Fast full-wave analysis of multistrip transmission lines based on MPIE and complex image theory

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 48, NO. 3, MARCH 2000 445 Fas Full-Wa e Analysis o Mul is ip T ansmission Lines Based on MPIE and Complex Image Theo y Joaquin Be nal, F ancisco Medina, Membe , IEEE, Ra ael R. Boix, Membe , IEEE, and Manuel Ho no, Membe , IEEE Abs ac —The mixed-po en ial elec ic- ield in eg al equa ion is used in conjunc ion wi h he Gale kin’s me hod and complex image heo y o analyzing a ansmission line wi h mul iple s ips embedded in di e en laye s o a mul ilaye ed uniaxially aniso opic dielec ic subs a e. The wo-dimensional G een’s unc ions o he scala and ec o po en ials a e analy ically ob ained in he space domain due o he app oxima ion o i s spec al-domain e sion wi h complex images, hus a oiding leng hy nume ical e alua ions. Double in eg als in ol ed in he compu a ion o Gale kin’s ma ix en ies a e quasi-analy ically ca ied ou o he chosen basis unc ions, which a e well sui ed o he p oblem. Index Te ms—Complex image me hod, in eg al equa ions, lay- e ed media, plana ansmission lines. I. INTRODUCTION THE analysis o a plana mul is ip sys em such as ha shown in Fig. 1 has been ca ied ou by using a a ie y o echniques du ing he pas h ee decades, including bo h quasi-TEM and ull-wa e o mula ions. Achie ing high nu- me ical e iciency has been he goal o many ecen pape s. A sample o his ype o wo k in he ame o he quasi-TEM anal- ysis can be ound in [1] and e e ences he ein. In his pape , emphasis is placed on he ull-wa e app oach. Ve y e icien algo i hms dealing wi h he ull-wa e analysis o plana lines ha e been also epo ed, including he singula in eg al-equa- ion me hod [2], [3] and he eigen alue app oach [4] o boxed s uc u es, he Wiene –Hop me hod [5], and a ious enhanced implemen a ions o he spec al-domain analysis (SDA) [6]–[9]. In his pape , he au ho s p opose a e y as analysis o he s uc u e in Fig. 1 based on he mixed-po en ial in eg al equa ion (MPIE) [10]–[12]. The nume ical pe o mance o his app oach is d as ically imp o ed by using a sui able wo-di- mensional (2-D) space-domain ep esen a ion o he po en ial G een’s unc ions and quasi-analy ical compu a ion o he eac ion in eg als appea ing when a Gale kin scheme is used o sol ing he MPIE o ind he su ace cu en s. This as and accu a e compu a ion o Gale kin’s ma ix en ies is he key Manusc ip ecei ed July 27, 1999; e ised Decembe 10, 1999. This wo k was suppo ed by he Comisión In e minis e ial de Ciencia y Tecnología, Spain unde P ojec TIC95-0447. J. Be nal is wi h he Depa men o Applied Physics, Uni e si y o Se ille, 41092 Se ille, Spain. F. Medina and R. R. Boix 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, 41012 Se ille, Spain (e-mail: [email p o ec ed]). M. Ho no, deceased, was wi h he Mic owa e G oup, Depa men o Elec- onics and Elec omagne ism, Uni e si y o Se ille, 41012 Se ille, Spain. Publishe I em Iden i ie S 0018-9480(00)02050-0. Fig. 1. C oss sec ion o he mul iconduc o ansmission line unde analysis. poin o ge e y high e iciency. In his way, he de e mina ion o space-domain G een’s unc ions is ca ied ou ia he com- plex images echnique [13]–[15], hus a oidingcommonly used nume ical spec al 2-D Somme eld- ype in eg a ion [16], [17]. This me hod, o iginally in ended o he analysis o a adia ing dipole in a h ee-dimensional (3-D) s a i ied medium, has been adap ed he e o ou 2-D p oblem. A his poin , i should be men ioned ha a co ec ion o he o mula ion o [15], in o- duced by Kipp and Chan in [18], mus be also applied in he 2-D case. As i is well known, i s - and second-kind Chebyshe polynomials weighed by he p ope s ip edge condi ion a e e y sui able basis unc ions o he cu en expansion [9]. The eac ion in eg als in ol ing hese unc ions and he closed- o m exp ession o he G een’s unc ions ob ained wi h he complex images me hod a e quasi-analy ically