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Unformatted text preview: 28 TM mn modes in rectangular waveguides When the operation frequency f in a parallelplate waveguide exceeds the cutoff frequency f c = c 2 a of the TE 1 mode, dual or multimode operations become unavoidable in the guide. Singlemode operation at high frequencies can be attained by turning off the guided TEM(=TM ) mode by introducing a pair of new plates on, say, y = 0 and y = b planes as shown in the margin. This configuration is known as the “rectangular waveguide”, which is the subject of the next set of lectures. • Briefly, the guided TEM mode is suppressed in the rectangular waveg uide, and propagation is only possible in terms of TM mn and TE mn modes. By definition: 1. H z = 0 for TM mn mode, for which the mode properties can be derived from a nonzero E z ( x, y, z ) = f ( x, y ) e jk z z ; 2. E z = 0 for TE mn mode, for which the mode properties can be derived from a nonzero H z ( x, y, z ) = f ( x, y ) e jk z z ; where the constraints on f ( x, y ) and k z are to be determined from Maxwell’s equations and the relevant boundary conditions. • Both TM mn and TE mn modes consist of the superposition of free propagating TEM wave fields reflecting from the guide walls and satis 1 fying the wellknown vector wave equations ∇ 2 ˜ E + ω 2 μ o o ˜ E = 0 and ∇ 2 ˜ H + ω 2 μ o o ˜ H = 0 derived from (see margin) Maxwell’s equations. Vector wave equation in pha sor form: Taking the curl of Faraday’s law ∇ × ˜ E = jωμ o ˜ H , and using ∇ × ∇ × ˜ E = ∇ ( ∇ · ˜ E ) ∇ 2 ˜ E , ∇ · ˜ E = 0 , ∇ × ˜ H = jω o ˜ E , it follows that ∇ 2 ˜ E + ω 2 μ o o ˜ E = 0 ....
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This note was uploaded on 09/27/2011 for the course ECE 450 taught by Professor Staff during the Fall '08 term at University of Illinois, Urbana Champaign.
 Fall '08
 Staff
 Electromagnet, Frequency

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