Capacitance Computation of a Charge Conducting Plate using Method of Moments
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1 Itertiol Jourl of Reserch d Scietific Iovtio (IJRSI olue, Issue I, Jury 8 ISS 3 75 Cpcitce Coputtio of Chrge Coductig Plte usig Method of Moets Kishore Mity Deprtet of Electricl Egieerig d Coputer Sciece, Syrcuse Uiversity, Syrcuse, Y, strct I this pper the cpcitce of chrge coductig plte hs ee clculted usig Method of Moets ith MTLB The result is lso copred ith the Lest Squre pproxitio to lyze the chrge distriutio of etllic surfce Keyords Cpcitce, Method of Moets, Lest Squre pproxitio T I ITRODUCTIO he field of Coputtiol Electrogetics hs ecoe populr d got rpid pce i the lst to decdes There ere severl techiques hs ee proposed i the pst to solve the Electrogetics prole due to the rditio of electrogetic ojects such s trsissio lie, ire tes d sctters Soe of this techiques re Methods of Moets (MOM,fiite differece ethod (FD, Mote Crlo ethod (MCM, fiite eleet ethod (FEM, d vritio ethods (M etc But MOM hs soe certi dvtges d disdvtges depedig upo the forultio of the proles Tcticlly MOM used to solve the differetil equtios The ethod of oet (MOM is uericl procedure for solvig lier opertor equtio y trsforig it ito syste of siulteous lier lgeric equtio, referred to s trix equtio My proles i electrogetics c e estlished i the for of itegrl equtios itegrl equtio is oe i hich the uko fuctio ppers i the itegrd The MOM provides y to solve such itegrl equtios for oth the sttic s ell s tie hroic electrogetic fields s e goes fro MHz to GHz of frequecy the Electrogetics Modellig(EM hs ecoe populr suject of iterest i the lst to decdes The high opertig frequecy of electrogetic ojects cuses siulteous electrogetics iterctio ithi the circuits So it is very iportt to evlute the cpcitce d chrge distriutio of the etllic structures very ccurtely i the itegrted circuits The MOM helps us to trsfor the itegrl equtio ito trix equtios sed o the expsio of the ukos i ters of the sis fuctio ith uko coefficiets such s chrge distriutios d sed o this cpcitce c e clculted This pper itroduced deep isight of the uericl solutio to solve the field desity distriutio d cpcitce of D chrge plte d D prllel plte usig MOM The covergece of the uericl solutios of the MOM hs lso ee copred ith Lest Squre Method II MTHEMTICL EQUTIO We c cosider Electrogetics proles i ters of ihoogeeous equtio L( f g ( here L is lier opertor, g is ko fuctio (excittio d f is the uko fuctio to e deteried We expd f i series of fuctios f f ( Where re costt The set f is clled the expsio fuctio or sis fuctio For exct solutio,, ut i prctice tructed to fiite vlue L ( f g (3 We defie set of eightig fuctios or testig fuctios, i the rge of L d tke the ier product of equtio (3 ith ech of the stisfyig, f g (4, sclr product, g is defied to e sclr, g g, (5 cg,, c g, (6 * g, g if g (7 * g, g if g (8 rsisitertiolorg Pge 78
2 Itertiol Jourl of Reserch d Scietific Iovtio (IJRSI olue, Issue I, Jury 8 ISS 3 75 Here d c re sclrs d * idictes coplex cojugtio The ier product correspodig to our previous exple is of the for g ( x, g( x, dxdy, (9 Siilrly, the testig fuctio for our previous exple is x x ( y y ( ( ie the testig fuctios re Dirc delt fuctios Such choice of testig fuctio is clled poit tchig The equtio (4 c e reduced to g ( Where If, Lf, Lf, Lf, Lf g g, g, is o-sigulr oe c rite g ( III FIELD DISTRIBUTIO OF CHRGE CODUCTIG SIGLE PLTE Method of Moets(MOM To strt ith, let us cosider the exple of deteriig the electrosttic potetil due to isolted chrged coductig plte, eters o side d lyig o the z ple ith its ceter t the origi s sho i Fig Fig Squre coductig chrged plte The plte is ssued to hve zero thickess Let ( x, represet the surfce chrge desity o the plte Further, the oudry coditio o the potetil is =costt o the plte For the chrged coductig plte, the potetil t y poit ( x, y, z c e ritte s: ( x, y, z ( x', d 4 ( x x' ( y ( z z' (3 If ( x, is ko (hich is ofte ssued i solvig siple electrosttic proles, i e chrged desity is specified, the potetil fuctio c e coputed directly But i prcticl proles, ofte the chrge distriutio ( x, is ot ko The equtio (3 is exple of itegrl equtio s the uko ( x, ppers uder the itegrl To solve the uko chrge distriutio e pply the ethod of oets The procedure is explied elo: We sudivide the plte ito squres of side s sho i Fig The th squre is deoted y S d We pproxite the chrge distriutio s: ( x', f ( x', (4 here f ( x', o S o S rsisitertiolorg Pge 79
