What is emissivity and why does it matter for pyrometers?

Emissivity (ε) is the ratio of radiation emitted by a real surface to that of a perfect black body at the same temperature (range 0–1). If the pyrometer emissivity setting differs from the actual material emissivity, a temperature correction is needed: T_true⁴ = T_measured⁴ × (ε_setting / ε_actual).

How is a potentiometer used as a voltage divider?

Connect the two end terminals across the supply voltage Vin. The wiper terminal gives an output voltage: Vout = Vin × (R₂/(R₁+R₂)), where R₂ is the resistance from the wiper to the ground terminal. Moving the wiper continuously varies the output from 0 V to Vin.

What are the practical applications of NTC thermistors?

NTC thermistors are used in digital thermometers, HVAC thermostat sensors, automotive coolant sensors, 3D printer hotend control, battery management systems, and medical patient monitoring. Their high sensitivity (large dR/dT) over −40°C to +150°C makes them ideal for precision temperature measurement.

How do you convert Celsius to Kelvin?

T(K) = T(°C) + 273.15. This conversion is essential for thermistor and pyrometer calculations, which require absolute temperature in Kelvin. For example, 25°C = 298.15 K and 0°C = 273.15 K.

Why do pyrometers use emissivity correction?

Pyrometers assume the object is a perfect black body (ε = 1) unless corrected. Real materials have ε < 1, meaning they emit less radiation than a black body at the same temperature. If ε_setting ≠ ε_actual, the temperature reading is wrong. Correction formula: T_true = T_measured × (ε_setting/ε_actual)^(1/4).

Potentiometers, Thermistors & Pyrometers – Definitions, Formulas, Calculators & Worked Examples

IGCSE Physics GCSE Science O Level Physics A Level Electronics

This comprehensive study guide covers three essential electrical measurement and sensing devices: potentiometers, thermistors, and pyrometers. Each section includes a precise definition, step-by-step explanation of the working principle, all key formulas rendered in proper mathematical notation, interactive calculators, fully worked exam-style examples, and a comparison table. Whether you are revising for IGCSE, GCSE, O Level, A Level, or any undergraduate electronics course, this is the complete reference you need.

Introduction: Electrical Sensors and Measurement Devices

Modern engineering, medicine, and industry depend on accurate measurement. Three of the most widely-used electronic measurement components are potentiometers, thermistors, and pyrometers. Each device exploits a different physical phenomenon to produce an electrical signal that encodes a measurable quantity:

⚡ Potentiometer

A variable resistor with a moving wiper that creates an adjustable voltage divider. Measures position, angle, or displacement as a voltage signal.

🌡 Thermistor

A semiconductor resistor whose resistance changes dramatically with temperature. Converts temperature into a resistance (and hence voltage) signal.

🔭 Pyrometer

A non-contact thermometer that detects thermal radiation from hot objects. Converts radiated energy into a temperature reading without physical contact.

Why learn about these three together? They all appear together in IGCSE, GCSE, and O Level physics syllabuses under the topic of electrical measurements and sensors. Understanding them gives you a complete picture of how engineers measure two of the most important physical quantities: position and temperature.

All three devices are transducers — they convert one form of energy or physical quantity into an electrical signal. The potentiometer converts mechanical position; the thermistor converts thermal energy; and the pyrometer converts electromagnetic radiation. Together, they represent the core toolkit of instrumentation engineering.

Potentiometers – Complete Guide

A potentiometer (colloquially called a "pot") is a three-terminal passive resistive component with a movable contact called the wiper. The wiper slides along a resistive track, dividing the total resistance into two series portions. Potentiometers are simultaneously one of the oldest and one of the most widely-used electronic components, appearing in everything from guitar volume knobs to precision industrial servo controls.

Working Principle

The potentiometer has a fixed resistive element (track) of total resistance R connected between terminals A and B. A third terminal — the wiper W — makes sliding contact with the track at some position α (0 ≤ α ≤ 1). This splits the total resistance into two parts:

Potentiometer Resistance Split

R_AW = R × α | R_WB = R × (1 − α)

Where α is the fractional wiper position (0 = fully at A, 1 = fully at B). Total: R_AW + R_WB = R at all positions.

