Why is the Power Equation Non-Linear

Motivation:

When studying circuits for the first time, Ohm's law is introduced and it is easy to grasp the linear relationship between voltage (V) and current (I). The resistor value (R) is a constant value and it is the slope (m) of the linear equation that we are familiar with.

\text{(Eq. 1)} \; \; V= R*I \longrightarrow y = mx

Then the power equation is introduce, and the literature claims that the power equation is non-linear.

\text{(Eq. 2)} \; \; P=V*I \not\equiv y = mx

This non-linear claim is easy to see if we substitute (Eq. 1) in (Eq. 2), leading to two different ways to write the power equation where it is easy to see why power is non-linear.

\text{(Eq. 3)} \; \; P=\frac{V^2}{R} \\
or \\
\text{(Eq. 4)} \; \; P=R*I^2

However, it can still be confusing to understand this claim in (Eq. 2) since there are no elements raised to the power of 2:

P=V*I \not\equiv y = mx

Explanation:

Let's clarify this confusion by color coding the equations above.

Let's start with the generic linear equation:

And relate it to Ohm's law:

If we color code (Eq. 3) and (Eq. 4):

Notice that all the equations that we color coded so far have 3 terms with different colors. 1 dependent variable in blue, 1 constant in red and 1 independent variable in green. For this example, the green variable is used to define if a function is linear or non-linear as long as we have one red and one blue term in the equation.

Let's color code (Eq. 2):

Since the term in green has an exponent of 1, we may be inclined to say that this equation is linear. However, remember that this is only true if the green term is being multiplied or divided solely by a red term and in this case we have a blue term instead.

We can't see from (Eq. 2) that power is non-linear until we substitute (Eq. 1) in (Eq. 2).

I hope this helps understanding why power is a non-linear equation from (Eq. 2) 🙂