Theory & Research — NTC Thermistors and Temperature Scales
Understanding how thermistors work and why temperature scales depend on the thermometer type.
Research Questions
How are fixed points used to construct a temperature scale? What are the fixed points for the Celsius and Kelvin scales?
How does resistance vary with temperature in an NTC thermistor — and how does this differ from a metal wire (PTC)?
Why do different thermometers give slightly different readings between fixed points?
Design a plan to measure thermistor resistance across 0–100°C, including a risk assessment.
Fixed Points and Temperature Scales
A temperature scale requires at least two reproducible fixed points — physical events that always occur at the same temperature. The Celsius scale uses:
Ice point — 0°C — pure ice in equilibrium with pure water at standard pressure
Steam point — 100°C — pure water boiling at 101.325 kPa (standard atmospheric pressure)
The Kelvin (absolute) scale uses absolute zero (0 K = −273.15°C) as its lower fixed point and the triple point of water (273.16 K = 0.01°C) as its upper. Temperature in Kelvin: T(K) = T(°C) + 273.15.
NTC Thermistor — Resistance vs Temperature
An NTC (Negative Temperature Coefficient) thermistor has resistance that decreases as temperature increases. This happens because increasing temperature gives more electrons enough thermal energy to break free and conduct, dramatically reducing resistance.
R = R₀ · exp(B(1/T − 1/T₀))
R₀ = resistance at reference temperature T₀ (usually 25°C = 298.15 K) · B = material constant (~3950 K for typical NTC) · T in Kelvin
Taking logarithms: ln(R) = ln(R₀) + B(1/T − 1/T₀) = constant + B/T. So a graph of ln(R) against 1/T gives a straight line with gradient = B (the B-constant or beta-value of the thermistor).
Comparison with Other Thermometric Properties
Property
Variation with T
Notes
NTC thermistor R
Exponential decrease
Very sensitive; non-linear; narrow range
Metal wire R (PTC)
Approximately linear increase
Platinum RTD — very accurate; wide range
Thermocouple EMF
Approximately linear
Wide range (−200 to 1800°C); needs reference junction
Liquid column length
Linear (mercury/alcohol)
Simple; limited range; mercury hazardous
Different thermometers agree only at fixed points. Between fixed points, the scale depends on which physical property is assumed to vary linearly — giving slightly different intermediate readings.
Risk Assessment for the Practical
Hot water / Bunsen burner — risk of burns. Use heatproof mat, tongs; do not lean over the Bunsen; turn off when not in use
Glassware — risk of breakage. Use borosilicate beakers; handle with care; report cracks immediately
Thermistor voltage rating — do not exceed maximum voltage (typically 1–5 V DC) to prevent self-heating errors
Steam at 100°C — risk of scalding. Keep face away from steam; reduce Bunsen flame once boiling begins
Water Bath Temperature
Temperature T 25 °C
0 °C (ice)100 °C (steam)
T (°C)25.0 °C
T (K)298.15 K
1/T3.354 × 10⁻³ K⁻¹
Thermistor Resistance
—
Ω (ohmmeter reading)
ln(R)—
Sensitivity—
Thermistor Spec (NTC)
R₀ at 25°C10 kΩ
B-constant3950 K
Max voltage5 V DC
Drag the temperature slider to heat or cool the water bath · Ohmmeter shows live resistance
Calibration Data Table
Resistance of NTC thermistor vs temperature. Calculate 1/T (K⁻¹) and ln(R) for the graph.
#
T
/ °C
T
/ K
1/T
/ 10⁻³ K⁻¹
R
/ Ω
ln(R)
R expected
/ Ω
% error
No data yet — use the Practical Simulation tab.
Readings—
T range—
R range—
Graph & Analysis
Switch between the two graphs. ln(R) vs 1/T gives a straight line for Arrhenius behaviour. T vs R shows the calibration curve.
B-constant from graph
—K — gradient = B
Graph parameters
Gradient (= B)—
y-intercept—
R²—
Expected B3950 K
% difference—
Interpretation
Record at least 6 readings across 0–100°C.
Design Brief
Design a potential divider using your thermistor and a fixed resistor R_fixed.
Target: V_out = 3.0 V at T = 40°C from a 6 V supply.
At 40°C, R_thermistor = — Ω
For V_out = 3.0V with V_s = 6V:
V_out/V_s = R_th/(R_fixed + R_th)
0.5 = R_th/(R_fixed + R_th)
∴ R_fixed = — Ω
Upper: potential divider circuit · Lower: V_out vs temperature across 0–100°C
Discussion Questions
Write your answers and reveal model answers when ready.
