RMS - what didn't I get?

[RossW]

A friend of mine has been building a power monitoring device and the issue of measuring RMS voltage with non-sinusoidal waveforms came up.
A simple test was conducted - using a function generator and Fluke 87 DMM.
We measured a 10V p-p sinewave (confirmed with calibrated DSO), and 10V p-p squarewave at 50%, 25% and 75% dutycycle.
Reading with sine wave: 3.524V (Close enough)
Reading with 50% duty square wave: 5.00V
Reading with 75% duty square wave: 4.33V
Reading with 25% duty square wave: 4.35V

I repeated the test with another function generator, a UT71E DMM and my bench DMM, which all claim to be "true RMS" reading.
Sine        3.5312V and 3.5390V
Sqr 50%. 5.000V  and 5.0009V
Sqr 25%. 4.329V  and 4.3377V
Sqr 75%. 4.332V  and 4.3149V

The output from the function generator is confirmed as being true bipolar output (+/- 5V reference ground, not 0V to 10V)

The DSO shows the RMS voltage as 3.53V for the sinewave, and 5.0V the squarewave regardless of dutycycle.
The UT71E has a selectable mode of "RMS" or "RMS+DC". If I select "RMS+DC" it still shows 3.53V for the sinewave but 5.00V for the squarewave regardless of dutycycle.

Logic tells me that a square wave (regardless of dutycycle) that switches hard from rail to rail should indeed have a RMS value of half the peak to peak.
That the DSO agrees gives me confidence in that, and the UT71E in RMS+DC agrees.
I can't however understand how 3 different meters, all of "reasonable" makes, agree remarkably closely with each other but seem to get it wrong with the squarewave with anything other than 50% dutycycle?

Can anyone explain (and/or confirm our results)?
What am I not understanding is why

[bdunham7]

Can anyone explain (and/or confirm our results)?
What am I not understanding is why

Your meters are not broken, they are correctly telling you the RMS value of the AC component of your signal.  The AC+DC mode is telling you something else, specifically the root of the sum of the squares of the AC and DC values.

To help you understand what the RMS values should be mathematically and why they aren't all 5V, ask yourself what you expect the AC RMS value of a 0% duty cycle signal (effectively -5VDC) to be?  Now how about a 0.000000001% duty cycle--pretty close to the previous answer, right?

As for how and why the meters read (correctly) that way, they have an input capacitor that will block the DC component  by charging up to the average value of the signal.  Thus for a solid -5V signal, the capacitor charges up to 5V and the meter sees zero.  For a very small duty cycle square wave, the capacitor charges up to almost 5V and the meter sees mostly short 10V pulses.  For a 50% duty cycle square wave, the meter essentially doesn't charge up much at all and the meter sees an even +/-5V.

[RossW]

ask yourself what you expect the AC RMS value of a 0% duty cycle signal (effectively -5VDC) to be?

Yes, I agree that the AC component of any DC value is 0.

Perhaps my error is in the interpretation of "RMS".

RMS = root mean square, RMS essentially calculates the equivalent direct current (dc) value of an ac waveform. More technically, it determines the "effective," or dc heating value, of any ac wave shape.

which is at variance to your

have an input capacitor that will block the DC component  by charging up to the average value of the signal.  Thus for a solid -5V signal, the capacitor charges up to 5V and the meter sees zero.  For a very small duty cycle square wave, the capacitor charges up to almost 5V and the meter sees mostly short 10V pulses.  For a 50% duty cycle square wave, the meter essentially doesn't charge up much at all and the meter sees an even +/-5V.

A solid -5V indisputably has a "dc heating potential".
A 1% dutycycle squarewave that spends 1% of the time at -5V, and 99% of the time at +5V will have a "dc heating equivalent" of 5V 100% of the time, not "almost none of the time".

Therefor, "AC RMS" seems to be a self-contradictory term? Either it's RMS, or it's not?

[bdunham7]

Therefor, "AC RMS" seems to be a self-contradictory term? Either it's RMS, or it's not?

You have a valid point, but rather than debate the usefulness of the terms I'll just restate that the AC RMS reading that most of your meters are giving you is a DC-blocked reading that only considers the AC component.  They will not indicate the value, heating or otherwise, of any DC component present.  That function is called AC+DC TRMS.

