Question

By what percentage the impedance in an AC circuit should be increased without changing the resistance so that the power factor changes from $\dfrac{1}{2}$ to $\dfrac{1}{4}$?
A. 200%
B. 100%
C. 50%
D. 400%

Answer 1

Hint: In AC circuits, power factor is defined as the ratio of actual power dissipation to the apparent power dissipation. Power factor of a circuit lies between 0 and 1.
It is also defined as the ratio of resistance to impedance in a circuit.

Formula used:
Power factor of an AC circuit, $\cos \phi =\dfrac{R}{Z}$

Complete step-by-step answer:
In an AC circuit, both electromotive force and electric current changes with respect to time. Due to this reason, we cannot calculate power for the circuit directly as a product of voltage and current. Mathematically, power factor
$\cos \phi =\dfrac{R}{Z}$
Where R is the resistance and Z is the impedance of the AC circuit.
Initial power factor,
$\cos {{\phi }_{1}}=\dfrac{R}{{{Z}_{1}}}=\dfrac{1}{2}$
$\Rightarrow {{Z}_{1}}=2R$
Similarly final power factor,
$\cos {{\phi }_{2}}=\dfrac{R}{{{Z}_{2}}}=\dfrac{1}{4}$
$\Rightarrow {{Z}_{2}}=4R$
Change in impedance is given by
${{Z}_{2}}-{{Z}_{1}}=4R-2R=2R$
Percentage change in reactance is,
$\dfrac{{{Z}_{2}}-{{Z}_{1}}}{{{Z}_{1}}}\times 100%=\dfrac{2R}{2R}\times 100%$
$\Rightarrow \dfrac{{{Z}_{2}}-{{Z}_{1}}}{{{Z}_{1}}}=100%$
Therefore, option B is correct.

So, the correct answer is “Option B”.

Additional Information: In an AC circuit the power dissipation is calculated as
${{P}_{avg}}=\dfrac{\int\limits_{0}^{T}{VIdt}}{\int\limits_{0}^{T}{dt}}$
Where, V and I are the instantaneous e.m.f. and current in the AC circuit. T is the time period of the AC circuit.
Instantaneous values of V and I can be given as
$V={{V}_{0}}sin(\omega t)$
$I={{I}_{0}}\sin (\omega t-\phi )$
Where, ${{V}_{0}}$ and ${{I}_{0}}$are peak values of e.m.f and current and $\omega$ is the frequency of AC circuit.
Instantaneous power can be given as
${{P}_{inst}}=VI={{V}_{0}}{{I}_{0}}\cos \phi $
${{V}_{0}}{{I}_{0}}$ is known as apparent power or virtual power.

Note: For a purely inductive circuit or purely capacitive circuit, $\phi ={{90}^{{}^\circ }}$. This implies that the power factor for a purely inductive or purely capacitive circuit is zero i.e. power of a purely inductive circuit is zero.
The value of power factor of an AC circuit always lies between 0 and 1.