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    Fachgebiet

    HochspannungstechnikOvervoltage Protection and Insulation Coordination / Chapter 5 a - 1 -

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    Fachgebiet

    HochspannungstechnikOvervoltage Protection and Insulation Coordination / Chapter 5 a - 2 -

    The Harmonic Case: Standing Waves

    Each electromagnetic wave (in the free space/on a line)

    has a certain velocity of propagation.

    Reflections at the end of the line result in

    standing waves on the line.

    Dependence on time and location

    from:

    H.- G. Unger

    Elektromagnetische Wellen auf Leitungen

    Hthig- Verlag, Heidelberg, 1980

    ISBN 3- 7785- 0601- 3

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    Fachgebiet

    HochspannungstechnikOvervoltage Protection and Insulation Coordination / Chapter 5 a - 3 -

    The Harmonic Case: Standing Waves

    Advancing wave

    Advancing wave

    Regressing wave

    Regressing wave

    Resultingstanding wave

    Resultingstanding wave

    1,0

    0,7

    - 1,0

    - 0,7

    z

    z

    z

    t

    t

    t

    1,0

    0,7

    1,0

    0,7

    1,0

    0,7

    - 0,7

    - 1,0

    - 0,7

    - 1,0

    - 0,7

    - 1,0

    a)

    a) b)

    b) c)

    c)

    a)

    a)

    b)

    b)

    c)

    c)

    As a function of location:

    As a function of location: As a function of time:

    As a function of time:

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    Fachgebiet

    HochspannungstechnikOvervoltage Protection and Insulation Coordination / Chapter 5 a - 4 -

    The Transient Case (Surges):Traveling Waves

    Each electromagnetic wave (in the free space/on a line)

    has a certain velocity of propagation.

    Changes of voltage and current result in

    traveling waves on the line.

    Dependence on time and location

    u

    t

    0 1 s 2 s 3 s

    u

    x

    0 300 m 600 m 900 m 1200 m

    Example: lightning overvoltage on an OHL

    Dependence on time

    at a certain location

    Dependence on location

    at a certain time instant

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    Fachgebiet

    HochspannungstechnikOvervoltage Protection and Insulation Coordination / Chapter 5 a - 5 -

    Traveling Waves Example: Surge Starts at t= 0 and z= 0

    t

    0 1 s 2 s 3 s

    z

    0 300 m 600 m 900 m

    u

    t= 1 st= 1 s

    t= 2 st= 2 s

    t= 3 st= 3 s

    t= 4 st= 4 s

    u

    4 s

    z= 0 mz= 0 m

    z= 300 mz= 300 m

    z= 600 mz= 600 m

    z= 900 mz= 900 m

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    Fachgebiet

    HochspannungstechnikOvervoltage Protection and Insulation Coordination / Chapter 5 a - 6 -

    Traveling Waves Example: Lightning Overvoltage on an OH Line

    300 m/s

    Span length (typ.): 300 mTime to crest (typ.): 1 s

    Time to half value (typ.): (10100) s

    0 20 40 60 80 100

    In the (theoretical) case of a standard lightning impulse voltage 1.2/50:

    time / s

    voltage

    /kV

    location / km

    0 6 12 18 24 30

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    Fachgebiet

    HochspannungstechnikOvervoltage Protection and Insulation Coordination / Chapter 5 a - 7 -

    Traveling Waves

    Traveling waves to be taken into account whenever the change

    in voltage or current takes place in a time duration of the same

    order of magnitude as the propagation time electrically long line

    Velocity of propagation in air: v= c0 = 300 m/s

    Time for traveling along one span of a HV-OHL (300 m): 1 s

    Time for traveling along an OHL of 300 km length: 1 ms

    Spatial length of a lightning overvoltage surge (100 s): 30 km

    Spatial length of the front of a lightning overvoltage surge (1s): 300 m

    Spatial length of a switching overvoltage surge (5 ms): 1500 km

    Spatial length of the front of switching overvolage surge (250 s): 75 km

    Spatial length of one half-period of 50-Hz voltage (10 ms): 3000 km

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    Fachgebiet

    HochspannungstechnikOvervoltage Protection and Insulation Coordination / Chapter 5 a - 8 -

    Traveling Waves

    Velocity of propagation in air: v= c0 = 300 m/sVelocity of propagation in a measuring cable: v= 150 m/s

