PT100 and PT1000 calculator: resistance to temperature and back

What a PT100 is, and the Callendar-Van Dusen equation

A PT100 is a resistance temperature detector, an RTD, made of platinum with a resistance of 100 Ω at 0 °C. A PT1000 is the same thing with 1000 Ω. TI's application note SBAA275, "A Basic Guide to RTD Measurements" by Joseph Wu, is the source for everything on this page. It describes RTDs as "resistive elements that change resistance over temperature", used "to make precision temperature measurements, with capability of making measurements with accuracies of well under 0.1°C", and built either as "a length of wire wrapped around a ceramic or glass core" or as "thick film resistors plated onto a substrate".

The resistance does not rise in a straight line. "The relationship between platinum RTD resistance and temperature is described by the Callendar-Van Dusen (CVD) equation", which SBAA275 writes as two polynomials: a quadratic, R₀(1 + A·T + B·T²), from 0 °C up, and the same with a quartic term C·T³·(T − 100) added below 0 °C. The reference note under the calculator typesets both. The coefficients "are defined by the IEC-60751 standard", and SBAA275 prints them as A = 3.9083·10⁻³, B = −5.775·10⁻⁷ and C = −4.183·10⁻¹². IEC 60751 itself is a paid standard and is not reproduced here; the coefficients on this page are SBAA275's.

The A term is the main slope: R₀·A is 0.39083 Ω/°C at 0 °C. B is negative and small, so the slope falls as the temperature rises, to 0.2927 Ω/°C at 850 °C. C matters only well below zero: at −200 °C its term is −1.004 Ω out of 18.52 Ω. SBAA275's Figure 1-2 draws the result of fitting one straight line from −200 °C to 850 °C, and "The results show a non-linearity greater than 16 Ω". The equation here bows 16.42 Ω above that line at 317 °C, the peak of SBAA275's plot.

SBAA275 gives the element size only as R₀, "the resistance of the RTD at 0°C", and notes that "RTD elements are also available with 0°C resistances of 200, 500, 1000, and 2000 Ω". The equation is R₀ times a function of temperature, so a PT1000 that follows the same coefficients has exactly ten times a PT100's resistance at every temperature, and ten times its slope in ohms. The calculator treats it so; a PT1000's datasheet should confirm that it names IEC 60751.

PT100 and PT1000 resistance table, −200 °C to 850 °C

Every value below is the equation evaluated with SBAA275's coefficients, in 25 °C steps. The slope column is how many ohms one degree is worth on a PT100 at that temperature, which is what turns a resistance error into a temperature error. The last two columns are the IEC Class A and Class B tolerances from SBAA275's Table 1-1 at that temperature. The calculator gives any temperature in between, and the inverse.

SBAA275 rounds the ends of this range: "With this temperature range, the RTD would have an equivalent resistance range of 20 Ω to 400 Ω." The equation gives 18.52 Ω and 390.48 Ω. For sizing a circuit the rounded figures are the safe side; for converting a reading they are not, since 20 Ω is −196.6 °C rather than −200 °C.

Converting resistance to temperature

Going from a measured resistance back to a temperature means solving the equation for T. SBAA275: "For temperatures greater than 0°C, temperatures can be determined by solving the quadratic from Equation 2. For temperatures lower than 0°C, the third order polynomial of Equation 1 may be difficult to calculate. Using simple microcontrollers, determining the temperature may be computationally difficult and using a look-up table to determine the temperature is common practice."

At and above 0 °C the calculator uses the quadratic's closed form, shown in the reference note, which is exact. Below 0 °C it takes the quadratic's answer as a first guess and refines it with Newton's method on the full equation, each step dividing the remaining resistance error by the slope, until the change is below 10⁻¹² °C. The first guess alone would be wrong by the C term: at −40 °C the quadratic gives −40.009 °C, at −100 °C −100.208 °C, and at −200 °C −202.42 °C. Firmware that applies the positive-side formula to a cold reading makes exactly that error, and it is larger than Class A allows from −120 °C down.

