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Unformatted text preview: 28 TM mn modes in rectangular waveguides When the operation frequency f in a parallel-plate waveguide exceeds the cutoff frequency f c = c 2 a of the TE 1 mode, dual- or multi-mode operations become unavoidable in the guide. Single-mode 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 non-zero 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 non-zero 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 Maxwells 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 well-known vector wave equations 2 E + 2 o o E = 0 and 2 H + 2 o o H = 0 derived from (see margin) Maxwells equations. Vector wave equation in pha- sor form: Taking the curl of Faradays law E =- j o H , and using E = ( E )- 2 E , E = 0 , H = j o E , it follows that 2 E + 2 o o E = 0 ....
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