A constant-gain twin of a motor holding a load certifies every load: what does thermal derating do to the real hold?

research/thermal-twin

A constant-gain twin of a motor holding a load certifies every load: magnet derating makes holding a positive thermal feedback with a fold at λ=4/27 (real temperature 1.33–2.25× the twin's near it, no hold above it, runaway in ≈4π/(9√ε) thermal time constants), short tests cannot see it, the twin overstates holdable torque 1.1–1.6×, and refitting a constant gain does not remove the fold.

8 tests

What it found

Holding a torque dissipates Ri²; heating lowers the torque constant, so the hold current rises and heats more. In scaled variables φ′=λ/(1−φ)²−φ the hold exists iff λ≤4/27 (fold at φ=1/3) while the twin says φ=λ at every load: the real temperature rise is 1.33× and 1.76× the twin’s at λ=0.10 and 0.14 (current 1.15×, 1.33×) and 2.25× at the fold; above it there is no hold and the torque constant collapses in finite time, ≈4π/(9√ε) thermal time constants near the fold (ratio 0.925 at ε=0.002, 0.994 at 10⁻⁵). Tests shorter than the thermal time constant cannot see it (0.1 time constants at λ=0.14: +1.4% current). Under a winding temperature limit the twin overstates holdable torque by 1/(1−φ_m) (1.11× at φ_m=0.1, 1.64× at 0.4); refitting a constant gain on a hot low-load run still under-predicts (−0.06 to −0.10) and keeps no fold; winding-resistance rise lowers the fold (0.148 to 0.113 at β/α=1). Companion to the windup and gravity twins. Stylised: one thermal node, linear derating, constant load.

A figure from its own output

Figure produced by research/thermal-twin: steady_phi — thermal-twin/experiments/steady_phi.svg, written by the project's own experiment code.

Paper, code and results

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Produced with AI assistance (Claude). Results are from the code in this repository and are simulated or stylised unless the abstract says otherwise; nothing here implies affiliation with or endorsement by any lab, researcher or company cited. MIT licensed.