Download Analysis of Electrode Potential and Dielectric Problems in Three-Dimensional Fields and more Assignments Mathematics in PDF only on Docsity!
-Ass i~nment- T Hi'~h voff;~ enBineerih0t lllGIAO~Oi ~ fr Pron·, ja~i,
- '.J' \lost.. (^) u _. note f,n,te element rnethcd +;. t1on. IDi.scoss the procedo. or e~lut1on of +'ield distr,bo. d,scoss ·t d re ossoc,cxted wrth this rrethru a""-' I s C\ VO.ntq}'cS C\nd 1;rni'totions- ro
e^ 8·^ +in·,te. e'-ctnenu^ +^ meth--' LO^. ,s w,^ 'd e~^ l^ tJSed. 1n the numcrical solution ©/.'
lectoic .rie k) probletnS; 0ind :ecorne versi r--oRJ !or, lf\e -finite e\ern:nt a-ro l~sis oF on~ problern ihvolves t:osiCDl l_y iDU, Steps;
(O.} f'ini te elernen-t di SC reti~ti on;
.1 I\Xl:f (^) nom~~ [l elen-x::nt nombe 0 I 5 ' '
I I. I ~ , ✓ ~ Actu~lbourtkxa~ A WWPoi rtire kw ndC17JCf
A tief Picol -f'in;t,e e[eme~ 1 IL, u_;r_,__,,v,sron~, ,1.....,.J· • • LJt -r cm · 1~!01-
domo·,n ,o stoat. w',th J 7ne rJholE' proble<n domo\h ·,s ficLkioOSI~ di\9'ided ·,nto Srm II o.reos /volumes col 1ecJ elements. 7h: i:otentio I , which i5 unknown 7Ff(jOUfl hoot the pro ivlem do mi, n; ·10 Owwx ·,!1')CTbed ·,n eoc h oF the S'.:' e \ements ·,n -te-om5 & -the potent i o \ o:t th e",o seatices co I led nodes• -A's o, .5eSlllt al' -this the pm:enticd -function w'ill be unkn0wo 1 ot-the rodes· me elements dei^1 1?e -bhei1r no.ITT?& lriw~h-theia tn ~ d'. C .__ • l,...r_ 'I{" e \ernerrtS in one 'f11".'rlS^ Ioh .1 D) -t er io.ng lJ loi onoft' , ,e .=u (^) , and i q,uodo'do;teiro. I elernem:.5 In @D, ond -tebcro. hedoOn o.nd re,:cetha:l'l(Jn eternerrtS -fo'o OD pi-o blernS · ( i:, q~ni~ e:i,tffe·1 ons **,r;e_** potential Ve within on e!emeni i5 -Firs-I: oppMOtirnl'.lted JdOd then irrten1rel~ted -t:ctre r~nt-ia I d,sti1bution':> in
voi,ou~ elernents ouch -tnm -the patent i o I i e, cont .I nuous , oc~o$S \nteo-elernen--t boo021o'oie s • me oppro~rrute ciofation -Po'n th::^ Laho^ le^ Ji~n^ then^ fceCJDmes. v(':x,fd) = [ Ve (x; 'ti) -) (i) e=I Ve C'X) ~ ) = a-+ bx-t C(j )@) Ve (X; CJ)_:: G-+ foj +c~ +d:x~^ --1@ 71tPica I -to1on9olor^ e!erneni^ ~^ -6-he^ locQ^ I^ mode^ numberin9^ 1-~-
3 must proceed CounterdoctwiS::' as ird,ccud^ °d^ the^ a1row
Ve,3 ( 'X:, ;C/)^ ) 3 !9er r lA C-tr;^ ct,)^ l Ve 1- C 1.21 G/"i ~ -;t Consider^ o^ -t..i;i-Pl^ ca^ -t'oio.niu^ lcxo^ e^ lemeni^ 0f-oL<9n^ 'in^ -Pi·~^ ..^ The Pote nti C\ Is Ve i i Vm 'cmd Ve 3 ot nodes 1 i 2 ond^3^ cxire obtoined -foam^ eq^ Ve^ C^ 'Y-1^ CJ)^ -^ ~-^ ·;^ 0is ve 1 1 'tr Cffr IA Ve2 -^1^ ?f 2.^ ~2-^ b^ -^ ~^ ® '{,3 1 X.3 Cef-3 C o. 1 b J c ooe de:teominfd .fo0rn the obo~ Tue co&-f2^ ·10.en^ t equ0t1on"' 0i s ~l -f Ve °'
