Question

For a two-phase network, the phase voltages \( V_p \) and \( V_q \) are to be expressed in terms of sequence voltages \( V_\alpha \) and \( V_\beta \) as: \[ \begin{bmatrix} V_p
V_q \end{bmatrix} = S \begin{bmatrix} V_\alpha
V_\beta \end{bmatrix}. \] The possible option(s) for matrix \( S \) is/are:

• A. \(\begin{bmatrix} 1 & 1
  1 & 1 \end{bmatrix}\)
• B. \(\begin{bmatrix} 1 & 1
  1 & 0 \end{bmatrix}\)
• C. \(\begin{bmatrix} -1 & 1
  1 & 1 \end{bmatrix}\)
• D.

Hint:

When solving matrix transformation problems in electrical networks: Check the structure of the transformation matrix to ensure it aligns with the physical and mathematical requirements of the network. For two-phase networks, symmetry and the ability to differentiate between sequence components (\( V_\alpha \) and \( V_\beta \)) are key. Invalid matrices often fail due to redundancy or lack of proper mapping (e.g., identical rows or zero elements).

Answer 1

The Correct Option is A

Step 1: General formulation of the two-phase network. The relationship between the phase voltages \( V_p, V_q \) and the sequence voltages \( V_\alpha, V_\beta \) is given by: \[ \begin{bmatrix} V_p
V_q \end{bmatrix} = S \begin{bmatrix} V_\alpha
V_\beta \end{bmatrix}. \] Here, \( S \) is a transformation matrix that determines how \( V_\alpha \) and \( V_\beta \) are mapped to \( V_p \) and \( V_q \). Step 2: Testing given options. To determine the valid forms of \( S \), we need to ensure that \( S \) preserves the linear relationship between \( V_p, V_q \) and \( V_\alpha, V_\beta \) while adhering to the structure of a two-phase network. 1. **Option (A):** \[ S = \begin{bmatrix} 1 & 1
1 & -1 \end{bmatrix}. \] This option is valid because it satisfies the linear transformation requirements for the two-phase network. 2. **Option (B):** \[ S = \begin{bmatrix} 1 & 1
1 & 1 \end{bmatrix}. \] This option is not valid, as the second row \( [1 \, 1] \) does not differentiate between \( V_\alpha \) and \( V_\beta \), which is required for the phase transformation. 3. **Option (C):** \[ S = \begin{bmatrix} 1 & 1
1 & 0 \end{bmatrix}. \] This option is invalid because it does not maintain symmetry or proper mapping between \( V_p, V_q \) and \( V_\alpha, V_\beta \). 4. **Option (D):** \[ S = \begin{bmatrix} -1 & 1
1 & 1 \end{bmatrix}. \] This option is valid because it also satisfies the linear transformation requirements for the two-phase network. Step 3: Conclusion. The possible matrices for \( S \) are: \[ (A) \, \begin{bmatrix} 1 & 1
1 & -1 \end{bmatrix}, \quad (D) \, \begin{bmatrix} -1 & 1
1 & 1 \end{bmatrix}. \] Thus, the correct options are (A) and (D).