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Say a motor is rated for 20 N·m continuous torque, and 35 N·m peak (peak = for no more than 30 seconds). Alternatively it may be specified at with continuous/peak currents, but if the motor efficiency is good overall, then this is almost an equivalent definition.

Say that I want to run the motor producing 25 N·m for some time. Is there a general mathematical model that approximates the safe run-time at that level? It's certainly more than 30 seconds, but I'm unsure how to derive an approximate value.

More specifics: it's a 3-phase brushless motor, and it does have a temperature sensor. However using the temperature alone (and disregarding the specs) does not seem to be a good approach. E.g. 30 seconds at 35 N·m does not heat it very much, and it doesn't get even close to the max temperature per specifications. I'm also constraining myself to the typical application RPM ranges, e.g. 50 to 100% of the rated speed.

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  • \$\begingroup\$ It probably comes down to information in the much needed data sheet. \$\endgroup\$ – Andy aka May 2 at 8:48
  • \$\begingroup\$ @AndyAka, It's not just a single motor - it has to work with variety of motors, they are changeable, including prototype ones that don't have a data sheet yet :) I'm designing just a very general oversight that provides some level of safety even if all other things, e.g. temperature sensors, fail. \$\endgroup\$ – anrieff May 2 at 8:53
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The power dissipated in the motor goes as I2, which is the same as torque squared. (35/25)2 is more or less 2, so if the motor heating was purely adiabatic, you would expect to get a minimum of 60 seconds from it at 25 Nm. However, during that time, there will be some cooling, so you would expect more. How much more? That's not possible to say without more measurements on the motor.

The temperature sensor is likely to lag the winding temperature. The sensor is probably there to guard against long term overheat. If you measure the winding temperature directly, you might get a better idea of what the peak temperature is.

How to measure the winding temperature? Measure the DC resistance of a winding when cold. They might be quite low, so you may have to use a 4-terminal (Kelvin) measurement to get reasonable resolution and stability, which you will need.

Run the motor at 35 Nm for 30 seconds. Detach from the ESC and quickly measure the resistance again. The tempco of copper is about 0.4% per degree C, about 10% in 25 °C, which is why your measurement needs reasonable resolution.

Repeat with 25 Nm. Increase the run time without exceeding the previous winding temperature rise you measured. Note that you don't need an exact conversion from resistance change to temperature, you're just targeting the same resistance change.

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  • \$\begingroup\$ Thank you, the heating ~ I² model would be fine for the conservative, worst-case safety modelling I'm trying to do. The more exact method for heating measurement is also intriguing to do and I'll try it tomorrow. The only issue is that it will be hard to disconnect the phase wires from the ESC very quickly, so the measurement would not be very precise. \$\endgroup\$ – anrieff May 3 at 9:07
  • \$\begingroup\$ Crucial to the interpretation of the temperature rise under 30 s of 35 Nm overload is what temperature it starts from. Is that from cold, or is that from a long time running at 20 Nm? 'Very quickly' is only in the context of the time constant of the motor. You can estimate a reading at time zero if you want, set a clock running when you stop the run, measure every minute for 10 minutes, and extrapolate the plot back, that's what I do for measuring the power output of my microwave oven for instance. But that's a level of accuracy probably unwarranted by the situation. \$\endgroup\$ – Neil_UK May 3 at 9:32
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For a brushless motor the length of time it can be run at peak torque/current is simply a thermal limitation issue, even down to very low speeds. Provided you don't exceed the safe operating temperature of the motor, there is no fundamental reason why it cannot be run at peak torque continuously.

If the motor has a temperature sensor then that would seem like a good solution to your problem. However, some things to keep in mind are exactly where this sensor is located, and what the thermal resistance is between the motor coils and the sensor. This could be considerable. You can do some test to determine it by measuring the coil resistance (which is a essentially a direct measurement of their temperature) after running it at your desired torque for a while, and comparing this with the temperature given by the sensor.

Keep in mind that motor manufacturers are usually quite conservative with the peak torque operating conditions, as they cannot know the physical environment the motor will be installed in. If this is a closed cabinet, with little air circulation and a plastic (thermally insulated) mounting, then it will be substantially more arduous for the motor compared with it being bolted to a large aluminium mount with good airflow around it. Also note that the power loss rises with the square of current/torque, so there will be substantially more heat (~3x) that needs to be removed at 1.75x torque levels. This is yet another reason why manufacturers will be very conservative.

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