Transcript Introduction

Balanced Three-Phase Circuits
and Ideal Transformers
ELEC 308
Elements of Electrical Engineering
Dr. Ron Hayne
Images Courtesy of Allan Hambley and Prentice-Hall
Three-Phase Sources
 Most common multiple-phase AC voltage
source configuration

Wye connected (Y connected)
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Three-Phase Sources
v an t   VY cost   Van  VY 0 o
v bn t   VY cost 120o Vbn  VY  120o
v cn t   VY cost  120o Vcn  VY 120o

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Wye-Wye Connection
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Wye-Wye Connection
 Lines

Wires a-A, b-B, c-C
 Neutral

Wire n-N
 Balanced Load

Three load impedances are equal
 Phase Voltage or Line-to-Neutral Voltage

Peak magnitude of one source, VY
 Line Currents

Values IaA, IbB, and IcC
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Wye-Wye Connection
I aA
I bB
I cC
Van
VY 0


 I L  
Z
Z
Vbn
VY   120


 I L   120  
Z
Z
Vcn VY 120


 I L 120  
Z
Z
I nN  I aA  I bB  I cC  0
 Neutral current is ZERO in a BALANCED threephase system

Can eliminate the neutral wire
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Cost Advantage
 Three (or four) wires needed to connect a
three-phase source with three loads
 Six wires needed to connect three singlephase sources with three loads
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Three-Phase Power
pt   van t iaA t   vbn t ibB t   vcn t icC t 
VY I L
3
cos 
2
 Total power is CONSTANT with respect to time
VY I L
Pavg  pt   3
cos   3VYrms I Lrms cos 
2
VY
IL
whereVYrms 
and I Lrms 
2
2
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Three-Phase Reactive Power
 The reactive power delivered to a balanced
three-phase load is given by
VY I L
sin    3VYrms I Lrms sin  
2
VY
IL
whereVYrms 
and I Lrms 
2
2
Q3
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Line-to-Line Voltages
Vab  Van  330o
Vbc  Vbn  330o
Vca  Vcn  330o

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Analysis of a Wye-Wye System
 Balanced wye-connected 60-Hz three-phase source has lineto-neutral voltages of VY = 1000 V.
 Balanced wye-connected load consisting of a 0.1H
inductance in series with a 50-Ω resistance.
 Find the line currents, the line-to-line voltages, the power,
and the reactive power delivered to the load.
 Draw a phasor diagram showing the line-to-neutral voltages,
the line-to-line voltages, and the line currents.
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Analysis of a Wye-Wye System
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Delta-Connected Sources
Vab  Vbc  Vca  0

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Delta-Connected Loads
Z   3ZY
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Delta-Delta Connection
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Power Transmission Losses
 Power Line Losses

2
Ploss  RlineI rms
 Large Voltages and Small Currents
 Smaller Line Loss
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Ideal Transformers
v2 (t ) 
V2 rms
N2
v1 (t )
N1
N2

V1rms
N1
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Ideal Transformers
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Example 15.10
 Determine the rms values of the currents and
voltages.
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Power Transmission
 Step-Up and Step-Down Transformers

99% Efficiency (vs. 50% with no transformers)
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U.S. Power Grid
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Summary
 Three-Phase Circuits


Wye-Wye Connection
Three-Phase Power
 Ideal Transformers
 U.S. Power Grid
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