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# Parallel-Plate Transmission Line

Model parallel-plate transmission line

## Library

Transmission Lines sublibrary of the Physical library

## Description

The Parallel-Plate Transmission Line block models the parallel-plate transmission line described in the block dialog box in terms of its frequency-dependent S-parameters. A parallel-plate transmission line is shown in cross-section in the following figure. Its physical characteristics include the plate width w and the plate separation d.

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

### Stubless Transmission Line

If you model a parallel-plate transmission line as a stubless line, the Parallel-Plate 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. 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

`$\begin{array}{l}R=\frac{2}{w{\sigma }_{cond}{\delta }_{cond}}\\ L=\mu \frac{d}{w}\\ G=\omega {\epsilon }^{″}\frac{w}{d}\\ C=\epsilon \frac{w}{d}\end{array}$`

In these equations:

• σcond is the conductivity in the conductor.

• μ is the permeability of the dielectric.

• ε is the permittivity of the dielectric.

• ε″ is the imaginary part of 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 $1/\sqrt{\pi f\mu {\sigma }_{cond}}$.

• f is a vector of modeling frequencies determined by the Output Port block.

### Shunt and Series Stubs

If you model the transmission line as a shunt or series stub, the Parallel-Plate Transmission Line block first calculates the ABCD-parameters at each frequency contained in the vector of modeling frequencies. 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

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

## Parameters

### Main Tab

Plate width (m)

Physical width of the parallel-plate transmission line.

Plate separation (m)

Thickness of the dielectric separating the plates.

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.