    Impact on measurement of changes in sub-microsecond range

    Example: fast voltage change voltage breakdown/flashover

    t= 100 ns t= 10 ns

    in the test circuit (air)

    along the cable

    Spatial length of voltage ramp (-du/dt)

    30 m 3 m

    15 m 1,5 m

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    Fachgebiet

    HochspannungstechnikOvervoltage Protection and Insulation Coordination / Chapter 5 a - 9 -

    Traveling Waves

    Occurrence of traveling waves / Making use of traveling wave effects

    energization of a unloaded line

    propagation of lightning overvoltages on lines

    propagation of very fast transients in GIS

    separation effects / protective zone of surge arresters

    generating and measuring of LI voltages generating rectangular current impulses (energy tests on surge arresters)

    fault location on cables

    fault location on light wave guides / optical fibers

    location of partial discharges in GIS

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    Fachgebiet

    HochspannungstechnikOvervoltage Protection and Insulation Coordination / Chapter 5 a - 10 -

    Traveling Waves - Laws of Propagation

    General electrical equivalent circuit of a line element

    R ... Resistance

    L ... Inductance

    G ... Parallel conductance

    C ... Capacitance

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    Fachgebiet

    HochspannungstechnikOvervoltage Protection and Insulation Coordination / Chapter 5 a - 11 -

    Traveling Waves - Laws of Propagation

    Electrical equivalent circuit of a loss-less line element

    ( d ) ' du i

    u u x L xx t

    + = '

    u i

    Lx t

    =

    ( d ) ' di u

    i i x C xx t

    + =

    '

    i uC

    x t

    =

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    Fachgebiet

    HochspannungstechnikOvervoltage Protection and Insulation Coordination / Chapter 5 a - 12 -

    Traveling Waves - Laws of Propagation

    'u i

    Lx t

    =

    'i u

    C

    x t

    =

    Partial derivative with respect tox:2 2

    2'

    u iL

    x t x

    =

    2 2

    2'

    i uC

    t x t

    =

    Partial derivative with respect to t:

    2 2

    2 2' 'u uL C

    x t =

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    Fachgebiet

    HochspannungstechnikOvervoltage Protection and Insulation Coordination / Chapter 5 a - 13 -

    Traveling Waves - Laws of Propagation

    'u i

    Lx t

    =

    'i u

    C

    x t

    =

    Partial derivative with respect to t:

    Partial derivative with respect tox:

    2 2

    2'

    u iL

    x t t

    =

    2 2

    2'

    i uC

    x x t

    =

    2 2

    2 2' '

    i iL C

    x t

    =

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    Fachgebiet

    HochspannungstechnikOvervoltage Protection and Insulation Coordination / Chapter 5 a - 14 -

    Traveling Waves - Laws of Propagation

    2 2

    2 2' '

    i iL C

    x t

    =

    2 2

    2 2' '

    u uL C

    x t

    =

    General wave equations of the loss-less line

    General solution acc. to dAlembert (1717-1783):

    1 2( , ) ( ) ( ) v ru x t f x vt f x vt u u= + + = +

    1 2

    1 1( , ) ( ) ( ) v ri x t f x vt f x vt i i

    Z Z= + = +

    uv ur

    iv ir

    1' '

    vL C

    =

    '

    '

    LZ

    C=

    Velocity of propagation

    Surge impedance

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    Fachgebiet

    HochspannungstechnikOvervoltage Protection and Insulation Coordination / Chapter 5 a - 15 -

    Traveling Waves - Laws of Propagation

    Both voltage and current are composed of a forward and a backward wave.

    A positive forward voltage wave is linked to apositive forward current wave:

    A positive backward voltage wave is linked tonegative backward current wave:

    uv

    ivx

    ur

    ir

    x

    1 2( , ) ( ) ( ) v ru x t f x vt f x vt u u= + + = +

    uv ur

    1 21 1( , ) ( ) ( ) v ri x t f x vt f x vt i i

    Z Z= + = +

    iv ir

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    Fachgebiet

    HochspannungstechnikOvervoltage Protection and Insulation Coordination / Chapter 5 a - 16 -

    Traveling Waves - Laws of Propagation

    Wanderwellenausbreitung beim pltzlichen Abflieen einer freigewordenen Influenzladung auf einer

    Freileitung; linke Bildhlfte: zeitliche Entwicklung der Felder; rechte Bildhlfte: Wanderwellen auf der Leitung

    Traveling waves after sudden release of influenced charges on an OHL - left: development with time of fields

    right: traveling waves on the line (Note: urand irhave the same traveling direction, but the measured current is negative.)

