How to convert Y parameters to ABCD parameters?
2 Answers
To convert Y parameters (admittance parameters) to ABCD parameters (transmission parameters), you can use the following formulas. Given a two-port network characterized by its Y parameters:
\[
\begin{bmatrix}
Y_{11} & Y_{12} \\
Y_{21} & Y_{22}
\end{bmatrix}
\]
The ABCD parameters are defined as:
\[
\begin{bmatrix}
A & B \\
C & D
\end{bmatrix}
\]
The relationships between Y parameters and ABCD parameters are given by:
\[
A = \frac{Y_{22}}{Y_{11}Y_{22} - Y_{12}Y_{21}}, \quad B = \frac{1}{Y_{11}} - \frac{Y_{12}}{Y_{11}(Y_{22}Y_{11} - Y_{12}Y_{21})}
\]
\[
C = \frac{1}{Y_{22}} - \frac{Y_{21}}{Y_{22}(Y_{22}Y_{11} - Y_{12}Y_{21})}, \quad D = \frac{Y_{11}}{Y_{11}Y_{22} - Y_{12}Y_{21}}
\]
### Step-by-Step Conversion
1. **Calculate \(A\):**
\[
A = \frac{Y_{22}}{Y_{11}Y_{22} - Y_{12}Y_{21}}
\]
2. **Calculate \(B\):**
\[
B = \frac{1}{Y_{11}} - \frac{Y_{12}}{Y_{11}(Y_{22}Y_{11} - Y_{12}Y_{21})}
\]
3. **Calculate \(C\):**
\[
C = \frac{1}{Y_{22}} - \frac{Y_{21}}{Y_{22}(Y_{22}Y_{11} - Y_{12}Y_{21})}
\]
4. **Calculate \(D\):**
\[
D = \frac{Y_{11}}{Y_{11}Y_{22} - Y_{12}Y_{21}}
\]
### Example
If you have the following Y parameters:
- \(Y_{11} = 1 + j2\)
- \(Y_{12} = 0.5 + j0.5\)
- \(Y_{21} = 0.5 - j0.5\)
- \(Y_{22} = 2 - j1\)
You can plug these values into the equations above to find the corresponding ABCD parameters.
Let me know if you need further explanation or a specific example!
\[
\begin{bmatrix}
Y_{11} & Y_{12} \\
Y_{21} & Y_{22}
\end{bmatrix}
\]
The ABCD parameters are defined as:
\[
\begin{bmatrix}
A & B \\
C & D
\end{bmatrix}
\]
The relationships between Y parameters and ABCD parameters are given by:
\[
A = \frac{Y_{22}}{Y_{11}Y_{22} - Y_{12}Y_{21}}, \quad B = \frac{1}{Y_{11}} - \frac{Y_{12}}{Y_{11}(Y_{22}Y_{11} - Y_{12}Y_{21})}
\]
\[
C = \frac{1}{Y_{22}} - \frac{Y_{21}}{Y_{22}(Y_{22}Y_{11} - Y_{12}Y_{21})}, \quad D = \frac{Y_{11}}{Y_{11}Y_{22} - Y_{12}Y_{21}}
\]
### Step-by-Step Conversion
1. **Calculate \(A\):**
\[
A = \frac{Y_{22}}{Y_{11}Y_{22} - Y_{12}Y_{21}}
\]
2. **Calculate \(B\):**
\[
B = \frac{1}{Y_{11}} - \frac{Y_{12}}{Y_{11}(Y_{22}Y_{11} - Y_{12}Y_{21})}
\]
3. **Calculate \(C\):**
\[
C = \frac{1}{Y_{22}} - \frac{Y_{21}}{Y_{22}(Y_{22}Y_{11} - Y_{12}Y_{21})}
\]
4. **Calculate \(D\):**
\[
D = \frac{Y_{11}}{Y_{11}Y_{22} - Y_{12}Y_{21}}
\]
### Example
If you have the following Y parameters:
- \(Y_{11} = 1 + j2\)
- \(Y_{12} = 0.5 + j0.5\)
- \(Y_{21} = 0.5 - j0.5\)
- \(Y_{22} = 2 - j1\)
You can plug these values into the equations above to find the corresponding ABCD parameters.
