Network Parameters in Circuit Analysis

This article presents S, Z, Y, ABCD parameters and their transformation.

Definition

Impedance Parameter

$\left[ \begin{matrix} V_1 \\ V_2 \end{matrix} \right] = \left[ \begin{matrix} Z_{11} & Z_{12} \\ Z_{21} & Z_{22} \end{matrix} \right] \left[ \begin{matrix} I_1 \\ I_2 \end{matrix} \right]$

Admittance Parameter

$\left[ \begin{matrix} I_1 \\ I_2 \end{matrix} \right] = \left[ \begin{matrix} Y_{11} & Y_{12} \\ Y_{21} & Y_{22} \end{matrix} \right] \left[ \begin{matrix} V_1 \\ V_2 \end{matrix} \right]$

ABCD Parameter

Also called A Parameter or chain parameter. Note that for convenience of cascading, the definition of current at port 2 is opposite to others.

Port definition of ABCD parameter

$\left[ \begin{matrix} V_1 \\ I_1 \end{matrix} \right] = \left[ \begin{matrix} A & B \\ C & D \end{matrix} \right] \left[ \begin{matrix} V_2 \\ I_2 \end{matrix} \right]$

Hybrid Parameter

$\left[ \begin{matrix} V_1 \\ I_2 \end{matrix} \right] = \left[ \begin{matrix} H_{11} & H_{12} \\ H_{21} & H_{22} \end{matrix} \right] \left[ \begin{matrix} I_1 \\ V_2 \end{matrix} \right]$

G Parameter

$\left[ \begin{matrix} I_1 \\ V_2 \end{matrix} \right] = \left[ \begin{matrix} G_{11} & G_{12} \\ G_{21} & G_{22} \end{matrix} \right] \left[ \begin{matrix} V_1 \\ I_2 \end{matrix} \right]$

Scattering Parameter

$\left[ \begin{matrix} b_1 \\ b_2 \end{matrix} \right] = \left[ \begin{matrix} S_{11} & S_{12} \\ S_{21} & S_{22} \end{matrix} \right] \left[ \begin{matrix} a_1 \\ a_2 \end{matrix} \right]$

For S parameter

$V=V^++V^-\\ I=I^+-I^-=\frac{1}{Z_0}(V^+-V^-)$

Normalized incident and reflective waves are

$a=\frac{V^+}{\sqrt{Z_0}}=\frac{V+Z_0 I}{2\sqrt{Z_0}}\\ b=\frac{V^-}{\sqrt{Z_0}}=\frac{V-Z_0 I}{2\sqrt{Z_0}}\\$

Power delivered is

$P_L=\frac{1}{2}\mathrm{Re}[VI^*]=\frac{1}{2Z_0}(a^2-b^2)$

Reflection coefficient

$\Gamma_{IN}=S_{11}+\frac{S_{12}S_{21}\Gamma_L}{1-S_{22}\Gamma_L}\\ \Gamma_{OUT}=S_{22}+\frac{S_{12}S_{21}\Gamma_S}{1-S_{11}\Gamma_S}$

Mason's rule

$G=\frac{y_{out}}{y_{in}}=\frac{\sum_{k=1}^NG_k\Delta_k}{\Delta}\\ \Delta=1-\sum L_i+\sum L_iL_j-\sum L_iL_jL_k+...$

where

$\Delta$ is the determinant of the graph.
$y_{in}$ is the input-node variable.
$y_{out}$ is the output-node variable.
$G$ is the complete gain.
$N$ is the total number of forward paths between $y_{in}$ and $y_{out}$.
$G_k$ is the path gain of the k th forward path between $y_{in}$ and $y_{out}$.
$L_i$ is the loop gain of a closed loop in the system.
$L_iL_j$ is the product of the loop gains of two non-touching closed loops.
$\Delta_k$ is the cofactor value of $\Delta$ for the k th forward path, with the loops touching the k th forward path removed.

It is defined that

Path: a continuous set of branches traversed in the direction that they indicate.
Forward path: A path from an input node to an output node in which no node is touched more than once.
Loop: A path that originates and ends on the same node in which no node is touched more than once.
Path gain: the product of the gains of all the branches in the path.
Loop gain: the product of the gains of all the branches in the loop.

To calculate the input reflection coefficient, the following signal graph is used

Input reflection coefficient signal graph

we have

$G_1=S_{11}$
- $\Delta_1=1-S_{22}\Gamma_L$
$G_2=S_{21}\Gamma_L S_{12}$
- $\Delta_2=1$, because the only loop in the graph touches $\Gamma_L$
$\Delta=1-S_{22}\Gamma_L$
$S_{11}=\frac{G_1\Delta_1+G_2\Delta_2}{\Delta}$

Transfer Scattering Parameter

$\left[ \begin{matrix} a_1 \\ a_2 \end{matrix} \right] = \left[ \begin{matrix} T_{11} & T_{12} \\ T_{21} & T_{22} \end{matrix} \right] \left[ \begin{matrix} b_1 \\ b_2 \end{matrix} \right]$

Properties

Reciprocal Network

Not containing any active devices or nonreciprocal media, such as ferrites or plasmas。

$\mathbf{Z}=\mathbf{Z}^T\\ \mathbf{Y}=\mathbf{Y}^T\\ \mathbf{S}=\mathbf{S}^T\\ \mathrm{Det}[\mathbf{ABCD}]=1$

Lossless Network

All the parameters of $\mathbf{Z}$ and $\mathbf{Y}$ are purely imaginary.

