TIME-DOMAIN EQUIVALENT EDGE CURRENT (EEC's) TECHNIQUE TO IMPROVE A TLM-PHYSICAL OPTICS HYBRID PROCEDURE

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1 TIME-DOMAIN EQUIVALENT EDGE CURRENT (EEC's TECHNIQUE TO IMPROVE A TLM-PHYSICAL OPTICS HYBRID PROCEDURE J. Lanoë*, M. M. Ny*, S. L Magur* * Laboraory of Elcroncs and Sysms for Tlcommuncaons (LEST, GET-ENST Bragn/Unvrsy of Wsrn Brany, CS 83818, 938 Brs Cdx 3, FRANCE hp://jrmy.lano@ns-bragn.fr Kywords: TLM mhod, hybrd chnqus, physcal opcs, Equvaln Edg Currns, m -doman, fraconal drvav. Absrac Th ransmsson-ln Marx (TLM mhod s coupld wh asympoc mhods such as Physcal Opcs (PO o compu h lcromagnc fld producd by small objcs wh complx gomry ha nrac wh lcrcally larg conducng srucurs. Th problm s dvdd no wo subproblms. Frs, on solvs for flds usng TLM and compu flds ousd h TLM compuaonal volum by usng Krchhoff's ngral. Thn, usng Tm-doman Physcal Opcs (TDPO, flds radad by h larg mallc srucurs locad ousd h TLM ar compud. Thn, a mor accura calculaon can b achvd by addng o h PO fld, h frng wav (FW fld whch aks no accoun h dffracon. Fnally, som smulaon rsuls show h mprovmn brough by h dg currn corrcon (EEC's. 1 Inroducon Th Transmsson Ln Marx (TLM mhod [1] has bn provn o b wll sud o analys complx lcromagnc srucurs. Howvr, as soon as h srucur bcoms lcrcally larg, h compuaon m and mmory sorag ncras dramacally. In sp of consan mprovmns nroducd n TLM, s sphr of applcaons sll rmans rsrcd o a maxmum volum of abou (1 wavlnghs 3. Byond ha sz, compur s xhausv and such volumc mhods ar no suabl for compur add dsgn ool. Howvr, hr ar suaons whr combnng mhods can b consdrd. For nsanc, for hgh gan rflcor annnas, prmary sourcs ar gnrally lcrcally much smallr han rflcors bu ar much mor complx n boh gomry and consun marals. Thus, an da ha can mrg s o us asympoc approachs for larg objcs and volumc fullwav mhods for complx small objcs. Th asympoc mhod such as PO can b asly appld o larg mallc rflcors ha ar ofn ncounrd n such annnas. As a rsul, h problm now s o coupl ha mhod wh h fullwav analyss appld o h rs of h srucur. Frs, hs papr xplans a m-doman couplng chnqu bwn PO and a volumc full-wav m -doman mhod namly, h TLM mhod. Th ponal advanag ovr comparabl chnqus such as FDTD s ha h mplmnaon of Krchhoff surfac, whch s rqurd o compu h fld ousd h compuaonal volum s sraghforward and nsghful wh h TLM. Th rason s ha h sx lcromagnc fld componns ar dfnd a h sam locaon n TLM clls, only on surfac s ndd o achv h mplmnaon of Krchhoff formulaon (nsad of hr for FDTD []. Thn, s found ha dg ffcs, whch canno b accound for by h PO, may play som sgnfcan conrbuons. Thus, an dg corrcon chnqu s proposd o nhanc h accuracy of h fld compuaon a h obsrvaon pon ha ncluds boh h larg mal srucur and h complx objc conrbuons (s Fgur 1. As a frs approxmaon, nracons bwn objcs ar nglcd so ha fr-spac Grn's funcon can b sll usd. Also, whou loss of gnraly, h obsrvaon pon P for radad fld compuaons s locad n h far-fld rgon of boh srucurs. Prmary sourc S M TLM compuaonal Volum Nar fld o far fld Krchhoff ransformaon Krchhoff surfac Fgur 1: Gomry of h problm. Thory Nar fld o far fld Krchhoff ransformaon Q Mallc rflcor In hs scon, h TLM prncpl and h hory of h Tm - Doman Physcal Opcs ar brfly prsnd frs. Thn, h chnqu o accoun for h dg ffcs namly, Equvaln n n P TDPO z O S y x

