Slide 1

Prof. David R. JacksonDept. of ECENotes 9ECE 5317-6351 Microwave EngineeringFall 2011Waveguides Part 6:Coaxial Cable

1To find the TEM mode fieldsWe need to solve

Zero volt potential reference location (0 = b).

zy

xba

Coaxial Line: TEM Mode2

Thus,

Coaxial Line: TEM Mode (cont.)Hencezy

xba

3

Coaxial Line: TEM Mode (cont.)zy

xba

HenceNote: This does not account for conductor loss.4Coaxial Line: TEM Mode (cont.)zy

xba

Attenuation:Dielectric attenuation:Conductor attenuation:

5(We remove all loss from the dielectric in Z0lossless.)Coaxial Line: TEM Mode (cont.)zy

xba

Conductor attenuation:

6Coaxial Line: TEM Mode (cont.)zy

xba

Conductor attenuation:

Hence we have

or7Coaxial Line: TEM Mode (cont.)zy

xba

Lets redo the calculation of conductor attenuation using the Wheeler incremental inductance formula.

Wheelers formula:In this formula, dl (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

Coaxial Line: TEM Mode (cont.)zy

xba

Hencesoor9Coaxial Line: TEM Mode (cont.)zy

xba

whereWe 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)Coaxial Line: Higher-Order Modeszy

xba

We look at the higher-order modes of a coaxial line. 11The lowest mode is the TE11 mode. Sketch of field lines for TE11 modexyCoaxial Line: Higher-Order Modes (cont.)zy

xba

TEz: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. Plot of Bessel Functions13

xJn (x)n = 0n = 1n = 2

Plot of Bessel Functions (cont.)14

xYn (x)n = 0n = 1n = 2

zy

xba

15We 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.)zy

xba

16Boundary Conditions:

Hence

Note: The prime denotes derivative with respect to the argument.Coaxial Line: Higher-Order Modes (cont.)zy

xba

17

HenceIn 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.)zy

xba

18Denote

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 haveCoaxial Line: Higher-Order Modes (cont.)19xn1 xn2 xn3 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.20Fig. 3.16 from the Pozar book.

n = 1Approximate solution:

Coaxial Line: Higher-Order Modes (cont.)Exact solutionCoaxial Line: Lossless Case21Wavenumber:

TE11 mode:Coaxial Line: Lossless Case (cont.)22

At the cutoff frequency, the wavelength (in the dielectric) is

soorCompare with the cutoff frequency condition of the TE10 mode of RWG:

ab

Example23

Page 129 of the Pozar book:RG 142 coax: