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# NUMERICAL METHODS AND ELECTROMAGNETICS EE 525

UK

GPA 3.81

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This 3 page Class Notes was uploaded by Adaline Pollich on Friday October 23, 2015. The Class Notes belongs to EE 525 at University of Kentucky taught by Stephen Gedney in Fall. Since its upload, it has received 8 views. For similar materials see /class/228316/ee-525-university-of-kentucky in Electrical Engineering at University of Kentucky.

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Date Created: 10/23/15

Evaluating the PVI of the MFIE The MFIE governing the induced electric current density on S is expressed as ingtltIjlmcFjFidngtltVgtltfGltF7F jF ds 1 S a where in is the outward unit normal to S at F r lies on 3 which is the closed surface just outside of S and r F dng 2 where 8 is a very small positive real number G is the free space Green s function de ned as 6W 139 Sing GR F JGI F 3 where R V 39 and the integral on the righthandside of 1 requires a principal value integral PVI Next apply the identity V X 3 Wow if F X va i 4 and note that a a R 8 R 1 763 e R I varr39 R la RU 73G 22 igR K R 3K 22 5 where RJF FF 7 F39 GR and G are de ned in 3 and kR sinltkR kR 39 kR 76R COS 7 KW smlt gt3wslt gt7 m g 6 km M then 1 is rewritten as 3 an X F H Z an gtlt fjF39XRKRR jKI 11ml ds39ds 7 7T 3 where we have assumed that in X RImc Fr in X TIME The dif culty lies when R gt 0 At this point 51quot XHS is discontinuous on S hence we can say that 51 X13 is dual valued In the limit that R gt0 K1012 713 and is a regular function On the other hand KRR is hypersingular in that in the limit as R gt 0 KRR a 00 as 1 R3 It is the presence and characteristic of this hypersingularity that leads to the dual value of 51quot X HS NeXtde ne F7FF ingiFFiF dng dng 8 Thus in the limit that 6 a 0 RI R Thus 7 can be rewritten as an X F i F k3 A V I E117 xii10 xRSKRR 31603 ds 9 lim an gtlt fi xeanlme jKIR ds39 5H0 47F S where the second term on the righthandside is referred to as the a principal value of the integral and the last term is the residual Since K1 is regular and bounded the contribution to the residual from this term is 0 in the limit that 6 a 0 On the other hand KR has a l R3 singularity Furthermore the triple cross product in X is X in is nonzero in the limit R a 0 Hence KR will have a nonzero contribution to the residual in the region near the hypersingularity Away from the singularity KR is smooth and consequently in the limit that 8 gt 0 will be zero Thus to evaluate the residual term we need only to evaluate this integral in the very near vicinity of R a 0 Here we will approximate S to be locally planar and will approximate the integral as that over a at disk of radius Ap centered by the origin at the eld point 7 Small argument approximations for the trigonometric functions are then applied shim M 0kR3 10 coskR 1 0kR2 The residual is then written as k3 AP 2quot A MR 1 Res 6113 ploq loean gtltJr39gtlt an kmg W pd gtdp 11 Next let R 422 62 and assume that jF39 z HF leading to 3 A k e l l Pv1 lim ed XJF39 gtltd pdp 6H0 2 n lt lt n jail kid2 62 k3ltp2 62 ilim gjFgti zi 1 AJZ 2 i quotAp2 2i 12 H02 l k kngAp2 2 J k3 a 1 1 1 a 1 7 7 J 39 0k 3 7511M Finally from 12 and 9 the MFIE is expressed as k3 A a R I I 13 Ean ng r XR K RK 1 ds where F is onSand R l 7 F39l Mor generally we can express the MFIE as A a a 1 a A e a aanmcr JrangtltJL VgtltGRJr39ds39 14 S where the integral representing the principal value is now de ned nonhypersingular and inte grabl e

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