A New Method For Temperature Measurement Using Thermistors

DOI : 10.17577/IJERTV2IS70260

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A New Method For Temperature Measurement Using Thermistors

A.V. Seryakov

Scientific laboratory, Special Relay System Design and Engineering Bureau, Nekhinskaya Street, 55, 173021, Velikiy Novgorod, Russia.

This article considers the question of improving the accuracy of temperature measurement using thermistors. Improvement is carried out by binding the temperature to the inflection point Tinf of the functional dependence of the logarithm of the thermistor resistance lnRC and measurement of the temporal drift of the decomposition coefficients Ai(d).

A new method is proposed to derive a precise temperature of the thermistor since it provides a closer approximation to the actual temperature then simpler equations, and is useful over the entire working temperature range of the sensor.

Keywords: temperature measurement, inflection point of the functional dependence of the logarithm of the thermistor resistance, drift coefficients of decomposition.

Thermistors refer to equipment with a strong dependence of electrical resistance to temperature. At limiting high temperatures the thermistor resistance becomes almost constant and independent of temperature. This means that thermistor calibration has a constant at value at infinity, or has a reference point at very high temperature. This causes inconvenience in use and leads to significant errors at measurement and calculation of temperature using thermistors, at low and medium temperatures.

We offer to make a second constant value point, or reference point of thermistor calibration, which was determined by us to be at the inflection point of the functional dependence of the logarithm of the thermistor resistance lnRC.

The difficulty of temperature measurement using thermistors, which are used in the research

of many thermal processes and in thermal equipment, is of high interest at this time, and improving the accuracy of measurements is considered very important.

In view of this, calibration of the thermistor T3-19 within the temperature range of 0-200°C was made by verifying indicated values recorded in equilibrium thermodynamic conditions using the model platinum resistance temperature sensor [1, 2].


Thermistors CT3-19 are represented as thermally sensitive elements [6-8], made of ceramic oxide materials based on nickel, magnesium and cobalt, denoted in composition by the formula (Ni0.2Mn0.7Co0.1)3O4, with negative temperature coefficient (NTC) and resistance of about 10 kOhm at room temperature. The thermistor bead of a CT3-19 is coated with a thin layer of molybdenum glass and welded on output traverses (0.3 mm) by thin platinum wires (0.01 mm).

The sensitive element of the standard Platinum Resistance Thermometer PRT-10 is used as a resistance temperature sensor. The quartz helicoid with platinum spiral is placed in a 4mm diameter thin-walled cylinder made of molybdenum glass. Before sealing, the cylinder with helicoid was filled with helium under a pressure a little less than atmospheric. After manufacturing, the resistance thermometer is calibrated again in temperature range 0-200°C at Siberian Scientific Research Institute of Metrology (SSRIM) in Novosibirsk. The absolute error of temperature measurement using the platinum resistance thermometer is ± 0.02 K.

The thermometer together with a thermistor are placed in a heavy copper cylinder, 100 mm in length and 70 mm in diameter, which is located in a vacuum chamber on rigid hangers fixed on the internal surface of upper flange of the chamber. Two pipes are welded to the upper flange of the

vacuum chamber, through one of which the chamber is evacuated, though the other one all sensing wires exit. The pressure in the chamber is maximum 1.3·10-3Pa (10-5torr).

The vacuum chamber and copper unit are placed in liquid thermal bath with a capacity of 40 litres, which has is allowed to reach equilibrium temperatures in the range of 0-200°C with temperature gradients of less than 1·10-3K/cm.

The cooling unit was a conditioner BK-1500, for which the standard flash-heat exchanger was replaced with a subdivided loop.

The working fluid in the thermal bath was silicone oil PES-V2, which allows operation at temperatures of 0-200°C. The temperature variation inside the bath over a period of several hours of monitoring was not more than 1·10-3K, and rate of temperature drift was less than 1·10-4K/h.


Calibration of the thermistor CT3-19 was carried out over more than two years. Measurements were made in stable conditions under an isothermal cover [9-10], with gradual increase of temperature from 0°C to 200°C in increments of 10°C. The duration of one continuous

cycle of temperature rise and measuring run was up to 48 hours. In total there were 21 cycles of measurements .