Let me know if you need further explanation or a specific example!
To convert \(Y\) parameters (admittance parameters) to \(ABCD\) parameters (transmission or chain parameters), you can use the following method. Here’s a step-by-step guide:
### Given
- The \(Y\) parameters are defined as:
\[
\begin{bmatrix}
I_1 \\
I_2
\end{bmatrix}
=
\begin{bmatrix}
Y_{11} & Y_{12} \\
Y_{21} & Y_{22}
\end{bmatrix}
\begin{bmatrix}
V_1 \\
V_2
\end{bmatrix}
\]
### \(Y\) to \(ABCD\) Parameters Conversion
1. **Express the \(V\) and \(I\) relationship in terms of \(Y\) parameters:**
\[
\begin{bmatrix}
V_1 \\
V_2
\end{bmatrix}
=
\begin{bmatrix}
Z_{11} & Z_{12} \\
Z_{21} & Z_{22}
\end{bmatrix}
\begin{bmatrix}
I_1 \\
I_2
\end{bmatrix}
\]
Where \(Z\) parameters are the inverse of the \(Y\) parameters matrix.
2. **Calculate the inverse of the \(Y\) parameters matrix to get the \(Z\) parameters:**
\[
\begin{bmatrix}
Z_{11} & Z_{12} \\
Z_{21} & Z_{22}
\end{bmatrix}
=
\begin{bmatrix}
Y_{11} & Y_{12} \\
Y_{21} & Y_{22}
\end{bmatrix}^{-1}
\]
To find the inverse:
\[
\text{Inverse} = \frac{1}{\text{det}(Y)} \begin{bmatrix}
Y_{22} & -Y_{12} \\
-Y_{21} & Y_{11}
\end{bmatrix}
\]
where \(\text{det}(Y) = Y_{11}Y_{22} - Y_{12}Y_{21}\).
3. **Identify \(ABCD\) parameters:**
The \(ABCD\) parameters can be directly derived from the \(Z\) parameters:
\[
\begin{bmatrix}
V_1 \\
V_2
\end{bmatrix}
=
\begin{bmatrix}
A & B \\
C & D
\end{bmatrix}
\begin{bmatrix}
I_1 \\
I_2
\end{bmatrix}
\]
where:
\[
A = Z_{11} \text{ (or } \frac{Z_{11}}{Z_{21}} \text{ depending on the formulation)}
\]
\[
B = Z_{12}
\]
\[
C = \frac{1}{Z_{21}} \text{ (or } Z_{21} \text{ if in series)}
\]
\[
D = \frac{1}{Z_{22}}
\]
### Example
Suppose you have the following \(Y\) parameters:
\[
\begin{bmatrix}
Y_{11} & Y_{12} \\
Y_{21} & Y_{22}
\end{bmatrix}
=
\begin{bmatrix}
0.2 & 0.1 \\
0.1 & 0.3
\end{bmatrix}
\]
1. **Calculate \(Z\) parameters:**
\[
\text{det}(Y) = (0.2 \times 0.3) - (0.1 \times 0.1) = 0.06 - 0.01 = 0.05
\]
\[
\text{Inverse}(Y) = \frac{1}{0.05} \begin{bmatrix}
0.3 & -0.1 \\
-0.1 & 0.2
\end{bmatrix}
=
\begin{bmatrix}
6 & -2 \\
-2 & 4
\end{bmatrix}
\]
2. **Determine \(ABCD\) parameters:**
\[
A = Z_{11} = 6
\]
\[
B = Z_{12} = -2
\]
\[
C = -2
\]
\[
D = 4
\]
So, the \(ABCD\) parameters are:
\[
\begin{bmatrix}
6 & -2 \\
-2 & 4
\end{bmatrix}
\]
### Summary
To convert \(Y\) parameters to \(ABCD\) parameters, you first need to compute the \(Z\) parameters by inverting the \(Y\) matrix, and then use these \(Z\) parameters to derive the \(ABCD\) parameters. The specific formulas might vary slightly depending on the exact context and conventions used, but this approach outlines the general process.
