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Homework10Transient

# Homework10Transient - ECE 3030 Electromagnetic Fields and...

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ECE 3030: Electromagnetic Fields and Waves Fall 2009 Homework 10 SOLUTIONS for GRADER: Transmission Line Transients and Dielectric and Metallic Waveguides Reminder: Date: Tuesday 11/10 1. Problem 15.19: Conductor-Silicon dielectric slab waveguide. Just a few thoughts on this one before we start the math. The odd (anti- symmetric) modes of dielectric slab waveguides have a null in the electric field in the middle of the core. If the top half of the core were sliced off and replaced with a conductor as in this problem, then the boundary condition for the core-conductor interface is already satisfied! This not true for the even (symmetric) cases. This means that only the equations with the cotangent function on the left-hand side of the transcendental equation apply. The apply the core-conductor boundary condition, it would be most convenient to define the x coordinate to be zero there, so let us make a change of variable such that x 0 = x - d/ 4. a. At x = d/ 4 ( x 0 = 0), the electric field component parallel to the metal surface, E y , must be zero. The sine function will satisfy this. At x 0 = 0, the continuity of the parallel electric field requires - E 0 sin( k x h/ 2) = E 1 , and the continuity of the parallel component of the magnetic field requires E 0 k x cos( k x h/ 2) = α x E 1 . Dividing these two equations yields - cot( k x h/ 2) = α x /k x , but α 2 x + k 2 x = ω 2 μ 0 ( ε 1 - ε 2 ) = - cot k x h 2 = s ω 2 μ 0 ( ε 1 - ε 2 ) k 2 x - 1 - cot k x h 2 = s ω 2

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Homework10Transient - ECE 3030 Electromagnetic Fields and...

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