laplace_transform.pdf
TRANSCRIPT
-
9/28/2012
1
The Laplace Transform
EE 602 CIRCUIT ANALYSIS
Laila Rosemaizura Binti Yaakop
The Laplace Transform
The Laplace Transform of a function, f(t), is defined as;
0
)()()]([ dtetfsFtfL st
The Inverse Laplace Transform is defined by
j
j
tsdsesF
jtfsFL )(
2
1)()]([1
*notes
Eq A
Eq B
Laila Rosemaizura Binti Yaakop
-
9/28/2012
2
The Laplace Transform
We generally do not use Eq B to take the inverse Laplace. However,
this is the formal way that one would take the inverse. To use
Eq B requires a background in the use of complex variables and
the theory of residues. Fortunately, we can accomplish the same
goal (that of taking the inverse Laplace) by using partial fraction
expansion and recognizing transform pairs.
*notes Laila Rosemaizura Binti Yaakop
The Laplace Transform
Example 1 : Laplace Transform of the unit step.
*notes
|0
0
11)]([ stst e
sdtetuL
stuL
1)]([
The Laplace Transform of a unit step is:
s
1Laila Rosemaizura Binti Yaakop
-
9/28/2012
3
The Laplace Transform
An important point to remember:
)()( sFtf
The above is a statement that f(t) and F(s) are
transform pairs. What this means is that for
each f(t) there is a unique F(s) and for each F(s)
there is a unique f(t).
Laila Rosemaizura Binti Yaakop
The Laplace Transform
Building a transform pairs:
eL(
e
tasstatatdtedteetueL
0
)(
0
)]([
asas
etueL
stat 1
)()]([ |
0
astue
at 1)(A transformpair Laila Rosemaizura Binti Yaakop
-
9/28/2012
4
The Laplace Transform
Building transform pairs:
0
)]([ dttettuL st
0 00| vduuvudv
u = t
dv = e-stdt
2
1)(
sttu
A transform
pairLaila Rosemaizura Binti Yaakop
The Laplace Transform
Building transform pairs:
22
0
11
2
1
2
)()][cos(
ws
s
jwsjws
dteee
wtLst
jwtjwt
22)()cos(
ws
stuwt A transform
pairLaila Rosemaizura Binti Yaakop
-
9/28/2012
5
Table of Laplace Transform
____________________________________
)()( sFtf
f(t) F(s)
1
2
!
1
1
1)(
1)(
n
n
st
s
nt
st
ase
stu
t
Laila Rosemaizura Binti Yaakop
f(t) F(s)
22
22
1
2
)cos(
)sin(
)(
!
1
ws
swt
ws
wwt
as
net
aste
n
atn
at
Laila Rosemaizura Binti Yaakop
-
9/28/2012
6
The Laplace Transform
Time Shift
0 0
)( )()(
,.,0,
,
)()]()([
dxexfedxexf
SoxtasandxatAs
axtanddtdxthenatxLet
eatfatuatfL
sxasaxs
a
st
)()]()([ sFeatuatfL asLaila Rosemaizura Binti Yaakop
Theorems of Laplace Transform
First Shift (Frequency Shift)
0
)(
0
)()(
)]([)]([
asFdtetf
dtetfetfeL
tas
statat
)()]([ asFtfeL at
Laila Rosemaizura Binti Yaakop
-
9/28/2012
7
Example: Using First Shift Theoerm(Frequency Shift)
Find the L[e-atcos(wt)]
In this case, f(t) = cos(wt) so,
22
22
)(
)()(
)(
was
asasFand
ws
ssF
22 )()(
)()]cos([
was
aswteL
at
Laila Rosemaizura Binti Yaakop
Time Integration:
The property is:
stst
t
st
t
es
vdtedv
and
dttfdudxxfuLet
partsbyIntegrate
dtedxxfdttfL
1,
)(,)(
:
)()(
0
0 00
Laila Rosemaizura Binti Yaakop
-
9/28/2012
8
The Laplace Transform
Time Integration:
Making these substitutions and carrying out
The integration shows that
)(1
)(1
)(00
sFs
dtetfs
dttfL st
Laila Rosemaizura Binti Yaakop
The Laplace Transform
Time Differentiation:
If the L[f(t)] = F(s), we want to show:
)0()(])(
[ fssFdt
tdfL
Integrate by parts:
)(),()(
,
tfvsotdfdtdt
tdfdv
anddtsedueustst
*note
Laila Rosemaizura Binti Yaakop
-
9/28/2012
9
The Laplace Transform
Time Differentiation:
Making the previous substitutions gives,
0
00
)()0(0
)()( |
dtetfsf
dtsetfetfdt
dfL
st
stst
So we have shown:
)0()()(
fssFdt
tdfL
Laila Rosemaizura Binti Yaakop
The Laplace Transform
Time Differentiation:
We can extend the previous to show;
)0(...
)0(')0()()(
)0('')0(')0()()(
)0(')0()()(
)1(
21
23
3
3
2
2
2
n
nnn
n
n
f
fsfssFsdt
tdfL
casegeneral
fsffssFsdt
tdfL
fsfsFsdt
tdfL
Laila Rosemaizura Binti Yaakop
-
9/28/2012
10
The Laplace Transform
Transform Pairs:
f(t) F(s)
22
22
1
2
)cos(
)sin(
)(
!
1
ws
swt
ws
wwt
as
net
aste
n
atn
at
Laila Rosemaizura Binti Yaakop
The Laplace Transform
Transform Pairs:
f(t) F(s)
22
22
22
22
sincos)cos(
cossin)sin(
)()cos(
)()sin(
ws
wswt
ws
wswt
was
aswte
was
wwte
at
at
Yes !
Laila Rosemaizura Binti Yaakop
-
9/28/2012
11
Laila Rosemaizura Binti Yaakop