2-D analysis of leakage in printed-circuit lines using discrete complex-images technique

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 50, NO. 8, AUGUST 2002 1895 2-D Analysis o Leakage in P in ed-Ci cui Lines Using Disc e e Complex-Images Technique Joaquín Be nal, F ancisco Mesa, Membe , IEEE, and F ancisco Medina, Senio Membe , IEEE Abs ac —The mixed-po en ial in eg al equa ion is combined wi h he disc e e complex-images echnique o analyze he com- ple e spec um o mul ilaye ed p in ed ansmission lines. A ele an con ibu ion o he p esen wo-dimensional app oach is i s abili y o s udy bo h he bound and leaky egimes in a e y simple, sys ema ic, and e icien way. Since he analysis is ca ied ou in he spa ial domain, his me hod makes i possible o analyze he leakage phenomenon o s uc u es wi h nonze o- hickness conduc o s. E icien quasi-analy ical echniques a e employed o sol e he in eg al equa ion. Index Te ms—Complex images, leaky modes, mixed-po en ial in eg al equa ion, p in ed-ci cui lines. I. INTRODUCTION INTEGRAL-EQUATION me hods a e widely ecognized as e y e icien ools o s udying he p opaga ion cha ac e - is ics o p in ed-ci cui lines. O en, hese in eg al equa ions a e posed in he spec al domain since he G een’s unc ions o he laye ed medium a e only known in closed- o m in his domain [1]. Thus, many wo ks ha e shown he e iciency o he spec al-domain app oach (SDA) o compu e he p opaga ion pa ame e s o bo h bound and/o leaky modes [2]–[5]. Ne e heless, a well-known limi a ion o he SDA is i s inadequacy o deal p ope ly wi h gene al nonze o hickness conduc o s. A possible al e na i e o he SDA, able o ea wi h nonze o- hickness conduc o s, is ound when he co - esponding in eg al equa ion is di ec ly sol ed in he spa ial domain [6]. This echnique equi es o compu e he spa- ial-domain G een’s unc ions om hei spec al coun e pa s h ough Fou ie - ans o m in e sion. Nume ically e icien in eg a ion schemes ha e been used in he pas o pe o m ha ask [7]. Howe e , a powe ul echnique p oposed a he end o he 1980s, i.e., he disc e e complex-images echnique (DCIT) [8], has become, in a a ie y o e sions [9]–[11], he s anda d echnique o ob ain spa ial-domain G een’s unc ions o laye ed s uc u es. Tha echnique, o iginally in ended o he analysis o h ee-dimensional (3-D) plana ci cui s and an- ennas, has been ecen ly adap ed o deal wi h wo-dimensional (2-D) guiding s uc u es, including s ip-like [12], coplana wa eguide [13], and a bi a y c oss-sec ional conduc o s [14]. The ele ance o a as gene a ion o he G een’s unc ion in Manusc ip ecei ed Ap il 4, 2001. This wo k was suppo ed in pa by he Comisión In e minis e ial de Cienciay Tecnología,Spain, unde P ojec TIC98- 0630, and by Jun a de Andalucía. J. Be nal is wi h he Depa men o Applied Physics III, Uni e si y o Se ille, 41092 Se ille, Spain. F. Mesa is wi h he Depa men o Applied Physics I, Uni e si y o Se ille, 41012 Se ille, Spain (e-mail: [email p o ec ed]). F. Medina is wi h he 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]). Publishe I em Iden i ie 10.1109/TMTT.2002.801320. 