Thermistors are temperature-sensitive semiconducting devices. There are several types used in a wide variety of temperature measurement, switching, and sensing applications. This article will focus primarily on negative temperature coefficient (NTC) thermistors designed specifically for temperature measurement. While NTC thermistors are used over a very wide range of temperatures, from 1 K to 1000°C, we will focus on the range −80 °C to 300 °C, where they are most commonly used and offer genuine advantages over platinum resistance thermometers for some applications.

NTC thermistors are suited to a number of specialized thermometry applications because they are smaller, faster, and more sensitive than almost all other contact thermometers. However, they are very nonlinear and have a narrow temperature range.

Traditionally, thermistors have also had a reputation for instability, but thermistors are now readily available for temperature ranges within −40 °C to 60 °C with stabilities of a few tenths of a milli-kelvin per year. Their short-term performance approaches that of standard platinum resistance thermometers, but at a much lower cost.

Thermistors are also available with selected performance characteristics, including interchangeability to 0.05 °C, aged and selected for stability, as well as qualification for military and space applications

# NTC Thermistor Working Principle

NTC thermistors are extrinsic semiconductors manufactured from mixtures of metal oxides heated to high temperatures to form a polycrystalline ceramic.

Ordinarily, the individual oxides have a large energy gap between the vacant conduction band and the fully occupied valence band, which makes them good electrical insulators. The mixture of oxides, however, has intermediate electronic states that make the ceramic a semiconductor.

As the temperature increases, increasing numbers of electrons jump to higher-energy states enabling conduction in the conduction bands, and, by creating holes, enabling conduction in the valence band. The total number of mobile charge carriers, n, depends on the energies of the electrons as determined by the Boltzmann distribution and on the density of electronic states near the band edges:

Where E_{g} is the excitation energy for the carriers k is Boltzmann’s constant T is temperature (K).

The leading T^{3/2 }term in Equation 1 is due to the density of states at the band edges and tends to be compensated by a similar term in the carrier mobilities (due to the density of states for phonons that scatter the carriers).

The combination of effects gives an overall resistance–temperature behavior that is roughly approximated by

Where, R_{0} is a constant.

Usually, this equation is rewritten in the form

Where, T_{0} is some convenient reference temperature, often 298.15 K (25 °C). The parameter β is a characteristic of the thermistor material with typical values in the range 2000 – 6000 K.

Figure 1 plots the resistance–temperature characteristic for a range of commercially available thermistors.

Figure 2 plots the β values for the same set of thermistors and shows the typical variation in β with temperature.

The sensitivity, S, of the thermistor is defined as the relative change in resistance for a change in temperature and is given by

This number is useful for evaluating measurement errors, as will be shown later.

Figure 3 shows the sensitivities for the thermistors of Figure 1.

At room temperature, they range between about −0.03/°C and −0.05/°C.

That is, thermistors are about 10 times more sensitive to temperature than platinum resistance thermometers: the sensitivity for platinum resistance thermometers is more nearly constant and approximately equal to the alpha value, α = 3.85 × 10^{−3}/°C.

## Calibration Equations of NTC

One of the notable features of Figure 2 is that the β value for a thermistor varies with temperature, which is characteristic of extrinsic semiconductors. This means that is only useful as a calibration equation for temperature ranges of no more than 20 °C – 30 °C.

For high-accuracy or wide temperature ranges, more complex equations are required. Although a wide range of calibration equations have been explored, for example, there are two extended series expansions of Equation 3 that are most commonly used:

Where, R_{0} is usually a convenient reference resistance, for example, 1 Ω or 1 kΩ, depending on the measurement units.

The number of terms in each series, N, is chosen according to the temperature range and the accuracy required, with more terms required as the range or accuracy increases. Most commonly, the equations are used with four terms, but occasionally up to five terms.

Equation 5 has the advantage of giving temperature directly from the measured resistance and tends to be used more frequently. Both equations are linear in their coefficients so are amenable to least-squares fitting using the linear algorithms found in many spreadsheets and graphing applications.

Figure 4 summarizes the performance of Equation 5 when different numbers of terms are used. Note that the N = 2 case corresponds to Equation 3.

Among a number of equations investigated by Steinhart and Hart is an alternative calibration equation,

Unfortunately, which is in common use and recommended by many thermistor manufacturers.

