Tolerance of resistors in series and parallel

[prenato]

Ever wondered how combining multiple, same value resistors in series or parallel affects the overall tolerance? I did, so I decided to write an article analyzing this. A fun little problem that actually required some mathematics:) Check it out if you are interested:

http://paulorenato.com/joomla/index.php?option=com_content&view=article&id=109&Itemid=4

Paulo

[Neilm]

Very interesting 

[alm]

Leslie Green discusses this in his freely available book Analog SEEKrets. He uses the more conservative flat distribution within the tolerance band.

[nctnico]

Ever wondered how combining multiple, same value resistors in series or parallel affects the overall tolerance? I did, so I decided to write an article analyzing this. A fun little problem that actually required some mathematics:) Check it out if you are interested:

http://paulorenato.com/joomla/index.php?option=com_content&view=article&id=109&Itemid=4

IMHO this is severely flawed because it assumes the error of the resistors has a Gaussian distribution which never will be the case especially since chances are the resistors come from the same batch and where produced together / shortly after each other. Remember Murphy's law!

[amspire]

I agree with nctnico. I would say that increasing the number of resistors does nothing to improve tolerance as it is rare for the average resistance value to equal the specified value.

In fact, with some brands, I have seen what looks like a deliberate strategy of making the value a little low, and my guess is that the manufacturer knows these resistors drift higher with age.

The main thing is say they make a batch of 100R 1% resistors and the average for the batch is 100.5 ohms. Will the manufacturer sell this batch?  Definitely. It is within specs.

In this case, the more resistors you average, the closer you get to +0.5% error. You will not decrease the error further by adding more resistors from this batch.

Richard.

[KJDS]

Don't forget the temp coeff either.

I did have an instance a couple of years ago where I designed a board for multiple similar applications. The users had a grid of three by three 2520 sized surface mount resistors to place in order to get the required value. (it needed to cope with high power occasionally in a fault condition and survive, hence the large size. One of those listening to the talk I gave about using the board was convinced that using nine 1% resistors would result in a 9% tolerance. He took a little convincing, at least if I'd had the full maths available I could have made him work through it.

[prenato]

Ever wondered how combining multiple, same value resistors in series or parallel affects the overall tolerance? I did, so I decided to write an article analyzing this. A fun little problem that actually required some mathematics:) Check it out if you are interested:

http://paulorenato.com/joomla/index.php?option=com_content&view=article&id=109&Itemid=4

IMHO this is severely flawed because it assumes the error of the resistors has a Gaussian distribution which never will be the case especially since chances are the resistors come from the same batch and where produced together / shortly after each other. Remember Murphy's law!

Hi,

I'm wondering if you read the "experimental Results" portion of the article? There I mentioned that indeed the distribution did not look normal. I also showed that nevertheless the approximation worked well (averages of several non-normal distributions can approach normal because of the central limit theorem). In the conclusions I also recommend leaving some margin over this result to be conservative.

Paulo

[prenato]

I agree with nctnico. I would say that increasing the number of resistors does nothing to improve tolerance as it is rare for the average resistance value to equal the specified value.

In fact, with some brands, I have seen what looks like a deliberate strategy of making the value a little low, and my guess is that the manufacturer knows these resistors drift higher with age.

The main thing is say they make a batch of 100R 1% resistors and the average for the batch is 100.5 ohms. Will the manufacturer sell this batch?  Definitely. It is within specs.

In this case, the more resistors you average, the closer you get to +0.5% error. You will not decrease the error further by adding more resistors from this batch.

Richard.

Hi Richard,

In "experimental" results I did notice the average was slightly lower. I did not know this was usually the case; that's an interesting observation. In my definition of tolerance, I was comparing the "average" value of the individual resistor to the "average" value of the total so it didn't matter. You should still see the variation relative to that (lower than nominal) value decrease. However, you do have a point that if you define tolerance relative to the nominal value, then the gain is not so high. My analysis is only relative to the average value of individual resistors (which if you go to the trouble, you can measure as I did).

