Electronics > Metrology

Tolerance stackup

**edba**:

I am currently working on a problem calculating the total tolerance error of a circuit for a test specification.

Most of the stuff on the internet concerning tolerance stackup is concerned with mechanical engineering, although I think it will apply equally to electrical/electronic engineering as well.

Ok as an example say I have a adjustable voltage regulator where the voltage is set by two resistors of +/-1% tolerance and say the voltage reference of the regulator is +/-1.5%. Now there are three independent variables. A worst case analysis would put the tolernce as 1+1+1.5 = 3.5%. This is quite a pessimistic view of the tolerance. A more realistic one is aquired by using the Root Sum Square (RSS) method. Here you square all the values, add them together, then take the square root so (1^2+1^2+1.5^)^0.5 = 2.06%.

My question is has anybody got experience of using RSS and no of a good tutorial and why does it work?

**Kleinstein**:

The RSS is the normal method in many cases, though there are usually additional factors to take into account on how much each part / parameter effects the result. With a resistor divider or gain network this can give an additional factor slightly smaller than 1. So in addition to the tolerance there is the factor on how much the ouput changes with the parameter. This factor is also squared, like the tolerance.

The addition as RSS is correct for the standard deviation of independent variables with normal distribution. It is not correct for something like the worst case. Electronic parts may be binned to be worst case inside a tolerance area so the RSS may be only an approximation than.

With many small contribution the oveally error will become normal distributed. So at least for the smaller parts the RSS form is usually OK.

**mendip_discovery**:

RSS is how uncertainty is done, so some info will be on the isobudgets site. I did attempt to show an example budget for the 121GW but it wasn't easy to get it over to people, not sure this forum is ready for the discussion.

If you want I can share a blank budget calculator you can use.

Square root is not your only option as you might be able to argue some things are distributed differently such as triangular.

But I have never been on the building side of things so I am not sure how the manufacturer's build-up characterisations of devices, lots of testing I would say.

If the resistance is at the extremes of the 1% resistors how much of an effect will it have on the final measurement at max measured voltage? You could possibly use a coefficient.

**bdunham7**:

--- Quote from: edba on October 25, 2021, 07:03:14 pm ---My question is has anybody got experience of using RSS and no of a good tutorial and why does it work?

--- End quote ---

Why it 'works', to the extent that it actually does, is simply math and it simply means that if you have two uncorrelated sets of data with n points each, say A and B, each with a standard distribution, then the sums of the respective data points An + Bn will also be in a standard distribution with a standard deviation that is the RSS of the SD of A and B.

Why it might not work for you is that your data sets may not actually be uncorrelated and/or they may not be in a standard distribution. I think it would be an egregious error to assume any component has a standard distribution centered around its nominal spec. Also, when you are making a product, or doing anything else for that matter, you have to decide how many standard deviations to set your tolerances. With a sigma of 1, about 1/3 of your results will be out of limits. So what does a "1% resistor" actually imply? In your example, is 3.5% really the worst case? Generally your choices are to either test yourself, pay a lot for better characterization of components or use greater margins.

**TimFox**:

One could simplify this answer to "RSS is probably right, but the worst-case answer is never wrong."

One thing to watch out for when doing RSS analysis is to avoid correlated variables--the statistical analysis assumes that each variable entering into the sum is statistically independent of the others.

When doing RSS sums, one can either add absolute values or fractional values, e.g. 1% resistors and +/- 0.5 V batteries. If you have both, then convert the errors to the same type (absolute or fractional).

It's a little tricker to estimate the error on a voltage divider: start by writing the explicit algebraic formula in terms of the two resistors.

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