2-D p oblems is ela i ely highe han in 3-D cases because he CPU ime de o ed o sol e he inal sys em o linea equa ions is usually negligible in he 2-D si ua ion and he G een’s unc ions mus be gene a ed many imes o di e en alues o he unknown p opaga ion cons an . Al hough [12]–[14] could deal wi h a g ea a ie y o mul- iconduc o and mul ilaye s uc u es, hey we e pu posely e- s ic ed o s udy only he bound egime. The impo an ques ion posed by he exis enceo bo h su ace and space leaky-wa e so- lu ions has been o en ea ed in he ame o he SDA [3], [5] in such a way ha only ze o- hickness s ip-like o slo -like s uc- u es ha e been conside ed in dep h. Mo eo e , accoun ing o leakage in he ame o SDA equi es a he sophis ica ed phys- ical/ma hema ical easoning o p ope ly choose he in eg a ion pa hs ha momen -me hod spec al in eg als ha e o un along. Thesame easoninghas o beused i hespa ial-domain G een’s unc ion is ob ained by means o a di ec in eg a ion in he spec- al domain [7]. Thus, he aim o his pape is o show how o ex end he me hod p oposed in [12] and [14] o also deal wi h he leaky egime. This new app oach u ns ou o be e ysimple and nume ically e y e icien o s ic ly plana s uc u es and capable o dealing wi h nonplana conduc ing s uc u es. These aluable ea u es can be also ad an ageously used o s udying he p ac ical exci a ion o leaky modes [15], [16] since his 3-D p oblem in ol es as one o i s c ucial s eps he compu a ion o he eac ion in eg als appea ing in he 2-D case. II. FORMULATION OF THE PROBLEM Thespa ial-domainmixed-po en ialin eg alequa ion(MPIE) will be applied o s udy he wa e p opaga ion in p in ed-ci cui lines such as ha shown in Fig. 1. The equi ed ke nel o he MPIE, namely, he spa ial-domain G een’s unc ions associa ed wi h he scala and ec o po en ials [6], ha e o be ob ained om hei co esponding spec al e sions. This calcula ion im- plies o pe o m Fou ie ans o m in e sions o he ollowing gene al ype: (1) whe e (2) wi h being he ans e se wa enumbe , being he ee- space wa enumbe , and being he assumed p op- aga ion cons an . (No e ha has been used ins ead o wi h he pu pose o se ing all hese a iables as wa enumbe s; 0018-9480/02$17.00 © 2002 IEEE Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on July 07,2020 a 14:13:31 UTC om IEEE Xplo e. Res ic ions apply. 1896 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 50, NO. 8, AUGUST 2002 Fig. 1. C oss sec ion o he s uc u e unde s udy. , whe e is he a iable used in [14] and o he pa- pe s.) I can be checked ha he spec al-domain G een’s unc- ions o he laye ed s uc u e a e ac ually unc ions o p o- ided ha he media a e iso opic and/o uniaxial aniso opic dielec ic. W i ing hese unc ions in his con enien way (in- s ead o as unc ions o ) is e y ele an as i will become appa en la e . The DCIT can now be ad an ageously used o a oid he a - duous compu a ional e o in ol ed in a di ec nume ical in e- g a ion o (1). The e iciency o his app oach basically lies in he ac ha he spa ial-domain coun e pa o an spec al ex- ponen ial e m is known in closed o m hanks o a 2-D e - sion o he Somme eld iden i y. This idea has been success- ully applied in [12]–[14] o cha ac e ize bound modes. Ou pu pose he e is o gene alize ha o mula ion o also accoun o su ace- and space-wa e leaky modes. This gene aliza ion equi es a p e ious ea men o he singula i ies (b anch poin s and poles) o he spec al G een’s unc ions since hese singu- la i ies a e di ec ly ela ed o adia ion in he o m o space and su ace wa es [17]. Conside ing ha he unde lying idea o he DCIT is o ind a sum o complex exponen ial unc ions ha i s p ope ly and each exponen ial e m is a complex analy ical unc ion, i is hen c ucial in his me hod o ind ha pa o he G een’s unc ion ha is ac ually