CR · Transients

RL Transient and the Time Constant

Even when you apply a voltage to an inductor, the current does not flow all at once; it rises slowly along a curve. Just as with the capacitor voltage, this time you learn to measure and handle how far the current lags behind with a single number, the time constant τ.

One time constant fills 63%

The gold curve shows the inductor current rising after the switch closes, heading for the final value on the white dashed line at the top, and the dot on the curve is the current at the present moment. Slide the time from 0 and find the instant the dot reaches the corner of the green dashed line, that is, 63% of the final value. That moment 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)

Far from one time constant

Why a curve

An inductor opposes changes in current. When the source tries to push the current up, the inductor fights back with a back emf proportional to di/dt. At the instant the switch closes the current is 0, so no voltage drops across the resistor and the whole source voltage sits across the inductor, making the current rise at its steepest. As the current nears its final value I = V/R, the resistor takes more of the voltage and the inductor's share, that is di/dt, shrinks, so the curve is not a straight line but an exponential 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 larger inductance opposes the change in current more strongly and slows things down, while a larger resistance makes the final current smaller, so it settles that much sooner. In RC the resistance sat in the numerator and a larger one meant slower; in RL it sits in the denominator and a larger one means faster, a mirror-image reversal. τ has units of time, and after one time constant about 63% of the remaining distance to the target is filled. Filling the same fraction of the remaining distance every τ is what makes it exponential.

When is it “done”?

After one time constant the current reaches 63%, after two 86%, and after three 95%. Past five time constants it exceeds 99% and is treated as done for practical purposes, so 5τ is the rule of thumb for how long a circuit takes to settle at its new current. The right end of the horizontal axis in the plot is exactly that point. All of this mirrors the RC transient: just as a capacitor opposes changes in voltage and charges with τ = RC, an inductor opposes changes in current and builds up with τ = L/R. The current also decays with the same τ, but if you simply open the switch the current has nowhere to go and the inductor produces a large voltage, so real circuits provide a separate path such as a diode.

Observei(t) = I(1 − e^−t/τ)

It approaches the target exponentially.

Chooseτ = ?

The time constant is inductance divided by resistance.

Fill int = τ → i = ? I

63% at one time constant.

On your own99% ≈ ? τ

After about five time constants it is nearly done.

Back to the first screen

As you slid the time, the dot on the gold curve set off steeply and gradually eased, and the moment it touched the corner of the green dashed line at 63% of the final value was exactly one time constant τ. After that the curve kept filling the same share of the remaining distance every τ and was practically flat by about five time constants, at the right end of the axis. The single number τ = L/R held the whole speed of this curve. The curve the capacitor voltage traced with RC is repeated, mirror-like, by the current in an inductor.

The time constant τ = L/R is a single number with 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), I = V/R; after one time constant about 63% of the remaining distance to the final value is filled, and after about 5τ it passes 99% and has practically settled. Just as a capacitor opposes changes in voltage, an inductor opposes changes in current, and this is the mirror of the RC transient with τ = RC.