NTC Thermistors Steinhart and Hart Equation

The Steinhart and Hart equation is an empirical expression that has been determined to be the best mathematical expression for resistance temperature relationship of NTC thermistors and NTC probe assemblies.

The most common equation is:

1/T= a+ b(LnR)+c(LnR)^3  (Eqn 5)
where: T = degrees Kelvin
a,b, and c = coefficients derived from measurement

To solve for a, b and c coefficients, measure the thermistor at three different temperatures. The temperatures should be evenly spaced and atleast 10 degrees apart. Use the three temperatures to solve three simultaneous equations.

1/T1 = a +b(LnR1)+c(LnR1)^3
1/T2 = a +b(LnR2)+c(LnR2)^3
1/T1 = a +b(LnR3)+c(LnR3)^3

These equations allow you to derive a, b and c for any temperature range. Knowing a, b and c for the thermistor allows you to use the Steinhart and Hart equation in two ways.

1) If resistance is known and temperature desired then use eqn 5.
2) If temperature is known and expected resistance is desired than use this equation:

R= e exp [(ß – (ά/2))^1/3 – (ß + (ά/2))^1/3]
Where
ά=(a-(1/T))/c  and ß=[((b/3c)^3) + ((ά^2)/4)]^1/2