Worked example: SBAA275's two-wire PT100 circuit

SBAA275's design section builds a ratiometric measurement: a precision ADC's current source drives the RTD and a reference resistor in series, and the voltage across the reference is the ADC's reference, so the code depends on the ratio R_RTD/R_REF and "not on the IDAC1 current value". Its example: "If the IDAC current is selected to be 1 mA, then the reference resistor could be chosen to be 1620 Ω. The measurement of the 400 Ω could be set to a PGA gain of 4. This would make the input voltage 1.6 V, while the reference voltage is set to 1.62 V. This would maximize the input voltage range of the ADC to 98.8% of the positive full-scale range." The left column is the calculator's arithmetic with a 24-bit ADC; the right, what SBAA275 prints.

range      R(−200 °C)  ·  R(850 °C)                     = 18.52 Ω  ·  390.48 Ω SBAA275: 20 Ω to 400 Ω
input      1 mA × 400 Ω × gain 4                        = 1.60 V       SBAA275: 1.6 V
reference  V_REF = 1 mA × 1620 Ω                        = 1.62 V       SBAA275: 1.62 V
share      1.6 V / 1.62 V                               = 98.8 %       SBAA275: 98.8 %
code       2²³ × 4 × 400 / 1620                         = 8,285,045   
AIN1       Eq 15: 1 mA × (400 Ω + 1620 Ω)               = 2.02 V       SBAA275: 2.02 V
850 °C     2²³ × 4 × 390.48 / 1620                      = 96.4 %      
−200 °C    2²³ × 4 × 18.52 / 1620                       = 4.6 %       
2400 Ω     1.6 V / (1 mA × 2400 Ω)                      = 67 %         SBAA275: 67 %

Everything agrees. SBAA275 designs for its rounded 400 Ω; the element itself only reaches 390.48 Ω at 850 °C, so the top of the range uses 96.4 % of the ADC, and the bottom, −200 °C, 4.6 %. It chose 1620 Ω rather than 1600 Ω because "a small gain error or resistance error may push a 400 Ω measurement out of the range of operation", and "Selecting a marginally larger resistance only reduces the resolution of the measurement": with 2400 Ω the same input uses 67 %.

The calculator's defaults are this circuit read at 100 °C, with one addition SBAA275's example leaves out: 5 m of 24 AWG cable to the element, which is an example value chosen here, and SBAA275's small thin-film self-heating coefficient of 2.5 mW/°C.

element    R(100 °C) = 100 × (1 + 0.39083 − 0.005775)   = 138.5055 Ω  
leads      5 m of 24 AWG, each                          = 0.4103 Ω    
heat       (1 mA)² × 138.51 Ω / 2.5 mW/°C               = +0.0554 °C  
ADC reads  R(100.055 °C) + 2 × 0.410 Ω                  = 139.3471 Ω  
code       2²³ × 4 × 139.347 / 1620                     = 2,886,243   
reads      Equation 2 solved for T                      = 102.220 °C  
           of which leads +2.164 °C, self-heating +0.055 °C
3-wire     same leads, two matched 1 mA sources         = 100.055 °C  
4-wire     same leads                                   = 100.055 °C  

One code of the 24-bit ADC is 0.000127 °C at 100 °C, so the resolution is not the limit. The leads are: 0.821 Ω of copper read as +2.16 °C, more than twice the ±0.80 °C an IEC Class B element is allowed at that temperature. The same cable on a three-wire element with matched current sources reads 100.055 °C, and on a four-wire element 100.055 °C; what is left in both is the self-heating.

Two-, three- and four-wire RTDs: the lead resistance error

SBAA275 on two-wire elements: "the lead wire resistances cannot be separated from the RTD resistance, adding an error that cannot be separated from the RTD measurement. Two-wire RTDs yield the least accurate RTD measurements and are used when accuracy is not critical or when lead lengths are short." Both leads are in series with the element, and its Equation 92 carries them into the code as R_RTD + 2·R_LEAD. At 0 °C every ohm of lead reads as 2.560 °C on a PT100; at 850 °C, where the slope is lower, as 3.419 °C.

