Consider the following circuit:

From what we know about AC voltage, the instantaneous voltage across the resistor is

V_{R}= V_{0}sin(ωt)

where ω is the angular frequency. If the instantaneous voltage is v, then the instantaneous current, i, is

i = v/R = V_{0}sin(ωt)/R

The maximum current, I_{0}, is when sin(ωt)=1, so

I_{0}= V_{0}/R

So,

i = I_{0}sin(ωt)

If p is instantaneous power, and P_{0} is maximum power, then

p = iv = i^{2}R = (I_{0}sin(ωt))^{2}R = I_{0}^{2}R sin^{2}(ωt) P_{0}= I_{0}^{2}R p = P_{0}sin^{2}(ωt)

The following graph shows the relationship between the instantaneous values of voltage (v), current (i), and power (p):

Fischer-Cripps. A.C., *The Electronics Companion.* Institute of Physics, 2005.

Copyright © 2014 Barry Watson. All rights reserved.