Laplace Transform Douglas Wilhelm Harder Department of Electrical and Computer Engineering...
-
Upload
moises-kemble -
Category
Documents
-
view
218 -
download
3
Transcript of Laplace Transform Douglas Wilhelm Harder Department of Electrical and Computer Engineering...
![Page 1: Laplace Transform Douglas Wilhelm Harder Department of Electrical and Computer Engineering University of Waterloo Copyright © 2008 by Douglas Wilhelm Harder.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649cc15503460f949884d4/html5/thumbnails/1.jpg)
Laplace Transform
Douglas Wilhelm Harder
Department of Electrical and Computer Engineering
University of Waterloo
Copyright © 2008 by Douglas Wilhelm Harder. All rights reserved.
ECE 250 Data Structures and Algorithms
![Page 2: Laplace Transform Douglas Wilhelm Harder Department of Electrical and Computer Engineering University of Waterloo Copyright © 2008 by Douglas Wilhelm Harder.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649cc15503460f949884d4/html5/thumbnails/2.jpg)
Laplace Transform
Outline
• In this talk, we will:– Definition of the Laplace transform– A few simple transforms– Rules– Demonstrations
![Page 3: Laplace Transform Douglas Wilhelm Harder Department of Electrical and Computer Engineering University of Waterloo Copyright © 2008 by Douglas Wilhelm Harder.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649cc15503460f949884d4/html5/thumbnails/3.jpg)
• Classical differential equations
Laplace Transform
Background
tttt xyyy 12
tt eet 2
2
1
2
1y
1x t
Time Domain
Solve differential equation
![Page 4: Laplace Transform Douglas Wilhelm Harder Department of Electrical and Computer Engineering University of Waterloo Copyright © 2008 by Douglas Wilhelm Harder.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649cc15503460f949884d4/html5/thumbnails/4.jpg)
• Laplace transforms
Laplace Transform
Background
tttt xyyy 12
tt eet 2
2
1
2
1y
s
t
sss
1X
23
1)H(
2
23
112 sss
1x t
Time Domain Frequency Domain
Solve algebraic equation
Laplace transform
Inverse Laplace transform
![Page 5: Laplace Transform Douglas Wilhelm Harder Department of Electrical and Computer Engineering University of Waterloo Copyright © 2008 by Douglas Wilhelm Harder.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649cc15503460f949884d4/html5/thumbnails/5.jpg)
Laplace Transform
Definition
• The Laplace transform is
• Common notation:
s
dtett st
F
ff0
L
st
st
Gg
Ff
L
L st
st
Gg
Ff
![Page 6: Laplace Transform Douglas Wilhelm Harder Department of Electrical and Computer Engineering University of Waterloo Copyright © 2008 by Douglas Wilhelm Harder.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649cc15503460f949884d4/html5/thumbnails/6.jpg)
Laplace Transform
Definition
• The Laplace transform is the functional equivalent of a matrix-vector product
0
fF dtets st
n
jjjii vm
1,Mv
n
jjjvu
1
vu
0
vuvu dttttt
![Page 7: Laplace Transform Douglas Wilhelm Harder Department of Electrical and Computer Engineering University of Waterloo Copyright © 2008 by Douglas Wilhelm Harder.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649cc15503460f949884d4/html5/thumbnails/7.jpg)
Laplace Transform
Definition
• Notation:– Variables in italics t, s
– Functions in time space f, g
– Functions in frequency space F, G
– Specific limits
t
t
t
t
flim0f
flim0f
0
0
![Page 8: Laplace Transform Douglas Wilhelm Harder Department of Electrical and Computer Engineering University of Waterloo Copyright © 2008 by Douglas Wilhelm Harder.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649cc15503460f949884d4/html5/thumbnails/8.jpg)
Laplace Transform
Existence
• The Laplace transform of f(t) exists if– The function f(t) is piecewise continuous– The function is bound by
for some k and M ktMet f
![Page 9: Laplace Transform Douglas Wilhelm Harder Department of Electrical and Computer Engineering University of Waterloo Copyright © 2008 by Douglas Wilhelm Harder.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649cc15503460f949884d4/html5/thumbnails/9.jpg)
Laplace Transform
Example Transforms
• We will look at the Laplace transforms of:– The impulse function (t)
– The unit step function u(t)
– The ramp function t and monomials tn
– Polynomials, Taylor series, and et
– Sine and cosine
![Page 10: Laplace Transform Douglas Wilhelm Harder Department of Electrical and Computer Engineering University of Waterloo Copyright © 2008 by Douglas Wilhelm Harder.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649cc15503460f949884d4/html5/thumbnails/10.jpg)