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    Fachgebiet

    HochspannungstechnikOvervoltage Protection and Insulation Coordination / Chapter 5 a - 17 -

    Velocity of propagation

    Traveling Waves - Laws of Propagation

    d

    r r

    0' lnrd

    Lr

    =

    0'

    ln

    rC

    dr

    =

    with 0 = 1.25610-6 Vs/Am Permeability of vacuum

    0 = 8.85410-12As/Vm Permittivity of vacuum

    c0 300 m/s Velocity of light

    0

    0 0

    1 1 1

    r r r r

    c

    = = 1

    ' '

    v

    L C

    =Velocity of propagation

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    Fachgebiet

    HochspannungstechnikOvervoltage Protection and Insulation Coordination / Chapter 5 a - 18 -

    Traveling Waves - Laws of Propagation

    As r= 1: 01

    r

    v c

    =

    Air: r= 1.0006 1 vair= c0 = 300 m/s

    Cable: r= 2.5 ... 4 vcable = 190 m/s ... 150 m/s

    exclusively dependent on dielectrics!

    Velocity of propagation

    with 0 = 1.25610-6 Vs/Am Permeability of vacuum

    0 = 8.85410-12As/Vm Permittivity of vacuum

    c0 300 m/s Velocity of light

    0

    0 0

    1 1 1

    r r r r

    c

    = = 1

    ' 'v

    L C=Velocity of propagation

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    Fachgebiet

    HochspannungstechnikOvervoltage Protection and Insulation Coordination / Chapter 5 a - 19 -

    Surge impedance

    Traveling Waves - Laws of Propagation

    d

    r r

    0' lnrd

    Lr

    =

    0'

    ln

    rC

    dr

    =

    Surge impedance0

    0

    1

    lnr

    r

    d

    r

    =

    depends on dielectrics!

    depends on geometry! does not depend on location!

    '

    '

    L

    Z C=

    with 0 = 1.25610-6 Vs/Am Permeability of vacuum

    0 = 8.85410-12As/Vm Permittivity of vacuum

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    Fachgebiet

    HochspannungstechnikOvervoltage Protection and Insulation Coordination / Chapter 5 a - 20 -

    Surge impedance

    Traveling Waves - Laws of Propagation

    0

    0

    1lnr

    r

    dZ

    r

    =

    Figures:

    OHL 420 kV, quadruple bundle: Z 250

    OHL 123 kV, single conductor: Z 400

    GIS, GIL: Z 60

    polymeric (XLPE) hv-cable: Z 40

    polymeric (XLPE) mv-cable: Z< 40 measuring (coaxial) cable (RG-58): Z 50

    power transformer winding: Z 102 ... 104

    with 0

    = 1.25610-6 Vs/Am Permeability of vacuum

    0 = 8.85410-12As/Vm Permittivity of vacuum

    FM antenna cable: Z 75

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    Fachgebiet

    HochspannungstechnikOvervoltage Protection and Insulation Coordination / Chapter 5 a - 21 -

    Traveling Waves - Reflection and Refraction

    uv

    iv

    Leitung 1 Leitung 2

    Z1 Z2

    uv

    iv

    Leitung 1 Leitung 2

    Z1 Z2

    uv = Z1iv

    uv and iv suffer changes at the location of discontinuity

    Refraction (forward waves proceed at increased or reduced amplitudes)

    Reflection (waves travel back from the location of discontinuity)

    line 1 line 2

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    Fachgebiet

    HochspannungstechnikOvervoltage Protection and Insulation Coordination / Chapter 5 a - 22 -

    Traveling Waves - Reflection and Refraction

    u1v, i1v

    i1

    Leitung 1 Leitung 2

    Z1 Z2

    i2

    u1 u2

    u1v, i1v

    i1

    Leitung 2

    i2

    u1 u2

    u1 = u2i1 = i2

    u1 = u1v + u1ri1 = i1v + i1r

    u2 = u2v + u2r= u2vi2 = i2v + i2r= i2v

    u1v + u1r= u2v

    i1v + i1r= i2v

    line 1 line 2

    =

    =

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    Fachgebiet

    HochspannungstechnikOvervoltage Protection and Insulation Coordination / Chapter 5 a - 23 -

    Traveling Waves - Reflection and Refraction

    u1v, i1v

    i1

    Leitung 1 Leitung 2

    Z1 Z2

    i2

    u1 u2

    u1v, i1v

    i1

    Leitung 2

    i2

    u1 u2

    u1 = u2i1 = i2

    u1 = u1v + u1ri1 = i1v + i1r

    u2 = u2v + u2r= u2vi2 = i2v + i2r= i2v

    u1v + u1r= u2v

    i1v + i1r= i2v1v 2 v1r

    1 1 2

    u uuZ Z Z

    = 11v 1r 2 v2

    Zu u uZ

    =

    2v 2u

    1v 1 2

    2u Zb

    u Z Z

    = =

    +

    line 1 line 2

    =

    =

    1.

