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 September 6th, 2012, 09:29 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, 10:55 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 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. 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.). September 6th, 2012, 12:22 PM #4 Global Moderator   Joined: Dec 2006 Posts: 20,935 Thanks: 2209 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, 01:31 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, 02:56 PM #6 Global Moderator   Joined: Dec 2006 Posts: 20,935 Thanks: 2209 Are you sure? 0.004308T� - 3.08T + 21024 works really well (except for T = -7). September 11th, 2012, 07:34 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, 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? Tags fit, function, math Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post lun123 Real Analysis 6 November 11th, 2013 10:21 AM benmuskler Calculus 1 January 4th, 2013 03:09 PM Niceguy890 Algebra 4 July 1st, 2010 12:31 PM Gelembjuk Computer Science 5 November 26th, 2008 08:45 PM Peter.Drawa75 Algebra 0 December 31st, 1969 04:00 PM

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