Temperature Sensing — Thermistors and Thermocouples: 22-Agric-A5 Principles of Instrumentation, May 2015, Q5

Question 5: Temperature Sensing — Thermistors and Thermocouples (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) Given. A voltage divider with the fixed load resistor $R_{Load}$ on top (connected to $V_{supply}$) and the thermistor $R_{Th}$ below it to ground, with $V_{out}$ tapped at their junction. Find. $V_{out}$ as a function of $R_{Load}$, $R_{Th}$; the value of $R_{Load}$ that maximizes the divider's sensitivity to changes in $R_{Th}$ at a given operating point.

Fig. 2 — Thermistor voltage divider: $V_{out}$ is tapped between the fixed load resistor and the thermistor.

Approach. Write $V_{out}$ from the divider ratio, define sensitivity as $|dV_{out}/dR_{Th}|$ since it is the thermistor's resistance that changes with temperature, then maximize that sensitivity over $R_{Load}$ for a fixed operating $R_{Th}$.

Divider voltage. $V_{out}$ is measured at the junction with respect to ground, so it is the drop across the thermistor arm: $$V_{out}=V_{supply}\cdot\dfrac{R_{Th}}{R_{Load}+R_{Th}}.$$
Sensitivity to the thermistor's resistance. Differentiating with respect to $R_{Th}$ (holding $R_{Load}$, $V_{supply}$ fixed): $$S(R_{Load})=\left|\dfrac{dV_{out}}{dR_{Th}}\right|=\dfrac{V_{supply}\,R_{Load}}{(R_{Load}+R_{Th})^2}.$$
Maximize over $R_{Load}$ at fixed operating $R_{Th}$. Setting $dS/dR_{Load}=0$: $$\dfrac{dS}{dR_{Load}}=V_{supply}\cdot\dfrac{(R_{Load}+R_{Th})-2R_{Load}}{(R_{Load}+R_{Th})^3}=V_{supply}\cdot\dfrac{R_{Th}-R_{Load}}{(R_{Load}+R_{Th})^3}=0.$$ The numerator vanishes, and $S$ changes from increasing to decreasing there (a genuine maximum, confirmed by scanning $S(R_{Load})$ either side of the stationary point), only when $$\boxed{R_{Load}=R_{Th}.}$$

Quantity	Result
$V_{out}$	$V_{supply}\cdot R_{Th}/(R_{Load}+R_{Th})$
Sensitivity $\|dV_{out}/dR_{Th}\|$	$V_{supply}R_{Load}/(R_{Load}+R_{Th})^2$
Load resistor for max sensitivity	$R_{Load}=R_{Th}$

This is the same "matched-source" result as maximum power transfer: the divider's sensitivity to a small change in $R_{Th}$ is greatest exactly when the two arms of the divider are balanced, so a practical design picks $R_{Load}$ equal to the thermistor's resistance at the middle of the expected operating temperature range, accepting reduced sensitivity toward the extremes of that range.

b) A basic thermocouple circuit runs the dissimilar-metal pair (metal A / metal B) from the hot (measurement) junction, through an isothermal terminal block where they meet copper instrument leads (this junction pair is the "cold" or reference junction), and on to the measuring instrument (amplifier + cold-junction compensation + ADC/DAQ):

Fig. 3 — Thermocouple measurement circuit: hot junction, isothermal reference block (with a local temperature sensor for compensation), copper leads to the amplifier/DAQ.

c) A thermocouple does not measure absolute temperature directly — the Seebeck EMF it generates is a function of the difference in temperature between the hot (measurement) junction and the reference junction, formed wherever the two dissimilar thermocouple wires connect to the (usually copper) leads going to the instrument. Standard thermocouple reference tables assume that reference junction is held at a known fixed temperature, historically 0°C in an ice bath. If the reference junction is instead sitting at some uncontrolled ambient temperature, the raw measured EMF corresponds to (T_hot − T_ambient), not the desired (T_hot − 0°C) the tables expect. Cold junction compensation adds back the EMF equivalent of the actual reference-junction temperature so the combined signal corresponds to the correct 0°C-referenced value; it is required because maintaining a physical ice bath at every reference junction in a real multi-channel system is impractical.

d) Replace the physical ice bath with an electronic reference: mount an independent, absolute-temperature sensor (a thermistor, RTD, or an integrated semiconductor sensor such as an LM35) directly on the same isothermal terminal block where the thermocouple wires join the copper leads, so it reads the true local temperature of that reference junction, $T_{ref}$. Convert $T_{ref}$ to the EMF the thermocouple would have produced if that junction were actually at 0°C, using the thermocouple's own characteristic (polynomial or look-up table for its type), and electronically (via an op-amp summing/adder stage) or digitally (in software, after the ADC) add that compensation voltage to the raw thermocouple signal before applying the standard 0°C-referenced conversion table. This synthesizes a "virtual" ice point without needing any ice, provided the compensation sensor genuinely shares the same temperature as the two dissimilar-metal junctions (hence mounting all of them on one isothermal block).

Concept & Theory

Topic: Thermistor voltage-divider sensitivity optimization; thermocouple Seebeck effect and cold-junction compensation.

Key relations: $V_{out}=V_{supply}R_{Th}/(R_{Load}+R_{Th})$; sensitivity maximized at $R_{Load}=R_{Th}$; thermocouple EMF is a function of $(T_{hot}-T_{reference})$, not of $T_{hot}$ alone.

Why this works: Both devices convert a temperature-dependent electrical property into a voltage through a circuit whose transfer function has one "knob" the designer controls (the load resistor value; the reference-junction temperature) — recognizing that knob and its effect on the output turns a fixed physical sensor into a properly matched, or properly referenced, measurement.

Common pitfall: Assuming maximum output voltage and maximum sensitivity occur at the same $R_{Load}$ — they don't; with $V_{out}$ taken across the thermistor, the output level keeps growing toward $V_{supply}$ as $R_{Load}\to0$, while sensitivity peaks at the finite value $R_{Load}=R_{Th}$ (where the output sits at only half the supply). A second trap is writing the divider ratio for the wrong arm: $V_{out}$ is measured from the junction to ground, so it is $R_{Th}/(R_{Load}+R_{Th})$, not $R_{Load}/(R_{Load}+R_{Th})$ — the sensitivity magnitude happens to be the same either way, but the transfer function and its trend with $R_{Load}$ are not. For thermocouples, forgetting that the isothermal block's compensation sensor must be at the exact same temperature as the reference junction itself (not just "nearby") reintroduces the very error compensation is meant to remove.

Source: Doebelin, Measurement Systems, 5th ed., Ch. 10 (resistive and thermoelectric temperature sensors); Fraden, Handbook of Modern Sensors, 5th ed., Ch. 4 (thermistors, thermocouples).

Question 5 of 7: Temperature Sensing — Thermistors and Thermocouples

Question 5: Temperature Sensing — Thermistors and Thermocouples (20 marks)