# 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

• View
7

1

Embed Size (px)

### Text of Why Laplace transforms? - University of California, San · PDF file Our Laplace Transforms...

• 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

Documents
Education
Documents
Documents
Education
Documents
Documents
Documents
Documents
Documents
Engineering
Documents
Documents
Documents
Documents
Documents
Documents
Documents
Engineering
Documents
Documents
Documents
Documents
Documents
Documents
Documents
Documents
Documents
Documents
Documents