compu ed. The e o e, he mos ime-consuming s ep in sea ching o he p opaga ion cons an s, which is he compu a ion o Gale kin’s ma ix, is d as ically accele a ed. II. FORMULATION OF THE INTEGRAL EQUATION Le us conside a ansmission line consis ing o in in- i ely hin s ips embedded in he a ious laye s o a mul ilay- e ed subs a e (see Fig. 1). Each laye is a uniaxial aniso opic dielec ic, wi h i s op ical axis pe pendicula o he in e aces be ween laye s. Since we a e in e es ed in modes ha p opaga e in he -di ec ion, we assume a common phase ac o o ields and cu en s, whe e is he unknown p opaga ion con- s an . By en o cing he bounda y condi ion o he angen ial elec ic ield a he su ace o he conduc o s, we ob ain an elec- ic- ield in eg al equa ion (EFIE). The ke nel o his EFIE has a se e e singula i y ha makes i unsui able o a di ec appli- ca ion o he me hod o momen s [10]. To o e come his di - icul y, we can ans o m his in eg al equa ion in o an MPIE o m, whose ke nel has a weake singula i y [10], [11], [19]. 0018–9480/00$10.00 © 2000 IEEE Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on July 15,2020 a 17:03:06 UTC om IEEE Xplo e. Res ic ions apply. 446 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 48, NO. 3, MARCH 2000 Since he sou ces in ou s uc u e a e pe pendicula o , he MPIE has he ollowing o m: (1) on he conduc ing s ips. and in (1) a e he magne ic ec o po en ial and elec ic scala po en ial due o he su ace cu en on he h conduc o (2) (3) In (1)–(3), s ands o he su ace o he h conduc o placed a he plane . I is well known ha o a ho izon ally di ec ed dipole, wo componen s o he ec o po en ial a e necessa y o sa is y he bounda y condi ions a he in e aces [20]. We ha e used he adi ional Somme eld’s o mula ion o he ec o po en ial [21] so he componen o he ec o po en ial is chosen oge he wi h he componen pa allel o he sou ce. This o mula ion is con enien o he analysis o plana s uc u es because . Mo eo e , he e olu ion symme y o ou p oblem subs a e a ound he -axis leads o . The e o e, only one spec al in eg al is necessa y o ob aining he 2-D G een’s unc ion o he magne ic ec o po en ial. III. KERNEL OF THE INTEGRAL EQUATION I is easible o ob ain a closed- o m exp ession o and in he spec al domain [21], [22], namely, and , being ( and a e he Ca esian spec al a iables and is he adial pola spec al a iable). I should be poin ed ou ha i he s uc u e has conduc o s placed a di e en le els, and mus be e alua ed o sou ce and obse a ion poin s a any o he le els. Taking in o accoun he ecip oci y o he G een’s unc ions, his lead o combina ions o sou ce and obse a ion planes. Once he spec al e sion o he ke nel o ou in eg al equa ion is known, i s 2-D spa ial coun e pa can be ob ained om he ollowing spec al in eg al: (4) In (4) and s and o he spa ial (2-D) and spec al ep e- sen a ions o any o he and unc ions. The -dependence is no explici ly shown since i will no play any ole in he de elopmen he ea e . The in eg and in (4) may ha e se e al poles in he eal axis o he -plane, which depend on he s uc u e and equency. Thesepolescanbeeasily emo ed,as will beexplainedla e on. Ano he impo an opological ea u e o he spec al-domain G een’s unc ions is heexis enceo b anchpoin s a . These b anch poin s a e ela ed o he ee-space unbounded uppe laye o he s uc u e, and hey will play an impo an ole in he de elopmen o he nume ical app oach. The compu a ion o (4) akes a signi ican pa o he o e all compu a ion ime since he in eg ands a e ypically oscilla o y and slowly decaying. Since he in eg ands depend on , ha in- eg al mus be ecalcula ed o e e y alue o he p opaga ion cons an in he oo sea ch p ocess. The e o e, a as me hod o e alua e (4) is o pa amoun impo ance. The complex image me hod al eady used in he analysis o plana ci cui s, an ennas, and sca e ing p oblems [13]–[15], [18] can be adap ed o ac- complish ha goal. The basic idea o his me hod is o ex ac om he spec al ke