laplace_transform.pdf

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9/28/2012 1 The Laplace Transform EE 602 CIRCUIT ANALYSIS Laila Rosemaizura Binti Yaakop The Laplace Transform The Laplace Transform of a function, f(t), is defined as; 0 ) ( ) ( )] ( [ dt e t f s F t f L st The Inverse Laplace Transform is defined by j j ts ds e s F j t f s F L ) ( 2 1 ) ( )] ( [ 1 *notes Eq A Eq B Laila Rosemaizura Binti Yaakop

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  • 9/28/2012

    1

    The Laplace Transform

    EE 602 CIRCUIT ANALYSIS

    Laila Rosemaizura Binti Yaakop

    The Laplace Transform

    The Laplace Transform of a function, f(t), is defined as;

    0

    )()()]([ dtetfsFtfL st

    The Inverse Laplace Transform is defined by

    j

    j

    tsdsesF

    jtfsFL )(

    2

    1)()]([1

    *notes

    Eq A

    Eq B

    Laila Rosemaizura Binti Yaakop

  • 9/28/2012

    2

    The Laplace Transform

    We generally do not use Eq B to take the inverse Laplace. However,

    this is the formal way that one would take the inverse. To use

    Eq B requires a background in the use of complex variables and

    the theory of residues. Fortunately, we can accomplish the same

    goal (that of taking the inverse Laplace) by using partial fraction

    expansion and recognizing transform pairs.

    *notes Laila Rosemaizura Binti Yaakop

    The Laplace Transform

    Example 1 : Laplace Transform of the unit step.

    *notes

    |0

    0

    11)]([ stst e

    sdtetuL

    stuL

    1)]([

    The Laplace Transform of a unit step is:

    s

    1Laila Rosemaizura Binti Yaakop

  • 9/28/2012

    3

    The Laplace Transform

    An important point to remember:

    )()( sFtf

    The above is a statement that f(t) and F(s) are

    transform pairs. What this means is that for

    each f(t) there is a unique F(s) and for each F(s)

    there is a unique f(t).

    Laila Rosemaizura Binti Yaakop

    The Laplace Transform

    Building a transform pairs:

    eL(

    e

    tasstatatdtedteetueL

    0

    )(

    0

    )]([

    asas

    etueL

    stat 1

    )()]([ |

    0

    astue

    at 1)(A transformpair Laila Rosemaizura Binti Yaakop

  • 9/28/2012

    4

    The Laplace Transform

    Building transform pairs:

    0

    )]([ dttettuL st

    0 00| vduuvudv

    u = t

    dv = e-stdt

    2

    1)(

    sttu

    A transform

    pairLaila Rosemaizura Binti Yaakop

    The Laplace Transform

    Building transform pairs:

    22

    0

    11

    2

    1

    2

    )()][cos(

    ws

    s

    jwsjws

    dteee

    wtLst

    jwtjwt

    22)()cos(

    ws

    stuwt A transform

    pairLaila Rosemaizura Binti Yaakop

  • 9/28/2012

    5

    Table of Laplace Transform

    ____________________________________

    )()( sFtf

    f(t) F(s)

    1

    2

    !

    1

    1

    1)(

    1)(

    n

    n

    st

    s

    nt

    st

    ase

    stu

    t

    Laila Rosemaizura Binti Yaakop

    f(t) F(s)

    22

    22

    1

    2

    )cos(

    )sin(

    )(

    !

    1

    ws

    swt

    ws

    wwt

    as

    net

    aste

    n

    atn

    at

    Laila Rosemaizura Binti Yaakop

  • 9/28/2012

    6

    The Laplace Transform

    Time Shift

    0 0

    )( )()(

    ,.,0,

    ,

    )()]()([

    dxexfedxexf

    SoxtasandxatAs

    axtanddtdxthenatxLet

    eatfatuatfL

    sxasaxs

    a

    st

    )()]()([ sFeatuatfL asLaila Rosemaizura Binti Yaakop

    Theorems of Laplace Transform

    First Shift (Frequency Shift)

    0

    )(

    0

    )()(

    )]([)]([

    asFdtetf

    dtetfetfeL

    tas

    statat

    )()]([ asFtfeL at

    Laila Rosemaizura Binti Yaakop

  • 9/28/2012

    7

    Example: Using First Shift Theoerm(Frequency Shift)

    Find the L[e-atcos(wt)]

    In this case, f(t) = cos(wt) so,

    22

    22

    )(

    )()(

    )(

    was

    asasFand

    ws

    ssF

    22 )()(

    )()]cos([

    was

    aswteL

    at

    Laila Rosemaizura Binti Yaakop

    Time Integration:

    The property is:

    stst

    t

    st

    t

    es

    vdtedv

    and

    dttfdudxxfuLet

    partsbyIntegrate

    dtedxxfdttfL

    1,

    )(,)(

    :

    )()(

    0

    0 00

    Laila Rosemaizura Binti Yaakop

  • 9/28/2012

    8

    The Laplace Transform

    Time Integration:

    Making these substitutions and carrying out

    The integration shows that

    )(1

    )(1

    )(00

    sFs

    dtetfs

    dttfL st

    Laila Rosemaizura Binti Yaakop

    The Laplace Transform

    Time Differentiation:

    If the L[f(t)] = F(s), we want to show:

    )0()(])(

    [ fssFdt

    tdfL

    Integrate by parts:

    )(),()(

    ,

    tfvsotdfdtdt

    tdfdv

    anddtsedueustst

    *note

    Laila Rosemaizura Binti Yaakop

  • 9/28/2012

    9

    The Laplace Transform

    Time Differentiation:

    Making the previous substitutions gives,

    0

    00

    )()0(0

    )()( |

    dtetfsf

    dtsetfetfdt

    dfL

    st

    stst

    So we have shown:

    )0()()(

    fssFdt

    tdfL

    Laila Rosemaizura Binti Yaakop

    The Laplace Transform

    Time Differentiation:

    We can extend the previous to show;

    )0(...

    )0(')0()()(

    )0('')0(')0()()(

    )0(')0()()(

    )1(

    21

    23

    3

    3

    2

    2

    2

    n

    nnn

    n

    n

    f

    fsfssFsdt

    tdfL

    casegeneral

    fsffssFsdt

    tdfL

    fsfsFsdt

    tdfL

    Laila Rosemaizura Binti Yaakop

  • 9/28/2012

    10

    The Laplace Transform

    Transform Pairs:

    f(t) F(s)

    22

    22

    1

    2

    )cos(

    )sin(

    )(

    !

    1

    ws

    swt

    ws

    wwt

    as

    net

    aste

    n

    atn

    at

    Laila Rosemaizura Binti Yaakop

    The Laplace Transform

    Transform Pairs:

    f(t) F(s)

    22

    22

    22

    22

    sincos)cos(

    cossin)sin(

    )()cos(

    )()sin(

    ws

    wswt

    ws

    wswt

    was

    aswte

    was

    wwte

    at

    at

    Yes !

    Laila Rosemaizura Binti Yaakop

  • 9/28/2012

    11

    Laila Rosemaizura Binti Yaakop