CR · Transients

RL Transient and the Time Constant

Apply a voltage to an inductor and the current does not flow at once but climbs slowly along a curve. Just as the capacitor’s voltage did, this time it is the current whose slowness you learn to measure and handle with one number, the time constant τ.

One time constant covers 63%

Push the time t up from zero. Find the moment the inductor current reaches 63% of its final value, and that instant is the time constant τ.

Time tt = 0.6 ms

Inductor current i

i = 0.52 A

26% of the final value (τ = 2 ms)

Just starting

Why a curve

An inductor opposes any change in current. As the source tries to push the current up, the inductor pushes back with a counter-emf proportional to di/dt. At first the current is zero, so nearly the whole source voltage lands on the inductor and drives the current up fast; as the current nears its final value I = V/R, the resistor takes more of the voltage and the voltage on the inductor — and so di/dt — shrinks, and the rise slows. So it is not a straight line but an exponential curve that keeps easing off.

The time constant τ = L/R

What sets the speed of the curve is the inductance divided by the resistance, τ = L/R. A large inductance opposes the change in current more strongly, so it is slow; a large resistance lets the circuit settle faster, so it is quick. In RC the resistance was on top, so larger meant slower; in RL the resistance is on the bottom, so larger means faster — flipped, mirror-like. 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 climbs to 63%, two to 86%, three to 95%. After five time constants it is past 99%, so in practice we call it done. That is why 5τ is the rule of thumb for how long a circuit takes to settle into a new current. This is the mirror of the RC transient: just as a capacitor resisted a change in voltage and filled with τ = RC, an inductor resists a change in current and climbs with τ = L/R. The decay when the current is cut follows the same τ.

Observei(t) = I(1 − e^−t/τ)

It approaches the target exponentially.

Chooseτ = ?

The time constant is inductance over resistance.

Fill int = τ → i = ? I

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 inductor current 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 = L/R held the entire speed of this curve. The story a capacitor told with RC, an inductor retells with current, mirror-like.

The time constant τ = L/R is a single number, in units of time, that sets how fast an RL circuit responds to a change in current. The rising current is i(t) = I(1 − e^−t/τ) with I = V/R: 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. Just as a capacitor resists a change in voltage, an inductor resists a change in current — this is the mirror of the RC transient, where τ = RC.