2 Edg Currn (EEC corrcon. Fnally, h mplmnaon of h corrcon n m-doman s prsnd n dals..1 TLM mhod basc prncpl Th basc (unloadd cubc SCN-TLM cll llusrad n Fgur wll b usd n h prsn work, I can b sn as a sx-arm dvc, ach conssng of wo orhogonal ransmsson-ln wh volag ha can b assocad o local plan wav. V 6 Y V 3 V 9 X V 1 V 5 V 7 V 8 V 1 Fgur : SCN-TLM afr Johns [1] V 4 V V 11 V 1 Hnc, h compuaonal doman s flld wh a nwork of nrconncd nods shown abov. Flds a h cnr of h cll can b shown as a lnar combnaon of h ncdn volags on h arms. Suppos ha som ncdn volag (du o som sourc mpngs on h nod. A scarng procss wll hn occur, producng rflcd volag a all arms. Hnc, h TLM cll corrsponds o a 1-por dvc wh scarng marx dfnd by: r [V] k+ 1 = [S] [V] k (1 r whr [V] k and [V] k + 1 s h vcor of ncdn and rflcd volags a all nod arms, rspcvly, and k h m ndx. Th nx sp s o ransfr rflcd volags o h nghborng clls a m (k+1. Thus, hy bcom nw ncdn volags for h nx m raon. No ha unlk FDTD, h TLM schm dos no xplcly solv Maxwll's quaons bu smula propagaon mchansm by mans of local wavs. Th prsnc of maral and/or paralllppd cll s accound for by addng subs o h basc wlv-por llusrad n Fg. 1. Thus, a complly loadd SCN ncluds ghn pors, losss bng ncludd n h marx [S] componns. Unlk FDTD, h conncon bwn nods n dffrn mda or sz (rrgular msh s rval for SCN- TLM. I should b srssd ha h TLM algorhm provds som mor nformaon as compard o Y's: Th sx fld componns a h sam m and locaon (cnr of h cll ar compud from h ncdn volags. Ths provds som sgnfcan advanags whn couplng TLM wh ohr chnqus s consdrd as sn furhr n hs papr. Fnally, for h gomry nvsgad hr, h TLM compuaonal doman mus b lmd by som absorbng condons snc s opn o fr-spac. PML (Prfcly Machd Layr chnqu [3] s usd s usd n hs cas.. Tm-Doman Physcal Opcs (TDPO Physcal Opcs s mployd o sudy h radaon of h mallc plan. Is formulaon s basd on Chu-Sraon quaons [4] usng four assumpons: h lcromagnc fld s non-xsn on h surfac non-drcly llumnad by h ncdn fld. Th man rad of curvaur of h surfac of h objc ar much largr han h wavlngh. Th obsrvaon pon P s locad n h far fld rgon of h plan consdrd as prfcly conducng. Ths assumpons lad us o h follow xprsson (s Fgur 1: 1 HPO( R, = n Hnc( R', ' n ds' ( 4πRc S' ' whr R ' s h poson vcor qual o OQ, c s h spd of lgh n fr spac, "r" ndcas ha h ngrand s valuad a h rardd m: ' = - R - R' /c (3 whr R s h modulus of R, H s h ncdn magnc fld on h plan producd by h sourc conand n h TLM compuaonal volum, n s h normal vcor of h surfac S', n rprsns h un vcor n h obsrvaon drcon and S' dnos h surfac of h mallc plan. Now, H nc has sll o b valuad from TLM compuaon. Ths s don by usng Krchhoff ransformaon [], rfrrd o as nar fld-far fld ransformaon. In fac, any fld componn ousd h TLM compuaonal volum can b compud n fr spac. For nsanc, h ncdn fld componns on h pla ar xprssd by: ' ' 1 R R R R ' ' Ψ( R, = Ψ( + Ψ( π ns R, R, 3 4 S R R cr R " " " Ψ( R, ds (4 R R r' whr R " s h poson vcor qual o OM, subscrp "r'" ndcas ha h ngrand s valuad a h rardd m: " = '- R'-R" /c (5 whr S dnos h Krchhoff surfac nclosng h radang sourc n h TLM volum and Ψ h sam fld componn on S. Ths ss up h foundaon of h hybrd mhod prncpl. Indd, h funcon of h Krchhoff surfac s fundamnal snc consss n nrfacng h wo compuaonal domans. No ha as h sx lcromagnc fld componns ar dfnd a h sam locaon n TLM clls, only on surfac s nc r

