Analysis of leakage in multilayered microstrip lines using complex images

Analysis o leakage in mul ilaye ed mic os ip lines using complex images J. Be nal, E Mesa , F. Medina GN~O de Mic oondas. Dep . Elec mica y Elec omagne ismo Facul ad de Fsica A da. Reina Me cedes s/n, 41012 Se illa (SPAIN) Fax: + 34 5 4239434 email: [email p o ec ed] Abs ac The Mixed Po en ial In eg al Equa ion (MPIE) is combined wi h he complcx iinagc me hod Lo analyze he comple e spec um o mul ilaye ed p in ed ansmission lines. A ele an con ibu- ion o his me hod IS i s abili y o s udy bo h he bound and leaky egimes in a e y simple and e icien way. Since he analysis is ca ied ou in he spa ial domain, his wo k also makes i possible o analyze he leakage phenomenon o s uc u es wi h nonze o hickness conduc o s. 1 In oduc ion The p opaga ion cha ac e is ics o guiding s uc u es in laye ed media a e e y e i- cien ly compu ed by means o in eg al equa ion me hods. Fo he laye ed geome y, he equi ed G een’s unc ions a e only known in closed- o m in he spec al domain. I a spa ial domain o mula ion we e used, he spa ial-domain G een’s unc ions can be ob ained om hei spec al e sions h ough Fou ie ans o m in e sion [I]. S ic ly ze o- hickness p in ed ansmission lines can be sol ed bo h in he spec al and he spa- ial domains wi h a he low compu a ional e o . Non-plana conduc o s a e be e ac- coun ed o by means o he spa ial-domain e sion o he in eg al equa ion [Z]. Du ing he las decade, a powe ul ool - he Disc e e Complex Images Technique (DCIT- was de eloped o ob ain he spa ial G een’s unc ions o laye ed s uc u es [3]. This echnique has been p edominan ly applied o 3D plana ci cui s and an ennas p oblems. Recen ly his me hod has been adap ed o deal wi h 2D guiding p oblems, whe e he ele ance o a as gene a ion o he G een’s unc ions is e en mo e signi ican han in he 3D con ex . Thus, plana s ip-like [4] and slo -like [5] s uc u es ha e been sol ed ollowing ha p ocedu e. Howe e , he impo an ques ion posed by he exis ence o bo h su ace and space leaky-wa e solu ions has no been conside ed in hose wo ks whe e only bound modes we e ea ed. Leaky-wa es ha e been comp ehensi ely s ud- ied in he ame o he Spec al Domain Analysis (SDA) [6]. Wo king in SDA implies o pe o m nume ical in eg a ion along a p ope ly chosen pa h in he spec al a iable complex plane. Va ious physical and ma hema ical a gumen s ha e been employed o de e mine he ea u es o his pa h. In his pape we will show how o adap he me hod in [4] o include he leaky egime. This new app oach has he ad an age o being e y simple and e icien and he capabili y o dealing wi h non-plana conduc ing s uc u es. 0-7803-7070-8/0 1 /$lO.OO 0200 1 IEEE 580 Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on July 09,2020 a 14:11:05 UTC om IEEE Xplo e. Res ic ions apply. 2 Analysis The p oblem o he de e mina ion o he p opaga ion cons an s o a mul is ip sys em embedded in a laye ed subs a e can be posed in e ms o a Mixed Po en ial In eg al Equa ion iis shown in [4]. The G een' unc ions o he scala and ec o po en ials, Ji"' and K,; , inus be ob ained oin hei spec al e bions. This Implies o pe o m Fou ie ~iiis o m in e sions o he ype: whe e A,J = m, A., b ing lie ans e w wa enuinbe and I;: = siiined p opaga ion cons an . - j.u he as- The DClT can be ad an ageously uaed in his p oble n o a oid he edious nume - iciil in eg a ion in ol ed in (I ). The unde lying idea ia o expand he spec al unc ion ii i a sum o complex exponen ial unc ions whose spa ial-domain coun e pa is known hanks o a 2D Som nc eld iden i y [4]. Howe e , his echnique was only applied o cha x e ize bound modes. Nex we will explain he guidelines o he me hod along wi h lie modi ica ions ha mus be in oduced o gene alize he app oach o accoun o leaky modes. - I is el1 known ha he spec al unc ions G(b:,) show wo ypes o singula i ies: b anch poin a a b., = and poles. They a e espec i ely ela ed o he adia- i~oii iii o i e space and o he inodcs o he backg ound wa eguide. Since he complex im,igcs cxlmision ciiii only i analy ical unc ions, he abo e singula i ies mus be p op- e ly haiidled. This IS done by i s in oducing a change o a iable (ILO = m) ha elimina es b anch poin s and second ex ac ing ou in analy ical o m he signi - icm polcs (in p x ice, hobe poles associa ed wi h abo e cu o wa eguide modes). . a ic con ibu ion IS also explici ly ex ac ed ou . Thus, he spec al domain G een' 'unc lons a e spli in o h ee con ibu ions: - -- whe e G,, ih he quasi-s a ic c in (Go = Ii~iia,+~ G), Gp ep esen s he con ibu ion o he u hce wa e e ms and LG, is he emaining ha is sui able o complex expo- nen~i;iI expansion. /! " The spa ial doinain e sion o Gp u ns ou o be [4]: whe e No i he numbe o signi ican poles, k, is he loca ion o hep- h pole, 4 i s eqidue and = k:: - k,$ In he bound egime (k: = l > k,,) his exp ession yields exponen ially decaying ields in he anb e se di ec ion. I we a e in e es ed in he leaky egime asocia ed o ii pa icula su ace-wa e pole, i can be p o ed ha i su ices o chenge lie sign o 6,> o ha pa icula pole in (3). 