RL circuit in transient operation

Formulating the Equation

Consider a series circuit consisting of a resistor R and an inductor L, powered by a DC voltage source E via a switch that is closed at t = 0. The loop law gives the differential equation:

E = R * i(t) + L * di/dt

Solution and time constant

Assuming i(0) = 0 (current continuity), solving this first-order linear equation gives:

i(t) = (E/R) * (1 - exp(-t/tau)), where tau = L/R (in seconds)

tau is the time constant of the circuit: it characterises the rate at which the current stabilises. After 5*tau, steady state is reached to within 1 per cent.

Notable values

Time	Value of i(t)
t = tau	0.63 * E/R
t = 2*tau	0.86 * E/R
t = 3*tau	0.95 * E/R
t = 5*tau	approximately E/R

Example

For R = 100 ohms and L = 0.5 H, we have tau = L/R = 5 ms, and steady state is practically reached after approximately 25 ms.

Common pitfall

Do not confuse tau = L/R (RL circuit) with tau = R*C (RC circuit): the position of R is reversed. Another pitfall: at t = 0+, we do indeed have i(0+) = 0 by continuity, but the voltage across the coil is u_L(0+) = E, because the entire source voltage is initially across the coil (the resistor is not yet carrying any current).