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Model coaxial transmission line
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_{cond} is the conductivity in the conductor.
μ is the permeability of the dielectric.μ = μ_{0} μ_{r}where:
μ_{0} is the permeability in free space.
μ_{r} 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′ = ε_{0}ε_{r}. ε″ is the imaginary part of complex dielectric constant ε, ε″ = ε_{0}ε_{r}tan δ where :
ε_{0} is the permittivity of free space.
ε_{r} is the Relative permittivity constant parameter value.
tan δ is the Loss tangent of dielectric parameter value.
δ_{cond} is the skin depth of the conductor, which the block calculates as $$1/\sqrt{\pi f\mu {\sigma}_{cond}}$$.
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
$$\begin{array}{c}A=1\\ B={Z}_{in}\\ C=0\\ D=1\end{array}$$
Radius of the outer conductor of the coaxial transmission line.
Radius of the inner conductor of the coaxial transmission line.
Relative permeability of the dielectric expressed as the ratio of the permeability of the dielectric to permeability in free space μ_{0}.
Relative permittivity of the dielectric expressed as the ratio of the permittivity of the dielectric to permittivity in free space ε_{0}.
Loss angle tangent of the dielectric.
Conductivity of the conductor in siemens per meter.
Physical length of the transmission line.
Type of stub. Choices are Not a stub, Shunt, or Series.
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.