A residual compensation method for the calibration equation of negative temperature coefficient thermistors

Thermochimica Acta 616 (2015) 27–32

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Short communication

A residual compensation method for the calibration equation of negative temperature coefficient thermistors Jae Pil Chung a,b,∗ , Se Woon Oh a,b a b

Medical Physics, Korea University of Science and Technology, Daejeon 305-350, Republic of Korea Center for Ionizing Radiation, Korea Research Institute of Standards and Science, Daejeon 305-340, Republic of Korea

a r t i c l e

i n f o

Article history: Received 26 February 2015 Received in revised form 13 July 2015 Accepted 27 July 2015 Available online 4 August 2015 Keywords: Temperature measurement NTC thermistor Calibration equation R–T relation Residual compensation

a b s t r a c t We propose a calibration equation for negative temperature coefficient thermistors. This equation is modified from the basic thermistor equation by a residual compensation method. The residual compensated equation includes a 2nd order error correction term. This equation is convenient for converting temperature to resistance and vice versa. Calibrations of various thermistors were performed in a high precision temperature bath to test the resistance–temperature relation. The thermistors were calibrated in glass tubes using a reference thermometer. Most calibrations are performed in a temperature span (15–35 ◦ C). From the resistance–temperature curves, the fitting function of the basic thermistor equation and the residual compensated equation were evaluated and their fitting errors were compared. Standard fitting errors (1) in temperature of the basic equation and the residual compensated basic equation were 8.131 mK and 0.151 mK respectively. The residual compensated equation showed about 1/54 smaller fitting errors than the basic thermistor equation. © 2015 Elsevier B.V. All rights reserved.

1. Introduction In the radiation measurement field, calorimeters are used to evaluate the absorbed dose, measuring the temperature rise of the irradiated body. When the temperature rise produced by irradiation is very small, a high sensitivity temperature sensor is needed to measure the temperature change. In such cases, thermally sensitive resistors – thermistor – are used for precise temperature measurement. Thermistors are temperature dependent resistors with a high resistance temperature coefficient [1,2]. Their high coefficient and very small size are good characteristics for a calorimeter system. In practice, thermistors with a negative temperature coefficient (NTC type) are frequently used for temperature measurement [3]. Because of their non-linear response, the thermistor resistanceto-temperature (R–T) relationship has been explained by several researchers. Considering the electrons in an n-type material and carrier mobility, it was shown that the resistivity of the n-type material is proportional to T−c e(k/T) where c is a constant and k is a material constant [4–6]. Another study showed that the best fit to

∗ Corresponding author at: Korea Research Institute of Standards and Science, Chemical research building (No. 306), Room No. 228, Ga-Jung st. 267, DaeJeon, YuSeong Gu [305-340], Republic of Korea. Tel.: +82 42 868 5830/+82 10 9319 3197; fax: +82 42 868 5042. E-mail address: [email protected] (J.P. Chung). http://dx.doi.org/10.1016/j.tca.2015.07.015 0040-6031/© 2015 Elsevier B.V. All rights reserved.

these R–T relationships gives a simple relation to the R–T property, RT = RT→∞ e(ˇ/T) [3]. RT is thermistor resistance at T, T is temperature in K, R∞ is the limit value of RT as T approaches to infinity, B is a constant depending on the thermistor material, in K. Because R∞ cannot be measured, this equation is modified in terms of resistance R0 at reference temperature T0 [7]. Eq. (1) is a widely used simple form for the thermistor response. RT = R0 eˇ(1/T −1/T0 )

(1)

The most frequently used reference resistance of thermistors R0 is determined at 25 ◦ C, i.e. 298.15 K. In a narrow temperature span, this simple exponential relationship will be a good approximation with its satisfactorily small residuals. But this equation still results in slight temperature errors of several mK, which is called the “plus-minus-plus” effect [8]. This error behavior of the basic equation already was shown by Bosson et al. [9]. If one needs a more precise fitting result with sub milli-Kelvin accuracy, especially for fine temperature control and measurement, this equation needs to be modified. Otherwise, other thermistor equations which include higher order terms are usually applied to the fitting equation. Unfortunately high order polynomial thermistor equations have difficulty in converting temperature to resistance or vice versa because the solutions of the high order equations are in a very complicated form. To address these issues, we introduce a method to compensate the errors of the basic thermistor equation using resistance and its

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Fig. 1. (a) The thermistor calibration water bath and the R–T measurement system. (b) An enlarged picture of the glass tube tip with a scaler. (c) A microscopic image of a thermistor bead for estimating its size.

residual values. This method uses just a 2nd order form and provides easy conversion of resistance to temperature and vice versa. This residual compensated basic (RCB) equation is modified from the basic equation, Eq. (1), using the compensation method of resistance residuals. We explain the procedures for researchers who are interested in this method.

