An Improved Iterative Technique for the Quasi-Tem Analysis of Generalized Planar Lines

652 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 40, NO. 4, APRIL 1992 An Imp o ed I e a i e Technique o he Quasi-TEM Analysis o Gene alized Plana Lines En ique D ake, F ancisco Medina, Membe IEEE, and Manuel Homo, Membe IEEE Abs ac -The Gene alized Bioconjuga e G adien Me hod (GBGM) and FFT algo i hms a e used o he quasi-TEM anal- ysis o gene alized mul is ip lines embedded in mul ilaye ed lossless/lossy, iso/aniso opic dielec ic and/o magne ic media. Impo an compu a ional imp o emen is achie ed by includ- ing asymp o ic ex ac ion echniques in he de e mina ion o he spa ial G een’s unc ion ma ix. Compa isons wi h o he i e a i e p ocedu es a e p esen ed. Se e al p ac ical s uc u es a e analyzed and nume ical esul s a e compa ed wi h p e i- ously published da a. I. INTRODUCTION N THE PAST decades, he quasi-TEM app oxima ion I has been ex ensi ely used o analyze plana mic os ip- like lines appea ing in MIC and MMIC. As i is well known, quasi-TEM analysis is use ul and easonably ac- cu a e a he lowe end o he equency spec um o many p ac ical lines in ol ing lossless/lossy dielec idmag- ne ic ma e ials [l]. Unde quasi-TEM assump ion, he p opaga ion p ob- lem can be educed o sol ing he wo dimensional La- place’s equa ion subjec ed o he app op ia e bounda y condi ions. A wide a ie y o echniques has been used o sol e ha p oblem (con o mal mapping, spec al and a ia ional me hods, in eg al equa ion me hod and so on). When one o hese s anda d me hods is applied o he analysis o plana s uc u es o a bi a y geome y, he ad- di ion o subs a e laye s and me alliza ions conside ably complica es he applica ion o he me hod. This also oc- cu s in he esolu ion o o he elec omagne ic p oblems (sca e ing, adia ion . . .) in which plana s uc u es a e in ol ed. Owing o his, se e al i e a i e p ocedu es ha e been ecen ly p oposed o deal wi h his ype o p oblems [2]-[8]. These i e a i e echniques, in conjunc ion wi h FFT algo i hms, p o ide an e icien way o sol e in eg al o ma ix con olu ional equa ions. In he case o la ge size ma ix ope a o s, he p ima y ad an age a ising om he use o ecu si e algo i hms is o ci cum en he ex- cessi e s o age p oblems inhe en in he Gaussian elimi- na ion o o he di ec in e sion me hods. Ano he a gu- men o i e a i ely sol ing an ope a o equa ion is he ob ious ac ha he p ocess can be s opped once a p e- Manusc ip ecei ed May 20, 1991 e ised Oc obe 28, 1991. This wo k was suppo ed by he DGICYT, Spain (P ojec PB87-0798-C03-01). The au ho s a e wi h he Mic owa e G oup, Depa men o Elec onics and Elec omagne ism, Uni e si y o Se ille, A da. Reina Me cedes sh. 41012 Se ille, Spain. IEEE Log Numbe 9106048. speci ied deg ee o accu acy in he solu ion is eached. This gene ally esul s in CPU ime sa ings. In addi ion, he choice o he ini ial es ima e (s a ing poin o he i - e a i e p ocess) is no c i ical. The e o e, i is no nec- essa y o ha e p e ious knowledge o he ea u es o he solu ion. The di e en e sions o he Conjuga e G adien Me hod (CGM) a e p obably he bes known i e a i e echniques [6]. In con as o he spec al i e a i e ech- niques [3], [7], he CGd o e s heo e ical con e gence o he exac solu ion in a ini e numbe o s eps (in absence o ound-o e o ). Ne e heless, in some p ac ical cases, he spec al i e a i e echniques (CCST [3], SIM [7]) ha e p o ed o ha e a highe a e o con e gence han he CGM . A modi ica ion o he CGM has been ecen ly de el- oped o enhance i s a e o con e gence: he Gene alized Biconjuga e G adien Me hod (GBGM) [8]. The GBGM simul aneously sol es bo h he ope a o equa ion and i s adjoin equa