Charge and current, RMS or aritmethic mean?

[hadoque]

I'm calculating current through a resistor by measuring a single current pulse, integrating it and multiplying it with its frequency. This would correspond to calculating an arithmetic mean.
I also tried calculating RMS of this waveform, using GNU Octave, and was a bit surprised by the difference.
My interpretation of RMS current is the DC-equivalent of the AC. Should be more or less the same, or should it? I started looking into this by looking at the definition of the two means. For the arithmetic mean, I use the absolute of the function, since I don't want negative charge to cancel positive.
I_a = ∫^T2_T1 √f(t)²dt/(T2-T1)
I_RMS = √( ∫^T2_T1 f(t)²dt/(T2-T1) )

If we use f(t)=Sin(t) as an example, we get:
I_a=2/π ≈ 0.64
I_RMS = 1/√2 ≈ 0.71

Using a square pulse wave with height 1, period 1 and time on T, we get
I_a= T
I_RMS = √T

Quite a difference.
So, I think the arithmetic mean is the most correct in this case, since it is equal to the actual definition of current. So why do we use RMS? Am I missing something?
I have searched the net, but haven't found a discussion about this.

 

[anorlunda]

RMS is like DC in the sense that the power in a resistor is the same for 1A RMS or 1A DC.

It differs from the arithmetic mean because P=I²R for a resistor, so 2x more current gives 4x more power, not 2x.

 

[essenmein]

To elaborate on the above, RMS is really only useful for calculating "heating equivalent" of an AC current or voltage source in a resistor, as mentioned due to the square of the current (or voltage) relationship with power. You would not use RMS value for basically anything else, for example calculating the losses in a diode, you would use the arithmetic average current, since diode heating is V*I.

 

[hadoque]

Great! When you all say it like that it seems really obvious. I was actually thinking that power equivalence could have something to do with it.
It’s good to think about these things sometimes...

 

[jim hardy]

It’s good to think about these things sometimes...

Amen. Good background work for the brain while raking leaves or watching sitting in front of the boob-tube.

old jim

 

[Windadct]

Actually in power electronics the DC RMS of the pulses is critical to consider. In high current devices the resistive elements do contribute significant heat. Luckily for square pulses it is relatively easy to keep track of.

Also MOSFET Rds on is a resistive characteristic - this has become a big factor in Silicon Carbide MOSFET where the die are becoming very small, so they can not, initially absorb(think mass), and then dissipate(Think Surface area) this I^2*R heat. (Just one of the many issues with SiC...)

Consider a DC Chopper - as the duty cycle (Vin/Vout) gets lower the portion of the losses ( heat) from the resistive elements becomes dominant.

 

[The Electrician]

Actually in power electronics the DC RMS of the pulses is critical to consider. In high current devices the resistive elements do contribute significant heat. Luckily for square pulses it is relatively easy to keep track of.

What is "DC RMS"?

 

[Windadct]

I was referring to a square wave pulse that is not "alternating", it is a dc pulse, then zero. So, we typically think of RMS as a way to get an effective value for AC, or sine.
In these cases we use both the mean, and RMS in the loss calculations.

 

[jim hardy]

Strictly speaking, DC just means it doesn't reverse polarity. So a zero to 10 volt square wave qualifies.

Non-varying current as from a battery is often called "Steady" ,

It's one of those little fine distinctions that's not respected in everyday communication..

I learned from looking at wall-wart plug supplies that DC might mean 'rectified but not filtered' as in battery charging applications.
Steady, or filtered DC, is often but not always indicated by a straight line adjacent the voltage marking.
You can tell them apart with a meter by how quickly the voltage decays after you unplug it.

old jim

 

[anorlunda]

I prefer other words. If a DC signal varies in time, you can calculate the RMS for any interval. Time varying does not make it AC.

 

[The Electrician]

I was referring to a square wave pulse that is not "alternating", it is a dc pulse, then zero. So, we typically think of RMS as a way to get an effective value for AC, or sine.
In these cases we use both the mean, and RMS in the loss calculations.

If we have a waveform such as you're describing:

How is the DC RMS calculated, as distinguished from ordinary RMS?

 

[jim hardy]

RMS is RMS.
Divide some period up into instantaneous readings at small time intervals
and use the mnemonic RMS is backward for " Square Mean Root ."
or if you prefer .forward, 'square Root of the Mean of the Squares".

