Full-wave analysis of a wide class of microstrip resonators fabricated on magnetized ferrites with arbitrarily oriented bias magnetic field

1510 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 50, NO. 6, JUNE 2002 Full-Wa e Analysis o a Wide Class o Mic os ip Resona o s Fab ica ed on Magne ized Fe i es Wi h A bi a ily O ien ed Bias Magne ic Field Ge mán León, Ra ael R. Boix, Membe , IEEE, and F ancisco Medina, Senio Membe , IEEE Abs ac —A nume ical code has been de eloped o he ull-wa e de e mina ion o he esonan equencies and quali y ac o s o mic os ip pa ches wi h igh -angle co ne s o a bi a y shape in he case in which he subs a e o he pa ches is a mag- ne ized e i e wi h a bi a ily o ien ed bias magne ic ield. The code is based on he solu ion o an elec ic- ield in eg al equa ion by means o Gale kin’s me hod in he spec al domain. The e alua ion o he in ini e in eg als a ising om he applica ion o he nume ical me hod is e icien ly ca ied ou by means o a echnique based on he in e pola ion o he spec al dyadic G een’s unc ion. The nume ical esul s ob ained indica e ha mic os ip pa ches ab ica ed on e i e subs a es p esen cu o equency egions in which esonances canno occu owing o he exci a ion o magne os a ic modes. The limi s o hese cu o egions a e shown o be dependen on he o ien a ion and he magni ude o he bias magne ic ield, on he shape o he pa ches, and e en on he na u e o e e y pa icula esonan mode. The nume ical esul s also show ha he esonan equencies o mic os ip pa ches on magne ized e i es can always be uned o e a wide equency ange p o ided he o ien a ion o he bias magne ic ield is sui ably chosen. Index Te ms—Magne ic uning, magne os a ic modes, mic o- s ip esona o s, mic owa e e i es, spec al-domain app oach. I. INTRODUCTION RESONANT mic os ip pa ches can be used ei he as an- ennas o as componen s o oscilla o s and il e s in mi- c owa ein eg a edci cui s.Al hough hemos con en ionalmi- c os ip pa ches a e he ec angula and ci cula pa ches, o he geome ies such as he ec angula ing [1], he H-shaped pa ch [1], and he meande -shaped pa ch [2] ha e p o en o be use ul because o hei size educ ion capabili ies (e.g., in he design o an enna a ays, an ennas o pe sonal communica ion sys ems, ). This means ha he algo i hms de eloped o he analysis o esonan mic os ip pa ches should co e a spec um o ge- ome ies as wide as possible. Apa om hei shape, he na- u e o he subs a e o mic os ip pa ches is ano he in e es ing deg ee o eedom o he designe o ci cui s and an ennas. Al hough he mos commonly used subs a e ma e ials a e di- elec ics, magne ized e i es ha e p o en o ha e po en ial ap- plica ion as subs a es o mic os ip pa ches. Fo ins ance, se - Manusc ip ecei ed Janua y 23, 2001. This wo k was suppo ed by he Comisión In e minis e ial de Ciencia y Tecnología, Spain, unde P ojec TIC98-0630. The au ho s a e wi h he Mic owa e G oup, Depa men o Elec onics and Elec omagne ism, School o Physics, Uni e si y o Se ille, 41012 Se ille, Spain (e-mail: [email p o ec ed]). Publishe I em Iden i ie S 0018-9480(02)05206-7. e al esea che s ha e epo ed ha esonan mic os ip pa ches p in ed on e i e subs a es can be used in he ab ica ion o unable band ejec ion il e s [3] and unable bandpass il e s [4], [5]. Also, measu emen s ha e shown ha he ope a ing es- onan equency o mic os ip an ennas p in ed on e i e sub- s a es can be a ied o e a wide equency ange by adjus ing he bias magne ic ield [6]. Apa om ha , e i e subs a es can be used o educing he ada c oss sec ion o mic os ip an ennas unde ce ain condi ions [7]–[9] and o achie ing lin- ea ly, as well as ci cula ly pola ized mic os ip an ennas wi h a single eed[10]–[12]. Finally, i should bepoin ed ou ha when e i e ma e ials a e used as subs a es o mic os ip phased a - ays, wide-angle impedance ma ching can be ob ained by dy- namicallyadjus ing hebiasmagne ic ieldwi hscanangle [11], [13]. In his pape , he au ho s p esen an algo i hm o he de e - mina ion o he esonan equencies and quali y ac o s o mi- c os ip pa ches wi h igh -angle co ne s o a bi a y shape in he case in which he pa ches a e ab ica ed on magne ized e - i eswi ha bi a ilyo ien edbiasmagne ic ield.Thisalgo i hm is based on an e icien applica ion o he spec al-domain ap- p oach (SDA) [14], [15]. The s udy ca ied ou in his pape is an ambi ious gene aliza ion o ha published in p e ious pa- pe s [3], [16], [17] whe e he SDA was applied o he ull-wa e analysis o mic os ip esona o s o bo h ci cula shape [3], [17] and ec angula shape [16] ab ica ed on no mally biased e i e subs a es. The e is also an addi ional ela ed pape in which ec angula mic