Application of the measured equation of invariance to radiation and scattering by flat surfaces

APPLICATION OF THE MEASURED EQUATION OF INVARIANCE TO RADIATION AND SCATTERING BY FLAT SURFACES Ra ael POUS: Ma k P ou y, and Kenne l, I<. Mei EECS Dep ., Uni . o Cali oinia, Be keley CA 94720 (R. Pouz is now wi h he TSC Dep ., Uni . Poli . de Ca alunya, Apdo. 30.002, 08080 Ba celona, Spain). Abs ac The ewn ly iii ocliic d concep o lie Measu ed Eq ia ion o In a iance (blE1) [l, 2. 31 is used ,o sol e .lic- p oblenl o adia io:] and sca e ing by la su aces. Becaus on la su aws li elec ic cu en s a e con ined o wo dimensions, a sinil)lc ec o po m .ial o mula ion can be used. The p oblem o adia ion and sca ing by ec aiigiila s ip dipoles is !,ol ed. including he ans e sal a ia ion o he cu m a osq he dipole wid .11. Also o in e es a lie cu eii s indu ed (111 an ennas wi h s ep a ia ion: in wid .h, and wi h hencl, and T-junc ioiis. The NE1 is ; i1 qua ioii lia caii be used o e mina e ,L Fini e Di e ence (FD) iiiesli a bi a ily close o lie objec o in e es . a oiding he use o abso bing bouiid- a y condi ions, n.hic1i cyiii e lie inesh o ex end beyond lie egion o in e es , wi h aii inc ease in onipu a ion ime aid s o age memo y. Tlie iiiexli us ~l o sol e a b ip diliolc- i.; 21 1e angula. mi sh which es ends only a ew cells ( ypi dly ! o 4) iii each di w ioii. The poin s o his mesh all in o h ee a ego ics: ex e m poin x o poiu s on lw iiiesh bounda y. poin s on he su ace o he dipole. and iii e ioi poin s no on lie su ace o lie dipole. Fo each ca ego y a di e en ype o FD eqiia ion will be used. The uiikiiowii i: cliosen o be he ec o po en ial .i de inc l as #.'"'' = d x d (p ac o ed in o simpiici y). I he la su ace is pa allel o he , 'y plaiie. only .AJ aiid d, need o be conL;ide ed, and hei alues a each mesli poin a c lw u~ikno~ iis o lie FD p oblem. Since wo unknowns a e associn pd wi h (wcli ni(4 poin wo eqiia ioiis need o he w i en also. Fo lie in e io poin s no on liu me al su ace, lie s ai da d FD app oxima ion o he wa e eqiia ioii is iis4 o pncli o lic, wo po en ials ,I = . . !) ("2 k, ) .1,1 = I1 (1) 'This esea ch wi s spon>o l I,? I lac CaIi1o ni.L Slaie hlICRO p og nm. xiid he indus ial Spon- >o Hughes .%i e a ('onlpnny 0-7003-1246-5/93/$3.00 0 1993 IEEE 540 Fo he poiii s on he me al su ace, he condi ion ha ESc = -E nc is used, yielding he equa ions This equa. ions caii easily be pu in FD o m. Special ca e needs o be aken a he edge o he la su ace. Fo he componen o he ec o po en ial ha is pa allel o he edge. a sligh a ia. ion o he abo e equa ions needs o be used, and o he no mal componen , we use he ac ha he no mal cu en anishes a he edge (see [4] o mo e de ails). Finally, o he poin s a he mesh bounda ies, we w i e an equa ion o he o m N-l .4&l - 1 (', ' -A,{, = 0 l = .U, 9 ( 4) j= I whe e Ad" is he alue o he ec o po en ial a he bound~y poin and Ad, a e he alues a he N -- 1 neigl ho ing poin s (X may be 4, 5. 