Steinhart–Hart equation

When applying a thermistor device to measure temperature, the equation relates a measured resistance to the device temperature, or vice versa.

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.

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.

To find the resistance of a semiconductor at a given temperature, the inverse of the Steinhart–Hart equation must be used. See the [https://web.archive.org/web/20110708192840/http://www.cornerstonesensors.com/reports/ABC%20Coefficients%20for%20Steinhart-Hart%20Equation.pdf Application Note], "A, B, C Coefficients for Steinhart–Hart Equation".

: $R\; =\; \backslash exp\backslash left(\backslash sqrt[3]\{y\; -\; x/2\}\; -\; \backslash sqrt[3]\{y\; +\; x/2\}\backslash right),$

where

: $\backslash begin\{align\}$

x &= \frac{1}{C}\left(A - \frac{1}{T}\right), \\

y &= \sqrt{\left(\frac{B}{3C}\right)^3 + \frac{x^2}{4}}.

\end{align}

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.

: $\backslash begin\{bmatrix\}$

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):

:$\backslash begin\{align\}$

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}

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, {{doi|10.1016/0011-7471(68)90057-0}}.

The most general form of the equation can be derived from extending the B parameter equation to an infinite series:

: $R\; =\; R\_0\; e^\{B\backslash left(\backslash frac\{1\}\{T\}\; -\; \backslash frac\{1\}\{T\_0\}\backslash right)\}$

: $\backslash frac\{1\}\{T\}\; =\; \backslash frac\{1\}\{T\_0\}\; +\; \backslash frac\{1\}\{B\}\; \backslash left(\backslash ln\; \backslash frac\{R\}\{R\_0\}\backslash right)\; =\; a\_0\; +\; a\_1\; \backslash ln\; \backslash frac\{R\}\{R\_0\}$

: $\backslash frac\{1\}\{T\}\; =\; \backslash sum\_\{n=0\}^\backslash infty\; a\_n\; \backslash left(\backslash ln\; \backslash frac\{R\}\{R\_0\}\backslash right)^n$

$R\_0$ 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.{{cite conference |last1=Matus |first1=Michael |title=Temperature Measurement in Dimensional Metrology – Why the Steinhart–Hart Equation works so well |url=https://oar.ptb.de/files/download/810.20130620D.pdf | conference=MacroScale 2011 |location=Wabern, Switzerland |date=October 2011}}

In the original paper, Steinhart and Hart remark that allowing $a\_2\; \backslash neq\; 0$ 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.{{cite journal |last1=Hoge |first1=Harold J. |title=Useful procedure in least squares, and tests of some equations for thermistors |journal=Review of Scientific Instruments |date=1 June 1988 |volume=59 |issue=6 |pages=975–979 |doi=10.1063/1.1139762 |url=https://aip.scitation.org/doi/10.1063/1.1139762 |issn=0034-6748|url-access=subscription }} Subsequent papers have found great benefit in allowing $a\_2\; \backslash neq\; 0$.

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 $n=5$.{{cite journal |last1=Rudtsch |first1=Steffen |last2=von Rohden |first2=Christoph |title=Calibration and self-validation of thermistors for high-precision temperature measurements |journal=Measurement |date=1 December 2015 |volume=76 |pages=1–6 |doi=10.1016/j.measurement.2015.07.028 |url=https://www.sciencedirect.com/science/article/pii/S0263224115003632 |accessdate=8 July 2020 |language=en |issn=0263-2241|url-access=subscription }} 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.{{cite web |title=Comments on the Steinhart–Hart Equation |url=http://www.bapihvac.com/wp-content/uploads/SHH_Equation_Comments.pdf |publisher=Building Automation Products Inc. |accessdate=8 July 2020 |date=11 November 2015}}

{{Reflist}}

• [https://www.thinksrs.com/downloads/programs/therm%20calc/ntccalibrator/ntccalculator.html Steinhart-Hart Coefficient Calculator Online]
• [https://github.com/Sanjit1/CalibratorJava/releases Steinhart-Hart Coefficient Calculator Java]

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Category:Semiconductors