Electronic – How many times of a frequency of a PWM can properly generate a sine wave to drive a motor

The 31.25kHz is 625x your 50Hz. So all you have to do is build a 625 entries sine lookup table, and at each interrupt set your output compare value to the next table entry. If you have a 32\$\mu\$s interrupt you'll have ran over the full table (=full sine cycle) every 625 x 32\$\mu\$s = 20ms, so that will be your 50Hz.

There are three main issues with the PWM frequency for driving a motor:

1. It must be fast enough so that the motor "sees" the average value and not the individual pulses. Motors have physical rotors that spin, the inertia of which will low pass filter the PWM. Usually 100 Hz or at most a few 100 Hz is good enough. Consider that many motors work fine when driven from a single phase of 50 or 60 Hz power.

2. It must be slow enough so that switching losses are a small fraction of the overall power. Transistors don't go instantly between on and off state where the power dissipation is zero (for a ideal switch). In between the dissipation in the transistor will be signigicant. This can be a problem due to the wasted power, but usually the problem of getting rid of the waste heat arises before that. For example, at 500 mW you can let a TO-220 package cool itself in free air. At 2 W you have to do the math and consider cooling carefully.

3. Depending on the application and the environment this motor will be installed in, you may need to consider whining. Even though 500 Hz may be plenty fast enough for the motor to average out, and 2 ms is a nice and slow switching time compared to the time of the transition region, it may cause audible whining. This can be quite irritating to humans, and is difficult to predict for any one motor. The magnetic fields caused by the coils will change with the current, which changes at the switching frequency. The force on individual wires of a winding is proportional to this magnetic field. Individual wires can vibrate much faster than the rotor as a whole can react. These winding and possibly other parts of the motor cause audible sound when vibrating. The sound is also proof of motion, which can eventually wear out insulation and the like.

   There is no way to know how audible a motor will be at a particular frequency without trying it. A lot of motor drivers switch at just above the human hearing limit for this reason. For example, 24 kHz is a common switching frequency, especially for off the shelf motor controllers that aren't matched to a particular motor and application.