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set5solution - ECE 222B Winter 2010 Solution to Homework...

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ECE 222B, Winter 2010 Solution to Homework Set 5 1. Problem 8-2, Balanis Solution: Calculate a number of guide parameters given the dimensions a = 0 . 02286 m and b = 0 . 1016 m, and relative permittivity ε r = 2 . 56. It follows that the speed of light in the waveguide is c = 2 . 99 × 10 8 / ε r m/s = 1 . 87 × 10 8 m/s. The intrinsic impedance for the medium is η = η 0 / ε r 235 Ohms. (a) Calculate the cutoff frequency for the TE 10 mode in GHz. ( f c ) 10 = c 2 a 4 . 1 GHz (b) Calculate the guide wavelength (z-directed wavelength) for the TE 10 mode at 10 GHz. Express answer in cm. Let f = 10 GHz. Then the total wavelength λ at frequency f within the guide is λ = c/f = 0 . 0187 m. It follows that ( λ g ) 10 = λ p 1 - ( f c ) 2 10 /f 2 = 0 . 0205 m = 2 . 05 cm (c) Calculate the wave impedance Z w for the TE 10 mode at 10 GHz. Z w = η p 1 - ( f c ) 2 10 /f 2 = 258 Ohms (d) Calculate the z-directed phase velocity v p for the TE 10 mode at 10 GHz in units of m/s. v p = f ( λ g ) 10 = 2 . 05 × 10 8 m/s (e) Calculate the z-directed group velocity v g for the TE 10 mode at 10 GHz in units of m/s.
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