[volt-nuts] Matched resistors

Tony vnuts at toneh.demon.co.uk
Wed Jul 23 21:36:52 EDT 2014


Have you considered using multiple identical resistors to reduce the 
variance? Depending on who you believe, you can reduce the variance of 
the overall resistance by SQRT(N) where N is the number of resistors in 
series/parallel. Its not that easy to create a good search query for 
this but here is one such explanation:


Ideally they should all come from the same batch - ie. manufactured by 
the same machine from the same batch of materials. Obviously there's no 
way to guarantee that without close liaison with the manufacturer (you 
did want 10 million parts at $.10 each didn't you!) but hopefully a set 
of resistors which come off the same reel would come close.

The absolute value isn't important however, but 'statistical gain' will 
also apply to the TCR and stability of the overall divider. The 
following assumes that both factors are similarly improved by SQRT(N), 
but in fact they may be rather better than that.

That80€ or $108 for one sealed Vishay foil divider will buy a lot of 
lower spec parts:

Approx 12558 x Susumu RR0510P .5%, 25ppm 0402 (Digikey, $86/10k). 6279 
in series and parallel in each leg of the 1:1 
might reduce the variance to 25ppm/SQRT(6279) = .32ppm. Can't see any 
spec for stability, but it may also improve similarly. Would take a 
while to solder them onto stripboard though!

Slightly more sensible might be 1078 x TE Connectivity RP73 1%, 10ppm 
1206 (Digikey, $100.18/1K).  Stability .5% (no qualifers in datasheet)
  => 10ppm/SQRT(539) = .43ppm, stability => 215ppm

Or 372 x KOA Speer RN731JTTD4021B5 .1%, 5ppm (Mouser, $29/100). 
Stability not on data sheet but typical endurance is +/- .02% for 1000 
hrs @ 70C on/off 1.5hours/.5hours.
=> 5ppm/SQRT(138) = .37ppm, endurance => 14.7ppm (Stability should be 
rather better than that). Note that the Mouser part no. is for a 25ppm 
part but their manufacturer's part number is the 5ppm part as is the 
description. Also, the price is way too high for 25ppm parts.

Or 28 x Susumu RG2012L .01%, 2ppm (Digikey, $39.6/10). Stability not 
quoted but typical Load Life is .01% (1000 x 1.5hours on/.5hours off at 85C)
=> 2ppm/SQRT(14) = .53ppm, endurance => 27ppm

You could also use multiple resistor networks. Eg:

104 x Susumu RM2012B-103/103-PBVW10 .1%, 5ppm tracking, 2 
resistors/device (Digikey $104/100). Stability not quoted, endurance 
500ppm (1000 x 1.5hours on/.5hours off at 85C)
=> 5ppm/SQRT(104) = .49ppm, endurance => 49ppm

35 x TT Electronics SFN08B4701CBQLF7, .25%, 5ppm tracking 7 
resistors/device (Digikey, $76/25) . Stability not quoted, high 
temperature exposure < 1000ppm
=> 5ppm/SQRT(122) = .52ppm

33 x TT Electronics 668A1001DLF .5%, 5ppm tracking 8resistors/device 
(Digikey, $82/25). Stability not quoted, load life < 1000ppm
=> 5ppm/SQRT(33 * 4) = .45ppm

16 x Vishay DFN .1%, 3ppm tracking with 4 resistors/device (Digikey, 
$5.24/1). Shelf life ratio stability is specced at 20ppm (1 year at 
25C). (That may be a typical rather than a maximum - your parts may all 
be much worse than typical). The 3ppm tracking TCR may also be a typical 
figure as its headlined in a section titled 'TYPICAL PERFORMANCE' but in 
the specification table its not qualified with '(typical)' as they 
sometimes do in other datasheets. Its hard to tell.
=> 3ppm/SQRT(32) = .53ppm shelf life stability => 3.5ppm

5 x Vishay DSMZ metal foil dividers, .5ppm tracking max (probably 
performs rather better than this over restricted temperature range, but 
don't believe the Vishay typical figure of < .1ppm/C) (Digikey, 
$22.93/1). Shelf life ratio stability not quoted but 'typical limit' for 
Load Life ratio stability is 50ppm (2000 hours at 70C). Who knows what a 
typical limit is? Again, probably best to treat Vishay 'typical' figures 
with a pinch of salt given the experience of another poster on volt-nuts.
=> .5ppm/SQRT(5) = .22ppm, load life => 22ppm

Interestingly Digikey quote a price of only $5400 for 1k parts for the 
similar DSM divider (1ppm tracking), which is a huge difference from 
$22.93. Might be worth considering a bulk buy if there enough volt-nuts 
with the same problem. They aren't stocked though so that price might 
not be 'real'. However:
20 x Vishay DSM dividers, 1ppm (Digikey, $5400/1000) Load life ratio 
stability 'typical limit' 50ppm
=> 1ppm/SQRT(20) = .22ppm, load life => 11ppm

Multiple LT5400 networks could also be used and may give the best 
results, but the much larger absolute tolerance, +/-15% would cause 
those with the highest value for series connected/lowest for parallel to 
dominate and reduce the statistical improvement. Do your own calculations.

