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Resonance: Series and Parallel: Complete Beginner's Guide

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Resonance occurs in an RLC circuit when the inductive reactance and capacitive reactance cancel each other exactly, so the circuit behaves as if it contains only resistance at that frequency. This special frequency is called the resonant frequency. At resonance, energy sloshes back and forth between the inductor's magnetic field and the capacitor's electric field, and the circuit draws energy from the source with unusual ease — or unusual difficulty, depending on the topology.

Resonance matters because it is the principle behind every tuner ever built. Radios select one station out of hundreds by resonating at that station's frequency and rejecting the rest. It also matters as a hazard: resonant voltage rise can destroy capacitors, and resonant current can overheat components, so power engineers actively avoid accidental resonance.

Key Concepts

  • Resonant frequency: f0 = 1 / (2 pi sqrt(LC)). At f0, XL = XC = 2 pi f0 L.
  • Series resonance: impedance is minimum (equal to R), current is maximum, and the voltages across L and C can each be Q times larger than the supply voltage.
  • Parallel resonance: impedance is maximum (the dynamic resistance L/(CR)), line current is minimum, while large circulating current flows between L and C.
  • Quality factor Q: measures sharpness of resonance; Q = (1/R) x sqrt(L/C) for series RLC. Higher Q means a sharper peak and larger voltage magnification.
  • Bandwidth: BW = f0 / Q, the frequency range where power is at least half the maximum (the -3 dB points).
  • Units: f0 in hertz, L in henries, C in farads, Q dimensionless, BW in hertz.

Formulas

  • f0 = 1 / (2 pi sqrt(LC)); omega0 = 1 / sqrt(LC) in rad/s.
  • Series Q = omega0 L / R = 1 / (omega0 C R) = (1/R) sqrt(L/C).
  • Bandwidth BW = R / (2 pi L) = f0 / Q (series RLC).
  • At series resonance: I0 = Vs / R; VL0 = VC0 = Q x Vs.
  • Parallel (high-Q) dynamic resistance: Rd = L / (C R). Assumes coil resistance R is small and Q is greater than about 10.

Worked Example

A series RLC circuit has R = 10 ohm, L = 100 mH and C = 100 microfarad, driven by a 10 V AC source. Find the resonant frequency, Q, bandwidth and inductor voltage at resonance.

  1. LC = 0.1 x 100 x 10^-6 = 1 x 10^-5. sqrt(LC) = 0.0031623 s.
  2. f0 = 1 / (2 pi x 0.0031623) = 1 / 0.019869 = 50.33 Hz.
  3. Q = (1/10) x sqrt(0.1 / 100 x 10^-6) = 0.1 x sqrt(1000) = 0.1 x 31.623 = 3.162.
  4. Bandwidth = f0 / Q = 50.33 / 3.162 = 15.92 Hz. Check: R/(2 pi L) = 10 / 0.6283 = 15.92 Hz. Matches.
  5. At resonance I0 = 10 / 10 = 1 A. XL0 = 2 pi x 50.33 x 0.1 = 31.62 ohm, so VL0 = 1 x 31.62 = 31.62 V — more than three times the 10 V supply, exactly Q x Vs = 3.162 x 10 = 31.62 V.

The circuit resonates near 50 Hz with a 16 Hz bandwidth, and the inductor and capacitor each see about 31.6 V even though the source is only 10 V.

Common Misconceptions

  • "Resonance creates extra energy." No energy is created. The inductor and capacitor simply trade the same stored energy back and forth; the source only tops up resistive losses.
  • "Current is maximum at resonance, always." Only for series resonance. In parallel resonance the current drawn from the source is minimum at f0.
  • "Higher Q is always better." Higher Q gives sharper tuning but also narrower bandwidth and dangerously high component voltages — a trade-off, not a free win.

Real-World Applications

  • Radio and TV tuners: a variable capacitor tunes an LC circuit to the desired station's frequency.
  • Wireless charging and RFID: transmitter and receiver coils are both resonated for efficient energy transfer.
  • Induction heating and Tesla coils: resonant circuits generate huge voltages and currents from modest sources.
  • Crystal oscillators: quartz resonators with extremely high Q set precise clock frequencies.
  • Power-factor correction: capacitor banks must be detuned to avoid resonating with transformer inductance at harmonic frequencies.

At resonance, XL cancels XC and the circuit is purely resistive at f0 = 1/(2 pi sqrt(LC)). Series circuits show minimum impedance and magnified component voltages; parallel circuits show maximum impedance and minimum line current. The Q factor sets both the sharpness and the voltage magnification — tune it deliberately, because resonance amplifies whatever it touches.

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