The $S$-parameters of a two-port network is given as
$$
[S]=\left[\begin{array}{ll}
S_{11} & S_{12} \\
S_{21} & S_{22}
\end{array}\right]
$$
with reference to $Z_0$. Two lossless transmission line sections of electrical lengths $\theta_1=\beta l_1$ and $\theta_2=\beta l_2$ are added to the input and output ports for measurement purposes, respectively. The $S$-parameters $\left[S^{\prime}\right]$ of the resultant two port network is
- $\left[\begin{array}{ll}S_{11} e^{-j 2 \theta_1} & S_{12} e^{-j\left(\theta_1+\theta_2\right)} \\ S_{21} e^{-j\left(\theta_1+\theta_2\right)} & S_{22} e^{-j 2 \theta_2}\end{array}\right]$
- $\left[\begin{array}{ll}S_{11} e^{j 2 \theta_1} & S_{12} e^{-j\left(\theta_1+\theta_2\right)} \\ S_{21} e^{-j\left(\theta_1+\theta_2\right)} & S_{22} e^{j 2 \theta_2}\end{array}\right]$
- $\left[\begin{array}{ll}S_{11} e^{j 2 \theta_1} & S_{12} e^{j\left(\theta_1+\theta_2\right)} \\ S_{21} e^{j\left(\theta_1+\theta_2\right)} & S_{22} e^{j 2 \theta_2}\end{array}\right]$
- $\left[\begin{array}{ll}S_{11} e^{-j 2 \theta_1} & S_{12} e^{j\left(\theta_1+\theta_2\right)} \\ S_{21} e^{j\left(\theta_1+\theta_2\right)} & S_{22} e^{-j 2 \theta_2}\end{array}\right]$