analy ic (i.e., ee o singula i- ies) wi hin he ange o in e es . In his sense, i is impo an o no e ha , as a unc ion o , is no a mul i alued unc- ion because i does no show b anch poin s in he complex plane. Ne e heless, is s ill me omo phic since i does ha e poles, which should be ex ac ed ou in o de o isola e he analy ic pa o . In ac , has an in ini e numbe o poles a , whe e (3) wi h being he wa enumbe o he h mode o he g ounded laye edsubs a e. Despi e he exis enceo in ini e poles,in mos p ac ical cases, i is only necessa y o ex ac ou hose poles appea ing on he eal axis, i.e., he poles associa ed wi h he abo e-cu o su ace wa es o he g ounded laye ed subs a e [14]. Fo con enience, he quasi-s a ic con ibu ion should be also ex ac ed ou explici ly, as o iginally epo ed in [9] and la e , in a 2-D con ex , in [12]. The combined MPIE–DCIT app oach was used in [14] o s udy he bound egimeins uc u eswi h nonplana conduc o s. This app oach will be b ie ly ou lined he e in o de o in oduce he non i ial changes equi ed o ex end he me hod o dealing wi h he leaky egime. When nonplana conduc o s a e consid- e ed, he ke nel o he in eg al equa ion con ains se e al e ms o he dyadic magne ic ec o -po en ial G een’s unc ion and some de i a i es plus he e m associa ed o he scala -po en- ial G een’s unc ion [14]. Thus, he G een’s unc ion p oblem can be educed o ob aining he space-domain e sion o he ol- lowing gene ic G een’s unc ion: (4) whe e is a equency-dependen cons an and (5) wi h being he spec al G een’s unc ion o (apa om ). The spec al unc ion can be spli in o he ollowing h ee di e en con ibu ions: (6) whe e accoun s o he quasi-s a ic pa (7) (wi h beingaknowncons an ); ep esen s hecon ibu ion o he signi ican su ace-wa e e ms (8) whe e is he pole associa ed wi h he h su ace wa e and is i s esidue. is he emaining quasi-analy ical pa ha can be accu a ely expanded as he ollowing sum o complex exponen ial unc ions: (9) whe e and a e, espec i ely, heampli ude anda gumen o each exponen ial e m ha expands he pa o no depending o and . A. T ea men o The spa ial-domain e sion o can be w i en as (10) whe e is gi en by (11) wi h . Fo he nonplana case, he abo e in e- g al and i s de i a i e wi h espec o can be e icien ly compu ed using in eg a ion con ou echniques, as explained in he Appendix. Howe e , o he plana case ( ), (11), which will be deno ed as , can be ob ained in closed o m.I ela ionship [18,(4.91)] is now adap edand analy ically ex ended o make con inuous in he en i e complex plane, can be con enien ly exp essed as (12) Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on July 07,2020 a 14:13:31 UTC om IEEE Xplo e. Res ic ions apply. BERNAL e al.: 2-D ANALYSIS OF LEAKAGE IN PRINTED-CIRCUIT LINES USING DCIT 1897 Fig. 2. Pa hs o in eg a ion in he complex k -plane. Fo simplici y, only a pai o su ace-wa e poles a e shown ( 222 ). in o de o emphasize he na u e o he associa ed ield. F om a physical poin -o - iew, he case implies ha he ields associa ed wi h he abo e-cu o h su ace wa e decay exponen ially in he ans e se di ec ion. Taking all he su - ace-wa e e ms in his way would be consis en wi h he bound egime. The su ace-wa e leaky egime is accoun ed o by he case, which gi es he cha ac e is ic exponen ially g owing- ield beha io in he ans e se di ec ion [17]. Taking he second choice o some o he abo e-cu o su ace wa es, he esul ing leaky egime would accoun o a