os ip esona o s on in-plane biased e i e sub- s a es we e analyzed by means o he ca i y model [18]. How- e e , he d awback o he ca i y model is ha i ails o explain how he esonances o mic os ip pa ches on e i e subs a es a e a ec ed by he exci a ion o magne os a ic modes along he subs a e[17]. Fo una ely, heSDA iscapableo accoun ing o he exci a ion o magne os a ic modes since he in o ma ion o hese modes is included in he spec al dyadic G een’s unc ion (in ac , he p opaga ion cons an s o he magne os a ic modes a e complex poles o he a o emen ioned spec al G een’s unc- ion [9], [17]). Conce ning he con en s o his pape , Sec ion II b ie ly desc ibes he applica ion o he SDA o he ull-wa e analysis o mic os ip esona o s ab ica ed on e i e subs a es. In his sec ion, a powe ul echnique is explained, which makes possible he as compu a ion o in ini e in eg als a ising om he applica ion o he SDA. In Sec ion III, nume ical esul s a e p esen ed o he esonan equencies and quali y ac o s o mic os ip pa ches o di e en shapes ( ec angula , H-, and 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 15:03:38 UTC om IEEE Xplo e. Res ic ions apply. LEÓN e al.: FULL-WAVE ANALYSIS OF WIDE CLASS OF MICROSTRIP RESONATORS 1511 (a) (b) Fig. 1. (a) Mic os ip pa ch wi h igh -angle co ne s o a bi a y shape on a e i e subs a e. (b) O ien a ion o he in e nal bias magne ic ield o he e i e subs a e wi h espec o he coo dina e axes shown in (a). meande shaped) p in ed on e i e subs a es. Special em- phasis is pu on showing ha he p opaga ion o magne os a ic modes along he e i e subs a es p e en s he pa ches om esona ing in ce ain cu o equency bands, and ha hese cu o equency bands a y as a unc ion o he magni ude and he o ien a ion o he bias magne ic ield. II. FORMULATION OF THE PROBLEM AND NUMERICAL PROCEDURE Fig. 1(a) shows a mic os ip pa ch wi h igh -angle co ne s o a bi a y shape p in ed on a e i e laye o pe mi i i y and hickness . Bo h he me allic pa ch and g ound plane a e assumed o be pe ec elec ic conduc o s (PECs) o negligible hickness, and bo h he e i e laye and he g ound plane a e assumed o be o in ini e ex en along he and coo dina es. I is also assumed ha he elec omagne ic ields exis ing in he pa ch egion show a ime dependence o he ype (whe e is complex o accoun o adia ion losses), which will be supp essed h oughou . Fig. 1(b) shows he o ien a ion o he dc-bias magne ic ield exis ing inside he e i e laye o Fig. 1(a) wi h espec o he coo dina e axes de ined in ha igu e. In gene al, he pe meabili y enso o his e i e laye can be w i en as (1) whe e he elemen s o he pe meabili y enso can be ob ained in e ms o he gy omagne ic a io C/kg, he sa u a ion magne iza ion o he e i e laye , he magni ude o hein e nal biasmagne ic ield , helinewid ho he e i e laye , and he angles and o Fig. 1(b), as shown in [19, eqs. (1)–(4)] and in [20, eqs. (10.37)–(10.40)]. Le be he me allic su ace occupied by he pa ch o Fig. 1(a), le be he su ace cu en densi y ex- is ing on he pa ch unde esonan condi ions, and le be a 2 2 ma ix, which s ands o he ans e se ( o ) dyadic G een’s unc ion [21] o he conduc o -backed e i e subs a e. The ans e se elec ic ield c ea ed by he cu en on he pa ch can be w i en in e ms o as (2) I we o ce ha he ans e se elec ic ield on he pa ch su ace is ze o, he ollowing elec ic- ield in eg al equa ion (EFIE) o is ob ained: (3) In o de o sol e he EFIE shown abo e, in his pape , he un- known ec o unc ion has been app oxima ed as a linea combina ion o known basis unc ions (4) whe e ha e been chosen o be subsec- ional oo op basis unc ions [22]. Wi h a iew o ob aining he unknown coe icien s , (4) has been in oduced in (3) and he Gale kin’s e sion o he me hod o momen s has been applied o he esul ing exp ession. The inal p oduc o hese ope a ions u ns ou o be a homogeneous sys em o linea equa ions o gi en by (5) whe e (6) No e ha he dependence o and on he angula e- quency has been explici ly shown in (5) and (6). The homo- geneous sys em o (5) o he coe icien s only has non i ial solu ions when (7) Equa ion (7) is an eigenequa ion o , om which he eso- nan equencies and quali y ac o s o he esonan modes o he mic os ip pa ch o Fig. 1(a) can be ob ained. In ac , le be he h complex oo o . In ha case, he quan i y s ands o he esonan equency o he h esonan mode o he pa ch and he quan i y s ands o he quali y ac o o ha esonan mode [14],[15].The oo so in hecomplex -planeha e been ob ained in his pape by means o Mulle ’s me hod. Al hough (2)–(7) seem o p o ide a s aigh o wa d way o hede e mina iono he esonan equencies andquali y ac o s Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on July 07,2020 a 15:03:38 UTC om IEEE Xplo e. Res ic ions apply. 