Measurement 44 (2011) 2054–2059

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Measurement

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Voltage divider resistance for high-resolution of the thermistortemperature measurement

Jongwon Kim a, Jong Dae Kim b,⇑a BiomedLab Co., Ltd., 7F, Gayang-Dong, Gangseo-Gu, 143-813 Seoul, Republic of Koreab Division of Information and Communication Engineering, Hallym University, 1 Okchon-dong, Chunchon 200-702, Republic of Korea

a r t i c l e i n f o

Article history:Received 4 November 2010Accepted 17 August 2011Available online 27 August 2011

Keywords:Negative Temperature Coefficient (NTC)thermistorTemperature measurementVoltage divider resistanceHigh-resolution

0263-2241/$ - see front matter � 2011 Elsevier Ltddoi:10.1016/j.measurement.2011.08.004

⇑ Corresponding author. Tel.: +82 33 248 2320; faE-mail address: kimjd@hallym.ac.kr (J.D. Kim).

a b s t r a c t

When measuring temperature with a voltage divider, and changing the variation of thethermistor resistance from the temperature to the voltage, the divider resistance greatlyimpacts the resolution of each ADC step. This work presents a method for determiningthe divider resistance to minimize the resolution’s maximum value in a given temperaturerange. Since the function of the resolution strongly depends on the derivative of the therm-istor resistance, we also investigated the effect on the resolution when the derivative wascalculated by forward and backward finite differences and the Stein–Hart calibration equa-tion. The results showed that the resolution’s maximum calculated by the three methodshad only a 5% difference, for the four types of commonly used NTC thermistors. Also, wedemonstrated that the divider resistance which minimizes the interval resolution’s maxi-mum can be determined by the thermistor resistance and its derivative at each end of thetemperature range.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Compared with many temperature measuring sensors,the NTC thermistor has greater sensitivity, faster thermalresponse, flexible sizes and shapes, and low cost. It alsohas a 10-fold higher temperature coefficient than platinumsensors which are used to measure the temperature pre-cisely. Due to these advantages, the NTC thermistor hasbeen adopted by the biomedical field, where only a limitedrange of temperatures are of interest [1–3].

While a bridge circuit or voltage divider is usually usedto measure the temperature with an NTC thermistor, vari-ous linearization methods have been developed to handletheir highly nonlinear behaviors [4,5]. However, we canalso obtain reasonably good resolution using a simple volt-age divider circuit, due to the thermistor’s high sensitivity.The voltage divider circuit converts the variation in thethermistor resistance to a variation in voltage, by dividinga constant voltage with the thermistor and a reference

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resistor. The divided voltage is digitized by the ADC, andthe linearization is obtained in the digital form [1]. Usingthe ADC with high resolution enables us to gain sufficienttemperature resolution; however, there is a decrease insystem speed and an increase in cost. Eventually, it is nec-essary to determine the minimum ADC resolution forachieving the required temperature resolution [1].

When the resistance variation is changed to a voltagevariation using a voltage divider, the amount of the varia-tion depends on both the divider resistance and the tem-perature. Consequently, the temperature resolution is afunction of both the temperature and the resistance ofthe divider; thus, knowledge of the divider’s resistance isrequired to obtain the optimum resolution. To date, therehas been no proposed quantitative method for determiningthe divider resistance, although previous work has pro-vided the subjective means to do so [1].

When a temperature range is known for a specific appli-cation, the maximum resolution per ADC step in the rangeis key for determining the ADC bit width, because theresolution for each temperature in the range would bethe same or smaller than the maximum resolution.

J. Kim, J.D. Kim / Measurement 44 (2011) 2054–2059 2055

Therefore, it is useful to determine the divider resistancethat minimizes the maximum resolution in that tempera-ture range [1]. Meanwhile, the resolution at any tempera-ture is related to the derivative of the resistancetemperature to the sensitivity; therefore, a method to esti-mate the derivative is needed. Since the thermistor manu-facturers provide the resistance values in units of 1 �C, wecan use the one-degree finite difference for estimating thedifferentiation. We also use several calibration equationsto estimate the derivative, since none of them can providethe true value of the differentiation. But, as we will presentin the next section, the bounds of the derivative can bedetermined knowing that the resistance of the thermistor,and its first and second differentiation value, decrease pro-portionally to the temperature. Our presented method usesthe above properties to determine the divider resistanceneeded to minimize the maximum resolutions per step,and we also propose the bounds for the estimated dividerresistance and the maximum resolution.

