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 H-parameters to S-parameters

In matrix form, the formula is

S=([k100−Z2 k2] H+[−Z1 k100k2]) ([k100Z2 k2] H+[Z1 k100k2])−1\mathbf{S} =\left(\begin{bmatrix} k_{1} & 0\\ 0 & -Z_{2}\,k_{2} \end{bmatrix}\,\mathbf{H}+\begin{bmatrix} -Z_{1}\,k_{1} & 0\\ 0 & k_{2} \end{bmatrix}\right)\,{\left(\begin{bmatrix} k_{1} & 0\\ 0 & Z_{2}\,k_{2} \end{bmatrix}\,\mathbf{H}+\begin{bmatrix} Z_{1}\,k_{1} & 0\\ 0 & k_{2} \end{bmatrix}\right)}^{-1}

While for each element, we obtain

S11=−Z1−H11−H11 H22 Z2+H12 H21 Z2+H22 Z1 Z2H11+Z1+H11 H22 Z2−H12 H21 Z2+H22 Z1 Z2S12=2 H12 Z1 k1k2 (H11+Z1+H11 H22 Z2−H12 H21 Z2+H22 Z1 Z2)S21=−2 H21 Z2 k2k1 (H11+Z1+H11 H22 Z2−H12 H21 Z2+H22 Z1 Z2)S22=H11+Z1−H11 H22 Z2+H12 H21 Z2−H22 Z1 Z2H11+Z1+H11 H22 Z2−H12 H21 Z2+H22 Z1 Z2\begin{align*}S_{11} &=-\frac{Z_{1}-H_{11}-H_{11}\,H_{22}\,Z_{2}+H_{12}\,H_{21}\,Z_{2}+H_{22}\,Z_{1}\,Z_{2}}{H_{11}+Z_{1}+H_{11}\,H_{22}\,Z_{2}-H_{12}\,H_{21}\,Z_{2}+H_{22}\,Z_{1}\,Z_{2}}\\S_{12} &=\frac{2\,H_{12}\,Z_{1}\,k_{1}}{k_{2}\,\left(H_{11}+Z_{1}+H_{11}\,H_{22}\,Z_{2}-H_{12}\,H_{21}\,Z_{2}+H_{22}\,Z_{1}\,Z_{2}\right)}\\S_{21} &=-\frac{2\,H_{21}\,Z_{2}\,k_{2}}{k_{1}\,\left(H_{11}+Z_{1}+H_{11}\,H_{22}\,Z_{2}-H_{12}\,H_{21}\,Z_{2}+H_{22}\,Z_{1}\,Z_{2}\right)}\\S_{22} &=\frac{H_{11}+Z_{1}-H_{11}\,H_{22}\,Z_{2}+H_{12}\,H_{21}\,Z_{2}-H_{22}\,Z_{1}\,Z_{2}}{H_{11}+Z_{1}+H_{11}\,H_{22}\,Z_{2}-H_{12}\,H_{21}\,Z_{2}+H_{22}\,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

[V1I2]=H[I1V2]\begin{bmatrix}V_1 \\I_2\end{bmatrix} = \mathbf{H}\begin{bmatrix}I_1 \\V_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.