When Al Gore was the vice president, he created the Hammer Award to recognize people in the U.S. government who helped eliminate inefficiency and waste. The award consisted of a $6 hammer wrapped with a red, white, and blue ribbon, and mounted in a glass encased frame. Apparently the idea for the award came at least in part from the discovery that the U.S. Navy once paid $436 for a single hammer.
I’m all for cutting government waste, but you don’t go messing with someone’s tools. You can have the $640 toilet seat and the $670 vinyl armrest pad, but don’t touch the $748 pliers or the $599 drill. You gotta have the right tools for the job, and sometimes the right tool costs a little more.
Take, for example, a voltmeter. You can pick up an analog multimeter at Radio Shack for as little as $16.99 or you can get a digital one for as little as $19.99. On the other hand, you can spend as much as $420 for a Fluke 83 volts or $549 for a Fluke 289 digital multimeter. Of course, there is a world of difference between these meters, but one of the most important differences is how they measure AC voltage. The cheaper meters in this example are voltage averaging, RMS calibrated meters, while the two more expensive ones are “true RMS” meters. In some instances the AC voltage they read will be the same but in others they will differ by as much as 40 percent. The reason for this difference has to do with the difference between the average voltage and the RMS voltage.
A single cycle of an AC waveform has a positive half and a negative half. In a pure sine wave they are mirror images of each other and the two halves average to zero. But that doesn’t mean it transfers no power. Obviously, the average value of an AC voltage waveform over one cycle doesn’t convey enough information.
A more useful way to convey the measure of AC voltage is to use the RMS value, which is the value of AC voltage that would transfer the same amount of power to a resistive load as a DC voltage. 120 VAC RMS, for example, would heat up a resistor just as much as 120 VDC. RMS is an acronym that stands for “root mean square,” or in plain English, if you take an average of the square of the voltage and then take the square root of the answer, you have the RMS. The reason we average the square of the voltage is because the squaring operation gets rid of the negative values by inverting the negative half cycle and it creates a direct relationship between voltage and the power. (P = V2 ÷ R).
The RMS voltage is different than the average voltage, even in a fully rectified sinewave where we flip the negative half cycle and made it positive. If you work out the numbers, you’ll find a fixed relationship between the peak voltage, the RMS voltage, and the average voltage in a sine wave. These relationships are illustrated by Fig. 1:
VRMS = 0.707 × Vpeak and VRMS = 0.636 × Vpeak
A true RMS voltmeter samples several points in a cycle and does a calculation before it spits out the reading. A voltage averaging, RMS calibrated voltmeter simply rectifies the waveform (flips the negative half cycle to make it positive) and averages the voltage over one cycle; then it uses a multiplier to scale it so that it will match the RMS value. The multiplier is the ratio of the RMS value to the average value, which is:
1.112 (0.707/0.636 = 1.112).
In North America, for example, most of the household AC is 169.7 volts peak; the RMS voltage is 120 volts and the average voltage of a fully rectified single cycle is 107.9 volts. If we took this average and multiplied by 1.112 then the result would be 120 volts. So far, so good…as long as the waveform we’re measuring is a pure sinewave.
If the waveform is not a pure sinewave — if it has any distortion — then the multiplier will give an incorrect reading. Just how incorrect it is depends on how much the waveform differs from a pure sinewave.
Take, for example, a sinewave that has been run through a conventional forward-phase dimmer, the most common variety in the entertainment lighting industry. Figure 2 shows the output of the dimmer at a level of about 50 percent (given a linear dimming curve).
If the peak is 169.7 volts, then the RMS voltage is 84.85 volts and the average voltage is 54 volts. The multiplier in this case should be 1.57 (84.85/54 = 1.57). But an averaging, RMS calibrated meter uses a multiplier of 1.112, so using this meter will give you a reading of 60 volts instead of 84.85 volts. That’s an error of 29.3 percent. The actual error could be even higher depending on the waveform.
{mosimage}We don’t typically go around measuring dimmed voltage levels, but this illustrates how a distorted sinewave can lead to incorrect measurements using an averaging, RMS calibrated meter. And every time you dim a load, current flows through the neutral back to the feeder transformer in a distorted waveform. Also, switch-mode power supplies (a.k.a. “electronic power supplies”) alter the waveform, leading to the possibility of inaccurate measurements of voltage and current. In fact, a non-sinusoidal waveform is probably much more common in our electrical systems than a pure sinewave, so it’s very important to use a true RMS voltmeter and to understand why you spent more money for it.
The truth is that you can buy a true RMS voltmeter at Radio Shack for as little at $89, but don’t tell my wife.
