What an NTC actually does

A negative-temperature-coefficient (NTC) thermistor is a sintered metal-oxide bead whose resistance falls exponentially as temperature rises. The exponential is set by the ceramic composition; resistance at a reference temperature (usually 25 °C) is set by the bead geometry. The two together give you the part number: a "10 kΩ / 3950 K" NTC has 10 kΩ at 25 °C and a B-value of 3 950 K.

The B-value, also written β, is just the average slope of ln(R) versus 1/T across a chosen two-temperature interval: 

β = ln(R₁ / R₂) / (1/T₁ − 1/T₂)         [T in kelvin]

Two temperatures, two resistance measurements, one curve. The most commonly quoted interval is B25/85 (25 °C to 85 °C); a less common but more accurate interval for HVAC work is B25/50. Always check which interval a datasheet is quoting — a part listed as "B = 3 950 K" with no interval declaration is ambiguous, and the value will shift by 20-50 K depending on whether the manufacturer used 25/50, 25/85 or 25/100.

The Beta equation — what it gets right, what it gets wrong

Once you have B and the reference resistance R0 at T0, the inverse equation gives you temperature from a measured resistance R: 

1/T = 1/T₀ + (1/β) · ln(R/R₀)

This is the equation that lives inside almost every thermostat, washing-machine controller and battery pack management circuit on the planet. It is cheap (one division, one logarithm), it is robust (no sensitivity to weird coefficients), and over a narrow temperature band it is accurate to a few tenths of a degree.

The catch is that B is itself a function of temperature. The exponential is not quite an exponential; there is a slight curve in ln(R) vs 1/T that a single-slope fit cannot capture. Within ±10 °C of the calibration midpoint the error is below 0.1 °C and ignorable. Push the same B fit out to a 100 °C span and the error grows to several degrees — bad for any precision use, catastrophic for medical-grade thermometry.

Rule of thumb: if the application stays inside a 50 °C window centred on the calibration midpoint, a single B-value is fine. Outside that, you need Steinhart-Hart.

The Steinhart-Hart equation

John S. Steinhart and Stanley R. Hart published their three-term polynomial fit for thermistor R(T) data in 1968. The form is: 

1/T = A + B · ln(R) + C · ln(R)³

Three coefficients (A, B, C) instead of one, and the ln(R)² term is intentionally omitted because empirical fits show it adds no real accuracy. With Steinhart-Hart, the residual error across the full operating range of a typical NTC drops below ±0.01 °C — limited by measurement noise rather than curve-fit quality. That is the headline reason precision instrumentation always uses Steinhart-Hart internally even when the datasheet quotes a B-value on the front page.

Generating the coefficients takes three resistance measurements at three well-separated temperatures (typically 0 °C, 25 °C and 70 °C for general use; 0 / 50 / 100 °C for high-temperature work). The three measurements give you three equations in three unknowns. Most thermistor vendors will provide the coefficients with their part if you ask — at Jianlu we ship them as a CSV alongside any production build that needs Steinhart-Hart calibration.

Converting between B-value and Steinhart-Hart

You can derive an approximate B-value from a Steinhart-Hart fit by evaluating the slope at your reference interval — but the conversion is lossy and not symmetric. Going the other way (B-value to Steinhart-Hart) requires inventing two extra data points, which guesses at curve detail that B alone cannot encode.

For mixed-tool workflows, the practical convention is: store the original three (T, R) calibration points alongside any derived coefficients. That way you can re-derive either form on demand without losing information.

R25 tolerance vs B-value tolerance

Two separate spec lines control how interchangeable your NTC build is from part to part:

  • R25 tolerance sets the resistance at 25 °C. Typical grades: ±1 %, ±2 %, ±3 %, ±5 %. For an HVAC sensor you want ±3 % or better; for a battery thermistor you want ±1 % to keep cell-to-cell asymmetry from corrupting the BMS.
  • B-value tolerance sets the slope. Typical grades ±0.5 %, ±1 %, ±3 %. Tight B-tolerance is what makes a part "fully interchangeable" — you can drop in any unit and the controller does not need a per-part recalibration.

Combined interchangeability grades (1 % R25 with 1 % B) translate to roughly ±0.5 °C accuracy across a 100 °C span without per-part calibration. That is the spec to ask for when you are designing a high-volume product that has no calibration step on the production line.

B-value choice and sensitivity

Higher B means more resistance change per degree, which means more signal for the ADC to resolve. A typical 10 kΩ / 3 977 K NTC drops from 32 kΩ at 0 °C to about 700 Ω at 100 °C — almost two decades of dynamic range. A 10 kΩ / 3 380 K part traverses a narrower range, so the curve is flatter at the extremes and sensitivity is lower at the limits.

In return, lower-B parts have better linearity inside their useful band. That trade-off — sensitivity versus range — is why the CWF series at Jianlu spans B-values from 3 380 K to 4 250 K with multiple R25 options. Send the operating range and target ADC resolution, and we will recommend the right combination.

Self-heating and Steinhart-Hart accuracy

A subtle trap when validating Steinhart-Hart accuracy: the calibration current itself heats the bead through I²R dissipation. The dissipation constant (typically 1 - 5 mW/°C in still air, much higher in flowing liquid) sets how much error this contributes.

For a 10 kΩ NTC at 25 °C measured with a 100 µA excitation, dissipation is 0.1 mW — well below the 0.01 °C accuracy limit. At a 1 mA excitation the same part dissipates 10 mW, which is enough to heat the bead by 5 °C in still air and 0.5 °C in a flowing fluid. Always check the test current that was used to extract the Steinhart-Hart coefficients before trusting them at high excitation.

Where this matters in product selection

For under-floor heating, HVAC duct probes, refrigerator and freezer thermistors, the B-value equation is sufficient — the controller works in a 50 °C window and a single β covers it. Spec R25 = 10 kΩ or 12 kΩ, B25/85 = 3 950 K, ±1 % interchangeability.

For BMS, medical instrumentation, calibration references and any application that needs < ±0.5 °C across > 50 °C of operating range, ask for Steinhart-Hart coefficients per lot. Our CWF series NTC thermistors can be supplied with per-lot Steinhart-Hart calibration data on request, or with ±0.5 % interchangeability for drop-in high-precision applications.