AC Source Connected to an Inductor

If V is the potential difference across the inductor,

V(t) = L dI(t)/dt = V_m cos ωt

dI = V_m/L cos ωt dt

Integrating,

∫ dI = V_m/L ∫ cos ωt dt

Since integral of cos x is sin x,

I(t) = V_m/ωL sin ωt + constant

When t = 0, I = 0. Hence constant of integration becomes zero. Thus

I(t) = Vm/ωL sin ωt

The inductor current and potential difference across it are not in phase. The potential difference peaks one-quarter cycle before the current. In case of an inductor, current lags the potential difference by π/2 rad (or 90°). 

The quantity ωL has units of resistance and is called inductive reactance. It is denoted by symbol X_L.

X_L = ωL = 2πvL

Like capacitive reactance, the inductive reactance, X_L, is expressed in ohm. Inductive reactance is a measure of the extent to which the inductor limits ac current in the circuit. It depends on the inductance and the frequency of the generator. Inductive reactance increases, if either frequency or inductance increases.