Symmetrical Components - eal.ei.tum.de .1 Symmetrical Components Prof. Dr.-Ing. Ralph Kennel...

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  • 1

    Symmetrical Components

    Prof. Dr.-Ing. Ralph Kennel

    (ralph.kennel@tum.de)

    Technische Universitt Mnchen

    Lehrstuhl fr Elektrische Antriebssysteme und Leistungselektronik

    Mnchen, 06. Mrz 2018

  • Symmetrical Components

    2

    symmetrical 3phase system

    U1

    U2

    U2 = U1 e-j2/3

    U3

    U3 = U1 e-j4/3

    versors

    U2 = U1 a

    U3 = U1 a

  • the basic idea is

    that an asymmetrical set of N phasors

    can be expressed

    as a linear combination

    of N symmetrical sets of phasors

    by means of a complex linear transformation

    3Dr. rer. nat. Erika Mustermann (TUM) | kann beliebig erweitert werden | Infos mit Strich trennen

    Symmetrical Components

    https://en.wikipedia.org/wiki/Phasorhttps://en.wikipedia.org/wiki/Linear_combinationhttps://en.wikipedia.org/wiki/Complex_numberhttps://en.wikipedia.org/wiki/Linear_transformation

  • usual strategies

    in electrical engineering

    Real World

    (time domain)initial situation

    Dream World

    (xxx-domain)initial situation

    transformation

    Dream World

    (xxx-domain)resultback

    transformation

    calculation(s)

    Real World

    (time domain)result

    calculation(s)

    might be

    possible

    as well

    usually

    it is easier

    this way!

    but more complex

  • Symmetrical Components

    5all components are rotating in the same direction (within the complex plane) !!!

  • 6Prof. Dr.-Ing. Ralph Kennel (TUM)

    any 3phase system

    can be split into the

    symmetrical components

  • 7Prof. Dr.-Ing. Ralph Kennel (TUM)

    what is the purpose ?

  • 8Prof. Dr.-Ing. Ralph Kennel (TUM)

    what is the purpose ?

    1st step :

    split the system into the

    symmetrical components

    2nd step :

    calculate the response for

    each of the

    symmetrical components

    3rd step :

    recombine the

    symmetrical components

    into real system

  • 9Prof. Dr.-Ing. Ralph Kennel (TUM)

    what is the purpose ?

    1st step :

    split the system into the

    symmetrical components

    2nd step :

    calculate the response for

    each of the

    symmetrical components

    3rd step :

    recombine the

    symmetrical components

    into real system

  • 10Prof. Dr.-Ing. Ralph Kennel (TUM)

    what is the purpose ?

    1st step :

    split the system into the

    symmetrical components

    2nd step :

    calculate the response for

    each of the

    symmetrical components

    3rd step :

    recombine the

    symmetrical components

    into real system

  • 11Prof. Dr.-Ing. Ralph Kennel (TUM)

    what is the purpose ?

    1st step :

    split the system into the

    symmetrical components

    2nd step :

    calculate the response for

    each of the

    symmetrical components

    3rd step :

    recombine the

    symmetrical components

    into real system

  • 12Prof. Dr.-Ing. Ralph Kennel (TUM)

    what is the purpose ?

    1st step :

    split the system into the

    symmetrical components

    2nd step :

    calculate the response for

    each of the

    symmetrical components

    3rd step :

    recombine the

    symmetrical components

    into real system

  • 13

    Symmetrical Components

  • 14Prof. Dr.-Ing. Ralph Kennel (TUM)

    what is the purpose ? electrical AC machines

    show different impedances

    in symmetrical components

  • Graphic Decomposition in Symmetrical Components15

  • 16Graphic Decomposition in Symmetrical Components

  • 17Graphic Decomposition in Symmetrical Components

  • 18Graphic Decomposition in Symmetrical Components

    all in one

  • 19Prof. Dr.-Ing. Ralph Kennel (TUM)

    Multiphase Systems in General

    the number of phases is not really fixed to 3

  • 20Prof. Dr.-Ing. Ralph Kennel (TUM)

    Symmetrical Components

    for 3 phase systems

  • 21Prof. Dr.-Ing. Ralph Kennel (TUM)

    Symmetrical Components

    for 4 phase systems

  • 22Prof. Dr.-Ing. Ralph Kennel (TUM)

    Symmetrical Components

    for 5 phase systems

  • 23Prof. Dr.-Ing. Ralph Kennel (TUM)

    Symmetrical Components

    for 6 phase systems

  • 24Prof. Dr.-Ing. Ralph Kennel (TUM)

    What about 2phase Systems ???

