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# Numerical Techniques in Electromagnetics EEL 6481

University of Central Florida

GPA 3.7

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This 26 page Class Notes was uploaded by Isaac Hauck on Thursday October 22, 2015. The Class Notes belongs to EEL 6481 at University of Central Florida taught by Staff in Fall. Since its upload, it has received 35 views. For similar materials see /class/227662/eel-6481-university-of-central-florida in Electrical Engineering at University of Central Florida.

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Date Created: 10/22/15

U CF Maxwell s Equations in Phasor Domain 6 Phasor Form Representation of UCF Time Harmonic Signal A COS D1 Time domain Re Ae WW Re Ae New I U A6 Jcp Phasor form in frequency domain To go back to time domain 625 ReUej 6 U CF Derive Equations in Phasor Form V x 6 a g3 at 6 ReEe m 93 ReBe m jwt 6 93 ReB 66 at at RejaBe Or gt ja ReVx Eela Rej03Beja VXE j0B UCF Maxwell s Equations in Phasor Form ngz a Q VXEZ jOJB 6t VXHJ D Vxfa J03 at VDZ pV VQZO VBTO Only the rst two phasor domain equations are independent Or the last two equations can be obtained from the rst two equations 6 Constitutive Relations for W Time ilarmonic Field In simple media Isotropic linear D 2 SE B H 8 permittivity or dielectric constant Fm Ll permeability Hm Also J c 2 GE CS 2 conduct1V1ty bm UCF Time Harmonic Generalized Maxwell s Equations Instantaneous and timeharmonic forms of Maxwell s equations and continuity equation in di erential and integral forms Instantaneous Time harmonic Differential form 6 g Vx Jl 59 at 69 V 39 9 7 V 39 9 7m ayequot V I jlt Integral form a gSC Ddl fL j ds Efj ds xd1ffsfdsffsfcdsgfLQds QQds3 Qds nl 6021 gjic 39 d5 7 67 vxE M ij vaIJCij VDq il V39Bqmu V Jic jwqev CEdl fjMidsijwfvBds HdljJidsfLJcds jwffSDds UCF Time Harmonic Boundary Conditions Boundary conditions on timeharmonic electromagnetic elds F inite Medium 1 of Medium 1 of conductivity media in nite in nite no sources or electric magnetic charges conductivity conductivity opoz oo aloo02oo Hl0 General J M 0 q 61 0 M 0 M 0 L 0 q 0 Tangential electric eld intensity hxE2 E1 Ms hgtltE2 E10 th20 th2 Mx Tangential magnetic eld intensity XH2 H1JS XH2 H10 hXH2Jx xH20 Normal electric uxdensity 39D2 D1qm Dz D10 39D2qw D20 Normal magnetic uxdensity BZ B1qm 1 139B2Bl0 BZ0 n39B2qm U CF x y z 3fx y z y DX9f y Poynting Vector 5V agtlt3f Eejw Eefw39 t Re E06 y Zejquotquot Hefw Hejw l 2 I Re Hx y zej quot Eam Ee j gtlt Hej quot Hef quot E x H E x H E x He12 E x He 1392 39 E x H E x H E x He E x Hef2 L 2 L 2 L 2 y Re E x H Re E x Hejzw 111116 average Poynting Vector LSQVSReEXH Power and Energy I VXE Mi jpr VXHJiJcjw8EJioEjweE 39 HVXEH39Mi jwuH H EV XHE JI0E39Ejw E E gt 1m xH HV x13 H39MiE JjoE39E jweE39E jpr H v 39E x H H M E J oE2 1200 le sE2 V E x H H M E J W2 12wuH 2 ME 6 UCF Power and Energy ll Conservation of v a x 11 H M E Jf aE2 j2w pH2 eE12 Power Law 4va Egtlt Hdv 5 9 Ex H 39ds 39Mi E39Jidv H01E2 dv j2w mmml elE dv fffVH Ml E thyk dv E X H ds fffVolE 2dv MULGMHIZ ilelz dv P5 Fe Pdj2wV Vm 1717 P ff H M E Ji dv supplied complex power W s V 1 Pe X H ds exiting complex power W Pd ILME do dissipated real power W Wm ffV lelz dv timeaverage magnetic energy J We H eE2 dv timeaverage electric energy J V U CF Wave Equations Wave Equations I UCF 39 39 V X v i p 31 669 V xXjia eW 32 33 9 VXVX a VXIi LVX7 VX i pEVXX Substituting 32 into the right side of 33 and using the vector identity v xv XFVVF V2F into the left side we can rewrite 3 3 as 6 vvamp v2 v X i p395 g 86quot l 0 Eat g 269 J agi M 3253 vv v vx m m 39 a at 862 veV6 evgtv a 1 86 32 V2 VXJtip V 0 8 79 139 E a 32 1 3111 ax 829 V2 V xjio i V mvs EM07M 817 Wave Equations ll UCF 265 a 1V 86 3245 V7 X a e 79 a at a 32 V29 v 1 1v 3 39 92329 x For sourcefree regions fi ya O and ymv O V26 66 826 I a 8797 V2 3 829 3 39ua at us 822 For source free f5 you 0 and will gm 0 and lossless media a O 326 V26 5 82 829 v23 a 32 UCF Uniqueness Theorem I Whenever a problem is solved it is always gratifying to know that the obtained solution is unique that is it is the only solution If so we would like to know under what conditions or what information is needed to obtain such solutions Given the electric and magnetic sources Ji and M 1 let us assume that the elds generated in a lossy