Transmission Line

A transmission line is usually described by its ABCD-matrix. (Note that in ABCD-matrices, i.e. the chain matrix representation, the current $i_2$ is defined to flow out of the output port.)

$$\left(\underline{A}\right) = \begin{pmatrix}\cosh{\left(\gamma\cd... ...t(\gamma\cdot l\right)} / Z_L & \cosh{\left(\gamma\cdot l\right)} \end{pmatrix}$$ (9.193)

These can easily be recalculated into impedance parameters.

$$Z_{11} = Z_{22} = \frac{Z_L}{\tanh{\left(\gamma\cdot l\right)}}$$ (9.194)

$$Z_{12} = Z_{21} = \frac{Z_L}{\sinh{\left(\gamma\cdot l\right)}}$$ (9.195)

Or in admittance parameter representation it yields

$$\begin{split}Y_{11} = Y_{22} &= \frac{1}{Z_L \cdot \tanh{\lef... ...rac{-1}{Z_L\cdot \sinh{\left(\gamma\cdot l\right)}} \end{split}$$ (9.196)

whence $\gamma$ denotes the propagation constant $\alpha + j\beta$ and $l$ is the length of the transmission line. $Z_L$ represents the characteristic impedance of the transmission line. The Y-parameters as defined by eq. (9.199) can be used for the microstrip line. For an ideal, i.e. lossless, transmission lines they write accordingly.

$$Z_{11} = Z_{22} = \frac{Z_L}{j\cdot\tan{\left(\beta\cdot l\right)}}$$ (9.197)

$$Z_{12} = Z_{21} = \frac{Z_L}{j\cdot\sin{\left(\beta\cdot l\right)}}$$ (9.198)

$$Y_{11} = Y_{22} = \frac{1}{j\cdot Z_L \cdot \tan{\left(\beta\cdot l\right)}}$$ (9.199)

$$Y_{12} = Y_{21} = \frac{j}{Z_L\cdot \sin{\left(\beta\cdot l\right)}}$$ (9.200)

The scattering matrix of an ideal, lossless transmission line with impedance $Z$ and electrical length $l$ writes as follows.

$$r = \frac{Z-Z_0}{Z+Z_0}$$ (9.201)

$$p = \exp\left(-j\omega\frac{l}{c_0}\right)$$ (9.202)

$$S_{11} = S_{22} = \frac{r\cdot(1-p^2)}{1-r^2\cdot p^2} \qquad,\qquad S_{12} = S_{21} = \frac{p\cdot(1-r^2)}{1-r^2\cdot p^2}$$ (9.203)

With $c_0$ = 299 792 458 m/s being the vacuum light velocity. Adding attenuation to the transmission line, the quantity $p$ extends to:

$$p = \exp\left(-j\omega\frac{l}{c_0} - \alpha\cdot l \right)$$ (9.204)

Another equivalent equation set for the calculation of the scattering parameters is the following: With the physical length $l$ of the component, its impedance $Z_L$ and propagation constant $\gamma$, the complex propagation constant $\gamma$ is given by

$$\gamma = \alpha + j\beta$$ (9.205)

where $\alpha$ is the attenuation factor and $\beta$ is the (real) propagation constant given by

$$\beta = \sqrt{\varepsilon_{r_{eff}}(\omega)} \cdot k_0$$ (9.206)

where $\varepsilon_{r_{eff}}(\omega)$ is the effective dielectric constant and $k_0$ is the TEM propagation constant $k_0$ for the equivalent transmission line with an air dielectric.

$$k_0 = \omega \sqrt{\varepsilon_0 \mu_0}$$ (9.207)

The expressions used to calculate the scattering parameters are given by

$$S_{11} = S_{22} = \dfrac{\left(z - y\right) \sinh{\gamma l}}{2\cosh{\gamma l} + \left(z + y\right) \sinh{\gamma l}}$$ (9.208)

$$S_{12} = S_{21} = \dfrac{2}{2\cosh{\gamma l} + \left(z + y\right) \sinh{\gamma l}}$$ (9.209)

with $z$ being the normalized impedance and $y$ is the normalized admittance.