Calibration of the thermistor is in the precise measuring of the thermistor resistance RC, (Ohm), using the standard potentiometric method, in a stationary state under fixed temperature T, (K), determined by the platinum resistance temperature thermometer PRT-10. A high quality voltage comparator 3003, accuracy class 0.0005, is used, coupled with standard resistance coil 321 accuracy class 0.01, located in a temperature-insulated box. A pack of batteries Backen in a grounded metal enclosure worked as a current source.

A total of 500 experimental points were obtained, which were formed into a source data array of temperatures T, and the logarithms of the resistance of the thermistor lnRC. The maximum random error of measurement of temperature with thermometer PRT-10 does not exceed (2-3)·10-3K, thermistor resistance 5·10-4Ohm.

All temperature measurements, conducted using platinum resistance thermometer PRT-10, including calibration at SSRIM, were made when measuring a current value of 1 mA (1·10-3), and sensor dissipation is WPRT = (10-17)·10-6 W.

Fig. 1 shows the results of one cycle of thermistor resistance RC measurement, against temperature. For easier viewing, the graphic is presented as a reciprocal temperature function 104/T, 1/, of lnRC. This relationship is close to linear, but at high temperatures around T~473K, where

lnRC ~4-5, and at low temperatures around T~273K, where lnRC~9-10, some deviations are noticed. At moderate temperature, at the range of lnRC ~6-8, an inflexion point possibly exists.

Fig.1. Experimental dependence of the logarithm of thermistor CT3-19's resistance on the reciprocal temperature 104/T, 1/K. The straight line is the calculation according to equation (6), the curve -calculation according to equation (9).

Considering the temperature detector as a system with lumped parameters, the thermistors equation is written as follows [1,2]

TC + C T C = WC + T0 ; C = CC



Where TC thermistor temperature, K; derivative with respect to thermistors time- temperature in dynamic conditions of measurement; C thermistors characteristic lag time, s;

WC electrical heating output, W ; KC heat-transfer coefficient between thermistor and copper unit, W/K; T0 measurement with thermometer PRT-10 temperature of copper unit or, for example, heat pipe; time, s; CC – thermistors heat capacity, J/K.

Solution of equation (1) is the following formula [3], in which at moment in time = * the

thermistors temperature is considered to be equal to .

T = T exp + WC + T 1 exp (2)


KC 0 C

The electrical heating output WC, generated on the thermistor by the act of measuring the current, s constant during all calibration tests and equals 20·10-6W, and the thermistors overheating TC is calculated by the formula:

TC = WC = C WC



To define the thermistors overheating special tests were performed: when in a stationary state at temperatures from 2°C to 195°C, the thermistors (and thermometer's) heating and cooling temperature was measured during a staged evolution of the electrical heating output WC. With the help of a measuring system liquid calorimeter, described in [9,10], detailed measurements of the relaxational characteristics of the sensors were made. Calculation of the thermistors response (lag) time C, according to equation (2) using ordinary least squares technique (OLS) [11-14], gave the results shown in Figure 2.

Fig.2 . Experimentally defined thermistors CT3-19 lag time C.

An approximation to the curve in Figure 2 is given by a polynomial formula

C(t) = 3.3963027·10-5·t2 – 1.39779·10-2·t + 2.261388 minutes (4)

where t temperature Celsius, experimental points mean squared departure ~ 0.18 min.

The heat capacity of the thermistor CC~0.3 J/K, weak dependence of the heat capacity of the thermistor temperature; heat-transfer coefficient KC~(2÷5)·10-3 W/K; and the overheat value caused by the measurement current not more than TC~(10-4)·10-3, and is considered in all measurements. The temperature difference between the copper cylinder and its isothermal cover, was not more than 0.05-0.1 K; temperature change during calibration was less than 10-7K/s. Therefore, the

thermodynamic state of the temperature sensors inside the copper unit during this period was quasi- stationary.


The functional dependence of the electrical resistance of the oxide semiconductor thermistor RC on temperature T is quite difficult, and at first approximation it is represented as a resistance of an

ideal semiconductor with strictly the same number of holes and charge carriers in the exponential form:



R = A exp B (5)


where RC electric resistance of thermistor, Ohm, at temperature T, ; A thermistors resistance value, Ohm, at infinite temperature;

B thermistors sensitivity parameter, dependent upon temperature in general way, .

