How to convert Z-parameters to y parameters?
2 Answers
Converting Z-parameters (impedance parameters) to Y-parameters (admittance parameters) involves using matrix operations. Here's a detailed explanation of the conversion process:
### Definitions
- **Z-parameters**: Represent the relationship between the voltages and currents at the terminals of a network in terms of impedance.
- **Y-parameters**: Represent the relationship between the currents and voltages at the terminals in terms of admittance.
### Z-Parameter Matrix
The Z-parameter matrix for a network with \( n \) ports is given by:
\[
\mathbf{Z} =
\begin{bmatrix}
Z_{11} & Z_{12} & \cdots & Z_{1n} \\
Z_{21} & Z_{22} & \cdots & Z_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
Z_{n1} & Z_{n2} & \cdots & Z_{nn}
\end{bmatrix}
\]
Where \( Z_{ij} \) represents the impedance seen at port \( i \) when port \( j \) is short-circuited.
### Y-Parameter Matrix
The Y-parameter matrix is given by:
\[
\mathbf{Y} =
\begin{bmatrix}
Y_{11} & Y_{12} & \cdots & Y_{1n} \\
Y_{21} & Y_{22} & \cdots & Y_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
Y_{n1} & Y_{n2} & \cdots & Y_{nn}
\end{bmatrix}
\]
Where \( Y_{ij} \) represents the admittance between port \( i \) and port \( j \) when the remaining ports are open-circuited.
### Conversion Formula
The relationship between the Z-parameters and Y-parameters can be expressed as:
\[
\mathbf{Y} = \mathbf{Z}^{-1}
\]
Here’s a step-by-step procedure for converting Z-parameters to Y-parameters:
1. **Form the Z-Matrix**: Create the Z-matrix using the impedance parameters provided.
2. **Invert the Z-Matrix**: Calculate the inverse of the Z-matrix to obtain the Y-matrix. The inversion of a matrix can be done using various methods, such as:
- **Analytical Inversion**: For a 2x2 matrix:
\[
\mathbf{Z} =
\begin{bmatrix}
Z_{11} & Z_{12} \\
Z_{21} & Z_{22}
\end{bmatrix}
\]
The inverse is:
\[
\mathbf{Y} = \mathbf{Z}^{-1} =
\frac{1}{\text{det}(\mathbf{Z})}
\begin{bmatrix}
Z_{22} & -Z_{12} \\
-Z_{21} & Z_{11}
\end{bmatrix}
\]
Where \(\text{det}(\mathbf{Z}) = Z_{11}Z_{22} - Z_{12}Z_{21}\).
- **Numerical Methods**: For larger matrices, numerical methods such as LU decomposition or using matrix inversion functions in software (e.g., MATLAB, Python’s NumPy) are typically employed.
3. **Verify the Conversion**: It’s a good practice to verify the conversion by checking if the original relationships between voltages and currents hold true with the calculated Y-parameters.
### Example
Consider a simple 2-port network with Z-parameters:
\[
\mathbf{Z} =
\begin{bmatrix}
10 & 2 \\
2 & 5
\end{bmatrix}
\]
To find the Y-parameters:
1. Calculate the determinant of \( \mathbf{Z} \):
\[
\text{det}(\mathbf{Z}) = (10 \cdot 5) - (2 \cdot 2) = 50 - 4 = 46
\]
2. Compute the inverse of \( \mathbf{Z} \):
\[
\mathbf{Y} = \mathbf{Z}^{-1} = \frac{1}{46}
\begin{bmatrix}
5 & -2 \\
-2 & 10
\end{bmatrix}
=
\begin{bmatrix}
\frac{5}{46} & -\frac{2}{46} \\
-\frac{2}{46} & \frac{10}{46}
\end{bmatrix}
=
\begin{bmatrix}
0.1087 & -0.0435 \\
-0.0435 & 0.2174
\end{bmatrix}
\]
So, the Y-parameters are:
\[
\mathbf{Y} =
\begin{bmatrix}
0.1087 & -0.0435 \\
-0.0435 & 0.2174
\end{bmatrix}
\]
This method can be generalized to larger matrices using similar principles.
### Definitions
- **Z-parameters**: Represent the relationship between the voltages and currents at the terminals of a network in terms of impedance.
- **Y-parameters**: Represent the relationship between the currents and voltages at the terminals in terms of admittance.
