A platinum RTD is not perfectly linear, and a straight line model quietly adds error as the temperature moves away from the ice point. The right equation with the right coefficients turns every ohm into a trustworthy temperature reading.
Every Pt100 transmitter in your plant converts ohms to degrees with a polynomial, whether you see it or not. Knowing that equation, and feeding it coefficients from a real calibration, can cut the measurement error by more than half.
What Is the Callendar Van Dusen Equation?
The Callendar Van Dusen equation is the polynomial that relates the resistance of a platinum resistance thermometer to its temperature, using three constants called A, B and C. It is the model behind the standard Pt100 table and behind the linearisation inside almost every temperature transmitter used with RTDs.
A platinum element changes resistance smoothly with temperature, as explained in what is an RTD and how it works. The change is close to linear, yet a small curvature grows large over a wide span, so a straight line is not good enough for accurate work.
Hugh Longbourne Callendar proposed the quadratic form in the late nineteenth century while studying platinum thermometers. Later, M. S. Van Dusen added a fourth order term that corrects the curve below 0 °C, which gave the equation its two part name.
The famous alpha of a Pt100, 0.00385 per °C, is simply A plus 100 times B from the equation. With the IEC 60751 values this works out to 0.0038506 per °C.
The Callendar Van Dusen Formula and IEC 60751 Coefficients
The equation has two forms that share the same A and B constants. Above 0 °C the C term is zero and the curve is a simple quadratic, while below 0 °C an extra term bends the curve to match real platinum behaviour.
For minus 200 °C to 0 °C: R(T) = R0 × (1 + A·T + B·T² + C·(T minus 100)·T³)
IEC 60751 values for industrial platinum:
A = 0.0039083 per °C
B = minus 0.0000005775 per °C²
C = minus 0.000000000004183 per °C⁴
Example, Pt100 at 150 °C:
A·T = 0.0039083 × 150 = 0.586245
B·T² = minus 0.0000005775 × 22500 = minus 0.012994
1 + 0.586245 minus 0.012994 = 1.573251
R(150) = 100 × 1.573251 = 157.33 Ω
R0 is the resistance at 0 °C, which is 100 Ω for a Pt100 and 1000 Ω for a Pt1000, as compared in Pt100 vs Pt1000 differences. The coefficients stay the same for both, because they describe the platinum and not the element size.
Texas Instruments, in its application note A Basic Guide to RTD Measurements, lists exactly these IEC 60751 values. The same note warns that a straight line fit across the full range can be wrong by more than 16 Ω, which is why linearisation matters.
Callendar Van Dusen Resistance Calculator
Enter R0, the temperature and the three coefficients from your sensor certificate. Enter B and C as positive magnitudes, because the calculator applies their negative sign for you.
The slope tells you how many ohms the sensor changes per degree at that point. It falls slowly as temperature rises, which is the nonlinearity that the equation captures and that a RTD resistance to temperature calculator must also handle in reverse.
Second Worked Example Below Zero
Take the same Pt100 at minus 100 °C, where the C term now applies. First, A·T = 0.0039083 × (minus 100) = minus 0.39083, and B·T² = minus 0.0000005775 × 10000 = minus 0.005775.
Next, C·(T minus 100)·T³ = minus 0.000000000004183 × (minus 200) × (minus 1000000) = minus 0.000837. Adding everything gives 1 minus 0.39083 minus 0.005775 minus 0.000837 = 0.602558, so R = 60.26 Ω, which matches the standard Pt100 table.
When you check a transmitter with a decade box, set the box to the table value from the Callendar Van Dusen curve, not a value from a straight line. At minus 100 °C the straight line answer with alpha 0.00385 is 61.50 Ω, which looks like a 3 °C fault.
Converting Resistance Back to Temperature
Transmitters must solve the equation in reverse, because they measure ohms and report degrees. Above 0 °C the quadratic has a direct solution, T = (minus A + √(A² minus 4·B·(1 minus R/R0))) ÷ (2·B).
