Power in a.c. circuits

Let a voltage v=Vm sinωt be applied to a circuit containing pure inductance (theoretical case). The resulting current is i = Im sin(ωt − (π /2)) since current lags voltage by 90◦ in a purely inductive circuit, and the corresponding instantaneous power, p, is given by:

p = vi = (Vm sinωt)Im sin(ωt − (π /2)) i.e. p = VmIm sinωt sin(ωt − (π /2)) However, sin(ωt−(π /2))=−cosωt Thus p = −VmIm sinωt cosωt

Rearranging gives: p=− 1 2 VmIm (2sinωt cosωt).