The RMS value of a time dependent voltage or current is based on the simple concept of equality of power dissipation; however, its theoretical calculation or instrument implementation may not be easy. Reference 1 explains the principles of RMS measurements and compares performance of 20 digital multimeters (DMMs). The results are alarming. Many units report lower RMS values than expected; for some instruments, numbers are almost half of what they should be.

Twenty years later, the traps of RMS measurements are still there. Technology made a large step forward and most people trust the readings of their true-RMS multimeters and simulation tools without verification. However, there are important details. This paper is aimed at improving awareness of RMS measurements.

The generic formula to calculate the RMS value of a signal is:

(1)

The calculation takes three steps: squaring the AC signal, finding the mean (also, average or DC) value of the squared signal and, getting the square root of the mean, thus the name RMS.

The easiest way to verify if an instrument does the right job is to apply a square-wave signal to it, as the expected output can be calculated with simple graphical analysis. **Figure 1** shows an example with a 5-V single polarity signal. Recall that the average value calculation transforms the time dependent signal into a box that spans the whole period of the signal and has the same area as the area under the signal; the height of the box is the average value. We end up with the RMS value of 3.535 V.

**Figure 1** A graphical calculation of the RMS value.

Let us check if the equipment produces the same result: **Figure 2** shows the performance of a simulation tool. The square wave signal is connected to a DC voltmeter at the left and a probe and an AC voltmeter at the right. The DC value is correct (be careful as this is the DC value of the input, not the squared input signal), the RMS value reported by the probe is correct; however the reading of the AC voltmeter is far below expectations.

**Figure 2** The RMS numbers reported by the probe and the AC voltmeter do not match in the simulation.

Things become clear when we remove the DC component in the input signal and calculate the expected RMS value. The graphical analysis in **Figure 3** yields the value of 2.5 V.

**Figure 3** RMS calculation when the input signal has no DC component in it.

Obviously, the AC voltmeter silently blocks the DC component of the input signal and defines the RMS value of the modified signal.

You can still get the correct RMS result by plugging the readings of the two meters in Figure 2 in the following formula:

(2)

The same problem appears with many digital multimeters and some digital oscilloscopes. They produce correct RMS values for DC-free signals and wrong values when the signal has a DC component in it. Some oscilloscopes offer two options for RMS measurements called DC RMS and AC RMS which can be confusing. How can one survive this?

The fastest way is to test the instrument with a square wave signal. Oscilloscopes have a built-in generator that provides a unipolar square wave signal to adjust probe compensation. Connect that signal to the input, select DC coupling and measure VMAX and VMIN. Using the two results, calculate the mean and the RMS value of the signal as presented in Figure 2. Then measure mean and RMS values. The two numbers should differ. Check if they match the calculated values. Now subtract the mean value from VMAX and VMIN and calculate the RMS value as presented in Figure 3. Select AC coupling—the mean and the RMS readings should be very close. Check if the numbers match the calculated values.

DMMs need an external signal to test DC and AC performance. The easiest way is to apply the probe compensation signal of a scope to the meter. The advantage is that you already know the expected results. Be careful: the unit may display slightly different numbers for DC and AC measurements despite it blocking the DC component of the signal. The reason for this is due to measurement errors, an inherent feature of every measurement.

If you want to skip manual calculations, you can use the simple circuit presented in **Figure 4**. Similar to the circuit in Reference 2, it generates three signals with different duty cycles. The table displays expected values of VDC, VAC and VRMS for each signal.

**Figure 4** A simple circuit can provide three test signals with a frequency of 1 kHz and different duty cycles (D). The values of VDC, VAC and VRMS change accordingly.

Use the circuit and the table to tell whether you can take the RMS reading of your instrument as is, or if you must measure VDC and VAC separately and use formula (2) to calculate VRMS.

You can also use a function generator to create these signals. Make sure the signal spans from 0 to 5 V, otherwise you cannot rely on the numbers in the table.

So, be careful with RMS measurements, as wrong results can lead to wrong conclusions and wrong decisions. As a rule, it generally takes a (very) short time to fix problems. In order to avoid sleepless nights, make sure you clarify how your equipment works before you start measuring.

A recent video (Reference 3) provides examples of RMS measurements with many instruments and various waveforms, crest factors and frequencies. Give it a watch!

**References**

- Williams J., T. Owen. Understanding and selecting rms voltmeters. EDN, May 11, 2000, 54-58.
- Stofka M. Waveform generator produces 25 and 75 % duty cycles. EDN, Mar 18, 2010.
- Measuring RMS Voltages: How true is your True RMS multimeter really https://www.youtube.com/watch?v=jhy_kfhwwbo

*–Jordan Dimitrov is an electrical engineer & PhD with 30 years of experience. He teaches electrical and electronics courses at a Toronto community college.*

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