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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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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

To find the TEM mode fields

We need to solve

b

a

z

x

Zero volt potential reference location (0 = b).


Coaxial Line: TEM Mode (cont.)

y

Hence

b

a

Thus,

z

x


Coaxial Line: TEM Mode (cont.)

y

b

a

z

x

Hence

Note: This does not account for conductor loss.


Coaxial Line: TEM Mode (cont.)

y

Attenuation:

b

a

Dielectric attenuation:

z

x

Conductor attenuation:

(We remove all loss from the dielectric in Z0lossless.)


Coaxial Line: TEM Mode (cont.)

Conductor attenuation:

y

b

a

z

x


Coaxial Line: TEM Mode (cont.)

Conductor attenuation:

y

b

a

z

x

Hence we have

or


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.


Coaxial Line: TEM Mode (cont.)

y

b

a

z

x

so

Hence

or


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)


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


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.


Plot of Bessel Functions

n =0

n =1

n =2

Jn (x)

x


Plot of Bessel Functions (cont.)

n =0

n =1

n =2

Yn (x)

x


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.


Coaxial Line: Higher-Order Modes (cont.)

y

Boundary Conditions:

b

a

z

x

Note: The prime denotes derivative with respect to the argument.

Hence


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


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.


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:


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