Let us have some basics.
All motors are rated on their output power, nothing else.
A VFD is rated on its output power, plus about 2000 other not important factors.
An induction motor is an amazingly complex electromagnetic machine. Pick up any book on motors from any charity shop to convince yourself. An induction motor is essentially a transformer where power is transferred from primary, stator, to secondary, rotor. With the huge air gap between the power factor of the motor just rotating, nothing whatever attached to the spindle, will be about 0.2. At full load the power factor of a 2.2kW motor will be 0.85. If you measure the current, note current, taken at these two points it will be near identical, about 5%, possibly 10%. Read up about circle diagrams.
To determine the current taken by an induction motor you take the rating plate Watts divided by voltage equals current, then divide by 0.8 for efficiency and again by 0.8 for power factor. This is pretty worst case. Note that efficiency and power factor improve as the motor gets larger. If you have a large machine with multiple kW motors then divide by 0.85.
This leads onto a little known fact about induction motors, they are very fussy about their voltage. The power output, that is torque, is proportional to the voltage squared. So if your mains is 216 volts, and your motor is old and rated at 240V then this is 0.9 down, so squared is 81% power!
Conduction angle. Oh, this has hurt many people! The VFD has a bridge rectifier stuck across the mains with the electrolytic caps after, the idea being to charge the caps. But the caps are also being discharged by the motor as it is driven. Now, the maximum voltage the caps can be charged to is the RMS line voltage times 1.41, root 2, the difference between the RMS and peak voltages. The conduction angle is the actual part of the sinewave that the mains is more positive than the cap, so current flows through the rectifier to charge the cap. If the cap is at 300V then it can be seen that this charging period is not 180 degrees, more like 40 degrees. The problem here is that all the power needed to run the motor, 370W, has to be passed through the rectifier in these 40 degrees. So whilst the motor is happily delivering 370W with a current of 1.1/2A for 180 degrees, the poor rectifier has to pass that power in a quarter of the time, so 6A nominally flows. We then have the problem that the current is actually peaky, more like 10A peak for very few degrees. This is why they are so infernally noisy. Need very very fast switching diodes to do it with minimum loaa and noise. Now factor in the usual case where the workshop is run from some lighting flex and it is obvious that current drop due both to the wire resistance and impedance start to become important.
Just to keep you all happy there were some posts about single phase in to 400V three phase out VFDs. Now these really are pushing the limits because they use a voltage doubling circuit to generate the 400V or so for the caps. So what is in the single phase VFD a full wave bride, in these it is half wave. One half wave charges the cap to 240V, the next half wave pushes the 240V on the cap to 450V. Not at all nice. Now you see what three phase is used, and wired to create 6 phases.
For SoD, more load means more REAL amps, not inductive amps. Power factor REALLY does vary widely as the current taken shifts from inductive to real amps. I don't mind at all, I make as many mistakes as anyone else!
Foe AJ, same as SoD, at no load the motor is a large leaky inductor with non0existant [power factor.
I said buy a VFD same size as the motor.
Buying a larger VFD really does nothing to improve the conduction angle or charge current.
The bottom line is that the electronics, particularly the rectifiers and caps, are seriously stressed. One 4kW VFD I have actually has a programme option to measure the impedance of the caps.