NTC and PTC Thermistors

The exponential physics of an NTC

A negative-temperature-coefficient thermistor is a ceramic semiconductor. The bulk material is a sintered powder of metal oxides — manganese, nickel, cobalt, copper, sometimes iron — pressed into a bead or chip and fired at 1,200 °C. The mix sets the band structure: a wide-bandgap insulator at absolute zero, with hopping-conduction carriers freed thermally at room temperature. The activation energy E_a divided by Boltzmann's constant gives the β-parameter, β = E_a / k_B, typically 3,000 to 4,500 kelvin. Heat the ceramic: more carriers escape their bound states; conductivity rises; resistance falls. The functional form is exponential because the population of free carriers follows a Boltzmann distribution.

The β-equation reproduces this physics in engineering form. R(T) = R_0 · exp(β(1/T − 1/T_0)) where R_0 is the resistance at the reference temperature T_0 (almost always 25 °C = 298.15 K). For β = 3950 K and R_0 = 10 kΩ at 25 °C, a 50 °C reading gives R = 10,000 · exp(3950 · (1/323.15 − 1/298.15)) ≈ 4,160 Ω. The same part at 0 °C reads ≈ 32,650 Ω, at 100 °C reads ≈ 970 Ω. A 100 °C span produces a 33:1 resistance change — that's the sensitivity advantage thermistors have over RTDs (where the change is only about 39 percent over the same span). The same advantage means thermistors are fundamentally nonlinear; software or hardware must linearize.

For better accuracy than β provides (which is good to roughly ±1 °C over a 50 °C window), the industry uses the three-coefficient Steinhart-Hart equation: 1/T = A + B·ln(R) + C·(ln(R))³. With three coefficients fitted to calibration points spanning the operating range, accuracy improves to better than 0.01 °C — better than most application demands. Medical-grade body thermometers and PCR-machine well sensors live at that accuracy level using Steinhart-Hart with three- or four-point calibration per device.

PTC: the Curie-point switch

A positive-temperature-coefficient thermistor is a very different beast. The classical PTC is barium titanate (BaTiO₃) doped with lanthanum or yttrium to make it semiconducting. At low temperature the material is ferroelectric and conductive — bulk resistance is only a few tens of ohms-centimeter. As temperature rises past the Curie point (110 to 130 °C for BaTiO₃, tunable to 60 to 200 °C by changing dopants and ratios), the ferroelectric domains depolarize and the grain-boundary potential barriers rise dramatically. Bulk resistance jumps by three to five orders of magnitude in a span of just 10 to 20 °C.

That sharp transition is the basis of two large product families. Self-regulating heaters are simply a PTC element across the supply. When cold, the PTC's low resistance lets significant current flow, dissipating I²R as heat. Self-heating raises the temperature; the resistance climbs; current falls; the steady state is just past the Curie point, where the dissipation balances the heat loss. The PTC sets its own temperature with no thermostat, no microcontroller, no feedback algorithm. Found in plug-in fan heaters, automotive cabin pre-heaters (especially in EVs where waste engine heat isn't available), heated mirrors, even some hair-styling tools.

Resettable fuses (polyfuses, often sold as PolySwitch or PolyFuse) use a polymeric PTC: polyethylene loaded with conductive carbon-black particles. Cold resistance is a fraction of an ohm; the device passes normal load current with negligible drop. An overcurrent fault heats the polymer past its melting transition (typically 100 to 125 °C). The polymer matrix expands, separating the carbon particles and breaking the percolation chains; resistance jumps by 4 to 5 orders of magnitude; current drops to a small leakage; the device "trips." Power dissipation falls to a fraction of a watt, holding the device latched in the high-resistance state. Remove the fault, let the device cool, and the polymer contracts, restoring the conductive chains. Hundreds to thousands of trip-reset cycles before the part wears out.

Worked example: NTC voltage-divider thermometer

Build the cheapest possible 10 to 90 °C digital thermometer for a hot-water heater.

Parts:
  NTC thermistor:  R_0 = 10 kΩ at 25 °C, β = 3950 K (Vishay NTCLE100E3)
  Fixed resistor:  R_fixed = 10 kΩ, 1% tolerance
  Supply:          V_cc = 3.3 V (microcontroller rail)
  ADC:             12-bit on microcontroller (4096 counts)

Divider:
  V_out = V_cc · R_fixed / (R_NTC + R_fixed)

Temperature-to-resistance lookup (β model):
  T (°C)    T (K)     R_NTC (Ω)        V_out (V)        ADC count
  10        283.15    18,800           1.156            1435
  25        298.15    10,000           1.650            2048
  50        323.15     4,162           2.337            2901
  90        363.15       865           3.039            3771

Sensitivity at the middle (50 °C):
  dV/dT ≈ 21 mV/°C → ADC count change ≈ 26 counts/°C
  ADC resolution: 0.04 °C/count
  Practical accuracy after β-fit: ±0.3 °C (limited by 5% NTC part tolerance)

