Lets talk about AC and in particular the mysterious RMS, or Root Mean Square value because I noticed
some strange behaviour with my True RMS multimeters that had me confused for a bit.
You see, it all started when I noticed that the RMS voltage produced by my AC converter, which will be
one of the next videos was rather sensitive to the duty cycle of the square wave but it shouldn’t be
because the theory says that the RMS of this wave form does not depend on duty cycle.
The blue meter is showing the AC RMS voltage and the red one the duty cycle.
You can see that the voltage drops as the duty cycle changes. It turns out my AC converter has nothing
to do with this. It is rather a feature of the multimeters.
Here is such a square wave from my function generator with 5V peak to peak and 50% duty cycle and the
RMS value calculated by the scope is 2.50V.
If you observe the RMS value while I change the duty cycle first to 60%,
and then to 70%
and then to 80%, you’ll see that it does not change much, despite the fact that wave looks now very different.
This is in line with the maths for RMS for this wave form.
Now I do the same test with two multimeters connected, both display AC volts true RMS. The small one
on the right is instead showing the duty factor.
At 50% the RMS values agree with each other and the scope value of about 2.5V. There is some noise on
the signal due to the long cables which cause a slightly higher value but for this test it does not matter.
At 60% the voltage has dropped already
At 70% both are reading about 2.33V
and 80% we are down to 2.03V
Repeating the same test with my two bench multimeters. The duty cycle is again displayed by the small
You see that the RMS values agree with each other and the values shown by the hand-held multimeters.
I would have understood if one multimeter got it wrong but they all do exactly the same. The odd one
out seems to be the scope.
But, in fact the scope is the only one that showed the correct value of about 2.5V RMS. Here is why 2.5V
is correct and independent of the duty cycle.
In a 50% duty cycle square wave,
the voltage is +Vpeak for half the time and -Vpeak for the rest.
If this voltage was driving a load we have a positive current here
and a negative current here.
Power is the product of voltage and current, so effectively the area under the curve
but because negative voltage times negative current is positive power, we end up with a value that is
the same as if we had a constant +Vpeak all the time.
Since the RMS value of an AC voltage is defined as being the value of the DC voltage that produces the
same power, in this case the RMS value is Vpeak.
In our case Vpeak was 2.5V so that should be the theoretical RMS value.
There is also a so called crest factor which is Vpeak over VRMS and therefore 1 in this case. I’ll cover
the importance of the crest factor later.
The key observation is that regardless of the duty cycle, the result of this operation
always ends in the same RMS value
which is Vpeak or 2.5V.
So the scope was right. The RMS value should not have changed when I changed the duty cycle. Are all
the other multimeters wrong?
But then they all agreed with each other which points to something else at play here. I decided to test
a collections of my multimeters against a couple of other wave forms to see if there was a pattern.
Lets start easy with a simple sine wave. The Vpeak is 2.5V, the RMS value is 1.8V and the average voltage
is of course effectively zero because the positive and negative half waves cancel each other.
All except one these multimeters agree with the scope and show the same 1.8V RMS. The three that agree
are all True RMS. The yellow one which is a cheap Chinese model uses what is called an averaging circuit
instead. In this instance it should actually show the same value as the others but its lowest AC range
is 200VAC, so it naturally struggles showing such low voltages as 1.8V correctly but at least shows 1.5V.
Both of my bench meters are of course True RMS and also agree on the 1.8V RMS value.
To try something more fancy, here is a nice triangular wave with Vpeak of about 2.5V and the scope claims
an RMS of 1.47 V
All three True RMS meters agree with the scope and show the same 1.47V RMS. The yellow one gets only
up to 1.1V
Both of my bench meters also agree on the 1.47V RMS.
Here is something a bit more exotic. This the output of a sine wave after a full-bridge rectifier. Basically
all negative half waves are flipped upwards to the positive side which also doubles the frequency. Note
that my function generator was set to 5V amplitude so in the previous case where the waves were symmetrical
around zero, we had a Vpeak of 2.5V but now Vpeak is 5V and we have a positive average of 3.19V and an
RMS value of 3.54V.
That is interesting. All true RMS multimeters agree on 1.52V while the yellow one shows a whopping 6.3V
AC. You can also see that frequency is now 100Hz.
Lets check what DC values we have.
All 4 meters agree on 3.18V which matches the average shown on the scope.
The BM869 can show AC+DC and that value of 3.53V actually matches the RMS value shown on the scope.
Both of my bench meters show the same strange RMS value as the hand held meters, 1.52V
They also agree on the 3.18V DC.
For good measure, I include a sine wave after a single diode rectifier. Basically the negative half-wave
is simply cut off. Vpeak is 5V and we have a positive average of 1.61V and an RMS value of 2.50V.
As with the full bridge rectifier before, all three true RMS multimeters agree on an RMS voltage of about
1.91V which is quite different from the scope value of 2.5V while the yellow multi meter shows 3.2V.
