The Steinhart–Hart equation is a model of the
resistance of a
semiconductor at different
temperatures. The equation is
:
where
:
is the temperature (in
kelvins),
:
is the resistance at
(in ohms),
:
,
, and
are the Steinhart–Hart coefficients, which vary depending on the type and model of
thermistor and the temperature range of interest.
Uses of the equation
The equation is often used to derive a precise temperature of a thermistor, since it provides a closer approximation to actual temperature than simpler equations, and is useful over the entire working temperature range of the sensor. Steinhart–Hart coefficients are usually published by thermistor manufacturers.
Where Steinhart–Hart coefficients are not available, they can be derived. Three accurate measures of resistance are made at precise temperatures, then the coefficients are derived by solving 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 ...
.
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):
:
Developers of the equation
The equation is named after John S. Steinhart and
Stanley R. Hart who first published the relationship 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, .] Professor Steinhart (1929–2003), a fellow of the
American Geophysical Union
The American Geophysical Union (AGU) is a 501(c)(3) nonprofit organization of Earth, atmospheric, ocean, hydrologic, space, and planetary scientists and enthusiasts that according to their website includes 130,000 people (not members). AGU's acti ...
and of the
American Association for the Advancement of Science
The American Association for the Advancement of Science (AAAS) is an American international non-profit organization with the stated goals of promoting cooperation among scientists, defending scientific freedom, encouraging scientific responsi ...
, was a member of the faculty of
University of Wisconsin–Madison
A university () is an institution of higher (or tertiary) education and research which awards academic degrees in several academic disciplines. Universities typically offer both undergraduate and postgraduate programs. In the United States, the ...
from 1969 to 1991. Dr. Hart, a Senior Scientist at
Woods Hole Oceanographic Institution since 1989 and fellow of the
Geological Society of America, the American Geophysical Union, the
Geochemical Society
The Geochemical Society is a nonprofit scientific organization founded to encourage the application of chemistry to solve problems involving geology and cosmology. The society promotes understanding of geochemistry through the annual Goldschmidt Co ...
and the
European Association of Geochemistry The European Association of Geochemistry (EAG) is a pan-European organization founded to promotes geochemical research. The EAG organizes conferences, meetings and educational courses for geochemists in Europe, including the Goldschmidt Conference w ...
, was associated with Professor Steinhart at the
Carnegie Institution of Washington
The Carnegie Institution of Washington (the organization's legal name), known also for public purposes as the Carnegie Institution for Science (CIS), is an organization in the United States established to fund and perform scientific research. Th ...
when the equation was developed.
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 modelled as an ''n''th degree polynomial in ''x''. Polynomial regression fi ...
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