Temperature measurement

Temperature is measured indirectly, through a property that changes with it: an emf, a resistance, a junction voltage, or emitted radiation. Absolute temperature $T$ is in kelvin (K). Any law with $T$ in an exponent, a reciprocal or a fourth power needs kelvin. Laws that are linear in a temperature difference can use degrees Celsius.

When two dissimilar metals A and B are joined at two junctions held at different temperatures, an emf appears in the loop. This is the Seebeck effect. The emf depends only on the metals and on the two junction temperatures. It is measured with one junction at a known reference temperature. Unless stated, tables and equations give the emf with the reference junction at 0 °C.

Reference-junction compensation: the instrument sees $E(T_{hot})-E(T_{ref})$, so the true hot-junction emf $E(T_{hot},0)$ is found by adding $E(T_{ref},0)$. An ice bath fixes the reference at 0 °C. Electronic compensation measures the terminal temperature with an RTD or semiconductor sensor and adds a voltage equal to $E(T_{ref},0)$. Extension or compensating cables carry the junction to the instrument, and must be of the same pair (or matching emf) up to the reference point.

A metal's resistance rises with temperature. Platinum is the standard material because it is stable and nearly linear. A Pt100 has $R_0=100\Omega$ at 0 °C. The temperature coefficient $\alpha$ is referred to 0 °C unless the question states otherwise, and for standard platinum it is about $0.00385{/}^{\circ }$C.

Self-heating: the excitation current dissipates $I^2{R}_T$ in the RTD. The temperature error is $\Delta T=I^2{R}_T/\delta$, where $\delta$ is the dissipation constant in W/°C. Keeping the current small limits it.

A thermistor is a semiconductor resistor. An NTC (negative temperature coefficient) thermistor's resistance falls steeply as temperature rises. It is far more sensitive than an RTD, but non-linear and useful over a narrow range. Temperature in these formulas is in kelvin.

Self-heating: power $I^{2}R$ raises the thermistor above the ambient by $\Delta T=I^{2}R/\delta$. This error is large because R is large and $\delta$ is small. Linearisation: a fixed resistor in parallel with the thermistor flattens its curve and lowers the sensitivity. A series resistor in a voltage divider is also used. Neither makes the response truly linear over a wide range.

Semiconductor (IC) temperature sensors use the junction of a transistor. The base-emitter voltage falls by about 2 mV per °C for silicon at room temperature (CTAT). The difference of the base-emitter voltages of two matched transistors run at a current-density ratio $n$ is proportional to absolute temperature (PTAT). The ratio $n$ comes from a current ratio $n$ with equal emitter areas, or from equal currents with an emitter-area ratio $n$. IC sensors give a voltage output (for example 10 mV/°C) or a current output (for example 1 µA/K).

A radiation (total radiation) pyrometer focuses the radiation from a hot surface on a detector, often a thermopile. The radiant emittance of a surface of emissivity $\epsilon$ obeys the Stefan-Boltzmann law. Emissivity is 1 for a black body and less than 1 for real surfaces. The instrument is calibrated against a black body, so a real surface appears cooler than it is. While the target fills the instrument's field of view, the reading does not depend on the distance to the target.

An optical pyrometer (disappearing filament type) compares the brightness of the target with a calibrated lamp filament through a narrow band near $\lambda \approx 0.65\mu$m. The operator adjusts the lamp current until the filament vanishes against the target. In a narrow band the Wien approximation $L_{\lambda }\propto {\epsilon }_{\lambda }{\lambda }^{-5}{e}^{-C_2/\lambda T}$ holds when $\lambda T$ is much smaller than $C_2=1.4388\times 10^{-2}$ m K, which is true at 0.65 µm over the usual pyrometer range.

A bolometer is a very small element (a thermistor, a metal film or a barretter wire) that absorbs radiant power, warms, and changes its resistance. It is placed in a bridge, and the unbalance gives the absorbed power, for infrared radiation or microwave power. Its sensitivity can be stated as $\Delta R/\Delta P$ (Ω/W) for the element, or as bridge output volts per watt. A thermistor bolometer has high sensitivity and a negative coefficient. A metal one has a positive coefficient and is more rugged.

A bare sensor in a fluid exchanges heat by convection. If the sensor is treated as a lumped mass with one thermal capacity, its response is first order: the rate of heat storage equals the heat entering by convection. The lumped model needs the temperature inside the sensor to be nearly uniform, that is, a small Biot number ($h{L}_c/k$ below about 0.1). A thermowell or sheath adds a second lag and the response becomes second order, but at GATE level a first-order response is the standard model.