    2.

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    Fachgebiet

    HochspannungstechnikOvervoltage Protection and Insulation Coordination / Chapter 5 a - 24 -

    Traveling Waves - Reflection and Refraction

    2v 2u

    1v 1 2

    2u Zb

    u Z Z

    = =

    +voltage refraction factorvoltage refraction factor

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    Fachgebiet

    HochspannungstechnikOvervoltage Protection and Insulation Coordination / Chapter 5 a - 25 -

    Traveling Waves - Reflection and Refraction

    2 v u 12 v 1v 1v u

    2 2 2

    u b Zi u i b

    Z Z Z= = = 2v 2

    u

    1v 1 2

    2u Zb

    u Z Z

    = =

    +

    2 v 1 1u i

    1v 2 1 2

    2i Z Zb b

    i Z Z Z

    = = =

    +

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    Fachgebiet

    HochspannungstechnikOvervoltage Protection and Insulation Coordination / Chapter 5 a - 26 -

    Traveling Waves - Reflection and Refraction

    2 v 1i

    1v 1 2

    2i Zb

    i Z Z

    = =

    +current refraction factorcurrent refraction factor

    2v 2u

    1v 1 2

    2u Zb

    u Z Z

    = =

    +voltage refraction factorvoltage refraction factor

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    Fachgebiet

    HochspannungstechnikOvervoltage Protection and Insulation Coordination / Chapter 5 a - 27 -

    Traveling Waves - Reflection and Refraction

    u1v, i1v

    i1

    Leitung 1 Leitung 2

    Z1 Z2

    i2

    u1 u2

    u1v, i1v

    i1

    Leitung 2

    i2

    u1 u2

    u1 = u2i1 = i2

    u1 = u1v + u1ri1 = i1v + i1r

    u2 = u2v + u2r= u2vi2 = i2v + i2r= i2v

    u1v + u1r= u2v

    i1v + i1r= i2v

    line 1 line 2

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    Fachgebiet

    HochspannungstechnikOvervoltage Protection and Insulation Coordination / Chapter 5 a - 28 -

    Traveling Waves - Reflection and Refraction

    1r 2 v 1v u 1v 1v 1v u 1v u( 1)u u u b u u u b u r = = = =

    1r 2 1u u

    1v 2 1

    1u Z Zr bu Z Z

    = = =+

    u1v + u1r= u2v

    2vu

    1v

    ub

    u=

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    Fachgebiet

    HochspannungstechnikOvervoltage Protection and Insulation Coordination / Chapter 5 a - 29 -

    Traveling Waves - Reflection and Refraction

    voltage reflection factorvoltage reflection factor1r 2 1

    u u

    1v 2 1

    1u Z Z

    r bu Z Z

    = = =

    +

    2 v 1i

    1v 1 2

    2i Zb

    i Z Z

    = =

    +current refraction factorcurrent refraction factor

    2v 2u

    1v 1 2

    2u Zb

    u Z Z

    = =

    +voltage refraction factorvoltage refraction factor

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    Fachgebiet

    HochspannungstechnikOvervoltage Protection and Insulation Coordination / Chapter 5 a - 30 -

    Traveling Waves - Reflection and Refraction

    u1v, i1v

    i1

    Leitung 1 Leitung 2

    Z1 Z2

    i2

    u1 u2

    u1v, i1v

    i1

    Leitung 2

    i2

    u1 u2

    u1 = u2i1 = i2

    u1 = u1v + u1ri1 = i1v + i1r

    u2 = u2v + u2r= u2vi2 = i2v + i2r= i2v

    u1v + u1r= u2v

    i1v + i1r= i2v

    line 1 line 2

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    Fachgebiet

    HochspannungstechnikOvervoltage Protection and Insulation Coordination / Chapter 5 a - 31 -

