The RC charge curve and time constant
How a capacitor charges along an exponential curve, what the 63%, 95% and 99% marks mean, and why five time constants counts as full.
The exponential charging curve
When a resistor and capacitor sit across a fixed supply, the capacitor voltage does not rise in a straight line. It follows an exponential curve that starts steep and flattens as the capacitor fills. The voltage at any time is the supply times the quantity one minus e raised to the negative time over tau. That single shape governs every RC charging problem, and the time constant tau is the knob that stretches or compresses it along the time axis.
What 63 percent really means
One time constant is defined as the moment the capacitor reaches about 63.2 percent of the supply voltage, which is one minus one over e. It is not the halfway point and it is not full charge, but it is a fixed, repeatable fraction that makes tau easy to measure on an oscilloscope. Because the curve is exponential, each additional time constant closes roughly 63 percent of the remaining gap, so progress toward the supply slows steadily even though it never quite stops.
Why five time constants counts as full
Following the curve, at 3 tau the capacitor sits at about 95 percent of the supply and at 5 tau at about 99 percent. Engineers treat 5 tau as fully charged because the last one percent is usually lost in noise and component tolerance. This calculator reports both the 3 tau and 5 tau times so you can judge settling for your circuit, whether you need a quick 95 percent for a rough threshold or the tighter 99 percent for precision timing.
The same constant sets the filter cutoff
Tau does double duty. Treat the same resistor and capacitor as a first-order filter and the cutoff frequency is f = 1 / (2 pi tau), the frequency where the output drops 3 dB and the phase shifts 45 degrees. A large tau gives slow charging and a low cutoff, while a small tau gives fast charging and a high cutoff. That link is why the time domain and frequency domain views of an RC circuit are two sides of one number.