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Note 5 for ECE 6341 with Professor Jackson at UH


Note 5 for ECE 6341 with Professor Jackson at UH

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This 25 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 6341 Spring 2009 Prof David R Jackson ECE Dept Notes 5 TMX SurfaceWave Solution h i R TEN 251M g 202M x I k k W x x Z 01 1 Z OZM 0 mg 080 kxl klz kzz kxo k02 kzz Zn jZOTlM tankx1h Z 203M TMX SurfaceWave Solution cont TRE Em 2m jZOTlM tankxlh ZoM k tankx1h km 081 080 Hence or 8r 2 j 11 jtanchlh x0 Note Assuming a real k2 a solution will only exist if kxo is imaginary TMx SurfaceWave Solution cont 1 2 2 5 Let kxO ax0 a axO k2 k0 Then we have 1 2 2 E l or 19216032 Note This must be solved numerically Properties of Surface Wave Solution Assumptions A lossless structure A bound surfacewave solution the fields decay at x 00 Properties of SW Solution cont Property 1 k2 is real Otherwise conservative of energy is violated Assume Properties of SW Solution cont Property 2 k2 210 Otherwise 8r imaginary no solution possible 8r 2 k jtaIKCxlh axO 1 1 k12 k22 axO kzz 02 x1 Properties of SW Solution cont Property 3 kzlt k1 1 Otherwise kx1 2 k12 32 j0x1 a 1 I th x t h n IS case 8r axo 311 Jaxl axO j a 0 tarmltax1hgt axl jtanhmxlh 050 x 2 negative number TEX Solution for Slab h 5 R 2 ZTE 00 Z TE ZTE 01 00 A TE r Z01 x TRE Z 011 010 tan k h J k J x1 k x1 x0 or i 39kxo J tankx1h r k x1 TEX Solution for Slab cont 12 Using kxo jax0 05x0 2 k22 02 we have ax J jg Ojtanwxlh x1 1w jtanchllz x1 or L r L r Graphical Solution for SW Modes Consider TMx 06x06 kxl tankx1h or axoh i kxlh tankx1h I Let then Graphical Solution cont We can develop another equation by relating u and v Hence u hk12 k2212 V hk22 k0212 2 h2k12 k22 v2 hz kz2 k02 add Graphical Solution cont u2 v2 142k12 koz 0th n12 1 Define R2 E koh2 n12 1 Note R is proportional to frequency Graphical Solution cont V K 112v2 2R2 R2 E kohym2 1 TM0 E 1 VII utanu Graphical Solution cont Graph for a Higher Frequency v TM1 57 37r2 u Improper SW v lt 0 Proper vs Improper v axoh If v gt 0 proper SW fields decrease in x direction If v lt 0 improper SW fields increase in x direction Cutoff frequency the transition between a proper and improper mode Note This definition is different from that for a closed waveguide structure where k2 O at the cutoff frequency Cutoff frequency TM1 mode V 0 16 TMx Cutoff Frequency v i i i TMI R7r 7 g u R a E a E For other TM modes Further Properties of Solutions obtained from the graphicalquot solrutiron Property 1 k2 2 k0 atfc Proof v axoh thj koz Atffc v20 so kZk0 Properties of SW Solutions cont Property 2 k2 gt k1 at f gt 00 Proof 7t 37 u gtconst etc 2 2 Hence u kxlh h k12 k22 gt const k 2 k2 SO klh 1 f gt const so gt 1 1 1 20 TMO Mode The TM0 mode has three special properties TM0 1 No cutoff fc gt 0 Proof see graphical solution TM0 Mode cont TM02 91213 f gt0 k0 1 1 2 Proof v utanu u 8 8 I 1 Hence hxkzz k02 wimk kgt 87 k 2 1 k 2 k0 2 z kohf 1112 2 k0 8r k0 k 2 1 k 2 2 1z kh 2 2 k0 5r 0 gt0 21 Dispersion Plot kx1 luraxO 1 r tankx1h axoh Icth TEX Modes cont No TE0 mode f6 0 TE1 cutoff frequency at R 7f 2 In general TEX Modes cont v ucotu u hk12 k2212 v hk22 k0212 At this frequency u 0 For lower frequencies u becomes imaginary If we wish to track the TE1 ISW for lower frequencies we need to re formulate the graphical solution 25


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