Root-Mean-Square (RMS)

There are different types of periodic waveforms and the main idea described in this article. applies to any periodic waveform. However, we will be focusing on the sinusoidal waveform because that is the one most common in circuits.

Let's start by exploring the average value for a full cycle.

Consider . In the graph below we can visually see that the average value of f(t) is zero.

Mathematically we can calculate the average value for a full cycle using the following formula: .

The average value in this form is not that useful. We need to find a different way to calculate the average value for periodic signals.

Since the positive part of a sinusoidal waveform is equal to the negative part, let's focus on calculating the average value of half a cycle using the same formula as in the previous section but changing the limits of integration accordingly.

Let's include the other half cycle by multiplying the result by 2.

Note: I am calling it so we can distinguish it from , but in this context means the same thing.

AC Power Calculations

Let's try to use the expression on the AC circuit from the motivation section to calculate the power dissipated by the resistor.

Measured values from Falstad simulation (Link):

The average voltage is and the average current is , which gives a power equal to .

In the motivation section, I made a claim and stated that the AC circuit and the DC circuit where equivalent. For that to be true, the power dissipated by the resistor needs to be the same for both circuits.

It is clear that using the average value (half a cycle) to calculate the power is not giving the same result ().

We need to approach this problem differently.

Instead of calculating the average value (half a cycle) of voltage and current and plug those values into , let's instead rewrite the equation to and work with the following average power equation:

Note: we can also work with , which will give . The idea is the same and will give the same answer.

The question that now rises is, what is this new expression or ?

In statistics there are different types of averages also called measures of central tendency. The most common types of averages are the mean, median and mode. However, inside the mean category, there are more options available.

It is outside of the scope of this article to cover statistics. If you want to read more about this subject you can check [REF2] [REF3] [REF4].

Equations (discrete form) for the different types of mean:

A particular interest relation to pay attention to, is between the arithmetic and the quadratic mean. Notice that the quadratic mean squares all the numbers, then takes the usual average, and then takes the square root.

Since we tried the arithmetic mean during the average power (half cycle), it would be interesting to explore if there is a way to apply the quadratic mean and see the results that we get. The cool part about this is that without too much effort the quadratic mean is already present in our new expression of the average power. Let's see how...

The continuous form representation of the quadratic mean is:

Relating the quadratic mean with the new average power equation, , we can see that this new approach of expression the average power uses the quadratic mean of voltages or currents.

In fact, the quadratic mean is also called root-mean-square (RMS).

We have a new expression for the average power:

With this new finding, let's calculate the average power using the value of a sinusoidal waveform.

AC Power Calculations

Let's now use this on the AC circuit again from the motivation section and calculate the power dissipated by the resistor.

The measured values from Falstad simulation are the same (Link):

The RMS voltage is which gives a power equal to .

We can apply the same process using the RMS current, which gives a power equal to .

It is clear that using the RMS values to calculate the power gives the same result ().

If we go back to the AC circuit from the motivation section, we can fix the measurement tools (voltmeter and ammeter) to reflect the RMS value of the circuit.

We are also in better shape now to understand why at the beginning I fixed the maximum voltage from 5V to 7.07V. If we simulate the circuit with a maximum voltage of 5V we don't get the equivalent AC circuit. We need to raise the maximum voltage of the AC voltage source to get the equivalent DC heating effect in the resistor.

Despite the fact that the average value (half a cycle) of an AC circuit doesn't correlate with the DC circuit at first glance, it is still an important parameter to calculate.

The relationship between the average value and the RMS value is called form factor: . The table below shows the RMS, AVG, and respective form factor for different types of periodic waveforms.

Why is form factor important?

AC measuring instruments are often built with specific waveforms in mind. For example, many multimeters on their AC ranges are specifically scaled to display the RMS value of a sine wave. Since the RMS calculation can be difficult to achieve digitally, the absolute average is calculate instead and the result is multiplied by the form factor of a sinusoid.

You are probably already guessing that if the sinusoidal waveform is not pure, using the average and the form factor to get the RMS is not going to be accurate.

This dictates the difference between when to use regular multimeters or true RMS multimeters. Topic for another article.

• [REF1] "RMS Voltage Tutorial", Electronics Tutorials [Article]
• [REF2] "Mean, Median, and Mode: How Visualizations Help Find What’s "Typical"", The Stats Ninja Website [Article]
• [REF3] "Visualizing the Geometric and Harmonic Means", Medium [Article]
• [REF4] "Four Kinds of “Mean”", The Math Doctors [Article]