Thermocouple and RTD conversion — mV, ohms and temperature per ITS-90 and IEC 60751

Q: Why does type K have an exponential term and the others do not?

Because Chromel goes through a magnetic ordering transition near 127 °C, and the EMF-versus-temperature curve picks up an inflection there that a ninth-order polynomial cannot reproduce. NIST handles it by adding a Gaussian term to the summation, valid only above 0 °C. That term is worth 108.8 µV at 100 °C and peaks at 118.6 µV — 2.6 and 2.9 °C respectively.

Q: What is the difference between an ice bath and cold-junction compensation?

They are two distinct benches. With an ice bath the reference junction is held at 0 °C and the terminals see the absolute EMF of the simulated temperature; it requires the instrument internal compensation to be off. With internal compensation the instrument measures its own cold junction and adds that share, so what you inject is the difference. Confusing the two is the most common field calibration mistake.

Q: Can I use the NIST inverse polynomial directly?

You can, provided you account for the error band it declares — up to 0.06 °C on type K. What the standard defines is the direct function; the inverse is a convenience approximation of it. Using the inverse only as a seed for a Newton iteration on the direct function drops the residual to the order of 10⁻¹¹ °C and makes the round trip close.

Q: Pt100 or Pt1000 — which should I specify?

The curve is the same; only the scale changes. Because the Pt1000 sensitivity is ten times larger in ohms per degree, each ohm of lead resistance is worth a tenth as many degrees. On long runs or 2-wire connections that is the decisive technical argument. In exchange, the Pt1000 dissipates more power at the same current, which tightens the self-heating criterion.

Q: Why does thermometer class B stop at 600 °C while resistor class 0.3 reaches 660 °C?

Because they are two different tables in IEC 60751: Table 1 covers platinum resistors, the bare sensing element, and Table 2 covers assembled thermometers. The tolerance values match, the ranges of validity do not. Buying the element and buying the assembled thermowell do not give you the same upper limit.

Q: How much error does a 3-wire connection actually cancel?

Exactly as much as the legs are matched. The compensation subtracts one leg from the other, so what remains is the imbalance between them — which comes from unequal lengths, a splice or a loose terminal, and is not zero. The electronics cannot see that residue, and a bench calibration does not reproduce it. With 4 wires the cable does not enter the measurement at all.

When to use

When calibrating or verifying a temperature transmitter, when simulating a temperature by injecting millivolts, when checking an instrument reading against the resistance measured at the terminals, and when specifying the tolerance class and wiring configuration of a resistance thermometer.

What the conversion actually does

No temperature sensor delivers temperature. A thermocouple delivers a potential difference on the order of tens of microvolts per degree; a resistance thermometer delivers an ohmic resistance. Between what the sensor produces and the number on the operator screen sits a standardised reference function — not a straight line, not a scale factor — and that function is what performs the conversion.

The distinction matters because the non-linearity is not small. On a type K thermocouple, sensitivity ranges from 39.5 µV/°C at zero to 41.4 µV/°C at 100 °C and falls back to 36 µV/°C near the top of the range. Approximating that with a constant costs several degrees mid-scale. On a Pt100 the curvature is gentler, but it is still worth about 10 °C across 0 to 500 °C if the conversion is treated as linear.

Thermocouple: the ITS-90 reference function

NIST publishes, in Monograph 175, a polynomial per sub-range for each of the eight letter-designated types — B, E, J, K, N, R, S and T. The order runs from five to fourteen depending on type and branch, and type K carries, above 0 °C, an additional exponential term accounting for the magnetic anomaly in Chromel near 127 °C.

That exponential term is the most common implementation trap in the conversion. It is not an optional refinement: the constant coefficient of the upper type K range is −17.6 µV and exists precisely to be cancelled by it at t = 0. Implementing only the summation creates an artificial step at zero and accumulates 2.6 °C of error at 100 °C.

NIST also publishes inverse polynomials, to go from millivolts to degrees directly. They are approximations, with a declared error band — up to 0.06 °C on type K, and with no coverage at all on the cryogenic tails of types E, K, N and T. The path most faithful to the standard is to use that inverse only as a starting point and converge by iteration on the direct function, which is what the standard defines.

The reference junction decides the number

The EMF appearing at an instrument’s terminals is always the difference between the measuring junction voltage and the reference junction voltage, both referred to 0 °C. Two distinct benches follow from that, and mixing them up is the most frequent field calibration mistake.

With an ice bath or zero block, the reference junction sits at 0 °C and its share is nil: the terminals see the absolute EMF of the simulated temperature. This requires the instrument internal compensation to be off — with it on, the correction is applied twice and the indication is wrong by the ambient temperature.

