1. MMC
  2. All Volumes and Issues
  3. Volume 12, Number 3, 2025
  4. Modeling the dependence of thermistor resistance on temperature

Modeling the dependence of thermistor resistance on temperature

MMC.
2025;
Volume 12, Number 3
: pp. 906–913
Received: July 31, 2025
Revised: September 16, 2025
Accepted: September 18, 2025

Malachivskyi R. P., Bun R. A., Majewski J., Medynskyi I. P.  Modeling the dependence of thermistor resistance on temperature.  Mathematical Modeling and Computing. Vol. 12, No. 3, pp. 906–913 (2025)

Authors:
  1. R. P. Malachivskyi
  2. R. A. Bun
  3. J. Majewski
  4. I. P. Medynskyi
1
Lviv Polytechnic National University
2
Lviv Polytechnic National University; WSB University
3
Lublin University of Technology
4
Lviv Polytechnic National University

The results of a study on the accuracy of models describing the dependence of thermistor resistance on temperature are presented.  The feasibility of using minimax (Chebyshev) approximation to calculate the model parameters is substantiated. Compared to the least squares method, the minimax approximation provides the smallest modeling error.  A model in the form of an exponential function of a rational expression is proposed to describe the thermistor resistance as a function of temperature.  The use of this model is based on considering the physical properties of the semiconductor resistance dependence on temperature.  For the studied calibration results, the model recommended by the Consultative Committee for Thermometry (CCT) under the International Committee for Weights and Measures (CIPM) provides higher accuracy in describing the thermistor resistance--temperature dependence compared to the exponential of a rational expression.  However, the accuracy of the model in the form of an exponential function of a rational expression is only slightly lower and practically comparable.  Additionally, the model in the form of an exponential of a rational expression allows the use of temperature in the Celsius scale.

Thermistor
Thermometric Characteristic
least squares method
Chebyshev approximation
exponential dependence
Rational Expression
temporal stability
  1. White D. R., Hill K., del Campo D., Izquierdo C. G.  Guide on Secondary Thermometry – Thermistor Thermometry, BIPM, Consultative Committee for Thermometry (CCT) (2014).
  2. Rudtsch S., von Rohden C.  Calibration and self-validation of thermistors for high-precision temperature measurements.  Measurement: Journal of the International Measurement Confederation.  76, 1–6 (2015).
  3. Liu G., Guo L., Liu C., Wu Q.  Evaluation of different calibration equations for NTC thermistor applied to high-precision temperature measurement.  Measurement.  120, 21–27 (2018).
  4. Mitin V. F., Kholevchuk V. V., Semenov A. V., Kozlovskii A. A., Boltovets N. S., Krivutsa V. A., Slepova A. S., Novitskii S. V.  Nanocrystalline SiC Film thermistors for cryogenic applications.  Review of Scientific Instruments.  89 (2), 025004-1–025004-5 (2018).
  5. Lan Y., Yang S., Chen G., Yang S.  Characteristics of a Type of NTC Thermistors for Cryogenic Applications.  Advances in Materials Physics and Chemistry.  10 (8), 167–177 (2020).
  6. Tang X., Gong X., Li X., Lin Z., You M., Liu J.  An optimal piecewise Chebyshev fitting method to calibrate cryogenic temperature sensors.  International Conference on Optoelectronic and Microelectronic Technology and Application.  11617, 116173D (2020).
  7. Gong X., Tang X., Li Y., Lin M. Y. Z., Liu J.  Investigation the Minimum Measurement Points for Calibration a High Precision NTC Thermistors in Cryogenic Field.  2021 IEEE 16th International Conference on Nano/Micro Engineered and Molecular Systems (NEMS).  1088–1091 (2021).
  8. Li T., Wang H., Sun J., Hao X., Wang C., Yu J.  Exploration of Calibration Methods for Ocean NTC Thermistor Thermometers.  Metrology Science and Technology.  65 (6), 55–61 (2021).
  9. Li J., Sun J., Li T, Wang H., Wang G., Pan J., Chen Z., Gao C.  Investigation of calibration equations for NTC thermistors utilized in the deep-ocean temperature range.  Measurement.  207, 112394 (2023).
  10. Jerotić A., Đokić D., Atanasijević P., Mihailovic P.  Non-bridge NTC thermistor anemometer with programmable sensitivity.  Flow Measurement and Instrumentation.  96, 102544 (2024).
  11. White D. R.  Interpolation Errors in Thermistor Calibration Equations.  International Journal of Thermophysics.  38, 59 (2017).
  12. Chen H.-Y., Chen C.  Evaluation of Calibration Equations by Using Regression Analysis: An Example of Chemical Analysis.  Sensors.  22 (2), 447 (2022).
  13. Chen C.  Evaluation of resistance–temperature calibration equations for NTC thermistors.  Measurement.  42 (7), 1103–1111 (2009).
  14. Chung J. P., Oh S. W.  A residual compensation method for the calibration equation of negative temperature coefficient thermistors.  Thermochimica Acta.  616, 27–32 (2015).
  15. Guide to Instruments and Methods of Observation (WMO-No. 8). 
  16. Remez E. Ya.  Fundamentals of the Numerical Methods of Chebyshev Approximation.  Naukova Dumka, Kyiv (1969), (in Russian).
  17. Collatz L., Krabs W.  Approximationstheorie. Tschebyscheffsche Approximation mit Anwendungen.  Teubner, Stuttgart (1973), (in German).
  18. Yatsuk V. A., Malachivskyy P. S.  Methods to Improve Measurement Accuracy. Beskid Bit (2008), (in Ukrainian).
  19. Malachivskyy P. S., Pizyur Ya. V., Danchak N. V., Orazov E. B.  Chebyshev approximation by exponential expression with relative error.  Cybernetics and Systems Analysis.  51 (2), 286–290 (2015).
  20. Malachivskyy P. S., Melnychok L. S., Pizyur Ya. V.  Chebyshev approximation of multivariable functions by the exponential expression.  Cybernetics and Systems Analysis.  57 (4), 429–435 (2021).
  21. Malachivskyy P. S., Matviychuk Y. N., Pizyur Y. V., Malachivskyi R. P.  Uniform approximation of functions of two variables.  Cybernetics and Systems Analysis.  53 (4), 426–431 (2017).
  22. Malachivskyy P. S., Melnychok L. S., Pizyur Y. V.  Chebyshev approximation of the functions of many variables with the condition.  2020 IEEE 15th International Conference on Computer Sciences and Information Technologies (CSIT).  54–57 (2020).
  23. Liu G., Guo L., Liu C. L., Wu Q.  Uncertainty propagation in the calibration equations for NTC thermistors.  Metrologia.  55 (4), 437–445 (2018).
  24. Malachivskyy P. S., Pizyur Y. V., Malachivskyi R. P.  Chebyshev approximation by a rational expression for functions of many variables.  Cybernetics and Systems Analysis.  56 (6), 811–819 (2020).
  25. Malachivskyi R. P., Bun R. A., Medynskyi I. P.  Chebyshev approximation by the exponent from a rational expression.  Mathematical Modeling and Computing.  12 (1), 233–240 (2025).
  26. Malachivskyy P. S., Melnychok L. S., Pizyur Y. V.  Chebyshev approximation of multivariable functions by a constrained rational expression.  Cybernetics and Systems Analysis.  59 (1), 146–159 (2023).