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# Class Note for ECE 6340 with Professor Jackson at UH

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This 36 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 13 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 8 8 vi 1 3 gnal Propagation on Line C C Z0 0 2aJ v0 Z Infinite line Introduce Fourier transform Signal Propagation on Line Goal To get this into a form that looks like v10 2 ZReltA77 emf Signal Propagation on Line cont We start by considering a useful property of the transform 1710 Cinch Welt 00 Since V10 Real 1 Next use v t w ejmda i 3170ejmda I 27x 0 I 27 1 Signal Propagation on Line cont 1 1 0 v10 viaemda j v1 0 era 270 27zoo 1 USea a 1 0 10 iaejatda iale jatdar 27100 2700 1 00 I 39a39t r l ae 1 da 270 1 EN v w39 e fa rtdw39 27 0 Signal Propagation on Line cont Hence v10 2 0 W 17 w W da 0 or v1 r 31142 w W da 0 or v10 Re 1171 dd em 7239 Signal Propagation on Line cont Signal Propagation on Line cont v10 2 Z ReAnejwquott Therefore v0 t Z ReAne7quotlejwquott 2 Re 171 0 de eyquotlejwquott n 7 or v00 JRei iweylejwtdaj 0 7239 where 7 ya Signal Propagation on Line cont Final result 10 Lossless Line 7 J jaw LC vO jwvp vp constant denote t V 1 Then V0 I Re 7239 f 0317 w ejm39da 0 Lossless Line cont Hence V0 I Vial A w Signal Propagation with Dispersion A A v v 4 Via G Lowloss line V00 n c v 20 Zl VP 2 VP dispersion Assume V10 50 COS wot SQ slowly varying envelope function Signal Propagation with Dispersion cont Signal Propagation with Dispersion cont The spectrum of St is very localized near zero frequency Sg OStej fdt Narrow pulse in Q domain Signal Propagation with Dispersion cont 1 00 Use v0 t Re Ivia a ejmda 72 0 v10 2 S t cos wot 39v w ism gtxlt y cos 001 ism gtxlt y lefa o le fa o 27 2 2 1 Z5agtxlt7z a a07r5dwo Note y efwot 27 60 mo Signal Propagation with Dispersion cont iw w 5w wo aa0 215w w0l aa0 221 Peak near 600 Peak near a0 00 1201 1R6 I i e Vlefmda The integration is 7 0 from O to Infinity Hence inside the 1 integral we can write Vi N 580 a0 Signal Propagation with Dispersion cont Hence 1 3 v00 z Re Sm mo e ylejmda 27 0 with 7aj Next use d z wod j 0 00 6 0 aaltwod 0 00 610 w 0 neglect Signal Propagation with Dispersion cont v0 t z iRe ISM a0eylejmda 27 0 d 0jwo 0 00 0 wo azao 050 aa0 1 3 1201 z Re 5a cooe le J ol 27 0 e ejmda Signal Propagation with Dispersion cont 1 1201 z Re 5a 600e le l 27 0 e ejmda Multiply and divide by exp jwol 1 3 1201 z Re 5a mo ole N301 27 0 d 1 e dwjw 00 efwolejwwotda Signal Propagation with Dispersion cont 1 3 1201 z Re 5a mo ole N301 27 0 d ejjw wo lejworefww0 d0 1 1 I 0 g v0tz Re e oe J oejwof j S e dd ej dg 27 w0 Signal Propagation with Dispersion cont 1 1 1 0 J51 votz Re e aoe J oejwof j S e dw elffdg 27 Extend the lower limit to minus infinity since the spectrum of the envelope function is concentrated near zero frequency 27 395 r 1 v0 t wiRc eaoleJ lemot ISQC e E d jdgc Signal Propagation with Dispersion cont w 395 r 1 vO t z ReLeO Olej ole woti 55 6 K d jag Hence 1201 z Reeaolej lejw tS t da Signal Propagation with Dispersion cont or v0 t z Sp ljel 005th ol da Define group velocity votzSt Zvg cosa0t lvp e aol Signal Propagation with Dispersion cont Summary vi t SQ COS wot votzzSt zvg cosa0t zvp 6 0501 25 v0 tz v0 Example g vgt0 26 Rectangular Waveguide Example 27 Example cont Now use 2 z 021080 a Z d 2001001080 3v dmz 1 2162 g d a uogo vp 2 Hence V V C 817 Note this property only holds for lossless waveguides Example cont 29 Example cont Graphical representation w diagram 30 Dispersion Theorem If there is no dispersion then VI vg Proof 12170 constant EXAMPLES Hence 2 a plane wave in free space 8 1 lossless TL 3 2 avg distortionless TL dd 2 ald d m2a1 d 31 Dispersion cont gt This means no distortion if the attenuation is also frequency independent 32 Dispersion cont ltV Normal Dis ersion v p g P Example waveguide gtV Anomalous Dis ersion V p g P Example lowloss transmission line 33 Backward Wave Definition of backward wave The group velocity has the opposite sign as the phase velocity This type of wave will never exist on a TEM transmission line but may exist on a periodic artificial transmission line Note do not confuse quotbackward wavequot with quota wave traveling in the backward directionquot 34 Energy Velocity Definition of energy velocity vE Note in many systems the energy velocity is equal to the group velocity 35 Signal Velocity Relativity vs lt C Note on Group Velocity Note sometimes vg gt c eg lowloss TL lossless Il

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