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Laplace Transform
T. D. Bui
Concordia University
• The Laplace transform of a function f(t) is defined as:
• The inverse Laplace transform is defined as:
where and the value of σ is determined by the singularities of F(s). And
dtetftfsF st−∞
∫==0
)()]([)( l
dsesFj
sFtfj
j
st
∫∞+
∞−
− ==σ
σπ)(
2
1)]([)( 1
l
1−=j
∫≡≡t
dtsdt
ds
0
1 ,
Is Laplace Transform Useful ?
• Model a linear time-invariant analog system as a transfer function.
• In control theory, Laplace transform converts linear differential equations into algebraic equations.
• This is much easier to manipulate and analyze.
An Example
• The Laplace transform of can be obtained by:
ate−
asas
edtedteesF
tastsastat
+=
+−=== ∞
+−∞+−−
∞−
∫∫1
|)( 0
)(
0
)(
0
Linearity property
• These are useful properties:
)()()]([)]([)]()([
)()]([)]([
212121 sFsFtftftftf
skFtfktkf
+=+=+==
lll
ll
Laplace TransformTime function f(t)Name
(s2-b2) /(s2+b2)2t cos(bt)Diverging cosine
2bs/(s2+b2)2t sin(bt)Diverging sine
(s+a)/((s+a)2+b2)e-at cos(bt)Damped cosine
b/((s+a)2+b2)e-at sin(bt)Damped sine
s/(s2+b2)cos(bt)Cosine
b/(s2+b2)sin(bt)Sine
n!/(s+a)n+1t n e-atnth-Order exponential
1/(s+a)e-atExponential
n!/sn+1t nnth-Order ramp
1/s2tUnit ramp
1/su(t)Unit Step
1δ(t)Unit Impulse
Find the Laplace transform of f(t)=5u(t)+3e -2t.
• Solution:
)2(
108
2
35)(
2
3][3]3[
5)]([5)](5[
22
++=
++=
+==
==
−−
ss
s
sssF
see
stutu
ttll
ll
Find the inverse Laplace transform of F(s)=5/(s2+3s+2).
Solution:
)()55()]([
:have weSo2
5
1
5)(
5|1
5|)()2(
5|2
5|)()1(
2123
5)(
21
222
111
212
tueesF
sssF
ssFsk
ssFsk
s
k
s
k
sssF
tt
ss
ss
−−−
−=−=
−=−=
−=
+−+
+=
−=+
=+=
=+
=+=
++
+=
++=
l
Find the inverse Laplace transform of F(s)=(2s+3)/(s3+2s2+s).
• Solution:
tt
s
ss
ss
ss
teetf
ssssF
s
ssk
s
s
ds
dsFs
ds
dk
s
ssFsk
s
sssFk
s
k
s
k
s
ksF
−−
−=
−=−=
−=−=
==
−−=+−+
+−+=
−=−−=+−=
+=+−
=
−=+=+=
=++==
++
++=
33)(
)1(
1
1
33)(
31
12|
)1)(32()2(
|]32
[|)]()1[()!12(
1
1|32
|)()1(
3|)1(
32|)(
)1(1)(
2
1221
112
21
112
22
0201
222211
Find the inverse Laplace transform ofF(s)=10/(s3+4s2+9s+10).
• This problem has complex poles.
