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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:
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| :<math>{1 \over T} = A + B \ln(R) + C (\ln(R))^3 \,</math>
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| where:
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| * <math>T</math> is the temperature (in kelvins)
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| * ''R'' is the resistance at ''T'' (in ohms)
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| * <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.)
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| ==Uses of the equation==
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| 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.
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| 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]].
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| ==Inverse of the equation==
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| 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".
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| :<math>R = \exp\left(\sqrt[3]{x - y} - \sqrt[3]{x + y}\right),</math>
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| where
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| :<math>y = {A - {1 \over T} \over 2C},</math>
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| :<math>x = \sqrt{\left({B \over 3C}\right)^3 + y^2}.</math>
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| ==Steinhart–Hart coefficients==
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| 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.
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| :<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>
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| 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):
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| <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>,
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| <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>,
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| <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>
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| :<math>\Rightarrow C=\left( \frac{ \gamma_3 - \gamma_2 }{ L_3 - L_2} \right) \left(L_1 + L_2 + L_3\right)^{-1}</math>
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| :<math>\Rightarrow B=\gamma_2 - C \left(L_1^2+L_1 L_2+L_2^2\right)</math>
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| :<math>\Rightarrow A=Y_1 - \left(B+L_1^2 C\right) L_1</math>
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| ==Developers of the equation==
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| 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.
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| ==References==
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| <references/>
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| {{DEFAULTSORT:Steinhart-Hart equation}}
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| [[Category:Condensed matter physics]]
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