ion, hus a oiding he esolu ion o he no - mal equa ion associa ed wi h non-He mi ian ope a o s- his is he case in his pape -, which is one o he main easons o he slow con e gence in he CGM. In he p esen pape , we in end o use he GBGM o analyzing a e y gene al class o plana ansmission lines unde quasi-TEM assump ion and o compa e he GBGM wi h o he i e a i e schemes. P io o sol ing he in eg al equa ion o he unknown ee cha ge densi y pe uni leng h (p.u.1.) on he con- duc ing s ips, i is necessa y o de e mine he spa ial G een’s unc ion ma ix co esponding o he s uc u e unde analysis. In his pape , we ha e also ocused ou a en ion on he e icien compu a ion o his quan i y. To achie e his goal, we ha e used an e icien asymp o ic ex ac ion echnique in he de e mina ion o he spa ial G een’s unc ion om i s spec al ep esen a ion. The spec al G een’s unc ion is eadily ob ained by using he heo y explained in [9], [ 11. This echnique, oge he wi h he FFT algo i hm, has made i possible o minimize memo y s o age and CPU ime. In o de o illus a e he alidi y and he s eng h o he me hod, nume ical esul s a e p esen ed and compa ed wi h published da a o some p ac ical s uc u es. 11. OUTLINE OF THE PROBLEM: QUASI-TEM ANALYSIS The c oss sec ion o he gene al plana mul iconduc o ansmission line o be analyzed is shown in Fig. 1. The 0018-9480/92$03.00 0 1992 IEEE Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on July 20,2020 a 15:24:58 UTC om IEEE Xplo e. Res ic ions apply. 653 DRAKE e al.: AN IMPROVED ITERATIVE TECHNIQUE FOR QUASI-TEM ANALYSIS Elec ic wall. magne ic wall o open bounda y o he ope a o equa ion appea ing in (1) when subs a e losses, o longi udinally magne ized semiconduc o s o e i es a e p esen . Ne e heless, in he p esen wo k we ha e checked ha he GBGM has a as e con e gence han he o dina y CGM e en i he ope a o o (1) is He - mi ian. To sol e (1) by means o he GBGM, i is necessa y o disc e ize ha con olu ional exp ession. Two possibili- ies a e a ailable o his pu pose: he use o he Me hod o Momen s (MM) [4], [lo] o he di ec applica ion o he GBGM. In he p esen pape , we choose he la e op- ion. The o al egion ha akes pa in he p oblem is di ided in o N, subin e als o wid h T. All he unc ions i-4 i=N1 i=N-2 i= n i=nl i=2 Y i=l i=O L Elec ic wall Fig. 1. C oss-sec ion o a gene al mul ilaye ed mul is ip line. sys em p esen s ansla ional symme y in he di ec ion pe pendicula o he x-y plane. The s a i ied medium is made o N laye s o lossy iso/aniso opic dielec ic o magne ic subs a es. The lowe bounda y o he con igu- a ion (in e ace 0) is an elec ic wall and he uppe bounda y (in e ace N) can be conside ed o be any one o hese h ee possibili ies: g ounded pla es, magne ic walls o open bounda ies. The ans e se pe mi i i y en- so [4, and he ans e se magne ic pe meabili y enso [pJ o each laye (i = 1, * , N) a e assumed o be complex in o de o accoun o subs a e losses in he analysis. The equi alen pe mi i i y enso [l] used o he de e mina ion o he induc ance ma ix, [L] , becomes non-symme ical i longi udinally magne ized semicon- duc o s o e i es a e in ol ed. In Fig. 1, M in e aces which a e de ined in ha egion and appea in he i e a i e p ocess (including he cha ge densi y and he G een’s unc ions) a e conside ed o be cons an in each subin e - al and a e assumed o be equal o hei alue a he cen e o he sub egion. Once he disc e iza ion p ocess has been ca ied ou , in each i e a ion, he con olu ions a e e al- ua ed a he same poin s a which he o iginal unc ions> a e sampled. This is wha a me hod o momen p ac i- ione would e m as del a unc ion expansion and weigh - ing. A his poin , i mus be no ed ha in o de o compu e a linea