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Advanced Electrical Engineering
Advanced Electrical Engineering
Michael E. Auer
Three Phase Circuits
Michael E.Auer
08.05.2012
AEE0x
Advanced Electrical Engineering
AEE Content Advanced Circuit Analysis •
Basic Concepts
•
Three-Phase Circuits
• •
Transforms Power Conversion and Management
Field Theory • • • •
Waves and Vector Fields Transmission Line Theory Electrostatics Magnetostatics
Applications • •
Michael E.Auer
Magnetic Field Applications Basics of Electrical Machines
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
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
AEE0x
Advanced Electrical Engineering
Single Phase Systems
two-wire type
Michael E.Auer
three-wire type
08.05.2012
AEE0x
Advanced Electrical Engineering
Two-Phase Three-Wire System
Michael E.Auer
08.05.2012
AEE0x
Advanced Electrical Engineering
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
Michael E.Auer
Four-wire system
08.05.2012
AEE0x
Advanced Electrical Engineering
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.
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
AEE0x
Advanced Electrical Engineering
Balanced Three-Phase Voltages (1) A three-phase generator consists of a rotating magnet (rotor) surrounded by a stationary winding (stator).
Three-phase generator
Michael E.Auer
Generated voltages
08.05.2012
AEE0x
Advanced Electrical Engineering
Balanced Three-Phase Voltages (2) Two possible configurations:
Y-connected
Michael E.Auer
∆-connected
08.05.2012
AEE0x
Advanced Electrical Engineering
Phase Voltages and ist Sequences Positive sequence (abc)
Negative sequence (acb)
Van Vbn Vcn 0 Van Vbn Vcn
Michael E.Auer
08.05.2012
AEE0x
Advanced Electrical Engineering
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 20010 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.
Michael E.Auer
08.05.2012
AEE0x
Advanced Electrical Engineering
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 through their respective maximum values. A balanced load is one in which the phase impedances are equal in magnitude and in phase
Michael E.Auer
08.05.2012
AEE0x
Advanced Electrical Engineering
Possible Load Configurations Four possible connections between source and load:
Michael E.Auer
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
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
AEE0x
Advanced Electrical Engineering
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
Michael E.Auer
08.05.2012
AEE0x
Advanced Electrical Engineering
Balanced Y–Y System (2) VL 3V p , where V p Van Vbn Vcn VL Vab Vbc Vca
Michael E.Auer
08.05.2012
AEE0x
Advanced Electrical Engineering
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 6.8198.2 A
Michael E.Auer
08.05.2012
AEE0x
Advanced Electrical Engineering
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
Michael E.Auer
08.05.2012
AEE0x
Advanced Electrical Engineering
Balanced Y–Δ System (2)
Transformation of the ∆-connected load to an equivalent Y-connected load Single Phase Equivalent
Michael E.Auer
08.05.2012
AEE0x
Advanced Electrical Engineering
Balanced Y–Δ System Example A balanced abc-sequence Y-connected source with Van 10010 is connected to a Δ-connected load (8+j4) per phase. Calculate the phase and line currents. Solution Using single-phase analysis,
Ia
Van 10010 33.54 16.57 A Z / 3 2.98126.57
Other line currents are obtained using the abc phase sequence
Michael E.Auer
08.05.2012
AEE0x
Advanced Electrical Engineering
Balanced Δ–Δ System A balanced ∆-∆ system is a three-phase system with a balanced ∆-connected source and a balanced ∆-connected load.
Michael E.Auer
08.05.2012
AEE0x
Advanced Electrical Engineering
Balanced Δ–Δ System Example A balanced ∆-connected load having an impedance 20-j15 is connected to a ∆-connected positive-sequence generator having Vab 3300 V . Calculate the phase currents of the load and the line currents. Answer: The phase currents
I AB 13.236.87 A; I BC 13.2 81.13 A; I CA 13.2156.87 A The line currents
I a 22.866.87 A; I b 22.86 113.13 A; I c 22.86126.87 A
Michael E.Auer
08.05.2012
AEE0x
Advanced Electrical Engineering
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.
