# GATE Papers >> EEE >> 2007 >> Question No 31

Question No. 31

Suppose we define a sequence transformation between ‘‘a-b-c’’ and ‘‘p-n-o’’ variables as follows:

$\left[\begin{array}{c}{f}_{a}\\ {f}_{b}\\ {f}_{c}\end{array}\right]=k\left[\begin{array}{ccc}1& 1& 1\\ {\alpha }^{2}& \alpha & 1\\ \alpha & {\alpha }^{2}& 1\end{array}\right]\left[\begin{array}{c}{f}_{p}\\ {f}_{n}\\ {f}_{o}\end{array}\right]$ where $\alpha ={e}^{j\frac{2\pi }{3}}$ and k is constant.

Now, if it is given that: $\left[\begin{array}{c}{V}_{p}\\ {V}_{n}\\ {V}_{o}\end{array}\right]=\left[\begin{array}{ccc}0.5& 0& 0\\ 0& 0.5& 0\\ 0& 0& 2.0\end{array}\right]\left[\begin{array}{c}{i}_{p}\\ {i}_{n}\\ {i}_{o}\end{array}\right]$ and $\left[\begin{array}{c}{V}_{a}\\ {V}_{b}\\ {V}_{c}\end{array}\right]=Z\left[\begin{array}{c}{i}_{a}\\ {i}_{b}\\ {i}_{c}\end{array}\right]$ then,