On the DC side, all meters agree with average value of around 1.6V shown on the scope.
Again the BM869’s AC+DC display of 2.49V matches the RMS value shown by the scope.
And the bench meters repeat what the hand-held meters were showing: 1.91V RMS
and a DC value of 1.6V
Homing in on DC pulses, here is one with 50% duty cycle going from 0 to 5V. Its average is of course
2.5V as you would expect, and the scope declares its RMS value to be 3.51V
All True RMS meters show the RMS as about 2.48 volts while the yellow multimeter shows 5.1V.
As for DC, all 4 hand held multimeters agree on 2.5V which matches the scope’s average voltage.
As before the AC+DC value of 3.52 agrees with RMS value on the scope.
And the bench meters follow exactly the hand-held meters by showing 2.48V RMS
and also agree on the DC value of 2.5V
Changing the duty cycle to 90%, and both the average has increased to 4.47 V, and the RMS value is now
about 4.71V. The RMS changes because this is a pulse not a +/- square wave.
All three true RMS meters read about 1.48V while the yellow one reads a massive 9.2V
but all four meters agree on a DC value of 4.49V which matches the scope’s average readout
The AC+DC value of 4.73V again agrees with the scope’s RMS value
There is no surprise that the bench meters show the same RMS of about 1.48V as the true RMS hand held
and also agree on the DC value of 4.49V
Last not least, lets try the rectangular wave from the beginning again. Its peak to peak value is 5V,
so the positive peak value is 2.5V. The scope measures its average as about 2.05V and the RMS as 2.55V.
We know it should really be 2.5V but there is a bit noise on the line as is evident from the scope trace.
All three true RMS meters read about 1.52V while the yellow one reads 4.4 V
but all four meters agree on a DC value of 2.03V which matches the scope
and the AC+DC value of 2.54V again agrees with the scope’s RMS value
As before, the bench meters show the same RMS of about 1.52V
and also agree on the DC value of 2.03V
You may have wondered why I picked those wave forms we just saw. The reason is they all have equations
that allow me to calculate the theoretical values so we can easily determine what’s wrong and what’s
The equations to calculate the RMS value
and crest factor for sine and triangle wave forms are shown here
and considering the limitations of my setup, theoretical values and my actual measurements are in agreement.
The only problem is the yellow averaging meter but as explained it only has a 200V range so that is to
I only included it because it is not True RMS and as we have seen some of its readouts are quite different.
More interesting are the full and half rectified sine waves.
For DC or the averaging value,
the measurements agree well with the theoretical value; and..
the scope RMS values also match the theoretical RMS values quite well,
but all my true RMS meters show a substantially too low value.
A clue to that behaviour is that BM869’s
AC+DC readout which shows the same value as the RMS .
What really happens here is that these two wave forms introduce a substantial DC offset, shown here as
the red line.
But when you switch your meter to measure AC, it is normally using a capacitor in the input to keep any
DC parts out. In effect the whole trace is shifted downwards and the red DC line becomes the new zero
We can see the effect in action if we simply switch the scope input to be AC coupled as well
The trace moves as predicted downwards with the average being close to zero and the RMS value is now
about the same value as the True RMS multimeters showed earlier.
So the seemingly wrong True RMS values have an explanation. What the multimeters show the AC part of
the true RMS. By simply adding the DC offset, you will get the even “truer” RMS value which then matches
the RMS equations you find in literature. However, the addition has to be done geometrically
and not just by adding the DC and AC values.
The DC offset is also what causes the averaging meter to show these crazy values but I am not sure what
mechanism produces this exact value.
Moving on to the DC pulses with 50% and 90% duty cycle.
The equations for RMS, Average and Crest Factor are slightly more complex.
The key parameter is the duty cycle which determines both the RMS and the average DC value.
If we look at the values , we see the same pattern. The DC values are in agreement across the board.
The AC coupled multimeters show a different RMS value to the scope but the AC+DC value matches both the
scope and the theoretical value.
In other words, we have again the problem the DC offset
needs to be added to get the correct value.
And the wave form that started it all?
It is clear from the equations
that as explained,the RMS value is constant but the average value is not.
So deviating from a 50% duty cycle will introduce a DC offset
and, as the RMS as the geometrical sum of AC and DC components remains constant,
this means the AC component must reduce in value to compensate for the increasing DC. Since my True RMS
meters only show the AC component, it appears
as if the RMS voltage is dropping when the duty cycle changes.
The proof of the pudding is running the same test again but this time the multimeter on the left is showing
AC+DC in the main display. The small multimeter on the right is again showing duty cycle.
At 50% duty cycle we have no DC component and everything is showing the same AC value within tolerances.
At 60% duty cycle we see the expected drop of the AC but the AC+DC value stays the same
The same at 70%
and 90%. The AC value has dropped by more that a volt but the combined AC+DC value is still the same.
