How many times have you poked around a circuit with a digital multimeter (DMM), read an AC voltage value, then gone on your merry way? Ever wonder if your meter is giving you peak value, average value, root mean square (rms) value, or something in between? It’s the something in-between part that can get you into trouble, and the trouble usually happens with rms measurements.
AC RMS Measurements: Their Deceptive Simplicity
Since measuring true rms values is rather complicated, it’s a good idea to remind ourselves why it’s worth the trouble. True rms is the only AC voltage reading that doesn’t depend on the shape of the signal, which means it often is the most useful measurement for real-world waveforms.
Often, rms is described as a measure of equivalent heating value, relating it to the amount of power dissipated by a resistive load driven by the equivalent DC value. For example, a 1-Vpk sine wave will deliver the same power to a resistive load as a 0.707-VDC signal. If you can get a reliable rms reading on a signal, you’ll have a better idea of the effect it will have in your circuit.
Figure 1 shows four common voltage parameters. Vpk and Vpk-pk are simple enough. Vavg is the average of all the instantaneous values in one complete cycle of the waveform.
For a sine wave, the negative half of the waveform would cancel out the positive half and average to zero over one cycle. This wouldn’t provide much insight into the signal’s effective amplitude, so most meters compute Vavg based on the absolute value of the waveform. For a sine wave, this works out to Vpk × 0.637 (Figure 2).
Vrms, the quantity of most interest here, can be derived by squaring every point in the waveform, finding the average (mean) value of these squares, then taking the square root of the average. Pure sine waves provide a couple of shortcuts: just multiply Vpk × 0.707 or Vavg × 1.11. This approach is taken in inexpensive peak-responding or average-responding meters.
These scaling factors apply only to pure sine waves. The result will be misleading answers for every other type of signal. And these aren’t always minor errors, either. Depending on the meter and the signal, it’s alarmingly easy to encounter errors of 20%, 30%, and 40% or more with a meter that’s not really designed for the task.
The relationship between Vpk and Vrms is a key factor in measurement accuracy. The ratio of Vpk to Vrms is the crest factor. The higher the crest factor, the more difficult it becomes to make an accurate AC measurement.
High crest factors present two sticky measurement problems. The first involves input range. Imagine a pulse train with a very low duty cycle but a relatively high peak amplitude. Signals like this force the meter to simultaneously measure a high peak value and a much lower rms value, possibly creating overload problems on the high end and resolution problems on the low end.
The second challenge is the amount of higher-frequency energy in the signal. In general, high crest factors indicate more harmonics, which can cause trouble for all meters. Those particularly affected are the peak- and average-responding meters trying to measure rms.
Three-Step Guide to Better RMS Measurements
Given the importance—and difficulty—of rms measurements, what are the best ways to proceed with your day-to-day measurement tasks? Adhering to the following three steps will help achieve better results.
Step 1: Understand what your meter is doing.
Multimeters are not created equal when it comes to measuring rms values. To better understand their differences, here is a brief look at the operational advantages and disadvantages of four major technologies used in multimeters. The first three operate by converting AC to DC; the last one digitizes the analog input signal and then computes rms.
Thermal AC-to-DC Converters. This older approach to rms measurements is based on the notion of the equivalent heating value. The AC signal heats a thermocouple, then the DC section of the meter reads the thermocouple output. The advantages are wide bandwidth and the capability to handle very high crest factors, meaning it can deliver true rms for a wide variety of real-world signals.
The disadvantages of this approach are cost and a lack of flexibility in trading off measurement speed with low-frequency accuracy. For these reasons, the technique isn’t being used in the latest-generation DMMs.
If you need to measure high bandwidth and high crest-factor signals with great accuracy, you may want to search for one of these thermal models. If you’re interested in a more contemporary instrument, you may find that the analog AC-to-DC converter technology offers acceptable bandwidth and crest-factor performance, coupled with superior responsiveness. If your needs lean more toward high accuracy, check out the digital sampling method.
Peak and Averaging AC-to-DC Converters. Inexpensive meters, in particular inexpensive hand-held meters, usually derive rms levels from either peak or average values. These approaches deliver true rms only for pure, undistorted sine waves. If you need true rms on real-world signals, these meters are not a viable option.
Analog AC-to-DC Converters. An analog computer here in the digital age seems a bit out of place, but that’s the approach used in many mid-range and high-end DMMs. They use a chain of analog circuits that compute the square, then the mean, then the square root of the mean to deliver true rms for nearly all signal types. Thanks to advances in integrated circuitry, these DMMs are small, accurate, and still relatively inexpensive.
Digital Sampling. The last method leverages sampling techniques similar to those used in digital oscilloscopes to create a set of data points that then are sent through an rms algorithm. Synchronous sampling uses multiple passes to capture a signal as shown in Figure 3. Each subsequent pass is delayed by a small amount, and with enough passes, the signal can be digitized with very high resolution.
The advantages of this technique are true rms on a wide range of signals, high accuracy, and the capability to create very fast, effective sampling rates and wider bandwidths, even with fairly slow analog-to-digital converters. This method, however, only works with repetitive signals.
You probably can sense from these descriptions that multimeters are a clear case of “you get what you pay for.” If accurate rms measurements are important and you’re likely to run into pulse trains and other complicated signals, a true rms meter is the only way to go. On the other hand, you can save some money with a peak- or average-responding meter. Just keep in mind what these meters can and cannot do.
