Steinhart-Hart equation

The stone-Hart equation describes the change of electric resistance with a semiconductor , according to a temperature change and is used, inter alia in the field of electrical measuring equipment for measuring temperature using thermistors (NTC) which is one type of thermistors represent. The formulation of the equation goes back to John S. Steinhart and Stanley R. Hart , who first published this connection in 1968.

General

The general form of the Steinhart-Hart equation, which expresses the temperature as a function of the measured electrical resistance , has the following form: ${\displaystyle T}$ ${\displaystyle R}$

${\displaystyle {{\frac {1}{\{T\}}}=\sum _{i=0}^{\infty }a_{i}\ln ^{i}(\{R\})=a_{0}+a_{1}\ln(\{R\})+a_{2}\ln ^{2}(\{R\})+a_{3}\ln ^{3}(\{R\})+a_{4}\ln ^{4}(\{R\})+\ldots }}$

with the coefficients depending on the thermistor material . In this numerical value equation , the numerical value of the temperature in the unit Kelvin and the numerical value of the electrical resistance in the unit ohms must be inserted. ${\displaystyle a_{i}}$ ${\displaystyle T}$ ${\displaystyle R}$

For practical applications, only the terms with the coefficients , and have a meaning; the remaining coefficients are so small in relation that they can be neglected for practical applications in the field of measurement technology in the measuring range from 0 ° C to +70 ° C. This reduces the Steinhart-Hart equation for practical applications to the simplified form: ${\displaystyle a_{0}}$ ${\displaystyle a_{1}}$ ${\displaystyle a_{3}}$

${\displaystyle {\frac {1}{\{T\}}}=a_{0}+a_{1}\ln(\{R\})+a_{3}\ln ^{3}(\{R\})}$

The three coefficients , and are referred to as Steinhart-Hart coefficients and are tabulated for different, commercially available types of NTC thermistor resistors in their technical data sheets. If the Steinhart – Hart coefficients are unknown, they can be determined by measuring three points spread over the measuring range. ${\displaystyle a_{0}}$ ${\displaystyle a_{1}}$ ${\displaystyle a_{3}}$

Inverse equation

The Steinhart-Hart equation, which is inverse for practical measurements, provides the electrical resistance for a given temperature value , with the following relationship: ${\displaystyle T}$ ${\displaystyle R}$

${\displaystyle \{R\}=\exp \left({\sqrt[{3}]{x-y}}-{\sqrt[{3}]{x+y}}\right)}$

With

${\displaystyle y={a_{0}-{1 \over \{T\}} \over 2a_{3}}}$

and

${\displaystyle x={\sqrt {\left({a_{1} \over 3a_{3}}\right)^{3}+y^{2}}}}$

Individual evidence

1. ^ John S. Steinhart, Stanley R. Hart: Calibration curves for thermistors . tape 15 , no. 4 . Deep Sea Research and Oceanographic Abstracts, 1968, ISSN  0011-7471 , pp. 497-503 , doi : 10.1016 / 0011-7471 (68) 90057-0 .
2. ↑ Comments on the Steinhart-Hart Equation. BAPI, company publication, accessed on September 28, 2014 . 
3. ↑ Comments on the Steinhart-Hart Equation. BAPI, company publication, accessed on September 28, 2014 (page 4, table values ​​for NTCs from 1.8kΩ to 100kΩ). 
4. ^ Coefficients for Steinhart – Hart Equation. Archived from the original on March 3, 2012 ; accessed on September 28, 2014 .