NTC thermistor calculator

An NTC thermistor's resistance falls a few per cent per kelvin, set by its R25 and its B value; reading it means putting current through it, and the current heats it. Enter the part's R25, B and tolerances, the temperature, and the divider it sits in to get the resistance and temperature coefficient, the tolerance as degrees at that temperature, the divider's output and sensitivity, the power in the part, the self-heating from the mounted dissipation factor, and whether it fits Vishay's rule that self-heating takes no more than half of the accuracy wanted.

Fig 1 — Left, the B-value curve R_T = R25·exp(B(1/T − 1/298.15)) for 10.0 kΩ and B = 3988 K, with the band the 1 % R25 and 0.3 % B tolerances allow — Vishay's "butterfly", narrowest at 25 °C. Right, what a 5.00 V divider through 10.0 kΩ makes of it: steepest where the series resistor equals R_T. The working point is 25 °C.

Calculate

R25 (kΩ)

B value (K)

Temperature (°C)

R25 tolerance (%)

B tolerance (%)

Divider supply (0 = none) (V)

Series resistor (kΩ)

NTC position

Dissipation factor (mW/K)

Accuracy wanted (°C)

R at 25 °C · temperature coefficient α: 10.0 kΩ · −4.49 %/K
Tolerance at 25 °C: resistance · as temperature: ±1.00 % · ±0.22 °C
Divider: output · current · sensitivity: 2.50 V · 250 µA · −56.1 mV/K
Power in the NTC · self-heating · allowed for the accuracy (50 % rule): 0.625 mW · +0.21 °C · 0.150 mW — over
Series resistor for maximum sensitivity at 25 °C: 10.0 kΩ

0.625 mW in the thermistor self-heats it 0.21 °C at 3 mW/K — more than the 0.150 mW Vishay's rule allows for ±0.1 °C ("3 mW/°C × 0.1 °C × 50 % = 0.15 mW"). A larger series resistor, a lower supply, or a pulsed measurement; TDK's comparison shows constant-voltage feed through a series resistor spreads the self-heating more evenly over temperature than a constant current does.

The ±0.22 °C is the part's tolerance alone, from Vishay's ΔR/R = ΔR25/R25 + ΔB·(1/T − 1/298.15): the 1 % on R25 is fixed, and the 0.3 % on B adds 0.00 % at 25 °C. The divider's resistor tolerance, the ADC's reference and the B model's own fit add to it.

How this is calculated

Standard: TDK/EPCOS NTC general technical information; Vishay 29053; Vishay 33001

$R_T = R_{25}\,e^{B\left(\frac{1}{T} - \frac{1}{298.15}\right)}, \qquad B = \frac{\ln(R_1/R_2)}{1/T_1 - 1/T_2}$: TDK formulas 1 and 3, Vishay eq 1 and 2; T in kelvin. A restricted-range model; datasheet tables or Steinhart-Hart beyond it.
$\alpha = -\frac{B}{T^2}$: TDK formula 5: the temperature coefficient, "2 %/K to 6 %/K".
$\frac{\Delta R}{R} = \frac{\Delta R_{25}}{R_{25}} + \Delta B\left|\frac{1}{T} - \frac{1}{298.15}\right|, \qquad \Delta T = \frac{\Delta R / R}{|\alpha|}$: Vishay 29053 eq 5, the tolerance butterfly; converted to temperature through α.
$P = \delta_{th}\,(T - T_A), \qquad P_{max} = \delta_{th}\cdot \Delta T_{acc} \cdot 50\,\%$: TDK formula 11 in equilibrium; Vishay 33001's budget, "3 mW/°C × 0.1 °C × 50 % = 0.15 mW".
$\frac{dV_{out}}{dT} = \frac{V_s R_s}{(R_T + R_s)^2}\,\alpha R_T$: Divider sensitivity, greatest where R_s = R_T.

Assumptions

The exponential B model throughout; its own error against the real curve is not included.
B is treated as constant over the range; the tolerance formula's third-order term (TDK formula 6) is dropped as TDK's formula 7 does.
Self-heating is the steady-state value with the dissipation factor entered; the mounted value differs from the datasheet's free-air figure.
The divider's resistor, supply and ADC contribute no error.

