EET 103 Three Phase System Chapter 3 (Lecture 1)

EET 103EET 103

Three Phase SystemThree Phase System

Chapter 3Chapter 3

((Lecture 1Lecture 1))

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

INTRODUCTION TO THREE PHASE SYSTEMINTRODUCTION TO THREE PHASE SYSTEM

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

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

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

Three phase voltages Three phase voltages

A 3 phase generator basically consists of a A 3 phase generator basically consists of a rotating magnet (called the rotor) surrounded rotating magnet (called the rotor) surrounded by a stationary winding (called the stator). by a stationary winding (called the stator). Three separate windings or coils with Three separate windings or coils with terminals terminals a - a’, b - b’ a - a’, b - b’ and and c - c’ c - c’ are physically are physically placed 120placed 120oo apart around the stator. apart around the stator.

Generated VoltagesGenerated Voltages

The three phase generator can supply power to The three phase generator can supply power to both single phase and three phase loadsboth 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 The sinusoidal expression for each of the phase voltages voltages

7

Phase expressionPhase expression

In phase expressionIn phase expression

WhereWhere

EEM M : peak value: peak value

EEAA, E, EBB and E and ECC : rms value : rms value

02

EE M

A 1202

EE M

B 1202

EE M

C

The phasor diagram 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 The effective value of each is determined each is determined by by

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

omCN

omBN

omAN

CNBNAN

VVV

VVV

1201200 )()()(

0)866.05.0866.05.00.1( jjVm

mCNBNAN VVVV ||||||

Where Where

10

Connection in Three Phase SystemConnection in Three Phase System

A 3 phase system is equivalent to three single A 3 phase system is equivalent to three single phase circuitphase circuit

Two possible configurations in three phase Two possible configurations in three phase systemsystem

1.1. Y - connection (star connection)Y - connection (star connection)

2.2. ∆ ∆ - connection (delta connection)- connection (delta connection)

11

Three phase Voltages SourceThree phase Voltages Source

Y-connected source ∆-connected sourceY-connected source ∆-connected source

12

Three phase LoadThree phase Load

Y - connected load ∆ - connected loadY - connected load ∆ - connected load

Generator and Load ConnectionsGenerator and Load Connections

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

Z

Z Z Z

Z

Z

Wye Connected Generator Wye Connected Generator

gL II

NBANBNANAB VVVVV

NCBNCNBNBC VVVVV

NACNANCNCA VVVVV

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

BAAB VVV

00 1200 VV

For line-to-line voltage VFor line-to-line voltage VABAB is given by is given by

VjVV

2

3

2

1

VjV2

3

2

3

2

1

2

33 jV

0303 V

Phasor DiagramPhasor Diagram

00 30330 ANABAB VVV

01503 CNCA VV

02703 BNBC VV

VVLL 3

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

The line voltages are shifted 30The line voltages are shifted 3000 with respect to with respect to the phase voltages. Phasor diagram of the line the phase voltages. Phasor diagram of the line and phase voltage for the Y connection is shown and phase voltage for the Y connection is shown below.below.

RearrangeRearrange

VAB

VAN

VBCVBN

VCA

VCN

Line-to-line voltagesPhase voltages

Delta Connected Generator Delta Connected Generator

VVLL

CAABA III

00 2400 II

For line-to-line voltage VFor line-to-line voltage VABAB is given by is given by

IjII

2

3

2

1

IjI2

3

2

3

2

1

2

33 jI

0303 I

IIL 3

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

The line currents are shifted 30The line currents are shifted 3000 relative to the relative to the corresponding phase current. Phasor diagram of corresponding phase current. Phasor diagram of the line and phase current for the Y connection is the line and phase current for the Y connection is shown below.shown below.

IA

IAB

IBC

IB

ICA

Line-to-line currentsPhase currents

IC

Phase sequencePhase sequence

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

VA

VB

VC

VA

VC

VB

abc abc phase sequencephase sequence acbacb phase sequencephase sequence

21

EXAMPLE 3.1EXAMPLE 3.1

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

22

Solution 3.1Solution 3.1

Single Phase Equivalent CircuitSingle Phase Equivalent Circuit

Phase ‘a’ equivalent circuitPhase ‘a’ equivalent circuit

23

21.86.8121.816.155

0110I

8.21155.16)810()25(Z;Z

VI

Aa

TT

ANAa jj

A2.986.811.8266.81

024II

A141.86.81

120II

AaCc

AaBb

24


A balanced delta connected load having an A balanced delta connected load having an impedance 20 - j15 impedance 20 - j15 is connected to a delta is connected to a delta connected, positive sequence generator connected, positive sequence generator having Vhaving VABAB = 330 = 33000 V. Calculate the phase V. Calculate the phase

currents of the load and the line currents.currents of the load and the line currents.

