Motor Thermal Resistance: Convection-Limited, Not Constant

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Yes, the published thermal resistance moves with temperature, and the part that moves is the housing-to-ambient term. On a naturally cooled motor the convection coefficient rises roughly 6% for a 50 °C rise in surface temperature, so the housing-to-ambient resistance drops by a similar few percent. Over a 100 K rise that is a low-double-digit shift in one branch of a two-branch model.

That is not what is wrecking your prediction. Copper resistance climbs about 39% over the same 100 K, and the torque constant falls as the magnets heat, so you need more current for the same torque. Fix the loss term, then worry about the thermal resistance.

Read the Symptom Before You Touch the Model

Match what the machine is doing to the term that is wrong. Do not start by rebuilding the convection correlation.

Split the Thermal Path and See Which Half Moves

Two resistances in series carry the loss out of a permanent-magnet motor:

• Rth1, winding to housing. Pure conduction through slot insulation, impregnation, entrapped air, the lamination stack, and the stack-to-frame interface. Carries copper loss plus most of the stator iron loss.
• Rth2, housing to ambient. Convection plus radiation off the external surface. Carries everything.

A catalog that lists a single figure such as 10.3 K/W almost always means Rth2. Confirm it before you build anything on top of it: multiply the number by the rated dissipation and see whether the result lands near the rated housing rise or near the rated winding rise. If P × Rth = 60-80 K against a Class F winding limit, it is the winding-to-ambient or housing-to-ambient figure, not the slot conduction path. Getting Rth1 and Rth2 backwards puts the whole copper loss across the wrong branch and shifts the predicted winding temperature by tens of kelvin.

Rth1 is nearly temperature-independent. Copper, electrical steel, and cured epoxy all change conductivity by only a few percent over a 100 K span, and the air trapped in the slot actually conducts slightly better hot, which pushes Rth1 down a fraction of a percent. Treat it as constant.

Rth2 is where the temperature dependence lives. For laminar natural convection the film coefficient scales close to h ∝ ΔT^0.25, because air conductivity rises with film temperature while expansion coefficient and viscosity work the other way. Radiation adds to that: h_rad = εσ(T_s² + T_a²)(T_s + T_a) grows with the cube of absolute temperature, so a painted or anodized housing sheds a meaningfully larger share of its heat by radiation at 120 °C than at 40 °C. Both effects push in the same direction: Rth2 falls as the motor heats. Holding the cold value is therefore conservative, not dangerous.

On a fan-cooled or through-flow machine the coefficient is set by Reynolds and Prandtl numbers rather than by ΔT, and Rth2 is far flatter with temperature. If the motor has forced air, stop chasing this effect.

Fix the Loss Term First

The nonlinearity that actually breaks predictions is on the heat-generation side, and it is an order of magnitude larger than the Rth drift.

• Copper. Temperature coefficient near 0.00393/K at 20 °C. Use the resistance-ratio form: R2 = R1 × (234.5 + T2) / (234.5 + T1). From 25 °C to 125 °C that is a factor of 1.39 on I²R.
• Magnets. Remanence falls with temperature at the reversible coefficient of B_r for that grade — read it off the magnet datasheet, not the motor datasheet. Torque constant falls in proportion, so holding torque means raising current, which raises I²R again.
• Compounding. Both effects feed the same loss term, so the loop is: hotter → lower k_t and higher R → more current, more loss → hotter.

Confirm whether the catalog resistance is per phase or line-to-line. In a wye winding the line-to-line value is twice the phase value, and using it in 3 × I² × R doubles your copper loss.

Run the Iteration

1. Collect at 25 °C reference: phase resistance, torque constant, Rth1, Rth2 (and the test fixture behind it), the reversible B_r coefficient of the magnet grade, and the insulation class limit.
2. Seed the winding and magnet temperatures at ambient.
3. Correct the torque constant for magnet temperature, then solve for the current the load torque demands: I = T_load / k_t(T).
4. Correct phase resistance for winding temperature and compute copper loss. Add iron loss at the operating speed and, on high-speed units, bearing and windage loss. Iron loss is speed-driven, not temperature-driven — treat it as a constant adder in the loop.
5. Housing: T_housing = T_ambient + P_total × Rth2. Winding: T_winding = T_housing + (P_cu + P_fe_stator) × Rth1.
6. Scale Rth2 for the new surface rise. Reduce it about 5-6% for each 50 °C the surface has climbed above the temperature at which the published value applies. Leave Rth1 alone.
7. Repeat from step 3 until the winding temperature changes by less than 1 K. Four to six passes is normal.

If the loop climbs without settling, do not add iterations. The loop gain Rth_total × dP/dT has reached 1 and the machine cannot hold that torque at that ambient in that mounting. Reduce continuous torque, improve the mounting heat sink, or add airflow.

Verify With a Two-Point Test

You cannot back the manufacturer's convection coefficient out of published material properties, because you do not know the surface area, orientation, emissivity, or fixture they used. Characterize it on your hardware instead.

1. Log housing temperature at the hottest external point, ambient air 25 mm from the surface, bus current, and speed.
2. Shut down and immediately measure winding resistance. Convert to average winding temperature: T_wdg = (R_hot / R_cold) × (234.5 + T_cold) − 234.5. Extrapolate back to the shutdown instant from two or three successive readings.
3. Compute Rth2 = (T_housing − T_ambient) / P_total and Rth1 = (T_wdg − T_housing) / P_winding_path.
4. Repeat at full rated dissipation and compare. Rth1 should hold within measurement noise. Rth2 should come out lower at the hot point — that difference is your measured temperature dependence, in K/W, for this mounting.

Two points give you a straight-line correction good enough for the whole operating envelope. Fold it into step 6 of the iteration and stop guessing.

One more sanity check: log the cooldown curve and pull the thermal time constant. If it disagrees badly with mass × specific heat × Rth, your Rth attribution is wrong, not your convection model.

Pitfalls That Waste a Day

• Mounting. Published Rth2 is measured bolted to a specified plate. Mount the same motor on a small aluminum bracket instead of a large steel plate and Rth2 shifts 30-50%. That single error dwarfs every temperature coefficient in this article. Settle the mounting before you refine anything else.
• Cabinet ambient. The sink is the air around the motor. Inside a closed housing that air is 10-25 K above room temperature and climbing with the machine.
• Hand-built convection correlations. Deriving h from a flat-plate correlation and substituting it for the datasheet number replaces a measured value with an unmeasured one. Use correlations only to scale the published figure with temperature, never to replace it.
• Radiation ignored. A bare machined-aluminum housing has low emissivity; a painted or black-anodized one is near 0.9 and sheds substantially more at high rise. If you swapped the surface finish from the datasheet unit, Rth2 is not the same number.
• Peak versus RMS current. Duty-cycled loads must be reduced to an RMS current over the full cycle before it goes into I²R. Feeding peak current into a steady-state thermal model overpredicts badly.
• Winding limit versus magnet limit. The model gives you both. Check the winding against its insulation class and the rotor against the magnet grade's demagnetization knee at temperature. A design can pass one and fail the other.

When to Escalate

Call the motor manufacturer when the datasheet does not state the reference points and test conditions behind the thermal resistance — ask specifically whether it is winding-to-ambient or housing-to-ambient, the plate material, size, and thickness used, the orientation, the ambient, and the dissipation level at which it was measured. Ask for the reversible temperature coefficient of B_r and the demagnetization curve family for the installed magnet grade. If they cannot supply those, stop modeling and run the two-point test on your own hardware; measured Rth on your mounting beats any published figure taken on someone else's fixture.

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