Connection

Parallel-Parallel

$\mathbf{Y}=\mathbf{Y_1}+\mathbf{Y_2}$

Series-Series

$\mathbf{Z}=\mathbf{Z_1}+\mathbf{Z_2}$

Series-Parallel

$\mathbf{H}=\mathbf{H_1}+\mathbf{H_2}$

Parallel-Series

$\mathbf{G}=\mathbf{G_1}+\mathbf{G_2}$

Cascade

Cascade-connection

$\mathbf{A}=\mathbf{A_1}\cdot\mathbf{A_2}\\ \mathbf{T}=\mathbf{T_1}\cdot\mathbf{T_2}$

Image Impedance

A pair of image impedance $Z_{i1},Z_{i2}$ is defined for reciprocal network as

$Z_{i1}$ is the input impedance at port 1 when port 2 is terminated with $Z_{i2}$
$Z_{i2}$ is the input impedance at port 2 when port 1 is terminated with $Z_{i1}$

With ABCD parameter, the input impedance at port 1 when port 2 is terminated with $Z_{i2}$ equals to

$Z_{in1}=\frac{AV_2+BI_2}{CV_2+DI_2}=\frac{AZ_{i2}+B}{CZ_{i2}+D}$

Similarly

$Z_{in2}=\frac{DV_1-BI_1}{CV_1+AI_1}=\frac{DZ_{i1}+B}{CZ_{i1}+A}$

Since $Z_{in1}=Z_{i1},Z_{in2}=Z_{i2}$, the solution is

$Z_{i1}=\sqrt{\frac{AB}{CD}}=\sqrt{\frac{Z_{11}}{Z_{22}}\mathrm{Det[\mathbf{Z}]}}\\ Z_{i2}=\sqrt{\frac{BD}{AC}}=\sqrt{\frac{Z_{22}}{Z_{11}}\mathrm{Det[\mathbf{Z}]}}\\$

When terminated with image impedance $Z_{i2}$, the voltage transfer ratio is

$\frac{V_2}{V_1}=\sqrt{\frac{D}{A}}(\sqrt{AD}-\sqrt{BC})$

For current

$\frac{I_2}{I_1}=\sqrt{\frac{A}{D}}(\sqrt{AD}-\sqrt{BC})$

Other

ABCD

For T

T Network

$\mathbf{Z}= \left[ \begin{matrix} Z_1+Z_2 & Z_2 \\ Z_2 & Z_2+Z_3 \end{matrix} \right]$

For Pi

Pi Network

$\mathbf{Y}= \left[ \begin{matrix} Y_1+Y_2 & -Y_2 \\ -Y_2 & Y_2+Y_3 \end{matrix} \right]$

Conversion

General

$\mathbf{Z}=\mathbf{Y}^{-1}\\ \mathbf{Y}=\mathbf{Z}^{-1}\\ \mathbf{Z}=\mathbf{G_{ref}}^{-1}\cdot(\mathbf{E}-\mathbf{S})^{-1}\cdot(\mathbf{S}+\mathbf{E})\cdot\mathbf{Z_{ref}}\cdot\mathbf{G_{ref}}\\ \mathbf{Y}=\mathbf{G_{ref}}^{-1}\cdot\mathbf{Z_{ref}}^{-1}\cdot(\mathbf{S}+\mathbf{E})^{-1}\cdot(\mathbf{E}-\mathbf{S})\cdot\mathbf{G_{ref}}\\ \mathbf{S}=\mathbf{G_{ref}}\cdot(\mathbf{Z}-\mathbf{Z_{ref}})\cdot(\mathbf{Z}+\mathbf{Z_{ref}})^{-1}\cdot\mathbf{G_{ref}}^{-1}\\ \mathbf{S}=\mathbf{G_{ref}}\cdot(\mathbf{E}-\mathbf{Z_{ref}}\cdot\mathbf{Y})\cdot(\mathbf{E}+\mathbf{Z_{ref}}\cdot\mathbf{Y})^{-1}\cdot\mathbf{G_{ref}}^{-1}\\ \mathbf{Z_{ref}}= \left[ \begin{matrix} Z_{0,1} & 0 & \ldots & 0\\ 0 & Z_{0,2} & \ldots & 0\\ \vdots & \vdots & \ddots & \vdots\\ 0 & 0 & \ldots & Z_{0,N} \end{matrix} \right]\\ \mathbf{G_{ref}}= \left[ \begin{matrix} G_{1} & 0 & \ldots & 0\\ 0 & G_{2} & \ldots & 0\\ \vdots & \vdots & \ddots & \vdots\\ 0 & 0 & \ldots & G_{N} \end{matrix} \right]\\ G_n=\frac{1}{\sqrt{\mathrm{Re}[Z_{0,n}]}}$

where $\mathbf{E}$ is an identity matrix, $Z_{0,n}$ is the reference impedance of port n. The renormalization of S parameter is as follows.