2. Materials and method 2.1. Materials and apparatuses We prepared the following thermistors: VECO (Victory Engineering Corporation) NTC thermistors (Part No.: 32A130, G32A173, 43A70, 33A83, R25 : 1.5–20 k, Temperature coefficient at 25 ◦ C (˛25 ): −3.4%/◦ C to −4.0%/◦ C), Measurement Specialties, Inc. NTC thermistors (Part No.: GAG22K7MCD419, R25 : 22 k, ˛25 : −3.9%/◦ C). We used a temperature control water bath, and its specifications were as follows; Model: HART SCIENTIFIC High Precision Bath R-12 system (Fig. 1). Control resolution: 0.002 ◦ C, Temperature Stability: ±0.001 ◦ C, Control probe: 100 , 4 wire, Platinum Resistance Thermometer (PRT). This PRT control probe in Fig. 1 is only used for temperature control of the water bath. We used two digital multi-meters (DMM), Keithley 2000, for the temperature and resistance measurement. The two DMM were calibrated with standard resistors in advance and each correction factor was applied to the raw measurement values. For auto data acquisition, the DMMs are connected to a PC through general purpose interface bus (GPIB) cables. The temperature and resistance data were measured with 5 s sampling time. The R–T measurement software was developed using LabVIEW which was fitted for our measurement system. For thermistor water sleeves, we used glass tubes with a sealed round tip which were 450 mm in length and 5 mm diameter. The glass tube setup used for the thermistor calibration consisted of a thermistor, 4 enamel copper wires and a Universal Serial Bus (USB) cable port. A USB cable was used for the signal cable because the cable consists of 5 wires including a guard cable that reduces noise signals. And the USB port is convenient when connecting the glass tube to the measurement apparatus. The 4-wire resistance measurement used separate pairs of current-carrying and

voltage-sensing electrodes to make more accurate measurements than the 2-wire measurement. Temperature sensing was done with a reference thermometer, and the reference thermometer was calibrated with a standard platinum resistance thermometer (SPRT, Rosemount 162CE, S/N: 4053) in Korea Research Institute of Standards and Science (KRISS). The resistance of the SPRT at the triple point of water is 25.554246  and the deviation coefficients of International Temperature Scale of 1990 (ITS-90) are (a) −2.09509 × 10−4 , (b) −3.8047 × 10−5 , and (c) 9.610 × 10−6 respectively. The temperature calibration system consists of a precision thermometry bridge (F700 Tinsley—ASL) with a high precision standard resistor (GUILD LINE, 7334-100, 99.99979 ), a high precision water bath (Hart Scientific, Model 7012), and the SPRT. Measuring Rt /Rs ratio (the SPRT resistance vs. the standard resistance) from the thermometry bridge, the water temperature is calculated. This system was used for the R–T calibration of the reference thermometer. The calibration temperature range was 10–40 ◦ C (11 measurement points with 3 ◦ C step) in the water bath. The measurement were done in the stable temperature range of ±0.5 mK. The resistance of the SPRT that will be converted to temperature and the resistance of the reference thermometer were recorded automatically to instantly sense any slight change of the temperature water bath every second. The calibrated reference thermometer had the following properties: the resistance value at 25 ◦ C (R25 ) was 1642.73 , temperature coefficient at 25 ◦ C (˛25 ) was −0.0345/◦ C. The reference thermometer was embedded and sealed in a glass tube. That tube had the identical size and shape as the glass tubes for test thermistors, so the reference thermometer underwent temperature changes close to the test thermistor. The two glass tubes for the reference thermometer and the test thermistors were closely arranged and contacted at each tip to make the temperature difference between the two tubes as small as possible. This temperature difference at every second between the two tubes was evaluated by using two calibrated thermistors which had smallest sizes for rapid responses. Although the temperature variation in the water bathe was about ±0.5 mK, the temperature difference between the two thermistor was very small (±0.126 mK). This uncertainty was included in Table 2 as the temperature gradient term. The above settings make the R–T curves more exact because the test thermistor simultaneously feels the temperature change when the reference thermometer feels the temperature change. The two combined glass tubes were located in the water bath, which was shown in Fig. 1.

