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Prof. David R. Jackson Dept. of ECE

ECE 5317-6351 Microwave Engineering. Fall 2011. Prof. David R. Jackson Dept. of ECE. Notes 9. Waveguides Part 6: Coaxial Cable. Coaxial Line: TEM Mode. y. T o find the TEM mode fields We need to solve. b. a. z. x. Zero volt potential reference location (  0 = b ).

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Prof. David R. Jackson Dept. of ECE

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  1. ECE 5317-6351 Microwave Engineering Fall 2011 Prof. David R. Jackson Dept. of ECE Notes 9 Waveguides Part 6: Coaxial Cable

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

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

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

  5. Coaxial Line: TEM Mode (cont.) y Attenuation: b a Dielectric attenuation: z x Conductor attenuation: (We remove all loss from the dielectric in Z0lossless.)

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

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

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

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

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

  11. Coaxial Line: Higher-Order Modes y We look at the higher-order modes of a coaxial line. b The lowest mode is the TE11 mode. a y z x x Sketch of field lines for TE11 mode

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

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

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

  15. Coaxial Line: Higher-Order Modes (cont.) y We choose (somewhat arbitrarily) the cosine function for the angle variation. b Wave traveling in +z direction: a z x 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.

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

  17. Coaxial Line: Higher-Order Modes (cont.) y b a 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. z x Hence

  18. Coaxial Line: Higher-Order Modes (cont.) y b Denote a z x The we have 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.

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

  20. Coaxial Line: Higher-Order Modes (cont.) Approximate solution: n = 1 Exact solution Fig. 3.16 from the Pozar book.

  21. Coaxial Line: Lossless Case Wavenumber: TE11 mode:

  22. Coaxial Line: Lossless Case (cont.) At the cutoff frequency, the wavelength (in the dielectric) is Compare with the cutoff frequency condition of the TE10 mode of RWG: b so a or

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

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