Lead resistance is 20 °C copper from the AWG definition, the same arithmetic as the wire gauge calculator. A PT1000 suffers a tenth of the error for the same cable, because the same ohms are a tenth of a degree's worth of its resistance. Copper's own temperature coefficient means the error also drifts with the temperature of the cable, which is why a one-time offset correction only holds while the cable stays where it was calibrated.

Three-wire. "In the three-wire configuration, the RTD is connected to a single lead wire on one end and two lead wires on the opposite end. Using different circuit topologies and measurements, lead resistance effects can effectively be cancelled ... Compensation for lead wire resistance assumes that the lead resistances match." In SBAA275's circuit a second, matched current source drives the second lead, so the ADC input is I₁(R_RTD + R_LEAD1) − I₂·R_LEAD2 and the two lead drops cancel. What is left is their difference: on 10 m of 24 AWG, each 1 % by which lead 1 exceeds lead 2 leaves 0.0082 Ω, or 0.0210 °C. The currents must match too. SBAA275: "if IDAC2 is larger than IDAC1 by 1%, the reference would be 0.5% larger than expected, resulting in a 0.5% gain error". At 100 °C that gain error is 0.50 % of 138.51 Ω, or −1.82 °C, which is why SBAA275 swaps the two sources between two conversions and averages them: "it is not important that IIDAC1 and IIDAC2 are not equal, it is only important that IIDAC1 and IIDAC2 are the same values after they are swapped".

Four-wire. "The RTD excitation is driven through one lead on either end, while the RTD resistance is measured with the other lead on either end. In this measurement, the RTD resistance is sensed without error contributed from the lead wire reacting with the sensor excitation. Four-wire RTDs yield the most accurate measurements, but are the most expensive RTD configuration." The leads still carry the excitation current, so they raise the voltage at the current source's pin, which has to stay within its compliance; the calculator shows it.

Self-heating

The excitation current dissipates I²R in the element and warms it above whatever it is measuring. SBAA275: "The change in temperature (ΔT) is determined by the power dissipation of the RTD divided by the self-heating coefficient E, in mW/°C", its Equation 14, and "The typical range of RTD self-heating coefficients is 2.5 mW/°C for small, thin-film elements and 65 mW/°C for larger, wire-wound elements." Its own check: "With 1-mA excitation at the maximum RTD resistance value and a larger self-heating coefficient, the power dissipation in the RTD is less than 0.4 mW and will keep the measurement errors due to self-heating to less than 0.01°C." At 400 Ω and 65 mW/°C Equation 14 gives 0.0062 °C; a small thin-film element at 2.5 mW/°C would rise 0.16 °C on the same current.

The power goes as the square of the current, so halving it quarters the error, and a PT1000 on the same current dissipates ten times as much. SBAA275 adds that "Self-heating coefficients will vary with RTD construction and the measurement medium (in air or in water, for example)". The error is also always in one direction: the element reads warm. Against that, a larger current lifts the signal above the noise: "For the best noise performance, maximize the excitation current ... However, most excitation currents should be kept lower than 1 mA because of self heating."

RTD tolerance classes

SBAA275's Table 1-1 lists the classes of the American standard, ASTM E1137, and of IEC 60751, "used world wide". "In both standards, the RTD has the tightest tolerance at 0°C. An absolute error is combined with a proportional error that has a temperature coefficient." The formulas are in degrees; the table below evaluates them, and the last two columns compare the ohms at 0 °C with the table's own column.

SBAA275's "Error at 100°C" column matches the formulas in every row. The "Resistance at 0°C" column is the tolerance times the slope at 0 °C, 0.39083 Ω/°C, in every row but one: Class C prints ±0.24 Ω where 0.6 °C × 0.39083 Ω/°C is 0.2345 Ω. Every value in the column, Class C's included, is the tolerance times a round 0.4 Ω/°C, so the table appears to use that slope. The difference is a hundredth of an ohm and does not change the class. 1/10 DIN "is not included in the IEC 60751 specification but is an industry accepted tolerance for performance demanding applications. It is 1/10th of the DIN IEC Class B specification."