The original recommendation for Equation 7 was based on a numerical error, and use of Equation 7 should be discouraged. The equation has some peculiar attributes that make its performance depend on the R(25 °C) value of the thermistor, the unit of measurement, R_{0}, and whether thermistors are connected in series or parallel.

Under very specific conditions, its performance can approach, but never exceed, that of Equation 5 with N = 4.

## Instrumentation

For the highest accuracy and for temperature ranges of more than a few degrees, the thermistor resistance is best measured directly using a good-quality digital voltmeter in combination with either Equation 5 or 6, for example.

Care should be taken to account for lead resistance effects and offset voltages and to make sure that the self-heating of the thermistor due to the sensing current is not excessive.

With care in the design of the measurement, selection of the thermistor, use, and calibration, accuracies approaching 0.001 °C are achievable.

Thermistor-based thermometers are commercially available with this level of accuracy. For narrow temperature ranges, and the highest resolution, purpose-built measurement circuits can be advantageous, and an example is shown in Figure 5.

The output voltage of the circuit in Figure 5 is

Where the resistors are as indicated. The condition for zero output voltage is

R_{in}/R_{f} = r/R(T_{o}) …..(Equation 9)

Where, T_{0} is the nominal operating temperature of the thermometer.

The output voltage is a nonlinear function of the thermistor resistance, and this nonlinearity is used to compensate for the thermistor nonlinearity. By replacing R(T) in Equation 8 by 3, and differentiating with respect to T, it is found that the nonlinearity is largely eliminated if r is chosen so that

The key feature of the linearization circuit is the resistance r_{in} series with the thermistor (the half-bridge in Figure 5). The same scheme can be applied when a single thermistor is used in more complex resistor networks by recognizing that r corresponds to the Norton or Thévenin equivalent resistance seen by the thermistor. Near room temperature, r is often found to be about 0.6 × R(T_{0}).

With the linearizing resistor, r, set according to Equation 10 and the gain setting resistors R_{in }and R_{f} set to the ratio given by Equation 9, the output of the thermometer in Figure 5 is approximately

The residual nonlinearity is almost purely a cubic function of temperature, as shown in Figure 6, with maximum errors of about ±0.1 °C for a 30 °C range. Equation 11 shows that the residual nonlinearity scales as cube of the temperature range.

Sapoff and Oppenheim describe refinements of this basic linearization technique and give a wide range of examples for different applications. One of the big advantages of purpose-built circuits like that of Figure 5 is, with care and appropriate signal conditioning, a resolution of better than 10 μK can be achieved relatively easily.

This enables thermistors to be used for narrow-range high-resolution applications such as differential thermometry, calorimetry, and precision temperature control.

For temperature ranges wider than a couple of tens of degrees, linearization of the thermistor response becomes more difficult.

Practical solutions include the use of logarithmic amplifiers and oscillators which work well for thermistors with a nearly constant β value. Some commercial thermistor thermometers use this technique and yield accuracies of a few tenths of a degree over a 100 °C range.

Another solution is to expand the linearizing scheme of Figure 5 to resistance networks in which two or more thermistors are used in combination with linearizing resistors (see Figure 7 for example).

Commercial thermistor networks optimized for specific temperature ranges are available with linearities as good as ±0.03 °C for a 40 °C temperature range and ±0.2 °C for a 100 °C temperature range.

A variation on the multi-thermistor scheme is to sum the outputs from several single-thermistor networks. While superficially the same as the type of network illustrated in Figure 7, the approach uses additional resistors and creates extra degrees of freedom in the design, which enables further reduction in the nonlinearities.

A three-thermistor network of this design has peak nonlinearities of about 16 mK over a 100 °C range, more than 10 times better than the network of Figure 7.

## Selection of Thermistor

The temperature range of commercially available thermistors varies from 1 K to 1000 °C with large differences in the R(25 °C) resistance, β values, the metal oxides used, and the encapsulation and sheath assemblies.

The most stable thermistors for temperature measurement have a much restricted range: glass-encapsulated thermistors are most useful in the range from about −40 °C to 100 °C, but some are usable from −80 °C to 250 °C. Epoxy-encapsulated thermistors have an upper temperature limit of about 150 °C.

Thermistors are made in many different forms. Bead thermistors are supplied with leads attached during the manufacturing process but may be further encapsulated in glass or epoxy, as well as diode like forms suitable for automatic insertion into printed circuit boards.

