Sensors & Measurement · Study deck
Thermistor Linearization
A thermistor channel reports a warmer room when the reference thermometer cools.
Physics Phoebe is your guide for this deck.
After studying this chapter
Learning objectives
The temperature result depends on both the sensor curve and the measuring circuit.
- An NTC thermistor has a curved resistance response.Its resistance falls as temperature rises, so equal resistance changes cannot represent equal temperature changes across the full operating range.
- Divider orientation determines the conversion from voltage to resistance.A channel can report warming while the reference cools if its resistance calculation assumes the opposite arrangement of thermistor and fixed resistor.
- The Beta equation needs a reference point and temperatures in kelvin.The worked model uses 10 kΩ at 25 °C, converted to 298.15 K, with the specified Beta value of 3950.
- Reference checks across the range can expose hidden model errors.A match at room temperature still leaves cold and warm behaviour unproven, so the calibration record needs several settled reference measurements.
Major section
Start With the Measurement Story
A curved sensor response needs a conversion suited to the required temperature range.
- Equal resistance steps do not imply equal temperature steps.The small sensing bead follows a curved response, making a single slope-and-offset conversion unsuitable for the whole temperature range.
- A narrow operating range may allow a simpler approximation.A straight-line rule can fit a narrow window, while a wider span may need a curve, lookup table, or multiple lines.
- The chosen model must meet the actual decision’s error limit.A cheap thermistor becomes a useful channel only when its conversion is checked against a trusted reference with the exact installed part.
- A detailed formula cannot remove physical measurement errors.Self-heating, poor contact, wire resistance, age, and part tolerance can shift the final temperature even when the calculation follows its model correctly.
Major section
Cold, middle, and warm reference checks
Calibration evidence must retain the raw reading as well as the final temperature.
- Cold, middle, and warm references test the intended temperature range.An ice-water point near 0 °C, a room-temperature reference, and warm water can reveal whether the conversion follows the curve beyond its midpoint.
- Repeated settled readings help separate random variation from model error.A consistent mismatch at the range ends points toward curve fit, while repeated raw readings show whether noise is also affecting the result.
- The record needs supply and reference details beside each reading.Raw ADC code, computed resistance, reference temperature, and model temperature belong together so later reviews can identify unit, divider, and calibration mistakes.
- The simplest acceptable conversion meets the application’s stated error limit.A model that passes only at room temperature cannot justify the full range, even if its midpoint arithmetic looks neat and repeatable.
Major section
Cheap Nonlinear Thermistors
This photo connects the NTC sensing bead with the resistance that firmware must interpret.
- The bead is the temperature-sensitive resistive body.The photo’s small sensing body changes resistance with heat but has no digital interface or built-in linearization to report a temperature directly.
- The two leads connect that resistance to the measurement circuit.The external divider supplies a measurable voltage, so the physical component needs circuit arithmetic before firmware can apply a temperature model.
- An NTC has lower resistance as its temperature rises.The falling resistance makes the bead sensitive, but a simple offset cannot straighten the response across the intended temperature range.
- The divider and model must produce a validated temperature together.Beta or Steinhart-Hart conversion interprets the recovered resistance, while reference measurements check whether the assembled channel actually meets its error limit.
Major section
Practitioner: The Beta Equation, Worked
This reference probe highlights a different sensing construction from the two-lead NTC.
- The metal probe has a characterized platinum sensing element.This PT100 reference construction provides a more nearly linear comparison with the inexpensive NTC bead whose changing resistance needs nonlinear conversion.
- The three leads support compensation for measurement wiring resistance.The extra conductor belongs to the probe’s measurement design, showing why the wires remain part of the error review around the sensor.
- The platinum probe differs from the two-lead NTC construction.The worked NTC model uses 10 kΩ at 25 °C and B = 3950, while the PT100 needs its own resistance-temperature model.
- The comparison separates sensor modelling from measurement-path errors.A characterized reference element still needs attention to lead compensation and excitation current, just as the thermistor needs correct divider orientation and coefficients.
Major section
Check Divider Direction before Applying the Curve
This workflow places divider arithmetic inside a reference-based calibration process.
- Reference standards provide known temperatures for the installed channel.The first workflow step establishes the reference before a plausible result from the 10 kΩ, B = 3950 model can be trusted.
- Raw measurements connect each reference with its voltage or resistance.The next step needs readings across the operating range, retaining ADC code and computed resistance beside the reference and model temperatures.
- Error analysis separates offset from gain and curve-shape problems.A matching midpoint with errors at cold and hot holdouts can expose a weak curve model rather than justify a room-temperature offset correction.