RMS literally means "root mean square" or "root of the mean of the squares".  So if you took a number of sequential samples of the signal, took the squares of all the values of those samples and then averaged them, the square root of that average would be your RMS value.  That happens to be the same thing as the equivalent DC voltage that would provide the same average power.

The reason AC RMS is useful for many common purposes is that many systems (mains power) will only have AC components that are signficant since the power all comes from transformers.  Any signficant DC bias on mains power is a serious problem and generally doesn't happen. 

[TimFox]

The actual RMS value of a waveform is defined as the square root of the long-term mean (average) of the square of the voltage, and corresponds to the heating capability of that voltage, since the instantaneous power is the voltage squared (positive-definite) divided by the load resistance.
However, for many applications such as audio or noise measurement, only the variation is important, not the mean value (DC component), so the RMS voltage is measured after an input capacitor (high-pass filter) for an AC voltage measurement.
In audio, you might have mV of AC signal superimposed on V of DC.
To get the true RMS, you can use the “AC + DC” mode (without capacitor), or separately measure the AC RMS and DC values, and add them in quadrature (root sum of squares).

[RossW]

Great, thanks guys...

So I think that it seems most meters on the market that claim to be "True RMS" are in fact not quite as they claim. TRUE RMS would not have any dc blocking.
I will grant that in the typical circumstance of measuring a mains supply that there shouldn't be any DC component, and that trying to measure (for example) an audio signal that may be superimposed on a DC bias, the blocking would be desirable.

I maintain that such meters are in fact not "TRUE" RMS but at least now I understand WHY they are giving an incorrect reading

[TimFox]

As terms of art, “true rms” means not “average-responding, rms calibrated”.
Back in the day, hp ac voltmeters usually stated the latter on the meter face for common voltmeters that passed the voltage through a rectifier into a filter capacitor and showed the dc voltage across the filter.

[RossW]

As terms of art, “true rms” means not “average-responding, rms calibrated”.

When "reputable" companies like fluke say at https://www.fluke.com/en-au/learn/blog/electrical/what-is-true-rms

Yet a true-RMS meter is widely preferred because it is the only one that can accurately measure both sinusoidal and non-sinusoidal ac waveforms.

    Sinusoidal (sine) waves: Pure, without distortion, with symmetrical transitions between peaks and valleys.
    Nonsinusoidal waves: Waves with distorted, irregular patterns—spikes, pulse trains, squares, triangles, sawtooth and any other ragged or angular waves.

They're acknowledging that it's the "DC-heating equivalent" and saying how their sophisticated instruments measure "true RMS" but completely omit to mention that they're crippling it by AC-coupling it, thus making it NOT really read "true RMS".

For someone working entirely in a grid-connected environment that's probably ok. But lots of folk are using these meters in more delicate work, instrumentation etc where the difference matters, yet they seem to be entirely silent on the matter?

Anyway, can put it all to bed now. The root cause is identified, and I'm sure me standing on a soapbox saying they're telling us only partial truth won't change anything!

[TimFox]

Also note that the “true rms” meters with input high-pass filters (capacitors) are usually called “ac voltmeters”.

[mawyatt]

Modern DSOs do a respectable job of displaying RMS values, both DC and AC versions, and may outperform RMS capable DMMs since the DSO can operate at much higher frequencies and capture the higher frequency content of non-sinusoidal waveforms.

Best,

[bdunham7]

I'm sure me standing on a soapbox saying they're telling us only partial truth won't change anything!

It's one of those things that "everyone knows", except if you don't, you don't--and now you do!  In the case of Fluke they do specify these things in great detail if you read the manual closely.  The details don't always make simple, intuitive sense and often they can't be simplified to fit in a pithy ad blurb.

[J-R]

The truth is don't peek behind the curtain because in reality there is no such thing as AC or DC voltage, just a voltage at an infinitesimally small point in time.   And the word "current" in the AC/DC abbreviation being combined with voltage is a further abomination since voltage simply means potential and is valid even without current flow.  Also, current flow is another long story since in reality the electrons don't really move very far or fast in a conductor.

