Magnetic energy and impedance in forced sinusoidal regime

Energy stored in the coil

The coil stores energy in magnetic form:

E_L = (1/2)Li^2 (in joules)

This energy is returned to the circuit when the current decreases (breaking the circuit), which explains the voltage surge and the electric arc. It cannot be dissipated instantaneously, hence the continuity of i(t).

Power balance in the transient regime

By multiplying the equation E = Ri + Ldi/dt by i(t), we obtain the power balance:

Ei = Ri^2 + Lidi/dt = P_Joule + d(E_L)/dt

The power supplied by the generator is split between Joule dissipation and magnetic storage.

Forced sinusoidal regime

In the steady sinusoidal regime of angular frequency omega, we use complex notation. The complex impedance of the RL dipole is:

Z = R + jLomega

Magnitude: |Z| = sqrt(R^2 + (L*omega)^2)

Argument (voltage/current phase shift): phi = arctan(L*omega/R)

The current always lags the voltage by a phase phi (0 <= phi <= pi/2): the RL circuit is said to be inductive.

Limiting cases

omega	Behavior
omega -> 0	Z -> R (coil = wire)
omega -> infinity	Z -> infinity (coil = open circuit)

Common pitfall

Do not mix up the transient-regime analysis (time differential equation, tau = L/R) with the forced sinusoidal-regime analysis (complex impedance, phase shift): these are two different tools for two different situations.