IJREAT International Journal of Research in Engineering & Advanced Technology, Volume 1, Issue 4, Aug-Sept, 2013 ISSN:

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1 IJREAT Itertiol Jourl of Reerch i Egieerig & Advced Techology, Volue, Iue 4, Aug-Sept, 3 ISS: 3-79 Method of Moet for Electrogetic Modelig of Chrge Ditributio ito Thi Wire Mohed Louzzi, El H Aroud, He Ytii 3, Merye El Moufri 4, Oufe Hdou 5,,3,4,5 Modelig d Siultio of Mechicl Syte Lbortory, Fculty of Sciece, Uiverity Abdellek Edi, BP., M hech, 93,Tetou, Morocco Abtrct Thi pper repreet the Method of Moet (MoM) for the evlutio of the lier chrge deity o fiite tright eget of thi chrged coductig wire. We ue tht the chrge deity piecewie cott over the legth d the electric potetil re i oe volt. The coductig tructure re odeled by plr ub doi. The Method of Moet i eployig pule d delt bi d tetig fuctio, repectively, i ued for lyi. The ect forultio for the tri eleet i evluted for ub doi, d we obti yetric tri of Toeplitz. Becue electrottic proble re reltively iple, the proble of fidig the potetil tht i due to give chrge ditributio i ofte coidered. They provide good cotet for itroducig lgorith ued to olve itegrl equtio. I thi ectio, we will coider itegrl equtio pproch to olve for the electric chrge ditributio, oce the electric potetil i pecified. The electric potetil t poit r due to electric chrge deity q i give by the itegrl. Keyword: Method of Moet, Chrge deity, Coductig Wire, Electric Potetil, Toeplitz Mtri... Itroductio The Electrogetic (EM) odelig d deterite of chrge deity yte h bee the ubject of eteive reerch i the lt three decde. While i the pt, deig d ditributio of chrge y hve bee coidered ecodry iue i overll yte deig, tody it ply criticl role i pcecrft [][][9]. The ccurte evlutio of chrge ditributio d cpcitce of etllic tructure i iportt tep i deig of high frequecy itegrted circuit. I thi pper, it h bee choe the qure hpe of the ub doi becue of it bility to cofor eily to y geoetricl urfce or hpe d t the e tie to iti iplicity of pproch copred to the other hped odelig. The MoM i bed upo the trfortio of itegrl equtio, ito tri equtio by eployig epio of the ukow i ter of kow bi fuctio with ukow coefficiet uch chrge ditributio i to be deteried. Much of wht the ethod of oet i ued for lyi d deig of te. A te i thi, it i poible to view how it c be repreeted lie.. Electrottic proble q( r ') V (r) = ource dl ' ch rge R If we kow V ( r ), we c obti the electric potetil everywhere. If we ited kow the electric potetil but ot the chrge deity, () becoe itegrl equtio for ukow chrge deity. We will ow olve thi proble uericlly for pir of prcticl eple, the chrged wire d plte [] [][3]. Where r '( ', y ', z ') deote the ource coordite, r(, y, z ) deote the obervtio coordite, dl ' i the pth of itegrtio, d R i the ditce fro y poit o the ource to the obervtio poit [][3][4], which i geerlly repreeted by Equtio () i ued to clculte the potetil tht re due to y kow lie chrge deity. The chrge ditributio o ot cofigurtio of prcticl iteret, cople geoetrie, i ot uully kow, eve whe the potetil o the ource i give. It i the otrivil proble of deteriig the chrge ditributio, for pecified potetil, tht i to be olved here uig itegrl Publihed by: PIOEER RESEARCH & DEVELOPMET GROUP ( ( ') ( ') ( ') R = + y y + z z () ()