nel i s quasi-s a ic and su ace-wa e con i- bu ions, and o app oxima e he emaining unc ion by a sum o complex exponen ials. In he 3-D case, he Somme eld iden i y can henbeemployed oe alua einclosed o m heSomme eld in eg als. This leads o a e y e icien algo i hm p o ided ha we ha e a mean o e alua e he quasi-s a ic and su ace-wa e con ibu ions in closed o m. In he 2-D case, as a as he au- ho s know, he spec al in eg als in (4) a e usually nume ically calcula ed [16], [17]. Al hough e icien nume ical in eg a ion algo i hms a e used, he p ocedu e is no as e icien as hose epo ed in [13]–[15], [18]. Wha we p opose in his pape is o adap he complex image me hod o ou p oblem. In o de o do his, i is ins uc i e o examine he spec al-domain e sion o bo h he ec o - and scala -po en ial G een’s unc ion o a a - eling-wa e line sou ce in he ee space a a heigh abo e a g ound plane. These o mulas can be w i en in he ollowing o m: (5) whe e ,( ), o he componen o he dyadic spec al G een’s unc ion o he ec o po en ial and o he scala po en ial. The i s e m in (5) co esponds o he e ec o he sou ce i sel , whe eas he second e m is he image con ibu ion. Hence, i seems easonable o hink ha i he sou ce is embedded in a s a i ied medium, he spec al G een’s unc ion is sui able o be exp essed as a quasi-s a ic e m (which accoun s o he nea - ield con ibu ion o he sou ce and has a singula space-domain coun e pa ) plus a numbe o images o he o m . Howe e , i is well known ha a s a i ied medium is also capable o p opaga ing su ace wa es, which a e independen o he sou ce. The in luence o hese p opaga ing modes in he spec al-domain G een’s unc ion is he exis ence o a ini e numbe o poles ha do no appea in he ee-space p oblem. These poles modi y he spec al-domain G een’s unc ion beha io ha can no longe be exclusi ely exp essed as a sum o exponen ial unc ions. As a mean o e alua ing (4) in a e icien way, we hen w i e he spec al G een’s unc ion in he ollowing app oxima ing o m: (6) Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on July 15,2020 a 17:03:06 UTC om IEEE Xplo e. Res ic ions apply. BERNAL e al.: FAST FULL-WAVE ANALYSIS OF MULTISTRIP TRANSMISSION LINES 447 The e m is he quasi-s a ic con ibu ion, ep esen s he su ace wa e e m, and is he emaining e m, which is o be expanded by a ini e se ies o complex exponen ial unc- ions. A. Quasi-S a ic Te m The quasi-s a ic ields a e dominan when he dis ance be ween he sou ce and ield poin s is small compa ed o he ee-space wa eleng h. In such a case, he complex exponen- ial appea ing in he in eg and o (4) oscilla es wi h a la ge pe iod. The e o e, he beha io o he spa ial G een’s unc ion is s ongly a ec ed by he asymp o ic alues o he spec al G een’s unc ion ( ). Since he in eg ands dec ease e y slowly, he ollowing asymp o ic beha io o and mus be ex ac ed ou : o he wise (7) o he wise (8) whe e is he posi ion o he h in e ace (see Fig. 1). On he o he hand, and a e he ela i e pe mi i i ies o he h laye in di ec ions pe pendicula and pa allel o he -axis, espec i ely. No e ha we will ha e nonze o asymp o ic e ms only when sou ce and obse a ion poin s a e a he same le el. F om (7) and (8), we can, in gene al, w i e (9) whe e κis a cons an ha is ze o i and whose alue depends on he cases ea ed in (7) and (8) i sou ce and obse - a ion poin s a e a he same le el ( ). The cons an was de ined in (4). I should be poin ed ou ha he b anch poin s appea ing in in he spec al G een’s unc ions a e also p esen in (9). The e o e, his e m does no in oduce any new b anch cu in he -plane opology [18]. In o de o calcula e he 2-D space-domain e sion o (9), he ollowing spec al in eg al mus be ca ied ou : (10) whe e .