40 Thermistor Linearization
A thermistor channel reports a warmer room when the reference thermometer cools. The temperature formula may be correct while the resistance calculation uses the opposite divider orientation. Check the physical circuit before adjusting calibration coefficients.
40.1 Start With the Measurement Story
Turn a Curved Reading Into a Useful Temperature
Picture a small bead whose electrical resistance changes with heat. This part is called a thermistor. It is cheap and sensitive, but its change is curved rather than even. The same resistance step does not mean the same temperature step across the full range.
Begin with the temperature range that matters. An analog-to-digital converter is a part that turns the sensor voltage into a number. It is often shortened to ADC. Take the measured resistance and apply a stated conversion. Compare the result with a trusted reference at several points, not only at room temperature. Keep the units and the test conditions beside each result.
Check a cold point. Check a warm point. Check the middle. Repeat each test. Let the part settle. Note the supply value. Note the reference tool. Keep the raw reading. Keep the final value. Set a clear pass limit.
A simple straight-line rule may work across a narrow range. A wider range needs a curve, a table, or more than one line. The best choice is the least complex method that still meets the error limit for the real decision.
The simple story has limits. Part tolerance, self-heating, wire resistance, age, and poor contact can shift the answer. A neat formula cannot remove those physical effects.
Use Practitioner to choose and test the conversion. Use Under the Hood to derive the curve and track the full error.
A thermistor gives a curved resistance story, not a neat straight line. Start by mapping temperature to resistance and ADC counts, then choose a linearization method that is accurate enough for the control decision.
40.2 Learning Objectives
After this page, you should be able to:
- Explain why NTC thermistors need nonlinear conversion instead of slope-and-offset calibration.
- Convert divider voltage into thermistor resistance before applying a temperature model.
- Use the Beta equation for a focused range and know when Steinhart-Hart or RTD-specific models are needed.
- Identify self-heating, ADC reference drift, lead resistance, leakage, and coefficient mismatch as temperature-measurement errors.
40.3 Why This Follows Temperature Sensor Labs
Temperature Sensor Labs introduces common temperature devices, wiring patterns, lab exercises, and debugging values. This page narrows in on analog temperature channels where the circuit produces resistance or voltage before firmware can report degrees Celsius.
Use it when a project uses an NTC thermistor, PT100/PT1000 RTD, divider, ADC, or reference resistor and the main risk is a plausible-looking temperature number from the wrong model.
40.4 Cheap Nonlinear Thermistors
Before modelling temperature, inspect Figure to anchor the nonlinear curve in the two-lead sensing element. The device exposes resistance only; the divider and firmware must create and interpret the measurable voltage.
Read Figure, begin with the bead that changes resistance, then follow its two leads into the divider implied by the next discussion. Heating an NTC lowers resistance nonlinearly, connecting a cheap physical component to the need for Beta or Steinhart-Hart linearisation rather than a simple slope and offset.
The NTC thermistor is the most common temperature sensor in low-cost electronics: it is a resistor whose resistance falls steeply as it warms. That steep response makes it sensitive — small temperature changes give large resistance changes — but it comes with a catch. The resistance-versus-temperature relationship is not a line; it is an exponential curve. A thermistor that reads 10 kΩ at 25 °C might be 4 kΩ at 50 °C and 30 kΩ near freezing.
Not every thermistor behaves this way. The “NTC” in the name marks it as negative-temperature-coefficient: resistance falls as it warms, which is the behavior modeled below. Its counterpart, the PTC (positive-temperature-coefficient) thermistor, does the reverse — resistance climbs as it warms — and that reversal changes its job entirely. A PTC is rarely read as a continuous temperature value; wired in series with a circuit, its resistance stays low until self-heating from excess current pushes it past a switching temperature, then it snaps to a high-resistance state and throttles the current down to a safe trickle. That makes a PTC a resettable fuse rather than a sensor, and it is why “thermistor” on a datasheet is not automatically synonymous with the NTC curve this page linearizes.
Because of that curvature, you cannot convert resistance to temperature with a simple slope-and-offset. You need a linearization equation that models the curve. This is the defining difference from an RTD or a silicon IC sensor, which are far more linear but less sensitive and more expensive. That behavioral gap traces back to materials: an NTC thermistor's sensing element is a sintered ceramic or polymer semiconductor, while an RTD's element is a coil or film of pure metal, almost always platinum. It is also why the precision comparison flips with range — within roughly -90 °C to 130 °C a well-calibrated NTC thermistor usually out-precisions an RTD, but push further outside that window and the RTD's more linear metal response becomes the safer choice.
