This entire forum of Practical Machinist basically discusses how to build or buy and use rotary phase converters and VFDs (variable frequency drives).
You might want to view a schematic I put together for a 30 hp rotary converter on that forum. It is under the thread called "Integrated panel rotary phase converter design", and the document can be viewed by clicking here:
http://www.practicalmachinist.com/phaseconverter.pdf
I'll not go through the whole process of designing a converter, but here's brief synopsis of how they work:
Basically, a rotary phase converter consists of taking a three phase induction motor and running it on single phase power using two legs, and then taking the third leg for three phase off of the running motor stator windings. You end up with the two 'live' legs from single phase 240 volt supply, and the third or 'manufactured' leg comes from the motor windings, where the third leg is generated in the motor windings by magnetic action of the rotating rotor. A rotary phase converter is quite literally a rotating transformer.
You then take these three wires and connect them to a three phase machine motor, which will start and run just as on regular three phase from the power company.
The biggest job in building a rotary converter is getting the converter motor (called the 'idler') started. A three phase motor will not start when supplied with single phase; you must either get the shaft up to speed mechanically, or use a starting capacitor for a second or two during starting. Once running, a three phase idler motor will run on single phase, and generate three phase in the process. Since the idler is not loaded mechanically, the energy comes out of it in the form of amps on the generated third leg at 240 volts.
The final stages of building a rotary converter involves wiring run capacitors between one or both of the single phase legs and the manufactured leg. This supplies reactive power to that leg, which increases its voltage and also improves power factor.