How to convert Z parameters to y parameters?
2 Answers
To convert Z parameters (impedance parameters) to Y parameters (admittance parameters) in electrical engineering, you can use the following relationship. Given a network with impedance parameters \( Z_{11} \), \( Z_{12} \), \( Z_{21} \), and \( Z_{22} \), the corresponding admittance parameters \( Y_{11} \), \( Y_{12} \), \( Y_{21} \), and \( Y_{22} \) can be found using the following formulas:
1. **Find the Admittance Matrix \( Y \):**
The admittance parameters are given by:
\[
Y_{11} = \frac{Z_{22}}{Z_{11}Z_{22} - Z_{12}Z_{21}}
\]
\[
Y_{22} = \frac{Z_{11}}{Z_{11}Z_{22} - Z_{12}Z_{21}}
\]
\[
Y_{12} = -\frac{Z_{12}}{Z_{11}Z_{22} - Z_{12}Z_{21}}
\]
\[
Y_{21} = -\frac{Z_{21}}{Z_{11}Z_{22} - Z_{12}Z_{21}}
\]
2. **Apply the Formulas:**
- **For \( Y_{11} \):**
\[
Y_{11} = \frac{Z_{22}}{Z_{11}Z_{22} - Z_{12}Z_{21}}
\]
- **For \( Y_{22} \):**
\[
Y_{22} = \frac{Z_{11}}{Z_{11}Z_{22} - Z_{12}Z_{21}}
\]
- **For \( Y_{12} \):**
\[
Y_{12} = -\frac{Z_{12}}{Z_{11}Z_{22} - Z_{12}Z_{21}}
\]
- **For \( Y_{21} \):**
\[
Y_{21} = -\frac{Z_{21}}{Z_{11}Z_{22} - Z_{12}Z_{21}}
\]
### Example:
If you have the following Z parameters for a network:
\[
Z = \begin{bmatrix}
Z_{11} & Z_{12} \\
Z_{21} & Z_{22}
\end{bmatrix}
\]
Let's say:
\[
Z_{11} = 2 \, \Omega, \quad Z_{12} = 1 \, \Omega, \quad Z_{21} = 1 \, \Omega, \quad Z_{22} = 3 \, \Omega
\]
You can find \( Y_{11} \), \( Y_{12} \), \( Y_{21} \), and \( Y_{22} \) as follows:
1. Compute the determinant of \( Z \):
\[
\text{det}(Z) = Z_{11}Z_{22} - Z_{12}Z_{21} = 2 \times 3 - 1 \times 1 = 6 - 1 = 5
\]
2. Compute \( Y_{11} \), \( Y_{12} \), \( Y_{21} \), and \( Y_{22} \):
- \( Y_{11} = \frac{Z_{22}}{\text{det}(Z)} = \frac{3}{5} \)
- \( Y_{22} = \frac{Z_{11}}{\text{det}(Z)} = \frac{2}{5} \)
- \( Y_{12} = -\frac{Z_{12}}{\text{det}(Z)} = -\frac{1}{5} \)
- \( Y_{21} = -\frac{Z_{21}}{\text{det}(Z)} = -\frac{1}{5} \)
Thus, the Y parameters matrix is:
\[
Y = \begin{bmatrix}
\frac{3}{5} & -\frac{1}{5} \\
-\frac{1}{5} & \frac{2}{5}
\end{bmatrix}
\]
This conversion allows you to work with admittance parameters, which are sometimes more convenient for circuit analysis, especially in parallel circuits.