con olu ion sum in an e icien way, i is sui able o app oxima e ha linea con olu ion by a cyclic disc e e con olu ion, hus aking ad an age o he use o FFT al- go i hms. A e doing his, (1) is educed o (nk, k = 1, * - , M) a e occupied by an a bi a y num- M be , N,, o in ini ely hin pe ec conduc ing s ips wi h ,(m) = TFFT-l[ J= c 1 ~;~(n) FFT {pj}] a bi a y loca ions. The de e mina ion o he quasi-TEM p opaga ion pa- ame e s o he line is en i ely based on he e alua ion o k/kTe D, i = 1, * - , M (2) he complex capaci ance ma ix pe uni leng h (p.u.l), [C], [ 13. This e alua ion implies he esolu ion o he ol- lowing sys em o in eg al equa ions ( o N, canonical ex- ci a ion p oblems): whe e Di is he egion occupied by me alliza ions a he i h me allized in e ace, pj(x) and K(x) a e he complex cha ge densi y and he ol age exci a ion a he i h me al- lized in e ace espec i ely, and Gij(x - x’) (i, J = 1, - , M) s and o he alues o he spa ial G een’s unc- ion a he me allized in e aces. 111. APPLICATION OF THE GBGM-FFT ALGORITHM The GBGM [8] is an i e a i e me hod used o sol e he ope a o equa ion AI = Y in which A is a gi en linea ope a o and I is he unknown o be ound o a pa icula exci a ion Y. As i is said in [8], he GBGM is specially i ed o he solu ion o he equa ion AI = Y when he ope a o A is non-He mi ian. In gene al, his is he case whe e K(kT) is he ol age (wi h alue 0 o 1) on he k h poin sampled on he s ips o he i h me allized in e ace, FFT { pj} is he Fas Fou ie T ans o m o he sampled cha ge densi y a he j h me allized in e ace including he ze o padding o he egions wi hou me alliza ions, and he G:,(n) (i, J = 1, - * - , M) a e ob ained as desc ibed in he ollowing sec ion. Once he disc e iza ion p ocess has been ca ied ou , he compu a ional implemen a ion o he GBGM is no longe a p oblem because (2) is jus a sys em o a linea algeb aic equa ions. I can be obse ed ha he use o FFT (co esponding o cyclic con olu ions) o compu e linea con olu ion sums implies ha he c oss sec ion o he line unde s udy p esen s a pe iodic na u e (in he x-axis di ec ion). In ac , i To is he o al wid h o he sampled egion (To = N,, T), he equa ion (2) co esponds o he s uc u e ob ained by he pe iodic epe i ion o ha egion wi h pe iod To. The e o e, he ape iodic sec ions mus be pe iodically simula ed by in oducing wo ic i ious side walls a away om he me allized egions. As we will see, he choice o he wid h (To) o an app op ia e simula ing pe iod is a unc ion o he geome ical cha ac e is ics o each line. Ob iously, eally pe iodic s uc u es a e aken in o ac- coun in an exac way. Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on July 20,2020 a 15:24:58 UTC om IEEE Xplo e. Res ic ions apply. 654 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 40, NO. 4, APRIL 1992 IV. TREATMENT OF THE GREEN’S FUNCTION MATRIX The compu a ion o he spa ial G een’s unc ion ma ix o a gene al mul ilaye ed con igu a ion canno be achie ed in closed o m. On he con a y, a e y simple sys ema ic algo i hm can be implemen ed o ob ain i s Fou ie ans o m. This has been done he e by using he ecu en scheme epo ed in [9]- alid o non-coplana conduc ing s ips-in conjunc ion wi h he heo y de el- oped in [ 11-which enables us o deal wi h lossy and mag- ne ic subs a es. This echnique has been ecen ly called he Equi alen Bounda y Me hod (EBM) [ 1 13. In p ac ice, he e icien compu a ion o he con olu ion sums is achie ed by using he Disc e e Con olu ion Theo- em and he FFT algo i hms. The applica ion o his ech- nique only equi es he knowledge o he spec al G een’s unc ion ma ix. Howe e , a compu a ional ques ion d i es us o build an app oxima ion o he spa ial G een’s unc