[volt-nuts] Matched resistors

[Tony Holt]

Randy,

On 24/07/2014 04:22, Randy Evans wrote:
> Tony,
>
> Your improvement factor of SQRT(n) assumes that each resistor in the group
> has random changes uncorrelated to all others in the group.  For similar
> type resistors, I would think that is not likely to be true.

Yes/,/ I'm well aware of that which is why I discussed that point 
further down in my post. It was a long post though so I don't blame you 
for getting bored and not getting that far!

> For shelf life
> stability it is likely that they all "age" in a similar way.  Unless the
> resistors are in a hermetic package, humidity would impact all the
> resistors in a similar manner.
>
> Randy
Exactly. Since they are being used in a 1:1 divider configuration, if 
they age in a similar way, the tracking ratio stability will be good. 
The reality however is that there will be some variance between 
components, and using multiple resisters will reduce that overall 
variance. Part of the variance between individual resisters will likely 
follow a Guassian distribution and thus the improvement factor for that 
element will be SQRT(N).

Some of the variance will likely be due to random factors which have a 
rather different distribution, probably highly skewed with long tails, 
and thus the improvement probably won't be SQRT(N). My conjecture (ok 
random speculation) is that factors such as stress differences due to 
microcracking in the ceramic substrate or at the terminations may cause 
some of the latter. Nevertheless, even though part of the variance 
doesn't follow SQRT(N) the variance will still reduce by using multiple 
identical resisters (if there are enough*). The problem is knowing how 
much - it probably can only be determined by lengthy experimentation, 
unless some good empirical data can be obtained from manufacturers or 
research papers.

Another complication is that I believe that thin film resistor stability 
and TCR characteristics improve as the resistance reduces. This is not 
usually reflected in the datasheet but using multiple resistors in 
series allows lower values to be used which may perform better. On the 
other hand, thermal EMF problems may increase proportionally.

TCR tracking is much easier to measure, so it might be interesting to 
see how it improves with increasing numbers of resisters. However, I 
understand that ratio stability is likely to be a bigger problem than 
TCR tracking.

The other end of the spectrum, using a single Vishay VHD foil divider is 
certainly the simplest; however bear in mind that Vishay's stated 
typical tracking TCR of < .1ppm is just that, and the one that you buy 
may be anything but typical. And if you can work out the maximum 
tracking TCR from the VHD144/200 datasheet, you're a better man than I. 
My guess is that its probably better than .5ppm which is likely good 
enough for your application. But would it perform better than, say $30 
worth of Vishay DFN, 3ppm 4-resister networks, 1 year shelf life ratio 
stability < 20ppm? I don't know.

Of course there's nothing to stop you using multiple VHDs if you can get 
them at a good price. Ebay maybe?

*) If all resisters are identical expcept that 1 in a 100 is markedly 
different, then any 10 will have a good chance (90%) of being identical; 
using 100 will have a good chance (64%) that at least one is different 
and thus the overall error would be at least 1/100 of the difference.

Tony H