[TimFox]

Current is defined as a flow of charge, which is a more general concept than electrons.
By simple algebra, one can express a given voltage as a function of time as a sum of ac and dc components, ignoring the pedantic quibble about the abbreviations.
The dc component is simply the long-term mean of the voltage; everything else is ac.

[IanB]

It is my understanding that the RMS voltage is defined over a time interval using the following formula:
$$V_{RMS}=\sqrt{\frac1{t_2 - t_1}\int_{t_1}^{t_2} v(t)^2\,\mathrm{d}t}$$
This is the formula that I believe a DSO would use when displaying an RMS value for a displayed voltage trace. There is no mention of "AC" or "DC" in this formula. It simply treats instantaneous voltage as a function of time and computes a time average over some interval.

If a handheld DMM does something different than this, it is no longer "true" RMS, but some derived value. If that is the case, then the documentation and user manual for the meter ought to make this very clear. It should not make the assumption that "DMM users know what is going on and won't get confused by it."

[bdunham7]

If a handheld DMM does something different than this, it is no longer "true" RMS, but some derived value. If that is the case, then the documentation and user manual for the meter ought to make this very clear. It should not make the assumption that "DMM users know what is going on and won't get confused by it."

If the meter had a range simply marked "RMS", I'd concede your point.  But they don't.  They say "AC RMS" or "RMS AC".  Or they separate them, saying TRMS in one location but AC on the selector.  But I've not yet seen one of these meters that doesn't make it quite clear on the face of it, without referring to the manual, that it is measuring AC volts. 

[IanB]

If the meter had a range simply marked "RMS", I'd concede your point.  But they don't.  They say "AC RMS" or "RMS AC".  Or they separate them, saying TRMS in one location but AC on the selector.  But I've not yet seen one of these meters that doesn't make it quite clear on the face of it, without referring to the manual, that it is measuring AC volts.

Granted, and depending on the model, the manual for the meter may provide the required explanation.

For example, here is a quote from the BM869s manual:

AC True RMS
AC True RMS, normally refers as True RMS, identifies a DMM function that is AC
coupled, and responds accurately only to the effective RMS AC component value
regardless of the waveforms. However, DC component plays an important role in the
distorted non-symmetrical waveforms, and will also be of interest sometimes.

DC+AC True RMS
DC+AC True RMS calculates both of the AC and DC components given by the
expression \$\sqrt{\mathrm{DC}^2+\mathrm{(AC\ rms)}^2}\$ when making measurements, and can responds
accurately to the total effective RMS value regardless of the waveform.

If I had a quibble, it would be that only the DC+AC case is "true" RMS. I do not think you can say "AC True RMS" is True RMS. By definition, if it is AC coupled, then it is not measuring the true RMS voltage anymore, but instead it is measuring a modified voltage.

[TimFox]

Again, the phrase “true rms” is applied to meters that actually calculate or measure the rms value of something, instead of the mean absolute value which is then multiplied by the conversion factor for a sinusoid to obtain an rms value.

[2N3055]

If the meter had a range simply marked "RMS", I'd concede your point.  But they don't.  They say "AC RMS" or "RMS AC".  Or they separate them, saying TRMS in one location but AC on the selector.  But I've not yet seen one of these meters that doesn't make it quite clear on the face of it, without referring to the manual, that it is measuring AC volts.

Granted, and depending on the model, the manual for the meter may provide the required explanation.

For example, here is a quote from the BM869s manual:

AC True RMS
AC True RMS, normally refers as True RMS, identifies a DMM function that is AC
coupled, and responds accurately only to the effective RMS AC component value
regardless of the waveforms. However, DC component plays an important role in the
distorted non-symmetrical waveforms, and will also be of interest sometimes.

DC+AC True RMS
DC+AC True RMS calculates both of the AC and DC components given by the
expression \$\sqrt{\mathrm{DC}^2+\mathrm{(AC\ rms)}^2}\$ when making measurements, and can responds
accurately to the total effective RMS value regardless of the waveform.

If I had a quibble, it would be that only the DC+AC case is "true" RMS. I do not think you can say "AC True RMS" is True RMS. By definition, if it is AC coupled, then it is not measuring the true RMS voltage anymore, but instead it is measuring a modified voltage.

