What the calculator does
A platinum resistance thermometer turns temperature into resistance along a curve that IEC 60751 fixes for every sensor with α = 0.00385: 100.00 Ω at 0 °C for a PT100, 138.51 Ω at 100 °C, 18.52 Ω at −200 °C. The calculator runs that curve, the Callendar–Van Dusen equation, in both directions: enter a temperature and get the resistance a good sensor should show, or enter a resistance from a multimeter or a transmitter's raw display and get the temperature. It also gives the slope of the curve at that point, which is what turns any resistance error into a temperature error.
Then it does the part that decides the accuracy of most installations: the wiring. Every metre of cable between the sensor and the input adds resistance that a 2-wire connection cannot tell from the sensor; a 3-wire connection lets the input subtract one lead and leaves only the mismatch between the wires; a 4-wire connection measures the sensor alone. The calculator gives the error of each in °C, what the input would display, and the sensor's own tolerance class at that temperature for comparison. Ahmedonics uses it when specifying temperature measurement for the automation systems it designs, and when a customer's reading is a few degrees off and the sensor gets the blame.
Formula
T ≥ 0 °C: R(T) = R0 (1 + A T + B T²)
T < 0 °C: R(T) = R0 (1 + A T + B T² + C (T − 100) T³)
A = 3.9083 × 10⁻³ °C⁻¹ B = −5.775 × 10⁻⁷ °C⁻² C = −4.183 × 10⁻¹² °C⁻⁴
Inverse, R ≥ R0: T = ( −A + √( A² − 4 B (1 − R ÷ R0) ) ) ÷ (2 B)
Inverse, R < R0: Newton iteration on the quartic from the linear estimate T ≈ (R ÷ R0 − 1) ÷ A
Sensitivity: dR/dT = R0 (A + 2 B T) (plus R0 C (4 T³ − 300 T²) below 0 °C)
Tolerance (IEC 60751:2008): AA ±(0.10 + 0.0017 |T|) A ±(0.15 + 0.002 |T|) B ±(0.30 + 0.005 |T|) C ±(0.60 + 0.01 |T|) °C
Wiring error: 2-wire ΔR = 2 RL 3-wire ΔR = RL × mismatch 4-wire ΔR = 0
ΔT = ΔR ÷ (dR/dT)
where R0 is 100 Ω for a PT100 and 1 000 Ω for a PT1000, T is in °C, and RL is the resistance of one lead. The C term only exists below 0 °C, which is why the inverse has a closed form above zero and needs iteration below it. The wiring error is linearised at the sensitivity at T, which is exact to better than 0.01 °C for lead resistances of a few ohms. The tolerance formulas are those of IEC 60751:2008; the earlier edition's "1/3 DIN" and "1/10 DIN" grades are manufacturers' fractions of the old class B and still appear on datasheets.
Worked example
A class A PT100 in a pasteuriser at 100 °C, wired 2-wire through 15 m of 0.5 mm² cable (0.5 Ω per wire) to a PLC RTD input.
- R(100) = 100 × (1 + 0.39083 − 0.005775) = 100 × 1.385055 = 138.51 Ω.
- Sensitivity dR/dT = 100 × (0.0039083 − 2 × 0.0000005775 × 100) = 100 × 0.0037928 = 0.3793 Ω/°C.
- Class A tolerance at 100 °C = ±(0.15 + 0.002 × 100) = ±0.35 °C, which is ±0.35 × 0.3793 = ±0.133 Ω of resistance.
- 2-wire: the input sees 2 × 0.5 = 1.000 Ω of lead in series with the sensor, so ΔT = 1.000 ÷ 0.3793 = +2.64 °C and the PLC displays 102.6 °C for a sensor that is at 100.0 °C. The wiring error is 7.5 times the sensor's tolerance.
- 3-wire with 5 % mismatch between the leads: the input subtracts one lead, leaving 0.5 × 0.05 = 0.025 Ω, so ΔT = +0.07 °C and the display reads 100.1 °C.
- 4-wire: ΔR = 0, the display reads 100.0 °C, and the sensor's ±0.35 °C and the input's own accuracy are all that remain.
