Skip to content

Self-heating and zero-power resistance

Available since: rtd-sensor 0.8.0.

Measuring an RTD requires current. That current dissipates electrical power in the sensing element and can raise the element temperature above its surroundings. The size of the effect depends on the thermometer construction and on the thermal environment, so self-heating is not an immutable property of a Pt100 or another RTD characteristic.

rtd_sensor.self_heating analyzes current/resistance observations supplied by the caller. It does not control current sources, bridges, ADCs, MAX31865 devices, or other acquisition hardware.

Two-current zero-power extrapolation

The first 0.8.0 API follows the resistance-thermometry method documented by the BIPM: under a stable external thermal condition, measured resistance is treated as linear in measurement-current squared over the two current levels. Extrapolating that line to zero current gives the zero-power resistance estimate.

For two observations (i1, R1) and (i2, R2), the relation is equivalent to:

R0 = (R1*i2² - R2*i1²) / (i2² - i1²)

The BIPM industrial PRT guide also describes the common 1 mA and √2 mA case. When the higher-current power is approximately doubled, the resistance increase between the two measurements equals the low-current self-heating resistance rise under the linear model.

import math

from rtd_sensor import self_heating

low = self_heating.SelfHeatingObservation(
    measurement_current_a=0.001,
    resistance_ohms=100.01,
)
high = self_heating.SelfHeatingObservation(
    measurement_current_a=math.sqrt(2.0) * 0.001,
    resistance_ohms=100.02,
)

result = self_heating.extrapolate_zero_power_resistance(low, high)

print(result.zero_power_resistance_ohms)  # approximately 100.0
print(result.low_current_resistance_rise_ohms)  # approximately 0.01

Fit three or more observations

With three or more observations, rtd-sensor can fit the same resistance-versus-current-squared relationship instead of forcing an exact line through only two points:

observations = [
    self_heating.SelfHeatingObservation(0.001, 100.010),
    self_heating.SelfHeatingObservation(2**0.5 * 0.001, 100.019),
    self_heating.SelfHeatingObservation(0.001, 100.011),
    self_heating.SelfHeatingObservation(2**0.5 * 0.001, 100.021),
]

context = self_heating.SelfHeatingExperimentContext(
    medium="flowing water",
    flow_condition="approximately 0.4 m/s",
    mounting="fully immersed probe",
)

fit = self_heating.fit_zero_power_resistance(
    observations,
    context=context,
)

print(fit.zero_power_resistance_ohms)
print(fit.resistance_slope_ohms_per_a2)
print(fit.evidence.residuals_ohms)
print(fit.evidence.residual_standard_deviation_ohms)

The fit is:

R(i) = R0 + k*i²

and uses ordinary least squares in resistance. The observation-level I²R power remains available for inspection, but it is not substituted as the independent fit coordinate. That keeps the larger-observation analysis on the same model as the documented two-current extrapolation and avoids putting measured resistance on both sides of the regression.

Repeated current cycles are allowed

Three observations do not require three different current levels. Repeated low/high cycles are useful evidence when the experiment needs multiple readings to estimate a stable difference. For example:

low → high → low → high

provides four observations at two current levels and two residual degrees of freedom. rtd-sensor preserves the observations and residuals in the order the caller supplied them so the sequence remains inspectable.

What the residuals tell you

The returned evidence reports:

  • every resistance residual (observed - fitted);
  • descriptive RMS residual;
  • maximum absolute residual;
  • residual standard deviation using the positive residual degrees of freedom;
  • the number of observations and distinct current levels; and
  • the sampled current and current-squared span.

A larger observation set can therefore expose scatter or departures from the assumed linear relation that two points cannot reveal. It still cannot prove that the external temperature was stable. Drift, incomplete settling, acquisition error, and genuine nonlinearity can all appear in the residuals.

