Signals and Networks: Two-port Networks

A four terminal network is called a two-port network when the current entering one terminal of a pair exits the other terminal in the pair.

Impedance parameters (z-parameters)

\begin{bmatrix} V_1 \\ V_2 \end{bmatrix} = \begin{bmatrix} z_{11} & z_{12} \\ z_{21} & z_{22} \end{bmatrix} \begin{bmatrix} I_1 \\ I_2 \end{bmatrix}

where

\begin{align} z_{11} \,&\stackrel{\text{def}}{=}\, \left. \frac{V_1}{I_1} \right|_{I_2 = 0} \qquad z_{12} \,\stackrel{\text{def}}{=}\, \left. \frac{V_1}{I_2} \right|_{I_1 = 0} \\ z_{21} \,&\stackrel{\text{def}}{=}\, \left. \frac{V_2}{I_1} \right|_{I_2 = 0} \qquad z_{22} \,\stackrel{\text{def}}{=}\, \left. \frac{V_2}{I_2} \right|_{I_1 = 0} \end{align}

(z-equivalent two port showing independent variables I₁ and I₂.)

Admittance parameters (y-parameters)

\begin{bmatrix} I_1 \\ I_2 \end{bmatrix} = \begin{bmatrix} y_{11} & y_{12} \\ y_{21} & y_{22} \end{bmatrix} \begin{bmatrix} V_1 \\ V_2 \end{bmatrix}

where

\begin{align} y_{11} \,&\stackrel{\text{def}}{=}\, \left. \frac{I_1}{V_1} \right|_{V_2 = 0} \qquad y_{12} \,\stackrel{\text{def}}{=}\, \left. \frac{I_1}{V_2 } \right|_{V_1 = 0} \\ y_{21} \,&\stackrel{\text{def}}{=}\, \left. \frac{I_2}{V_1} \right|_{V_2 = 0} \qquad y_{22} \,\stackrel{\text{def}}{=}\, \left. \frac{I_2}{V_2 } \right|_{V_1 = 0} \end{align}

(Y-equivalent two port showing independent variables V₁ and V₂.)

Hybrid parameters (h-parameters)

\begin{bmatrix} V_1 \\ I_2 \end{bmatrix} = \begin{bmatrix} h_{11} & h_{12} \\ h_{21} & h_{22} \end{bmatrix} \begin{bmatrix} I_1 \\ V_2 \end{bmatrix}

where

\begin{align} h_{11} \,&\stackrel{\text{def}}{=}\, \left. \frac{V_1}{I_1} \right|_{V_2 = 0} \qquad h_{12} \,\stackrel{\text{def}}{=}\, \left. \frac{V_1}{V_2} \right|_{I_1 = 0} \\ h_{21} \,&\stackrel{\text{def}}{=}\, \left. \frac{I_2}{I_1} \right|_{V_2 = 0} \qquad h_{22} \,\stackrel{\text{def}}{=}\, \left. \frac{I_2}{V_2} \right|_{I_1 = 0} \end{align}

(H-equivalent two-port showing independent variables I₁ and V₂₎

ABCD-parameters

\begin{bmatrix} V_1 \\ I_1 \end{bmatrix} = \begin{bmatrix} A & B \\ C & D \end{bmatrix} \begin{bmatrix} V_2 \\ -I_2 \end{bmatrix}

For reciprocal networks

\scriptstyle AD-BC=1

. For symmetrical networks

\scriptstyle A=D

. For networks which are reciprocal and lossless, A and D are purely real while B and C are purely imaginary.

\begin{bmatrix} V_2 \\ I'_2 \end{bmatrix} = \begin{bmatrix} A' & B' \\ C' & D' \end{bmatrix} \begin{bmatrix} V_1 \\ I_1 \end{bmatrix}

\begin{align} \left\lbrack\mathbf{a}\right\rbrack &= \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix} = \begin{bmatrix} A & B \\ C & D \end{bmatrix} \\ \left\lbrack\mathbf{b}\right\rbrack &= \begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{bmatrix} = \begin{bmatrix} A' & B' \\ C' & D' \end{bmatrix} \end{align}

