Self-heating in resistance thermometers

A resistance thermometer is read by passing a current through it and measuring the voltage across it. That current dissipates power in the sensing element, and the element sits slightly above the temperature of whatever surrounds it. You are then measuring the thermometer, not the thing you wanted to measure.

How much power

The dissipated power is I²R, which makes it quadratic in the current and linear in the sensor's resistance. Both terms matter, and the second one catches people out.

A Pt1000 gives ten times the sensitivity of a Pt100, which is why it is the better choice on long two-wire runs. At the same measuring current it also dissipates ten times the power. That trade is usually worth making, but it should be a decision rather than a surprise — and it is an argument for running a Pt1000 at a lower current rather than at the 1 mA a Pt100 datasheet suggested.

How much temperature rise

Power becomes temperature through the self-heating coefficient, in millikelvin per milliwatt, and that coefficient is a property of the sensor in its environment rather than of the sensor alone. It is dominated by how well heat leaves the element. Typical figures, applied to the 0.1 mW a Pt100 dissipates at 1 mA:

The span across that table is a factor of seventy, from an effect you can ignore to one that dominates a good budget. Nothing about the sensor changed — only what surrounds it. This is why a calibration certificate that does not state the medium and the current tells you less than it appears to.

The common failure is calibrating in a stirred bath and using the same thermometer in an air-filled chamber or a cold room. The certificate's self-heating was 0.3 mK; in service it is 20 mK, uncorrected and unbudgeted, and it biases every reading the same way.

Measuring it: the √2 method

Self-heating scales with power, and power scales with the square of current. So if you read the thermometer at current I and again at √2·I, the power exactly doubles and so does the temperature rise. The difference between the two indicated temperatures is therefore equal to the self-heating at the original current I.

• Let the thermometer reach equilibrium at the normal current I and record the indication.
• Switch to √2·I — 1.414 mA if you normally use 1 mA — wait for equilibrium again, and record.
• The difference is the self-heating at I. Either correct for it, or enter it in the budget as a known bias with an uncertainty.
• Do it in the medium you will actually use, not the one that is convenient.

For the lowest uncertainty, repeat at several currents and extrapolate to zero power. Most instruments that offer a choice of sense current also offer this as an automated function.

Keeping it out of the budget

The cleanest approach is to make self-heating cancel rather than to correct it: use the same current, the same sensor and the same kind of medium in calibration and in use, and the bias is common to both and largely divides out. That is rarely achievable in full, but it is the target. Where it cannot be achieved, the residual belongs in the budget as a rectangular component bounded by the difference between the two conditions.

In resistance terms the effect is small but not negligible: a Pt100 at 0 °C has a sensitivity of 0.3908 Ω/°C, so 10 mK of self-heating is 3.91 mΩ. Any bridge capable of resolving a millikelvin is resolving this comfortably, which means it is a real measured quantity and not a rounding artefact.

For most industrial and data-logger applications self-heating sits well below the other terms and can be bounded rather than measured. For reference-grade work below 10 mK it is often the third or fourth largest line in the budget, and treating it as negligible is how a budget comes to understate its own uncertainty.