The multi-observation fit deliberately does not assign a universal "good" or "bad" residual threshold. Such a threshold depends on the experiment, measurement uncertainty, thermometer, and intended use. It also does not infer a weighting or dependence model: resistance/current uncertainties must be supplied explicitly for the WLS, GLS, or York paths; dependence is never inferred from the observation sequence or shared instrumentation.

Ask what the observations can actually support

Introduced in rtd-sensor 0.8.0.

After either a two-current extrapolation or a larger fit, you can ask for a threshold-free evidence assessment:

assessment = self_heating.assess_zero_power_extrapolation(fit)

print(assessment.warning_codes)
print(assessment.minimum_to_maximum_current_ratio)
print(assessment.zero_power_extrapolation_distance_in_current_squared_spans)

This does not turn residuals into an arbitrary green/red score. It reports structural limitations that can be stated without knowing the experiment's required uncertainty or acceptance tolerance. For example:

  • two observations exactly define a two-parameter line, so they have no residual check;
  • repeated observations at only two current levels can show scatter at those levels but cannot test line shape across a third level;
  • three or more distinct current levels without repeats can test line shape but do not show within-level repeatability; and
  • a zero or negative R-versus-I² slope does not show the positive resistance rise expected for ordinary self-heating.

The assessment also exposes geometry rather than silently choosing a threshold. minimum_to_maximum_current_ratio shows how separated the sampled currents are. zero_power_extrapolation_distance_in_current_squared_spans reports how far zero current lies beyond the lowest sampled I² point relative to the sampled I² span. A larger value means zero lies farther outside the observed span, but rtd-sensor intentionally does not define a universally acceptable maximum.

These diagnostics are evidence checks, not proof of thermal stability. BIPM/CCT guidance still requires the external temperature to remain constant and readings to become steady; when drift is present, repeated current cycles may be needed. The software has no external-temperature or time-history evidence unless the experiment records it separately.

Estimate fit-parameter uncertainty from residual scatter

For a 3+ observation fit, the parameter-covariance model depends on how the resistance observations were fitted. An unweighted fit uses the usual residual- scatter ordinary-least-squares estimate of uncertainty in the fitted zero-power resistance and slope:

fit_uncertainty = self_heating.estimate_zero_power_fit_uncertainty(fit)

print(fit_uncertainty.zero_power_resistance_standard_uncertainty_ohms)
print(fit_uncertainty.resistance_slope_standard_uncertainty_ohms_per_a2)
print(fit_uncertainty.parameter_covariance_matrix)

The estimate uses the residual variance SSE / residual_degrees_of_freedom and the two-parameter ordinary-least-squares information matrix. The covariance matrix order is:

zero_power_resistance_ohms
resistance_slope_ohms_per_a2

This is a conditional regression uncertainty estimate, not a complete measurement uncertainty budget. It assumes the sampled I² values are effectively known, and that the resistance-domain errors about the fitted line are independent and zero-mean with a common variance. That unknown variance is estimated from the retained residuals. This makes the result useful for repeated-current experiments whose scatter reasonably matches those assumptions.

If every resistance observation instead has a defensible absolute standard uncertainty, use inverse-variance weighted least squares:

weighted_fit = self_heating.fit_zero_power_resistance(
    observations,
    resistance_standard_uncertainties_ohms=(0.002, 0.002, 0.005, 0.005),
)
weighted_uncertainty = self_heating.estimate_zero_power_fit_uncertainty(weighted_fit)

print(weighted_fit.evidence.effective_weights)
print(weighted_fit.evidence.chi_squared)
print(weighted_fit.evidence.reduced_chi_squared)
print(weighted_uncertainty.parameter_covariance_matrix)

The effective fit weights are proportional to 1/u(R)². They are normalized for numerical stability, but the original absolute uncertainties are retained and used for chi-square and parameter covariance. That covariance is not multiplied by reduced chi-square or otherwise rescaled to make the observed residuals agree with the supplied uncertainties. An exact weighted line can therefore have zero chi-square residual while still retaining nonzero parameter uncertainty.

