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I
+ 
and 
I
-
in terms of sequence components of delta phase-source voltages. The resulting Y-connected equivalent 
circuit is shown in circuit Fig. 8.5-11(b).
Now, we have determined the positive sequence and negative sequence components of phase 
voltages of the Y-connected equivalent. They are 
V
V
−
∠ − °
∠ °
3
3
q
q
and 
, respectively. The zero 
sequence component is zero. Note that there is non-zero zero sequence content in delta-connected 
sources. 
V
V
V
a
a
a
a
V
V
V
0
2
2
1
2
3
1
3
1
1
1
1
1
+
−










=




















Substituting for 
V
1
, 
V
2
, 
V
3
and 
a
, we get 
V
0
=
19.62
∠-
69.76
°
V rms, 
V
+ 
=
190.04
∠
12.6
°
V rms, 
and 
V
-
=
36.051
∠-
39.69
°
V rms.
Y
B
R
I
+
I
+
I
+
I
–
I
–
a
a
2
a
2
a
I
–
I
0
I
0
I
0
Z
Z
Z
Fig. 8.5-10 

Resolutionofthree-phasecurrentsourceinFig.8.5-9intermsofitssequence
components
Fig. 8.5-11 
Y-equivalentofcircuitinFig.8.5-10
Y
B
R
+
+
+
–
– –
–
–
–
+
+
+
(b)
3
3
∠
–30°
∠
–30°
V
–
V
+
3
Z
3
Z
3
Z
R
Y
B
(a)
3 
I
+
∠
–30°
3 
I
–
∠
–30°
3
Z
3
Z
3
Z

---

8.36


SinusoidalSteady-StateinThree-PhaseCircuits
The star equivalent phase voltage are obtained by
V
V
V
a
a
a
a
V
V
RN
YN
BN










=










∠ − °
∠ °
+
−
1
1
1
1
1
1
30
3
1 30
3
2
2
00
130 38
16 17
103 63
148 11
98 38 11


















=
∠ −
°
∠ −
°
∠
.
.
.
.
.
22 23
.
.
°










V rms
8.6 
summary
• A set of three sinusoidal quantities, all at the same frequency, with equal peak (and hence rms) 
values and shifted successively by 120
°
in phase is defined as a Balanced Three-Phase Quantity. 
Therefore, if x
1
(t), x
2
(t) and x
3
(t) is a three-phase set, then,
x
1
(t) 
=
X
m
cos
w
t, x
2
(t) 
=
X
m
cos(
w
t 
- 
120
°
) and x
3
(t) 
=
X
m
cos(
w
t 
+ 
120
°
) or 
x
1
(t) 
=
X
m
cos
w
t, x
2
(t) 
=
X
m
cos(
w
t 
+ 
120
°
) and x
3
(t) 
=
X
m
cos(
w
t 
- 
120
°
)
and x
1
(t) 
+ 
x
2
(t) 
+ 
x
3
(t) 
=
0 for all t.
If the peak values are unequal and/or successive phase shifts are different from 120
°
, the set will 
be called an Unbalanced Three-Phase Quantity. A three-phase source is specified by specifying 
line-to-line voltage rms values.
• Each limb or branch of a three-phase system (source or load) is termed as a phase. R, Y and B are 
used to designate the line terminals of a three-phase source or load.
• A balanced three-phase system comprising balanced three-phase sources and balanced three-phase 
loads is superior to a single-phase system due to the following facts: (i) Power loss in transmission 
system is lower in three-phase system (ii) Copper utilisation is superior in three-phase system. 
(iii) Power delivered to a balanced three-phase load by a balanced three-phase supply is free of 
pulsation. (iv) Electrical equipment designed for three-phase operation is more efficient than their 
single-phase counter parts.
• The sum of line voltages in any three-phase system is zero since these three voltages are voltages 
in a loop formed by R to Y, Y to B and B to R.The sum of line currents in any three-phase three-
wire system is zero since these three currents have to satisfy KCL.
• The line voltages in a balanced Y-connected source form a balanced three-phase set of voltages 
that are 
√
3 times in magnitude and 30
°
ahead in phase with respect to phase voltages.
• The line currents delivered by a balanced 
D
-connected source form a balanced three-phase set of 
currents that are 
√
3 times in magnitude and 30
°
behind in phase with respect to phase currents 
delivered by the phase sources.
• The three-phase complex power delivered by a three-phase source is given by 
S
=
√
3 V
L
I
L
∠
q
=
[
√
3 V
L
I
L
cos
q
+ 
j 
√
3 V
L
I
L
sin
q
] VA where 
q
is the angle by which the current 
phasor delivered by a phase-source lags behind its voltage phasor. This relationship is independent 
of whether the source is Y-connected or 
D
-connected. 
q
is not the angle between line voltage 
phasor and line current phasor.
• Any balanced three-phase circuit can be equivalenced to a Y-Y balanced three-phase circuit by 
replacing 
D
-connected sources and loads (if any) by their Y-connected equivalents. The equivalence 
is valid as far as line voltages, line currents and complex power flow are concerned.

---

Problems

8.37
• A balanced Y-Y circuit can be solved by solving its R-phase and employing three-phase symmetry 
to arrive at the solution for Y-phase and B-phase. The R-phase circuit is solved by using the single-
phase equivalent circuit.
• The sets of three-phase balanced source components and possible single-phase components of an 
unbalanced source are called its Symmetrical Components.
• There are three symmetrical components for an unbalanced three-phase source function. Each 
symmetrical component is a set of three source functions. The first set – called the positive 
sequence component – is a balanced three-phase set of source functions that has positive phase 
sequence. The second set – called the negative sequence component – is a balanced three-phase 
set of source functions that has negative phase sequence. The third set – called the zero sequence 
component – is a set of three cophasal (i.e., of same phase) single-phase source functions. It is not 
a three-phase set at all. 
• The zero sequence component is the average of the three three-phase quantities. Line voltages 
in a three-phase system cannot have a zero sequence content anywhere in the system. Line 
currents in a three-phase three-wire system cannot have a zero sequence content anywhere in the
system. 
• Active power carried by sequence components obeys superposition principle.
• Positive sequence component of voltage can send active power only through positive sequence 
component of current; negative sequence component of voltage can send active power only 
through negative sequence component of current and zero sequence component of voltage can 
send active power only through zero sequence component of current.
• Balanced impedance is incapable of bringing about a sequence conversion involving translation 
of positive sequence current into negative sequence voltage and vice versa. An unbalanced load 
produces sequence coupling between positive sequence components and negative sequence 
components 
• The most energy-efficient way to draw active power from a balanced three-phase source is to draw 
it through balanced three-phase currents at unity power factor.