αis supposed o be posi i e since we a e in e es ed only in he bound egime (as opposi e o he leaky egime). The in eg al (10) can be analy ically calcula ed by using he ollowing esul [23]: (11) whe e is he ze o h-o de modi ied Bessel unc ion o he second kind. The in eg al in (10) can be seen as he limi o (11) when , hence, he con ibu ion o he quasi-s a ic e m in he space domain is (12) No e ha since o small , he 2-D space- domain G een’s unc ions ha e a loga i hmic singula i y when and he ield poin app oaches he sou ce poin ( ). B. Su ace-Wa e Poles Con ibu ion The complex image scheme can be applied o e a complex pa h, hus a oiding p oblems ela ed o he p esence o poles on he eal axis o he complex plane [24]. Howe e , he e a e heo e ical and nume ical easons ha make i ad isable o e- mo e hepolecon ibu ions om hespec alG een’s unc ions. Complex exponen ial unc ions canno ep oduce accu a ely in hespec aldomain hebeha io associa ed o hesepoles. Since he spec al unc ions a e e en unc ions o , he poles always appea in pai s. Consequen ly, we can w i e [14], [15] (13) whe e is he numbe o poles, is he loca ion o he h pole in he -plane, and is i s esidue The e o e, he space-domain con ibu ion o he su ace wa e poles is (14) whe e is supposed o be posi i e (bound egime). Ananaly icalexp ession o hein eg alin(14)isa ailable om [23], in such a way ha (15) No e ha , in con as wi h he 3-D case [13], [14], he con- ibu ion om he su ace-wa e poles in ou 2-D si ua ion does no in oduce any singula i y. Thus, he ea men s epo ed in [25] o [26] o deal wi h his p oblem is no equi ed inou case. The e o e, we can di ec ly ex ac he su ace-wa e con ibu ion om he complex image expansion, ob aining a well-beha ed app oxima ion o any alue o he spa ial a iable. This makes Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on July 15,2020 a 17:03:06 UTC om IEEE Xplo e. Res ic ions apply. 448 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 48, NO. 3, MARCH 2000 an impo an di e ence be ween he ansmission and adia ion p oblems. C. Applica ion o he Complex Image Me hod We ha e ound closed- o m exp essions ha allow us o ex- ac he asymp o ic and su ace wa e e ms o he spec al-do- main G een’s unc ion and o eco e hem in he 2-D space do- main.The emainingspec al—domain unc ionisnowsui able o be expanded as a ini e sum o complex exponen ial unc ions by using, o ins ance, he gene al pencil o unc ion (GPOF) me hod [27] (16) whe e is he numbe o employed complex images. I is expec ed ha a sho numbe o images is enough o p ac ical pu poses because o he sui abili y o he expanding unc ions: hey a e he spec al e sion o cylind ical wa es. In o de o ob ain he unknown coe icien s and appea ing in (16), we ha e sampled ou spec al unc ions on a pa h in he -plane, which a oids he poles and b anch poin singula i y. To ensu e an op imum esul , we ha e applied a wo-s ep p ocedu e ha makes i possible o ake mo e samples nea he o igin, whe e, owing o he p oximi y o he b anch poin , he spec al unc ion p esen s as a ia ions. This wo-s ep app oach has been p oposed by Aksun in [24] o app oxima ing 3-D G een’s unc ions (al hough his au ho does no ex ac he su ace-wa e con ibu ion). The use o he wo-s ep algo i hm is also ad an ageous o deal wi h s uc u es ha ing e y hin laye s. The easons o ha a e explained in [28] in he con ex o he quasi-s a ic analysis o coplana wa eguide (CPW) s uc u es. On he o he hand, i is wo h no ing ha al hough we could also expand he e m as a sum o complex exponen ials in he o a iables, his would in oduce a new b anch poin in he plane opology and, he e o e, he expansion would ail o app oxima e he ac ual unc ion in a co ec way [18]. Now, he las e m in (6) can be ans o med in o he 2-D spa- ial domain by using (11). The e o e, wecan w i e he ollowing exp ession o he whole 2-D space domain G een’s unc ion: (17) The 2-D spa ial-domain G een’s unc ions a e ob ained as a sum o adial wa es (plus he su ace-wa e con ibu ion) in he samewayassphe icalwa esa eob ained o he spa ial-domain 3-D G een’s unc ions [13]. No e ha (17) has he impo an ad- an ageo beinganexplici unc iono hep opaga ioncons an h ough and ( ). In he oo sea ching p ocess