# RC time constant and RLC circuit calculator

> RC and RL time constant and the time to charge or discharge to any percent; series RLC resonant frequency, impedance, phase, Q factor and bandwidth.

Interactive version: https://www.calcopenly.com/science/rc-rl-rlc-circuit-calculator
Subject: Science calculators

An RC circuit responds with time constant τ = RC and an RL circuit with τ = L/R: after one τ the capacitor voltage or inductor current has covered 63.2% of the way to its final value, and reaching a fraction p takes t = −τ ln(1 − p). For a series RLC circuit the calculator finds the resonant frequency f₀ = 1/(2π√(LC)), the impedance and phase at your signal frequency from the reactances 2πfL and 1/(2πfC), and Q = (1/R)√(L/C).

The default, 10 kΩ with 100 µF on a 5 V supply, gives τ = 1 s and 2.30 s to reach 90%. RC time constants set timer delays, switch debounce periods and filter cut-off frequencies, 1/(2πRC). With 10 Ω, 10 mH and 1 µF in series, f₀ = 1,591.55 Hz and Q = 10.

Components are ideal and the input is a step: no capacitor leakage and no inductor winding resistance beyond R. After 5τ the response is within 0.67% of its final value (e⁻⁵), the usual rule for "fully charged".

## Inputs

- **Circuit** (options: RC, RL, Series RLC)
- **Resistance R**: Accepts 4.7k
- **Resistance unit** (options: mΩ, Ω, kΩ, MΩ)
- **Capacitance C**
- **Capacitance unit** (options: pF, nF, µF, mF, F)
- **Inductance L**
- **Inductance unit** (options: nH, µH, mH, H)
- **Process** (options: Charging / rising, Discharging / decaying)
- **Source voltage**
- **Time to reach this level**: Percent of the final value (charging) or of the starting value (discharging)
- **Show times in**: Auto picks s, ms, µs or ns from the size of τ (options: Auto, s, ms, µs, ns)
- **Value at time**
- **Signal frequency**

## Results

- Time constant τ (s) — main result
- Time to reach the level (s)
- Value at the chosen time
- Final value
- Energy stored when fully charged (J)
- Resonant frequency f₀ (Hz)
- Impedance |Z| at the signal frequency (Ω)
- Phase angle (voltage leads current) (°)
- Inductive reactance X_L (Ω)
- Capacitive reactance X_C (Ω)
- Quality factor Q
- Bandwidth (−3 dB) (Hz)
- Current amplitude at the signal frequency (A)
- Damping

## Formula

$$
\tau_{RC} = RC,\ \tau_{RL} = \frac{L}{R},\quad v(t) = V(1 - e^{-t/\tau});\qquad f_0 = \frac{1}{2\pi\sqrt{LC}},\ |Z| = \sqrt{R^2 + (X_L - X_C)^2},\ Q = \frac1R\sqrt{\frac{L}{C}}
$$

## Worked examples

### 10 kΩ and 100 µF charging to 90%

- Circuit: RC
- Resistance R: 10
- Resistance unit: kΩ
- Capacitance C: 100
- Capacitance unit: µF
- Process: Charging / rising
- Source voltage: 5 V
- Time to reach this level: 90%
- **Time constant τ: 1 s**
- **Time to reach the level: 2.30259 s**
- **Energy stored when fully charged: 0.00125 J**
- Checked against: Python 3.8 math: τ = 10⁴ × 10⁻⁴ = 1 s; t = τ ln 10; E = ½CV² = ½·10⁻⁴·25

### Capacitor voltage after one τ

- Circuit: RC
- Resistance R: 10
- Resistance unit: kΩ
- Capacitance C: 100
- Capacitance unit: µF
- Process: Charging / rising
- Source voltage: 5 V
- Time to reach this level: 90%
- Value at time: 1 s
- **Value at the chosen time: 3.1606 V**
- Checked against: Python 3.8 math: 5(1 − e⁻¹) = 3.1606028

### Discharging to half: t = τ ln 2

- Circuit: RC
- Resistance R: 10
- Resistance unit: kΩ
- Capacitance C: 100
- Capacitance unit: µF
- Process: Discharging / decaying
- Source voltage: 5 V
- Time to reach this level: 50%
- **Time to reach the level: 0.693147 s**
- Checked against: Python 3.8 math: ln 2 = 0.6931472 (τ = 1 s)

### RL: 100 Ω and 10 mH

- Circuit: RL
- Resistance R: 100
- Resistance unit: Ω
- Inductance L: 10
- Inductance unit: mH
- Process: Charging / rising
- Source voltage: 5 V
- Time to reach this level: 90%
- Show times in: s
- Value at time: 50 µs
- **Time constant τ: 0.0001 s**
- **Final value: 0.05 A**
- **Value at the chosen time: 0.019673 A**
- Checked against: Python 3.8 math: τ = L/R = 10⁻⁴ s, I = 5/100, i(50 µs) = 0.05(1 − e^−0.5) = 0.0196735