Laplace Transform
Example Transforms
• While deriving these, we will examine certain properties:– Linearity– Damping– Time scaling– Time differentiation– Frequency differentiation
![Page 11: Laplace Transform Douglas Wilhelm Harder Department of Electrical and Computer Engineering University of Waterloo Copyright © 2008 by Douglas Wilhelm Harder.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649cc15503460f949884d4/html5/thumbnails/11.jpg)
Laplace Transform
Impulse Function
• The easiest transform is that of the impulse function:
1
δδ
0
0
s
st
e
dtettL
1δ t
![Page 12: Laplace Transform Douglas Wilhelm Harder Department of Electrical and Computer Engineering University of Waterloo Copyright © 2008 by Douglas Wilhelm Harder.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649cc15503460f949884d4/html5/thumbnails/12.jpg)
• Next is the unit step function
s
es
es
dte
dtett
s
st
st
st
1
10
1
uu
0
0
0
0
L
11
00u
t
tt
st
1u
Laplace Transform
Unit Step Function
![Page 13: Laplace Transform Douglas Wilhelm Harder Department of Electrical and Computer Engineering University of Waterloo Copyright © 2008 by Douglas Wilhelm Harder.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649cc15503460f949884d4/html5/thumbnails/13.jpg)
Laplace Transform
Integration by Parts
• Further cases require integration by parts
• Usually written as
b
a
b
a
b
a
dfgfgdgf
![Page 14: Laplace Transform Douglas Wilhelm Harder Department of Electrical and Computer Engineering University of Waterloo Copyright © 2008 by Douglas Wilhelm Harder.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649cc15503460f949884d4/html5/thumbnails/14.jpg)
Laplace Transform
Integration by Parts
• Product rule
• Rearrange and integrate
t
dt
dttt
dt
dtt
dt
dgfgfgf
b
a
b
a
b
a
b
a
b
a
dtttdt
dtt
dtttdt
ddttt
dt
ddtt
dt
dt
ttdt
dtt
dt
dt
dt
dt
gfgf
gfgfgf
gfgfgf
![Page 15: Laplace Transform Douglas Wilhelm Harder Department of Electrical and Computer Engineering University of Waterloo Copyright © 2008 by Douglas Wilhelm Harder.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649cc15503460f949884d4/html5/thumbnails/15.jpg)
Laplace Transform
Ramp Function
• The ramp function
2
0
0
00
0
111
10
111
u
se
ss
dtes
dtes
es
t
dttett
st
st
stst
st
L
2
1u
stt
t
t
ddf
f
st
st
es
te
1
g
ddg
![Page 16: Laplace Transform Douglas Wilhelm Harder Department of Electrical and Computer Engineering University of Waterloo Copyright © 2008 by Douglas Wilhelm Harder.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649cc15503460f949884d4/html5/thumbnails/16.jpg)
Laplace Transform
Monomials
• By repeated integration-by-parts, it is possible to find the formula for a general monomial for n ≥ 0
1
!u
nn
s
nttL
1
!u
nn
s
ntt
![Page 17: Laplace Transform Douglas Wilhelm Harder Department of Electrical and Computer Engineering University of Waterloo Copyright © 2008 by Douglas Wilhelm Harder.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649cc15503460f949884d4/html5/thumbnails/17.jpg)
Laplace Transform
Linearity Property
• The Laplace transform is linear
• If and then
sbsatbta
sbsatbta
GF)g()f(
GF)g()f(
L
st F)f( L st G)g( L
![Page 18: Laplace Transform Douglas Wilhelm Harder Department of Electrical and Computer Engineering University of Waterloo Copyright © 2008 by Douglas Wilhelm Harder.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649cc15503460f949884d4/html5/thumbnails/18.jpg)
Laplace Transform
Initial and Final Values
• Given then
• Note sF(s) is the Laplace transform of f(1)(x)
st Ff
ss
ss
s
s
Flimf
Flim0f
0
![Page 19: Laplace Transform Douglas Wilhelm Harder Department of Electrical and Computer Engineering University of Waterloo Copyright © 2008 by Douglas Wilhelm Harder.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649cc15503460f949884d4/html5/thumbnails/19.jpg)
Laplace Transform
Polynomials
• The Laplace transform of the polynomial follows:
n
kkk
n
k
kk s
katta
01
0
!uL
![Page 20: Laplace Transform Douglas Wilhelm Harder Department of Electrical and Computer Engineering University of Waterloo Copyright © 2008 by Douglas Wilhelm Harder.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649cc15503460f949884d4/html5/thumbnails/20.jpg)
Laplace Transform
Polynomials
• This generalizes to Taylor series, e.g.,
1
1
1
!
