EE757 Numerical Techniques in Electromagnetics Lecture 9
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1 EE757 uericl Techiques i Elecroeics Lecure 9
2 EE Dr. Mohed Bkr Diereil Equios Vs. Ierl Equios Ierl equios ke severl ors e.. b K d b K d Mos diereil equios c be epressed s ierl equios e.. b F d d / C d F d d / C C C d F C
3 Gree s Fucios Gree s ucios oer sseic o coveri Diereil Equio DE o Ierl Equio IE A Gree s ucio is he soluio o he DE correspodi o ipulsive ui eciio Cosider he diereil equio L = here L is diereil operor is he uko ield d is he ko ive eciio For his proble he Gree s ucio Grr is he soluio o he DE LG =r subec o he se boudr codiios For rbirr eciio e hve Φ r G r r dv EE Dr. Mohed Bkr eciio volue 3
4 Gree s Fucios: Eples Obi he Gree s ucio or he DE subec o = o he boudr B The Gree s ucio is he soluio o G G c be decoposed io priculr ierl d hooeeous soluio G=F+U ih F d U sisi F U 0 F Sichi o polr or e e 0 F Al C F A is obied usi li dl 0 A= R EE Dr. Mohed Bkr / / Φ R 4
5 Gree s Fucio: Eples Co d The ehod o ies c lso be pplied o obi iiie series epsio o Gree s ucios Cosider he cse o lie chre beee o coduci ples G represes he poeil due o lie chre o vlue.0 c/ loced h- +q Oriil proble EE Dr. Mohed Bkr 5
6 Gree s Fucio: Eples Co d +q h+ h- h- h- -q +q -q A iiie uber o chres is required o ii he se boudr codiios +q EE Dr. Mohed Bkr 6
7 7 EE Dr. Mohed Bkr The poeil cused b c/ lie chre i ubouded ediu is ive b Usi he iure e coclude h he Gree s ucio is ive b he iiie series Specil heicl echiques re usull uilized o su such slol covere series l 4 V Gree s Fucio: Eples Co d l l l l l l 4 h h h h G
8 Gree s Fucio: Eples Co d The Gree s ucio c lso be epded i ers o he eieucios o he hooeeous proble As eple cosider he ve equio 0 k Subec o 0 or 0 Le he eievlues d eieucios be k d k 0 The se is orhoorl se i.e. * dd S i i 0 i o B EE Dr. Mohed Bkr 8
9 9 EE Dr. Mohed Bkr Gree s Fucio: Eples Co d We he epd he Gree s ucio i ers o he eieucios Bu s he Gree s ucio sis G G k k k Subsiue or G * * ds k k i S i Mulipl b d iere * i k k i i i *
10 0 EE Dr. Mohed Bkr Gree s Fucio: Eples Co d Usi Gree s ucios cosruc he soluio or he Poisso s equio Subec o V0 =V =V o=v b=0 Sho h V V si si b b b b b b A si si b d d G V 0 0
11 Ddic Gree s Fucios Ddic Gree s ucios re used o epress he siuio here source i oe direcio ives rise o ields i diere direcios I eerl ddic Gree s ucio ill hve 9 copoes G G i z z G G G k EE Dr. Mohed Bkr z iig G z ki ig G For ui source i he direcio e obi he ield E=G.J = G z z k ik G For eerl source rbirr disribuio d orieios E z G z z. J V i J G zz kk i z z G z dv z k
12 Ier Producs The ier produc o o ucios is sclr h us sis he olloi codiios: couive h h h disribuive * 0 i 0 * 0 i 0 Eple: d 0 EE Dr. Mohed Bkr
13 Adoi Operors For operor L e soeies deie doi operor L deied b L L For he DE d /d 0==0 We uilize he ier produc L / d d d L d d d d d 0 d d d 0 0 d d i 0==0 e hve L d L 0 d L=L 0 d EE Dr. Mohed Bkr 3
14 Mehod o Moes MoM MoM is obii soluio o he ihooeeous equio L = here L is ko lier operor is ko eciio d is uko Le be epded i series o ko bsis ucios Subsiui i he equio e e L L We deie se o eihi ucios Oe equio i ukos EE Dr. Mohed Bkr 4
15 5 EE Dr. Mohed Bkr MoM Co d L equios i ukos Tki he ier produc o boh sides ih he h eihi ucio e obi I ri or e c rie l L L L L L L L L L l
16 6 EE Dr. Mohed Bkr MoM Co d The uko coeicies re hus ive b l The uko ucio c o be epressed i he copc or l - ~ ~
17 MoM Eple Solve d /d =+4 0==0 usi MoM We choose he bsis ucios s = + = is hus pproied b Also e choose = = Glerki s pproch our ier produc is We hve L =d /d =+ - Sho h l =< L >=/++ d / 4 EE Dr. Mohed Bkr 7
18 8 EE Dr. Mohed Bkr MoM Eple Co d For = e hve l =/3 =/30 =/0 For = e hve For =3 e hve Ec soluio is obied or =3! 7 / / 30 4 / 5 / / / 3 3 / /0 70 5/ 7 / / / 5 3/ 5 4 / / 5 3/ / 3 / 3 3 / 0 / 3
19 Tpes o Bsis Fucios Eire doi bsis ucios re deied or he eire doi o he ucio Subseciol bsis ucios re deied ol over subsecio o he doi o he ucio P Pulse ucios i D EE Dr. Mohed Bkr 9
20 Tpes o Bsis Fucios Co d T Triulr ucios i D EE Dr. Mohed Bkr 0
21 Tpes o Weihi Fucios Recll h EE Dr. Mohed Bkr L I e choose = = Glerki chi L I e choose =r-r = Poi chi r r L r r L r r The o sides o he sse equio re ched uber o spce pois
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