Root mean square, or RMS, is a statistic that is calculated from a set of numbers. Other common statistics, that may be more familiar, are averages and standard deviation. Each of these statistics is able to tell you something about the set of numbers, which can sometimes be more important than knowing each number in the set.

It is prudent to understand what an RMS value is, how it is calculated, and why it is useful, before tackling a specific example. Once these concepts are clear, the calculation can be demonstrated with a specific example of calculating RMS power for an electronic circuit or device.

#### TL;DR (Too Long; Didn't Read)

An RMS value for a sinusoidal function is calculated by multiplying the peak or maximum value by the square root of 1/2. Thus the RMS value is higher in magnitude than the average value.

## How is a Root Mean Square Statistic Calculated?

The name of the quantity is very conveniently telling you exactly what to calculate: the square root of the mean of the set, after squaring each element in the set. A general procedure for calculating RMS values will likely help you understand the statistic.

To calculate the RMS for the set *A*, which has *N* elements in it, called *a _{i}*. The steps are:

Step 1: Individually square each number in the set of numbers, such that the elements are now *a*_{i}^{2}.

Step 2: Calculate the mean or average of the set. The general formula for average an average, *B _{av}* is:

Because we are calculating RMS, the elements have been squared, in step 1. Thus, the average *A _{av}* is:

Step 3: The RMS value of the set A can be calculated very easily:

## Why Calculate an RMS Value?

There are many reasons for calculating the RMS value of a set or function, instead of a simple average. Specifically, for distributions that oscillate around zero, calculating an RMS value is a superior statistic and more informational.

Consider a sine function; sine is defined to oscillate at unit amplitude about 0. That means that the average of a sine function is 0, if you average over a full period or any integer number of full periods.

This is very easy to see if you plot the sine function over a full period; from 0 to π, the function is positive, and from π to 2π, it is identical in value, but negative. If you add a set of values that are identical but have opposite signs, the sum is o, and thus the average is 0.

However, the RMS value of a sine function is not 0. Therefore, **the RMS value is able to tell you information about the magnitude of the elements in a set, or amplitude of some function**, regardless of the sign of the element values.

## RMS Values for Electronics and Circuit Design

By now, the way RMS values are calculated should be clear. The use of RMS values is prevalent in electronics and circuit design, due to the use of alternating current. Alternating current is a sinusoidal function of time, such that in some period of time *T*, the sine wave completes one full cycle.

To calculate the RMS power in units of watts. To calculate RMS power, it is necessary to determine how to calculate the power from a circuit.

For a simple circuit, the power dissipated by the circuit is calculated: * P = I^{2}R*, where

*I* is the current through the circuit, in units of Amperes, or Coulomb/sec, and

*R* is the resistance in Ohms.

For a DC current, the power is very easy to calculate because the current is constant, and the resistance is known. However, how are peak, average, and RMS power values calculated for alternating current?

## Calculating RMS Values for Sinusoidal Continuous Functions

In order to calculate the RMS value for a sinusoidal current that varies with time, **I(t) = I**_{0} **sin(t),** the period of the function is needed. For the given current, the period is 2π. For a current of the form I(t) = I_{0}sin(ωt), the period is 2π/*ω*.

Just like the procedure for calculating an average of a set up numbers, the elements of the set must be added up, and then divided by the number of elements in the set. The same can be done for a continuous function, by integrating the function over some period, and then dividing resulting value by the period.

However, for calculating an RMS value, you need to square the elements in the set. Therefore, simply calculate the integral of the squared function:

Just as before, the RMS value is simply

For a typical sinusoidal function, the period is 2π, therefore *A _{av}* simplifies to

*I*/2. Because the amplitude, or maximum value of the function, of a sinusoidal function is simply the coefficient, it is clear why the RMS value of any continuous function is the peak value multiplied by the square root of 1/2.

_{0}The square root of 1/2 is approximately 0.7071.

## What is a Peak Power to RMS Calculator?

As we calculated above, an RMS value is related to the maximum value that the function can reach, or the peak value. Therefore a peak power to RMS calculator would determine the RMS power from a power function.

Peak power can either be calculated by determining the peak current, and then calculating the peak power using the power equation: *P = I ^{2}R.*

For a sinusoidally varying current, we determined that a peak Power to RMS calculator would simply multiply the peak power by 0.7071.

For any other current distribution, the RMS value must be determined by determining the square mean (by integrating the square of the function over a full period and dividing by the period), and then taking the square root of the resulting value.

## How to Amplify your Favorite Music

So you've bought some new speakers and are ready to listen to your music with the sound turned up. However, the receiver that you may be using to provide the source of music to the speakers may not provide enough power to the speakers. An amplifier is a device that takes the original signal and converts it to a higher power, in order to maintain the sound quality.

An amplifier RMS calculator could help you determine the correct audio set up.

In general, the RMS power that the amplifier generates in watts will be listed on the amplifier, and tells you how much continuous power it supplies. If it is not listed, but the current is, you can calculate the RMS power of the amplifier as described earlier. This is your amplifier RMS calculator.

Subwoofers require more power, and may require a separate amplifier than the rest of your speakers for this reason.

The RMS power of the amplifier should match the power rating on the speaker. If the RMS power of the amplifier does not match the power rating on the speaker, this could cause overheating of the speaker, or damage to the speakers.

References

Tips

- For greater accuracy, use an oscilloscope fitted with an AC current probe instead of a multimeter when current measurements are small. The oscilloscope should also be used in cases where the circuit cannot be opened to insert a multimeter for the purpose of taking RMS current readings.

About the Author

Lipi Gupta is currently pursuing her Ph. D. in physics at the University of Chicago. She earned her Bachelor of Arts in physics with a minor in mathematics at Cornell University in 2015, where she was a tutor for engineering students, and was a resident advisor in a first-year dorm for three years. With this experience, when not working on her Ph. D. research, Gupta participates in STEM outreach activities to promote young women and minorities to pursue science careers.