Question

At resonance, power factor in a parallel RLC circuit is

• A. Unity
• B. Zero
• C. Infinite
• D. 0.5

Hint:

Regardless of whether an RLC circuit is arranged in a series or parallel configuration, the baseline condition for resonance always means that all inductive and capacitive reactances cancel each other out. Consequently, the input impedance becomes purely real, and the power factor is always unity (1) at resonance!

Answer 1

The Correct Option is A

Concept: A parallel RLC circuit consists of a resistor ($R$), an inductor ($L$), and a capacitor ($C$) connected in parallel across an AC voltage source. In parallel circuit analysis, it is mathematically more convenient to work with Admittance ($Y$), which is defined as the reciprocal of total impedance ($Y = \frac{1}{Z}$). The total complex admittance of a parallel RLC network is the sum of its individual branch admittances: \[ Y = G + jB = \frac{1}{R} + j\left(\omega C - \frac{1}{\omega L}\right) \] Where:
• $G = \frac{1}{R}$ is the conductance.
• $B = B_C - B_L = \omega C - \frac{1}{\omega L}$ is the net susceptance. The condition for electrical resonance occurs when the net reactive susceptance component drops to zero, making the circuit appear entirely resistive to the source: \[ B = 0 \quad \Rightarrow \quad \omega_0 C = \frac{1}{\omega_0 L} \quad \Rightarrow \quad \omega_0 = \frac{1}{\sqrt{LC}} \] Step-by-Step Power Factor Calculation at Resonance:

Step 1: Determine the phase angle of the admittance. The phase angle $\theta$ between the total line current and the node voltage is determined by the ratio of the imaginary part to the real part of the admittance: \[ \theta = \tan^{-1}\left(\frac{B}{G}\right) = \tan^{-1}\left( \frac{\omega C - \frac{1}{\omega L}}{\frac{1}{R}} \right) \]

Step 2: Evaluate the phase angle at the resonant frequency $\omega_0$. Substituting the resonant condition ($\omega_0 C = \frac{1}{\omega_0 L}$) into the phase equation: \[ \theta = \tan^{-1}\left( \frac{0}{\frac{1}{R}} \right) = \tan^{-1}(0) = 0^\circ \] A phase angle of $\theta = 0^\circ$ means that the total line current and the applied system voltage are perfectly in phase with each other.

Step 3: Calculate the Power Factor (PF). The power factor is defined as the cosine of the phase angle ($\theta$) between voltage and current: \[ \text{PF} = \cos(\theta) = \cos(0^\circ) = 1 \] A power factor of exactly 1 is referred to as Unity Power Factor. Thus, the power factor at resonance in a parallel RLC circuit is unity, corresponding to Option (1).