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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 : \frac = A + B \ln R + C (\ln R)^3, where : T 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), : R is the resistance at T (in ohms), : A, B, and 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 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". : R = \exp\left(\sqrt - \sqrt right), where : \begin x &= \frac\left(A - \frac\right), \\ y &= \sqrt. \end


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. : \begin 1 & \ln R_1 & \ln^3 R_1 \\ 1 & \ln R_2 & \ln^3 R_2 \\ 1 & \ln R_3 & \ln^3 R_3 \end\begin A \\ B \\ C \end = \begin \frac \\ \frac \\ \frac \end With R_1, R_2 and R_3 values of resistance at the temperatures T_1, T_2 and T_3, one can express A, B and C (all calculations): :\begin L_1 &= \ln R_1, \quad L_2 = \ln R_2, \quad L_3 = \ln R_3 \\ Y_1 &= \frac, \quad Y_2 = \frac, \quad Y_3 = \frac \\ \gamma_2 &= \frac, \quad \gamma_3 = \frac \\ \Rightarrow C &= \left( \frac \right) \left(L_1 + L_2 + L_3\right)^ \\ \Rightarrow B &= \gamma_2 - C \left(L_1^2 + L_1 L_2 + L_2^2\right) \\ \Rightarrow A &= Y_1 - \left(B + L_1^2 C\right) L_1 \end


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: : R = R_0 e^ : \frac = \frac + \frac \left(\ln \frac\right) = a_0 + a_1 \ln \frac : \frac = \sum_^\infty a_n \left(\ln \frac\right)^n R_0 is a reference (standard) resistance value. The Steinhart–Hart equation assumes R_0 is 1 ohm. The curve fit is much less accurate when it is assumed a_2=0 and a different value of R_0 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 a_2 \neq 0 degraded the fit. This is surprising as allowing more freedom would usually improve the fit. It may be because the authors fitted 1/T instead of T, and thus the error in T increased from the extra freedom. Subsequent papers have found great benefit in allowing a_2 \neq 0. 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 n=4 has been found to be accurate within 1 mK over the calibrated range. Some authors recommend using n=5. 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 OnlineSteinhart-Hart Coefficient Calculator Java
{{DEFAULTSORT:Steinhart-Hart equation Semiconductors