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FUNDAMENTALS OF ELECTRICAL
ENGINEERING
[ ENT 163 ]
LECTURE #9
THREE – PHASE CIRCUITS
HASIMAH ALI
Programme of Mechatronics,
School of Mechatronics Engineering, UniMAP.
Email: [email protected]
Contents
Introduction
Balanced Three-Phase Voltages
Balanced Wye – Wye Connection
Balanced Wye – Delta Connection
Balanced Delta – Delta Connection
Introduction
 Single – phase ac power system consists of a generator connected
through a pair of wires (transmission line ) to a load.
a)
b)
Single-phase systems: a) two-wire type, b) three-wire type.
Introduction
 Polyphase: circuits or systems in which the ac sources operate at the same
frequency but different phases.
Two-phase three-wire system.
Three-phase four-wire system.
Introduction
 Importance of three – phase system:
 Nearly all electric power is generated and distributed in three –
phase.
 The instantaneous power in a three – phase system can be
constant – results in uniform power transmission and less vibration
(constant torque).
 More economical than the single – phase.
 The power delivery capacity tripled (increased by 200%) by
increasing the number of conductors from 2 to 3 (increased by
50%)
Balanced Three – Phase Voltages
 Often produced with a three – phase ac generator (alternator)
A three-phase
generator
 Generator:
a
b
c
n
 Consists of rotating magnet (rotor) surrounded by a stationary winding
(stator)
 Three separate winding/ coils with terminal a-a’, b-b’ and c-c’ are
physically placed 120° apart around the stator.
 As the rotor rotates, its magnitude field cuts the flux from the three coils
and induces voltages in the coils.
Balanced Three – Phase Voltages
 Because the coils are placed 120° apart, the induced voltages in the coils
are equal magnitude but out of phase by 120 °.
The generated voltages are 120° apart from each other
Balanced Three – Phase Voltages
Vca
Vab
Vbc
 Typical three – phase system:
 From the above figure, the voltages Van, Vbn, Vcn are respectively between
the lines a, b, c and the neutral line n. These voltages are called phase
voltage.
 If the voltage source have same amplitude and frequency ω and are out of
phase with each other by 120°, the voltages are said to be balanced.
Balanced Three – Phase Voltages
This implies that,
Van  Vbn  Vcn  0,
Van  Vbn  Vcn
Balanced phase voltages are equal in magnitude and are out of phase
with each other by 120°
 Since the three–phase voltages are 120 ° out of phase with each other,
there are two possible combinations.
 abc sequence or positive sequence
 acb sequence or negative sequence
Balanced Three – Phase Voltages
1. abc sequence or positive sequence:
Van  V p 0
Vbn  V p   120
Vcn  V p   120  Vbn  V p   240
Where,
• Vp is the effective or rms value of the phase
Balanced Three – Phase Voltages
1. For negative sequence:
Van  V p 0
Vcn  V p   120
Vbn  V p   120  Vbn  V p   240
The phase sequence is the time order in which the voltage pass
through their respective maximum values.
A balanced load is one in which the phase impedances are equal in
magnitude and in phase.
Balanced Three – Phase Voltages
Two possible three-phase load configuration:
a) A Y-connected load,
b) a- ∆ connected load
Balanced Three – Phase Voltages
 For balanced wye – connected load,
Z1  Z 2  Z3  ZY
 For balanced delta – connected load,
Z a  Zb  Zc  Z 
 Recall that,
Z   3Z Y
1
ZY  Z 
3
Balanced Three – Phase Voltages
 4main elements in three – phase circuit:
 Phase Voltage (VØ): measured between the neutral and any line , i.e
line to neutral voltage.
 Line Voltage (VL): measured between ant two of the three lines, i.e
line to line voltage.
 Line current (IL): current in each line of the source or load.
 Phase current (IØ): current in each phase of the source or load.
 Four possibilities connections:
 Wye – wye (Y-Y) connection
 Wye – delta (Y-∆)connection
 Delta – delta (∆ - ∆) connection
 Delta – wye (∆ -Y) connection
Balanced Wye – Wye Connection
A balanced Y-Y system is a three-phase system with a balanced Yconnected source and a balance Y-connected load.
A balanced Y-Y system
Balanced Three – Phase Voltages
•
Phase and line voltages/ currents for balanced Y-Y system (assuming
positive/ abc sequence):
Phase voltages/currents
Line voltages/ currents
Van  V p 0
Vab  3V p 30
Vbn  V p   120
Vbc  3V p   90  Vab  120
Vcn  V p   120
Vca  3V p 150  Vab  120
Van
Ia 
ZY
Van
Ia 
ZY
I b  I a   120
I c  I a   120
I b  I a   120
I c  I a   120
Balanced Wye – Wye Connection
Phasor diagrams illustrating the relationship between line
voltages and phase voltages
Balanced Wye – Wye Connection
•
All phase and line voltages have the same magnitude,
V p  Van  Vbn  Vcn
VL  Vab  Vbc  Vca
Where they are out of phase with each other by 120°, and
VL  3 VP
VL  VP  30
A single-phase equivalent circuit.
Balanced Wye – Wye Connection
Example:
1. A Y-connected balanced three-phase generator with an impedance of
0.4+j0.3 Ω per phase is connected to a Y-connected balanced load
with an impedance of 24+j19 Ω per phase. The line joining the
generator and the load has an impedance of 0.6+j0.7 Ω per phase
.Assuming a positive sequence for the source voltages and that

