What is the time constant of an RC circuit?
τ = R × C. With R in ohms and C in farads, τ is in seconds: 10 kΩ × 100 nF = 1 ms. It is the time a capacitor charging through the resistor takes to reach 63.2 % of its final voltage.
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RC time constant calculator
From a resistor and a capacitor, this RC time constant calculator gives τ = RC and the time to charge or discharge to any percentage. It also works back from a target time to the resistor or capacitor that sets it. See the charge and discharge curves and the voltage at each time constant.
Solve for
Examples
Ω
F
%
Percent of the final voltage. 63.2 % is one τ, 99.3 % is five.
V
Optional: shows volts, peak current and stored energy.
Time constant τ = RC
1 ms
- Time to charge to 63.2 %
- 999.7 µs 0.9997 τ, or discharge to 36.8 %
- Time to 99.3 % (5 τ)
- 5 ms
- Rise time 10–90 %
- 2.197 ms 2.197 τ
- As a low-pass filter, fc
- 159.2 Hz
Assumes a clean voltage step from a source with negligible resistance (add any source resistance to R) and nothing else loading the capacitor.
Tolerance moves every time in proportion: a 5 % resistor with a 10 % capacitor puts τ anywhere from −14.5 % to +15.5 % of nominal.
ChargingDischarging
Charge level after each time constant
| Time | Elapsed | Charged to | Discharged to |
|---|---|---|---|
| 1 τ | 1 ms | 63.2 % | 36.8 % |
| 2 τ | 2 ms | 86.5 % | 13.5 % |
| 3 τ | 3 ms | 95.0 % | 5.0 % |
| 4 τ | 4 ms | 98.2 % | 1.8 % |
| 5 τ | 5 ms | 99.3 % | 0.7 % |
How to calculate RC time constants
Multiply the resistance by the capacitance: τ = R × C. A 10 kΩ resistor with a 100 nF capacitor gives τ = 1 ms. After one τ the capacitor has charged to 63.2 % of the supply, after five τ to 99.3 %, and the time to any level p is −τ × ln(1 − p).
RC time constant formula
When a voltage step is applied through a resistor, the capacitor’s current is largest at the start and falls as the capacitor charges, so the voltage approaches the supply exponentially.
τ = R × C
- τ
- time constant in s
- R
- resistance the capacitor charges through, in Ω
- C
- capacitance in F
Charging from 0 V toward a supply Vs, and discharging from V0 toward 0 V:
Vc(t) = Vs × (1 − e^(−t/τ)) Vc(t) = V0 × e^(−t/τ)
Solved for the time to reach a fraction p of the final value, and for R or C given that time:
t = −τ × ln(1 − p) R = t ÷ (−C × ln(1 − p)) C = t ÷ (−R × ln(1 − p))
- p
- fraction of the final voltage, 0 to 1 (0.5 for 50 %)
- t
- time to reach p when charging, or to fall to 1 − p when discharging
Useful special cases: 50 % takes ln 2 = 0.693 τ, 90 % takes ln 10 = 2.303 τ, 99 % takes 4.605 τ and 99.9 % takes 6.908 τ.
Worked examples
Example 1: 10 kΩ and 100 nF
τ = 10,000 Ω × 100 × 10⁻⁹ F = 1 ms. The capacitor is at 50 % after 693.1 µs, at 63.2 % after 1 ms and at 99.3 % after 5 ms. The 10–90 % rise time is 2.197 ms.
Example 2: the ESP32 CHIP_PU (enable) delay
Espressif recommends an RC delay on the ESP32’s CHIP_PU pin, usually 10 kΩ and 1 µF, so the chip comes out of reset after the supply has settled. τ = 10 ms. From a 3.3 V rail the pin reaches 2.086 V after one τ and 3.136 V after 30 ms. Espressif notes the values should be adjusted to the actual supply’s power-up behavior.
Example 3: a capacitor for a 100 ms delay to 50 % with 100 kΩ
C = 0.1 s ÷ (100 kΩ × ln 2) = 1.44 µF. The nearest E12 value, 1.5 µF, gives 104 ms. With R = 100 kΩ, keep the capacitor’s leakage current well below the resistor current near the threshold, or the delay stretches.
