Temperature Sensors and Thermometer Corrections

Thermal and fluid calibration requires precise management of thermodynamic state variables, heat transfer mechanisms, and phase transitions. Temperature realization rests on the International Temperature Scale of 1990 (ITS-90) and defined intrinsic fixed points. Environmental hygrometry, fluid flow metering, and volumetric dispensing demand mathematical corrections for ambient density, buoyancy, and thermal expansion to ensure defensible measurement uncertainty.

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Temperature Standards and the ITS-90 Scale

Thermodynamic temperature ($T$) is an SI base quantity measured in Kelvins ($\text{K}$), with the Celsius scale defined by $t / ^\circ\text{C} = T / \text{K} - 273.15$. Because primary thermodynamic gas thermometers are cumbersome, international metrology operates on the International Temperature Scale of 1990 (ITS-90).

ITS-90 defines temperatures through 17 fixed-point phase equilibria of high-purity chemical substances and specified interpolating instruments.

• Triple Point of Water (TPW): The cornerstone of ITS-90, realizing $273.16\text{ K}$ ($+0.0100^\circ\text{C}$) in a sealed borosilicate glass cell containing vapor, liquid, and solid phases of isotopic VSMOW water. Achieves realization uncertainties $< 0.1\text{ mK}$ ($0.0001^\circ\text{C}$).
• ITS-90 Interpolating Instruments:
  • From $13.8033\text{ K}$ to $1234.93\text{ K}$ ($961.78^\circ\text{C}$): Standard Platinum Resistance Thermometer (SPRT), requiring high-purity strain-free platinum wire ($W(29.7646^\circ\text{C}) \ge 1.11807$).
  • Above $1234.93\text{ K}$ ($961.78^\circ\text{C}$): Spectral Radiation Pyrometer referenced to Planck's blackbody radiation law.

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Temperature Sensors and Calibration Techniques

Liquid-in-Glass (LIG) Thermometers

LIG thermometers rely on differential volumetric thermal expansion between a reservoir bulb fluid (mercury or dyed organic liquid) and capillary glass stem (ASTM E1 / ASTM E77).

Emergent Stem Correction

For a total-immersion liquid-in-glass thermometer used with an emergent liquid column, a common correction is $C=kN(T_b-T_s)$, where $N$ is the exposed column length in scale degrees, $T_b$ the bulb temperature, and $T_s$ the mean exposed-stem temperature. For a partial-immersion thermometer, the relevant temperature difference is between the specified calibration stem condition and the actual stem condition. Do not use the bath temperature in that second case without checking the method.

$$C = K \cdot N \cdot (T - t)$$

For the total-immersion model above:

• $K$ = differential thermal expansion coefficient of thermometric fluid relative to glass ($0.00016\ ^\circ\text{C}^{-1}$ for mercury in borosilicate glass; $\approx 0.001\ ^\circ\text{C}^{-1}$ for organic liquids)
• $N$ = number of scale degrees emergent from the liquid bath
• $T$ = indicated bath temperature
• $t$ = average temperature of the emergent mercury column, measured using an auxiliary stem thermometer

An ice-point check can detect a zero shift in a liquid-in-glass thermometer. Applying that shift to other temperatures requires support from the thermometer’s calibration model and method. A one-point check does not demonstrate the entire scale is accurate.

Resistance Temperature Detectors (RTDs / PRTs)

Resistance Temperature Detectors operate on the positive temperature coefficient of pure metals. Platinum is the industry standard due to exceptional chemical inertness, high resistivity, and a stable, monotonic, generally nonlinear resistance–temperature relation.

Platinum Resistance Formulations

• Pt100 Standard: $R_0 = 100.00\ \Omega$ at $0^\circ\text{C}$.
• IEC 60751 / DIN Curve: $\alpha = 0.003851\ ^\circ\text{C}^{-1}$ (industrial standard globally).
• US Industrial Standard: $\alpha = 0.003923\ ^\circ\text{C}^{-1}$.
• SPRT (High Purity): $\alpha \ge 0.003926\ ^\circ\text{C}^{-1}$.

