DC motor temperature estimation

Coreless motors have several benefits, such as low inductance because of absence of iron core around the winding, low inertia for the same reason and no cogging torque, the residual torque due to the attraction between the winding and the magnet. This provide a fast response of current variations, with corresponding high dynamics, fast acceleration and smooth rotation during operation.

The operational point is typically given by motor speed and torque. In coreless motors these two parameters are linearly related, starting from the no-load speed condition to the zero-speed at stall torque, the dashed-black line below. Let's consider a DC motor with graphite commutation and Neodymium magnet 2642W024CR running at 5000rpm at load of 20mNm.

At the working point the motor current (the white dot on the red line), about I = 660mA. From the data-sheet the motor resistance is R = 5.78Ω. So the power losses in the motor, P = R · I^2, are responsible for the winding heating.

In the data-sheet the thermal resistance against the thermal flow between winding and housing is indicated with Rth1 (2.1K/W), and the thermal resistance against the thermal flow between the housing and the ambient is indicated with Rth2 (11K/W).

With these data it's possible to estimate the temperature of winding and housing, simulating the motor with an equivalent circuit, where the thermal losses is like a current, the thermal resistance are considered as resistors and the temperature is like a voltage across the resistor.

With the above data we can calculate the temperature difference of winding and housing:

ΔTw = Tw - Ta = R · I^2 · (Rth1 + Rth2) = 33°C

ΔTh = R · I^2 · Rth2 = 27.6°C

If we consider the ambient temperature of 22°C, we have

Tw = Ta + ΔTw = 22°C + 33°C = 55°C

Th = Ta + ΔTh = 22°C + 2.5°C = 49.6°C

In reality the winding reaches 55°C after five time constant (so 5 Tw1 = 50 s). The resistance value changes with the temperature with the formula R' = R · (1 + α · ΔTw) with α = 0.0039K^-1 thermal constant for copper.

With 55°C the new resistance will be then R' = 6.52Ω, 13% higher than the value at 22°C.

With this new value we must re-calculate the temperatures, obtaining slightly higher temperatures:

ΔTw' = Tw - Ta = R' · I^2 · (Rth1 + Rth2) = 37.2°C

ΔTh' = R' · I^2 · Rth2 = 31.2°C

This leads to the new temperature of winding and housing

Tw' = Ta + ΔTw' = 22°C + 37.2°C = 59.2°C

Th' = Ta + ΔTh' = 22°C + 31.2°C = 53.2°C

This is a simple way to estimate the motor winding and housing temperatures, with this model valid up to a winding temperature of 70°C. In effect, the Neodymium magnet losses (-0.1%/K) are not negligible above 70°C, and a more sophisticated motor model must be used to get better temperature calculation.

I hope this can help to understand how to estimate the winding and housing temperature of a DC motor. Brushless motors are slightly different, because at high speed the bearing friction and the effect of the eddy current increase the thermal losses. But we will see this difference in another article.

Thank you for reading until the end of this article!