How it works
An inductor resists sudden changes in current. When you switch a voltage across an inductor and resistor in series, the current does not jump to its final value; it climbs exponentially. The time constant τ = L ÷ R sets how fast.
After one τ the current is 63.2% of its final value V ÷ R; after five τ it is above 99%. The energy the coil holds at the final current is ½ L I². That stored energy is why switching an inductive load off can make a voltage spike.
Worked example
A 10 mH coil in series with 100 Ω, switched onto 12 V, with a 95% target:
- τ = L ÷ R = 10 mH ÷ 100 Ω = 100 µs
- Time to 95% = −τ × ln(1 − 0.95) = 299.573 µs
- Final current = 12 V ÷ 100 Ω = 120 mA
- Stored energy = ½ × 10 mH × (120 mA)² = 72 µJ
- Cutoff f = R ÷ (2π L) = 1.592 kHz
| Input | Value |
|---|---|
| Inductance | 10 |
| Inductance unit | mH |
| Total series resistance | 100 Ω |
| Supply voltage | 12 V |
| Target current level | 95 % |
| Result | Value |
|---|---|
| Time constant τ | 100 µs |
| Time to reach target current | 299.573 µs |
| Final (steady-state) current | 120 mA |
| Stored energy at final current | 72 µJ |
| Cutoff frequency (−3 dB) | 1.592 kHz |
Assumptions and limits
- Ideal inductor with all resistance lumped into one series resistor, including coil winding resistance and the source.
- Constant-voltage step input and an inductor that starts with zero current.
- Iron-core coils saturate at high current, which makes the real inductance drop and the current rise faster than this ideal model.
Common questions
What is the time constant of an RL circuit?
τ = L ÷ R in seconds. A 10 mH coil with 100 Ω total resistance has τ = 100 µs.
How long until an inductor is fully energized?
About five time constants, when the current is within 1% of its final value.
Why does a relay coil make a spark when it turns off?
The coil's stored energy has nowhere to go, so the current change induces a high voltage across the switch. A flyback diode across the coil gives the current a path.
Sources
- Standard first-order RL circuit theory: τ = L/R, i(t) = (V/R)(1 − e^(−t/τ)), energy = ½LI².
Updated 2026-09-30