Code snippet to convert ADC count to temperature:
  R_NTC = R_fixed · (4095 / count − 1)
  ln_ratio = ln(R_NTC / R_0)
  T_kelvin = 1 / (1/T_0 + ln_ratio / β)
  T_celsius = T_kelvin − 273.15

Power dissipation in NTC:
  Worst case (R_NTC = 865 Ω at 90 °C):
  I = V_cc / (R_NTC + R_fixed) ≈ 304 µA
  P = I² · R_NTC ≈ 80 µW (well below the 1 mW self-heating budget)

That's a complete sensor design. Total cost: maybe twenty cents of components. The hardest piece is the calibration step at one or two reference temperatures to remove the NTC's 5 percent initial tolerance — without it, ±2 °C uncertainty; with one-point calibration at 25 °C, ±0.5 °C; with two-point calibration at 25 °C and 80 °C (Steinhart-Hart fit), ±0.05 °C.

Thermistor family comparison

Type	Mechanism	R(T) shape	Typical range	Accuracy	Cost	Use
NTC bead / chip	Metal-oxide ceramic, exponential	R drops smoothly with T	−55 to +150 °C	±0.1 to ±2 °C	$0.05 – $1	Temperature sensing, compensation
NTC high-temperature	Refractory metal-oxide	R drops smoothly with T	0 to +300 °C	±1 to ±5 °C	$1 – $5	Engine, exhaust, industrial
PTC ceramic (BaTiO₃)	Curie-point grain-boundary barrier	R is low then jumps at T_C	+60 to +200 °C switch	±5 °C at switch	$0.20 – $2	Self-regulating heater
PTC polymer (polyfuse)	Polyethylene + carbon black, melt transition	R is low then trips at I_hold	Trips ~110 °C internal	n/a (binary)	$0.05 – $0.50	Resettable overcurrent fuse
RTD (Pt100)	Pure platinum, linear	R rises linearly at 3.85 Ω/°C	−200 to +850 °C	±0.1 °C	$5 – $50	Process control, lab
Thermocouple (K-type)	Seebeck voltage at junction	μV per °C, nonlinear	−200 to +1,260 °C	±1 to ±2 °C	$2 – $20	Furnace, exhaust, broadest range

The selection rule is straightforward. For consumer-grade temperature measurement below 150 °C — laptop and CPU sensors, battery packs, white-goods controllers, medical thermometers, automotive cabin temperature, refrigerator thermostats — NTCs win on cost and sensitivity. For process-control applications wanting linearity, traceable accuracy, and wider range, RTDs are the standard. For furnaces and combustion above 500 °C, thermocouples are the only practical option. PTCs are non-sensing devices — they're functional elements (heater, fuse) that exploit the same physics in the opposite direction.

Where thermistors appear

• Battery packs. NTC sensors at every cell or every cluster, feeding the battery management system; PTC current limiters in cell-balancing circuitry.
• Medical thermometers. Tympanic, oral, rectal — NTCs deliver ±0.1 °C with two-point factory calibration; cheap enough for disposable probes.
• Inrush current limiters. Big-body NTCs in series with the AC supply of capacitor-input power supplies; cold resistance limits inrush, then NTC self-heats and gets out of the way.
• 3D printer extruders. 100 kΩ NTCs at the hot-end, reading 0 to 300 °C; the firmware lookup table converts ADC counts to temperature.
• Self-regulating heaters. Pipe-freeze prevention, automotive cabin heaters, electric water-bed pads — PTC ceramics with intrinsic temperature limiting.
• USB and consumer-electronics protection. Polyfuse on every USB downstream port to survive shorts without blowing a glass fuse.
• Liquid-cooled servers. Manifold and inlet/outlet NTC sensors monitor coolant; the BMS reads them through a 16-bit ADC at the chassis controller.

Common misconceptions

• The β-parameter is constant. β drifts with temperature; the β-equation is a two-point local fit, not a universal law. Use Steinhart-Hart for high-accuracy work.
• Higher β means better sensor. Higher β gives a steeper response, but narrower useful range and worse linearity at extremes.
• NTCs are linear over short ranges. They're approximately linear in a few-degree window if you squint, but full microcontroller code should always use the proper equation.
• Self-heating is negligible. A 10 kΩ NTC with 3.3 V supply burns 1 mW — enough to raise its temperature 0.5 °C in still air. Drive the bridge with a low-duty-cycle pulse or pick a higher resistance to avoid the error.
• Polyfuses are fuses. They limit current; they don't open. After tripping they're still a high-resistance path, not an open circuit.
• Cheap thermistors are accurate. 1% NTC parts are 1% of their nominal R_0, which translates to ±2 °C without calibration; the high sensitivity is wasted unless the part is calibrated.