Root Sum Squared Error Analysis

[edba]

Hi

If you have two resistors in series that are say 100 ohm and 10 ohm with tol 1% then RSS analysis suggests it will have a resistance of 110 ohm +/- SQRT(1%^2 + 1%^2) = 110 ohm +/- 1.414%. This makes sense as it is less than worst case analysis of +/-2%. What if the resistors are in parallel? What form does the RSS calculations take?

Edba

[moffy]

If 100 is +1% high = 101
if 10 is +1% high = 10.1
result: (101 + 10.1)/110 = 111.1/110 = 1.01 or a +1% error

[Tation]

Revise the calculations because 100 + 10 (1 % tol.) gives 110 with 0.913 % tol. Already discussed here.

Also discussed, in the same thread (and post), the parallel case.

[SteveThackery]

I'm really struggling with this. Assuming the OP's 100 ohm and 10 ohm 1% resistors, in series, then it is completely obvious that the maximum possible value of resistance when both resistors are 1% high is 111.1 ohms, which is 110 ohms + 1%.

So it is surely the case that putting 1% resistors in series MUST give a tolerance of 1% on the summed value.

It seems to me that RSS analysis gives a probability curve of some kind. That is, RSS says the probability of both resistors being a full 1% high is lower than the probability of an individual resistor being 1% high.  That feels intuitively true.

However, it is surely wrong to claim that a pair of 1% resistors has a tolerance of less than 1%.  The definition of tolerance is the worst case situation: 1% tolerance means the worst case error is 1% high (or low). The worst case error for a pair of 1% resistors is - must be - 1% high or low.

So I think it is misleading to say that - thanks to RSS - the tolerance of a pair of 1% resistors is less than 1%.  It is still 1%, but the probability curve of the combined values is narrower - that is, the combined value is more likely to be close to the centre of the tolerance band.

Am I wrong?

I also suspect another shortcoming of RSS: it doesn't (can't) take into account the probability curve of the individual resistors' tolerance band, because it doesn't know it.

[bdunham7]

The definition of tolerance is the worst case situation:

Not necessarily, you'd have to actually define the term for any of this discussion to make any sense.  If the resistor values are a standard distribution and the tolerance is defined as x std devs, then the RMS calculation will give you a new tolerance also as x std devs.  If the distribution is rectangular, not random or even completely unknown then the math changes.  In any case, when it comes to these matters many assumptions are often made and not all of those have solid support.  If the spec is absolutely guaranteed--say by selection--and you want the result to also be absolutely guaranteed then you just add the tolerances as your intuition suggests.

[Tation]

If:

• the resistance of the resistors distributes normally
• and the tolerance is defined as being proportional to the standard deviation
• and the tolerance is small
• and the resistances of different resistors are not correlated

then the series or parallel combination of 2 or more resistors also distributes normally and its tolerance (defined as above) is less than that of the original resistors.

Under other sets of assumptions the result may be other, of course.

[edba]

Hi I think I have worked it out:

For two resistors in series each 1 % tolerance:

Worst case gives:

100 Ω + 10 Ω +/- (1 Ω + 0.1 Ω) = 110 Ω +/-1.1 Ω or 110 Ω +/- 1 %

RSS gives:

100 Ω + 10 Ω +/- √((1 Ω)^2 + (0.1 Ω)^2) = 110 Ω +/-1.005 Ω or 110 Ω +/- 0.91 %

Now two resistors in parallel each 1 % tolerance:

First of all convert to admittances:

1/100 Ω = 0.01 S +/- 0.0001 S, 1/10 Ω = 0.1 S +/- 0.001 S

Now use the same formula:

Worst case gives:

0.01 S + 0.1 S +/- (0.0001 S + 0.001 S) = 0.11 S +/-0.0011 S
= 9.09 Ω +/- (0.0011 S/0.11 S)
= 9.09 Ω +/- 1%

RSS gives:

0.01 S + 0.1 S +/- √0.0001 S^2 + 0.001 S^2 = 0.11 S +/-0.000100499 S
= 9.09 Ω +/- (0.000100499 S/0.11 S)
= 9.09 Ω +/- 0.92 %

With the formulas you can have resistors of different values and tolerances,

Brgds

Edba

[SteveThackery]

.If the resistor values are a standard distribution and the tolerance is defined as x std devs, then the RMS calculation will give you a new tolerance also as x std devs. 

