CR · Transients

RC Transient and the Time Constant

Apply a voltage to a capacitor and it does not fill at once; it creeps toward the target along a curve. This unit shows how to measure and handle that slowness with a single number, the time constant τ.

One time constant covers 63%

The gold curve is the capacitor voltage rising as it charges, and the dot on the curve is the voltage at the present moment. Increase the time t from 0 and find the instant when the voltage reaches 63% of its final value. That instant, where the dot meets the dashed vertical line, is the time constant τ.

Time tt = 0.6 ms

Capacitor voltage v

v = 2.6 V

26% of the final value (τ = 2 ms)

Far from 63%

Why a curve

For the capacitor voltage to rise, current has to flow in. But that current comes in through the resistor, and its size is proportional to the voltage difference still remaining. So the capacitor fills quickly while the difference is large and slows down as it nears the target, and the graph is not a straight line but an exponential curve that keeps easing off.

The time constant τ = RC

What sets the speed of the curve is the product of resistance and capacitance, τ = RC. A large resistance lets in little current, so charging is slow; a large capacitance means a bigger bucket to fill, so it is slow too. τ has units of time, and after one time constant about 63% of the remaining distance to the target has been covered. The circuit in the plot has τ = 2 ms, so at t = 2 ms the voltage reaches 6.3 V, 63% of 10 V. Covering the same fraction of the remaining distance every τ is the defining property of an exponential.

When is it “done”?

One time constant fills it to 63%, two to 86%, three to 95%. After five time constants it is over 99% full and counts as full for practical purposes; the right end of the graph's horizontal axis marks exactly that point. That is why 5τ is used to estimate how long a circuit takes to settle into a new state. Discharging follows the same τ, only in the opposite direction.

Observev(t) = V(1 − e^−t/τ)

It approaches the target exponentially.

Chooseτ = ?

The time constant is resistance times capacitance.

Fill int = τ → v = ? V

63% at one time constant.

On your own99% ≈ ? τ

About five time constants and it is nearly done.

Back to the first screen

As you increased the time, the capacitor voltage started out steeply and then eased off, and the moment it reached 63% of its final value was exactly one time constant τ. After that it kept covering the same share of the remaining distance every τ, and by about five time constants the curve had all but flattened. A single number, τ = RC, held the entire speed of the curve.

The time constant τ = RC is a single number, in units of time, that sets how quickly an RC circuit responds to change. The charging voltage follows v(t) = V(1 − e^−t/τ): after one time constant about 63% of the remaining distance to the final value is covered, and after about 5τ it is over 99% full and effectively settled. Discharging follows the same τ.