in ol ed in he esolu ion o he eigen alue p oblem, he p opaga ion cons an is changed many imes, bu hose changes a e au oma ically aken in o accoun by (17). I a nu- me ical in eg a ion scheme is applied o sol e o (4), he in e- g als ha e o be ecompu ed o each new guess alue o . The use o ou app oach ob iously implies impo an cen al p o- cessing uni (CPU) ime sa ings. IV. APPLICATION OF GALERKIN’S METHOD Once he space-domain ke nel o he in eg al equa ion has been e icien ly ob ained, we can apply he Gale kin’s me hod. A well-es ablished se o basis unc ions o plana - ype s uc- u es has been chosen. In ac , o a plana s ip o wid h and whose cen al poin coo dina e is , he basis unc ions o he componen s o he cu en densi y ha e been aken o be (18) (19) whe e is he numbe o basis unc ions employed o he ans e se componen o he cu en densi y (one mo e basis unc ion mus be used o he axial componen o he cu en densi y o ensu e ha he o al cu en ul ills he con inui y equa ion). and s and o i s - and second-kind Chebyshe polynomials, espec i ely. These unc ions mimic he eal beha io o he cu en s nea he edge o he conduc ing s ips and a e quasi- o hogonal o he space ke nel we a e using. This allows us o a ain accu a e esul s while using e y ew basis unc ions. Thenex s epis ocalcula e hecon olu ionandinne p oduc in eg als by using he basis unc ions in (18) and (19) and he ke nel in (17). Thanks o he ela ionship be ween i s - and second-kindChebyshe polynomials[29],weonlyneed ocom- pu ein eg alsin ol ing i s -kindpolynomials.The unc ion p esen s a loga i hmic singula i y ha migh cause p oblems in he con olu ion in eg al. Fo una ely, he con ibu ion o he con olu ionin eg also his singula i ycan behandled in closed o m, such as explained in [1]. The es o he ke nel is egula and does no gene a e in eg a ion p oblems. Due o he ype o singula i y p esen in he basis unc ions, low-o de Chebyshe quad a u es a e sui able o accu a ely ca y ou he in eg a ions in ol ing he egula pa o he ke nel. In his way, he elemen s o Gale kin’s ma ix a e gene a ed bo h e y accu a ely and e - icien ly. V. NUMERICAL RESULTS The i s s ep o checking he pe o mance o he p oposed app oach is o e i y ha he app oxima ion o he 2-D space-domain G een’s unc ions is co ec . These unc ions show an exponen ial decay wi h he dis ance be ween sou ce and ield poin s, which is qui e di e en om ha ob ained in he 3-D case. This decaying is as e o la ge alues o . Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on July 15,2020 a 17:03:06 UTC om IEEE Xplo e. Res ic ions apply. BERNAL e al.: FAST FULL-WAVE ANALYSIS OF MULTISTRIP TRANSMISSION LINES 449 We ha e compa ed in his pape he space-domain G een’s unc ions compu ed by di ec in eg a ion a combina ion o he Rombe g’s me hod, and he weigh ed a e ages me hod e- po ed in [11] has been used agains hose ob ained by using he echnique in his pape . In his way, Fig. 2 shows o he wo-laye s uc u e depic ed in he igu e. The ela i e di e ence be ween nume ical and GPOF esul s is plo ed. The GPOF has been applied wi h and wi hou pole ex ac- ion (bu always wi h ex ac ion o he quasi-s a ic e m). Fo he 9-GHz case [see Fig. 2(a)], we can see ha he GPOF app oxima ion is e y accu a e (bu ai ly be e i pole ex- ac ion is applied) in he whole ange o in e es . Rela i e e o is la ge only o hose egions whe e he alues o he app oxima ed unc ion is negligible. In Fig. 2(b), simila da a a e plo ed o a equency o 33 GHz ( wo poles a e in ol ed in his case). No e ha i poles a e no emo ed, la ge e o s a e ob ained, while a e y good app oxima ion is achie ed a e emo ing hem. The e o e, emo ing he poles is s ongly ad ised, a e all, i is nei he di icul no ime consuming o ind hem [30], while nume ical bene i s a e impo an . I is wo h men ioning ha he example conside ed in Fig. 2 co esponds o a con igu a ion ha ing a e y hin dielec ic laye . This could cause se ious nume ical p oblems, which ha