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

676 Chapter 21 The Laplace transform
The Laplace transformof δ(t) follows by settingd = 0:
L{δ(t)} =1
Example21.32 For a particular circuititcan be shown that the transferfunction,G(s), isgiven by
G(s) = V o (s)
V i
(s) = 1
s +2
where V i
(s) and V o
(s) are the Laplace transforms of the input and output voltages
respectively.
(a) Find v o
(t) when v i
(t) = δ(t).
(b) Usethe convolution theorem tofind v o
(t) when
{
e
−t
t0
v i
(t) =
0 t<0
Solution (a) We aregiven the transfer function
G(s) = V o (s)
V i
(s) = 1
s +2
When v i
(t) = δ(t),V i
(s) = 1andhence
and
V o
(s) = 1
s +2
{ } 1
v o
(t) = L −1 = e −2t
s +2
Thisisknownastheimpulseresponseofthesystem,g(t).Iftheimpulseresponse,
g(t),isknownthentheresponse, v o
(t),toanyotherinput, v i
(t),canbeobtainedby
convolution, thatis v o
(t) =g(t) ∗ v i
(t).
(b) The response toan input, v i
(t) =u(t)e −t , isgiven by
v o
(t) =g(t)∗v i
(t) =
=
∫ t
0
∫ t
0
e −2λ e −(t−λ) dλ
∫ t
= e −t e −λ dλ
0
[ ] e
= e −t −λ t
−1
0
=e −t (1−e −t )
= e −t −e −2t
g(λ)v i
(t − λ)dλ