350lect29 - 29 TE mn modes in rectangular waveguides • The analysis of TE mn modes starts with the wave equation for H z that is TE mode fields H

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Unformatted text preview: 29 TE mn modes in rectangular waveguides • The analysis of TE mn modes starts with the wave equation for H z , that is TE mode fields: H x =- jk z ∂H z ∂x k 2- k 2 z , H y =- jk z ∂H z ∂y k 2- k 2 z , E y = jωμ o ∂H z ∂x k 2- k 2 z , E x =- jωμ o ∂H z ∂y k 2- k 2 z . ∇ 2 H z + k 2 H z = 0 . By analogy to the TM mn case, and using separation of variables, we have H z ( x, y, z ) = ( A cos k x x + B sin k x x )( C cos k y y + D sin k y y ) e- jk z z . Pertinent boundary conditions need to be applied in terms of E y and E x on waveguide walls at x = 0 and a , and y = 0 and b , respectively: 1. E y = 0 at x = 0 and a requires ∂H z ∂x = 0 at the same locations, implying B = 0 and k x a = mπ . 2. E x = 0 at y = 0 and b requires ∂H z ∂y = 0 at the same locations, implying D = 0 and k y b = nπ . Hence, H z ( x, y, z ) = H o cos( k x x ) cos( k y y ) e- jk z z , with k x = mπ a , k y = nπ b , k z = ω c 1- k 2 x + k 2 y k 2 = ω c 1- f 2 c f 2 , 1 where f c = ( mc 2 a ) 2 + ( nc 2 b ) 2 is the pertinent cutoff frequency of the TE mn mode. – Note that m = 0 or n = 0 — but not both zero — are permitted since these choices do not lead to trivial H z ....
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This note was uploaded on 09/27/2011 for the course ECE 450 taught by Professor Staff during the Fall '08 term at University of Illinois, Urbana Champaign.

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350lect29 - 29 TE mn modes in rectangular waveguides • The analysis of TE mn modes starts with the wave equation for H z that is TE mode fields H

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