Step 2: Understand how the signal affects the quality of the measurement.
Let’s explore several different signals, starting with a sine wave. The crest factor for a pure sine wave is 1.414, and an average-responding meter can provide accurate rms simply by scaling the value of Vpk. With a Vpk value of 500 mV, we should expect an rms value in the neighborhood of 350 to 357 mV (the range accounts for the inaccuracy of the signal generator used). Sure enough, a true rms meter reads the signal as 353.53 mV. Its less expensive average-responding cousin reads the signal as 351 mV.
Unlike the pure sine wave, the triangle wave in Figure 4 has some higher-frequency energy, so the crest factor of 1.732 comes as no surprise. Dividing the peak value by the crest factor yields an expected rms value of roughly 290 mV. Now, the average-responding meter starts to get into trouble, reading the signal as 276 mV, a 4% error compared to the true rms meter’s reading of 288.68 mV.
Now let’s make it really interesting by looking at pulse trains, where the crest factor depends on the duty cycle. You can get a close approximation of crest factor with the formula:
where: CF = the crest factor
T = the period of the waveform
t = the on portion of that period
This also is equal to the square root of the reciprocal of the duty cycle. So, for the pulse train in Figure 5, which has a 2% duty cycle, the crest factor is the square root of 50, or 7.071.
Unlike sines and triangles, where the rms value is simply Vpk divided by the crest factor, computing the AC rms value for a pulse train is a bit more complicated:
As a result, the theoretical rms value of our 2-Vpk pulse train with 2% duty cycle in Figure 5 is roughly 280 mV. Even in this case, which is outside its specified performance range, the true rms meter reads 275.9 mV. On the other hand, the average-responding meter reads 73 mV, a 74% error. This is an extreme example, but it provides a clear picture of what high crest factors can do to your measurements.
Let’s consider one more waveform—the noisy, messy sine wave shown in Figure 6. The true rms meter pegs it at 348.99 mV, which is close to the digital scope’s measurement of 345 mV. The average-responding meter puts the value at 273 mV, an error of more than 20%. Again, this is a crest-factor problem, with the signal presenting a fair amount of high-frequency energy that the average-responding meter doesn’t take into account.
Step 3: Avoid these common measurement traps.
Now that you know how your meter and your signals are behaving, the final step is to avoid some common traps that can affect rms measurements. We’ve touched on some of these already, and you’ve probably run into many of them before.
Measurements below full scale. Most meters specify AC inputs down to 5% or 10% of full scale (some go as low as 1% of full scale). For maximum accuracy, measure as close to full scale as you can. You might need to override autoscaling in some cases if a manual setting will help maximum the input range.
AC and DC Coupling. This is a simple issue that can be easy to overlook when you are in a hurry. If your meter is AC coupled (or has selectable AC coupling), it inserts a capacitor in series with the input signal which blocks the DC component in your signal. This might be desirable or it might not be, depending on the signal and what you are trying to accomplish.
If you are expecting to include the DC component, but the meter is AC coupled, the results can be dramatically wrong. By the way, if you need to measure a small AC signal riding on a large DC offset but your meter doesn’t provide AC + DC directly, you can measure the AC component using AC coupling and measure the DC component separately. Then add the two using rms addition:
Saturation Problems With High Crest-Factor Signals. In addition to the problems they cause with high-frequency content, high crest-factor signals also can wreak havoc on your input range. Think back to that pulse train with a 2% duty cycle. Its 7+ crest factor means that the peak value is more than seven times greater than the rms value. That means your meter needs to provide adequate amplitude resolution for the low rms value without saturating on the high peak value.
To make this worse, you generally don’t get an overload indication with crest-factor saturation, either. Bottom line: check your meter’s specifications for maximum crest factor and don’t exceed them.
Bandwidth Errors. Signals that are rich in harmonics can yield low-reading measurements if the more significant of these components aren’t included in the measurement. Again, the instrument’s data sheet will let you know how much bandwidth you have to work with. Then you need to make sure your signals don’t exceed this.
Self-Heating Errors. High voltages can heat up the meter’s signal-conditioning components, leading to offset measurement values. Pay attention to the maximum input voltage; if you exceed it, give the meter time to cool down before making another measurement.
Settling Time. By definition, rms measurements require time averaging over multiple periods of the lowest frequency being measured. Consequently, if you are not concerned about low frequencies in a particular measurement and your DMM has selectable averaging filters, switch to a faster filter.
The More You Know, the Better Off You’ll Be
So, the bad news is that AC rms measurements are more complicated than they might seem at first glance. The good news is that a little bit of learning goes a long way. If you haven’t already, verify the crest factor, bandwidth, and other limitations noted in your DMM’s data sheet. Stay within those limits, at least as much as possible based on how well you know the signal.
If an AC rms reading doesn’t make sense, don’t automatically assume there’s something wrong with your circuit—the trouble might be in the measurement. On the bright side, a quality meter used within its limits should deliver consistently dependable measurements.
About the Author
Barry Scott has spent more than nine years with Hewlett-Packard in various support and marketing roles. Currently, he is a digital multimeter/data acquisition product manger for HP’s Electronic Measurements Division. Mr. Scott graduated from Oregon State University with a B.S. in electrical engineering. Hewlett-Packard, Electronic Measurements Division, 815 SW 14th St., Loveland, CO 80537, (970) 679-3551
Copyright 1999 Nelson Publishing Inc.
February 1999