1512 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 50, NO. 6, JUNE 2002 o he mic os ip pa ch o Fig. 1(a), he calcula ion o he quan- i ies ia (6) poses p oblems because he nume ical compu a ion o he ans e se dyadic G een’s unc- ion o e e ypai o alues o and is a ask ha equi es e y high CPU imes. In o de o o e come his incon enien , he au ho s o his pape ha e compu ed ia an al e na i e exp es- sion in he Fou ie ans o m domain since he de e mina ion o in ha domain can be ca ied ou in a as , accu a e, and s ableway o heconduc o -backed e i e subs a eo Fig.1(a) by using he ideas desc ibed in [21]. The al e na i e exp ession o he compu a ion o in he Fou ie ans o m domain is ob ained by applying Pa se al’s heo em o (6) and is gi en by (8) whe e s and o he wo-dimensional Fou ie ans o ms (2D-FTs) o wi h espec o and , and s ands o he 2D-FT o (see [21, eq. (11)]). As explained abo e, he compu a ional expense equi ed o he de e mina ion o ia (8) u ns ou o be much smalle han ha equi ed when (6) is used. In spi e o his ac , he CPU ime in ol ed in he b u e- o ce nume ical compu a ion o ia (8) is s ill oo high because he in eg ands o he double in ini e in eg als show an oscilla o y slowly decaying beha io as he spec al a iables and g ow. In o de o sa e CPU ime, he au ho s o his pape ha e applied a echnique o accele a ing he nume ical compu a ion o he a o emen ioned double in eg als. The i s s ep o his echnique is o exp ess (8) in e ms o pola spec al a iables and and )as ollows: (9) whe e o e e y alue o is an in eg a ion pa h in he com- plex -plane loca ed abo e he complex poles and he com- plex b anch poin o (see [23, Fig. 2]). The ans o ma ion om (8) o (9) is e y ad an a- geous because i makes i possible o educe om wo o one he numbe o in ini e in eg als in ol ed in he nume ical compu a- ion o each quan i y [24]. The second s ep o he echnique o he as compu a ion o he in eg als o (8) is based on he in e pola ion o he asymp o ic beha io o he spec al dyadic G een’s unc ion o la ge in e ms o Chebyshe polynomials o he a iable [21]. In ac , i has been ound ha o p ac ical alues o and can be app oxima ely ex- p essed in he in e al by means o he in e pola ing exp ession (10) In (10), s and o Cheby- she polynomials o he i s kind, and he unknown ma ices mus be ob ained o e e y pai o ixed alues o and by making he ma ix unc ions and exac ly coincide a he ze os o in o de o minimize in e pola ion e o s (see [21] o mo e de ails). Nume ical expe imen s ha e shown ha he use o in (10) su ices o make and co- incide a leas wi hin six signi ican igu es in he whole in e al when . Howe e , a alue o equal o i e is necessa y o eaching he same p ecision in he in e pola ion o ia (10) when (in ac , i has been checked ha , in his la e ange o alues o , he choice in (10) may lead o an inaccu a e in e pola ion o ). Bea ing in mind his ac , in his pape , he exp ession o (10) has always been used wi h a alue o equal o i e as a way o ensu e ha he men ioned exp ession p o ides an accu a e in e pola ion o in he whole in e al o any alue o (no e ha his choice di e s om ha ollowed in [23], whe e a alue o equal o h ee was ound o be enough in a simila in e pola ing exp ession). Once he ma ix unc ion has been sui ably chosen o p o iding an accu a e app oxima ion o in he in e al o any alue o , his ac can be used o ew i ing (10) as (11) The implemen a ion o (11) is he inal and c ucial s ep o he echnique o he as compu a ion o in he spec al domain. I has been ound ha he ini e in eg als wi h espec o o (11) can be wo ked ou o a high p ecision by means o a single 160-poin Gauss–Legend e quad a u e o - Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on July 07,2020 a 15:03:38 UTC om IEEE Xplo e. Res ic ions apply. LEÓN e al.: FULL-WAVE ANALYSIS OF WIDE CLASS OF MICROSTRIP RESONATORS 1513 (a) (b) (c) Fig. 2. Top iews o mic os ip pa ches o di e en shapes. (a) Rec angula shape. (b) H shape. (c) Meande shape. The coo dina e axescoincide wi h hose shown in Fig. 1. mula,which means ha hecompu a ion o eachquan i y e- duces o he compu a ion o 160 in eg als wi h espec o .In (11),eacho hesein eg alswi h espec o isspli in oa ini e in eg al and one in ini e in eg al. Whe eas he ini e in eg als in- ol ea ela i elyna ow in e al andcan be nume icallyca ied ou wi hin sho CPU imes, he in ini e in eg als can be com- pu ed wi hin e en sho e CPU imes o he pa icula choice o basis unc ions made in his pape by using he o mulas sup- plied in he Appendix. As an es ima e o he ad an ages o he use o (11), a compa ison