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 S-parameters to B-parameters

In matrix form, the formula is

B=([012 k2012 Z2 k2] S+[012 k20−12 Z2 k2]) ([12 k10−12 Z1 k10] S+[12 k1012 Z1 k10])−1\mathbf{B} =\left(\begin{bmatrix} 0 & \frac{1}{2\,k_{2}}\\ 0 & \frac{1}{2\,Z_{2}\,k_{2}} \end{bmatrix}\,\mathbf{S}+\begin{bmatrix} 0 & \frac{1}{2\,k_{2}}\\ 0 & -\frac{1}{2\,Z_{2}\,k_{2}} \end{bmatrix}\right)\,{\left(\begin{bmatrix} \frac{1}{2\,k_{1}} & 0\\ -\frac{1}{2\,Z_{1}\,k_{1}} & 0 \end{bmatrix}\,\mathbf{S}+\begin{bmatrix} \frac{1}{2\,k_{1}} & 0\\ \frac{1}{2\,Z_{1}\,k_{1}} & 0 \end{bmatrix}\right)}^{-1}

While for each element, we obtain

B11=k1 (S22−S11−S11 S22+S12 S21+1)2 S12 k2B12=−Z1 k1 (S11+S22+S11 S22−S12 S21+1)2 S12 k2B21=k1 (S11+S22−S11 S22+S12 S21−1)2 S12 Z2 k2B22=Z1 k1 (S11−S22−S11 S22+S12 S21+1)2 S12 Z2 k2\begin{align*}B_{11} &=\frac{k_{1}\,\left(S_{22}-S_{11}-S_{11}\,S_{22}+S_{12}\,S_{21}+1\right)}{2\,S_{12}\,k_{2}}\\B_{12} &=-\frac{Z_{1}\,k_{1}\,\left(S_{11}+S_{22}+S_{11}\,S_{22}-S_{12}\,S_{21}+1\right)}{2\,S_{12}\,k_{2}}\\B_{21} &=\frac{k_{1}\,\left(S_{11}+S_{22}-S_{11}\,S_{22}+S_{12}\,S_{21}-1\right)}{2\,S_{12}\,Z_{2}\,k_{2}}\\B_{22} &=\frac{Z_{1}\,k_{1}\,\left(S_{11}-S_{22}-S_{11}\,S_{22}+S_{12}\,S_{21}+1\right)}{2\,S_{12}\,Z_{2}\,k_{2}}\\\end{align*}

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

Definitions

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

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