If measurement-current uncertainty is material, the independent coordinate itself is uncertain. Supply current standard uncertainties together with resistance standard uncertainties to use York errors-in-variables regression:

eiv_fit = self_heating.fit_zero_power_resistance(
    observations,
    resistance_standard_uncertainties_ohms=(0.002, 0.002, 0.005, 0.005),
    measurement_current_standard_uncertainties_a=(2e-7, 2e-7, 5e-7, 5e-7),
)
eiv_uncertainty = self_heating.estimate_zero_power_fit_uncertainty(eiv_fit)

print(eiv_fit.evidence.current_squared_standard_uncertainties_a2)
print(eiv_fit.evidence.chi_squared)
print(eiv_uncertainty.parameter_covariance_matrix)

The fit transforms each current uncertainty to the actual fitted coordinate using the first-order relation u(I²) = 2 I u(I). That local linearization is appropriate only when the supplied standard uncertainty makes first-order propagation a defensible description of the squared-current coordinate. The method does not claim that an arbitrarily broad current distribution becomes Gaussian after squaring.

If current and resistance errors for the same observation are correlated, pass one coefficient per observation with current_resistance_error_correlations. With positive measurement currents, the first-order I -> I² transformation preserves the sign and magnitude of that correlation coefficient. Omitting the argument records zero within-observation correlation.

For correlation between different resistance observations with exact current coordinates, supply a full covariance matrix and use generalized least squares:

resistance_covariance = self_heating.ResistanceObservationCovariance(
    covariance_matrix_ohms_squared=(
        (4e-6, 1e-6, 0.0, 0.0),
        (1e-6, 4e-6, 1e-6, 0.0),
        (0.0, 1e-6, 25e-6, 2e-6),
        (0.0, 0.0, 2e-6, 25e-6),
    )
)
gls_fit = self_heating.fit_zero_power_resistance(
    observations,
    resistance_observation_covariance=resistance_covariance,
)

GLS uses the full supplied covariance rather than reducing it to marginal error bars. The matrix must be positive definite; an exactly singular common-mode model needs a more explicit source/constrained treatment rather than an arbitrary pseudoinverse. Because the matrix already contains the marginal resistance variances, do not also pass separate resistance standard uncertainties. Cross-observation covariance is never inferred from residuals, replicate identity, or shared instrumentation. Measurement-current uncertainty cannot be combined with the GLS covariance path; doing so would require a more general correlated measurement-error model than the York or GLS paths implemented here.

A perfectly fitted unweighted finite dataset can produce zero residual-based covariance. That does not prove the experiment has zero physical uncertainty; it only means the residual-scatter estimator has no scatter from which to estimate a nonzero common resistance variance.

A positive fitted slope is the direction ordinarily expected for self-heating. Zero or negative slopes are retained and reported as evidence rather than being rejected, because software cannot determine from the sign alone whether the cause was negligible heating, drift, measurement noise, or an invalid experiment.

Interpret the larger fit as temperature and power

The 3+ observation fit can be converted through an RTD model without replacing its resistance-domain evidence:

fit_temperatures = self_heating.evaluate_zero_power_fit_temperatures(
    fit,
    model=model,
)

print(fit_temperatures.zero_power_temperature_c)
print(fit_temperatures.fitted_temperature_rises_c)
print(fit_temperatures.observed_dissipated_powers_w)
print(fit_temperatures.fitted_dissipated_powers_w)

The same model is used for the fitted zero-power resistance, each observed resistance, and the fit-predicted resistance at each sampled current. The result therefore keeps both views:

  • observed temperatures and I²R_observed powers show the measured data;
  • fitted temperatures and I²R_fitted powers show the retained linear resistance-versus-current-squared fit at those same current coordinates.

The difference between observed and fitted temperatures is also retained as a temperature-domain residual. Negative fitted temperature rises remain visible when the fitted resistance slope is negative; the API does not relabel such a dataset as valid physical self-heating.