has been made be ween he o al CPU ime needed o he compu a ion o (see (7)) when using (11) and ha needed when using b u e- o ce nume ical in eg a ion ia (9). Nume ical expe imen s ha e shown ha o he s uc u es analyzed in Sec ion III, he CPU imes in ol ed in he compu a ion o by means o (11) a e on a e age woo de s o magni udesmalle han hoseob ainedwhenusing (9) when an accu acy o h ee signi ican igu es in he alue o is equi ed. III. NUMERICAL RESULTS Fig. 2(a)–(c) shows he op iews o he h ee ypes o mic os ip pa ches wi h igh -angle co ne s p in ed on e i e subs a es ( ec angula -, H-, and meande -shaped pa ches) o which esul s a e p esen ed in his sec ion. Addi ional esul s o mic os ip pa ches wi h igh -angle co ne s o any o he shape can be easily gene a ed by means o he nume ical algo i hm implemen ed by he au ho s. In Table I(a) and (b), he con e gence o he nume ical me hod desc ibed in he p e ious sec ion is checked wi h espec o he numbe o oo op basis unc ions used in he app oxima ion o he cu en densi y. In pa icula , esul s a e p esen ed o he i s i e esonan modes o a ec angula mic os ip pa ch p in ed on bo h a no mally biased e i e subs a e and an in-plane biased e i e subs a e. No e ha whe eas he di e ences be ween he esul s ob ained o he esonan equencies when and hose ob ained when a e wi hin 5%, he di e ences be ween he esul s ob ained o he esonan equencies when and hose TABLE I CONVERGENCE PATTERN OF THE COMPLEX RESONANT FREQUENCIES OF THE FIRST FIVE RESONANT MODES OF A RECTANGULAR PATCH ON (a) NORMALLY AND (b) IN-PLANE BIASED FERRITE SUBSTRATES WITH RESPECT TO THE NUMBER OF BASIS FUNCTIONS USED IN THE APPROXIMATION OF THE CURRENT DENSITY ( w = 4 mm, l =5 : 5 mm, h =0 : 6 mm,  =12 : 8 ; H =0 : 036 T,  M =0 : 178 T,  1 H =0 T). THE MODES ARE NAMED AFTER THE NOTATION USED IN THE CAVITY MODEL (a) (b) ob ained when a e always wi hin 0.8%. This seems o indica e ha oughly 100 oo op basis unc ions should su ice o p o ide eliable esul s o he esonan equencies o he i s esonan modes o a mic os ip pa ch on a e i e subs a e. Table I(a) and (b) also shows ha , o a gi en alue o , he highe he o de o a esonan mode, he wo se he accu acy o he esul ob ained o he esonan equency, which is a ibu ed o he ac ha he cu en densi ies o he esonan modes become mo e oscilla ing and di icul o app oxima e as he o de o he modes inc eases. Conce ning he quali y ac o s, hey a e ob ained wi h less accu acy han he esonan equencies because he imagina y pa s o he esul s ob ained o he complex esonan equencies in Table I(a) and (b) a e usually wo o h ee o de s o magni ude smalle han he eal pa s and, he e o e, hese imagina y pa s a e compu ed wi h less p ecision han he eal pa s. In pa icula , he disc epancies be ween he esul s de i ed om Table I(a) and (b) o he quali y ac o s when and hose de i ed when lie wi hin 4%. Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on July 07,2020 a 15:03:38 UTC om IEEE Xplo e. Res ic ions apply. 1514 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 50, NO. 6, JUNE 2002 TABLE II RESONANT FREQUENCIES AND QUALITY FACTORS OF THE FIRST FIVE RESONANT MODES OF A RECTANGULAR PATCH ON (a) NORMALLY AND (b) IN-PLANE BIASED FERRITE SUBSTRATES ( w = 4 mm, l =5 : 5 mm, h =0 : 6 mm,  =12 : 8 ; H =0 : 036 T,  M =0 : 178 T,  1 H =0 T). THE MODES ARE NAMED AFTER THE NOTATION USED IN THE CAVITY MODEL.OUR RESULTS FOR THE RESONANT FREQUENCIES ARE COMPARED WITH THE NUMERICAL RESULTS OBTAINED IN [25] FOR THE RCS RESONANT PEAKS OF THE PATCH (a) (b) In Table II(a) and (b), he au ho s es he alidi y o he nu- me ical me hod desc ibed in Sec ion II by compa ing hei nu- me ical esul s o he esonan equencies o he mic os ip pa ch analyzed in Table I(a) and (b) [ has been used in Table II(a) and (b)] wi h nume ical esul s published in [25] o he RCS esonan peaks o he same pa ch. I can be e i ied ha he di e ences be ween he wo se s o esul s o he es- onan equencies a e always wi hin 1.5%. Table II(a) and (b) also shows he nume ical esul s ob ained o he quali y ac o s o he esonan modes analyzed. In Fig. 3, esul s a e p esen ed o he esonan equencies o he i s h ee esonan modes o a ec angula mic os ip pa ch p in ed on a no mally biased e i e as a unc ion o he magni- ude o he bias magne ic ield. This igu e is analogous o ha p esen edin[17] o he esonan equencieso he i s i e es- onan modes o a ci cula mic os ip pa ch on a no mally biased e i e. As in [17], in Fig. 3, he e is a cu o equency egion ( Fig. 3. Resonan equencies o he i s h ee esonan modes o a ec angula mic os ip pa ch on a no mally biased e i e subs a e ( w =5 mm, l = 6 : 5 mm, h =1 : 27 mm,  =15 ; M =0 : 178 T,  1 H =0 : 001 T,  =  =0 ). The modes a e named a e he no a ion used in he ca i y model.Thes iped equencybands ands o hep opaga ion egiono FMSVW modes. Fig. 4. Resonan equencies o he i s i e esonan modes o a ec angula mic os ip pa ch on an in-plane biased e i e subs a e ( w =5 mm, l = 6 : 5 mm, h =1 : 27 mm,  =15 ; M =0 : 178 T,  1 H =0 : 001 T,  =  =90 ). The modes a e named a e he no a ion used in he ca i y model. The s iped equency bands s and o he p opaga ion egions o BMSVW and MSSW modes. and ) in which esonances canno occu because o he exci a ion in ha equency egion o an in ini e numbe o o wa dmagne os a ic olume-wa e(FMSVW) modesalong he e i e subs a e in all he di ec ions o he plane - o Fig. 2(a) [26]. Also, he esonan equencies o all esonan modes appea abo e and below he cu o equency egion as i happens in [17]. Fig. 3 shows ha he esonan equencies o he h ee esonan modes analyzed can be uned o e a wide e- quency ange by means o he bias magne ic ield as p edic ed in [6]. In pa icula , he esonan equency o he undamen al mode changes mo e han 50% when is a ied om 0 o 0.4 T bo h below he cu o equency egion and abo e ha equency egion. In Fig. 4, he au ho s plo he esonan equencies o he i s i e esonan modes o a ec angula mic os ip pa ch p in ed on an in-plane biased e i e e sus he magni ude o he bias magne ic ield. No e ha , in his case, some esul s o he eso- nan equencies ha e been ob ained wi hin he p opaga ion e- gions o magne os a ic modes. Since he p opaga ion o mag- ne os a ic modes in ol es he appea ance o unbounded poles in he spec al G een’s unc ion Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on July 07,2020 a 15:03:38 UTC om IEEE Xplo e. Res ic ions apply. LEÓN e al.: FULL-WAVE ANALYSIS OF WIDE CLASS OF MICROSTRIP RESONATORS 1515 o (9), e i e losses ha e been included so as o keep hese poles below he in eg a ion pa h o (9) o e e y alue o [9]. I can be no iced ha whe eas he esonan e- quencies o he modes and analyzed in Fig. 4 can be uned by means o he bias magne ic ield, as i happens in Fig. 3, he o he wo eso- nan modes and a e p ac ically una ec ed by he a ia ions in he magni ude o he bias mag- ne ic ield. The explana ion o his la e beha io is ha he in e ac ion be ween he dc-bias magne ic ield and he ac mag- ne ic ield exis ing a ound he pa ch o Fig. 4 is e y weak in he esonan modes and because he ac magne ic ield o hese wo modes is mainly pa allel o he dc magne ic ield [di ec ed along he axis o Fig. 2(a)] [27]. Howe e , his does no happen in he case o he modes and analyzed in Fig. 4 and in he case o he modes analyzed in Fig. 3. In ac , maximum in e ac ion be ween he dc and ac magne ic ields is expec ed o he modes and o Fig. 4 and he modes o Fig. 3 because he ac magne ic ield o all hese modes is mainly pe pendicula o he dc-bias mag- ne ic ield [27]. Fig. 4 also shows ha he esonan equencies o he esonan modes and oc- cupy pa o he egion o p opaga ion o backwa d magne o- s a ic olume-wa e (BMSVW) modes , bu do no occupy he egion o p opaga ion o magne os a ic su ace wa e (MSSW) modes [26]. This is because he cu en s o hese wo esonan modes a e mainly di- ec ed along he axis o Fig. 2(a)—which is he di ec ion o he bias magne ic ield—and, he e o e, hese modes may ex- ci e MSSW modes p opaga ing in a di ec ion pe pendicula o ha o he cu en s (as i happens wi h he MSSW ansduce s desc ibed in [28] and [29]), bu canno exci e BMSVW modes p opaga ing in he di ec ion o he cu en s. The opposi e holds o he esonan modes and whose esonan equencies occupy he egion o p opaga ion o MSSW modes, bu do no occupy he egion o p opaga ion o BMSVW modes. In his la e case, he cu en s o he wo es- onan modes a e mainly di ec ed along he axis o Fig. 2(a) (i.e., pe pendicula o he di ec ion o he bias magne ic ield) and, he e o e, he esonan modes may exci e BMSVW modes p opaga ing in a di ec ion pe pendicula o ha o he cu en s (as i happens wi h he magne os a ic backwa d olume wa e ansduce desc ibed in [30]), bu canno exci e MSSW modes p opaga ing in he di ec ion o he cu en s. Finally, he cu en o he esonan mode has bo h - and -com- ponen s and, he e o e, his esonan mode may exci e magne- os a ic modes p opaga ing in a di ec ion making an angle be- ween 0 and 90 wi h he di ec ion o he bias magne ic ield. These magne os a ic modes may be BMSVW and/o MSSW modes depending on he di ec ion o p opaga ion and he mag- ni ude o he bias magne ic ield [26], which jus i ies ha he esonan equencies o he mode may occupy he egion o p opaga ion o BMSVW modes and/o he egion o p opaga ion o MSSW modes. In Fig. 5, ou nume ical esul s o he esonan