J.P. Chung, S.W. Oh / Thermochimica Acta 616 (2015) 27–32

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2.2. Residual compensation method and equations Considering R0 and T0 to be constants, the basic equation Eq. (1) is reduced to a simpler form, Eq. (2). R = e(A+B/T )

(2)

This equation is convenient for fitting the thermistor’s R–T curve because Eq. (2) can be expressed as a simple fitting form, lnR = A + B/T. The fitting constant A includes R0 , T0 , and ˇ in Eq. (1), and is changed when T is expressed as K or ◦ C. Constant B includes ˇ, and the constant is dependent on the thermistor materials. Eq. (2) is easily converted to Eq. (3) to obtain temperature from the observed resistance value. T=

B ln R − A

(3)

Fitting the measured R–T curve using Eq. (2), the fitting constants A and B are determined. And from this result, the fitting residuals, i.e. errors in ohm, can be related to the corresponding resistance value. Considering the fact that the basic equation has residuals which tend to have plus-minus-plus values [8], the measured resistance can be divided into two parts; the resistance from the fitting equation and from the error part, that is, residuals. R = Req + r(R)

Fig. 2. R–T curve of thermistor No. 39 (10K3A1B) over the temperature range 15–30 ◦ C. The side enlarged plots show that the measured R–T points (+) deviated slightly from the basic equation fitting result.

(4)

R is the observed resistance which is considered to be a real resistance value and Req is the resistance value from the basic equation fitting. r(R) is the residual function expressed by the observed resistance R, which acts as a temperature error correction term. Through our many previous tests of thermistor R–T relations, when the basic thermistor equation is applied to the fitting curve, we found an interesting result, in that the resistance residual curves (R vs r(R) ) are approximately fitted to the 2nd order polynomial form (see Fig. 3) in a narrow temperature span. So, r(R) can be expressed as the following equation: r(R) ≈ a + b · R + c · R2

(5)

Req also can be expressed as a function of R using Eqs. (4) and (5). Req = −a + (1 − b) · R − c · R2

(6)

Fig. 3. Resistance and corresponding residual (+) plot of the thermistor (No. 39). Note that this plot is not for the temperature residuals, but for resistance residuals.

The constants a, b and c are determined from 2nd order fitting or from solving simultaneous equations. Since all fitting constants were determined, the measured resistance can be converted to temperature using Eq. (7), which corrects the temperature error of the basic equation. Eq. (3) can be converted to Eq. (7) using Eq. (6). T=

B ln(−a + (1 − b)R − cR2 )) − A

(7)

For a certain temperature T, the corresponding resistance value RT also can be calculated; the solution of the second order equation which is expressed about R. Using Eqs. (3)–(6), the observed resistance RT at temperature T is expressed as: RT = e(A+B/T ) + a + bRT + cRT2

(8)

After rearranging Eq. (8) about RT , we have a solution by solving Eq. (8).



RT =

1−b−

2



(b − 1) − 4c a + e(A+b/T ) 2c

 (9)

Only the (−) sign of the root term is the solution of Eq. (8). Eqs. (7) and (9) are the RCB equations modified from the basic thermistor formula Eq. (2).

Fig. 4. Upper plot shows temperature error comparison of basic equation and RCB equation. The lower plot shows an enlarged plot of the region around 0 mK error.

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Table 1 Fitting error (1) comparison of the basic Eq. and RCB Eq. of various thermistors. ˛ [%/K]

Thermistors

Standard fitting error (1)[mK]

S/N

Part no.

Basic Eq.

RCB Eq.

SH–H Eq.

No. 1 No. 2 No. 3 No. 4 No. 5 No. 6 No. 7 No. 8 No. 9 No. 10 No. 11 No. 12 No. 13 No. 14 No. 15 No. 16 No. 17 No. 18 No. 19 No. 20 No. 21 No. 22 No. 23 No. 24 No. 25 No. 26 No. 27 No. 28 No. 29 No. 30 No. 31 No. 32 No. 33 No. 34 No. 35 No. 36 No. 37 No. 38 No. 39 No. 40 No. 41 No. 42

G32A173 G32A173 G32A173 G32A173 G32A130 G32A130 G32A130 G32A130 G32A130 G32A130 G32A130 G32A130 G32A130 G32A130 G32A130 33A83 33A83 33A83 33A83 33A83 43A70 43A70 43A70 43A70 43A70 43A70 43A70 43A70 43A70 43A70 GAG22K7MCD419 GAG22K7MCD419 GAG22K7MCD419 GAG22K7MCD419 GAG22K7MCD419 GAG22K7MCD419 3K3A1B 5K3A1B 10K3A1B 30K5A1B 50K6A1B 100K6A1B

