Exercise set

Thermocouple, RTD and Temperature Sensor Exercises

Solved temperature sensor exercises for thermocouples, cold-junction compensation, RTDs, lead-wire bias, self-heating, response lag and release gates.

Branch: Engineering Physics
Content: Exercise set
Updated: Jul 03, 2026
Revision: v1.0.0 · reviewed

These exercises focus on practical temperature measurement with thermocouples, RTDs and temperature transmitters. The goal is to connect sensor physics to installed bias, wiring, compensation, response time, self-heating, calibration and release evidence.

Assume simplified linear coefficients when stated. Real temperature measurements should also check immersion depth, thermal contact, sheath conduction, cold-junction location, lead resistance, excitation current, environmental gradients, calibration bath stability, transmitter scaling, drift history and uncertainty.

Release Evidence Notes

Temperature is easy to display and hard to prove. A release package should state the measured point, sensor construction, mounting method, response time, calibration standard, wiring configuration, compensation method and whether the displayed value represents the process, the sensor tip, the sheath, the surface or the transmitter terminals.

Engineering Boundary Notes

Temperature measurements often fail because the sensor is calibrated but not thermally coupled to the intended measurand. A thermocouple bead, RTD element, surface patch, thermowell and transmitter terminal can all be at different temperatures. The boundary should state whether the decision concerns fluid bulk temperature, wall temperature, winding temperature, component case temperature, junction temperature estimate or ambient condition.

Thermocouples are robust and broad-range, but they depend on cold-junction compensation, polarity, extension wire and thermal-gradient control. RTDs are usually more accurate at moderate temperatures, but lead resistance, excitation current and self-heating can dominate installed error. A release decision should therefore compare the sensor type with the required uncertainty, response time and installation environment rather than choosing from habit.

Dynamic temperature events need special care. A slow probe can pass calibration and still miss a short overtemperature event. If the decision protects equipment, products or patients, the response time and mounting lag should be tested or bounded, not inferred from a catalog value alone.

Common Release Mistakes

Do not treat transmitter calibration as proof of process temperature. The transmitter may be correct while the sensor is poorly immersed, mounted on the wrong surface, affected by radiation, cooled by airflow or delayed by a thermowell. Do not mix thermocouple extension wire with copper junctions away from the compensation terminal. Do not use two-wire RTDs for tight remote measurements unless lead resistance is included in the decision.

Temperature release should also avoid false precision. Displaying tenths of a degree does not mean the installed uncertainty is tenths of a degree. If the tolerance is tight, include calibration uncertainty, installation bias, drift, response time and environmental gradients before accepting the reading.

Validation Package Checklist

Minimum evidence should include sensor type, calibration range, wiring configuration, compensation method, mounting detail, response-time basis, environmental gradient check, drift history and guarded acceptance rule. If any of these are missing, the temperature reading should be treated as an indication rather than final release evidence.

For commissioning, include at least one comparison point against an independent reference or a controlled bath when the sensor affects acceptance, alarms or energy-performance claims.

Scenario Map

Scenario	Exercises	Primary check	Engineering decision
Thermocouple voltage	1, 2, 3, 4	Seebeck voltage, cold junction and polarity	Decide whether the thermocouple signal is interpreted correctly.
RTD resistance and wiring	5, 6, 7, 8	Pt100 resistance, lead-wire bias and three-wire correction	Decide whether wiring error is acceptable.
Error and dynamics	9, 10, 11, 12, 13	self-heating, conduction, time constant, drift and transmitter scaling	Decide whether the installed value is physically valid.
Release gates	14, 15, 16, 17, 18	uncertainty, guard band, sensor selection and final acceptance	Release, derate, recalibrate or relocate the sensor.

Exercise 1: Thermocouple Voltage Estimate

A thermocouple sensitivity is approximately $41\ \mu\text{V}/^\circ\text{C}$ . The hot junction is $120^\circ\text{C}$ and the cold junction is $20^\circ\text{C}$ . Estimate voltage.

Solution

Temperature difference:

\Delta T=120-20=100^\circ\text{C}

Voltage:

V=41(100)=4100\ \mu\text{V}=4.10\ \text{mV}

Engineering Comment

Thermocouple signals are small. Offset, noise and cold-junction error can matter.

Plausibility Check

Tens of microvolts per degree over one hundred degrees gives millivolts.

Exercise 2: Cold-Junction Compensation Error

The true cold junction is $28^\circ\text{C}$ , but the transmitter assumes $22^\circ\text{C}$ . Estimate temperature bias.