What sets an NTC thermistor's reading

An NTC thermistor is, in TDK's definition from IEC 60539, a "thermally sensitive semiconductor resistor which show[s] a decrease in resistance as temperature increases", and the size of that decrease is what makes it useful: "with 2 %/K to 6 %/K, the negative temperature coefficients of resistance are about ten times greater than those of metals". Two numbers describe one. R25, "the resistance of the sensor in Ω at the reference temperature of 25 °C", and the B value, "a material constant, expressed in Kelvin" that "represents the slope of the R/T curve", so that R_T = R25 · exp(B(1/T − 1/298.15)). B is itself defined from two points — B25/85 or B25/100 — and TDK is candid about the model's reach: it "is only suitable for describing a restricted range around the rated temperature or resistance with sufficient accuracy"; for more, "either more complicated approaches (e.g. the Steinhart-Hart equation) are used or the resistance/temperature relation is given in tabulated form". Vishay's parts come with a third-order polynomial that fits "with an error smaller than 0.1 %".

The part's accuracy is two tolerances that add. Vishay: "the total tolerances of the NTC sensor over its operating temperature range is a combination of the tolerances on R25 and on B-value", ΔR/R = ΔR25/R25 + ΔB·(1/T − 1/298.15), which "shows a minimum at 25 °C since this is the temperature at which the sensor is calibrated. Above and below this temperature, the tolerances increase due to the increasing tolerances on B-value, giving the graph a 'butterfly' shape." Dividing that resistance tolerance by the temperature coefficient, α = −B/T² (TDK formula 5), gives the temperature error.

Reading it puts current through it, and current heats it. TDK: "when a current flows through the thermistor, the device will heat up more or less by power dissipation. This self-heating effect depends not only on the load applied, but also on the thermal dissipation factor δ_thand the geometry of the thermistor itself" — in equilibrium P = δ_th(T − T_A). Vishay's selection note defines the same quantity as the dissipation constant, "the amount of power (expressed in milliwatts) required to self-heat the thermistor … 1 °C above its environment", warns that it depends on "the lead length and type of lead, the type of encapsulating material … the mounting … the medium", and gives the budget: "if the D.C. of a thermistor assembly had been determined as 3 mW/°C … and it was desired to measure the oil bath to an absolute temperature accuracy of ± 0.1 °C, the maximum power that should be developed in the thermistor by the measuring current is 0.15 mW. This is to keep the self-heat factor to 50 % or less of the measurement accuracy."

NTC thermistor chart: TDK's 10 kΩ part across temperature

The worked example's part and divider, computed by the calculator above at the temperatures a board actually sees. Three columns tell the story. The resistance spans more than two decades, which is why the divider is the wrong shape at both ends; the tolerance in degrees grows away from 25 °C because the B tolerance is a slope error; and the sensitivity collapses at the hot end, where a 10 kΩ series resistor is far too large for a part that has fallen below 1 kΩ.

T	R	Coefficient α	Tolerance	V_out	Sensitivity	Self-heating
-20 °C	108 kΩ	-6.22 %/K	±0.28 °C	4.58 V	-24 mV/K	+0.06 °C
0 °C	34.0 kΩ	-5.35 %/K	±0.26 °C	3.86 V	-47 mV/K	+0.15 °C
25 °C	10.0 kΩ	-4.49 %/K	±0.22 °C	2.50 V	-56 mV/K	+0.21 °C
50 °C	3.55 kΩ	-3.82 %/K	±0.34 °C	1.31 V	-37 mV/K	+0.16 °C
85 °C	1.06 kΩ	-3.11 %/K	±0.54 °C	0.48 V	-14 mV/K	+0.07 °C
125 °C	348 Ω	-2.52 %/K	±0.80 °C	0.17 V	-4 mV/K	+0.03 °C

Worked example: TDK's 10 kΩ, B = 3988 K part in a 5 V divider

The defaults are TDK's example sensor — 10 kΩ ± 1 %, B = 3988 K ± 0.3 % — at 25 °C, below a 10 kΩ series resistor from 5 V, with 3 mW/K of dissipation and ±0.1 °C wanted.