AC RMS measures True RMS value of AC component separately. AC+DC measures DC coupled total RMS value.
AC coupled RMS is important and useful. If you have single supply audio amplifier, there is a capacitor on output to couple only AC component into speaker. Since that signal is complex you need TRMS to actually measure it.

Scopes have 2 separate RMS measurements (AC only and AC+DC) specifically because both are useful.
One typical measurement is measuring ripple on power supply rail. You need only AC component and waveform won't be sinusoidal...

[bdunham7]

If I had a quibble, it would be that only the DC+AC case is "true" RMS. I do not think you can say "AC True RMS" is True RMS. By definition, if it is AC coupled, then it is not measuring the true RMS voltage anymore, but instead it is measuring a modified voltage.

You are interpreting the word "true" in an entirely different way than is intended by the manufacturers that are printing it on their devices and in their manuals.  It might help to consider that the modern "TRMS" mark is a short version of "true RMS responding", meaning simply that the AC to DC conversion circuit responds in a 'true' manner to the signal as opposed to estimating it in a different manner.

Other options include average responding and peak responding circuits that would output either the average absolute value or the peak value of the signal.  To display the RMS of a sine wave--the only signal for which these methods is accurate--the result are scaled by either 1.11 for averaging or 0.707 for peak responding.  For any non-sinusoidal signal, these methods are inaccurate.  A true RMS responding converter is accurate for any shape of signal provided bandwidth and crest factor limitations are observed.

In short, the term 'true' is intended to convey that the conversion process uses a method that yields accurate RMS values for any signal shape, not that the result includes both DC and AC components.  If you choose to attach a different meaning to the word 'true', then your conclusions will be different.  But that's like me saying that the two front tires of a truck are the "steering wheels" because they are attached to the steer axle and they are, in fact, the wheels that actually steer the truck.  That thing in the cab?  Oh, that's the operator's directional control circular assembly.

You could also complain that meters carrying the TRMS logo use averaging to measure DC instead of RMS, but that would be a different discussion. 

[IanB]

Again, the phrase “true rms” is applied to meters that actually calculate or measure the rms value of something, instead of the mean absolute value which is then multiplied by the conversion factor for a sinusoid to obtain an rms value.

And on reflection, I realize this is what is going on. "AC True RMS" is chosen to mean the true RMS value of the AC component of the voltage, and the intended meaning of "true" is to distinguish it from the alternative of assuming a sinusoid and applying a simple factor.

[boB]

What was the frequency of the square wave, any duty cycle, that you were measuring with the meter ?

True-RMS reading meters will normally have a maximum frequency they can run with.

Fluke meters used to be 50 kHz.  Not sure these days ?

Should read really close with waveforms at power line frequencies at fundamental F.

[garrettm]

As far as I understand, "true RMS" implies "AC+DC RMS": sqrt(V_ACrms^2 + V_DCavg^2). Which, while useful in certain situations, is not terribly helpful when taking general measurements. I often only care about the isolated DC and AC components: e.g, AC ripple on a DC rail, or the DC offset of an AC signal. Of course, power measurements for complex signals must be done as AC+DC RMS.

As for why your meters display different values for each signal, despite having the same "RMS" value, is that the AC component being measured is limited to the pass band of the measuring instrument. Most RMS reading meters have maybe 5 or 10 Hz out to 100 kHz or 10 to 30 MHz bandwidth -- depending on if it is a modern DMM or old-school instrument like the HP 3400A/B, Racal Dana 5002, Marconi 2610, Fluke 8922A and similar broadband AC VMs. You also have to consider crest factor (V_pk / V_rms = CF) -- high crest factor signals give many DMMs trouble. In your case, I believe CF is the main issue.

The easiest way to compute the correct true RMS value is to use a "digitizer" (ADC) to sample a signal and perform the RMS calculation using the samples collected over the region of interest. This method even works for fast broadband signals when sampling *below* its nyquist frequency (e.g. Clark Hess 2330 and most Yokogawa power meters). A zero-crossing detector is needed to compute an integer multiple of the waveform's period -- which minimize errors in computing the RMS. The (below nyquist) sampling method statistically reconstructs a fast signal into something within the bandwidth of the measuring instrument. The sampling method is also the best way to compute RMS measurements of slow AC signals (say less than 5 to 10 Hz).