Now to the arithmetic

Square each individual reading,
take their Mean(average)
take square Root of that mean(average),,
and you have the RMS of your wave over the period you chose.

i know you know that but for high school readers who are likely not fluent with integration (i know i wasn't)
...
proper math is to get the mean by integration, the infamous Studebaker sign ∫ ,
which just means divide your interval into an infinite number of time slices, very narrow ones of course
.....

anyhow if you do that to a steady DC voltage you'll get that exact voltage for a result.

Squaring each reading before averaging gives any taller ones more weight in the average than if they weren't squared
and that's why a 'peaky' waveform has a higher heating value than a sine shaped one.
That's significant in power supply filter capacitors where the charging current flows only in short gulps near the sinewave voltage peaks so is very peaky.
Every capacitor has a rating for RMS ripple current .

image courtesy of http://www.cde.com/resources/technical-papers/selectinvcap.pdfold jim

 

[essenmein]

I was referring to a square wave pulse that is not "alternating", it is a dc pulse, then zero. So, we typically think of RMS as a way to get an effective value for AC, or sine.
In these cases we use both the mean, and RMS in the loss calculations.

We typically think of RMS as exactly that, square root of the mean of the squares, it is a calculated value based on the wave shape, regardless of that shape, for example in DC-DC you often get a step ramp (inductive power transfer), if zero current switched, then its a portion of a triangle wave, RMS still applies as long as its a resistive component (wire, FET rdson, ESR of capacitors etc).

Note I would call the waveform you are describing an AC waveform with a DC offset.

 

[The Electrician]

RMS is RMS.

But what is DC RMS? I was hoping Windadct would explain how it's different enough from ordinary RMS to merit a special name.

 

[jim hardy]

But what is DC RMS?

I never heard that term before either..

Something to ponder though.

Imagine you have a True RMS Voltmeter ,
and use it to measure a signal composed of AC with DCoffset

Does the meter have a series capacitor to block the DC component and report RMS of just the sine wave?
Or does its report include the additional heating effect of the DC component too ?

I can see how someone in a hurry might ascribe the name "DC RMS" to the latter behavior
but he should define his term unambigupusly.
It's important to choose names that lead the mind to a correct mental picture, and DC RMS doesn't really take me anywhere.

Tonight i'll connect my True RMS Fluke to a flashlight battery and see what it reports on AC scale.

The venerable Simpson 260 includes a capacitor that you can use to block the DC component of such a signal..
You plug the red lead into the "Output" jack.

old jim

 

[anorlunda]

Thanks Jim. Very clear.

Squaring each reading before averaging gives any taller ones more weight in the average than if they weren't squared

You should also mention that it makes negative values positive, so that a signal with an average value of zero has nonzero RMS.

 

[jim hardy]

You should also mention that it makes negative values positive, so that a signal with an average value of zero has nonzero RMS.

Yes,

The Squaring step in RMS dictates a positive result because, as we learned in grade school, (-X)² = +X²,

so the Mean step can only produce a positive result because there's not a single solitary negative value among those X² 's that it's about to average

which is a good thing because the square Root step would bomb if it were handed a negative argument...

Hope that's not too un-academic for PF
i just think we need simple mental pictures to help struggle up the learning curve.

Wiki explains more eloquently than i could, thus

and every point on that sin² curve is positive so it'll have a nonzero average
unlike a sinewave with no offset which DOES have an average value of zero...
more at https://calculus.subwiki.org/wiki/Sine-squared_function

 

[jim hardy]

That squaring a sine doubles its frequency is well known from our trig identities
but it's not quite intuitive to me yet.

Seems i should be able to get there via Euler's equation

because when we square we double the exponent
so squaring left side yields e^j2ωt

when I've successfully expanded the right side i'll be back.
(You all know I'm math challenged...Coming to grips with Euler will be a major milestone for me. )

old jim

 

[jim hardy]

BTW i did measure a flashlight battery with my Fluke 80 True RMS DMM set to AC.

As expected it obviously has a capacitor to block DC .
At the instant of connection its reading changes from ~0 to OL(overrange) and drifts back down to zero over a second or so.
That's the capacitor charging transient, which allows brief current flow into the measuring circuit.

Somewhere in my life i have encountered a True RMS DMM that offered the choice whether to include or block the DC content of an AC measurement.
One should be aware of how his test equipment works because he could lose days of experimental work by unawareness of such a feature..
I use my Fluke so seldom that i wanted to refresh my memory as to whether it has that option. It does not (Unlike the analog Simpson 260 above).

Thanks folks, for putting up with this detail-obsessed old guy.

old jim