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Difficulty: MediumAlternating Current (AC) Circuits

An alternating current (AC) circuit consists of a 40 Ω40\ \Omega resistor, an inductor with a reactance of 70 Ω70\ \Omega, and a capacitor with a reactance of 40 Ω40\ \Omega connected in series across an AC supply. What is the power factor of the circuit?

  1. 0.80Answer
  2. B
    0.60
  3. C
    0.27
  4. D
    1.00

Answer

0.80
The total impedance of the series RLC circuit is found using phasor addition: Z=402+(70−40)2=50 ΩZ = \sqrt{40^2 + (70 - 40)^2} = 50\ \Omega. The power factor is the ratio of resistance to impedance: cos⁡θ=4050=0.80\cos \theta = \frac{40}{50} = 0.80.

Step-by-Step Solution

1
Calculate the net reactance (XnetX_{\text{net}}) of the circuit
Xnet=XL−XC=70 Ω−40 Ω=30 ΩX_{\text{net}} = X_L - X_C = 70\ \Omega - 40\ \Omega = 30\ \Omega
In a series RLC circuit, inductive and capacitive reactances oppose each other in phase.
2
Calculate total impedance (ZZ) using phasor addition
Z=R2+(XL−XC)2=402+302=1600+900=50 ΩZ = \sqrt{R^2 + (X_L - X_C)^2} = \sqrt{40^2 + 30^2} = \sqrt{1600 + 900} = 50\ \Omega
Resistance and net reactance act at right angles in the impedance phasor diagram.
3
Calculate the power factor (cos⁡θ\cos \theta)
cos⁡θ=RZ=4050=0.80\cos \theta = \frac{R}{Z} = \frac{40}{50} = 0.80
The power factor is defined as the cosine of the phase angle, which equals the ratio of resistance to total impedance.

Key Concept

Power factor of a series RLC AC circuit
Estimated Time:1m 30s
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Topics

Alternating Current (AC) Circuits

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