By returning to the original S-parameter definitions by Kurokawa 1 and taking some care to preserve the noncommutative properties of the matrix expressions then relatively simple expressions valid for an arbitrary number of ports may be derived.

The \( [\mathbf{a}, \mathbf{b}] \) parameters that define S-parameters are fundamentally related to the \( [\mathbf{i}, \mathbf{v}] \) parameters by the expressions originally derived by Kurokawa. The expressions are restated here in matrix form which emphasises the noncommutative properties used to combine the various column vector parameters.

\begin{equation} \mathbf{a} = \frac{1}{2} \left[ \mathbf{K} \right] ^{-1} ( \mathbf{v} + \mathbf{Z_{p}} \mathbf{i} ) \end{equation}

\begin{equation} \mathbf{b} = \frac{1}{2} \left[ \mathbf{K} \right] ^{-1} ( \mathbf{v} - \mathbf{Z_{p}^{*}} \mathbf{i} ) \end{equation}

Where \( \mathbf{i} \) and \( \mathbf{v} \) are the respective column vectors of current and voltage at each port and \( \mathbf{Z_{p}} \) is the diagonal matrix of the complex termination impedances for each port. Finally,

\[ \mathbf{K} = \sqrt{\left| \Re (\mathbf{Z_{p}}) \right|} \]

Note that when \( \mathbf{Z_{p}} \) are complex impedances then \( \mathbf{a} \) and \( \mathbf{b} \) are implicitly in the frequency domain. If \( \mathbf{Z_{p}} \) are purely resistive (real) then the equations may be either time or frequency domain due to linearity.

The respective terminal equations for S and Y parameters are

\begin{equation} \mathbf{b} = \mathbf{S} \mathbf{a} \end{equation}

\begin{equation} \mathbf{i} = \mathbf{Y} \mathbf{v} \end{equation}

Substituting \( \mathbf{a} \) ,\( \mathbf{b} \) and \( \mathbf{i} \) into \( \mathbf{b} = \mathbf{S} \mathbf{a} \).

\begin{equation} \left[ \mathbf{K} \right]^{-1} \mathbf{v} - \left[ \mathbf{K} \right]^{-1} \mathbf{Z_{p}^{*}} \mathbf{Y} \mathbf{v} = \mathbf{S} \left( \left[ \mathbf{K} \right]^{-1} \mathbf{v} + \left[ \mathbf{K} \right]^{-1} \mathbf{Z_{p}} \mathbf{Y} \mathbf{v} \right) \end{equation}

The \( v \) terms cancel, leaving

\begin{equation} \left[ \mathbf{K} \right]^{-1} - \left[ \mathbf{K} \right]^{-1} \mathbf{Z_{p}^{*}} \mathbf{Y} = \mathbf{S} \left[ \mathbf{K} \right]^{-1} + \mathbf{S} \left[ \mathbf{K} \right]^{-1} \mathbf{Z_{p}} \mathbf{Y} \end{equation}

Solving for \( \mathbf{Y} \) in terms of \( \mathbf{S} \) yields

\begin{equation} \mathbf{Y} = \left[ \mathbf{S} \left[ \mathbf{K} \right]^{-1} \mathbf{Z_{p}} + \left[ \mathbf{K} \right]^{-1} \mathbf{Z_{p}^{*}} \right] ^{-1} (\mathbf{1_{N}} - \mathbf{S}) \left[ \mathbf{K} \right]^{-1} \end{equation}

Solving for \( \mathbf{S} \) in terms of \( \mathbf{Y} \) yields

\begin{equation} \mathbf{S} = \left[ \mathbf{K} \right] ^{-1} (\mathbf{1_{N}} - \mathbf{Z_{p}^{*}} \mathbf{Y}) \left[ \mathbf{1_{N}} + \mathbf{Z_{p}} \mathbf{Y} \right] ^{-1} \mathbf{K} \end{equation}

A similar argument can be made based on the terminal equations for conversion between S and Z parameters.

  1. K. Kurokawa, “Power waves and the scattering matrix,” IEEE Trans. Microw. Theory Tech., pp194-202, 1965.