Why we use 3 phase power supply, not 4/5/6 phase

Single-phase motors have unambiguous rotation (with a capacitor start), and with a center tap neutral, a single-phase system can be balanced.

I'm not sure regarding the economics of multi-phase systems. It seems to me that for each phase added, the total power capacity is increased by one line-ground voltage times one line current so there is no natural less-costly number of phases. There have been "economic" arguments for six-phase systems decades ago, but the reason for six-phases was that power transfer could be doubled using a single six-phase transmission line on a constrained right-of-way. It is interesting that in a six-phase system, the line-to-line voltage is the same as the line-to-ground voltage.

I think the real reason we use three-phases lies with the torque produced by other systems. For instance with a single-phase system, the torque imposed on motors and generators pulsates from zero to the maximum. This is OK for small motors, but not for large generators or motors.

With a balanced three-phase system, the motor or generator torque is a constant and the shaft does not constantly deal with oscillating torques.

You may recall the problems with sub-synchronous resonance in the 1960's; the phenomena caused two large generator shafts to fail due to oscillating shaft torques.

An economic argument for three-phase systems works when considering the cost and complexity of wiring motors and generators for more than three phases.

I have trouble with the argument that that the 3-phase system requires less cross-section of wire to transmit power. Let us look at the recently proposed reason with additional number of phases:

Imagine that we have 2/3/4/5/6 similar single-phase generators and 2/3/4/5/6 similar single-phase loads. Total cross-section area of wires will be 2A/6A/8A/10A/12A (2/3/4/5/6 direct and 2/3/4/5/6 back-going lines). If we unify going back lines into one, there will be total current 2I/3I/4I/5I/6I, where I is the current of single direct line. If loads are the same, then providing [a] shift [of] 180/120/90/72/60 degree between voltages we get the same shifts between equal currents. The sum of returning currents will be equal to zero and we theoretically have no need in going back lines!!! Thus … we get power supply via cross-section area of 3/4/5/6 instead of 6/8/10/12.

Therefore, in each case, we reduce the cross-section area by a factor of two. There is nothing special about 3-phases when considering conductor cross-section area. In addition, if we were to choose the simplest system, it would be 2-phase (+/- V across the neutral).

However, all phase conductors share the return conductor. Thus, a 2-phase system would use three conductors to transmit 2pu power; a three-phase power system would use four conductors to transmit 3pu power; and a 6-phase system would use seven conductors to transmit 6pu power. That is, the more phases, the cheaper cross-section area of conductors, because more phases share the cost of the single return conductor.

As I mentioned earlier, we choose 3-phase systems, because the torque of large 3-phase motors and generators is constant, avoiding oscillating torques, which can ruin large motor or generator shafts. Constant torque comes with triple phase systems (3, 6, 9 …), but 3-phase is the simplest (smallest number of phases) system that provides constant motor/generator torque.