01 G depending 011 whe he he poin is 011 a side. ail 4ge, o a o i c o he mesh,. The coe icien s c, a e calcula ed o inake Eqn. 1 a leas squa e e o i o he hl po en ials (which we call measu ing iinc ion.~) whe e .J;, J; a c wo se s o linea ly independen cu en i, which we call me ons. Fo a ec angula pla e o leng h I and wid h u . hese me ons a e ])La 713 (my - 1)ay Jyly(.T, y) = 4111 ___ 15 m, 5 M, 1s my 5 My (6) 1 5 my 5 ~4~ (7) 1 c 5 W ( 7zz - . sin Jym=n'qX,y) = CO5 1 5 ma 5 M, I W Figu e 1 shows he longi udid cu en on a s ip dipole o leng h and wid h I = 18.5~ = 1.42X. when ed o -cen e a 1/4, and he inpu admi ance o a cen e - ed hin s ip dipole. Figu e 2 shows he cu en s in he wo di ec ions when he wis ed dipole is ed a he cen e . Bo h he longi udinal and ans e sal a ia ions o he cu en s a e ob ained. as well as a clea pic u e o how he cu en s beha e lieax he 90 deg ee bend. Figu e 3 shows he monos a ic ada c oss sec ion o a squa e pla e o side n, coinpa cd o he ineasu emen s epo ed in [GI. Re e ences [l] K. I<. Mei. R. Pous, Z. Chen, Y. W. Liu, and M. D P ou y, "The Measu ed Equa ion o In a iance: a new concep in ield compu a ion," IEEE IPmns. An- ennas P opaga ., submi ed o publica ion. 541 mA/1,’ V 10 8 1G 14 12 10 S F 4 2 0 Re . [3] )- 0 0.2 0.4 0.G 0.8 1 1.2 1.4 1.6 1.8 [/A Figu e 1. Cu en on an o - en e ed s lip dipole an enna wi h 1 = 18.5~ = 1.42X, ed a 1/4 ( op) niid inpu admi ance o a en e - ed s ip dipole an enna wi h I = 18.5~ e sus equen ). compa ed wi h he MOM esul p esen ed in [5] (bo om). [2] R. Pous, ‘The Measu ed Equa. ion o In a iance: a new concep in ield compu- a ion,” Ph.D. diss a ion, Uni . o Cali o nia a Be kdey, 1992. (31 I<. K. Mei, R. Pous. h.1. D. P ou y, and Y. U’. Liu, “Fu he insigh in o he Mea- su ed Equa ion o In a iance,” IEEE An ennas and P paga . In l. Sumposium, Ann A bo , Michigan, 1993. [4] M. D. P ou y, R. Pous, and I<. K. hlei, “Applica ion o he Measu ed Equa ion o In a iance o ansmission lilies and discon inui ies.” IEEE An ennas and P opaga . h l. Synqi~siil m, Ann h bo , Michigan, 1993 151 R. F. Ha sii g on, Field Conipi n ion by Momen Me hods, Robe E. K iege Publishiiig Company. hlalaba , Flo ida, 1952. [GI S. M. Rao. D. R. Wil oii. and A. W. Glisson, “Elec omagne ic sca e ing by su aces o a. bi a y shape,” IEEE Z’ uiis. An ennas and P opaga ., ol. 30, pp. 409-418, May 1982. 542 Figu e 2: Cu eli 3 on a wis ed dipole. o X = In ( e ical w eii on he le , and lio izoii al cu en on li igli J. 10 1 0.1 a jxz 0.01 0.001 0.0001 0 0.2 O.G 0,5 1 1.2 l/X Figu e 3: I Iono ;i ic ad;iI C~OS~ scsc ioii s. equen y o :I squa e pla e o side a, o no mal iiic.idcn e. c.ompa ed o lw measu emen s epo ,d in [GI ( he cdcula ioii is o a ze o- liicli~iehs pla e. and he iiieaw enieii s a e o a pla e o liickiiess 0 00012iX). 543