Series and Parallel Resonance
A passive circuit developing 500 V from a 10 V source, with no amplifier in it — and KVL never violated.
Skip to the animationAt ω₀ = 1/√(LC) the inductive and capacitive reactances cancel exactly, leaving a purely resistive circuit — which in series means minimum impedance and voltages Q times the source, and in parallel means maximum impedance and a circulating current Q times the line current.
Why cancellation happens
Inductive reactance grows with frequency and capacitive reactance shrinks, and they carry opposite signs. At ω₀ = 1/√(LC) they are equal and opposite. Note what is absent from that expression: resistance does not set the resonant frequency.
Series resonance
Cancelling reactances leave only R, so impedance dips to its minimum and the circuit is purely resistive — voltage and current in phase. Minimum impedance means maximum current, making it an *acceptor* circuit.
That current flows through both reactances, and each develops Q times the source voltage across it. At Q = 50, a 10 V source produces 500 V across the inductor and 500 V across the capacitor. KVL holds — they are in antiphase and cancel — but each component individually sees the full magnified voltage, which is what destroys capacitors.
Q and bandwidth
Q is the ratio of energy stored to energy lost per cycle — ω₀L/R for a series circuit. It sets how sharp the resonance is, and bandwidth = ω₀/Q, so selectivity and bandwidth are one number expressed two ways.
| Q | Bandwidth | Character |
|---|---|---|
| High (low R) | Narrow | Selective, slow to settle, large magnification |
| Low (high R) | Broad | Tolerant, quick, little magnification |
Parallel resonance
The exact dual. Impedance is maximum at ω₀, so line current is minimum — a *rejector* circuit that blocks one frequency and passes the rest. And the dual of voltage magnification appears: the current circulating inside the LC tank is Q times the current drawn from the supply.
Where it is wanted
A tuned circuit selects one radio station and attenuates its neighbours; turning the dial changes C to move ω₀. Selectivity is Q — but too high a Q narrows the passband enough to cut the sidebands carrying the audio, so a tuner is a deliberate compromise between selectivity and fidelity.
Where it is not
- Cable capacitance resonates with transformer inductance somewhere, and if a harmonic lands there the voltage magnifies.
- Power-factor correction capacitors resonate with supply inductance — which is why they are fitted with detuning reactors.
- Any accidental LC in a layout has an ω₀ and a Q whether anyone designed them or not.
The same mathematics governs bridges, rotors and wings, with mass for L, compliance for C and damping for R. Resonance is not an electrical phenomenon — it is what any second-order system does.
The numbers you will be asked for
- Resonant frequency
ω₀ = 1/√(LC) · f₀ = 1/(2π√(LC))
- Series Q
Q = ω₀L / R = (1/R)·√(L/C)
- Parallel Q
Q = R / ω₀L = R·√(C/L)
- Bandwidth
BW = ω₀ / Q
- Voltage magnification
V_L = V_C = Q · V_source
- Impedance at resonance
series: Z = R · parallel: Z = L/(RC)
Watch it work
Check yourself
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