actual
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A Hamon divider. Yes, very cool!
actual
NEGATIVE,
A Hamon divider. Yes, very cool!
-- John Devereux
da
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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) =
1 ----------------------------------------------------- 1 + (3G)(1-(a+b+c)/3 + sumsq/9 + h.o.t) ------------------------------------------------- (G/3)(1-(a+b+c)/3 + sumsq/9 + (2/9)*cross + h.o.t)which is
1 Ratio = -------------------------- 1 + 9 (1+2/9*cross +h.o.t)= (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
-- Dr Philip C D Hobbs Principal Consultant ElectroOptical Innovations LLC Optics, Electro-optics, Photonics, Analog Electronics 160 North State Road #203 Briarcliff Manor NY 10510 845-480-2058 hobbs at electrooptical dot net http://electrooptical.net
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hI fished out the article itself -- "Calibrator Brings Record Accuracy Even to Production and Repair," Eccleston, et al., Electronics, Sep.
8, 1982, p121-127 It described the Fluke 5440A voltage calibrator.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
kinda
and
Nowadays, you could just use a coarse/fine DAC and a 24-bit delta-sigma ADC, and close the loop.
-- John Larkin Highland Technology, Inc jlarkin at highlandtechnology dot com http://www.highlandtechnology.com Precision electronic instrumentation Picosecond-resolution Digital Delay and Pulse generators Custom laser controllers Photonics and fiberoptic TTL data links VME thermocouple, LVDT, synchro acquisition and simulation
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Or even PWM, with care.
-- Cheers, James Arthur
kinda
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[...] (Snipped Phils derivation)
These techniques can achieve 0.1ppm linearity - better than the best sigma delta as far as I know.
-- John Devereux
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and
At a roughly 100:1 cost ratio.
Maintaining 0.1 PPM over time and temperature will be really, really hard.
-- John Larkin Highland Technology, Inc jlarkin at highlandtechnology dot com http://www.highlandtechnology.com Precision electronic instrumentation Picosecond-resolution Digital Delay and Pulse generators Custom laser controllers Photonics and fiberoptic TTL data links VME thermocouple, LVDT, synchro acquisition and simulation
REALLY
You guys are no fun anymore. ;)
Cheers
Phil Hobbs
-- Dr Philip C D Hobbs Principal Consultant ElectroOptical Innovations LLC Optics, Electro-optics, Photonics, Analog Electronics 160 North State Road #203 Briarcliff Manor NY 10510 845-480-2058 hobbs at electrooptical dot net http://electrooptical.net
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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.
-- John Larkin Highland Technology, Inc jlarkin at highlandtechnology dot com http://www.highlandtechnology.com Precision electronic instrumentation Picosecond-resolution Digital Delay and Pulse generators Custom laser controllers Photonics and fiberoptic TTL data links VME thermocouple, LVDT, synchro acquisition and simulation
kinda
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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
1:N**2+1, i.e. 1:5, 1:10, 1:17, 1:26, .... If you have a fancier calibrator, you can do any ratio you like, but of course then you don't need the trick!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
-- Dr Philip C D Hobbs Principal Consultant ElectroOptical Innovations LLC Optics, Electro-optics, Photonics, Analog Electronics 160 North State Road #203 Briarcliff Manor NY 10510 845-480-2058 hobbs at electrooptical dot net http://electrooptical.net
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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
-- Dr Philip C D Hobbs Principal Consultant ElectroOptical Innovations LLC Optics, Electro-optics, Photonics, Analog Electronics 160 North State Road #203 Briarcliff Manor NY 10510 845-480-2058 hobbs at electrooptical dot net http://electrooptical.net
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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.
14 13 12 11 10 9 8 | | | | | | | R R R R R R R | | | | | | | 1 2 3 4 5 6 7Hey, 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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Oh that's OK, cute circuit tricks are still fun.
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Oh that's OK, cute circuit tricks are still fun.
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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.
-- John Larkin Highland Technology Inc www.highlandtechnology.com jlarkin at highlandtechnology dot com Precision electronic instrumentation Picosecond-resolution Digital Delay and Pulse generators Custom timing and laser controllers Photonics and fiberoptic TTL data links VME analog, thermocouple, LVDT, synchro, tachometer Multichannel arbitrary waveform generators
That is a 5 cent gate though isn't it?
Is there some way to cancel the transition time?
-- John Devereux
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
-- Dr Philip C D Hobbs Principal Consultant ElectroOptical Innovations LLC Optics, Electro-optics, Photonics, Analog Electronics 160 North State Road #203 Briarcliff Manor NY 10510 845-480-2058 hobbs at electrooptical dot net http://electrooptical.net
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