Determine the Z parameter of the given network.

Explanation:

Determination of Z Parameters

Definition: Z parameters (impedance parameters) are a set of parameters used in electrical network analysis to describe the relationship between voltages and currents at the ports of a network. In a two-port network, the Z parameters are defined as:

\[ V_1 = Z_{11}I_1 + Z_{12}I_2 \]

\[ V_2 = Z_{21}I_1 + Z_{22}I_2 \]

where:

• \(V_1\) and \(V_2\): Voltages at port 1 and port 2 respectively
• \(I_1\) and \(I_2\): Currents flowing into port 1 and port 2 respectively
• \(Z_{11}, Z_{12}, Z_{21}, Z_{22}\): Impedance parameters

Correct Option Analysis:

The correct option is:

Option 1: \(\rm Z=\left[\begin{array}{cc}32 & 8 \\ 8 & 16\end{array}\right]\)

This option provides the correct Z parameter matrix for the given network. To determine the Z parameters, we analyze the network by considering open-circuit conditions at each port (i.e., \(I_2 = 0\) for port 1 and \(I_1 = 0\) for port 2). The Z parameters are calculated as follows:

1. Calculation of \(Z_{11}\):

\(Z_{11}\) represents the impedance looking into port 1 when port 2 is open (\(I_2 = 0\)). Under this condition, the voltage \(V_1\) is proportional to the current \(I_1\), and the impedance is given by:

\[ Z_{11} = \frac{V_1}{I_1} \quad \text{(with \(I_2 = 0\))} \]

From the network analysis, \(Z_{11} = 32 \, \Omega\).

2. Calculation of \(Z_{12}\):

\(Z_{12}\) represents the transfer impedance between port 2 and port 1. It is defined as the ratio of voltage \(V_1\) at port 1 to the current \(I_2\) at port 2, with port 1 open (\(I_1 = 0\)):

\[ Z_{12} = \frac{V_1}{I_2} \quad \text{(with \(I_1 = 0\))} \]

From the network analysis, \(Z_{12} = 8 \, \Omega\).

3. Calculation of \(Z_{21}\):

\(Z_{21}\) represents the transfer impedance between port 1 and port 2. It is defined as the ratio of voltage \(V_2\) at port 2 to the current \(I_1\) at port 1, with port 2 open (\(I_2 = 0\)):

\[ Z_{21} = \frac{V_2}{I_1} \quad \text{(with \(I_2 = 0\))} \]

From the network analysis, \(Z_{21} = 8 \, \Omega\).

4. Calculation of \(Z_{22}\):

\(Z_{22}\) represents the impedance looking into port 2 when port 1 is open (\(I_1 = 0\)). Under this condition, the voltage \(V_2\) is proportional to the current \(I_2\), and the impedance is given by:

\[ Z_{22} = \frac{V_2}{I_2} \quad \text{(with \(I_1 = 0\))} \]

From the network analysis, \(Z_{22} = 16 \, \Omega\).

Thus, the Z parameter matrix for the given network is:

\[ \rm Z = \left[\begin{array}{cc}32 & 8 \\ 8 & 16\end{array}\right] \]

Important Information

To further understand the analysis, let’s evaluate the other options:

Option 2: \(\rm Z=\left[\begin{array}{cc}8 & 32 \\ 32 & 16\end{array}\right]\)

This matrix is incorrect as it does not match the calculated Z parameters. The values in the matrix are swapped, which does not represent the correct impedance characteristics of the given network.

Option 3: \(\mathrm{Z}=\left[\begin{array}{ll}32 & 8 \\ 16 & 8\end{array}\right]\)

This matrix is incorrect as the values of \(Z_{21}\) and \(Z_{22}\) are swapped, which is inconsistent with the calculated Z parameters. \(Z_{21}\) and \(Z_{12}\) should be equal, but this matrix does not satisfy that condition.

Option 4: \(\mathrm{Z}=\left[\begin{array}{cc}16 & 8 \\ 8 & 32\end{array}\right]\)

This matrix is incorrect as the values of \(Z_{11}\) and \(Z_{22}\) are swapped. The calculated Z parameters clearly indicate that \(Z_{11} = 32 \, \Omega\) and \(Z_{22} = 16 \, \Omega\), which is not reflected in this matrix.

Conclusion:

By carefully analyzing the Z parameter matrix, we conclude that the correct matrix is:

\[ \rm Z = \left[\begin{array}{cc}32 & 8 \\ 8 & 16\end{array}\right] \]

This matrix accurately represents the impedance parameters of the given network, as derived through standard network analysis techniques. Understanding the calculation and significance of Z parameters is essential for electrical network analysis and design.