RL Circuit Calculator (Time Constant, Current & Voltage vs Time)

How it works

τ = L / R · build-up: i(t) = (V/R)·(1 − e^(−t/τ)), v_L(t) = V·e^(−t/τ) · decay: i(t) = (V/R)·e^(−t/τ), v_L = −V·e^(−t/τ) · energy E = ½·L·I² · f_c = R/(2πL)

An inductor opposes any change in its current, so when a DC source is switched onto a series RL circuit the current cannot jump: it rises exponentially toward V/R while the coil’s voltage falls from V to zero. The time constant τ = L/R is the time to reach 63.2% of the final current; after 5τ the transient is over for practical purposes (99.3%). When the source is removed and the coil discharges through R, the current decays by the same factor and the coil voltage reverses — the reason relay coils need a flyback diode. Equations follow OpenStax University Physics Vol. 2, section 14.4.

Worked example

A 50 mH coil in series with 100 Ω is switched onto 12 V. τ = 0.050 / 100 = 0.5 ms (500.0 µs). The final current is 12 / 100 = 120.0 mA. After 1 ms (2τ) the current is 0.12 × (1 − e⁻²) = 103.8 mA, the inductor voltage has fallen to 12 × e⁻² = 1.624 V and the resistor holds 10.376 V. The coil stores 360.0 µJ at full current and settles after 5τ = 2.500 ms.

Sources: OpenStax University Physics Vol. 2 — RL Circuits · HyperPhysics — Inductor current rise and decay

Frequently asked questions