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September 6th, 2012, 09:29 AM   #1
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Math function to fit into the

Dear All,

Could anyone please suggest a function which would describe the following data set.
I am trying to calibrate my pyrometer (non-contact, infrared temp. meter) and I want to interpolate the Voltage data within the given range of -7 to 600 deg. Obviously, I need R^2 as close to 1 as possible.

I just need a mathematical function which describes these values (power function? exponential?).

Measurement results:

Temp [deg. C] Output voltage [uV]
-7 20494
600 20755
700 20974
800 21285
900 21696
1000 22282
1100 22871
1200 23534
1300 24333
1400 25161
1500 26078
1600 27074
1680 28031
1700 28251

I cannot use polynomials. I have to stick to the theory (Planck's law).
Would prefer a power function, exponential growth function or any other function resembling Planck's Law.

Thank you very much in advance for any comments/suggestions/criticism/anything.

Peter Drawa
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September 6th, 2012, 10:55 AM   #2
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Re: Math function to fit into the

The "slope" seems to increase sharply as temperature increases so I certainly think an expoential would be appropriate. If you want to use something like then a "quick and dirty" solution would be to use the lowest and highest temperatures to get two equations to solve for C and a. That is, with T= -7, V= 20494 so we have the equation . With T= 1700, V= 28251, so we have the equation . Of course, there is no way of telling whether it will fit the other points. Another thing you could do is use, say, , which gives you three values to determine and so you can fit it to three points. I recommend using three points as far apart as possible: the first, T= -7, V= 20494, the middle, T= 1200, F= 23534, and the last, T= 1700, V= 28251. The hardest thing to do would be a "least squares" exponential, which would not necessarily exactly match any point but would be close to all of them.
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September 6th, 2012, 11:43 AM   #3
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Re: Math function to fit into the

Quote:
Originally Posted by Peter.Drawa75
I cannot use polynomials. I have to stick to the theory (Planck's law).
I suggest computing the radiated energy via Stefan–Boltzmann (or Planck) and graphing energy vs. voltage. It looks sort of like a power relationship to me, but this would give you a polynomial... (linear -> quartic, sqrt -> quadratic, etc.).
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September 6th, 2012, 12:22 PM   #4
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This doesn't look like exponential growth or a power function to me. The data (except for (-7, 20494)) seem to correspond to a parabola (as the "slope graph" is roughly linear), but it would be unwise to use that parabola to interpolate for temperatures between -7° and 600°. Is at least some data available for temperatures within that range?
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September 6th, 2012, 01:31 PM   #5
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Re: Math function to fit into the

I think a parabola is too low-degree. A cubic fits much better. In any case you should convert to Kelvins and set the constant term to something like 0.
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September 6th, 2012, 02:56 PM   #6
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Are you sure? 0.004308T˛ - 3.08T + 21024 works really well (except for T = -7).
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September 11th, 2012, 07:34 AM   #7
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Re: Math function to fit into the

Thank you all for your comments (very useful!).

I have carried out some curve fitting over the last few days.

Here is what I discovered:
- modified Stefan-Boltzman (y(x) = a*T^b + c) fits well for T < 300 deg. of C.
- I have come across a function based on Planck's Law, which has a form y(x) = A/(exp(B/T) - C) and it becomes very accurate for T > 400 deg. of C. (So one can use it complementary with Stefan-Boltzman for Temps > 400 deg. C).
- surprisingly, the formula for the cumulative distribution function of the Weibull distribution (y(x) = a - b*exp(-c*T^d)) fits very well especially for lower temperatures (T < 500 deg. of C). What I also like about this function is that it is sigmoidal so in contrast to the previous function (the one related to Planck's law), it flattens out for T >> inf. instead of breaking down. I am not sure if it is legitimate to use such a function to calibrate my pyrometer, but I guess I can treat it as a purely empirical function.

Thank you again for all your comments.

Peter
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October 14th, 2012, 04:56 AM   #8
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Quote:
Originally Posted by Peter.Drawa75
. . . fits well for T < 300 deg. of C.
How can you tell, given your lack of data for such values of T?
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