Average Power

The average power (aka active power, real power) is defined as:

$$\begin{array}{rl}P& =\frac{1}{T}{\int }_{t_1}^{t_1+T}p(t)dt\\ & =\frac{1}{T}{\int }_{t_1}^{t_1+T}v(t)i(t)dt\end{array}$$

where $T=1/f$ is the period.

Alternatively, this can also be written in the frequency domain with:

$$P=\frac{1}{2\pi }{\int }_{\omega t_1}^{\omega t_1+2\pi }p(\omega t)d\omega t$$

If voltage and current are defined as:

$$\begin{array}{rl}v(t)& =\sqrt{2}V\cos (\omega t+{\theta }_v)\\ i(t)& =\sqrt{2}I\cos (\omega t+{\theta }_i)\end{array}$$

then:

$$P=VI\cos ({\theta }_v-{\theta }_i)$$

Active power: Active power or real power is the power whose generation, transmission, distribution, and consumption are the main objectives in power systems. Active power leads to useful work in heaters, pumps, hoists, etc.

This has units of $\text{W}$, watts.

Resistors

The average power in a resistor is given by:

$$\begin{array}{rl}{P}_R& =VI\cos ({\theta }_v-{\theta }_i)\\ & =VI\cos (0°)\\ & =VI=I^{2}R=V^2/R\end{array}$$

This is because $I$ is in phase with $V$, so ${\theta }_v-{\theta }_i=0$.

Inductors

The average power in an inductor is:

$$\begin{array}{rl}{P}_L& =VI\cos ({\theta }_v-{\theta }_i)\\ & =VI\cos (90°)\\ & =0\end{array}$$

$V$ leads $I$ by $90°$, so ${\theta }_v-{\theta }_i=90°$.

Capacitors

The average power in an capacitor is:

$$\begin{array}{rl}{P}_C& =VI\cos ({\theta }_v-{\theta }_i)\\ & =VI\cos (-90°)\\ & =0\end{array}$$

$V$ lags behind $I$ by $90°$, so ${\theta }_v-{\theta }_i=-90°$.