What does the RC time constant τ mean?

τ = RC, in seconds (1 kΩ × 1 μF = 1 ms). While charging, v_C = V(1 − e^(−t/τ)): after 1τ the capacitor reaches 63.2% of its final voltage, 86.5% after 2τ, 95% after 3τ and 99.3% after 5τ, which is usually taken as fully charged. Discharging, 36.8% remains after 1τ. The time to reach a fraction x is t = −τ·ln(1 − x), so reaching half takes 0.693τ.

When does an RLC circuit ring?

It depends on the damping ratio ζ = (R/2)√(C/L). With ζ 1 (overdamped) creeps up without oscillating. The critical resistance is R = 2√(L/C). When underdamped, the overshoot is e^(−πζ/√(1−ζ²)) and the ringing frequency is ω_d = √(ω₀² − α²), with α = R/(2L) and ω₀ = 1/√(LC).

Why can the inductor and capacitor voltages exceed the source at resonance?

At f₀ = 1/(2π√(LC)) the inductive and capacitive reactances are equal and opposite and cancel, leaving only R, so the current peaks. Each of V_L and V_C is then Q times the source voltage, with Q = (1/R)√(L/C), but they are 180° apart and cancel around the loop. A higher Q gives a sharper peak and a narrower bandwidth BW = f₀/Q, which is how a radio’s tuned circuit picks out one station.

How do I calculate an RC low-pass filter’s cutoff frequency?

f_c = 1/(2πRC). At the cutoff the output is 1/√2 of the input (−3 dB) and lags by 45°; well above f_c the gain falls at −20 dB per decade and the phase approaches −90°. With R = 1 kΩ and C = 100 nF, f_c ≈ 1.59 kHz. Swap R and C and you have a high-pass filter with the same cutoff; for RL filters f_c = R/(2πL).

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🔌RC, RL & RLC Circuit Simulator - Time Constant, Resonance, Bode Plot

The first- and second-order circuits from every circuits course, as simulations you can play with. Transient mode covers RC charging and discharging, RL current rise and decay, and the series RLC step response: voltage and current are plotted against time with τ to 5τ marked, you can hover or press play to read any instant, and dots on the schematic flow with the current. For RLC the tool classifies the response as under-, critically or overdamped and reports overshoot, peak time, rise time and settling time. AC mode covers series RLC resonance and RC and RL low-pass and high-pass filters: a Bode plot of impedance or gain and phase, the resonant frequency, Q and bandwidth or the cutoff frequency, and a rotating phasor diagram showing how each component’s voltage lines up in phase. Every input accepts engineering notation such as 10k, 4.7u, 100n.

Free · No sign-up · Runs in your browserLast updated: 2026-10-26

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How to use RC / RL / RLC Circuit Simulator

1Choose transient or AC mode, then the circuit type (or click a preset)
2Enter R, L, C and the source voltage — 10k, 4.7u, 100n and 2m all work
3In transient mode press play or hover over the curves; in AC mode drag the frequency slider or the Bode plot
4Read the time constant, damping type, resonant frequency, Q, cutoff frequency and the formulas used

FAQ

What does the RC time constant τ mean?: τ = RC, in seconds (1 kΩ × 1 μF = 1 ms). While charging, v_C = V(1 − e^(−t/τ)): after 1τ the capacitor reaches 63.2% of its final voltage, 86.5% after 2τ, 95% after 3τ and 99.3% after 5τ, which is usually taken as fully charged. Discharging, 36.8% remains after 1τ. The time to reach a fraction x is t = −τ·ln(1 − x), so reaching half takes 0.693τ.
When does an RLC circuit ring?: It depends on the damping ratio ζ = (R/2)√(C/L). With ζ < 1 (underdamped) the capacitor voltage overshoots and oscillates as it decays; ζ = 1 (critically damped) settles fastest with no overshoot; ζ > 1 (overdamped) creeps up without oscillating. The critical resistance is R = 2√(L/C). When underdamped, the overshoot is e^(−πζ/√(1−ζ²)) and the ringing frequency is ω_d = √(ω₀² − α²), with α = R/(2L) and ω₀ = 1/√(LC).
Why can the inductor and capacitor voltages exceed the source at resonance?: At f₀ = 1/(2π√(LC)) the inductive and capacitive reactances are equal and opposite and cancel, leaving only R, so the current peaks. Each of V_L and V_C is then Q times the source voltage, with Q = (1/R)√(L/C), but they are 180° apart and cancel around the loop. A higher Q gives a sharper peak and a narrower bandwidth BW = f₀/Q, which is how a radio’s tuned circuit picks out one station.
How do I calculate an RC low-pass filter’s cutoff frequency?: f_c = 1/(2πRC). At the cutoff the output is 1/√2 of the input (−3 dB) and lags by 45°; well above f_c the gain falls at −20 dB per decade and the phase approaches −90°. With R = 1 kΩ and C = 100 nF, f_c ≈ 1.59 kHz. Swap R and C and you have a high-pass filter with the same cutoff; for RL filters f_c = R/(2πL).