!
1
u!
1u
01
01
0
s
s
s
k
k
ttk
te
n
kk
n
kk
n
k
kt LL
1
1u
stet
![Page 21: Laplace Transform Douglas Wilhelm Harder Department of Electrical and Computer Engineering University of Waterloo Copyright © 2008 by Douglas Wilhelm Harder.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649cc15503460f949884d4/html5/thumbnails/21.jpg)
Laplace Transform
The Sine Function
• Sine requires two integration by parts:
ttss
dtetss
dtetss
stets
dtets
dtets
stets
dtettt
st
st
st
st
st
usin11
sin11
sin11
cos1
cos1
0
cos1
sin1
sinusin
22
022
02
0
0
00
0
L
L
1 of 2
![Page 22: Laplace Transform Douglas Wilhelm Harder Department of Electrical and Computer Engineering University of Waterloo Copyright © 2008 by Douglas Wilhelm Harder.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649cc15503460f949884d4/html5/thumbnails/22.jpg)
Laplace Transform
The Sine Function
• Consequently:
1
1usin
1usin1
usin11
usin
2
2
22
stt
tts
ttss
tt
L
L
LL
1
1usin
2
stt
2 of 2
![Page 23: Laplace Transform Douglas Wilhelm Harder Department of Electrical and Computer Engineering University of Waterloo Copyright © 2008 by Douglas Wilhelm Harder.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649cc15503460f949884d4/html5/thumbnails/23.jpg)
Laplace Transform
The Cosine Function
• As does cosine:
ttss
dtetss
dtetss
stetss
dtetss
dtets
stets
dtettt
st
st
st
st
st
ucos11
cos1
01
cos11
sin11
sin11
sin1
cos1
cosucos
2
022
02
0
0
00
0
L
L
1 of 2
![Page 24: Laplace Transform Douglas Wilhelm Harder Department of Electrical and Computer Engineering University of Waterloo Copyright © 2008 by Douglas Wilhelm Harder.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649cc15503460f949884d4/html5/thumbnails/24.jpg)
Laplace Transform
The Cosine Function
• Consequently:
1
ucos
ucos1
ucos11
ucos
2
2
2
s
stt
stts
ttss
tt
L
L
LL
1
ucos2
s
stt
2 of 2
![Page 25: Laplace Transform Douglas Wilhelm Harder Department of Electrical and Computer Engineering University of Waterloo Copyright © 2008 by Douglas Wilhelm Harder.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649cc15503460f949884d4/html5/thumbnails/25.jpg)
Laplace Transform
Periodic Functions
• If f(t) is periodic with period T then
• For example,
sT
Tst
e
dtet
t
1
f
f 0L
s
s
s
st
es
sse
e
dtet
t
11
1
cos
f2
0L
![Page 26: Laplace Transform Douglas Wilhelm Harder Department of Electrical and Computer Engineering University of Waterloo Copyright © 2008 by Douglas Wilhelm Harder.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649cc15503460f949884d4/html5/thumbnails/26.jpg)
Laplace Transform
Periodic Functions
• Here cos(t) is repeated with period
tfL
tcos
tcosL
tf
![Page 27: Laplace Transform Douglas Wilhelm Harder Department of Electrical and Computer Engineering University of Waterloo Copyright © 2008 by Douglas Wilhelm Harder.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649cc15503460f949884d4/html5/thumbnails/27.jpg)
• Consider f(t) below:
Laplace Transform
Periodic Functions
s
s
s
s
s
st
es
e
es
e
e
dte
t222
1
0
1
1
1
1
1f
L
tf tu
s
t1
u L tfL
![Page 28: Laplace Transform Douglas Wilhelm Harder Department of Electrical and Computer Engineering University of Waterloo Copyright © 2008 by Douglas Wilhelm Harder.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649cc15503460f949884d4/html5/thumbnails/28.jpg)
