# Reading temperature through a signal

These are hypothetical analytical teaching models. No measured sensor, named crystal or published material curve is represented. Run `python generate.py` with matplotlib to regenerate the two CSVs, metadata and original PNG/SVG/PDF figure.

The linear example uses S = 0.2 + 0.01*T volts, with T expressed in Celsius over 0–100 C (201 samples). Inversion gives T=(S−0.2)/0.01. An assigned readout standard uncertainty of 0.001 V contributes 0.1 K temperature standard uncertainty when calibration coefficients are exact. Celsius and kelvin differences have equal numerical size; their absolute zeros differ. This is a readout contribution, not a calibrated thermometer accuracy specification.

The transition comparator is s=1/(1+exp(−(T−4)/0.02)), with T in kelvin, sampled from 3.8 to 4.2 K in 0.001 K increments (401 samples). The midpoint and width are invented. The logistic curve is a smooth phenomenological response and never reaches exact zero or one; it is not a microscopic superconductivity model. The inverse is 4+0.02*ln(s/(1−s)), defined only for 0<s<1. Invalid endpoints are rejected rather than clipped to artificial temperatures.

Its slope is s*(1−s)/0.02 per kelvin. With assigned normalized readout standard uncertainty 0.001, the first-order temperature contribution is 0.001/slope. This is shown only between normalized signals 0.05 and 0.95. Elsewhere the CSV leaves this value blank and metadata records the restriction. At the midpoint the slope is 12.5/K and the contribution is 0.00008 K (0.08 mK). That small number is solely a result of hypothetical parameters, not achieved accuracy. A threshold-only device gives an inequality; an analog response can supply local temperature information after calibration.

Checks: monotonicity; inverse reconstruction of every row (maximum 3.82e-14 K); independent central finite differences for slope (maximum relative disagreement 4.37e-8); midpoint values; six invalid inverse inputs; uncertainty-domain endpoints. Over the restricted range, exact inverse half-spans for plus/minus one readout standard uncertainty differ from the linearized value by at most 0.0117%. These spans illustrate local approximation only, not confidence intervals or an assumed noise distribution.

Excluded effects include reference uncertainty, uncertain calibration coefficients, drift, thermal gradients, contact, response delay, self-heating, field and bias shifts, hysteresis and feedback. No total uncertainty budget or material performance follows. Sources in study.json establish calibration and real transition-edge sensing as scientific contexts; their measured performance is not transferred to these curves.