')(l Ve^ ) (5)
b
C 'X3^ lit3 Ve
C
Ce) I l C (e) 2 I c(e) JI Ce) C c(e) 23 C Cc') 33 7he rnotoi 1- ~f1 \9en D@©tJe, ,·e rotfY)ail I~ Co. lled QS e lernerrt Coe#i Cien-t rnotaix : 7he O"XA--tiri1 elerren-t Cu (e)^ o+^ -the^ coe#ici- -en-t tnatoiJ. lo Considered ()-b -tJhe CDuplih~ b/w rrrJe0^ i'x--J
(c ) -AsseMbling ~ AlI elements; ,
-Hovin3 cons·1cered^ o^ --fidPiOA^ I^ element)^ -bhe^ rext^ oto~e^ io-to o.ssem ble oJl 8och elerrx:nts ·,h -the oolution --91ejion^ · "1 :T We ~ ~ we =- 1/ 2 i; [ vJ CcJ CvJ e= I [v] ~ v v') I 1 I Vn Cd) 60 \vin9^ the^ F?eso^ 1-t·1^ n~^ equCA+1ons; 8w/0v,:: Sw/6v 2 :^ ___^ _ _^ Sl;J/Ovn=O 6w/8vr..-=O i-fk= 112; -~- Jn In caene'oO r J 8 (A) I 8 vk =-0 \eods -io 0:: f Vi C; t e= 1 k= 1/2) - -- n me odvo.ntoges ©+^ -the^ +inite^ e^ lernent^ method 1· mode \in~ ~· -Ad(A pto 011 ·,t ~ 3· Accuroc~ 4 · Enond()r ie S :J,^ ··n^ rn^ e-^ de^ perce^ rrl^ Si^ ML)^ lcxti^ On €· viSuD'i~on.
· e~p\oin brie~l<:J ino-R:wstePS -the procedure o.ssocioted .. th Cho.~ 8im ulafi On rr-rthoGl · J;) i Sc USS its od vorwra e5 'k
t tM itati on&
➔ When Conductot) io ect:"C.°ttecl b~ o.n o.p19l'1ed \9©1t()cae) crxXt?feS oppeo.^0 (rt) the 8ur+oce @12 C-©nductoa ➔ lnese cho~s procloces e\ect~ic +'ield O\Jtside Condoctmi, o.t SG1rne -!;' tn ()'1nto.1ns cordctoo cx6 eqpipotenti°'l· -> oim,do.rly, c,0hen dielectoic 10 etcited ~ 0in exteoro\ +,'eld) it iets Polorl-y:d Ji ,e. t cmr~ed PAot'1c\es 0+-ite m(s)fecules @-P-the d,ele:ta-; ~ 8 hl-i'te d -f' ~ n, ·tf:d'il ne u-to GI I suxte to 'P rodlX.e 0. vo I ome c,.fl dipoles · -> in essence) ,t ·,s poss',ble -lo"1eploce this volume ~\o.ritymiot, 0:d ihe c ho"~7JBCY.· &JatoiCQJ , -> cstv1 is one ~ oornerico.l o.no,l~is to co.lcolote electo'1c +;e\d U'ii~ boundo.r~ rnethcd · -> 1he basic C(Oocept d -bhe CSM i6 to replo.ce d1stcr,buted C~- -rie ef2 C(Onduct065 ond the p;IOOiyt ·,on c\ronws (On the die\eC;boic '1ntertoces be& o \CXcfijt' number~ f't'ct'rtlous discrete cm~~s (^). ➔ lne Sho.~ ef' ·1rdividLU 1 +ict·1orou5 choa,geS CJo' ''simolotiw cb.(x~es'' eon Ice cha.sen nS p0in-t, line C+'i'n·,tt:) cro 'oin~ C Ct ico lo r )-foi M fv y0 i :- ~ PJ· Q. ::- j: Q I j
