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.

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

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

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

$$\mathbf{b} = \mathbf{S} \mathbf{a}$$

$$\mathbf{i} = \mathbf{Y} \mathbf{v}$$

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

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

The $$v$$ terms cancel, leaving

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

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

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

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

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

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.