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Model parallel-plate transmission line

Transmission Lines sublibrary of the Physical library

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.

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:

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

*Z*_{0} 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:

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

where

$$\begin{array}{l}R=\frac{2}{w{\sigma}_{cond}{\delta}_{cond}}\\ L=\mu \frac{d}{w}\\ G=\omega {\epsilon}^{\u2033}\frac{w}{d}\\ C=\epsilon \frac{w}{d}\end{array}$$

In these equations:

*σ*is the conductivity in the conductor._{cond}*μ*is the permeability of the dielectric.*ε*is the permittivity of the dielectric.*ε″*is the imaginary part of*ε*,*ε″*=*ε*_{0}*ε*tan_{r}*δ*, where:*ε*_{0}is the permittivity of free space.*ε*is the_{r}**Relative permittivity constant**parameter value.tan

*δ*is the**Loss tangent of dielectric**parameter value.

*δ*is the skin depth of the conductor, which the block calculates as $$1/\sqrt{\pi f\mu {\sigma}_{cond}}$$._{cond}*f*is a vector of modeling frequencies determined by the Output Port block.

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.

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.

*Z _{in}* is the input impedance
of the shunt circuit. The ABCD-parameters for the shunt stub are calculated
as

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

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.

*Z _{in}* is the input impedance
of the series circuit. The ABCD-parameters for the series stub are
calculated as

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

**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

*μ*_{0}.**Relative permittivity constant**Relative permittivity of the dielectric expressed as the ratio of the permittivity of the dielectric to permittivity in free space

*ε*_{0}.**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`

.

For information about plotting, see Create Plots.

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