Motor Thermal Protection, Stall Time & Voltage Unbalance

Separate thermal memories

Currents are normalized to rated full-load current. Let $m=I_1/I_{FL}$ and $j=I_2/I_{FL}$. The stator memory $x$ follows $dx/dt=((m^2+j^2)/SF^2-x)/\tau_{run}$. A constant-current segment uses the exact exponential update. Trip is at $x=1$. Its rated-hot initial value is $1/SF^2$. The overload time is $\tau_{run}\ln[(h-x_0)/(h-1)]$ when $h>1$.

The rotor start budget is $B=m_{LR}^2t_{cold}$ in per-unit-current-squared seconds, and rated-hot memory is $b=1-t_{hot}/t_{cold}$. During each programmed start, $dy/dt=(m^2+Kj^2)/B$; this reproduces cold and hot rotor stall times for balanced constant locked-rotor current. Running rotor memory tends to $b(m^2+Kj^2)$ with $\tau_{run}$. This running replica is an explicit simplification, not SEL's slip-dependent rotor algorithm. Stopped memories decay to zero with $\tau_{stop}$.

The first-order stator and I²t rotor foundations follow Zocholl's open SEL paper, printed pp. 3–6 and Appendix A. GE 469 brochure p. 3 supports separate cooling constants and unbalance bias. Each start uses 80 subintervals. The rotor integrates squared linear current exactly; the stator uses its analytic exponential convolution with linear current squared (stable series for small time steps). History plots, peak summaries and restart headroom use the indicated samples. Trip timing is independent of chart samples: each constant-current memory crossing is solved analytically, and separate bracketed root solves locate stator and rotor crossings during starting ramps. The stator's interior maximum is found first so a crossing cannot be missed when the start ends below the limit. The earliest individual crossing over all phases is reported. Constant-current plotting nodes follow the thermal time constants and include exact memory crossings. The animated cursor evaluates the continuous equations instead of interpolating chart samples. The current-time envelope uses rotor adiabatic heating at currents of at least 2.5 pu and the running replica below that value. Its sweep starts just above the lower of the unbalance-adjusted stator and running-rotor pickups, with a small positive floor for the logarithmic current axis. The time axis expands to contain all plotted cold, hot and starting times; the legend remains outside the plotting area.

Unbalance and derating

NEMA unbalance is the maximum deviation from the average line voltage divided by that average. Independently, the voltage triangle gives $\beta=\sum V^4/(\sum V^2)^2$, $g=\sqrt{3-6\beta}$, and $V_2/V_1=\sqrt{(1-g)/(1+g)}$. With rated positive-sequence voltage, the estimate is $j=m_{LR}(V_2/V_1)$. Measured $I_2$ overrides this estimate.

The current derating factor $D=\sqrt{\max(0,1-Kj^2)}$ keeps $D^2+Kj^2=1$ when feasible. It is a heat-budget estimate, not an IEC/NEMA curve. The separate DOE empirical rise increment is $\Delta T_{bal}\,2u_{NEMA}^2/100$ with unbalance entered as a percentage. It is informational and is not added to the memories, avoiding double counting.

Restart cooling checks the entire sampled start, including its interior maximum. It finds the allowable initial stator and rotor memories, then solves their exponential stopped decay. A start that exceeds cold capacity has no finite cooling solution.

• Three-phase cage induction motor with sinusoidal fundamental currents and correct phase rotation. Current ratios are positive-sequence current ratios, not mechanical load percentages.
• Voltage magnitudes form a closed three-wire phasor triangle; phase rotation must be verified independently. The voltage-based current estimate assumes positive-sequence voltage at the motor's rated value. Measured negative-sequence current is preferred.
• Stator loss weight is 1 for each sequence; the user-supplied rotor weight K is applied in both starting and running phases. Fixed K and locked-rotor resistance during the start are conservative scenario assumptions for many accelerating motors, not a universal guaranteed bound.
• Initial “hot” means balanced rated-load equilibrium. Supplied unbalance is then applied throughout each energized interval, including both starts; it is zero while stopped.
• The same entered running time constant is used for the stator and the simplified rotor replica. Independent stopped cooling is allowed to be faster or slower.
• SF is a thermal current-limit ratio, not an instruction to operate continuously above the nameplate rating. The illustrative defaults combine published motor parameters with assumed duty and stopped cooling.
• Prospective heating continues after a limit crossing; no trip opening, reset, or restart permission is simulated.

For preliminary coordination and teaching within 0–5% voltage unbalance, with motor-specific safe stall times and cooling data. No claim of implementing a particular relay firmware or reproducing standard derating tables is made. The 2.5 pu transition in the displayed envelope is a model convention; actual acceleration and stall protection may differ.

The thermal memories are fractions of separately defined stator and rotor limits, not measured temperatures or physical heat capacities in J/K. The model derating limits positive-sequence current at the original rated rotor heating; it is not an automatic conversion to allowable shaft power. The DOE rise estimate is empirical and independent.

Confirm motor thermal damage and acceleration curves, ambient/altitude/ventilation conditions, service-factor applicability, CT behavior, relay filtering, protection margins, phase-loss protection, and process restart constraints. High-inertia or reduced-voltage starts require manufacturer data and a suitable slip-dependent relay model. Field measurements and installation work belong to a qualified person.

References are identified as of 29 October 2026: IEEE C37.96-2012; IEC 60034-26:2026, edition 2; ANSI/NEMA MG 1-2021. Check against the standards and the relay manual in force.