In the following section, we demonstrate the method todetermine the divider resistance, and derive the bounds forthe true maximum resolution. Section 3 presents simula-tion results for four types of commercial thermistors, andconclusions are provided in Section 4.

2. The divider resistance to minimize the maximumresolution for a given temperature range

Fig. 1 shows a schematic for a circuit that measures thetemperature using a voltage divider with an NTC thermis-tor. As the temperature increases, the resistance decreases,resulting in an increase in the divider voltage. The ratio ofthe ADC bias voltage and the input voltage uðx; tÞ satisfiesthe following:

uðx; tÞ ¼ xxþ rðtÞ ; 1 6 uðx; tÞ 6 1; ð1Þ

where rðtÞ, x, and uðx; tÞ represent the resistance of thethermistor at absolute temperature t, the resistance ofthe voltage divider, and the divided voltage at t,respectively.

If we linearly digitize the constant voltage V using an n-bit ADC, the resolution per step at a given temperature

ADC

r(t)

x

V

u(x,t) y

signal processor

Fig. 1. Temperature measurement circuit using an NTC thermistor.

f ðx; tÞ, and the maximum resolution at the temperaturerange [a,b], gðxÞ, can be expressed as follows:

f ðx; tÞ ¼ � 12n½xþ rðtÞ�2

xr0ðtÞ ; ð2Þ

gðxÞ ¼maxt2½a;b�ff ðx; tÞg; ð3Þ

where r0ðtÞ represents the derivative of the thermistorresistance to temperature.

The purpose of our work is to obtain the voltage dividerresistance x that minimizes gðxÞ: i.e., the maximum resolu-tion at the given temperature range, as provided in Eq. (3).To understand the behavior of gðxÞ, we need to determinechanges in the thermistor resistance due to temperature,along with the change in the derivative. The thermistorresistance dependence on the temperature changesfollows:

rðtÞ ¼ r0 exp b1t� 1

t0

� �� �; ð4Þ

where r0 is the thermistor resistance at absolute tempera-ture t0, and b is the thermistor material constant.

In Eq. (4), b changes slowly with temperature and has asufficiently high value, enabling us to approximate thethermistor’s derivative value as follows:

r0ðtÞ ffi � brt2 : ð5Þ

From Eqs. (2) and (5), the resolution per step f ðx; tÞtakes the following form:

f ðx; tÞ ffi 12n

t2½xþ rðtÞ�2

bxr: ð6Þ

f ðx; tÞ is determined by linearly combining t2=r, t2, andt2r with respect to t. t2=r and t2r are in the form oft2e�1=t , which is convex excluding the break point att ¼ 0. In other words, all three functions are convex forany positive t. Therefore, the sum of the functions is alsoconvex, and the maximum exists at the domain boundary[6]. Eventually, the maximum resolution gðxÞ at a giventemperature range can be written as:

gðxÞ ¼maxff ðx; aÞ; f ðx; bÞg: ð7Þ

Since f ðx; tÞ is convex for x, gðxÞ is also convex, indicat-ing there is only one minimum value at the intersection off ðx; aÞ and f ðx; bÞ [6]. Then, the divider resistance x0 thatminimizes the maximum of the resolution in [a,b] will sat-isfy the following equation:

½x0 þ rðaÞ�2

x0r0ðaÞ ¼ ½x0 þ rðbÞ�2

x0r0ðbÞ :

Eventually, the divider resistance x0 that minimizes themaximum of the resolution can be expressed as a functionof the thermistor resistances and their derivatives at theboundaries of the given range.

x0 ¼prðaÞ � rðbÞ

1� p; 0 < p ¼

ffiffiffiffiffiffiffiffiffiffir0ðbÞr0ðaÞ

s< 1: ð8Þ

2056 J. Kim, J.D. Kim / Measurement 44 (2011) 2054–2059

As shown by Eq. (5), r0ðtÞ also decreases monotonically,satisfying 0 < p < 1. The maximum resolution will be theresolution at each end temperature, as shown here:

