Converting Resistance (ohms) to Temperature (K) : Results

The equations to convert resistance to temperature for three different resistors Resistor 2 (the still), Resistor 14 (the cold plate), and Resistor 15 (the mixing chamber with one damaged wire) were found using data tables supplied by Rich Schmitt.  Microsoft Excel was used to extract the equations.  Data of resistance and temperature were entered in, graphs were made from the data, and trendlines and equations were found for the graphs.  The data for temperatures of 4.2 K and below came from the Oxford Generic Data.  On Resistors 2 and 14, for temperatures of 4.2 K and above, data from the Oxford Test Results was used.  On Resistor 15, for temperatures of 4.2 K and above, 200 ohms were added to the resistance in the data for Resistor 14. 

It was too difficult to get one equation per resistor; as a result, the data was broken into two (Resistor 2) or three (Resistors 14 and 15) sections to graph.  For the data of temperatures of 4.2 K and above, there were only three points, which were not enough to get an accurate trendline and equation.  To remedy this, those three points were graphed by hand, connected using a French Curve, and then other points were found on the curve to add to the original data.  These points helped to find a better-fitted trendline and a more accurate equation.  For Resistor 2, a power equation was used.  For Resistors 14 and 15, exponential equations were used.  Even with the extra data, accuracy is not very good for temperatures of 4.2 K.  The error for each resistor is in the plots below.

For Resistor 2, the coldest data was not essential, so it was cut to make it easier to find a well-fit trendline.  Then a graph was made of the natural log of the resistance and the natural log of the temperature for temperatures of 4.2 K and below.  Using the natural log made it easier to find a trendline that fit.  A third degree polynomial equation was used.  The accuracy for this equation is significantly better than the data for temperatures of 4.2 K and above.  Again, the error is given on the plots below.

For Resistors 14 and 15, the cold data (4.2 K and below) was broken into two pieces.  For Resistor 14, second degree polynomials were used for both the mid-range temperatures data and the coldest temperatures data.  For Resistor 15, a third degree polynomial was used for the mid-range temperatures data, and a second degree polynomial was used for the coldest temperatures data.  All these equations have good accuracy the error can be seen below.

On all of the polynomial equations (all of the data at 4.2 K and below), if the degree was increased, Excel showed a trendline that supposedly fit closer to the data.  However, when the equations of the fourth, fifth, or sixth degree were tested by putting the natural log of resistance into them and seeing how close they came out to the natural log of the temperature, they were found to actually be less accurate.  When the equations of the second or third degree were tested, things came out much closer to their actual values. 

 

Resistor Two - The Still:

 

The equation used for warmer temperatures (4.2 - 300 K) was:
T = (5*10^54)R^-15.703

The equation used for colder temperatures (4.2 K and below) was:
lnT = -.7749(lnR)^3 + 20.992(lnR)^2 - 190.97(lnR) + 582.24

 

Intercept is at 2760 Ohms

Error:

At 300 K, it is off 181.1991 K
From 77 - 17 K it is off by approximately 10.80171 K
From 4.2 - 1 K it is off by approximately .029758 K
From .5 - .25 K it is off by approximately .0034818 K

 

 

 

 

 

 

 

 

Resistor Fourteen - The Cold Plate:

 

The equation used for warmer temperatures (4.2 - 300 K) was:
T = (5*10^8)e^(-.0068R)

The equation used for the middle range of  temperatures (4.2 K - 0.3 K) was:
lnT = 1.1537(lnR)^2 - 21.882(lnR) + 102.44

The equation used for the coldest temperartues (.25 K and below) was
lnT = 0.1071(lnR)^2 - 3.1229(lnR) + 18.315

 

Intercept 1: 2718 Ohms
Intercept 2 (closest intercept between middle and cold graphs):
8100 Ohms

Error:

At 300 K it is off by 186.69099 K
At 175 K it is off by 61.69099 K
From 77 - 5 K it is off by approximately 10.164402 K
From 4.2 - 1 K it is off by approximately .079126
From .5 - .3 K it is off by approximately .0130434 K
From .25 - .08 K it is off by approximately .0019342 K
From .07 - .02 K it is off by approximately .0003936 K

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Resistor Fifteen - The Mixing Chamber With One Damaged Wire:

 

The equation used for the warmest temperatures (4.2 - 300 K) was:
T = (7*10^9)e^(-.0072R)

 

The equation used for the middle range of temperatures (0.3 - 4.2 K) was:
lnT = -.9584(lnR)^3 + 25.936 (lnR)^2 - 235.48(lnR) + 716.19

 

The equation used for the coldest temperatures (.25 K and below) was:
lnT = .111(lnR)^2 - 3.215(lnR) + 18.854

 

Intercept 1 : 2920 Ohms
Intercept 2 : 10550 Ohms

Error:

At 300 K it is off by 147.192954 K
At 200 K it is off by 72.364827 K
From 77 - 5 K it is off by approximately 9.1838972 K
From 4.2 - 1 K it is off by approximately .0216436 K
From .5 - .3 K it is off by approximately .0031686 K
From .25 - .08 K it is off by approximately .0020506 K
From .07 - .02 K it is off by approximately .000513 K

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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