# 3 to n-Phase System Comparisons.

#### PhilCorso

This is in response to the direct requests for the answer to the question:

"Why Are 3-Phase Power Systems more Common Than 2, 4, 5, ..., n-Phase Systems?"

Most of you were correct. It is, of course, the most economical. BUT... will be addressed at the end.

All systems will be compared based on the following:

a) The amount of power to be transmitted or carried is the same.
b) Transmission distance is the same.
c) The phase to phase voltage of each system is the same.
d) Wire material (Cu, Al, ACSR, etc) and their subsequent losses are the same.
e) Hardware, i.e., isolators, connectors, tower structure, etc, are ignored
f) Wire weights (W) are directly proportional to the number of wires (N) since distance is the same.
g) Wire weights (W) are inversely proportional to their resistance (R).

First compare 1 with 3-phase:

P1=VxI1xPF, and P3=Sqrt(3)xVxI3xPF.

Since P1=P3, then I1=SQRT(3)xI3

and, since losses are equal,

I1^2xR1x2=I3^2xR3x3

or R1/R3=(3xI3^2)/(2x3xI3^2)=1/2

then, (W3/W1)=(N3/N1)x(R1/R3)=(3/2)x(1/2)=0.75

Conclusion: the weight of wire for a 3-ph system is 3/4 that of the
1-ph system.

One more. Now compare 3 with 4-phase.

P3=SQRT(3)xVxI3xPF and P4=SQRT(4)xVxI4xPF.

Since P3=P4, then I3/I4 =2/SQRT(3)

and, since losses are the same 3xI3^2xR3=4xI4^2xR4

or, R4/R3=(3/4)x(4/3)=1

then, W3/W4=(N3/N4)x(R4/R3)=(3/4)x1

Conclusion: The result is the same as the single-phase case. I leave additional cases for the reader to do.

Now the BUT. The above problem considered n-phase systems having N=n wires. But, for systems where the voltage to neutral is fixed, then there is no difference. For a mathematical proof of this
situation, contact me.

In closing, all of the above was known over 100 years ago! Well before computers. It is certainly proof that Electrical Power Systems are more of a science, than the "black art" called computers.

Regards,
Phil Corso, PE
Trip-A-Larm Corp
(Deerfield Beach, FL)