Voltage Divider Formula

The most fundamental use of a potentiometer is as a voltage divider. When a supply voltage V_in is applied across terminals A and B, the wiper taps off a fraction of that voltage:

Voltage Divider Output

V_out = V_in × R_WBR_AW + R_WB = V_in × R₂R₁ + R₂

R₁ = resistance from high-voltage terminal to wiper; R₂ = resistance from wiper to ground terminal. V_out varies linearly from 0 V to V_in for a linear-taper pot.

Important: This formula assumes no load is connected to the wiper terminal. When a load resistance R_L is connected, it forms a parallel combination with R₂, reducing the effective resistance and decreasing V_out. Always account for loading effects in real circuits.

Potentiometer Loading Effect

When load resistance R_L is connected to the wiper, the actual output voltage becomes:

Loaded Voltage Divider

V_out(loaded) = V_in × R₂ ∥ R_L R₁ + (R₂ ∥ R_L)

Where R₂ ∥ R_L = (R₂ × R_L) / (R₂ + R_L). For negligible loading, R_L ≫ R₂.

Types of Potentiometers

Linear (Type B)

Resistance changes proportionally to wiper position. Used in test equipment, position sensors, and general-purpose applications.

Logarithmic (Type A)

Resistance changes logarithmically. Matches human hearing perception — used in audio volume controls, where equal rotation steps "sound equal" to the ear.

Anti-Log (Type C / Reverse Audio)

The reverse of logarithmic taper. Used in some audio mixing consoles and gain controls.

Wire-Wound

Resistive element is a coil of precision resistance wire. Very low temperature coefficient; used in precision measurement. Limited resolution (step-wise).

Cermet & Carbon Film

Surface-deposited resistive layers: cermet (ceramic + metal, high stability) and carbon film (cost-effective, general purpose). Both provide continuous resolution.

Digital Potentiometer

An IC that simulates a potentiometer with discrete resistance steps under microcontroller control. Resistant to mechanical wear; used in programmable amplifiers, digital audio, and calibration circuits.

Key Potentiometer Specifications

Total resistance (R): Typically 100 Ω to 10 MΩ. Choose based on circuit impedance.
Power rating: Range from 0.1 W (trimpot) to 5 W (industrial), determines maximum safe current.
Resolution: Wirewound pots have discrete steps; film types are continuous (infinite resolution).
Linearity error: The deviation of actual resistance taper from the ideal. Precision pots have <0.1% linearity error.
Temperature coefficient: How much resistance changes with ambient temperature (ppm/°C). Critical for precision applications.

Real-World Applications

Audio volume & tone controls: Adjusting speaker volume and EQ in amplifiers
Joysticks & game controllers: Two-axis position sensing using two linear pots
Industrial position sensors: Linear pots measure actuator travel; rotary pots measure shaft angle
Calibration and trimming: Trimpots set the gain of amplifiers or offset of ADCs
Voltage reference adjustment: Setting output voltage of adjustable power supplies (e.g., with the LM317 IC)

Thermistors – Complete Guide

A thermistor (contraction of "thermal resistor") is a type of resistor made from semiconducting metal oxide material, whose electrical resistance changes significantly and predictably with temperature. Unlike ordinary resistors (which change resistance only slightly with temperature), thermistors exploit large resistance-temperature sensitivity for sensing and control applications.

NTC vs PTC Thermistors

🔴 NTC (Negative Temperature Coefficient)

Resistance decreases as temperature increases. Made from sintered metal oxides (Mn, Ni, Co, Fe). Most common thermistor type. High sensitivity: a 10°C rise can halve the resistance. Operating range: −55°C to +200°C.

🔵 PTC (Positive Temperature Coefficient)

Resistance increases as temperature increases. Two sub-types: Switching PTC (BaTiO₃ — sharp resistance jump at Curie temperature) and Silistor (silicon — linear characteristic). Used for circuit protection and self-regulating heaters.

The Beta (β) Equation for NTC Thermistors

The relationship between NTC thermistor resistance and temperature is described by the Beta equation (also called the simplified B-parameter equation). It is derived from the Arrhenius equation for semiconductor conduction:

NTC Beta Equation — Resistance from Temperature

R(T) = R₀ × e^{β(1/T − 1/T₀)}

R(T) = resistance at temperature T (Kelvin) | R₀ = resistance at reference temperature T₀ (usually 298.15 K = 25°C) | β = Beta constant (Kelvin), a material property typically 2000–6000 K | e = Euler's number ≈ 2.71828

NTC Beta Equation — Temperature from Resistance

1T = 1T₀ + 1β × ln RR₀ ⇒ T = βln(R/R₀) + β/T₀

Rearrangement of the Beta equation. All temperatures must be in Kelvin (T(K) = T(°C) + 273.15). This is the formula used in the thermistor calculator below.

