082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

628 Chapter 21 The Laplace transform
blocksofthesystem,correspondingtothesystemelements,areconnectedtogetherand
the result is a block diagram for the whole system. By breaking down a system in this
way it is much easier to visualize how the various parts of the system interact and so
a transfer function model is complementary to a time domain model and is a valuable
way of viewing an engineering system. Transfer functions are useful in many areas of
engineering, butareparticularly important inthe design of control systems.
21.2 DEFINITIONOFTHELAPLACETRANSFORM
Let f (t) be a function of time t. In many real problems only values of t 0 are of
interest.Hence f (t)is given fort 0,and forallt < 0, f (t) istaken tobe 0.
The Laplace transform of f (t)isF(s), defined by
F(s) =
∫ ∞
0
e −st f (t)dt
Note that to find the Laplace transform of a function f (t), we multiply it by e −st and
integrate between the limits 0 and ∞.
TheLaplacetransformchanges,ortransforms,thefunction f (t)intoadifferentfunctionF(s).Notealsothatwhereas
f (t)isafunctionoft,F(s)isafunctionofs.Todenote
the Laplace transform of f (t) we write L{f (t)}. We use a lower case letter to represent
the time domain function and an upper case letter to represent thesdomain function.
Thevariablesmayberealorcomplex.Astheintegralisimproperrestrictionsmayneed
tobe placed onstoensure thatthe integral does not diverge.
Example21.1 Find the Laplace transformsof
(a) 1
(b) e −at
∫ ∞
[ ] e
Solution (a) L{1} = e −st −st ∞
1dt = = 1
0 −s
0
s =F(s)
This transformexists provided the real partofs,Re(s), ispositive.
(b) L{e −at } =
=
∫ ∞
0
∫ ∞
0
e −st e −at dt
e −(s+a)t dt
[ e
−(s+a)t ∞
=
−(s +a)]
0
( )
1
= 0 −
−(s +a)
= 1
s +a =F(s)
This transformexists provideds +a > 0.
Throughoutthe chapteritisassumed thatshas a valuesuchthatall integrals exist.