Interrelation of parameters

	$\mathbf{[z]}$	$\mathbf{[y]}$	$\mathbf{[h]}$	$\mathbf{[g]}$	$\mathbf{[a]}$	$\mathbf{[b]}$
$\mathbf{[z]}$	$\begin{bmatrix} z_{11} & z_{12} \\ z_{21} & z_{22} \end{bmatrix}$	$\begin{bmatrix} \dfrac{y_{22}}{\Delta \mathbf{[y]}} & \dfrac{-y_{12}}{\Delta \mathbf{[y]}} \\ \dfrac{-y_{21}}{\Delta \mathbf{[y]}} & \dfrac{y_{11}}{\Delta \mathbf{[y]}} \end{bmatrix}$	$\begin{bmatrix} \dfrac{\Delta \mathbf{[h]}}{h_{22}} & \dfrac{h_{12}}{h_{22}} \\ \dfrac{-h_{21}}{h_{22}} & \dfrac{1}{h_{22}} \end{bmatrix}$	$\begin{bmatrix} \dfrac{1}{g_{11}} & \dfrac{-g_{12}}{g_{11}} \\ \dfrac{g_{21}}{g_{11}} & \dfrac{\Delta \mathbf{[g]}}{g_{11}} \end{bmatrix}$	$\begin{bmatrix} \dfrac{a_{11}}{a_{21}} & \dfrac{\Delta \mathbf{[a]}}{a_{21}} \\ \dfrac{1}{a_{21}} & \dfrac{a_{22}}{a_{21}} \end{bmatrix}$	$\begin{bmatrix} \dfrac{-b_{22}}{b_{21}} & \dfrac{-1}{b_{21}} \\ \dfrac{- \Delta \mathbf{[b]}}{b_{21}} & \dfrac{-b_{11}}{b_{21}} \end{bmatrix}$
$\mathbf{[y]}$	$\begin{bmatrix} \dfrac{z_{22}}{\Delta \mathbf{[z]}} & \dfrac{-z_{12}}{\Delta \mathbf{[z]}} \\ \dfrac{-z_{21}}{\Delta \mathbf{[z]}} & \dfrac{z_{11}}{\Delta \mathbf{[z]}} \end{bmatrix}$	$\begin{bmatrix} y_{11} & y_{12} \\ y_{21} & y_{22} \end{bmatrix}$	$\begin{bmatrix} \dfrac{1}{h_{11}} & \dfrac{-h_{12}}{h_{11}} \\ \dfrac{h_{21}}{h_{11}} & \dfrac{\Delta \mathbf{[h]}}{h_{11}} \end{bmatrix}$	$\begin{bmatrix} \dfrac{\Delta \mathbf{[g]}}{g_{22}} & \dfrac{g_{12}}{g_{22}} \\ \dfrac{-g_{21}}{g_{22}} & \dfrac{1}{g_{22}} \end{bmatrix}$	$\begin{bmatrix} \dfrac{a_{22}}{a_{12}} & \dfrac{-\Delta \mathbf{[a]}}{a_{12}} \\ \dfrac{-1}{a_{12}} & \dfrac{a_{11}}{a_{12}} \end{bmatrix}$	$\begin{bmatrix} \dfrac{-b_{11}}{b_{12}} & \dfrac{1}{b_{12}} \\ \dfrac{\Delta \mathbf{[b]}}{b_{12}} & \dfrac{-b_{22}}{b_{12}} \end{bmatrix}$
$\mathbf{[h]}$	$\begin{bmatrix} \dfrac{\Delta \mathbf{[z]}}{z_{22}} & \dfrac{z_{12}}{z_{22}} \\ \dfrac{-z_{21}}{z_{22}} & \dfrac{1}{z_{22}} \end{bmatrix}$	$\begin{bmatrix} \dfrac{1}{y_{11}} & \dfrac{-y_{12}}{Y_{11}} \\ \dfrac{y_{21}}{y_{11}} & \dfrac{\Delta \mathbf{[y]}}{y_{11}} \end{bmatrix}$	$\begin{bmatrix} h_{11} & h_{12} \\ h_{21} & h_{22} \end{bmatrix}$	$\begin{bmatrix} \dfrac{g_{22}}{\Delta \mathbf{[g]}} & \dfrac{-g_{12}}{\Delta \mathbf{[g]}} \\ \dfrac{-g_{21}}{\Delta \mathbf{[g]}} & \dfrac{g_{11}}{\Delta \mathbf{[g]}} \end{bmatrix}$	$\begin{bmatrix} \dfrac{a_{12}}{a_{22}} & \dfrac{\Delta \mathbf{[a]}}{a_{22}} \\ \dfrac{-1}{a_{22}} & \dfrac{a_{21}}{a_{22}} \end{bmatrix}$	$\begin{bmatrix} \dfrac{-b_{12}}{b_{11}} & \dfrac{1}{b_{11}} \\ \dfrac{-\Delta \mathbf{[b]}}{b_{11}} & \dfrac{-b_{21}}{b_{11}} \end{bmatrix}$
$\mathbf{[g]}$	$\begin{bmatrix} \dfrac{1}{z_{11}} & \dfrac{-z_{12}}{z_{11}} \\ \dfrac{z_{21}}{z_{11}} & \dfrac{\Delta \mathbf{[z]}}{z_{11}} \end{bmatrix}$	$\begin{bmatrix} \dfrac{\Delta \mathbf{[y]}}{y_{22}} & \dfrac{y_{12}}{y_{22}} \\ \dfrac{-y_{21}}{y_{22}} & \dfrac{1}{y_{22}} \end{bmatrix}$	$\begin{bmatrix} \dfrac{h_{22}}{\Delta \mathbf{[h]}} & \dfrac{-h_{12}}{\Delta \mathbf{[h]}} \\ \dfrac{-h_{21}}{\Delta \mathbf{[h]}} & \dfrac{h_{11}}{\Delta \mathbf{[h]}} \end{bmatrix}$	$\begin{bmatrix} g_{11} & g_{12} \\ g_{21} & g_{22} \end{bmatrix}$	$\begin{bmatrix} \dfrac{a_{21}}{a_{11}} & \dfrac{-\Delta \mathbf{[a]}}{a_{11}} \\ \dfrac{1}{a_{11}} & \dfrac{a_{12}}{a_{11}} \end{bmatrix}$	$\begin{bmatrix} \dfrac{-b_{21}}{b_{22}} & \dfrac{-1}{b_{22}} \\ \dfrac{\Delta \mathbf{[b]}}{b_{22}} & \dfrac{-b_{12}}{b_{22}} \end{bmatrix}$
$\mathbf{[a]}$	$\begin{bmatrix} \dfrac{z_{11}}{z_{21}} & \dfrac{\Delta \mathbf{[z]}}{z_{21}} \\ \dfrac{1}{z_{21}} & \dfrac{z_{22}}{z_{21}} \end{bmatrix}$	$\begin{bmatrix} \dfrac{-y_{22}}{y_{21}} & \dfrac{-1}{y_{21}} \\ \dfrac{-\Delta \mathbf{[y]}}{y_{21}} & \dfrac{-y_{11}}{y_{21}} \end{bmatrix}$	$\begin{bmatrix} \dfrac{-\Delta \mathbf{[h]}}{h_{21}} & \dfrac{-h_{11}}{h_{21}} \\ \dfrac{-h_{22}}{h_{21}} & \dfrac{-1}{h_{21}} \end{bmatrix}$	$\begin{bmatrix} \dfrac{1}{g_{21}} & \dfrac{g_{22}}{g_{21}} \\ \dfrac{g_{11}}{g_{21}} & \dfrac{\Delta \mathbf{[g]}}{g_{21}} \end{bmatrix}$	$\begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}$	$\begin{bmatrix} \dfrac{b_{22}}{\Delta \mathbf{[b]}} & \dfrac{-b_{12}}{\Delta \mathbf{[b]}} \\ \dfrac{-b_{21}}{\Delta \mathbf{[b]}} & \dfrac{b_{11}}{\Delta \mathbf{[b]}} \end{bmatrix}$
$\mathbf{[b]}$	$\begin{bmatrix} \dfrac{z_{22}}{z_{12}} & \dfrac{- \Delta \mathbf{[z]}}{z_{12}} \\ \dfrac{-1}{z_{12}} & \dfrac{z_{11}}{z_{12}} \end{bmatrix}$	$\begin{bmatrix} \dfrac{-y_{11}}{y_{12}} & \dfrac{1}{y_{12}} \\ \dfrac{\Delta \mathbf{[y]}}{y_{12}} & \dfrac{-y_{22}}{y_{12}} \end{bmatrix}$	$\begin{bmatrix} \dfrac{1}{h_{12}} & \dfrac{-h_{11}}{h_{12}} \\ \dfrac{-h_{22}}{h_{12}} & \dfrac{\Delta \mathbf{[h]}}{h_{12}} \end{bmatrix}$	$\begin{bmatrix} \dfrac{-\Delta \mathbf{[g]}}{g_{12}} & \dfrac{g_{22}}{g_{12}} \\ \dfrac{g_{11}}{g_{12}} & \dfrac{-1}{g_{12}} \end{bmatrix}$	$\begin{bmatrix} \dfrac{a_{22}}{\Delta \mathbf{[a]}} & \dfrac{-a_{12}}{\Delta \mathbf{[a]}} \\ \dfrac{-a_{21}}{\Delta \mathbf{[a]}} & \dfrac{a_{11}}{\Delta \mathbf{[a]}} \end{bmatrix}$	$\begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{bmatrix}$

