1. Three Phase Circuits.pdf

Advanced Electrical Engineering

Advanced Electrical Engineering Engineering

Michael E. Auer

Three Phase Circuits

Michael E.Auer

08.05.2012

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AEE Content Ad A d v an anc c ed Ci Cirr c u i t An A n al aly ysis •

Basic Concepts

•

Three-P Three-Phase hase Cir Circu cuit its s

• •

Transforms Power Conversion and Management

Field Theory • • • •

Waves and Vector Fields Transmission Line Theory Electrostatics Magnetostatics

Ap A p p l i c at atii o n s • •

Michael E.Auer

Magnetic Field Ap Applications Basics of of Electrical Machines

08.05.2012

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AEE Content Ad A d v an anc c ed Ci Cirr c u i t An A n al aly ysis •

Basic Concepts

•

Three-P Three-Phase hase Cir Circu cuit its s

• •

Transforms Power Conversion and Management

Field Theory • • • •

Waves and Vector Fields Transmission Line Theory Electrostatics Magnetostatics

Ap A p p l i c at atii o n s • •

Michael E.Auer

Magnetic Field Ap Applications Basics of of Electrical Machines

08.05.2012

AEE0x


Chapter Content

Michael E.Auer

•

Introduction

•

Balanced Three-Phase Voltages

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Balanced Th Three-Phase Co Connections

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Power in Balanced Systems

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Unbalanced Three-Phase Systems

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Chapter Content

Michael E.Auer

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Introduction

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Balanced Three-Phase Connections

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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:

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1.

Most of the electric power is generated and distributed in threephase. Operating frequency 50Hz (Europe) or 60Hz (US).

2.

The instantaneous power in a three-phase system can be constant (not pulsating!).

3.

For the same amount of power, the three-phase system is more economical that the single-phase.

4.

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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Chapter Content

Michael E.Auer

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Introduction

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Balanced Three-Phase Voltages (1)

A three-phase generator consists of a rotating magnet (rotor) surrou stationary winding (stator).


Balanced Three-Phase Voltages (2) Two possible configurations:

∆-connected

Y-connected

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Phase Voltages and ist Sequences Positive sequence (abc)

Negative sequence (acb)

Van Van

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 

Vbn Vbn



Vcn  0



Vcn

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Phase Voltage Sequences Example Determine the phase sequence of the set of voltages.

Solution: The voltages can be expressed in phasor form as

Van



20010 V

Vbn



200  230 V

Vcn



200  110 V

van



200 cos( t  10)

vbn



200 cos( t  230)

vcn



200 cos( t  110)

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:

1.

Y-Y connection (Y-connected source with a Y-connected load)

2.

Y- ∆ connection (Y-connected source with a ∆-connected load)

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08.05.2012

3.

∆- ∆

connection

4.

∆-Y

connection

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Chapter Content

Michael E.Auer

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Introduction

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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

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 Z S  Z l 

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Z L

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Balanced Y–Y System (2) V L

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

3V p , where

V p



Van



Vbn



Vcn

V L



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 :

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Ia



6.81  21.8 A

Ib



6.81  141.8 A

Ic



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

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

3 I p , where

I L



Ia

I p



I AB



Ib





Ic

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 Z / 3



10010 2.98126.57

 33.54  16.57 A

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

Ia

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

22.866.87 A; I b



22.86  113.13 A; I c

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

22.86126.87 A

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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.

Michael E.Auer

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Balanced Δ –Y System (2) Transformation of the ∆-connected source to a Y-connected one.

V P

Van  Vbn 

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V P

3 V P 3

I a    30

  150

Vcn 

V P

3

  30

3 Z Y

  90

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Chapter Content

Michael E.Auer

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Introduction

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Power in Balanced Systems For a Y-connected load the phase voltages are:

For:

v AN 

2 V P cos  t

vCN 

2 V P cos ( t  120)

v BN 

2 V P cos ( t  120)

Z Y  Z 

The phase currents lag behind their corresponding phase voltages by θ ia 

2 I P cos ( t   )

ic 

2 I P cos ( t    120)

ib 

2 I P cos ( t    120)

p  pa  pb  pc  3V P I P cos 

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08.05.2012

 f (t )

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Comparison of Power Loss Three Phase System

Single Phase System

2

P 'loss  2 R

P L

V L2

, single - phase

P'loss  R '

P L2 2

V L

, three - phase

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

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Chapter Content

Michael E.Auer

•

Introduction

•


•


•


•


08.05.2012

AEE0x

1. Three Phase Circuits.pdf

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