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**Unformatted text preview: **ECE 222B, Winter 2010 Solutions for Homework Set 6 1. Assume that you are given a dielectric covered ground plane as shown in Figure 8-30 of Balanis. Assume that the height of the dielectric covering is h , the dielectric permittivity is ² d , and the permeability μ is the same for both the dielectric and the vacuum regions. We wish to examine the propagation properties of the ideal TM z mode; that is, the mode with a zero-frequency cutoff and vector potential A z ( y,z ) = A d sin( β yd y )exp( jβ z z ) for y < h , and A z ( y,z ) = A exp(- αy )exp( jβ z z ) for y > h , and where A d and A are constants. For the following plots, assume that ² r = 2 . 56. (a) Since β yd → 0 and α → 0 as ω → 0, use a small argument expansion in frequency to demonstrate that β yd h ’ ω ω c 1 ¶ π 2 and αh ’ 1 ² r ω ω c 1 ¶ 2 ‡ π 2 · 2 where ² r = ² d /² , and where ω c 1 = π/ ( 2 h √ ² μ √ ² r- 1 ) is the cutoff of the TM z 1 mode for the slab dielectric waveguide. Solution: Using the normalizations x = β yd h , y = αh , and z = β z h , and the definitions β = ω √ ² μ and β d = ω √ ² d...

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