Below 0 °C the fourth order term has no neat closed form, so firmware uses a lookup table, a fitted polynomial or a few Newton iterations. Your Pt100 RTD calibration guide values should agree with any of these methods to within a few millikelvin.
Alpha, Delta and Beta: The Older Callendar Form
Some transmitters, including several Emerson models, ask for alpha, delta and beta instead of A, B and C. The two sets are exactly equivalent and can be converted with three short formulas.
B = minus α × δ ÷ 10000
C = minus α × β ÷ 100000000
IEC 60751 platinum: α = 0.00385055, δ = 1.4999, β = 0.10863
A = 0.00385055 × 1.014999 = 0.0039083
B = minus 0.00385055 × 1.4999 ÷ 10000 = minus 0.0000005775
Alpha is the average slope between 0 °C and 100 °C divided by R0, delta describes the curvature above zero and beta the extra bend below zero. Beta is zero when the sensor is used only above 0 °C.
Where Callendar Van Dusen Meets ITS 90
The International Temperature Scale of 1990, ITS 90, defines temperature with fixed points and reference functions for standard platinum resistance thermometers. Earlier scales such as IPTS 68 used a modified form of the Callendar Van Dusen equation, but ITS 90 dropped it for primary work.
Industrial sensors still use the IEC 60751 form because it is simple and accurate enough for process work. Laboratories accredited under ISO 17025 requirements calibrate against ITS 90 references and then fit custom Callendar Van Dusen coefficients to the results.
A calibration certificate often states the fitted coefficients together with a residual error at each test point. Small residuals show that the three coefficient model describes that particular sensor well.
5 Smart Steps for Sensor Transmitter Matching
Endress Hauser explains that a Pt100 class AA sensor with a high accuracy transmitter gives about ±0.4 °C at 150 °C. When the same sensor is matched to the transmitter with its own coefficients, the total drops to about ±0.18 °C.
Endress Hauser also notes that at least three calibration points are needed to derive A, B and C, and that matching gives a more or less constant tolerance over the calibrated range. Use a stable dry block calibrator or liquid bath with a reference probe for those points.
Loading the coefficients is part of normal temperature transmitter configuration and ranging. Most modern devices from Endress Hauser, Yokogawa, Emerson and ABB accept the values directly, often under a menu called sensor matching or CVD.
Write the sensor serial number into the transmitter tag description after matching. When an RTD is replaced during a shutdown, that note warns the technician to reload new coefficients.
IEC 60751 Tolerance Classes and the Standard Curve
IEC 60751 sets tolerance classes that limit how far a real sensor may stray from the standard Callendar Van Dusen curve. The allowed error grows with temperature, which creates the funnel shape that matching removes.
| Class | Tolerance in °C | Error at 150 °C |
|---|---|---|
| AA | ±(0.1 + 0.0017 × |t|) | ±0.355 °C |
| A | ±(0.15 + 0.002 × |t|) | ±0.45 °C |
| B | ±(0.3 + 0.005 × |t|) | ±1.05 °C |
| C | ±(0.6 + 0.01 × |t|) | ±2.1 °C |
Thin film elements usually meet class A or B, and wire wound elements can reach AA, as discussed in thin film RTD and its advantages. Matching a class B sensor can often beat an unmatched class AA sensor at much lower cost.
Transmitter uses standard Callendar Van Dusen values.
Transmitter uses fitted values from the sensor certificate.
Software uses many calibration points directly.
Installation Factors That Still Matter
Callendar Van Dusen coefficients only correct the sensor curve, so wiring errors remain. Lead resistance must be cancelled with 3 wire or 4 wire connections, explained in 2 wire, 3 wire and 4 wire RTD, and terminals should be clean and tight.
Immersion depth, stem conduction and self heating also add error that no equation can remove. A summary of these effects is given in temperature measurement errors, and good thermowell design reduces most of them.
- Matches the real curve of platinum closely.
- Built into almost every RTD transmitter.
- Custom values cut sensor error by more than half.
- Simple to calculate by hand or in a PLC.
