You don't. Every circuit needs a return wire back to the source.
A source of 208 volts can be obtained from any two legs of a three phase four wire 208 volt system. The two legs are classed as single phase 208 volts. The lead tags can be L1-L2, L2-L3, or L3-L1 all of which will give you 208 volts. Any of these lead tags to the grounded star point (wye) will give you 120 volts.
A three phase panel will not give you 110 and 220 volts. A three phase four wire panel will, but not at these voltages. The nearest voltages will be 120 and 208 volts. The 120 volt is the wye voltage of 208 volts. 208/1.73 = 120 volts. A single phase three wire panel will give you 110 and 220 volts.
208 v is a three-phase supply voltage used in North America, carried on a 4-wire system. The voltage between one of the three live wires and the neutral wire is 120 v and a single-phase supply can be taken by using one live and the neutral. If in doubt about the connection consult an electrician.
A 208/230 volt single phase motor only has two wires out. It sounds like you are trying to pick up a feed from a 120/240 source that needed a neutral wire to operate. The black and red wire will give you the voltage you need. Make sure that you use a starter and overload protection for the motor.See discuss question below.
A 208 v three-phase system has 120 v between each live wire and the neutral. It is used in the USA as an alternative to the 120/240 v single-phase system, offering 50% more power for the same current.
That is 20.8 kVa but a 208 v single phase supply is very rare. 208 v is usually three-phase, with 120v between neutral and the live wires.
A device that operates on 208 volts single phase power requires a power source that provides 208 volts of electricity in a single phase. It is important to ensure that the device is compatible with this specific voltage to prevent damage or malfunction.
#8 copper
As with any electrical installation, the wire size used depends on the expected maximum current the wire will carry. That cannot be determined by voltage alone.
A 208/230 volt single phase appliance requires a dedicated electrical circuit with a voltage of 208 to 230 volts and a single phase power supply. The appliance should be connected to a properly rated outlet or wiring to ensure safe and efficient operation.
208
Another name that can be used for a single-strand wire is thread or strand.