--- title: "Parallel resonance with voltage and current waveforms" subject: "Electrical Circuit Theory" chapter: 1 key: "semester_3/electrical_circuit_theory:parallel-resonance-waveforms" source: https://www.bctengineer.com/q/semester_3/electrical_circuit_theory/parallel-resonance-waveforms site: BCT Engineering --- # Parallel resonance with voltage and current waveforms Electrical Circuit Theory · Chapter 1: Network Analysis of AC circuit & Dependent Sources - Asked in: 2074 Chaitra, 8 marks - Video: [S02_PR01 - Series and Parallel Resonance](https://www.youtube.com/watch?v=VtKlmxsL-ds) — Dept. of Electrical Engineering | UoM - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_3/electrical_circuit_theory/parallel-resonance-waveforms - Whole chapter: https://www.bctengineer.com/md/semester_3/electrical_circuit_theory/chapter-1 ## Question Explain the phenomenon of resonance in an RLC parallel circuit. Also derive the expression for resonance frequency and draw the wave form of instantaneous voltage and current at resonance. ## Short answer At parallel resonance frequency f0 = 1/(2π√(LC)), the inductor and capacitor currents are equal and opposite, cancelling. The supply voltage and total line current are in phase, and the circuit appears purely resistive with minimum line current. ## Answer **Resonance** in a parallel RLC circuit occurs at the frequency where the inductor current and capacitor current cancel each other, so the supply voltage and supply current are in phase and the circuit acts as a pure resistor R. #### Phenomenon For R, L, C connected across a source v = Vm sin ωt: ```text I_R = V/R (in phase with v) I_L = V/(ωL) (lags v by 90°) I_C = V·ωC (leads v by 90°) Y = 1/R + j(ωC − 1/ωL) ``` I_L and I_C are 180° out of phase. When ωC = 1/(ωL), they are equal and cancel, so the source supplies only I_R. The admittance is a minimum (1/R), the impedance is a maximum (R), and the line current is minimum. #### Resonance frequency ```text Im(Y) = 0 → ω0C = 1/(ω0L) ω0 = 1/√(LC) rad/s f0 = 1/(2π√(LC)) ``` Q-factor: Q0 = R/(ω0L) = ω0CR, bandwidth BW = ω0/Q0 = 1/(RC). #### Waveforms at resonance ```text v, i ^ v and i (total) coincide in phase | __ __ | / \ / \ --+--/----\----/----\---> ωt | \__/ \__/ | i_L: lags v by 90° (peak at ωt = π) i_C: leads v by 90°, i_C = −i_L at every instant ``` | ωt | v | i_R (= i) | i_L | i_C | i_L + i_C | |---|---|---|---|---|---| | 0 | 0 | 0 | −Im_L | +Im_C | 0 | | π/2 | Vm | Vm/R | 0 | 0 | 0 | | π | 0 | 0 | +Im_L | −Im_C | 0 | | 3π/2 | −Vm | −Vm/R | 0 | 0 | 0 | Here Im_L = Vm/(ω0L) and Im_C = ω0C·Vm are equal at resonance, so the two reactive currents have equal peaks and mirror each other. The total current is therefore a sine wave in phase with v, with peak Vm/R only. **Answer: f0 = 1/(2π√(LC)); at f0, v and the line current i are in phase, and i_L and i_C are equal and opposite.** ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. How are i_L and i_C related at parallel resonance? A. Equal in magnitude and 180° out of phase, so they cancel B. Equal and in phase, so they add C. In quadrature with each other D. Equal to the resistor current Answer: A. i_L lags v by 90° and i_C leads by 90°, giving i_C = −i_L at every instant. (Key point: Reactive currents) 2. What does the line current waveform look like at resonance? A. A sine wave lagging v by 90° B. A sine wave in phase with v, with peak Vm/R C. A sine wave leading v by 90° with peak Vm/(ω0L) D. Zero at all times Answer: B. Only the resistor current remains, which is in phase with v and has peak Vm/R. (Key point: Line current) 3. Which condition gives the parallel resonance frequency? A. Re(Y) = 0 B. ω0L = R C. Im(Y) = 0, i.e. ω0C = 1/(ω0L) D. ω0C = R Answer: C. Resonance requires the admittance to be real, so the imaginary part ωC − 1/ωL must be zero. 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