Absolutely right. But that is not how resistor tolerances are given.

When you buy a 100 ohm 1% resistor:

1/ All you know is that it is somewhere in the range 99 to 101 ohms.

2/ You are given no information on the shape of the distribution curve.

3/ If the resistors are assigned to tolerance bins after manufacture, you can be virtually certain that the values do not follow a normal distribution curve, or anything like it.

This is the problem with mathematics - the hidden assumptions built into it may be far away from reality. I'm guessing the RSS method assumes a normal distribution curve around the nominal value?

[SteveThackery]

Worst case gives:

RSS gives:

What is the point of the RSS calculation when it is wrong? Or rather, the assumptions it makes (normal distribution?) are untested.

If you are designing a circuit in which resistors are not individually hand-picked for their value, then you must design the circuit to work over the full range of possible values: from -1% to +1%.  If you design for the RSS value then some of your circuits won't work.  Why would you do that?

(I'm assuming for the sake of argument that the resistor values are critical to circuit function; I realise that is rare.)

[gf]

I'm guessing the RSS method assumes a normal distribution curve around the nominal value?

My understanding is that variance(a) + variance(b) = variance(a + b) holds for random variables a and b with arbitrary probability distributions (even for different distributions for a and b), but a and b must be uncorrelated.

[Conrad Hoffman]

I've seen RMS used for many things because people crave some reasonable sounding information. I think we use it because it's easy and we don't have enough information to use anything better. Does it actually predict reality? I suspect it does poorly at best.

[Siwastaja]

Just remember that resistor value errors definitely can be correlated! Assumptions of errors averaging out to better accuracy are wrong because manufacturers sometimes do binning of parts: they sell parts that fail the tighter tolerance as lower tolerance parts: ten 1% resistors can all be off by 0.5% to 1% all in the same direction!

Besides, even if they were completely randomly distributed, statistical analysis makes no sense at N=2 or N=3 or so. The chances that two resistors are both +1%, is just too high for any practical use.

So I fail to see what is the practical purpose of such analysis. Just do the good old worst-case analysis instead.

[bdunham7]

1/ All you know is that it is somewhere in the range 99 to 101 ohms.

2/ You are given no information on the shape of the distribution curve.

3/ If the resistors are assigned to tolerance bins after manufacture, you can be virtually certain that the values do not follow a normal distribution curve, or anything like it.

I'm guessing the RSS method assumes a normal distribution curve around the nominal value?

1/ You don't really even know that for sure

2/ Most of the time, but some precision lines do give you additional information.

3/ Binning is one issue and IDK how common it is nowadays.  I think modern process control methods are more likely to give you batches of components that are all very close in value to each other but only within the tolerance of the nominal value. 

Yes, a standard distribution around the nominal value is assumed.  I don't know that this assumption is always wrong, but I suspect it usually is.  But I also think it is a mistake to replace it with another unsupported assumption such as the specified tolerance indicating the "worst case".  If your needs are critical then you should investigate further or characterize the parts on your own.

[SteveThackery]

1/ All you know is that it is somewhere in the range 99 to 101 ohms.

1/ You don't really even know that for sure

What? Of course you do! 1% tolerance means plus or minus 1% of the nominal value. What else can it mean?

[eugene]

The problem I'm having with this discussion is the assumption that the actual values of components in a batch will fall in a normal distribution around the nominal value. If that's truly the case then RSS does make sense and you could put four 1% resistors in series/parallel and end up with the same nominal resistance with a tolerance better than 1%. In this case tolerance really is a matter of probabilities because the claim is that any random sample will fall within, for example, 6 sigma of the entire distribution.

I would guess that a more likely situation with modern manufacturing is that the components in a single batch fall in a normal distribution around the mean value of the batch. That mean value might not line up with the nominal value. However, presumably some minimum percentage of the batch will still fall within the tolerance of the nominal value. But all of this is just a guess and is not based on any actual knowledge of the industry. (I did once work at a company that made thick film resistors, but there every resistor was laser trimmed by ablating any of a finite number of 'fuses.' I don't know what the final distribution looked like.)

