Question 3 of 7: Thermocouples and Thermistors

Question 3: Thermocouples and Thermistors (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) A thermocouple junction is a passive, unpowered metal-to-metal contact — it has very low source impedance and generates no self-noise beyond the small Johnson (thermal) noise of its own low resistance, so although the wanted Seebeck signal is small in absolute terms ($52\,\mu V/^{\circ}\text{C}$), the noise floor it competes against is also extremely small. What matters for signal-to-noise ratio is the ratio of signal to noise, not the absolute signal size: a low-noise low-impedance source can have excellent SNR even at microvolt signal levels, provided the downstream amplifier itself is a low-noise design and the leads are properly shielded/twisted (Q2i) against externally coupled interference.

b) A thermocouple only measures a voltage proportional to the temperature difference between its two junctions (the measuring/hot junction and the reference/cold junction) — the Seebeck effect produces zero net voltage around a loop of uniform-temperature dissimilar metals; the EMF only appears because the two junctions are at different temperatures. To convert the measured voltage into an absolute temperature at the hot junction, the temperature of the other (reference/cold) junction must be independently known and held fixed (or measured and compensated), which is the cold junction reference — without it, the same output voltage could correspond to many different hot-junction temperatures depending on what the unknown cold-junction temperature happens to be.

c) Replace the physical ice bath with electronic cold-junction compensation: measure the actual local temperature of the reference terminal block (where the thermocouple wires connect to the copper instrument leads) with a separate, independently-calibrated local temperature sensor (e.g. a thermistor, RTD, or semiconductor IC temperature sensor mounted right at that block), then use its reading to compute and add the exact voltage the thermocouple would have generated between that actual block temperature and a true $0^{\circ}\text{C}$ reference (using the same Seebeck relation), correcting the raw measured EMF electronically instead of physically forcing the reference junction to $0^{\circ}\text{C}$ with ice.

d) Given. Voltage divider $V_{out}=V_{supply}\,R_{load}/(R_{load}+R_{th})$, thermistor resistance $R_{th}$ at the mid-range temperature of interest.

Find. The value of $R_{load}$ that maximizes sensitivity $|dV_{out}/dR_{th}|$.

Approach. Differentiate the divider output with respect to $R_{th}$, then maximize the resulting sensitivity expression with respect to $R_{load}$.

1. Sensitivity to a thermistor resistance change. $$\left|\dfrac{dV_{out}}{dR_{th}}\right|=\dfrac{V_{supply}\,R_{load}}{(R_{load}+R_{th})^2}.$$
2. Maximize over $R_{load}$. Treating $R_{th}$ (its value at the centre of the expected range) as fixed and differentiating the sensitivity expression with respect to $R_{load}$, then setting the result to zero, gives $$\boxed{R_{load}=R_{th}}$$ — the same mathematical form as the maximum-power-transfer condition.

Matching $R_{load}$ to the thermistor's resistance at the middle of the expected temperature range (rather than at either end) keeps the divider close to its peak sensitivity across the whole span the sensor will actually see in service, since sensitivity falls off symmetrically as $R_{load}$ and $R_{th}$ diverge in either direction.

e) Self heating is the temperature rise of the sensor itself caused by the electrical power it dissipates while carrying the measurement excitation current, $P=I^2R_{th}$ (equivalently $P=V_{supply}^2R_{th}/(R_{load}+R_{th})^2$ for the divider circuit of part d). This raises the thermistor's own temperature above the true temperature of the medium being measured, introducing a systematic (upward-biased) error that grows with excitation power. It is minimized by keeping the supply voltage/current as low as practical for the required signal level, by mounting the thermistor with good thermal contact to a medium with high heat capacity/flow (so the heat is carried away quickly), and/or by exciting the sensor only in short, low-duty-cycle pulses and reading it before it has time to self-heat significantly, rather than continuously.