minx

gðxÞ ¼ gðx0Þ ¼ f ðx0; aÞ ¼ f ðx0; bÞ: ð9Þ

Since the optimal divider resistance and the minimumof its maximum resolution, expressed in Eqs. (8) and (9),are functions of derivatives, we need to estimate the valuefrom the differentiation. Since thermistor manufacturersprovide the resistance value in 1 �C units, we can use aone-degree finite difference or various types of calibrationequations for the differentiation estimate, although wecannot obtain a precise value. Instead, we can obtain thebounds of the minimum at the optimal resistance, by usingthe property of the thermistor’s R–T Eq. (4). First, we canapproximate the optimal divider resistance x0 through for-ward and backward finite difference. As the absolute valueof the derivative of the thermistor temperature–resistancefunction monotonically decreases, as shown in Eq. (5), thesecond derivative r00ðtÞ also decreases monotonically.

r00ðtÞ ffi brðbþ 2tÞt4 : ð10Þ

Therefore, the following absolute inequalities aresatisfied:

jrf ðaÞj > jr0ðaÞj > jrbðaÞj; jrf ðbÞj > jr0ðbÞj > jrbðbÞj; r00ðaÞ> r00ðbÞ;

where rf and rb are the forward and backward differences,respectively.

Let pf and pb be calculated p in (8), using the forwardand backward difference, respectively. Then, the followinginequalities are also satisfied:

r0ðbÞr0ðaÞ ffi

rbðbÞ þ r00ðbÞrbðaÞ þ r00ðaÞ <

rbðbÞrbðaÞ ;

r0ðbÞr0ðaÞ ffi

rf ðbÞ � r00ðbÞrf ðaÞ � r00ðaÞ

>rf ðbÞrf ðaÞ ; pf < p < pb:

Since x0 in Eq. (8) increases monotonically with respectto p, the optimal resistances xf and xb obtained through thefinite differences also satisfy the following inequality:

xf > xo > xb: ð11Þ

On the other hand, if we set the acquired functions byapplying rf or rb in place of r0 in f ðx; tÞ, and set gðxÞ to beff , fb, gf , and gb; then as f decreases monotonically and isinversely proportional to r0, the following relations aresatisfied:

ff ðx; tÞP f ðx; tÞP fbðx; tÞ; 8t;

gf ðxÞP gðxÞP gbðxÞ; 8x:

From the above inequalities, the optimal resistanceminimizing gf and gb will be xf and xb, which are used inthe following:

gf ðxf Þ > gðxoÞ > gbðxbÞ: ð12Þ

Consequently, the optimal divider resistance whichminimizes the maximum resolution in the given tempera-ture interval, as well as the maximum resolution at thatresistance, has the bounds shown in Eqs. (11) and (12).

It is impossible to calculate the interval maximum reso-lution gðxÞprecisely, because it requires the true value of thederivative. However, if the derivative of the thermistorresistance obtained by the calibration equation existsaccording to the backward and forward differences, andthe first and second derivatives monotonically decrease asin Eqs. (5) and (10), the following is also considered true:

xf > xc > xb; ð13Þ

gf ðxf Þ > gcðxcÞ > gbðxbÞ; ð14Þ

where gcðxcÞ and xc are the interval maximum resolutionand the resistance minimizing it, respectively; both arecalculated by the calibration equation.

The derivatives from the calibration equation can beconsidered to be reasonably close to the true value if thebounding interval of Eqs. (13) and (14) is sufficiently small.In this work, the interval maximum resolution, expressedas Eq. (3), is calculated as the derivative from the backwardand forward difference and the calibration equation, andverifies whether the calculated minimum satisfies Eqs.(13) and (14). Also, the optimal resistance calculated usingEq. (8) with the thermistor resistance and its derivative ateach end point of the temperature interval are compared.The notations rb, rf , and rc used in the next section repre-sent the derivatives calculated by the backward and for-ward finite difference and the Stein-Hart equation,respectively. The corresponding step resolution f , the max-imum resolution g, and the optimal divider resistance cal-culated with the boundary values xB are similarly notatedas follows:

xB are notated similarly as follows:

fiðx; tÞ ¼ �12n½xþ rðtÞ�2

xriðtÞ ; i ¼ b; c; f ;

giðxiÞ ¼minx� 1

2n maxt2½a;b�ffiðx; tÞg

� �; i ¼ b; c; f ;

xBi ¼

pirðaÞ � rðbÞ1� pi

; pi ¼

ffiffiffiffiffiffiffiffiffiffiriðbÞriðaÞ

s; i ¼ b; c; f :

3. Experiments and results

For the experiment, four types of thermistors were cho-sen that are interchangeable within ±0.2 �C, and they wereprovided with 1 �C unit resistances. The chosen sensorswere: 44006RC (formerly YSI temperature Co. product) ofMeasurement Specialties Co.; 10K4A1W (formerly Beta-THERM Co. product); B57863 (whole product name:B57863S0103F040) from EPCOS Co.; and PT103J2 of USsensor.

