What is the power factor of a series LCR circuit
The Correct Answer and Explanation is:
The power factor (PF) of a series LCR circuit can be defined as the cosine of the phase angle (θ) between the total voltage and the total current. It is a measure of how effectively the circuit converts the supplied electrical power into useful work. Mathematically, the power factor is given by: Power Factor (PF)=cos(θ)\text{Power Factor (PF)} = \cos(\theta)Power Factor (PF)=cos(θ)
Where:
- θ\thetaθ is the phase angle between the voltage and current in the circuit.
Explanation:
In a series LCR circuit, the total impedance ZZZ is the sum of the inductive reactance XLX_LXL, capacitive reactance XCX_CXC, and the resistance RRR: Z=R+j(XL−XC)Z = R + j(X_L – X_C)Z=R+j(XL−XC)
The phase angle θ\thetaθ is the angle between the total voltage and current, and it is determined by the impedance components. The phase angle is calculated as: θ=tan−1(XL−XCR)\theta = \tan^{-1} \left( \frac{X_L – X_C}{R} \right)θ=tan−1(RXL−XC)
When the circuit is resonant, meaning the inductive reactance equals the capacitive reactance (XL=XCX_L = X_CXL=XC), the impedance of the circuit is purely resistive, and the phase angle θ\thetaθ is zero. At resonance, the power factor becomes: PF=cos(0)=1\text{PF} = \cos(0) = 1PF=cos(0)=1
This indicates that the current and voltage are in phase, and all the supplied power is converted into useful work (real power).
Outside resonance, when XL≠XCX_L \neq X_CXL=XC, the phase angle becomes non-zero, and the power factor is less than 1. The power factor will be a value between 0 and 1, reflecting a mixture of real and reactive power in the circuit.
In summary, at resonance, the power factor is 1, meaning the circuit operates at maximum efficiency, while at other frequencies, it will be less than 1.
