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 September 6th, 2012, 10:29 AM #1 Newbie   Joined: Sep 2012 Posts: 2 Thanks: 0 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
 September 6th, 2012, 11:55 AM #2 Math Team   Joined: Sep 2007 Posts: 2,409 Thanks: 6 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 $V= Ce^{aT$ 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 $20494= Ce^{-7a}$. With T= 1700, V= 28251, so we have the equation $28251= Ce^{1700a}$. Of course, there is no way of telling whether it will fit the other points. Another thing you could do is use, say, $V= Ce^{at}+ b$, 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.
September 6th, 2012, 12:43 PM   #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.).

 September 6th, 2012, 01:22 PM #4 Global Moderator   Joined: Dec 2006 Posts: 20,310 Thanks: 1981 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?
 September 6th, 2012, 02:31 PM #5 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms 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.
 September 6th, 2012, 03:56 PM #6 Global Moderator   Joined: Dec 2006 Posts: 20,310 Thanks: 1981 Are you sure? 0.004308T² - 3.08T + 21024 works really well (except for T = -7).
 September 11th, 2012, 08:34 AM #7 Newbie   Joined: Sep 2012 Posts: 2 Thanks: 0 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
October 14th, 2012, 05:56 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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