The Beta Conversion and the RTD Comparison
The Beta Conversion and the RTD Comparison
Ada re-derives this chapter’s own numbers step by step, at full precision
ADA · CALCULATION AUDIT
The Beta Conversion and the RTD Comparison
The firmware result is only credible if the units survive the trip from resistance physics to kelvin arithmetic and back to Celsius.
A 10 kΩ NTC thermistor with Beta constant 3950 is the chapter’s worked example: it reads 10 kΩ at 25°C but only 4.0 kΩ once warmed, and the chapter’s Beta equation is supposed to convert that resistance back into a temperature. Elsewhere the same chapter contrasts this exponential curve with a PT100 RTD, whose resistance is expected to rise almost linearly with temperature. This audit asks the question that contrast invites: does the Beta equation actually recover a sensible temperature from the 4.0 kΩ reading, and does the RTD’s response really stay as linear as the chapter claims?
Companion to the chapter Thermistor Linearization — every number here comes from that chapter.
See the relationship before changing it
The figure reads from left to right. The blue card is thermistor resistance. The middle card applies this page's rule. The green card is beta-model temperature. Walk the arrows once: set the input, apply the rule, then read the result with its unit.
The retained audit below checks several chapter fixtures. This model keeps those stated values fixed and changes only thermistor resistance, so the numeric fixture does not switch without explanation.
Derive the baseline in four named moves
- 1
Name the input. The chapter baseline is 4000 ohm.
- 2
Name the relationship. temperature = 1 / (1 / 298.15 K + ln(R / 10,000) / 3,950) - 273.15
- 3
Substitute with units. 1 / (1 / 298.15 + ln(4,000 / 10,000) / 3,950) - 273.15 = 47.2 degrees C
- 4
Read the result. Keep the unit beside the value. Use it only inside the technical boundary on this page.
Predict, then change thermistor resistance
Try Predict the direction of temperature = 1 / (1 / 298.15 K + ln(R / 10,000) / 3,950) - 273.15. Test another thermistor resistance, then compare beta-model temperature.
Observe Lower NTC resistance maps to a higher temperature under the stated Beta model. Reset thermistor resistance to 4000 and compare beta-model temperature.
Explain Lower NTC resistance maps to a higher temperature under the stated Beta model.
Check yourself
What should you do before trusting a moved-control result?
What does this small model leave out?
Start with measured R = 4.0 kΩ, R0 = 10 kΩ, B = 3950, and T0 = 298.15 K; carry the Beta conversion through kelvin before subtracting 273.15.
The temperature ledger rises from the 25 °C reference to about 47.2 °C, while the PT100 comparison reaches 138.5 Ω at 100 °C.
An NTC's exponential resistance curve makes ln(R/R0) negative as it warms, so the reciprocal-temperature term falls and the recovered absolute temperature rises.
Thermistor check
Use the chapter’s values: R0 = 10 kΩ at T0 = 25 °C = 298.15 K, B = 3950, and measured R = 4.0 kΩ.
1/T = 1/298.15 + (-0.9163/3950) = 0.0033540 - 0.0002320 = 0.0031220
T = 1/0.0031220 = 320.3 K; Celsius = 320.3 - 273.15 = 47.2 °C
Physics check
The negative logarithm is expected: an NTC thermistor has lower resistance when warmer. A falling resistance must produce a temperature above the 25 °C reference, so 47.2 °C has the right direction before the number is trusted.
Linear sensor contrast. For the PT100 comparison in this page, R = R0(1 + αT) = 100(1 + 0.00385×100) = 100(1 + 0.385) = 138.5 Ω. That near-linear result is why the RTD calculation feels like a calibration line while the thermistor needs the exponential model.
Every number above is taken from the chapter’s own thermistor and RTD examples and re-derived step by step.
Technical boundaries: This fixed-point Beta and linear-PT100 audit omits thermistor self-heating, Beta and resistor tolerance, lead resistance, ADC error, and any multi-point linearisation used by real firmware.