ECSE 486B Power Laboratory
Review of Three-Phase Circuits
1 Balanced Three-phase Circuits
1.1 Introduction
The use of polyphase circuits, mainly three-phase circuits, is quite common in industry and in electrical energy
production and transmission. The preference given to three-phase circuits over single-phase circuits comes from the
fact that the instantaneous power is constant and independent of time in balanced three-phase circuits while it is
pulsating in single-phase circuits. In addition, for rotating machines, and more speciﬁcally for three-phase induction
machines, their operating principle is more simple than that of a single-phase machine. The manufacturing and
utilization cost is also less, since the power in a three-phase context is constant. Moreover, the magnetic circuit of
a single-phase machine can only be used to a fraction of its full capacity
1
.
In this unit, we will look mainly at three-phase circuits because of their practical importance and widespread
use. However, be aware that there exist other types of polyphase circuits. For instance, the ﬁrst polyphase circuits
were 2-phase circuits with voltages phasors in quadrature with respect to each other (
i.e.
90
◦
out of phase). There
exist practical applications which use circuits with up to 36 phases.
1.2 Deﬁnitions and Generalities
1. A three-phase circuit is balanced when the source and the load are both balanced.
2. A three-phase source is balanced when the three generated voltages have the same amplitude and are phase
shifted by 120
◦
one with respect to each other.
3. A three-phase load is balanced when the impedances (admittances) of each of the three phases is equal in
magnitude and angle.
4. It results that in a balanced three-phase circuit, the three line currents have the same amplitude and are
phase shifted by 120
◦
one with respect to each other.
A three-phase machine is essentially made up of three windings that are electrically independent. We then have
three pairs of wires, that is six wires corresponding to the ends
a
,
a
0
,
b
,
b
0
,
c
and
c
0
(Figure 1). We can represent
the three voltages as phasors (Figure 2). For now, the only relationship existing between these phasors is their
respective orientation and frequency, which is the same for all three. The way by which these phasors are ordered
with respect to a given reference corresponds to the phase sequence.
1.3 Y (Star)-Connection
Instead of taking the six wires from the source to the load, we can connect together the points
a
0
,
b
0
and
c
0
, thus
forming a common node,
n
; this permits a reduction in the number of wires from six to four. A
line
wire or
conductor is one of the wires connected to either points
a
,
b
or
c
, while the
neutral
conductor is connected to
n
.
Through this connection, the three sources become electrically linked; we represent them by the phasors