mode adia ing in he o m o only hose su ace wa es hus conside ed. The abo e esul s can be ela ed o he SDA by means o con- ou in eg a ion echniques. I , o simplici y, only one su ace wa e is assumed o be abo e cu o , he bound- egime esul in (12) is equi alen o ha ing used he con en ional eal-axis in eg a ion pa h when pe o ming he Fou ie - ans o m in e - sion (pa h in Fig. 2). The su ace-wa e leaky- egime esul can be iewed as a consequence o employing an in eg a ion con ou de ou ing a ound he poles as pa h in Fig. 2. Taking in o accoun ha he in eg and in (11) has no b anch poin s, in- eg a ion along pa h is equi alen o in eg a ion along pa h in Fig. 2, whe eas he in eg a ion pa h is equi alen o he in eg a ion pa h . The e o e, swi ching be ween and pa hs simply implies changing he pole ha is cap u ed by he in eg a ion pa h. This swi ch is equi alen o choose he pole acco ding o he sign o i s imagina y pa . The abo e discussion has shown ha he selec ion o he egime o be deal wi h (bound o su ace-wa e leakage) is simply imposed by he p ope choice in (12). Following he heo y gi en in [15], he momen -me hod de e minan unc ion whose ze os a e he p opaga ion cons an s o he line de ines a Riemann su ace in he longi udinal wa enumbe complex -plane wi h mul iple b anch poin s. Thus, he choice made in can be ela ed o he p ope /imp ope shee ha he p opaga ion cons an will be loca ed on when sea ching o he oo s o he dispe sion equa ion o he line. B. T ea men o Taking in o accoun he complex exponen ial expansion (9), he space-domain con ibu ion associa ed wi h can be exp essed as (13) Fig.3. In eg a ionpa hs C and C in hecomplex k -plane.In eg a ionpa hs on he p ope /imp ope shee wi h espec o he 6 k b anch poin s a e display insolid/dashed lines.Fo simplici y,only apai o su ace-wa epoles a eshown ( 222 ). whe e (14) Followinga simila a ionale as ha p e iouslyused o , each o hein eg al ep esen a ionsgi enby hein e seFou ie ans- o m will be exp essed in an app op ia e closed o m. Thus, ela ionship [18, (4.156)] is now adap ed and analy ically con- inued in he en i e plane o w i e (15) whe e a e he b anch poin s o he G een’s unc ion in he complex -plane, , is he ze o h-o de modi ied Bessel unc ion o he second kind, and is he ze o h-o de Hankel unc ion o he second kind. Taking in o accoun ha bo h he quasi-s a ic e m and he i s e m in (4) can be conside ed as pa icula cases o he exponen ial e ms in expansion (9) (when ), i s spa ial-domain coun e pa can be eadily ob ained om (15). Looking a (15), he i s op ion, i.e., ,gi es place o exponen ially decaying ields in he no mal -di ec ion, asi ua ion ha is compa iblewi h hebound egime.The second op ionin(15)p o idesexponen iallyg owing ieldsin heuppe hal -space, hus accoun ing o he space-wa e leaky egime. The wo- alued (15) can be ela ed o he di e en in eg a ion pa hs used in he SDA o space-wa e leaky modes [3], [16]. The con en ional eal-axis in eg a ion pa h in Fig. 3 is ound o be equi alen o he choice in (15), whe eas he in eg a ion pa h in his igu e would e lec he choice. C. Sol ing he In eg al Equa ion A ele an ea u e o he p esen app oach is ha e y ac- cu a e closed- o m exp essions a e ob ained o he ke nels o he in eg al equa ion o bo h he bound and leaky egimes. The egime o be s udied (bound, su ace wa e, and space wa e) is simply selec ed by he con enien choice in (12) and (15). This p o ides a sys ema ic, e y e icien , and a he simple way o Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on July 07,2020 a 14:13:31 UTC om IEEE Xplo e. Res ic ions apply. 