2 IJREAT Itertiol Jourl of Reerch i Egieerig & Advced Techology, Volue, Iue 4, Aug-Sept, 3 ISS: 3-79 equtio pproch [][3][4]. Ad our equtio i Fredhol itegrl equtio, i geerl, thee equtio re writte b g( t) = K(, t) Φ( ) d Where the fuctio K(, t) d g( t ) the liit db re kow. The ukow fuctio Φ( ) i to be deteried; the fuctio K(, t) i clled the kerel of the equtio. The oet ethod i coo uericl techique ued i olvig itegrl equtio uch i equtio 3 [][] [5]. (3) he obervtio poit i choe log the wire i d the chrge deity i L repreeted log the f( ')d ' = f( ) + Lf( ) = f ( ) = urfce of the wire to void R(, ') =, which would itroduce igulrity i the itegrd of (4). 4. Thi wire egettio We erch to trfor equtio (4) ito lier yte of equtio, d pplied the ethod of oet. Let ubdivide the wire ito ub eget ech of legth = L /, how i figure [][3][4]. We ue tht the chrge deity h cott vlue d piecewie over the legth of the wire [][][3]. 3. Fiite tright wire Coider tright coductig wire of rdiu, d legth L ( L >> ) locted i free pce log the y i, how i figure. The wire i give orlized cott electric potetil of V. Our gol i to deterie the chrge deity λ log the wire uig the oet of ethod. Oce we deterie λ, relted field qutitie c foud [][3] [4][5]. Fig. Thi wire of cott potetil. Fig. Thi coductig wire held t cott potetil. Mtheticlly, we write thi q( ') α f ( ') Where f ( ') = = (6) re ukow weightig coefficiet, d i et of pule fuctio tht re cott o oe eget but zero o ll other eget ote tht equtio () i vlid everywhere, icludig o the wire itelf ( V = V ). Thu, chooig the obervtio wire log the wire i ( = z = ) d repreetig the chrge deity o the urfce of wire, t y poit o the wire [][4], equtio () reduce to itegrl equtio of the for L λ( ') = dl ' L (4) 4 πε < < R(, ') Where ( ) ( ) ( ) R(, ') = ' + ( ') + z ' = ' + (5) ' < ( ) f( ') = ( ) ' ' > Sice equtio (4) pplie for obervtio poit everywhere oe the wire, t fied poit kow the tch poit. L λ( ') = d ' 4 πε ' Publihed by: PIOEER RESEARCH & DEVELOPMET GROUP ( (7) ()

3 IJREAT Itertiol Jourl of Reerch i Egieerig & Advced Techology, Volue, Iue 4, Aug-Sept, 3 ISS: 3-79 The itegrtio eetilly fidig i the re uder curve, if i ll, the itegrtio of f ( ') over < < L i give by (9) Where the itervl L h bee divided ito uit of ech legth. Ad the wire re divided ito eget of equl legth how i figure, equtio () becoe πε 4 = + + L + () Where = L / d we uig i equtio i tht the ukow chrge deity λ o the th eget i cott. I equtio () we hve ukow λ, λ, K, λ cott. Equtio () ut hold t ll poit o the wire, we obti iilr equtio by chooig tch poit t,, K, k, K o the wire. Thu we obti [ V ] = [ ][ λ ] Where ech ter i equl to L f ( ') = d ' = d ' ( ') + ( ') + ( ) [ V ] = [ ] [ ] (5) M With M M M M (4) 3 3 = 3 () (3) L = L = M M M M = + + L + 5. Mtri of chrge deity () We re gurteed to fid uique olutio for ll vlue of the equtio (), we y therefore epre the bove yte tri vector equtio with = b. Ad the,, K,, K re plced t the tch poit k ceter of ech eget [5]. Equtio () c be writte = [ λ] Publihed by: PIOEER RESEARCH & DEVELOPMET GROUP ( λ λ = M λ (6) (7) I equtio (), [ λ ] i the tri whoe eleet re ukow. We c deterie [ λ [ ] λ] = [ ] [ V ] fro equtio () uig Crer 3

4 IJREAT Itertiol Jourl of Reerch i Egieerig & Advced Techology, Volue, Iue 4, Aug-Sept, 3 ISS: 3-79 rule, tri iverio, or Gui eliitio techique, uig tri iverio. Where [ ] () i the ivere of tri[ ], i evlutig the digol eleet of tri [ ] i equtio () or (5), cutio ut be eercied. Sice the wire i coductig, urfce chrge deity σ i epected over the wire urfce. Hece t the ceter of ech eget Equtio () ow becoe λ log L λ log L = M M M M log λ () V ( ceter) = = Auig Where σ = π σ. Thu, the elf ter = re L π / 4 πε / ( ) σ dφd π / 4 πε / ( ) / ( / ) σ + + = log 4ε / ( / ) σ d + π σ V ( ceter) = log σ = log (9) () 6. Reult Coider thi coductive wire with legth L = d rdiu, uig equtio () d with vritio of ( = L / ), Mtlb code c be developed, the plot i how i figure 3. It hould be epected tht ller vlue of would give le ccurte reult d lrger vlue of would yield ore ccurte reult. However, if i too lrge, we y hve the coputtio proble of. The cpcity of the ivertig the qure tri [ ] coputig fcilitie t our dipol c liit the ccurcy of the uericl eperiet. Chrge Deity (PC/) Chrge Deity of Thi Stright Lie for = Legth (c) () = log () Publihed by: PIOEER RESEARCH & DEVELOPMET GROUP ( 4

