Question

The power factor is given as $\frac{x}{1000}$ in lag. Find $x$

Answer 1

$Z_1=\frac{115}{3}=38.33\Omega$
$\cos {\varphi }_1=\frac{R_1}{Z_1}$
$\Rightarrow R_1=Z_1\cos {\varphi }_1$
$=\left(38.33\right)\left(0.6\right)=23\Omega$
$X_L=\sqrt{Z_1^2-R_1^2}=30.67\Omega$
$Z_2=\frac{115}{5}=23\Omega$
$R_2=Z_2\cos {\varphi }_2=\left(23\right)\left(0.707\right)$
$=16.26\Omega$
When connected in series
$R=R_1+R_2=39.26\Omega$
$X_L-X_C=14.41\Omega$
$\therefore Z=\sqrt{R^2+(X_L-X_C{)}^2}$
$=41.82\Omega$
Power factor=$\cos \varphi =\frac{R}{Z}=\frac{39.26}{41.92}=0.939$
Since $X_L>X_C$, this power factor is lagging.