Laplace Transform
Damping Property
• Time domain damping ⇔ frequency domain
shifting
as
dtet
dtetete
tas
statat
F
f
ff
0
)(
0
L
aste at Ff
![Page 29: Laplace Transform Douglas Wilhelm Harder Department of Electrical and Computer Engineering University of Waterloo Copyright © 2008 by Douglas Wilhelm Harder.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649cc15503460f949884d4/html5/thumbnails/29.jpg)
Laplace Transform
Damping Property
• Damped monomials
A special case:
1
1
!u
!u
nnat
nn
as
ntte
s
ntt
as
te
st
at
1u
1u
![Page 30: Laplace Transform Douglas Wilhelm Harder Department of Electrical and Computer Engineering University of Waterloo Copyright © 2008 by Douglas Wilhelm Harder.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649cc15503460f949884d4/html5/thumbnails/30.jpg)
• Consider cos(t)u(t)
1
ucos2
s
stt
Laplace Transform
Damping Property
![Page 31: Laplace Transform Douglas Wilhelm Harder Department of Electrical and Computer Engineering University of Waterloo Copyright © 2008 by Douglas Wilhelm Harder.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649cc15503460f949884d4/html5/thumbnails/31.jpg)
• Time scale by = 2
22 2
1u2sin
stt
Laplace Transform
Damping Property
![Page 32: Laplace Transform Douglas Wilhelm Harder Department of Electrical and Computer Engineering University of Waterloo Copyright © 2008 by Douglas Wilhelm Harder.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649cc15503460f949884d4/html5/thumbnails/32.jpg)
• Time scale by = ½
4
1221
1usin
stt
Laplace Transform
Damping Property
![Page 33: Laplace Transform Douglas Wilhelm Harder Department of Electrical and Computer Engineering University of Waterloo Copyright © 2008 by Douglas Wilhelm Harder.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649cc15503460f949884d4/html5/thumbnails/33.jpg)
Laplace Transform
Time-Scaling Property
• Time domain scaling ⇔ attenuated frequency domain scaling
a
s
a
adea
da
e
dteatat
as
as
st
F1
)f(1
1)f(
)f()f(
0
0
0
L
dta
d
at
a
s
aat F
1f
![Page 34: Laplace Transform Douglas Wilhelm Harder Department of Electrical and Computer Engineering University of Waterloo Copyright © 2008 by Douglas Wilhelm Harder.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649cc15503460f949884d4/html5/thumbnails/34.jpg)
• Time scaling of trigonometric functions:
22
2
1
11usin
s
sttL
22
2
1
1ucos
s
s
s
s
ttL
1
1usin
2
stt
1ucos
2
s
stt
22
usin
s
tt 22
ucos
s
stt
Laplace Transform
Time-Scaling Property
![Page 35: Laplace Transform Douglas Wilhelm Harder Department of Electrical and Computer Engineering University of Waterloo Copyright © 2008 by Douglas Wilhelm Harder.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649cc15503460f949884d4/html5/thumbnails/35.jpg)
• Consider sin(t)u(t)
1
1usin
2
stt
Laplace Transform
Time-Scaling Property
![Page 36: Laplace Transform Douglas Wilhelm Harder Department of Electrical and Computer Engineering University of Waterloo Copyright © 2008 by Douglas Wilhelm Harder.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649cc15503460f949884d4/html5/thumbnails/36.jpg)
• Time scale by = 2
22 2
1u2sin
stt
Laplace Transform
Time-Scaling Property
