Lecture5 - Lecture 5 Notes, 95.658, Spring 2012,...

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Lecture 5 Notes, 95.658, Spring 2012, Electromagnetic Theory II Dr. Christopher S. Baird, UMass Lowell 1. Waveguides Continued - In the previous lecture we made the assumption that the wave is free in the z dimension but stays general in the other dimensions, and has a harmonic time dependence. - From this simple set-up, we were able to find the transverse fields in terms of the parallel fields: t E t =− i k E z t B t =− i k B z ̂ z ⋅(∇ t × E t )= i ω B z ̂ z ⋅(∇ t × B t )=− i μ ϵω E z E t = i μ ϵω 2 k 2 ( k t E z −ω ̂ z ×∇ t B z ) B t = i μ ϵω 2 k 2 ( k t B z +μ ϵω ̂ z ×∇ t E z ) - Looking at the two bottom equations, we realize that each is a superposition of two particular solutions. - If the parallel component of the electric field is zero, E z = 0, but the transverse components of the electric field E t are non-zero, and all components of the magnetic field are non-zero, then the wave is known as a “Transverse Electric” wave or “TE” wave. - If the parallel component of the magnetic field is zero, B z = 0, but the transverse components of the electric field B t are non-zero, and all components of the electric field are non-zero, then the wave is known as a “Transverse Magnetic” wave or “TM” wave. - The solution presented in the equations above can be thought of as a superposition of a TM wave and a TE wave. TRANSVERSE ELECTRIC (TE) WAVE: E t = i ω μ ϵω 2 k 2 ( ̂ z ×∇ t B z ) B t = i k μ ϵω 2 k 2 ( t B z ) TRANSVERSE MAGNETIC (TM) WAVE: E t = i k μ ϵω 2 k 2 ( t E z ) B t = i μ ϵω μ ϵω 2 k 2 ( ̂ z ×∇ t E z ) z x y B z B t E t z x y B t E z E t
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- There is a special case where both the electric field and the magnetic field are completely transverse, that is B z = 0 and E z = 0. This is called a transverse electromagnetic (TEM) wave. - A TEM wave is not supported in simple hollow metal waveguides. A plane wave in free space or in an infinite uniform material exists in the TEM 00 mode. TEM waves are the dominant ones in coaxial cables. - The problem has been reduced to finding the parallel components of the fields, and then the transverse components follow immediately. - Let us now use the divergence equations at the top of the first page and plug in the solution for the transverse fields: TM: t i k  2 k 2 t E z =− i k E z TE: t i k  2 k 2 t B z =− i k B z - After multiplying through: TM: t 2 E z =−(μϵω 2 k 2 ) E z TE: t 2 B z =−(μ ϵω 2 k 2 ) B z - Solving for the waves inside any waveguide reduces to solving the above equations and applying appropriate boundary conditions. - The wave inside the waveguide can be thought of as propagating along the waveguide and
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This note was uploaded on 02/13/2012 for the course PHYSICS 95.658 taught by Professor Staff during the Spring '11 term at UMass Lowell.

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Lecture5 - Lecture 5 Notes, 95.658, Spring 2012,...

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