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Circuit Analysis: AC, Linear Circuits and Transients

In the series RLC circuit in the figure, R = 10 Ω, L = 10 mH and C = 1 µF, and the frequency of a 1 V RMS sinusoidal voltage is set to the resonant frequency. Which of the following is the RMS voltage VC across the capacitor?

Series RLC circuit driven by a sinusoidal AC source V I R L C VR VL VC Z = R + j(ωL − 1/(ωC))
Series RLC circuit driven by a sinusoidal AC source · View in diagrams
  1. A0.1 V
  2. B1 V
  3. C10 V
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Answer: C. 10 V

Method 1 (step-by-step from the resonance condition): the resonant angular frequency is ω0 = 1/√(LC) = 1/√(10×10⁻³ × 1×10⁻⁶) = 1×10⁴ rad/s (about 1.6 kHz). At resonance the reactances of L and C cancel, leaving only R, so I = 1 V/10 Ω = 0.1 A. The capacitor reactance is 1/(ω0C) = 100 Ω, so VC = 0.1 A × 100 Ω = 10 V. Method 2 (using the Q factor): a series resonant circuit has Q = (1/R)√(L/C) = (1/10) × √(10⁴) = 10, and at resonance the voltages across L and C are Q times the source voltage. VC = Q × 1 V = 10 V, which is larger than the source voltage, so the voltage ratings of the parts need attention.

  • A. This is V/Q = 0.1 V, dividing where you should multiply by Q.
  • B. This is the voltage across R at resonance (equal to the 1 V source voltage). The voltages across L and C cancel each other, but neither one is 0.
  • C. Correct. VC = Q × V = 10 × 1 V = 10 V.
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