The Steinhart-Hart Equation (Advanced)

For higher accuracy across a wide temperature range, the Steinhart-Hart equation is used instead of the simpler Beta equation:

Steinhart-Hart Equation

1T = A + B · ln(R) + C · [ln(R)]³

A, B, C = Steinhart-Hart coefficients (from datasheet or derived by measuring R at three known temperatures). T is in Kelvin. Provides accuracy better than ±0.02°C over a 200°C range, vs ±0.5°C for the Beta equation.

Thermistor Sensitivity

The key advantage of a thermistor over other temperature sensors is its high sensitivity. The sensitivity (or temperature coefficient of resistance) of an NTC thermistor is:

NTC Sensitivity (Temperature Coefficient of Resistance)

α = 1R × dRdT = − βT²

α is negative for NTC thermistors (resistance decreases with temperature). At 25°C (T = 298.15 K) with β = 3950 K: α = −3950/298.15² = −0.0444 = −4.44%/°C. This is 10× more sensitive than a platinum RTD (−0.385%/°C).

Thermistor Applications

Medical thermometers: Digital oral, ear (tympanic), and rectal thermometers all use NTC thermistors for fast, accurate temperature measurement.
HVAC thermostat sensors: Monitoring room or duct temperature for climate control system feedback.
Automotive sensors: Engine coolant temperature (ECT), outside air temperature, automatic climate control.
Battery management systems (BMS): Monitoring lithium-ion battery temperature to prevent overheating and thermal runaway.
3D printer hotend control: Precise temperature control of the print nozzle (typically ±1°C) using an NTC thermistor in a PID feedback loop.
Inrush current limiting (PTC): PTC thermistors placed in series with motor or transformer primary windings to limit startup current surge.
Resettable fuses (PTC): Self-resetting overcurrent protection — resistance increases sharply at a threshold current, cutting power; cools and resets automatically.
Temperature compensation: NTC thermistors stabilize crystal oscillator frequency drift with temperature in timing circuits.

Reading a Thermistor in a Voltage Divider Circuit

The standard way to measure a thermistor with a microcontroller (e.g., Arduino) is to place it in a voltage divider with a known series resistor R_s:

Thermistor Voltage Divider — Read Resistance from ADC

R_thermistor = R_s × V_cc − V_outV_out

Thermistor connected between V_cc and the measurement node; R_s connected from node to GND. V_out is the ADC voltage reading. Choose R_s ≈ R_thermistor at mid-range temperature for best sensitivity.

Pyrometers – Complete Guide

A pyrometer is a non-contact temperature measurement instrument that infers the temperature of an object by detecting and measuring the thermal electromagnetic radiation it emits. The word comes from the Greek pyr (fire) + metron (measure). Pyrometers are indispensable in industries where the measured object is too hot, too distant, too delicate, or moving too fast for physical contact thermometers.

Physical Basis: Thermal Radiation

Every object with a temperature above absolute zero (0 K = −273.15°C) emits electromagnetic radiation. The total power and spectral distribution of this radiation depend entirely on the object's absolute temperature and surface emissivity. Two fundamental laws govern this:

Stefan-Boltzmann Law

Stefan-Boltzmann Law — Total Radiated Power (Black Body)

j* = σ · T⁴

j* = total radiated power per unit area (W m⁻²) | σ = Stefan-Boltzmann constant = 5.670 × 10⁻⁸ W m⁻² K⁻⁴ | T = absolute temperature in Kelvin. Note the T⁴ dependence: doubling the temperature increases radiation by 16×.

Stefan-Boltzmann Law — Real Surface (with Emissivity)

j* = ε · σ · T⁴

ε (emissivity) is a dimensionless factor 0 < ε ≤ 1, where ε = 1 is a perfect black body emitter and ε = 0 is a perfect reflector. Most engineering materials have ε between 0.1 and 0.95.

