# GATE Papers >> EEE >> 2017 >> Question No 10

Question No. 10

$A\;3$-bus power system is shown in the figure below, where the diagonal elements of $Y$-bus matrix are: $Y_{11}=-j12\;pu,\;Y_{22}=-j15\;pu$ and $Y_{33}=-j7\;pu.$

The per unit values of the lines reactances $p,q$ and $r$ shown in the figure are

##### Answer : (B) $p=0.2,q=0.1,r=0.5$

Solution of Question No 10 of GATE 2017 EEE Paper

$\begin{array}{l}{\mathrm Y}_{11}=-\frac{\mathrm j}{\mathrm q}-\frac{\mathrm j}{\mathrm r}=-\mathrm j12\Rightarrow\frac1{\mathrm q}+\frac1{\mathrm r}=12.....(\mathrm i)\\{\mathrm Y}_{22}=-\frac{\mathrm j}{\mathrm q}-\frac{\mathrm j}{\mathrm p}=-\mathrm j15\Rightarrow\frac1{\mathrm q}+\frac1{\mathrm p}=15.....(\mathrm{ii})\\{\mathrm Y}_{33}=-\frac{\mathrm j}{\mathrm p}-\frac{\mathrm j}{\mathrm r}=-\mathrm j7\Rightarrow\frac1{\mathrm p}+\frac1{\mathrm r}=7\;.....(\mathrm{iii})\end{array}$

2 $\left(\frac1{\mathrm p}+\frac1{\mathrm q}+\frac1{\mathrm r}\right)$ = 34

$\frac1{\mathrm p}+\frac1{\mathrm q}+\frac1{\mathrm r}$ =17  ......(iv)

Subtract one by one (i), (ii) & (iii) from (iv)

$\frac1{\mathrm p}=5\;\;\frac1{\mathrm q}=10\;\;\frac1{\mathrm r}=2$

Hence p = 0.2, q = 0.1, r = 0.5