![Page 37: Laplace Transform Douglas Wilhelm Harder Department of Electrical and Computer Engineering University of Waterloo Copyright © 2008 by Douglas Wilhelm Harder.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649cc15503460f949884d4/html5/thumbnails/37.jpg)
• Time scale by = ½
4122
11
usin
s
tt
Laplace Transform
Time-Scaling Property
![Page 38: Laplace Transform Douglas Wilhelm Harder Department of Electrical and Computer Engineering University of Waterloo Copyright © 2008 by Douglas Wilhelm Harder.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649cc15503460f949884d4/html5/thumbnails/38.jpg)
Laplace Transform
Damping Property
• Damped time-scaled trigonometric functions are also shifted
22
22
usin
usin
astte
stt
at
22
22
ucos
ucos
as
astte
s
stt
at
![Page 39: Laplace Transform Douglas Wilhelm Harder Department of Electrical and Computer Engineering University of Waterloo Copyright © 2008 by Douglas Wilhelm Harder.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649cc15503460f949884d4/html5/thumbnails/39.jpg)
Laplace Transform
Time Differentiation Property
• The Laplace transform of the derivative
0fF
f0f
ff
ff
0
00
0
11
ss
dtets
dtetset
dtett
st
stst
stL
![Page 40: Laplace Transform Douglas Wilhelm Harder Department of Electrical and Computer Engineering University of Waterloo Copyright © 2008 by Douglas Wilhelm Harder.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649cc15503460f949884d4/html5/thumbnails/40.jpg)
Laplace Transform
Time Differentiation Property
• The general case is shown with induction:
0f0f
0f0f0f
Ff
12
23121
nn
nnn
nn
s
sss
sst
L
![Page 41: Laplace Transform Douglas Wilhelm Harder Department of Electrical and Computer Engineering University of Waterloo Copyright © 2008 by Douglas Wilhelm Harder.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649cc15503460f949884d4/html5/thumbnails/41.jpg)
Laplace Transform
Time Differentiation Property
• If g(t) = f(t)u(t) then 0 = g(0+) = g(1)(0+) = ···
• Thus the formula simplifies:
• Problem:– The derivative is more complex
sst nn Fg L
ttttdt
dδ0fufg )1(
![Page 42: Laplace Transform Douglas Wilhelm Harder Department of Electrical and Computer Engineering University of Waterloo Copyright © 2008 by Douglas Wilhelm Harder.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649cc15503460f949884d4/html5/thumbnails/42.jpg)
Laplace Transform
Time Differentiation Property
• Example: if g(t) = cos(t)u(t) theng(0–) = 0
g(1)(t) = sin(t)u(t) + (t)
![Page 43: Laplace Transform Douglas Wilhelm Harder Department of Electrical and Computer Engineering University of Waterloo Copyright © 2008 by Douglas Wilhelm Harder.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649cc15503460f949884d4/html5/thumbnails/43.jpg)
Laplace Transform
Time Differentiation Property
• We will demonstrate that– The Laplace transform of a derivative is the
Laplace transform times s– The next six slides give examples that
f(1)(t) = g(t) implies sF(s) = G(s)
1 of 7
![Page 44: Laplace Transform Douglas Wilhelm Harder Department of Electrical and Computer Engineering University of Waterloo Copyright © 2008 by Douglas Wilhelm Harder.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649cc15503460f949884d4/html5/thumbnails/44.jpg)
Laplace Transform
Differentiation of Polynomials
• We now have the following commutative diagram when n > 0
1
!
u
n
n
s
ns
ttsL
ttn n u1L tt ndtd uL
ns
n!