ion ma ix. When he pe iodic simula ion o an ape - iodic s uc u e is pe o med, all he disc e ized unc ions mus be usually padded wi h a la ge numbe o ze os. This ze o padding may o ce us o s o e an excessi e amoun o samples wi h he consequen p oblems o CPU ime and memo y s o age limi a ion. The knowledge o an ap- p oxima ion o he spa ial G een’s unc ion ma ix would allow us o o e come his d awback by keeping only he pa o i which is in ol ed in he con olu ion p ocess, i.e., a middle egion whose wid h is wice he o al wid h o he egion wi h me alliza ions. As a i s possibili y, we migh sample he spec al G een’s unc ions { Gl, (a)} = and apply he adequa e in e se FFT’s. Howe e , he band-unlimi ed cha ac e o hese spec al unc ions, specially when i = j, would o ce us o keep a high numbe o samples o educe he inhe - en e o associa ed wi h he spec al unca ion. In he p esen pape , a new asymp o ic ex ac ion echnique has been applied o he diagonal spec al G een’s unc ion {Gii(a)) n= I o minimize he s o age equi emen s and he CPU ime o hese in e se FFT’s. O -diagonal elemen s ha e no been ea ed since hey exponen ially app oach o ze o when he spec al a iable, a, app oaches in ini y. F om he s udies p esen ed in [9] and [l], i is easy o check ha he asymp o ic beha io o he diagonal spec- al G een’s unc ion associa ed wi h each me allized in- e ace (i = 1, * * - , M) is K’, IaI G,<a) + - o a + +03 whe e j being he imagina y uni G, and (3) (4) Obse e ha when any subs a e adjacen o he me al- liza ions has a complex non-symme ical pe meabili y enso and, he e o e, a complex non-symme ical equi - alen pe mi i i y enso , he asymp o ic beha io o he co esponding G een’s unc ion has no any symme y wi h espec o he spec al a iable a (in his sense, we ha e in gene al a non-symme ical spec al G een’s unc- ion). In he ollowing, we a e going o de ine auxilia y unc- ions e;(,) associa ed wi h he diagonal spec al G een’s unc ions &(a). The unc ions &(CY) and Gll(a) mus ha e he same asymp o ic beha io in he spec al domain o a gi en alue o i. In addi ion, he spa ial coun e pa o eL(a) mus be analy ically known. In he applica ion o he asymp o ic ex ac ion echnique he spec al G een’s unc ion ma ix is i s ob ained by using he EBM. Then, he auxilia y e&(,) a e subs ac ed om he diagonal Le Gh(a) be he spec al G een’s unc ion a he i h me allized in e ace co esponding o he s uc u e ob- ained om he o iginal line by emo ing he uppe bounda y and eplacing he o iginal subs a es by an iso- opic and homogeneous medium wi h dielec ic pe mi - i i y CL. The analy ical exp ession o &,(a) may be eas- ily ob ained om he EBM [9]: G’ ,(a) = [~&((a( + a co h (ah&))]-’ (5) whe e E& mus be chosen in such a way ha he possible non-symme ical asymp o ic beha io o Glj (a) is accom- moda ed, i.e.: Gjj(a) o i = 1, * 9 M. a>o (.:=.; 1 and he e ec i e subs a e heigh h&, al hough a bi a y o some ex en , has been chosen in such a way ha he condi ion Re {G~,(O)} = Re {eii(0)} is ul illed. Wi h his choice, GL(a) and Gjj(a) a e no e y di e en in he su oundings o a = 0, hus a oiding nume ical p oblems as we will see la e on. A his poin , we can ob ain a unc ion ma ix [Gj(a)] (i, j = 1, , M) de ined as ollows: Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on July 20,2020 a 15:24:58 UTC om IEEE Xplo e. Res ic ions apply. DRAKE e al.: AN IMPROVED ITERATIVE TECHNIQUE FOR QUASI-TEM ANALYSIS A disc e e app oxima ion o he co esponding spa ial unc ion ma ix [Gd(x - x’)] can be buil by aking Np samples (wi h pe iod equal o l/To) o [Gd(a)] and ap- plying in e se FFT: I Ga(mT) = TFFT-’ {Ga(n/To)} m, n = -Np/2, * - , Np/2 - 1 i,j = 1, ,M. (8) As a consequence o he asymp