Its interesting that all these different components end up providing 
pretty much the same performance for the same cost - in other words the 
cost is inversely proportional to the TCR^2

My gut feeling is that the tracking TCR will improve rather better than 
the SQRT(N) calculated, if they do indeed come from the same batch, as I 
would expect them to have similar absolute TCRs. Thus you might be able 
to get away with rather less parts to achieve < 1ppm. The SQRT(N) factor 
comes from assuming that the variation in the value is random, and I 
believe, has a particular distribution (Guassian or normal?). Component 
specifications are often derived from the distribution parameters 
measured from a large set of production samples, with the max/min values 
determined from a multiple (typically 6?) of the standard deviations of 
the distribution? The worst case specifications for TCR and stability 
may (I don't know, just hypothesizing) be derived very differently. For 
example, the TCR may be affected not only by the characteristics of the 
bulk resistive material, but also due to stresses on the element due to 
thermal expansion of the substrate/packaging. It may be that the former 
is almost identical for all components from the batch, but the latter is 
less predictable. The specification max/min would have to allow for the 
worst cases which might be due to a  relatively few which for some 
reason (microcracking in the substrate perhaps) have much larger 
variance from the majority. The distribution of TCRs from a set of 
resistors could be very skewed with long tails and the SQRT(N) reduction 
in variance may be well off the mark.

Stability is more difficult because the shelf life stability is rarely 
specified, but is likely to be the closest to your usage. For reference, 
the Vishay DFSMZ datasheet specifies ratio stability of .015% for 2000 
hour at 70C and .002% for shelf life ratio stability. The 7.5X 
difference might be useful for estimating shelf life stability for 
resistors that only quote load life or endurance specs. But it might 
not! I'm not sure that the endurance spec is very useful either as it 
subjects the resistor to a large number of large temperature cycles 
which won't be anywhere near your usage.

I would expect the long term tracking stability to be much better than 
(worst case datasheet stability)/SQRT(N) as I would expect the vast 
majority to age in similar ways, if not by the same magnitude. Whilst 
the specs show stability to be +/- xx% I would expect that most will age 
in the same way - probably slowly increasing resistance over time. I 
also expect there are experienced posters here who know otherwise! 
Similarly to TCR, it could be that for example, the stability of most 
resistors in a batch may be quite good, but the specs reflect that a few 
may be much worse due to random faults in individual samples - such as 
defects in the protective coating of the element allowing corrosion to 
occur in a few samples. You'd need a very good understanding of the 
factors that determine the resistor stability to calculate the overall 
stability of multiple resistors.

I would expect similar factors to apply to ratio tracking due to 
humidity changes. No doubt there is some useful information out their in 
application notes/research papers on the variance in long term stability 
between resistors of various types (and maybe even for parts taken from 
the same batch) just waiting for some interested volt-nut to discover?

The fewer the parts, the more chance of statistical outliers reducing 
the improvement over a single part, but you could test each divider for 
the best matching, if you've got a decent meter, fairly easily by 
applying a voltage from a stable, low noise source (a battery would be 
good if its temperature is kept very stable), and measure the voltage at 
the centre tap. Then put the resistor network in a plastic bag  and 
immerse it in boiling water to raise the temperature by 75C or so; .5ppm 
tracking would give 9.4uV/V maximum change; you'd probably need to 
reverse the meter leads a few times to null out thermal EMFs. 
Alternatively measure the voltage difference between the divider under 
test and another driven by the same voltage source and kept at a stable 
temperature - ie. in a bridge configuration. A simple high gain 
amplifier (say 1000x) with adjustable offset would allow testing with a 
more realistic lower temperature difference of say 20C and/or a cheap meter.

Accuracy is not particularly important - you probably don't need to know 
the temperature tracking coefficient to better than 20%.

Component layout would need to ensure any thermal gradients apply 
equally to both legs of the divider by interleaving upper and lower 

Tony H

On 17/07/2014 16:26, Randy Evans wrote:
> Frank,
> The high cost is my concern, although high performance demands high price
> typically.  I am trying to double the voltage reference from either an
> LM399 or LTZ1000, hence the need for precision matched resistors for a x2
> non-inverting amplifier (using a LT1151 precision op amp).  An alternative
> I am investigating is using the LTC1043 in a voltage doubling circuit as
> shown in Linear Technology app note AN 42, page 6, Figure 16.  It states
> that Vout = 2xVin +/- 5 ppm.  I am less concerned about the absolute
> accuracy than I am about the long term stability.  I assume that a high
> quality capacitor is required (low leakage, low ESR, low dielectric
> absorbtion, etc.) but the circuit does not appear to be dependent on the
> absolute value of the capacitors.  I'm not sure if the two 1uF caps  need
> to be matched.  If they do then that would be a show stopper.
> Does anyone have any experience using the LTC1043 in such a circuit?
> Thanks,
> Randy
> On Wed, Jul 16, 2014 at 9:40 PM, Frank Stellmach <frank.stellmach at freenet.de
>> wrote:
>> Randy,
>> resistor matched in T.C. are extremely expensive, as the manufacturer (or
>> yourself) would have to select these from a batch of many samples.
>> reistors with very small T.C. (<1ppm/K) would do the job also, but they
>> also need to be stable over time, in shelf life opereation mode, i.e.
>> P<10mW.
>> That means, you need those hermetically sealed VHP202Z from Vishay, T.C.
>> is typically < 1ppm/K and they are stable to < 2ppm over 5years. But they
>> cost already 80€ each, depending on tolerance.
>> I made a longterm observation of these and found these parameters
>> confirmed.
>> Frank
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