The calibration equation was used to confirm the ten-dency of the first and second derivatives. Table 1 showsthe standard deviation of the error and its maximum, whenthe Stein-Hart (S-H), Hoge-2 (4 parameter), and Hoge-3 (5parameter) were fitted to the data of the temperature–resistance table within 0–100 �C [3]. All approximationsshowed an error of no more than 50 mK. In this work,

Table 1Standard deviation of the error and the maximum error of the threecalibration equations at 0–100 �C (unit: mK).

S–H Hoge-2 Hoge-5

Std Max Std Max Std Max

44006RC 5.23 15.02 4.43 11.82 4.13 13.9910K3A1AW 4.72 17.07 4.81 18.53 4.70 17.35B57863 8.01 22.68 6.70 22.50 6.61 25.77PT103J2 6.94 24.43 7.03 24.94 5.26 19.68

0 20 40 60 80 1000

0.2

0.4

0.6

0.8

1

temperature (oC)

(rc -rb )/(

rf -rb )

Fig. 4. Relative position of the derivative calculated through the S–Hequation to the backward and forward differenceq ¼ rc ðtÞ�rb ðtÞ

rf ðtÞ�rb ðtÞ ; 44006RC� �

.

J. Kim, J.D. Kim / Measurement 44 (2011) 2054–2059 2057

the Stein–Hart equation, which had the smallest maximumerror, was used to approximately express the tendency ofthe derivatives of the four thermistors.

Figs. 2 and 3 illustrate the transition of the first and sec-ond derivatives according to the temperature of the44006RC thermistor. The same tendencies were also ob-served for the other two calibration equations. As predictedthrough Eqs. (5) and (10), the first and second derivativesdecreased monotonically with the temperature.

We also investigated the relationship between thederivative calculated through the calibration equationand the finite differences in 1 �C units. Fig. 4 shows the rel-ative position of the derivative from the calibration equa-tion to the forward and backward differences. It can beobserved that the position of the derivative obtained fromthe calibration equation was in the middle of the finite dif-ferences. This central position of the derivatives occurs onaverage for all four thermistors (Table 2). The only temper-ature exception was ‘10K4A1W’, which had minimum

0 20 40 60 80 100-1500

-1000

-500

0

temperature (oC)

r' (t)

Fig. 2. Derivative of the thermistor resistance (S–H equation, 44006RC).

0 20 40 60 80 1000

2000

4000

6000

8000

10000

12000

temperature (oC)

r'' (t

)

Fig. 3. Second derivative of the thermistor resistance (S–H equation,44006RC).

positions at 94�, 95�, and 98�. This was because both thebackward and forward differences were the same as thederivative from the calibration equation.

Fig. 5 illustrates the function of the interval maximumresolution with respect to the divider resistance, whenapplying the three speculated derivatives of thermistor44006RC and the position of the divider resistances thatminimizes them. As foretold, the inequalities (13) and(14) were satisfied. Tables 3 and 4 provide the minimumof the interval maximum resolution, as calculated by thethree speculated derivatives on the temperature intervals[40,95] and [0,100] and the divider resistances that mini-mize them. The definitions of the relative variations belowwere established to illustrate their differences:

dx ¼maxðxb; xc; xf Þ �minðxb; xc; xf Þ

minðxb; xc; xf Þ� 100;

Table 2Relative position of the approximate expression through the S–H equationto the backward and forward difference (unit: %).

44006RC 10K3A1AW B57863 PT103J2

Dev. fromxb

Mean 49 47 49 50

Max 84 85 76 80Min 18 0 7 26

103.1 103.2 103.3

35

40

x (Ω)

g(x)

(mK) gb(x)

gf(x)

gc(x)

xbxc

xf

Fig. 5. Trend of the interval maximum resolution according to the dividerresistance (44006RC).

Table 3Minimums of the interval maximum resolution of the temperature interval [40,95] and the optimum divider resistances (unit: X, mK, %).

