Adam Cooman

YZSHGABT
Y Y→Z Y→S Y→H Y→G Y→A Y→B Y→T
Z Z→Y Z→S Z→H Z→G Z→A Z→B Z→T
S S→Y S→Z S→H S→G S→A S→B S→T
H H→Y H→Z H→S H→G H→A H→B H→T
G G→Y G→Z G→S G→H G→A G→B G→T
A A→Y A→Z A→S A→H A→G A→B A→T
B B→Y B→Z B→S B→H B→G B→A B→T
T T→Y T→Z T→S T→H T→G T→A T→B

From S-parameters to Z-parameters

In matrix form, the formula is

Z=K−1(IN+S)(IN−S)−1KZ0\mathbf{Z} =K^{-1} \left(I_N + \mathbf{S}\right)\left(I_N - \mathbf{S}\right)^{-1} K Z_0

When dealing with 2 ports, we obtain

Z11=−Z1 (S11−S22−S11 S22+S12 S21+1)S11+S22−S11 S22+S12 S21−1Z12=−2 S12 Z2 k2k1 (S11+S22−S11 S22+S12 S21−1)Z21=−2 S21 Z1 k1k2 (S11+S22−S11 S22+S12 S21−1)Z22=−Z2 (S22−S11−S11 S22+S12 S21+1)S11+S22−S11 S22+S12 S21−1\begin{align*}Z_{11} &=-\frac{Z_{1}\,\left(S_{11}-S_{22}-S_{11}\,S_{22}+S_{12}\,S_{21}+1\right)}{S_{11}+S_{22}-S_{11}\,S_{22}+S_{12}\,S_{21}-1}\\Z_{12} &=-\frac{2\,S_{12}\,Z_{2}\,k_{2}}{k_{1}\,\left(S_{11}+S_{22}-S_{11}\,S_{22}+S_{12}\,S_{21}-1\right)}\\Z_{21} &=-\frac{2\,S_{21}\,Z_{1}\,k_{1}}{k_{2}\,\left(S_{11}+S_{22}-S_{11}\,S_{22}+S_{12}\,S_{21}-1\right)}\\Z_{22} &=-\frac{Z_{2}\,\left(S_{22}-S_{11}-S_{11}\,S_{22}+S_{12}\,S_{21}+1\right)}{S_{11}+S_{22}-S_{11}\,S_{22}+S_{12}\,S_{21}-1}\\\end{align*}

The formulas are obtained with the methods explained here. The MATLAB implementation can be found in circuitconversions on Gitlab.

Definitions

[B1⋮BN]⏟B=S[A1⋮AN]⏟A\underbrace{ \begin{bmatrix}B_1 \\ \vdots \\ B_N \end{bmatrix} }_{\mathbf{B}} = \mathbf{S} \underbrace{ \begin{bmatrix}A_1 \\ \vdots \\ A_N \end{bmatrix} }_{\mathbf{A}}

[V1⋮VN]⏟V=Z[I1⋮IN]⏟I\underbrace{ \begin{bmatrix}V_1 \\ \vdots \\ V_N \end{bmatrix} }_{\mathbf{V}} = \mathbf{Z} \underbrace{ \begin{bmatrix}I_1 \\ \vdots \\ I_N \end{bmatrix} }_{\mathbf{I}}

The incident and reflected waves are defined as

A=K(V+Z0I)B=K(V−Z0I)\mathbf{A} = K \left( \mathbf{V} + Z_0 \mathbf{I} \right)\qquad \mathbf{B} = K \left( \mathbf{V} - Z_0 \mathbf{I} \right)

where KK and Z0Z_0 are defined as

K=[k1⋱kN]Z0=[Z0,1⋱Z0,N]K = \begin{bmatrix}k_1 & & \\ & \ddots & \\ & & k_N \end{bmatrix}\qquad Z_0 = \begin{bmatrix}Z_{0,1} & & \\ & \ddots & \\ & & Z_{0,N} \end{bmatrix}

with Z0,iZ_{0,i} the reference impedance for port ii and kik_i defined as

ki=12ℜ(Z0,i)ki=αℜ(Z0,i)2∣Z0,i∣k_i=\frac{1}{2\sqrt{\Re\left( Z_{0,i}\right) }}\qquad k_i=\alpha\frac{\sqrt{\Re\left( Z_{0,i} \right) }}{2\left|Z_{0,i}\right|}

depending on whether you are using power- or pseudowaves.