Accurate Analysis of Dielectric Backed Planar Conducting Layers of Arbitrarily Shaped in a Rectangular Waveguide

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1 Accurte Anss of Deectrc Bcked Pnr Conductng Lers of Artrr Sped n Rectngur Wvegude Downoded from eee.ust.c.r t 5:59 IRST on ond Octoer st 8. ornned Foumn* nd. K-Amrossen* Astrct: Te crcterstcs of deectrc cked pnr conductng ers of rtrr sped n rectngur wvegude re ccuted mens of couped ntegr equton tecnque (CIET wc ccurte tkes ger order mode nterctons. Spectr ddc green s functons re derved for tese equvent structures. A couped mgnetc fed ntegr equton formuton s proposed wc s soved usng metod of moment (o. Ten one mtr equton tt nvoves te mgnetc currents of non metc prts (perture prts of dontnutes s derved. Ts snge mtr formuton repces te procedure of cdng ndvdu S s of ec ock. B dvdng te re of cross secton n dontnutes nto gven numer of susectons, t s posse to mode n spe of metc prts. Te proposed tecnque permts modeng of vret of structures suc s cvt-cked mcro strp ntenn, frequenc seectve surfces (FSS s, wvegude fters wt prnted rses nd gener mutered med wt prnted crcuts emedded etween deectrc ers n wvegude. Te usefuness of te proposed metod nd ts performnce re verfed ccutng nd smutng of gven structure. Kewords: Deectrc Bcked Pnr Conductng Lers, Rectngur Wvegude, Couped Integr Equton Tecnque, etod of oment. Introducton n wvegude sed mcrowve nd mmeter wve sstems re constructed of deectrc cked pnr conductng ers of rtrr sped wc s octed trnsverse n wvegude. A wde rnge of prctc structure suc s wvegude-sed fters, perodc grd structures n wvegudes, wvegude-sed strp nd mcro strp fters, ptc rrs, dense pckged pssve eements nd wvegude-sed spt power comners re mong te structures tt cn e ctegored s pnr conductve ers n guded wve envronment. A of mentoned structures consst of numer of ptterned deectrc cked conductng ers seprted wvegude sectons. One w to eectromgnetc crctere tese structures s usng of muter metod of moments (o tecnque wc resuts n ver rge mtr proem. owever genered tterng mtr (S pproc s good souton. In te S metod ec ock Irnn Journ of Eectrc & Eectronc Engneerng,. Pper frst receved 5 Apr nd n revsed form Oct.. * Te Autors re wt te Deprtment of Eectrc Engneerng, Irn Unverst of Scence nd Tecnoog, Tern 6846, Irn. E-ms: gornned@gun.c.r nd k@ust.c.r represented mtr tt retes te coeffcents of forwrd nd ckwrd propgtng wvegude modes t te two sdes of ec ock. Tese mtrces re cded to otn over response of woe structure wc s cded of numer of ocks [-]. In S pproc propgtng nd evneent TE nd T modes re e consdered tt provde n ccurte smuton of nterctons etween dcent ocks. A proem occurs wen te seprton etween tese dcent ocks ecomes eectrc sm. Ts necessttes sgnfcnt ncrese n te numer of wvegude modes wc n turn resuts n te se of S nd susequent ncreses te memor nd computton requrements. B now muc effort ve een done to reduce te se of mcrowve devces, suc s compct ndpss fter utng deectrc fed wvegudes [-4] nd ndpss fters usng frequenc seectve surfces [5-6]. In of teses mntured structures seprton etween negorng deectrc cked conductng ers re sm compred to engt wve. Te souton s to mode te woe cose spced ers s snge ock rter tn s te cde of numer of ocks. In [7] couped ntegr equton tecnque ve een pped for te nss of -pne Irnn Journ of Eectrc & Eectronc Engneerng, Vo. 8, No., rc

