1 Answer
The condition of symmetry for the Z-parameters (impedance parameters) of a two-port network means that the network behaves the same when the ports are interchanged. Mathematically, for the two-port network to be symmetric, the Z-parameters must satisfy the following condition:
\[
Z_{12} = Z_{21}
\]
This condition means that the mutual impedance between port 1 and port 2 is the same in both directions. In other words, the impedance seen by port 1 due to the current injected at port 2 should be equal to the impedance seen by port 2 due to the current injected at port 1.
For a two-port network with Z-parameters represented as:
\[
\begin{bmatrix}
V_1 \\
V_2
\end{bmatrix}
=
\begin{bmatrix}
Z_{11} & Z_{12} \\
Z_{21} & Z_{22}
\end{bmatrix}
\begin{bmatrix}
I_1 \\
I_2
\end{bmatrix}
\]
The symmetry condition means that \( Z_{12} = Z_{21} \), so the Z-parameter matrix becomes:
\[
\begin{bmatrix}
Z_{11} & Z_{12} \\
Z_{12} & Z_{22}
\end{bmatrix}
\]
When this condition holds, the network is said to be symmetric, meaning it behaves the same regardless of which port is considered the input or output.
\[
Z_{12} = Z_{21}
\]
This condition means that the mutual impedance between port 1 and port 2 is the same in both directions. In other words, the impedance seen by port 1 due to the current injected at port 2 should be equal to the impedance seen by port 2 due to the current injected at port 1.
For a two-port network with Z-parameters represented as:
\[
\begin{bmatrix}
V_1 \\
V_2
\end{bmatrix}
=
\begin{bmatrix}
Z_{11} & Z_{12} \\
Z_{21} & Z_{22}
\end{bmatrix}
\begin{bmatrix}
I_1 \\
I_2
\end{bmatrix}
\]
The symmetry condition means that \( Z_{12} = Z_{21} \), so the Z-parameter matrix becomes:
\[
\begin{bmatrix}
Z_{11} & Z_{12} \\
Z_{12} & Z_{22}
\end{bmatrix}
\]
When this condition holds, the network is said to be symmetric, meaning it behaves the same regardless of which port is considered the input or output.