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.