    Traveling Waves - Reflection and Refraction

    1r 2v 1v i 1v 1v 1v i 1v i( 1)i i i b i i i b i r = = = =

    1r 1 2i i

    1v 1 2

    1i Z Z

    r bi Z Z

    = = =+

    i1v + i1r= i2v

    2 vi

    1v

    ib

    i

    =

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    Fachgebiet

    HochspannungstechnikOvervoltage Protection and Insulation Coordination / Chapter 5 a - 32 -

    Traveling Waves - Reflection and Refraction

    current reflection factorcurrent reflection factor1r 1 2i i1v 1 2

    1i Z Z

    r bi Z Z

    = = =

    +

    voltage reflection factorvoltage reflection factor1r 2 1

    u u

    1v 2 1

    1u Z Z

    r bu Z Z

    = = =

    +

    2 v 1i

    1v 1 2

    2i Zb

    i Z Z

    = =

    +current refraction factorcurrent refraction factor

    2v 2u

    1v 1 2

    2u Zb

    u Z Z

    = =

    +voltage refraction factorvoltage refraction factor

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    Fachgebiet

    HochspannungstechnikOvervoltage Protection and Insulation Coordination / Chapter 5 a - 33 -

    Traveling Waves - Reflection and Refraction at End of Line

    u1v, i1v

    Leitung 1

    Z1

    Ri u

    u1v, i1v

    Leitung 1

    Z1

    Ri uline 1

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    Fachgebiet

    HochspannungstechnikOvervoltage Protection and Insulation Coordination / Chapter 5 a - 34 -

    a) end = open circuit R

    ru = 1 u1r= u1v u = 2u1v

    ri = 1 i1r= i1v i= 0

    Doubling of voltage at lines end, current = zero

    u1v, i1v

    Leitung 1

    Z1

    Ri u

    u1v, i1v

    Leitung 1

    Z1

    Ri uline 1

    Traveling Waves - Reflection and Refraction at End of Line

    u

    t11

    21

    1u1v

    2u1v

    t11 21

    1i1v

    2i1v

    i

    0

    0

    line entranceu

    t11

    21

    1u1v

    2u1v

    t11 21

    1i1v

    2i1v

    i

    0

    0

    u

    t11

    21

    1u1v

    2u1v

    t11 21

    1i1v

    2i1v

    i

    0

    0

    line entrance u

    t11

    21

    1u1v

    2u1v

    t11 21

    1i1v

    2i1v

    i

    0

    0

    end of lineu

    t11

    21

    1u1v

    2u1v

    t11 21

    1i1v

    2i1v

    i

    0

    0

    u

    t11

    21

    1u1v

    2u1v

    t11 21

    1i1v

    2i1v

    i

    0

    0

    end of line

    l fl d f E d f

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    Fachgebiet

    HochspannungstechnikOvervoltage Protection and Insulation Coordination / Chapter 5 a - 35 -

    b) end = short-circuit R= 0

    ru = 1 u1r= u1v u = 0

    ri= 1 i1r= i1v i= 2i1v

    Doubling of current at lines end, voltage = zero

    u1v, i1v

    Leitung 1

    Z1

    Ri u

    u1v, i1v

    Leitung 1

    Z1

    Ri uline 1

    Traveling Waves - Reflection and Refraction at End of Line

    u

    t11

    21

    1u1v

    2u1v

    t11 21

    1i1v

    2i1v

    i

    0

    0

    line entranceu

    t11

    21

    1u1v

    2u1v

    t11 21

    1i1v

    2i1v

    i

    0

    0

    u

    t11

    21

    1u1v

    2u1v

    t11 21

    1i1v

    2i1v

    i

    0

    0

    line entrance u

    t11

    21

    1u1v

    2u1v

    t11 21

    1i1v

    2i1v

    i

    0

    0

    end of lineu

    t11

    21

    1u1v

    2u1v

    t11 21

    1i1v

    2i1v

    i

    0

    0

    u

    t11

    21

    1u1v

    2u1v

    t11 21

    1i1v

    2i1v

    i

    0

    0

    end of line

    T li W R fl i d R f i E d f Li

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    Fachgebiet

    HochspannungstechnikOvervoltage Protection and Insulation Coordination / Chapter 5 a - 36 -

