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When a voltage is applied to an inductor and resistor in series, the current can't jump instantly — the inductor opposes the change. It rises exponentially toward its final value V/R, and the time constant τ = L/R sets how fast. After one time constant the current reaches 63.2% of its final value, and after about 5τ it is essentially complete (99.3%).
| Quantity | Formula |
|---|---|
| Time constant | τ = L / R |
| Final current | Ifinal = V / R |
| Current at time t | i(t) = Ifinal(1 − e−t/τ) |
| Time to reach X% | t = −τ·ln(1 − X/100) |
Milestones: 1τ = 63.2%, 2τ = 86.5%, 3τ = 95%, 4τ = 98.2%, 5τ = 99.3%.
It is τ = L/R, the time for the current in an inductor-resistor circuit to reach about 63.2% of its final value when energizing (or fall to 36.8% when de-energizing).
An inductor opposes changes in current by generating a back-EMF, so the current builds up exponentially rather than jumping instantly to its final value.
Once fully energized the inductor looks like a plain wire, so the current settles at I = V/R, limited only by the resistance.
About 5 time constants (5τ), by which the current reaches 99.3% of its final value — close enough to be considered complete.
1τ = 63.2%, 2τ = 86.5%, 3τ = 95.0%, 4τ = 98.2%, 5τ = 99.3% of the final current.
Use t = −τ·ln(1 − X/100), where X is the target percentage of final current. For 90% that is about 2.3τ.
A larger L increases τ (slower current rise); a larger R decreases τ (faster rise), since τ = L/R.
The exponential shape is the same, but when de-energizing the current decays as e^(−t/τ) — and if the circuit opens abruptly, a large voltage spike appears.
Both are exponential, but RC = R×C governs a capacitor charging voltage, while RL = L/R governs an inductor current. Note R multiplies in RC but divides in RL.
The inductor tries to keep its current flowing; if the path opens quickly, the rapid change produces a large back-EMF (V = L·di/dt). A freewheeling diode absorbs it safely.
No — τ depends only on L and R. Voltage only scales the final current (V/R), not how fast it gets there.
Use the total series resistance in the loop, including the inductor's own winding resistance and any external resistor, since that is what limits and shapes the current.
RC Time Constant • Reactance (XL) • Inductor Energy • All Calculators