Wien's Displacement Law

Wien's Displacement Law — Peak Wavelength

λ_max × T = b

λ_max = peak wavelength of emitted radiation (metres) | T = absolute temperature (Kelvin) | b = Wien's displacement constant = 2.898 × 10⁻³ m·K. At 1000 K: λ_max = 2.898 × 10⁻³ / 1000 = 2.9 μm (infrared). At 6000 K (Sun): λ_max ≈ 0.48 μm (visible green).

Emissivity and Correction

The most critical parameter in pyrometry is emissivity. When a pyrometer is set to ε = 1.0 (black body assumption) but the actual material emissivity is ε < 1.0, the pyrometer reads a lower temperature than the actual temperature. The correction formula (derived from the Stefan-Boltzmann law) is:

Pyrometer Emissivity Correction Formula

T_true⁴ = T_measured⁴ × ε_settingε_actual ⇒ T_true = T_measured × ε_setting^1/4ε_actual^1/4

All temperatures in Kelvin. T_true > T_measured when ε_setting > ε_actual (the most common situation when ε setting is left at 1.0 for a non–black body surface). Always convert °C to K (+273.15), apply correction, then convert back.

Emissivity Reference Values

Material / Surface	Emissivity (ε)	Notes
Blackened / oxidised steel	0.85–0.95	Good for pyrometry
Mild steel (oxidised)	0.70–0.80	Common in furnaces
Polished stainless steel	0.10–0.20	Needs large correction
Aluminium (polished)	0.04–0.09	Very reflective; difficult
Aluminium (anodized)	0.55–0.85	Depends on coating
Human skin	0.95–0.98	Near-ideal emitter
Liquid water	0.95–0.96	Near-ideal emitter
Wood (oak)	0.90	Good emitter
Graphite	0.85–0.95	High temperature furnaces
Perfect black body	1.00	Theoretical limit

Types of Pyrometers

Total Radiation Pyrometer

Collects radiation over a broad spectrum. Uses a thermopile or bolometer detector. Range: 0°C to 4000°C. Subject to emissivity error.

Optical (Disappearing Filament) Pyrometer

Compares the brightness of the target with an internal calibrated tungsten filament at a specific wavelength (655 nm). Range: 700°C to 3000°C. Temperature when filament "disappears" against target = target temperature. Requires operator skill.

Infrared (IR) Pyrometer

Measures radiation in a specific infrared wavelength band using a thermopile or photodiode. Range: −50°C to 3000°C. Fast response; compact; widely used in industrial and consumer applications (IR thermometers).

Ratio (Two-Colour) Pyrometer

Measures radiation at two wavelengths and takes the ratio. Less sensitive to target emissivity, dust, and smoke than single-wavelength types. Ideal for fluctuating emissivity targets.

Pyrometer Applications

Steel mills: Measuring molten steel temperature (1400–1700°C) in converters and ladles
Glass manufacturing: Monitoring glass melt and forming temperatures
Gas turbine monitoring: Measuring blade temperature in jet engines during operation
Semiconductor fabrication: Wafer temperature during rapid thermal annealing processes
Medical / clinical: Non-contact forehead and ear thermometers (consumer IR pyrometers)
Electrical maintenance: Thermal cameras (imaging pyrometers) detect overheating joints and cables
Food safety: Non-contact surface temperature measurement in food processing

Interactive Calculators

⚡ Potentiometer Voltage Divider Calculator

Formula: V_out = V_in × (R₂ / (R₁ + R₂)) = V_in × wiper position

Total Resistance R (Ω)

Wiper Position (0–100%)

Input Voltage V_in (V)

Results

R₁ (terminal A → wiper): 5000.00 Ω

R₂ (wiper → terminal B): 5000.00 Ω

Output Voltage V_out: 2.50 V

Voltage Ratio V_out/V_in: 50.00%

🌡 NTC/PTC Thermistor Temperature Calculator (Beta Equation)

Formula: T = β / (ln(R/R₀) + β/T₀), where T₀ = 298.15 K (25°C)

Thermistor Type

R₀ — Resistance at 25°C (Ω)

Beta Constant β (K)

Measured Resistance R (Ω)

Results

Temperature: °C

Temperature (Kelvin): K

Resistance Ratio R/R₀:

🔭 Pyrometer Emissivity Correction Calculator

Formula: T_true = T_measured × (ε_setting/ε_actual)^1/4 (all temps in Kelvin)