    c) matched end R= Z

    ru = 0 u1r= 0 u = u1v

    ri= 0 i1r= 0 i= i1v

    Neither refraction nor reflection

    u1v, i1v

    Leitung 1

    Z1

    Ri u

    u1v, i1v

    Leitung 1

    Z1

    Ri uline 1

    Traveling Waves - Reflection and Refraction at End of Line

    u

    t11

    21

    1u1v

    2u1v

    t11 21

    1i1v

    2i1v

    i

    0

    0

    line entranceu

    t11

    21

    1u1v

    2u1v

    t11 21

    1i1v

    2i1v

    i

    0

    0

    u

    t11

    21

    1u1v

    2u1v

    t11 21

    1i1v

    2i1v

    i

    0

    0

    line entrance u

    t11

    21

    1u1v

    2u1v

    t11 21

    1i1v

    2i1v

    i

    0

    0

    end of lineu

    t11

    21

    1u1v

    2u1v

    t11 21

    1i1v

    2i1v

    i

    0

    0

    u

    t11

    21

    1u1v

    2u1v

    t11 21

    1i1v

    2i1v

    i

    0

    0

    end of line

    T li W R fl ti d R f ti t E d f Li

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    Fachgebiet

    HochspannungstechnikOvervoltage Protection and Insulation Coordination / Chapter 5 a - 37 -

    d) end = any real-valued resistance Z1 R

    ru = 0 1 u1r= 0 u1v u = (12)u1v

    ri= 0 -1 i1r= 0 -i1v i= 0 i1v

    Increase of voltage and decrease of current at lines end

    u1v, i1v

    Leitung 1

    Z1

    Ri u

    u1v, i1v

    Leitung 1

    Z1

    Ri uline 1

    Traveling Waves - Reflection and Refraction at End of Line

    u

    t11

    21

    1u1v

    2u1v

    t11 21

    1i1v

    2i1v

    i

    0

    0

    line entranceu

    t11

    21

    1u1v

    2u1v

    t11 21

    1i1v

    2i1v

    i

    0

    0

    line entrance u

    t11

    21

    1u1v

    2u1v

    t11 21

    1i1v

    2i1v

    i

    0

    0

    end of lineu

    t11

    21

    1u1v

    2u1v

    t11 21

    1i1v

    2i1v

    i

    0

    0

    end of line

    T li W R fl ti d R f ti t E d f Li

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    Fachgebiet

    HochspannungstechnikOvervoltage Protection and Insulation Coordination / Chapter 5 a - 38 -

    e) end = any real-valued resistance 0 R Z1

    ru = 0 -1 u1r= 0 -u1v u = 0 u1v

    ri= 0 1 i1r= 0 i1v i= (12)i1v

    Decrease of voltage and increase of current at lines end

    u1v, i1v

    Leitung 1

    Z1

    Ri u

    u1v, i1v

    Leitung 1

    Z1

    Ri uline 1

    Traveling Waves - Reflection and Refraction at End of Line

    u

    t11

    21

    1u1v

    2u1v

    t11 21

    1i1v

    2i1v

    i

    0

    0

    line entranceu

    t11

    21

    1u1v

    2u1v

    t11 21

    1i1v

    2i1v

    i

    0

    0

    line entrance u

    t11

    21

    1u1v

    2u1v

    t11 21

    1i1v

    2i1v

    i

    0

    0

    end of lineu

    t11

    21

    1u1v

    2u1v

    t11 21

    1i1v

    2i1v

    i

    0

    0

    end of line

    T lin W s R fl ti n nd R f ti n t End f Lin

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    Fachgebiet

    HochspannungstechnikOvervoltage Protection and Insulation Coordination / Chapter 5 a - 39 -

    f) capacitor at line's end Z from 0 to

    Exponential functions of voltage and current at lines end

    u1v, i1v

    Leitung 1

    Z1

    Ri u

    u1v, i1v

    Leitung 1

    Z1

    Ri uline 1

    Traveling Waves - Reflection and Refraction at End of Line

    C

    ( )11( ) 2 1 e t Z Cvu t u = 11( ) 2 et Z C

    vi t i

    =

    u

    t11

    21

    1u1v

    2u1v

    t11 21

    1i1v

    2i1v

    i

    0

    0

    line entranceu

    t11

    21

    1u1v

    2u1v

    t11 21

    1i1v

    2i1v

    i

    0

    0

    u

    t11

    21

    1u1v

    2u1v

    t11 21

    1i1v

    2i1v

    i

    0

    0

    line entrance u

    t11

    21

    1u1v

    2u1v

    t11 21

    1i1v

    2i1v

    i

    0

    0

    end of lineu

    t11

    21

    1u1v

    2u1v

    t11 21

    1i1v

    2i1v

    i

    0

    0

    u

    t11

    21

    1u1v

    2u1v

    t11 21

    1i1v

    2i1v

    i

    0

    0

    end of line

    Traveling Waves Reflection and Refraction at End of Line

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    Fachgebiet

    HochspannungstechnikOvervoltage Protection and Insulation Coordination / Chapter 5 a - 40 -