Temperature-resistance conversion above $0^\circ\text{C}$ follows the Callendar-Van Dusen Equation:

$$R(T) = R_0 \left[ 1 + A T + B T^2 + C (T - 100^\circ\text{C}) T^3 \right]$$

Where $C = 0$ for $T \ge 0^\circ\text{C}$. For IEC 60751 platinum:

• $A = 3.9083 \times 10^{-3}\ ^\circ\text{C}^{-1}$
• $B = -5.7750 \times 10^{-7}\ ^\circ\text{C}^{-2}$
• $C = -4.1830 \times 10^{-12}\ ^\circ\text{C}^{-4}$ (for $T < 0^\circ\text{C}$)

RTD Wiring Topologies

• 2-Wire: Carries both excitation current and sensing voltage on the same two leads. In Pt100 sensors where sensitivity is $\approx 0.39083\ \Omega/^\circ\text{C}$ near zero, a lead resistance of just $1\ \Omega$ creates a massive systematic positive error of $\approx 2.56^\circ\text{C}$ for 1 ohm total series lead resistance near zero.
• 3-Wire: Employs a bridge or dual-ADC circuit that subtracts lead resistance under the assumption that all three leads have identical resistance ($R_{L1} = R_{L2} = R_{L3}$). Fails if connections oxidize or temperature gradients along wires differ.
• 4-Wire (Kelvin): Two leads force constant excitation current; two independent high-impedance leads measure voltage directly at the sensor head. Minimizes lead-resistance effects in the ideal circuit; finite sense current, leakage, contact behavior, and instrument limitations remain.

Select RTD excitation from the manufacturer guidance, thermal contact, and required uncertainty. One milliampere is a common example for a Pt100, not a universal maximum; resistance, construction, and medium determine self-heating. Evaluate it by measurements at suitable currents and an appropriate power model. A higher-resistance sensor can require a different current. Account for uncertainty in any zero-current extrapolation.

Thermocouples (Thermoelectric Sensors)

Thermocouples operate via the Seebeck Effect: a temperature gradient along a conductor generates an open-circuit thermoelectric voltage (EMF):

$$V = \int_{T_1}^{T_2} S_{AB}(T) \, dT$$

Where $S_{AB}(T) = S_A(T) - S_B(T)$ is the relative Seebeck coefficient ($dE / dT$, typically $10\text{ to }68\ \mu\text{V}/^\circ\text{C}$).

Cold Junction Compensation (CJC)

Thermocouple voltage depends on both junction temperatures through the temperature-dependent Seebeck response; it is not generally proportional to temperature difference over a wide range. NIST reference polynomials use a 0°C reference junction. Evaluate or compensate the actual reference junction and use the appropriate nonlinear relation.

Modern digital calibrators employ electronic CJC: an internal precision thermistor or RTD measures terminal block temperature $T_{\text{block}}$, computes the equivalent thermoelectric voltage $V_{\text{CJC}}$, and algebraically adds it to the measured sensor EMF before evaluating temperature.

Thermistors

Thermistors are ceramic metal oxide semiconductor resistors with large Negative Temperature Coefficients (NTC), changing resistance by $-3\%\text{ to }-5\%\text{ per }^\circ\text{C}$. Calibration curves across broad spans are modeled using the Steinhart-Hart Equation:

$$\frac{1}{T} = A + B \ln(R) + C [\ln(R)]^3$$

Where $T$ is in Kelvins and $A, B, C$ are unique curve-fit coefficients determined by calibration at three known temperatures.

Temperature Calibration Sources

• Dry Block Calibrators: Portable metal blocks with drilled insertion sleeves. Convenient for field calibration, but vulnerable to axial immersion gradients, radial uniformity errors, and stem conduction errors (heat draining up the sheath of large-diameter probes). Use the required immersion depth and check stem conduction for the particular probe and source; a diameter-based rule is only a starting estimate.
• Stirred Liquid Baths: Metrology gold standard for comparison calibration. Utilize vigorous fluid circulation and deep immersion. Stabilities reach $\pm 0.002^\circ\text{C}$. Fluids: chilled alcohol ($-80^\circ\text{C}$ to $0^\circ\text{C}$), water ($+1^\circ\text{C}$ to $+80^\circ\text{C}$), silicone oil ($+80^\circ\text{C}$ to $+250^\circ\text{C}$), molten salts (up to $+550^\circ\text{C}$).

Technical reference checked October 10, 2026: BIPM industrial platinum resistance thermometer guide.