Adam Cooman

YZSHGABT
YY→ZY→SY→HY→GY→AY→BY→T
ZZ→YZ→SZ→HZ→GZ→AZ→BZ→T
SS→YS→ZS→HS→GS→AS→BS→T
HH→YH→ZH→SH→GH→AH→BH→T
GG→YG→ZG→SG→HG→AG→BG→T
AA→YA→ZA→SA→HA→GA→BA→T
BB→YB→ZB→SB→HB→GB→AB→T
TT→YT→ZT→ST→HT→GT→AT→B

From A-parameters to S-parameters

In matrix form, the formula is

S=([k1−Z1 k100] A+[00k2Z2 k2]) ([k1Z1 k100] A+[00k2−Z2 k2])−1\mathbf{S} =\left(\begin{bmatrix} k_{1} & -Z_{1}\,k_{1}\\ 0 & 0 \end{bmatrix}\,\mathbf{A}+\begin{bmatrix} 0 & 0\\ k_{2} & Z_{2}\,k_{2} \end{bmatrix}\right)\,{\left(\begin{bmatrix} k_{1} & Z_{1}\,k_{1}\\ 0 & 0 \end{bmatrix}\,\mathbf{A}+\begin{bmatrix} 0 & 0\\ k_{2} & -Z_{2}\,k_{2} \end{bmatrix}\right)}^{-1}

While for each element, we obtain

S11=A12+A11 Z2−A22 Z1−A21 Z1 Z2A12+A11 Z2+A22 Z1+A21 Z1 Z2S12=2 Z1 k1 (A11 A22−A12 A21)k2 (A12+A11 Z2+A22 Z1+A21 Z1 Z2)S21=2 Z2 k2k1 (A12+A11 Z2+A22 Z1+A21 Z1 Z2)S22=A12−A11 Z2+A22 Z1−A21 Z1 Z2A12+A11 Z2+A22 Z1+A21 Z1 Z2\begin{align*}S_{11} &=\frac{A_{12}+A_{11}\,Z_{2}-A_{22}\,Z_{1}-A_{21}\,Z_{1}\,Z_{2}}{A_{12}+A_{11}\,Z_{2}+A_{22}\,Z_{1}+A_{21}\,Z_{1}\,Z_{2}}\\S_{12} &=\frac{2\,Z_{1}\,k_{1}\,\left(A_{11}\,A_{22}-A_{12}\,A_{21}\right)}{k_{2}\,\left(A_{12}+A_{11}\,Z_{2}+A_{22}\,Z_{1}+A_{21}\,Z_{1}\,Z_{2}\right)}\\S_{21} &=\frac{2\,Z_{2}\,k_{2}}{k_{1}\,\left(A_{12}+A_{11}\,Z_{2}+A_{22}\,Z_{1}+A_{21}\,Z_{1}\,Z_{2}\right)}\\S_{22} &=\frac{A_{12}-A_{11}\,Z_{2}+A_{22}\,Z_{1}-A_{21}\,Z_{1}\,Z_{2}}{A_{12}+A_{11}\,Z_{2}+A_{22}\,Z_{1}+A_{21}\,Z_{1}\,Z_{2}}\\\end{align*}

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

Definitions

[V1I1]=A[V2−I2]\begin{bmatrix}V_1 \\I_1\end{bmatrix} = \mathbf{A}\begin{bmatrix}V_2 \\-I_2\end{bmatrix}

[B1B2]=S[A1A2]\begin{bmatrix}B_1 \\B_2\end{bmatrix} = \mathbf{S}\begin{bmatrix}A_1 \\A_2\end{bmatrix}

The incident and reflected waves are defined as

A1=k1 (V1+I1 Z1)B1=k1 (V1−I1 Z1)A2=k2 (V2+I2 Z2)B2=k2 (V2−I2 Z2)\begin{align*}A_1 &= k_{1}\,\left(V_{1}+I_{1}\,Z_{1}\right)\qquad B_1 &= k_{1}\,\left(V_{1}-I_{1}\,Z_{1}\right)\\A_2 &= k_{2}\,\left(V_{2}+I_{2}\,Z_{2}\right)\qquad B_2 &= k_{2}\,\left(V_{2}-I_{2}\,Z_{2}\right)\\\end{align*}

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.