Introduction to Thermister

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A thermistor is a type of resistor whose resistance varies significantly with temperature, more so than in standard resistors. The word is a portmanteau of thermal and resistor. Thermistors are widely used as inrush current limiters, temperature sensors, self-resetting overcurrent protectors, and self-regulating heating elements.

Thermistors differ from resistance temperature detectors (RTD) in that the material used in a thermistor is generally a ceramic or polymer, while RTDs use pure metals. The temperature response is also different; RTDs are useful over larger temperature ranges, while thermistors typically achieve a higher precision within a limited temperature range, typically ?90 °C to 130 °C.[1]

Basic operation

Assuming, as a first-order approximation, that the relationship between resistance and temperature is linear, then:


\Delta R=k\Delta T \,


\Delta R = change in resistance
\Delta T = 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, or 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 a k as close to zero as possible, so that their resistance remains nearly constant over a wide temperature range.

Instead of the temperature coefficient k, sometimes the temperature coefficient of resistance \alpha_T (alpha sub T) is used. It is defined as[2]

\alpha_T = \frac{1}{R(T)} \frac{dR}{dT}.

This \alpha_T coefficient should not be confused with the a parameter below.

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:


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

R=e^{{\left( x-{y \over 2} \right)}^{1\over 3}-{\left( x+{y \over 2} \right)}^{1\over 3}}


y={{a-{1\over T}}\over c} and x=\sqrt{{{{\left({b\over{3c}}\right)}^3}+{{y^2}\over 4}}}

The error in the Steinhart–Hart equation is generally less than 0.02 °C in the measurement of temperature over a 200 °C range.[3] As an example, typical values for a thermistor with a resistance of 3000 ? at room temperature (25 °C = 298.15 K) are:

a = 1.40 \times 10^{-3}

b = 2.37 \times 10^{-4}

c = 9.90 \times 10^{-8}

B or ? parameter equation

NTC thermistors can also be characterised with the B (or ?) parameter equation, which is essentially the Steinhart Hart equation with a = (1/T_{0}) - (1/B) \ln(R_{0}), b = 1/B and c = 0,

\frac{1}{T}=\frac{1}{T_0} + \frac{1}{B}\ln \left(\frac{R}{R_0}\right)

Where the temperatures are in kelvins and R0 is the resistance at temperature T0 (25 °C = 298.15 K). Solving for R yields:

R=R_0e^{B(\frac{1}{T} - \frac{1}{T_0})}

or, alternatively,

R=r_\infty e^{B/T}

where r_\infty=R_0 e^{-{B/T_0}}.

This can be solved for the temperature:

T={B\over { {\ln{(R / r_\infty)}}}}

The B-parameter equation can also be written as \ln R=B/T + \ln r_\infty. This can be used to convert the function of resistance vs. temperature of a thermistor into a linear function of \ln R vs. 1/T. The average slope of this function will then yield an estimate of the value of the B parameter....
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