a •' I ...-. ,... - + - I .',., '"'"" ,.. ..,_ ,-. ....a..J_ - • • ~ - (^) fl '°' ~ C~lfun ~rie..\ll1t Gin ·1mpro~ ve~iM ~ cr.0'1f~e$ Oimo\crtion ~) th(x:I <Jr d,scu.ss ·rts odvo.n-w.9eg o._nd nm·,~H©ns• &Jr.fovc.e, cho~ c§imLJla.ti on rnethru (SSM) c,r &'DmPAr'lf- eleme~ rrethtd (B~M) ~ e lectoiC +\e Id: lf)o~h --tne cho.~ s,·mulo.tion ~eth©? io known -fmr rts o.cx:oroc~ and opeed 1 it r"s not veryJ e#ic1ent rn Cose ~ mu1t·1 - dielec-Lo--ic. PhD!oletn 5 ond verc.; TFiin electoi,des, vSfiich me Mten u.sed to Cmt;mr,/ IAl1d ~ -the electoic -Herd stoet1ffUh m condenser bushing) t~ons+'('.Ylmers etc- Soch p"Oblerns a,.n be SCJIIJ€.d usi ni the O()?J.fuce cho.~e t.Sr'm uta-ti«m llr6hod ( S0 M) Which is odso CC\ I I ed 800 ndmrcq- e f e me t1t method CB~ M) · (p (~) = i (1Dittfo) f 8 (?1) ¢' (ft) dF{;.) {[)(~)" I '; (2 cnrto ) J fs ( 7 ) V ¢ 1 ( ~ :x) jd PC-x ) 7Fe. y. (^) ' 31esoltm~ qs-tem ~ eq,Lnfi on obtttlned us·,n~ --&he Source -furno lati(t)n u9H I o.~n ic)e 0..Stt /Y) IY)~i C. Btt ,So/Vi "1/ This cS~ote,l'Yl ~ f'4uaHmns, the un/GMl<!)n \IQ lueo. ~ **~he chr>~e dens;tlef Coin be_** -ti) 00
d. roce ~ ch,oJ!,ff cti~sto, '8u"t1 on ,s khown , RJte nt-i o I on d
elechic ~eld v0<loes CDh be Co.lcolcrbE'd ToOO~hotJtthe doMDin USi' n~ fA,, s· cgt'l9etrv ~~. .. A::hK\nto.~.s rx dis.ad VD nto. 8e& o-f 13G M (o.)--Mv~rrt ~es: (BJ ~iSC\dvarrtages:
l•J te$JJ do.ta ( ·, ) un~M ·, f io y M~thervtt&c_s
(ri J less Time ('ri J Non- { ,·rmr -f2 (ou5 foo©~E'P'6.
Lii1· ) Less un W)nbe.d .1 ntor-n1crtion c ri'i') -Pu11 ~ popolcrted mat,. •i-i t,v ) proc~ MdiSc'3B·t iicxn·on (V) tu3 t Cof:>t l lef ( "') Or-,eh dorro.in
4) 1\us-toate +1hi-te iJi#erence -M;h~d -R,-~e~;lc.mo~ ~--f!~I; d~ibution, £Jiscoss -t-he proce,dut-e ossC,CWted w·,th -thi.s rn~ o.rd discuss its o.dvonto~es o.nd fi mitctt\Dn.5 · A· Fin,te i<Jift?eJlence t--1\ethod ( FDM):
J-e;f; us ~U/M.e. tn;at vC9(~ vx,',a.tions is o-two-dimensiohaf
pro bl e rn i, e .it voo-ie s in 'J-Cj plo.n e o. rd rt cbe<:, not vmtj o.l©n'ij ;i-Co -o?dinote o.rd \et us di\9ide the ,nteoim Mo. co~J:> ,secti©n ~ t re 3ie~ i O n where the i=ute nt i ~ 1 dish", butf on is Jieqoi"1!-e::l into &:i,uOJle-3 oP len~th h on 6 Si de os .S.hoLvh in-tJ·i · .._ ~
. -Hen~· £0-place)s e9tLJDJ:;10n^ I^ n CD?Jte.sio.n CoO?J-di-
tS ~\JBn OiS i }'l V ::- Jv + _Q°_ + a, ::: 0 -°"'[1] Q){, 2 OLf 2 01-2 7 ~h~ 3 p ~t
J;
4 y ( v')_ /_J C - b °' V. d Vo V . A ?&rlion 0"¥ o --'ieaio n C'orrta·,n·1':) °' @-dir,m.