Paulo

[Ferroto]

Dave covered this in the decade resistor substitution box tutorial. Tolerances don't add up although i'm not sure what would happen if you mix resistors of different tolerances.

[alm]

I'm wondering if you read the "experimental Results" portion of the article? There I mentioned that indeed the distribution did not look normal. I also showed that nevertheless the approximation worked well (averages of several non-normal distributions can approach normal because of the central limit theorem). In the conclusions I also recommend leaving some margin over this result to be conservative.

The average of a large number of resistors from the same batch will approach a normal distribution, since they are likely independent identically distributed. There is no guarantee that their mean is their nominal value. In amspire's example, the tolerance of the sum of those resistors will be certainly better than 1%, but the expectation value will be n * 100.5 Ohms.

The general case is a bit better, since the circuit will generally consist of different components. You won't usually have a string of ten identical resistors in series. These different components will not be from the same batch, so will usually have a different bias.

Leslie Green discusses most of the issues raised in this thread in chapter 3 of his book, which I highly recommend anyone interested in this to read.

[prenato]

Another observation:

If, as you noted, manufacturers deliberately make the resistors slightly lower than their nominal value in "average" expecting that they will drift higher in age and also that the normal operation temperature will be higher, then this means their average resistance during normal operation would be closer to the nominal than when measured like I did. If this is true, then the formulas could be applied to the "nominal" value correct? Of course, you would always want to add some conservative margin on top of that.

Paulo

[Rufus]

We have been here before

https://www.eevblog.com/forum/blog/gaussian-resistors/

https://www.eevblog.com/forum/blog/gaussian-resistors-redux/

[JackOfVA]

Attached plot shows distribution of measured values for 5 resistor types, 4.7K ohm nominal value.

1/4 watt 5% carbon film
1/2 watt 5% carbon composition
1/4 watt 5% carbon composition
1/4 watt 5% surface mount 1206 thick film
1/4 watt 1% surface mount 1206 thick film

Bin width in the histogram is 0.5%.

The plot is from an article I wrote a few years ago published in the amateur radio journal QEX.

For amusement purposes, a second plot is attached, providing max, min and mean resistance values measured with a box of 5% carbon composition resistors manufactured around 1960 and measured in 2007. As typical, almost all have drifted high in value over time, some more than others.

[amspire]

I just do not see where the concept of adding multiple devices to improve tolerance on average has a lot of practical benefit.

If you take the case of something truly random - Low Frequency noise on a precision voltage reference, you can reduce the noise statistically by adding parallel references, but the catch is the peak noise does not change. You get a lower average, but the peak error is no better then a single reference.  All you can say with multiple parallel references is that the result form one measurement is probably more precise then the measurement with a single reference, but you do not know. So multiple noisy references is no replacement for a single low noise reference, despite whatever statistics tell you.

In commercial design, you have to almost go in the opposite direction. You have to ask if the circuit still works to specs if every part is the absolute worst possible value within their tolerance range. If the answer is yes, you have a very good design. If you base a design on the basis that the circuit works as long as you assume a standard bell curve variability of part values in the circuit, then I can tell you happens as I have seen it. You make the device for years without a problem, and then suddenly you are finding that you are getting near 100% failure on the production line even though every part is within specs, and there are no errors in production. This is one of the worse problems to fix, as nothing is faulty - it is just a bad design that does not allow for the fact that current batches of components could just happen to produce the worse possible accumulations of errors.

You see the results of good design practice in quality multimeters - it is not uncommon for the accuracy to be much better the a tenth of the specified accuracy. The designers have assumed the worse, but on average, component errors will cancel to a degree actually improving accuracy.

I think the fundamental rule is that multiple low performance components is not a substitute for one high performance component.

Richard.

[alm]

So multiple noisy references is no replacement for a single low noise reference, despite whatever statistics tell you.

