Measuring non-sinusoidal signals
A multimeter that is not true RMS reads a square wave 11% too high and a 10% pulse train one third too low. With the sine wave, the only shape it is calibrated for, it gets it right.
01Three values for the same signal
Several things can be measured from a periodic voltage, and each answers a different question:
Two ratios summarize the shape of the waveform: the form factor, \( V_{rms}/V_{rect} \), and the crest factor, \( V_{peak}/V_{rms} \). For a sine wave they are 1.111 and 1.414. The sine-wave values and their use on the mains are in AC measurements, in the technician program.
02Average-responding instruments calibrated in RMS
The simplest AC voltmeter rectifies the signal and measures its average value. For it to display the RMS value, its scale is multiplied by the form factor of the sine wave, 1.111. With a sine wave the reading is correct; with any other shape, the instrument applies a factor that does not correspond:
Many inexpensive multimeters still work this way. The specification gives it away when it says “average-responding” or “calibrated for sine wave” instead of “true RMS”.
03True RMS and its limits
A true-RMS instrument calculates the root of the mean of the square: with a logarithmic analog converter, with a thermal sensor that compares the heating, or by sampling and calculating digitally. It measures any waveform correctly, but within two limits that the specification states:
- Bandwidth. Harmonics above its cutoff frequency are not added. A 20 kHz square wave has harmonics at 60, 100, 140 kHz…; a 100 kHz instrument sees only the first three and reads 3.4% low.
- Maximum crest factor. Narrow pulses have very high peaks compared with their RMS value, which saturate the input. A crest factor of 3 at full scale is typically allowed.
Another detail: almost all multimeters measure AC with capacitive coupling, without the DC component. The total RMS value is \( \sqrt{V_{avg}^2 + V_{rms,AC}^2} \), and some instruments have an “AC + DC” mode that measures it directly.
A signal of 1 V peak. Two voltmeters in AC (without the DC): one average-responding, calibrated for sine wave, and one true-RMS with 100 kHz of bandwidth and a maximum crest factor of 3. At the top, the signal, with its average value as a dashed line.
04Peak, peak-to-peak and the oscilloscope
For highly distorted signals, pulses or transients, no single reading is enough: the shape has to be seen. The oscilloscope gives the peak and peak-to-peak values directly, and digital ones also calculate the average and RMS values over the acquired waveform, taking care to do so over a whole number of periods. For the harmonic content, the appropriate instrument is the spectrum analyzer or the oscilloscope’s FFT.
05Where it appears in electronics
| Case | Why it matters |
|---|---|
| Current of a switching power supply or a drive | Narrow pulses with a high crest factor; an average-responding instrument reads far too low |
| Output voltage of a dimmer | Chopped sine wave; the RMS value depends on the firing angle |
| Power supply ripple | A small AC on top of a lot of DC: only the AC component has to be measured |
| Neutral current with nonlinear loads | Third harmonics add up and the neutral can carry more current than the phases |
06In the laboratory
Measure a sine wave, a square wave, a triangle wave and 10% and 5% pulses with an average-responding multimeter and a true-RMS one. Compare with the lab values.
Sweep the frequency of a square wave from 1 kHz to 200 kHz and record the true-RMS reading. Find at what frequency it starts to read low.
Measure the input current of a cell phone charger with a true-RMS clamp and with an average-responding multimeter, and view its shape with the oscilloscope.
07Common mistakes
- Measuring non-sinusoidal shapes with an average-responding instrument. The error can exceed 30%.
- Forgetting that the AC range removes the DC. A square wave from 0 to 5 V reads as 2.5 V, not 3.5 V.
- Exceeding the crest factor. The true-RMS meter saturates and reads low without warning.
- Ignoring the bandwidth. A handheld multimeter usually has only a few kilohertz.
08Self-assessment
What does an average-responding voltmeter read for a ±2 V square wave?
\( 1.111 \cdot 2 = 2.22 \) V, when the RMS value is 2 V.
What is the RMS value of a triangle wave of 3 V peak?
\( 3/\sqrt3 = 1.73 \) V.
What crest factor does a pulse train from 0 to 1 V with a 10% duty cycle have, without the DC?
Peak of 0.9 V over 0.3 V RMS: 3.
A signal has 2 V of DC and 1.5 V RMS of AC. What is its total RMS value?
\( \sqrt{4 + 2.25} = 2.5 \) V.
09Further reading
- Analog Devices. RMS-to-DC Converters Application Guide. 2nd ed., Analog Devices, 1986. How true RMS is measured, with the bandwidth and crest-factor limits.
- Fluke Corporation. Calibration: Philosophy in Practice. 2nd ed., Fluke, 1994. AC measurement in calibration practice, with the treatment of non-sinusoidal waveforms.
- Albert D. Helfrick and William D. Cooper. Modern Electronic Instrumentation and Measurement Techniques (Spanish edition: “Instrumentación electrónica moderna y técnicas de medición”). Prentice Hall, 1990. AC voltmeters with average, peak and RMS response.