### Z-Parameter Matrix
The Z-parameter matrix for a network with \( n \) ports is given by:
\[
\mathbf{Z} =
\begin{bmatrix}
Z_{11} & Z_{12} & \cdots & Z_{1n} \\
Z_{21} & Z_{22} & \cdots & Z_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
Z_{n1} & Z_{n2} & \cdots & Z_{nn}
\end{bmatrix}
\]
Where \( Z_{ij} \) represents the impedance seen at port \( i \) when port \( j \) is short-circuited.
### Y-Parameter Matrix
The Y-parameter matrix is given by:
\[
\mathbf{Y} =
\begin{bmatrix}
Y_{11} & Y_{12} & \cdots & Y_{1n} \\
Y_{21} & Y_{22} & \cdots & Y_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
Y_{n1} & Y_{n2} & \cdots & Y_{nn}
\end{bmatrix}
\]
Where \( Y_{ij} \) represents the admittance between port \( i \) and port \( j \) when the remaining ports are open-circuited.
### Conversion Formula
The relationship between the Z-parameters and Y-parameters can be expressed as:
\[
\mathbf{Y} = \mathbf{Z}^{-1}
\]
Here’s a step-by-step procedure for converting Z-parameters to Y-parameters:
1. **Form the Z-Matrix**: Create the Z-matrix using the impedance parameters provided.
2. **Invert the Z-Matrix**: Calculate the inverse of the Z-matrix to obtain the Y-matrix. The inversion of a matrix can be done using various methods, such as:
- **Analytical Inversion**: For a 2x2 matrix:
\[
\mathbf{Z} =
\begin{bmatrix}
Z_{11} & Z_{12} \\
Z_{21} & Z_{22}
\end{bmatrix}
\]
The inverse is:
\[
\mathbf{Y} = \mathbf{Z}^{-1} =
\frac{1}{\text{det}(\mathbf{Z})}
\begin{bmatrix}
Z_{22} & -Z_{12} \\
-Z_{21} & Z_{11}
\end{bmatrix}
\]
Where \(\text{det}(\mathbf{Z}) = Z_{11}Z_{22} - Z_{12}Z_{21}\).
- **Numerical Methods**: For larger matrices, numerical methods such as LU decomposition or using matrix inversion functions in software (e.g., MATLAB, Python’s NumPy) are typically employed.
3. **Verify the Conversion**: It’s a good practice to verify the conversion by checking if the original relationships between voltages and currents hold true with the calculated Y-parameters.
### Example
Consider a simple 2-port network with Z-parameters:
\[
\mathbf{Z} =
\begin{bmatrix}
10 & 2 \\
2 & 5
\end{bmatrix}
\]
To find the Y-parameters:
1. Calculate the determinant of \( \mathbf{Z} \):
\[
\text{det}(\mathbf{Z}) = (10 \cdot 5) - (2 \cdot 2) = 50 - 4 = 46
\]
2. Compute the inverse of \( \mathbf{Z} \):
\[
\mathbf{Y} = \mathbf{Z}^{-1} = \frac{1}{46}
\begin{bmatrix}
5 & -2 \\
-2 & 10
\end{bmatrix}
=
\begin{bmatrix}
\frac{5}{46} & -\frac{2}{46} \\
-\frac{2}{46} & \frac{10}{46}
\end{bmatrix}
=
\begin{bmatrix}
0.1087 & -0.0435 \\
-0.0435 & 0.2174
\end{bmatrix}
\]
So, the Y-parameters are:
\[
\mathbf{Y} =
\begin{bmatrix}
0.1087 & -0.0435 \\
-0.0435 & 0.2174
\end{bmatrix}
\]
This method can be generalized to larger matrices using similar principles.
To convert Z-parameters (impedance parameters) to Y-parameters (admittance parameters), you can use a matrix transformation. Here’s a step-by-step guide to converting Z-parameters to Y-parameters:
### Z-Parameters and Y-Parameters Overview
- **Z-parameters (Impedance Parameters)** describe the relationship between the voltages and currents at the ports of a network. For a two-port network, the Z-parameters are defined as:
\[
\begin{bmatrix}
V_1 \\
V_2
\end{bmatrix}
=
\begin{bmatrix}
Z_{11} & Z_{12} \\
Z_{21} & Z_{22}
\end{bmatrix}
\begin{bmatrix}
I_1 \\
I_2
\end{bmatrix}
\]
Where:
- \( V_1 \) and \( V_2 \) are the voltages at the ports.
