Im interested in getting some high precision resistors for a project, however they are very expensive.
I was wondering how the Accuracy of a resistor changes when you hook them up in parallel or series.
For example, if I have 2x 1% accuracy resistors and I hook them up in Series or Parallel, does the accuracy of the result double to 0.5%?

 

There is no certainty because the resistors could both have errors with the same sign. e.g. two resistors of 100 ohms ±1% could both be 99 ohms or both be 101 ohms.

In general, resistors from reputable manufacturers will be considerably closer tolerance than nominal, however it would be unwise to count on this. Simply soldering a resistor can change its value a little.

One of the significant limitations of semi-precision resistors is the temperature coefficient of resistance. Most low-cost ±1% tolerance SMD types have a tempco of ±100 ppm/°C, This is also true for through-hole types, though better quality parts will have a tempco of ±50 ppm.

In surface mount, 0.1% 25 ppm/°C are quite reasonably priced. If you require closer tolerance or better tempco, the price jumps to nearly a dollar each in small quantities. True high-precision low-tempco types, such as Vishay bulk metal foil, start at several dollars each.

 

If you put two 1% resistors in series, the tolerance becomes 2%. This is a gross simplification, assuming that the resistors are of equal values. If the resistances are vastly different, you need to add the absolute errors.

For resistances in parallel, the combined tolerance is more complicated because the parallel combination is Rtot = (R1 x R2)/(R1 + R2), but will still be greater than 1%.

You could go the absolute and relative error method or you can use the root-mean-square method.

 

If you put two 1% resistors in series, the tolerance becomes 2%.

Try that again.
Two 1% resistors in series will still have a tolerance of 1% for the total resistance.
Same as if they are in parallel.

 

Try that again.
Two 1% resistors in series will still have a tolerance of 1% for the total resistance.
Same as if they are in parallel.

Ah! You got me. I was thinking that you add the errors.
I ought to be thinking that you add the absolute errors.

When multiplying and dividing, you add the relative errors.

 

So, there is no way to combine resistors to improve accuracy?
I was hoping that accuracy followed the same rules as parallel resistor values... Guess not.

Yes, im a total amateur...

 

So, there is no way to combine resistors to improve accuracy?
I was hoping that accuracy followed the same rules as parallel resistor values... Guess not.

Yes, im a total amateur...

If you have an accurate enough ohm meter and you're willing to hand trim each circuit, you can manually choose and/or combine resistors to better tolerances by measuring them.

But if you're talking about being able to just design a circuit, order parts, and slap it together, then you have to pay for tighter tolerances.

How tight of tolerances do you actually need, and at what price points? Are you sure you need to search for a way to beat the system? (Cause I don't think you're going to find one!)

 

I was looking for sub 1% tol for working with op-amps.
Trying to build some circuits I found, and they made a strong point of saying that the voltage divider had to be very accurate.

Maybe im over-reacting, but I don't really know how it will work and was just trying to get the best parts I could to try and avoid any issues. I was hoping there was a short-cut you could take to improve accuracy...

 

Are you trying to build an IA (Instrumentation Amplifier) ?

If so generally speaking IAs are laser trimmed on production line in
final test.

There is another way using low accuracy devices to achive high accuracy.
See attached.

Regards, Dana.

 

You can take a carbon composition resistor and raise it's value by filing and then covering with nail polish when your done. Use a higher wattage resistor to start with. I know it works for carbon composition resistors.

 

I'll make a bet -- winner gets a beer:

Someone solder 100 randomly chosen resistors of the same value (parallel or series, I don't care).

If the final accuracy is not much better (say by a factor of 10) than the tolerance of the individual resistors, you win.

The law of large numbers says the error must converge to zero as N approaches infinity.

 

The simple solution would be to buy 100 ($3 worth) or 1000 ($12 worth) 1% resistors of the same value and select resistors by testing with an Ohmmeter.

 

But all the 1% and 0.1% resistors are picked over.

 

But all the 1% and 0.1% resistors are picked over.

Doesn't matter. You just want the resistors that have the same value.

 

The last time I did a project where they had to be close, we found that none of the 10% resistors were within 5%, which it seems comes from the sorting process, where the ones within 5%were all sold as 5% resistors while those within 10% were then relegated to the 10% bin. The trick to have close resistors is to use a resistor array chip, where they are all very close to each other. So you find a 16 pin array with 15 resistors all very close to each other. That is one way to do it. And the temp coefficients match as well.

 

Long ago they used to sort out the 1% and 5% resistors and sold them at a premium. Hence 10% or 20% had a double-humped distribution.
They don't do that any more because process control is much improved.

 

So, there is no way to combine resistors to improve accuracy?
I was hoping that accuracy followed the same rules as parallel resistor values... Guess not.

Yes, im a total amateur...

There are a number of concepts that come into play here. There is a difference between accuracy and precision, which plays a role. There is also a difference between tolerance and standard deviation, which plays a big role. But perhaps the biggest role is the difference between an assumed distribution of resistor values and the actual distribution.

First, when we talk about tolerance we are saying only that the resistor value, under some specified set of conditions (such as room temperature when new) are essentially guaranteed to be within the tolerance limit of the nominal value. We are saying absolutely nothing about the distribution of values about the nominal value. So if we buy a bag of a thousand 1 kΩ resistor that are 10% tolerance, all we know is that we have a reasonable expectation that every resistor in that bag has a high likelihood of being somewhere between 900 Ω and 1100 Ω. If it turns out that all of them are between 1063 Ω and 1091 Ω, then that bag of resistors meets spec. If it turns out that every last one of them is 902 Ω, that bag of resistors meets spec. If 30% of the resistors are between 920 Ω and 932 Ω while 50% are between 1094 Ω and 1098 Ω and the rest are scattered all of the place but there aren't any that are between 987 Ω and 1053 Ω, then that bag of resistors meets spec.

