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    1

    Chapter 3

    Three-PhaseSystem

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    2

    Three-Phase CircuitsChapter 3

    3.1 What is a Three-Phase Circuit?

    3.2 Balance Three-Phase Voltages

    3.3 Balance Three-Phase Connection3.4 Power in a Balanced System

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    3

    It is a system produced by a generator consisting ofthree sources having the same amplitude and

    frequency but out of phase with each other by 120.

    3.1 What is a Three-Phase Circuit?(1)

    Three sources

    with 120 out

    of phaseFour wired

    system

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    3.2 Balance Three-Phase Voltages (1)

    A three-phase generator consists of a rotating

    magnet (rotor) surrounded by a stationarywinding (stator).

    A three-phase generator The generated voltages

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    3.2 Balance Three-Phase Voltages (2)

    Two possible configurations:

    Three-phase voltage sources: (a) Y-connected ; (b) -connected

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    3.2 Balance Three-Phase Voltages (3)

    Ba lanced ph ase vo l t ages are equal inmagnitude and are out of phase with each otherby 120.

    The ph ase sequ enceis the time order in which

    the voltages pass through their respectivemaximum values.

    A ba lan ced load is one in which the phaseimpedances are equal in magnitude and in phase

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    3.2 Balance Three-Phase Voltages (4)

    Example 1

    Determine the phase sequence of theset of voltages.

    )110cos(200)230cos(200

    )10cos(200

    =

    =

    +=

    tvtv

    tv

    cn

    bn

    an

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    3.2 Balance Three-Phase Voltages (5)

    Solution:

    The voltages can be expressed in phasor form

    as

    We notice that Van leads Vcn by 120 and Vcn inturn leads Vbn by 120.

    Hence, we have an acb sequence.

    V110200V

    V230200V

    V10200V

    =

    =

    =

    cn

    bn

    an

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    Phase Sequence Order in which the voltages in the individual phases peak

    Negative and positive sequence

    Positive sequence (a-b-c)

    Negative sequence (a-c-b)

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    3.3 Balance Three-Phase Connection (1)

    Four possible connections

    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

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    3.3 Balance Three-Phase Connection (2)

    A balanced Y-Y system is a three-phase system with a

    balanced y-connected source and a balanced y-connected

    load.

    cabcabL

    cnbnanp

    pL

    V

    V

    VV

    VVV

    VVV

    where,3

    ===

    ===

    =

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    120

    120

    120 RNV

    YNV

    BNV

    Schematic diagram for WYE-

    connectionPhase voltage phasor diagram

    Therefore, for phase voltage:

    o

    YN VV 120=o

    BN VV 120=

    oRN VV 0=

    For line voltage:

    opRY VV 303 =

    opYB VV 903 =

    opBR VV 1503 =

    PL II =

    13

    3.3 Balance Three-Phase Connection (3)

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    Therefore, for phase voltage:

    o

    YN VV 120=o

    BN VV 120=

    oRN VV 0=

    For line voltage:

    opRY VV 303 =

    opYB VV 903 =

    opBR VV 1503 =

    BRV RYV

    YBV

    RNV

    YNV

    BNV

    303090

    Phasor diagram illustrating the relationship between line and phasevoltages

    14

    3.3 Balance Three-Phase Connection (4)

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    3.3 Balance Three-Phase Connection (5)

    Example 2

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

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    3.3 Balance Three-Phase Connection (6)

    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

    ===

    ===

    =

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    120

    120

    120 RNV

    YN

    V

    BNV

    Schematic diagram for DELTA-connection Phase voltage phasor

    diagram

    Therefore, for phase voltage:

    oRNRY VVV 0==o

    YNYB VVV 120==o

    BNBR VVV 120==

    PL VV =

    opAa II 303 =

    For line current:

    opBb II 1503 =

    opCc II 903 =

    17

    3.3 Balance Three-Phase Connection (7)

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    Therefore, for phase voltage:

    o

    RNRY VVV 0==o

    YNYB VVV 120==o

    BNBR VVV 120==

    o

    pAa II 303 =

    For line current:

    opBb II 1503 =

    o

    pCc II 903 =

    30

    CcIbcI

    caI

    abI

    AaI

    caI

    BbI

    bcI

    abI

    Phasor diagram illustrating the relationship between line and phase

    currents

    18

    3.3 Balance Three-Phase Connection (8)

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    3.3 Balance Three-Phase Connection (9)

    Example 3

    A balanced abc-sequence Y-connected source with( ) is connected to a -connected load

    (8+j4) per phase. Calculate the phase and linecurrents.

    Solution

    Using single-phase analysis,

    Other line currents are obtained using the abc phase

    sequence

    = 10100Van

    A57.1654.3357.26981.210100

    3/ZVI an =

    ==

    a

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    3.4 Power in a Balanced System (1)

    Real power,

    Reactive power,

    Apparent power,

    cos3

    cos3

    LLIVP

    IVP

    =

    =

    sin3

    sin3

    LLIVQ

    IVQ

    =

    =

    LL

    IVS

    IVS

    3

    3

    =

    =

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    Three- phase Balanced System

    ExampleA balanced three phase system as shown in Figure below , maintainedat constant voltage and frequency is to supply a three- phase load,with three single-phase impedance. IfVab = 440 30 V, find:

    1) The load phase currents IAB, IBC, and ICA.2) The real power, reactive power, apparent power and the powerfactor of the load.

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    Three- phase Balanced System

    ExampleA balanced star- connected load of ZY=8+j6per phase is connected to a balanced 3- phase

    400V supply as shown in Figure below. Finethe line current, power factor, real power,reactive power and total volt-amperes.


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