At temperature T=1°C thermistors CT3-19 resistance RC~30 kOhm; at temperature T=200°C RC~50 hm; B~4000 , A~0.013 Ohm. Taking logs in equation (5):

1 = 1 lnA + 1 lnRC (6)


To clarify the question of inflection of the experimental curve in Fig.1, the derivative was calculated d(1/T)/d(lnRC), and analyzed with the whole array of experimental points. Derivative value d(1/T)/d(lnRC), calculated according to the results of one cycle of thermistors resistance measurement dependent upon lnRC, are shown in Fig.3.

According to the results of numerical differentiation [15] of the entire array of experimental points, curve minimum was defined for the value lnRC min=7.63±0.01, which corresponds to the temperature of inflection point, Tinf = 336.34K or 63.19°C.

Fig.3. Calculated derivative dependence d(1/T)/d(lnRC)·104, 1/ dependent upon logarithm of thermistors CT3-19 resistance lnRC.

Expanding the derivative d(1/T)/d(lnRC) to a series form at a point minimum:

1 2 3

1 2 3

1/ = + 7.63 2 + 7.63 3 + (7)

where ai the expansion coefficients.

After integration the expansion (7) leads to the following form



1 = A0 d + A1 d lnRC 7.63 + A2 d lnRC 7.63 3 + A3 d lnRC 7.63 4 (8)

where Ai(d) the expansion coefficients, d time drift of the coefficients.

Thus, the calibration of the thermistor CT3-19 essentially consists of determining the numerical values of polynomial (8) Ai(d) coefficients and taking account of their drift through time d.

The main feature of the calculation according to the equation (8) is in binding the temperature to the

inflection point Tinf of the functional dependence of the logarithm of thermistors resistance.

Calculation of coefficients Ai(d) was made using ordinary least squares technique (OLS) [11-14]. The absolute errors of calculation coefficients Ai(d) are following:

A0~1·10-3, A1~1·10-3, A2~1·10-4, A3~1·10-5.

For quality control and calibration and long term stability, after each of the 21 defining set of coefficients Ai(d) from formula (8), the derivatives d(1/T)/d(lnRC) were calculated. With dispersion of not more than ~5·10-7 all the values of derivatives lie on the curve in Fig. 3. In the low

temperature area at values of the logarithms lnRC~9-10, the dispersion of calibration is a bit higher, and reaches ~(5-7)·10-7. This is due to the higher thermistor resistance RC, and the steeper temperature dependence dRC/dT, then the lower density of experimental points, since the calibration of the thermistor was mostly performed at intervals of 10°C.

Fig.4 shows the time history of coefficients Ai(d), and the temporal drift of the thermistors calibration. The first eight measurement cycles were made when heating the thermistor up to 200°C. As a result of such heating there was significant calibration drift and change of coefficient Ai(d) , for example: dA0/d~6·10-4 1/month. After the limiting high temperature was decreased to 190°C,

temporal drift of coefficients decreased notably, for example the rate of change of coefficient A0(d) became only dA0/d~1.5·10-4 1/month. Values of temporal coefficients drift Ai(d) are shown below, d months.

Fig.4. Time history of coefficients Ai(d) at heating up to 200°C left part of graphics, heating up to 190°C right part of graphics.

Coefficients A0(d)

A0 = -6.21997·10-4·d + 29.824488 at heating up to 200°C A0 = -2.3075444·10-4·d + 29.8213 at heating up to 190°C

Coefficients A1(d)

A1 = 4.051207·10-5·d + 2.4893 at heating up to 200°C A1 = 1.5876991·10-5·d + 2.4895 at heating up to 190°C

Coefficients A2(d)

A2 = -2.2266277·10-5·d + 0.00227 at heating up to 200°C A2 = -1.0559017·10-5·d + 0.00218 at heating up to 190°C

Coefficients A3(d)

A3 = – 3.98635·10-8 · (d)2 + 1.771915·10-6 ·d + 6.3241 ·10-5 at heating up to 190°C and to 200°C

Substitution of the coefficients Ai(d), calculated when heating the thermistor up to 190°C, into equation (8) reduces the dispersion of derivatives d(1/T)/d(lnRC) of the curve in Fig.3 to the value ~(2-3)·10-7, and allows to define the minimum point more accurately: lnRC min = 7.632±0.01, thus reducing the calculation error when temperature measurement using thermistor CT3-19.

