# Make Better RMS Measurements with Your DMM

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1 Make Better RMS Measurements with Your DMM Application Note Introduction If you use a digital multimeter (DMM) for AC voltage measurements, it is important to know if your meter is giving you peak value, average value, root-mean-square (rms) value, or something else. If the answer is something else, you may be in trouble, and the trouble usually happens with rms measurements. This application note will help you understand the different techniques DMMs use to measure rms values, how the signal affects the quality of your measurements, and how to avoid common measurement mistakes.

2 Measuring RMS Measuring rms values is more complicated than it appears at first glance. If it is complicated, why do we bother? Because true rms is the only AC voltage reading that does not depend on the shape of the signal. It often is the most useful measurement for real-world waveforms. Often, rms is described as a measure of equivalent heating value, with a relationship to the amount of power dissipated by a resistive load driven by the equivalent DC value. For example, a 1V pk sine wave will deliver the same power to a resistive load as a 0.707V dc signal. A reliable rms reading on a signal will give you a better idea of the effect the signal will have in your circuit. The scaling factors apply only to pure sine waves. For every other type of signal, using this approach produces misleading answers. If you are using a meter that is not really designed for the task, you easily can end up with significant error as high as 40 percent or more depending on the meter and the signal. The ratio of V pk to V rms known as the crest factor, is important to measurement accuracy. The crest factor is a measure of how high the waveform peaks, relative to its RMS value. The higher the crest factor, the more difficult it is to make an accurate AC measurement. Two measurement challenges are associated with high crest factors. 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. Peak- and average-responding meters that are trying to measure rms have a particularly hard time. Figure 1 shows four common voltage parameters. Peak voltage (V pk ) and peak-to-peak voltage (V pk-pk ) are simple. V avg is the average of all the instantaneous values in one complete cycle of the waveform. You will learn how we calculate V rms below. For sine waves, the negative half of the waveform cancels out the positive half and averages to zero over one cycle. This type of average would not provide much insight into the signal s effective amplitude, so most meters compute V avg based on the absolute value of the waveform. For a sine wave, this works out to V pk x (Figure 2). You can derive V rms by squaring every point in the waveform, finding the average (mean) value of the squares, then finding the square root of the average. With pure sine waves, you can take a couple of shortcuts: just multiply V pk x or V avg x Inexpensive peak-responding or average-responding meters rely on these scaling factors. Volume Time Figure 1. Common voltage parameters Volume V avg V rms V pk V pk-pk V avg Time Figure 2. V avg is calculated based on the absolute value of the waveform. V pk 2

4 Digital sampling This last method uses sampling techniques similar to those in digital oscilloscopes to create a set of data points that 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. This technique has several advantages: 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. If accurate rms measurements are important to you and you are likely to run into pulse trains and other complicated signals, a true rms meter is the only solution. 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. Tip 2: Understand how the signal affects the quality of your measurement. Let s look at several different signals, starting with a sine wave. The crest factor for a pure sine wave is 1.414, and a peak-responding meter can provide accurate rms simply by scaling the value of V pk. With a V pk value of 500 mv, we expect an rms value around 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 mv. A less-expensive averageresponding meter reads the signal as 351 mv. Figure 3. Digital sampling 2nd Trigger Point 1st Trigger Point Unlike the pure sine wave, the triangle wave in Figure 4 has some higherfrequency energy, so the crest factor of comes as no surprise. Dividing the peak value by the crest factor yields an expected rms value of roughly 290 mv. Now, the averageresponding meter starts to get into trouble, reading the signal as 276 mv, a 4 percent error compared to the true rms meter s reading of mv. Now let s look at pulse trains, where the crest factor depends on the duty cycle. You can get a close approximation of crest factor with the formula: CF ` t T where: CF = the crest factor T = the period of the waveform t = the on portion of that period Figure 4. Measuring rms on a triangle wave 4

5 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 percent duty cycle, the crest factor is the square root of 50, or Computing the rms value for sines and triangles is quite simple; the rms value is V pk divided by the crest factor. However, computing the AC rms value for a pulse train is a bit more complicated: V V pk 1 rms = x (1- ) 2 CF CF Figure 5. Measuring rms on a low-duty-cycle pulse train Using the formula, the theoretical rms value of our 2-V pk pulse train with 2 percent 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 mv. On the other hand, the average-responding meter reads 73 mv, a 74 percent 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 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 percent. This error is due to the limited bandwidth of the average-responding meter. The signal contains high-frequency energy that the average-responding meter does not take into account. Figure 6. Measuring rms on a noisy sine wave 5

7 Glossary Crest factor a measure of how high the waveform peaks, relative to its rms value Related Agilent Literature Agilent Truevolt Series of DMMs Datasheet EN DMM digital multimeter rms an abbreviation for root-meansquare True rms The term true rms is used to distinguish meters that actually measure the rms value, from meters that derive rms levels from either peak or average values V avg average voltage, using the absolute value of the waveform (the negative portion of the cycle is treated as if it were positive) V pk peak voltage V pk-pk peak-to-peak voltage V rms the rms value of AC voltage 7

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