When a resistor (R) and a capacitor (C) are connected in series, it takes time for the capacitor voltage to follow the supply voltage. The unit of that time is the time constant (τ, tau), and τ = R × C. Delay circuits, filters, switch debounce, and even the timing of a 555 timer all start from this one equation. The calculator above works out the time constant, the cutoff frequency, and the time for charging to reach a ratio you want from R and C.
Charging and discharging
When a supply voltage Vs is connected to C through R, the capacitor voltage changes as follows.
Charging: Vc(t) = Vs × (1 - e^(-t/τ)) Discharging: Vc(t) = V0 × e^(-t/τ)
The charge ratio after multiples of τ is as follows.
| Elapsed time | Charge reached | Remaining on discharge |
|---|---|---|
| 1τ | about 63.2% | about 36.8% |
| 2τ | about 86.5% | about 13.5% |
| 3τ | about 95.0% | about 5.0% |
| 4τ | about 98.2% | about 1.8% |
| 5τ | about 99.3% | about 0.7% |
In general, after 5τ it is considered "fully charged (or discharged)". The time to reach a ratio you want is found with t = -τ × ln(1 - ratio); for example, 90% takes about 2.3τ.
Cutoff frequency
An RC low-pass filter (a resistor followed by a capacitor to ground) reduces signals above the cutoff frequency.
f_c = 1 / (2π × R × C)
With R = 10 kΩ and C = 100 nF, τ = 1 ms and the cutoff frequency is about 159 Hz. Swap the positions of the same parts, with the capacitor in the signal path and the resistor to ground, and you get a high-pass filter.
Uses
- Switch debounce: 10 kΩ and 100 nF (τ = 1 ms) filter out the brief jitter of a contact. To block jitter of 5–20 ms, you usually set τ to 1–5 ms.
- Reset delay: when you keep a reset signal for a set time after power comes on, match the R·C value to the time.
- LED fade: if you drive a transistor with a capacitor voltage that charges and discharges through a resistor, the LED can fade on and off gradually.
The concept is explained in the guide Understanding RC Time Constants and Capacitor Charging, and an application is in 555 Timer Monostable Circuit. To calculate the timing of a 555 timer with the same R and C values, use the 555 timer calculator.