CR · Transients

RC Transient and the Time Constant

When you apply a voltage to a capacitor, it does not jump there at once but eases toward it along a curve. Learn to measure and handle that slowness with one number, the time constant τ.

One time constant covers 63%

Push the time t up from zero. Find the moment the capacitor voltage reaches 63% of its final value, and that instant 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)

Just starting

Why a curve

Voltage on a capacitor rises only as current flows into it. That current comes through the resistor, and its size is proportional to the voltage gap still left. When the gap is large the filling is fast; as it nears the target it slows. So the rise 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 it is slow; a large capacitance is a bigger vessel 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 is covered. Filling the same fraction of what is left every τ is the signature of an exponential.

When is it “done”?

One time constant fills to 63%, two to 86%, three to 95%. After five time constants it is past 99%, so in practice we call it full. That is why 5τ is the rule of thumb for how long a circuit takes to settle into a new state. Discharge follows the same τ, just draining the other way.

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 is nearly done.

Back to the first screen

As you pushed time forward, the capacitor voltage started steeply and kept easing off, and the moment it touched 63% of the final value was exactly one time constant τ. Beyond that it filled the same share of the remaining distance every τ, and by about five time constants the curve had all but flattened. The single number τ = RC held the entire speed of this curve.

The time constant τ = RC is a single number, in units of time, that sets how fast an RC circuit responds to change. The charging voltage is 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 past 99% and essentially settled. Discharge follows the same τ.