equencies o some o he esonan modes o a squa e mic os ip pa ch on anin-plane biased e i ea e compa edwi hmeasu emen s pub- Fig. 5. Resonan equencies o h ee o he esonan modes o a squa e mic os ip pa ch on an in-plane biased e i e subs a e ( w = l = 9 : 525 mm, h =0 : 508 mm,  =11 : 41 ; M =0 : 0985 T,  1 H =0 : 001 T,  =  =90 ). The modes a e named a e he no a ion used in he ca i y model. The s iped equency bands s and o he p opaga ion egions o BMSVW and MSSW modes. Ou esul s (solid, dashed, and do ed lines) a e compa ed wi h hose ob ained in [18] ( 2 ; + and  ). lished in [18] wi h a iew o alida ing he gene ic beha io o pa o he esul s plo ed in Fig. 4. The ag eemen be ween nu- me ical and expe imen al esul s in Fig. 5 is good. The di e - encesa eona e agewi hin2% o hemodes and , and wi hin 4% o he mode . No e ha he expe imen al esonan equencies o he h ee esonan modes analyzed in Fig. 5 occupy he egion o p opaga ion o BMSVW modes, bu do no occupy he egion o p opaga ion o MSSW modes, which is cohe en wi h he heo e ical p edic ion p o ided by he nume ical esul s o Fig. 4. Also, no e ha he expe imen al esonan equencies o he h ee esonan modes appea abo e and below he egion o p opaga ion o MSSW modes, which is again in ag eemen wi h he beha io appea ing in Fig. 4. I should be poin ed ou ha , al hough he esonan equencies o o he esonan modes we e nume ically ound in be ween he esonan equencies plo ed in Fig. 5, hese o he esonan equencies we e no expe imen ally de ec ed in [18], which is a ibu ed o he ype o exci a ion mechanism used in he mea- su emen s o his la e pape . In Fig. 6, esul s a e p esen ed o he esonan equencies o he i s h ee esonan modes o a ec angula mic os ip pa ch p in ed on a e i e subs a e wi h bias magne ic ield o ien ed in a di ec ion which makes an angle o 45 wi h he no mal o he subs a e. I can be ecognized ha among he h ee eso- nan modes, he mode is he mos sensible o a ia ions in he magni ude o he bias magne ic ield since his is he only mode o which he ac magne ic ield is mainly pe - pendicula o he dc magne ic ield. Fo he o ien a ion o he bias magne ic ield conside ed in Fig. 6, he e i e subs a e suppo s he p opaga ion o BMSVW modes in he equency egion , he p opaga- ion o FMSVW modes in he equency egion , and he p opaga ion o MSSW modes in he equency egion (whe e ). No e ha he esonan equen- cies o he h ee esonan modes analyzed in Fig. 6 occupy pa o he egions o p opaga ion o BMSVW modes and MSSW modes, jus as i happens in Fig. 4. Howe e , esonances ne e occu in he egion o p opaga ion o FMSVW modes, which is in ag eemen wi h he esul s ob ained in Fig. 3. Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on July 07,2020 a 15:03:38 UTC om IEEE Xplo e. Res ic ions apply. 1516 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 50, NO. 6, JUNE 2002 Fig. 6. Resonan equencies o he i s h ee esonan modes o a ec angula mic os ip pa ch on a e i e subs a e wi h inclined bias magne ic ield ( w = 5 mm, l =6 : 5 mm, h =1 : 27 mm,  =15 ; M =0 : 178 T,  1 H = 0 : 001 T,  =45 ; =90 ). The modes a e named a e he no a ion used in he ca i y model. The s iped equency bands s and o he p opaga ion egions BMSVW, FMSVW, and MSSW modes. (a) (b) Fig. 7. (a) Resonan equencies and (b) quali y ac o s o he undamen al esonan mode o h ee mic os ip pa ches ( ec angula , H-, and meande shaped) on a no mally biased e i e subs a e ( w =2 : 5 mm, l =3 : 25 mm, h =0 : 635 mm; e i e subs a e:  =15 ; M =0 : 068 T,  1 H =0 : 001 T,  =  =0 ; H-shaped pa ch: l =(9 = 11) l ; w =(4 = 9) w ; meande -shaped pa ch: l =(1 = 11) l;w =(1 = 9) w ). In Fig. 7(a), he s iped equency band s ands o he p opaga ion egion o FMSVW modes and, in Fig. 7(b), AMMR s ands o esonances abo e he egion o FMSVW modes and BMMR s ands o esonances below he egion o FMSVW modes. In Fig. 7(a) and (b), esul s a e plo ed o he esonan e- quencies and quali y ac o s o he undamen al esonan mode o ec angula mic os ip, an H-shaped, and a meande -shaped pa ches. The pa ches a e assumed o be ab ica ed on he same no mally biased e i e subs a e occupying he same o e all Fig. 8. Resonan equencies o he undamen al esonan mode o h ee mic os ip pa ches ( ec angula , H-, and meande shaped) on an in-plane biased e i e subs a e ( w =2 : 5 mm, l =3 : 25 mm, h =0 : 635 mm; e i e subs a e:  =15 ; M =0 : 068 T,  1 H =0 : 001 T,  =  =90 ; H-shaped pa ch: l =(9 = 11) l;w =(4 = 9) w ; meande -shaped pa ch: l =(1 = 11) l;w =(1 = 9) w ). The s iped equency bands s and o he p opaga ion egions o BMSVW and MSSW modes. subs a e size (gi en by ). No e ha he esonan equen- cies o he meande -shaped pa ch a e always