Inductors resist change
An inductor is a coil of wire. When current flows through it, it builds a magnetic field, and any change in that current induces a voltage that fights the change. Put an inductor in series with a resistor and a voltage source, and the current cannot jump to its final value at once: it rises along a smooth curve. Switch the source off, and the current dies away the same way.
The formulas
The time constant sets the pace:
τ = L ÷ R
When switching on, with a source voltage V:
- Current: i(t) = (V ÷ R) × (1 − e^(−t/τ))
- Inductor voltage: v(t) = V × e^(−t/τ)
When the source is removed and the coil discharges through the resistor:
- Current: i(t) = (V ÷ R) × e^(−t/τ)
The energy stored in the magnetic field is W = ½ L i².
Worked example
A 500 mH coil in series with 100 Ω on a 12 V supply:
- Time constant: 0.5 ÷ 100 = 5 ms.
- Final current: 12 ÷ 100 = 120 mA.
- 2 ms after switching on: i = 120 × (1 − e^(−0.4)) = 39.56 mA, or 33% of the final value.
- The inductor still holds 8.044 V at that moment; the resistor takes the remaining 3.956 V.
- Stored energy at 2 ms: 0.39 mJ, rising to 3.6 mJ at full current.
- The current reaches 99% of its final value after 4.6τ, or 23 ms.
Milestones
| Time | Switching on | Switching off |
|---|---|---|
| 1τ | 63.2% | 36.8% |
| 2τ | 86.5% | 13.5% |
| 3τ | 95.0% | 5.0% |
| 5τ | 99.3% | 0.7% |
Where RL circuits matter
- Relays and solenoids take a few milliseconds to pull in because their current needs time to build up.
- Switching off an inductive load abruptly creates a large voltage spike, v = L di/dt. A flyback diode across the coil, or a snubber, protects the switch. The same effect fires a car’s spark plugs through the ignition coil.
- Switching power supplies store energy in an inductor on each cycle and release it to the output.
- Filters: an RL circuit passes low frequencies and blocks high ones, with a cutoff at R ÷ (2πL).
Practical notes
Real inductors have their own winding resistance, which adds to R and shortens τ. Iron-core coils also saturate at high current, which makes L drop. For an RC circuit, where a capacitor charges through a resistor, the time constant is R × C instead, and the voltage rather than the current follows the same curve.