The Study of Thermistors and their Characteristics

Workstation 13Raffi Tachjian 260330829, Marc Rousseau 260247753

14 April 2010

Abstract

The experiment consisted of measuring the resistance of a thermistor as a function oftemperature. This was accomplished by applying a voltage across the thermistor via acustom built current source, whilst the thermistor was at a given temperature. The exper-iment was conducted for two negative coefficient thermistors. A characteristic value β of3771.3 ± 0.3K was found for Thermistor A and a β value of 2869.0 ± 0.8K was calcu-lated for Thermistor B. These values were characterized with certainty for the temperaturesrange of −50C to +50C.

1 IntroductionA thermistor is a resistor whose resistance varies with temperature. Its uses can rangefrom an ordinary current limiter to a fully functioning thermometer[1]. There are twotypes of thermistors, positive and negative coefficient. This indicates sign of the slope ofthe resistance function, thus a negative coefficient thermistor’s resistance will decrease astemperature increases and vice versa.

Next, all thermistors possess a unique characteristic value depending on the materialfrom which it is made. The β value of a thermistor is an indication of the slope of thecurve which represents the relationship between the resistance and the temperature of aparticular thermistor measured under zero power conditions[2]. In other words, the largera β value, the greater the change in resistance per 1C. To add, the β value of a thermistoris temperature dependant within a small range of operating temperatures. Thus, it is not arigorous constant.

The essential part of the experiment consisted of changing the temperature of the en-vironment that the thermistor was in, thus changing the temperature of the thermistor. Toaccomplish this, the thermistor was placed inside an aluminium block which could be easilycooled by liquid nitrogen and heated via a power source.

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2 Theory

2.1 Ohm’s LawSince the resistance of the thermistor was considered in this experiment, ohm’s law wasessential. It states that the current through a conductor between two points is directlyproportional to the voltage across the two points, provided that the temperature remainsconstant[3]. Thus, the resistance of a thermistor was found using:

R =V

I(1)

where V represents the potential difference in Volts, I symbolizes the current in Amperesand R determines the resistance of the thermistor in Ohms.

2.2 Operational Amplifier

2.2.1 Theoretical Function

Operational amplifiers, also known as op-amps, are typically used in circuits as voltageamplifiers. They consist of a DC-coupled high-gain electronic voltage amplifier with a dif-ferential input and a single-ended output[3]. Figure1 displays the physical shape of a 741op-amp as used in the experiment.

Figure 1: Figure showing the type of op-amp used in the experiment.[1]

To end, all ideal op-amps are governed by two golden rules:

1. Both inputs have the same voltage.

2. No current flows through the op-amp.

In this experiment, the op-amps used were assumed to be ideal.

2.2.2 Experimental Function

In this experiment, a circuit was designed to supply enough current to the thermistor. Fig-ure 2 shows the circuit used to perform the experiment. In the circuit, one op-amp was usedas a current buffer while the other allowed us to observe a determined voltage at the outputand is considered to act as a ”current converter”, thus converting a current function to avoltage function.

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Figure 2: Figure displaying the circuit used to perform the experiment.

A current buffer is capable of transferring a current from a first circuit, having a lowoutput impedance level, to a second circuit with a high input impedance level[3]. In thisexperiment, it was used to increase the current sent to the thermistor since the DAC couldnot supply a sufficient current. To find the gain function in the circuit shown in figure 2, thegolden rules of op-amps were used and yielded two key equations:

VADC = −IR and VDAC = IRT

Both VDAC and VADC represent signals sent and received by the Labmaster, I representsthe current flowing through the circuit, R is the resistance of the resistor used in the circuitand RT is the resistance of the thermistor. Solving for the current in both equations yieldsthe gain function:

VADC

VDAC= − R

RT(2)

2.3 ThermistorFirst, assuming the relationship between a change in resistance and a change in temperaturein a thermistor is linear, it can be shown as[3]:

∆R

∆T= k (3)

where k symbolizes the first-order approximation temperature coefficient of resistance. Toadd, this coefficient is positive for positive temperature coefficient thermistors and vice-versa. Figures 3 and 4 display its shape and circuit notation.

Figure 3: Figure showing the shape of the thermistors used in the experiment.[1]

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Next, the linear first-order approximation is only consistent over a very small tempera-ture range. The Steinhart-Hart equation is used to obtain and predict accurate temperaturemeasurements in thermistors. Thus, it represents a third-order approximation and statesthat:

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T= a+ bln(R) + cln3(R) (4)

where a, b and c are the Steinhart-Hart parameters unique to each device, R represents theresistance in Ohms and T determines the temperature in Kelvins.

Figure 4: Figure displaying the circuit notation of a thermistor.