1.590 2.220 1.798 3.579 4.317 3.337 4.118 3.847 3.569 4.287 3.903 3.175 3.654 3.718 3.762 7.564 5.439 2.668 4.063 8.399 17.84 16.16 18.67 15.93 19.14 20.84 1.943 3.141 4.294 2.971 10.19 33.14 12.92 13.11 15.99 15.93 3.968 4.014 15.54 6.100 5.309 5.374

0.115 0.339 0.261 0.053 0.076 0.135 0.098 0.092 0.080 0.093 0.115 0.099 0.096 0.096 0.107 0.165 0.120 0.206 0.108 0.436 0.107 0.091 0.123 0.087 0.109 0.150 0.051 0.083 0.100 0.178 0.076 0.199 0.193 0.203 0.280 0.230 0.179 0.193 0.300 0.272 0.126 0.114

0.118 0.342 0.263 0.036 0.083 0.135 0.101 0.093 0.079 0.101 0.116 0.101 0.097 0.101 0.107 0.165 0.123 0.210 0.107 0.443 0.106 0.177 0.236 0.274 0.224 0.261 0.051 0.090 0.103 0.172 0.129 0.430 0.270 0.258 0.360 0.296 0.179 0.213 0.184 0.267 0.112 0.113

Total average

8.131

0.151

0.177

−3.466 −3.468 −3.493 −3.462 −3.466 −3.458 −3.475 −3.494 −3.465 −3.472 −3.490 −3.481 −3.410 −3.494 −3.469 −4.013 −4.059 −4.013 −4.007 −4.031 −3.991 −4.008 −3.996 −3.969 −4.020 −4.064 −4.009 −3.988 −4.029 −4.009 −3.879 −3.874 −3.885 −3.885 −3.879 −3.884 −4.386 −4.385 −4.381 −4.297 −4.669 −4.669

Constants of basic Eq. & RCB Eq. A

B

a

b

c

−3.0024 −2.9549 −3.0757 −2.9573 −2.9715 −2.9249 −2.9603 −3.0617 −2.9604 −3.0279 −3.0554 −2.9532 −2.8050 −3.0571 −2.9454 −3.9991 −4.0192 −3.9815 −4.0132 −3.9365 −1.9089 −2.0049 −2.0229 −1.9254 −2.1027 −2.1512 −1.9857 −2.0263 −1.9972 −2.0754 −1.6341 −1.6054 −1.5046 −1.6093 −1.5309 −1.6166 −5.0696 −4.5583 −3.8524 −2.5016 −3.1005 −2.4075

3081.4 3083.2 3105.4 3077.2 3080.8 3073.7 3088.8 3105.5 3079.7 3086.6 3102.4 3094.0 3031.3 3106.0 3083.5 3567.3 3608.1 3567.1 3561.5 3583.6 3548.1 3562.5 3552.5 3527.9 3573.2 3612.2 3563.6 3545.5 3581.2 3564.1 3448.0 3443.5 3453.8 3453.8 3448.1 3453.0 3898.5 3898.4 3894.5 3819.4 4150.3 4150.3

−1.056E + 01 −1.531E + 01 −1.169E + 01 −1.620E + 01 −1.852E + 01 −1.768E + 01 −1.906E + 01 −1.875E + 01 −1.811E + 01 −1.843E + 01 −1.837E + 01 −1.925E + 01 −1.942E + 01 −1.775E + 01 −1.821E + 01 −4.791E + 01 −4.153E + 01 −4.028E + 01 −4.414E + 01 −4.708E + 01 −3.221E + 02 −3.080E + 02 −2.780E +02 −2.740E + 02 −2.682E + 02 −3.057E + 02 −2.762E + 02 −2.599E + 02 −3.150E + 02 −2.207E + 02 −2.675E + 02 −2.944E + 02 −2.934E + 02 −2.540E + 02 −2.844E + 02 −2.615E + 02 −2.817E + 01 −4.711E + 01 −1.004E + 02 −4.444E + 02 −5.925E + 02 −1.189E + 03

