Establishment of current in an RL circuit

The RL dipole: modelling and differential equation

The RL dipole

An RL circuit consists of a resistor R (in ohms) and an inductor L (in henrys) connected in series, powered by a DC voltage source E via a switch K.

Mesh Theorem

When the switch is closed (t=0), the mesh theorem gives:

E = Ri(t) + Ldi/dt

This is a first-order linear differential equation with constant coefficients and a constant right-hand side. The unknown is i(t), the instantaneous current in the circuit.

Voltage across the coil

The coil opposes changes in current: u_L(t) = L*di/dt. This voltage is zero in steady state (di/dt = 0) but can become very large during sudden changes in current.

Time constant

We define τ = L/R (in seconds), which is a time constant. It characterises the speed of the transient response: the larger tau is, the slower the current stabilises.

Common pitfall

Do not confuse the transient state (di/dt ≠ 0) with the steady state (di/dt = 0, i = E/R). In steady-state DC operation, the coil behaves like a simple wire (zero resistance), but at the initial instant the current is determined by continuity (i(0)=0 if i(0-)=0).