In many cases, for example for an integrating ADC, the average noise is more important than peak noise.

In commercial design, you have to almost go in the opposite direction. You have to ask if the circuit still works to specs if every part is the absolute worst possible value within their tolerance range.

This is fine in simple systems, but in complex systems it can be quite tricky to figure out the worst possible combination of values. This may not be the highest or lowest value. This is where Monte-Carlo methods are useful. Any engineering solution can only cover so many nines of reliability, no solution is 100% reliable, so what's wrong with applying the same to component tolerances?

[Psi]

As you add more and more resistors to the series/parallel array the likelihood of having a totally wrong value resistor mixed in with the correct values increases.

[amspire]

So multiple noisy references is no replacement for a single low noise reference, despite whatever statistics tell you.

In many cases, for example for an integrating ADC, the average noise is more important than peak noise.

True. But for such an application, you will get about the same performance from a single reference as with several paralleled references.

In commercial design, you have to almost go in the opposite direction. You have to ask if the circuit still works to specs if every part is the absolute worst possible value within their tolerance range.

This is fine in simple systems, but in complex systems it can be quite tricky to figure out the worst possible combination of values. This may not be the highest or lowest value. This is where Monte-Carlo methods are useful. Any engineering solution can only cover so many nines of reliability, no solution is 100% reliable, so what's wrong with applying the same to component tolerances?

It is often easier then you may think at first. You do not have to look at every part - for most circuits, the performance is defined by certain critical parts and for the majority of parts, variations over the full tolerance range is no issue. For the critical parts, the worst case is often minimum or maximum value and it is often not hard to find worst case situations. I am talking abstractly, and yes, there are always counter examples. in the case of a circuit that may become unstable, the worst case could be with parts that are not at their tolerance extremes.

I have used Monte-Carlo methods to check a circuit about once - probably about 30 years ago. Never seen a need since. It can waste a lot of time. I tend instead to run multiple simulations based on a list of my chosen values all selected based on my understanding of the circuit. Again, I am sure there are examples where the Monte-Carlo method is fabulously useful, but I can't help thinking it is more useful in convincing others (eg in a technical paper) then actually helping the designer. A designer usually wants to vary something and to see the result. A statistical result of randomly chosen values is comforting, but you do not learn much about the circuit behavior and characteristics.

[alm]

I agree with your take on Monte-Carlo methods. It does not provide any insight and is essentially a last-ditch effort to get the information you want.

Keep in mind that any figure the manufacturer will quote is based on statistics. Those 1% resistors will likely be sample tested and guardbanded to so at least 99.9% (or whatever the number is) falls within specs if the statistics work out . Even if they test each individual component, then the calibration of their equipment will also be based on statistics. They take components, test a sample of them for drift, multiply the specs by some factor, and out comes the accuracy. Physicists also make heavy use of statistics to derive fundamental quantities.

If you go down to the physics level, there is a tiny probability that all electrons will flow to the negative terminal of your power source. This is extremely unfavorable, however, so Boltzmann statistics tells us that this configuration is extremely improbable. Assuming that it won't happen is relying on statistics, however.

Guaranteed max only exists on paper, and may help in shifting the blame, but does not reflect reality. This does not mean that I would apply the central limit theorem to two identical series resistors, however.

[prenato]

This does not mean that I would apply the central limit theorem to two identical series resistors, however.

Hi Alm,

Just one small clarification: You make a good point that for small number of resistors the central limit theorem doesn't apply. However, you will notice that I didn't use the CLT in the derivation at all:) I just mentioned it as it provides an alternative way to look at it (for a large number of resistors only) as you well noted. The analysis uses a linear approximation that is valid provided the variance is much smaller than the average *regardless of distribution*. What we cannot say though, is that the resulting combined distribution is 'normal' as the CLT would imply, so that the "confidence" interval for a small number of resistors will naturally be larger than for a large number or resistors combined...