1.518E − 02 2.085E − 02 1.669E − 02 2.025E − 02 2.538E − 02 2.364E − 02 2.487E − 02 2.575E − 02 2.414E − 02 2.587E − 02 2.508E − 02 2.402E − 02 2.617E − 02 2.430E − 02 2.370E − 02 3.364E − 02 2.596E − 02 2.422E − 02 2.789E − 02 2.583E − 02 2.830E − 02 2.856E − 02 2.744E − 02 2.697E − 02 2.735E − 02 2.906E − 02 2.894E − 02 2.989E − 02 3.179E − 02 2.555E − 02 2.334E − 02 2.497E − 02 2.403E − 02 2.447E − 02 2.463E − 02 2.472E − 02 1.862E − 02 1.868E − 02 1.939E − 02 2.939E − 02 2.348E − 02 2.356E − 02

−5.411E − 06 −7.047E − 06 −5.912E − 06 −6.248E − 06 −8.591E − 06 −7.817E − 06 −8.016E − 06 −8.731E − 06 −7.954E − 06 −8.952E − 06 −8.451E − 06 −7.424E − 06 −8.731E − 06 −8.213E − 06 −7.608E − 06 −5.805E − 06 −3.981E − 06 −3.605E − 06 −4.340E − 06 −3.430E − 06 −5.936E − 07 −6.333E − 07 −6.287E − 07 −6.298E − 07 −6.595E − 07 −6.452E − 07 −7.503E − 07 −8.524E − 07 −7.925E − 07 −7.300E − 07 −4.880E − 07 −4.780E − 07 −4.796E − 07 −5.534E − 07 −5.057E − 07 −5.523E − 07 −3.020E − 06 −1.818E − 06 −8.749E − 07 −4.772E − 07 −2.277E − 07 −1.143E − 07

3. Results

3.2. Resistance residual curve

3.1. The R–T curve and temperature differences of the basic equation

Fig. 3 shows resistance residuals and the fitting result of the thermistor. This plot was plotted from the fitting results of Fig. 2 in terms of residuals. Since we knew that the resistance residual curves were close to 2nd order equation form, Eq. (5) was used for the fit curve of the resistance vs residual plot (Fig. 3). The 2nd order polynomial curve fit showed a good approximation to the residual curve as presented in Fig. 3. This is the key point of the residual compensation method. The error correction term (i.e. residual compensation term) was included in the modified equations, Eqs. (7) and (9). Another advantage is that the outlier is easily detected when the residual fitting is performed. This allows users to easily determine whether the thermistor calibration was performed exactly or not.

Thermistor resistances were experimentally measured over several temperature steps by changing the water temperature. Fig. 2 shows an R–T measurement curve and its fitting curve of a thermistor over several temperature steps. The basic equation was used as the fitting equation and the fitting constants A and B were determined to be −3.85239, 3894.47 respectively. The coefficient of determination R2 was 0.999995 in this fitting. This R2 value appears to show the fitting curve is well fitted to the measurement data. But, the enlarged plot inside Fig. 2 shows that the temperature differences between the fitting curve and measurement data are 40 mK and 16 mK. This small discrepancy came from the incomplete approximation property of the basic thermistor equation, Eq. (2). Considering that the basic equation fitting in Fig. 2 was done with a temperature span, T ≈ 20 ◦ C, the errors will increase when the measurement and fitting are performed under a wider temperature span. From the R–T fitting result, the resistance values and residuals can be plotted as shown in Fig. 3.

3.3. Temperature errors from fitting results Using the two equations, temperature differences from the measurement data were compared, and they are shown in Fig. 4. To convert temperature from the measured resistance value, Eqs. (3) and (7) were used for the basic equation and RCB equation respectively.

J.P. Chung, S.W. Oh / Thermochimica Acta 616 (2015) 27–32 Table 2 Uncertainty budgets of standard uncertainties (1) in the thermistor calibration procedures. Uncertainty Fitting results (RCB Eq.) Resistance measurement (DMM) - DMM resolution (NPLC 10) - DMM accuracy (after the calibration) Reference thermometer calibration Temperature gradients Combined standard uncertainty

0.15 mK (5.0 × 10-5 ) 0.04 mK (1.4 × 10-5 ) 0.03 mK (1.0 × 10-5 ) 0.03 mK (1.0 × 10-5 ) 1.0 mK (3.4 × 10-4 ) 0.13 mK (4.4 × 10-5 ) 1.03 mK (3.5 × 10-4 )

