Instrumentation & Measurement Magazine 26-2 - 17

Fig. 4. Comparison of fitting equations.
residuals were 3.4 mK and 3.8 mK), which means the fourthorder
term of the Hoge equation had little influence on the
fitting curve. Since the equation with higher-order terms
will increase the computational cost, we recommend the
Hoge-2 equation be used to fit the resistance-temperature
relationship of NTCs.
Effect of Temperature on Fitting Results
The variation of fitting residuals with temperature is shown in
Fig. 3. The residual changing trend of different types of thermistors
was inconsistent, and the inconsistent fitting quality
among NTCs was caused by the missing second-order term in
S-H equation.
The residuals obtained by Hoge-2 and Hoge-3 equations
did not change significantly with temperature. Such a
pattern indicated that the fitting results of the Hoge-2 and
Hoge-3 equations were not greatly affected by temperature.
The Hoge-2 and Hoge 3 equations were expected to better describe
the resistance-temperature relationship. The residual of
R-C equation changed significantly with temperature change,
especially in temperatures above 70 K; however, the temperature
had little effect on the residual below 20 K. As a result,
the R-C equation was only suitable for a narrow temperature
range of 20 K.
April 2023
Effect of Total Experiment Data on
Fitting Results
Fitting results become more accurate as the amount of data increases,
but in practice, it is difficult to obtain a large amount of
data. Therefore, it is necessary to choose an equation that can
obtain accurate fitting results with less data.
Thermistors were measured every 20 K, the number of data
points was reduced to six, and the obtained fitting residuals were
compared with the full data fitting residuals. As shown in Fig. 4,
the fitting results of the S-H equation were seriously affected by
the amount of data. When 21 data points were used for fitting,
the maximum residual was about 20 mK, but when data points
were reduced to six, the fitting residual could reach more than
30 mK. In contrast, the fitting results of the Hoge-2 and Hoge-3
equations were hardly affected by the amount of data, which
further verified that these two equations better describe the resistance-temperature
relationship of the thermistors. For the R-C
equation, the residuals became smaller as the amount of data
decreased. It indicated that the model did not have the proper
mathematic form to fit the temperature-resistance relationship.
Conclusion
In this paper, four equations were used to fit the resistancetemperature
relationship of thermistors in the temperature
IEEE Instrumentation & Measurement Magazine
17

Instrumentation & Measurement Magazine 26-2

Table of Contents for the Digital Edition of Instrumentation & Measurement Magazine 26-2

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