We are not justified in making ANY assumptions about the distribution of actual resistors in that bag. In particular, we are not justified in assuming either that the distribution is remotely normal or that the mean value of all the resistors is particularly close to the nominal value. These two assumptions used to be especially poor in years past when manufacturing tolerances were poor and component grading was used to sort and classify parts for sale. The tolerances are a lot better now so these two assumptions aren't nearly as dangerous, but they still need to be taken with a huge grain of salt.

Second, when we talk about the tolerance of combinations of parts, such as ten resistors in series or ten resistors in parallel, the fact that we are talking about tolerance means that we have to come up with a number that, assuming all of the individual resistors meet spec, the combination of parts is guaranteed to meet. So if you put ten of our resistors in series, the tolerance is still 10% because I can only guarantee that the total resistance will be within 10% of 10 kΩ. But the likelihood that it will be that far out generally goes down because of the averaging effect -- the expected value will probably be within about 3% of 10·Ravg where Ravg is the average value of all the resistors in the bag -- but remember that Ravg might be 8% greater than the nominal value. Without knowing the actual distribution (or having a sound basis for approximating it somehow), that's about all I can say.

IF the resistors are normally distributed (and this actually applies for most non-normal distributions as well), then the uncertainty in the actual value relative to the expected value falls as the square root of the number of measurements. This means that if you have 100 resistors in series, then if the uncertainty (which is NOT the same as the tolerance) is 10% for a single resistor, then the uncertainty will fall to about 1% for the string of 100 resistors -- but remember that the expected value is 100 times the average value of the resistors and NOT the nominal value.

Another way of saying this is that if you take your 1000 resistors and built 10 strings of 100 resistors each, then you now expect the fractional difference between the total resistance of each string to be about ten times smaller than you would have expected between pairs of individual resistors. This is a reflection of improved precision, not accuracy. Precision is the ability to repeat a measurement, accuracy is the ability to get the measurement close to the design value. The accuracy is dictated by the closeness of the average value of all the resistors relative to the nominal value.

Putting the resistors in parallel introduces a complicating factor because the expected value of 100 resistors in parallel is NOT 1/100 of the average value, even if the resistors are normally distributed. This is because the distribution of the conductances of N resistors is different from the distribution of the resistances of those same N resistors and the average conductance is not the reciprocal of the average resistance -- the average conductance is a bit higher than the reciprocal of the average resistance, and so the resistance of the parallel combination of resistors drawn from that distribution will be biased downward.

To see that this is the case, consider three resistors of 9 Ω, 10 Ω, and 11 Ω. Clearly the average is 10 Ω and the series resistance is 30 Ω. or three times the average. But what about the parallel resistance? Is it 3.333 Ω? Nope. It's 3.311 Ω, a difference of about 0.7%. So if you are trying to achieve better accuracy by combining resistors in parallel, you have to be very careful because you can easily inject a systematic error that quickly guarantees you can never get within the desired reach of the desired value unless you take it into account.

EDIT: Fix a few typos.

 

I'll make a bet -- winner gets a beer:

Someone solder 100 randomly chosen resistors of the same value (parallel or series, I don't care).

If the final accuracy is not much better (say by a factor of 10) than the tolerance of the individual resistors, you win.

The law of large numbers says the error must converge to zero as N approaches infinity.

You are confusing accuracy with precision.

The law of large numbers would also only say that the uncertainty converges to zero around the mean, not the nominal value.

 

I have found that the solution is to design the circuits to work with a bigger spread of values. Depending on tight control of component values is a good way to encourage production problems. In those rare instances when a very specific value is needed then a tighter tolerance part may be the only choice, although adjustable resistors are a very convenient means of calibrating a system, and allowing re-calibration as components age.

 

I was looking for sub 1% tol for working with op-amps.
Trying to build some circuits I found, and they made a strong point of saying that the voltage divider had to be very accurate.

Maybe im over-reacting, but I don't really know how it will work and was just trying to get the best parts I could to try and avoid any issues. I was hoping there was a short-cut you could take to improve accuracy...

You can get 0.1% tolerances starting at $0.46 (single unit pricing, improves with quantities) from DigiKey. Of course, other distributors will have similar options - this isn't an endorsement, just an example:

https://www.digikey.com/products/en...tock=on&pv7=2&pv3=431&pv3=357&pv3=380&pv3=370

Like several others here, I doubt that you need extreme precision for "total amateur" projects, for a variety of reasons. If this is for individual projects you'll build by hand, as opposed to mass production, there are many ways to account for resistor tolerances, including measuring and hand selection, using trim pots to fine tune total resistance, changing calibration coefficients in code (if it's a microprocessor project, like Arduino, Raspberry Pi, etc.) etc.

For *most* projects, only a few resistors will need high accuracy to begin with, and even those can be easily worked around. At work, with surface mount parts, it's easy to source 0.1% parts, which is close enough for nearly everything I need. One process requires manual calibration after construction, but everything else works as is. For personal projects using through hole parts, I usually just get 1% parts and then either measure-and-match, or just tweak code calibrations to account for actual values vs. spec'd values.