1898 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 50, NO. 8, AUGUST 2002 s udy he p opaga ion phenomena, supe seding he possible in- con eniences o he SDA ela ed o he nume ical e alua ion o he spec al in eg als along di e en in eg a ion pa hs in he complex plane. Once he space-domain ke nels a e ob ained, he in eg al equa ion is sol ed by using he Gale kin momen me hod. Chebyshe polynomials o he i s and second kinds weighed by he edge condi ion a e used as basis unc ions o analyze s uc u es wi h in ini ely hin conduc o s. S uc u es wi h a bi a y c oss-sec ional conduc o s a e analyzed employing nonuni o mly dis ibu ed piecewise linea unc ions as basis unc ions. In bo h cases, a quasi-analy ical e alua ion o he ma ix en ies is ca ied ou . Mo e de ails can be ound in [12] and [14]. III. NUMERICAL RESULTS The p esen me hod has been alida ed by compa ing ou e- sul s wi h p e iously published esul s and wi h hose ob ained by means o a well-es ablished SDA code [19] de eloped o ze o- hickness s ip-like s uc u es. The ag eemen o bo h bound and leaky egimes is excellen o all he cases consid- e ed. As a i s example, Fig. 4(a) and (b) shows he no malized phase and a enua ion cons an s o a space-wa e leaky mode suppo ed by a mic os ip line p e iously s udied by Michalsky and Zheng [7] only o he ze o- hickness case. The compa ison be ween he da a epo ed in [7] wi h ou MPIE–DCIT esul s shows an excellen ag eemen . Fig. 4(a) and (b) also includes esul s o wo nonze o hickness cases, i.e., ec angula and apezoidal c oss-sec ional conduc o s. These cu es show how he shape o he conduc o can in luence he dispe sion cu es. To gi e an idea o he o e all compu a ional e o equi ed by ou me hod, 100 alues o he p opaga ion cons an o he ze o- hickness case we e compu ed in 6 s wi h a Pen ium II 450-MHz compu e . Rec angula o apezoidal s uc u es equi e a ew seconds o ge a single alue o using he same pla o m. Fig. 5 shows ano he compa ison wi h he dispe sion cu es o a ci cula -wi e ansmission line epo ed in [7, Fig. 7]. Again, he ag eemen ound bo h o he bound () and he space-wa e leaky ( ) modes is e y good. I is in e es ing o men ion ha , in he compu a ion o he abo e esul s, because o he p esence o se e al slab wa eguide modes ( , , and ), he explici con ibu ion o up o h ee su ace-wa e poles had o be conside ed in he calculus. Finally, and o show he capabili y o ou me hod o deal wi h mul iconduc o and mul ilaye nonplana s uc u es, a wo-laye LIGA mic omachined ansmission line has been analyzed. The LIGA p ocess, desc ibed in [20], gi es ise o me alliza ions wi h a high hickness/wid h a io. This makes i possible o use he conduc o hickness as a new a iable in he design o il- e s and couple s. The analyzed s uc u e consis s o a pai o squa e c oss-sec ional conduc o s (200 200 m) sepa a ed by a 120- m dis ance and placed on a laye ed subs a e. The dispe sion cu es o he no malized p opaga ion and a enua- ion cons an s o a su ace-wa e leaky mode appea ing in his s uc u ea eshowninFig.6.Tos udy hee ec o heme alliza- ion hickness, he dispe sion cu es o he same s uc u e as- suming ze o- hickness conduc o s a e alsoshown. I can be seen (a) (b) Fig. 4. No malized: (a) phase and (b) a enua ion cons an o a space-wa e leaky mode in