In a lab, the sensor choice starts with the temperature window and the kind of error that matters. A 10 kΩ NTC is excellent for battery packs, enclosures, and short-range environmental checks because it is cheap, small, and responsive around room temperature. A PT100 or PT1000 RTD is slower and needs a cleaner analog front end, but it is easier to calibrate over a wide range. A digital IC is often the best choice when you want repeatable board temperature without doing analog conversion at all.
Reference points turn that physics into usable numbers. An ice-water bath near 0 °C, a room-temperature reference, and a warm-water point expose whether the curve is being modeled or merely guessed. If the reading is right at 25 °C but wrong at both ends, the wiring is probably fine and the equation is the weak link. If every point is offset by the same amount, the issue is more likely a reference, divider, or ADC calibration error.
Intuition: a thermistor is a magnifying glass for temperature — wonderfully sensitive in a narrow window, but the view is warped. The linearization equation is the prescription that un-warps it back into real degrees.
40.4.1 Overview Knowledge Check
40.5 Practitioner: The Beta Equation, Worked
Before applying the specification, inspect the real pt100 platinum rtd below: its package, terminals, scale, and installation context are part of the engineering evidence.
Carry those visible constraints into the surrounding analysis; the abstract symbol or capability name does not capture mounting, wiring, protection, or service access.
Before working the Beta equation, inspect Figure to contrast the NTC model with a more nearly linear platinum reference sensor. The three-wire construction also makes lead resistance part of the measurement design.
Read Figure, trace the metal probe to its three leads and note that the extra conductor supports compensation for wiring resistance. That construction contrasts with the two-lead NTC and clarifies the running choice: inexpensive sensitivity with nonlinear modelling versus a characterised platinum element and cleaner wide-range calibration.
The simplest usable model is the Beta (B-parameter) equation, which needs one reference point (R0 at T0) and the datasheet Beta constant:
1/T = 1/T0 + (1/B) × ln(R / R0) (all temperatures in kelvin)
Worked example: 10 kΩ NTC, B = 3950
Datasheet: R0 = 10 kΩ at T0 = 25 °C = 298.15 K, B = 3950
You measure R = 4.0 kΩ. Find T.
ln(R/R0) = ln(4000/10000) = ln(0.4) = -0.9163
1/T = 1/298.15 + (1/3950)(-0.9163)
= 0.0033540 - 0.0002320
= 0.0031220
T = 1 / 0.0031220 = 320.3 K = 47.2 °C
Compare that with an RTD, which barely needs linearizing at all. A PT100 follows R = R0(1 + αT) with R0 = 100 Ω and α ≈ 0.00385/°C, so at 100 °C it reads 100(1 + 0.385) = 138.5 Ω — almost a straight line.
The Beta equation can produce a plausible temperature even when the divider or coefficients are wrong. Before trusting the 10 kΩ, B = 3950 example, use Figure 40.1 to place its calculated points inside a calibration-and-validation record.
In Figure 40.1, 1. Reference Standards supplies known temperatures, and 2. Collect Raw Measurements pairs them with divider or resistance readings across the full range. 3. Error Analysis distinguishes offset and gain from non-linearity; that matters because changing the Beta coefficient alters curve shape rather than merely shifting every result. The Calibration Curve compares Ideal and Actual responses before 4. Correction Model selects a polynomial or lookup table when the single-parameter model is insufficient. Applied to the arithmetic below, the 25°C nominal point checks the reference resistance, while additional cold and hot holdouts test whether B = 3950 remains defensible over the intended span.
The electrical measurement comes before the temperature equation. In a common divider, the thermistor and a fixed resistor split the supply voltage; the ADC measures the middle node. With the fixed resistor tied to VCC and the thermistor tied to ground, R_thermistor = R_fixed * Vout / (Vin - Vout). Reversing the divider reverses the formula, so the first debug step is to confirm the schematic before blaming the math.
A practical calibration run records the raw ADC code, computed resistance, reference temperature, and model temperature in the same table. That table shows whether the error is random noise, a single-point offset, or a curve-fit problem. It also catches unit mistakes: the Beta equation uses kelvin, while lab notes and most datasheets report degrees Celsius.
Field tip: the Beta value itself is specified between two temperatures (e.g. 25/85 °C). Using a Beta calibrated for one range far outside it adds error, because Beta is only an approximation of the true curve.
40.5.1 Practitioner Knowledge Check
40.6 Under the Hood: When Beta Isn’t Enough
The Beta equation is a two-parameter fit, so it matches the true curve well near its calibration range and drifts away from it at the extremes. When you need accuracy across a wide span, you move to a richer model.