2 of 7
![Page 45: Laplace Transform Douglas Wilhelm Harder Department of Electrical and Computer Engineering University of Waterloo Copyright © 2008 by Douglas Wilhelm Harder.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649cc15503460f949884d4/html5/thumbnails/45.jpg)
Laplace Transform
Differentiation of Trigonometric Functions
• We now have the following commutative diagram
1
1
usin
2
ss
ttsL
tt ucosL ttdtd usinL
12 s
s
3 of 7
![Page 46: Laplace Transform Douglas Wilhelm Harder Department of Electrical and Computer Engineering University of Waterloo Copyright © 2008 by Douglas Wilhelm Harder.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649cc15503460f949884d4/html5/thumbnails/46.jpg)
Laplace Transform
Differentiation of Trigonometric Functions
• We now have the following commutative diagram
1
ucos
2
s
ss
ttsL
11
1
δusin
2
s
tttL ttdtd ucosL
12
2
s
s
4 of 7
![Page 47: Laplace Transform Douglas Wilhelm Harder Department of Electrical and Computer Engineering University of Waterloo Copyright © 2008 by Douglas Wilhelm Harder.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649cc15503460f949884d4/html5/thumbnails/47.jpg)
Laplace Transform
Differentiation of Exponential Functions
• We now have the following commutative diagram
ass
tes at
1
uL
1
δu
as
a
ttae atL te atdtd uL
as
s
5 of 7
![Page 48: Laplace Transform Douglas Wilhelm Harder Department of Electrical and Computer Engineering University of Waterloo Copyright © 2008 by Douglas Wilhelm Harder.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649cc15503460f949884d4/html5/thumbnails/48.jpg)
Laplace Transform
Differentiation of Trigonometric Functions
• We now have the following commutative diagram
22
usin
ss
ttsL
22
ucos
s
s
ttL ttdtd usin L
22 s
s
6 of 7
![Page 49: Laplace Transform Douglas Wilhelm Harder Department of Electrical and Computer Engineering University of Waterloo Copyright © 2008 by Douglas Wilhelm Harder.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649cc15503460f949884d4/html5/thumbnails/49.jpg)
Laplace Transform
Differentiation of Trigonometric Functions
• We now have the following commutative diagram
22
ucos
s
ss
ttsL
1
δusin
22
s
tttL ttdtd ucos L
22
2
s
s
7 of 7
![Page 50: Laplace Transform Douglas Wilhelm Harder Department of Electrical and Computer Engineering University of Waterloo Copyright © 2008 by Douglas Wilhelm Harder.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649cc15503460f949884d4/html5/thumbnails/50.jpg)
Laplace Transform
Frequency Differentiation Property
• The derivative of the Laplace transform
)f(
f
f
fF
0
0
0
)1(
tt
dtett
dtetds
d
dtetds
ds
st
st
st
L stt )1(Ff
![Page 51: Laplace Transform Douglas Wilhelm Harder Department of Electrical and Computer Engineering University of Waterloo Copyright © 2008 by Douglas Wilhelm Harder.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649cc15503460f949884d4/html5/thumbnails/51.jpg)
Laplace Transform
Frequency Differentiation Property
• Consider monomials
1
!
nn
s
nt
2
1 !1
nnn
s
nttt
2
21
!1
!1
!
n
nn
s
ns
nn
s
n
ds
d
![Page 52: Laplace Transform Douglas Wilhelm Harder Department of Electrical and Computer Engineering University of Waterloo Copyright © 2008 by Douglas Wilhelm Harder.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649cc15503460f949884d4/html5/thumbnails/52.jpg)
Laplace Transform
Frequency Differentiation Property
• Consider a sine function
• We have that
but what is ?
