One Time Constant Is 63 % of the Way, Not the End

The same constant says how long a supply stays dangerous

Discharge follows the same curve downward, so a large capacitor behind a large resistance empties slowly. A 470 µF smoothing capacitor with a 1 MΩ bleeder resistor across it has a time constant of 470 seconds — nearly eight minutes — and reaches the conventional five constants after thirty-nine. That is the whole reason those resistors exist and the reason a warning label on a power supply gives a waiting time rather than saying the unit is safe once unplugged. Where no bleeder is fitted, the only path is leakage, the constant is far longer, and the charge can survive the walk back to the bench.

The prefixes are where the factor of a million hides

The formula wants ohms and farads, and nobody writes farads. A microfarad is a millionth, a nanofarad a thousandth of that, a picofarad a thousandth again, so a component marked 100 with no unit could be six orders of magnitude away from what the calculation assumes. Two combinations are worth memorising because they cover most of the bench: kilohms with microfarads give milliseconds, and megohms with microfarads give seconds. Anything outside those, write the powers of ten out in full before dividing — the mental arithmetic that skips them is the one that produces a filter tuned three decades from where it was meant to be.

Frequently asked questions

How does this relate to a filter's cut-off frequency?

They are the same fact in two units. The corner frequency of a simple RC filter is one divided by two pi times the time constant, so a constant of 0.1 second corresponds to about 1.6 hertz. A slow circuit in the time domain is a low-frequency filter in the frequency domain, and multiplying the resistance by ten moves both by the same factor — ten times slower, ten times lower. Designing in whichever domain the requirement is written in and converting once is quicker than trying to hold both pictures at the same time.

Does the supply voltage change the time constant?

No. It sets where the curve ends up, not how fast it gets there: a 5 V supply and a 50 V supply through the same resistor and capacitor both reach 63.2 % of their own target in the same 100 ms. This is what makes the constant a property of the circuit rather than of the signal, and it is why the same filter works on inputs of any amplitude. The voltage does matter for two things the formula does not model — the current at the start, which is highest when the capacitor is empty, and whether the capacitor is inside its voltage rating at the end.