--- title: "Parallel RLC resonance and resonance frequency" subject: "Electrical Circuit Theory" chapter: 1 key: "semester_3/electrical_circuit_theory:parallel-rlc-resonance-frequency" source: https://www.bctengineer.com/q/semester_3/electrical_circuit_theory/parallel-rlc-resonance-frequency site: BCT Engineering --- # Parallel RLC resonance and resonance frequency Electrical Circuit Theory · Chapter 1: Network Analysis of AC circuit & Dependent Sources - Asked in: 2071 Chaitra, 8 marks; 2068 Baisakh, 6 marks (asked 2 times) - Video: [Resonance Circuits - Frequency Behaviour, RLC Series/Parallel Resonance Circuit, Mechanical Analogy](https://www.youtube.com/watch?v=YfpMKsmD-4Y) — Institute of Electronics - TU Graz - 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-rlc-resonance-frequency - Whole chapter: https://www.bctengineer.com/md/semester_3/electrical_circuit_theory/chapter-1 ## Question Explain the phenomenon of resonance in a parallel RLC circuit and derive the expression for the resonance frequency. ## Short answer Parallel RLC resonance happens when inductive and capacitive susceptances cancel, so admittance is minimum (1/R) and the line current is in phase with voltage. The resonant frequency is f0 = 1/(2π√(LC)), independent of R; impedance is maximum at resonance. ## Answer A parallel RLC circuit is **in resonance** when the supply voltage and the total line current are in phase. This happens when the inductive susceptance equals the capacitive susceptance, so the circuit behaves as a pure resistance. ```text +-----+-----+-----+ I -->| | | | R L C | V | | | | +-----+-----+-----+ ``` #### Explanation of the phenomenon Total admittance of the parallel circuit (R, L and C all across the same voltage V): ```text Y = 1/R + 1/(jωL) + jωC = 1/R + j(ωC − 1/ωL) ``` - At low frequency the inductor takes a large lagging current, so the circuit is inductive. - At high frequency the capacitor takes a large leading current, so the circuit is capacitive. - At one frequency the two reactive currents are equal and opposite (I_L = I_C). They cancel, and the source supplies only the resistor current I_R = V/R. At resonance Y = 1/R is minimum, so the impedance is maximum (Z = R), the line current is minimum, and the power factor is unity. A large current circulates between L and C (I_L = I_C = Q × I), so this is also called a current-magnification circuit. #### Derivation of resonance frequency Resonance needs the imaginary part of Y to be zero: ```text ω0C − 1/(ω0L) = 0 ω0² = 1/(LC) ω0 = 1/√(LC) rad/s f0 = 1/(2π√(LC)) Hz ``` The resonant frequency is independent of R. #### Other results ```text Q0 = R/(ω0L) = ω0CR = R√(C/L) BW = ω0/Q0 = 1/(RC) rad/s ``` Note: for a practical coil (resistance r in series with L) in parallel with C, the resonant frequency becomes ω0 = √(1/LC − r²/L²). **Answer: f0 = 1/(2π√(LC)), the impedance is maximum (= R) and the line current is minimum at f0.** ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. When is a parallel RLC circuit in resonance? A. When supply voltage and line current are in phase (susceptances equal) B. When the line current is maximum C. When R equals the inductive reactance D. When the inductor current is zero Answer: A. Resonance means the inductive and capacitive susceptances cancel so the circuit behaves as pure resistance. (Key point: Meaning of resonance) 2. At parallel resonance, what happens to the impedance and line current? A. Impedance is minimum and line current is maximum B. Impedance is maximum (= R) and line current is minimum C. Both impedance and line current are maximum D. Impedance is zero and line current is infinite Answer: B. I_L and I_C cancel, so the source supplies only V/R; Y is minimum, so Z is maximum. (Key point: Behaviour at resonance) 3. What is the resonant frequency of an ideal parallel RLC circuit? A. 1/√(RC) B. √(1/LC − R^2/L^2) C. 1/√(LC), independent of R D. R/(2πL) Answer: C. Setting Im(Y) = ωC − 1/ωL = 0 gives ω0 = 1/√(LC); R does not appear. (Key point: Resonant frequency) 4. Below resonance, why is a parallel RLC circuit inductive? A. The capacitor takes the larger current B. The resistor current dominates C. The inductor current leads the voltage D. The inductor takes a larger lagging current than the capacitor leading current Answer: D. At low ω, 1/ωL is large and ωC is small, so the lagging inductor current dominates. 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