Question

The rms readings of voltmeters $V_{1}, V_{2}, V_{3}$ are $150 V , 130$ $V$ and $120 V$, respectively. If the power factor of the circuit is $\frac{a}{\sqrt{b}}$, then find $(a b)$.

Answer 1

$V_{L}^{2}+R_{R}^{2}=150^{2}$
$\Rightarrow V_{L}^{2}=150^{2}-120^{2}$
$\Rightarrow V_{L}=90 V$
Also, $V_{C}^{2}+V_{R}^{2}=130^{2}$
$\Rightarrow V_{C}=50 V$
$\therefore V=\sqrt{V_{R}^{2}+\left(V_{L}-V_{C}\right)^{2}}$
$=\sqrt{120^{2}+40^{2}}=40 \sqrt{10} V$
Power factor, $\cos \phi=\frac{R}{Z}=\frac{V_{R}}{V}$
$=\frac{120}{40 \sqrt{10}}=\frac{3}{\sqrt{10}}$