Thermistors

Why it matters

Thermistors are the cheapest way to get high temperature resolution: every phone battery, laptop, 3-D printer bed, car coolant circuit and medical thermometer has one. PTC thermistors also protect motors and transformers from overcurrent. Their huge sensitivity comes with strong non-linearity and self-heating, both of which you must handle numerically.

Key ideas

A thermistor (thermal resistor) is a semiconducting ceramic whose resistance changes strongly with temperature.

• NTC thermistors are sintered mixtures of metal oxides (Mn, Ni, Co, Cu, Fe). Raising temperature frees more charge carriers, so resistance falls roughly exponentially. Typical coefficient at 25 °C is about −3 to −6 %/°C — roughly ten times a platinum RTD's +0.385 %/°C. Useful range is about −50 °C to 150 °C (some glass-encapsulated types to 300 °C).
• PTC thermistors (doped barium titanate ceramics) have a small or negative coefficient up to a switching (Curie) temperature, above which resistance rises by several orders of magnitude over a few degrees. They are used as resettable overcurrent protectors, motor-winding over-temperature sensors, inrush limiters and self-regulating heaters — not usually for measurement. Silicon "linear PTC" sensors are a separate family.
• Forms: beads (small, fast), discs, chips, glass-encapsulated probes.

R–T models for NTCs.

• Beta (B) equation: R_T = R0·exp[B(1/T − 1/T0)], with absolute temperatures. B (typically 3000–5000 K) is quoted between two temperatures, e.g. B25/85, and is accurate to about ±1 °C over 50–100 °C spans.
• Steinhart–Hart equation: 1/T = A + B·ln R + C·(ln R)³ (different A, B, C from the beta form) fits to about ±0.01 °C over wide ranges; the three constants are found from three calibration points.
• The local temperature coefficient follows from the beta form: α = (1/R)(dR/dT) = −B/T², so sensitivity falls as temperature rises.

Linearisation and readout. A thermistor in a voltage divider gives an S-shaped but much more linear output than R_T itself. A fixed resistor in parallel flattens the curve; making the parallel resistor R_p = R_Tm·(B − 2Tm)/(B + 2Tm) puts the inflection point at the mid-range temperature Tm. Microcontrollers often skip hardware linearisation and use a look-up table or the Steinhart–Hart equation.

Self-heating. Measuring current dissipates power P in the thermistor, raising its temperature by P/δ, where δ is the dissipation constant (mW/°C, depends on the medium and mounting; about 1–2 mW/°C for a small bead in still air). Keep the power to microwatts for accurate work. Self-heating is used deliberately in thermistor flowmeters, liquid-level detectors and inrush limiters.

Advantages: very high sensitivity, small size, fast response (bead types), low cost, high resistance so lead resistance is negligible. Limitations: strongly non-linear, narrow range, less interchangeable and less stable than RTDs (unless glass-encapsulated and aged), self-heating, possible drift at high temperature.

Formulas

R_T = R0·exp[B·(1/T − 1/T0)] (beta equation, NTC) R_T = resistance at T (Ω); R0 = resistance at reference T0 (Ω); B = material constant (K); T, T0 = absolute temperatures (K).

B = ln(R1/R2) / (1/T1 − 1/T2) (B from two calibration points)

α = −B / T² (local temperature coefficient, per K)

1/T = A + B·ln R + C·(ln R)³ (Steinhart–Hart; constants from calibration)

Vo = Vs·R_T / (R_T + R_s) (thermistor at the bottom of a divider) Vs = supply (V); R_s = series resistor (Ω).

ΔT_SH = P / δ P = dissipated power (W); δ = dissipation constant (W/°C).

Worked examples

Example 1 — beta equation. An NTC has R = 10 kΩ at 25 °C and B = 3950 K. Find R at 50 °C and α at 25 °C.

1. Convert: T0 = 298.15 K; T = 323.15 K.
2. 1/T − 1/T0 = 1/323.15 − 1/298.15 = −2.5948 × 10⁻⁴ K⁻¹.
3. B·(1/T − 1/T0) = 3950 × (−2.5948 × 10⁻⁴) = −1.0249.
4. R = 10 000 × e^(−1.0249) = 10 000 × 0.3588 = 3588 Ω.
5. α = −B/T0² = −3950/298.15² = −0.0444 K⁻¹ = −4.44 %/°C.

Answer: R(50 °C) ≈ 3.59 kΩ; α(25 °C) = −4.44 %/°C (about 11 times a Pt100's coefficient, opposite sign).

Example 2 — calibration and inversion (GATE level). A thermistor reads 10.0 kΩ at 25 °C and 3.60 kΩ at 50 °C. (a) Find B. (b) Using this B, find the temperature when it reads 15.0 kΩ.

1. B = ln(10.0/3.60)/(1/298.15 − 1/323.15) = 1.0217/2.5948 × 10⁻⁴ = 3937 K.
2. Invert the beta equation: 1/T = 1/T0 + ln(R/R0)/B = 1/298.15 + ln(1.5)/3937.
3. 1/T = 3.3540 × 10⁻³ + 1.0299 × 10⁻⁴ = 3.4570 × 10⁻³ → T = 289.27 K.
4. T = 289.27 − 273.15 = 16.1 °C.

Answer: B ≈ 3937 K; T ≈ 16.1 °C.

Example 3 — divider and self-heating. The 10 kΩ (at 25 °C) thermistor is the lower arm of a divider with a 10 kΩ series resistor on 5 V. At 25 °C, Vo = 5 × 10/(10 + 10) = 2.5 V; power in the thermistor P = 2.5²/10 000 = 0.625 mW. With δ = 2 mW/°C, self-heating error = 0.625/2 = 0.31 °C — too much for precise work, so reduce the supply or raise R_s.

Common mistakes

• Using °C in the beta equation; it needs kelvin.
• Expecting NTC resistance to rise with temperature.
• Treating B as a universal constant; it depends on the temperature pair used to define it.
• Assuming a linear coefficient over a wide range — α changes as 1/T².
• Ignoring self-heating; a few milliwatts can cause errors of degrees in still air.
• Using a PTC switching thermistor as a measuring sensor.

For GATE IN

NAT problems: R at another temperature from the beta equation, B from two readings, temperature from a measured R, α = −B/T², divider or bridge output with a thermistor arm, and self-heating error. MCQs compare thermistors with RTDs and thermocouples (sensitivity, range, linearity) and test NTC/PTC behaviour. Practise the logarithm inversions quickly on a calculator.

Quick check

1. An NTC has B = 4000 K. What is α at 300 K?
2. Which equation fits thermistor data to about ±0.01 °C?
3. If R at 25 °C is 10 kΩ and B = 3950 K, what is R at 35 °C?
4. Why is lead resistance usually negligible for thermistors? Answers: 1. −4000/300² = −4.44 %/K; 2. Steinhart–Hart; 3. about 6.51 kΩ; 4. their resistance and dR/dT are large (kΩ and hundreds of Ω/°C), so an ohm of lead is a tiny fraction of a degree.