S. Boyd EE102
Lecture 3
The Laplace transform
•definition & examples
•properties & formulas
–linearity
–the inverse Laplace transform
–time scaling
–exponential scaling
–time delay
–derivative
–integral
–multiplication by t
–convolution
3–1
Prepare-se para as provas
Estude fácil! Tem muito documento disponível na Docsity
Ganhe pontos para baixar
Ganhe pontos ajudando outros esrudantes ou compre um plano Premium
Guias e Dicas
Prepare-se para as provas
Estude fácil! Tem muito documento disponível na Docsity
Encontrar documentos
Prepare-se para as provas com trabalhos de outros alunos como você, aqui na Docsity
Encontra documentos específicos para os exames da tua universidade
Encontra documentos específicos para os exames da tua universidade
Videoaulas
Prepare-se com as videoaulas e exercícios resolvidos criados a partir da grade da sua Universidade
Quiz
Responda perguntas de provas passadas e avalie sua preparação.
Ganhe pontos para baixar
Ganhe pontos ajudando outros esrudantes ou compre um plano Premium
Transformada de Laplace: Definição, Propriedades e Exemplos, Notas de aula de Cálculo
Este documento aborda a transformada de laplace, sua definição, propriedades e exemplos. A transformada de laplace é uma ferramenta matemática utilizada para analisar sinais e sistemas, permitindo a transformação de sinais no domínio do tempo para o domínio da frequência. O documento abrange conceitos básicos, como a linearidade, a transformada inversa, escalonamento temporal e espacial, escalonamento exponencial e aplicação a circuitos elétricos e sistemas de controle.
Tipologia: Notas de aula
2021
1 / 33
Esta página não é visível na pré-visualização
Não perca as partes importantes!
Documentos relacionados
Pré-visualização parcial do texto
Baixe Transformada de Laplace: Definição, Propriedades e Exemplos e outras Notas de aula em PDF para Cálculo, somente na Docsity!
S. Boyd
EE
Lecture 3
The Laplace transform
•^
definition & examples
-^
properties & formulas –^
linearity
-^
the inverse Laplace transform
-^
time scaling
-^
exponential scaling
-^
time delay
-^
derivative
-^
integral
-^
multiplication by
t
–^
convolution
Idea
the Laplace transform converts
integral
and
differential
equations into
algebraic
equations
this is like phasors, but^ •
applies to general signals, not just sinusoids • handles non-steady-state conditions allows us to analyze^ •
LCCODEs • complicated circuits with sources, Ls, Rs, and Cs • complicated systems with integrators, differentiators, gains The Laplace transform
The Laplace transform
we’ll be interested in signals defined for
t
the
Laplace transform
of a signal (function)
f
is the function
F
L
(f
defined by
F
(s
∫^
∞ 0
f^ (
t)
−e
st
dt
for those
s
C
for which the integral makes sense
•^
F
is a complex-valued function of complex numbers
-^
s^
is called the (complex)
frequency variable
, with units sec
−
t
is called
the
time variable
(in sec);
st
is unitless
•^
for now, we assume
f
contains no impulses at
t
common notation convention:
lower case letter denotes signal; capital
letter denotes its Laplace transform,
e.g.
,^
U
denotes
L
(u
V
in
denotes
L
(v
in
), etc.
The Laplace transform
Example
let’s find Laplace transform of
f
(t
e
t:
F
(s
∫^
∞ 0
t e e
−
st
dt
∫^
∞ 0
(1 e
−
s)
t^ dt
s
(1e
−
s)
t
∣^ ∞∣∣ ∣^0
s^
provided we can say
e
(
−
s)
t^ →
as
t
, which is true for
s >
∣ ∣e∣
(
−
s)
∣ ∣t∣
∣ ∣e∣
−
j(
=s
)t
︸^
=
∣ ∣e∣
(
−<
s)
∣ ∣t∣
e
(
−<
s)
t
•^
the
integral
defining
F
makes sense for all
s
C
with
s >
(the
‘region of convergence’
of
F
•^
but the resulting
formula
for
F
makes sense for all
s
C
except
s
we’ll ignore these (sometimes important) details and just say that
L
(e
t) =
s^
The Laplace transform
sinusoid:
first express
f
(t
) = cos
ωt
as
f^
(t
jωte
−e jωt
now we can find
F
as
F
(s
)^
∫^
∞ 0
− e
st
jωte
−e
jωt
)^
dt
∫^
∞ 0
( e −
s+
jω
)t
dt
∫^
∞ 0
( e −
s−
jω
)t
dt
s^
jω
s^
jω
s (^2) s
ω
2
(valid for
s >
; final formula OK for
s
jω
The Laplace transform
powers of
t
:^
f^ (
t) =
t
n^
(n
we’ll integrate by parts,
i.e.