Where

\Delta \mathbf{[x]}

is the determinant of [x].

Combinations of two-port models

Series connection of two 2-port networks: ${\bf Z}={\bf Z}_1+{\bf Z}_2$
Parallel connection of two 2-port networks: ${\bf Y}={\bf Y}_1+{\bf Y}_2$
Cascade connection of two 2-port networks: ${\bf A}={\bf A}_1 \cdot {\bf A}_2$

Example:

Find the Z-model and Y-model of the circuit shown.

$\begin{displaymath}\left\{ \begin{array}{l} V_1=Z_{11}I_1+Z_{12}I_2 \\ V_2=Z_{21}I_1+Z_{22}I_2 \end{array} \right. \end{displaymath}$

First assume , we get

$\begin{displaymath}Z_{11}=V_1/I_1=j\omega L+1/j\omega C,\;\;\;\;\;Z_{21}=V_2/I_1=1/j\omega C \end{displaymath}$
Next assume , we get

$\begin{displaymath}Z_{22}=V_2/I_2=1/j\omega C,\;\;\;\;\;\;Z_{12}=V_1/I_2=1/j\omega C \end{displaymath}$

The parameters of the Y-model can be found as the inverse of

$\begin{displaymath}\left[\begin{array}{cc}Y_{11}&Y_{12} Y_{21}&Y_{22}\end{arra... ...a L\\ -1/j\omega L & j\omega C+1/j\omega L\end{array}\right] \end{displaymath}$

Resources:

Examples with solutions