Other forms include flakes, disks, surface-mount, and rod types, which may be supplied with or without leads. When supplied without leads, they have metallized surfaces for the contacts.

There are many large companies manufacturing thermistors, and comparing specifications from different suppliers can be very helpful to highlight limitations in the specifications, manufacturing tolerances, and sheath assemblies that might be relevant to any application. Most of the manufacturers also provide extensive application information.

Thermistors are available to meet military specifications, space qualification, and can be selected for long-term stability and interchangeability. Military standard MIL-PRF-23648 [30] not only provides specifications for thermistors, but also gives the measurement conditions for the various electrical and thermal properties of thermistors.

## Other Applications of Thermistors

The term thermistor is applied generically to any semiconducting device with a high temperature coefficient of resistance. While NTC thermistors are the best of the devices for thermometry, once other fields of application are considered, other types of thermistors, including positive temperature coefficient (PTC) ceramic and silicon thermistors, become useful.

**NTC Thermistors: **When used for thermometry, thermistors are operated with a low current to avoid self-heating errors. However, for many applications, NTC thermistors are deliberately operated at very high currents to exploit the thermal feedback that occurs.

If a thermistor is operated at a nominal ambient temperature T_{a} and with a sensing current I_{0}, then the thermistor temperature can be written as

Where P is any additional power applied to the thermistor (e.g., microwave or optical) and ρ is the thermal resistance between the thermistor and its environment.

The essential feature of the model equation is that any power applied to the thermistor, either through Joule heating I_{o}^{2}R (T) or external power P, causes the temperature of the thermistor to rise and, hence, its resistance to fall.

Equation 12 can be solved numerically for T and then the voltage I_{0}R(T) plotted against the current to give a voltage–current characteristic for the thermistor. Example curves are shown in Figure 8.

There are three distinct operating regimes in Figure 8. At low currents, the power dissipated is small compared with the dissipation constant, and the self-heating effect is negligible.

Under these conditions, the thermistor resistance depends only on the temperature, and the voltage across the thermistor is directly proportional to the current.

Consequently, at low currents, the ratio of the voltage to the current is equal to the zero-power resistance of the thermistor. This is the regime used for temperature measurement, as described in the previous sections.

As the current increases, the effects of self-heating become more evident, the thermistor temperature rises, its resistance begins to decrease, and the slope of the voltage–current characteristic decreases. This trend continues until the slope of the voltage–current characteristic is zero.

The temperature, voltage, and current at the peak of the curve are

Beyond the peak voltage, the slope of the curve continues to decrease, and the thermistor exhibits a negative resistance characteristic, that is, dV/dI_{0} is negative.

Also shown in Figure 8 is a straight line corresponding to the voltage–current characteristic of a Thévenin voltage source. The characteristic is usually called a “load line” and has the equation

V = V_{Thev} – IR_{Thev} ……….(Equation 16)

Where, V_{Thev} and R_{Thev} are the voltage and resistance in the Thévenin equivalent circuit for the circuit around the thermistor.

The load line intersects with the thermistor’s characteristic at the points indicated. Applications based on self-heated thermistors work by detecting changes in the operating point of the thermistors.

For example, Figure 8 shows three voltage–current curves for the same thermistor but with different values of the dissipation constant. The changes in voltage and current between the points on the load line indicate the circuit’s response to the changes in the dissipation constant. The circuit could be used in a wide variety of applications where, by measuring changes in the dissipation constant, the circuit can be used for leak detection, flow measurement, and level detection.

Once the operating point of the thermistor is chosen, the sensitivity of the circuit in the vicinity of the operating point can be expressed as a partial derivative of the voltage to all of the other variables in the circuit. For example,

Where, the sensitivity coefficients μ_{T0}, μ_{I0}, μ_{ρ}, μ_{V0}, μ_{P} …, are the respective derivatives of the voltage while all other variables are constant. For example,

In some cases, mathematical expressions can be determined for the sensitivity coefficients. The key to using the thermistor as a sensor is to design the circuit so that the sensitivity of the circuit to the variable of interest is as large as practicable while all other sensitivity coefficients are as small as practicable.

It is not possible to have all other coefficients equal to zero, so the circuit is often designed as a differential system using two identical thermistor sub-circuits with one measuring the variable of interest and the other serving as a reference.

