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The '''Steinhart–Hart equation''' is a model of the [[Electrical resistance|resistance]] of a [[semiconductor]] at different [[temperature]]s. The equation is:

:<math>{1 \over T} = A + B \ln(R) + C (\ln(R))^3 \,</math>

where:
* <math>T</math> is the temperature (in kelvins)
* ''R'' is the resistance at ''T'' (in ohms)
* <math>A</math>, <math>B</math>, and <math>C</math> are the '''Steinhart–Hart coefficients''' which vary depending on the type and model of [[thermistor]] and the temperature range of interest. (The most general form of the applied equation contains a <math>(\ln(R))^2</math> term, but this is frequently neglected because it is typically much smaller than the other coefficients, and is therefore not shown above.)

==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]].

==Inverse of the equation==
To find the resistance of a semiconductor given the temperature the inverse of the Steinhart–Hart equation must be used. See the [http://www.cornerstonesensors.com/reports/ABC%20Coefficients%20for%20Steinhart-Hart%20Equation.pdf Application Note], "A, B, C Coefficients for Steinhart–Hart Equation".
:<math>R = \exp\left(\sqrt[3]{x - y} - \sqrt[3]{x + y}\right),</math>
where
:<math>y = {A - {1 \over T} \over 2C},</math>
:<math>x = \sqrt{\left({B \over 3C}\right)^3 + y^2}.</math>

==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.
:<math>\begin{cases} A + \left(\ln R_1 \right) B + \left(\ln R_1 \right)^3 C=\frac{1}{T_1} \\ A + \left(\ln R_2 \right) B + \left(\ln R_2 \right)^3 C = \frac{1}{T_2} \\ A + \left(\ln R_3 \right) B + \left(\ln R_3 \right)^3 C = \frac{1}{T_3} \end{cases}</math>

With <math>R_1</math>, <math>R_2</math> and <math>R_3</math> values of resistance at the temperatures <math>T_1</math>, <math>T_2</math> and <math>T_3</math>, one can express <math>A</math>, <math>B</math> and <math>C</math> (all calculations):

<math>L_1 = \ln\left(R_1\right)</math>,   <math>L_2=\ln\left(R_2\right)</math>   and   <math>L_3=\ln\left(R_3\right)</math>,

<math>Y_1=\frac{1}{T_1}</math>,  <math>Y_2=\frac{1}{T_2}</math>   and   <math>Y_3=\frac{1}{T_3}</math>,

<math>\gamma_2=\frac{Y_2-Y_1}{L_2-L_1}</math>, <math>\gamma_3=\frac{Y_3-Y_1}{L_3-L_1}</math>

:<math>\Rightarrow C=\left( \frac{ \gamma_3 - \gamma_2 }{ L_3 - L_2} \right) \left(L_1 + L_2 + L_3\right)^{-1}</math>

:<math>\Rightarrow B=\gamma_2 - C \left(L_1^2+L_1 L_2+L_2^2\right)</math>

:<math>\Rightarrow A=Y_1 - \left(B+L_1^2 C\right) L_1</math>

==Developers of the equation==
The equation is named after [[John S. Steinhart]] and [[Stanley R. Hart]] who first published the relationship in 1968.<ref>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, {{doi|10.1016/0011-7471(68)90057-0}}.</ref> Professor Steinhart (1929–2003), a fellow of the [[American Geophysical Union]] and of the [[American Association for the Advancement of Science]], was a member of the faculty of [[University of Wisconsin–Madison]] from 1969 to 1991.[http://www.secfac.wisc.edu/senate/2004/0405/1775(mem_res).pdf] 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]] and the [[European Association of Geochemistry]], [http://www.whoi.edu/science/GG/people/shart/cv.htm] was associated with Professor Steinhart at the [[Carnegie Institution of Washington]] when the equation was developed.

==References==
<references/>

{{DEFAULTSORT:Steinhart-Hart equation}}
[[Category:Condensed matter physics]]

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