he mic os ip line shown in he inse : " = 9 : 8 , w =3 mm, h =0 : 635 mm. Fig. 5. No malized phase and a enua ion cons an s o he undamen al mode and he i s highe mode in a ci cula -wi e ansmission line. Lines: ou esul s. Symbols: da a in [7]. S uc u al pa ame e s: a=h =0 : 25 , " =4 . ha hespli ing poin ( he poin whe e acomplexleakysolu ion me geswi hacomplexconjuga esolu ion,and he wosolu ions hen spli apa as wo imp ope eal modes) appea s a lowe equencies in he nonplana case. Resul s o he squa e-shape conduc o s ha e been ob ained wi h 15 basis unc ion o e each Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on July 07,2020 a 14:13:31 UTC om IEEE Xplo e. Res ic ions apply. BERNAL e al.: 2-D ANALYSIS OF LEAKAGE IN PRINTED-CIRCUIT LINES USING DCIT 1899 Fig. 6. No malized phase and a enua ion cons an s o a su ace-wa e leaky mode in a wo-conduc o and wo-laye ansmission line wi h w = 200  m, s = 120  m, h = h = 210  m, " =3 , and " =4 : 6 . conduc o . The CPU ime was much la ge o he nonplana case because o he use o many basis unc ions (in compa - ison wi h he ze o- hickness s ip case) and he need o com- pu ing mo e ime-consuming in eg als. Rega dless, he me hod p o ides he possibili y o e icien ly de e mining in a quan i a- i e way he in luence o he hickness o he shape o he con- duc o s in he leaky egime. IV. CONCLUSIONS A new app oach based on he combina ion o he MPIE and DCIT has been de eloped o compu e in a simple and sys em- a ic way he p opaga ion cha ac e is ics o bo h bound, su ace-, and space-wa e leaky modes in p in ed-ci cui lines. One o he main goals o his app oach has been he ob aining o accu a e exp essions o he ke nels o he co esponding in eg al equa- ioninaquasi-analy icalway.Thishasa oided hein ol ed ask o in eg a ing along di e se in eg a ion pa hs in he ans e se wa enumbe complex plane equi ed in he SDA. The use o he complex images echnique has also p o ided high accu acy and e iciency o he analysis. Wo king in he spa ial domain has allowed o a di ec (al hough non i ial) ex ension o he anal- ysis o nonplana conduc o s. Some esul s ha e been epo ed o show he accu acy and e iciency o he compu e code de- eloped. APPENDIX An e icien me hod o compu e in eg al (11) and i s de i a- i e wi h espec o will be shown he e. To simpli y he no a ion, he ollowing new a iables will be used: and . The in eg and in (11) has wo poles a and wo b anch poin s a . I an in eg a ion pa h unning om o along he eal axis is con enien ly closed in he lowe hal -plane, he esul o his in eg al is de e mined by he su ace-wa e pole enclosed by such a pa h. Conside ing ha he closed pa h mus de ou a ound he b anch cu , (11) can be exp essed as (16) whe e (17) The i s e m in (16) is a wo- alued unc ion accoun ing o he su ace-wa e pole con ibu ion wi h —as in (12), he sign o de e mines he bound/leaky egime unde conside a ion. The second e m in (16) gi es he con ibu ion o he in eg a ion along he b anch cu (assuming ha ) and, mo e speci ically, i comes om an in eg a ion along a e - icalb anchcu wi h .Ashappensin(15), he sign o in (17) se s he egime (bound/space-wa e leaky) o be ea ed. The change o a iable causes in eg al o be quickly calcula ed by using, o in- s ance, Gauss quad a u es. The de i a i e o can be eadily compu ed om he abo e exp essions o gi e (18) whe e (19) No e ha , o la ge alues o , he in eg ands in (17) and (19) become highly oscilla o y, hus gi ing