Homework5 - Homework 5 Answers, 95.658 Spring 2011,...

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Homework 5 Answers, 95.658 Spring 2011, Electromagnetic Theory II Dr. Christopher S. Baird, UMass Lowell Problem 1 Consider a waveguide that is uniform in the z dimension and has a cross-sectional shape of a square with lengths a , placed so that one corner sits at the origin and the opposite corner sits at ( x = a , y = a ). The walls of the waveguide are perfect conductors and the entire interior is free space. Considering only Transverse Electric (TE) modes, apply boundary conditions and solve for all components of the electric and magnetic field. What is the width of the frequency window for which single-mode traveling waves are guaranteed? (Because of the symmetry, the 1,0 and 0,1 modes are identical, apart from a coordinate rotation, and so can be thought of as degenerate instances of the same mode.) SOLUTION: The equation to be solved is 2 B z x 2 2 B z y 2 =− 2 B z with the boundary conditions [ B z y ] y = 0 = 0 , [ B z y ] y = a = 0 , [ B z x ] x = 0 = 0 , [ B z x ] x = a = 0 Try cosines as solutions: B z = B 0 cos A x cos B y Plugging into the differential equation, we discover: A 2 B 2 = 2 Apply boundary conditions to find: A = m a , B = n a where m and n = 0, 1, 2, … so that 2 = 2 a 2 m 2 n 2 The final solution becomes: B z = B 0 cos m x a cos n y a e i k z i t where k = 0 0 2 2 a 2 m 2 n 2 We now apply the waveguide equations to find the transverse components: E t = a B 0  m 2 n 2 − x n cos m x a sin n y a
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Homework5 - Homework 5 Answers, 95.658 Spring 2011,...

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