58 1 Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on July 09,2020 a 14:11:05 UTC om IEEE Xplo e. Res ic ions apply. - The G, unc ion can be app op ia ely expanded as a siini o coinplcx exponen ial unc ions: ! I,:, G;, c u*c-y~I(o (4) 1=I The space domain con ibu ion o he e ms in he abo e expansion can be analy ically ob ained by using he ollowing iden i y: whe e ( = ,/(:I; - :I:')~ + y2, A'" ib he ze o o de modi ied Bessel unc ion o he sec- ond kind and Hi2) hc ze o o de Haenkel unc ion o he wcond kind. The i s op ion gi e5 place o g owing ields o accoun ing o leaky egime whe eas lic second op- ion p o ides he bound solu ions. In [6, 71 he space-wa leaky modes a c ob ained by imposing an al e na i e in eg a ion pa h in he I;, complex plane o he nume ical e alua ion o :he spec al in eg als (I). This ciilcula ion is supe seded by he p esen app oach. The e m in (I) can be conside ed a pa icula case o (5). To summa ize, he p oposed me hod allow us o analy ically ob ain he spacc doinai ke nel o he in eg al equa ion in bound and lcaky egimes. Once he space domain ke nels ha e been ob ained, he in eg al equa ion is sol ed by using he Gale kin me hod wi h adequa e basis unc ions leading o quasi-analy ical e illua ion o ma ix en ies 141. 3 Nume ical esul s Ou me hod has been alida ed by compa ing ou esul s wi h he ones ob ained by means o a well es ablished SDA code [7] de eloped o ze o- hickiie~h mip-like s iic- u es. The ag eemen in bo h bound and leaky egimes is excellen b - id1 he cascs conside ed. As an example, Figu e 1 shows p opaga ion cons an da a o a space-wa e leaky mode suppo ed by a mic os ip line. Fo he ze o- hickness case, his igu e coin- pa es SDA esul s e sus ou esul s o B/k0 and o/kU. An excellen ag eeinen is obse ed. This s uc u e was also analyzed in [I], showing ou esul s good ag eemen wi h he da a epo ed in [I]. Al hough he ex ension o ou app oach o nonze o hick- ness conduc o s equi es u he ma hema ical s eps han hose p esen ed in his pape , his ask has been done and some nume ical esul s ha e been include in he igu e. 4 Conclusions The MPIE app oach combined wi h he complex iinage me hod has been adap ed o analyzing mic os ip ansmission lines in leaky egime hy he i s ime. Bo h su ace- wa e and space-wa e leaky modes a e inco po a ed o he analysis in a e y simple way. The complex image echnique p o ides high accu acy and e iciency o he analysis. Wo king in he spa ial domain allows o an ex ension o nonplana conduc o s. Al- hough his gene aliza ion is no s aigh o wa d, he heo y epo ed in his pape o e s he necessa y backg ound. 582 Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on July 09,2020 a 14:11:05 UTC om IEEE Xplo e. Res ic ions apply. I2l 1 .o 06 A- CO , 4 6 8 io 12 i4 F equency (GHz) Figu e 1 : No malized p opaga ion (P/ko) and a enua ion (culko) cons an s o a space- wa e leaky mode in a mic os ip line: E, = 9.8, w = 3 mm, h = 0.635 mm, = s ip hickness. Re e ences [I] K. A. Michal ky and D. Zheng, "Rigu ous analysis o open mic os ip lines o a bi a y c oss sec ion in hound and leaky egimes," IEEE T ans. Mic owa e Theo y Tech., ol. 37, pp. 2005-2010, Dec. 1989. [2] C-I.G.Hsu, R.F.Ha ing on, K.A.Michalski and D.Zheng "Analysis o mul iconduc o ansmission lines o a bi a y c oss sec ion in mul ilaye ed uniaxial media," IEEE T ans. Mic owa e Theo y Tech., ol. 41, pp. 10-18, Jan. 1993. [3] 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 . Eh. Eng., ol. 135, pp, 297-302, Oc . 1988. [4] J.Bc nal, EMedina, R.R.Boix and M.Ho no, "Fas ull wa e analysis o mul is ip ans- mission lines based on MPIE and complex images,'' IEEE T ans. Mic owa e Theo y Tech., ol. 48, pp.445452, Ma . 2000. [51 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 ansmission lines in laye ed media - aplica ion o mul islo in MCM-D echnology," IEEE T ans. Mic owa e Theo y Tech., ol. 47, pp. 1782-1787, Sep. 1999. [61 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. 2000. [71 R. Ma quCs, F. Mesa, "Spec al domain analysis o highe o de leaky modes in mic os ip lines: a new spec al-gap e ec ," Jou nal o Elec omagne ic Wa es and Applica ions, ol. II, pp. 1367-1384.1997. 583 Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on July 09,2020 a 14:11:05 UTC om IEEE Xplo e. Res ic ions apply.