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# 161_1_Class4 - EE161 Electromagnetic Waves Spring 2010 Prof...

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Unformatted text preview: EE161 Electromagnetic Waves Spring, 2010 Prof. Y. Ethan Wang Electrical Engineering Dept. UCLA Lesson 4 • Waves in Lossy Medium • Electromagnetic Power Density • Transmission Line Theory Review Wave Propagation in Lossy Media ~ ~ ~ ) ( ~ ~ 2 2 2 2 = ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ − ∇ ⇒ = − ∇ z y x E E E γ γ E E ) ( 2 ωε σ ωμ γ j j + = where β α γ j + = The solution is, Considering the wave propagating in z axis with the electric field component in x only, The wave equation for lossy media is, z x z x x e E e E z E γ γ or ) ( ~ − = α: attenuation constant β: phase constant z j z x x e e E z E β α − − = ) ( ~ Thus ) ( ~ z x x e E z E α − = α δ 1 = s : Skin depth Amplitudes relate only to the attenuation constant Wave Parameters in Lossy Media 2 2 2 2 ) ( ) ' ' ' ( ) ( β α γ ε ε μ ω ωε σ ωμ γ j j j + = − − = + = E H ~ ) ( ~ ωε σ j + = × ∇ E H ~ ~ c j ωε = × ∇ ' ' ' ε ε ω σ ε ε j j c − = − = Define complex permittivity: For lossy media, Ampere’s law gives: E H ~ ) ( ~ ε ω σ ω + = × ∇ j j c ε 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 The propagation constant: Comparing the real part and imaginary part of the above two yields, 2 / 1 2 1 ' ' ' 1 2 ' ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + = ε ε με ω α 2 / 1 2 1 ' ' ' 1 2 ' ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + = ε ε με ω β , μσ π ω μσ ω με ω α f = = ≅ 2 2 ' ' Lossy Media Classification Low loss dielectric Good conductor Quasi-conductor ) 10 ( 1 2 ' ' ' − < << ε ε ) 10 ( 1 2 ' ' ' > >> ε ε 2 2 10 10 < < − ' '' ε ε For low loss dielectric For good conductor ε μ σ ε μ ωε α 2 2 ' ' = ≅ με ω με ω β = ≅ ' ε μ η ≅ c σ α σ μ π ε μ η ) 1 ( ) 1 ( ' ' j f j j c + = + = ≅ μσ π α β f ≅ = For a lossy medium, the ratio plays an important role in determining...
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## This note was uploaded on 05/10/2010 for the course EE 190776201 taught by Professor Yuanxunethanwang during the Spring '10 term at UCLA.

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161_1_Class4 - EE161 Electromagnetic Waves Spring 2010 Prof...

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