Steinhart-Hart equation
The three-term form 1/T = A + B·ln(R) + C·(ln R)^3 fits the thermistor curve across a wide range far better than Beta, typically to a small fraction of a degree once A, B, C are calibrated from three points.
RTDs curve too, just gently
Even a PT100 is not perfectly linear. High-accuracy work uses the Callendar–Van Dusen equation, adding second- and higher-order terms to the simple R0(1 + αT) approximation.
Sensitivity is not uniform
A thermistor is most sensitive near its nominal temperature and flattens far from it. Choosing the divider resistor near the thermistor's resistance at your target temperature keeps the reading in the sensitive region.
Self-heating shifts the reading
The current you push through the thermistor to read it dissipates power and warms the element slightly, biasing it upward. Keep the sense current low, especially in still air where heat cannot escape.
The coefficients also depend on the exact part, not just the sensor family. Two nominal 10 kΩ NTCs can share the same resistance at 25 °C yet diverge outside that point because their Beta values and tolerances differ. That is why production systems either buy a specified tolerance class, store per-device calibration constants, or calibrate the whole assembled channel rather than the bare component alone.
Model error and measurement error stack. Lead resistance is usually negligible for a 10 kΩ thermistor but can matter for low-resistance RTDs. ADC reference drift changes the computed divider ratio. Moisture or board leakage creates parallel resistance paths that look like a warmer thermistor. Long cables add pickup, so filtering can make the trace quieter without making the physics more correct.
The through-line is that a temperature reading is only as good as its model of the sensor's curve and its measurement chain. A cheap thermistor with a well-calibrated Steinhart-Hart fit can beat a nominally better sensor read through a careless linear approximation, while a high-grade RTD still fails if its excitation current, reference resistor, or lead compensation is ignored.
40.6.1 Under-the-Hood Knowledge Check
40.7 Check Divider Direction before Applying the Curve
Assume a 3.3 V supply, a 10 kΩ fixed resistor above the sense node and the thermistor below it to ground. If the node measures 1.1 V, resistance is 10 kΩ × 1.1 V divided by (3.3 V − 1.1 V) = 5 kΩ. The voltage units cancel, leaving resistance. Swapping which element sits above the node changes the conversion; the same formula cannot be kept without checking the schematic.
Use the chapter’s ideal Beta model with reference resistance 10 kΩ at 25 °C and Beta 3,950 K. Convert the reference temperature to 298.15 K. The reciprocal-temperature term is 1/298.15 plus ln(5/10) divided by 3,950, in inverse kelvin. Since ln(0.5) is about −0.69315, the result is about 0.00317854 per kelvin. Inverting gives about 314.61 K, or 41.46 °C. Rounding is shown so the calculation can be re-derived.
The result fits the curve’s direction: an NTC element has lower resistance when warmer. Predict what resistance the same divider gives at half the supply voltage, 1.65 V. It gives 10 kΩ, so the ideal model returns the 25 °C reference point. That midpoint check catches several wiring and scale mistakes before a wider calibration trial.
Read Figure 40.1 from reference standards to raw measurements and error analysis. A matching midpoint is one check, while cold and hot holdouts test curve shape. The correction model comes after those residuals are visible. A single offset fitted at room temperature cannot prove that the chosen Beta coefficient describes the whole range.
Next, increase the sense current and observe a slow upward temperature bias in still air. Self-heating can warm the actual bead, so a more elaborate curve fit does not remove the physical cause. The module’s linearization lesson joins circuit orientation, units, model parameters and calibration evidence. The ideal arithmetic needs validation with the exact part, reference and installed thermal conditions.
40.8 Release Checklist
Before relying on an analog temperature channel, confirm these points:
Start by the schematic orientation is documented so firmware converts ADC voltage into the correct resistance. Then the Beta value or Steinhart-Hart coefficients belong to the exact part and temperature range in use. Next reference measurements include the operating range, not only room temperature. After that the ADC reference, fixed resistor tolerance, lead resistance, leakage paths, and filtering are included in the error budget. Continue by sense current is low enough to avoid self-heating in still air or inside the enclosure. Finally validation records raw ADC code, computed resistance, reference temperature, model output, and residual error.
40.9 See Also
Start by Temperature Sensor Labs for DS18B20, DHT22, thermocouple, humidity, and lab workflows. Then Resistive Dividers and ADC Loading for divider-to-ADC measurement boundaries. Next Sensor Calibration Techniques for reference points, coefficient storage, and validation. Finally Signal Conditioning for Sensors for analog filtering and ADC preparation.
40.10 Next
Return to Temperature Sensor Labs once the analog conversion model is bounded, then continue to Motion and Environmental Sensor Labs for the next lab family.