In this experiment, negative coefficient thermistors were used. As mentioned before,they are characterised with a β value. The equation containing this value is obtained bysimply substituting β = 1

b and c = 0 in equation 4[3]. This yields:

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T=

1

T0+

1

βln(

R

R0) (5)

where R0 symbolizes the resistance of the thermistor in Ohms at the temperature T0 whichis typically room temperature. They both represent set conditions. To end, the equationused to find β and describe the curves found was:

R = R0eβ[ 1

T − 1T0

] (6)

3 Experimental MethodsThe experiment consisted of measuring the resistance and temperature of a thermistor si-multaneously, while varying the temperature using two distinct procedures. To perform theexperiment, the thermistor analysed was placed in an aluminium cylindrical block with ap-proximate dimensions of 2 cm as its diameter and about 4 cm as its length. The aluminiumblock was heated by supplying power to a wire wrapped around it. Due to the high thermalconductivity of aluminium( 250 W

m.k ), the experimenter can be confident that the systemwill be in equilibrium in terms of temperature[4]. The resistance of the thermistor was thenmeasured by supply a voltage to the circuit via the Labmaster. In order to measure thethermistor’s temperature, a thermocouple, connected to a Labmaster’s ADC channel, wasalso inserted into the aluminium block. A diagram of the whole set-up is shown in figure 5.

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Figure 5: Figure showing the apparatus and setup of the experiment.

As for the second procedure, the aluminium block was cooled to 77K by being placedin liquid nitrogen bath. The temperature and resistance of the thermistor was then measuredas the system slowly warmed up. Figures 6 and 7 show pictures of the experiment.

Figure 6: Figure displaying the amplifying cir-cuit. The orange wires connect the thermistor tothe circuit as shown in the next figure.

Figure 7: Figure showing an overview of theapparatus used including the aluminium block.

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4 Data and ResultsThe β values of the thermistors were obtained after fitting an exponential curve to thecollected data, as seen in Figures 8 and 9.

Figure 8: Resistance as a function of temperature for Thermistor A, fit to an exponential curve.β = 3771.3± 0.3K with rχ2 of 0.9897.

For thermistor A, the resistance function was determined to be:R(T ) = 4.674± 0.006KΩexp((3771.3± 0.3K)( 1

T − 1296K )).

Thermistor B’s characterisation function is:R(T ) = 992.2± 0.0Ωexp((2869.0± 0.8K)( 1

T − 1295K ))

Figure 9: Resistance as a function of temperature for Thermistor B, fit to an exponential curve.β = 2869.0± 0.8K with rχ2 of 0.9992.

Data taken between 250K to 290K and 175K and 225K was then plotted semi-logarithmicallyas seen in Figures 10 and 11 respectively.

Figure 10: Figure showing the exponential relationship of resistance and temperature for Ther-mistor A, rχ2 = 0.9989

Figure 11: Figure showing the exponential relationship of resistance and temperature for Ther-mistor B, rχ2 = 0.9965

The errors on these data points are incredibly small as this is the range in which thecircuit best behaved.

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5 DiscussionFor temperatures above room temperature, it was possible to hold the aluminium block at anearly constant temperature. This allowed for a large sequence of data points to be taken,thus reducing the error. Difficulty arose when the block was allowed to warm up after itsliquid nitrogen bath as it was not possible to maintain at a constant temperature. To slowthe warming process, the block was placed inside a styrofoam box in order to insulate it.Fewer sequences were used per reading since fluctuations in temperature during acquisitionwould result in significant errors.

Notice the first few error bars on Figures 8 and 9 are incredibly large, to the order of13GΩ. This can be explained by the circuit’s limitations. The op-amp circuit, figure 2,has an ADC as input and is thus limited to providing at most 10V. When the thermistoris very cold, its resistance is very large, thus the current supplied will be incredibly small.For a resistance of 1MΩ, the current would be 0.01mA. This causes problems since theelectronics in the circuit need a certain current threshold to work. A simple fix would be toincrease the input voltage by swapping for an DAC with a larger voltage range.The linearity of Figures 10 and 11 show that the exponential fit is relevant for the rangespecified, the typical range in which thermistors are employed. By extrapolating data fromthis fit, we expect a much more precise β value since the range being studied has negligibleerror. Indeed the fits were calculated to have rχ2 values of 0.9989 for thermistor A and0.9965 for thermistor B.

6 ConclusionThe characteristics of the two unknown thermistors were successfully determined. This wasaccomplished by simply sticking them into an aluminium block and varying the temperatureof the block while reading the resistance of the thermistor and temperature. The temperaturewas recorded via a thermocouple output to the LabMaster’s ADC while the resistance wascalculated using the output voltage of a circuit in which the thermistor was apart. The βvalue for thermistor A was 3771.3± 0.3K and 2869.0± 0.8K for thermistor B.

References[1] Karl K. Turekian John H. Steele, Steve A. Thorpe. Measurement Techniques, Platforms

Sensors: A derivative of the Encyclopedia of Ocean Sciences. Academic Press, 2010.

[2] CANTHERM. How to choose and work with ntc thermistors. http://www.cantherm.com/products/thermistors/choosing_ntc.html#3, Jan-uary 2009.

[3] Forrest M. Mims III. Getting Started in Electronics. Master Publishing, Inc., 2003.

[4] LeMay Brown and Bursten. Chemistry, The Central Science. Prentice Hall, 2006.

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