Derivation of expressions

Question

Derive an expression for the instantaneous current, instantaneous power and average power drawn by a pure inductive circuit on application of a sinusoidal voltage and show that the current lags behind the applied voltage by quarter of a cycle.

Answer 1

Assume the voltage applied is v(t)= Asin(wt) so the current through the inductance will be i(t)=Ldv/dt=LAwcos(wt). The power delivered to the inductance will be P(t)=v(t)i(t)=LA²wsin(wt)cos(wt). The average power =1/T∫P(t)dt from 0 to T where T = period =2π/w or 2π radians per cycle. The integral =A2L/Tsin²wt from 0 to T =0. i(t)=LAwcoswt=LAwsin(wt+π/2) (trig identity thus i(t) lags v(t) by T/4.