smalle han hose o he H-shaped and ec angula pa ches, which indica es ha he meande -shaped pa ch shows he bes pe o mance o size- educ ion applica ions. Howe e , he quali y ac o s o he me- ande -shapedpa cha eusuallyla ge han hoseo heH-shaped pa ch and he ec angula pa ch and, he e o e, he bandwid hs o he meande -shaped pa ch a e usually smalle han hose o heH-shapedand ec angula pa ches, whichisaclea disad an- age when he pa ches a e used as an ennas. Fig. 7(a) shows ha he esonan equencies o he h ee pa ches analyzed can be uned o e a wide equency ange by a ying he magni ude o he bias magne ic ield. The meande - and he H-shaped pa ches u n ou o be mo e unable han he ec angula pa ch abo e he egion o p opaga ion o FMSVW modes (in ac , he esonan equency o he undamen al esonan mode o he meande - and H-shaped pa ches changes mo e han 70% when is a ied om 0 o 0.5 T), bu he ec angula pa ch is mo e un- able han he o he wo pa ches below he egion o p opaga ion o FMSVW modes (in his la e case, he esonan equency o he undamen al esonan mode o he ec angula pa chchanges oughly 50% when is a ied om 0 o 0.5 T). In Figs. 8 and 9, he au ho s plo he esonan equencies o he undamen al esonan mode o he h ee pa ches analyzed in Fig. 7(a) and (b) in he cases in which he e i e subs a e is in-plane biased along he and axes shown in Figs. 2(a)–(c). In Fig. 8, he e i e subs a e is biased along he axis and, in his igu e, he undamen al esonan mode o he meande - shaped pa ch is mo e a ec ed by a ia ions in he magni ude o he bias magne ic ield han he esonan modes o he H-shaped and ec angula pa ches. The explana ion o his beha io is ha , whe eas he cu en o he esonan mode o he meande - shaped pa ch is mainly di ec ed along he axis, he cu en s o bo h he H-shaped and he ec angula pa ches a e mainly di ec ed along he axis and, he e o e, whe eas he ac mag- ne ic ield o he esonan mode o he meande -shaped pa ch is mainly pe pendicula o he dc-bias magne ic ield, he ac mag- ne ic ields o he esonan modes o he H-shaped and ec - angula pa ches a e mainly pa allel o he dc magne ic ield. Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on July 07,2020 a 15:03:38 UTC om IEEE Xplo e. Res ic ions apply. LEÓN e al.: FULL-WAVE ANALYSIS OF WIDE CLASS OF MICROSTRIP RESONATORS 1517 Fig. 9. Resonan equencies o he undamen al esonan mode o h ee mic os ip pa ches ( ec angula , H-, and meande shaped) on an in-plane biased e i e subs a e ( w = 2 : 5 mm, l =3 : 25 mm, h =0 : 635 mm; e i e subs a e:  =15 ; M =0 : 068 T,  1 H =0 : 001 T,  =90 ;  =0 ; H-shaped pa ch: l =(9 = 11) l;w =(4 = 9) w ; meande -shaped pa ch: l =(1 = 11) l;w =(1 = 9) w ). The s iped equency bands s and o he p opaga ion egions o BMSVW and MSSW modes. The opposi e holds o Fig. 9. In his igu e, he e i e sub- s a e is biased along he axis and, in his case, whe eas he ac magne ic ield o he meande -shaped pa ch is mainly pa allel o he dc magne ic ield, he ac magne ic ield o he H-shaped and ec angula pa ches a e mainly pe pendicula o he dc magne ic ield, which jus i ies ha he esonan modes o he H-shaped and ec angula pa ches a e much mo e unable by he magni ude o he bias magne ic ield han he esonan mode o he meande -shaped pa ch. As shown in he commen s o Fig. 4, he pene a ion o he esonan equencies o he h ee esonan modes analyzed in Figs. 8 and 9 inside he egions o p opaga ion o magne os a ic modes can be explained in e ms o he di ec ion o he cu en s o hose h ee esonan modes. In ac , whe eas in Fig. 8 he esonan equencies o he me- ande -shaped pa ch mainly occupy he egion o p opaga ion o BMSVW modes (which canno be exci ed by cu en s pa allel o he bias magne ic ield) and he esonan equencies o he H-shaped and ec angula pa ches mainly occupy he egion o p opaga ion o MSSW modes (which canno be exci ed by cu - en s pe pendicula o he bias magne ic ield), in Fig. 9, he es- onan equencies o he meande -shaped pa ch mainly occupy he egion o p opaga ion o MSSW modes, and he esonan equencies o he H-shaped and ec angula pa ches mainly oc- cupy he egion o p opaga ion o BMSVW modes. IV. CONCLUSION Gale kin’s me hod in he spec al domain has been used o he de e mina ion o he esonan equencies and quali y ac- o so he esonan modeso mic os ippa cheswi h igh -angle co ne s o a bi a y shape in he case in which he subs a e o he pa ches is a magne ized e i e wi h a bi a ily o ien ed bias magne ic ield. A special echnique has been de eloped o ac- cele a ing he nume ical compu a ion o he double in ini e in- eg als a ising om he applica ion o Gale kin’s me hod in he spec aldomain.The esul sob ained show ha he esonan e- quencies o a mic os ip