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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 1-D volume equations for the only nonzero E y or H z component; the surface and volume integral equations for E z in the 2-D TM z case; and surface and volume integral equations for H z in the 2-D TE z case. In general, 3-D EM problems are vectorial, and thus require vector basis functions. Similarly, in the 2-D 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 2-D 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, divergence-conforming basis functions should be used. 9.1 Surface Integral Equations In this chapter we develop the two-dimensional integral equation methods. The first discuss the surface integral equations based on the 3-D surface integral equations given in Chapter 1 and recast here. The 3-D 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