450fall09hw11_Sol - ECE 450 Fall 2009 Homework 11 Solutions...

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ECE 450 Fall 2009 Homework 11 Solutions Due: Tuesday, Nov 10 th , 2009 1. Given TE m mode: < 0 ˆ 2s i n ( ) f o r 0 x jyE e k x x a =< E z jk z ± a. From Faraday’s Law: 0 0 j j ωμ ∇× ∇× =− = E EH H ±± ± ± () 0 0 00 0 0 0 0 ˆˆ ˆ 1 1 ˆ ˆ 1 ˆ ˆ i n ( )2s i n ( 1 ˆ ˆ 2 s i n ()2 c o s 1 ˆ i n ( ) zz z xyz yy jk z jk z xx jk z jk z zx x x jk z jx y z EEE xEzE jz x ) x jE e k x z jE e k x x xkEe kx z j kEe kx j xj −− ∂∂∂ = −∂ ∂∂ ⎛⎞ =− + ⎜⎟ ⎝⎠ + + H ± 0 0 0 ˆ 2c o s ( ) 2 ˆ ˆ sin( ) cos( ) z z jk z jk z x x zkEe E xjk k x zk k x e =+ H ± b. Boundary condition: ˆ s n ⎡⎤ ×−= ⎣⎦ HH J +− ±± ± Let ˆ n = ˆ x at x = 0 and ˆ n = ˆ x at x = a , then by convention we can find the surface currents on the metallic PEC conductor. x = 0: 1
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() 0 0 0 0 0 0 2 ˆˆ ˆ sin( ) cos( ) 0 ˆ 2 z z jk z zx x x x jk z x s x E xx j k k x z k k x e k Ey e ωμ = = ⎡⎤ ×+ ⎢⎥ ⎣⎦ =♠ J J ± ± s = x = a: 0 0 0 0 0 0 0 0 2 ˆ sin( ) cos( ) 0 2 ˆ cos( ) ˆ 2c o s ( ) ˆ o s ( ) z z z z jk z sz x x x x a jk z sx xa jk z x s jk z x x E x x jk kx zk kx e E m yk a e a k m e k kae π = = = =− × + ⎛⎞ = ⎜⎟ ⎝⎠ =− J J J ± ± ± c. {} * Re 2 H SE ±± 1 0 0 0 0 0 0 2 2 0 0 2 ˆ 2 sin( ) sin( ) cos( ) 2 ˆ 2 sin( ) sin( ) cos( ) 4 ˆ ˆ sin ( ) sin( )cos( ) 2 1 Re 2 1 Re 2 Re z z z z jk z jk z xz x x x jk z jk z x x x x x E jyE e k x xjk k x zk k x e E jyE e k x xjk k x zk k x e E xj + ⎧⎫ ⎪⎪ ⎨⎬ ⎩⎭ ×− + + = S * 22 0 0 0 0 ˆ 2s i n ( ) ˆ i n ( ) m z x z k zE k m x a = = S 2
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For the TE 2 mode, E 0 =1 mV/m, f = 12 GHz and the given problem constraints, () ( ) 22 0 0 2 2 2 0 0 2 9 2 2 8 2 32 97 2 2 2 9 ˆ 2s i n ( ) 2 2 ˆ i n ( ) 21 2 1 0 2 0.03 31 0 2 ˆ210 s in( ) 0.03 2 1 041 0 138.926 2 s ) 0.001466 0.03 2 ˆˆ 2.933 10 sin ( ) 2.93 0.03 x x kk zE kx ca x a zx z ωμ ωπ π ππ ⎛⎞ ⎜⎟ ⎝⎠ ×× × = × = = = = S 92 2 W 3 10 sin (209.44 ) m x × 2. Given TM m mode: 0 ˆ 2c o s ( ) f o r 0 x z jk z yHe x a =< ± < H a. Using Amperer’s Law: 0 0 j j ω ∇× = = H HE E ± ±±
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This note was uploaded on 02/21/2010 for the course ECE 450 taught by Professor Staff during the Spring '08 term at University of Illinois, Urbana Champaign.

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450fall09hw11_Sol - ECE 450 Fall 2009 Homework 11 Solutions...

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