### Given
- The \(Y\) parameters are defined as:
\[
\begin{bmatrix}
I_1 \\
I_2
\end{bmatrix}
=
\begin{bmatrix}
Y_{11} & Y_{12} \\
Y_{21} & Y_{22}
\end{bmatrix}
\begin{bmatrix}
V_1 \\
V_2
\end{bmatrix}
\]
### \(Y\) to \(ABCD\) Parameters Conversion
1. **Express the \(V\) and \(I\) relationship in terms of \(Y\) parameters:**
\[
\begin{bmatrix}
V_1 \\
V_2
\end{bmatrix}
=
\begin{bmatrix}
Z_{11} & Z_{12} \\
Z_{21} & Z_{22}
\end{bmatrix}
\begin{bmatrix}
I_1 \\
I_2
\end{bmatrix}
\]
Where \(Z\) parameters are the inverse of the \(Y\) parameters matrix.
2. **Calculate the inverse of the \(Y\) parameters matrix to get the \(Z\) parameters:**
\[
\begin{bmatrix}
Z_{11} & Z_{12} \\
Z_{21} & Z_{22}
\end{bmatrix}
=
\begin{bmatrix}
Y_{11} & Y_{12} \\
Y_{21} & Y_{22}
\end{bmatrix}^{-1}
\]
To find the inverse:
\[
\text{Inverse} = \frac{1}{\text{det}(Y)} \begin{bmatrix}
Y_{22} & -Y_{12} \\
-Y_{21} & Y_{11}
\end{bmatrix}
\]
where \(\text{det}(Y) = Y_{11}Y_{22} - Y_{12}Y_{21}\).
3. **Identify \(ABCD\) parameters:**
The \(ABCD\) parameters can be directly derived from the \(Z\) parameters:
\[
\begin{bmatrix}
V_1 \\
V_2
\end{bmatrix}
=
\begin{bmatrix}
A & B \\
C & D
\end{bmatrix}
\begin{bmatrix}
I_1 \\
I_2
\end{bmatrix}
\]
where:
\[
A = Z_{11} \text{ (or } \frac{Z_{11}}{Z_{21}} \text{ depending on the formulation)}
\]
\[
B = Z_{12}
\]
\[
C = \frac{1}{Z_{21}} \text{ (or } Z_{21} \text{ if in series)}
\]
\[
D = \frac{1}{Z_{22}}
\]
### Example
Suppose you have the following \(Y\) parameters:
\[
\begin{bmatrix}
Y_{11} & Y_{12} \\
Y_{21} & Y_{22}
\end{bmatrix}
=
\begin{bmatrix}
0.2 & 0.1 \\
0.1 & 0.3
\end{bmatrix}
\]
1. **Calculate \(Z\) parameters:**
\[
\text{det}(Y) = (0.2 \times 0.3) - (0.1 \times 0.1) = 0.06 - 0.01 = 0.05
\]
\[
\text{Inverse}(Y) = \frac{1}{0.05} \begin{bmatrix}
0.3 & -0.1 \\
-0.1 & 0.2
\end{bmatrix}
=
\begin{bmatrix}
6 & -2 \\
-2 & 4
\end{bmatrix}
\]
2. **Determine \(ABCD\) parameters:**
\[
A = Z_{11} = 6
\]
\[
B = Z_{12} = -2
\]
\[
C = -2
\]
\[
D = 4
\]
So, the \(ABCD\) parameters are:
\[
\begin{bmatrix}
6 & -2 \\
-2 & 4
\end{bmatrix}
\]
### Summary
To convert \(Y\) parameters to \(ABCD\) parameters, you first need to compute the \(Z\) parameters by inverting the \(Y\) matrix, and then use these \(Z\) parameters to derive the \(ABCD\) parameters. The specific formulas might vary slightly depending on the exact context and conventions used, but this approach outlines the general process.