161_1_Class10

161_1_Class10 - EE161 Electromagnetic Waves Spring, 2010...

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Unformatted text preview: EE161 Electromagnetic Waves Spring, 2010 Prof. Y. Ethan Wang Electrical Engineering Dept. UCLA Lesson 10 • Wave Equations for Ez and Hz • Rectangular Waveguides – TM Modes • Rectangular Waveguides – TE Modes TM Waves (E-Wave) TM waves: , = ≠ z z H E k H E H E Z x y y x TM βη ωε β = = − = = Wave Impedance: x E k j E z c x ∂ ∂ − = 2 β y E k j E z c y ∂ ∂ − = 2 β y E k j H z c x ∂ ∂ = 2 ωε x E k j H z c y ∂ ∂ − = 2 ωε ) ( 2 x H y E k j H z z c x ∂ ∂ − ∂ ∂ = β ωε ) ( 2 y H x E k j H z z c y ∂ ∂ + ∂ ∂ − = β ωε ) ( 2 x E y H k j E z z c x ∂ ∂ + ∂ ∂ − = β ωμ ) ( 2 y E x H k j E z z c y ∂ ∂ − ∂ ∂ = β ωμ Previously we have transverse – longitudinal relationship described by, Substitute H z =0, Once we know E z , we know everything else! different from η ! TE Waves (H-Wave) TE waves: , ≠ = z z H E β η β ωμ k H E H E Z x y y x TE = = − = = Wave Impedance: , 2 x H k j H z c x ∂ ∂ − = β y H k j H z c y ∂ ∂ − = 2 β , 2 y H k j E z c x ∂ ∂ − = ωμ x H k j E z c y ∂ ∂ = 2 ωμ ) ( 2 x H y E k j H z z c x ∂ ∂ − ∂ ∂ = β ωε ) ( 2 y H x E k j H z z c y ∂ ∂ + ∂ ∂ − = β ωε ) ( 2 x E y H k j E z z c x ∂ ∂ + ∂ ∂ − = β ωμ ) ( 2 y E x H k j E z z c y ∂ ∂ − ∂ ∂ = β ωμ Substitute E z =0, Once we know H z , we know everything else! Waves Equations for E z or H z , ≠ c k ⇒ = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + ∂ ∂ + ∂ ∂ + ∂ ∂ 2 2 2 2 2 2 2 z E k z y x 2 2 2 2 2 = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + ∂ ∂ + ∂ ∂ z c e k y x 2 2 2 c k k − = β 2-D Wave equation or Helmholtz equation !! Cutoff wave number Original 3D wave equation: 2 β − z j z z e y x e z y x E β − = ) , ( ) , , ( where, For the same reason, we have the Helmholtz equation for H z : 2 2 2 2 2 = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + ∂ ∂ + ∂ ∂ z c h k y x z j z z e y x...
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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_Class10 - EE161 Electromagnetic Waves Spring, 2010...

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