2 Downoded from eee.ust.c.r t 5:59 IRST on ond Octoer st 8 wve gude fters n wc te ntercton etween dontnutes re ccurte dered. In ts pper fu wve ntegr equton tecnque s deveoped for eectromgnetc modeng of structures n wc deectrc cked pnr conductng ers of rtrr sped re octed trnsverse n wvegude. A couped set of mgnetc ntegr equtons ve een pped for unknown mgnetc current denstes t te nonmet prts (perture prts of te nterfces. Tese equtons s soved metod of moment nd usng pecewse trngur or pecewse snusod overppng ss nd testng functons for te mgnetc current denstes. Couped Integr Equton Tecnque Formuton Consder rectngur wvegude sown n Fg. wt tree deectrc cked pnr conductng ers. Artrr sped metton S m, S nd S m m re octed on te nterfces, nd respectve. Retve permttvt of deectrc ers re ε, r ε nd r ε r wt tckness d,d nd d respectve. Te oss of conductors nd deectrcs s not een consdered for te smpct. Accordng to equvence teorem te eectrc nd mgnetc fed n te regon < s te tot of ncdent eectrc nd mgnetc fed (drect from n mpressed current source nd refected from te nterfce wen repced perfect conductng pne nd ttered (refected fed generted n nduced mgnetc current on te non metton prts (perture prts of nterfce. Fed n te regons < < nd < < s on due to nduced mgnetc current on te perture prts of nterfces, nd nterfces, respectve. In te sme w te eectrc nd mgnetc fed n te regon > s due to nduced mgnetc current on te perture prts of nterfce on, ut te contnut of te mgnetc feds cross te nterfces provdes te ntercton of regons n necessttes te formuton of te proem n terms of couped mgnetc ntegr equtons. Fg. sows te equvent structure for otnng feds n vres regons. Aperture prts of nterfce (octed t te = pne s repced perfect conductng pne (sorted perture, wt te orgn tngent eectrc fed t te perture prts restored t = nd = pproprte mgnetc surfce currents nd ( n te reverse drecton, respectve. Aperture prts of nterfce octed t te = pne s repced perfect conductng pne (sorted perture, wt te orgn tngent eectrc fed t te perture prts restored t = nd = pproprte mgnetc surfce currents nd, respectve. ε r ( d r d d r d r d ε r r ε r ( Fg. eometr of rectngur wvegude consstng deectrc cked pnr conductng ers of rtrr sped. ( Tree dmenson vew. ( Sde vew wt equvent mgnetc current. ( d r d r ε r ε r ( d ε r d (c (d Fg. Equvent structures for otnng feds of structure sown n Fg.. ( vd n te regon < ( vd n te regon < < (c vd n te regon < < (d vd n te regon >. Smr nd cn e repced nd perfect conductng pne t = ( ( =, respectve. As sown n Fg. ( te ttered feds t = s rdted n te ornned & K-Amrossen: Accurte Anss of Deectrc Bcked Pnr Conductng

3 Downoded from eee.ust.c.r t 5:59 IRST on ond Octoer st 8 presence of te conductng pne octed t = nd te envronment n te regon <. Te tot feds t = nd = s rdted nd n te presence of two conductng pnes, octed t = nd = nd te envronment n te regon < <. In te sme w te tot feds t nd ( = = s rdted nd n te presence of two conductng pnes, octed t = nd = nd te envronment n te regon < <. Fn te tot feds t = ( s rdted n te presence of conductng pne, octed t = nd te envronment n te regon >. B usng te conventon spectr domn mmttnce pproc [8-9] one cn drve te pproprte spectr domn reen s functons for tese equvent structures. Te enforcement of contnut of trnsverse mgnetc feds cross te perture prts provdes te ntercton of regons. A couped set of mgnetc fed ntegr equtons s otned enforcng tngent component of mgnetc feds on te perture prt of nterfces. In te foowng, te fn epressons for te mgnetc feds re gven nd te det of spectr domn mmtnce pproc s omtted. Wt te pproprte vres defned n te Append, tngent mgnetc feds t =, = nd =, pprocng from eter sde of te pne, tke te foowng forms. Qunttes wt tde ( re Fourer trnsforms of correspondng qunttes wtout tde. = = = = = 4 = 4 5 = 5 6 = = = = = ( ( ( (4 (5 (6 7 7 = 7 7 = 8 8 ( = 8 8 = 9 9 ( = 9 9 = = = ( (7 (8 (9 ( were e = sn θ cos θ ( e ( snθ cosθ = ( = ( e cos θ = sn θ (4 for =,, K,. After enforcng contnut of te tngent mgnetc fed cross te perture prts of nterfces t =, = nd = n te spt domn, we otn: = = = = (5 = = = = = (6 = = ( ( = ( nc, nc, = (7 B epressng te feds n term of ter spectr green s functons, te mtr equton cn e otned. To sove tese set of couped ntegr equtons (CIE s, we epnd mgnetc currents n seres of ss functons usng trdton -nd -drected pece-wse snusod or roof-top ss nd testng functons. Ten pp erkn s metod to ec one of tem. Let B denote te -t eement of set of B ( nd ( ss functons for te functons (, nd (,, respectve. We ve N = = ( (, C B (8 Irnn Journ of Eectrc & Eectronc Engneerng, Vo. 8, No., rc

4 Downoded from eee.ust.c.r t 5:59 IRST on ond Octoer st 8 N = ( (, = C B (9 for =, nd nd so B ( nd ( B cn e pecewse snusod ss functon or pecewse trngur (roof-top ss functons. At ere te re oc roof-top ss functons defned s B (, = c oterwse, < c & < d (, < d & < c B ( = d oterwse, ( As n empe, rectngur ces wt te -drected overppng roof-top ss functons re sown n Fg.. If tese epnson re used n te ntegr equtons nd erkn s metod s pped to ec one of tem, set of couped ne equtons n te epnson coeffcents C nd C resuts. Te tree ner equtons cn e put n te foowng form A C A C = U ( A C A 4C A5C = ( A 6 C A 7C = (4 Ec mtr n (-(4 s een mde of 4 or su mtrces. It s note tt some of tese mtrces m e equ n smmetrc condtons, fct wc, s used to reduce te computton effort. Aso t s wort notng tt (-(4 foow cer pttern wc ows te numer of deectrc cked conductng ers to e vred retve smpe. Empes nd Numerc Resuts As n empe resonnt ptc rr supported deectrc s n rectngur wvegude (Fg. 4 wc s seected from [] s ned for ppcton n g frequenc E nd qus-optc trnsmttng nd recevng sstems. Ts desgn s een performed usng rectngur wvegude wose dmensons re =.87 mm, =.86 mm nd snge deectrc cked conductng er. Te structure cn e mode consderng =.86 mm, =.87 mm, d =5 mm, d = d = nd ε r =. n te Fg.. Te mgntude of S nd S ve een sown n Fg. 5 nd compred wt reference [] nd FSS [] resuts. Tere s good greement etween tem. In te second empe compct ndpss fter s ned wc s desgned te utors. Fg. 6( sows ts ndpss fter wc s een desgned usng deectrc cked ptterned conductng pnes (PCPs wc re octed trnsverse n dstnct seprtons n WR-9 rectngur wvegude. For -order cesev tpe ndpss fter wt center frequenc f =, te retve ndwdt percent nd equ rppes.5 db, te geometr of ec ptterned conductng pne s een otned usng genetc gortm. Fg. 6( sows te spes of tese ptterned conductng pnes n wc nk pes correspond to nonmet prt nd tced pes correspond to met prts of nterfce pnes. Te desred fter cn e reed coosng, ε r = ε r = ε r = for retve deectrc permttvt, r = r =, d = nd d = d = = =9.9 mm. Fg. 7 sows te trnsmsson crcterstc of trnsmsson te fter wc s compred FSS resuts. Tere s good greement etween proposed metod nd FSS resuts. Te est resuts w e otned f our computer word-processor s sever font ses. Te mn font used trougout te document s Tmes New Romn. Tr to foow te font ses specfed n Te, s est s ou cn. d / d / c c c Fg.. Rectngur ces wt te -drected overppng pecewse roof-top ss functons. Fg. 4. eometr of ptc rr supported deectrc s n rectngur wvegude. =.87 mm, =.86 mm, = 5 mm, ε r =., d = 4.57 mm, c =.49 mm, τ =.4 mm nd τ =.86 mm []. APPENDIX Te vres tt pper n te spectr ddc reen s functons n ( to (4 re gven s e, = (A ornned & K-Amrossen: Accurte Anss of Deectrc Bcked Pnr Conductng