25


V 0330V

87.3625 j1520Z

AB

Δ

26

Phase CurrentsPhase Currents

A87.15613.2120II

A13.83-13.2120II

A36.8713.238.8725

0330

Z

VI

abca

abbc

Δ

abab

27

A 87.12686.22120II

A 13.311-86.22120II

A 87.686.22

30336.8713.2

303II

AaCc

AaBb

abAa

Line CurrentsLine Currents

28

∆ ∆ - Connected Generator with a Y - - Connected Generator with a Y - Connected LoadConnected Load

29


A balanced Y - connected load with a phase A balanced Y - connected load with a phase impedance 40 + j25 impedance 40 + j25 is supplied by a balanced, is supplied by a balanced, positive-sequence positive-sequence ΔΔ-connected source with a -connected source with a line voltage of 210V. Calculate the phase line voltage of 210V. Calculate the phase currents. Use Vcurrents. Use VABAB as reference. as reference.

30


the load impedance, Zthe load impedance, ZYY and the source voltage, V and the source voltage, VABAB are are

V 0210V

3217.47 j2540Z

AB

Y

31

When the ∆ - connected source is transformed When the ∆ - connected source is transformed to a Y - connected source, to a Y - connected source,

V 30-121.2

3013

0210

303

VV AB

an

32

The line currents areThe line currents are

A 582.57120II

A 182-2.57120II

A 62-2.573247.17

03121.2

Z

VI

AaCc

AaBb

Y

anAa

33

Summary of Relationships in Y and Summary of Relationships in Y and ∆ - connections∆ - connections

Y-connectionY-connection ∆∆-connection-connection

Voltage Voltage magnitudesmagnitudes

Current Current magnitudesmagnitudes

Phase Phase sequencesequence

VVLL leads leads VVφφ by by

30°30°IILL lags lags IIφφ by 30° by 30°

φV3VL φVVL

φI3IL φIIL

EET 103EET 103

Three Phase SystemThree Phase System

Chapter 3Chapter 3

((Lecture 2Lecture 2))

PowerPower

Y - Connected Balanced LoadY - Connected Balanced Load

Average PowerAverage Power

The average power delivered to each phaseThe average power delivered to each phase

The total power to the balanced load isThe total power to the balanced load is

Reactive PowerReactive Power

The reactive power of each phase isThe reactive power of each phase is

The total reactive power of the load isThe total reactive power of the load is

Apparent PowerApparent Power

The apparent power of each phase is The apparent power of each phase is

The total apparent power of the load isThe total apparent power of the load is

Power FactorPower Factor

The power factor of the system isThe power factor of the system is

∆ ∆ - Connected Balanced Load- Connected Balanced Load

Average PowerAverage Power

Reactive PowerReactive Power

Apparent PowerApparent Power

Power FactorPower Factor

EXAMPLE 3.4 EXAMPLE 3.4

Determine the total power (P), reactive power Determine the total power (P), reactive power (Q) and complex power (S) at the source and at (Q) and complex power (S) at the source and at the load. the load.

Single Phase Equivalent CircuitSingle Phase Equivalent Circuit

Phase Phase ‘a’‘a’ equivalent circuit equivalent circuit

Known quantitiesKnown quantitiesVg =VAN= 1100 V

ZY = 10 + j8

Zline = 5 - j2


Line / Phase CurrentsLine / Phase Currents

A 21.86.8121.816.155

0110I

ZZ

VI

A

Yline

ANA

Source & Load PowerSource & Load Power

VAR 834.6Q W,2087P

j834.6)VA(2087

I3VS

ss

φφsource

VAR 1113Q W,1392P

j1113)VA(1392

ZI3S

LL

2

φLoad


A three phase motor can be regarded as a A three phase motor can be regarded as a balanced Y - load. A three phase motor balanced Y - load. A three phase motor draws 5.6 kW when the line voltage is 220 V draws 5.6 kW when the line voltage is 220 V and the line current is 18.2 A. Determine the and the line current is 18.2 A. Determine the power factor of the motorpower factor of the motor

