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 T-parameters to Z-parameters

In matrix form, the formula is

Z=([12 k112 k100] T+[0012 k212 k2]) ([12 Z1 k1−12 Z1 k100] T+[00−12 Z2 k212 Z2 k2])−1\mathbf{Z} =\left(\begin{bmatrix} \frac{1}{2\,k_{1}} & \frac{1}{2\,k_{1}}\\ 0 & 0 \end{bmatrix}\,\mathbf{T}+\begin{bmatrix} 0 & 0\\ \frac{1}{2\,k_{2}} & \frac{1}{2\,k_{2}} \end{bmatrix}\right)\,{\left(\begin{bmatrix} \frac{1}{2\,Z_{1}\,k_{1}} & -\frac{1}{2\,Z_{1}\,k_{1}}\\ 0 & 0 \end{bmatrix}\,\mathbf{T}+\begin{bmatrix} 0 & 0\\ -\frac{1}{2\,Z_{2}\,k_{2}} & \frac{1}{2\,Z_{2}\,k_{2}} \end{bmatrix}\right)}^{-1}

While for each element, we obtain

Z11=Z1 (T11+T12+T21+T22)T11+T12−T21−T22Z12=2 Z2 k2 (T11 T22−T12 T21)k1 (T11+T12−T21−T22)Z21=2 Z1 k1k2 (T11+T12−T21−T22)Z22=Z2 (T11−T12−T21+T22)T11+T12−T21−T22\begin{align*}Z_{11} &=\frac{Z_{1}\,\left(T_{11}+T_{12}+T_{21}+T_{22}\right)}{T_{11}+T_{12}-T_{21}-T_{22}}\\Z_{12} &=\frac{2\,Z_{2}\,k_{2}\,\left(T_{11}\,T_{22}-T_{12}\,T_{21}\right)}{k_{1}\,\left(T_{11}+T_{12}-T_{21}-T_{22}\right)}\\Z_{21} &=\frac{2\,Z_{1}\,k_{1}}{k_{2}\,\left(T_{11}+T_{12}-T_{21}-T_{22}\right)}\\Z_{22} &=\frac{Z_{2}\,\left(T_{11}-T_{12}-T_{21}+T_{22}\right)}{T_{11}+T_{12}-T_{21}-T_{22}}\\\end{align*}

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

Definitions

[A1B1]=T[B2A2]\begin{bmatrix}A_1 \\B_1\end{bmatrix} = \mathbf{T}\begin{bmatrix}B_2 \\A_2\end{bmatrix}

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