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How do you find the Z-parameters from Y parameters?

2 Answers

 
Best answer
To find the Z-parameters (impedance parameters) from Y-parameters (admittance parameters), you can use the relationships between these two sets of parameters. Here's a step-by-step guide to converting Y-parameters to Z-parameters:

### Definitions

1. **Y-parameters (Admittance Parameters)**:
   - \( Y_{11} \): Input admittance with the output shorted.
   - \( Y_{12} \): Transfer admittance from output to input.
   - \( Y_{21} \): Transfer admittance from input to output.
   - \( Y_{22} \): Output admittance with the input shorted.

   The relationships in terms of voltages and currents are:
   \[
   \begin{bmatrix}
   I_1 \\
   I_2
   \end{bmatrix}
   =
   \begin{bmatrix}
   Y_{11} & Y_{12} \\
   Y_{21} & Y_{22}
   \end{bmatrix}
   \begin{bmatrix}
   V_1 \\
   V_2
   \end{bmatrix}
   \]
   where \( I_1 \) and \( I_2 \) are the currents, and \( V_1 \) and \( V_2 \) are the voltages.

2. **Z-parameters (Impedance Parameters)**:
   - \( Z_{11} \): Input impedance with the output open.
   - \( Z_{12} \): Transfer impedance from output to input.
   - \( Z_{21} \): Transfer impedance from input to output.
   - \( Z_{22} \): Output impedance with the input open.

   The relationships in terms of voltages and currents are:
   \[
   \begin{bmatrix}
   V_1 \\
   V_2
   \end{bmatrix}
   =
   \begin{bmatrix}
   Z_{11} & Z_{12} \\
   Z_{21} & Z_{22}
   \end{bmatrix}
   \begin{bmatrix}
   I_1 \\
   I_2
   \end{bmatrix}
   \]

### Conversion Formula

To convert from Y-parameters to Z-parameters, use the following formulas:

1. **Calculate \( Z_{11} \)**:
   \[
   Z_{11} = \frac{1}{Y_{11}} - \frac{Y_{12}Y_{21}}{Y_{11}Y_{22} - Y_{12}Y_{21}}
   \]

2. **Calculate \( Z_{12} \)**:
   \[
   Z_{12} = -\frac{Y_{12}}{Y_{11}Y_{22} - Y_{12}Y_{21}}
   \]

3. **Calculate \( Z_{21} \)**:
   \[
   Z_{21} = -\frac{Y_{21}}{Y_{11}Y_{22} - Y_{12}Y_{21}}
   \]

4. **Calculate \( Z_{22} \)**:
   \[
   Z_{22} = \frac{1}{Y_{22}} - \frac{Y_{12}Y_{21}}{Y_{11}Y_{22} - Y_{12}Y_{21}}
   \]

### Derivation of the Formulas

To understand why these formulas work, let’s briefly outline the derivation process:

1. **Inverse Matrix Approach**: The Z-parameters can be derived by inverting the matrix of Y-parameters. Given the matrix equation:
   \[
   \begin{bmatrix}
   I_1 \\
   I_2
   \end{bmatrix}
   =
   \begin{bmatrix}
   Y_{11} & Y_{12} \\
   Y_{21} & Y_{22}
   \end{bmatrix}
   \begin{bmatrix}
   V_1 \\
   V_2
   \end{bmatrix}
   \]

   Rearranging, you get:
   \[
   \begin{bmatrix}
   V_1 \\
   V_2
   \end{bmatrix}
   =
   \begin{bmatrix}
   Y_{11} & Y_{12} \\
   Y_{21} & Y_{22}
   \end{bmatrix}^{-1}
   \begin{bmatrix}
   I_1 \\
   I_2
   \end{bmatrix}
   \]

   The inverse of the admittance matrix gives the impedance matrix. The formulas above are derived by inverting the 2x2 matrix of Y-parameters.

### Example

Let’s consider an example with arbitrary values for the Y-parameters:

\[
Y =
\begin{bmatrix}
2 & -0.5 \\
-0.5 & 1
\end{bmatrix}
\]

Using the formulas:

1. **Calculate \( Z_{11} \)**:
   \[
   Z_{11} = \frac{1}{Y_{11}} - \frac{Y_{12}Y_{21}}{Y_{11}Y_{22} - Y_{12}Y_{21}}
   = \frac{1}{2} - \frac{(-0.5)(-0.5)}{2 \cdot 1 - (-0.5)(-0.5)}
   = 0.5 - \frac{0.25}{2 - 0.25}
   = 0.5 - \frac{0.25}{1.75}
   = 0.5 - 0.1429
   = 0.3571
   \]

2. **Calculate \( Z_{12} \)**:
   \[
   Z_{12} = -\frac{Y_{12}}{Y_{11}Y_{22} - Y_{12}Y_{21}}
   = -\frac{-0.5}{2 \cdot 1 - (-0.5)(-0.5)}
   = \frac{0.5}{1.75}
   = 0.2857
   \]

3. **Calculate \( Z_{21} \)**:
   \[
   Z_{21} = -\frac{Y_{21}}{Y_{11}Y_{22} - Y_{12}Y_{21}}
   = -\frac{-0.5}{1.75}
   = 0.2857
   \]

4. **Calculate \( Z_{22} \)**:
   \[
   Z_{22} = \frac{1}{Y_{22}} - \frac{Y_{12}Y_{21}}{Y_{11}Y_{22} - Y_{12}Y_{21}}
   = \frac{1}{1} - \frac{(-0.5)(-0.5)}{1.75}
   = 1 - \frac{0.25}{1.75}
   = 1 - 0.1429
   = 0.8571
   \]

Thus, the Z-parameter matrix is:

\[
Z =
\begin{bmatrix}
0.3571 & 0.2857 \\
0.2857 & 0.8571
\end{bmatrix}
\]

This process ensures that you can convert Y-parameters to Z-parameters and vice versa for any two-port network.
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### 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:

---

### 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.
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