In this way, unwanted influences affect both subcircuits equally and the differential output signal does not change. Note too that the load line in Figure 8 could be placed to intersect the curve at one, two, or three points.

In some cases, the circuit will exhibit switching behavior as it jumps from one operating point to another in response to external changes. Sapoff cataloged a wide range of applications for thermistors and describes many examples of circuits using different operating principles:

- Variation in dissipation constant and time constant: vacuum manometers, flow meters, velocity meters, thermal conductivity, leak detectors, gas chromatography, liquid level, surge suppression, filament protection, overload protection, and switching.
- Variation in the load line (over a fixed current–voltage curve): oscillator amplitude and/or frequency control, gain control, level stabilization, equalization, voltage regulators, signal limiters, signal expansion and compression, switching.
- Variation in ambient temperature: principally thermometry including measurement, control, and alarm applications.
- Power measurement: bolometry for optical and microwave measurement, ac-to-dc conversion, and true rms measurement.

**PTC Thermistors**: The PTC thermistor refers to another family of semiconducting ceramic devices with an extremely variable temperature coefficient that is dominated by a temperature region in which the resistance rises extremely rapidly.

This makes the thermistors ideally suited to switching applications, and typically they are manufactured in families with specified switching temperatures.

Figure 9 shows a sample of the resistance–temperature characteristics for a family of PTC thermistors.

At low temperatures, the resistance islow, and the thermistor exhibits a small NTC of resistance on the order of −1%/°C. As the temperature increases, the temperature coefficient of resistance becomes positive and the resistance begins to rise, and then rise rapidly.

At the switching or“threshold” temperature, the rate of rise becomes very rapid and the resistance–temperature characteristic becomes very steep, sometimes as high as 100%/°C. At temperatures above the switching range, the resistance reaches a maximum value beyond which the temperature coefficient becomes negative again.

The resistance–temperature characteristic of the PTC thermistor can be exploited in a number of ways. A simple application is to use the thermistor as a heating element.

The power dissipated is V^{2}/R(T), and as the temperature rises in response to the heat, the applied power falls so that the thermistor can be used as a temperature controller or simply to limit the current flowing in the circuit.

If the joule heating is sufficient to maintain the thermistor temperature when in its high-resistance state, the thermistor may be used for delaying the switching of start-up mechanisms in electronic equipment such as the auxiliary windings in electric motors. When used as a liquid level or flow sensor, the thermistor is forced back into its low-resistance state by changing its dissipation coefficient.

These simple examples show that the thermistor can have a wide range of different behaviors, which makes them suitable for heating, current limiting, motor start, over current protection, temperature control, alarm, time delay, level, and flow sensing applications.

PTC thermistors are available in many different forms with the form usually designed for specific applications. The switching temperature can be varied between about 70 °C and 240 °C by altering the chemical composition of the ceramic. As with the NTC thermistors, most manufacturers provide extensive literature supporting their selection and application.

**Silicon Thermistors** : Silicon PTC thermistors, also known as “silistors” or “sensistors,” are typically single-crystal silicon devices that have been heavily doped to give the silicon a resistance–temperature characteristic similar to that of a metal.

The resistance–temperature characteristic is predominantly linear with an additional weak quadratic dependence that gives the device an almost constant sensitivity of about 0.7%/°C for all temperatures. It is about twice that of platinum resistance thermometers.

The main advantages of the silicon thermistors are the higher sensitivity, higher resistance (typically 100 Ω to 5 kΩ, but some up to 40,000 kΩ), and very much lower cost (below US$1 in large numbers). Although less accurate and less stable than NTC thermistors and platinum resistance thermometers, the low cost and ease of use make them the sensor of choice in many low-accuracy high-production applications.

The most common types of silicon PTC have the same package as small glass diodes with diameters of 1.8 – 2.5 mm and lengths of 3.8 – 7.5 mm, and the molded epoxy package with diameters of 3.6 – 6.0 mm and lengths of 10.4 – 15.0 mm. Such units also have axial leads, so are suited for electronic applications with automated component insertion.

Epoxy-coated chips and disks with radial leads are also available, with diameters of approximately 3 mm.

Silicon PTC thermistors that comply with MIL-T-23648 have an operating temperature range of −55 °C to 125 °C. Commercial versions can be obtained with a range of −65 °C to 300 °C.