3 ndd o achv h mplmnaon of Krchhoff formulaon (nsad of hr for FDTD. Thn, (4 s approxmad by a smpl summaon. No ha nsans '' should corrspond o TLM samplng ms..3 Equvaln Edg Currns (EEC's Alhough combnng TDPO wh TLM ylds good rsuls, n many ralsc cass [5], h dffracon causd by dgs rqurs o b accound for as shown by h rsuls llusrad n Fgur 3. Th fgur llusras h far-fld parn of a shor dpol ornd prpndcularly o a mallc squar pla of 1-m dmnson a a dsanc of 7.5 cm. As can b obsrvd up o 5 db dscrpancs occur nar normal whl som good agrmn s found a grazng angls. f =.5GHz Magnud (db ha (dgr Fgur 3: Comparson bwn and BEM (rfrnc. Cas of a vrcal dpol ovr a squar mal rflcor. Radaon parn n h xoz plan (Fgur 1. A mor accura calculaon can b achvd by addng o h PO fld, h frng wav (FW fld whch aks no accoun h dffracon. Basd on h Physcal Thory of Dffracon, an approxmaon o h FW fld can b workd ou from a ln ngral along h llumnad par of h dgs of h srucur by usng Mchal s quvaln dg currns (EEC s [Mc 1]. Closd-form xprssons hav bn drvd frs for unruncad ncrmnal wdgs srps bu hy yld sngulars and dsconnus. In ordr o avod hs problms, runcad srps hav bn mployd ladng o runcad EEC s. Mchal has drvd closd-form xprsson for runcad EEC s for a wdg wh arbrary angls [7]. Howvr, appars ha Mchal s EEC s conan also sngulars whch ar causd by h mahmacal procdur appld o oban closd-form xprssons. Johansn [8] has proposd a nw runcad EEC s whch ar wll bhavd for all drcons of ncdnc and obsrvaon pons, apar from h drcons of ncdnc and obsrvaons whch ar srcly followng h dg. Lar, Johansn [9] has drvd h whol procdur n h m doman. L us consdr a prfcly conducng wdg llumnad by a wav as llusrad n Fgur 4. Ladng dg s β β z y Fgur 4: Gomry for EEC mplmnaon. β In h far fld of h srucur, a hgh-frquncy approxmaon o h FW fld s calculad from a ln ngral along h llumnad par C of h dgs of h srucur. Th lcrc FW fld s calculad from h magnc currn M T and h lcrc currn I T by : fw 1 I T E ( R, ( R, s ( s 4 C πrc M (6 T + ( R, ( s dl R R = c Hrn, R s h obsrvaon poson vcor, s h m, R s h dg poson vcor, c s h fr spac spd of lgh, s h fr-spac mpdanc, s s h un vcor of h dffracd fld, s s h un vcor of h ncdnc fld and s h ladng dg un angn vcor. In h local Carsan sysm (, x y, z a h pon of ngraon, = z and y s h ouward normal un vcor, on can xprss s and s as: s = snβα x + snβsnα y + βz (7 s = snβα x snβsnα y + β z (8 whr α and α s h projcon angl bwn s and s, rspcvly, wh h xoy plan (s Fgur 4. Th runcad EEC s ar drmnd by a dffrnc bwn h unruncad EEC s and h corrcd EEC s [9]: I = I I (9 T UT COR M = M M (1 T UT COR Th unruncad EEC s can b xprssd xacly by a closdform, gvn by : I UT ( R, = E ( R, UT1 + H ( R, UT (11 M UT ( R, = H ( R, UT 3 (1 wh : csn( α ( snβ ( µ+ α ( UT1= α 1 µ x l A s Mal pla Tralng dg (13

4 ( ( c UT α α = sgn α + snβ an an µ+ α β β + ( α α µ anβ an β 1 µ (14 ( α sgn( ( / c snα UT 3 = α sn β sn β ( µ + α 1 µ (15 whr ( ( snβ snβ snβα+ β β β µ= E ( R, = E ( R, H ( R, = H ( R, n whch ndcas h scalar produc.4 Nw m-doman mplmnaon of corrcd EEC's Rvsng h work of Johansn n frquncy doman, on can no ha s possbl o ransform n a sraghforward mannr h corrcon n h m doman by usng h concp of fraconal drvav [1]. Indd, h frquncy doman xprsson ncluds rms such (jω -1/ whch corrsponds o fraconal drvav of ordr 1/. Thus, can b shown ha corrcon EEC s can b wrn as: I COR ( R, = D H ( R, A COR1 (16 M COR ( R, = D H ( R, A COR (17 whr sgn ( / c c α COR1 = A ( snβ ( µ+ α πl ( α COR ( α α µ an an β β α α + anβ anβ ( 1 µ = A ( ( ( α ( α sgn c snα c snβ snβ µ+ α ( α + ( 1 µ π l A ( snβ ( 1 µ l A = c whr A dno a m dlay and D = dnos a fraconal drvav of ordr ½. Th concp of h fraconal drvav s o approach hs opraor usng a dgal flr proposd by Ousaloup [11]. I allows on o nforc h valdy of D -1/ ovr any frquncy rang. For nsanc, suppos ha on slcs a frquncy rang sarng from? A o? B. Thn, on chooss wo angular frquncs? l and? h such ha: ω l << ω A and ω h >> ω (18 For a gvn ordr NP, h opraor s xprssd n h Laplac ransform doman by: ' NP 1 s +ω D ( s = lm DNP ( s = B (19 NP = s +ω whr: NP NP NP 1 ω ' ω B =ωl, h ω h ω = ω, ω = ω ' l l = ω ω l ω l For nsanc, Fgur 5 shows h comparson bwn h ransfr funcon of D -1/ compud wh h abov procdur and h analycal soluon. Th frquncy rang s s by h paramrs ω l = (ω max /61-1 and ω h = (ω max /11. Magnud 8 x Thory DF(-1/ frquncy (GHz Fgur 5: Transfr funcon of h opraor D -1/ for NP = 7, ω max = π 1 9. On can obsrv h xclln agrmn ovr h frquncy rang ha s usd furhr n h smulaons. Ousd ha frquncy rang, som ncrasng rror can b sn a low frquncs du o h sngular bhavour of h funcon. Howvr, som adjusmn can b don f on wans o xnd h accuracy o lowr frquncs. In ordr o mplmn h abov ransfr funcon n h m doman, a blnar ransform can b usd.

