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 B-parameters to T-parameters

In matrix form, the formula is

T=[k1Z1 k1k1−Z1 k1] B−1 [k2Z2 k2k2−Z2 k2]−1\mathbf{T} =\begin{bmatrix} k_{1} & Z_{1}\,k_{1}\\ k_{1} & -Z_{1}\,k_{1} \end{bmatrix}\,{\mathbf{B}}^{-1}\,{\begin{bmatrix} k_{2} & Z_{2}\,k_{2}\\ k_{2} & -Z_{2}\,k_{2} \end{bmatrix}}^{-1}

While for each element, we obtain

T11=−k1 (B12−B11 Z1−B22 Z2+B21 Z1 Z2)2 Z2 k2 (B11 B22−B12 B21)T12=k1 (B12−B11 Z1+B22 Z2−B21 Z1 Z2)2 Z2 k2 (B11 B22−B12 B21)T21=−k1 (B12+B11 Z1−B22 Z2−B21 Z1 Z2)2 Z2 k2 (B11 B22−B12 B21)T22=k1 (B12+B11 Z1+B22 Z2+B21 Z1 Z2)2 Z2 k2 (B11 B22−B12 B21)\begin{align*}T_{11} &=-\frac{k_{1}\,\left(B_{12}-B_{11}\,Z_{1}-B_{22}\,Z_{2}+B_{21}\,Z_{1}\,Z_{2}\right)}{2\,Z_{2}\,k_{2}\,\left(B_{11}\,B_{22}-B_{12}\,B_{21}\right)}\\T_{12} &=\frac{k_{1}\,\left(B_{12}-B_{11}\,Z_{1}+B_{22}\,Z_{2}-B_{21}\,Z_{1}\,Z_{2}\right)}{2\,Z_{2}\,k_{2}\,\left(B_{11}\,B_{22}-B_{12}\,B_{21}\right)}\\T_{21} &=-\frac{k_{1}\,\left(B_{12}+B_{11}\,Z_{1}-B_{22}\,Z_{2}-B_{21}\,Z_{1}\,Z_{2}\right)}{2\,Z_{2}\,k_{2}\,\left(B_{11}\,B_{22}-B_{12}\,B_{21}\right)}\\T_{22} &=\frac{k_{1}\,\left(B_{12}+B_{11}\,Z_{1}+B_{22}\,Z_{2}+B_{21}\,Z_{1}\,Z_{2}\right)}{2\,Z_{2}\,k_{2}\,\left(B_{11}\,B_{22}-B_{12}\,B_{21}\right)}\\\end{align*}

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

Definitions

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