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Electrical Circuits and Sensors · Module 8 of 10

Sensors and Transducers

A sensor turns a physical quantity into an electrical one you can read. Strain becomes resistance, temperature becomes voltage, and the engineer's job is to know the conversion and its limits.

01

Readiness check

This module turns physics into voltage. Tick only what you can do closed-notes.

  • Recall that resistance depends on length, area, and resistivity.
  • Compute a small fractional change ΔR/R.
  • Recall the meaning of strain ε as ΔL/L.
  • Read a linear relation y = mx + b.
  • Convert between Ω, mΩ, and parts per million.
0 or 1 weak itemsContinue with this module.
2 weak itemsRevisit resistance and Ohm's law in Module 1.
3 or more weak itemsConnect to measurement practice in Robotics and Mechatronics.
02

The core idea

A sensor maps a measurand to an electrical output through a known characteristic. For a strain gauge that map is the gauge factor; for an RTD it is a temperature coefficient. Knowing the map both ways is the measurement.

strain gauge: ΔR/R = GF · εRTD: R(T) = R0(1 + αT)output = sensitivity × measurand

Every sensor has a characteristic that links the physical input to the electrical output. A metallic strain gauge stretches with the surface it is bonded to; its resistance rises in proportion to strain, with the gauge factor GF (about 2 for foil gauges) as the constant. A resistance temperature detector (RTD) uses the steady rise of a metal's resistance with temperature, R(T) = R0(1 + αT), where α is the temperature coefficient. A thermocouple instead generates a small voltage from a temperature difference between two junctions. In each case the slope of the characteristic is the sensitivity, and a good measurement means knowing that slope, its linearity, and the range over which it holds.

The skill works when: you write the sensor characteristic and use its slope to convert between measurand and output.
The skill breaks down when: a sensitivity is used outside its linear range, or a self-heating or lead resistance is ignored.
The concept. A transducer sits between the physical world and the circuit, converting strain or temperature into a resistance or voltage through a characteristic whose slope is the sensitivity.
03

The skills, taught in order

Five skills cover the measurement chain, the two resistive sensors, the thermocouple, and the static characteristics that describe them all.

8.1 The measurement chain

A measurement runs from the measurand to a sensor, then through signal conditioning, to a display or data logger. Each stage can add error, so the chain is only as good as its weakest link. The sensor sets the sensitivity; later stages amplify and digitise.

8.2 The strain gauge

A strain gauge is a fine resistive grid bonded to a surface. When the surface strains by ε, the grid's resistance changes by ΔR/R = GF · ε, where the gauge factor GF is near 2 for metal foil. Strains are tiny, microstrain, so the resistance change is a fraction of a percent and must be read carefully.

SensorMeasurandOutputCharacteristic
Strain gaugestrainresistanceΔR/R = GF · ε
RTDtemperatureresistanceR0(1 + αT)
Thermocoupletemperature differencevoltageSeebeck coefficient
Thermistortemperatureresistancenonlinear, large slope

Four common sensors and the characteristic that maps their measurand to an electrical output.

8.3 Resistance temperature detectors

An RTD exploits the near-linear rise of a metal's resistance with temperature: R(T) = R0(1 + αT). A platinum Pt100 has R0 = 100 Ω at 0 °C and α ≈ 0.00385 per °C. RTDs are accurate and stable but need care with lead resistance and self-heating.

8.4 Thermocouples and thermistors

A thermocouple produces a voltage from the temperature difference between a measuring and a reference junction, via the Seebeck effect; it covers wide ranges but gives small, nonlinear outputs. A thermistor is a semiconductor whose resistance changes steeply with temperature, sensitive but markedly nonlinear.

8.5 Static characteristics

Sensors are described by sensitivity (output per unit input), range, linearity, hysteresis, and resolution. These static characteristics tell you how to convert a reading and how far to trust it, and they govern the choice of sensor for a task.

Engineering connection: the tiny ΔR/R of a strain gauge is read by placing it in a Wheatstone bridge, the subject of Module 9.

04

Worked example 1: a strain-gauge bridge output

A 120 Ω strain gauge with gauge factor 2 forms one arm of a quarter-bridge with 5 V excitation. The surface strains by 500 microstrain. Find the resistance change and the bridge output voltage.

Figure 1. The strained gauge unbalances the bridge by a tiny amount. A 500 microstrain input changes the 120 ohm gauge by 0.12 ohm and produces a 1.25 millivolt output: small, but readable after amplification.
  1. ProblemFind ΔR and the bridge output for the strain gauge in Figure 1.
  2. Given / findR = 120 Ω, GF = 2, ε = 500 µε = 500×10−6, Vs = 5 V, quarter bridge. Find ΔR and Vo.
  3. AssumptionsOne active gauge; small change, so the linear quarter-bridge approximation holds.
  4. ModelΔR = GF · ε · R; for a quarter bridge Vo ≈ (Vs/4)(ΔR/R).
  5. EquationsΔR/R = GF · εΔR = (GF · ε) RVo = (Vs/4)(ΔR/R)
  6. SolveΔR/R = 2 × 500×10−6 = 1×10−3. ΔR = 0.001 × 120 = 0.12 Ω. Vo = (5/4) × 0.001 = 1.25 mV.
  7. CheckThe fractional change is 0.1 percent, and a quarter-bridge gives a quarter of Vs times that, 1.25 × 0.001 V = 1.25 mV. The output is far below the supply, so amplification is essential.
  8. ConclusionEven a healthy strain of 500 microstrain shifts the gauge by only a tenth of an ohm and the bridge by a millivolt, which is why bridges and amplifiers exist.
Result. ΔR = 0.12 Ω and Vo = 1.25 mV.
05

Worked example 2: an RTD reading

A platinum Pt100 RTD has R0 = 100 Ω at 0 °C and a temperature coefficient α = 0.00385 per °C. Find its resistance at 150 °C.

