What is the value of power factor at resonance in LCR circuit?
The Correct Answer and Explanation is:
The value of the power factor at resonance in an LCR circuit is 1.
Explanation:
In an LCR circuit (comprising an inductor LLL, capacitor CCC, and a resistor RRR), resonance occurs when the reactances of the inductor and capacitor cancel each other out. This happens at a specific frequency called the resonant frequency, denoted as:f0=12πLCf_0 = \frac{1}{2 \pi \sqrt{LC}}f0=2πLC1
At resonance, the inductive reactance XL=2πfLX_L = 2 \pi f LXL=2πfL and the capacitive reactance XC=12πfCX_C = \frac{1}{2 \pi f C}XC=2πfC1 are equal in magnitude but opposite in phase. As a result, their effects cancel each other, and the total reactance X=XL−XC=0X = X_L – X_C = 0X=XL−XC=0.
Power Factor:
The power factor in an AC circuit is defined as the cosine of the phase angle ϕ\phiϕ between the applied voltage and the current. It is given by:Power Factor=cos(ϕ)\text{Power Factor} = \cos(\phi)Power Factor=cos(ϕ)
At resonance, because the total reactance is zero, the circuit behaves purely resistively. The current and voltage are in phase, meaning the phase angle ϕ=0\phi = 0ϕ=0. Since:cos(0)=1\cos(0) = 1cos(0)=1
the power factor at resonance is 1. This means that all the power supplied by the source is being used effectively in the circuit, and there is no reactive power (no phase difference between current and voltage).
Conclusion:
At resonance, the LCR circuit is purely resistive, and the power factor reaches its maximum value of 1. This signifies that the circuit is operating with maximum efficiency, with all the energy being converted into useful power without any energy stored in the inductor or capacitor.
