Derivation of the Algorithm in `gentherm2`

Here's the Steinhart-Hart equation, which gives a thermistors characteristic curve that's within 0.02 C over the temperature range of interest (this, of course, assumes that your resistance measurements are that accurate!). Thanks to David Parrish for finding this.

   1/T = A + B*ln(R) + C*ln(R)**3

The Steinhart-Hart coefficients for the GM sensor are A=0.00150664 B=0.00022484 C=1.25794e-07.

To solve it, first make it look like a matrix equation.

   A + c1*B + c2*C = c3

where

      c1 = ln(R)
      c2 = ln(R)**3
      c3 = 1/T

So, given three points (Tn, Rn):

   for n = 1..3:
      cn1 = ln(Rn)
      cn2 = ln(Rn)**3
      cn3 = 1/Tn

   A*1 + B*c11 + C*c12 = c13
   A*1 + B*c21 + C*c22 = c23
   A*1 + B*c31 + C*c32 = c33

Do Gaussian elimination in a generic fashion. Subtract the top from each of the bottom two, resulting in:

   B*(c21-c11) + C*(c22-c12) = c23-c13
   B*(c31-c11) + C*(c32-c12) = c33-c13

Multiply the bottom one by (c21-c11)/(c31-c11):

   B*(c21-c11) + C*(c32-c12)*(c21-c11)/(c31-c11) = (c33-c13)*(c21-c11)/(c31-c11)

Now subtract this from the top one:

   C*((c22-c12) - (c32-c12)*(c21-c11)/(c31-c11)) = (c23-c13) - ((c33-c13)*(c21-c11)/(c31-c11))

And get C by itself:

       (c23-c13) - (c33-c13)*(c21-c11)/(c31-c11)
   C = -----------------------------------------
       (c22-c12) - (c32-c12)*(c21-c11)/(c31-c11)

Now that C is known, solve for B:

       c33-c13 - C*(c32-c12)
   B = ---------------------
            c31-c11

Then A,

   A = c13 - B*c11 - C*c12

Now we need the original equation rewritten as T(R):

   1/T = A + B*Ln(R) + C*(Ln(R))**3

   T(R) = 1 / (A + B*ln(R) + C*ln(R)**3)

Here's a source for some test data (in fact it is encoded in gentherm2.c as the default values):

   Date: Wed, 20 Feb 2002 16:42:22 -0500
   From: "Camden Lindsay" 
     To: megasquirt@yahoogroups.com

just found this spec for an 88 audi quattro temp sensor (engine computer)

Degrees C            Ohms
  0                  5 - 6.5K
 20                  2 - 3 K
 50                  700 - 900
100                  175 - 210

hope this helps

camden

Peter Florance told us about an old L-Jet-equipped BMW 528: http://www.firstfives.org/faq/ljet/528i_Ljet_schematic.PDF

 70f  1950-3000 ohms
100    950-1600
160    250- 550

Here is a plot of the Audi raw data against a curve generated by gentherm2. Also included are the raw data from Bruce's original gentherm program, in yellow (I assumed they were the GM sensor data, but I'm not sure now after comparing the output of gentherm with the airdenfactor.inc and thermfactor.inc files).

Derivation of the Algorithm in gentherm2

Derivation of the Algorithm in `gentherm2`