Figure 2. The platinum RTD rises almost linearly with temperature. From 100 ohms at 0 degrees, a 150 degree rise adds about 58 ohms, giving 157.75 ohms.
  1. ProblemFind the resistance of the Pt100 RTD at 150 °C in Figure 2.
  2. Given / findR0 = 100 Ω, α = 0.00385 per °C, T = 150 °C. Find R(150).
  3. AssumptionsLinear characteristic over this range; lead resistance and self-heating neglected.
  4. ModelApply the RTD characteristic R(T) = R0(1 + αT).
  5. EquationsR(T) = R0(1 + αT)
  6. SolveR(150) = 100(1 + 0.00385 × 150) = 100(1 + 0.5775) = 100 × 1.5775 = 157.75 Ω.
  7. CheckThe rise is αT = 0.5775, so 57.75 Ω above 100 Ω, a 57.75 percent increase over 150 °C, consistent with platinum's roughly 0.385 percent per degree.
  8. ConclusionReading 157.75 Ω corresponds to 150 °C. Inverting the same linear law turns a measured resistance back into a temperature.
Result. R(150 °C) = 157.75 Ω.
06

Misconceptions and diagnostics

MistakeSymptomDiagnostic questionCorrection
Microstrain mishandledΔR off by 106"Is ε in strain, not microstrain?"500 µε means ε = 500×10−6.
Sensitivity outside rangeLinear law used far from calibration"Is the input within the linear range?"Use the characteristic only where it is linear.
Ignoring lead resistanceRTD reads a few degrees high"Did I account for the lead wires?"Use three- or four-wire connections for RTDs.
Thermocouple needs no referenceTemperature read from voltage alone"Where is the reference junction?"A thermocouple measures a difference, needing a known reference.
07

Practice ladder

Level 1 · Direct skill

A 350 Ω gauge with GF = 2 sees 1000 microstrain. Find ΔR.

Show answer

ΔR = GF · ε · R = 2 × 0.001 × 350 = 0.70 Ω.

Level 2 · Mixed concept

A Pt100 RTD (α = 0.00385) reads 119.4 Ω. What temperature is that, approximately?

Show answer

From R = R0(1 + αT): 119.4 = 100(1 + 0.00385T), so 0.00385T = 0.194, T = 50.4 °C.

Level 3 · Independent problem

For the Example 1 gauge (120 Ω, GF = 2, 5 V), what strain gives a 2.5 mV quarter-bridge output?

Show answer

Vo = (Vs/4)(GF · ε), so 0.0025 = 1.25 × 2 × ε, giving ε = 0.0025/2.5 = 0.001 = 1000 µε.

Transfer task | Real engineering

You must measure both fast vibration strain and slow oven temperature. Argue which sensor suits each, citing one static characteristic.

What good work looks like

A strain gauge for vibration (fast response, linear over its strain range) and an RTD or thermocouple for the oven (wide range, stable sensitivity), each justified by range, linearity, or response time.

08

Working with AI, and proving it yourself

Use AI as an examiner, not a solver

"Check that I kept microstrain and strain straight in this calculation."
"Give me three sensors; I will state each one's measurand and characteristic."
"Which sensor should I use?" Matching the static characteristics to the task is the skill.
"Convert this resistance to temperature." Inverting the characteristic yourself is the point.

Portfolio task

Pick a real sensor, write its characteristic, and convert one reading to a measurand and back, noting the range over which the conversion holds.

Must include: the characteristic, a sensitivity, a two-way conversion, and a stated range.
09

Retrieval and spaced review

Closed notes. Answer out loud, then reveal.

1. Write the strain-gauge characteristic.

ΔR/R = GF · ε, with GF near 2 for foil gauges.

2. Write the RTD characteristic.

R(T) = R0(1 + αT), linear over a wide range.

3. What does a thermocouple measure?

A voltage from the temperature difference between two junctions.

4. What is sensitivity?

The slope of the characteristic: output per unit measurand.

5. Why is a strain-gauge output so small?

Strains are microstrain, so ΔR/R is a fraction of a percent, giving a millivolt-level signal.

TodayFinish this quiz and Levels 1 and 2 of the ladder.
+1 dayRe-derive the gauge and RTD readings from a blank page.
+3 daysConvert three sensor readings to measurands.
+7 daysCarry the bridge into signal conditioning, Module 9.
+30 daysReuse sensor characteristics when building a measurement system.
10

Textbook mapping

This module draws on Alexander and Sadiku, Fundamentals of Electric Circuits, 4th edition for the resistive elements, with sensor characteristics following standard measurement texts.

Topic in this moduleWhere to read more
Resistance and resistive sensorsAlexander & Sadiku, Chapter 2
Strain gauges, RTDs, thermocouplesFigliola & Beasley, measurement systems
Static characteristics of sensorsFigliola & Beasley, measurement systems

Circuit fundamentals follow the 4th edition of Alexander and Sadiku; sensor characteristics use standard values from measurement texts such as Figliola and Beasley.