### Series RLC 10 Ω, 10 mH, 1 µF at 1 kHz

- Circuit: Series RLC
- Resistance R: 10
- Resistance unit: Ω
- Capacitance C: 1
- Capacitance unit: µF
- Inductance L: 10
- Inductance unit: mH
- Source voltage: 5 V
- Signal frequency: 1 kHz
- **Resonant frequency f₀: 1,591.55 Hz**
- **Quality factor Q: 10**
- **Impedance |Z| at the signal frequency: 96.8408 Ω**
- **Phase angle (voltage leads current): -84.07 °**
- **Bandwidth (−3 dB): 159.155 Hz**
- Checked against: Python 3.8 math: f₀ = 1/(2π√(10⁻⁸)), X_L = 62.83185, X_C = 159.15494, |Z| = 96.8407852, φ = atan(−96.3231/10)

### Critically damped RLC (R = 2√(L/C))

- Circuit: Series RLC
- Resistance R: 200
- Resistance unit: Ω
- Capacitance C: 1
- Capacitance unit: µF
- Inductance L: 10
- Inductance unit: mH
- Source voltage: 5 V
- Signal frequency: 1 kHz
- **Quality factor Q: 0.5**
- **Damping: Critically damped**
- Checked against: Q = ½ exactly when R = 2√(L/C) = 200 Ω (OpenStax UP2 §14.6)

## Questions

### What is the RC time constant?

τ = R × C, in seconds when R is in ohms and C in farads. It is the time a charging capacitor takes to reach 63.2% (1 − e⁻¹) of the supply voltage, or a discharging one to fall to 36.8%. 10 kΩ with 100 µF gives τ = 1 s; 1 kΩ with 1 µF gives 1 ms.

### How long does it take a capacitor to fully charge?

About five time constants. After 5τ the voltage is 99.33% of the supply (1 − e⁻⁵), which engineers treat as fully charged; in theory it never quite gets there. Reaching 90% takes τ ln 10 ≈ 2.303τ and 99% takes τ ln 100 ≈ 4.605τ, so a 10 kΩ, 100 µF circuit reaches 99% in 4.6 s.

### How do you calculate the resonant frequency of an RLC circuit?

f₀ = 1/(2π√(LC)), with L in henries and C in farads; in a series circuit R does not shift it. 10 mH with 1 µF resonates at 1,591.55 Hz, and 100 µH with 100 pF at 1.59 MHz. At f₀ the inductive and capacitive reactances are equal and cancel, so the impedance falls to R alone and the current peaks.

### What is the Q factor of a series RLC circuit?

Q = (1/R)√(L/C), the ratio of either reactance at resonance to the resistance. It sets the −3 dB bandwidth, Δf = f₀/Q: 10 Ω, 10 mH and 1 µF give Q = 10 and a 159 Hz band around 1,592 Hz. Above Q = 0.5 the circuit rings (underdamped); exactly 0.5 is critically damped, reached here at R = 2√(L/C) = 200 Ω.

### What is the time constant of an RL circuit?

τ = L/R. It is the time for the current to rise to 63.2% of its final value V/R after the switch closes, or to fall to 36.8% once the source is removed and the coil discharges through R. 10 mH with 100 Ω gives τ = 100 µs, so on 5 V the current reaches 99% of 50 mA in about 460 µs.

### How accurate is the RC time constant and RLC circuit calculator?

Accuracy depends on your inputs and the method's assumptions. Decimal arithmetic uses 50 significant digits, but estimates, numerical methods and source data can be less precise; the displayed rounding does not remove those limits. It is checked against 7 worked examples whose answers come from independent sources; for example, “10 kΩ and 100 µF charging to 90%” is checked against Python 3.8 math: τ = 10⁴ × 10⁻⁴ = 1 s; t = τ ln 10; E = ½CV² = ½·10⁻⁴·25.

### Where does the method come from?

OpenStax University Physics Volume 2, §10.5 RC circuits; §14.4 RL circuits; §15.3 RLC series circuits with AC; HyperPhysics — Series RLC resonance and Q.

## Sources

- [OpenStax University Physics Volume 2, §10.5 RC circuits; §14.4 RL circuits; §15.3 RLC series circuits with AC](https://openstax.org/books/university-physics-volume-2/pages/15-3-rlc-series-circuits-with-ac)
- [HyperPhysics — Series RLC resonance and Q](http://hyperphysics.phy-astr.gsu.edu/hbase/electric/serres.html)