I'm not quite sure how they determine these tables but they must ultimately be referenced back to a handful of reliable triple point references. I did check the fit using Chebeshev polynomials on the range
-270 to 0 and the results are suggestive that the published polynomials were derived as Chebeshev fits and converted to divergent polynomials!
-- Regards, Martin Brown
May I now take this great and very educating discussion back to the RTD topic?
Everywhere I look, I see the same Callendar van Dusen equations, but with cryptic references to more accurate equations used by the standards bodies.
Then we have the two RTD types: 385 and 392. 385 seems to be commercial ones and the 392 coefficient seem to be lab standards. This covers that topic well
The first thing I don't get is which is supposed to be the exact resistance-temperature relationship - the tables or the equations?
I reckon it must be the tables, so simply interpolating them would be the safest way.
Otherwise, there are two forms of the CVD equation: one for below 0C and one for above 0C. It's easy to select which one; you switch them above/below 100 ohms.
Is it really the case that a simple cubic equation represents the behaviour of the metal?
Presumably they are about the same. The polynomials were no doubt derived from a finite number of point measurements. But then so were the tables.
It's easy.
No, but it's close enough. Even 2nd order is pretty good in the human-compatible temperature range. Super-accurate RTDs are expensive and generally not practical; there are lots of other errors in temperature measurement.
-- John Larkin Highland Technology, Inc lunatic fringe electronics
This is the sort of case where the rational Chebyshev approximation is super useful. The estimable Forman Acton, in "Numerical Methods that Work", talks about the folly of trying to fit functions that don't have polynomial-type geometries, e.g. things with asymptotes. Even if the difficult place is outside the fit range, it still drives polynomials batty.
The procedure I outlined upthread will often produce nearly the minimax rational function.
Cheers
Phil Hobbs
-- Dr Philip C D Hobbs Principal Consultant ElectroOptical Innovations LLC / Hobbs ElectroOptics Optics, Electro-optics, Photonics, Analog Electronics Briarcliff Manor NY 10510 http://electrooptical.net https://hobbs-eo.com
The tables are derived from a master calibration done against however many precisely defined reference temperatures the standards body uses.
The cubic is good enough for most practical purposes and if you intend to use anything higher order you really need to know what you are doing because some of the stuff in the reference documents is gibberish.
On 0 to 1372 Type-K linear error +0.8C/-0.6C quadratic error +0.45C/-0.45C * cubic error +/-0.1C
10th order fit +/-0.05C (that's a lot of extra work for no real gain in accuracy)By a strange quirk of fate the residual error on the quadratic fit for this is an almost perfect sine wave with zeroes at 0,686,1372. Fitting that instead of a cubic term gets within 0.05C for the full range.
Interpolating on the tables is safe enough and is probably what a lot of kit does since anything that relies on evaluating those crazy divergent
10th order polynomial fits would be doomed to fail spectacularly.-- Regards, Martin Brown
snipped-for-privacy@highlandsniptechnology.com wrote
I guess one could check this by evaluating the equation but I read of a 3K error at +3K which would be highly relevant in a cryogenic application.
Interpolating the table would be a lot better - if one assumes the table is definitive.
RTDs tend to go to hell at liquid helium temps, just where you want milliKelvin accuracy. All sorts of things get weird below about 20K.
We used Lakeshore silicon diodes for the Cebaf cryo stuff. They have some magical recipe. Below 20K the carriers "freeze out" and the voltage drop skyrockets, which is great.
I use type T for magnetic applications because it's the least magnetic of the TC pairs. Using thin wire helps too obviously.
George H. I derived this from the NIST
Another blast from the past. I used to like that book too.
In this case a considerable improvement at the low temperature end is possible simply by a shift of temperature origin from Centigrade to Kelvin so that the rather curved non-linear bit near absolute zero is not being computed as the small difference between large numbers.
I'm a little surprised by just how effective the shift of origin is as it gains nearly an order of magnitude decrease in the residuals using the standard spreadsheet precision against Chebeshev. RTD still isn't the method of choice down there but that's another story.
Moral of the story which seems not to have been followed in this case is always look at a plot of the residuals after fitting any function.
-- Regards, Martin Brown
The equation as cast in Centrigrade gets pretty ropey down at T=-270 and the RTD doesn't help things as its dV/dT gets smaller as you approach absolute zero which magnifies a small error in voltage converted to temperature. You can do slightly better at low temperatures by recasting the problem so that you fit a polynomial in absolute T ie in Kelvin.
Fitting T/Centigrade the values are subject to numerical cancellation in the cryogenic regime just where things get interesting.
The difference is not huge but if you are intending to use it for cryogenincs could matter. Better sensors are available for the task.
Chebeshev fit 0 to -270C chi-squared 0.0125 max error 22.5mV Kelvin polynomial fit 3 to 273K chi-squared 0.000428 max error 3.2mV
It gets the error in temperature down to 2.5K worst case at the expense of being subject to cancellation errors near normal fridge temperatures.
Not an entirely safe assumption. The table is derived. -- Regards, Martin Brown
Good point. Our thermocouples were located close to the sample, near the center of an X-MHz superconductive NMR magnet.
We controlled the heater that blew air on to the sample, and the t/c sampled the air close to the sample, all in a dewar.
-- John Larkin Highland Technology, Inc lunatic fringe electronics
I know little of various curve fitting schemes. But having some basis in 'real' physics would seem like an obvious first step. (Such as using absolute temperature and not some arbitrary zero.)
George H.
With infinite machine precision they would all give the same answer but in practice even at double precision how you evaluate things matters.
Numerical analysis of data fitting for calibration can be fickle.
There are always trade-offs to be made in practice.
-- Regards, Martin Brown
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