Reading a Temperature Sensor

A thermistor's resistance drops as it warms, following a known curve. A thermometer measures resistance, so it needs the inverse function: given the resistance, what temperature? Graphically, the inverse is the same curve reflected across the line y = x.

How it works

Functions and inverse functions

In one sentence

A one-to-one function $f$ has an inverse $f^{-1}$ with $f^{-1}(f(x)) = x$ and $f(f^{-1}(y)) = y$ ; the graph of $f^{-1}$ is the reflection of the graph of $f$ across the line $y = x$ .

General formula

{\htmlData{sym=0}{R(T)}} = {\htmlData{sym=1}{R_0}}\,e^{{\htmlData{sym=2}{B}}\left(\frac1T - \frac1{{\htmlData{sym=4}{T_0}}}\right)} \quad\Longleftrightarrow\quad {\htmlData{sym=3}{T(R)}} = \left(\frac{1}{{\htmlData{sym=4}{T_0}}} + \frac{1}{\htmlData{sym=2}{{B}}}\ln\frac{R}{{\htmlData{sym=1}{R_0}}}\right)^{-1}

An inverse runs the machine backwards: put the output in, get the input out. Swapping the axes of the graph does exactly that.

What the symbols mean

$R(T)$
Sensor curve
Resistance in ohms at temperature T.
$R_0$
Reference resistance
The resistance at the reference temperature T₀ (25 °C = 298 K).
$B$
Sensitivity
How strongly resistance changes with temperature, in kelvin.
$T(R)$
Inverse
The temperature that produces resistance R: what the thermometer computes.
$T_0$
Reference temperature
298 K, room temperature.

The sensor's curve $R(T)$ gives resistance for each temperature; it is one-to-one, so it has an inverse.
To invert, swap roles and solve for $T$ : take logs to undo the exponential, then rearrange.
The graph of the inverse is the original graph reflected across $y = x$ : every point $(T, R)$ becomes $(R, T)$ .
Composing them gets you back where you started: $T(R(T)) = T$ .

Learn the math

Functions and Inverse FunctionsCalc I · Lesson 1, Review: The Functions of Calculus Current, Resistance and Ohm's LawE&M · Lesson 13, Current and DC Circuits Calibration CurvesEng Stats · Lesson 20, Regression for Engineering Linearizing Engineering ModelsEng Stats · Lesson 23, Regression for Engineering

Learn the math

Functions and Inverse FunctionsCalc I · Lesson 1, Review: The Functions of Calculus

Current, Resistance and Ohm's LawE&M · Lesson 13, Current and DC Circuits

Calibration CurvesEng Stats · Lesson 20, Regression for Engineering

Linearizing Engineering ModelsEng Stats · Lesson 23, Regression for Engineering