    g) inductor at line's end Z from to 0

    Exponential functions of voltage and current at lines end

    u1v, i1v

    Leitung 1

    Z1

    Ri u

    u1v, i1v

    Leitung 1

    Z1

    Ri uline 1

    Traveling Waves - Reflection and Refraction at End of Line

    L

    ( )11( ) 2 1 e tL Zvi t i = 11( ) 2 e tL Zvu t u =

    u

    t11 21

    1u1v

    2u1v

    t11 21

    1i1v

    2i1v

    i

    0

    0

    line entranceu

    t11 21

    1u1v

    2u1v

    t11 21

    1i1v

    2i1v

    i

    0

    0

    u

    t11 21

    1u1v

    2u1v

    t11 21

    1i1v

    2i1v

    i

    0

    0

    line entrance u

    t11 21

    1u1v

    2u1v

    t11 21

    1i1v

    2i1v

    i

    0

    0

    end of lineu

    t11 21

    1u1v

    2u1v

    t11 21

    1i1v

    2i1v

    i

    0

    0

    u

    t11 21

    1u1v

    2u1v

    t11 21

    1i1v

    2i1v

    i

    0

    0

    end of line

    Traveling Waves Reflection and Refraction at End of Line

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    Fachgebiet

    HochspannungstechnikOvervoltage Protection and Insulation Coordination / Chapter 5 a - 41 -

    Traveling Waves - Reflection and Refraction at End of Line

    Traveling Waves Reflection and Refraction at End of Line

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    Fachgebiet

    HochspannungstechnikOvervoltage Protection and Insulation Coordination / Chapter 5 a - 42 -

    matched: R= Z

    open circuit

    short-circuit

    Traveling Waves - Reflection and Refraction at End of Line

    Traveling Waves - Reflection and Refraction at End of Line

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    Fachgebiet

    HochspannungstechnikOvervoltage Protection and Insulation Coordination / Chapter 5 a - 43 -

    Traveling wave equivalent electrical circuit

    2uv

    Z1

    R L C2uv

    Z1

    R L C

    ik = 2uv/Z1 = 2iv2uv

    2iv

    u

    i

    Z1

    Traveling Waves - Reflection and Refraction at End of Line

    Traveling Waves Bewley Diagram

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    Fachgebiet

    HochspannungstechnikOvervoltage Protection and Insulation Coordination / Chapter 5 a - 44 -

    Traveling Waves Bewley Diagram

    2

    3

    4

    2

    3

    4

    local axislocal axis

    time axis at location "A"time axis at location "A"

    Factors of reflectionand refractionFactors of reflectionand refraction

    Incoming

    voltage surge

    Incoming

    voltage surge

    11 22 33loc

    al-timeaxes

    local-timeaxes

    local-tim

    e axes

    local-tim

    e axes

    time axis at location "B"time axis at location "B"

    z

    Traveling Waves Bewley Diagram

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    Fachgebiet

    HochspannungstechnikOvervoltage Protection and Insulation Coordination / Chapter 5 a - 45 -

    Traveling Waves Bewley Diagram

    Traveling Waves Bewley Diagram

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    Fachgebiet

    HochspannungstechnikOvervoltage Protection and Insulation Coordination / Chapter 5 a - 46 -

    Traveling Waves Bewley Diagram

    Traveling Waves Bewley Diagram

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    Fachgebiet

    HochspannungstechnikOvervoltage Protection and Insulation Coordination / Chapter 5 a - 47 -

    Traveling Waves Bewley Diagram

    Traveling Waves Bewley Diagram

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    Fachgebiet

    HochspannungstechnikOvervoltage Protection and Insulation Coordination / Chapter 5 a - 48 -