Two things the formula alone does not say. The band widens with |t|, so a Class B element is ±4.55 °C at 850 °C, and since the slope has fallen there that is ±1.332 Ω. And the classes do not hold everywhere: "The specified temperature range of each RTD class tolerance becomes smaller with more accurate grades and classes. Additionally, the range varies with the RTD construction type. For more details about tolerance values and temperature ranges, consult the data sheets of the RTD manufacturer." The calculator applies the formula at whatever temperature is entered; whether the element's class is valid there is a datasheet question.

Where the model stops being valid

Outside −200 °C to 850 °C. SBAA275 treats that span as "the full measurement range of a PT100 RTD", and the calculator refuses temperatures and resistances beyond it rather than extrapolate the polynomials.

The equation is not the element. The CVD equation describes the nominal curve; a real element sits within its tolerance class of it. SBAA275 also notes: "Newer calibration standards allow for more calculation accuracy using higher order polynomials over segmented temperature ranges, but the Callendar-Van Dusen equation remains a commonly used conversion standard."

The reference resistor. In a ratiometric reading the result is R_REF times the code ratio, so "Any error in the reference resistance becomes a gain error in the measurement", and "Assuming the ADC has a low gain error, R_REF is often the largest source of error." A 0.1 % reference resistor is 0.1 % of the RTD's resistance: 0.139 Ω, or 0.37 °C at 100 °C on a PT100. The calculator treats R_REF as exact.

Leakage and mismatch. SBAA275: "if there are leakage currents in the measurement (from TVS or other protection diodes for example), then the leakage contributes to the error." Current that leaves the loop through a protection diode flows in the RTD or in R_REF but not both, and the ratio no longer cancels it. The three-wire cancellation also assumes the two current sources and the two leads match; the calculator takes the currents as matched and lets the lead mismatch be entered.

PT1000 in a PT100 circuit. Switching the element without the circuit breaks the design. At 850 °C a PT1000 through SBAA275's 1620 Ω reference at gain 4 asks for 965 % of full scale, and even at 100 °C it is 343 %. A PT1000 wants a tenth of the current or ten times the reference, and on the same current it self-heats ten times as much.

Common PT100 mistakes

• Using the linear formula R = R₀(1 + α·t) over a wide range. The straight line through 0 °C and 100 °C has α = A + 100·B = 0.00385055 per °C and is exact at those two points only. Read back through it, a PT100 is −1.15 °C out at −50 °C, −3.00 °C at 200 °C and −95.6 °C at 850 °C. The table has the rest; the Class B column is there for scale.
• Forgetting the leads. On a two-wire element they add directly to the reading: 5 m of 24 AWG on a PT100 is +2.16 °C. Three- or four-wire, or a PT1000, cut it down.
• Using the quadratic below 0 °C. It drops the C term, which is −2.42 °C at −200 °C.
• Assuming a PT1000 needs different coefficients. In SBAA275's equation the element size is R₀ alone, and the calculator uses the same A, B and C for both; what changes is ten times the ohms, the slope, the signal and the self-heating on the same current, and a tenth of the lead error.
• Treating SBAA275's rounded range as the element's. 20 Ω to 400 Ω is right for sizing the reference and gain; the element spans 18.52 Ω to 390.48 Ω.
• Raising the excitation current for a cleaner signal without checking Equation 14. A 2 mA current in a thin-film PT100 at 100 °C is 0.2216 °C of self-heating.

Further reading

• TI SBAA275, A Basic Guide to RTD Measurements (Joseph Wu) — the Callendar-Van Dusen equation and IEC 60751 coefficients, the tolerance classes, two-, three- and four-wire circuits with their conversion equations, self-heating, and the ratiometric design example this page works through.
• NTC thermistor calculator — the other resistive temperature sensor: a few per cent per kelvin against a PT100's 0.391 % per °C at 0 °C, exponential rather than nearly straight, and with the same self-heating problem.
• Thermocouple calculator — the temperature sensor that makes a voltage instead of changing a resistance, by the NIST ITS-90 reference functions.
• Wheatstone bridge calculator — the other way to read a resistance change, with the bridge's own nonlinearity.
• ADC resolution calculator — what one code is worth, for the ADC that reads the RTD.
• Wire gauge calculator — the resistance of the lead wires by gauge and length.