Answer to the Question 12/03

THREE-PHASE CURRENTS

The question was:



Why do power companies produce three-phase currents. What are the advantages over single-phase, two-phase, four-phase,... currents?

Solution:

Since there are numerous aspects to this problem we will publish partial answers as they arrive. (Credits to people that provided that particular answer will be given at the bottom of this page.)

1. If the number of phases N is greater or equal 3, then the total instantaneous power is constant, i.e. the generator runs under a constant load. (This assumes that all phases are equally loaded.)
To prove that we note that instantenous load (up to a numerical prefactor) is
I=SUMn=0N-1cos2(wt+2{pi}n/N)
Note that
dI/dt=2 SUMn=0N-1cos(wt+2{pi}n/N)sin(wt+2{pi}n/N)=
SUMn=0N-1sin(2wt+4{pi}n/N)=0.
for N equal to or greater than 3.

2. If the potential difference (amplitude) between 0 and one of the "phase wires" is Vo, then for 2-phase current the potential difference between the wires of two phases will be 2Vo. For a larger number of phases (N=3,4,...) the potential difference between two phases depends on which phase we choose. For the case of even N there will be pairs of phases that have potential difference 2Vo. For N=3, the potential difference between any pair will be 31/2Vo. For other even Ns, there will always be a pair that has a potential difference larger than 31/2Vo! In fact for large N the maximal potential difference will always be close to 2Vo. Thus, in the three-phase case we have the smallest maximal potential difference between the wires. This enables, to use less insulation when the wires are close together. Only, three-phase system has this advantage!

3. It should be noted that 3-phase motors (as opposed to motors using 1-phase current) are both simpler, because it is easy to create a "rotating magnetic field" with 3-phases available, and do not need auxillary device to start the motor.




Credits:
1. This item was solved (in part) by (28/3/04) by Luca Pappalardo, Mechanical Engineering student at the University of Salerno (Italy) (e-mail pappalardo.luca@virgilio.it).

2. This observation was made by D. J. Bergman.

3. This observation was made (29/7/2004) an anonymous correspondent (e-mail rotem13@netvision.net.il)


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