A capacitor charging through a resistor does not fill at a steady rate. The current depends on the difference between the supply and whatever the capacitor already holds, so as the voltage rises the current falls and the charging slows. The result is an exponential curve that approaches the supply without ever quite arriving.
The time constant
The product RC has units of seconds and is written τ (tau). It is the single number that characterises the circuit: 10 kΩ with 100 µF gives τ = 1 second, and so does 1 kΩ with 1000 µF. The shape of the curve is identical in both cases.
| Time | Charged to | Discharged to |
|---|---|---|
| 1τ | 63.2% | 36.8% |
| 2τ | 86.5% | 13.5% |
| 3τ | 95.0% | 5.0% |
| 4τ | 98.2% | 1.8% |
| 5τ | 99.3% | 0.7% |
Five time constants is the working definition of “finished”. The remaining 0.7% is below the noise floor of most circuits, and waiting for mathematical completion means waiting forever.
Solving for time instead of voltage
Rearranging the charging equation gives the time to reach any particular voltage:
The logarithm is why the last stretch takes so long. Getting to half the supply takes 0.69τ; getting to 99% takes 4.6τ — nearly seven times as long for the second half of the journey.
Where this shows up
- Power-on reset. An RC network holds a reset pin low until the supply has stabilised. Sizing it is exactly this calculation.
- Debouncing. A mechanical switch chatters for a few milliseconds. An RC with τ around 10 ms smooths it into a single clean transition.
- Supply decoupling. How long a bulk capacitor can hold a rail up during a current spike is a discharge problem.
- Timing circuits. The 555 timer sets its period from exactly these charge and discharge curves against fixed threshold voltages.
Practical cautions
Real electrolytic capacitors leak, so they will not hold charge indefinitely, and their tolerance is often −20%/+80% — the timing you calculate may be well off with a cheap part. The initial current at the instant of switch-on is V_s/R with nothing limiting it but the resistor, which matters when charging large capacitors from a low-impedance supply. And a large charged capacitor stays dangerous after power is removed: energy is ½CV², so a 1000 µF capacitor at 400 V holds 80 joules, which is more than enough to kill.