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 Y-parameters to S-parameters

In matrix form, the formula is

S=K(IN−Z0Y)(IN+Z0Y)−1K−1\mathbf{S}=K \left( I_N -Z_0 \mathbf{Y} \right) \left( I_N + Z_0 \mathbf{Y} \right)^{-1} K^{-1}

When dealing with 2 ports, we obtain

S11=Y22 Z2−Y11 Z1−Y11 Y22 Z1 Z2+Y12 Y21 Z1 Z2+1Y11 Z1+Y22 Z2+Y11 Y22 Z1 Z2−Y12 Y21 Z1 Z2+1S12=−2 Y12 Z1 k1k2 (Y11 Z1+Y22 Z2+Y11 Y22 Z1 Z2−Y12 Y21 Z1 Z2+1)S21=−2 Y21 Z2 k2k1 (Y11 Z1+Y22 Z2+Y11 Y22 Z1 Z2−Y12 Y21 Z1 Z2+1)S22=Y11 Z1−Y22 Z2−Y11 Y22 Z1 Z2+Y12 Y21 Z1 Z2+1Y11 Z1+Y22 Z2+Y11 Y22 Z1 Z2−Y12 Y21 Z1 Z2+1\begin{align*}S_{11} &=\frac{Y_{22}\,Z_{2}-Y_{11}\,Z_{1}-Y_{11}\,Y_{22}\,Z_{1}\,Z_{2}+Y_{12}\,Y_{21}\,Z_{1}\,Z_{2}+1}{Y_{11}\,Z_{1}+Y_{22}\,Z_{2}+Y_{11}\,Y_{22}\,Z_{1}\,Z_{2}-Y_{12}\,Y_{21}\,Z_{1}\,Z_{2}+1}\\S_{12} &=-\frac{2\,Y_{12}\,Z_{1}\,k_{1}}{k_{2}\,\left(Y_{11}\,Z_{1}+Y_{22}\,Z_{2}+Y_{11}\,Y_{22}\,Z_{1}\,Z_{2}-Y_{12}\,Y_{21}\,Z_{1}\,Z_{2}+1\right)}\\S_{21} &=-\frac{2\,Y_{21}\,Z_{2}\,k_{2}}{k_{1}\,\left(Y_{11}\,Z_{1}+Y_{22}\,Z_{2}+Y_{11}\,Y_{22}\,Z_{1}\,Z_{2}-Y_{12}\,Y_{21}\,Z_{1}\,Z_{2}+1\right)}\\S_{22} &=\frac{Y_{11}\,Z_{1}-Y_{22}\,Z_{2}-Y_{11}\,Y_{22}\,Z_{1}\,Z_{2}+Y_{12}\,Y_{21}\,Z_{1}\,Z_{2}+1}{Y_{11}\,Z_{1}+Y_{22}\,Z_{2}+Y_{11}\,Y_{22}\,Z_{1}\,Z_{2}-Y_{12}\,Y_{21}\,Z_{1}\,Z_{2}+1}\\\end{align*}

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

Definitions

[I1⋮IN]⏟I=Y[V1⋮VN]⏟V\underbrace{ \begin{bmatrix}I_1 \\ \vdots \\ I_N \end{bmatrix} }_{\mathbf{I}} = \mathbf{Y} \underbrace{ \begin{bmatrix}V_1 \\ \vdots \\ V_N \end{bmatrix} }_{\mathbf{V}}

[B1⋮BN]⏟B=S[A1⋮AN]⏟A\underbrace{ \begin{bmatrix}B_1 \\ \vdots \\ B_N \end{bmatrix} }_{\mathbf{B}} = \mathbf{S} \underbrace{ \begin{bmatrix}A_1 \\ \vdots \\ A_N \end{bmatrix} }_{\mathbf{A}}

The incident and reflected waves are defined as

A=K(V+Z0I)B=K(V−Z0I)\mathbf{A} = K \left( \mathbf{V} + Z_0 \mathbf{I} \right)\qquad \mathbf{B} = K \left( \mathbf{V} - Z_0 \mathbf{I} \right)

where KK and Z0Z_0 are defined as

K=[k1⋱kN]Z0=[Z0,1⋱Z0,N]K = \begin{bmatrix}k_1 & & \\ & \ddots & \\ & & k_N \end{bmatrix}\qquad Z_0 = \begin{bmatrix}Z_{0,1} & & \\ & \ddots & \\ & & Z_{0,N} \end{bmatrix}

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.