Inductor Formulas: Energy and L/R Time Constant
Inductor formulas: stored energy, L/R time constant, current rise after switch-on, series and parallel inductance, with worked examples and a table.
The formulas
| To find | Formula |
|---|---|
| Stored energy | E (J) = ½ × L × I² |
| Time constant | τ (s) = L / R |
| Current rise on a step of voltage | i(t) = (V / R) × (1 − e^(−t/τ)) |
| Final current | I = V / R |
| Inductors in series | L = L1 + L2 + … (no coupling) |
| Inductors in parallel | 1 / L = 1 / L1 + 1 / L2 + … (no coupling) |
L is in henries, I in amperes, R in ohms and t in seconds. The current reaches about 63% of its final value after one time constant and about 99% after five. The series and parallel rules assume no magnetic coupling between the inductors.
Worked example: energy in a coil
A 0.5 H coil carries 4 A.
- E = ½ × 0.5 × 4² = 4 J
Worked example: switching on an L-R circuit
A 100 mH coil in series with 20 Ω is switched onto 24 V.
- τ = 0.1 / 20 = 5 ms and the final current is 24 / 20 = 1.2 A
- After 10 ms, i = 1.2 × (1 − e^(−10/5)) = 1.038 A
Current rise in the 100 mH, 20 Ω circuit
| Time | Elapsed | Share of final current | Current |
|---|---|---|---|
| 1 τ | 5 ms | 63.21% | 0.7585 A |
| 2 τ | 10 ms | 86.47% | 1.038 A |
| 3 τ | 15 ms | 95.02% | 1.14 A |
| 5 τ | 25 ms | 99.33% | 1.192 A |
Frequently asked questions
What happens when I open the circuit of a coil carrying current?
The current tries to continue and the voltage across the coil rises sharply. A diode across a relay coil, or a snubber, limits that voltage.
Is the coil resistance included in R?
Yes. R is the total resistance in the loop, including the winding resistance of the coil.
How does this compare with a capacitor?
A capacitor stores energy in the electric field as ½CV² and has a time constant RC. A coil stores it in the magnetic field as ½LI² and has a time constant L/R.
Does iron saturation change the result?
Yes. A cored inductor saturates at high current and its inductance falls, so the simple formulas no longer apply.
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