The Steinhart–Hart equation is a model relating the varying electrical resistance of a semiconductor to its varying temperatures. The equation is
1 | |
T |
=A+BlnR+C(lnR)3,
T
R
T
A
B
C
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 Application Note, "A, B, C Coefficients for Steinhart–Hart Equation".
R=\exp\left(\sqrt[3]{y-x/2}-\sqrt[3]{y+x/2}\right),
\begin{align} x&=
1 | |
C |
\left(A-
1 | |
T |
\right),\\ y&=\sqrt{\left(
B | |
3C |
\right)3+
x2 | |
4 |
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&lnR1&ln3R1\\ 1&lnR2&ln3R2\\ 1&lnR3&ln3R3 \end{bmatrix}\begin{bmatrix} A\\ B\\ C \end{bmatrix}=\begin{bmatrix}
1 | |
T1 |
\\
1 | |
T2 |
\\
1 | |
T3 |
\end{bmatrix}
With
R1
R2
R3
T1
T2
T3
A
B
C
\begin{align} L1&=lnR1, L2=lnR2, L3=lnR3\\ Y1&=
1 | |
T1 |
, Y2=
1 | |
T2 |
, Y3=
1 | |
T3 |
\\ \gamma2&=
Y2-Y1 | |
L2-L1 |
, \gamma3=
Y3-Y1 | |
L3-L1 |
\\ ⇒ C&=\left(
\gamma3-\gamma2 | |
L3-L2 |
\right)\left(L1+L2+
-1 | |
L | |
3\right) |
\\ ⇒ B&=\gamma2-C
2 | |
\left(L | |
1 |
+L1L2+
2\right) | |
L | |
2 |
\\ ⇒ A&=Y1-\left(B+
2 | |
L | |
1 |
C\right)L1 \end{align}
The equation was developed by John S. Steinhart and Stanley R. Hart, who first published it in 1968.[1]
The most general form of the equation can be derived from extending the B parameter equation to an infinite series:
R=R0
| ||||||||
e |
1 | |
T |
=
1 | |
T0 |
+
1 | |
B |
\left(ln
R | |
R0 |
\right)=a0+a1ln
R | |
R0 |
1 | |
T |
=
infty | |
\sum | |
n=0 |
an\left(ln
R | |
R0 |
\right)n
R0
R0
a2=0
R0
In the original paper, Steinhart and Hart remark that allowing
a2 ≠ 0
1/T
T
T
a2 ≠ 0
The equation was developed through trial-and-error testing of numerous equations, and selected due to its simple form and good fit.[1] 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
n=5