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Model coaxial transmission line

Transmission Lines sublibrary of the Physical library

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.

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:

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

*Z*_{0} 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:

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

where

$$\begin{array}{l}R=\frac{1}{2\pi {\sigma}_{cond}{\delta}_{cond}}\left(\frac{1}{a}+\frac{1}{b}\right)\\ L=\frac{\mu}{2\pi}\mathrm{ln}\left(\frac{b}{a}\right)\\ G=\frac{2\pi \omega {\epsilon}^{\u2033}}{\mathrm{ln}\left(\frac{b}{a}\right)}\\ C=\frac{2\pi {\epsilon}^{\prime}}{\mathrm{ln}\left(\frac{b}{a}\right)}\end{array}$$

In these equations:

*a*is the radius of the inner conductor.*b*is the radius of the outer conductor.*σ*is the conductivity in the conductor._{cond}*μ*is the permeability of the dielectric.*μ*=*μ*_{0}*μ*where:_{r}*μ*_{0}is the permeability in free space.*μ*is the_{r}**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′*=*ε*_{0}*ε*._{r}*ε″*is the imaginary part of complex dielectric constant*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}

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.

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

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

**Outer radius (m)**Radius of the outer conductor of the coaxial transmission line.

**Inner radius (m)**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

*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.

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