- \SiOh() I p:tenti o. l ~ee,d dll9e{QQ,&. --As~orni n~ 7he 3ie~eTu to be c~~ -Po-ee vQ:: o oo v E::: 0 0nd ~^0 t0Jo-dimension0 I Si tuoti on
.1} '.Fin•t ' e D&enm rnettx,& ( r:DM) @Jr· inite element method C t=fM) "3) c~ 0i rrKJ I()j;i on rriethod ( C3M ) ; ) Oor-fac.e, ch~ 3 i m lJ ta.ti on rrd0@c;Q ( SS M.) C 61) J3~oncfo,i-u ervt0vit--M~ { J3-f M). (^) - -a-.
Fi h '1-ie ~·&en_gz, method; -
~ \he rneth©& i c, -f'oond Su 'rt:rli le On I Cf --161 0/ di rne n 8iorJci I^3 &tmrneiiri cal, --A'e£dl u9heJ1P, (), cJJ.oecJ;_ -So I uti on i.s ~ t 6 le. -> 'r n 0oc:l e,i -to vJ 6'!-k ~ ·,~ I lX71 3 - ~ --?-i e I d so ·-rr-o.+ Tho.Jae. YlOdei 00i2, --fi;t'W- uperu Wun~
1 lxco~^ e')[toernel.!:J^ di#l-
-co 1-t · f I n ·rte demerr+ fY';ei-hod. ;- ➔ 7n 1 ·s tv\eihod is mt bOcSed on SEel:::i n.9 7ne d ioecLSoluticn ~ Jnp\o.CQ, eq,U'.Won cu, in Cetle ~ f'Di/1 1 lnsteod ·in .+'.ini .je element method l.}58 rs ~ ~ -f;f.,vz, --f'e,.c& Th~ in an electoosta-ti c -l\eid -the -tDtal ene,i~ encloscb' in me whole --Pie Id 31eyon o.cq,LJi~ o. TYJinif'YlUfYl Wice· c~ c5\mu\aflt!Jn Mcth~- ->-k - ~dl- b'lf -t\1Q. no.trJe \tSeQt I in -this MdtiOd J the (^) d isloi 6~ ch:XJigu, on -tne su~ce. 0'0" 0 cordoc:Ml /el~ m cUe2ei2hlc tnteii~aJ.J i.S ,Sirnoloieol b:J 31e-plocin9 7h!l.be ch~ ·t::Jtt Y] diSC)]ei: ...pi ct itious \rd i vi dt,o. I c ~ oO-'ll°' niae/9 3oit-o. bI{lJ- insick, the C©ncluc,W11, &i, ()Ui;c)icQ_a. -tho- SjXICQ. in which -the -Pield is iD be c~ SJ'i)+O ce., ~ 8 i rvi u lro;io n McthC>d ·, --- -1 7n oo 9 h The L hcwge 8 i ff1U lcui on Mefhod io kr(JLO n -fa r ·r1:i o.ccorC1c~ 'X Speed) rt is not \3E'A~ efficieni in CO.Se oi= mol-ti- clielectaic problems '¼- ve,i ~ ""Thin eleciocJc:W>J which OJ!e o/'ten (^) osetl to contw I cwd ~ t-he e k:crt11c -Pie Id strergth in Co ndm2&1 husi ra et'·
':l-b) eT plo.i n with neo-t d 1' o 'a'I 0 tn s -the 0pp^ Ii^ cctti^ on^ ~^ cm elect-mtf(tic^ -to.nk^ -!'or^ evC1lootin9^ ele"Ctoic^ -field^ distribution^ in dielect~i c med i um or e\ect?ri _c C_ I eqjJ ·, pme ni-5 · t~too rc.;tiC Tt1nk · d. 4-,...,,..: b t · · n corn ple:x 3-D ~ o.~in9 e!ectoi c +,eid^ hS,^ o'^ u^ ,on^ ' 8 i tuations, o.mlcttica^ I^ rneth:i^ c/5^ 0~^ ~,nsl'iiDlole^ ·^ Tcvci^ ~ 0 P~-
- h. eJ:=imentCI^ I^ cm.,\o~^ C1nd^ rvmbo,m^ iiecr{lq,Ll"i
oc e.& ,n use 01re) , .... v d^ ·^ "^ ......^ ,^ ·^ -th?