o ic ex ac ion p ocess, he unc ions ma ix [ed(a)] has a na owe ange o al- ues signi ican ly di e en om ze o, hus making possible he d as ic educ ion o he numbe Np o samples. This educ ion o Np and he consequen diminu ion o he size o he sampled spec al egion (Np/To = 1/T) imply a la ge sepa a ion (T) be ween he con iguous samples in he spa ial domain. Thi d o de spline in e pola ion is now used o inc ease he disc e iza ion le el. Once he samples o [Gd] in (8) ha e been in e pola ed wi h an in e pola ion ac o o Ni, we ha e Npi = NiNp poin s o [Gd(x - x’)] sepa a ed by a pe iod = T/Ni. A his poin , i is im- po an o emembe ha only he N, samples co espond- ing o a middle in e al-whose wid h is wice he o al wid h o he egion wi h me alliza ions-a e going o be in ol ed in he con olu ions. Hence, only N, samples o he disc e ized spa ial G een’s unc ions Gij(mTi) m = -N,/2, * - - ? N,/2 - 1 mus be compu ed om Gj(m6) and G&(m&): G2(m&) + GL(m&) o i = j Gy (mTJ o i # j (9) Gij(m&) = whe e GL(m&) a e samples o G’,(x - x’ ) which a e he in e se Fou ie ans o ms o in ini e combs o samples o GL(a) aken wi h pe iod equal o l/To. No e ha he unc ions GL(x - x’) compu ed in his way a e he spa ial G een’s unc ions o he asymp o ic equi alen s uc u es keeping he spa ial pe iodici y (wi h pe iod To) in he x-di ec ion. The unc ions GL(x - x‘) (i = 1, * , M) ha e been analy ically ob ained as GL(x - x’) whe e Go-- -+- ’ - ;i (€!+ E!) ~ 655 X-X‘ (m) (b) Fig. 2. (a) The spec al G een’s unc ion and he emainde spec al unc ions a e asymp o ic ex ac ion GA (wi h ha, = h) and Gd (wi h h., as in his wo k). No e he nonsymme ical na u e o he G een’s unc ion wi h espec o a. (b) The middle egion o he spa ial G een’s unc ion G and i s analy ical pa Go. o a mic os ip con igu a ion on sa u a ed FMS (h = 100 pm, w = 200 pm, e = 1%,, U = 5( lm)-’, 4 M, = 2000 G, H, = 1500 Oe, AH = I5 Oe). The singula i ies o Gij(0) ha e been eplaced by he nume ically compu ed in eg al a e ages o Gij (x - x’ ) in he cen al in e al [-T/2, T/2]. Finally, he alues o o (2) a e wo ked ou om he N, samples o he spa ial G een’s unc ions Gij(m) (whe e he p ime ma k deno es he eplacemen o Gij (0)) by di ec FFT’s: ei(n) = FFT {G:j(mTJ} m, n = -Nc/2, - - , N,/2 - 1 i,j= 1, --- ,M (1 1) Once he unc ions G:j(n) ha e been compu ed, (2) is eady o be sol ed by GBGM. Since an in e pola ion p o- cess is assumed, we mus subs i u e T by T, in (2). V. NUMERICAL RESULTS To illus a e he asymp o ic ex ac ion echnique de- sc ibed abo e, he Fig. 2(a) shows he absolu e alue o he eal pa o he no malized spec al G een’s unc ion @a) o a mic os ip con igu a ion on sa u a ed FMS sub- s a e longi udinally magne ized. No ice ha he p esence o an ex e nal longi udinal magne ic ield H, makes he spec al G een’s unc ion be non-symme ical wi h e- Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on July 20,2020 a 15:24:58 UTC om IEEE Xplo e. Res ic ions apply. 656 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 40, NO. 4, APRIL 1992 h, h2eo K IY IY- W W 0 2 3 W IT 0 5 10 SIMUIAT. PERIOD/METALLlZED WIDTH (b) Fig. 3. (a) Se e al mic os ip con igu a ions on a dielec ic subs a e o e = 15. (b) Rela i e e o in he sel -capaci ance o a conduc ing s ip wi h he con igu a ions o Fig. 3(a) e sus he a io be ween he simula ing pe- nod and he wid h o he me allized egion. (4 . I wi h w/h, = 1, hz/h, = 20; (-) o (I) wi h w/h, = 1, hz/h, = 1; (-A--A--A--A--A-) o (11) wi h w/h = 1, s/h = 0.1; (-H--D--H--H-) o (111) wi h w/h,, = 1, h,/h, = 0.1. I ) o (1) spec o a. I we apply he asymp o ic ex ac ion scheme wi h he choice ha, = h, he dashed line is ob ained o he emainde G een's unc ion Gi(a). I mus be e- ma