)4.1532cos(23.22)(
4.153/118.1)90/4)(4.63/236.2(
10
)4)(21(
10
)2121)(221(
10
|)21)(2(
10)()21(
2|4)1(
10|)()2(
21212)(
]2)1)[(2(
10
1094
10)(
2
2
2
212
2221
*221
2223
ott
ooo
js
ss
teetf
k
jjjjjk
jsssFjsk
ssFsk
js
k
js
k
s
ksF
ssssssF
−+=
=−−
=
−−=
−+−−+−−=
−++=++=
=++
=+=
−++
+++
+=
+++=
+++=
−−
−−=
−=−=
Laplace Transform Theorems
τττ
τττ
ττ
dtffsFsF
dftfsFsF
asFtfe
sFettuttf
s
sFdf
ffssFsdt
fd
fssFdt
df
t
t
at
st
t
nnnn
n
)()()]()([:integral nalConvolutio
)()()]()([:integral nalConvolutio
)()]([:hiftFrequencyS
)()]()([:Shifting
)(])([ :Integral
)0(...)0()(][
)0()(][ :Derivative
2
0
1211
2
0
1211
00
0
11
0
−=
−=
+=
=−−
=
−−=
−=
∫
∫
∫
−
−
−
−
+−+−
+
l
l
l
l
l
l
l
Laplace transforms of the differential equations in Lecture #2
• Resistance circuit:
• Inductance circuit:
RsIsV
Rtitv
)()(
)()(
==
))0()(()(
)()(
+−=
=
issILsV
dt
tdiLtv
• Capacitance circuit:
s
v
s
sI
CsV
vdiC
tvt
)0()(1)(
)0()(1
)(0
+=
+= ∫ ττ
• Kirchkoff’s Laws:
)()(1
)(
)()(1
)()(
)()(1
)(
)()(1
)()(
2
1
0
0
2
21
22
121
sVs
sI
CsIR
sVs
sI
CsIRsIR
tvdiC
tiR
tvdiC
tiRtiR
t
t
=+
=++
=+
=++
∫
∫
ττ
ττ
• R-L series circuit, step voltage:
• At the moment the switch is closed I=0.
• E = step voltage
s
EsRIissIL
EtRidt
tdiL
=+−
=+
+ )())0()((
)()(
• R-L series circuit, impulse voltage source:
• R-C circuit step voltage source:
0))0()(()(
0)(
)(
=−+
=+
+issILsRI
dt
tdiLtRi
s
sI
CsRI
s
E
dttiC
tRitEu
)(1)(
)(1
)()(
+=
+= ∫
• R-C circuit , impulse voltage source:
• R-L-C series circuit, impulse voltage source:
0)(1
)(
0)(1
)(
=+
=+ ∫
s
sI
CsRI
dttiC
tRi
0)(1
))0()(()(
0)(1)(
)(
=+−+
=++
+
∫
s
sI
CissILsRI
dttiCdt
tdiLtRi
• Model of an RLC circuit:
• Model of a mass-spring-damper system:
)()(1
))0()(()(
)()(1)()(
0
sIs
sV
LvssVC
Rs
sV
tidttvLdt
tdvC
R
tv t
=+−+
=++
+
∫
)())0()(()())0(')0()((
)()(
)()(
2
2
2
sKXxssXBsFxsxsXsM
tKxdt
tdxBtf
dt
txdM
−−−=−−
−−=
+
• Simplified automobile suspension system:
)())()((
)]0()()0()([)()]0(')0()([
))()((
)]0()()0()([)]0(')0()([
)())()(())()(
()()(
))()(())()(
()(
22121
11222222
2
211
22111112
1
21212
22
2
2121
21
2
212
11
sXKsXsXK
xssXxssXBsFxsxsXsM
sXsXK
xssXxssXBxsxsXsM
txKtxtxKdt
tdx
dt
tdxBtf
dt
txdM
txtxKdt
tdx
dt
tdxB
dt
txdM
−−−+−−−=−−
−−+−−−=−−
−−−−−=
−−−−=
• Model of a torsional pendulum (pendulum in clocks inside
glass dome);
Moment of inertia of pendulum bob denoted by J
Friction between the bob and air by B
Elastance of the brass suspension strip by K
)()]0()([)()]0()0()([
)()(
)()(
'2
2
2
sKssBssssJ
tKdt
tdBt
dt
tdJ
Θ−−Θ−=−−Θ
−−=
θτθθ
θθτθ
• Model of electromechanical systems.
• Model of a servomotor:
)]0()([)(
)]0()([)]0()0()([
)()()(
21
'2
212
2
θθθθ
θθθ
−Θ−=−Θ+−−Θ
−=+
ssksEk
ssBsssJ
dt
dktek
dt
tdB
dt
tdJ
a
a
