Because 3 phases are sufficient.
For the vast majority of applications of electric power there are no scientific, engineering, business or economic reasons to justify spending the extra money which would be needed to generate electricity in 4 phases or more.
There are several advantages of 3-phase power over single phase power. First, with 3-phase power, more power can be transferred with the same amount of wire (meaning the amount of copper used in all of the conductors transferring that power) as with single or 2-phase power. Also, power is transferred more "evenly" with 3-phase power because the "next peak" is closer than it is with single and 2-phase power. The 3-phase motors run more smoothly than those run on fewer phases. Also, there are motor design considerations (one being control of the direction of rotation) regarding the motor that make 3-phase power more attractive. Use the link to the Wikipedia article on 3-phase power and skim it. It isn't long and it is quite readable.
Power is generated in three phases because it allows for the generation of a more constant and smooth flow of power compared to a single-phase system. Three-phase power also allows for greater efficiency and power transmission over long distances.
You need three phase power to get three phase power. If you only have one phase, you will need some kind of inverter or motor-generator set. Generally, however, the demands of a three phase motor will exceed the capability and efficiency of these "tricky" solutions. It is best to simply contact your local power company and have them provide the proper three phase power to your facility.
Most electricity for the power grid is generated as 3-phase because this can be most economically sent long distances on high voltage power transmission lines. While the additional wire and equipment is more expensive than 1-phase, resistive losses are minimized by using high voltage and radiative losses are minimized because the different phases partially cancel each other. Most power transmission lines use 3-phase-delta as it requires no separate return/neutral line (as required by 3-phase-wye, 2-phase, and 1-phase) thus saving some money on these very long lines. Using more than 3 phases would be uneconomical as the costs for additional wire, insulators, poles in switches, transformers, and other equipment would increase without any additional savings beyond the use of 3-phase. By use of transformers the electricity generated, transmitted, and distributed as 3-phase can be converted to whatever must be delivered to the customer according to their needs: 3-phase-delta, 3-phase-wye, 2-phase, or 1-phase. Most industries have 3-phase delivered as either delta or wye because large motors start and run easier and more efficiently on 3-phase than on 2-phase or 1-phase. Also if the industry needs DC, 3-phase is easier to rectify (using a 3-phase bridge rectifier) and filter to get DC with less ripple than using 2-phase or 1-phase. Most homes have 2-phase delivered and use both phases for appliances, and one or the other phase for devices needing only 1-phase power. Most small generators/alternators generate only 1-phase to keep them simpler and less expensive.
The power circuit is only active between two phases - the live and neutral. Thwe third phase (Earth) is simply a safety device designed to blow a fuse in case of short-circuit. The power is delivered down the live wire, and returned by the neutral to complete the circuit.
As only one phase of a 3-phase supply can ever be properly connected to a piece of 1-phase equipment, such as a motor or whatever, no damage will result if that connection is done correctly.As most household 1-phase ac power is derived from power generated and distributed as 3-phase power, this question may just be an exam question designed to catch out students who don't really know what they are talking about...For more information please see the answer to the Related Question shown below.
You cannot run a three phase motor on single phase power. You'll burn the motor up within minutes. By the way, I've never seen or heard of a 380 volt motor. Do you mean a 480 volt motor? If so, the only way you can power it is with the correct full voltage and a 3 phase power supply.
The Scott T Connection is what you are considering. There is an article in wikipedia.org, but they say there that this concept is of historic importance only.
The one wattmeter method will only measure the power of the phase to which it is connected. So, by reconnecting it to measure each phase separately, you can measure the power in each phase in turn, and add them up to give you the total power.
This is only true of three phase power. Two phase power has a phase factor of 1.414, and single phase power has no differing factor (1). These different power factors are due to the fact that generators are designed with multiple stator coils placed at angles to one another, with three phase generators having a number of poles (each end of a coil is attached to a pole that is directly opposite its opposing pole) that is a multiple of 6. Thus, the least number of coils is 3, next 6, then 9 and so on. Three phase generators generate alternating current, shown visually by a sine wave, created by the rotation of the field coils through the stator coils, each coil peaks in power (on a 6 pole generator) exactly 120 degrees of rotation before and after the next or previous coil peaks. Because of this the power output is maintained a bit more stable than a single phase alternator, and the average power output of all three coils added together is equal to the number of phases square rooted, 1.732 (sq.rt. of 3). Power output in volt-amperes (watts) from the generator is affected by this phase factor.
Your question asked "Which is the cheaper to run?" For a residence, since you only pay for the energy you consume, then it will make no difference whatsoever which system is installed. However, it is more expensive to install a three-phase system.
To measure the power of a three-phase load, you need to use one less wattmeter than there are conductors supplying that load.