5 3 Smulad rsuls Th cas usd for h rsuls shown n Fgur 3, s llusrad n Fgur 6 wh EEC's mplmnaon for dffrn frquncs. f =.5GHz Magnud (db Magnud (db Magnud (db EEC ha (dgr f = 1. GHz +EEC ha (dgr f =. GHz +EEC ha (dgr Fgur 6: Cas of Fgur 3: Comparson of radaon parns (BEM, TLM -PO and wh EEC's corrcon. h frquncy ncrass, dg hav lss ffcs for h cas shown hr and h dffrn chnqus nd o agr, alhough +EEC s always closr o h rfrnc. 3 Concluson Th ransmsson-ln Marx (TLM mhod was coupld wh Physcal Opcs (PO o compu h lcromagnc fld producd by small objcs wh complx gomry ha nrac wh lcrcally larg conducng srucurs n m - doman. Thn, a mor accura calculaon was achvd by addng o h PO fld, h frng wav (FW fld whch aks no accoun h dffracon. Fnally, som smulaon rsuls show h mprovmn brough by quvaln dg currn (EEC's corrcon mp lmnd n m-doman usng h concp of fraconal drvav. Currn work concrns corrcon procdurs o accoun for nar-fld couplng. Rfrncs [1] P.B. Johns. "A symmrcal condnsd nod for h TLM mhod", IEEE Trans. on Mcrowav Thory and Tch., 35, pp , (1987. [] M. Ny, S. L Magur. "Th Transmsson-Ln Marx (TLM mhod: An ffcn ool for passv componn fld modlng", Nw rnd n Mcrowav Thory and Tchnqus, Ed. By H. Baudrand, Rsarch Sgnpos, pp , (3. [3] L Magur and M.M Ny. "Exndd PML-TLM Nod: An Effcn Approach For Full-Wav Analyss Of Opn Srucurs", Inrnaonal Journal of Modllng Elcronc Nworks, Dvcs and Flds, 14, pp , (1. [4] Chu and Srao [5] J. Lanoë, N. Jacquy, S. L Magur, M.M. Ny. " A Hybrd Tchnqu Combnng Transmsson Ln Marx and h Physcal Opcs n Tm Doman ", NUMELEC' 6, Llls, Franc, (6. [6] Mchal 1 [7] A. Mchal. Elmnaon of nfns n Equvaln Edg Currns, Par I: Frng Currn Componns, IEEE Trans. On Annnas Prop., AP-34, pp , (1986. [8] John 1 [9] P. M. Johansn. Tm -Doman Vrson oh h Physcal Thory of Dffracon, IEEE Trans. Annnas Prop., 47, pp. 61-7, (1999. [1] J. Lanoë, S. L Magur and M. M. Ny. "A Fraconal Drvav Opraor for Surfac Impdanc TLM Modlng", IEEE Mcrowav and Wrlss Componns Lrs, Spmbr (7. [11] A. Ousaloup, F. Lvron, B. Mahu and F. Nano. Frquncy-Band Compl x Non ngr Dffrnaor: Characrzaon and Synhss, IEEE Trans. on crcus and sysms, 47, pp. 5-39, (. On can obsrv h subsanal mprovmn brough by h EEC's corrcon nar h normal drcon. Also no ha as

Consider a system of 2 simultaneous first order linear equations

Consider a system of 2 simultaneous first order linear equations Soluon of sysms of frs ordr lnar quaons onsdr a sysm of smulanous frs ordr lnar quaons a b c d I has h alrna mar-vcor rprsnaon a b c d Or, n shorhand A, f A s alrady known from con W know ha h abov sysm