$\mathbf{S'}=\mathbf{A}^{-1} \cdot (\mathbf{S}-\mathbf{R}) \cdot (\mathbf{E}-\mathbf{R} \cdot \mathbf{S})^{-1} \cdot \mathbf{A}\\ \mathbf{R}= \left[ \begin{matrix} r(Z_1) & 0 & \ldots & 0\\ 0 & r(Z_2) & \ldots & 0\\ \vdots & \vdots & \ddots & \vdots\\ 0 & 0 & \ldots & r(Z_N) \end{matrix} \right]\\ r(Z_n)=\frac{Z_n-Z_{n,before}}{Z_n+Z_{n,before}}\\ \mathbf{A}= \left[ \begin{matrix} A_1 & 0 & \ldots & 0\\ 0 & A_2 & \ldots & 0\\ \vdots & \vdots & \ddots & \vdots\\ 0 & 0 & \ldots & A_N \end{matrix} \right]\\ A_n=\sqrt{\frac{Z_n}{Z_{n,before}}}\cdot\frac{1}{Z_n+Z_{n,before}}$

where $\mathbf{S}$ is the original S parameter matrix, and $\mathbf{S'}$ is the recalculated scattering matrix, $Z_n$ is the reference impedance of port n after the normalizing process, and $Z_{n,before}$ is the reference impedance of port n before the normalizing process.

Two Port

Two Port Parameter Conversion

Two Port Parameter Conversion 2

Appendix

Mathematica functions

(* Conversion between Z and Y parameters *)
Z2Y[Y_] := Inverse[Y];
Y2Z[Z_] := Inverse[Z];

(* Conversion between Y and S parameters *)
Y2S[Y_, Zref_] := Module[{E0, G0, Z0},
  E0 = IdentityMatrix[Length[Y]];
  G0 = DiagonalMatrix[(1/Sqrt[Re[#1]]) & /@ Zref];
  Z0=DiagonalMatrix[Zref];
  G0.(E0 - Z0.Y).Inverse[(E0 + Z0.Y)].Inverse[G0]
  ];

S2Y[S_, Zref_] := Module[{E0, G0, Z0},
  E0 = IdentityMatrix[Length[S]];
  G0 = DiagonalMatrix[(1/Sqrt[Re[#1]]) & /@ Zref];
  Z0=DiagonalMatrix[Zref];
  Inverse[G0].Inverse[S.Z0 + Z0].(E0 - S).G0
  ];

(* Conversion between Z and S parameters *)
Z2S[Z_, Zref_] := Module[{E0, G0, Z0},
  E0 = IdentityMatrix[Length[Z]];
  G0 = DiagonalMatrix[(1/Sqrt[Re[#1]]) & /@ Zref];
  Z0=DiagonalMatrix[Zref];
  G0.(Z - Z0).Inverse[(Z + Z0)].Inverse[G0]
  ];

S2Z[S_, Zref_] := Module[{E0, G0, Z0},
  E0 = IdentityMatrix[Length[S]];
  G0 = DiagonalMatrix[(1/Sqrt[Re[#1]]) & /@ Zref];
  Z0=DiagonalMatrix[Zref];
  Inverse[G0].Inverse[E0 - S].(S.Z0 + Z0).G0
  ];

(* Renormalization of S parameters *)
S2S[S_, Zn_, Znb_] := Module[{R, A, E},
	R = DiagonalMatrix[(Zn-Znb)/(Zn+Znb)];
	A = DiagonalMatrix[Sqrt[Zn/Znb]/(Zn+Znb)];
	E = IdentityMatrix[Length[S]];
	Inverse[A].(S-R).Inverse[E-R.S].A
	];

Examples

Y2S[({
    {0.2, 0.8},
    {0.8, 0.2}
   }), {50, 50}] // N

Another way of stating the Mason's rule

$T=\frac{P_1[1-\sum L(1)^{(1)}+\sum L(2)^{(1)}\ldots]+\\P_2[1-\sum L(1)^{(2)}+\sum L(2)^{(2)}\ldots]\ldots}{1-\sum L(1)+\sum L(2)-\sum L(3)+\ldots}$

where

$P_n$ is a path that connects the beginning and the ending. And node can be in the path only once.
$L(1)$ is the first-order loop. It is a closed loop made of branches in the same direction.
$L(n)$ is the product of $n$ separate $L(1)$.
$L(n)^{(m)}$ is the n th order loop that is not in contact with path $P_m$.

Reference

For the basic concepts of Z, Y, S, ABCD parameters, refer to

D. M. Pozar, Microwave Engineering. NJ, Hoboken: Wiley, 1998.

For a detailed conversion process of different parameters, refer to

S. Jahn, M. Margraf, V. Habchi and R. Jacob, Qucs Technical Papers, available at http://qucs.sourceforge.net/tech/technical.html.