Instrumentation & Measurement Magazine 26-2 - 16

Fig. 3. Effect of temperature on fitting residuals.
parameter calculation, and the least-squares method was used
for parameter optimization. The four different fitting equations
mentioned above were used to fit all of the experiment
data, and the corresponding coefficients are listed in Table 2.
The four equations mentioned above were used to fit the
measurements of eight NTCs and calculated corresponding
residuals. To verify that the method of evaluating the fitted
equation with residuals is feasible, we calculate the uncertainty
of temperature and determination of coefficient (R2)
separately. The uncertainty of results mainly originated from
four primary aspects: uniformity and stability of the temperature
field, accuracy of the SPRT, accuracy of the electrical
measuring device, and measurement repeatability. The expanded
uncertainty (k=2) of temperature was less than 10 mK.
R2 is a value between 0~1 used to measure the reliability of the
regression equation fitting results; the larger the value of R2,
the closer the fitting curve is to the theoretical value. By calculation,
the worst R2 value of four equation fitting results are
0.99999995, 0.99999996, 0.99999977, 0.99999484; all are greater
than 0.99999, indicating that the four equations we choose can
16
accurately describe the actual curve, and that the Hoge-2 equation
has the best goodness of fit.
Fig. 2 shows the residual distributions of the fits using the
above four equations for each of the four thermistor models. To
show the reproducibility of the fitting results, two thermistors
were selected from each manufacturer for parallel experiments,
different individuals of the same manufacturer were
not much different, which means the thermistor had good consistency
and little influence on the fitting results.
The residual obtained by the R-C equation was the largest
among the four equations, and for the three types of
MF5802, 44007 and 55104, the maximum residual reach
more than 100 mK. For S-H equation, the maximum fitting
residuals of the four thermistors were above 20 mK, meaning
that it was not an ideal fitting equation in the range of
0~100 °C. The maximum fitting residuals obtained by the
Hoge-2 and Hoge-3 equations were all within 20 mK. Specifically
for 44007, 55104 and P60BA, the fitting residuals
were within 10 mK. There was little difference between the
fitting results of Hoge-2 and Hoge-3 (the square root of the
IEEE Instrumentation & Measurement Magazine
April 2023

Instrumentation & Measurement Magazine 26-2

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