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

636 Chapter 21 The Laplace transform
Another usefulresultis
{ ∫ }
t
L f(t)dt = 1 s F(s)
0
Example21.10 The Laplace transform ofx(t) isX(s). Givenx(0) = 2 andx ′ (0) = −1, write expressionsforthe
Laplace transformsof
(a) 2x ′′ −3x ′ +x (b) −x ′′ +2x ′ +x
Solution L{x ′ } =sX(s) −x(0) =sX(s) −2
L{x ′′ } =s 2 X(s)−sx(0)−x ′ (0) =s 2 X(s)−2s+1
(a) L{2x ′′ −3x ′ +x} = 2(s 2 X(s) −2s +1) −3(sX(s) −2) +X(s)
= (2s 2 −3s+1)X(s)−4s+8
(b) L{−x ′′ +2x ′ +x} = −(s 2 X(s) −2s +1) +2(sX(s) −2) +X(s)
= (−s 2 +2s+1)X(s)+2s−5
Engineeringapplication21.1
Voltageacrossacapacitor
Thevoltage, v(t), acrossacapacitorofcapacitanceC isgiven by
v(t) = 1 C
∫ t
0
i(t)dt
Taking Laplace transforms yields
V(s)= 1 Cs I(s)
whereV(s) = L{v(t)}andI(s) = L{i(t)}.
Engineeringapplication21.2
Frequencyresponseofasystem
The Laplace variablesis sometimes referred to as the generalized or complex frequency
variable. It consists of a real and an imaginary part, wheres = σ + jω. If
onlythesinusoidalsteady-stateresponsetoasinewaveinputforasystemisrequired
then we can obtain this by putting σ = 0 into the expression for s and sos = jω.
Thusitispossibletomakethissubstitutioninanytransferfunctiongivenintermsof
the Laplace variablesto obtain the sinusoidal steady-state frequency response. We
willnotprovethisassertionhereforreasonsofbrevity.Insteadwewilldemonstrate
its usefulness.