BASIC ELECTRICAL TECHNOLOGY

DET 211/3

Chapter 3 - Three Phase System

• Thinner conductors can be used to transmit the same kVA at the same voltage, which reduces the amount of copper required (typically about 25% less) and turn reduces construction and maintenance costs.

• The lighter lines are easier to install, and the supporting structures can be less massive and farther apart.

• In general, most larger motors are three phase because they are essentially self starting and do not require a special design or additional starting circuitry.

INTRODUCTION TO THREE PHASE SYSTEM

In general, three phase systems are preferred over single phase systems for the transmission of the power system for many reasons, including the following:

Three phase voltages

A 3-phase generator basically consists of a rotating magnet (called the rotor) surrounded by a

stationary winding (called the stator). Three separate windings or coils with terminals a-a’, b-b’ and c-c’ are physically placed 120o apart around

the stator.

Generated Voltages

The three phase generator can supply power to both single phase and three phase loads

tVv ANmAN sin)(

)120sin()(o

BNmBN tVv

)120sin()240sin( )()(o

CNmo

CNmCN tVtVv

The sinusoidal expression for each of the phase voltages

The phasor diagram of the phase voltages

)AN(m)AN(m

AN V.V

V 70702

)()( 707.0

2BNm

BNmBN V

VV

)()( 707.0

2CNm

CNmCN V

VV

o)m(ANAN VV 0

o)m(BNBN VV 120

o)m(CNCN VV 120

The effective value of each is determined by

If the voltage sources have the same amplitude and frequency ω and are out of the phase with each other by 120o, the voltages are said to be balanced. By rearranging the phasors as shown in figure below, so

omCN

omBN

omANCNBNAN VVVVVV 1201200 )()()(

0)866.05.0866.05.00.1( jjVm

mCNBNAN VVVV ||||||

Where

Generator and Load Connections

Each generator in a 3-phase system maybe either Y- or -connected and loads may be mixed on a power

system.

Z

Z Z Z

Z

Z

Wye Connected Generator

gL II

NBANBNANAB VVVVV

NCBNCNBNBC VVVVV

NACNANCNCA VVVVV

Applying KVL around the indicated loop in figure above, we obtain

Wye Connected Generator

BAAB VVV

00 1200 VV

For line-to-line voltage VAB is given by:

VjVV

2

3

2

1

VjV2

3

2

3

2

1

2

33 jV

0303 V

Phasor Diagram

00 30330 ANABAB VVV

01503 CNCA VV

02703 BNBC VV

Wye Connected Generator

Wye Connected Generator

VVLL 3

The relationship between the magnitude of the line-to-line and line-to-neutral (phase) voltage is:

The line voltages are shifted 300 with respect to the phase voltages. Phasor diagram of the line and phase voltage for the Y

connection is shown below.

RearrangeVAB

VAN

VBCVBN

VCA

VCN

Line-to-line voltagesPhase voltages

Delta Connected Generator

VVLL

CAABA III

00 2400 II

For line-to-line voltage VAB is given by:

IjII

2

3

2

1

IjI2

3

2

3

2

1

2

33 jI

0303 I

Delta Connected Generator

IIL 3

The relationship between the magnitude of the line and phase current is:

The line currents are shifted 300 relative to the corresponding phase current. Phasor diagram of the line and phase current for the Y connection is shown

below.

IA

IAB

IBC

IB

ICA

Line-to-line currentsPhase currents

IC

Phase sequence

The phase sequencephase sequence is the order in which the voltages in the individual phases peak.

VA

VB

VC

VA

VC

VB

abc phase sequence acb phase sequence

Power relationship-phase quantities

The power equations applied to Y-or load in a balanced 3-phase system are:

cosIVP 3

sinIVQ 3

IVS 3

cosZIP 23

sinZIQ 23

ZIS 23

Real power

Unit=Watts(W)

Apparent power

Unit=Volt-Amps (VA)

Reactive power

Unit=Volt-Amps-Reactive (VAR)

- angle between voltage and current in any phase of the load

Power relationship-Line quantities

The power equations applied to Y-or load in a balanced 3-phase system are:

cosIVP LLL3

sinIVQ LLL3

LLL IVS 3

Real power

Apparent power

Reactive power

- angle between phase voltage and phase current in any phase of the load

Since both the three-phase source and the three-phase load can be either Y- or - connected, we

have 4 possible connections:

i) Y-Y connections (i.e: Y-connected source with a Y-connected load).

ii) Y- connection.