In reverse: a sensor whose leads have been subtracted (or a 4-wire measurement) reads 119.40 Ω, so T = (−0.0039083 + √(0.0039083² − 4 × (−5.775 × 10⁻⁷) × (1 − 1.194))) ÷ (2 × (−5.775 × 10⁻⁷)) = 50.0 °C, where the sensitivity is 0.3851 Ω/°C. Below zero: a cold-store PT100 at −40 °C is 100 × (1 − 0.156332 − 0.000924 − 4.183 × 10⁻¹² × (−140) × (−64 000)) = 84.27 Ω, and its sensitivity is 0.3958 Ω/°C; the C term contributes only −0.0037 Ω there, but about −1.0 Ω (2.3 °C worth) at −200 °C, which is why it cannot be dropped for cryogenic work. A PT1000 in the same 2-wire installation at 100 °C would be 1 385.05 Ω with a sensitivity of 3.7928 Ω/°C, and the same 1.000 Ω of lead would cost +0.26 °C instead of +2.64 °C.
PT100 resistance and sensitivity by temperature (IEC 60751, α = 0.00385)
| T (°C) | R (Ω) | dR/dT (Ω/°C) | T (°C) | R (Ω) | dR/dT (Ω/°C) |
|---|---|---|---|---|---|
| −200 | 18.52 | 0.4323 | 200 | 175.86 | 0.3677 |
| −150 | 39.72 | 0.4166 | 250 | 194.10 | 0.3620 |
| −100 | 60.26 | 0.4053 | 300 | 212.05 | 0.3562 |
| −50 | 80.31 | 0.3971 | 400 | 247.09 | 0.3446 |
| 0 | 100.00 | 0.3908 | 500 | 280.98 | 0.3331 |
| 50 | 119.40 | 0.3851 | 600 | 313.71 | 0.3215 |
| 100 | 138.51 | 0.3793 | 700 | 345.28 | 0.3100 |
| 150 | 157.33 | 0.3735 | 800 | 375.70 | 0.2984 |
Tolerance classes (IEC 60751:2008)
| Class | Tolerance | At 0 °C | At 100 °C | At 200 °C | Typical use |
|---|---|---|---|---|---|
| AA | ±(0.10 + 0.0017 |T|) °C | ±0.10 °C | ±0.27 °C | ±0.44 °C | Laboratory and calibration reference; specified over a narrow temperature range |
| A | ±(0.15 + 0.002 |T|) °C | ±0.15 °C | ±0.35 °C | ±0.55 °C | Process measurement where accuracy matters; specified over a narrower range than B |
| B | ±(0.30 + 0.005 |T|) °C | ±0.30 °C | ±0.80 °C | ±1.30 °C | The industrial default; the full sensor range |
| C | ±(0.60 + 0.01 |T|) °C | ±0.60 °C | ±1.60 °C | ±2.60 °C | Where the reading is indicative only |
Assumptions and limitations
- The IEC 60751 curve with α = 0.00385. This is the world standard and what almost every transmitter and PLC input expects. Sensors to the older JIS and US industrial curve with α = 0.003916 exist; their resistance changes 1.7 % more per degree, so one read on a 0.00385 input shows about 1.7 °C high at 100 °C (39.16 Ω instead of 38.51 Ω across 0–100 °C). Check the sensor's marking if a new probe reads consistently high.
- Self-heating. The input passes a current through the sensor to measure it, and the power heats the element. At 1 mA through 138.5 Ω that is 0.14 mW; a small probe in still air can rise by a few tenths of a degree from that, a probe in flowing water by far less. Keep the excitation at or below 1 mA (most industrial inputs use 0.1–1 mA) and do not calibrate in still air.
- The sensor reads its own temperature, not the process's. Immersion depth (at least ten times the probe diameter plus the length of the sensing element, as a rule of thumb), stem conduction along the sheath to a cold head, a thermowell that touches the pipe wall, and a slow response in a heavy well all produce errors the calculator cannot see. A perfectly wired class A sensor can still be 2 °C wrong in a poor installation.
- Lead resistance changes with temperature. Copper's resistance rises about 0.39 % per °C, so the 1.0 Ω loop of the example changes by 0.12 Ω, about 0.3 °C of reading, between a cable at 10 °C in January and one at 40 °C in a June cable tray. Entering an offset in the transmitter fixes a 2-wire loop at one cable temperature only; 3-wire compensates continuously.
- The transmitter's own accuracy adds. A good head-mounted transmitter contributes about ±0.1–0.2 °C; a PLC RTD input channel is often ±0.5 °C or more of span. Add it to the sensor class and the wiring error to get what the loop really delivers.