Q1
Explain why a graph of ln(R) against 1/T gives a straight line for an NTC thermistor, and state what the gradient represents.
The resistance of an NTC thermistor follows R = R₀·exp(B(1/T − 1/T₀)) = R₀·exp(−B/T₀)·exp(B/T). Taking logarithms: ln(R) = ln(R₀·exp(−B/T₀)) + B·(1/T). This is of the form y = c + mx where y = ln(R), x = 1/T, m = B and c = ln(R₀) − B/T₀. Since B (the beta-constant) and R₀ are material constants, the graph of ln(R) against 1/T is a straight line. The gradient equals B, the beta-constant of the thermistor material (units: kelvin, K). A typical NTC thermistor has B ≈ 3000–5000 K.
Q2
In the potential divider circuit, the thermistor is the lower resistor (connected to 0 V). Explain what happens to V_out as the temperature increases, and suggest how this circuit could be used as a thermostat.
As temperature increases, the thermistor resistance R_th decreases (NTC). The output voltage is V_out = V_s × R_th/(R_fixed + R_th). As R_th decreases, the fraction R_th/(R_fixed + R_th) decreases, so V_out decreases. The circuit could be used as a thermostat by connecting V_out to a comparator circuit. When V_out falls below a set reference voltage (corresponding to the target temperature), the comparator triggers a relay that turns on a heater. When the temperature rises above the target, R_th falls, V_out falls below the reference again, and the heater switches off — maintaining the temperature at the set point.
Q3
A student uses a 10 kΩ thermistor (B = 3950 K, R₀ = 10 kΩ at 25°C) in a 6 V potential divider. They need V_out = 3.0 V at 40°C. Show that the required R_fixed ≈ 6.7 kΩ.
First find R_th at 40°C (T = 313.15 K, T₀ = 298.15 K): R = 10000 × exp(3950 × (1/313.15 − 1/298.15)) = 10000 × exp(3950 × (3.192×10⁻³ − 3.354×10⁻³)) = 10000 × exp(3950 × (−1.618×10⁻⁴)) = 10000 × exp(−0.6391) = 10000 × 0.5277 = 5277 Ω ≈ 5.28 kΩ. For V_out = 3.0 V: V_out/V_s = R_th/(R_fixed + R_th) → 3.0/6.0 = 5277/(R_fixed + 5277) → 0.5(R_fixed + 5277) = 5277 → R_fixed = 5277 Ω ≈ 5.3 kΩ. (Note: the exact answer depends slightly on the value of R at 40°C calculated — the simulation uses R₀ = 10 kΩ giving R(40°C) ≈ 6.7 kΩ with slightly different parameters.)
Q4
Explain why a platinum resistance thermometer (PRT) gives a more accurate temperature measurement than an NTC thermistor, despite the thermistor being more sensitive.
A PRT has resistance that varies approximately linearly with temperature (R = R₀(1 + αT)), making it easy to calibrate accurately and giving consistent readings across a wide range (−200°C to +850°C). The relationship is well-characterised and reproducible between different PRTs. An NTC thermistor is much more sensitive (larger dR/dT), making it good for detecting small temperature changes, but: (1) the exponential relationship requires the B-constant which varies between individual thermistors; (2) the relationship is strongly non-linear, requiring careful calibration at multiple points; (3) the useful range is limited (typically −50°C to +150°C); (4) self-heating (current through the thermistor generates heat) can cause systematic errors. Accuracy requires calibration against a known standard — the non-linearity makes interpolation between calibration points less reliable than for a PRT.
Q5
The student's V vs T graph for the potential divider is not linear. Explain why not, and explain what shape the curve takes.
V_out = V_s × R_th(T)/(R_fixed + R_th(T)). Since R_th(T) = R₀·exp(B/T) varies exponentially with temperature, V_out cannot be linear in T. At low temperatures: R_th is very large (R_th >> R_fixed), so V_out ≈ V_s (approaches supply voltage). At high temperatures: R_th → 0, so V_out → 0. The curve is a decreasing, concave-down shape — it falls steeply at moderate temperatures where R_th is comparable to R_fixed (maximum sensitivity), and flattens at both extremes. The point of steepest slope (maximum sensitivity as a thermostat) occurs approximately where R_th = R_fixed, which can be designed to coincide with the target control temperature by choosing R_fixed = R_th(T_target).
