Tod suggested that I use my background applying variable speed drive to write a tutorial. I have been doing this for over 30 years so I felt that other could benefit from my experience. This will be long and I hope not to boring. Keep in mind that I was a Physics major in college .
When applying variable speed controls one to machinery on must keep in mind the application. Turning is one of the few applications where more torque is needed a slow speed then at high speed. It requires more torque to cut material the further you are away from the center on the turning axis. A step pulley system is well suited for this since as you slow down the speed by changing pulley ratios you are actually increasing the torque. This is true for most mechanical variable speed drive systems. DC motors with permanent magnet motors and the lesser expensive inverters with 3 phase motors are constant torque devices so great care should be taken when applying them on lathes. A 1 HP motor designed to run at 1750 RPM will produce 36 in.lb. of torque throughout its speed range. A 1 HP motor designed to run at 3450 RPM will produce 18 in.lb. of torque through out it's speed range. Using my little Rikon lathe as an example the following table will tell you the speed and torque that the present drive system has. It has a 1/2 HP 1750 RPM motor. So the motor produces 18 IN.LB. of torque at the motor shaft.
Speed RPM Torque produced at spindle
430 73 in.lb.
810 39 in.lb.
1230 25 in.lb.
1810 17 in.lb.
2670 11 in.lb.
3900 8 in.lb.

If I wanted to replace the motor step pulley arrangement with an inverter or DC system I would need to figure an equivalent HP and then select a motor pulley ratio for the top speed. Lets say I wanted to use a 3450 RPM so that I can have the motor speed close to the 3900 RPM top speed. Then I take the spindle speed of 3900 and divide it by the motor top speed. This would mean I would use a 1.13:1 ratio (3900/3450). Now to figure the HP needed by taking the low speed torque multiplied by the highest speed and divide by 63025. HP=TorquexRPM/63025. In this example it tells me that I need 4.5 HP to have an equal DC or inverter based system. This would be a real waste of energy and my mini lathe would have a vary big motor hanging off of it. I would need a bigger bench to set it on also. Another way around t would be to limit the speed range of the lathe. I could for instance give up the highest two speed and use a 1750 RPM motor an belt it in so that the top spindle speed was 1810. This would mean using a 1.03: pulley ratio . The and would only require a 2.1 HP motor. 73x1810/63025. Do able but still not that practical.