ked ha he sha p na u e o eJ(a) would o ce us o ha e ine sampling. Because o his, he choice o ha, ha makes Re (e,(O)) = Re (QO)) is mo e sui able (solid line). Ano he ad an age o he asymp o ic ex ac ion echnique is ha he cen al egion o he spa ial G een's unc ion G(x - x'), in ol ed in he con olu ion p ocess, is almos analy ically buil up (see Fig. 2(b)). The alues o he G een's unc ion in ha egion a e mainly a ec ed by he spec al asymp o ic beha io , which is analy ically aken in o accoun . In a p e ious sec ion, we ha e jus i ied he need o se- lec a simula ing pe iod o he analysis o ape iodic s uc- u es. In he p esen wo k, we ha e in es iga ed he e- la ion be ween he sui able size o he pe iodic window and he ea u es o he line s udied. Fig. 3(b) shows he ela i e e o in oduced o he pe iodic simula ion in he sel -capaci ance o a conduc ing s ip unde di e en con- igu a ions (see Fig. 3(a)) as a unc ion o he a io be- ween he wid h o he simula ing pe iodic window and he wid h o he me allized egion. The e o is ela i e o he alue o he sel -capaci ance when he simula ing pe- 0 5 10 (a) NUMBER OF ITERATIONS II'" CCST / El 0""""'~~ -0 10 20 NUMBER OF ITERATIONS (b) Fig. 4. Ra es o con e gence o di e en i e a i e algo i hms o he cal- cula ion o (a) he capaci ance o a symme ical s ipline on alumina (E, = 9.6, w/h = 1) wi h 20 o 40 samples on he s ip and (b) he sel -capaci- ance o he lowe s ip o a b oadside con igu a ion (E, = 15, w/h = 1). iod app oaches ini ini y. No ice ha he ela i e p ox- imi y be ween conduc o s (specially, g ounded pla es) implies a close con inemen o he elec omagne ic ield o he me allized egion, hus allowing us o educe he wid h o he pe iodic window. Ano he impo an aspec is o check he imp o emen in oduced in he a e o con e gence o he i e a i e p o- cess by he use o he GBGM ins ead o he o dina y CGM alga i hm. A spec al i e a i e echnique success ully used in [3] (named CCST) has been also p og ammed o com- pa ison. In Fig. 4(a), we compa e he a es o con e - gence o he CGM, he GBGM and he CCST in he com- pu a ion o he capaci ance pe uni leng h o a sym- me ical s ipline. These esul s co espond o bo h 20 and 40 samples on he s ip. The 50% educ ion in he numbe o i e a ions ob ained by using he GBGM ins ead o CGM is a e y ypical esul in he s uc u es analyzed. Any- way, he highes a e o con e gence co esponds o he CCST. Ne e heless, Fig. 4(b) epea s he compa ison o a pai o b oadside coupled s ips, showing he s agna ion p ocess (one-s ep imp o emen is less han he compu e p ecision) in he CCST. The s agna ion p oblem in he spec al i e a i e echniques was obse ed in [7]. This led he au ho s o ha pape o modi y he algo i hm. In e- la ion o he CPU ime, he GBGM p esen ed an a e age Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on July 20,2020 a 15:24:58 UTC om IEEE Xplo e. Res ic ions apply. DRAKE e al.: AN IMPROVED ITERATIVE TECHNIQUE FOR QUASI-TEM ANALYSIS > 0.5 W/b -+ 1 Fig. 5. E ec i e dielec ic cons an s o a b oadside, edge-coupled mic o- s ip wi h in e ed dielec ic in [4] (s/b = d/b = 0.2, c/b = 10, E, = 10). - his wo k - = measu ed - LL 11 w IIIIIIIIII 5 6 7 8 9 101112 FREQUENCY (GHz) (a) ‘‘I 0.2 2 10 100 FREQ (GHz) Fig. 6. (a) E ec i e ela i e pe meabili y o a mic os ip on a la ched ga - ne subs a e in [12](W/h = 0.5, 4 ? M, = 1780G. 4 M, = 1030 G). (b) Modal slow-wa e ac o s and a enua ion cons an s o a pai o asym- me ical s ips on wo laye s in he pa ially demagne ized s a e (see [13]) (hl = 100 pn, h2 = 100 pm, wI = 160 pm, s = 100 pn, w2 = 100 pm, e,, = = 14.9, 4 M, = 870 G, 4sM, = 550 G, dielec ic losses: U = 0.001 (Q “.)