Linear Circuit Analysis

1. Introduction
2. Basic Concepts
3. Simple Circuits
4. Nodal and Mesh Analysis
5. Additional Analysis Techniques
6. AC Analysis
7. Operational Amplifiers
8. Laplace Transforms
9. Time-Dependent Circuits
10. Two-port networks

Two-port parameter conversion formulas

Table 1 gives the coversion formulas between the 4 two-port networks presented in this chapter. $\Delta_Y$, $\Delta_Z$, $\Delta_H$, and $\Delta_T$ refer to determinants of the $Y$, $Z$, $H$ and transmittance parameters matrices (e.g. $\Delta_Y=y_{11}y_{22}-y_{12}y_{21}$, ...).

Table 1. Two-port conversion formulas.
Y Z H T
Y $\begin{bmatrix}y_{11} & y_{12}\\y_{21} & {y_{22}}\end{bmatrix}$ $\frac{1}{\Delta_Z}\times \begin{bmatrix}z_{22} & -z_{12}\\-z_{21} & z_{11}\end{bmatrix}$ $\frac{1}{h_{11}}\times \begin{bmatrix}1 & -h_{12}\\h_{21} & \Delta_H\end{bmatrix}$ $\frac{1}{B}\times \begin{bmatrix}D & -\Delta_T\\-1 & A\end{bmatrix}$
Z $\frac{1}{\Delta_Z}\times \begin{bmatrix}y_{22} & -y_{12}\\-y_{21} & y_{11}\end{bmatrix}$ $\begin{bmatrix}z_{11} & z_{12}\\z_{21} & {z_{22}}\end{bmatrix}$ $\frac{1}{h_{22}}\times \begin{bmatrix}\Delta_H & h_{12}\\-h_{21} & 1\end{bmatrix}$ $\frac{1}{C}\times \begin{bmatrix}A & \Delta_T\\1 & D\end{bmatrix}$
H $\frac{1}{y_{11}}\times \begin{bmatrix}1 & -y_{12}\\y_{21} & \Delta_Y\end{bmatrix}$ $\frac{1}{z_{22}}\times \begin{bmatrix}\Delta_Z & z_{12}\\-z_{21} & 1\end{bmatrix}$ $\begin{bmatrix}h_{11} & h_{12}\\h_{21} & {h_{22}}\end{bmatrix}$ $\frac{1}{D}\times \begin{bmatrix}B & \Delta_T\\-1 & C\end{bmatrix}$
T $\frac{1}{y_{21}}\times \begin{bmatrix}-y_{22} & -1\\-\Delta_Y & -y_{11}\end{bmatrix}$ $\frac{1}{z_{21}}\times \begin{bmatrix}z_{11} & \Delta_Z\\1 & z_{22}\end{bmatrix}$ $\frac{1}{h_{21}}\times \begin{bmatrix}-\Delta_H & -h_{11}\\-h_{22} & -1\end{bmatrix}$ $\begin{bmatrix}A & B\\C &D\end{bmatrix}$
How to convert from one set of parameters to another

To convert from the set of parameters to another, we first write the current-voltage characteristics of the network in the first set and define the variable of the second set as new unknowns. Then, we solve the system of two equtions in the new unknowns.

For instance, assume we know the y-parameters of a nework and want to find the h-parameters. First we write the current-voltage characteristics of the network using the y-parameters $$\begin{equation}I_1=y_{11}{\color{red}V_1} + y_{12}V_2\end{equation}$$ $$\begin{equation}{\color{red}I_2}=y_{21}{\color{red}V_1} + y_{22}V_2\end{equation}$$ Then, since we want compute the values of the h-parameters, we need to solve the above system of equations for ${\color{red}V_1}$ and ${\color{red}I_2}$. We obtain $$\begin{equation}{\color{red}V_1}=\frac{I_1}{y_{11}} -\frac{y_{12}}{y_{11}}V_2\end{equation}$$ $$\begin{equation}{\color{red}I_2}=\frac{y_{21}}{y_{11}}I_1 + (y_{22}-\frac{y_{12}y_{21}}{y_{11}})V_2\end{equation}$$ which can be written as $$\begin{equation}\begin{bmatrix}{\color{red}V_1}\\{\color{red}I_2}\end{bmatrix} = \begin{bmatrix}\frac{1}{y_{11}} & -\frac{y_{12}}{y_{11}}\\\frac{y_{21}}{y_{11}} & \frac{y_{22}y_{11}-y_{12}y_{21}}{y_{11}}\end{bmatrix} \begin{bmatrix}I_1\\V_2\end{bmatrix}\end{equation}$$ The $2\times 2$ matrix in the right hand side of the last equation, can be identified as the matrix of hybrid parameters (see also Table 1).

Sample Solved Problems
The examples below are randomly generated.
Sample Solved Problems
The examples below are randomly generated.
See also
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Two-port networks