Measured Temperature (°C)

Pyrometer Emissivity Setting (ε)

Actual Material Emissivity (ε)

Results

True Temperature: °C

True Temperature (Kelvin): K

Temperature Error (measured vs true): °C

Radiated Power Ratio (actual/ideal):

🔄 Temperature Conversion Calculator

Temperature Value

From Unit

To Unit

Results

Result:

In Kelvin: K

In Celsius: °C

In Fahrenheit: °F

Worked Examples – Exam-Style Solutions

Example 1 — Potentiometer Voltage Divider

Problem: A linear potentiometer with total resistance R = 5 kΩ is connected across V_in = 12 V. The wiper is set at 30% from the ground terminal. Calculate (a) R₁, (b) R₂, and (c) V_out.

Identify wiper position fraction: α = 30/100 = 0.30 (measured from ground terminal)

Calculate R₂ (wiper to ground): R₂ = R × α = 5000 × 0.30 = 1500 Ω

Calculate R₁ (supply to wiper): R₁ = R × (1 − α) = 5000 × 0.70 = 3500 Ω

Apply voltage divider formula: V_out = V_in × R₂/(R₁+R₂) = 12 × 1500/5000 = 12 × 0.30 = 3.6 V

Answer: R₁ = 3500 Ω, R₂ = 1500 Ω, V_out = 3.6 V

Example 2 — NTC Thermistor Temperature from Resistance

Problem: An NTC thermistor has R₀ = 10 kΩ at T₀ = 25°C and β = 3950 K. The measured resistance is R = 25.4 kΩ. Find the temperature.

Convert T₀ to Kelvin: T₀ = 25 + 273.15 = 298.15 K

Apply the inverse Beta equation: 1/T = 1/T₀ + (1/β) × ln(R/R₀)

Calculate ln(R/R₀): ln(25400/10000) = ln(2.54) = 0.9322

Calculate 1/T: 1/T = 1/298.15 + (1/3950) × 0.9322 = 0.003354 + 0.000236 = 0.003590 K⁻¹

Find T: T = 1/0.003590 = 278.55 K

Convert to °C: T = 278.55 − 273.15 = 5.40°C

Answer: The temperature is approximately 5.4°C

Example 3 — Pyrometer Emissivity Correction

Problem: A radiation pyrometer is set to ε = 1.0 (black body) and reads T_measured = 850°C when aimed at a steel surface. The actual emissivity of the steel is ε_actual = 0.75. Find the true temperature.

Convert measured temperature to Kelvin: T_measured = 850 + 273.15 = 1123.15 K

Apply emissivity correction: T_true = T_measured × (ε_setting/ε_actual)^1/4

Calculate ratio: ε_setting/ε_actual = 1.0/0.75 = 1.3333

Fourth root: (1.3333)^1/4 = 1.3333^0.25 = 1.0746

True temperature in K: T_true = 1123.15 × 1.0746 = 1206.9 K

Convert to °C: T_true = 1206.9 − 273.15 = 933.8°C ≈ 934°C

✓

Error: 934 − 850 = 84°C — a significant under-reading if emissivity is not corrected!

Answer: True temperature ≈ 934°C (not 850°C). Emissivity correction adds 84°C.

Example 4 — Thermistor in a Voltage Divider (Find Vout)

Problem: A 10 kΩ NTC thermistor (β = 3950 K, R₀ = 10 kΩ at 25°C) is in series with a 10 kΩ fixed resistor from 5 V to GND. The NTC is connected between 5 V and the measurement node. Find V_out at 0°C.

Find R at 0°C: T = 273.15 K. R = R₀ × e^(β(1/T − 1/T₀)) = 10000 × e^(3950 × (1/273.15 − 1/298.15))

1/273.15 = 0.003661 K⁻¹ | 1/298.15 = 0.003354 K⁻¹ | Difference = 0.000307 K⁻¹

3950 × 0.000307 = 1.2127 | e^1.2127 = 3.364 | R(0°C) = 10000 × 3.364 = 33,640 Ω ≈ 33.6 kΩ

Apply voltage divider (series resistor R_s = 10 kΩ from node to GND): V_out = 5 × 10000 / (33640 + 10000) = 5 × 10000/43640

V_out = 5 × 0.2291 = 1.146 V ≈ 1.15 V

Answer: V_out ≈ 1.15 V at 0°C (at 25°C it would be 2.5 V — the voltage drops as the NTC resistance rises with cooling).

