Steinhart–Hart equation
The Steinhart–Hart equation is a model of the resistance of a semiconductor at different temperatures. The equation is where 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 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". 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 formula_11, formula_12 and formula_13 values of resistance at the temperatures formula_14, formula_15 and formula_16, one can express formula_5, formula_6 and formula_7 (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. 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. 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, was associated with Professor Steinhart at the Carnegie Institution of Washington  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: formula_24 is a reference (standard) resistance value. The Steinhart–Hart equation assumes formula_25 is 1 ohm. The curve fit is much less accurate when it is assumed formula_26 and a different value of formula_24 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 formula_28 degraded the fit. This  is surprising as allowing more freedom would usually improve the fit. It may be because the authors fitted formula_29 instead of formula_2, and thus the error in formula_2 increased from the extra freedom. Subsequent papers have found great benefit in allowing formula_28. 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 formula_33 has been found to be accurate  within 1 mK over the calibrated range. Some authors recommend using formula_34. If there are many data points, standard polynomial regression can also generate accurate curve fits. Some manufacturers have begun providing regression coefficients as an alternative to Steinhart–Hart coefficients.