161_1_Lecture_9

161_1_Lecture_9 - EE161 El EE161 Electromagnetic Waves...

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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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ecture Lecture 9 • Wave Equations for Ez and Hz • Rectangular Waveguides TM Modes • Rectangular Waveguides TE Modes
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Relationship between Transverse Components and Longitudinal Components ) ( 2 x H y E k j H z z c x  Therefore, e have: Observations: nce we know the ) ( 2 y H x E k j H z z c y E H j z z We have: Once we know the longitudinal components, we know everything else. ) ( 2 x y k E c x  ) ( E H j E z z y , c k Very important !!! need to be pre- determined 2 y x k c 2 2 2 utoff wave number: 2 2 2 k k c Cutoff wave number: z k y x c k k k Conclusion: we need determine first, then everything else. y x z z k k H E , , ,
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Waves Equations for E or H z z , 0 c k 2 2 2 c k k Cutoff wave number 0 2 2 2 2 2 2 2 z E k z y x 0 2 2 2 2 2 z c e k y x Original 3D wave equation: 2-D Wave equation or Helmholtz equation !! 2 z j z z e y x e z y x E ) , ( ) , , ( where, For the same reason, we have the Helmholtz equation for H z : 2 2 2 z j nd 0 2 2 z c h k y x z z e y x h z y x H ) , ( ) , , ( and One only needs to solve a wave equation that is only defined in the cross-section!! yq y
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Waveguide Boundary Conditions On top of the PEC On top of the PMC (Perfect Magnetic Conductor) 0 t E C 0 t H C S n J H i 0 n H B.C. 0 n E B.C. How about the interface between two perfect dielectrics? rom T- relationship assume H = 0: (implicit boundary conditions) ) ( 2 x H y E k j H z z c x  From T L relationship, assume H z 0: on PEC 0 x H 0 y E z And so on…….
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TM Waves (E-Wave) TM waves: 0 , 0 z z H E Previously we have transverse – longitudinal relationship described by, ) ( 2 x H y E k j H z z c x  Substitute H z =0, x E k j E z c x 2 y E k j E z c y 2 E j   ) ( 2 y H x E k j H z z c y E H j z z y E k j H z c x 2 x k H z c y 2 ) ( 2 x y k E c x  ) ( 2 E H j E z z y y x k c Once we know E z , we know everything else!
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161_1_Lecture_9 - EE161 El EE161 Electromagnetic Waves...

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