Motor Thermal Management Analysis — Electromagnetic-Thermal Coupling Theory and Practice
Theory: losses become heat, heat changes losses
Overview
An EV motor datasheet says 'peak torque for up to 30 seconds'. How is that time determined?
Winding temperature decides it. Most motor loss is copper loss from winding current, proportional to current squared — roughly torque squared. High torque makes heat jump, but winding heat capacity delays the temperature rise. The time until winding insulation reaches its limit (say 155 °C) is the allowable overload duration. A thermal network is handy for estimating it: feed in losses from electromagnetic analysis and compute temperature over time.
Two-node thermal network
$T_w, T_h$ are winding and housing temperatures, $C_w, C_h$ heat capacities, $R_{wh}$ the winding-to-housing resistance (including slot liner and impregnation), $R_{hc}$ housing-to-coolant resistance and $P_{fe}$ iron loss. Real motor networks usually add stator teeth and yoke, rotor, magnets and bearings for tens of nodes.
Temperature dependence of copper loss
$k$ is torque relative to rating (assumed proportional to current) and $P_0$ copper loss at rating and reference temperature $T_{ref}$. Copper resistance rises about 0.39% per kelvin, so the same current produces more heat as temperature climbs.
Insulation classes and the ten-degree rule
Motor and transformer insulation is classified by temperature capability: class A (105 °C), E, B (130 °C), F (155 °C), H (180 °C) and so on. Behind this lies the rule of thumb, rooted in Arrhenius reaction kinetics, that insulation life roughly halves for every 10 K rise. Running continuously at a 155 °C limit rather than holding 145 °C nearly halves insulation life — which is why estimating peak temperature within a few kelvin is worth the effort.
Worked example: an illustrative water-cooled motor
Conditions
| Item | Value |
|---|---|
| Rated copper loss (at 120 °C) | 400 W |
| Iron loss | 200 W |
| Winding heat capacity (5 kg Cu) | 1,925 J/K |
| Housing heat capacity (15 kg Al) | 13,500 J/K |
| Winding→housing resistance | 0.08 K/W |
| Housing→coolant resistance | 0.04 K/W |
| Coolant temperature | 65 ℃ |
Example 1: rating and continuous torque
At rating the winding is 121 °C and the housing 89 °C. With a 155 °C (class F) winding limit, continuous torque is about 1.25 times rated. A 25% torque increase raises copper loss by over 56%, and resistance grows with temperature too, so the margin is smaller than it looks.
Example 2: allowable overload time
| Torque (× rated) | Time to 155 °C |
|---|---|
| 1.5 | about 200 s |
| 2.0 | about 61 s |
| 3.0 | about 21 s |
About a minute at twice the torque, about 20 seconds at three times.
Those are from rated steady state. The winding time constant (capacitance × resistance) is about 150 s, so for shorter times the housing barely warms and the winding alone stores the heat. Datasheet figures like '30 seconds' come from such calculations plus a safety margin. Starting cold allows longer, but real driving considers worst sequences — overtaking partway up a hill — so the warm state is the usual basis. These network values are illustrative; real motors need thermal resistances calibrated by test.
How to approach the analysis
Procedure
- From electromagnetic analysis, obtain copper, iron and magnet eddy-current losses at each operating point (as for efficiency maps).
- Build a thermal network and calibrate key resistances (winding–core, housing–coolant) by test or detailed thermal analysis.
- Compute temperatures over time-varying duty cycles such as drive cycles.
- Update losses with temperature (copper resistance, magnet properties) and iterate; magnet temperature also feeds the demagnetisation check.
Cautions
- Temperature within windings: slot centres and end windings run hottest; split nodes accordingly.
- Contact resistance: the stator–housing fit is a major resistance.
- Rotor cooling: rotor and magnets are hard to cool and heat up from magnet eddy losses at high speed.
“Passed on the bench, derated on the hill”
A motor passed rated and peak-torque tests on the bench, yet in the vehicle it was derated partway up a long hill. Bench tests each started cold, but on the hill high torque lasted minutes, after high-speed driving had already warmed the magnets and windings. Running the thermal network through the full drive cycle would have predicted the temperature at the foot of the hill. Allowable overload time depends heavily on the starting temperature.
Common mistakes
Mistakes and fixes
| Mistake | Effect | Fix |
|---|---|---|
| Copper loss at fixed temperature | Hot side underestimated | Include resistance vs temperature |
| Overloads from cold only | Overheating in service | Simulate the full duty cycle |
| Winding as one node | Hot spot missed | Split slot and end windings |
| Ignoring fit contact resistance | Temperature underestimated | Calibrate by test |
| Not tracking magnet temperature | Demagnetisation risk missed | Include rotor nodes |
I'd like to try motor design and thermal calculations myself.
Try the electric motor design tool for motor design and the motor torque calculator for torque. Thermal networks are explained in thermal resistance networks, and related topics in efficiency maps, demagnetisation analysis, PMSM design and axial flux motors.