Interval Divider resistance (kX) Min. of max. resolution (mK)

[4095] xb xc xf dx (%) gb(xb) gc(xc) gf(xf) dg (%)

44006RC 1.634 1.646 1.661 1.7 36.6 37.4 38.1 3.910K4A1AW 1.643 1.646 1.721 4.8 36.7 37.4 37.2 1.3B57863 1.464 1.470 1.472 0.5 35.1 35.8 36.5 4.0PT103J2 1.460 1.467 1.474 1.0 35.2 35.8 36.6 4.2

Table 4Minimum of the interval maximum resolution calculated by three speculated derivatives of the temperature interval [0,100], and its minimizing dividerresistance (unit: X, mK, %).

Interval Divider resistance (kX) Min. of max. resolution (mK)

[0100] xb xc xf dx (%) gb(xb) gc(xc) gf(xf) dg (%)

44006RC 3.387 3.404 3.422 1.0 55.7 56.7 57.7 3.710K4A1AW 3.393 3.404 3.410 0.5 55.4 56.7 58.1 4.9B57863 3.211 3.237 3.256 1.4 57.1 58.3 59.6 4.4PT103J2 3.196 3.235 3.261 2.0 57.1 58.3 59.6 4.3

103

40

50

60

70

80

x (Ω)

g(x)

(mK)

Fig. 6. Trend of the interval maximum resolution according to the dividerresistance (+: x = r(40), r(95), thermistor: 44006RC, temperature interval:[40,95], derivative: from S–H equation).

2058 J. Kim, J.D. Kim / Measurement 44 (2011) 2054–2059

dg ¼maxðgbðxbÞ; gcðxcÞ; gf ðxf ÞÞ �minðgbðxbÞ; gcðxcÞ; gf ðxf ÞÞ

minðgbðxbÞ; gcðxcÞ; gf ðxf ÞÞ� 100:

The minimum of the interval maximum resolution ofthe thermistor 10K4A1AW on the temperature interval[40,95] did not satisfy the inequality (14). But since thedisparity of the minimum was only 1.3%, similar resultswould be observed with any derivative. Generally, theminimum of the interval maximum resolution using anyof the three derivatives gives a similar result, with a differ-ence less than 5%.

The relative deviation indicated below was calculatedto compare the minimum resistances obtained with thethermistor resistances and the derivatives at the ends ofthe interval, as well as those shown in Tables 3 and 4.

ri ¼jxi � xB

i jxi

; i ¼ b; c; f :

The relative deviation showed no more than 0.1% with-out distinguishing the temperature interval, types of thethermistors, and the estimate of the derivative. Conse-

quently, the divider resistance minimizing the intervalmaximum resolution could be calculated with the deriva-tive and the thermistor resistance at the boundary of thegiven interval.

Fig. 6 demonstrates the difference in the interval maxi-mum resolution when the divider resistance was improp-erly selected. The ‘+’ marks show the interval maximumresolutions when the divider resistances were selected asthe thermistor resistances at the boundary temperatures.If we selected the thermistor resistance at 95� (one of thetemperature boundaries) as the divider resistance, thenthe interval maximum resolution was two times greaterthan that with the optimum divider resistance.

4. Discussion

A method was proposed to minimize the resolution perADC step at a given temperature interval using the therm-istor resistance and its derivative at the boundary. It wasshown that the one-degree finite differences provided thelower and upper bounds to the optimum divider resis-tance, where the maximum resolution over the given tem-perature interval was minimized. The Stein–Hartcalibration equation can be employed for the derivativeestimation, using the results that the optimized dividerresistance resided within the bounds in most cases. Butsince the variation of the interval maximum resolution cal-culated with the backward and forward difference was lessthan 5%, it would be more thorough to use the centered fi-nite difference instead of the calibration equation.

The estimation and effect of the derivative of the therm-istor resistance with respect to the temperature at anypoint was also investigated. Using the same method, thedivider resistance minimizing the interval average resolu-tion can be calculated. This method could also be usedfor any research investigating the temperature sensitivityof the NTC thermistor.

J. Kim, J.D. Kim / Measurement 44 (2011) 2054–2059 2059

Acknowledgment

This research was supported by the Basic ScienceResearch Program through the National Research Foundationof Korea (NRF), funded by the Ministry of Education,Science and Technology (No. 2010-0015951).

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