iii)- connection

iv)-Y connection

(i) Y connected generator/source with Y connected load

LLg III

EV

VEL 3

321 ZZZ

(ii) Y-Connection

3

ZZY

Z

Z

Z

Z/3

Z/3Z/3

must consists of three equal impedances

A balanced Y- system consists of a balanced Y-connected source feeding a balanced -connected load

(iii) ∆-∆Connection

Z

Z

Z

A balanced ∆- system consists of a balanced ∆-connected source feeding a balanced -connected load

Z

Z

Z

(iv) YConnection

Z/3

Z/3Z/3

A balanced -Y system consists of a balanced -connected source feeding a balanced Y-connected

load

Z

Z

Z

Example 1

A 208V three-phase power system is shown in Figure 1. It consists of an ideal 208V Y-connected three-phase generator connected to a three-phase transmission line to a Y-connected load. The transmission line has an impedance of 0.06+j0.12per phase, and the load has an impedance of 12+j9per phase. For this simple system, find

(a) The magnitude of the line current IL

(b) The magnitude of the load’s line and phase voltages VLL and VL

(c) The real, reactive and apparent powers consumed by the load

(d) The power factor of the load

(e) The real, reactive and apparent powers consumed by the transmission line

(f) The real, reactive and apparent powers supplied by the generator

(g) The generator’s power factor

Example 1

Figure 1

Z

ZZ

Z=12+ i9

-

+

+

0.06

_

0.06

0.06

i0.12

i0.12

i0.12

V

Van=12000

Vbn=120-1200

Vcn=120-2400

208V

Solution Example 1

(a)The magnitude of the line current IL

A

j

jj

V

ZZ

VI

loadline

lineline

1.3794.7

1.3712.15

0120

12.906.12

0120

)912()12.006.0(

0120

So, the magnitude of the line current is thus 7.94 A

Solution Example 1

(b) The magnitude of the load’s line and phase voltages VLL and VL

V

A

jA

ZIV LLL

2.01.119

)9.3615)(1.3794.7(

)912)(1.3794.7(

VV L 1.119

VVV LLL 3.2063

The phase voltage on the load is the voltage across one phase of the load. This voltage is the product of the phase impedance and

the phase current of the load:

Therefore, the magnitude of the load’s phase voltage is

and the magnitude of the load’s line voltage is

Solution Example 1

W

AV

IVPLoad

2270

9.36cos)94.7)(1.119(3

cos3

var1702

9.36sin)94.7)(1.119(3

sin3

AV

IVQLoad

VA

AV

IVSLoad

2839

)94.7)(1.119(3

3

(c) The real power consumed by the load is

The reactive power consumed by the load is

The apparent power consumed by the load is

Solution Example 1

(d) The load power factor is

lagging

PFLoad

8.0

9.36cos

cos

A 1.3794.7 )12.006.0( j 4.63134.0

W

A

ZIPLine

3.11

4.63cos)134.0()94.7(3

cos32

2

var7.22

4.63sin)134.0()94.7(3

sin32

2

A

ZIQLine

VA

A

ZISLine

3.25

)134.0()94.7(3

32

2

(e) The current in the transmission line is

and the impedance of the line is or

per phase. Therefore, the real, reactive and apparent powers consumed in the line are:

Solution Example 1

W

WW

PPP loadlinegen

2281

22703.11

var1725

var1702var7.22

loadlinegen QQQ

VA

QPS gengengen

2860

22

(f) The real and reactive powers supplied by the generator are the sum of the powers consumed by the line and the load:

The apparent power of the generator is the square root of the sum of the squares of the real and reactive powers:

Solution Example 1

1.372281

1725tan

tan

1

1

W

VAR

P

Q

gen

gengen

laggingPFgen 798.01.37cos

(g) From the power triangle, the power factor angle is

Therefore, the generator’s power factor is

Assignment 2

A 208V three-phase power system is shown in Figure 2. It consists of an ideal 208V Y-connected three-phase generator connected to a three-phase transmission line to a -connected load. The transmission line has an impedance of 0.06+j0.12per phase, and the load has an impedance of 12+j9per phase. For this simple system, find

(a) The magnitude of the line current IL

(b) The magnitude of the load’s line and phase voltages VLL and VL

(c) The real, reactive and apparent powers consumed by the load

(d) The power factor of the load

(e) The real, reactive and apparent powers consumed by the transmission line

(f) The real, reactive and apparent powers supplied by the generator

(g) The generator’s power factor

Assignment 2

Figure 2