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What is the G parameter of two-port network?

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Best answer
The \( G \) parameter, also known as the **transmission conductance** parameter, is one of the parameters used to describe a two-port network in the context of electrical engineering and circuit analysis. It is part of the **Y-parameters** (admittance parameters) which are used to characterize linear electrical networks.

In a two-port network, the \( G \) parameter specifically represents the relationship between the output current and the input voltage while keeping the output port voltage constant. The \( G \) parameter is defined as:

\[
G = \frac{I_1}{V_2} \bigg|_{V_1 = 0}
\]

where:
- \( I_1 \) is the current flowing into port 1,
- \( V_2 \) is the voltage at port 2,
- \( V_1 \) is the voltage at port 1.

### Key Points:
- **Units**: The unit of \( G \) is siemens (S), which is the same as ohms\(^{-1}\).
- **Significance**: A high \( G \) value indicates that the network allows a large current to flow with a small voltage, meaning it has low impedance at the output.
- **Use**: The \( G \) parameter is particularly useful in analyzing and designing circuits, especially in high-frequency applications.

In summary, the \( G \) parameter is crucial for understanding how the two-port network behaves under various electrical conditions, particularly in terms of conductance and current flow.
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The \(G\) parameter, or **admittance parameter**, is one of the ways to represent a two-port network. It describes the relationship between the voltages and currents at the two ports of the network using admittance rather than impedance. In other words, it characterizes how the network behaves in terms of admittances, which are the reciprocals of impedances.

For a two-port network, the \(G\) parameters (or admittance parameters) are defined as follows:

### Definitions

Given a two-port network with:
- \( V_1 \) and \( V_2 \) as the voltages at port 1 and port 2, respectively.
- \( I_1 \) and \( I_2 \) as the currents entering port 1 and port 2, respectively.

The relationship between these quantities in terms of the \(G\) parameters is given by:

\[ \begin{bmatrix}
I_1 \\
I_2
\end{bmatrix}
=
\begin{bmatrix}
G_{11} & G_{12} \\
G_{21} & G_{22}
\end{bmatrix}
\begin{bmatrix}
V_1 \\
V_2
\end{bmatrix} \]

Where:
- \( G_{11} \) is the admittance from port 1 to port 1 when port 2 is open-circuited.
- \( G_{12} \) is the admittance from port 2 to port 1 when port 2 is open-circuited.
- \( G_{21} \) is the admittance from port 1 to port 2 when port 1 is open-circuited.
- \( G_{22} \) is the admittance from port 2 to port 2 when port 1 is open-circuited.

### Interpretation

1. **\( G_{11} \)**: Represents the self-admittance of port 1. It tells you how much current flows into port 1 for a given voltage \(V_1\) when port 2 is open (i.e., no current flows into port 2).

2. **\( G_{12} \)**: Represents the mutual admittance from port 2 to port 1. It tells you how much current flows into port 1 due to a voltage \(V_2\) applied at port 2 while port 1 is open.

3. **\( G_{21} \)**: Represents the mutual admittance from port 1 to port 2. It tells you how much current flows into port 2 due to a voltage \(V_1\) applied at port 1 while port 2 is open.

4. **\( G_{22} \)**: Represents the self-admittance of port 2. It tells you how much current flows into port 2 for a given voltage \(V_2\) when port 1 is open.

### Applications

The \(G\) parameters are particularly useful in high-frequency applications and in analyzing networks where admittances are more convenient than impedances. They are often used in the analysis and design of RF and microwave circuits, as well as in situations where network analysis involves linear and reciprocal networks.

### Example

Consider a simple example of a two-port network with the following \(G\) parameters:

\[ \begin{bmatrix}
G_{11} & G_{12} \\
G_{21} & G_{22}
\end{bmatrix}
=
\begin{bmatrix}
0.5 & 0.2 \\
0.3 & 0.4
\end{bmatrix} \]

If you apply \(V_1 = 1 \text{V}\) and \(V_2 = 0 \text{V}\), then:

\[ \begin{bmatrix}
I_1 \\
I_2
\end{bmatrix}
=
\begin{bmatrix}
0.5 & 0.2 \\
0.3 & 0.4
\end{bmatrix}
\begin{bmatrix}
1 \\
0
\end{bmatrix}
=
\begin{bmatrix}
0.5 \\
0.3
\end{bmatrix} \]

So, \(I_1 = 0.5 \text{A}\) and \(I_2 = 0.3 \text{A}\) when \(V_1 = 1 \text{V}\) and \(V_2 = 0 \text{V}\).

This illustrates how the \(G\) parameters provide a straightforward way to relate voltages and currents in a network using admittance matrices.
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