These temperature-rise-versus-power pairs are useful evidence for later dissipation analysis, but they are not automatically called a self-heating coefficient or dissipation constant. The BIPM industrial-PRT guidance notes that such coefficients depend on the sensor/probe construction and the specific thermal environment, and manufacturers commonly specify them for a stated medium and flow condition.

Propagate the larger fit covariance into temperature rises

The retained intercept/slope covariance can be propagated through the same RTD model:

fit_temperature_uncertainty = (
    self_heating.propagate_zero_power_fit_temperature_uncertainty(fit_temperatures)
)

print(fit_temperature_uncertainty.zero_power_temperature_standard_uncertainty_c)
print(fit_temperature_uncertainty.fitted_temperature_rise_standard_uncertainties_c)

At each sampled current-squared coordinate x, the fitted resistance is R0 + k*x. Its sensitivity to the fitted parameters (R0, k) is therefore (1, x). Multiplying by the model's local dT/dR gives the fitted-temperature sensitivity vector. The zero-power sensitivity is subtracted before propagating a temperature rise, so the shared fitted intercept and its covariance with slope are not discarded.

This propagation uses whichever intercept/slope covariance the retained fit provides: residual-scatter covariance for unweighted OLS, covariance defined by supplied absolute resistance standard uncertainties for inverse-variance weighted least squares, or York parameter covariance when current and resistance coordinate uncertainties were supplied. For the York path, current uncertainty has therefore influenced the fitted-parameter covariance, but the sampled current coordinate used for each reported fitted temperature is still treated as the retained nominal coordinate. No additional direct current-coordinate term is added to those reported fitted-temperature uncertainties. The RTD model is treated as fixed, and model- parameter covariance and any experimental effects not represented by the supplied WLS/GLS/York uncertainty model remain separate.

Convert the result to zero-power temperature

Once the resistance extrapolation is complete, the same result can be interpreted through any supplied RTD model:

from rtd_sensor import catalog, self_heating

model = catalog.get_model("pt100")
temperatures = self_heating.evaluate_two_current_temperatures(
    result,
    model=model,
)

print(temperatures.zero_power_temperature_c)
print(temperatures.low_current_temperature_rise_c)
print(temperatures.high_current_temperature_rise_c)

The function converts three resistance values through the same model:

  • the extrapolated zero-power resistance;
  • the low-current observed resistance; and
  • the high-current observed resistance.

The reported self-heating temperature rise at each operating point is the difference between that observed temperature and the extrapolated zero-power temperature. The result also retains the exact supplied model object so the model used for those conversions remains inspectable. The calculation does not modify the RTD model, and model range or conversion errors propagate normally.

The zero-power temperature is the RTD-model interpretation of the extrapolated zero-current resistance under the experiment's stable-condition assumption. It should not be described as an independently measured ambient temperature.

Propagate measurement uncertainty

If standard uncertainties are available for the two measurement currents and two measured resistances, they can be propagated through the extrapolation without changing the original observations:

inputs = self_heating.TwoCurrentInputStandardUncertainties(
    low_current_standard_uncertainty_a=1e-6,
    low_resistance_standard_uncertainty_ohms=0.002,
    high_current_standard_uncertainty_a=1e-6,
    high_resistance_standard_uncertainty_ohms=0.002,
)

zero_power_uncertainty = self_heating.propagate_two_current_zero_power_uncertainty(
    result,
    input_standard_uncertainties=inputs,
)
print(zero_power_uncertainty.zero_power_resistance_standard_uncertainty_ohms)

temperature_uncertainty = self_heating.propagate_two_current_temperature_uncertainty(
    temperatures,
    input_standard_uncertainties=inputs,
)
print(temperature_uncertainty.zero_power_temperature_standard_uncertainty_c)
print(temperature_uncertainty.low_current_temperature_rise_standard_uncertainty_c)