Known QuantitiesKnown Quantities

• PPLoad Load = 5600 W= 5600 W

• VVLL = 220 V = 220 V

• IILL = 18.2 A = 18.2 A

Power factor = cos Power factor = cos

VA 6935.13

IV 3

I3VS

LL

φφ

0.86935.13

5600

S

Pθ cos

θ cosSP

|S|Q

P


Example 3.6Example 3.6

For the Y - connected load in FigureFor the Y - connected load in Figure

a)a) find the average power to each phase and find the average power to each phase and the total loadthe total load

b)b) determine the reactive power to each phase determine the reactive power to each phase and the total reactive power and the total reactive power

c)c) find the apparent power to each phase and find the apparent power to each phase and the total apparent power the total apparent power

d)d) find the power factor of the loadfind the power factor of the load

Figure


a) The average power to each phase isa) The average power to each phase is

Total loadTotal load

W1200

13.53cos20100

cos

VIIVP

W3600W120033 PPT

b) The reactive power to each phase isb) The reactive power to each phase is

Total reactive powerTotal reactive power

VAR1600

13.53sin20100

sin

VIIVQ

VAR4800160033 QQT

c) The apparent power to each phase isc) The apparent power to each phase is

Total apparent powerTotal apparent power

VA2000

20100

IVS

VA6000200033 SST

d) The power factord) The power factor

lagging

FP

6.0VA 6000

W3600

S

P

T

T

Power relationship - Phase quantitiesPower relationship - Phase quantities

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

cosIVP 3

sinIVQ 3

IVS 3

cosZIP 23

sinZIQ 23

ZIS 23

Real powerReal power

Watts (W)Watts (W)

Apparent powerApparent power

Volt-Amps (VA)Volt-Amps (VA)

Reactive powerReactive power

Volt-Amps-Reactive (VAR)Volt-Amps-Reactive (VAR)

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

Power relationship - Line quantitiesPower relationship - Line quantities

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

cosIVP LLL3

sinIVQ LLL3

LLL IVS 3

Real powerReal power

Apparent powerApparent power

Reactive powerReactive power

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

Since both the three phase source and the three Since both the three phase source and the three phase load can be either Y or phase load can be either Y or connected, we connected, we have 4 possible connectionshave 4 possible connections

1.1.Y - Y connections Y - Y connections (Y - connected source with (Y - connected source with Y - connected load)Y - connected load)

2.2.Y - Y - connection connection (Y - connected source with (Y - connected source with - connected load)- connected load)

3.3. - - connection connection (( - connected source with - connected source with - connected load)- connected load)

4.4. - Y connection - Y connection (( - connected source with Y - connected source with Y - connected load)- connected load)

1. Y connected generator / source with Y 1. Y connected generator / source with Y connected loadconnected load

LLg III

EV

VEL 3

321 ZZZ

2. Y - 2. Y - ConnectionConnection

3

ZZY

Z

Z

Z

Z/3

Z/3Z/3

must consists of three equal impedancesmust consists of three equal impedances

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

3. 3. ∆ ∆ - - ∆∆ConnectionConnection

Z

Z

Z

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

Z

Z

Z

4. 4. YYConnectionConnection

Z/3

Z/3Z/3

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

Z

Z

Z


Each transmission line of the 3 wire, three phase Each transmission line of the 3 wire, three phase system in Figure has an impedance of 15 Ω + j system in Figure has an impedance of 15 Ω + j 20 Ω. The system delivers a total power of 160 20 Ω. The system delivers a total power of 160 kW at 12,000 V to a balanced three-phase load kW at 12,000 V to a balanced three-phase load with a lagging power factor of 0.86.with a lagging power factor of 0.86.

a.a. Determine the magnitude of the line voltage Determine the magnitude of the line voltage EEABAB of the generator. of the generator.

b.b. Find the power factor of the total load Find the power factor of the total load applied to the generator.applied to the generator.

c.c. What is the efficiency of the system?What is the efficiency of the system?

