Question 1 of 6: Thermocouple Transfer Function, Accuracy vs. Precision & Alternative Sensors

Question 1: Thermocouple Transfer Function, Accuracy vs. Precision & Alternative Sensors (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

(a) and (b) are answered as a calculation; (c) and (d) are short essay responses.

Given. Two calibration points on the thermocouple's linear output/input (voltage/temperature) curve: $V_1=82\ \mu\text{V}$ at $T_1=20\,{}^{\circ}\text{C}$, and $V_2=32.6\ \text{mV}$ at $T_2=800\,{}^{\circ}\text{C}$.

Find. (a) the linear transfer function $V(T)=mT+b$; (b) the output voltage at $T=100\,{}^{\circ}\text{C}$.

Approach. Two calibration points fully determine a straight line, so the sensitivity (slope) and offset (intercept) follow directly from the two-point data.

1. Sensitivity (slope). Working in microvolts throughout ($32.6\ \text{mV}=32600\ \mu\text{V}$), $$m=\frac{V_2-V_1}{T_2-T_1}=\frac{32600-82}{800-20}\ \frac{\mu\text{V}}{{}^{\circ}\text{C}}=\frac{32518}{780}=41.69\ \frac{\mu\text{V}}{{}^{\circ}\text{C}}.$$
2. Intercept. $$b=V_1-mT_1=82-41.69(20)=82-833.8=-751.8\ \mu\text{V}.$$
3. Transfer function. $$\boxed{V(T)=41.69\,T-751.8\ \ [\mu\text{V}],\quad T\ \text{in}\ {}^{\circ}\text{C}}$$
4. Part (b) — voltage at 100°C. $$V(100)=41.69(100)-751.8=4169-751.8=3417.2\ \mu\text{V}=\boxed{3.42\ \text{mV}}.$$

(c) Accuracy vs. precision. Accuracy describes how close the measured value is to the true 100°C reference, and is degraded by systematic (bias) error — an uncompensated cold-junction reference, a thermocouple that has drifted from the factory calibration curve used above, or wire-to-wire inhomogeneity along the thermocouple leads. A biased instrument can be perfectly repeatable and still read consistently wrong. Precision describes how closely repeated readings of the same 100°C target agree with one another, and is governed by random error — thermal-EMF noise in the microvolt-level signal, data acquisition/ADC quantization, and contact-resistance fluctuation at the junction — quantified by the standard deviation (spread) of the repeat readings, independent of whether that cluster is centred on 100°C. A thermocouple can therefore be precise but inaccurate (a tight cluster offset from 100°C by a calibration bias) or accurate but imprecise (a widely scattered set of readings that happens to average to 100°C). The two properties are diagnosed differently: accuracy requires comparison against a traceable reference standard, while precision is assessed purely from the statistics of the repeat-reading data itself.

(d) Alternative temperature sensors. A resistance temperature detector (RTD, e.g. a platinum Pt100) uses the near-linear rise of a pure metal's electrical resistance with temperature; it is more accurate and stable long-term than a thermocouple but needs an excitation current and responds more slowly. A thermistor is a semiconductor whose resistance changes far more steeply with temperature than an RTD — more sensitive, but non-linear and usable over a narrower span, so it needs a linearizing circuit or look-up table. A non-contact infrared pyrometer measures the blackbody radiation the target itself emits, making it the practical choice for very high temperatures, moving parts, or surfaces that cannot be touched.