page 78

2.8 LAPLACE TRANSFORMS

•The Laplace transform allows us to reverse time. And, as you recall from before the inverse of time is frequency. Because we are normally concerned with response, the Laplace transform is much more useful in system analysis.

•The basic Laplace transform equations is shown below,

∞	–stdt
F(s ) = ∫ f(t )e
0
where,
f(t ) =	the function in terms of time t
F(s ) =	the function in terms of the Laplace s

2.8.1 Laplace Transform Tables

• Basic Laplace Transforms for operational transformations are given below,

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TIME DOMAIN

FREQUENCY DOMAIN

Kf(t )

Kf(s )

f1(t )+ f2(t )–f3(t )+ …

f1(s )+ f2(s )–f3(s )+ …

df(t )

sf(s )– f(0

–

)

-----------

dt

d2f(t )

s

2

f(s )– sf(0

–

df(0–

)

-------------

)– -----------------

dt

2

dt

dnf(t )

s

n

f(s )– s

n – 1

f(0

–

)– s

n – 2 df(0–

)

dnf(0–

)

-------------

----------------- – …

– -------------------

dt

n

dt

dt

n

t

(t )dt

f(s )

∫f

--------

0

s

f(t – a )u(t – a ),a >0

e–asf(s )

e–atf(t )

f(s – a )

f(at ),a >0

1

s

--f

--

a

a

tf(t )

–df(s )

---------------

ds

t

n

f(t )

n dnf(s )

(–1 ) --------------

∞

dsn

f(t )

∫

-------

f(u )du

t

s

• A set of useful functional Laplace transforms are given below,

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TIME DOMAIN

FREQUENCY DOMAIN

A

A

--

s

t

1

----

s2

e

–at

1

s-----------+ a

sin (ω

t )

ω

-----------------

s2 + ω

2

cos (ω

t )

s

s-----------------2 + ω

2

te

–at

1

------------------

2

(s + a )

e

–at

sin (ω

t )

ω

-------------------------------

2

+ ω

2

(s + a )

e

–at

cos (ω

t )

s + a

-------------------------------

2

+ ω

2

(s + a )

Ae

–at

A

----------

s – a

Ate

–at

A

-----------------

2

(s – a )

2

A

e–α t cos (β t + θ )

A

Acomplex conjugate

s-----------------------+ α –β j +

-------------------------------------

s + α +β j

2t

A

e–α t cos (β t + θ )

A

+ A

complex conjugate

------------------------------

2

2

(s + α –β

j )

(s + α +β

j )

• Laplace transforms can be used to solve differential equations.