R at 25 °C          10.0 kΩ;  α = −3988 / 298.15²             = −4.49 %/K
tolerance at 25 °C  1 % on R25 + 0.3 % × 3988 K × 0            = ±1.00 %  → ±0.22 °C
tolerance at 85 °C  1 % + 0.003 × 3988 × (1/358.15 − 1/298.15) = ±1.67 %  → ±0.53 °C
divider             5 V × 10 k / 20 k = 2.50 V;  250 µA;  0.625 mW in the NTC
sensitivity         dV/dR × αR = 1.25e-4 × (−449 Ω/K)           = −56 mV/K
self-heating        0.625 mW / 3 mW/K                          = +0.21 °C
allowed             3 mW/K × 0.1 °C × 50 %                     = 0.15 mW  — 0.625 mW is over

The 1 % part reads 0.2 °C wrong at its best temperature, half a degree at 85 °C, and the divider that reads it adds another 0.2 °C of self-heating on top — more than the whole tolerance. The fix is not a better thermistor: drop the supply to 2 V (0.1 mW), or raise the series resistor, or pulse the excitation. TDK's comparison is the reason to prefer a voltage and a series resistor over a constant current: with a constant current "the self-heating strongly depends on the ambient temperature (there is a steep gradient at T_A = 25 °C and below), whereas in the case of constant voltage the self-heating is better distributed over the whole temperature range".

NTC beta calculator: B from two measurements

Set the calculator to B value from two measurements to fit B to a part you have measured. It is TDK's formula 3, B = ln(R₁/R₂) / (1/T₁ − 1/T₂) with both temperatures in kelvin, and it is why a datasheet names the pair: B25/85 and B25/100 are the same equation run on different second points. The fit passes exactly through both readings, so the pair to choose is the one that brackets the range the design reads.

The hot reading needs care, because B is a logarithm of a ratio divided by a small difference of reciprocals. For TDK's 10 kΩ, B = 3988 K part, a reading 1 % high at the second point moves B by the amounts below, and the closer the two temperatures, the larger the error:

Pair	R at the hot point	B returned	B if the hot reading is 1 % high
B25/50	3.55 kΩ	3988 K	−38 K
B25/85	1.06 kΩ	3988 K	−18 K
B25/100	680 Ω	3988 K	−15 K

Set the mode to Temperature from a measured resistance for the inverse, T = 1/(1/T₂₅ + ln(R/R₂₅)/B), which is the same B model solved for temperature, with the part's tolerance carried through as a temperature band.

Where the B-value model stops being valid

Far from 25 °C the exponential drifts. The B value is measured between two points and the real curve bends between them; the datasheet's R/T table or Steinhart-Hart coefficients are the accurate description, and TDK recommends the table.
The dissipation factor is a mounting property. Vishay measures it "suspended by its two-inch leads in still air"; a part soldered to a plane or immersed in oil sinks heat several times better, an SMD part in still air worse. "A prototype should be tested under actual operating conditions."
Steady state only. Self-heating and the thermal time constant (Vishay: the time "to change through 63.2 % of the difference between its initial and final body temperatures") are the same physics; a pulsed measurement shorter than the time constant heats less than the steady-state figure.
The divider's own errors are not included. The series resistor's tolerance and TCR, the supply or reference, and the ADC's offset all add; at high current the self-heating is usually the largest, at low current the ADC is.
Linearisation is a choice. The series resistor makes the output nearly linear around the temperature where it equals R_T, and nowhere else; a lookup table in firmware removes the constraint entirely.

Common thermistor mistakes

Reading B as if it were the whole curve. It is B25/85 or B25/100 — two points — and a "3950" part from two vendors can differ by a degree at −20 °C.
Sizing the divider for signal. A 10 kΩ part on 5 V through 10 kΩ is 0.6 mW and a fifth of a degree of self-heating; the accuracy budget, not the ADC's full scale, sets the current.
Trusting the 1 % at 100 °C. The butterfly widens with |1/T − 1/298|; at the ends of the range the B tolerance dominates.
Constant-current excitation for a wide range. TDK's curves: the self-heating "strongly depends on the ambient temperature" with constant current, and is "better distributed" with a voltage and series resistor.
Measuring the resistance with a multimeter and calling it R25. The meter's test current self-heats a small part; the datasheet value is "zero-power resistance".