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RL transients describe current changes when inductors energize or de-energize through resistors. The current in a series RL circuit follows i(t) = I_f + (I_i - I_f)·exp(-t/τ), where τ = L/R is the time constant. Inductors oppose current changes, resulting in exponential approach to steady-state current. RL transients appear in switching power supplies, motor control, and relay circuits.
You already know from inductor analysis that an inductor opposes changes in current — its terminal voltage is v = L · di/dt, so an instantaneous current jump would require infinite voltage. This means the current through an inductor cannot change instantaneously. That single physical fact is the entire source of RL transient behavior. When you flip a switch, the circuit must negotiate a smooth transition from its initial current to its final steady-state current, and that negotiation plays out over time.
Consider a series RL circuit where a voltage source V is connected at t = 0 with the inductor initially carrying no current. KVL gives V = i·R + L·di/dt. The steady-state solution (di/dt → 0) is simply i_f = V/R — the inductor looks like a wire at DC. But the initial condition forces i(0) = 0. The solution that satisfies both is: i(t) = I_f · (1 − e−t/τ), where τ = L/R is the time constant. After one time constant, the current has reached 63.2% of its final value. After 5τ, it's within 1% of I_f and the transient is effectively over. The time constant τ has a clean physical interpretation: larger inductance means more "inertia" against current change; larger resistance means faster dissipation and quicker approach to steady state.
The general formula i(t) = I_f + (I_i − I_f) · e−t/τ handles all cases, including those where current starts at a nonzero value. The three quantities you need are the initial current I_i (determined by continuity — the current just before the switch event), the final current I_f (determined by DC steady state with the new circuit), and the time constant τ = L/R (determined by the Thévenin resistance seen by the inductor after the switch event). Once you have these three, the entire transient waveform follows. This "three-element recipe" is the universal method for first-order RL transients.
When a current-carrying inductor is suddenly disconnected from its source, a voltage spike appears at the inductor terminals. The inductor tries to maintain current through whatever path is available — if none exists, the voltage climbs until an arc occurs or a protective clamp absorbs the energy. Relay coils, motor windings, and solenoids routinely produce these spikes, which can destroy switching transistors. The classic protection solution is a freewheeling diode placed in parallel with the inductor: it provides a safe current path and allows the stored magnetic energy (½LI²) to dissipate harmlessly in the resistance during turn-off. Recognizing and managing inductive voltage spikes is one of the most practical skills from RL transient analysis.