    the so-called 2phase system

    is not really a 2phase system

    because the phase shift is not 180

    based on the phase shift of 90 between the phases

    it is a 4phase system with only 2 phases used

    symmetrical components for 4phase systems

  • 25Prof. Dr.-Ing. Ralph Kennel (TUM)

    Symmetrical Components

    Example: Transformer

    U1 = Z1 I1

    U2 = Z2 I2

    U0 = Z0 I0

    in rotating electrical machines Z1, Z2 and Z0 are different

    in stationary electrical machines at least Z1 and Z2 are equal

    in a transformer Z1 = Z2 = ZK2

    equivalent circuit

  • 26Prof. Dr.-Ing. Ralph Kennel (TUM)

    Symmetrical Components

    Example: Transformer

    in a transformer the zero impedance Z0 depends on the design of the transformer

    U0 = Z0 I0

    in case of a star connection on primary and secondary side

    there cannot be any zero sequence current on the primary side (due to Krichhoffs law)

    the equivalent circuit contains the secondary side only

    the quantity of the mutual inductance depends on the design

    3-leg-core mutual inductance is low

    (magnetic flux has to go through the air)

  • 27Prof. Dr.-Ing. Ralph Kennel (TUM)

    Symmetrical Components

    Example: Transformer

    in a transformer the zero impedance Z0 depends on the design of the transformer

    U0 = Z0 I0

    in case of a star connection on primary and secondary side

    there cannot be any zero sequence current on the primary side (due to Krichhoffs law)

    the equivalent circuit contains the secondary side only

    the quantity of the mutual inductance depends on the design

    5-leg-core mutual inductance is higher

    (magnetic flux goes through the outer legs)

  • 28Prof. Dr.-Ing. Ralph Kennel (TUM)

    Symmetrical Components

    Example: Transformer

    in a transformer the zero impedance Z0 depends on the design of the transformer

    U0 = Z0 I0

    in case of a delta connection on primary and a star connection secondary side

    a zero sequence current on the primary side is possble

    the equivalent circuit contains primary and secondary side

    The quantity of the mutual inductance is negligible

    in comparison to the leakage inductances

  • 29Prof. Dr.-Ing. Ralph Kennel (TUM)

    Symmetrical Components

    Example: single phase load of a transformer

    Uu = Uz = Iz Z

    Iu = - Iz

    Iu = I1 + I2 + I0 = - Iz

    Iv = Iw = 0

    Uu = U1 + U2 + U0 = Uz

    U1 = U20u + Z1 I1

    U2 = Z2 I2

    U1 = Z0 I0

    we assume, that the

    grid source voltage contains

    a U20u component only

  • 30Prof. Dr.-Ing. Ralph Kennel (TUM)

    Symmetrical Components

    Example: single phase load of a transformer

    Uu = Uz = Iz Z

    Iu = - Iz

    Iu = I1 + I2 + I0 = - Iz

    Iv = Iw = 0

    Uu = U1 + U2 + U0 = Uz

    U1 = U20u + Z1 I1

    U2 = Z2 I2

    U1 = Z0 I0

    we assume, that the

    grid source voltage contains

    a U20u component only

    I1 = 1/3 (Iu + Iv a + Iv a)

    I1 = I2 = I0 = 1/3 Iu = - 1/3 Iz

    U1 = Z1 I1

    U2 = Z2 I2

    U0 = Z0 I0

    U1 = Uz = U20u + Z1 I1 + Z2 I2 + Z0 I0

    U1 = Uz = U20u + 1/3 (Z1 + Z2 + Z0) Iu

    Iu = - IzU1 = Uz = U20u - 1/3 (Z1 + Z2 + Z0) Iz

  • 31Prof. Dr.-Ing. Ralph Kennel (TUM)

    Symmetrical Components

    Example: single phase load of a transformer

    U1 = Uz = U20u + 1/3 (Z1 + Z2 + Z0) Iu

    U1 = Uz = U20u - 1/3 (Z1 + Z2 + Z0) Iz

    in case of a delta connection : Z1 = Z2 Z0

    in case of a double star connection :

    Z1 = Z2 Z0 and consequently :

    Z1 = Z2 1/3 (Z1 + Z2 + Z0)

    with respect to high 1/3 (Z1 + Z2 + Z0)

    U1 = Uz = U20u - 1/3 (Z1 + Z2 + Z0) Izwill be very low

  • 32Prof. Dr.-Ing. Ralph Kennel (TUM)

    Symmetrical Components

    Conclusion

    symmetrical components

    simplify the investigation

    in unsymmetrical loads

  • Thank you !