Steinhart–Hart equation

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\left(\sqrt[3]{y - x/2} - \sqrt[3]{y + x/2}\right), where : \begin{align} x &= \frac{1}{C}\left(A - \frac{1}{T}\right), \ y &= \sqrt{\left(\frac{B}{3C}\right)^3 + \frac{x^2}{4}}. \end{align}

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{bmatrix} 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{bmatrix}\begin{bmatrix} A \ B \ C \end{bmatrix} = \begin{bmatrix} \frac{1}{T_1} \ \frac{1}{T_2} \ \frac{1}{T_3} \end{bmatrix}

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{align} L_1 &= \ln R_1, \quad L_2 = \ln R_2, \quad L_3 = \ln R_3 \ Y_1 &= \frac{1}{T_1}, \quad Y_2 = \frac{1}{T_2}, \quad Y_3 = \frac{1}{T_3} \ \gamma_2 &= \frac{Y_2 - Y_1}{L_2 - L_1}, \quad \gamma_3 = \frac{Y_3 - Y_1}{L_3 - L_1} \ \Rightarrow C &= \left( \frac{ \gamma_3 - \gamma_2 }{ L_3 - L_2} \right) \left(L_1 + L_2 + L_3\right)^{-1} \ \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{align}

History

The equation was developed by John S. Steinhart and Stanley R. Hart, who first published it in 1968.

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^{B\left(\frac{1}{T} - \frac{1}{T_0}\right)} : \frac{1}{T} = \frac{1}{T_0} + \frac{1}{B} \left(\ln \frac{R}{R_0}\right) = a_0 + a_1 \ln \frac{R}{R_0} : \frac{1}{T} = \sum_{n=0}^\infty a_n \left(\ln \frac{R}{R_0}\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. 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. 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.

References

References

1. 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.
2. (October 2011). "Temperature Measurement in Dimensional Metrology – Why the Steinhart–Hart Equation works so well".
3. (1 June 1988). "Useful procedure in least squares, and tests of some equations for thermistors". Review of Scientific Instruments.
4. (11 November 2015). "Comments on the Steinhart–Hart Equation". Building Automation Products Inc..