Lect27RectWaveguides - ECE 3030 Electromagnetic Fields and...

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1 ECE 3030 Electromagnetic Fields and Waves Fall 2009 Lecture 27 2009/10/30 Rectangular Metal Waveguides TE and TM Guided Modes 1 Swartz 06/5/25 Electromagnetic Fields and Waves – Fall 2009 Lecture 27: Instructor: Dr. Wesley E. Swartz Metal Waveguides Parallel Metal Plate Waveguide from Lecture 26: d x y z W 3 Swartz 06/5/25 Electromagnetic Fields and Waves – Fall 2009 Lecture 27: Rectangular Metal Waveguide Somewhat like a parallel plate metal waveguide that is closed by metal walls on the remaining two sides. a b x y z o μ ε Parallel Plate Metal Waveguide Theory TE Modes: Dispersion relation: () { , 3 , 2 , 1 sin ˆ = = m e x d m E y r E z k j o z π d z x o m = 1 m = 2 E y E y 2 2 = d m k o z ω 4 Swartz 06/5/25 Electromagnetic Fields and Waves – Fall 2009 Lecture 27: d z o m = 1 m = 2 H y H y x { , 3 , 2 , 1 , 0 cos ˆ = = m e x d m H y r H z k j o z 2 2 = d m k o z TM Modes: Dispersion relation: TE Guided Modes (1) The fields of the guided wave satisfy the complex wave equations: We look for solutions of the equation: r H j r E o = × r E j r H = × r E r E o 2 2 = r H r H o 2 2 = ( ) ( ) r E r E r E r E o o 2 2 2 2 2 2 = + + = b y 5 Swartz 06/5/25 Electromagnetic Fields and Waves – Fall 2009 Lecture 27: Try separation of variables (with E field pointing in directions transverse to the direction of propagation): where: z y x 2 2 2 a x z () () [] z k j o z e y D x C x y B x A y E r E + = ˆ ˆ x k x k x C x A x x cos or sin , ( ) ( ) y k y k y D y B y y cos or sin , o z y x k k k 2 2 2 2 = + + TE Guided Modes (2) Boundary condition : Components of E-field parallel to the metal walls must go to zero at the metal walls: b a y x z z k j o z e y D x C x y B x A y E r E + = ˆ ˆ 0 , 0 = = = b y y x E 0 , 0 = = = a x x y E This implies: 6 Swartz 06/5/25 Electromagnetic Fields and Waves – Fall 2009 Lecture 27: x k x A x sin = ( ) y k y D y sin = , 3 , 2 1 , 0 : where , m a m k x = = , 3 , 2 1 , 0 : where , n b n k y = = ()() z k j y x o z e y k x C x y B x k y E r E + = sin ˆ sin ˆ So we have: What do we do now? TE Guided Modes (3) Another Boundary Condition : Components of H-field normal to the metal walls must go to zero at the metal walls: Turns out that the above solution form already satisfies the H-field boundary conditions: b a y x z 0 , 0 = = = a x x x H 0 , 0 = = = b y y y H z k j y x o z e y k x C x y B x k y E r E + = sin ˆ sin ˆ 7 Swartz 06/5/25 Electromagnetic Fields and Waves – Fall 2009 Lecture 27: We use something that we never had to use before in the context of guided waves:
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Lect27RectWaveguides - ECE 3030 Electromagnetic Fields and...

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