- \( I_1 \) and \( I_2 \) are the currents through the ports.
- \( Z_{11} \), \( Z_{12} \), \( Z_{21} \), and \( Z_{22} \) are the Z-parameters.
- **Y-parameters (Admittance Parameters)** describe the relationship between the currents and voltages at the ports of a network. For a two-port network, the Y-parameters are defined as:
\[
\begin{bmatrix}
I_1 \\
I_2
\end{bmatrix}
=
\begin{bmatrix}
Y_{11} & Y_{12} \\
Y_{21} & Y_{22}
\end{bmatrix}
\begin{bmatrix}
V_1 \\
V_2
\end{bmatrix}
\]
Where:
- \( Y_{11} \), \( Y_{12} \), \( Y_{21} \), and \( Y_{22} \) are the Y-parameters.
### Conversion Formula
The relationship between Z-parameters and Y-parameters can be expressed through matrix inversion. Specifically, the Y-parameters are the inverse of the Z-parameter matrix. Here's how you can find them:
1. **Write down the Z-parameter matrix:**
\[
\mathbf{Z} =
\begin{bmatrix}
Z_{11} & Z_{12} \\
Z_{21} & Z_{22}
\end{bmatrix}
\]
2. **Find the inverse of the Z-matrix:**
The inverse of a 2x2 matrix \( \mathbf{Z} \) is given by:
\[
\mathbf{Y} = \mathbf{Z}^{-1} =
\frac{1}{\text{det}(\mathbf{Z})}
\begin{bmatrix}
Z_{22} & -Z_{12} \\
-Z_{21} & Z_{11}
\end{bmatrix}
\]
where \(\text{det}(\mathbf{Z})\) is the determinant of the Z-matrix, calculated as:
\[
\text{det}(\mathbf{Z}) = Z_{11} \cdot Z_{22} - Z_{12} \cdot Z_{21}
\]
3. **Calculate the Y-parameter matrix:**
Substituting the determinant and the cofactor matrix into the formula gives:
\[
\mathbf{Y} =
\frac{1}{Z_{11} \cdot Z_{22} - Z_{12} \cdot Z_{21}}
\begin{bmatrix}
Z_{22} & -Z_{12} \\
-Z_{21} & Z_{11}
\end{bmatrix}
\]
### Example
Let’s work through an example to make it clearer.
Suppose you have the following Z-parameters:
\[
\mathbf{Z} =
\begin{bmatrix}
10 & 2 \\
3 & 5
\end{bmatrix}
\]
1. **Calculate the determinant:**
\[
\text{det}(\mathbf{Z}) = (10 \cdot 5) - (2 \cdot 3) = 50 - 6 = 44
\]
2. **Find the inverse matrix:**
\[
\mathbf{Z}^{-1} =
\frac{1}{44}
\begin{bmatrix}
5 & -2 \\
-3 & 10
\end{bmatrix}
\]
3. **Compute the Y-parameters:**
\[
\mathbf{Y} =
\frac{1}{44}
\begin{bmatrix}
5 & -2 \\
-3 & 10
\end{bmatrix}
=
\begin{bmatrix}
\frac{5}{44} & -\frac{2}{44} \\
-\frac{3}{44} & \frac{10}{44}
\end{bmatrix}
=
\begin{bmatrix}
0.1136 & -0.0455 \\
-0.0682 & 0.2273
\end{bmatrix}
\]
So, the Y-parameter matrix is:
\[
\mathbf{Y} =
\begin{bmatrix}
0.1136 & -0.0455 \\
-0.0682 & 0.2273
\end{bmatrix}
\]
### Summary
To convert Z-parameters to Y-parameters:
1. Write down the Z-parameter matrix.
2. Compute the determinant of this matrix.
3. Find the cofactor matrix of the Z-matrix.
4. Multiply the cofactor matrix by the reciprocal of the determinant to get the Y-parameter matrix.
This process can be applied similarly for networks with more ports, although the calculations will be more complex for larger matrices.
