If I may, I think what ActionDave meant in asking what type of compressor was the mechanical design, as in reciprocal, screw, centrifugal etc. If it is a 7-1/2HP basic shop compressor, I'd guess there is a 90+% chance it is recip. You can reduce the speed of a recip, but you can quickly run into trouble if it uses a mechanical oil pump or splash lubrication, because with reduced speed you get reduced oil flow and can damage the machine. So you must thoroughly investigate your compressor before reducing the speed and if it does have an electric oil pump, make sure it can be powered separately from the compressor motor as well.
For 7-1/2HP you must double the size of the VFD for the reasons Besoeker mentioned, so you need to buy a 15HP 230V drive to do phase conversion. Be careful in that there are a few brands of VFD that do not allow for single phase input because they have phase loss protection that cannot be disabled. If your vendor doesn't know, you can look in the operating manual to see if there is a fault code for phase loss and if there is, then is there a way to disable it. If you can't determine it, don't use that brand.
Also keep in mind that most compressors start unloaded via an unload valve but again if that valve is powered from the same line power as the compressor, ramping the compressor voltage may cause that valve to fail to open and your compressor will not start.
So bottom line it can be done, but you must do it carefully with full knowledge of what you have and all of the ramifications.
Second issue, a VFD as a phase converter has very little waste (losses) compared to an RPC, which can have 20% losses or more. Also a capacitor type phase adder reduces the motor capacity by about 30% so you would have to start off with a 10HP motor at least if you wanted to go that route.
