Types of Power

There is a huge amount of false information and misunderstanding regarding types of electrical power, i.e. instantaneous, active, reactive, distortion and apparent. The following article aims to address any misconceptions and provide a solid base for any future analysis on power.

Let’s define what power is!

Instantaneous Power $p(t)$ , measured in Watts [W], is given by (1) and it is the product of voltage and current, in the time domain.

(1) $\begin{equation*} p(t) = v(t) \cdot i(t) \hspace{5mm} [W]\end{equation*}$

This equation is independent of $v(t), i(t)$ waveforms.

Let’s consider that current $i(t)$ , exits from the positive pole of the source. Positive value of $p(t)$ means that power flow is from the power source to the load, while negative means the opposite, as can be seen in the figure below.

But how is possible for a load to transfer power to a source?

There are some mechanism for this:

Energy stored in electric or magnetic fields, generated by the load, i.e. capacitors and inductors
Power transformed to electrical from another form, i.e. an electrical machine that rotates
- In this category we can include batteries, with store chemical energy and with some proper peripherals can either be charged or supply power
Power Electronic Converters that use non-linear components, such as diodes, thyristors and transistors and distort the signals $v(t), i(t)$ , resulting to power oscillations

So, how do we know how much energy or power was transferred from the source to the load, during a time interval? We integrate!

Active Power $P$ , is the integral of instantaneous power $p(t)$ , during some time interval, as defined in equation (2)

(2) $\begin{equation*} P = \frac{1}{t} \int_0^t p(t) dt = \frac{1}{t} \int_0^t v(t) \cdot i(t) dt \hspace{5mm} [W]\end{equation*}$

Now, if $p(t)$ is a periodic signal, if for example it is the product of two sinusoidal voltage and current waves, then we define $P$ , as shown in equation (3)

(3) $\begin{equation*} P = \frac{1}{T} \int_0^t p(t) dt = \frac{1}{T} \int_0^t v(t) \cdot i(t) dt \hspace{5mm} [W]\end{equation*}$

Does this makes sense from dimensions perspective? Well, sure. $\int_0^t p(t) dt$ is measured in Joules [J] and divided by time [s], $P$ is measured in $\frac{[J]}{[s]} = [W]$ .