Figure


0V V

A 94.886.042.69363

W160000

cos3

V

PI T

a.a. VVøø (load) = (load) =

PPTT (load) = 3 (load) = 3 VVøø IIøø cos cos θθ

and

Since Since θθ = cos = cos-1-1 0.86 = 30.68 0.86 = 30.68oo (lagging) (lagging)

And assigning , a lagging power And assigning , a lagging power factor results in factor results in 68.30A94.8I

V 42.69361.73

V12000

3LV

0VZIE lineAN

V 26.12358)V5.7143)(73.1(3 gAB EE

For each phase, the system will appear as shown For each phase, the system will appear as shown in figure below.in figure below.

Or

Then

0.687143.5V

85.35V7142.98V

V42.6936V35.85V56.206

0V42.693645.22V5.223

0V42.693613.532530.68-A94.8

VZIE

j

j

lineAN

b.b.

W55.596,163

W55.3596W000,160

15A94.83kW160

)(3kW1602

2

lineL

linesloadT

RI

PPP

TLLT IVP cos3

A94.8V26.358,121.73

W55.596,163

3cos

LL

TT

IV

P

856.0pF

%8.97978.0 W3596.55kW 160

kW 160

losseso

o

i

o

PP

P

P

P

And

< 0.86 of loadAnd

c.c.


A 208V three phase power system is shown in Figure 1. It consists A 208V three phase power system is shown in Figure 1. It consists of an ideal 208V Y - connected three phase generator connected to of an ideal 208V Y - connected three phase generator connected to a three phase transmission line to a Y - connected load. The a three phase transmission line to a Y - connected load. The transmission line has an impedance of transmission line has an impedance of 0.06 + j0.120.06 + j0.12per phase and per phase and the load has an impedance of the load has an impedance of 12 + j912 + j9per phase. For this simple per phase. For this simple system, findsystem, find

(a)(a) The magnitude of the line current The magnitude of the line current IILL

(b)(b) The magnitude of the load’s line and phase voltages The magnitude of the load’s line and phase voltages VVLLLL and and VVLL

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

(d)(d) The power factor of the loadThe power factor of the load

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

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

(g)(g) The generator’s power factorThe generator’s power factor

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


(a)(a)The magnitude of the line current The magnitude of the line current IILL

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

(b) The magnitude of the load’s line and phase (b) The magnitude of the load’s line and phase voltages voltages VVLLLL and and VVLL

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 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 load. This voltage is the product of the phase impedance and the phase current of the loadphase current of the load

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

and the magnitude of the load’s line voltage isand the magnitude of the load’s line voltage is

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(c) The real power consumed by the load is

The reactive power consumed by the load isThe reactive power consumed by the load is

The apparent power consumed by the load isThe apparent power consumed by the load is

(d) The load power factor is (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 (e) The current in the transmission line is transmission line is

TThe impedance he impedance per phase of the line is of the line is or

Therefore, the real, reactive and apparent powers consumed in the line are

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

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

1.372281

1725tan

tan

1

1

W

VAR

P

Q

gen

gengen

laggingPFgen 798.01.37cos

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

Therefore, the generator’s power factor isTherefore, the generator’s power factor is

Assignment 3.1Assignment 3.1

A 208V three phase power system is shown in Figure 2. It consists of 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 an ideal 208V Y - connected three phase generator connected to a three phase transmission line to a three phase transmission line to a - connected load. The - connected load. The transmission line has an impedance of transmission line has an impedance of 0.06 + j0.120.06 + j0.12per phase and per phase and the load has an impedance of the load has an impedance of 12 + j912 + j9per phase. For this simple per phase. For this simple system, findsystem, find

a.a. The magnitude of the line current The magnitude of the line current IILL

b.b. The magnitude of the load’s line and phase voltages The magnitude of the load’s line and phase voltages VVLLLL and and VVLL

c.c. The real, reactive and apparent powers consumed by the loadThe real, reactive and apparent powers consumed by the load

d.d. The power factor of the loadThe power factor of the load

e.e. The real, reactive and apparent powers consumed by the The real, reactive and apparent powers consumed by the transmission linetransmission line

f.f. The real, reactive and apparent powers supplied by the The real, reactive and apparent powers supplied by the generatorgenerator

g.g. The generator’s power factorThe generator’s power factor

Figure 2

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EET 103 Three Phase System Chapter 3 (Lecture 1)