Comparison Table – Potentiometer vs Thermistor vs Pyrometer

Feature	Potentiometer	Thermistor (NTC)	Pyrometer
Measured quantity	Position / displacement / angle	Temperature	Temperature (high or inaccessible)
Principle	Resistive voltage division	Resistance vs temperature (semiconductor)	Stefan-Boltzmann thermal radiation law
Contact required?	Physical contact with moving part	Physical contact with measured object	No — non-contact
Temperature range	N/A	−55°C to +200°C	−50°C to +4000°C (type-dependent)
Typical accuracy	±0.1% (precision), ±1% (standard)	±0.1°C to ±0.5°C	±1°C to ±2% of reading
Key equation	V_out = V_in × R₂/(R₁+R₂)	T = β/(ln(R/R₀) + β/T₀)	j* = εσT⁴
Response time	Instantaneous	1–10 seconds (thermal mass)	Milliseconds
Cost	Very low (£0.10 – £50)	Low (£0.20 – £10)	Moderate–High (£30 – £5000+)
Key industry	Audio, robotics, automotive	Medical, HVAC, automotive, consumer electronics	Steel, glass, semiconductor, aerospace

Frequently Asked Questions

What is a potentiometer and how does it work?+

A potentiometer is a three-terminal variable resistor. A movable contact (wiper) slides along a resistive track, dividing the total resistance R into two parts: R₁ (from supply terminal to wiper) and R₂ (from wiper to ground terminal). When a voltage V_in is applied across the full track, the wiper produces an output voltage V_out = V_in × R₂ / (R₁ + R₂). Moving the wiper continuously varies V_out from 0 V to V_in.

What is the difference between NTC and PTC thermistors?+

NTC (Negative Temperature Coefficient) thermistors decrease in resistance as temperature rises — they are the most common type, used for temperature measurement and sensing. PTC (Positive Temperature Coefficient) thermistors increase in resistance as temperature rises. They are used primarily for circuit protection (PTC resettable fuses) and self-regulating heaters, where the rising resistance at high temperature limits current automatically.

What is the Beta equation for a thermistor?+

The Beta (β) equation models NTC thermistor resistance as a function of temperature: R(T) = R₀ × e^(β × (1/T − 1/T₀)), where R₀ = resistance at reference temperature T₀ (usually 25°C = 298.15 K), β = Beta constant (material property, typically 3000–5000 K), and T is absolute temperature in Kelvin. To find temperature from a measured resistance: T = β / (ln(R/R₀) + β/T₀).

How does a pyrometer measure temperature without contact?+

A pyrometer detects thermal electromagnetic radiation emitted by the target object. By the Stefan-Boltzmann Law, all objects above 0 K emit radiation with total power j* = εσT⁴. The pyrometer focuses this radiation onto a detector (thermopile, bolometer, or photodiode), converts the intensity signal to a temperature reading, and corrects for the set emissivity value. No physical contact with the target is needed.

What is emissivity and why does it matter?+

Emissivity (ε) is the ratio of radiation emitted by a real surface to that of a perfect black body at the same temperature. It ranges from 0 (perfect mirror) to 1 (perfect black body). A polished aluminium surface has ε ≈ 0.05; human skin has ε ≈ 0.97. If the pyrometer emissivity setting differs from the true material emissivity, the temperature reading will be incorrect. Correction: T_true = T_measured × (ε_setting/ε_actual)^(1/4), with temperatures in Kelvin.

What is the Stefan-Boltzmann Law?+

The Stefan-Boltzmann Law states that the total power radiated per unit area by a perfect black body is j* = σT⁴, where σ = 5.670 × 10⁻⁸ W m⁻² K⁻⁴ is the Stefan-Boltzmann constant and T is absolute temperature in Kelvin. For real surfaces: j* = εσT⁴. This T⁴ dependence means doubling the temperature multiplies the emitted radiation by 2⁴ = 16. This law is the physical basis for all pyrometry.

How is a potentiometer used as a position sensor?+

A linear potentiometer is mechanically linked to a moving component (e.g., a piston or robot arm). As the component moves, it moves the wiper, changing R₂ and hence V_out. Since V_out is proportional to position for a linear-taper pot, an ADC reading V_out directly gives position information. This is used in joysticks, automotive throttle pedals, and industrial actuator feedback systems.