But my BM869 that can read AC+DC is a fairly recent addition to my lab and not all will have a meter
like that. So what is one to do with a True RMS multimeter that doesn’t have an AC+DC option?
You read the AC value, 1.510V and then switch to DC
If there is a substantial DC voltage as in this case, 2.035V, there is no choice other than to whip out
And use the formula to add AC and DC manually
So square root of the sum of the AC, 1.51 squared + the DC, 2.035 squared
and we get 2.534 which is the real RMS value we wanted.
Sometimes however this isn’t as straight forward as you’d hope. Here I am measuring a pulse with 10%
duty cycle. Both bench meters show about the same AC value.
Lets check the DC voltage.
Both are very different from each other.
Moreover all 4 hand-held meters in the same situation show yet another DC voltage which is slightly above
But if I switch to manual ranging and select the next higher range, both bench meters now display the
correct DC value.
In this case the relative high AC voltage messed up the auto-ranging for the comparatively small DC value.
The problem of auto ranging if DC and AC values are sufficiently different is a problem for hand held
Here all meters are showing the DC part of 219mV of a pulse with 4% duty cycle.
That is such a low voltage that you might be tempted to switch to mV instead, but watch what happens
The BM869 briefly shows Overflow and then happily displays nonsense for both DC and AC values
The BM235 also simply shows a wrong millivolt value
In this case I much prefer the response of the 10-dollar AN8008 which just shows overload.
Another pitfall to watch out for is the crest factor. I mentioned the crest factor before when I explained
the different wave forms I used for testing. In that little table here
is a quick summary ordered by increasing Crest Factor values.
The crest factor is important because it messes up the way most True RMS circuitry works. There are exceptions.
Some very expensive bench multimeters sample the wave form into memory and determine RMS graphically
just like a digital storage oscilloscope.
But if you have a reputable multimeter with True RMS that isn’t of the sampling kind, it will have a
crest factor statement or table in its specification. It could be an additional error percentage like
in the case of the bench meters
or a crest factor range
expected by the meter to stay within its rated accuracy. In either case if your signal is outside, all
bets on what it displays are off.
Sometimes you see crest factor written as a ratio, like 2.1:1
or 4.2:1. That is just a posh way of saying 2.1 or 4.2
The fact that the AN8008
spec doesn’t mention the crest factor in a posh form or any other way makes it questionable for use with
non sinusoidal wave forms and frankly for pure sine wave forms you don’t need True RMS anyway.
The last thing I want to mention in this long video is frequency. So far all measurements were at 50
Hz, but what happens if the frequency is higher? To check, I enabled the frequency counter on the scope
in the upper right corner and changed the frequency in stages up to 1 MHz, adjusting the time base accordingly
to maintain a decent wave form display. I selected a sine wave because all meters should have no problem
with the RMS value for a sine wave and I increased the amplitude to +-10V to give the averaging meter
a better chance to keep up.
You can see that the scope has no problem in determining RMS in this large frequency range.
Repeating the same sweep for the hand-held meters. The frequency is 50 Hz as you can see in the upper
display of the left meter.
You can see that all meters are displaying the RMS value except for the averaging one which is still
300mV short but that is probably a limit of the 200V range.
100 Hz is fine for all meters
At 1KHz the blue BM235 is starting to show a lower voltage
At 2KHz it drops further and the AN8008 is starting to drop as well.
Rising the frequency further reduces the voltage more and more and at 10Khz both meters are showing zero.
The averaging meter has dropped slightly but is still going strong at 3V. Sometimes the simpler circuit
of an averaging meter has its advantages over True RMS magic.
At 50Khz it has dropped to 2.3V and at 100KHz to just 1.7
At 200Khz the BM369 is starting to drop and
at 300 Khz the frequency display stops but even at 500 Khz the display was still showing 3.13V
Anyway, time to wrap things up
Ok, so here is the key lesson that I have learned from all this is. It maybe surprising to you, it certainly
was to me. Just because your meter says True RMS doesn’t mean you can probe whatever wave forms and it
will always show you the true RMS.
The underlying reason is that most affordable multimeters are AC coupled when measuring AC, so they only
show the AC part of the RMS. Although it is quite obvious when you think about it, a wave form may contain
a significant DC content but we tend to think wave equals AC. So that DC part needs to be added to the
AC part using the formula shown earlier.
And then there are all the gotchas that are out there just waiting for you to get a completely wrong
The most prominent is that in DC mode, lots of AC gets into your multimeter too and happily confuses
the auto ranging if given half a chance. And while those meters with AC+DC modes are generally brilliant,
they open the door for DC to do the same for your AC measurements.
Lastly, there is the crest factor out there to get you by degrading the accuracy
and of course the bandwidth limits of your meter reducing or even completely zeroing the readings at
Lots of things to keep in mind when doing such a simple thing as measuring AC.
Thanks for watching