Module 5 b

# Module 5 b - Module 5 Lecture 2 Modes of hollow rectangular...

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page 1 Module 5 Lecture 2 Modes of hollow rectangular conductors

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page 2 Contents (Sadiku 12-3, 12-4, 12-6) TM mode solutions Mode nomenclature and associated properties Mode cut-off Phase velocity of guided modes Group velocity of guided modes Intrinsic impedance of guided modes TE modes Dominant mode of a waveguide Power loss formulae (summary)
page 3 Boundary conditions for TM modes Try a solution where H z = 0, for the hollow rectangular conductor First, find E z . Then 4 other components. E z must be zero at x=0 and a y=0 and b Because E z is tangential to a perfect conductor at those boundaries

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page 4 Use the general solution (Mod.5 Lect.1), at x=0 and y=0 E z = (A 1 cosk x x + A 2 sink x x)(A 3 cosk y y + A 4 sink y y)exp(- g z)
page 5 Two more TM boundary conditions Then we must have sink x a = 0 which implies k x a = m p (m=1,2,3,. ...) The same argument holds for the boundary condition at y=b From which we get that sink y b = 0 which implies k y b = n p (m=1,2,3,. ...)

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page 6 E z field of TM solutions IMPORTANT NOTE: m OR n CANNOT BE ZERO, OTHERWISE THERE IS NO SOLUTION
page 7 The other field components of this TM solution

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page 8 Different (m,n) pairs correspond to different TM “modes” The values of m and n dictate the shape of the field solution in the x-y plane (through the cosine functions). The propagation constant g of each mode is different (depends on m and n). Which proves that several different waves can be propagating at the same frequency but with different shapes and propagation constants.
page 9 1D field profiles for TM modes TM 11 TM 21

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page 10 1D field profiles for more TM modes TM 12 TM 23
page 11 3D Fields of the TM 11 mode From Ullaby, Chapter 8

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## Module 5 b - Module 5 Lecture 2 Modes of hollow rectangular...

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