5 -5 S - Downoded from eee.ust.c.r t 5:59 IRST on ond Octoer st 8 gntude S [db] S -5 our code Ref. [9] FSS FREQUENC [] Fg 5. gntude of S nd S for te ptc rr emedded n rectngur wvegude. ( ( Fg. 6. ( Wvegude ndpss fter usng ptterned conductng pnes (PCPs. ( Desgned PCPs 4. Concuson Deectrc cked pnr conductng ers of rtrr sped n rectngur wvegude ve een ned. Te nss metod s sed of couped ntegr equton tecnque wc ccurte tkes ger order mode nterctons. Te ect ntc formus nd equtons ve een etrcted wc cn e used n vret of structures. Te proposed metod foow cer pttern wc ows te numer of deectrc cked conductng ers of rtrr sped to e vred retve smpe. S [db] LC Prototpe o FSS Frequenc [] Fg. 7. Trnsmsson crcterstc of te fter e, e, 4e, = = d d d d = tn d tn tn( γ dd tn( γ r ( γ r tn( γ d d cos ( γ d cos( γ r d d (A (A ( γ r tn( γ dd tn( γ dd tn( γ r tn( γ d tn( γ r d d (A4 5e, e, = (A5 6e, d tn( γ d d tn( γ r = d d tn( γ r tn( γ d d 7e, 8e, = d = d tn d cos ( γ d cos( γ r d (A6 ( γ r tn( γ d d tn( γ d d tn( γ r tn( γ d tn( γ r d d (A7 (A8 9e, 7e, = (A9 e, d tn( γ d d = (A d d tn( γ d d were TE γ d d =, ωμ =,,. (A T ωε ε r d =, γ d =,,. (A TE γ = ωμ (A 4 Irnn Journ of Eectrc & Eectronc Engneerng, Vo. 8, No., rc

6 Downoded from eee.ust.c.r t 5:59 IRST on ond Octoer st 8 T ωε = (A4 γ = k k με (A5 γ ω d γ = k k ω μ ε ε, =,,. r (A6 k sn θ = (A7 k k k cosθ = (A8 k k n wc k = mπ, k = nπ (A9 References [] K A. I., kovev A. B. nd Steer. B., Effcent metod of moments formuton for te modeng of pnr conductve ers n seded guded-wve structure, IEEE Trntons on crowve Teor nd Tecnques, Vo. 47, No. 9, pp. 7-76, Sep [] kovev A. B., K A. I., cks C. W., ortw A. nd Steer. B., Te genered tterng mtr of cose spced strp nd sot ers n wvegude, IEEE Trntons on crowve Teor nd Tecnques, Vo. 48, No., pp. 6-7, Jn.. [] ornned. nd K-Amrossen., Compct ndpss fters utng deectrc fed wvegudes, Progress n Eectromgnetcs Reserc B, Vo. 7, pp. 5-5, 8. [4] ot Ksn F. nd Smned R., A nove nd ccurte metod for te desgn of deectrc resontor fters, Progress In Eectromgnetcs Reserc B, Vo. 7, pp. 5-5, 8. [5] Or., Deguc., Tsu. nd Sgesw., Nove wvegude fters wt mutpe ttenuton poes usng du-evor resonnce of frequenc-seectve surfces, IEEE Trntons on crowve Teor nd Tecnques, Vo. 5, No., pp. -6, Nov. 5. [6] Or., Deguc., Tsu. nd Sgesw., A new du-evor FSS resontor for wvegude fter wt mutpe ttenuton poes, Europen crowve Conference, Vo., Oct. 5. [7] Amr S., Bornemnn J. nd Vdeck R., Fst nd ccurte nss of wvegude fters te couped ntegr equtons tecnque IEEE Trntons on crowve Teor nd Tecnques, Vo. 45, No. 9, Sep [8] Ito T., Spectr domn mmtnce pproc for dsperson crcterstcs of genered prnted trnsmsson nes, IEEE Trntons on crowve Teor nd Tecnques, Vo.8, No., pp. 7-76, Ju 98. [9] Scmdt L. P. nd Ito T., Spectr domn nss of domnnt nd ger order modes n fn-nes, IEEE Trntons on crowve Teor nd Tecnques, Vo. 8, No., pp , Sept. 98. [] FSS Reese 9., Ansoft Corp... ornned-foumn ws orn n gun, Irn n 97. e receved s B. Sc. degree from un Unverst n nd s. Sc. nd P.D. degrees from Irn Unverst of Scence nd Tecnoog (IUST n 5 nd respectve, n Eectrc Engneerng. e s current ssstnt professor t engneerng fcut of un Unverst. s entfc feds of nterest re eectromgnetc proems nd mcrowve devces.. k Amrossen ws orn n Tern, Irn n 969. e receved s B. Sc.,. Sc nd P.D. degrees from Irn Unverst of Scence nd Tecnoog (IUST n 99, 994 nd 998 respectve, n Eectrc Engneerng. e s current n Assocte Professor t Coege of Eectrc Engneerng of IUST. s entfc feds of nterest re eectromgnetc drect nd nverse proems ncudng mcrowves, ntenns nd eectromgnetc comptt. ornned & K-Amrossen: Accurte Anss of Deectrc Bcked Pnr Conductng 5