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Chapter 3
Three-PhaseSystem
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Three-Phase CircuitsChapter 3
3.1 What is a Three-Phase Circuit?
3.2 Balance Three-Phase Voltages
3.3 Balance Three-Phase Connection3.4 Power in a Balanced System
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It is a system produced by a generator consisting ofthree sources having the same amplitude and
frequency but out of phase with each other by 120.
3.1 What is a Three-Phase Circuit?(1)
Three sources
with 120 out
of phaseFour wired
system
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3.2 Balance Three-Phase Voltages (1)
A three-phase generator consists of a rotating
magnet (rotor) surrounded by a stationarywinding (stator).
A three-phase generator The generated voltages
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3.2 Balance Three-Phase Voltages (2)
Two possible configurations:
Three-phase voltage sources: (a) Y-connected ; (b) -connected
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3.2 Balance Three-Phase Voltages (3)
Ba lanced ph ase vo l t ages are equal inmagnitude and are out of phase with each otherby 120.
The ph ase sequ enceis the time order in which
the voltages pass through their respectivemaximum values.
A ba lan ced load is one in which the phaseimpedances are equal in magnitude and in phase
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3.2 Balance Three-Phase Voltages (4)
Example 1
Determine the phase sequence of theset of voltages.
)110cos(200)230cos(200
)10cos(200
=
=
+=
tvtv
tv
cn
bn
an
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3.2 Balance Three-Phase Voltages (5)
Solution:
The voltages can be expressed in phasor form
as
We notice that Van leads Vcn by 120 and Vcn inturn leads Vbn by 120.
Hence, we have an acb sequence.
V110200V
V230200V
V10200V
=
=
=
cn
bn
an
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Phase Sequence Order in which the voltages in the individual phases peak
Negative and positive sequence
Positive sequence (a-b-c)
Negative sequence (a-c-b)
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3.3 Balance Three-Phase Connection (1)
Four possible connections
1. Y-Y connection (Y-connected source
with a Y-connected load)
2. Y- connection (Y-connected source
with a -connected load)
3. - connection
4.-Y connection
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3.3 Balance Three-Phase Connection (2)
A balanced Y-Y system is a three-phase system with a
balanced y-connected source and a balanced y-connected
load.
cabcabL
cnbnanp
pL
V
V
VV
VVV
VVV
where,3
===
===
=
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120
120
120 RNV
YNV
BNV
Schematic diagram for WYE-
connectionPhase voltage phasor diagram
Therefore, for phase voltage:
o
YN VV 120=o
BN VV 120=
oRN VV 0=
For line voltage:
opRY VV 303 =
opYB VV 903 =
opBR VV 1503 =
PL II =
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3.3 Balance Three-Phase Connection (3)
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Therefore, for phase voltage:
o
YN VV 120=o
BN VV 120=
oRN VV 0=
For line voltage:
opRY VV 303 =
opYB VV 903 =
opBR VV 1503 =
BRV RYV
YBV
RNV
YNV
BNV
303090
Phasor diagram illustrating the relationship between line and phasevoltages
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3.3 Balance Three-Phase Connection (4)
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3.3 Balance Three-Phase Connection (5)
Example 2
Calculate the line currents in the three-wire Y-Ysystem shown below:
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3.3 Balance Three-Phase Connection (6)
A balanced Y- system is a three-phase system with a
balanced y-connected source and a balanced -connected
load.
CABCABp
cbaL
pL
I
I
II
III
III
where,3
===
===
=
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120
120
120 RNV
YN
V
BNV
Schematic diagram for DELTA-connection Phase voltage phasor
diagram
Therefore, for phase voltage:
oRNRY VVV 0==o
YNYB VVV 120==o
BNBR VVV 120==
PL VV =
opAa II 303 =
For line current:
opBb II 1503 =
opCc II 903 =
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3.3 Balance Three-Phase Connection (7)
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Therefore, for phase voltage:
o
RNRY VVV 0==o
YNYB VVV 120==o
BNBR VVV 120==
o
pAa II 303 =
For line current:
opBb II 1503 =
o
pCc II 903 =
30
CcIbcI
caI
abI
AaI
caI
BbI
bcI
abI
Phasor diagram illustrating the relationship between line and phase
currents
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3.3 Balance Three-Phase Connection (8)
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3.3 Balance Three-Phase Connection (9)
Example 3
A balanced abc-sequence Y-connected source with( ) is connected to a -connected load
(8+j4) per phase. Calculate the phase and linecurrents.
Solution
Using single-phase analysis,
Other line currents are obtained using the abc phase
sequence
= 10100Van
A57.1654.3357.26981.210100
3/ZVI an =
==
a
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3.4 Power in a Balanced System (1)
Real power,
Reactive power,
Apparent power,
cos3
cos3
LLIVP
IVP
=
=
sin3
sin3
LLIVQ
IVQ
=
=
LL
IVS
IVS
3
3
=
=
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Three- phase Balanced System
ExampleA balanced three phase system as shown in Figure below , maintainedat constant voltage and frequency is to supply a three- phase load,with three single-phase impedance. IfVab = 440 30 V, find:
1) The load phase currents IAB, IBC, and ICA.2) The real power, reactive power, apparent power and the powerfactor of the load.
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Three- phase Balanced System
ExampleA balanced star- connected load of ZY=8+j6per phase is connected to a balanced 3- phase
400V supply as shown in Figure below. Finethe line current, power factor, real power,reactive power and total volt-amperes.