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Vfd design for capacitor run single phase motor

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MuznaSF

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I am designing a vfd module for capacitor run motor (like the ones typically used in AC compressors). I want to start the motor at 50Hz and then bring it to 45Hz once the ambient temperature cools down.
Is it necessary that I design two inverters for this purpose- one to run the main winding and the other to run the auxiliary winding with a 90 degree phase difference between the two?
Considering the small difference between the two frequencies that the motor is supposed to run at, can a single inverter suffice while making use of the auxiliary winding's internal capacitor to provide the phase delay?
 

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