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ELECTRICAL CIRCUIT ET 201

Become familiar with the operation of a three phase generator and the magnitude and phase relationship.

Be able to calculate the voltages and currents for a three phase Wye and Delta connected generator and load.

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THREE PHASE THREE PHASE SYSTEMSSYSTEMS

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23.1 – IntroductionIf the number of coils on the rotor is increased

in a specified manner, the result is a polyphase ac generator, which develops more than one ac phase voltage per rotation of the rotor

An ac generator designed to develop a single sinusoidal voltage for each rotation of the shaft (rotor) is referred to as a single-phase ac generator

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

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Introduction1. 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 in turn reduces construction and maintenance costs.

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

3. Three-phase equipment and motors have preferred running and starting characteristics compared to single-phase systems because of a more even flow of power to the transducer than can be delivered with a single-phase supply.

4. 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.

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IntroductionThe frequency generated is determined by the

number of poles on the rotor (the rotating part of the generator) and the speed with which the shaft is turned.

Throughout the United States the line frequency is 60 Hz, whereas in Europe (incl. Malaysia) the chosen standard is 50 Hz.

On aircraft and ships the demand levels permit the use of a 400 Hz line frequency.

The three-phase system is used by almost all commercial electric generators.

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IntroductionMost small emergency generators, such as the

gasoline type, are one-phased generating systems.The two-phase system is commonly used in

servomechanisms, which are self-correcting control systems capable of detecting and adjusting their own operation.

Servomechanisms are used in ships and aircraft to keep them on course automatically, or, in simpler devices such as a thermostatic circuit, to regulate heat output.

The number of phase voltages that can be produced by a polyphase generator is not limited to three. Any number of phases can be obtained by spacing the windings for each phase at the proper angular position around the stator.

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23.2 – Three-Phase Generator

The three-phase generator has three induction coils placed 120° apart on the stator.

The three coils have an equal number of turns, the voltage induced across each coil will have the same peak value, shape and frequency.

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Three-Phase GeneratorAt any instant of time, the algebraic sum of

the three phase voltages of a three-phase generator is zero.

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Three-Phase GeneratorThe sinusoidal expression for each of the

induced voltage is:

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Phase expression

• In phase expression:

• Where:

EM : peak value

EA, EB and EC : rms value

02

EE M

A 1202

EE M

B 1202

EE M

C

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Connection in Three Phase System

• A 3-phase system is equivalent to three single phase circuit

• Two possible configurations in three phase system:

1. Y-connection (star connection)

2. ∆-connection (delta connection)

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Three-phase Voltages Source

Y-connected source ∆-connected source

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Three-phase Load

Y-connected load ∆-connected load

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23.3 – Y-Connected GeneratorIf the three terminals denoted N are connected

together, the generator is referred to as a Y-connected three-phase generator.

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Y-Connected Generator

The point at which all the terminals are connected is called the neutral point.

Two type of Y-connected generator:

1. Y-connected, three-phase, three-wire generator

(a conductor is not attached from this point to the load)

2. Y-connected, three-phase, four-wire generator

(the neutral is connected)

The three conductors connected from A, B and C to the load are called lines.

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Y-connected, 3-phase, 3-wire generator

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Y-connected, 3-phase, 4-wire generator

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Y-Connected GeneratorThe voltage from one line to another is called

a line voltage The magnitude of the line voltage of a

Y-connected generator is:

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Definition of Phase Voltage

• In 3-phase system, for Y-connected, the voltage from line to neutral point is called a phase voltage.

EAN – phase A voltage

EBN – phase B voltage

ECN – phase C voltage

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Definition of Line Voltage

• In 3-phase system, for Y-connected, the voltage from one line to another is called a line voltage.

