The Steinhart–Hart equation is a model relating the varying electrical resistance of a semiconductor to its varying temperatures. The equation is
1 T = A + B ln R + C ( ln R ) 3 , {\displaystyle {\frac {1}{T}}=A+B\ln R+C(\ln R)^{3},}
where
T {\displaystyle T} is the temperature (in kelvins),
R {\displaystyle R} is the resistance at T {\displaystyle T} (in ohms),
A {\displaystyle A} , B {\displaystyle B} , and C {\displaystyle C} 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 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 procedure applied to three measurements at various known temperatures. Given the three temperature-resistance observations, the coefficients are solved from three simultaneous equations.
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 the Application Note, "A, B, C Coefficients for Steinhart–Hart Equation".
R = exp ( y − x / 2 3 − y + x / 2 3 ) , {\displaystyle R=\exp \left({\sqrt[{3}]{y-x/2}}-{\sqrt[{3}]{y+x/2}}\right),}
where
x = 1 C ( A − 1 T ) , y = ( B 3 C ) 3 + x 2 4 . {\displaystyle {\begin{aligned}x&={\frac {1}{C}}\left(A-{\frac {1}{T}}\right),\\y&={\sqrt {\left({\frac {B}{3C}}\right)^{3}+{\frac {x^{2}}{4}}}}.\end{aligned}}}
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.
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