The calculation uses first-order propagation from the original four measured inputs in this order:

low current, low resistance, high current, high resistance

When no additional dependence information is supplied, those four standard uncertainties are treated as independent. If a defensible correlation model is available, retain it explicitly instead of hiding it inside an independence assumption:

correlations = self_heating.TwoCurrentInputCorrelationMatrix(
    correlation_matrix=(
        (1.0, 0.0, 0.0, 0.0),
        (0.0, 1.0, 0.0, 1.0),
        (0.0, 0.0, 1.0, 0.0),
        (0.0, 1.0, 0.0, 1.0),
    )
)

correlated = self_heating.propagate_two_current_zero_power_uncertainty(
    result,
    input_standard_uncertainties=inputs,
    input_correlation_matrix=correlations,
)

print(correlated.input_covariance_matrix)
print(correlated.propagation_method)

This example represents perfectly correlated low/high resistance errors while the other cross-correlations are zero. The matrix order is still I_low, R_low, I_high, R_high. rtd-sensor requires a finite, symmetric, positive-semidefinite correlation matrix with unit diagonal and combines it with the supplied standard uncertainties to form the covariance matrix used in J Cov(x) Jᵀ.

Correlation is not inferred merely because readings use the same current source, bridge, meter, calibration chain, or acquisition sequence. A shared instrument can create dependence, but its sign and magnitude require an actual measurement/error model. Because this remains a local first-order propagation, the supplied uncertainties should also be small enough for the local linearization to be meaningful. A current uncertainty that is large relative to the separation between the two current levels needs more careful treatment.

For temperature results, the local dT/dR sensitivity comes from the same RTD model used for the temperatures. Fitted-model covariance is not added automatically; it remains a separate contribution because its dependence on the measurement inputs cannot be assumed in general.

Why temperature-rise uncertainty is propagated directly

The low-current temperature and the zero-power temperature are not independent: the low-current resistance is one of the measurements used to calculate the zero-power resistance. The same is true at the high-current point.

For that reason, rtd-sensor propagates each temperature rise directly from the original current/resistance inputs. It does not simply combine the uncertainty of T_observed and T_zero_power by root-sum-square, which would discard the shared input dependence and can give the wrong result. When an input correlation matrix is supplied, the same full covariance matrix is used here as well; common-mode terms can therefore cancel or reinforce according to the retained sensitivity vectors.

Report a context-bound self-heating coefficient

A self-heating coefficient is meaningful only with the thermal environment that produced it. For the 3+ observation fit, retain that information when fitting and then derive the coefficient from the model-based temperature result:

coefficient = self_heating.evaluate_self_heating_coefficient(fit_temperatures)

print(coefficient.self_heating_coefficient_c_per_mw)
print(coefficient.dissipation_constant_mw_per_c)
print(coefficient.pointwise_self_heating_coefficients_c_per_w)
print(coefficient.coefficient_fit_residuals_c)

The scalar is a through-origin fit of fitted temperature rise versus fitted I²R power at the distinct sampled current levels. It is a finite-range coefficient over those levels, not the zero-power differential d(ΔT)/dP. Repeated measurements at one current level still influence the resistance fit, but that level appears only once in the coefficient fit so replicate count does not create an extra secondary weight.

The finite-range distinction matters even for idealized data. If the resistance fit is exactly R = R0 + kI², then fitted power is P = I²(R0 + kI²). The kI⁴ term means pointwise ΔT/P can change across the current range even with no measurement noise and, for a linear RTD model, no model curvature. The result therefore retains the pointwise ΔT/P values and coefficient-fit residuals. Their RMS and maximum absolute residual are descriptive shape diagnostics of the fitted relationship, not a second statistical error model. Use those diagnostics to judge whether one scalar describes the sampled range; rtd-sensor does not invent a universal acceptance threshold. A zero or negative resistance slope is retained by the resistance fit but is not promoted into a named positive self-heating coefficient. The two-current correction path is intentionally not used for this named characterization because two points leave no residual degrees of freedom.

York errors-in-variables fits are intentionally not accepted for this coefficient calculation yet. Once current uncertainty is material, the fitted power itself has direct current-coordinate uncertainty and shares dependence with the fitted line; propagating only the intercept/slope covariance would understate that uncertainty. The EIV result remains valid for zero-power extrapolation, but the finite-range coefficient stays limited to fixed-current OLS/WLS/GLS fits for now.

