Question

A sinusoidal voltage of peak value 283V and frequency 50Hz is applied to a series LCR circuit in which $R=3\Omega ,L=25.48mH$ and $C=786\mu F$.
Find :
a) Impedance of the circuit
b) The phase difference between the voltage across the source and the current
c) The power factor.

Answer 1

Angular frequency of the ac signal $w=2\pi \nu$

$\therefore$ $w=2\pi \left(50\right)=100\pi$

Capacitive reactance $X_c=\frac{1}{wC}$

$\therefore$ $X_c=\frac{1}{100\pi \times 786\times {10}^{-6}}=4\Omega$

Inductive reactance $X_L=wL$

$\therefore$ $X_L=100\pi \times (25.48\times {10}^{-3})=8\Omega$

(a) Impedance of the circuit $Z=\sqrt{R^2+(X_L-X_c{)}^2}$

$\therefore$ $Z=\sqrt{3^2+(8-4{)}^2}=5\Omega$

(b) Phase difference $\varphi ={\tan }^{-1}(\frac{X_L-X_c}{R})$

Or $\varphi ={\tan }^{-1}(\frac{8-4}{3})={\tan }^{-1}\left(\frac{4}{3}\right)$

$⟹$ $\varphi ={53.13}^o$

(c) Power factor $\cos \varphi =\cos {53.13}^o=0.6$