What is the Steinhart-Hart equation and when is it used?+

The Steinhart-Hart equation is a more accurate model for NTC thermistors: 1/T = A + B·ln(R) + C·[ln(R)]³, where A, B, C are coefficients determined from datasheet data or measurements at three known temperatures. It provides accuracy better than ±0.02°C over a 200°C range, compared to ±0.5°C or more for the simpler Beta equation. Use it wherever precision better than ±0.5°C is required.

What is Wien's Displacement Law?+

Wien's Displacement Law states that λ_max × T = b, where λ_max is the peak wavelength of thermal radiation, T is absolute temperature in Kelvin, and b = 2.898 × 10⁻³ m·K is Wien's constant. Hotter objects emit at shorter (bluer) wavelengths. At 3000 K (incandescent lamp), λ_max ≈ 966 nm (near infrared); at 6000 K (Sun), λ_max ≈ 483 nm (visible green). This determines the appropriate sensor wavelength for different pyrometer designs.

Why does a thermistor give a non-linear output in a voltage divider?+

Because the NTC thermistor's resistance-temperature relationship is exponential (from the Beta equation), not linear. Even though the voltage divider equation V_out = V_cc × R_s/(R_thermistor + R_s) is itself non-linear, the dominant non-linearity comes from the exponential R(T) relationship. For precise temperature measurement, either a look-up table, the Beta/Steinhart-Hart equation computed in firmware, or a linearisation circuit (e.g., parallel resistor) is used.

How do you convert temperature between Celsius, Fahrenheit, and Kelvin?+

Three conversions: (1) °C to K: T(K) = T(°C) + 273.15. (2) °C to °F: T(°F) = T(°C) × 9/5 + 32. (3) K to °C: T(°C) = T(K) − 273.15. For thermistor and pyrometer calculations, always work in Kelvin and convert the result back to °C at the end. Use the temperature conversion calculator above for quick reference.

What are the main sources of error in pyrometer measurements?+

The main error sources are: (1) Emissivity error — incorrect emissivity setting; (2) Atmospheric absorption — water vapour and CO₂ absorb infrared radiation between the target and detector; (3) Field of view errors — if the target does not completely fill the field of view, background radiation enters; (4) Reflected radiation — particularly for polished low-emissivity surfaces, reflected radiation from the environment adds to the signal; (5) Dirty optics — dust or condensation on the lens reduces signal; (6) Detector calibration drift — aging of the detector element.

Disclaimer: All calculations use standard mathematical models (Beta equation, Stefan-Boltzmann law, ideal voltage divider). Real-world results may differ due to component tolerances, loading effects, emissivity variations, and measurement uncertainty. Always consult component datasheets and calibration records for critical applications. This content is intended for educational use at IGCSE, GCSE, O Level, and undergraduate level.

Potentiometers, Thermistors & Pyrometers – Definitions, Formulas, Calculators & Worked Examples

Introduction: Electrical Sensors and Measurement Devices

Potentiometers – Complete Guide

Working Principle

Voltage Divider Formula

Potentiometer Loading Effect

Types of Potentiometers

Key Potentiometer Specifications

Real-World Applications

Thermistors – Complete Guide

NTC vs PTC Thermistors

The Beta (β) Equation for NTC Thermistors

The Steinhart-Hart Equation (Advanced)

Thermistor Sensitivity

Thermistor Applications

Reading a Thermistor in a Voltage Divider Circuit

Pyrometers – Complete Guide

Physical Basis: Thermal Radiation

Stefan-Boltzmann Law

Wien's Displacement Law

Emissivity and Correction

Emissivity Reference Values

Types of Pyrometers

Pyrometer Applications

Interactive Calculators

Results

Results

Results

Results

Worked Examples – Exam-Style Solutions

Example 1 — Potentiometer Voltage Divider

Example 2 — NTC Thermistor Temperature from Resistance

Example 3 — Pyrometer Emissivity Correction

Example 4 — Thermistor in a Voltage Divider (Find Vout)

Comparison Table – Potentiometer vs Thermistor vs Pyrometer

Frequently Asked Questions

Related Calculators & Study Notes

Related Posts

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