Single Phase or 3-Phase?

This is not correct. You lose horsepower with a static phase converter, not a VFD and not with a rotary phase converter.

Err, no::

A single phase motor has 3 windings (just like a 3-phase motor) and operates like a 3-phase motor but with the phases running 0 degrees, 180 degrees and 90 degrees and the voltage at 90 degrees is only 71% that of the other two. In this orientation, the motor produces 86% of the input power as output power, whereas the 3-phase motor with 0 degrees, 120 degrees, and 240 degrees, runs at 100% input=output power. {Both neglection thermodynamic losses,...}

It is possible to make a static power converter that happens to develop 0 degrees, 120 degrees, and 240 degrees, but only if you use the motor at constant HP. This typically requires a box full of big capacitors an a few inductors.
 
Vfd allows for rpm as well mpr in back gear.

Makes threading easier as it is easier to stop.

Vfd needs to be proper size for motor.

Ours is 2 hp motor and 3 hp vfd.

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You can see a discernible difference in surface finish between single phase and 3 phase lathe motors, the degree is highly variable even within the same model of lathe. See these old threads, where the issue went away when converting to 3 phase with a VFD or modifing the single phase motor/mounts. There are may be other variables such as anchoring, rigidity, oscillation, etc. that exacerbate the phenomenon.

A VFD is a practical option with motors up to 3 Hp, beyond that Hp the cost goes up significantly and typically one has to buy a 3 phase input VFD and derate it for single phase use above 5 Hp. They do have many benefits, including controlled acceleration and deceleration, the latter usually requires a braking resistor in this application. I have installed VFDs on quite a few machines and lathes up to 1640, braking times of around 1-2 seconds are easily achieved, but spin a very heavy chuck/material at high RPM and try to brake too quickly and you will most likely get an over voltage error and the VFD will free wheel to a stop. Some VFDs have the ability to circumvent this by dynamically modifying the braking rate to prevent an over voltage error. Cheap VFDs are often lacking the braking circuity despite having the connection terminals for an external braking resistor. A VFD only need to be rated for the Hp/amps for a particular motor, so up sizing a VFD is not needed if it meets the motor specs.

As far as Hp output of motors, my understanding is that a 2 Hp rated motor is 2 Hp independent of being single phase or 3 phase. The efficiency/PF for a single phase motor is less (typically around 65%, but newer motors are higher) vs. 3 phase which are in the 80 - 90%, so although both motors have the same rated Hp, the single phase will draw more power. Out of curiosity I checked out the performance data for two Baldor 2 Hp motors, one single phase the other 3 phase, the major difference is at 100% rated output (2 Hp) the single phase motor is pulling 2400 watts, the 3 phase is 1700 watts. Given that an ideal motor 100% efficiency would be ~1500 watts. The 3 phase motor has more break-down, pull-up and locked rotor torque.
 
It is possible to make a static power converter that happens to develop 0 degrees, 120 degrees, and 240 degrees, but only if you use the motor at constant HP. This typically requires a box full of big capacitors an a few inductors.


So for practical use on a lathe we are basically talking about a spherical cow?

Milk production at a dairy farm was low, so the farmer wrote to the local university, asking for help from academia. A multidisciplinary team of professors was assembled, headed by a theoretical physicist, and two weeks of intensive on-site investigation took place. The scholars then returned to the university, notebooks crammed with data, where the task of writing the report was left to the team leader. Shortly thereafter the physicist returned to the farm, saying to the farmer, "I have the solution, but it works only in the case of spherical cows in a vacuum".

 
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