# Coaxial Transmission Line

Model coaxial transmission line

## Library

Transmission Lines sublibrary of the Physical library

## Description

The Coaxial Transmission Line block models the coaxial transmission line described in the block dialog box in terms of its frequency-dependent S-parameters. A coplanar waveguide transmission line is shown in cross-section in the following figure. Its physical characteristics include the radius of the inner conductor a and the radius of the outer conductor b.

The block lets you model the transmission line as a stub or as a stubless line.

### Stubless Transmission Line

If you model a coaxial transmission line as a stubless line, the Coaxial Transmission Line block first calculates the ABCD-parameters at each frequency contained in the modeling frequencies vector. It then uses the `abcd2s` function to convert the ABCD-parameters to S-parameters.

The block calculates the ABCD-parameters using the physical length of the transmission line, d, and the complex propagation constant, k, using the following equations:

$\begin{array}{l}A=\frac{{e}^{kd}+{e}^{-kd}}{2}\\ B=\frac{{Z}_{0}*\left({e}^{kd}-{e}^{-kd}\right)}{2}\\ C=\frac{{e}^{kd}-{e}^{-kd}}{2*{Z}_{0}}\\ D=\frac{{e}^{kd}+{e}^{-kd}}{2}\end{array}$

Z0 and k are vectors whose elements correspond to the elements of f, a vector of modeling frequencies, determined by the Output Port block. Both can be expressed in terms of the resistance (R), inductance (L), conductance (G), and capacitance (C) per unit length (meters) as follows:

$\begin{array}{c}{Z}_{0}=\sqrt{\frac{R+j\omega L}{G+j\omega C}}\\ k={k}_{r}+j{k}_{i}=\sqrt{\left(R+j\omega L\right)\left(G+j\omega C\right)}\end{array}$

where

$\begin{array}{l}R=\frac{1}{2\pi {\sigma }_{cond}{\delta }_{cond}}\left(\frac{1}{a}+\frac{1}{b}\right)\\ L=\frac{\mu }{2\pi }\mathrm{ln}\left(\frac{b}{a}\right)\\ G=\frac{2\pi \omega {\epsilon }^{″}}{\mathrm{ln}\left(\frac{b}{a}\right)}\\ C=\frac{2\pi {\epsilon }^{\prime }}{\mathrm{ln}\left(\frac{b}{a}\right)}\end{array}$

In these equations:

• a is the radius of the inner conductor.

• b is the radius of the outer conductor.

• σcond is the conductivity in the conductor.

• μ is the permeability of the dielectric.μ  = μ0 μrwhere:

• μ0 is the permeability in free space.

• μr is the Relative permeability constant parameter value.

• The is a complex dielectric constant given by ε   =  ε′ − јε″= ε′ ( 1 − јtanδ)

• ε′ is the real part of complex dielectric constant ε, ε′  = ε0εr. ε″ is the imaginary part of complex dielectric constant ε, ε″  = ε0εrtan δ where :

• ε0 is the permittivity of free space.

• εr is the Relative permittivity constant parameter value.

• tan δ is the Loss tangent of dielectric parameter value.

• δcond is the skin depth of the conductor, which the block calculates as $1/\sqrt{\pi f\mu {\sigma }_{cond}}$.

### Shunt and Series Stubs

If you model the transmission line as a shunt or series stub, the Coaxial Transmission Line block first calculates the ABCD-parameters at each frequency contained in the modeling frequencies vector. It then uses the `abcd2s` function to convert the ABCD-parameters to S-parameters.

### Shunt ABCD-Parameters

When you set the Stub mode parameter in the mask dialog box to `Shunt`, the two-port network consists of a stub transmission line that you can terminate with either a short circuit or an open circuit as shown here.

Zin is the input impedance of the shunt circuit. The ABCD-parameters for the shunt stub are calculated as

$\begin{array}{c}A=1\\ B=0\\ C=1/{Z}_{in}\\ D=1\end{array}$

### Series ABCD-Parameters

When you set the Stub mode parameter in the mask dialog box to `Series`, the two-port network consists of a series transmission line that you can terminate with either a short circuit or an open circuit as shown here.

Zin is the input impedance of the series circuit. The ABCD-parameters for the series stub are calculated as

$\begin{array}{c}A=1\\ B={Z}_{in}\\ C=0\\ D=1\end{array}$

## Dialog Box

### Main Tab

Radius of the outer conductor of the coaxial transmission line.

Radius of the inner conductor of the coaxial transmission line.

Relative permeability constant

Relative permeability of the dielectric expressed as the ratio of the permeability of the dielectric to permeability in free space μ0.

Relative permittivity constant

Relative permittivity of the dielectric expressed as the ratio of the permittivity of the dielectric to permittivity in free space ε0.

Loss tangent of dielectric

Loss angle tangent of the dielectric.

Conductivity of conductor (S/m)

Conductivity of the conductor in siemens per meter.

Transmission line length (m)

Physical length of the transmission line.

Stub mode

Type of stub. Choices are `Not a stub`, `Shunt`, or `Series`.

Termination of stub

Stub termination for stub modes `Shunt` and `Series`. Choices are `Open` or `Short`. This parameter becomes visible only when Stub mode is set to `Shunt` or `Series`.

### Visualization Tab

For information about plotting, see Create Plots.

## References

[1] Pozar, David M. Microwave Engineering, John Wiley & Sons, Inc., 2005.