### Z-Parameters and Y-Parameters Overview
- **Z-parameters (Impedance Parameters)** describe the relationship between the voltages and currents at the ports of a network. For a two-port network, the Z-parameters are defined as:
\[
\begin{bmatrix}
V_1 \\
V_2
\end{bmatrix}
=
\begin{bmatrix}
Z_{11} & Z_{12} \\
Z_{21} & Z_{22}
\end{bmatrix}
\begin{bmatrix}
I_1 \\
I_2
\end{bmatrix}
\]
Where:
- \( V_1 \) and \( V_2 \) are the voltages at the ports.
- \( I_1 \) and \( I_2 \) are the currents through the ports.
- \( Z_{11} \), \( Z_{12} \), \( Z_{21} \), and \( Z_{22} \) are the Z-parameters.
- **Y-parameters (Admittance Parameters)** describe the relationship between the currents and voltages at the ports of a network. For a two-port network, the Y-parameters are defined as:
\[
\begin{bmatrix}
I_1 \\
I_2
\end{bmatrix}
=
\begin{bmatrix}
Y_{11} & Y_{12} \\
Y_{21} & Y_{22}
\end{bmatrix}
\begin{bmatrix}
V_1 \\
V_2
\end{bmatrix}
\]
Where:
- \( Y_{11} \), \( Y_{12} \), \( Y_{21} \), and \( Y_{22} \) are the Y-parameters.
### Conversion Formula
The relationship between Z-parameters and Y-parameters can be expressed through matrix inversion. Specifically, the Y-parameters are the inverse of the Z-parameter matrix. Here's how you can find them:
1. **Write down the Z-parameter matrix:**
\[
\mathbf{Z} =
\begin{bmatrix}
Z_{11} & Z_{12} \\
Z_{21} & Z_{22}
\end{bmatrix}
\]
2. **Find the inverse of the Z-matrix:**
The inverse of a 2x2 matrix \( \mathbf{Z} \) is given by:
\[
\mathbf{Y} = \mathbf{Z}^{-1} =
\frac{1}{\text{det}(\mathbf{Z})}
\begin{bmatrix}
Z_{22} & -Z_{12} \\
-Z_{21} & Z_{11}
\end{bmatrix}
\]
where \(\text{det}(\mathbf{Z})\) is the determinant of the Z-matrix, calculated as:
\[
\text{det}(\mathbf{Z}) = Z_{11} \cdot Z_{22} - Z_{12} \cdot Z_{21}
\]
3. **Calculate the Y-parameter matrix:**
Substituting the determinant and the cofactor matrix into the formula gives:
\[
\mathbf{Y} =
\frac{1}{Z_{11} \cdot Z_{22} - Z_{12} \cdot Z_{21}}
\begin{bmatrix}
Z_{22} & -Z_{12} \\
-Z_{21} & Z_{11}
\end{bmatrix}
\]
### Example
Let’s work through an example to make it clearer.
Suppose you have the following Z-parameters:
\[
\mathbf{Z} =
\begin{bmatrix}
10 & 2 \\
3 & 5
\end{bmatrix}
\]
1. **Calculate the determinant:**
\[
\text{det}(\mathbf{Z}) = (10 \cdot 5) - (2 \cdot 3) = 50 - 6 = 44
\]
2. **Find the inverse matrix:**
\[
\mathbf{Z}^{-1} =
\frac{1}{44}
\begin{bmatrix}
5 & -2 \\
-3 & 10
\end{bmatrix}
\]
3. **Compute the Y-parameters:**
\[
\mathbf{Y} =
\frac{1}{44}
\begin{bmatrix}
5 & -2 \\
-3 & 10
\end{bmatrix}
=
\begin{bmatrix}
\frac{5}{44} & -\frac{2}{44} \\
-\frac{3}{44} & \frac{10}{44}
\end{bmatrix}
=
\begin{bmatrix}
0.1136 & -0.0455 \\
-0.0682 & 0.2273
\end{bmatrix}
\]
So, the Y-parameter matrix is:
\[
\mathbf{Y} =
\begin{bmatrix}
0.1136 & -0.0455 \\
-0.0682 & 0.2273
\end{bmatrix}
\]
### Summary
To convert Z-parameters to Y-parameters:
1. Write down the Z-parameter matrix.
2. Compute the determinant of this matrix.
3. Find the cofactor matrix of the Z-matrix.
4. Multiply the cofactor matrix by the reciprocal of the determinant to get the Y-parameter matrix.
This process can be applied similarly for networks with more ports, although the calculations will be more complex for larger matrices.