Electronic – Difference between standard and extended Steinhart-Hart equation

thermistor

I looked up the data table for a thermistor I purchased and noticed it uses an 'extended' Steinhart-Hart equation. I'd like to understand the difference between these equations, such as how the D coefficient is acquired, whether more coefficients add accuracy, and so on.

The standard Steinhart-Hart equation I had seen is:

standard Steinhart-Hart eqn

On the thermistor data sheet, page 4 includes the 'extended' Steinhart-Hart equation:

extended Steinhart-Hart equation

Is the inclusion of a D coefficient here simply an instance of resistance data having been measured for a third temperature? Does the inclusion of an additional coefficient relate in some way to the material properties of the thermistor? Or did the manufacturer simply choose to include it for further accuracy?

Best Answer

Actually the general form of the Steinhart-Hart equation is an infinite sum (see German Wikipedia article of Steinhart-Hart equation):

\$\frac{1}{T} = \sum_{k=0}^{\infty} c_k \ln^k \frac{R}{R_{ref}} \$

In most practical cases it is enough to approximate by using only 3 of the first 4 coefficients \$c_0, c_1, c_3\$ (in your case called \$A, B, C\$) and throw away the rest (as \$c_2\$ often is small enough to neglect and the terms with higher powers of \$\ln \frac{R}{R_{ref}}\$ become less and less significant).

In your case, obviously, in the "extended" equation just some extra effort was taken to also use coefficient \$c_2\$ in order to get a better approximation.

Probably the coefficients are obtained not only by measuing at a 4th temperature but by doing many (>4) measurements at many temperatures in the interesting range and doing least-square curve fitting using a 4th order polynomial as model function.