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

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

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

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

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

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

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

8. 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.

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

10. 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)

11. 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

12. 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.

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

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

15. 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.

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

17. 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

18. 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.

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 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: