NTC thermistors: the B value, and why reading one heats it

An NTC loses 2–6 % of its resistance per kelvin, and two numbers describe it: R25 and B. What B is, the tolerance butterfly, and self-heating.

The cheapest accurate temperature sensor is a bead of metal-oxide ceramic whose resistance falls by a few per cent for every degree it warms. TDK’s general note starts from the definition — “as defined by IEC 60539, NTC (Negative Temperature Coefficient) thermistors are thermally sensitive semiconductor resistors which show a decrease in resistance as temperature increases” — and from the number that makes them worth using: “with 2 %/K to 6 %/K, the negative temperature coefficients of resistance are about ten times greater than those of metals.” A platinum RTD gives 0.4 %/K; a thermistor gives ten times the signal for a fraction of the price, at the cost of a curve that is not a straight line and a part that heats itself when read. This article is what the vendor notes say about both, and the NTC thermistor calculator does the arithmetic for any part in any divider.

R25 and B: two numbers for a curve

A thermistor’s datasheet leads with two parameters. Vishay’s application note: R25, “the resistance of the sensor in Ω at the reference temperature of 25 °C”, which “provides a convenient reference point”, and the B value, “a material constant, expressed in Kelvin”, that “controls the slope of the RT characteristic”. With them, “at least to a first approximation”, the resistance at any temperature is

$R_T = R_{25}\,\exp\!\left[B\left(\frac{1}{T} - \frac{1}{298.15}\right)\right]$

with T in kelvin. Turned around, B is what two measurements give: TDK’s formula 3, B = ln(R₁/R₂) / (1/T₁ − 1/T₂), and “the specifications in this databook refer in most cases to resistance values at temperatures of 25 °C (T₁) and 100 °C (T₂); i.e. B_25/100 is stated. For SMD NTCs B_25/50 and B_25/85 values are additionally given.” Which pair of temperatures matters, because “in practice, B varies somewhat with temperature” (Vishay): the same part has a slightly different B_25/50 and B_25/100, and two vendors’ “3950” parts are 3950 between different points.

Three curves of resistance against temperature on a log axis, from minus 40 to 125 degrees, for B values of 3000, 3988 and 5000 kelvin, all passing through 10 kilohms at 25 degrees. The higher the B value the steeper the curve: at 125 degrees the 5000 kelvin part is near 100 ohms and the 3000 kelvin part near 700 ohms. — Fig 1 — The B-value model R_T = R25·exp(B(1/T − 1/298.15)) for a 10 kΩ part with three B values spanning TDK's "2000 through 5000 K". B "is determined by the ceramic material and represents the slope of the R/T curve"; at 25 °C the slope is α = −B/T², -4.49 %/K for B = 3988 K, "about ten times greater than those of metals". The exponential is "only suitable for describing a restricted range around the rated temperature"; the datasheet's table is the real curve.

The slope at any temperature follows from the exponential. TDK’s formula 5, the temperature coefficient α = −B/T², is −4.5 %/K at 25 °C for B = 3988 K, and it is the number the rest of the error budget divides by.

Where the exponential stops

TDK is direct about the model’s limits: “the actual characteristic of an NTC thermistor can be roughly described by the exponential relation. This approach, however, is only suitable for describing a restricted range around the rated temperature or resistance with sufficient accuracy. For practical applications a more precise description of the real R/T curve is required. Either more complicated approaches (e.g. the Steinhart-Hart equation) are used or the resistance/temperature relation is given in tabulated form.” Its own answer is the table: “standardized curves … experimentally determined with utmost accuracy over the whole specified temperature range”, “available for temperature increments of 1 degree”. Vishay’s is a polynomial: “each NTC material curve is defined by a 3rd order polynomial” whose two forms — R from T, and T from R — “represent the real material curves with an error smaller than 0.1 % at any given temperature”, with coefficients published per material. The B-value formula is for the calculation at the bench and a range of a few tens of degrees; the firmware that reads the sensor over −40 to +125 °C wants the table or the polynomial.

The tolerance butterfly

A “1 %” thermistor is 1 % at one temperature. 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 given by the formula” 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.” TDK’s formula 7 is the same expression, and its explanation: “the resistance tolerance at a certain temperature is influenced by two variables: the manufacturing tolerance of the rated resistance and the variation of the B value.”

Two curves of temperature error against temperature, both V-shaped with the minimum at 25 degrees. The 1 percent R25 with 0.3 percent B part rises from 0.22 degrees at 25 to about 0.5 degrees at 85 and 0.6 at minus 40. The 1 percent with 1 percent B part rises much faster, to over a degree at the ends. — Fig 2 — Vishay's "butterfly", as temperature error: ΔR/R = ΔR25/R25 + ΔB·|1/T − 1/298.15| divided by |α|. For TDK's 1 % / 0.3 % part the error is ±0.22 °C at 25 °C and ±0.54 °C at 85 °C; with a 1 % B tolerance it is ±1.04 °C at 85 °C. "Above and below this temperature, the tolerances increase due to the increasing tolerances on B-value" — which is why a tight B, not a tight R25, is what an accurate sensor buys.