Going back many decades to the origin of E6, E12, E24, etc series', the story I heard was that they would simply make a whole bunch if carbon comp resistors with essentially random values, then bin them into 6, 12, 24, etc equal size bins. Obviously, if this story is true then the distribution of values around nominal would not be Gaussian, but uniform. This also debunks what I used to hear a lot (either when those resistors were still in use, or by people that still remembered using them) that if you bought a bag of 100k, 10% resistors, then none of them would be within 5% of 100k because all of those went into the 5% bin. In any case, that particular fear couldn't be justified once you realize that they also needed 110k 5% resistors. Where did they come from if all of the 109k resistors went into the 100k 10% bin?

My final conclusion: there are too many plausible possibilities to make any predictions unless the manufacturer provides information about the probability distribution of those particular components.

[eugene]

1/ All you know is that it is somewhere in the range 99 to 101 ohms.

1/ You don't really even know that for sure

What? Of course you do! 1% tolerance means plus or minus 1% of the nominal value. What else can it mean?

It could mean that some minimum portion of the distribution of values falls within +/- of the tolerance. If they're manufacturing to 6 sigma then there is still a 0.0000002% chance that any random sample will fall outside of tolerance. If they're manufacturing to 3 sigma then as many as 0.3% could be outside tolerance. I don't know anything about any particular series by any particular manufacturer, but it's certainly possible.

[SteveThackery]

But I also think it is a mistake to replace it with another unsupported assumption such as the specified tolerance indicating the "worst case". 

1% means the resistor will be within +/- 1% of the nominal value. We agree that does not say anything about the distribution curve, so any assumptions about that are just that - assumptions.

As I see it, 1% must be the worst case. If a resistor is outside those limits then it is defective. If it is at or inside those limits, then it is in specification.  The worst case is at the tolerance limits. Where else could it be?

Assuming the resistor will not exceed +/- 1% is not an "unsupported assumption" - it is a published specification by the manufacturer.

[SteveThackery]

If they're manufacturing to 6 sigma then there is still a 0.0000002% chance that any random sample will fall outside of tolerance.

In which case it is defective. Do you think they don't test the value of each resistor? I think they are bound to, in order to assign a tolerance band to it. And that test could and should exclude even that 0.0000002%.

If you are arguing that resistors are not individually tested, but the manufacturer relies on the accuracy of the manufacturing process, then I totally agree: we should expect to see something close to a normal distribution with some outside the claimed 1% (say).

But I would point out that this appears to make the RSS calculation even less reliable, because it assumes none of the resistors exceed their tolerance. Doesn't it?

[eugene]

If they're manufacturing to 6 sigma then there is still a 0.0000002% chance that any random sample will fall outside of tolerance.

In which case it is defective. Do you think they don't test the value of each resistor? I think they are bound to, in order to assign a tolerance band to it. And that test could and should exclude even that 0.0000002%.

We like to make that assumption, and it might be true, but I am not prepared to assume that every manufacturer tests every every resistor. Even though it's a comparatively easy thing to do. many of these resistors cost a small fraction of a penny each. I cannot make the assumption that there aren't manufacturers that get their line going so that all but 0.0000002% are within tolerance and ship without testing each part. It's economics. And lets face it, the vast majority of my designs will continue to work just fine even if one of the nominally 1% resistors happens to be 1.00000001% outside of tolerance. If I really need that kind of tolerance then I go with 0.1% in the first place.

Again, I get that we want to make the assumption, and most of us have, but in this context I can't.

[bdunham7]

In which case it is defective.

Unless the manufacturer supplies a confidence interval or gives you a 1000% money-back guarantee that absolutely none of their parts will ever exceed tolerances, you don't know the expected defect rate.  If you declare that any resistor outside of tolerance is defective, then by your definition all non-defective resistors will meet their specified tolerance.  But that is circular reasoning and the lack of any defects by your definition was not included in the original statement about buying a 1% resistor.

If you are arguing that resistors are not individually tested, but the manufacturer relies on the accuracy of the manufacturing process, then I totally agree: we should expect to see something close to a normal distribution with some outside the claimed 1% (say).

I think you'll more likely see a small distribution of some kind centered around a mean value that is within the tolerance specification of the nominal value.  Dave Jones did a video on this and that was what he found with his samples.  YMMV (your manufacturer may vary) and I wouldn't make this a universal assumption. The easiest way to meet a 1% tolerance is to use a manufacturing process that gives you 0.1% or less variations.  But there's always a chance that something goes wrong.