At any rate, each measuring technique (thermal, LOG, sampling, and so on) has its advantages and disadvantages.

[Conrad Hoffman]

Can't remember exactly where it is, but there's a switch on the back of my HP3455A that switches the AC coupling in and out. I've actually had to use this when looking at half wave rectified signals and power dissipation to get the actual heating value.

[TimFox]

The bandwidth discussion above is important, for both high frequency measurement and noise measurement.
Some DMMs have high-frequency limits appropriate for low harmonics of the 60 Hz line, others are reasonable for audio.
I have three true-rms meters that are good up to roughly 10 MHz:  two have dc switches, the hp 3400 analog unit is ac only.
My hp 3403C is good up to 100 MHz, with a 3.5 digit readout.
The even older hp 3406 uses incoherent sampling to get a low-frequency signal for the voltmeter.
The analog panel meter is average-responding, but the rear panel output can be sent to an rms meter.

[donlisms]

Wouldn't the easiest way to verify all of these things be to use one of those (expensive) thermal converter gizmos? I thought that's what they were for.  RMS defined as "equivalent heat?"

[bdunham7]

Wouldn't the easiest way to verify all of these things be to use one of those (expensive) thermal converter gizmos? I thought that's what they were for.  RMS defined as "equivalent heat?"

You'd still have the issue of whether or not the input amplifier is AC or DC coupled.  I have a Thermal RMS meter and it give me both options--so I have to choose.  RMS is not defined as equivalent heat, it is just commonly explained that way.  RMS is a term that defines itself (root of the mean of the squares) but it works out to be the equivalent of DC for heating a perfectly resistive load.

[TimFox]

As mentioned above, rms has a simple mathematical definition.
If you square an rms voltage, you have the mean of the voltage squared.
Since power into a resistive load is proportional to the square of the voltage, you then get the mean power into that load, which heats it.
Note that this is the mean (or average) power, not the rms power (despite popular usage).

[horo]

But I've not yet seen one of these meters that doesn't make it quite clear on the face of it, without referring to the manual, that it is measuring AC volts.

My Fluke has two ranges: AC and DC, while my Gossen Metrahit29s offers AC, ACDC and DC. So for the Fluke I know that I have to add both values (quadratic) w/o reading the manuals.

[garrettm]

[...]
Note that this is the mean (or average) power, not the rms power (despite popular usage).

Correct. The quantity calculated by multiplying RMS voltage and RMS current is *average* power. Or, using the squared RMS voltage or current, is *proportional* to the average power (assuming a purely *resistive* load). While it can be computed, the RMS value for a given power signal has no physical meaning and should never be used in practice. So if anyone ever says "RMS power" be wary: they are either confused, or were taught wrong. Also, and I believe I saw someone mention this possibly by accident, DC is measured/computed as an average value not RMS. Hence there is no such thing as "DC RMS" (as that would simply compute the true (AC+DC) RMS value -- at least when using the sampling method).

For those who don't believe me about RMS power not having physical meaning please see ADI's Analog Dialogue article "RMS Power vs. Average Power":

https://www.analog.com/en/analog-dialogue/raqs/raq-issue-177.html

[r6502]

Hello All,

Bandwidth is very importand. I took just a look on the specs of my hendheld multimeter Fluke 179, that has a badwidth of only 45Hz to 500Hz in the AC rasnges till 600V and is avertised at true RMS meter.  My HP34401 bench top multimeter as a bandwidth of 3 Hz to 300kHz, but error is increasing in the low and very high frequencys, see page 218 of the linked manual.

A good explanation on "True RMS AC Measurements" can be found in the manual of the HP34401a, chapter 7, page 206ff,, see link below. Here everything you need is explained.

When you want to measure a PWM signal you need a meter that ia able to measure signals with an high Crest factor, like the good old HP 3400a or the HP3403 series as TimFox wrote in his post. Modern DSO may do it as well, when set up correct.

link to HP34401 manual

Guido