Once you have a VFD in the picture, all bets are off. It is _probably_ the case that you would connect the motor in delta, and use a motor with the same nominal voltage as the supply voltage, but this is not necessarily true.
The 'star' and the 'delta' connection are simply two different ways to drive the motor. In the star connection, the voltage required to properly magnetize the motor is higher, but the current required to run the motor is lower, relative to the delta connection. But either will run the motor just fine if the proper voltage at the proper frequency is supplied.
_Normally_ the way a start/delta system is used, the motor is first started in the star connection. Since the star connection requires higher voltage for normal operation, the motor is effectively being operated at substantially reduced voltage. This will tremendously reduce the available torque from the motor, and can only be used with mechanical loads that don't require much starting torque. But this has the great benefit of reducing the starting current of the motor. Then once the motor has gotten up to speed, it is switched over to delta operation in order to operate efficiently.
But with a VFD, you can adjust the output voltage and frequency in an arbitrary fashion, at least within the limits set by the available voltage and the current handling capacity of the VFD itself. This means, for example, that you could select a voltage and frequency combination that would properly run the motor when star connected. As long as you correctly match the V/Hz ratio set on the drive to the particular motor connection, then the VFD will run the motor properly.
I expect that the motor should be connected Delta at your supply voltage, because this lets the motor get to nominal full speed with full available torque. However if the motor/drive system is intended to operate at other speeds (either only at reduced speed, or at higher than nominal speed), then other connections might be used.
What is the nominal HP rating of the motor?
What is the nominal speed rating of the motor?
What are the voltage ratings of the motor?
What is the nominal HP rating of the drive?
At what speeds will the motor normally be operated?
-Jon
