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# Class Note for ECE 6340 with Professor Jackson at UH 2

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This 16 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Houston taught by a professor in Fall. Since its upload, it has received 20 views.

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Date Created: 02/06/15
ECE 6340 Intermediate EM Waves Fall 2008 Prof David R Jackson Dept of ECE Notes 10 Wave Equation Assume Ji0 MiQ 5 WE jaw Vx ngc 3 VgtltVgtltE jauVgtlt W jwsc Use k2 wzlu 50 Vector Wave Hence Equahon Wave Equation cont Similarly VxVx k2 g Taking the divergence VgtltVgtltE k2EQ VVxVxE k2VEO Hence pv O in the timeharmonic steady state for a homogeneous material The zerodivergence is built in to the vector wave equation Vector Laplacian VZKEVVK VXVgtltK In rectangular coordinates VZK gszx 2V2Vy V2VZ The vector wave equation can be written as VV V2E k2 20 Vector Helmholtz Equation Vector Laplacian cont Take the rectangular components VZEX kZEx 0 2 2 V Ey k Ey 0 VZEZ kZEZ 0 Note VE O is not implied from the vector Helmholtz Equation see the example later IQ 91 31 W z ZL IJ ZHQ Z3lf 239 3mf H W 531 31 49 z z39zyJ ZHQ 22 ZHQ 30f H IQ 531 31 W 91 35 239 Z3lf ZHQ mf H W 521 35 IQ Sly gt 239 Z3lf ZHQ mf X H se slueuodwoo leugpmg uol em 40 sums u uenum eq New eAem pepgn e lo spleu eq LUSJOSLn eAempepgnti sq LUOJ suonenba epgn eAeM Waveguide Equations cont These may be written more compactly as Waveguide Equations cont For the special case of TEz and TMz fields we can combine these as follows Waveguide Equations cont 2 2 E1 2X k kz Ht 7621622 jkz E A k2 NM Define the wave impedance ZTE Waveguide Equations cont Summary for TEz and TMz cases 11 Waveguide Equations cont For the special case of a TEM wave kzzk ZTEZQlzwluz 4 77 k2 k a u8 ZTM k2 k a ug77 we we we 12 Exam ple Rectangular Waveguide VZEZ kZEZ 0 VZHZ k2HZ O In this example we examine the modes and illustrate that solutions to the scalar Helmholtz equation may be nonphysical since they violate the divergence condition 13 pie Solution from separation of variables given later 14 Example cont Question What is the lowestorder mode smallest m and n for a nonJHWal dd Tan1 E 2 E0 Sincmrxjsin 6 1792 a TEmn HZ H0 coscmrxjcos e jkzz a TM m1n1 mn TE m0n0 mn 15 Example cont TEmn mode m7rx mr HZ ZHOCOS jcos yj e 1 92 kzka2 a b TEO0 mode HZ H0 e J39kz Hx H y 0 Q 0 on PEC no variation inside 16 Example cont VH6A6HZ ax ay 82 jk H 0 6 ij V 39 7g 0 invalid mode The TE10 mode is the mode with the lowest cutoff frequency Whenever we solve the vector Helmholtz equation we should check to make sure that the divergence condition is satisfied

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