EE 221
CIRCUITS II
Chapter 12
Chapter 12
Three
Three-
-Phase Circuit
Phase Circuit
1
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Three-Phase Circuits: Understanding Balanced Systems - Prof. Olivera K. Notaros, Study notes of Electrical and Electronics Engineering
An in-depth exploration of three-phase circuits, focusing on balanced systems. Topics covered include the definition of three-phase circuits, balanced three-phase voltages, balanced three-phase connections, and power in a balanced system. The document also discusses unbalanced three-phase systems and their applications, such as residential wiring.
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EE 221 CIRCUITS II
Chapter 12
Chapter 12
Three
Three
Phase Circuit
Phase Circuit
1
THREE-PHASE CIRCUITS CHAPTER 12
What is a Three-Phase Circuit?
12.2 Balanced Three-Phase Voltages 12.
Balanced Three-Phase Connection
Power in a Balanced System
Unbalanced Three-Phase Systems
Application – Residential Wiring
Advantages:
Most
of the electric power is generated and distributed
in three-phase.
The instantaneous power in a three-phase system canbe constant
.
The amount of power in a three-phase system is moreeconomical
that the single-phase.
In fact, the amount of wire required for a three-phasesystem is less than
that required for an equivalent
single-phase system.
12.1 What is a Three 12.1 What is a Three-
-Phase Circuit?
Phase Circuit?
|
A three-phase generator consists of a rotatingmagnet (in the rotor) surrounded by stationarywindings (in the stator).
12.2 Balanced Three 12.2 Balanced Three-
-Phase Voltages
Phase Voltages
A three-phase generator
The generated voltages
|
Balanced phase voltages
are equal in magnitude
and are out of phase
with each other by 120°.
|
The
phase sequence
is the time order
in which the
voltages pass through their respective maximumvalues.
|
A
balanced load
is one in which the phase
impedances are equal in magnitude and in phase
12.2 Balance Three 12.2 Balance Three-
-Phase Voltages
Phase Voltages
Example 1
Determine the phase sequence of the set of
voltages.
cos(
cos(
cos(
t
v
t
v
t
v
cn bn an
ω ω ω
12.2 Balance Three 12.2 Balance Three-
-Phase Voltages
Phase Voltages
|
Four possible connections
Y-Y connection (Y-connected source witha Y-connected load)
Y-
Δ
connection (Y-connected source with
a
Δ
-connected load)
Δ
Δ
connection
Δ
-Y connection
12.3 Balanced Three 12.3 Balanced Three-
-Phase Connection
Phase Connection
ca
bc
ab
L
cn
bn
an
p
p
L
V V
V
V
V
V
V
V
V
V
where
,
3
=
=
=
=
=
=
=
11
12.3 Balance Three 12.3 Balance Three-
-Phase Connection
Phase Connection
A balanced Y-Y system is a three-phase system with a
balanced y-connected source and a balanced y-connectedload.
I
I
I
c
b
a
n
I
CA
BC
AB
p
c
b
a
L
p
L
I I
I
I
I
I
I
I
I
I
where
,
3
=
=
=
=
=
=
=
12.3 Balanced Three 12.3 Balanced Three-
-Phase Connection
Phase Connection
A balanced Y-
Δ
system is a three-phase system with a
balanced y-connected source and a balanced
Δ
-connected
load.
Example 3
A balanced
abc
-sequence Y-connected source with
(
) is connected to a
Δ
-connected load (8+j4)
Ω
per
phase. Calculate the phase and line currents.
Solution
Using single-phase analysis,Other line currents are obtained using the
abc
phase sequence
°
∠
=
10
100
V
an
A
57
.
16
54
.
33
57
.
26
981
.
2
10
100
3
/
Z
V
I
an
° − ∠ = ° ∠
°
∠
=
=
Δ
a
14
12.3 Balanced Three 12.3 Balanced Three-
-Phase Connection
Phase Connection
*Refer to in-class illustration, textbook
Example 4
A balanced
Δ
-connected load having an impedance 20-j
Ω
is
connected to a
Δ
-connected positive-sequence generator having
(
). Calculate the phase currents of the load and
the line currents.
Ans:
The phase currentsThe line currents
V
0
330
V
ab
°
∠
=
A
87
.
156
2
.
13
I
A;
13
.
81
2
.
13
I
A;
87
.
36
2
.
13
I
° ∠ = ° − ∠ = ° ∠ =
AB
BC
AB
A
87
.
126
86
.
22
I
A;
13
.
113
86
.
22
I
A;
87
.
6
86
.
22
I
°
16
12.3 Balance Three 12.3 Balance Three-
-Phase Connection
Phase Connection
∠ = ° − ∠ = ° ∠ =
c
b
a
*Refer to in-class illustration, textbook
12.3 Balanced Three 12.3 Balanced Three-
-Phase Connection
Phase Connection
A balanced
Δ
-Y system is a three-phase system with a
balanced y-connected source and a balanced y-connectedload.
)
cos(
3
)
cos(
3
I
V
I
V
P
L
p
=
=
12.4 Power in a Balanced System 12.4 Power in a Balanced System
)
sin(
3
)
sin(
3
I
V
I
V
Q
L
p
=
=
L
p
V
I
V
S
3
3
=
=
Example 6
Determine the total average power, reactive power, and complex
power at the source and at the load
Balanced Three Balanced Three-
-Phase Systems
Phase Systems
Ans At the source: S
s
= (2087 + j834.6) VA
P
s
= 2087W
Q
s
= 834.6VAR
At the load: S
L
= (1392 + j1113) VA
P
L
= 1392W
Q
L
= 1113VAR
*Refer to in-class illustration, textbook