Skip to content
LearnEE
Sign inGet started

Electronics

RL circuit time constant explained

Switch on a coil, an inductor, and its current doesn't jump: it builds up along a curve. The RL time constant, τ = L ÷ R, says how fast: after one τ the current has reached 63% of its final value, V ÷ R, and after about 5τ it's there.

Why the current ramps

A coil fights any change in its current. At switch-on the current is zero, so the whole supply goes into fighting the change, and the current grows at its fastest rate. As the current rises, the resistance takes more of the supply, less is left across the coil, and the rise slows. Once the current is steady, the coil is just a piece of wire and only the resistance limits it.

It's like pushing a heavy cart against friction: it speeds up gradually until the friction balances your push.

The two numbers

τ = L over Rtotal
Ifinal = V over Rtotal

Rtotal includes the coil's own wire. Henries divided by ohms give seconds, and the curve is I = Ifinal × (1 − e−t/τ), the same shape as a charging capacitor's voltage.

Example: a 1 H coil with 100 Ω of wire, on 5 V.

Ifinal = 5 V over 100 Ω = 50 mA, τ = 1 H over 100 Ω = 10 ms

Now use 10 V with an extra 100 Ω in series, 200 Ω in all. The final current is 10 ÷ 200 = 50 mA, the same, but τ = 1 ÷ 200 = 5 ms, twice as fast.

Live experiment

Time a relay coil

A relay coil, 0.5 H with 200 Ω of wire, pulls in at 20 mA and lights Pip. Choose the battery and a series resistor, switch on, and watch the current climb in slow motion. On 6 V alone it settles at 30 mA with τ = 2.5 ms.

Try it yourself: free lesson

“RL circuits” is lesson 5 of 5 in Capacitor Caves, Electronics World 3. Free with an account. Lessons unlock in order, so you’ll do the 19 before it first.

Relays and solenoids

A relay pulls in once its coil current reaches a set level, so a faster rise means a faster relay. A 6 V relay coil of 0.5 H and 200 Ω settles at 30 mA with τ = 2.5 ms; if it pulls in at 20 mA, two thirds of the way, that takes a little over one τ, about 2.75 ms.

Solenoid valves and fuel injectors wait for their current the same way. Old stepper drivers used a high voltage plus series resistors, an L/R drive, to speed their coils up, at the cost of heat in the resistors. Modern drivers apply a high voltage and chop it with PWM to hold the current, wasting far less.

Switching off

Cut the current and the coil keeps it flowing for a moment from the energy in its magnetic field. With nowhere to go, it jumps the opening switch as a spark, which can destroy a transistor. A flyback diode across the coil gives it a path, and the current then dies away through it, roughly with τ = L ÷ R of that loop.

Free lesson

Now build it yourself

Each lesson comes in three short parts: see the idea, try it on a live board, then build it yourself, with quick checks as you go.

Try it yourself: free lesson

“RL circuits” is lesson 5 of 5 in Capacitor Caves, Electronics World 3. Free with an account. Lessons unlock in order, so you’ll do the 19 before it first.

Or start from the very first lesson, no account needed

Work it out

Practice questions

Three quick questions from the lessons. Pick an answer to check it.

  1. Question 1 of 3After one τ, a coil's current has reached what share of its final value?
    Show the answer

    Answer: 63%. The same rule as RC charging. In each τ it closes 63% of the gap to its final value.

    • Why not “37%”? 37% is what's left of the gap after one τ, not how far the current has got.
    • Why not “100%”? It takes about five time constants to get close to the final value, not one.
  2. Question 2 of 3Why does a coil spark when you cut its current?
    Show the answer

    Answer: The energy in its magnetic field has to go somewhere. The field's energy keeps the current flowing for a moment. With no path, the current jumps the opening switch as a spark.

    • Why not “It holds extra charge”? That's a capacitor's trick. A coil stores its energy in a magnetic field, carried by its current.
    • Why not “Its resistance is zero”? A coil's wire does have some resistance, and zero resistance wouldn't cause a spark anyway.
  3. Question 3 of 3An inductor mainly resists changes in…
    Show the answer

    Answer: current. An inductor fights any change in its current. A capacitor is the one that fights changes in voltage.

    • Why not “voltage”? Fighting changes in voltage is what a capacitor does.
    • Why not “resistance”? A coil has some resistance in its wire, but what it opposes is a change in its current.
Keep practising: free lesson

“RL circuits” is lesson 5 of 5 in Capacitor Caves, Electronics World 3. Free with an account. Lessons unlock in order, so you’ll do the 19 before it first.

Questions people ask

What is the time constant of an RL circuit?

τ = L ÷ R, the inductance in henries divided by the total resistance in ohms, including the coil's own wire. It gives seconds. After one τ the current has reached 63% of its final value, and after about 5τ it's there.

Why does more resistance make an RL circuit faster?

Because R divides in τ = L ÷ R. But the final current, V ÷ R, falls too. To get a fast coil at the same current, raise the voltage and add series resistance together, as old stepper-motor drivers did.

What is the difference between RC and RL time constants?

RC is τ = R × C, so more resistance is slower. RL is τ = L ÷ R, so more resistance is faster. Both follow the same curve: each τ closes 63% of the gap to the final value.

What happens when you switch off an RL circuit?

The coil's current can't stop at once. Given a path, like a flyback diode, it dies away with τ = L ÷ R of that path. With no path, it forces its way across the opening switch as a high-voltage spike.

Related topics

Every topic