But I would point out that this appears to make the RSS calculation even less reliable, because it assumes none of the resistors exceed their tolerance. Doesn't it?

No, a standard distribution, even if the specification is many std devs like "six sigma", allows for a mathematically defined number of out-of-tolerance results.  So if your component or device has a given standard deviation and you set the specified tolerance at 3X that amount (k=3) then there is a 99.5% confidence level--but still a 0.5% chance of a particular unit being out of tolerance.  Six sigma (k=6) reduces that to 0.0004%, but not zero.

[iMo]

The distribution of the resistor's values is uniform, my bet. Otherwise the manufacturers will not use the wording "1k 5% tolerance", but "1k x_sigma" instead. Also they do not guarantee the "batches with certain distribution", afaik. Moreover, a guy who once did the binning of a larger "batch" of resistors told me the distribution he saw was uniform (any value between +/- x% from the nominal value has got the same probability) ..

[Siwastaja]

Unless the manufacturer supplies a confidence interval or gives you a 1000% money-back guarantee that absolutely none of their parts will ever exceed tolerances, you don't know the expected defect rate.  If you declare that any resistor outside of tolerance is defective, then by your definition all non-defective resistors will meet their specified tolerance.  But that is circular reasoning and the lack of any defects by your definition was not included in the original statement about buying a 1% resistor.

This. Specifications of low-cost commodity components are rarely absolute guarantees. Some better datasheets even specify confidence numbers for min/max and tolerances.

One can try asking money back if out-of-tolerance resistor is found... the whole 0.1 cents.

Anything of extraordinary accuracy and reliability requirements need working with manufacturers and buying special grade parts, and that is going to be at least 100 times more expensive than working with normal components off Digikey.

And if I buy 100 000 pcs of 1% resistors 0.1 cents each and one of them is 1.05% off nominal value, I will let it be. Then again, if it's a systematic and significant failure of keeping within spec, I will definitely complain and name the manufacturer to others.

[Victorman222]

For illustration here are some real resistor value distibution "graphs" i made over the years
Plot 1 is of Tesla TR191 5% resistors, i was surprised how almost all are within 1%
Plot 2 is of S2-29V 1% precision thin film resistors, 1% is the loosest spec for them, can see that all resistors within 0.1% or even 0.2% of value were taken out. Right when i needed a 0.1% one...
Plot 3 is of S5-25 1% precision wirewounds, worth noting this isn't one batch but 6x 10pcs boxes
Plot 4 is of S2-29V 0.05% PTF resistors, interestingly here all resistors are in range 0...+0.05%
Plot 5 is of S5-50 0.05% PWW resistors

[Conrad Hoffman]

Let me offer an example of something I can't prove, but very strongly suspect. I suspect electrolytic capacitor manufacturers can now control their process to a much higher precision than their standard tolerance. Since they can do that, they save a lot of money on etched foil by building all their capacitors on the low side of nominal. You very rarely, if ever, find one at or above the nominal value. At least this has been my experience with higher values for the last decade or so.

[SteveThackery]

No, a standard distribution, even if the specification is many std devs like "six sigma", allows for a mathematically defined number of out-of-tolerance results.  So if your component or device has a given standard deviation and you set the specified tolerance at 3X that amount (k=3) then there is a 99.5% confidence level--but still a 0.5% chance of a particular unit being out of tolerance.  Six sigma (k=6) reduces that to 0.0004%, but not zero.

You've been describing the maths around devices with a normal distribution, and the idea of standard distributions and confidence levels.

But you are not describing RSS. Please read the very first post. The input to the RSS calculation is the nominal value and the specified tolerance limit (1%, in our case) for each resistor. The calculation produces an answer that is tighter tolerance than the worst case. No distribution curve is input to the calculation.

So please answer these questions:

1/ What distribution curve does the RSS assume for the resistor values?  What justification is there for that answer?

2/ Does the RSS assume some resistors will be outside the specified tolerance? If so, how many?

3/ The RSS calculation produces one number as its answer.  What grounds are there for assuming any particular distribution curve around that value?

When you can answer those fundamental questions we'll be getting somewhere. Otherwise we are vigorously debating something we have no data for: the distribution curve of resistor values. Probably they differ from one manufacturer to another, and probably they differ over time as new manufacturing processes are adopted.