- Matched values are valid only for that one sensor.
- Needs a calibration with at least three points.
- Does not correct wiring, immersion or drift errors.
- Below 0 °C the reverse calculation needs iteration.
Common Mistakes With RTD Coefficients
- Confirm R0 is 100 Ω or 1000 Ω before entering coefficients.
- Check the sign of B and C, both are negative for platinum.
- Do not mix A, B, C with alpha, delta, beta fields.
- Match the decimal scaling the transmitter menu expects.
- Reload values whenever the sensor is replaced.
- Verify one point after loading the coefficients.
- Keep the calibration certificate with the loop file.
A wrong exponent is the most frequent error, because a B value entered a thousand times too large produces a reading that drifts badly at high temperature. The same care applies when you calibrate a temperature transmitter and compare it with a reference.
Applications of Matched RTD Curves
For general guidance on where RTDs suit better than thermocouples, see thermocouple vs RTD. In each of the above uses, the equation is the bridge between a certified sensor and a trusted reading.
TI Guide to RTD Measurements
Sensor Transmitter Matching Video
Callendar Van Dusen FAQ
It is the polynomial that links the resistance of a platinum RTD to its temperature. It uses the ice point resistance R0 and three coefficients called A, B and C as constants.
Above 0 °C only A and B are used, while below zero the extra C term is added. IEC 60751 gives standard values for every industrial Pt100 and Pt1000 sensor.
A is 0.0039083 per °C and B is minus 0.0000005775 per °C squared. C is minus 0.000000000004183 per °C to the fourth power and applies only below zero.
These three values generate the standard Pt100 table from minus 200 °C up to 850 degrees. Custom values from a calibration certificate replace them whenever sensor matching is used.
Real platinum bends slightly more below 0 °C than a simple quadratic can follow. Van Dusen added the fourth order term in 1925 to correct that cold region of the curve.
Above 0 °C the quadratic already fits within the tolerance needed for industrial work. Setting C to zero there also keeps the reverse calculation simple for transmitter firmware.
It means loading the measured coefficients of one specific RTD into its own transmitter. The transmitter then uses the real curve of that sensor instead of the standard one.
Endress Hauser reports that this can improve total accuracy at 150 °C from about ±0.4 °C to about ±0.18 degrees. The sensor and transmitter must then stay together as a documented pair.
At least three points are needed to fit R0, A and B above zero. A fourth point below 0 °C is added when the C coefficient must also be found from the data.
Choose points that cover the real operating range of the loop rather than the full sensor range. Extra points improve confidence and reveal the residual error of the fit.
They are an older way of writing exactly the same equation with different constants. A equals alpha times one plus delta over 100, and B equals minus alpha times delta over 10000.
C equals minus alpha times beta divided by one hundred million. Use whichever set your transmitter menu asks for, but never mix the two sets in one device.
No, it only describes how the platinum element itself behaves with temperature. Lead resistance, poor immersion, stem conduction and self heating must be handled separately in the installation.
Use 3 wire or 4 wire connections and a correct immersion depth for good results. Then the fitted coefficients can deliver their full accuracy benefit in the loop.
Related Articles
- What Is RTD and How It Works
- RTD Pt100 vs Pt1000 Differences
- Pt100 RTD Calibration Guide
- RTD Resistance to Temperature Calculator
- Temperature Transmitter Configuration and Ranging
External References
- A Basic Guide to RTD Measurements, Texas Instruments
- Sensor Transmitter Matching With CVD Coefficients, Endress Hauser
- Callendar and Van Dusen Equation, Wikipedia
What We Learn Today
- The Callendar Van Dusen equation links platinum RTD resistance to temperature using R0 and the IEC 60751 coefficients A, B and C.
- Above 0 °C only A and B apply, so a Pt100 at 150 °C reads 157.33 Ω, while the C term adds a correction below zero.
- Loading fitted coefficients from a three point calibration into the transmitter can cut total error at 150 °C from about ±0.4 to ±0.18 °C.