pa ch p in ed on a e i e subs a e can always be uned by a ying he magni ude o he bias magne ic ield. Maximum unabili y is achie ed when he dc-bias mag- ne ic ield is mainly pe pendicula o he ac magne ic ield ex- is inga ound he esonan pa ch,andminimum unabili y esul s when he dc magne ic ield is mainly pa allel o he ac magne ic ield. The au ho s ha e ound ha esonances a e no possible inside he equency egion o p opaga ion o FMSVW modes along he e i e subs a e. Howe e , esonances may occu in- side he equency egions o p opaga ion o BMSVW modes and MSSW modes p o ided hese magne os a ic modes a e no exci ed by he cu en s o he esonan modes o he pa ch. Fi- nally, he meande - and he H-shaped pa ches ha e p o en o ha e size educ ion capabili ies wi h espec o he ec angula pa ch a he expense o showing smalle bandwid h. APPENDIX When subsec ional oo op basis unc ions a e used in (4) o app oxima ing he cu en densi y on he mic os ip pa ch o Fig. 1(a) and a alue o equal o i e is used in (10) in he in e pola ing exp ession o he asymp o ic spec al dyadic G een’s unc ion, a e some cumbe some manipula ions, i can be shown ha he in ini e in eg als wi h espec o appea ing in (11) can all be exp essed as linea combina ions o in eg als o he ype (12) (13) whe e he alues o he a iable , which a e equi ed o he e alua ion o e e y in ini e in eg al o (11), a e s ongly depen- den on he alues o and in ol ed in ha in eg al. In p inciple, he in eg als o (12) and (13) can be ob ained by means o he o wa d ecu ence exp essions [23] (14) (15) which a e ini ialized by means o he equa ions Si (16) Ci (17) whe e Si and Ci a e sine and cosine in eg als, which can be ob ained wi h easonable accu acy by means o he sub ou- ine “CISIA” published in [31]. The p oblem a ising om he ecu ence exp essions o (14) and (15) is ha hey a e uns able and lose accu acy as in- c eases. Also, he sub ou ine CISIA used o he calcula ion o Si and Ci has been ound o lose some accu acy as in- c eases. As a consequence o he combined e ec o hese wo la e ac s, nume ical expe imen s ha e shown ha (14)–(17) u n ou o be sligh ly inaccu a e o he calcula ion o he en in eg als o (12) and (13) when ( he la ge he alue o , he la ge he inaccu acies). Also, he au ho s ha e ound ha hese inaccu acies may be c i ical when he esul s p o ided Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on July 07,2020 a 15:03:38 UTC om IEEE Xplo e. Res ic ions apply. 1518 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 50, NO. 6, JUNE 2002 by (14)–(17) a e used in he calcula ion o he in ini e in eg als o (11) in hose cases in which . The e o e, he exp essions (14)–(17) should no be used o he compu a ion o he in ini e in eg als o (11) when . The ollowing exp essions should be used ins ead: (18) (19) Equa ions (18) and (19) ha e been ob ained by means o he Cauchy–Riemann esidue heo em. Nume ical expe imen s ha e shown ha he in ini e in eg als o (18) and (19) can be compu ed by means o a en-poin Gauss–Lague e quad a u e wi h high accu acy independen ly o he alue o , which indi- ca es ha (18) and (19) basically lead o accu a e closed- o m exp essions o he compu a ion o (12) and (13) when . Al hough he echnique o he compu a ion o (12) and (13) based on he use o (18) and (19) is eliable when , he au ho s ha e ound an al e na i e echnique o he compu a ion o (12) and (13), which is e en mo e accu a e and as e han he o me echnique when . This al e na i e echnique is based on he backwa d ecu ence exp essions (20) (21) These ecu ence exp essions a e ini ialized by he unca ed asymp o ic expansions o and o when , which a e gi en by (22) (23) whe e (24) (25) As s a ed abo e, he use o (20)–(25) wi h has been ound o p o ide a echnique o he compu a ion o (12) and (13), which is p e e ed o (18) and (19) when . To sum up, we should say ha he mos sui able echnique o he compu a ion o he en in eg als o (12) and (13) is ha based on (14)–(17) in he in e al , ha based on he exp essions (18) and (19) in he in e al , and inally, ha based on (20)–(25) (wi h ) in he in e al . REFERENCES [1] V. Palanisamy and R. 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Wil on, “Simple and e icien nume ical me hods o p oblems o elec omagne ic adia ion and sca e ing om sou ces,” IEEE T ans. An ennas P opaga ., ol. AP-28, pp. 593–603, Sep . 1980. [23] V. Losada, R. R. Boix, and M. Ho no, “Resonan modes o ci cula mic os ip pa ches in mul ilaye ed subs a es,” IEEE T ans. Mic owa e Theo y Tech., ol. 47, pp. 488–498, Ap . 1999. Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on July 07,2020 a 15:03:38 UTC om IEEE Xplo e. Res ic ions apply.