# Thermistor

A thermistor is a type of resistor used to measure temperature changes, relying on the change in its resistance with changing temperature. Thermistor is a portmanteau word from the words thermal and resistor.

If we assume that the relationship between resistance and temperature is linear (i.e. we make a first-order approximation), then we can say that:

[itex]\Delta R=k\Delta T[itex]

where

[itex]\Delta R[itex] = change in resistance
[itex]\Delta T[itex] = change in temperature
[itex]k[itex] = first-order temperature coefficient of resistance

Thermistors can be classified into two types depending on the sign of [itex]k[itex]. If [itex]k[itex] is positive, the resistance increases with increasing temperature, and the device is called a positive temperature coefficient (PTC) thermistor, or posistor. If [itex]k[itex] is negative, the resistance decreases with increasing temperature, and the device is called a negative temperature coefficient (NTC) thermistor. Resistors that are not thermistors are designed to have the smallest possible [itex]k[itex], so that their resistance remains almost constant over a wide temperature range.

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## Steinhart Hart equation

In practice, the linear approximation (above) works only over a small temperature range. For accurate temperature measurements, the resistance/temperature curve of the device must be described in more detail. The Steinhart-Hart equation is a widely used third-order approximation:

[itex]T={1\over{a+b\ln{R}+c\left(\ln R \right)^3}}[itex]

where a, b and c are called the Steinhart-Hart parameters, and must be specified for each device. T is the temperature in kelvins and R is the resistance in ohms. To give resistance as a function of temperature, the above can be rearranged into:

[itex]R=e^{{\left( \beta-{\alpha \over 2} \right)}^{1\over 3}-{\left( \beta+{\alpha \over 2} \right)}^{1\over 3}}[itex]

where

[itex]\alpha={{a-{1\over T}}\over c}[itex] and [itex]\beta=\sqrt{{{{\left({b\over{3c}}\right)}^3}+{{\alpha^2}\over 4}}}[itex]

## Conduction model

Many NTC thermistors are made from a thin coil of semiconducting material such as a sintered metal oxide. They work because raising the temperature of a semiconductor increases the number of electrons able to move about and carry charge - it promotes them into the conducting band. The more charge carriers that are available, the more current a material can conduct. This is described in the formula:

[itex]

I = n \cdot A \cdot v \cdot e [itex]

[itex]I[itex] = electric current (ampere)
[itex]n[itex] = density of charge carriers (count/m³)
[itex]A[itex] = area of the material (m²)
[itex]v[itex] = velocity of charge carriers (m/s)
[itex]e[itex] = charge of an electron ([itex]e=1,602 \cdot 10^{-19} \mbox{ C} [itex] (coulomb))

The current is measured using an ammeter. Over large changes in temperature, calibration is necessary. Over small changes in temperature, if the right semiconductor is used, the resistance of the material is linearly proportional to the temperature. There are many different semiconducting thermistors and their range goes from about 0.01 kelvin to 2000 kelvins (-273.14 °C to 1700 °C).

Most PTC thermistors are of the "switching" type, which means that their resistance rises suddenly at a certain critical temperature. The devices are made of a doped polycrystalline ceramic containing barium titanate (BaTiO3) and other compounds. The dielectric constant of this ferroelectric material varies with temperature. Below the Curie point temperature, the high dielectric constant prevents the formation of potential barriers between the crystal grains, leading to a low resistance. In this region the device has a small negative temperature coefficient. At the Curie point temperature, the dielectric constant drops sufficiently to allow the formation of potential barriers at the grain boundaries, and the resistance increases sharply. At even higher temperatures, the material reverts to NTC behaviour. The equations used for modelling this behaviour were worked by W. Heywang and G. H. Jonker in the 1960s.

Another type of PTC thermistor is the polymer PTC, which is sold under brand names such as "Polyfuse", "Polyswitch" and "Multiswitch". This consists of a slice of plastic with carbon grains embedded in it. When the plastic is cool, the carbon grains are all in contact with each other, forming a conductive path through the device. When the plastic heats up, it expands, forcing the carbon grains apart, and causing the resistance of the device to rise rapidly. Like the BaTiO3 thermistor, this device has a highly nonlinear resistance/temperature response and is used for switching, not for proportional temperature measurement.

## Applications

• Thermistors are used in smart camera flash guns which adjust for proper film exposure according to the light reflected.
• PTC thermistors can be used as current-limiting devices for circuit protection, as replacements for fuses. Current through the device causes a small amount of resistive heating. If the current is large enough to generate more heat than the device can lose to its surroundings, the device heats up, causing its resistance to increase, and therefore causing even more heating. This creates a self-reinforcing effect that drives the resistance upwards, reducing the current and voltage available to the device.
• NTC thermistors can be used as inrush-current limiting devices in power supply circuits. They present a higher resistance initially which prevents large currents from flowing at turn-on, and then heat up and become much lower resistance to allow higher current flow during normal operation. These thermistors are usually much larger than measuring type thermistors, and are purpose designed for this application.

## References

I.S. Steinhart & S.R. Hart in "Deep Sea Research" vol. 15 p. 497 (1968) - in which the Steinhart-Hart equation was first published.

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