e been o e come hanks o he applica ion o he wo-s ep scheme used in ou s udy. Al hough we ha e concen- a ed ou a en ion on , simila conclusions a e alid o . This s udy has been ca ied ou o many combina ions o subs a es and sou ce and ield poin loca ions (coplana and noncoplana ). The o e all conclusion is ha he wo-s ep app oach in conjunc ion wi h he quasi-s a ic e m and pole ex ac ion p o ide an excellen space-domain ep esen a ion o he equi ed 2-D G een’s unc ions. Once we a e ce ain abou he accu acy o he 2-D space-do- main G een’s unc ions compu ed ia(6), we ha e e alua ed he global pe o mance o ou me hod. Fi s o all, we ha e checked he accu acy and con e gence p ope ies o he eac ion in e- g als de ining he en ies o Gale kin’s ma ix. We ha e con- i med ha hese en ies a e compu ed wi h ex eme accu acy (mo e han six co ec igu es) using e y low-o de Chebyshe quad a u es and closed- o m e alua ion o he loga i hmic sin- gula i y con ibu ion. On he o he hand, we ha e ca ied ou exhaus i e compa isons wi h p opaga ion cons an s compu ed using nume ical e alua ion o he G een’s unc ion and using enhanced e sions o he SDA [31]. The ag eemen be ween he a ious esul s is o al and we only de ec di e ences in he compu a ional e o (CPU ime). I has been e i ied ha pole ex ac ion is necessa y o many cases because o he wise he e o in he space G een’s unc ions meaning ully a ec s he inal esul o he p opaga ion cons an s. As an example, some nume ical esul s o he undamen al and i s wo highe o de modes o a simple mic os ip line a e included in Table I. Those esul s ha e been ob ained by using ou basis unc ions o he longi udinal cu en and h ee unc ions o he ans e se one ( h ee and wo a e enough o he undamen al mode) wi h and wi hou pole ex ac ion. Ex ac ion o poles is clea ly necessa y o equencies abo e 35 GHz. O he wise esul s a e no eliable because hey a e s ongly dependen on he numbe o images, sample poin s, and quad a u e poin s. Mo eo e , spu ious solu- (a) (b) Fig. 2. Magni ude o K (solid line) and ela i e di e ence be ween nume ical in eg a ion compu a ion and complex images compu a ion wi h (black do s) and wi hou (whi e do s) pole ex ac ion o : (a) 9 GHz and (b) 33 GHz. Da a: d = 1 mm, d =0 : 01 mm, " = " =10 , " =2 : 25 , and " =1 : 5 . TABLE I =k FOR THE FUNDAMENTAL AND TWO FIRST HIGHER MODES OF THE MICROSTRIP IN THE TOP FIGURE. w =3 : 0 mm, h =0 : 635 mm, AND " =9 : 8 .LEFT-HAND-SIDE COLUMN:WITH SURFACE POLE EXTRACTION.RIGHT-HAND-SIDE COLUMN:NOPOLE EXTRACTION Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on July 15,2020 a 17:03:06 UTC om IEEE Xplo e. Res ic ions apply. 450 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 48, NO. 3, MARCH 2000 ions may appea . Eigh complex images in he app oxima ion o he G een’s unc ions and ou quad a u e poin s in he e al- ua ion o he eac ion in eg als ha e been used in his able. In o de o illus a e CPU ime sa ing, we ha e compa ed he echnique p oposed in his pape wi h nume ical e al- ua ion o he space-domain G een’s unc ion. E en hough he in eg a ion scheme we ha e used o gene a e he nu- me ical samples o he G een’s unc ion is e y e icien , he applica ion o he me hod p oposed he e s ill yields an impo an educ ion o CPU ime. Mo eo e , complex im- ages ha e no o be ecompu ed when is changed in he oo sea ching p ocess, whe eas new nume ical in eg a ions would be equi ed o new alues o . Hence, he ela i e impac o using his app oach in he analysis o ansmission lines (eigen alue 2-D p oblem) is p obably s onge han he impac o using a simila echnique in a 3-D plana p oblem since, in he la e case, he gene a ion o he G een’s unc- ions is a small ac ion o he o al nume ical e o . As an example, Fig. 3 shows he a io o CPU imes using nu- me ical in eg a ion agains he me hod epo ed he e as a unc ion o he numbe o s ips ( he same numbe o basis unc ions has been used in each s ip so as o keep he same accu acy le el). CPU ime