Divide by α and the resistance tolerance is a temperature error: for TDK’s example part, 10 kΩ ± 1 % with B = 3988 K ± 0.3 %, about ±0.22 °C at 25 °C and ±0.5 °C at 85 °C. A part with the same 1 % on R25 but 1 % on B is worse than a degree at the ends of the range. Vishay’s sales point is exactly this — “the exceptionally low ΔB-value … gives a flatter ΔR/R ‘butterfly’ curve” — and it is the right one: for a sensor used away from room temperature, the B tolerance is the specification to buy.

Reading it: the divider

The usual circuit is a resistor from a supply to the thermistor to ground, and an ADC on the junction. Its output is V_s·R_T / (R_T + R_s), an S-curve that is steepest where the series resistor equals the thermistor — which is where to put it: choose R_s equal to R_T at the middle of the range that matters, and the divider is nearly linear there and flattens at the ends. For the 10 kΩ part through 10 kΩ from 5 V, the output at 25 °C is 2.5 V and the slope about 56 mV/K, which the ADC resolution calculator turns into LSB. The resistor’s tolerance and TCR and the reference’s accuracy add to the thermistor’s own; the op amp error budget covers a buffer if one is needed.

The reading heats the sensor

The divider’s current flows through the thermistor and warms 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_th and the geometry of the thermistor itself”, with the rule “the smaller the device, the smaller is the permissible maximum load and the measuring load (zero power)”. In steady state its formula 11 is simply P = δ_th(T − T_A): the temperature rise is the power over the dissipation factor.

Vishay calls the same figure the dissipation constant, “the amount of power (expressed in milliwatts) required to self-heat the thermistor suspended by its two-inch leads in still air 1 °C above its environment”, and lists what changes it: “the lead length and type of lead, the type of encapsulating material … the mounting of the NTC thermistor / assembly, the medium of the surrounding environment (flowing gas, still air, water, oil, etc.)”. Hence “it is recommended that a prototype should be tested under actual operating conditions”. And it gives the budget to design to: “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.”

The 5 V / 10 kΩ divider above puts 0.625 mW into the part at 25 °C — four times that budget, a fifth of a degree of self-heating, and more than the thermistor’s own tolerance. The current has to be set by the accuracy, not by the ADC’s full scale.

Self-heating in degrees on a log axis against ambient temperature from minus 40 to 125. The constant-current 200 microamp line is high at cold temperatures, over 5 degrees at minus 40, and falls steeply as the thermistor resistance falls. The three constant-voltage lines for 5 volts through 5, 10 and 20 kilohms are lower and flatter, each peaking near the temperature where the series resistor equals the thermistor. — Fig 3 — TDK's figure 5 recomputed for its example part at 3 mW/K: "the constant current is 200 µA and the constant voltage is 5 V … for three different series resistors (R_S = 5 kΩ, R_S = 10 kΩ and R_S = 20 kΩ)". "In the case of 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." A voltage divider peaks at ¼·V²/R_S where R_T = R_S and falls either side of it.

How the current is supplied matters as much as how much. TDK compares a constant current with a constant voltage through a series resistor, “the constant current is 200 µA and the constant voltage is 5 V … for three different series resistors (R_S = 5 kΩ, R_S = 10 kΩ and R_S = 20 kΩ)”, and finds that “in the case of 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”. A constant current into a resistance that grows tenfold in the cold dissipates tenfold; a divider’s power peaks where R_T = R_S and falls either side. The trade the note names: “the higher the series resistor, the smaller will be the voltage that declines at the NTC thermistor” — less self-heating, less signal.

The other NTC

The same part run the other way is a different component. TDK’s V/I curve has four sections, and the sensor lives in the first, “the straight rise section where the dissipation power only produces negligible self-heating … Use of this curve section is made when NTC thermistors are employed as temperature sensors”. Section 4, “the falling-edge section where the decrease in resistance is greater than the relative increase in current”, is “the operating area of NTC thermistors when a self-heating effect is desired (e.g. inrush current limiters, liquid level sensors)” — the inrush limiter, which is an NTC chosen to heat itself into a low resistance. One physics, two parts, and the sensor is the one that must never get there.

Sources

TDK/EPCOS, NTC thermistors — General technical information, January 2018. The definition and 2–6 %/K, formulas 1–5 (R/T, B from two points, α), the tolerance formulas, zero-power measurement, self-heating and the V/I curve, and the 200 µA against 5 V comparison on the B57861S0103F045.
Vishay BCcomponents, NTC Thermistors application note, document 29053, 2021. R25, B and α; B25/85; the third-order polynomial and its 0.1 % fit; the tolerance butterfly.
Vishay Dale, Selecting NTC Thermistors, document 33001, 2015. The dissipation constant, its measurement, the 50 % self-heat budget and the thermal time constant.