EAB – voltage between line A and B

EBC – voltage between line B and C

ECA – voltage between line C and A

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Y-connected system

• Line voltage:

VAB ; VBC ; VCA

• Phase voltage:

VAN ; VBN ; VCN

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Voltage in Y-connected system

For 3-phase Y-connected system, if the phase voltage VAN is taken as the reference, so

0VV ANAN

012VV BNBN

012VV CNCN

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Voltage in Y-connected system

• By applying Kirchhoff’s Voltage Law, the line voltage can be written as

)30(1.732V

j0.866)(1.5V

j0.866))0.5(j0)((1V

)12010(1V

120V0V

VVV

AN

AN

AN

AN

BNAN

BNANAB

30V3V ANAB

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Voltage in Y-connected system• With the same method,

and

• The relationship between the line voltage and the phase voltage can be represented as

VL : line voltage Vφ : phase voltage

90V3

VVV

BN

CNBNBC

150V3

VVV

CN

ANCNCA

30V3VL φ

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Current in Y-connected system• For the Y-connected system, it should be

obvious that the line current equals the phase current for each phase; that is

IL : line current Iφ : phase

current

φIIL

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23.4 – Phase Sequence (Y-Connected Generator)

The phase sequence can be determined by the order in which the phasors representing the phase voltages pass through a fixed point on the phasor diagram if the phasors are rotated in a counterclockwise direction.

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3.4 – Phase Sequence (Y-Connected Generator)

In phasor notation,

Line voltage:

Phase voltage:

120VV

120VV

)reference(0VV

CACA

BCBC

ABAB

120VV

120VV

)reference(0VV

CNCN

BNBN

ANAN

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23.5 – Y-Connected Generator with a Y-Connected Load

Loads connected with three-phase supplies are of two types: the Y and the ∆.

If a Y-connected load is connected to a Y-connected generator, the system is symbolically represented by Y-Y.

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Y-Connected Generator with a Y-Connected Load

If the load is balanced, the neutral connection can be removed without affecting the circuit in any manner; that is, if Z1 = Z2 = Z3 , then IN will be zero, IN = 0 .

Since IL = V / Z the magnitude of the current in each phase will be equal for a balanced load and unequal for an unbalanced load. In either case, the line voltage is

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EXAMPLE 1• Calculate the line currents in the three-wire Y-Y

system as shown below.

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Solution:Single Phase Equivalent Circuit

Phase ‘a’ equivalent circuit

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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

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23.6 – Y-Connected Generator with a ∆-Connected Load

There is no neutral connection for the Y-∆ system shown below.

Any variation in the impedance of a phase that produces an unbalanced system will simply vary the line and phase currents of the system.

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Y-Connected Generator with a ∆-Connected Load

For a balanced load, Z1 = Z2 = Z3. The voltage across each phase of the load is equal to the line

voltage of the generator for a balanced or an unbalanced load: V = EL.

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Y-Connected Generator with a ∆-Connected Load

Kirchhoff’s current law is employed instead of Kirchhoff’s voltage law.The results obtained are:

The phase angle between a line current and the nearest phase current is 30°.

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EXAMPLE 2

A balanced positive sequence Y-connected source with VAN=10010 V is connected to a -connected balanced load (8+j4) per phase. Calculate the phase and line currents.

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Solution: Balanced WYE source, VAN = 10010 V

Balanced DELTA load, Z = 8 + j4

Phase and line currents = ??

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Phase Currents

A 43.1336.19j48

40173.2I

V 402.731V

30V 3V

Z

VI

ab

ab

ANAB

Δ

abab

Vab= voltage across Z = VAB= source line voltage

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Phase Currents

A 43.13336.19I

12043.1336.19I

A 57.10636.19I

12043.13II

A 43.1336.19I

ca

ca

bc

abbc

ab

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Line Currents

A 43.103 53.33120 II

A 57.136 53.33120 II

A 57.16 53.33I

3043.13 (19.36) 3

30 I 3I

AaCc

AaBb

Aa

abAa

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23.7 – ∆-Connected Generator

In the figure below, if we rearrange the coils of the generator in (a) as shown in (b), the system is referred to as a three-phase, three-wire.

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∆-Connected Generator∆-connected ac generator

In this system, the phase and line voltages are equivalent and equal to the voltage induced across each coil of the generator:

or EL = Eg

Only one voltage (magnitude) is available instead of the two in the Y-Connected system.