educ ion is signi ican o any case, becoming mo e impo an as he complexi y o he mul is ip sys em inc eases. As a inal example, Fig. 4 shows he dispe sion cu es o he undamen al modes o he i e conduc o mic os ip ans- mission line depic ed in he igu e. Two cases a e conside ed: in Case A, he s ips a e in he ai –dielec ic in e ace; in Case B, he cen e conduc o esides on he op in e ace o a e y hin co e laye . Dispe sion cu es o he con igu a ion (A) we e published by Ki azawa in [32] and la e ep oduced by Hsu in [33]. Resul s o he con igu a ion B a e gi en in [33]. Ki azawa uses a a ia ional me hod, whe eas Hsu employs an MPIE scheme sol ed in he space domain by using he me hod o momen s wi h piecewise linea basis unc ions and nume ical compu a ion o he spec al in eg als. The ag eemen be ween ou esul s and hose p esen ed in [32] and [33] is e y good, as can been seen in he g aphical ep esen a ion. In o de o ep oduce hose da a, we ha e used h ee longi udinal and wo ans e se basis unc ions along wi h ou quad a u e poin s and eigh complex images o app oxima ing he egula pa o he spec al-domain G een’s unc ions. Many o he esul s epo ed in he li e a u e ha e been ep oduced wi h ou me hod, bu hey a e no included he e o he sake o b e i y. As a inal commen on he accu acy and obus ness o he p oposed me hod, we ha e o say ha e y accu a e esul s a e also ob ained o he cu en dis ibu ion. A sys ema ic inc ease o he numbe o basis unc ions does no in oduce nume - ical ins abili ies and all he coe icien s o he cu en expan- sion a e compu ed wi h e y good accu acy ( i e co ec igu es a e easily ob ained o he expansion coe icien s). This is mo e signi ican ega ding he quali y o he employed echnique han p opaga ioncons an esul s[9].Tosumup,ou manynume ical expe imen s con i m ha he de eloped me hod wo ks p ope ly, p o iding e yaccu a e esul s and impo an compu a ional-e - o sa ings. Fig. 3. CPU ime a io o a mic os ip analysis using nume ical gene a ion o he G een’s unc ion and he echnique in his pape as a unc ion o he numbe o s ips. Longi udinal and ans e se cu en s ha e been app oxima ed by means o h ee and wo basis unc ions, espec i ely. Subs a e: hickness = 0 : 635 mm,  =9 : 8 . S ip wid h =3 mm. S ip sepa a ion: 1.5 mm. Fig. 4. Dispe sion cu es o he i e undamen al modes o he s uc u e o he igu e. Dielec ic da a as in Fig. 2. w =1 mm, s =0 : 2 mm. Case (A): d =0 . Black squa es: esul s in [32] and [33], solid line: ou esul s. Case (B): d =0 : 01 d . Whi e squa es: esul s in [33], dash line: ou esul s. VI. CONCLUSIONS A new me hod has been p oposed o he compu a ion o he dispe sion cu es o mul ile el mul iconduc o plana ans- mission lines embedded in a uniaxially aniso opic s a i ied medium.Theapp oachisbasedon hecompleximage echnique and MPIE o mula ion. We ake ad an age o a closed- o m de i a ion o he 2-D space-domain G een’s unc ion and o he use o a sui able se o basis unc ions o ob ain a as and ac- cu a e compu e code. The e alua ion o Gale kin’s ma ix en- iesis pe o medin a e ye icien way. Nume ical esul sha e Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on July 15,2020 a 17:03:06 UTC om IEEE Xplo e. Res ic ions apply. BERNAL e al.: FAST FULL-WAVE ANALYSIS OF MULTISTRIP TRANSMISSION LINES 451 been p esen ed and compa ed wi h da a a ailable in he li e a- u e and supplied by o he me hods. 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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, pp. 48–55, Feb. 1995. [28] J. Be nal, F. Medina, and M. Ho no, “Quasi-s a ic analysis o mul i- conduc o CPW by using he complex images me hod,” In . J. RF Mi- c owa e Compu e –Aided Eng., ol. 8, no. 5, pp. 405–416, 1998. [29] G. A ken, Ma hema ical Me hods o Physicis s. New Yo k: Aca- demic, 1985. [30] F. Mesa and M. Ho no, “Compu a ion o p ope and imp ope modes in mul ilaye ed bianiso opic wa eguides,” IEEE T ans. Mic owa e Theo y Tech., ol. 43, pp. 233–235, Jan. 1995. [31] F. Mesa, R. Ma ques, and M. Ho no, “An e icien nume ical spec al domain me hod o analyze a la ge class o non ecip ocal plana ansmission lines,” IEEE T ans. Mic owa e Theo y Tech., ol. 40, pp. 1630–1641, Aug. 1992. [32] T. Ki azawa, “Va ia ional me hod o mul iconduc o coupled s iplines wi h s a i ied aniso opic media,” IEEE T ans. Mic owa e Theo y Tech., ol. 37, pp. 484–491, Ma . 