1. **Find the Admittance Matrix \( Y \):**
The admittance parameters are given by:
\[
Y_{11} = \frac{Z_{22}}{Z_{11}Z_{22} - Z_{12}Z_{21}}
\]
\[
Y_{22} = \frac{Z_{11}}{Z_{11}Z_{22} - Z_{12}Z_{21}}
\]
\[
Y_{12} = -\frac{Z_{12}}{Z_{11}Z_{22} - Z_{12}Z_{21}}
\]
\[
Y_{21} = -\frac{Z_{21}}{Z_{11}Z_{22} - Z_{12}Z_{21}}
\]
2. **Apply the Formulas:**
- **For \( Y_{11} \):**
\[
Y_{11} = \frac{Z_{22}}{Z_{11}Z_{22} - Z_{12}Z_{21}}
\]
- **For \( Y_{22} \):**
\[
Y_{22} = \frac{Z_{11}}{Z_{11}Z_{22} - Z_{12}Z_{21}}
\]
- **For \( Y_{12} \):**
\[
Y_{12} = -\frac{Z_{12}}{Z_{11}Z_{22} - Z_{12}Z_{21}}
\]
- **For \( Y_{21} \):**
\[
Y_{21} = -\frac{Z_{21}}{Z_{11}Z_{22} - Z_{12}Z_{21}}
\]
### Example:
If you have the following Z parameters for a network:
\[
Z = \begin{bmatrix}
Z_{11} & Z_{12} \\
Z_{21} & Z_{22}
\end{bmatrix}
\]
Let's say:
\[
Z_{11} = 2 \, \Omega, \quad Z_{12} = 1 \, \Omega, \quad Z_{21} = 1 \, \Omega, \quad Z_{22} = 3 \, \Omega
\]
You can find \( Y_{11} \), \( Y_{12} \), \( Y_{21} \), and \( Y_{22} \) as follows:
1. Compute the determinant of \( Z \):
\[
\text{det}(Z) = Z_{11}Z_{22} - Z_{12}Z_{21} = 2 \times 3 - 1 \times 1 = 6 - 1 = 5
\]
2. Compute \( Y_{11} \), \( Y_{12} \), \( Y_{21} \), and \( Y_{22} \):
- \( Y_{11} = \frac{Z_{22}}{\text{det}(Z)} = \frac{3}{5} \)
- \( Y_{22} = \frac{Z_{11}}{\text{det}(Z)} = \frac{2}{5} \)
- \( Y_{12} = -\frac{Z_{12}}{\text{det}(Z)} = -\frac{1}{5} \)
- \( Y_{21} = -\frac{Z_{21}}{\text{det}(Z)} = -\frac{1}{5} \)
Thus, the Y parameters matrix is:
\[
Y = \begin{bmatrix}
\frac{3}{5} & -\frac{1}{5} \\
-\frac{1}{5} & \frac{2}{5}
\end{bmatrix}
\]
This conversion allows you to work with admittance parameters, which are sometimes more convenient for circuit analysis, especially in parallel circuits.
To convert Z-parameters (impedance parameters) to Y-parameters (admittance parameters), you need to use the following relationships between the two parameter sets. This conversion is particularly useful in circuit analysis and design when switching between impedance and admittance representations.
### Definitions:
- **Z-parameters** (Impedance parameters):
- \( V_1 = Z_{11}I_1 + Z_{12}I_2 \)
- \( V_2 = Z_{21}I_1 + Z_{22}I_2 \)
- **Y-parameters** (Admittance parameters):
- \( I_1 = Y_{11}V_1 + Y_{12}V_2 \)
- \( I_2 = Y_{21}V_1 + Y_{22}V_2 \)
Here, \( V_1 \) and \( V_2 \) are the voltages at the ports, and \( I_1 \) and \( I_2 \) are the currents through the ports.
### Conversion Formulae:
1. **Find \( Y_{11} \):**
\[
Y_{11} = \frac{Z_{22}}{Z_{11} Z_{22} - Z_{12} Z_{21}}
\]
2. **Find \( Y_{12} \):**
\[
Y_{12} = -\frac{Z_{12}}{Z_{11} Z_{22} - Z_{12} Z_{21}}
\]
3. **Find \( Y_{21} \):**
\[
Y_{21} = -\frac{Z_{21}}{Z_{11} Z_{22} - Z_{12} Z_{21}}
\]
4. **Find \( Y_{22} \):**
\[
Y_{22} = \frac{Z_{11}}{Z_{11} Z_{22} - Z_{12} Z_{21}}
\]
### Explanation of the Conversion:
- **Matrix Form:**
In matrix form, the Z-parameter matrix is:
\[
\mathbf{Z} = \begin{bmatrix}
Z_{11} & Z_{12} \\
Z_{21} & Z_{22}
\end{bmatrix}
\]
and the Y-parameter matrix is:
\[
\mathbf{Y} = \begin{bmatrix}
Y_{11} & Y_{12} \\
Y_{21} & Y_{22}
\end{bmatrix}
\]
- **Relationship:**
The relationship between Z-parameters and Y-parameters is given by the inverse of the Z-parameter matrix. Mathematically:
\[
\mathbf{Y} = \mathbf{Z}^{-1}
\]
where \(\mathbf{Z}^{-1}\) is the inverse of the impedance matrix \(\mathbf{Z}\).
