RLC Resonance
See how inductive and capacitive effects cancel at one frequency and why the response can become sharply selective.
Like pushing a swing at its natural rhythm, energy transfers efficiently at resonance. Resistance determines how quickly that stored energy is lost.
Capacitors store energy in an electric field and inductors store energy in a magnetic field. In an RLC circuit, energy can move back and forth between these fields.
At the resonant frequency, inductive and capacitive reactances have equal magnitude and opposite sign. Their reactive effects cancel, leaving the resistance to dominate a series circuit.
Lower resistance means less energy is lost each cycle, producing a higher Q and narrower bandwidth. Real coils add winding resistance, so measured Q is always finite.
Resonance is used for tuning radios, selecting frequencies, impedance matching and creating oscillators.
Core ideas
- At resonance, XL and XC have equal magnitude.
- Series resonance gives minimum impedance; parallel resonance gives maximum impedance in the ideal model.
- Quality factor Q describes how narrow and selective the resonance is.
f0 = 1 / (2π√LC)Series Q = ω0L / RBandwidth: BW = f0 / QWorked example — resonant frequency and bandwidth
1Given L = 10 mH and C = 100 nF.
2f0 = 1/(2π√(0.01 × 100×10⁻⁹)) ≈ 5.03 kHz.
3If the series resistance is 10 Ω, Q = 2π × 5030 × 0.01 / 10 ≈ 31.6.
4Bandwidth ≈ 5030/31.6 ≈ 159 Hz.
A very high Q can create unexpectedly large internal voltages even when the applied source voltage is modest.