Solution

Cold-junction error:

e=28-22=6^\circ\text{C}

The indicated hot-junction temperature is biased by approximately:

6^\circ\text{C}

Engineering Comment

Cold-junction error adds directly to the inferred hot-junction temperature in the simplified model.

Plausibility Check

If compensation is six degrees wrong, the result should be about six degrees wrong.

Exercise 3: Thermocouple Polarity Check

A heated junction should create a positive voltage, but the transmitter reports negative temperature change. What is the likely wiring fault?

Solution

The likely issue is reversed thermocouple polarity:

V_{measured}\approx -V_{expected}

Engineering Comment

Polarity reversal is a commissioning fault. It can look like a control problem when the loop responds in the wrong direction.

Plausibility Check

The sign is wrong while the magnitude may be plausible, which fits reversed leads.

Exercise 4: Extension-Wire Thermal Gradient

A junction between thermocouple wire and copper wire sits $15^\circ\text{C}$ away from the compensated terminal temperature. Estimate bias with $41\ \mu\text{V}/^\circ\text{C}$ sensitivity.

Solution

Equivalent voltage:

V_e=41(15)=615\ \mu\text{V}

Equivalent temperature bias:

15^\circ\text{C}

Engineering Comment

Uncontrolled junctions create unintended thermocouples. Use correct extension wire to the compensation point.

Plausibility Check

The thermal gradient maps directly to an equivalent temperature error.

Exercise 5: Pt100 Resistance at Temperature

Use:

R=R_0(1+\alpha T)

with $R_0=100\ \Omega$ , $\alpha=0.00385/^\circ\text{C}$ and $T=80^\circ\text{C}$ . Find resistance.

Solution

R=100(1+0.00385(80))=130.8\ \Omega

Engineering Comment

The linear formula is a useful screen, but precision RTD work uses the specified calibration curve.

Plausibility Check

Pt100 resistance should rise by about $0.385\ \Omega/^\circ\text{C}$ .

Exercise 6: RTD Temperature from Resistance

A Pt100 reads $119.25\ \Omega$ . Estimate temperature using the same linear coefficient.

Solution

\displaystyle T=\frac{R/R_0-1}{\alpha}

\displaystyle T=\frac{119.25/100-1}{0.00385}=50.0^\circ\text{C}

Engineering Comment

This assumes the measured resistance is only sensor resistance. Lead resistance must be removed or compensated.

Plausibility Check

$19.25\ \Omega$ above $100\ \Omega$ is fifty increments of $0.385\ \Omega$ .

Exercise 7: Two-Wire Lead Resistance Bias

A two-wire RTD has total lead resistance $1.54\ \Omega$ . Estimate temperature bias for a Pt100.

Solution

Pt100 slope:

0.385\ \Omega/^\circ\text{C}

Bias:

\displaystyle e_T=\frac{1.54}{0.385}=4.0^\circ\text{C}

Engineering Comment

Two-wire RTDs are poor for accurate remote measurement unless lead resistance is known and stable.

Plausibility Check

Four Pt100 degrees produce about $1.54\ \Omega$ .

Exercise 8: Three-Wire Mismatch Error

A three-wire RTD has unmatched lead resistance of $0.12\ \Omega$ . Estimate residual temperature bias.

Solution

\displaystyle e_T=\frac{0.12}{0.385}=0.31^\circ\text{C}

Engineering Comment

Three-wire compensation reduces common lead resistance but not mismatch.

Plausibility Check

The bias is much smaller than the two-wire case because only mismatch remains.

Exercise 9: RTD Self-Heating

A Pt100 at $120\ \Omega$ is excited with $1.0\ \text{mA}$ . Find power.

Solution

P=I^2R=(0.001)^2(120)=0.00012\ \text{W}

So:

P=0.12\ \text{mW}

Engineering Comment

Low current helps avoid self-heating, but low signal level can increase noise sensitivity.

Plausibility Check

Milliamp squared times hundreds of ohms gives tenths of a milliwatt.

Exercise 10: Self-Heating Temperature Error

If RTD self-heating coefficient is $0.4^\circ\text{C/mW}$ , find error from $0.12\ \text{mW}$ .

Solution

\Delta T=0.4(0.12)=0.048^\circ\text{C}

Engineering Comment

This error is small in a well-coupled fluid but can be larger in stagnant gas or poor contact.

Plausibility Check

The power is much less than a milliwatt, so the temperature rise is small.

