# Temperature Resistance Equation

Unlike RTDs, whose resistance is linearly proportional to absolute temperature, the resistance of a thermistor changes dramatically and sensitively with temperature, satisfying an exponential relationship. To convert the measured resistance to temperature, you can use a chart, or a portion of it, as a lookup table in your software and interpolate between adjacent degrees, or you can use an equation. The thermistor’s resistance varies exponentially with the inverse of absolute temperature, approximately as,

where the absolute temperature, *T _{abs}*, is in Kelvin. The values of

*α*and

*β*depend on the thermistor used.

We’ll use as an example a Yellow Springs Type YSI 400 thermistor. This particular thermistor has a calibration point resistance of *R _{o}* = 2252 Ω at

*T*= 25 °C. Its characteristic is available in chart form at http://www.advindsys.com/ApNotes/YSI400SeriesProbesRvsT.htm. Transforming the above equation to a ratio of thermistor resistance

_{o}*R*at temperature

_{TH}*T*, to calibration resistance

*R*at a reference temperature

_{o}*T*, and adding the absolute temperature offset of 273.15 Kelvin, cancels the coefficient

_{o}*α*and produces,

where *R _{TH}* is the resistance at a temperature

*T*(now measured in °C) and

*R*is the calibration resistance (2252 ohms) at the calibration temperature,

_{o}*T*, of 25 °C.

_{o}The *β* in the equation is a parameter that represents the semiconductor's temperature slope. If the above equation is fitted to the chart data at the two temperatures of 25 °C and 45 °C to find the best slope between those two points, then the equation fits just about perfectly for 25 °C through 45 °C, and fits within ±0.1 °C for the 15–55 °C range, using *β* = 3930.

Because *R _{o}* and

*T*are constants, we can group them together in a single coefficient

_{o}*r*(this combined coefficient happens to be the limit of thermistor resistance as temperature goes to infinity) for a simpler computation,

_{∞}Inverting the equation produces the equation you could use in an instrument to convert the measured resistance value to temperature in °C,

or using the specified *β* and *R _{o}* values for the YSI 400 thermistor,

The absolute accuracy of thermistors is whatever you pay for. They range in price from pennies each for surface-mount uncalibrated thermistors (calibrated to only ± a few °C) to ten dollars or more for thermistor beads hand sorted for accuracy, such as the US Sensor "PR" series at ±0.05 °C, to hundreds of dollars for custom probe assemblies. The YSI probes are specified at ±0.2 °C from -1 to 60 °C, and ±0.1 °C from 25 to 45 °C. That's very, very good!

Highly accurate thermistors or an application involving a wide temperature range may result in the exponential equation above introducing greater errors than the inherent accuracy of the thermistor, necessitating a more complex model of the relationship between resistance and temperature. A commonly used formula to fit thermistor data uses a fifth order polynomial in the logarithm of the resistance, as,

The parameters, *a*, *b*, *c*, and *d*, are found by curve fitting the equation to the thermistor data over the temperature range of interest. Afterward, the above equation is used at run-time to compute temperature from the thermistor resistance.

Finally, if thermistors are used in a voltage-divider circuit as shown in Measuring Thermistors, the relationship between voltage and temperature is somewhat more linear than the resistance-temperature relationship of the thermistor. This means that a curve fit can be applied directly to expected voltage in analog-to-digital converter counts at a range of temperatures, as explained in Efficient Thermistor Temperature Computation.

See also → Temperature Resistance Equation