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

> Simulate RC charging and discharging, RL current and RLC step response with τ marks, plus resonance, Q, bandwidth, filter cutoff, Bode plots and live phasors.

- URL: https://qilingtools.com/en/math/rc-circuit
- Category: Physics Tools (https://qilingtools.com/en/physics)
- Tags: [Physics](https://qilingtools.com/en/tags/physics)
- Last updated: 2026-10-26
- Price: Free, no sign-up
- Privacy: Runs locally in the browser; files are never uploaded

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.

## How to use

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

## Features

- Exact curves for RC, RL and RLC step and natural responses
- τ markers and time to reach any percentage
- Under/critical/overdamped classification with overshoot and settling time
- Series RLC impedance, phase, resonance, Q and bandwidth
- RC/RL low-pass and high-pass Bode plots with cutoff
- Animated phasors synced to waveforms; engineering-notation inputs

## 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).

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