.. ,, , ,e2, f--o. \Je (^) 0 1'.,ie evil~ been ,^ seas= 1 n 7ne numeorco [ -tee hni--11'-" , , st;LtJu c,.00l09 technrqpes. (^) espec,o.11~• Previous Sectimn. lt)e^ nol,U^ .::] the U5e ~ electoE>l0tic -tonk.· e\ectwl~e e\ectoa^ deS h .,.-:J I =- E;I^ I^ ----^ ----=----=-=--=- f:2- I
l0) Sirnulot ion of 0 cylirder,. plCt re crnf i901ntion e\ectw\Yt i c -tC\nk · I L _____ _
r--- r- - - , ...___- -^ -
ins_, !Mi n<a--
rY1C\ t e'7ill^ I
in the
3-D -fie\ ds (;.){;h roto.t i ono^ l^ Sid^ fY"^ rnet'lf~^ C0h^ be^ .?iecd^ i^ I^ t:1^ 0imo^ b-
- ted in o. wediae s ho.~ d um k w her e0..<i, -ft,l fie Id:i t'9tt h^ no .o/Otof1 on~ I S~ mrnet o <j one ho. 5 -to -'le sci"rt to rnuc 1-, (T('re c0rn fo~m5 oP 3- Q 0·1 MU \CTT '10 () · (:9Pi pot.en-ti _o._ boundD.ile5 et"'1f^ .:>icprE'Sented^ in^ the tonk b:J SpeciC1 II~ -foorned sheets t9f^ Mete^ I,^ --tt"'l^ e:i<0mp~ o.^ 61~^ le^ dielectoic^ pwbleM^ Ouch^ M^ Ci^ --thoec^ c&ie^ Chble In'.)^ '=1^ be^ .9iep1~^ nted^ mi^ Y1)^ _C_^ ~^ --/:t1h^ k^ C't6^ 0h!>c.vn^ in^ -A^3
m\tlra v 1 o let ~°'di an O n lren. ex. 't ~ :::- noe n ) (/..,1,.. ~ :: (n 0 +n+^ e ~ ho.ve eq,uoti on -too rn -rown3erds ~lrncx~-~ ,oni^ D°t1on^ C€ ~ = t"b e (){d > CD
'De eciu ctu~ -to
'1l ~ ( n+ f\ Jeoicl ~(2) e ~ + n+ - 03le^ -bhe^ homber^ of^ e\ectoonS^ ot^ -the
m+hodc
n+ = [ ~ - Cn +n+ ')] 75 7 @ Y"l+ l 1+-8) '.:C ~ • i-(1-- ~ =~(~-no~ 0~1) Put- vo1ue^ of^ n~ 1 1n @. nd^ ::^ [no+-^ ,&(~-ho)]^ ~d ( 1+1) e Qi= (6.+ ~ ·] otd e Q +i) Y'ld ( I .+ -£ -j. Po< d ) ::: no. ectd nd : Yb , ectd
[ 1 + 1 (r- eotd )J fD\Jerj(2 by_ +i r1 e. t'1J ho ,e~ T:: [ i+'IP (l ~~d )/t I :: , ectd -o -^ ~(T) I - 1 r <°old '7- \ J I I
♦»B. (a) is ~nM-h:13 c~n+ ~wn ecilXtf-ion c:Jue -t© pri~~ l)nd &'ccnic{Y:J Ci) ,oni~on · 9:}-, Q ) en,__ n • h 1 ~ lo w ~ de:!uG2 -fie CRJrd ffi01")6 --f'a, ~ne ~ sc enu J:>o. Sc_rerl'.S l C\ tu · ·fa~hen\s low- '' -At eonsto.n-t -:f.m peocrtore in o. Lm i.focr rn -fie id -the Spo.obn.9 r=ott:nti ar Of f?S daperd Dn ly upon +he pro:lo ~ gct°:J pr~or e <Y<- electrode sS po,ci l<l C3 'P- d n_
-Per breokdown ·
' X ( ecxd _ 1) ::- I -~ I. € cl is _l_ -Puncllori c,l'-'f¥e.9,otJVe p on d -the so.Tl O p So (^) ; ~::~( i) --H2J --A-Y)d & ,s o-funcnon cl-the "1cm'o £ p ZS. "'i( i) ~(3) ~~ onrferc:ir1 -fie td o-t breotdoLJ n; Vs =- ~s d s
,·, ~s:: Vs -A"1) Oub in(2)~(3) Then SobDII in{,)
d ',' ~ ( ~dJ ( e + ( Pis ). pds - I ) ~ I v 5 ::- +c p,d ) th: JJelotion Vs ~ -re P·d) is known Cl) pnschen 1 s bw
P.d I I --h 0^321 ('^1 '
N2.^ 2S^ l
o, £, I
02. 450 (}^
'
