# Power in ac circuits

The power consumed in any circuit is given by the equation:

Power = voltage x current = vi

Now for a resistor: v = vo sin (wt) and i = io sin(wt)

The power dissipated in a resistor is therefore:

P = iv = io vo sin (wt) sin(wt) = ½ io vo sin2(wt)

Now for a capacitor: v = vo sin (wt) and i = io cos(wt)

The power dissipated in a capacitor is therefore:

P = iv = io vo sin (wt) cos(wt) = ½ io vo sin (2wt)

However the average value of sin (2wt) is zero and therefore the power dissipated in a purely capacitative circuit is also zero.

The same type of argument can be used to show that the power dissipated in a purely inductive circuit is also zero. For this reason capacitors and inductors are often used in a.c circuits to limit the current since they do not waste any energy.

Example problem
A 220 W resistor is connected across an AC Voltage source where V = 150 x sin[2w60t]
What is the average power delivered to this circuit?

For an a.c signal the formula is:

V = Vosin(2wft) Also power = V2/R
Therefore: The peak power is given by:
Power = Vo2/R = 1502/220 = 102.3 W
The ROOT MEAN SQUARE POWER is given by Power = Vrms2/R

Power= 1502/[2x220] = 51.14 W