Fig. 4 shows the temperature errors of the basic equation and RCB equation. Temperature differences from fitting errors of RCB Eq. showed very low levels smaller than ±0.5 mK. The standard fitting errors of the basic and RCB equation were 15.5 mK and 0.3 mK respectively; the basic equation error of this thermistor (No. 39) was reduced to 1/55 by this method. The calibration results of the basic equation and the RCB equation of various thermistors are presented in Table 1, and the standard fitting errors, i.e. standard deviation values between the measurement data and the fitting results, are also presented. To be compared with another recommended thermistor equation, fitting results of Steinhart–Hart equation (SH–H Eq. in Table 1) are also presented and will be discussed in Section 4. Table 2 shows the combined standard uncertainty of the thermistor calibration system. Figures in parentheses represent the relative uncertainty. The combined standard uncertainty was estimated as 1.03 mK which is much larger than the fitting uncertainty, because this estimation includes possible uncertainties during the calibration procedures of the test thermistors.

4. Discussion One may wonder why we used the resistance residuals instead of the temperature residuals. From our previous study, we found that the resistance residual curve showed much less deviation than the temperature residual curve when the 2nd order fitting was applied. This made it more sensible to use the resistance residual curve instead of the temperature residual curve. Another reason is that when the temperature residual is used, the equation is expressed as two terms; temperature and resistance. Then, the equation becomes recursive and becomes more complicated to deal with. Because the raw data from the DMM is resistance, not temperature, the resistance residual can be easily and instantly expressed as a function of resistance. The 2nd order equation fitting gives a useful result. The R–T relation has a simpler solution than other higher order equations; R and T are easily converted to each other.

31

In the case of PID (proportional-integral-derivative) temperature control, for example, it is preferred that the set value should be resistance rather than temperature value, because the raw data from the thermistor are resistance, not temperature. If higher order equations are used, the R–T mutual conversion is not easy or there will be no solution. Table 3 shows the equations comparison among the basic equation, RCB equation, and the Steinhart–Hart equation (“SH–H Eq.” in Table 3). The basic equation has very simple forms, which is convenient to use for temperature measurement. But this equation has large errors as shown in Table 1. If one needs more exact relations, it has to be modified or another equation is needed. Steinhart–Hart equation is a simple temperature calculation form that uses only three calibration constants as seen in Table 3. Also this equation gives very accurate fitting results. The calibration results of Steinhart–Hart equation were presented in Table 1. The fitting error of Steinhart–Hart equation is estimated 0.177 mK which is slightly larger than the RCB equation (Table 1). The inverse relation (T to R) of the Steinhart–Hart equation is also presented in Table 3. Because the R to T form of the Steinhart–Hart equation includes 3rd order term of ln(Rt ), this T to R form is expressed as a rather complicated form. On the other hand, RCB equation presents a simpler form in the inverse relation because the RCB equation uses 2nd order terms. Also the RCB equation showed better fitting results than Steinhart–Hart equation in the narrow temperature span.

5. Conclusion We introduced an interesting approach to calibrate NTC thermistors using the residual compensation method. This method increased the temperature accuracy of the basic equation to sub milli-Kelvin. The 2nd order fitting of residuals showed a considerably smaller error and was well fitted to the resistance residual curve of the measured temperature. This method provides an advantage, in that the equation is easily modified from the basic equation by adding just the fitting constants a, b, and c. The RCB equation gave a more exact calibration result.

Acknowledgements The present work has been supported by the Korea Research Institute of Standards and Science (KRISS) under the project ‘Development of Measurement Standards for Medical Radiation,’ grant 15011045 and supported in part by the research project ‘Expansion of measurement standard infrastructure for the radiation use in medicine’ with contract number 14102016 for the development of the nuclear energy granted by the Ministry of Education, Science and Technology (MEST). We would like to extend our cordial and grateful regards to Dr. Yi, Dr. Kim, and Dr. Chun.

Table 3 Comparison of three thermistor equations, the basic equation, the RCB equation, and the Steinhart–Hart equation. R to T form Basic Eq.

T = B/ (ln R − A)

RCB Eq.

T = B/ ln −a + (1 − b) R − cR

SH–H Eq.

T=1/(A + B · ln(RT ) + C · (ln RT )3 )

 

T to R form

 2



)−A



R = e(A+B/T )



RT =

1−b−

 

R = exp



3

B/3C

 −

3





2

(b − 1) − 4c a + e(A+b/T )

B/3C

3  +

3  +



/ (2c)

  2 

A − 1/T /C /2

−

  2 

A − 1/T /C /2

+

 

A − 1/T /C /2

 

A − 1/T /C /2



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