An a.c. voltage, V = Vm sin ωt, is applied across a
(i) series RC circuit in which the capacitative impedance is 'a' times the resistance in the circuit.
(ii) series RL circuit in which the inductive impedance is 'b' times the resistance in the circuit. Calculate the value of the power factor of the circuit in each case.

Solution

Given, A.C voltage, V = Vm sin ωt

Capacitive reactance is 'a' times the resistance.
$χ_{C} = a R$
Inductive impedence is 'b' times the resistance.
$χ_{L} = bR$

(i) Power factor in RC circuit is
$\cos ϕ = \frac{R}{Z} = \frac{R}{\sqrt{R^{2} + X_{C}^{2}}}$

Here $X_{C} = aR$

$∴$ $\cos ϕ = \frac{R}{\sqrt{R^{2} + a^{2} R^{2}}} = \frac{1}{\sqrt{1 + a^{2}}}$

(ii) Power factor in RL circuit is
$\cos ϕ = \frac{R}{Z} = \frac{R}{\sqrt{R^{2} + X_{L}^{2}}}$
Here, $X_{L} = bR$

$∴$ $\cos ϕ = \frac{R}{\sqrt{R^{2} + b^{2} R^{2}}} = \frac{1}{\sqrt{1 + b^{2}}} .$

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