3 Itertiol Jourl of Reserch d Scietific Iovtio (IJRSI olue, Issue I, Jury 8 ISS 3 75 Here, the fuctios ( x', f re clled the expsio fuctios or sis fuctios d re the coefficiets Bsis fuctios c e of differet types; here e hve cosidered pulse sis fuctios, hich hve uit gitude over soe doi d re zero everyhere else With sufficietly lrge, equtio (4 closely pproxites the ctul ( x', To solve for the chrge distriutio ( x', pproxitely, the uko coefficiets re to e deteried Fro the give oudry coditio e ote tht o the surfce of the plte ( x, y, d this coditio c e used to deterie the uko coefficiets Usig the pproxite chrge distriutio give y equtio (86, let us o evlute the equtio (3 t the id poit ( x, y, of ech of the S The potetil t the idpoit of S is give y: f ( x', d 4 ( x x' ( y The ove equtio c e ritte s (5 (6 here is the potetil t the ceter of S due to uit chrge plced o S For For, 4 x' d l(, tretig the uit chrge o S, y S, locted t the id poit ( x of x x y y (7 s poit chrge (8 s, cosiderig the potetils t ll the su sectios e c rite Or, The uko coefficiets (9 s c e coputed s ( Fig d Fig 3 shos the chrge distriutio otied usig the MOM techique for, pproxite chrge desity log the side of the plte other preter of iterest is the cpcitce of the plte With pproxited s: Q C ( x, dxdy ( ko, the cpcitce of the plte c e C s ( I this exple e hve used pulse type sis fuctio d poit tchig, tht is, Dirc delt fuctio s testig or eightig fuctio B Lest Squre pproxitioi The chrge desity d the vlue of the true cpcitce of the ove prole c e foud usig Lest Squre pproxitio I the ethod of lest squres, the solutio of = is ttepted y iiizig the fuctiol F ( hich c e give y The optil solutio of F ( = - (3 is give y = ( T - ( T (4 I EXPERIETL RESULT With the referece of the Figure 3 the chrge desity for differet uer of sectio hs ee pproxited For rsisitertiolorg Pge 8
4 Itertiol Jourl of Reserch d Scietific Iovtio (IJRSI olue, Issue I, Jury 8 ISS 3 75 coputig the ore ccurte result xiu uer of susectios hs ee cosidered Usig the (, Fig pproxite chrge desity o ech susectio djcet to the ceterlie of uit squre coductig plte ddig the ko oudry coditio(, the uko chrge distriutio hs ee clculted ith MTLB progr hich helps to fid the cpcitce of the chrge coductig plte Fro thefigure the chrge desity of ech uer of susectio hs ee deteried Fig pproxite chrge desity o susectio djcet to the ceterlie of uit squre coductig plte Fig 3 Cpcitce ith differet uer of susectio Figure 3 shos the pproxite chrge desity o susectio djcet to the ceterlie of uit squre coductig plte hich is syptotic i ture Figure 4 shos tht fter icresig the certi o of susectio (i this cse the cpcitce of the uit squre coductig coverges The Tle- shos the vlue of the cpcitce clculted y ( d (4 Tle- shos the true vlue of cpcitce hs ee foud i Method of Moet d Lest Squre pproxitio for differet uer of susectio The vlue of the cpcitce foud i oth the cses re exctly se But lest squre pproxitio tkes ore coputtiol lod to oti the strog covergece of the residuls It is lso evidet tht eve through is uouded = coverge i the e to zero ootoiclly uer of Susectios ( Cpcitce(pF Usig Method of Moets Cpcitce(pF Usig Lest Squre pproxitio Tle- Copriso of the experietl result of Method of Moets d Lest Squre pproxitio rsisitertiolorg Pge 8
5 Itertiol Jourl of Reserch d Scietific Iovtio (IJRSI olue, Issue I, Jury 8 ISS 3 75 It is iportt to ote tht the sequece of solutio of ioth the ethod coverge syptoticlly I cse of syptotic covergece the solutio ted to coverge ore d ore s uch s sis fuctio re chose s the ethod of lest squre gurtees ootoic covergece, it is etter to use lest squre pproxitio he the exct solutios is ot ko I this scerio it is esier to estlished reltive errors estites the solutio COCLUSIO I this pper to differet ethod hs ee proposed to evlute the chrge distriutio d cpcitce of chrge coductig sigle plte to lyze the coputtiol lod of the ech ethod The experietl result of the cpcitce for differet uer of susectio deostrtes tht the vlue of the cpcitce coverges ith icresig the uer of susectio of the uit squre plte s oth of the ethods- Method of oets d Lest Squre pproxitio covey exctly se result, ore coputtiol ork is eeded for the Lest Squre pproxitio to oti the strog covergece of the residuls y iiizig the reltive error The steeper slope of the chrge distriutio of the coductig surfce exhiit tht the free chrge distriutio fetch up to the edge of the etllic surfce d coprtively flt t the iddle sectio of the coductig plte The Coverge vlue of the cpcitce is 469 pf i oth cses KOWLEDGEMET The uthor ould like to express his pprecitio tords Prof Tp Kur Srkr t Deprtet of Electricl Egieerig d Coputer Sciece, Syrcuse Uiversity, Y, US for his vlule support REFERECES [] Roger F Hrrigto, Field Coputtio y Moet Methods, IEEE PRESS Series of Electrogetics, e York, 99 [] Xioo Liu, Zhoxi Zeg, Jigsi Zhg, Rui Lu, Wei Li, Xioli Dog, d xue Zhg, Chrge Mppig Method for the Cpcitce of Coductig Plte, IEEE Microve d Wireless Copoets Letters, ol 7, o 5, My 7 [3] MDhodr, Meer, RDhsekr, Efficiet Cpcitce Coputtio for Coputtiol Electrogetics, IEEE Coferece o Couictio d Sigl Processig, pril 3-5, 4, Idi rsisitertiolorg Pge 8
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