- Thermocouples are different. Their errors come from the cold-junction compensation, extension-cable type and the alloy's stability, none of which this calculator covers.
- Range. The equation is defined from −200 °C to 850 °C; a thin-film PT100 is usually limited to about 500 °C by its construction, and a wire-wound element to 600–850 °C depending on the sheath.
Frequently asked questions
Why a PT100 and not a thermocouple?
Accuracy and stability below about 500 °C. A class A PT100 is ±0.35 °C at 100 °C and drifts very little; a type K thermocouple is ±1.5 °C at best (class 1) and its alloy ages. A thermocouple wins on range (to 1 200 °C and beyond), on speed (a bare junction responds in a fraction of a second) and on cost of the sensor itself, and it needs no excitation current. For a pasteuriser, a reactor jacket, a cold store or an HVAC plant, a PT100 is the right choice; for a furnace or a kiln, a thermocouple.
Two, three or four wires: which should I buy?
Three-wire is the industrial default and what most PLC inputs and transmitters expect; it removes the lead resistance to within the mismatch between the wires, which is a few hundredths of a degree on a sound installation. Four-wire is for laboratory and calibration work and for long or thin cable runs where even the mismatch matters; it needs a 4-wire input. Two-wire is acceptable only when the leads are very short, when the sensor is a PT1000, or when a head-mounted transmitter sits at the sensor and the two wires carry 4–20 mA instead of resistance.
What is a head-mounted transmitter, and why does it fix the wiring problem?
A small converter in the sensor's connection head that measures the element (usually 3- or 4-wire, over a few centimetres of lead) and sends the temperature out as a 4–20 mA current, or a digital signal, on two wires. Current is immune to the resistance of the cable, so the cable length no longer matters, the PLC needs an ordinary analogue input instead of an RTD card, and the transmitter's linearisation and range are set at the sensor. The current loop guide and the 4–20 mA scaling calculator cover the loop side.
PT1000 or PT100?
A PT1000 has ten times the resistance at every temperature (1 000 Ω at 0 °C, 1 385.05 Ω at 100 °C), so the same lead resistance is a tenth of the temperature error: 1 Ω of lead is +2.64 °C on a PT100 at 100 °C and +0.26 °C on a PT1000. That makes 2-wire connections workable and suits battery-powered and low-excitation devices, since the same measuring current gives ten times the signal voltage. Its element is smaller and a little more sensitive to self-heating per milliampere, and the input must be configured for it; a PT1000 on a PT100 input shows an over-range fault, not a wrong number.
How do I check a sensor with a multimeter?
Disconnect it and measure across the two element wires (on a 3-wire sensor, the two of the same colour are the pair; the third joins one of them at the element and should read near 0 Ω to its partner). At room temperature, about 23 °C, a PT100 reads about 109 Ω; in a cup of crushed ice and water it reads 100.0 Ω within its class; in boiling water at sea level about 138.5 Ω, a little less at altitude. An open circuit, a short, or a resistance that jumps when the cable is flexed means the sensor or its cable is dead. A reading a few ohms high that will not settle usually means moisture in the head.
Why does my reading drift or jump?
Moisture in the connection head is the most common cause: it puts a leakage resistance in parallel with the element and reads low, and it changes with the weather. Then corroded or loose terminals, which add resistance and read high on a 2-wire connection and produce mismatch on a 3-wire one; vibration fatiguing the element leads; a cable run alongside a drive output picking up noise; and a thin-film element that has been over-temperature. A reading that steps by 2–3 °C when the plant starts up is usually a 3-wire sensor on an input configured as 2-wire, with the cable warming up.
References
- IEC 60751:2008, Industrial platinum resistance thermometers and platinum temperature sensors — the resistance–temperature relationship, coefficients A, B, C, the reference table and tolerance classes AA to C
- ASTM E1137/E1137M, Standard Specification for Industrial Platinum Resistance Thermometers — the US specification, with the same α = 0.00385 curve
- Callendar, H. L., "On the practical measurement of temperature", Philosophical Transactions of the Royal Society A, vol. 178, 1887; and Van Dusen, M. S., "Platinum-resistance thermometry at low temperatures", Journal of the American Chemical Society, vol. 47, 1925 — the origin of the quadratic and of the sub-zero C term
- Manufacturer documentation for the RTD input or head-mounted temperature transmitter in use — input accuracy, excitation current, supported wiring and the sensor types it linearises