1
1sin
2
st
2221
2
1
1
s
s
sds
d
tt sinL
1 of 3
![Page 53: Laplace Transform Douglas Wilhelm Harder Department of Electrical and Computer Engineering University of Waterloo Copyright © 2008 by Douglas Wilhelm Harder.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649cc15503460f949884d4/html5/thumbnails/53.jpg)
Laplace Transform
Frequency Differentiation Property
• Applying integration by parts
00
000
cossin1
cossin1
sin1sin
dtettdtets
dtettts
etts
dtett
stst
ststst
00
000
sincos1
sincos1
cos1cos
dtettdtets
dtettts
etts
dtett
stst
ststst
2 of 3
![Page 54: Laplace Transform Douglas Wilhelm Harder Department of Electrical and Computer Engineering University of Waterloo Copyright © 2008 by Douglas Wilhelm Harder.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649cc15503460f949884d4/html5/thumbnails/54.jpg)
Laplace Transform
Frequency Differentiation Property
• Substituting
0000
sincos1
sin1
sin dtettdtets
dtets
dtett stststst
1
1
1
2sin
1
21sin
sin
1
1
1
1
sin1
1
1
11sin
222
22
2
222
22
sds
d
s
stt
sss
stt
s
tt
ssss
tts
s
ssstt
L
L
L
LL
3 of 3
![Page 55: Laplace Transform Douglas Wilhelm Harder Department of Electrical and Computer Engineering University of Waterloo Copyright © 2008 by Douglas Wilhelm Harder.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649cc15503460f949884d4/html5/thumbnails/55.jpg)
Laplace Transform
Time Integration Property
• The Laplace transform of an integral
s
s
des
tdde
tdde
dtedd
st
st
st
sttt
F
f1
f
f
ff
0
0
0
0 00
L
![Page 56: Laplace Transform Douglas Wilhelm Harder Department of Electrical and Computer Engineering University of Waterloo Copyright © 2008 by Douglas Wilhelm Harder.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649cc15503460f949884d4/html5/thumbnails/56.jpg)
Laplace Transform
Time Integration Property
• We will demonstrate that– The Laplace transform of an integral is the
Laplace transform over s– The next six slides give examples that
implies
t
dt0
f)g( s
ss
FG
1 of 7
![Page 57: Laplace Transform Douglas Wilhelm Harder Department of Electrical and Computer Engineering University of Waterloo Copyright © 2008 by Douglas Wilhelm Harder.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649cc15503460f949884d4/html5/thumbnails/57.jpg)
Laplace Transform
Integration of Polynomials
• We now have the following commutative diagram
n
n
s
n
s
s
t
!1
L
11
1
nt
nL
tnd
0
L
1
!ns
n
2 of 7
![Page 58: Laplace Transform Douglas Wilhelm Harder Department of Electrical and Computer Engineering University of Waterloo Copyright © 2008 by Douglas Wilhelm Harder.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649cc15503460f949884d4/html5/thumbnails/58.jpg)
Laplace Transform
Integration of Exponential Functions
• We now have the following commutative diagram
ass
s
e at
11
L
assa
ea
at
111
11L
ass 1
t
a de0
L
3 of 7
![Page 59: Laplace Transform Douglas Wilhelm Harder Department of Electrical and Computer Engineering University of Waterloo Copyright © 2008 by Douglas Wilhelm Harder.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649cc15503460f949884d4/html5/thumbnails/59.jpg)
Laplace Transform
Integration of Trigonometric Functions
• We now have the following commutative diagram
1
11
sin
2
ss
s
tL
1
1
cos1
2
s
s
s
tL
112 ss
t
d0
sin L
4 of 7
![Page 60: Laplace Transform Douglas Wilhelm Harder Department of Electrical and Computer Engineering University of Waterloo Copyright © 2008 by Douglas Wilhelm Harder.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649cc15503460f949884d4/html5/thumbnails/60.jpg)
Laplace Transform
Integration of Trigonometric Functions
• We now have the following commutative diagram
)sin(tL
1
12 s
t
d0
cos L
1
1
cos
2
s
s
s
s
tL
5 of 7
![Page 61: Laplace Transform Douglas Wilhelm Harder Department of Electrical and Computer Engineering University of Waterloo Copyright © 2008 by Douglas Wilhelm Harder.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649cc15503460f949884d4/html5/thumbnails/61.jpg)
