Alternating Current Analysis 261
R.3.69 The transformation from ∆ to Y is accomplished by the following set of equations:Z
ZZ
A ZZZ
12(^123)
Z
ZZ
B ZZZ
13(^123)
Z
ZZ
C ZZZ
23(^123)
Note that the ∆ to Y and the Y to ∆ transform equations for the AC case can
be extended to the DC case, where the (Rs) resistances are replaced by (Zs)
impedances.
R.3.70 Polyphase systems consist of two or more AC voltage sources with the same
frequency, and with fi xed differences in phase, connected in either a ∆ or Y
confi guration.
R.3.71 A single-phase generator outputs a sinusoidal voltage for each rotation of its rotor.
When more than one sinusoidal is generated by the system, then it is referred as a
polyphase system. Multiple-phase voltages are in general more effi cient to generate,
transmit, and distribute electrical power than single phase, which results in copper
reduction (thinner conductors) of up to 25%, and substantial saving in transmission
lines and structures (transmission towers). The most common multiphase system is
the three-phase system, denoted by 3 Φ.
R.3.72 A generator consists of a rotating shaft or rotor with coils around it that moves
in a constant magnetic fi eld. This arrangement results in an induced voltage in
each coil. Due to the location of the windings as well as the number of turns, A 3 Φ
system is a system that outputs three sinusoidal waves, 120 ° apart in phase, with
the same magnitudes. The generating frequency depends on the number of poles,
as well as the angular speed of the rotor. The standard commercial and residential
frequencies are 60, 50, and 400 Hz in the United States; Europe; and insulated, inde-
pendent large-scale systems such as ships, aircrafts, and satellites, respectively.
R.3.73 The power distribution of a 3 Φ system if implemented independently would need
six wires as transmission lines, one per phase. It is far more effi cient to interconnect
the winding in a way as to reduce the number of transmission lines into two struc-
tures—∆ or Y (also known as star) confi guration, reducing the number of transmis-
sion lines, and the resulting implementation cost.
R.3.74 A monophase generator is one that results in one or more outputs, but all of them
with the same frequency and with the same phase.
R.3.75 A single-phase system that is often used in residential installations consists of
three (secondary) wires and a transformer is illustrated in Figure 3.39.
R.3.76 Another popular connection consists of a two-phase (2 Φ) system with three wires
connected to two loads Z 1 and Z 2 as shown in Figure 3.40, with its corresponding
phasor diagram shown in Figure 3.41.
Where IN = I 1 − I 2 and E 12 = E √
__
2 ∠− 45°.