rms Voltage: what is it and why do we care? Chris Greacen DC voltages aren't ambiguaous. If you measure the voltage of a resting battery, the voltage stays put from one fraction of a second to the next. On the other hand, if you watch the voltage in an outlet of an American grid connected home, you'll see the voltage vary form 165 volts to -165 volts and back again, forming a sine wave sixty times a second. Mr. Josephson's article, (page XX), looks a little at why anyone would want to transmit electricity by sloshing it back and forth like this. But with the decision to use ac power comes, among other things, the question "how do we measure the voltage?" We could say the voltage an American household line is 165 volts, since the peak of the sine wave reaches 165 volts. Or just as easily we could say the voltage is -165 volts, since that's as low as it goes. Maybe better yet, we could say the voltage on a regular household line is 330 volts since there's 330 volts difference between the peak and the trough of each wave. Of course, most of the time the voltage isn't at the positive or negative peak, it's increasing or decreasing somewhere inbetween. How about taking the average? Well the average is zero: every bulge above zero in the sinewave is matched exactly by a bulge below zero. If you add them up, you get zero. What, then, is "the voltage" household ac current? Usually the figure "110 vac" or "120 vac" or more precisely, "117 vac" (volts alternating current) is used to describe household current. Why 117 vac? The 117 vac refers to the "root mean square" or rms voltage of the waveform. What does "root mean square mean? INSERT SinewavePICT The Effective Voltage The simplest answer is that 117 vac rms is the "effective" voltage of the sinewave. Imagine two identical electrical resistance heaters (excuse this wattage wasting example). One heater is powered by regular household 117 vac rms current, and other one runs on direct current whose voltage you can adjust. The two heaters will put out the exactly the same amount of heat, and draw the exact same power, only when the direct current voltage is adjusted to 117 Volts DC. What is rms Really? It's a bit cryptic, but the name contains it all: take the (square) root of the mean (average) of the square of the waveform. Here's how it works. Pick a wave form: INSERTArbwavechart Square it. At each point on the graph, multiply the value times itself and plot the result. The deep negative dips are now strong positive ones since a negative number times a negative number is a positive number. Notice that the units on the vertical axis are now volts squared, not volts. INSERTArbWaveSqrdChrt.1 Take the mean (average) value of this new wave. As long as the original wave was not zero everywhere, the average of its square will always be greater than zero. Finally, take the square root of this average. Notice that units are back to volts. 94.28 vac is the rms voltage of this weird waveform. You can see that the rms value depends a lot on the shape of the waveform. For a perfect sine wave, the rms value is 1/(sqrt 2) of the peak value (117 = 165/(sqrt 2)) INSERTArbWaveWithAveChrt Why Go Through All This Trouble? We go through all this trouble because the rms voltage is the "effective" voltage. Similarly rms current is the "effective" current. Using rms values lets us use Ohm's Law "V = I R" (voltage = current times resistance) and Joule's heating law "P = I V" (Power = current times voltage) to talk about the entire waveform, just as cleanly as they work for the unambiguous case of DC voltages and currents. [Of course Ohm's law and Joule's law could be applied to any instantaneous voltage and current in the ac waveform, but this only gives us information about what's happening at that instant. For example, just knowing and working with peak voltages and currents would only tell us what is happening the tiny fraction of a second when the peaks are happening. The heat at these peaks: Ppeak = IpeakVpeak is considerably more than you'll actually feel coming off the heater. The heat you'll feel is the average heat: Pave = IrmsVrms. Want proof? Here's a demonstration that the rms values are "effective values" Ä in other words, the average power consummed is Irms Vrms. How Are You Sure the rms Value is the Effective Value Now you've seen how you get an rms value from a waveform, but so far I've asked you to take it on faith that the rms values are the "effective" values. Here's a demonstration that the heat you'll feel coming off a heater is equal to the rms current times the rms voltage.