Question

A current of $4Amp$ flows in a coil when connected to a $12Volt$ dc source. If the same coil is connected to a $12Volt$ $50Hz$ ac source a current of $2.4Amp$ flows in the circuit, Determine the inductance of the coil. Also, find the power developed in the circuit if a $2500\mu F$ capacitor is connected in series with the coil?

Answer 1

$I=4AV=12V.dc$
$I=2.4AV=12V,50Hz$
$h=?$
$p=?$
Resistance of coil, $R=\frac{V_{dc}}{Idc}=\frac{12}{4}=3$
Ac.
$I_{rms}=\frac{V_{rms}}{z}$
$z=\frac{V_{rms}}{I_{rms}}$
squaring both sides
$z^2={\left(\frac{V_{rms}}{I_{rms}}\right)}^2$
$R^2+(\omega L{)}^2={\left(\frac{V_{rms}}{I_{rms}}\right)}^2$

$\frac{1}{\omega }\sqrt{{\left(\frac{V_{rms}}{I_{rms}}\right)}^2-(R{)}^2}$

$\frac{1}{2\pi 50}\sqrt{{\left(\frac{12}{2.4}\right)}^2-(3{)}^2}$
$=0.0127H$

$\left(ii\right)$ $C=2500\mu F$
$z=\sqrt{R^2+(X_L-X_C{)}^2}$
$X_L=2\pi f.h=4\Omega$
$X_C=\frac{1}{C{O}_C}=8\Omega$
$Z=\sqrt{(3{)}^2+(4{)}^2}=5\Omega$
$P=V_{rms}{I}_{rms}\cos \varphi$
$I_{rms}=\frac{V_{rms}}{z}$
$=\frac{12}{5}Amp$
$\cos \varphi =\frac{R}{z}=\frac{3}{5}$
$P=12\times \frac{12}{5}\times \frac{3}{5}$
$=17.28W$
$L\frac{1}{10}\sqrt{{\left(\frac{V_{rms}}{I_{rms}}\right)}^2-R^2}$
if $w=50rad/s$ instead of $f=50$
$L=0.08H$