1. Three Phase Circuits.pdf

Advanced Electrical Engineering

Michael E.Auer 08.05.2012 AEE0x


Michael E. Auer

Three Phase Circuits



AEE ContentAdvanced Circuit Analysis

• Basic Concepts• Three-Phase Circuits• Transforms• Power Conversion and Management

Field Theory• Waves and Vector Fields• Transmission Line Theory• Electrostatics• Magnetostatics

Applications• Magnetic Field Applications• Basics of Electrical Machines



Chapter Content

• Introduction

• Balanced Three-Phase Voltages

• Balanced Three-Phase Connections

• Power in Balanced Systems

• Unbalanced Three-Phase Systems



Chapter Content

• Introduction• Balanced Three-Phase Voltages






Single Phase Systems

two-wire type three-wire type



Two-Phase Three-Wire System



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 Four-wire system

What is a Three-Phase Circuit?



Advantages:

1. Most of the electric power is generated and distributed in three-phase. 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.

Reasons for the Use of Three-Phase Circuits



Chapter Content

• Introduction

• Balanced Three-Phase Voltages• Balanced Three-Phase Connections





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

Three-phase generator Generated voltages

Balanced Three-Phase Voltages (1)



Two possible configurations:

Balanced Three-Phase Voltages (2)

Y-connected ∆-connected



Phase Voltages and ist Sequences

Positive sequence (abc)

Negative sequence (acb)

cnbnan

cnbnan

VVV 0VVV



Phase Voltage Sequences ExampleDetermine the phase sequence of the set of voltages.

)110cos(200)230cos(200

)10cos(200

tvtvtv

cn

bn

an

Solution:

The voltages can be expressed in phasor form as

We notice that Van leads Vcn by 120° and Vcn in turn leads Vbn by 120°.

Hence, we have an acb sequence.

V 110200VV 230200V

V 10200V

cn

bn

an



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

Properties of Three-Phase Systems



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)

3. ∆-∆ connection

4. ∆-Y connection



Chapter Content

• Introduction


• Balanced Three-Phase Connections• Power in Balanced Systems




Balanced Y–Y System (1)

A balanced Y-Y system is a three-phase system with a balanced y-connected source and a balanced y-connected load.

LlSYwhere ZZZZ



Balanced Y–Y System (2)

cabcabL

cnbnanp

pL

V

V

VV

VVV

VVV

where, 3



Balanced Y–Y System Example

Calculate the line currents in the three-wire Y-Y system shown below:

A 2.9881.6IA 8.14181.6I

A 8.2181.6I:Answer

c

b

a



Balanced Y–Δ System (1)

A balanced Y-∆ system is a three-phase system with a balanced y-connected source and a balanced ∆-connected load.

CABCABp

cbaL

pL

I

I

II

III

III

where, 3



Balanced Y–Δ System (2)

Transformation of the ∆-connected load to an equivalent Y-connectedload

Single Phase Equivalent



Balanced Y–Δ System ExampleA balanced abc-sequence Y-connected source with is

connected to a Δ-connected load (8+j4) per phase. Calculate the phase and line currents.

Solution

Using single-phase analysis,

Other line currents are obtained using the abc phase sequence

10100Van

A 57.1654.3357.26981.2

101003/Z

VI an

a



BalancedΔ–Δ System

A balanced ∆-∆ system is a three-phase system with a balanced ∆-connected source and a balanced ∆-connected load.



BalancedΔ–Δ System ExampleA balanced ∆-connected load having an impedance 20-j15 is connected to a ∆-connected positive-sequence generator having . Calculate the phase currents of the load and the line currents.

Answer:The phase currents

The line currents

V 0330Vab

A 87.1562.13IA; 13.812.13IA; 87.362.13I CABCAB

A 87.12686.22IA; 13.11386.22IA; 87.686.22I cba



BalancedΔ–Y System (1)

A balanced ∆-Y system is a three-phase system with a balanced Y-connected source and a balanced y-connected load.



BalancedΔ–Y System (2)

Transformation of the ∆-connected source to a Y-connected one.

903

V1503

V

303

V

Pcn

Pbn

Pan

VV

V Y

P

a Z

V

I

30

3



Chapter Content

• Introduction



• Power in Balanced Systems• Unbalanced Three-Phase Systems



Power in Balanced Systems

)120(cos2

)120(cos2cos2

tVv

tVvtVv

PCN

PBNPAN

For a Y-connected load the phase voltages are:

For: ZYZThe phase currents lag behind their corresponding phase voltages by θ

)120(cos2

)120(cos2)(cos2

tIi

tIitIi

Pc

PbPa

)(cos3 tfIVpppp PPcba



Comparison of Power Loss

phase-single ,2' 2

2

L

Lloss V

PRP phase- three,'' 2

2

L

Lloss V

PRP

If same power loss is tolerated in both system, three-phase system use only 75% of materials of a single-phase system!

Single Phase System Three Phase System



Chapter Content

• Introduction







An unbalanced system is due to unbalanced voltage sources or an unbalanced load.

)III(I

,ZVI ,

ZVI ,

ZVI

cban

C

CNc

B

BNb

A

ANa

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.

Unbalanced Three-Phase Systems

Documents

1. Three Phase Circuits.pdf