5 IJREAT Itertiol Jourl of Reerch i Egieerig & Advced Techology, Volue, Iue 4, Aug-Sept, 3 ISS: 3-79 Chrge Deity (PC/) Chrge Deity (PC/) 9 Chrge Deity of Thi Stright Lie for = Legth (c) (b) Chrge Deity of Thi Stright Lie for = 4 6 Legth (c) 6 4 (c) Chrge Deity of Thi Stright Lie for =4 Chrge deity(pc/) Chrge deity(pc/) Legth() Methode of oet =, =L/7 (e) Methode of oet =, =L/ Legth() (f) Chrge Deity (PC/) Legth (c) (d) Fig. 3 Chrge ditributio o tright wire t V () =. (b) =5. (c) =, (d) =4 C hrge de ity (P C / ) 4 5 Methode of oet =, =L/ Legth() (g) Publihed by: PIOEER RESEARCH & DEVELOPMET GROUP ( 5

6 IJREAT Itertiol Jourl of Reerch i Egieerig & Advced Techology, Volue, Iue 4, Aug-Sept, 3 ISS: 3-79 Chrge deity(pc/) 6 4 Methode of oet =, =L/ Legth() (h) Fig. 3 Chrge ditributio ito wire with vritio of rdiu. (e) =L/7. (f) =L/9. (g) =L/3. (h) =L/4 7. Cocluio I ll reult, we how the coputed chrge deity o the wire uig, 5, d 4 eget, repectively. The repreettio of the chrge t the lower level of dicretiztio i oewht crude, epected. The icree to 4 ukow gretly icree the fidelity of the reult. Uig the coputed chrge deity, we the copute the potetil t poit log the wire. The potetil uig chrge eget h voltge er the epected vlue of V; it i ot of cott vlue, epecilly er the ed of the wire. Our how the potetil obtied uig 4 chrge eget. The voltge i ow erly cott cro the etire wire, ecept t the edpoit. Becue we ued uifor eget ize for the wire, the chrge deity ted to be oewht overpled i the iddle of the wire d uder pled er the ed. A reult, the vritio of the chrge er the ed of the wire i ot repreeted ccurtely i the ceter, d the coputed voltge ted to diverge fro the true vlue. Relitic hpe hve irregulr urfce feture uch crck, gp d corer tht give rie to ore rpid vritio i the olutio t thoe poit. I ttept to icree ccurcy, it i ofte dvtgeou to eploy deer level of dicretiztio i the re we epect the ot vritio. Referece [] Mohed Louzzi, El H Aroud, He Ytii, Fti hr Bghli, Yie Lkhl Vritio of Chrge Ditributio d Cpcitce o Thi Wire Uig the Method of Moet Itertiol Jourl of Mechtroic, Electricl d Coputer Techology, Vol. 3(9), Oct, 3 [] Priyk D. Ptel, Puit G. Ptidr Study of Method of Moet for Electrogetic Modelig of Coductig Structure tiol Coferece o Recet Tred i Egieerig & Techology, 3-4 My B.V.M. Egieerig College, V.V.gr,Gujrt,Idi [3] Wlto C. Gibo The Method of Moet i Electrogetic by Tylor & Frci Group, LLC [4] C. A. Bli Advced Egieerig Electrogetic Joh Wiley d So,99. [5] Elbert Hubbrd uericl Method Chpter 5 [6] D. J. Griffith d Y. Li, Chrge deity o coductig eedle, A. J. Phy. 64, 76 (996). [7] R. H. Good, Coet o Chrge deity o coductig eedle, A. J. Phy. 65, 55 (997). [] J. D. Jcko, Chrge deity o thi tright wire, reviited, A. J. Phy. 6, 79 (). [9] Je R. gel Solutio to the Sttic Chrge Ditributio o Thi Wire Uig the Method of Moet Deprtet of Electricl d Coputer Egieerig,Uiverity of Uth, Slt Lke City, Uth, Avril. [] O. F. de Alctr Bofi d Dvid Griffith Chrge deity o thi tright wire, reviited Reed College, Portld, Orego 97, Septeber. Publihed by: PIOEER RESEARCH & DEVELOPMET GROUP ( 6