Laplace Transform
Integration of Trigonometric Functions
• We now have the following commutative diagram
22
1
sin
ss
s
tL
22
11
)cos(11
s
s
s
tL
22 ss
t
d0
sin L
6 of 7
![Page 62: Laplace Transform Douglas Wilhelm Harder Department of Electrical and Computer Engineering University of Waterloo Copyright © 2008 by Douglas Wilhelm Harder.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649cc15503460f949884d4/html5/thumbnails/62.jpg)
Laplace Transform
Integration of Trigonometric Functions
• We now have the following commutative diagram
22
1
cos
s
s
s
s
tL
22
1
)sin(1
s
tL
22
1
s
t
d0
cos L
7 of 7
![Page 63: Laplace Transform Douglas Wilhelm Harder Department of Electrical and Computer Engineering University of Waterloo Copyright © 2008 by Douglas Wilhelm Harder.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649cc15503460f949884d4/html5/thumbnails/63.jpg)
Laplace Transform
The Convolution
• Define the convolution to be
• Then
dt
dtt
gf
gfgf
ssj
tt
sst
GF2
1gf
GFgf
![Page 64: Laplace Transform Douglas Wilhelm Harder Department of Electrical and Computer Engineering University of Waterloo Copyright © 2008 by Douglas Wilhelm Harder.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649cc15503460f949884d4/html5/thumbnails/64.jpg)
Laplace Transform
Integration
• As a special case of the convolution
s
s
ss
sttsdt
F1F
uff0
LL
![Page 65: Laplace Transform Douglas Wilhelm Harder Department of Electrical and Computer Engineering University of Waterloo Copyright © 2008 by Douglas Wilhelm Harder.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649cc15503460f949884d4/html5/thumbnails/65.jpg)
Laplace Transform
Summary
• We have seen these Laplace transforms:
1
2
!u
1u
1u
1δ
nn
s
ntt
stt
st
t
1
ucos
1
1usin
1
1u
2
2
s
stt
stt
stet
![Page 66: Laplace Transform Douglas Wilhelm Harder Department of Electrical and Computer Engineering University of Waterloo Copyright © 2008 by Douglas Wilhelm Harder.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649cc15503460f949884d4/html5/thumbnails/66.jpg)
Laplace Transform
Summary
• We have seen these properties:– Linearity– Damping– Time scaling
– Time differentiation– Frequency differentiation– Time integration
sbsatbta GF)g()f( aste at Ff
a
s
aat F
1f
sstt nn Fuf
sttt nn )(Fuf
s
sd
t Ff
0
![Page 67: Laplace Transform Douglas Wilhelm Harder Department of Electrical and Computer Engineering University of Waterloo Copyright © 2008 by Douglas Wilhelm Harder.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649cc15503460f949884d4/html5/thumbnails/67.jpg)
Laplace Transform
Summary
• In this topic:– We defined the Laplace transform– Looked at specific transforms– Derived some properties– Applied properties
![Page 68: Laplace Transform Douglas Wilhelm Harder Department of Electrical and Computer Engineering University of Waterloo Copyright © 2008 by Douglas Wilhelm Harder.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649cc15503460f949884d4/html5/thumbnails/68.jpg)
Laplace Transform
References• Lathi, Linear Systems and Signals, 2nd Ed., Oxford
University Press, 2005.• Spiegel, Laplace Transforms, McGraw-Hill, Inc., 1965.• Wikipedia,
http://en.wikipedia.org/wiki/Laplace_Transform
![Page 69: Laplace Transform Douglas Wilhelm Harder Department of Electrical and Computer Engineering University of Waterloo Copyright © 2008 by Douglas Wilhelm Harder.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649cc15503460f949884d4/html5/thumbnails/69.jpg)
Usage Notes
• These slides are made publicly available on the web for anyone to use
• If you choose to use them, or a part thereof, for a course at another institution, I ask only three things:– that you inform me that you are using the slides,– that you acknowledge my work, and– that you alert me of any mistakes which I made or changes which
you make, and allow me the option of incorporating such changes (with an acknowledgment) in my set of slides
Sincerely,
Douglas Wilhelm Harder, MMath