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Prof. David R. Jackson ECE Dept. Spring 2014 Notes 3 ECE 6341 1

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Grounded Dielectric Slab Goal: Determine the modes of propagation and their wavenumbers. Assumption: There is no variation of the fields in the y direction, and propagation is along the z direction. x z h 2

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Dielectric Slab TM x & TE x modes: x z H E TM x z E H TE x x 3

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Surface Wave The internal angle is greater than the critical angle, so there is exponential decay in the air region. z x Exponential decay The surface wave is a “slow wave”. 4 h

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Surface Wave The wave must also satisfy a “consistency condition”: This forces the angle 1 to be a discrete value, depending on n. or 5 z x h

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TM x Solution Assume TM x B.C. : (see TM x -TE x tables in Appendix) Assume 6

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Hence Denote TM x Solution (cont.) 7

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Applying boundary conditions at the ground plane, Note: Since the surface wave is a slow wave, we have: TM x Solution (cont.) 8

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Boundary Conditions BC 1) BC 2) 9

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Boundary Conditions (cont.) These two equations yield: Divide second by first: or 10

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Final Result: TM x This may be written as: This is a transcendental equation for the unknown wavenumber k z. Note: The choice of square root for k x1 is not important, but it is for k x0 : 11

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Appendix 12 TM x Note: There is a factor difference with the Harrington text.

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Appendix (cont.) 13 TE x Note: There is a factor difference with the Harrington text.

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