161_1_Lecture_4

# 161_1_Lecture_4 - EE161 El EE161 Electromagnetic Waves Fall...

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EE161 Electromagnetic Waves all 2010 Fall, 2010 Instructor: Dr. Shenheng Xu Electrical Engineering Dept., UCLA gg p , © Prof. Y. Ethan Wang

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Lecture 4 • Waves in Lossy Medium • Electromagnetic Power Density • Wave Reflection & Transmission at Normal Incidence ppendix: Transmission Line Theory • Appendix: Transmission Line Theory
Wave Propagation in Lossy Media ~ ~ ~ ~ x E where The wave equation for lossy media is, 0 ~ ) ( 0 2 2 2 2 z y E E E E ) ( 2   j j The solution is, Considering the wave propagating in z axis with the electric field component in x only, z x x e E z E 0 ) ( ~ j  attenuation constant hase constant z j z x x e e E z E 0 ) ( ~ Thus  phase constant ) ( 0 z x x e E z E 1 : Skin depth s p Amplitudes relate only to the attenuation constant 37%

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Wave Parameters in Lossy Media E H ~ ) ( ~  j For lossy media, Ampere’s law gives: ' ' ' j j c Define complex permittivity: E H ~ ) ( ~ j j E H ~ ~ c j c Therefore, the Maxwell equations will have the same form of solution for the lossy and lossless case, assuming the permittivity is a complex number in general he propagation constant: 2 2 2 2 ) ( ) ' ' ' ( ) (  j j j j The propagation constant: Comparing the real part and imaginary part of the above two yields, 2 / 1 2 ' ' '  2 / 1 2 ' ' ' 1 ' 1 2 1 ' 1 2 ,
Lossy Media Classification For a lossy medium, the ratio plays an important role in determining how lossy a medium is. Three category of medium can be divided according to this  / ' ' ' Low loss dielectric ood conductor ) 10 ( 1 2 ' ' '  0 2 ' ' ' ratio, Good conductor Quasi-conductor ) 10 ( 1  2 2 10 10 ' '' ε ε   ' ' For low loss dielectric For good conductor ' ' f 2 2 2 2 ' f  c ) 1 ( ) 1 ( ' ' j f j j c

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Electromagnetic Power Density Power density Poynting Vector: H E S (W/m 2 ) epresents instantaneous Represents instantaneous power flow per unit area Total power flow through the s ˆ A A here pg aperture A is, , cos SA dA n P A S S S where * ~ ~ e 1 Average power density: Analogue in Circuits ) , ( ) , ( ) , ( t z i t z v t z P   Re 2 H E S av   ) ( ~ ) ( ~ Re 2 1 ) ( * z I z V z P av
Poynting Vector for Plane Waves In lossless medium In lossy medium E H ~ ˆ 1 ~ n   ~ ~ Re 2 1 * av H E S  * ~ ~ Re 2 1 H E S av   ~ ˆ / ~ ~ Re 2 1 2 * * c z  E E   ~ ˆ / ~ ~ Re 2 1 2 * E E z ) 1 Re( 2 ) ( ˆ * c z z E 2 ˆ E z ) 1 Re( 2 ˆ * 2 2 0 c z e z E 2 ˆ 2 0 E z 2 E If , j c c e cos 2 ˆ ) ( 2 0 z c av e z z S (W/m 2 )

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Decibel (dB) Scale for Power Ratios
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## This note was uploaded on 02/09/2011 for the course EE 161 taught by Professor Huffaker during the Spring '08 term at UCLA.

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161_1_Lecture_4 - EE161 El EE161 Electromagnetic Waves Fall...

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