-I, magne ic losses: A = 0.01, N = 1.5). (b) o 20 ms pe i e a ion (in he case o 40 samples) on a VAX-11/785 compu e , while he o he wo algo i hms p esen ed 60 ms pe i e a ion. Finally, in Figs. 5 and 6, we include he analysis o some p ac ical s uc u es o compa ison pu pose. The e- sul s a e compa ed wi h da a epo ed in he bibliog aphy, 651 and e y good ag eemen is ound. The s uc u e analyzed in Fig. 5 is an example o mul iconduc o con igu a ion wi h non-coplana me alliza ions. Symme y o his s uc- u e is no aken in o accoun because ou aim is o check he e iciency o he algo i hm p og ammed o he case o se e al me allized in e aces. In Fig. 6, a pai o con- igu a ions wi h gy omagne ic subs a es a e conside ed (non-symme ical spec al G een’s unc ions a e in- ol ed). The me hod has been exhaus i ely checked by compa - ing i wi h many o he da a epo ed in he li e a u e wi h simila esul . The con e gence o he me hod in complex cases in ol ing mul ilaye ed, lossy and aniso opic ma- e ials has been also e i ied. We conclude ha he esul s ob ained wi h he compu e p og ams based on he heo y in his pape a e accu a e and eliable as long as quasi- TEM app oxima ion emains alid. So, his me hod is an e icien al e na i e o o he me hods ( o example he me hod o momen s) applied o he quasi-TEM analysis o e y gene al plana s uc u es. VI. CONCLUSION In his pape , we ha e p esen ed he quasi-TEM anal- ysis o a wide class o plana mul iconduc o ansmission lines by employing he GBGM and FFT algo i hms. P in ed conduc o s a e embedded in a laye ed s uc u e including dielec ics, semiconduc o s o magne ic ma e- ials. Na u al aniso opy and aniso opy p oduced by lon- gi udinal magne izing ields a e accoun ed o in he anal- ysis. The spa ial G een’s unc ion ma ix is used in he o - mula ion o he p oblem o educe memo y s o age and CPU ime. This ma ix is ob ained o he mul ilaye ed s uc u e om i s spec al domain ep esen a ion, which can be eadily compu ed by means o a simple ecu en scheme (EBM). This p ocess has been signi ican ly ac- cele a ed by using an asymp o ic ex ac ion echnique in he spec al domain. The singula beha io o he spa ial domain G een’s ma ix is analy ically aken in o accoun in such a way ha he emainde spec al ma ix is a na - ow band unc ion. In pa icula , he p esence o longi- udinally magne ized e i es o semiconduc o s-which esul s in non-symme ical spec al G een’s unc ions- can be accommoda ed by using his me hod. Se e al aspec s ela ed o he con e gence baha io o he me hod ha e been in es iga ed. The choice o simu- la ing pe iods o analyze ape iodic lines has been ound o be s ongly ela ed o he geome ical ea u es o he lines. The supe io i y (in he sense o a as e a e o con- e gence) o he GBGM o e he o dina y CGM algo- i hm has also been checked. In spi e o he ac ha CCST has p o ed o ha e he highes a e o con e gence, i p e- sen s some s agna ion p oblems. Some examples ha e been included o illus a e he s eng h and he e sa ili y o he me hod. Compa isons wi h published da a indica e ha he me hod p esen ed yields accu a e esul s, hus o - e ing an e icien al e na i e echnique o he quasi-TEM analysis o plana lines. Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on July 20,2020 a 15:24:58 UTC om IEEE Xplo e. Res ic ions apply. 658 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 40, NO. 4, APRIL 1992 REFERENCES [l] M. Homo, F. L. Mesa, F. Medina, and R. Ma quCs, “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. [2] T. K. Sa ka and S. M. Rao, “An i e a i e me hod o sol ing elec- os a ic p oblems,” IEEE T ans. An ennas P opaga ., ol. AP-30, no. 4, pp. 611-616, 1982. [3] P. M. an den Be g, “I e a i e compu a ional echniques in sca e ing based upon he in eg a ed squa e e o c i e ion,” IEEE T ans. An- ennas P opaga ., ol. AP-32, pp. 1063-1071, Oc . 