With internal compensation, the instrument measures its own cold junction and adds that share before converting. To simulate 100 °C on a type K whose cold junction is at 25 °C, you inject 3.0960 mV — the difference — not the 4.0962 mV of the absolute EMF.

A second-order effect is worth recording: an error in the cold-junction reading does not translate one-for-one into the indication. The ratio between sensitivity at the reference junction and sensitivity at the measuring junction is the gain of that error. On a type K near ambient it stays close to 1, which validates the usual rule of thumb; on a type S at 1400 °C it exceeds 1.5, and the rule of thumb starts to understate.

For the most searched case — millivolts read on a type K, with the table ready and the calculator preconfigured — see type K thermocouple: mV to temperature.

RTD: Callendar-Van Dusen and the two class tables

On the resistance thermometer side, IEC 60751 fixes the relationship between resistance and temperature in two halves: a quadratic above 0 °C and a quartic below, with the C(t − 100)t³ term acting only on the negative branch. The third edition of the standard, from 2022, inverted the basis: the formula became the specification and the numerical table in the annex became informative. Computing from the relationship, rather than interpolating the table, is now the normative route.

Above 0 °C the relationship is a quadratic in t, and the inverse has a closed form — it is the exact root, not an approximation. Below 0 °C no closed form exists, and the inverse polynomial that circulates in catalogues errs by tens of thousandths of a degree; iterating on the relationship itself resolves it with a negligible residual.

Tolerance deserves separate attention. The standard carries two tables: Table 1 for platinum resistors and Table 2 for assembled thermometers. The tolerance values match, the ranges of validity do not — a class 0.3 resistor is valid to +660 °C while a class B thermometer stops at +600 °C. And outside its range of validity a class does not degrade to the next one: it simply stops applying, and no tolerance is guaranteed.

What the cable and the wiring cost

On an RTD, the lead resistance enters the measurement and converts into degrees through the sensitivity at the point. In 2-wire both conductors enter, out and back; in 3-wire the compensation cancels one leg against the other and the imbalance between them remains; in 4-wire nothing enters, because two conductors carry the current and two sense the voltage without carrying it.

The arithmetic is direct and the result usually surprises. On a Pt100 at 0 °C, one ohm of cable is worth 2.56 °C, and at 850 °C it is worth 3.42 °C, because dR/dt falls with temperature. Two hundred metres of 1.5 mm² cable at 40 °C add just over 5 Ω in a 2-wire connection — nearly 14 °C of systematic error that a bench calibration cannot reproduce, because on the bench the cable is not there.

This is why clause 5.5 of IEC 60751 requires 3 or 4 wires for classes better than B, recommending 4. A class A sensor bought with a ±0.45 °C tolerance at 150 °C and installed 2-wire delivers, in the field, an accuracy that has nothing to do with the class that was paid for.

Resolution, and why it lives in the fourth decimal

One last bench note. A type K thermocouple EMF changes 41 µV per degree; a tenth of a degree is 4 µV. A table or an indication with three decimals in millivolts declares 1 µV of resolution, or 24 m°C — right at the edge of what you want to resolve. With four decimals, the declared resolution is 0.1 µV. The trailing zero is not decoration: it is what tells you how much of the number is significant.

Formulas and fundamentals

Thermocouple reference function (ITS-90) E(t) = c₀ + c₁t + c₂t² + … + cₙtⁿ [+ a₀·exp(a₁(t − a₂)²) for type K above 0 °C]

Thermoelectric voltage in mV of a junction at t °C, referred to a reference junction at 0 °C. Each type has two or three sub-ranges with distinct coefficient sets, and the order reaches fourteen on the cryogenic branch of type T.

Reference junction correction E_terminals = E(t_measuring) − E(t_reference)

Both terms referred to 0 °C. With an ice bath, E(t_reference) = 0 and the terminals see the absolute EMF; with internal compensation, the instrument measures its own cold junction and adds E(t_reference) before converting.

Seebeck coefficient S(t) = dE/dt

Derivative of the reference function, in µV/°C. It converts instrument uncertainty into temperature uncertainty, and varies widely with type and point: about 41 µV/°C on a type K at 100 °C against about 6 µV/°C on a type S near ambient.

Callendar-Van Dusen (IEC 60751) t ≥ 0: R = R₀(1 + At + Bt²) | t < 0: R = R₀(1 + At + Bt² + C(t − 100)t³)

A = 3.9083×10⁻³ °C⁻¹, B = −5.775×10⁻⁷ °C⁻², C = −4.183×10⁻¹² °C⁻⁴, valid from −200 to +850 °C for a resistor with α = 3.851×10⁻³ °C⁻¹. Above 0 °C the relationship is quadratic and its inverse has a closed form; below, it does not.

RTD class tolerance Δt = ±(a + b·|t|)

Class AA: ±(0.1 + 0.0017|t|). Class A: ±(0.15 + 0.002|t|). Class B: ±(0.3 + 0.005|t|). Class C: ±(0.6 + 0.01|t|). Each class applies only inside a declared range, which changes depending on whether the resistor is wire wound or thin film.

Lead error in a 2-wire connection Δt = R_lead / (dR/dt)

R_lead is the resistance of BOTH conductors, out and back. On a Pt100 at 0 °C, dR/dt is 0.39083 Ω/°C, so each ohm of cable shifts the indication by about 2.56 °C. On a Pt1000 the same ohm is worth a tenth of that.

Standards & methods

ITS-90 · NIST Monograph 175 — reference functions of the eight thermocouple types
NIST Temperature Scale Database (SRD 60), Version 3.0 — DOI 10.18434/T4S888
IEC 60751 Ed. 3.0 2022-01 — industrial platinum resistance thermometers
IEC 60584-1 — thermocouple tolerances (outside the scope of this tool)
IEC 60228 — maximum resistance of copper conductors at 20 °C

Typical reference values

Quantity	Typical range	Note
Type K Seebeck at 100 °C	41.4 µV/°C	falls to about 39.5 µV/°C at zero and to 36 µV/°C near 1300 °C
Type S Seebeck at 25 °C	≈ 6 µV/°C	there 1 µV of noise is already worth 0.17 °C — noble metal is insensitive
Type K range	−270 to 1372 °C	the inverse published by NIST only reaches down to −200 °C
IEC 60751 range	−200 to 850 °C	thermometer classes B and C stop at +600 °C; resistor class 0.3 reaches +660 °C
Pt100 at 100 °C	138.51 Ω	and 18.52 Ω at −200 °C, 390.48 Ω at 850 °C
2-wire error on a Pt100	2.56 to 3.42 °C per ohm	worsens with temperature, because dR/dt falls
Usual Pt100 measuring current	up to 1 mA	a clause 5.3 reference; the normative criterion is 25 % of the tolerance

Worked example

Simulating 100 °C on a type K transmitter with a 25 °C cold junction

Inputs

Thermocouple type: K Chromel/Alumel
Temperature to simulate: 100.0 °C
Reference junction: Internal CJC at 25.0 °C

Results

Measuring junction EMF: 4.0962 mV
Reference junction EMF: 1.0002 mV
EMF to inject at the terminals: 3.0960 mV
Sensitivity at the point: 41.369 µV/°C
Cold-junction error gain: 0.979 —

The calibrator must inject 3.0960 mV, not 4.0962 mV: the instrument will add its own cold-junction share back. The 41.4 µV/°C sensitivity means 1 µV of calibrator uncertainty is worth 24 m°C, so a calibrator with 1 µV resolution supports the intended accuracy comfortably. The 0.979 gain shows that at this point and on this type, a 1 °C error in the cold-junction reading is worth 0.98 °C in the indication — the usual rule of thumb holds here, but it does not hold on a type S at 1400 °C.

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Common mistakes

Applying cold-junction compensation twice. The classic bench mistake: an ice bath is declared, the absolute EMF is injected, and the instrument internal CJC is left on. The indication is then wrong by the ambient value.
Implementing type K without the exponential term. It is worth 108.8 µV at 100 °C, or 2.6 °C, and peaks near 127 °C — the magnetic ordering point of Chromel. Simplifying the polynomial matches the table and misses the bench.
Treating the NIST inverse polynomial as exact. It is an approximation with a declared error of up to 0.06 °C. As a seed for an iteration on the direct function it serves; as a final answer it carries an error nobody accounted for.
Inverting a type B EMF near ambient. Type B EMF is not monotonic: it dips to −2.6 µV at 21 °C and only returns to zero near 42 °C. Below that, one voltage matches two temperatures.
Assuming a sensor outside its class range of validity "falls back" to the next class. It does not: the marking simply stops applying, and no tolerance is guaranteed there.
Calibrating a class A Pt100 on the bench and installing it 2-wire. The lead resistance is not on the bench and enters in full in the field — 200 m of 1.5 mm² is worth nearly 14 °C of systematic error.
Ignoring self-heating. The measuring current dissipates power in the element itself; beyond 25 % of the class tolerance the sensor starts measuring itself, and the error is invisible to a calibration run at a different current.

Frequently asked questions

Why does type K have an exponential term and the others do not?