Exercise 11: First-Order Temperature Lag

A temperature probe has time constant $\tau=12\ \text{s}$ . After a step change, what fraction of final change is reached after $24\ \text{s}$ ?

Solution

f=1-e^{-t/\tau}=1-e^{-24/12}=1-e^{-2}=0.865

Engineering Comment

Slow probes can under-report short events. The process may have exceeded a limit before the sensor catches up.

Plausibility Check

After two time constants, first-order systems reach about $86\%$ .

Exercise 12: Surface Sensor Conduction Bias

A surface sensor reads $76^\circ\text{C}$ while an embedded reference reads $82^\circ\text{C}$ . Find bias.

Solution

b=76-82=-6^\circ\text{C}

Engineering Comment

Surface sensors can read low because of contact resistance, ambient cooling and adhesive thickness.

Plausibility Check

The surface reading is below the embedded reference, so bias is negative.

Exercise 13: Transmitter Scaling

A $4$ to $20\ \text{mA}$ temperature transmitter spans $0$ to $200^\circ\text{C}$ . What temperature corresponds to $12\ \text{mA}$ ?

Solution

Current fraction:

\displaystyle f=\frac{12-4}{20-4}=0.5

Temperature:

T=0+0.5(200)=100^\circ\text{C}

Engineering Comment

Scaling checks catch wiring, range and configuration errors before control tuning begins.

Plausibility Check

Mid-current corresponds to mid-range.

Exercise 14: Expanded Uncertainty

Standard uncertainty terms for a temperature channel are $0.30^\circ\text{C}$ , $0.20^\circ\text{C}$ and $0.10^\circ\text{C}$ . Find expanded uncertainty with $k=2$ .

Solution

u_c=\sqrt{0.30^2+0.20^2+0.10^2}=0.374^\circ\text{C}

U=2u_c=0.748^\circ\text{C}

Engineering Comment

Calibration, installation and repeatability should all be represented in the budget.

Plausibility Check

Expanded uncertainty should be larger than the largest standard term.

Exercise 15: Guarded Temperature Limit

A component limit is $85^\circ\text{C}$ . Measured temperature is $83.8^\circ\text{C}$ with $U=0.9^\circ\text{C}$ . Check guarded acceptance.

Solution

Guarded temperature:

T_g=83.8+0.9=84.7^\circ\text{C}

Since:

84.7<85

it passes with $0.3^\circ\text{C}$ guarded margin.

Engineering Comment

The nominal margin is $1.2^\circ\text{C}$ , but the guarded margin is much smaller.

Plausibility Check

Adding uncertainty moves the result toward the limit.

Exercise 16: Sensor Choice by Accuracy

A process needs $\pm 0.5^\circ\text{C}$ decision uncertainty. A thermocouple method has $U=1.2^\circ\text{C}$ and an RTD method has $U=0.35^\circ\text{C}$ . Which can support the decision?

Solution

Thermocouple:

1.2>0.5

fails. RTD:

0.35<0.5

passes.

Engineering Comment

The correct sensor is determined by the decision, not by convenience alone.

Plausibility Check

Only the lower-uncertainty method fits inside the required uncertainty.

Exercise 17: Drift Since Calibration

A temperature transmitter drifted from $0.1^\circ\text{C}$ error to $0.7^\circ\text{C}$ error over $12$ months. Estimate monthly drift.

Solution

\displaystyle r=\frac{0.7-0.1}{12}=0.05^\circ\text{C/month}

Engineering Comment

Drift rate supports interval decisions and impact review when a channel returns out of tolerance.

Plausibility Check

Twelve months at $0.05^\circ\text{C/month}$ gives $0.6^\circ\text{C}$ change.

Exercise 18: Temperature Sensor Release Gate

A channel has:

Check	Result	Gate
Cold-junction error	$0.4^\circ\text{C}$	$\le 0.5^\circ\text{C}$
RTD lead bias	$0.31^\circ\text{C}$	$\le 0.2^\circ\text{C}$
Guarded limit margin	$0.3^\circ\text{C}$	$>0$
Response time	$12\ \text{s}$	$\le 10\ \text{s}$
Calibration status	current	current

Decide release status.

Solution

Cold-junction error passes and guarded margin passes. Lead bias fails:

0.31>0.2

Response time also fails:

12>10

The channel should not be released for fast final acceptance until wiring bias and response time are corrected or the use is restricted.

Engineering Comment

Temperature release depends on installation and dynamics, not only sensor calibration.

Plausibility Check

Two technical gates fail, so restricted release is appropriate.

REF