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# 350lect29 - 29 TEmn modes in rectangular waveguides The...

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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 = j ω μ o H z x k 2 - k 2 z , E x = - j ω μ 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 = m π . 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 = n π . Hence, H z ( x, y, z ) = H o cos( k x x ) cos( k y y ) e - jk z z , with k x = m π a , k y = n π b , k z = ω c 1 - k 2 x + k 2 y k 2 = ω c 1 - f 2 c f 2 , 1

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where f c = ( mc 2 a ) 2 + ( nc 2 b ) 2 is the pertinent cuto ff 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 . However, m = n = 0 is not permitted, because in that case H z becomes independent of x and y , and leads to zero transverse fields.
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