∫∫∫∫ How to Calculate RMS

Current, Voltage, and Power all flow through electrical lines as sine waves, and hence engineers frequently refer to the RMS of a sine wave. Several people have asked us for a clear explanation of why the RMS of a sine wave is simply the peak divided by the square root of two, so here it is!

RMS stands for the square Root of the Mean of the Square. Why do we want that instead of a simple average? Well, sine waves alternate around zero, so their average is zero yet they still clearly deliver power, energy, and electric shocks if you are uncareful or unlucky. There are two ways to deal with this problem: 1) take the average of the absolute value, or 2) square everything, take the average, and then take the square root again. Absolute value functions end up being remarkably tricky in equations and formulas, so everyone uses the second method, leading us to RMS. Here’s how to calculate it.

The “thing” we’re interested in is the RMS of . Let’s follow the instructions. First, we square it, getting: . Next, we take the mean. The mean is the sum of all values divided by the number of values. A sine wave is continuous, so to get the sum of all values we have to integrate, then divide by the total number of values over which we evaluated the integral: . (The period of a sine wave is 2π). Lastly, we take the square root of all of that:

Okay, so now we know a complicated formula for what we’re after, we just need to solve it. Here’s how.

First, drop the zero and remove the constants from the integral to make it simpler:

Now we need to deal with the integral. We would need a fancy computer to integrate sin2(x), so we will use some good old fashioned trigonometry identities to put it in a form we can easily do ourselves instead. First, . Secondly, . Putting these together gives us  , which we can now substitute into the integral.

Integrating the first half of the integral gives us: , which we can then substitute into the rest of the expression, getting:

The remaining integral within the square root is zero since cosine integrated is sine, and the value of sine at both bounds is the same: , so now we have our final formula with no integrals left in it:

And we’re done! The RMS of a sine wave is simply equal to the amplitude of the sine wave divided by the square root of 2. This is a critical result for understanding the power flowing through our electric grid.

2 Responses to ∫∫∫∫ How to Calculate RMS

  1. geoff hardy says:

    I do enjoy the way you have framed this particular problem and it does present us some fodder for thought. Thank you for this exceptional piece.


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