ise o nume ical d awbacks. In ha case, can be calcula ed by means o a di- ec in eg a ion o (11). This nume ic in eg a ion mus be pe - o med along pa h in Fig. 2 o accoun o he bound egime, whe eas pa h / mus be chosen i we a e looking o a su - ace-/space-wa e leaky egime. The in eg al along he eal-axis pa h, deno ed as , can be ew i en in he ollowing o m: (20) (), which can be e icien ly calcula ed by using Gauss quad a u es. No e ha la ge alues o lead o a as con e gence o . The in eg a ion along pa h , i.e., , can be spli in o in e- g a ion along he eal axis ( ) in addi ion o in eg a ion along and pa hs in Fig. 2, namely, (21) whe e (22) wi h and . Acco ding o [16], he in eg al along pa h , i.e., , can be exp essed as in eg al (21) plus an in eg al unning along a loop, as shown in [16] (23) Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on July 07,2020 a 14:13:31 UTC om IEEE Xplo e. Res ic ions apply. 1900 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 50, NO. 8, AUGUST 2002 whe e (24) wi h and . Thede i a i eo wi h espec o canbe eadily ob ained om (20) and (22) yielding he ollowing exp essions: (25) (26) (27) REFERENCES [1] D. Mi sheka -Syahkal, Spec al Domain Me hod o Mic owa e In e- g a ed Ci cui s. No wood, MA: A ech House, 1990. [2] J. Boukamp and R. H. Jansen, “Spec al domain in es iga ion o su ace wa eexci a ionand adia ionbymic os iplines andmic os ipdisk es- ona o s,” in P oc. 13 h Eu . Mic owa e Con ., Sep . 1983, pp. 721–726. [3] N. K. Das and D. M. Poza , “Full-wa e spec al-domain compu a ion o ma e ial, adia ion, and guided wa e losses in in ini e mul ilaye ed p in ed ansmission lines,” IEEE T ans. Mic owa e Theo y Tech., ol. 39, pp. 54–63, Jan. 1991. [4] F. Mesa, R. Ma qués, 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. [5] D. Nghiem, J. T. Williams, D. R. Jackson, and A. A. Oline , “P ope and imp ope dominan mode solu ions o a s ipline wi h an ai gap,” Radio Sci., ol. 28, no. 6, pp. 1163–1180, No .–Dec. 1993. [6] C.-I.G.Hsu, R.F.Ha ing on, K.A.Michalski, andD. Zheng,“Analysis o mul iconduc o ansmission lines o a bi a y c oss sec ion in mul i- laye ed uniaxial media,” IEEE T ans. Mic owa e Theo y Tech., ol. 41, pp. 70–78, Jan. 1993. [7] K. A. Michalsky and D. Zheng, “Rigo ous analysis o open mic os ip lineso a bi a yc oss sec ion inbound andleaky egimes,”IEEE T ans. Mic owa e Theo y Tech., ol. 37, pp. 2005–2010, Dec. 1989. [8] D. G. Fang, J. J. Yang, and G. Y. Delisle, “Disc e e image heo y o ho izon al elec ic dipoles in a mul ilaye ed medium,” P oc. Ins . Elec . Eng., ol. 135, pp. 297–302, Oc . 1988. [9] Y. L. Chow, J. J. Yang, D. G. Fang, and G. E. Howa d, “A closed- o m spa ial G een’s unc ion o he hick mic os ip subs a e,” IEEE T ans. Mic owa e Theo y Tech., ol. 39, pp. 588–592, Ma . 1991. [10] M. I. Aksun, “A obus app oach o he de i a ion o closed- o m G een’s unc ions,” IEEE T ans. Mic owa e Theo y Tech., ol. 44, pp. 651–658, May 1996. [11] R.-C. Hsieh and J.-T. Kuo, “E alua ion o spa ial G een’s unc ions o mic os ips: Fas Hankel ans o m algo i hm and complex image me hod,” Elec on. Le ., ol. 34, no. 11, pp. 1110–1111, May 1998. [12] J. Be nal, F. Medina, R. R. Boix, and M. Ho no, “Fas ull wa e analysis o mul is ip ansmission lines based on MPIE and complex images,” IEEE T ans. Mic owa e Theo y Tech., ol. 48, pp. 445–452, Ma . 