RLC Resonance — Series, Parallel & Harmonics

At resonance, inductive and capacitive reactances cancel and the circuit's behavior flips: a series RLC becomes a short (current spikes, L and C voltages magnify), while a parallel RLC becomes an open (impedance peaks, branch currents circulate). Sweep the components, find the resonant frequency, and see why a capacitor bank that resonates on a harmonic is a power-system hazard.

Topology

Components

Resistance R

Source

Source voltage

Scope drive frequency

Harmonic source

Fundamental

Circuit

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Impedance vs. Frequency

|Z| magnitude Resonance f₀ Half-power band

Voltage & Current Magnification

Waveform Scope

Readout

Resonant freq f₀

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Quality factor Q

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Bandwidth

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Z at resonance

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The Physics

Both topologies resonate when the reactances are equal and opposite, ωL = 1/ωC:

f₀ = 1 / (2π·√(LC)) · Q_series = (1/R)·√(L/C) · Q_parallel = R·√(C/L) · BW = f₀ / Q

Series RLC (voltage source). At f₀ the reactances cancel, leaving only R, so impedance is minimum and current is maximum. The catch: the L and C voltages don't cancel internally — each rises to Q × source voltage. High Q means dangerous voltage magnification across the capacitor.

Parallel RLC (current source). At f₀ the branch susceptances cancel, so impedance is maximum (= R). A small source current produces a large node voltage, and the L and C branches carry a circulating current of Q × source current — even though the line current is small. Note Q rises with R here (light damping = sharp, dangerous peak), the opposite of the series case.

⚡ Why this matters: harmonic resonance. A shunt power-factor capacitor bank C sits in parallel with the system/transformer inductance L. Seen from a harmonic-injecting load (a VFD, a rectifier), that's a parallel resonant tank. If f₀ falls near a characteristic harmonic — 5th (300/360 Hz), 7th (420 Hz), 11th, 13th — the impedance peak amplifies that harmonic current into large harmonic voltages and capacitor overcurrents. Turn on the harmonic source and tune C until f₀ lands on a harmonic line — watch the response stem explode. The fix in practice: detune the bank with a series reactor (a tuned filter) to move f₀ below the lowest harmonic.