Variable speed drives...

Think of it this way...

To the VFD, the incoming power is just the "raw material" from which it recreates a NEW output power source for the motor, one with variable frequency and variable voltage.

A VFD as we see them is essentially 3 different sections: A Converter, or rectifier section where the AC is converted to raw DC; a Conditioner, or DC bus section where that raw DC is smoothed with capacitors and possibly an inductor; and finally an Inverter section, the part that creates the new AC pseudo-sine wave output using transistors. Since the Inverter is fed with smooth clean DC from the Conditioner section, it has no real issue with how the DC gets there from the Converter section, i.e. from a single phase source or a 3 phase source. In fact, there are some types of drives that are intended to be used where there is NO Converter or Conditioner specific to that one drive, but rather a single large DC source feeding multiple Inverters.

When feeding a VFD with single phase power however, the Converter and the Conditioner sections WILL indeed have issues to deal with. On the Converter section, the rectifier components (Diodes or SCRs) will have to handle 1.732 times the current of what they would have been designed for if they only had to deal with a 3 phase input. In addition, the 1 phase power will create a significantly higher ripple content when rectified to DC, so you will need more capacitance on the bus in the Conditioner section. For that reason, it is usually a requirement to double the size of the VFD compared to the motor in order to get the added current capacity of the rectifier and the added bus capacitance.