    Traveling Waves Bewley Diagram

    Traveling Waves Bewley Diagram

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    Fachgebiet

    HochspannungstechnikOvervoltage Protection and Insulation Coordination / Chapter 5 a - 49 -

    g y g m

    Traveling Waves Application Example: Oscillations

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    Fachgebiet

    HochspannungstechnikOvervoltage Protection and Insulation Coordination / Chapter 5 a - 50 -

    g pp p

    line with surge impedance Z2and propagation time

    u1 u2Ri

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    Fachgebiet

    HochspannungstechnikOvervoltage Protection and Insulation Coordination / Chapter 5 a - 51 -

    line with surge impedance Z2and propagation time

    u1 u2Ri

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    Fachgebiet

    HochspannungstechnikOvervoltage Protection and Insulation Coordination / Chapter 5 a - 52 -

    line with surge impedance Z2and propagation time

    g pp p

    u1 u2Ri

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    Fachgebiet

    HochspannungstechnikOvervoltage Protection and Insulation Coordination / Chapter 5 a - 53 -

    line with surge impedance Z

    and propagation time u1 u2Ri

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    Fachgebiet

    HochspannungstechnikOvervoltage Protection and Insulation Coordination / Chapter 5 a - 54 -

    Occurrence of traveling waves / Making use of traveling wave effects

    energization of a unloaded line

    propagation of lightning overvoltages on lines propagation of very fast transients in GIS

    separation effects / protective zone of surge arresters

    generating and measuring of LI voltages

    generating rectangular current impulses (energy tests on surge arresters)

    fault location on cables

    fault location on light wave guides / optical fibers

    location of partial discharges in GIS

    Making Use of Traveling Waves Effects

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    Fachgebiet

    HochspannungstechnikOvervoltage Protection and Insulation Coordination / Chapter 5 a - 55 -

    Long duration current impulse generator with LC distributed network

    t [ms]

    U[

    kV]

    -4-3-2

    -1012345

    6

    0 0,5 1 1,5 2 2,5 3 3,5 4

    -0,200,2

    0,40,60,811,21,41,6

    1,8

    I[kA]

    Long duration current impulse(2,4 ms, 1200 A)

    =

    Making Use of Traveling Waves Effects

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    Fachgebiet

    HochspannungstechnikOvervoltage Protection and Insulation Coordination / Chapter 5 a - 56 -

    Rdut = ZZ,

    Rdut = ZZ,

    t= 0

    U

    I

    Long duration current impulse generator with LC distributed network

    Ucharge

    Ucharge

    Making Use of Traveling Waves Effects

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    Fachgebiet

    HochspannungstechnikOvervoltage Protection and Insulation Coordination / Chapter 5 a - 57 -

    t> 0

    Uv = U0/2

    Iv = I0/2Z

    Rdut = ZZ,

    Rdut = ZZ,

    U

    I

    Long duration current impulse generator with LC distributed network

    Ucharge

    Ucharge

    Making Use of Traveling Waves Effects

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    Fachgebiet

    HochspannungstechnikOvervoltage Protection and Insulation Coordination / Chapter 5 a - 58 -

    t=

    Uv = U0/2

    Iv = I0/2Z

    Rdut = ZZ,

    Rdut = ZZ,

    U

    I

    Long duration current impulse generator with LC distributed network

    Ucharge

    Ucharge

    Making Use of Traveling Waves Effects

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    Fachgebiet

    HochspannungstechnikOvervoltage Protection and Insulation Coordination / Chapter 5 a - 59 -

    t>

    Uv = U0/2

    Iv = I0/2Z

    Rdut = ZZ,

    Rdut = ZZ,

    U

    I

    Long duration current impulse generator with LC distributed network

    Ucharge

    Ucharge

    Making Use of Traveling Waves Effects

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    Fachgebiet

    HochspannungstechnikOvervoltage Protection and Insulation Coordination / Chapter 5 a - 60 -

    t= 2

    Uv = U0/2

    Iv = I0/2Z

    Ucharge Rdut = ZZ,

    Rdut = ZZ,

    U

    I

    Long duration current impulse generator with LC distributed network

    Ucharge

    Making Use of Traveling Waves Effects

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    Fachgebiet

    HochspannungstechnikOvervoltage Protection and Insulation Coordination / Chapter 5 a - 61 -

    t [ms]

    U[

    kV]

    -4-3

    -2-1012

    3456

    0 0,5 1 1,5 2 2,5 3 3,5 4

    -0,20

    0,20,40,60,81

    1,21,41,61,8

    I[kA]

    Long duration current impulse(2.4 ms, 1200 A)

    Long duration current impulse generator with LC distributed network

    Traveling Waves Line Discharge

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    Fachgebiet

    HochspannungstechnikOvervoltage Protection and Insulation Coordination / Chapter 5 a - 62 -