24. Thermistors, the Beta Equation & Calibration

Start from the divider

Lesson 23 turned an LDR's resistance into a voltage with a fixed resistor. A thermistor uses exactly the same divider and the same resistance formula. What is new here is the second step: converting that resistance into a temperature, then checking the result against a real thermometer.

Keep the course order - 10 kΩ to 5 V, thermistor to GND, junction to A0 - so the formula and the SimpleThermistor library in Lesson 30 both match your wiring.

How an NTC thermistor works

A thermistor is a resistor whose resistance depends strongly on temperature. NTC means Negative Temperature Coefficient: as temperature rises, resistance falls.

Inside a typical bead NTC, a metal-oxide semiconductor material changes how easily charge carriers move. Warmer material usually conducts better, so resistance drops. Classroom kits often use a 10 kΩ NTC: about 10 kΩ at 25 °C (298.15 K), with a datasheet Beta (B) value often near 3950 K.

PTC thermistors (Positive Temperature Coefficient) do the opposite and are less common in this course. Always check the marking or datasheet: wrong type or wrong R0 ruins the Beta maths.

Divider calculation (course wiring)

Course wiring uses a fixed pull-up to 5 V and the sensor to GND (same order as the SimpleThermistor library in Lesson 30):

5 V → 10 kΩ pull-up → junction to A0 → NTC → GND

Voltage at the junction: Vout = 5 V × Rsensor / (Rfixed + Rsensor).

The Uno ADC maps about 0-5 V to 0-1023. From the ADC count you can recover resistance:

Rsensor = Rfixed / (1023.0 / ADC - 1.0)

Equivalently: Rsensor = Rfixed × ADC / (1023.0 - ADC).

If you swap sensor and fixed resistor, both the voltage direction and this formula change. Always match the code to the physical order.

Worked resistance example

Suppose Rfixed = 10000 Ω and analogRead returns 512.

Rsensor = 10000 / (1023.0 / 512 - 1.0) ≈ 10000 / (2.00 - 1.0) ≈ 10000 Ω.

So the sensor is about equal to the pull-up - for a 10 kΩ NTC that is near room temperature if R0 is 10 kΩ at 25 °C (exact Celsius still needs the Beta step).

If ADC = 0 or 1023, do not trust the formula: you risk division by zero or a meaningless result. The worked sketch treats those as a divider fault.

Beta equation: resistance to temperature

The Beta model converts NTC resistance to absolute temperature in kelvin:

1/T = 1/T0 + ln(R / R0) / B

T and T0 are in kelvin. T0 is usually 25 °C → 25 + 273.15 = 298.15 K. R0 is the resistance at T0. B is the datasheet Beta constant.

Then Celsius = T - 273.15.

Use float maths and the C log() natural logarithm from math.h. Wrong B or R0 values produce smoothly wrong temperatures - always copy constants from your part datasheet.

#include <math.h>

const float seriesR = 10000.0;      // Rfixed pull-up
const float nominalR = 10000.0;     // R0 at T0
const float nominalK = 25.0 + 273.15;
const float beta = 3950.0;          // B from datasheet

float resistance = seriesR / (1023.0 / raw - 1.0);
float invT = 1.0 / nominalK + log(resistance / nominalR) / beta;
float celsius = 1.0 / invT - 273.15;

Calibration and hysteresis

Real parts have tolerance. Breadboard contact resistance, supply voltage and self-heating add error. Calibrate against a known reference (a room thermometer, or an ice-water bath for about 0 °C) and record the sketch reading, the reference reading and the difference in your logbook.

For on/off control (fan, lamp, alarm), do not use a single threshold. Noise around that point makes the output chatter. Use hysteresis: turn ON above one value and OFF only below a lower value (or the reverse for a heater / night light).

const float onC = 30.0;
const float offC = 27.0;
bool fanOn = false;

if (!fanOn && celsius >= onC) {
  fanOn = true;   // rising trip
} else if (fanOn && celsius <= offC) {
  fanOn = false;  // falling trip
}
// Drive MOSFET/fan from Lesson 16 using fanOn - never from an Uno pin directly

Record the calibration

Take at least three reference points - for example ice water, room air and a warm hand around the sensor - and let each reading settle for a minute. If the error is roughly the same at every point, add a fixed offset in code. If it grows with temperature, check R0 and Beta against the datasheet before adding any correction.

Practical Task B asks for exactly this evidence: the formula, the reference points and the error you accepted.

Your own library

Once the thermistor maths works in a sketch, package it as a reusable library with a .h header and .cpp source file in Documents/Arduino/libraries/. See the course extension: How to create a SimpleThermistor library.

Wiring and safe build sequence

1. 5 V -> 10 kΩ pull-up -> A0 junction -> NTC -> GND
2. Drive any fan through the MOSFET circuit from Lesson 16 (not directly from an I/O pin)