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That's it! That guy's got a massively cool website too.

The original article I read in one of the trade mags did the math, developing equations to show exactly how much better the divider was than the individual resistors.

Neat stuff.

-- Cheers, James Arthur

That is fun, and the math is simple too--it just needs a second-order binomial expansion. Say you have three resistors,

R(1+a), R(1+b), and R(1+c). Their conductances are

G(1 - a + a**2 - a**3...) and similarly for the others.

If you wire them in parallel, you get a total conductance of

Gpar = G(3 - a - b - c + a**2 + b**2 + c**2 + higher order terms), which is = 3G(1 - a/3 - b/3 - c/3 + sumsq/9 + higher order terms),

where sumsq is the sum of the squares of a, b, and c.

If you then make a voltage divider with a fourth resistor, and trim it to exactly half, you have

Gbot = Gpar

Then when you wire the same three resistors in series, you get

Rser = R(3 + a + b + c) and a conductance of

Gser = (G/3)( 1 - (a+b+c)/3 + sumsq/9 + 2*(ab+bc+ac)/9 + h. o. t)

The only difference is in the cross terms between a, b, and c. So the final voltage divider ratio is

Ratio = 1/(Gpar/Gser + 1) =

which is

= (1/10)(1-1/5*(ab+bc+ac) + h. o. t.).

If the magnitudes of a, b, and c are all less than delta, the relative error bound becomes

RelError < 0.6 delta**2.

Thus if you want a 1 ppm divider, you need individual resistors that match to

delta < 1.3*sqrt(1 ppm) = 0.13%.

Of course the tempcos and the temperatures need to track as well.

Cute trick.

Cheers

Phil Hobbs

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I fished out the article itself -- "Calibrator Brings Record Accuracy Even to Production and Repair," Eccleston, et al., Electronics, Sep.

Hey, pretty good memory, eh? But, I was wrong about the authors showing the equations--I did that myself contemporaneously, not them. I just wrote a whole divider equation and took partial derivatives for each delta(r), for a loose approximation.

-- Cheers, James Arthur

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Nowadays, you could just use a coarse/fine DAC and a 24-bit delta-sigma ADC, and close the loop.

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Or even PWM, with care.

-- Cheers, James Arthur

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These techniques can achieve 0.1ppm linearity - better than the best sigma delta as far as I know.

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At a roughly 100:1 cost ratio.

Maintaining 0.1 PPM over time and temperature will be really, really hard.

REALLY

You guys are no fun anymore. ;)

Cheers

Phil Hobbs

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Sweet! Thanks for that. Say there's nothing special about three R's, is there? I mean you could do this trick with more R's and get a higher series/ parallel ratio.

(The factor of ten is nice.)

George H.

REALLY

Sooner or later, any really good, clever circuit gets blown away by some smart-alec IC designer.

You get a phenomenal amount of technology from a cheap 24-bit delta-sigma ADC. 3 PPM for $3.

The other amazing parts are pipelined ADCs, stuff like 12 bits at 250 MHz.

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Doing the tweak at 1:2 is convenient because you can turn the thing upside down and take out the error of the opposite side of the bridge. With that approach, always wind up with a voltage divider ratio of

There are probably more involved methods that would get you different values, but I expect they'd rapidly get more cumbersome than just using the classical bridge approach to calibrating each resistor individually.

Cheers

Phil Hobbs

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One other point is that you can make a 1:10 divider common centroid using a 14-pin resistor array: use resistors 1, 5, and 7 for one branch and 3, 4, and 6 for the other. (One set in parallel and the other in series, of course.)

That isn't as symmetrical as even-order networks, of course, so it won't get rid of even-order terms as a 2:2 or 4:4 network will.

It would also be a nice way to make very accurate 1-2-5 dividers.

Cheers

Phil Hobbs

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This is so cool. I'd never seen the Kelvin Varley Divdier either.

First (if I'm understanding you correctly) doesn't using different resistors for the series and parallel circuits negate the ratio-metric accuracy improvement?

And second for the common centroid thing, why not 1, 3, 5 and 2,4,6. I'm assuming the array looks like this.

Hey, a seven resistor array gives a 1:50 ratio. The error of course goes up with more resistors.

I was thinking more about sumusu 0.1/0.05% R's but you've still got to tweak the one resistor.

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Hey, we're lots of fun! And an elegant PWM--super accuracy for dirt cheap--that's fun too!

-- Cheers, James Arthur

PWM is difficult. To get to PPMs, the rise/fall time have to be absurdly fast. 1 PPM of a millisecond is 1 ns. Then you have to filter the ripple out.

That is a 5 cent gate though isn't it?

Is there some way to cancel the transition time?

The killer for PWM will be supply ripple, which will be slightly code-dependent and therefore hard to verify.

The edges will be steep enough that the edge contribution will hardly vary with code. (It's sort of like using a trombone line for a capacitor standard--the edge effects are constant, so stretching it produces a very constant dC/dx.)

Cheers

Phil Hobbs