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# Class Note for ECE 6340 with Professor Jackson at UH

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This 29 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 9 views.

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Date Created: 02/06/15

ECE 6340 lntermedlate EM Waves Fall 2005 Prof Donald R Wilton ECE Dept Notes 11 Based on notes of D R Jackson uides Waveguide geometry Assumptions 1 waveguide boundary is PEG 2 material inside is homogeneous may be lossy 3 A traveling wave exists in the z direction expjkZ z Mode Theorem No TEMz mode Proof Assume TEM mode EOC y Z Em x9 ye jkzz k2 EtO x9 Vqx9 VZCD O No TEM Mode cont C VZCIDO ES CI constant K e C Uniqueness theorem of electrostatics DOC y constant trivial field Helmholtz Equation for WG Assume TMZ VZEZ k2EZ O 2 2 2 V2 182 62 8x 8y 82 L Y VIZZVIOVI vz23 8x 8y so VIZEZ k2 k 2EZ 0 Z Helmholtz Equation for WG cont VfEZ k2 k22EZ 0 then VleZ kczEZ 0 Factoring out the z dependence similarly for H20 Property of kc Theorem kc real number even if so is complex Proof Let Kxy E0V E Z tZO 2D divergence theorem J VIZdS z dz S C n C Property of kc cont zM vtEw 0 Hence Vt dS 0 S Therefore Vt 39EO VIEzo d5 0 S or JUVZE VlEzo E vaZO dS 0 S Property of kc cont Hence J S Vl EZO 2 2 dS jEZOkcEZOdS 0 S 2 VZEZo dS or k2 S c JEzozdS S Hence 02 real positive number Note This is also a variational equation for 02 proof omitted Eigenvalue Problem PEC Assume TMz Vlez kaZ 0 Let Then WXyEzox9y V12Wx9y wx y 12 122 Wx y0lc Note x is independent of frequency and permittivity from the form of the eigenvalue problem WG Eigenvalue Problem cont Assume TEz Let wxy Hzoxy z Then Vlzwxy Myxy Note BC follows from Faraday s Law proof on next slide WG Eigenvalue Problem cont I amp 2 local coordinates PEC x a z 8H Hence 8y 82 a t Q 5 aHx 0 an 3x 82 Note Hn O on PEC Theorem on Field Assume a TMz mode similar proof for TEz mode where Rxy is a real function c is a complex constant PrOOf V1 2E20 kCZEZO 0 Note that R6VI2E20 VIZ Re E20 Hence VIZ Re EZ kcz Re EZ 0 ReEZ O on C Similarly VIZ 1m EZ 02 1m EZ 0 ImEZ 0 on C Theorem on Field cont Assume solution of eigenvalue problem is unique to within a multiplicative constant Hence 1m EZO x y A Re E20 x y Therefore EZO x y Re Ezox y1 Rx yc Corollary where R2 Xy is a real vector function m Ex ijz 2 8E2 k k2 ax Assume a TMz mode similar E 8E2 proof for TEz mode y kZ kf ay Corollary cont J79 Er k2 kZ2 VZEZ 39kZ EZO k2 k2 jvthO ka 2 k2 k2 jVZ cRxy C 632 1 Wavenumber 2 2 2 independent of kc k kz Real number frequency and material k2 wzyac m This is written as 2 8 ja i Attenuation constant Phase constant Wavenumber Lossless Case 8E88ramp ymm kz fys kqm Z C Conclusion k2 either real or imaginary Lossless Case Cutoff Frequency k2 w2y8 k212 C 2 2 Note Cutoff frequency k2 0 we u 8 kc 1 k we 0 Lu 8 c This is a physical interpretation of kc Lossless Case Wavenumber Above cutoff k2 k2 k3 fgtfc a zyg wczw 2 a ug 1 wcj a 2 Lossless Case Wavenumber Below cutoff k2 2 k2 k3 12 fe LIW quot quotc k l2 CV Lossless Case Wavenumber a Lossless k real Lossless Case Phase Velocity Above cutoff only fgtfc SO Lossless Case Group Velocity Above cutoff only fgtfc 822k2 kfzw2y8 k62 28d82aday8 da 1 8 2i v c g d uga dvp SO Example Rectangu er Waveguie Assume TEZ b Eigenvalue Problem Vfw 1W Mac y H2006 y 610 23kg an C Rectangular Waveguide cont Separation of variables Assume ryxy so XHYXYHZZXY H 2 Hence X constant kx Rectangular Waveguide cont X x A sinkxx B coskxx X3900 A20 X39a0 gt sinkxaO kxam7r m7r nkx a Hence COS m x a Rectangular Waveguide cont Yquot Similarly k2 k Y y y b Yy 00 Hence Rectangular Waveguide cont Summary H Z X y 2 H 0 cos m x cos e szz a

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