True RMS, and what an averaging meter does to a distorted waveform

As an Amazon Associate, Akermin earns from qualifying purchases. Product links on this page are affiliate links.

What root mean square is an average of

The root mean square value of a time varying signal is defined by its name read backwards: square the instantaneous value at every moment, take the mean of those squares over a whole number of cycles, and take the square root of that mean. The definition exists because it is the value that delivers the same heating in a resistance as a steady direct quantity of the same number. That equivalence is the reason the quantity is used at all. It is what sizes a conductor, what determines whether an element reaches its working temperature, and what every rating on a nameplate refers to.

Squaring is the essential step and the expensive one. It weights large instantaneous values far more heavily than small ones, so a waveform that spends most of its cycle near zero and briefly reaches a high peak has a root mean square value much closer to its peak than its average appearance suggests. Any measurement technique that does not square the signal is estimating rather than computing, and the quality of the estimate depends entirely on whether the waveform matches the shape the estimate assumed. The general background on what true RMS means on a multimeter covers the vocabulary. The point here is the size of the error when the assumption fails.

The fixed multiplier that only suits one shape

Building a circuit that squares a signal in real time, averages it and takes a root is harder than building one that rectifies a signal and smooths it. For most of the history of portable instruments the cheap path was the common one, and the trick that made it acceptable was calibration against a sine.

For a sine of peak value one, the root mean square is one divided by the square root of two, which is 0.7071. The mean of the rectified wave is two divided by pi, which is 0.6366. The ratio between them, the form factor, is 1.1107. Build a meter that rectifies, averages and multiplies by 1.1107, print root mean square volts on the scale, and on mains frequency sine supplies it is correct. The assumption is invisible because it is baked into a constant in the scaling, and the instrument gives no indication when it is violated.

Every other waveform has a different form factor, and the error is exactly the ratio between that waveform form factor and 1.1107. This is a deterministic error, not noise: it does not average out over time, it does not improve with a better display, and taking more readings does not help. The relationship with display resolution is worth keeping straight, since the count on a digital meter describes how finely the result is presented and has nothing to say about whether the result is the right quantity.

Waveforms and what an averaging meter shows

The table expresses everything as a fraction of the peak value, so the numbers can be applied to voltage or current at any scale. The phase controlled rows represent a sine switched on part way through each half cycle, which is what a dimmer, a soft starter or a resistive load controller produces.

The last row is the one worth sitting with. A load controlled late in the half cycle is reported at less than half its actual heating value. If that figure is being used to decide whether a circuit is adequately sized, the conclusion is not conservative, it is dangerously optimistic in the direction of under sizing. The square wave row matters for a different reason: it is the only common case where the averaging instrument reads high, so an operator who learns the habit of adding margin for an averaging meter gets that case backwards.

Where distorted current comes from on a real circuit

Thirty years ago an averaging instrument was defensible because most loads were resistive or motor driven and the current followed the voltage reasonably well. That is no longer the typical case. A switched mode power supply draws current only near the peaks of the voltage waveform, when its input capacitor is actually charging, which produces a narrow pulse with a high ratio of peak to root mean square. An electronic driver for lighting does the same. A phase controlled load chops the waveform deliberately. A variable frequency drive synthesises its output from a high frequency switching carrier and draws rectifier current at its input. A soft starter spends its whole ramp in the phase controlled region.

On a circuit carrying any of those, the voltage can still be reasonably sinusoidal while the current is heavily distorted, which is why current measurement is where the error usually bites. Solar installations add their own version of the problem on the direct current side, where the question is ripple on an otherwise steady quantity. The KPS DCM8500PV clamp meter is built around that work, stating true RMS measurement with direct voltage to 2000 V, alternating voltage to 1500 V, alternating and direct current to 600 A, plus resistance, capacitance, frequency, temperature, continuity, diode and microamp ranges, high frequency rejection for electrically noisy sites, and Bluetooth logging. The FLIR CM78-PV clamp meter covers similar ground with a measurement category rating at 1500 V, true RMS readings, a low impedance mode and a drive mode. That drive mode is the explicit acknowledgement of the problem: it low pass filters the input so the instrument reports the fundamental the motor responds to rather than the switching carrier.

True RMS is still bounded by bandwidth and crest factor

Buying a true RMS instrument removes the form factor error and leaves two limits that listings rarely mention.

Bandwidth is the first. A converter responds to the frequency components within its passband and ignores what lies above. Distorted waveforms are distorted precisely because they contain harmonics, and a pulse narrow enough to matter contains energy well above the fundamental. An instrument whose alternating response rolls off at a few kilohertz computes a correct root mean square of the part of the signal it can see and reports a value low by whatever the missing harmonics contributed.

Crest factor is the second. It is the ratio of peak to root mean square, one and a half for a triangular wave, about 1.414 for a sine, and three or more for the current drawn by a rectifier front end. An instrument specifies a maximum crest factor it can handle at full scale, because the peak has to fit inside the input range without clipping even though the displayed value is much smaller. Exceed it and the peaks are flattened before the squaring stage, which again reads low.

Four instruments, four true RMS claims, and not one of them publishes the two conditions that bound the claim. The figures exist in the manuals, and the listings are where the decision usually gets made.

Choosing a meter for the waveform you expect

Decide what you will be measuring before you look at the specification. If the work is clean sinusoidal supplies and resistive loads, an averaging instrument is correct and nothing is gained by changing it. If it includes any electronic load, any speed controller, any dimmer or any rectifier front end, the form factor error is present on every current reading and a true RMS converter is the minimum. The Fluke 177 multimeter states true RMS with a 6000 count display updating four times a second, a bar graph updating forty times a second and minimum, maximum and average memory, which is a handheld specification aimed at exactly that mixed work. For repeatable bench measurements the OWON XDM1241 bench multimeter states a 55,000 count true RMS display with internal recording and remote control over a standard instrument command set, where logging a changing waveform over time matters more than portability.

Two habits finish the job. Look up the bandwidth and crest factor in the manual rather than the listing, and compare them against the waveform you expect rather than against the other instrument you were considering. And treat any automatic correction with the same suspicion you now apply to the 1.1107 constant, because the structure of the mistake is identical: a fixed assumption applied silently to a situation that no longer matches it. The same pattern appears in automatic carbon dioxide calibration going wrong, where a routine assumes the sensor periodically sees clean outdoor air. It also explains why a sound level meter publishes its time weighting alongside its readings: it too reports a root mean square, and the interval over which that square is averaged changes the answer. A measurement is only as honest as the assumption buried in its scaling, and the whole value of a true RMS converter is that it removes one of them.

Frequently Asked Questions