How the temperature measurement error calculator works
A temperature reading in a control system passes through a chain: the sensor, its connection, a transmitter that converts it to 4–20 mA, and the input that reads the current. Each link has its own error, and datasheets state each one separately. The calculator puts them on one scale, kelvin at the process, and combines them.
The terms are:
- Sensor tolerance. The class of the element: AA, A, B or C for platinum resistance thermometers to IEC 60751, class 1, 2 or 3 for thermocouples to IEC 60584-1. This is the limit for an uncalibrated sensor at the process temperature.
- Connection. For an RTD, the error the lead resistance adds with 2, 3 or 4 wires. For a thermocouple, the error of the cold-junction compensation.
- Transmitter input. The digital accuracy on the datasheet, in kelvin.
- Analog output. The digital-to-analog accuracy, as a percentage of the calibrated span.
- Ambient effect. The change of the transmitter’s reading with its own temperature, as a percentage of span per 10 K, times the change since calibration.
- Other errors. Anything else, such as the input card of the controller or a calibration uncertainty.
The total probable error is the root-sum-square of all terms; the worst case adds them.
A worked example
A PT100, class A, at 100 °C, three-wire on 10 m of 0.5 mm² cable with a 5 % lead mismatch. The transmitter is ranged 0 to 200 °C, with an input accuracy of ±0.1 K, an output accuracy of ±0.03 % of span and a temperature effect of ±0.1 % of span per 10 K, in a cabinet 20 K warmer than at calibration.
| Term | Error |
|---|---|
| Sensor, class A at 100 °C | ±0.35 K |
| Lead wires, 3-wire | ±0.045 K |
| Transmitter input | ±0.1 K |
| Analog output, 0.03 % of 200 K | ±0.06 K |
| Ambient effect, 0.1 % × 200 K × 20/10 | ±0.4 K |
| Total probable error | ±0.55 K |
| Worst case | ±0.96 K |
With the same sensor on two wires, the lead term becomes 1.81 K and the total probable error 1.89 K. The RTD lead-wire error calculator shows that term in detail.
What usually dominates
The example shows a pattern that holds for many installations: the transmitter’s ambient effect is as large as the sensor class. It is quoted as a percentage of span, so it grows with a wide range. A range of 0 to 200 °C where 0 to 150 °C would do, or a transmitter in a field housing that sees the sun, costs more than a better sensor saves. Narrowing the span, or a transmitter with a smaller temperature coefficient, is often the cheapest improvement.
The second pattern is two-wire RTDs. Lead resistance adds to the reading directly, and on more than a few metres of cable it outweighs everything else.
Thermocouples: the cold junction
A thermocouple measures the difference between the measuring junction and the terminals. The instrument measures the terminal temperature and adds the matching EMF back. An error of ΔT in that measurement shifts the EMF by the Seebeck coefficient at the terminals times ΔT, which the instrument reads as a temperature error at the process scaled by the ratio of the Seebeck coefficients at the two temperatures. For type K the ratio is close to 1: 0.95 between 25 and 500 °C. For type S at 1000 °C it is 0.52, and for type B near room temperature it is close to zero. The thermocouple calculator gives the coefficients for every type.
What the calculation leaves out
- Installation. Immersion depth, heat conduction along the sheath, response time and radiation are often larger than every term above, and no datasheet states them.
- Drift. Sensors and transmitters drift with time and temperature; the tolerance class describes a new sensor.
- Statistics. Datasheet values are usually limits, not standard deviations, so the total probable error is an engineering estimate, not a confidence level.
Frequently asked questions
Should I use the total probable error or the worst case?
Use the total probable error to judge whether a design is reasonable, and the worst case when an error beyond the requirement is not acceptable at all, for example for a safety limit or a release temperature that must be proven.
How do I enter a calibrated sensor?
A calibration replaces the class tolerance with the calibration uncertainty, often far smaller. Enter that uncertainty under other errors and choose the class the sensor was bought as; the class term then overstates the error, so the result is on the safe side. The PT100 calculator converts calibration points from ohms to temperature.
My transmitter quotes accuracy as a percentage of span only. What do I enter?
Enter zero for the input accuracy and the percentage under output accuracy. If it quotes “±0.1 K or 0.05 % of span, whichever is greater”, enter the larger of the two as input accuracy.