Van=120 30° V. Find:

a) The line voltage
b) The line currents
Balanced Wye – Delta Connection
A balanced Y – ∆ system consists of a balanced Y-connected source
feeding a balanced ∆ -connected load.
ICA
Balanced Wye – Delta Connection
•
Assuming positive sequence, the phase voltage again:
Van  V p 0
Vbn  V p   120,
Vbn  V p   120
From previous section, the line voltage are
Vab  3V p 30  VAB
Vbc  3Vp   90  VBC
Vca  3Vp   210  VCA
The line
voltages are
equal to the
voltages
across the load
impedances
Balanced Wye – Delta Connection
•
From these voltage, the phase current can be obtained by:
I AB
•
VBC
VAB

I BC 
Z
Z
I CA
VCA

Z
The line currents can be obtain from phase currents by
applying KCL at node A, B and C. Thus,
These
currents have
same
magnitude but
out of phase
with each
other by 120°
I a  I AB  I CA , I b  I BC  I AB , I c  I CA  I BC ,
Since,
I CA  I AB  240
I a  I AB  I CA  I AB (1  1  240)
 I AB (1  0.5  j 0.866)    240)
Balanced Wye – Delta Connection
•
Phase and line voltages/ currents for balanced Y-∆ system (assuming
positive/ abc sequence)
Phase voltages/currents
Line voltages/ currents
Van  V p 0
Vab  VAB  3V p 30
Vbn  V p   120
Vcn  V p   120
I AB
V AB

Z
I BC
V
 BC
Z
I CA 
VCA
Z
Vbc  VBC  Vab  120
Vca  VCA  Vab  120
I a  I AB 3  30
I b  I a   120
I c  I a   120
Balanced Wye – Delta Connection
Phasor diagram illustrating the relationship between phase and line currents.
Balanced Wye – Delta Connection
•
Another alternative way of analyzing the Y-∆circuit is to transform the
connected load to an equivalent Y-connected load, using:
1
ZY  Z 
3
•
After this transformation, we now have a Y-Y system
A single-phase equivalent circuit of a balanced Y- ∆ circuit.
Balanced Delta – Delta Connection
A balanced ∆- ∆ system is one which both the balanced source and
balanced load are ∆-connected.
Vca
c
Balanced Delta – Delta Connection
•
Phase and line voltages/ currents for balanced ∆ -∆ system (assuming
positive/ abc sequence)
Phase voltages/currents
Vab  V p 0
Vbc  V p   120
Vca  V p   120
I AB
Vab

Z
I BC
V
 bc
Z
I CA 
Vca
Z
Line voltages/ currents
Vab  V p 0
Vbc  V p   120
Vca  V p   120
I a  I AB 3  30
I b  I a   120
I c  I a   120
Balanced Delta – Wye Connection
A balanced ∆- Y system consists of balanced ∆-connected source feeding
a balanced Y-connected load.
Vca
c
Balanced Delta – Wye Connection
•
Phase and line voltages/ currents for balanced ∆ -Y system
(assuming positive/ abc sequence)
Phase voltages/currents
Line voltages/ currents
Vab  V p 0
Vab  V p 0
Vbc  V p   120
Vbc  V p   120
Vca  V p   120
Vca  V p   120
Ia 
V p   30
3Z Y
Ia 
V p   30
3Z Y
I b  I a   120
I b  I a   120
I c  I a   120
I c  I a   120
Balanced Delta – Wye Connection
The single-phase equivalent circuit.
Further Reading
Fundamentals of electric circuit. (2th Edition), Alexander, Sadiku, McGrawHill.
(chapter 12).
Electric circuits.8th edition, Nilsson &Riedel, Pearson. (chapter 11).