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Three phase motors are actually simpler and better than single phase motors, because the three phase power provides a rotating field enabling the motor to start without any other circuitry. The problem is, they need three phase power, which requires a special hookup from the power company or a phase converter to synthesize three phase output from single phase input.. Single phase motors need a start winding to synthesize a lagging voltage phase, to "pull" the rotor from a dead stop. Once up to speed, the start winding is taken out of the circuit by a centrifugally activated switch. Typically, fractional horsepower motors of this design are described as "split phase". Bigger motors are described as "capacitor start". Both designs have the start winding and centrifugal switch, but the "capacitor start" version adds a capacitor in series with the start winding. This capacitor improves the starting torque and reduces the starting current, but costs a little more than the simpler "split phase" design. For an air compressor of decent capability, you are looking at multi-horsepower motors which will be described as "capacitor start" or maybe "split phase capacitor start". The practical economic limit of such motors is about 7.5 horsepower. Make sure they are designed for belt drive. These will cost $350 or more, and use 30 amps at 240 volts at 7.5 horsepower. There's no such thing as a "two phase" motor. The 240 volts sent to your house is a single phase with a center tapped transformer. When connecting 120 volt loads, people talk about "Phase A" vs. "Phase B", but this is really only relative to the center tapped neutral. Connected at 240 volts, your motor doesn't use the neutral, and is driven from the 240 volt single phase.
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