Anyway, point taken that the smaller the number of resistors, the less "confidence" you can have in the average result. Perhaps I should make that clearer in the article.

I should also note that these statistical results may be more applicable in the context of an analog chip design where the resistors are not "sorted" like in the discrete resistor manufacturing process? Maybe some chip designer around here can comment:)

Paulo

[alm]

The reference to CLT was not a comment on your work, but about the variance of the tolerance that could be expected for small numbers of resistors. So the tolerance of two 1% series resistors might turn out to be 0.999% at some sufficiently high confidence level, as opposed to the 0.7% you might expect. Constructing a confidence interval will by the way require you to assume some sort of distribution.

I'm not an IC designer, but I believe that the tolerance of on-die resistors is quite lousy, but they tend to be well matched within a die. So the assumption that they're independent identically distributed would definitely break down. Of course that assumption is problematic to begin with if resistors on a PCB are from the same batch.

[chickenHeadKnob]

Hello all:

I was reading the vishay website some time ago and they mention that the near zero tempco metal film series (expen$hiv!) is composed of elements with counterbalancing tempco, +/- I assume. I am a charter member of the cheap bastard hobbyist posse - (expression shamelessly stolen from MrFlibble) I was wondering can we construct such countervailing balanced resisters from off the shelf available discretes? All of the standard cheap resisters seem to be PTC and combining those with a typical NTC would appear to be an exercise in frustration. I guess if it was easy those vishay parts wouldn't  command the premium they do. Trimming cheap parts  in the home workshop is straight forward, like that bodge trim in the decade box that Dave exposed in one of his tear-downs, but then you have to  accept the lousy tempco.

Edit: just see that peterthenovice has started a similar question, but he is looking at an active digitally controled compensation, I was just considering passives, something that could be used with precision op-amps. 

[amspire]

All of the standard cheap resisters seem to be PTC and combining those with a typical NTC would appear to be an exercise in frustration.

This is a myth that keeps coming up. Metal film can be positive or negative temperature coefficient. Check the datasheets - it is +/- 50ppm for a typical resistor, and if you test some, you will get positive and negative tempcos. Typically, a single batch of 50ppm/C metal film resistors will mostly match to +/- 5ppm, but there will be some that may be +/-10ppm. To get a 1ppm/C match, you would definitely have to measure the coefficient of each resistor and that is a long, slow job.

I guess if it was easy those vishay parts wouldn't  command the premium they do. Trimming cheap parts  in the home workshop is straight forward, like that bodge trim in the decade box that Dave exposed in one of his tear-downs, but then you have to  accept the lousy tempco.

Edit: just see that peterthenovice has started a similar question, but he is looking at an active digitally controlled compensation, I was just considering passives, something that could be used with precision op-amps.

I have tried standard negative temperature coefficient resistors matched with an extra series copper wire winding (positive tempco) to give a net zero temperature coefficient. This could work if you put a constant current through the resistor and let it stabilize for half an hour, but under transient conditions, the performance is pretty disappointing. It is a lot of work, and ideally you have to be able to precisely match the thermal mass of the copper with the thermal mass of the resistors element. It is near impossible to get the exact fine copper guage you need.

The effect of this transient problems can be seen if you connect a 10K compensated resistors to a precision DMM ohm meter.  If the ohm meter is testing at 1V, the heat can easily make a cheap compensated resistor drift while you are watching by 200ppm until all the temperatures stabilize. The errors can be higher then you would expect from the resistor tempco, as the tempco specification applies to the case when the resistor temperature has had time to stabilize throughout the resistor. A Vishay metal foil resistor will probably not budge on a 6 1/2 digit meter.

Then you have to look at other factors that can change the value of cheap resistors like humidity. I think to have a chance of working, you would want to put the resistors in a mineral oil bath.

