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GATE 2024 EE – Question 46

Power Systems · Symmetrical components, Symmetrical and unsymmetrical fault analysis · 2 marks · Multiple select

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 $\begin{bmatrix}V_p\\V_q\end{bmatrix}=\mathbf S\begin{bmatrix}V_\alpha\\V_\beta\end{bmatrix}$. The possible option(s) for matrix $\mathbf S$ is/are

  1. $\begin{bmatrix}1&1\\1&-1\end{bmatrix}$
  2. $\begin{bmatrix}1&1\\1&1\end{bmatrix}$
  3. $\begin{bmatrix}1&1\\1&0\end{bmatrix}$
  4. $\begin{bmatrix}-1&1\\1&1\end{bmatrix}$

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Correct answer: (A) $\begin{bmatrix}1&1\\1&-1\end{bmatrix}$; (D) $\begin{bmatrix}-1&1\\1&1\end{bmatrix}$

Explanation

A sequence transformation must be invertible, and (as for the symmetrical-component transformation) its columns should be orthogonal so that the sequence components are independent. $\begin{bmatrix}1&1\\1&1\end{bmatrix}$ is singular. $\begin{bmatrix}1&1\\1&0\end{bmatrix}$ is invertible but its columns are not orthogonal. $\begin{bmatrix}1&1\\1&-1\end{bmatrix}$ and $\begin{bmatrix}-1&1\\1&1\end{bmatrix}$ have orthogonal columns, so they are valid.