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Why Laplace transforms? - University of California, San · PDF file Our Laplace Transforms will consist of rational functions (ratios of polynomials in s) and exponentials like e-st

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  • 133MAE40 Linear Circuits 133

    Why Laplace transforms? First-order RC cct

    KVL

    instantaneous for each t Substitute element relations

    Ordinary differential equation in terms of capacitor voltage

    Laplace transform

    Solve

    Invert LT

    +_

    R

    VA

    i(t)t=0

    CvcvS vR+ + +

    -

    -

    - 0)()()( =−− tvtvtv CRS

    dt tdvCtitRitvtuVtv CRAS )()(),()(),()( ===

    )()()( tuVtv dt tdvRC ACC =+

    ACCC Vs sVvssVRC 1)()]0()([ =+−

    RCs v

    RCss RCVsV CAC /1

    )0( )/1(

    /)( +

    + +

    =

    Volts)()0(1)( tueveVtv RC t

    CRC t

    AC ! "

    # $ %

    & +'' (

    ) ** +

    , −=

    −−

  • 134MAE40 Linear Circuits 134

    Laplace transforms

    The diagram commutes Same answer whichever way you go

    Linear cct

    Differential equation

    Classical techniques

    Response signal

    Laplace transform L

    Inverse Laplace transform L-1

    Algebraic equation

    Algebraic techniques

    Response transform

    Ti m

    e do

    m ai

    n (t

    do m

    ai n) Complex frequency domain(s domain)

  • 135MAE40 Linear Circuits 135

    Laplace Transform - definition

    Function f(t) of time Piecewise continuous and exponential order

    0- limit is used to capture transients and discontinuities at t=0 s is a complex variable (s+jw)

    There is a need to worry about regions of convergence of the integral

    Units of s are sec-1=Hz A frequency

    If f(t) is volts (amps) then F(s) is volt-seconds (amp-seconds)

    btKetf

  • 136MAE40 Linear Circuits 136

    Laplace transform examples Step function – unit Heavyside Function

    After Oliver Heavyside (1850-1925)

    Exponential function After Oliver Exponential (1176 BC- 1066 BC)

    Delta (impulse) function d(t)

    î í ì

    ³ <

    = 0for,1 0for,0

    )( t t

    tu

  • 137MAE40 Linear Circuits 137

    Laplace transform examples Step function – unit Heavyside Function

    After Oliver Heavyside (1850-1925)

    Exponential function After Oliver Exponential (1176 BC- 1066 BC)

    Delta (impulse) function d(t)

    0if1)()( 0

    )(

    000 >=

    + −=−===

    ∞+−∞−∞

    − ∞

    − ∫∫ σωσ

    ωσ

    sj e

    s edtedtetusF

    tjst stst

    î í ì

    ³ <

    = 0for,1 0for,0

    )( t t

    tu

  • 138MAE40 Linear Circuits 138

    Laplace transform examples Step function – unit Heavyside Function

    After Oliver Heavyside (1850-1925)

    Exponential function After Oliver Exponential (1176 BC- 1066 BC)

    Delta (impulse) function d(t)

    0if1)()( 0

    )(

    000 >=

    + −=−===

    ∞+−∞−∞

    − ∞

    − ∫∫ σωσ

    ωσ

    sj e

    s edtedtetusF

    tjst stst

    î í ì

    ³ <

    = 0for,1 0for,0

    )( t t

    tu

    F(s) = e−αte−stdt = e−(s+α )tdt = −e −(s+α )t

    s+α 0

    = 1

    s+α if σ > −α

    0

    ∫ 0

  • 139MAE40 Linear Circuits 139

    Laplace transform examples Step function – unit Heavyside Function

    After Oliver Heavyside (1850-1925)

    Exponential function After Oliver Exponential (1176 BC- 1066 BC)

    Delta (impulse) function d(t)