1984. [4] C. H. Chan and R. Mi a, “Analysis o MMIC s uc u es using an e icien i e a i e app oach,” IEEE T ans. Mic owa e Theo y Tech., ol. 36, pp. 96-105, Jan. 1988. [5] P. M. an den Be g, “I e a i e schemes based on he mi imiza ion o he e o in ield p oblems,” Elec omagn., ol. 5, pp. 237-262, 1985. [6] T. K. Sa ka and E. A as, “On a class o ini e s ep i e a i e me h- ods (conjuga e di ec ions) o he solu ion o an ope a o equa ion a ising in Elec omagne ics,” IEEE T ans. An ennas P opaga ., ol. [7] R. Mi a and C. H. Chan, “I e a i e app oaches o he solu ion o elec omagne ic bounda y alue p oblems,” Elec omagn., ol. 5, no. [8] T. K. Sa ka , “On he applica ion o he Gene alized Biconjuga e G adien Me hod,” J. Elec omagn. Wa . Appl., ol. 1, no. 3, pp. [9] F. Medina and M. Homo, “De e mina ion o G een’s unc ion ma ix o mul iconduc o and aniso opic mul idielec ic plana ansmission lines: a a ia ional app oach,’’ IEEE T ans. Mic owa e Theo y Tech., [lo] C. H. Chan and R. Mi a, “Analysis o a class o cylind ical mul- iconduc o ansmission lines using an i e a i e app oach,” IEEE T ans. Mic owa e Theo y Tech., ol. MTT-35, pp. 415-424, Ap . 1987. [ll] F. Mesa, R. Ma qu s, and M. Homo, “A gene al algo i hm o com- pu ing he bidimensional spec al G een’s dyade in mul ilaye ed com- plex bianiso opic media: he Equi alen Bounda y Me hod (EBM),” IEEE T ans. Mic owa e Theo y Tech., ol. MTT-39, pp. 1640-1649, Sep . 1991. [I21 D. J. Masse and R. A. Pucel, “Mic os ip p opaga ion on magne ic subs a es. Pa 11: Expe imen ,” IEEE T ans. Mic owa e Theo y Tech., ol. MTT-20, pp. 309-313, May 1972. [13] F. Mesa and M. Homo, “Quasi-TEM and ull wa e app oaches o coplana mul is ip lines including gy omagne ic media longi udinally magne ized.” Mic owa e Op . Tech. Le ., ol. 4, pp. 531-534, No . 1991. AP-33, pp. 1058-1066, Oc . 1985. 2-3, pp. 123-146, 1985. 223-242, 1987. ol. MTT-33, pp. 933-940, Oc . 1985. En ique D ake was bo n in Mon illa, Chdoba, Spain, in Sep embe 1966. He ecei ed he deg ee o Licenciado in physics in Sep embe 1990 om he Uni e si y o Se ille, Spain. He is cu en ly ollowing a Ph.D. p og am in mic owa es wi h a schola ship o he Spanish Go e nmen . His esea ch in e es s ocus on i e - a i e me hods o plana s uc u es and mul icon- duc o lines. F ancisco Medina (M’90) was bo n in Pue o Real, Cbdiz, Spain, in No embe 1960. He e- cei ed he Licenciado deg ee in Sep embe 1983 and he 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 ic- i y and Magne ism in he Depa men o Elec on- ics and Elec omagne ics, Uni e si y o Se ille. His esea ch deals mainly wi h analy ical and nu- me ical me hods o plana s uc u es and mul i- conduc o lines. Manuel Ho no (M’75) was bo n in To e del Campo, JaCn, Spain. He ecei ed he deg ee o Licenciado in physics in June 1969, and he de- g ee o Doc o in physics in Janua y 1972, bo h om he Uni e si y o Se ille, Spain. Since Oc obe 1969 he has been wi h he De- pa men o Elec onics and Elec omagne ism a he Uni e si y o Se ille, whe e he became an As- sis an P o esso in 1970, Associa e P o esso in 1975, and P o esso in 1986. He is a membe o Elec omagne ism Academy o MIT (Cam- b idge). His main ields o in e es include boundaj alue p oblems in elec omagne ic heo y, wa e p opaga ion h ough aniso opic media, and mic owa e in eg a ed ci cui s. He is p esen ly engaged in he analysis o plana ansmission lines embedded in aniso opic ma e ials, mul iconduc- o ansmission lines, and plana slow-wa e s uc u es. Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on July 20,2020 a 15:24:58 UTC om IEEE Xplo e. Res ic ions apply.