2000. [13] E. A. Soliman, P. Pie e s, E. Beyne, and G. A. E. Vandenbosch, “Nume - ically e icien spa ial-domain momen me hod o mul islo ansmis- sion lines in laye ed media—Applica ion o mul islo in MCM-D ech- nology,” IEEE T ans. Mic owa e Theo y Tech., ol. 47, pp. 1782–1787, Sep . 1999. [14] J. Be nal, F. Medina, R. R. Boix, and M. Ho no, “Full-wa e analysis o nonplana ansmission lines on laye ed medium by means o MPIE and complex image heo y,” IEEE T ans. Mic owa e Theo y Tech., ol. 49, pp. 177–185, Jan. 2001. [15] C. Di-Nallo, F. Mesa, and D. R. Jackson, “Exci a ion o leaky modes on mul ilaye s ipline s uc u es,” IEEE T ans. Mic owa e Theo y Tech., ol. 46, pp. 1062–1071, Aug. 1998. [16] F. Mesa, C. Di-Nallo, and D. R. Jackson, “The heo y o su ace-wa e andspace-wa eleaky-modeexci a iononmic os iplines,”IEEET ans. Mic owa e Theo y Tech., ol. 47, pp. 207–215, Feb. 1999. [17] A. A. Oline , “Leakage om highe modes on mic os ip line wi h appli- ca ion o an ennas,” Radio Sci., ol. 22, no. 6, pp. 907–912, No . 1987. [18] D. G. Dudley, Ma hema ical Founda ions o Elec omagne ic Theo y. Pisca away, NJ: IEEE P ess, 1994. [19] R.Ma quésandF.Mesa,“Spec aldomainanalysiso highe o de leaky modes in mic os ip lines: A new spec al-gap e ec ,” J. Elec omagn. Wa es Applica ., ol. 11, pp. 1367–1384, 1997. [20] T. L. Willke and S. S. Gea ha , “LIGA mic omachined plana ansmis- sion lines and il e s,” IEEE T ans. Mic owa e Theo y Tech., ol. 45, pp. 1681–1688, Oc . 1997. Joaquín Be nal was bo n in Se ille, Spain, in 1971. He ecei ed he Licenciado and Doc o deg ees om he Uni e si y o Se ille, Se ille, Spain, in 1994 and 2000, bo h in physics. 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 Depa men o Applied Physics III a he same uni- e si y. 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 and high-speed e y la ge scale in eg a ion (VLSI) in e connec s. D . Be nal was he ecipien o a Jun a de Andalucía schola ship. F anciscoMesa(M’94)wasbo ninCádiz,Spain,on Ap il 1965. He ecei ed he Licenciado and Doc o deg ees om heUni e si yo Se ille,Se ille,Spain, in 1989 and 1991, espec i ely, bo h in physics. He is cu en ly an Associa e P o esso in he De- pa men o Applied Physics I, Uni e si y o Se ille. His esea ch in e es ocuses on elec omagne ic p opaga ion/ adia ion in plana lines wi h gene al aniso opic ma e ials. F ancisco Medina (M’90–SM’01) was bo n in Pue o Real, Cádiz, Spain, in No embe 1960. He ecei ed he Licenciado and 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 wi h he Labo a oi ede Mic oondes de l’ENSEEIHT, Toulouse, F ance. F om 1985 o 1989, he was a P o- eso Ayudan e(Assis an P o esso )wi h heDepa - men o Elec onics and Elec omagne ism, 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 ism. He is also cu en ly Head o he Mic owa es G oup, Uni e si y o Se ille. His esea ch in e es includes an- aly ical and nume ical me hods o guidance, esonan and adia ing s uc u es, passi e plana ci cui s, and he in luence on hese ci cui s o aniso opic ma e- ials. D . Medina was a membe o bo h he Technical P og am Commi ee (TPC) o he 23 d Eu opean Mic owa e Con e ence, Mad id, Spain, 1993, and he TPC o ISRAMT’99, Malaga, Spain. He is on he Edi o ial Boa d o he IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES. He has been a e- iewe o o he IEEE and Ins i u ion o Elec ical Enginee s (IEE), U.K., pub- lica ions. Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on July 07,2020 a 14:13:31 UTC om IEEE Xplo e. Res ic ions apply.