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.
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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
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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
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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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