Charge and discharge after each time constant
The same percentages apply to every RC circuit; only τ changes the time scale.
| Elapsed time | Charged to | Discharged to |
|---|---|---|
| 0.5 τ | 39.3 % | 60.7 % |
| 1 τ | 63.2 % | 36.8 % |
| 2 τ | 86.5 % | 13.5 % |
| 3 τ | 95.0 % | 5.0 % |
| 4 τ | 98.2 % | 1.8 % |
| 5 τ | 99.3 % | 0.7 % |
| 7 τ | 99.9 % | 0.1 % |
RC circuit time constant for common R and C values
τ = R × C for standard values. Five times the entry is the time to about 99.3 %.
| R \ C | 1 nF | 10 nF | 100 nF | 1 µF | 10 µF |
|---|---|---|---|---|---|
| 1 kΩ | 1 µs | 10 µs | 100 µs | 1 ms | 10 ms |
| 4.7 kΩ | 4.7 µs | 47 µs | 470 µs | 4.7 ms | 47 ms |
| 10 kΩ | 10 µs | 100 µs | 1 ms | 10 ms | 100 ms |
| 47 kΩ | 47 µs | 470 µs | 4.7 ms | 47 ms | 470 ms |
| 100 kΩ | 100 µs | 1 ms | 10 ms | 100 ms | 1 s |
| 1 MΩ | 1 ms | 10 ms | 100 ms | 1 s | 10 s |
What changes the time constant of RC circuits in practice
The resistance in τ is everything the capacitor charges through: add the driver’s output resistance to R. A resistive load across the capacitor changes both the final voltage (a divider with R) and the time constant, which becomes (R ∥ Rload) × C. Part tolerances move τ directly, and Class 2 ceramics (X5R, X7R) lose capacitance with DC bias, so a 1 µF part at its working voltage can give a noticeably shorter delay than the label suggests. With large resistors, capacitor leakage and the input current of the pin that reads the node act as extra paths for charge. Finally, a slowly rising RC voltage into an ordinary logic input can switch more than once as it crosses the threshold; feed it to a Schmitt-trigger input or a supervisor when the timing has to be clean.
Glossary
- Time constant (τ)
- R × C in seconds: the time to reach 63.2 % of the final voltage when charging, or fall to 36.8 % when discharging.
- Exponential charge
- Vc(t) = Vs (1 − e^(−t/τ)): fast at first, slowing as the capacitor approaches the supply.
- Exponential decay
- Vc(t) = V0 e^(−t/τ) for a capacitor discharging through a resistor.
- Rise time (10–90 %)
- The time for a step response to go from 10 % to 90 % of its final value; τ ln 9 ≈ 2.2 τ for one RC.
- Settling time
- The time to get within a given error of the final value: 4.6 τ for 1 %, 6.9 τ for 0.1 %.
Related guides
- Capacitor charge and discharge calculator: voltages from any starting point.
- RC low pass filter calculator: the same RC in the frequency domain.
Frequently asked questions
Why is one time constant 63.2 %?
Because the charging voltage follows 1 − e^(−t/τ), and at t = τ that is 1 − e⁻¹ = 63.2 %. A discharging capacitor has 36.8 % (e⁻¹) of its starting voltage left after one τ.
How long does a capacitor take to fully charge?
In theory it never reaches 100 %; each τ closes 63.2 % of the remaining gap. In practice five time constants is treated as fully charged: 99.3 %. Reaching 99.9 % takes 6.908 τ.
How do I calculate the time to charge to a given voltage?
t = −τ × ln(1 − V/Vs). OpenStax works an example with τ = 5.05 s charging toward 100 V: reaching 80 V takes −5.05 s × ln(0.2) = 8.13 s.
What is the time constant of 10 kΩ and 1 µF?
10 ms. The capacitor is at 50 % after 6.931 ms, 63.2 % after 10 ms and 99.3 % after 50 ms.
How do you find the time constant of an RC circuit with a load?
Use the resistance the capacitor actually charges through. A resistive load across the capacitor makes τ = (R ∥ Rload) × C and also lowers the final voltage (a divider with R), and a driver’s output resistance adds to R.
How is the time constant related to the cutoff frequency?
The same R and C used as a low-pass filter have fc = 1 / (2πτ). A 1 ms time constant corresponds to 159.2 Hz, and the 10–90 % rise time is τ ln 9 ≈ 2.2 τ.
Is the discharge time constant the same as the charge time constant?
Yes, if the capacitor discharges through the same resistance. If it discharges through a different path (a load, a transistor, a bleed resistor), use that resistance: τ is always the resistance the capacitor sees times its capacitance.
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