Average Power of Sinusoidal Voltage

For any load in a sinusoidal ac network, the voltage across the load and the current through the load will vary in a sinusoidal nature. The questions then arise, How does the power to the load determined by the product v-i vary, and what fixed value can be assigned to the power since it will vary with time? If we take the general case depicted in [Fig. 1] and use the following for v and i: then the power is defined by Using the trigonometric identity the function $\sin(wt + \theta_v)$ $\sin(wt + \theta_i)$ becomes so that A plot of v, i, and p on the same set of axes is shown in [Fig. 2]. the voltage or current. The average value of this term is zero over one cycle, producing no net transfer of energy in any one direction. The first term in the preceding equation, however, has a constant magnitude (no time dependence) and therefore provides some net transfer of energy. This term is referred to as the average power, the reason for which is obvious from[ Fig. 2]. The average power, or real power as it is sometimes called, is the power delivered to and dissipated by the load. It corresponds to the power calculations performed for dc networks. The angle ($\theta_v - \theta_i$) is the phase angle between v and i. Since $\cos(-\alpha) = \cos \alpha$, Defining v as equal to |$\theta_v - \theta_i$|, where | | indicates that only the magnitude is important and the sign is immaterial, we have where P is the average power in watts. This equation can also be written or, since Let us now apply Eqs. (1) and (2) to the basic R, L, and C elements.

Resistor

In a purely resistive circuit, since v and i are in phase, |$\theta_v - \theta_i$| $= \theta = 0$, and $\cos 0 = 1$, so that Or, since then

Inductor

In a purely inductive circuit, since v leads i by $90^\circ$, |$\theta_v - \theta_i$| $= \theta = 90^\circ$. Therefore,

Capacitor

In a purely capacitive circuit, since i leads v by $90^\circ$, |$\theta_v - \theta_i$| $= \theta = |-90^\circ|$. Therefore,