Michael E.Auer
08.05.2012
AEE0x
Advanced Electrical Engineering
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
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
AEE0x
Advanced Electrical Engineering
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 ) Michael E.Auer
08.05.2012
AEE0x
Advanced Electrical Engineering
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
AEE0x
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. Michael E.Auer
08.05.2012
AEE0x
Advanced Electrical Engineering
Michael E. Auer
Three Phase Circuits
Michael E.Auer
08.05.2012
AEE0x
Advanced Electrical Engineering
AEE Content Advanced Circuit Analysis •
Basic Concepts
•
Three-Phase Circuits
• •
Transforms Power Conversion and Management
Field Theory • • • •
Waves and Vector Fields Transmission Line Theory Electrostatics Magnetostatics
Applications • •
Michael E.Auer
Magnetic Field Applications Basics of Electrical Machines
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
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
AEE0x
Advanced Electrical Engineering
Single Phase Systems
two-wire type
Michael E.Auer
three-wire type
08.05.2012
AEE0x
Advanced Electrical Engineering
Two-Phase Three-Wire System
Michael E.Auer
08.05.2012
AEE0x
Advanced Electrical Engineering
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
Michael E.Auer
Four-wire system
08.05.2012
AEE0x
Advanced Electrical Engineering
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.
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
AEE0x
Advanced Electrical Engineering
Balanced Three-Phase Voltages (1) A three-phase generator consists of a rotating magnet (rotor) surrounded by a stationary winding (stator).
Three-phase generator
Michael E.Auer
Generated voltages
08.05.2012
AEE0x
Advanced Electrical Engineering
Balanced Three-Phase Voltages (2) Two possible configurations:
Y-connected
Michael E.Auer
∆-connected
08.05.2012
AEE0x
Advanced Electrical Engineering
Phase Voltages and ist Sequences Positive sequence (abc)
Negative sequence (acb)
Van Vbn Vcn 0 Van Vbn Vcn
Michael E.Auer
08.05.2012
AEE0x
Advanced Electrical Engineering
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 20010 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.
Michael E.Auer
08.05.2012
AEE0x
Advanced Electrical Engineering
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 through their respective maximum values. A balanced load is one in which the phase impedances are equal in magnitude and in phase
Michael E.Auer
08.05.2012
AEE0x
Advanced Electrical Engineering
Possible Load Configurations Four possible connections between source and load:
Michael E.Auer
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
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
AEE0x
Advanced Electrical Engineering
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
Michael E.Auer
08.05.2012
AEE0x
Advanced Electrical Engineering
Balanced Y–Y System (2) VL 3V p , where V p Van Vbn Vcn VL Vab Vbc Vca
Michael E.Auer
08.05.2012
AEE0x
Advanced Electrical Engineering
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 6.8198.2 A
Michael E.Auer
08.05.2012
AEE0x
Advanced Electrical Engineering
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
Michael E.Auer
08.05.2012
AEE0x
Advanced Electrical Engineering
Balanced Y–Δ System (2)
Transformation of the ∆-connected load to an equivalent Y-connected load Single Phase Equivalent
Michael E.Auer
08.05.2012
AEE0x
Advanced Electrical Engineering
Balanced Y–Δ System Example A balanced abc-sequence Y-connected source with Van 10010 is connected to a Δ-connected load (8+j4) per phase. Calculate the phase and line currents. Solution Using single-phase analysis,
Ia
Van 10010 33.54 16.57 A Z / 3 2.98126.57
Other line currents are obtained using the abc phase sequence
Michael E.Auer
08.05.2012
AEE0x
Advanced Electrical Engineering
Balanced Δ–Δ System A balanced ∆-∆ system is a three-phase system with a balanced ∆-connected source and a balanced ∆-connected load.
Michael E.Auer
08.05.2012
AEE0x
Advanced Electrical Engineering
Balanced Δ–Δ System Example A balanced ∆-connected load having an impedance 20-j15 is connected to a ∆-connected positive-sequence generator having Vab 3300 V . Calculate the phase currents of the load and the line currents. Answer: The phase currents
I AB 13.236.87 A; I BC 13.2 81.13 A; I CA 13.2156.87 A The line currents
I a 22.866.87 A; I b 22.86 113.13 A; I c 22.86126.87 A
Michael E.Auer
08.05.2012
AEE0x
Advanced Electrical Engineering
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.
Michael E.Auer
08.05.2012
AEE0x
Advanced Electrical Engineering
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
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
AEE0x
Advanced Electrical Engineering
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 ) Michael E.Auer
08.05.2012
AEE0x
Advanced Electrical Engineering
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
AEE0x
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. Michael E.Auer
08.05.2012
AEE0x
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