)120sin(2 and

)120sin(2 and

sin2 and

tEeEE

tEeEE

tEeEE

CNCNCNCA

BNBNBNBC

ANANANAB

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∆-Connected Generator Unlike the line current for the Y-connected generator, the

line current for the ∆-connected system is not equal to the phase current. The relationship between the two can be found by applying Kirchhoff’s current law at one of the nodes and solving for the line current in terms of the phase current; that is, at node A,

IBA = IAa + IAC

or

IAa = IBA - IAC = IBA + ICA

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∆-Connected GeneratorThe phasor diagram is shown below for a balanced load.In general, line current is:

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Definition of Phase Current

• In 3-phase system, for ∆-connected, the current that flow from one phase to another is called a phase current.

IBA – phase A current

ICB – phase B current

IAC – phase C current

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Definition of Line Current

• In 3-phase system, for ∆-connected, the current that flow through the line is called a line current.

IAa – line A current

IBb – line B current

ICc – line C current

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∆-connected system (generator)

• Line current:

IAa ; IBb ; ICc

• Phase current:

for generator:

IBA ; IAC ; ICB

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∆-connected system (load)

• Line current:

IAa ; IBb ; ICc

• Phase current:

for load:

Iab ; Ibc ; Ica

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Current in ∆-connected system (Generator side)

For 3-phase ∆-connected system (generator), if the phase current IBA is taken as the reference, so

0II BABA

012II CBCB

012II ACAC

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Current in ∆-connected system (Generator side)

• By applying Kirchhoff’s Current Law, the line current can be written as

)30(1.732I

j0.866)(1.5I

j0.866))0.5(j0(1I

)12010(1I

120I0I

III

BA

BA

BA

BA

BABA

ACBAAa

A 30I3I BAAa

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• With the same method,

and

Current in ∆-connected system (Generator side)

150I3

III

CB

BACBBb

90I3

III

AC

CBACCc

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Current in ∆-connected system (Load side)

For 3-phase ∆-connected system (load), if the phase current Iab is taken as the reference, so

0II abab

012II bcbc

012II caca

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Current in ∆-connected system (Load side)

• By applying Kirchhoff’s Current Law, the line current can be written as

)30(1.732I

j0.866)(1.5I

j0.866))0.5(j0(1I

)12010(1I

120I0I

III

ab

ab

ab

ab

abab

caabAa

A 30I3I abAa

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• With the same method,

and

Current in ∆-connected system (Load side)

150I3

III

bc

abbcBb

90I3

III

ca

bccaCc

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• The relationship between the line current and the phase current can be represented as

Where;

IL : line current Iφ : phase current

Relationship between the phase current and the line current

(∆-connected system)

30I3IL φ

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Voltage in ∆-connected system• For the ∆-connected system, it should be obvious

that the line voltage equals the phase voltage for each phase; that is

VL : line voltage

V : phase voltage

φVVL

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23.8 – Phase Sequence (∆- Connected Generator)

Even though the line and phase voltages of a ∆ -connected system are the same, it is standard practice to describe the phase sequence in terms of the line voltages

In drawing such a diagram, one must take care to have the sequence of the first and second subscripts the same

In phasor notation,

VAB = VAB 0o

VBC = VBC 120o

VCA = VCA 120o

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23.9 - ∆-Connected Generator with a ∆-Connected Load

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EXAMPLE 3

A balanced delta connected load having an impedance 20 - j15 is connected to a delta connected, positive sequence generator having VAB = 3300 V. Calculate the phase currents of the load and the line currents.

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Solution:

V 0330V

87.3625 j1520Z

AB

Δ

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Phase Currents

A87.15613.2120II

A13.83-13.2120II

A36.8713.238.8725

0330

Z

VI

abca

abbc

Δ

abab

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A 87.12686.22120II

A 13.311-86.22120II

A 87.686.22

30336.8713.2

303II

AaCc

AaBb

abAa

Line Currents

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

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EXAMPLE 4

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

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Solution:

• the load impedance, ZY and the source voltage, VAB are

V 0210V

3217.47 j2540Z

AB

Y

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Solution:

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

V 30-121.2

3013

0210

303

VV AB

an

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Solution:

• The line currents are

A 582.57120II

A 182-2.57120II

A 62-2.573247.17

03121.2

Z

VI

AaCc

AaBb

Y

anAa

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Summary of Relationships in Y and ∆-connections

Y-connection ∆-connection

Voltage magnitudes

Current magnitudes

Phase sequence

VL leads Vφ by 30° IL lags Iφ by 30°

φV3VL φVVL

φI3IL φIIL