200 THREE-PHASE CIRCUITS AND RESIDENTIAL WIRING
120 °120 °120 °120 °(^0120) ° 240 °
(a)
(b)
Volts
eaa′ ebb′
Ebb′
Ecc′
c b
a
Eaa′
a′
c′
b′
ecc′
ωt
(c)
Ecc′
Ebb′
b c
a
Eaa′
a′
c′
b′
Figure 4.1.2(a)Waveforms in
time domain. (b) Positive-
sequence (a–b–c) phasors.(c)Ne-
gative-sequence (a–c–b) phasors.
From the nature of the connections shown in Figure 4.1.3 it can be seen that the line-to-line
voltages (VL-LorVL) are equal to the phase voltageVphfor the delta connection, and the line
current is equal to the phase current for the wye connection. A balanced wye-connected three-
phase source and its associated phasor diagram are shown in Figure 4.1.4, from which it can be
seen that the line-to-line voltage is equal to
√
3 times the phase voltage (or the line-to-neutral
voltage). The student should be able to reason on similar lines and conclude that, for the balanced
delta-connected three-phase source, the line current will be equal to
√
3 times the phase current.
The notation using subscripts is such thatVABis the potential at pointAwith respect
to pointB,IABis a current with positive flow from pointAto pointB, andIA,IB,andIC
are line currents with positive flow from the source to the load, as shown in Figure 4.1.5.
The notation used is rather arbitrary. In some textbooks a different notation for the voltage
is adopted such that the order of subscripts indicates the direction in which the voltage rise
is taken. The student should be careful not to get confused, but try to be consistent with
any conventions chosen. The rms values are usually chosen as magnitudes of the phasors
for convenience. It is customary to use the letter symbolEfor generated emf andV for
terminal voltage. Sometimes the two are equal, but sometimes not. If we should neglect the
existence of the generator winding impedance, the generated emf will be equal to the terminal
voltage of the generator. Although the three-phase voltages are generated in one three-phase
alternator, for analytical purposes this is modeled by three identical, interconnected, single-
phase sources.
The one-line equivalent circuit of the balanced wye-connected three-phase source is shown
in Figure 4.1.4(c). The line-to-neutral (otherwise known as phase) voltage is used; it may be taken