Power (AC)

The discussion of power in Chapter 12 (Average Power) included only the average power delivered to an ac network. We will now examine the total power equation in a slightly different form and will introduce two additional types of power: apparent and reactive. For any system such as in Fig. 1, the power delivered to a load at any instant is defined by the product of the applied voltage and the resulting current; that is, In this case, since v and i are sinusoidal quantities, let us establish a general case where The chosen v and i include all possibilities because, if the load is purely resistive, $\theta = 0$. If the load is purely inductive or capacitive, $\theta = 90$ or $\theta = -90$, respectively. For a network that is primarily inductive, v is positive (v leads i), and for a network that is primarily capacitive, v is negative (i leads v). Substituting the above equations for v and i into the power equation will result in If we now apply a number of trigonometric identities, e.g, replacing $a + b$ by $wt+\theta$, we get now multiplying eq(1) by $\sin wt$, Where and Hence eq(3) becomes, Now the power equation can be written as, where $V$ and $I$ are the rms values. It would appear initially that nothing has been gained by putting the equation in this form. However, the usefulness of the form of Eq. (4) will be demonstrated in the following sections.