Turn the dial of an old analog radio and, somewhere along its travel, one station snaps into focus while everything around it fades. Nothing mechanical moved inside the antenna. What changed is which frequency a small loop of inductor and capacitor was willing to accept. That selectivity — the ability of a circuit to favour one frequency and reject its neighbours — is the practical face of RLC resonance.
This article explains where the resonance condition comes from, works through a full numerical example, and points out the mistakes that most often distort a resonance calculation.
Why this calculation matters
Resonance is not a curiosity reserved for radios. A series RLC combination sits at the heart of RF front ends, antenna matching networks, induction heaters, switching-converter snubbers, and the EMC filters that keep noisy electronics legal. In every one of those, a designer wants the circuit to respond strongly at one frequency and quietly elsewhere.
The reason the calculation deserves care is that two circuits can share the same resonant frequency and still behave nothing alike. One might pass a wide band of frequencies; the other a razor-thin slice. That difference is set by resistance, and it is captured by the quality factor. Get the resonant frequency right but ignore Q, and you can build a filter that technically tunes the correct station yet also lets the two adjacent ones through.
The core formula
A series RLC circuit contains three reactive characters. The resistor dissipates energy. The inductor stores it in a magnetic field and opposes changes in current. The capacitor stores it in an electric field and opposes changes in voltage. Their reactances pull in opposite directions:
X_L = 2 * pi * f * L (grows with frequency)
X_C = 1 / (2 * pi * f * C) (falls with frequency)
Resonance is the frequency where these two exactly cancel. Setting X_L equal to X_C and solving for f gives the resonant frequency:
f_0 = 1 / (2 * pi * sqrt(L * C))
At f_0 the inductive and capacitive reactances annihilate each other, the circuit's impedance collapses to just the resistance R, and the current through a series circuit reaches its maximum. The circuit looks purely resistive — voltage and current are back in phase.
How sharply the circuit peaks is the quality factor:
Q = (1 / R) * sqrt(L / C)
A high Q means a narrow, tall response; a low Q means a broad, gentle one. The width of the response follows directly:
BW = f_0 / Q
BW is the bandwidth between the two half-power (-3 dB) points. The chain of logic is worth holding onto: L and C set where the peak sits, R sets how sharp it is, and the sharpness sets how wide the usable band becomes.
A worked example
Take a series RLC circuit with inductance L = 10 mH, capacitance C = 100 nF, and resistance R = 50 ohm.
Step 1 — resonant frequency.
f_0 = 1 / (2 * pi * sqrt(L * C))
f_0 = 1 / (2 * pi * sqrt(1e-2 * 1e-7))
f_0 = 5.03 kHz
The product L*C is 1e-9, its square root is about 3.16e-5, and dividing into 1/(2*pi) lands at roughly 5030 Hz.
Step 2 — quality factor.
Q = (1 / R) * sqrt(L / C)
Q = (1 / 50) * sqrt(1e-2 / 1e-7)
Q = 0.02 * 316 = 6.3
The ratio L/C is 1e5, whose square root is about 316. Multiplying by 1/R = 0.02 gives Q = 6.3 — a moderately selective circuit.
Step 3 — bandwidth.
BW = f_0 / Q = 5030 / 6.3 = 796 Hz
So this circuit resonates near 5.03 kHz and passes a band roughly 796 Hz wide between its half-power points. At resonance the inductive and capacitive reactances cancel and the circuit looks purely resistive — the impedance seen by the source is simply the 50 ohm resistor. Lower that resistance and Q rises, the peak narrows, and the bandwidth shrinks; raise it and the response broadens.
Common mistakes
Treating f_0 as the whole answer. The resonant frequency tells you where the circuit responds, not how cleanly. Two designs with identical f_0 can have wildly different selectivity. Always compute Q alongside it.
Confusing series and parallel behaviour. At resonance a series RLC circuit shows minimum impedance and maximum current; an ideal parallel RLC circuit shows the opposite — maximum impedance and minimum line current. Reaching for the series result when the topology is parallel inverts the conclusion.
Forgetting hidden resistance. Real inductors have winding resistance, and capacitors have a small loss term. These add to any resistor you placed deliberately, so the measured Q is often lower than the textbook value computed from R alone.
Mixing up angular and ordinary frequency. The formula f_0 = 1/(2*pi*sqrt(LC)) gives hertz. The companion expression omega_0 = 1/sqrt(LC) gives radians per second. Dropping or adding a factor of 2*pi is one of the most common slips in resonance arithmetic.
Assuming high Q is always the goal. A sharp peak rejects interference well but also rings, settles slowly, and is sensitive to component tolerance. A power-factor-correction network and a narrowband receiver want very different Q values.
Try the interactive NovaSolver calculator
Resonance is far easier to feel than to read about. The RLC Circuit Resonance Simulator on NovaSolver lets you switch between series and parallel topologies, set R, L, C, and the source voltage, and immediately see the resonant frequency, Q factor, bandwidth, and impedance at resonance update. It plots impedance and current against frequency on log scales and shows the phasor diagram, so you can watch the reactances cancel as you cross f_0 — and watch the peak sharpen the moment you reduce resistance.
Related calculators
- AC impedance of an RLC circuit — to see how total impedance and phase angle vary with frequency away from resonance.
- LCR resonance calculator — a focused tool for the resonant frequency and Q of an LCR combination.
- Resonant frequency calculator — for a quick f_0 from L and C when that single number is all you need.
The full set is collected in the electromagnetics tools hub.
Closing note
RLC resonance reduces to three linked numbers: a resonant frequency set by L and C, a quality factor set by resistance, and a bandwidth that follows from both. Once you see them as a chain rather than three separate formulas, tuned-circuit design stops feeling like guesswork. Decide how selective the circuit needs to be, pick Q to match, then choose L, C, and R to deliver it. Resonance rewards engineers who respect the role of resistance as much as the role of reactance.
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