, use
∫^
b^ u a
(t
)v
t)
dt
u
(t
)v
(t
b a
∫^
b^ v a
(t
)u
t)
dt
with
u
(t
t
n,
v
t) =
e
−
st
,^
a^
,^
b^
F
(s
∫^
∞ 0
n t
−e
st
dt
nt
−e
st s
n s
∫^
∞ 0
n t −
1 e
−
st
dt
n^ s^
L
(t
n−
provided
t
ne
−
st
if
t
, which is true for
s >
applying the formula recusively, we obtain
F
(s
n! n+1 s
valid for
s >
; final formula OK for all
s
The Laplace transform
Linearity
the Laplace transform is
linear
: if
f
and
g
are any signals, and
a
is any
scalar, we have
L
(af
aF,
L
(f
g
F
G
i.e.
, homogeneity & superposition hold example:
L
3 δ
(t
)^ −
te
)^
3 L
(δ
(t
L
(e
t)
s^
3 s
s^
The Laplace transform
One-to-one property
the Laplace transform is
one-to-one
: if
L
(f
L
(g
)^
then
f
g
(well, almost; see below)^ •
F
determines
f
•^
inverse Laplace transform
L
−
1
is well defined
(not easy to show) example
(previous page):
L
−
3 s
s^
δ(
t)
te
in other words, the
only
function
f
such that
F
(s
3 s
s^
is
f
(t
δ(
t)
te
The Laplace transform
Inverse Laplace transform
in principle we can recover
f
from
F
via
f^
(t
(^12) πj
∫^
σ+
j∞ σ−
j∞
F
(s
)e
st
ds
where
σ
is large enough that
F
(s
)^
is defined for
s^
σ
surprisingly, this formula isn’t really useful! The Laplace transform
Time scaling
define signal
g
by
g
(t
f
(at
), where
a >
; then
G
(s
/a
)F
(s/a
makes sense: times are scaled by
a
, frequencies by
/a
let’s check: G
(s
∫^
∞ 0
f^ (
at
)e
−
st
dt
/a
∞ 0
f^ (
τ^ )
−e (s/a
)τ
dτ
/a
)F
(s/a
where
τ
at
example:
L
(e
t) = 1
s^
so
L
(e
at
/a
)^
(s/a
)^
s^
a
The Laplace transform
Time delay
let
f
be a signal and
T >
; define the signal
g
as
g(
t) =
t < T
f^ (
t^ −
T
)^
t^
T
(g
is
f
, delayed by
T
seconds & ‘zero-padded’ up to
T
ag replacements
t
t^
t^
=
T
f^ (
t)
g(
t)
The Laplace transform
then we have
G
(s
e
−
sT
F
(s
derivation:
G
(s
∫^
∞ 0
− e
st
g(
t)
dt
∫^
∞ T
− e
st
f^ (
t^ −
T
)^
dt
∫^
∞ 0
− e
s(
τ^ +
T^ )
f^
(τ
)^
dτ
−e
sT
F
(s
The Laplace transform
Derivative
if signal
f
is continuous at
t
, then
L
(f
sF
(s
)^
f
•^
time-domain differentiation becomes multiplication by frequencyvariable
s
(as with phasors)
•^
plus
a term that includes initial condition (
i.e.
,^ −
f^ (0)
higher-order derivatives: applying derivative formula twice yields
L
(f
sL
(f
f
s(
sF
(s
)^
f
f
(^2) s
F
(s
)^
sf
f
similar formulas hold for
L
(f
(k
The Laplace transform
examples^ •
f
(t
e
t, so
f
t) =
e
t^
and^ L
(f
L
(f
s^
using the formula,
L
(f
s
(^
s^
)^
, which is the same
•^
sin
ωt
(^1) ω
d dt^
cos
ωt
, so
L
(sin
ωt
(^1) ω
s^
s (^2) s
ω
2
ω (^2) s
ω
2
•^
f^
is unit ramp, so
f
′^ is unit step^ L
(f
s
1 2 s
/s
The Laplace transform