The reciprocal is reported as the dissipation constant. The convenient unit forms follow the metrology guidance:

self-heating coefficient: °C/mW
dissipation constant:     mW/°C

Do not transfer a coefficient measured in one medium, flow condition, mounting, or setup to another without evidence. Treat it as local to the fitted zero-power temperature and sampled power/current range as well. The BIPM/CCT guidance notes that self-heating depends on thermal contact with the environment and can also vary with temperature; manufacturer coefficients are therefore measured under stated conditions.

Propagate the fit covariance into the coefficient

coefficient_uncertainty = self_heating.propagate_self_heating_coefficient_uncertainty(
    coefficient
)

print(coefficient_uncertainty.self_heating_coefficient_standard_uncertainty_c_per_mw)
print(coefficient_uncertainty.dissipation_constant_standard_uncertainty_mw_per_c)

This propagates the retained covariance of the fitted zero-power resistance and slope into the finite-range coefficient. That covariance can come from residual- scatter OLS or from supplied absolute resistance standard uncertainties in a weighted fit. It does not include the deterministic difference between that scalar and a zero-power differential coefficient. It also does not add coefficient-fit residual scatter, model parameter covariance, current uncertainty, or correlated environmental/acquisition effects. An exact resistance fit can therefore produce zero covariance-derived coefficient uncertainty without proving that the physical self-heating behavior is exact or range-independent.

What the observation retains

Each SelfHeatingObservation keeps the two measured quantities used by the analysis:

  • measurement-current magnitude in amperes; and
  • measured RTD resistance in ohms.

It also exposes current_squared_a2 and the observed electrical power dissipated_power_w = I²R for inspection and later analysis.

Both the two-current extrapolation and the multi-observation fit intentionally use the documented resistance versus current-squared relationship rather than substituting I²R as the independent coordinate. Observation-level power remains available for later dissipation/self-heating analysis.

Experimental assumptions matter

The calculation cannot prove that the physical experiment was valid. In particular, the source guidance requires the external temperature to remain stable and enough time to be allowed for readings to become steady. When temperature is drifting, repeated current cycles may be needed to estimate the self-heating change reliably.

Two observations exactly determine a line, so the first API has zero residual degrees of freedom. The returned evidence therefore exposes the current ratio, resistance change, current-squared span, and slope, but it does not label the data as experimentally stable.

What 0.8.0 deliberately does not infer

The 0.8.0 self-heating analysis now covers OLS, resistance-only WLS, fixed-current GLS with a supplied cross-observation resistance covariance matrix, and York EIV with per-observation current/resistance correlation. It still does not invent a correlation model from instrument identity, repeated measurements, residual shape, or thermal context. A model that combines uncertain current coordinates with covariance across different observations is more general than the implemented York or GLS paths and remains outside this release.

Likewise, there is no automatic residual or reduced-chi-square pass/fail threshold. The package reports the diagnostics; whether they satisfy an experiment depends on the required uncertainty, measurement procedure, thermal stability, and intended use.

Sources

The implementation basis is the BIPM/CCT guidance already retained in the project references:

  • Guide to the Realization of the ITS-90 — Platinum Resistance Thermometry, section 5.3.3, especially Equation 34 for two-current zero-current resistance;
  • Guide on Secondary Thermometry — Industrial Platinum Resistance Thermometers, section 4.1 and Equation 4.1.1 for linear extrapolation of resistance versus current squared and the requirement for stable external temperature.
  • Reeve (1988), NBS Technical Note 1246, for generalized least squares with a supplied observation covariance matrix;
  • York et al. (2004), Unified equations for the slope, intercept, and standard errors of the best straight line, for the errors-in-variables treatment with uncertainty in both fitted coordinates and within-observation correlated errors.

See the project docs/REFERENCES.md for the retained source record.