If you measure a DC voltage, and want to get some idea of how “big” it is over time, it’s pretty easy: just take a number of measurements and take the average. If you’re interested in the average power over the same timeframe, it’s likely to be pretty close (though not identical) to the same answer you’d get if you calculated the power using the average voltage instead of calculating instantaneous power and averaging. DC voltages don’t move around that much.
Try the same trick with an AC voltage, and you get zero, or something nearby. Why? With an AC waveform, the positive voltage excursions cancel out the negative ones. You’d get the same result if the flip were switched off. Clearly, a simple average isn’t capturing what we think of as “size” in an AC waveform; we need a new concept of “size”. Enter root-mean-square (RMS) voltage.
To calculate the RMS voltage, you take a number of voltage readings, square them, add them all together, and then divide by the number of entries in the average before taking the square root: . The rationale behind this strange averaging procedure is that the resulting number can be used in calculating average power for AC waveforms through simple multiplication as you would for DC voltages. If that answer isn’t entirely satisfying to you, read on. Hopefully we’ll help it make a little more sense.
When it comes to averages, the ideas of “big” and “little” for AC and DC voltages are fundamentally different. DC waveforms are roughly constant, and what matters is the distance from zero. AC waveforms are always wiggling around a center point, and this is often ground. If the waveform is symmetric, and you take enough samples, it’s going to average out to zero.
One way to measure the size of AC voltages is to take the maximum and minimum over time: the peak-to-peak voltage. Another possibility would be to take the absolute value of each voltage and average them together. That works too. A third choice is to square all of the individual voltage measurements before adding them up. This has the same effect as taking the absolute value — all of the individual terms are positive now and don’t cancel out — and has the additional side-effect of making the big values bigger and the small values smaller. Which do we choose?
Using the squared voltages in the average gets the physics right. If you’re interested in the power that you can get out of the AC signal, it’s the squares of the voltage that are relevant anyway. Let’s pretend you’re driving a resistive load for now — maybe you’re heating your apartment or using an electric stove — and do a tiny bit of algebra.
Remember that power is equal to the current flowing through our imaginary device times the voltage being dropped across it:
P = IV. And who could forget Ohm’s Law?
V = IR or
I = V / R. Put them together, and
P = V² / R. The power in the system, at any given instant, is proportional to the voltage squared. The average power over time is thus proportional to the average of the squared voltages. Sounding familiar? Since the average of squared instantaneous voltages is in units of volts squared, taking the square root at the end (“root of the mean of the squares”) brings it on home.
The same logic holds for RMS current measurements as well. Substituting Ohm’s Law the other way, you get
P = I² R and power is proportional to current squared. Average current in a balanced AC waveform is zero, but RMS-averaged current, squared, is proportional to power.
Again, the big takeaway is that RMS voltage is the measure of average AC voltage or current that lets you pretend it was a DC average to get the average power. By doing the squaring inside the average, you avoid voltages of opposite signs cancelling, and by taking the square root at the end, it gets the units right.
If you have an AC voltage that’s riding on top of a DC component, the RMS value still delivers. In that case, the squared DC component adds up
n times before dividing by
n again, and you get something like this: , where
v is just the pure AC voltage.
Rules of Thumb
One place you’ll see RMS voltages is in mains power. Indeed, the 120 V in the US (or 230 V in the EU) coming out of your walls right now is an RMS figure. For sine waves, like what you get from the electrical company, the peak voltage is a factor of
sqrt(2) higher than the RMS voltage. The peak voltage in the States is something like
120 V * sqrt(2) = 170 V, and the peak-to-peak is 340 V. That’s 650 V peak-to-peak in Europe; yikes!
This also means that if you’re lacking an RMS meter and need a quick-and-dirty estimate of something that’s sine-wave-like, you can take the amplitude and divide by 1.414, or take the peak-to-peak and divide by twice that.
Another waveform you might care about is the PWM’ed square wave that we often use to drive motors from microcontrollers. Clearly, if you alternate between zero volts and twelve volts, it’s only supplying power to the motor when it’s at twelve volts. Correspondingly, you won’t be surprised to hear that the RMS voltage of a PWM waveform is the square root of the duty cycle times the on-voltage.
Wikipedia has you covered for triangle waves and other funny waveforms.
It turns out that you’re often concerned with squared quantities. Kinetic energy is proportional to speed squared, for instance, so RMS speed is used in calculating temperature from the average velocity of molecules in a gas. If you have a measurement procedure that may be right on average, but you’re worried about the spread of the results as well, you might like to minimize RMS error. The statistician’s concept of standard deviation is similar, with the average value subtracted off beforehand.
You even calculate the hypotenuse of a triangle by the same procedure, just without dividing by
n. (OK, that’s a stretch, but square roots of sums of squares are everywhere!) I’m going to leave it to the mathematical philosophers among you to duke it out in the comments as to why the L2 norm appears so often. For the electrical hackers out there, it’s enough to remember the Ohm’s law rationale: when you’re interested in power, you’re interested in squares.