thermistor - Steinhart Hart equation, B parameter equation, Conduction model, Self-heating effects

A temperature sensor constructed from semiconductor material whose electrical resistance falls rapidly as the temperature rises; an abbreviation of thermal resistor. It is made from cobalt, nickel, and magnesium oxides mixed with finely divided copper. Thermistors are used in electronic circuits measuring or controlling temperature, in wave-guides to measure the transmitted power, and in time-delay circuits.

A thermistor is a type of resistor used to measure temperature changes, relying on the change in its resistance with changing temperature.

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:

ΔR = kΔT

where

ΔR = change in resistance ΔT = change in temperature k = first-order temperature coefficient of resistance

Thermistors can be classified into two types depending on the sign of k. If k is positive, the resistance increases with increasing temperature, and the device is called a positive temperature coefficient (PTC) thermistor, Posistor. If k 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 k, so that their resistance remains almost constant over a wide temperature range.

Steinhart Hart equation

In practice, the linear approximation (above) works only over a small temperature range. To give resistance as a function of temperature, the above can be rearranged into:

where

and

The error in the Steinhart-Hart equation is generally less than 0.02°C in the measurement of temperature. As an example, typical values for a thermistor with a resistance of 3000 Ω at room temperature (25°C = 298.15 K) are:

B parameter equation

NTC thermistors can also be characterised with the B parameter equation, which is essentially the Steinhart Hart equation with c=0. Using the expansion only to the first order yields:

or

or

where

R0 is the resistance at temperature T0 (usually 25 °C=298.15 K)

Conduction model

Many NTC thermistors are made from a pressed disc or cast chip of a semiconductor such as a sintered metal oxide. This is described in the formula:

I = electric current (ampere)
n = density of charge carriers (count/m³)
A = cross-sectional area of the material (m²)
v = velocity of charge carriers (m/s)
e = charge of an electron ( coulomb)

The current is measured using an ammeter. Over small changes in temperature, if the right semiconductor is used, the resistance of the material is linearly proportional to the temperature.

Most PTC thermistors are of the "switching" type, which means that their resistance rises suddenly at a certain critical 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. 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. Like the BaTiO3 thermistor, this device has a highly nonlinear resistance/temperature response and is used for switching, not for proportional temperature measurement.

Self-heating effects

When a current flows through a thermistor, it will generate heat which will raise the temperature of the thermistor above that of its environment. If the thermistor is being used to measure the temperature of the environment, this self-heating effect will introduce an error if a correction is not made.

The electrical power input to the thermistor is just

where I is current and V is the voltage drop across the thermistor. The rate of transfer is well described by Newton's law of cooling:

where T(R) is the temperature of the thermistor as a function of its resistance R, T0 is the temperature of the surroundings, and K is the dissipation constant, usually expressed in units of milliwatts per °C. As a simple example, if the voltage across the thermistor is held fixed, then by Ohm's Law we have I = V / R and the equilibrium equation can be solved for the ambient temperature as a function of the measured resistance of the thermistor:

The dissipation constant is a measure of the thermal connection of the thermistor to its surroundings. If the temperature of the environment is known beforehand, then a thermistor may be used to measure the value of the dissipation constant. For example, the thermistor may be used as a flow rate sensor, since the dissipation constant increases with the rate of flow of a fluid past the thermistor. Thermistors are also commonly used in modern digital thermostats and to monitor the temperature of battery packs while charging.