Uncertainty budget for a thermocouple calibration

A comparison calibration of a thermocouple has the usual terms — reference thermometer, source stability, repeatability — plus three that belong to thermocouples specifically: the reference junction, the emf measurement, and inhomogeneity of the wire. The third dominates almost every real budget, and it is the one that behaves least like an uncertainty and most like a property of the thermocouple's history.

A worked budget: type K at 400 °C

Comparison against a reference PRT in a furnace, ice-point reference junction, a voltmeter good to 2 µV, and a used base-metal thermocouple whose inhomogeneity has been evaluated by moving it through the gradient. Combined by the GUM law of propagation for uncorrelated inputs.

That gives u_c = 0.376 °C and U = 0.75 °C at k = 2. One line carries 84.9 % of the variance. Remove inhomogeneity entirely — a brand-new, never-cycled thermocouple — and the same budget gives U = 0.29 °C. Everything else in the table is noise beside it.

Why inhomogeneity dominates, and why you cannot calibrate it away

A thermocouple generates emf wherever a temperature gradient crosses the wire, not at the junction. If the wire is homogeneous, only the junction temperature matters and where the gradient sits is irrelevant. Thermal cycling changes the wire's composition along its length — oxidation, cold work, contamination — and once it is inhomogeneous the emf depends on where the gradient falls along it.

That is fatal to the usual logic of calibration. A calibration measures the thermocouple with the gradient in one place. Move it to a different immersion depth, a different furnace, a different lagging arrangement, and the gradient moves with it and the correction no longer applies. You can quantify inhomogeneity by withdrawing the thermocouple through a gradient in steps and watching the reading move, but you cannot subtract it.

The reference junction is worth more than the voltmeter

Every value in a thermocouple table is against a 0 °C reference junction, so however you realise that junction enters the result directly. A well-made ice point is 0.01–0.05 °C. Electronic cold-junction compensation in an indicator is often specified at 0.5 °C or worse, because it is measuring the terminal block with a small sensor in a box that has its own thermal gradients.

Swap the 0.03 °C ice point in the budget above for 0.5 °C electronic compensation and U goes from 0.75 °C to 0.95 °C — the second-largest term in the budget, from a component many people never write down. If the instrument will be used with its own compensation, calibrate it that way, as a system.

Converting voltage terms to temperature

Voltmeter uncertainty, resolution and noise are all in volts and have to be divided by the Seebeck coefficient at the calibration point to enter a temperature budget. That coefficient is a function of temperature: for type K it is 40.3 µV/°C at 20 °C and 42.2 µV/°C at 400 °C, so the same 2 µV is worth 0.050 °C at the low end and 0.047 °C at the high end. For a type N, at 26.6 µV/°C near room temperature, the same 2 µV is worth 0.075 °C — half again as much, which is the cost of type N's lower output.

The rest of the chain

• Extension or compensating cable: if the cable is part of the calibrated item, its tolerance class contributes; if it is not, say so on the certificate so nobody assumes it was included.
• Immersion: sheath conduction along the thermocouple is usually worse than for a PRT of the same diameter. Evaluate it by varying depth, not by assuming the manufacturer's minimum.
• Source gradient: a furnace is not a stirred bath. The ±0.20 °C above is optimistic for many block and furnace sources and should be measured, not adopted.
• Repeatability: Type A from repeated readings, with its real degrees of freedom. Five readings give ν = 5, and that does affect the coverage factor once a Type A term matters.

The practical lesson is uncomfortable: a type K thermocouple calibrated to 0.1 °C in a stirred bath is not a 0.1 °C thermometer in a furnace. If you need better than about half a degree below 600 °C, the honest answer is usually a platinum resistance thermometer rather than a better thermocouple budget.

For comparison, IEC 60584-1 class 1 for type K at 400 °C is ±1.6 °C on new wire. The 0.75 °C calibrated uncertainty above is better than that, which is the whole point of calibrating — but only for as long as the thermocouple's thermal history stays close to what it was when the certificate was written.