    0if1)()( 0

    )(

    000 >=

    + −=−===

    ∞+−∞−∞

    − ∞

    − ∫∫ σωσ

    ωσ

    sj e

    s edtedtetusF

    tjst stst

    î í ì

    ³ <

    = 0for,1 0for,0

    )( t t

    tu

    F(s) = e−αte−stdt = e−(s+α )tdt = −e −(s+α )t

    s+α 0

    = 1

    s+α if σ > −α

    0

    ∫ 0

    sdtetsF st allfor1)()( 0

    == ∫ ∞

    −δ

  • 140MAE40 Linear Circuits 140

    Laplace Transform Pair Tables Signal Waveform Transform impulse step

    ramp

    exponential

    damped ramp

    sine

    cosine

    damped sine

    damped cosine

    )(tδ

    22)( βα

    α

    ++

    +

    s

    s

    22)( βα

    β

    ++s

    22 β

    β

    +s

    22 β+s

    s

    1

    s 1

    2 1

    s

    α+s 1

    2)(

    1

    α+s

    )(tu

    )(ttu

    )(tue tα−

    )(tutte α−

    ( ) )(sin tutβ

    ( ) )(cos tutβ

    ( ) )(sin tutte βα−

    ( ) )(cos tutte βα−

  • 141MAE40 Linear Circuits 141

    Laplace Transform Properties

    Linearity – absolutely critical property Follows from the integral definition

    Example

    { } { } { } )()()()()()( 2121 sBFsAFtBtAtBftAf +=+=+ 21 fLfLL

    L(Acos(βt)) =

  • 142MAE40 Linear Circuits 142

    Laplace Transform Properties

    Linearity – absolutely critical property Follows from the integral definition

    Example

    { } { } { } )()()()()()( 2121 sBFsAFtBtAtBftAf +=+=+ 21 fLfLL

    L(Acos(βt)) = L A 2 e jβt + e − jβt( )

    $ % &

    ' ( ) =

    A 2

    L e jβt( ) + A 2

    L e − jβt( )

    = A 2

    1 s− jβ

    + A 2

    1 s+ jβ

    = As

    s 2 +β 2

  • 143MAE40 Linear Circuits 143

    Laplace Transform Properties

    Integration property

    Proof

    ( ) s sFdf

    t =

    ! " #

    $ % & ∫ 0

    )( ττL

    dtste t

    df t

    df −∫ ∞

    ∫=∫ $ %

    & ' (

    )

    * + ,

    - . /

    0 0 )(

    0 )( ττττL

  • 144MAE40 Linear Circuits 144

    Laplace Transform Properties

    Integration property

    Proof

    Denote

    so

    Integrate by parts

    ( ) s sFdf

    t =

    ! " #

    $ % & ∫ 0

    )( ττL

    dtste t

    df t

    df −∫ ∞

    ∫=∫ $ %

    & ' (

    )

    * + ,

    - . /

    0 0 )(

    0 )( ττττL

    )(and,

    0 )(and,

    tf dt dy

    e dt dx

    t dfy

    s

    ste x

    st ==

    ∫= −−

    =

    ττ

    ∫∫∫ ∞

    − ∞

    − +

    $ $ %

    &

    ' ' (

    )− =

    $ $ %

    &

    ' ' (

    )

    0000 )(

    1 )()( dtetf

    s df

    s edf st

    tstt ττττL

  • 145MAE40 Linear Circuits 145

    Laplace Transform Properties

    Differentiation Property

    Proof via integration by parts again

    Second derivative

    )0()()( −−= " # $

    % & ' fssF dt tdf

    L

    L df (t) dt

    " # $

    % & '

    = df (t) dt

    e−stdt 0−

    ∞ ∫ = f (t)e−st[ ]0−

    ∞ + s f (t)e−stdt 0−

    ∞ ∫

    = sF(s)− f (0−)

    )0()0()(2

    )0()()(2 )(2

    −"−−−=

    −− # $ %

    & ' (=

    # $ %

    & ' (

    )* +

    ,- .=

    /#

    / $ %

    /&

    / ' (

    fsfsFs

    dt df

    dt tdfs

    dt tdf

    dt d

    dt

    tfd LLL

  • 146MAE40 Linear Circuits 146

    Laplace Transform Properties General derivative formula

    Translation properties s-domain translation

    t-domain translation

    L dm f (t)

    dtm " # $

    % & '

    = s m F(s) − sm−1 f (0−) − sm−2 ) f (0−) −− f (m−1)(0−)

    )()}({ αα +=− sFtfe tL

    { } 0for)()()( >=−− − asFeatuatf asL

  • 147MAE40 Linear Circuits 147

    Laplace Transform Properties

    Initial Value Property

    Final Value Property

    Caveats: Laplace transform pairs do not always handle

    discontinuities properly Often get the average value

    Initial value property no good with impulses Final value property no good with cos, sin etc

    )(lim)(lim 0

    ssFtf st ∞→+→

    =

    )(lim)(lim 0

    ssFtf st →∞→

    =

  • 148MAE40 Linear Circuits 148

    Rational Functions We shall mostly be dealing with LTs which are

    rational functions – ratios of polynomials in s

    pi are the poles and zi are the zeros of the function K is the scale factor or (sometimes) gain

    A proper rational function has n³m A strictly proper rational function has n>m An improper rational function has n

  • 149MAE40 Linear Circuits 149

    A Little Complex Analysis

    We are dealing with linear ccts Our Laplace Transforms will consist of rational functions

    (ratios of polynomials in s) and exponentials like e-st These arise from • discrete component relations of capacitors and inductors • the kinds of input signals we apply

    – Steps, impulses, exponentials, sinusoids, delayed versions of functions

    Rational functions have a finite set of discrete poles e-st is an entire function and has no poles anywhere

    To understand linear cct respons

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