- The correction model follows the observed calibration errors.The workflow’s ideal and actual curves help determine whether the single-parameter model needs a polynomial or lookup table for the intended span.
Major section
Divider voltage and the reference midpoint
The divider midpoint offers a quick check before a wider calibration trial.
- A thermistor below the sense node gives lower voltage when warmer.With the fixed resistor above the node, falling NTC resistance lowers the divider voltage; swapping the components requires checking the formula again.
- The 1.1 V reading gives a thermistor resistance of 5 kΩ.The example uses 10 kΩ × 1.1 V divided by (3.3 V − 1.1 V), leaving resistance after the voltage units cancel.
- The 1.65 V midpoint gives the 10 kΩ reference resistance.The same divider calculation uses half the 3.3 V supply, recovering a thermistor resistance equal to the fixed 10 kΩ resistor.
- The ideal model returns 25 °C at the reference resistance.The logarithm term becomes zero when resistance equals its reference, making the midpoint useful for checking wiring and scaling before wider calibration.
Activity 1 · Work it out
✎ Check the divider midpoint
I want you to check the circuit direction before you trust the temperature curve.
Draw the chapter’s divider: 3.3 V supply, 10 kΩ fixed resistor above the node, thermistor below. At 1.65 V, calculate thermistor resistance using Rfixed × Vout ÷ (Vin − Vout). State the ideal temperature at that resistance.
4 minutes · Pen and paper · Answer: Activity 1
Major section
Kelvin, holdouts, and self-heating
A plausible computed temperature still needs physical and full-range validation.
- The Beta calculation needs kelvin before conversion to degrees Celsius.The reference is 298.15 K, and the 5 kΩ worked point gives about 314.61 K, or 41.46 °C, under the ideal model.
- Cold and hot holdouts test curve shape beyond the midpoint.Two nominal 10 kΩ NTCs can agree at 25 °C yet diverge elsewhere because their Beta values and tolerances differ.
- Higher sense current can warm the bead and bias its reading.The chapter’s still-air observation shows a slow upward temperature shift, so a richer curve fit cannot remove the physical cause of self-heating.
- The exact part and installation determine whether ideal arithmetic is acceptable.The reference, thermal contact, supply, and assembled measurement path all need validation before the computed temperature can support the real decision.
Deck summary
Key takeaways
A dependable temperature channel connects circuit orientation, model parameters, and reference evidence.
- The divider formula must recover resistance before temperature conversion.A reversed component arrangement can make the room appear warmer as the reference cools, even when the later Beta calculation is internally correct.
- Model coefficients must match the actual part and temperature span.A Beta value specified for one range can add error outside that range, while wide-span work may need calibrated Steinhart-Hart coefficients.
- Correct arithmetic does not remove self-heating or reference drift.Sense current can warm the bead, and ADC reference drift changes the computed divider ratio, leaving errors that a neat formula cannot repair.
- Raw readings and residual errors make calibration decisions reviewable.The record needs ADC code, computed resistance, reference temperature, and model output together so later checks can distinguish noise, offset, and curve-fit problems.
Retrieval practice
Recall check 1 of 3
Physics Phoebe says: answer from memory, then check your reasoning.
Q1Why can't you convert an NTC thermistor's resistance to temperature with a simple linear (slope + offset) formula?
Show answer
Answer: B The steep, curved response is why thermistors need a Beta or Steinhart-Hart equation.
Retrieval practice
Recall check 2 of 3
Physics Phoebe says: answer from memory, then check your reasoning.
Q2Using the Beta equation with R0 = 10 kΩ at 25 °C, a thermistor now measures exactly 10 kΩ. What temperature does the equation give, and why?
Show answer
Answer: A At the reference resistance the logarithm term vanishes and the equation returns the reference temperature.
Retrieval practice
Recall check 3 of 3
Physics Phoebe says: answer from memory, then check your reasoning.
Q3A thermistor read with the Beta equation is accurate near 25 °C but drifts off by more than a degree near the ends of a wide -20 to +120 °C range. What is the best fix?
Show answer
Answer: A Beta is a local approximation; Steinhart-Hart fits the whole curve and is the standard wide-range fix.
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Answers
Answer key.
- B · The steep, curved response is why thermistors need a Beta or Steinhart-Hart equation.
- A · At the reference resistance the logarithm term vanishes and the equation returns the reference temperature.
- A · Beta is a local approximation; Steinhart-Hart fits the whole curve and is the standard wide-range fix.
Print reference
Activity 1 answer
Model answer.
Work it out: 10 kΩ × 1.65 V ÷ (3.3 V − 1.65 V) = 10 kΩ. This equals the model’s reference resistance at 25 °C. Swapping the two components requires checking the conversion formula again.