The formula this page uses
f₀ = 1 / (2π√(LC)) ζ = R / (2√(L/C)) Q = (1/R)·√(L/C) bandwidth = f₀ / Q
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RLC Circuit Calculator
Solve rlc circuit problems with clear steps, notation, and a final check.
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Applied math
RLC Circuit Calculator explained
The short version
- An RLC circuit has a resistor, an inductor and a capacitor, and it can ring like a struck bell.
- The resonant frequency f₀ = 1/(2π√(LC)) depends only on L and C. The resistor decides how fast the ringing dies.
- Compare R against 2√(L/C): below it the circuit rings, above it the circuit just sags back.
What each part means
SymbolWhat it means
R — ResistanceIn ohms. It is the only part that removes energy, so it controls the damping.
L — InductanceIn henries. 100 mH is 0.1 H.
C — CapacitanceIn farads. 1 µF is 0.000001 F.
ζ — Damping ratioA plain number. Below 1 the circuit oscillates, at 1 it is critically damped, above 1 it is overdamped.
Show your work: a full example
- The circuita series RLC with R = 100 Ω, L = 0.1 H and C = 1 µF
- Product of L and C0.1 × 0.000001 = 0.0000001, and √0.0000001 = 0.0003162
- Resonant frequencyf₀ = 1 ÷ (2π × 0.0003162) = 503.3 Hz
- Critical resistance2√(L/C) = 2√(0.1 ÷ 0.000001) = 2√100000 = 2 × 316.23 = 632.46 Ω
- CompareR = 100 Ω is well below 632.46 Ω, so the circuit is underdamped and will ring
- Quality factorQ = 316.23 ÷ 100 = 3.162, and the damping ratio is ζ = 100 ÷ 632.46 = 0.158
- Bandwidth of the resonance peak503.3 ÷ 3.162 = 159.2 Hz wide
A second, different case
- A different case: keep L and C, raise the resistanceR = 2000 Ω, L = 0.1 H, C = 1 µF
- f₀ is unchangedstill 503.3 Hz, because R never appears in that formula
- Critical resistance is unchanged toostill 632.46 Ω
- Compare2000 Ω is above 632.46 Ω, so this circuit is overdamped
- Damping ratioζ = 2000 ÷ 632.46 = 3.162, more than three times critical
- Quality factorQ = 316.23 ÷ 2000 = 0.158, so there is no ringing at all
- Set R exactly to 632.46 Ω insteadζ = 1 exactly, which is critical damping: the fastest return with no overshoot
Copy-ready example
a series RLC with R = 100 Ω, L = 0.1 H and C = 1 µF
The circuit
Questions about the RLC Circuit Calculator
What does resonance actually mean here?
At f₀ the inductor's reactance and the capacitor's reactance are equal and opposite, so they cancel. A series circuit is then left with only R, which is why its current peaks sharply at 503.3 Hz.
What is Q in plain words?
It is how sharp and how long-lived the ringing is. Q = 3.162 means the resonance peak is 159 Hz wide out of 503 Hz, and the circuit rings for roughly three cycles before fading.
Why is critical damping useful if it does not ring?
Because it is the fastest possible return to rest without ever overshooting. Car suspensions and instrument needles are tuned near ζ = 1 for exactly that reason: you want the bump gone quickly, not bouncing.
How does a parallel RLC differ from a series one?
The resonant frequency formula is identical, but the roles of R invert. In series, a small R gives a high Q; in parallel, a large R gives a high Q, because the resistor is now a leak across the tank rather than a brake in the loop.