### How to Find Z-Parameters from Y-Parameters
Z-parameters (impedance parameters) and Y-parameters (admittance parameters) are both used to describe the behavior of electrical networks, particularly in terms of voltages and currents. These parameters are related by simple matrix inversions, as Z and Y are inverse matrices of each other.
Given the Y-parameters of a two-port network, you can calculate the Z-parameters using the following steps:
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### 1. **Matrix Representation of Y-parameters and Z-parameters**
#### Y-parameter equations:
For a two-port network, the Y-parameter matrix relates the port currents \( I_1 \) and \( I_2 \) to the port voltages \( V_1 \) and \( V_2 \):
\[
\begin{bmatrix}
I_1 \\
I_2
\end{bmatrix}
=
\begin{bmatrix}
Y_{11} & Y_{12} \\
Y_{21} & Y_{22}
\end{bmatrix}
\begin{bmatrix}
V_1 \\
V_2
\end{bmatrix}
\]
#### Z-parameter equations:
Similarly, the Z-parameter matrix relates the port voltages \( V_1 \) and \( V_2 \) to the port currents \( I_1 \) and \( I_2 \):
\[
\begin{bmatrix}
V_1 \\
V_2
\end{bmatrix}
=
\begin{bmatrix}
Z_{11} & Z_{12} \\
Z_{21} & Z_{22}
\end{bmatrix}
\begin{bmatrix}
I_1 \\
I_2
\end{bmatrix}
\]
To find Z-parameters from Y-parameters, we need to invert the Y-parameter matrix.
---
### 2. **Inverting the Y-parameter Matrix**
The Z-parameter matrix is the inverse of the Y-parameter matrix:
\[
\mathbf{Z} = \mathbf{Y}^{-1}
\]
For a 2x2 matrix, the inverse of the matrix:
\[
\mathbf{Y} =
\begin{bmatrix}
Y_{11} & Y_{12} \\
Y_{21} & Y_{22}
\end{bmatrix}
\]
is given by:
\[
\mathbf{Y}^{-1} =
\frac{1}{\Delta_Y}
\begin{bmatrix}
Y_{22} & -Y_{12} \\
-Y_{21} & Y_{11}
\end{bmatrix}
\]
Where \( \Delta_Y \) is the determinant of the Y-parameter matrix:
\[
\Delta_Y = Y_{11}Y_{22} - Y_{12}Y_{21}
\]
Thus, the Z-parameters are:
\[
Z_{11} = \frac{Y_{22}}{\Delta_Y}
\]
\[
Z_{12} = \frac{-Y_{12}}{\Delta_Y}
\]
\[
Z_{21} = \frac{-Y_{21}}{\Delta_Y}
\]
\[
Z_{22} = \frac{Y_{11}}{\Delta_Y}
\]
---
### 3. **Step-by-Step Process**
#### Step 1: Write the Y-parameter matrix
\[
\mathbf{Y} =
\begin{bmatrix}
Y_{11} & Y_{12} \\
Y_{21} & Y_{22}
\end{bmatrix}
\]
#### Step 2: Calculate the determinant of the Y-parameter matrix
\[
\Delta_Y = Y_{11}Y_{22} - Y_{12}Y_{21}
\]
#### Step 3: Apply the formula for the inverse of a 2x2 matrix
\[
\mathbf{Z} =
\frac{1}{\Delta_Y}
\begin{bmatrix}
Y_{22} & -Y_{12} \\
-Y_{21} & Y_{11}
\end{bmatrix}
\]
#### Step 4: Write down the Z-parameters from the inverse matrix
\[
Z_{11} = \frac{Y_{22}}{Y_{11}Y_{22} - Y_{12}Y_{21}}
\]
\[
Z_{12} = \frac{-Y_{12}}{Y_{11}Y_{22} - Y_{12}Y_{21}}
\]
\[
Z_{21} = \frac{-Y_{21}}{Y_{11}Y_{22} - Y_{12}Y_{21}}
\]
\[
Z_{22} = \frac{Y_{11}}{Y_{11}Y_{22} - Y_{12}Y_{21}}
\]
---
### 4. **Example Calculation**
Let’s consider an example where:
\[
\mathbf{Y} =
\begin{bmatrix}
4 & 2 \\
1 & 3
\end{bmatrix}
\]
#### Step 1: Calculate the determinant
\[
\Delta_Y = (4 \times 3) - (2 \times 1) = 12 - 2 = 10
\]
#### Step 2: Invert the matrix
\[
\mathbf{Z} =
\frac{1}{10}
\begin{bmatrix}
3 & -2 \\
-1 & 4
\end{bmatrix}
=
\begin{bmatrix}
0.3 & -0.2 \\
-0.1 & 0.4
\end{bmatrix}
\]
#### Step 3: Write down the Z-parameters
\[
Z_{11} = 0.3, \quad Z_{12} = -0.2, \quad Z_{21} = -0.1, \quad Z_{22} = 0.4
\]
---
### Summary
To convert Y-parameters to Z-parameters for a two-port network:
1. Write down the Y-parameter matrix.
2. Calculate its determinant.
3. Use the formula for the inverse of a 2x2 matrix to find the Z-parameter matrix.
4. Extract the Z-parameters from the matrix.
These calculations help in analyzing the behavior of networks in terms of either impedances (Z-parameters) or admittances (Y-parameters), depending on the need.