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Unformatted text preview: Chapter 9 Vector Integral Equation Methods Electromagnetic integral equations are in general vectorial. Only under some special conditions can we reduce these vectorial integral equations into scalar integral equations. For example, the 1D volume equations for the only nonzero E y or H z component; the surface and volume integral equations for E z in the 2D TM z case; and surface and volume integral equations for H z in the 2D TE z case. In general, 3D EM problems are vectorial, and thus require vector basis functions. Similarly, in the 2D TM z case, surface and volume integral equations for H = xH x + yH y are vectorial. The same is true for E = xE x + yE y in the 2D TE z case. Since the integral equations are for the current densities, the normal component of the unknown vector should be continuous accross adjacent elements. Therefore, divergenceconforming basis functions should be used. 9.1 Surface Integral Equations In this chapter we develop the twodimensional integral equation methods. The first discuss the surface integral equations based on the 3D surface integral equations given in Chapter 1 and recast here. The 3D surface integral for homogeneous objects are the electric field integral equations (EFIEs) r 1 L 1 I 1 + K 1 r 2 L 2 I 2 + K 2 J s M s = n 1 E inc 1 n 2 E inc 2 (9.1) and the magnetic field integral equations (MFIEs) ( I 1 + K 1 ) r 1 L 1 ( I 2 + K 2 ) r 2 L 2 J s M s = n 1 H inc 1 n 2 H inc 2 (9.2) where J s = J s , H inc i = H inc i , and for i = 1 , 2 I i [ J s , M s ] = ( n i n ) i 4 [ J s , M s ] = i 4 [ J s , M s ] (9.3) L i [ J s , M s ] = jk n Z S g i ( r , r )[ J s ( r ) , M s ( r )] ds + k 2 i Z S g i ( r , r ) s [ J s ( r ) , M s ( r )] ds (9.4) K i [ J s , M s ] = n  Z S g i ( r , r ) [ J s ( r ) , M s ( r )] ds (9.5) 191 CHAPTER 9. VECTOR INTEGRAL EQUATION METHODS where n = n 2 = n 1 is the objects unit outward normal; R represents the Cauchy principal integral, and i being the internal solid angle in region i ; note 2 = . For an electric impedance boundary, since M = n e J s on the surface S , the only the first equations in the above are needed. Thus, the EFIE for an electric impedance boundary is [ ri L i ( I i + K i ) n e ][ J s ] = n i E inc (9.6) and the MFIE for a PEC object is [( I i + K i ) + ri L i n e ][ J s ] = n i H inc i (9.7) When e = 0, the object becomes a PEC. Note that while the EFIE is applicable to both closed PEC objects and PEC shells, the MFIE only applies to closed PEC objects. This is because for a PEC shell, the electric current density on the shell is no longer given by J s = n H , but by J s = n ( H 2 H 1 ) where H 1 and H 2 are the magnetic fields on both sides of the shell....
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This note was uploaded on 01/16/2011 for the course ECE 277 taught by Professor Qingliu during the Spring '09 term at Duke.
 Spring '09
 QingLiu
 Electromagnet

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