Thus, the recommended equation for calculating temperature using the thermistor CT3-19, taking account of both inflection point Tinf = 336.34K to functional dependence of the logarithm lnRC of thermistors resistance, and temporal drift of the polynomial decomposition coefficients Ai(d), at periodic heating thermistor up to 190°C, is as follows:



1 = A0 d + A1 d lnRC 7.63 + A2 d lnRC 7.63 3 + A3 d lnRC 7.63 4 (9)


There is the well-known cubic polynomial Steinhart-Hart equation [4], designed to calculate



temperature using thermistors, by incorporating linear and cubic components using the logarithm of the resistance lnRC:



= A + B lnRC + C lnR3 (10)

where A, B, C the expansion coefficients.

The useful temperature range of this equation with one set of coefficients is not more than 50-75K [4,5]. Using sets of numerical coefficients of the Steinhart-Hart equation, available in internet publications, a comparison of temperature calculation data was made based on the Steinhart-Hart equation and on our biquadratic equation of logarithms of thermistor resistance (9). Fig.5 shows

temperature relative differences = (TSH T)/T·100% , dependent upon lnRC. Comparison f results shows that temperature differences , depending upon lnRC, have an alternating-sign nature. Minimum values of temperature differences on the order of 0.25% are observed at a value lnRC~6 of thermistor CT3-19. Maximum values of differences , reaching 0.8-1%, occur at the edges of applicability of the equation interval (9).

Fig.5. Temperature relative difference , calculated according to the Steinhart-Hart equation and biquadratic equation under point of inflection of thermistor characteristic (9).

The error, when defining temperature (temperature differences) using thermistor T3-19 with the biquadratic equation (9) subject to the inflection point at lnRC = 7.632, is not more than (3-5)·10-4 K.


The use of the biquadratic polynomial equation (9) with temperature binding to the inflection point of the functional dependence logarithm of thermistor resistance, can extend the range and increase accuracy of the temperature definition.

Account of temporal drift decomposition coefficients polynomial (9) Ai(d), allows to improve the accuracy of the minimum point definition lnRC min=7.632±0.01 of thermistor characteristics, and thus the accuracy of the temperature definition.

The use of Steinhart-Hart cubic polynomial equation increases temperature calculation error, using thermistor, at range limits 273K – 473K to 1.5-2K.

List of references

  1. Lineveg F. Temperature measurement in engineering. oscow .: Metallurgia». 1980. 544p.

  2. Kondratyev G.M. Thermal measurements. oscow .: «Mashgiz». 1957. 210p.

  3. Kamke E. Manual of Ordinary Differential Equations. oscow. .: «Nauka». 1976. 576p.

  4. Steinhart J. S., Hart S. R. Calibration curves for thermistors. Deep Sea Research and Oceanographic Abstracts. 1968, v. 15, 4, p. 497-503.

  5. CORNERSTONE SENSORS. A,B,C Coefficients for Steinhart-Hart Equation. Temperature Sensors for Health, Science and Industry. 2010. 2p. USA, California 92083.

  6. Macklen E.D. Thermistors. oscow.: 1983. 208p.

  7. Gendin G.S. All about resistors. Reference Manual. oscow.: 2000. 192p.

  8. Scheftel I.T. Thermistors. oscow.: 1973. 416p.

  9. Seryakov A.V. Automated constant-volume calorimeter for research of electrolyte solutions / / Thermophysical properties of solutions, melts and composites. Collection of scientific papers

    .Institute of Thermophysics. Novosibirsk. 1991. p.139-153.

  10. Gruzdev V.A., Seryakov A.V. Liquid calorimeter // Thermophysical properties of materials: Proceedings of the 8th All-Union Conference on Thermophysics. Novosibirsk. 1989. pt.1. p.256-261.

  11. Demidovich B.P., Maron I.A. Foundations Numerical Mathematics. – oscow: «Nauka», 1966., 664p.

  12. Amosov A.A., Dubinsky U.A., Kopchenova N.V. Computational Methods for Engineers. oscow: «Vysshaya shkola», 1994. 544p.

  13. Hamming R.W. Numerical Methods. – oscow.: «Nauka», 1968. 400p.

  14. Davenport J.,Sire I., Tournier E. Computer Algebra: Systems and algorithms for algebraic computation – oscow .: «Mir», 1991. 352p.

  15. Mudrov A.E. Numerical methods for the PC in BASIC, FORTRAN and Pascal – Tomsk: «Rasko». 1991. 270p.

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