medium of complex constitutive parameters is and 11 within S are E H and Eb Hb Each set must satisfy Maxwell s equations V X EMijpr V XHJij Jc jw E 7 1 01 v x Ea Mi jwaH V x H J Jcarijw E 7 13 V x E M1 ngub V x H Ji Jcb jw E 7 1b Subtracting 7 1b from 7 1a we have that v x 120 E ngiIIquot Hb v x H 11 ajw Ea Eb 7 2 or v x 8E jwaaH 5M within S 7 2a V x 8H jw SE 81 Thus the difference elds satisfy the source free eld equations within S The conditions for uniqueness are those for which SE 6H O or E E and H H U CF Uniqueness Theorem 6 Uniqueness Theorem ll UCF Let us new apply the conservation of energy equation 155a using S as the boundary and 8E 6H 8J and 8M as the sources 1 For a timeharmonic eld 155a can be written as gExHdsffVEJHMdv 0 73 which for our case must be SE x 8H ds 8E a jw 8E 8H jwil6H dv 0 s V 74 0139 SE x 8H ds fffyko jw 8E2 jwp6H2 dv 0 74a where a jw a jwe j 0 we jw3 a we jwe 74b jw 1quot M J39M wit jwu 740 6 Uniqueness Theorem Ill U CF If 1 A eld EH is unique when I X E is speci ed on S then r z X SE 0 over S This results from exact speci cation of the tangential components of E and satisfaction of 78 No speci cation on the normal components is necessary A eld EH is unique when it X H is speci ed on S then I X 8H 0 over S This results from exact speci cation of the tangential component of H and satisfaction of 78 3 A eld EH is unique when I X E is speci ed over part of S and X H is speci ed over the rest of S N we can show that 6012 x 8H ds 0 75 G UCF Uniqueness Theorem IV then the volume integral must also be zero or fffVKo jw 5El2 jamsalel dv Re quot8E2 39w39 8H2 dv HI0 stl I J MI I Im 0 jw 8E2 jwil16H2 dv 0 76 V Using 7 4b and 7 40 reduces 7 6 to 0 we 6E2 cop39ISHIZ dv 0 76a V we398E12 wu 8H2 dv 0 76b V Since a we and my are positive for dissipative media the only way for 76a to be zero would be for 6E2 8H2 O or SE 8H 0 Therefore we have proved uniqueness However all these were based upon the premise that 7 5 applies 1 Using the vector identity ABXCBCXACAXB 7 7 we can write 75 as 8E x 311 o da 95502 x 8E 8Hda wirk x a 8Eda 0 S S S If we can state the conditions under which 78 is satis ed then we have proved uniqueness This however will only be applicable for dissipative media However we can treat lossless media as special cases of dissipative media as the losses diminish 6 UCF Uniqueness Theorem V SUMMARY A eld in a lossy region created by sources J and M i is unique within the region when one of the following alternatives is speci ed 1 The tangential components of E over the boundary 2 The tangential components of H over the boundary 3 The former ouer part of the boundary and the latter over the rest of the boundary Note In general the uniqueness theorem breaks down for lossiess media To justify uniqueness in this case the elds in a lossless medium as the dissipation approaches zero can be considered to be the limit of the corresponding elds in a lossy medium In some cases however unique solutions for 10551635 problems can be obtained on their own merits without treating them as special cases of lossy solutions i g i OLCW39NJ Opeytdw I n gyg a opmm or Mewoustww Oman 1 1 vs Wm i 42 421 i 1 t 11 i 01 Witch PWP0 5 14 391 Than I 1 i b O I l 50 MW be Cb 1 1 I I IV I If 44 ng ell ucdvm kas a 39 0f N Mm 0 Cb s q galaHam FD l l I I Mas kl M2 quot 1quot MMCS quot k MKS 7 O MLM 9 MM M5 kl ML 0 My 39 O N SEQ0 MKS o MWU 0 AIM5 N Alexg quotLu 037th lt 134 nvltltJA u lto 9wd n AV US w 4 A l I II 3 3y outst vii xm 439s a 66 51 er Us ge wuw 33435 Vik vwsoz 23125 5 OZ owg Sokuk wl Regan S quot943 doscl or Lomded walem open or MnkowM Prune I a 7 6W 5 Ind 2 0rd D39 f rm39H UQ 57M H MS 2 0 Dr D if 3 aid 314 395 AS E Jisrew 76 a 9x any 97 CUAWC b 446 lt O 5amp LY5Me 96 knferbob c 144 gt0 4 km Pantwuc Lz 44C 0 g quot WWW Mu 5 61M 9 azcb D a 7 39 1 bo C 2 ax ay I 54 4ltO a WM EMF 6 PFC a cb L 22 o b 397 lax t O 3x M H a 39 C w 52 4 W r 7o kYWhWC 9 WWW WW CHM 5W 31 33 E at 0 63 b i0 Liqac 0 2mquot i0 eL o3o PWALOHC 5 de mums 49 3 14gt M449 kost Lomdwy andMm s I Dirvowet Eamolmy waL39i M CE0 r on S CZX VBMMAMIA tounolwy nouhvn 3 an 39 0 i r on S 3 MCXCA BOMJM Webym Yr 2Lkcrro fans 3n kL f CS kWIL kamst Eomdowy WOL MS J DCrCclx e E Loa dwy Wakwh rams L2 NCMMMM Loundwy Wu 9213 3 53 tons 3 M3de 13 va c r I at 3quot a z g ktr EC39 Wir r MS

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