### Example:
Suppose you have the following Z-parameter matrix:
\[
\mathbf{Z} = \begin{bmatrix}
10 & 5 \\
5 & 15
\end{bmatrix}
\]
1. **Calculate Determinant of \(\mathbf{Z}\):**
\[
\text{det}(\mathbf{Z}) = Z_{11}Z_{22} - Z_{12}Z_{21} = 10 \cdot 15 - 5 \cdot 5 = 150 - 25 = 125
\]
2. **Calculate \( Y_{11} \):**
\[
Y_{11} = \frac{15}{125} = 0.12
\]
3. **Calculate \( Y_{12} \):**
\[
Y_{12} = -\frac{5}{125} = -0.04
\]
4. **Calculate \( Y_{21} \):**
\[
Y_{21} = -\frac{5}{125} = -0.04
\]
5. **Calculate \( Y_{22} \):**
\[
Y_{22} = \frac{10}{125} = 0.08
\]
So, the Y-parameter matrix is:
\[
\mathbf{Y} = \begin{bmatrix}
0.12 & -0.04 \\
-0.04 & 0.08
\end{bmatrix}
\]
By using these conversion formulas and steps, you can effectively convert Z-parameters to Y-parameters for any linear network.
### Definitions:
- **Z-parameters** (Impedance parameters):
- \( V_1 = Z_{11}I_1 + Z_{12}I_2 \)
- \( V_2 = Z_{21}I_1 + Z_{22}I_2 \)
- **Y-parameters** (Admittance parameters):
- \( I_1 = Y_{11}V_1 + Y_{12}V_2 \)
- \( I_2 = Y_{21}V_1 + Y_{22}V_2 \)
Here, \( V_1 \) and \( V_2 \) are the voltages at the ports, and \( I_1 \) and \( I_2 \) are the currents through the ports.
### Conversion Formulae:
1. **Find \( Y_{11} \):**
\[
Y_{11} = \frac{Z_{22}}{Z_{11} Z_{22} - Z_{12} Z_{21}}
\]
2. **Find \( Y_{12} \):**
\[
Y_{12} = -\frac{Z_{12}}{Z_{11} Z_{22} - Z_{12} Z_{21}}
\]
3. **Find \( Y_{21} \):**
\[
Y_{21} = -\frac{Z_{21}}{Z_{11} Z_{22} - Z_{12} Z_{21}}
\]
4. **Find \( Y_{22} \):**
\[
Y_{22} = \frac{Z_{11}}{Z_{11} Z_{22} - Z_{12} Z_{21}}
\]
### Explanation of the Conversion:
- **Matrix Form:**
In matrix form, the Z-parameter matrix is:
\[
\mathbf{Z} = \begin{bmatrix}
Z_{11} & Z_{12} \\
Z_{21} & Z_{22}
\end{bmatrix}
\]
and the Y-parameter matrix is:
\[
\mathbf{Y} = \begin{bmatrix}
Y_{11} & Y_{12} \\
Y_{21} & Y_{22}
\end{bmatrix}
\]
- **Relationship:**
The relationship between Z-parameters and Y-parameters is given by the inverse of the Z-parameter matrix. Mathematically:
\[
\mathbf{Y} = \mathbf{Z}^{-1}
\]
where \(\mathbf{Z}^{-1}\) is the inverse of the impedance matrix \(\mathbf{Z}\).
### Example:
Suppose you have the following Z-parameter matrix:
\[
\mathbf{Z} = \begin{bmatrix}
10 & 5 \\
5 & 15
\end{bmatrix}
\]
1. **Calculate Determinant of \(\mathbf{Z}\):**
\[
\text{det}(\mathbf{Z}) = Z_{11}Z_{22} - Z_{12}Z_{21} = 10 \cdot 15 - 5 \cdot 5 = 150 - 25 = 125
\]
2. **Calculate \( Y_{11} \):**
\[
Y_{11} = \frac{15}{125} = 0.12
\]
3. **Calculate \( Y_{12} \):**
\[
Y_{12} = -\frac{5}{125} = -0.04
\]
4. **Calculate \( Y_{21} \):**
\[
Y_{21} = -\frac{5}{125} = -0.04
\]
5. **Calculate \( Y_{22} \):**
\[
Y_{22} = \frac{10}{125} = 0.08
\]
So, the Y-parameter matrix is:
\[
\mathbf{Y} = \begin{bmatrix}
0.12 & -0.04 \\
-0.04 & 0.08
\end{bmatrix}
\]
By using these conversion formulas and steps, you can effectively convert Z-parameters to Y-parameters for any linear network.