain me hod,” IEEE T ans. Mic owa e Theo y Tech., ol. 42, pp. 249–255, Feb. 1994. [19] J.-T. Kuo, “Accu a e quasi-TEM spec al domain analysis o single and mul iple coupled mic os ip lines o a bi a y me alliza ion hickness,” IEEE T ans. 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Pe ei a, “Using he ma ix pencil me hod o es ima e he pa ame e s o a sum o complex exponen ials,” An ennas P opaga . Mag., ol. 37, no. 1, pp. 48–55, Feb. 1995. [25] H. J. Rible , “An app oxima ion o he cha ac e is ic impedance o shielded-slab lines,” IEEE T ans. Mic owa e Theo y Tech., ol. MTT-27, pp. 557–559, June 1979. [26] J. R. No ie , “Calcula ion o he in e ac ion be ween he inging capaci ance o symme ical s ipline using he ini e elemen me hod,” IEEE T ans. Mic owa e Theo y Tech., ol. MTT-34, pp. 191–193, Jan. 1986. [27] G. G. Gen ili and A. Melloni, “The inc emen al induc ance ule in quasi- TEM coupled ansmission lines,” IEEE T ans. Mic owa e Theo y Tech., ol. 43, pp. 1276–1280, June 1995. Joaqu´ın Be nal was bo n in Se illa, Spain, in 1971. He ecei ed he Licenciado deg ee in physics om he Uni e si y o Se ille, Spain, in 1994, and is cu en ly wo king owa d he Ph.D. deg ee. His esea ch in e es s ocus on he analysis o plana s uc u es o in eg a ed mic owa e ci cui s. F ancisco Medina was bo n in Pue o Real, C´adiz, Spain, in No embe , 1960. He ecei ed he Licenciado and he Doc o deg ees, bo h in physics, om he Uni e si y o Se ille, Se ille, Spain, in 1983 and 1987, espec i ely. F om 1986 o 1987, he spen he academic yea a he Labo a oi e de Mic oondes de l’ENSEEIHT, Toulouse, F ance, on schola ship om MEC-MRT. F om 1985 o 1989, he was an Assis an P o esso in he Depa men o Elec onics and Elec omagne ics, Uni e si y o Se ille, and since 1990, he has been a P o eso Ti ula (Associa e P o esso ) o elec omagne ics. His esea ch deals mainly wi h analy ical and nume ical me hods o plana s uc u es and ci cui applica ions o mul iconduc o lines. D . Medina was a membe o he Technical P og amme Commi ee o he 23 d Eu opean Mic owa e Con e ence, Mad id, Spain, in 1993. Manuel Ho no (M’75) was bo n in To e del Campo, Ja´en, Spain. He ecei ed he Licenciado and he Doc o deg ees, bo h in physics, om he Uni e si y o Se ille, Se ille, Spain, in 1969, and 1972, espec i ely. Since 1969, he has been wi h he Depa men o Elec onics and Elec omagne ism, Uni e si y o Se ille, whe e he became an Assis an P o esso in 1970, Associa e P o esso in 1975, and Full 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 complex ma e ials, mul iconduc o ansmission lines, and p in ed an ennas. D . Ho no is a membe o he Elec omagne ism Academy, Massachuse s Ins i u e o Technology (MIT), Camb idge. Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on June 08,2020 a 15:49:53 UTC om IEEE Xplo e. Res ic ions apply.