Microwave Engineering 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 z x b a To find the TEM mode fields We need to solve Zero volt potential reference location (0 = b).

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

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

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

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

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

Coaxial Line: TEM Mode (cont.) Let’s redo the calculation of conductor attenuation using the Wheeler incremental inductance formula. z y x b a Wheeler’s formula: 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.

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

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

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

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

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

Plot of Bessel Functions (cont.) Yn (x) x

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

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

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

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

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

Coaxial Line: Lossless Case Wavenumber: TE11 mode:

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

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