You will never get a compensated metal film resistor to even get close to a Vishay metal foil resistor. The Vishay resistor is extremely carefully constructed using annealed foil, so it has no internal stresses, mounted on a precisely matched substrate. This is why it has great long term stability. This makes them superior to wirewound resistors which do have problems caused by the stresses in the wire resulting from the winding. These stresses can make the metal in the wire recrystallize at the stress points over time, and this results in a resistance change. Metal film resistors just do not have the kind of rigorous construction for long term stability.

[chickenHeadKnob]

This is a myth that keeps coming up. Metal film can be positive or negative temperature coefficient. Check the datasheets - it is +/- 50ppm for a typical resistor, and if you test some, you will get positive and negative tempcos. Typically, a single batch of 50ppm/C metal film resistors will mostly match to +/- 5ppm, but there will be some that may be +/-10ppm. To get a 1ppm/C match, you would definitely have to measure the coefficient of each resistor and that is a long, slow job.

I have tried standard negative temperature coefficient resistors matched with an extra series copper wire winding (positive tempco) to give a net zero temperature coefficient. This could work if you put a constant current through the resistor and let it stabilize for half an hour, but under transient conditions, the performance is pretty disappointing. It is a lot of work, and ideally you have to be able to precisely match the thermal mass of the copper with the thermal mass of the resistors element. It is near impossible to get the exact fine copper guage you need.

The effect of this transient problems can be seen if you connect a 10K compensated resistors to a precision DMM ohm meter.  If the ohm meter is testing at 1V, the heat can easily make a cheap compensated resistor drift while you are watching by 200ppm until all the temperatures stabilize. The errors can be higher then you would expect from the resistor tempco, as the tempco specification applies to the case when the resistor temperature has had time to stabilize throughout the resistor. A Vishay metal foil resistor will probably not budge on a 6 1/2 digit meter.

Thank you for relating your experience, it actually gives me some faint hope. I had a similar idea to your copper coil but with meandering copper tracks or short precisely cut bits of nichrome. But now I see that I would have to create some kind of automated sorting machine and an environmental test chamber and stick to using metal film parts only. At my age I don't have much patience for hand logging and sorting tasks anymore however the challenge of automating the process is  more intriguing. Batch selecting resistors to be connected in series for complementary tempco over a reasonable temperature range then binning would seem possible. There is still one more disadvantage to home selected parts over the best vishay/dale; the whole axial versus radial leads problem and metal can encapsulation advantage of the real thing.

[mikes]

Ever wondered how combining multiple, same value resistors in series or parallel affects the overall tolerance?

Statistics only show what the value is likely to be. Usually, when dealing with tolerances, you want to know that it will be in a range. Statistics do not offer such guarantees, so aren't particularly useful, and certainly don't apply to how overall tolerances (exact guarantees) are affected by random choice.

If you want to combine resistors to get a tighter tolerance, measure them, that's the only way.

[prenato]

Statistics only show what the value is likely to be. Usually, when dealing with tolerances, you want to know that it will be in a range. Statistics do not offer such guarantees, so aren't particularly useful, and certainly don't apply to how overall tolerances (exact guarantees) are affected by random choice.

If you want to combine resistors to get a tighter tolerance, measure them, that's the only way.

Depends how you defined "tolerance". You are correct in the strict sense of tolerance relative to a nominal value. If you read the article, you will see this is not how I defined "tolerance"; I defined it relative to the average value in a given batch of resistors.  Maybe I should have titled this "how 'variance' improves by combining multiple resistors"
Variance *does* reduce relative to the mean when you combine multiple resistors. This is a fact. Whether that is very useful given all the sorting / biasing introduced in production is questionable, I agree. But I didn't claim that tolerance relative to the nominal would improve. The example I provide by measuring 50 resistors shows their average is 997 instead of 1000. The variance reduces relative to the 997 (/50)  average value, but not relative to the 1000 'nominal' value. Hope this clarifies the confusion.

Paulo