Question

Power factor is unity for the following condition:
A. Average power is not equal to apparent power.
B. When the circuit is purely resistive.
C. for resonant condition
D. Both ‘b’ and ‘c’.

Answer 1

Hint: In alternating current circuits, power factor is defined as the ratio of actual power dissipated by an AC circuit to the product of r.m.s values of current and voltage. This difference between the two is caused by the reactance in the circuit and represents power that does no useful work.

Formula used:
 $power\ factor = cos\phi = \dfrac{R}{Z}$

Complete answer:
Resonating frequency – It is a special type of frequency at which, the inductive reactance and capacitive reactance of the circuit becomes numerically equal. At this frequency, the graph of both reactance cut each other and maximum current is flown in the circuit.
Also, power factor is the ratio of impedance of the circuit to the resistance of the circuit i.e. $power\ factor = cos\phi = \dfrac{R}{Z}$.
Now, power factor unity means $cos\phi = 1$ or$R = Z$. This means that the total impedance of the ac circuit equals the resistance. Hence we can say that the circuit is purely resistive and no effect of reactance in the circuit.
In other words, we can say that the net reactance of that circuit is zero. Thus $\chi = 0$.
Also, $\chi = \chi_L - \chi_C = \omega L - \dfrac{1}{\omega C} = 0$
Or we can write $\omega = \dfrac{1}{\sqrt{LC}}$, which is the condition of resonance.

So, the correct answer is “Option D”.

Note:
What type of frequency is this? This is a special type of angular frequency which is called ‘Resonating frequency’. At this frequency, the current in the circuit achieves its maximum value in the circuit. Mathematically it is equal to $\omega = \dfrac{1}{\sqrt{LC}}$. This is a very important frequency for a circuit. To obtain the maximum output from the radio, we adjust the frequency of the radio waves. Only at a certain frequency, we’re able to hear the clear voice. The frequency corresponding to that region is called resonance frequency.