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 Z-parameters to T-parameters

In matrix form, the formula is

T=([Z1 k10−Z1 k10]+[k10k10] Z) ([0−Z2 k20Z2 k2]+[0k20k2] Z)−1\mathbf{T} =\left(\begin{bmatrix} Z_{1}\,k_{1} & 0\\ -Z_{1}\,k_{1} & 0 \end{bmatrix}+\begin{bmatrix} k_{1} & 0\\ k_{1} & 0 \end{bmatrix}\,\mathbf{Z}\right)\,{\left(\begin{bmatrix} 0 & -Z_{2}\,k_{2}\\ 0 & Z_{2}\,k_{2} \end{bmatrix}+\begin{bmatrix} 0 & k_{2}\\ 0 & k_{2} \end{bmatrix}\,\mathbf{Z}\right)}^{-1}

While for each element, we obtain

T11=k1 (Z11 Z22−Z12 Z21+Z11 Z2+Z22 Z1+Z1 Z2)2 Z21 Z2 k2T12=k1 (Z12 Z21−Z11 Z22+Z11 Z2−Z22 Z1+Z1 Z2)2 Z21 Z2 k2T21=−k1 (Z12 Z21−Z11 Z22−Z11 Z2+Z22 Z1+Z1 Z2)2 Z21 Z2 k2T22=k1 (Z12 Z21−Z11 Z22+Z11 Z2+Z22 Z1−Z1 Z2)2 Z21 Z2 k2\begin{align*}T_{11} &=\frac{k_{1}\,\left(Z_{11}\,Z_{22}-Z_{12}\,Z_{21}+Z_{11}\,Z_{2}+Z_{22}\,Z_{1}+Z_{1}\,Z_{2}\right)}{2\,Z_{21}\,Z_{2}\,k_{2}}\\T_{12} &=\frac{k_{1}\,\left(Z_{12}\,Z_{21}-Z_{11}\,Z_{22}+Z_{11}\,Z_{2}-Z_{22}\,Z_{1}+Z_{1}\,Z_{2}\right)}{2\,Z_{21}\,Z_{2}\,k_{2}}\\T_{21} &=-\frac{k_{1}\,\left(Z_{12}\,Z_{21}-Z_{11}\,Z_{22}-Z_{11}\,Z_{2}+Z_{22}\,Z_{1}+Z_{1}\,Z_{2}\right)}{2\,Z_{21}\,Z_{2}\,k_{2}}\\T_{22} &=\frac{k_{1}\,\left(Z_{12}\,Z_{21}-Z_{11}\,Z_{22}+Z_{11}\,Z_{2}+Z_{22}\,Z_{1}-Z_{1}\,Z_{2}\right)}{2\,Z_{21}\,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

[V1V2]=Z[I1I2]\begin{bmatrix}V_1 \\V_2\end{bmatrix} = \mathbf{Z}\begin{bmatrix}I_1 \\I_2\end{bmatrix}

[A1B1]=T[B2A2]\begin{bmatrix}A_1 \\B_1\end{bmatrix} = \mathbf{T}\begin{bmatrix}B_2 \\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.