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Prof. David R. Jackson Dept. of ECE Notes 9 ECE Microwave Engineering Fall 2011 Waveguides Part 6: Coaxial Cable 1

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To find the TEM mode fields We need to solve Zero volt potential reference location ( 0 = b ). z y x b a Coaxial Line: TEM Mode 2

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Thus, Coaxial Line: TEM Mode (cont.) Hence z y x b a 3

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Coaxial Line: TEM Mode (cont.) z y x b a Hence Note: This does not account for conductor loss. 4

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Coaxial Line: TEM Mode (cont.) z y x b a Attenuation: Dielectric attenuation: Conductor attenuation: 5 (We remove all loss from the dielectric in Z 0 lossless.)

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Coaxial Line: TEM Mode (cont.) z y x b a Conductor attenuation: 6

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Coaxial Line: TEM Mode (cont.) z y x b a Conductor attenuation: Hence we haveor 7

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Coaxial Line: TEM Mode (cont.) z y x b a Lets redo the calculation of conductor attenuation using the Wheeler incremental inductance formula. Wheelers formula: In this formula, d l (for a given conductor) is the distance by which the conducting boundary is receded away from the field region. The formula is applied for each conductor and the conductor attenuation from each of the two conductors is then added. 8

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Coaxial Line: TEM Mode (cont.) z y x b a Hence so or 9

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Coaxial Line: TEM Mode (cont.) z y x b a where We can also calculate the fundamental per-unit-length parameters of the coaxial line. From previous calculations: 10 (Derived as a homework problem) (Formulas from Notes 1)

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Coaxial Line: Higher-Order Modes z y x b a We look at the higher-order modes of a coaxial line. 11 The lowest mode is the TE 11 mode. Sketch of field lines for TE 11 mode x y

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Coaxial Line: Higher-Order Modes (cont.) z y x b a TE z : We look at the higher-order modes of a coaxial line. 12 The solution in cylindrical coordinates is: Note: The value n must be an integer to have unique fields.

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Plot of Bessel Functions 13 x J n ( x ) n = 0 n = 1 n = 2

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Plot of Bessel Functions (cont.) 14 x Y n ( x ) n = 0 n = 1 n = 2

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z y x b a 15 We choose (somewhat arbitrarily) the cosine function for the angle variation. Wave traveling in +z direction: The cosine choice corresponds to having the transverse electric field E being an even function of, which is the field that would be excited by a probe located at = 0. Coaxial Line: Higher-Order Modes (cont.)

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z y x b a 16 Boundary Conditions: Hence Note: The prime denotes derivative with respect to the argument. Coaxial Line: Higher-Order Modes (cont.)

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z y x b a 17 Hence In order for this homogenous system of equations for the unknowns A and B to have a non-trivial solution, we require the determinant to be zero. Coaxial Line: Higher-Order Modes (cont.)

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z y x b a 18 Denote For a given choice of n and a given value of b/a, we can solve the above equation for x to find the zeros. The we have Coaxial Line: Higher-Order Modes (cont.)

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19 x n1 x n2 x n3 x A graph of the determinant reveals the zeros of the determinant. Coaxial Line: Higher-Order Modes (cont.) Note: These values are not the same as those of the circular waveguide, although the same notation for the zeros is being used.

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20 Fig from the Pozar book. n = 1 Approximate solution: Coaxial Line: Higher-Order Modes (cont.) Exact solution

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Coaxial Line: Lossless Case 21 Wavenumber: TE 11 mode:

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Coaxial Line: Lossless Case (cont.) 22 At the cutoff frequency, the wavelength (in the dielectric) is so or Compare with the cutoff frequency condition of the TE 10 mode of RWG: a b

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Example 23 Page 129 of the Pozar book: RG 142 coax:

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