Advanced Electrical Engineering
Advanced Electrical Engineering
Michael E. Auer
Three Phase Circuits
Michael E.Auer
08.05.2012
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Advanced Electrical Engineering
AEE Content
Advanced Circuit Analysis
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Basic Concepts
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Three-Phase Circuits
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Transforms
Power Conversion and Management
Field Theory
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Waves and Vector Fields
Transmission Line Theory
Electrostatics
Magnetostatics
Applications
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Magnetic Field Applications
Basics of Electrical Machines
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Chapter Content
Michael E.Auer
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Introduction
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Balanced Three-Phase Voltages
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Balanced Three-Phase Connections
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Power in Balanced Systems
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Unbalanced Three-Phase Systems
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Advanced Electrical Engineering
Chapter Content
Michael E.Auer
•
Introduction
•
Balanced Three-Phase Voltages
•
Balanced Three-Phase Connections
•
Power in Balanced Systems
•
Unbalanced Three-Phase Systems
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Single Phase Systems
two-wire type
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three-wire type
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Two-Phase Three-Wire System
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What is a Three-Phase Circuit?
It is a system produced by a generator consisting of three sources having
the same amplitude and frequency but out of phase with each other by 120°.
Three sources
with 120° out
of phase
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Four-wire
system
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Reasons for the Use of Three-Phase Circuits
Advantages:
1.
2.
3.
4.
Michael E.Auer
Most of the electric power is generated and distributed in threephase. Operating frequency 50Hz (Europe) or 60Hz (US).
The instantaneous power in a three-phase system can be
constant (not pulsating!).
For the same amount of power, the three-phase system is
more economical that the single-phase.
In fact, the amount of wire required for a three-phase system is
less than that required for an equivalent single-phase system.
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Advanced Electrical Engineering
Chapter Content
Michael E.Auer
•
Introduction
•
Balanced Three-Phase Voltages
•
Balanced Three-Phase Connections
•
Power in Balanced Systems
•
Unbalanced Three-Phase Systems
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Balanced Three-Phase Voltages (1)
A three-phase generator consists of a rotating magnet (rotor) surrounded by a
stationary winding (stator).
Three-phase generator
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Generated voltages
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Balanced Three-Phase Voltages (2)
Two possible configurations:
Y-connected
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∆-connected
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Phase Voltages and ist Sequences
Positive sequence (abc)
Negative sequence (acb)
Van  Vbn  Vcn  0
Van  Vbn  Vcn
Michael E.Auer
08.05.2012
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Phase Voltage Sequences Example
Determine the phase sequence of the set of voltages.
van  200 cos(t  10)
vbn  200 cos(t  230)
Solution:
vcn  200 cos(t  110)
The voltages can be expressed in phasor form as
Van  20010 V
Vbn  200  230 V
Vcn  200  110 V
We notice that Van leads Vcn by 120° and Vcn in turn leads Vbn by 120°.
Hence, we have an acb sequence.
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Properties of Three-Phase Systems
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 maximum values.
A balanced load is one in which the phase impedances are
equal in magnitude and in phase
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Possible Load Configurations
Four possible connections between source and load:
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08.05.2012
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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Advanced Electrical Engineering
Chapter Content
Michael E.Auer
•
Introduction
•
Balanced Three-Phase Voltages
•
Balanced Three-Phase Connections
•
Power in Balanced Systems
•
Unbalanced Three-Phase Systems
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Balanced Y–Y System (1)
A balanced Y-Y system is a three-phase system with a balanced yconnected source and a balanced y-connected load.
where Z Y  Z S  Z l  Z L
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Balanced Y–Y System (2)
VL  3V p , where
V p  Van  Vbn  Vcn
VL  Vab  Vbc  Vca
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Balanced Y–Y System Example
Calculate the line currents in the three-wire Y-Y system shown below:
Answer :
I a  6.81  21.8 A
I b  6.81  141.8 A
I c  6.8198.2 A
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Balanced Y–Δ System (1)
A balanced Y-∆ system is a three-phase system with a balanced yconnected source and a balanced ∆-connected load.
I L  3I p , where
I L  I a  Ib  Ic
I p  I AB  I BC  I CA
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Balanced Y–Δ System (2)
Transformation of the
∆-connected load to an
equivalent Y-connected
load
Single Phase Equivalent
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Balanced Y–Δ System Example
A balanced abc-sequence Y-connected source with Van  10010 is
connected to a Δ-connected load (8+j4) per phase. Calculate the phase
and line currents.
Solution
Using single-phase analysis,
Ia 
Van
10010

 33.54  16.57 A
Z  / 3 2.98126.57
Other line currents are obtained using the abc phase sequence
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Balanced Δ–Δ System
A balanced ∆-∆ system is a three-phase system with a balanced
∆-connected source and a balanced ∆-connected load.
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Balanced Δ–Δ System Example
A balanced ∆-connected load having an
impedance 20-j15  is connected to a
∆-connected positive-sequence
generator having Vab  3300 V .
Calculate the phase currents of the
load and the line currents.
Answer:
The phase currents
I AB  13.236.87 A; I BC  13.2  81.13 A; I CA  13.2156.87 A
The line currents
I a  22.866.87 A; I b  22.86  113.13 A; I c  22.86126.87 A
Michael E.Auer
08.05.2012
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Balanced Δ–Y System (1)
A balanced ∆-Y system is a three-phase system with a balanced Yconnected source and a balanced y-connected load.
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Balanced Δ–Y System (2)
Transformation of the ∆-connected source to a Y-connected one.
VP
V
Van  P   30
3
V
Vbn  P   150
3
Michael E.Auer
Ia 
Vcn 
3
  30
ZY
VP
  90
3
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Advanced Electrical Engineering
Chapter Content
Michael E.Auer
•
Introduction
•
Balanced Three-Phase Voltages
•
Balanced Three-Phase Connections
•
Power in Balanced Systems
•
Unbalanced Three-Phase Systems
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Power in Balanced Systems
For a Y-connected load the phase voltages are:
v AN  2 VP cos t
vBN  2 VP cos (t  120)
vCN  2 VP cos (t  120)
For:
ZY  Z 
The phase currents lag behind their corresponding phase voltages by θ
ia  2 I P cos (t   )
ib  2 I P cos (t    120)
ic  2 I P cos (t    120)
p  pa  pb  pc  3VP I P cos   f (t )
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08.05.2012
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Comparison of Power Loss
Three Phase System
Single Phase System
PL2
P'loss  2 R 2 , single - phase
VL
PL2
P'loss  R' 2 , three - phase
VL
If same power loss is tolerated in both system, three-phase system
use only 75% of materials of a single-phase system!
Michael E.Auer
08.05.2012
AEE0x
Advanced Electrical Engineering
Chapter Content
Michael E.Auer
•
Introduction
•
Balanced Three-Phase Voltages
•
Balanced Three-Phase Connections
•
Power in Balanced Systems
•
Unbalanced Three-Phase Systems
08.05.2012
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Advanced Electrical Engineering
Unbalanced Three-Phase Systems
An unbalanced system is due to unbalanced voltage sources or an
unbalanced load.
Ia 
V
V
V AN
, I b  BN , I c  CN ,
ZC
ZB
ZA
I n  (I a  Ib  Ic)
 To calculate power in an unbalanced three-phase system requires that we find
the power in each phase.
 The total power is not simply three times the power in one phase but the sum of
the powers in the three phases.
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08.05.2012
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