1989. [33] C. G. Hsu, “Analysis o a mul iconduc o ansmission line embedded in a laye ed uniaxial medium using a mixed po en ial in eg al equa ion app oach,” Ph.D. disse a ion, Dep . Elec . Eng., Sy acuse Uni ., Sy a- cuse, NY, Sep . 1991. Joaquin Be nal was bo n in Se ille, Spain, in 1971. He ecei ed he Licenciado deg ee in physics om he Uni e si y o Se ille, Se ille, Spain, in 1994, and is cu en ly wo king owa d he Ph.D. deg ee a he Uni e si y o Se ille. In 1995, he joined he Depa men o Elec onic and Elec omagne ism, Uni e si y o Se ille. Since 1998, he has been an Assis an P o esso in he De- pa men o Applied Physics, Uni e si y o Se ille. 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. M . Be nal was he ecipien o a 1995 schola ship p esen ed by Jun a de Andalucía. F ancisco Medina (M’90) was bo n in Pue o Real, Cádiz, Spain, in No embe 1960. He ecei ed he Li- cenciado and he doc o deg ees om he Uni e si y o Se ille, Se ille, Spain, in 1983 and 1987, espec- i ely, bo h in physics. 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. F om 1985 o 1989, he was a P o eso Ayudan e wi h he Depa men o Elec onics and Elec omag- ne ism, Uni e si y o Se ille, and since 1990, he has been P o eso Ti ula o elec omagne ism. He is cu en ly he Head o he Mi- c owa es G oup, Uni e si y o Se ille. His esea ch in e es s include analy ical and nume ical me hods o plana s uc u es and ci cui s and he in luence on hese ci cui s o aniso opic ma e ials. D .Medinawasamembe o heTechnicalP og ammeCommi eeo he23 d Eu opean Mic owa e Con e ence, Mad id, Spain (1993). He is on he edi o ial boa d o he IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES. He was he ecipien o a Minis e io de Educacion y Ciencia/Minis e e de la Reche che e la Technologie Schola ship. Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on July 15,2020 a 17:03:06 UTC om IEEE Xplo e. Res ic ions apply. 452 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 48, NO. 3, MARCH 2000 Ra ael R. Boix (M’97) was bo n in Melilla, Spain, in 1962. He ecei ed he Licenciado and doc o deg ees in physics om he Uni e si y o Se ille, Se ille, Spain, in 1985 and 1990, espec i ely. Since 1985, he has been wi h he Elec onics and Elec omagne ics Depa men , Uni e si y o Se ille, whe e he became an Associa e P o esso in 1994. Du ing he summe s o 1991 and 1992, he was wi h he Elec ical Enginee ing Depa men , Uni e si y o Cali o nia a Los Angeles, as a Visi ing Schola . Du ing he summe o 1996, he was wi h he Elec ical and Compu e Enginee ing Depa men , Sy acuse Uni e si y, Sy acuse, NY, as a Visi ing Schola . His cu en esea ch in e es is ocused on he analysis o he e ec s o complex subs a es on he pe o mance o plana ansmission-line discon inui ies, plana passi e mic owa e ci cui s, plana esona o s, and p in ed ci cui an ennas. ManuelHo no(M’75)wasbo n inTo edel Campo, Jaén, Spain, and died in Sep embe 1998, in Se ille, Spain. He ecei ed he Licenciado and he doc o de- g ees om he Uni e si y o Se ille, Spain, in 1969 and 1972, espec i ely, bo h in physics. In Oc obe 1969, he joined 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 P o esso in 1986. His main ields o in e es included 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. Du ing his inal yea s, he was engaged in he analysis o plana ansmission lines embedded in aniso opic ma e ials, mul iconduc o ansmission lines, and plana an ennas. He was 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 July 15,2020 a 17:03:06 UTC om IEEE Xplo e. Res ic ions apply.