Power (AC)

The discussion of power in Chapter 13 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.
Defining the power delivered to a load.
Fig. 1: Defining the power delivered to a load.
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, $$ p = vi$$ In this case, since v and i are sinusoidal quantities, let us establish a general case where $$v = V_m \sin (wt + \theta)$$ $$i = I_m \sin (wt)$$ 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 $$ p = V_m I_m \sin (wt + \theta) \sin (wt)$$ If we now apply a number of trigonometric identities, e.g, $$ \sin(a+b) = \sin a \cos b + \sin b \cos a \tag{1}$$ replacing $a + b$ by $wt+\theta$, we get $$ \sin(wt+\theta) = \sin wt \cos \theta + \sin \theta \cos wt \tag{2}$$ now multiplying eq(1) by $\sin wt$, $$ \sin wt \sin(wt+\theta) = \sin^2 wt \cos \theta + \sin \theta \sin wt \cos wt \tag{3}$$ Where $$ \sin^2 wt = {1 - \cos 2wt \over 2}$$ and $$ \sin wt \cos wt = {1 \over 2} \sin 2wt$$ Hence eq(3) becomes,
$$ \sin wt \sin(wt+\theta) = {1 - \cos 2wt \over 2} \cos \theta + \sin \theta {1 \over 2} \sin 2wt \tag{4}$$
Now the power equation can be written as, $$ \begin{split} p &= V_m I_m \sin (wt) \sin (wt + \theta)\\ &= V_m I_m [{1 - \cos 2wt \over 2} \cos \theta + \sin \theta {1 \over 2} \sin 2wt]\\ &= {V_m I_m \over 2} [(1 - \cos 2wt) \cos \theta + \sin \theta \sin 2wt]\\ &= VI(1 - \cos 2wt) \cos \theta + VI \sin \theta (\sin 2wt)\\ \end{split} $$ $$ \bbox[10px,border:1px solid grey]{p = VI(1 - \cos 2wt) \cos \theta + VI \sin \theta (\sin 2wt)} \tag{4}$$ 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.