The Steinhart–Hart equation is a model relating the varying electrical
resistance of a
semiconductor
A semiconductor is a material with electrical conductivity between that of a conductor and an insulator. Its conductivity can be modified by adding impurities (" doping") to its crystal structure. When two regions with different doping level ...
to its varying
temperature
Temperature is a physical quantity that quantitatively expresses the attribute of hotness or coldness. Temperature is measurement, measured with a thermometer. It reflects the average kinetic energy of the vibrating and colliding atoms making ...
s. The equation is
:
where
:
is the temperature (in
kelvin
The kelvin (symbol: K) is the base unit for temperature in the International System of Units (SI). The Kelvin scale is an absolute temperature scale that starts at the lowest possible temperature (absolute zero), taken to be 0 K. By de ...
s),
:
is the resistance at
(in ohms),
:
,
, and
are the Steinhart–Hart coefficients, which are characteristics specific to the bulk semiconductor material over a given temperature range of interest.
Application
When applying a
thermistor
A thermistor is a semiconductor type of resistor in which the resistance is strongly dependent on temperature. The word ''thermistor'' is a portmanteau of ''thermal'' and ''resistor''. The varying resistance with temperature allows these devices ...
device to measure temperature, the equation relates a measured resistance to the device temperature, or vice versa.
Finding temperature from resistance and characteristics
The equation model converts the resistance actually measured in a thermistor to its theoretical bulk temperature, with a closer approximation to actual temperature than simpler models, and valid over the entire working temperature range of the sensor. Steinhart–Hart coefficients for specific commercial devices are ordinarily reported by thermistor manufacturers as part of the device characteristics.
Finding characteristics from measurements of resistance at known temperatures
Conversely, when the three Steinhart–Hart coefficients of a specimen device are not known, they can be derived experimentally by a
curve fitting
Curve fitting is the process of constructing a curve, or mathematical function, that has the best fit to a series of data points, possibly subject to constraints. Curve fitting can involve either interpolation, where an exact fit to the data is ...
procedure applied to three measurements at various known temperatures. Given the three temperature-resistance observations, the coefficients are solved from three
simultaneous equations
In mathematics, a set of simultaneous equations, also known as a system of equations or an equation system, is a finite set of equations for which common solutions are sought. An equation system is usually classified in the same manner as single e ...
.
Inverse of the equation
To find the resistance of a semiconductor at a given temperature, the inverse of the Steinhart–Hart equation must be used. See th
Application Note "A, B, C Coefficients for Steinhart–Hart Equation".
:
where
:
Steinhart–Hart coefficients
To find the coefficients of Steinhart–Hart, we need to know at-least three operating points. For this, we use three values of resistance data for three known temperatures.
:
With
,
and
values of resistance at the temperatures
,
and
, one can express
,
and
(all calculations):
:
History
The equation was developed by John S. Steinhart and
Stanley R. Hart, who first published it in 1968.
[John S. Steinhart, Stanley R. Hart, Calibration curves for thermistors, Deep-Sea Research and Oceanographic Abstracts, Volume 15, Issue 4, August 1968, Pages 497–503, ISSN 0011-7471, .]
Derivation and alternatives
The most general form of the equation can be derived from extending the
B parameter equation to an infinite series:
:
:
:
is a reference (standard) resistance value. The Steinhart–Hart equation assumes
is 1 ohm. The curve fit is much less accurate when it is assumed
and a different value of
such as 1 kΩ is used. However, using the full set of coefficients avoids this problem as it simply results in shifted parameters.
In the original paper, Steinhart and Hart remark that allowing
degraded the fit.
[ This is surprising as allowing more freedom would usually improve the fit. It may be because the authors fitted instead of , and thus the error in increased from the extra freedom. Subsequent papers have found great benefit in allowing .][
The equation was developed through trial-and-error testing of numerous equations, and selected due to its simple form and good fit.][ However, in its original form, the Steinhart–Hart equation is not sufficiently accurate for modern scientific measurements. For interpolation using a small number of measurements, the series expansion with has been found to be accurate within 1 mK over the calibrated range. Some authors recommend using .] If there are many data points, standard polynomial regression
In statistics, polynomial regression is a form of regression analysis in which the relationship between the independent variable ''x'' and the dependent variable ''y'' is modeled as a polynomial in ''x''. Polynomial regression fits a nonlinear ...
can also generate accurate curve fits. Some manufacturers have begun providing regression coefficients as an alternative to Steinhart–Hart coefficients.
References
External links
Steinhart-Hart Coefficient Calculator Online
Steinhart-Hart Coefficient Calculator Java
{{DEFAULTSORT:Steinhart-Hart equation
Semiconductors