Chapter3 laplace
49
S ignals & S ystem s Chapter 3 The Laplace Transform NC212 Signals and Systems : 2 / 2554
Transcript of Chapter3 laplace
Signals & Systems
Chapter 3
The Laplace Transform
INC212 Signals and Systems : 2 / 2554
Laplace Transform of unit-step Function
01)(
)()(
dteX
dtetxX
tj
tj
0
)()(
)(
dteX
dteeX
tj
tjt
ssX
sj
jjX
ej
jX
ej
jX
dtejX
j
tt
tj
tj
1)(
1)(
]0[1
)(
][1
)(
)(
0)(
0)(
0
)(
Chapter 3 The Laplace TransformINC212 Signals and Systems : 2 / 2554
Laplace Transform of Signals
jsdtetxsX st
;)()(
0)()( dtetxsX st
One-side transform 0;0)( ttx
Chapter 3 The Laplace TransformINC212 Signals and Systems : 2 / 2554
Laplace Transform of Signals
0,0
0,1)()(
t
ttutx
0
)()( dtetxsX st
0,1
)]([
0),1
(0
)1()()]([
0
0
ss
tuL
sss
e
dtedtetutuL
st
stst
Chapter 3 The Laplace TransformINC212 Signals and Systems : 2 / 2554
Relationship between the FT and ℒT
jssXX
)()(
0)()( dtetxX tj
0
)()( dtetxsX st
)]([)(;)]([)(
)()(1 sXtxtxsX
sXtx
LL
One-side transform or Forward transform0;0;0)( jsttx
Chapter 3 The Laplace TransformINC212 Signals and Systems : 2 / 2554
Common ℒT Pairs
Chapter 3 The Laplace TransformINC212 Signals and Systems : 2 / 2554
Region of Convergence (ROC) 0);()(1 tuAetg t
0
)(
0
)(1 )()( dteeAdteAdtetuAesG tjttsstt
Chapter 3 The Laplace TransformINC212 Signals and Systems : 2 / 2554
)Re()(1 ss
AsG
Region of Convergence (ROC) 0);()()( 12 tgtuAetg t
0
)(0
)(2 )()( dteeAdteAdtetuAesG tjttsstt
Chapter 3 The Laplace TransformINC212 Signals and Systems : 2 / 2554
)Re()()( 12 ssGs
AsG
Region of Convergence (ROC) Example:
)()()( 2 tuetuetx tt
0
)2()1(2 ][)]()([)()( dteeAdtetuetuesXtx tstsstttL
Chapter 3 The Laplace TransformINC212 Signals and Systems : 2 / 2554
12
1
1
1)(
sssX
Region of Convergence (ROC) Example:
)()()( 2 tuetuetx tt
11
1)(
stue t
L
Chapter 3 The Laplace TransformINC212 Signals and Systems : 2 / 2554
212
1
1
1)()( 2
sstuetue tt
L
22
1)(2
stue t
L
Properties of the ℒT Linearity
)()(and)()( sVtvsXtxLL
)()()()( sbVsaXtbvtax L
Chapter 3 The Laplace TransformINC212 Signals and Systems : 2 / 2554
Properties of the ℒT)]()([ tuetuL t
)1(
12)()(
1
11)()(
1
1)(and
1)(
ss
stuetu
sstuetu
stue
stu
t
t
t
Chapter 3 The Laplace TransformINC212 Signals and Systems : 2 / 2554
Example: Linearity
Properties of the ℒT Right Shift in Time
)()( sXtx
)()()( sXectuctx cs
Chapter 3 The Laplace TransformINC212 Signals and Systems : 2 / 2554
Properties of the ℒT Example: Right Shift in Time
t
cttx
other all,0
0,1)(
)()()( ctututx
s
e
s
e
sctutu
cscs
11)()(
Chapter 3 The Laplace TransformINC212 Signals and Systems : 2 / 2554
Properties of the ℒT
Chapter 3 The Laplace TransformINC212 Signals and Systems : 2 / 2554
Computation of the Inverse ℒT
jc
jc
stdsesXj
tx )(2
1)(
Chapter 3 The Laplace TransformINC212 Signals and Systems : 2 / 2554
011
1
011
1
)(
)(
)(
)()(
asasasasA
bsbsbsbsB
sA
sBsX
NN
NN
MM
MM
)())((
)()(
)())(()(
0)(
21
21
NN
NN
pspspsa
sBsX
pspspsasA
sA
Let p1, p2, …, pN denote the roots of the equation
The pi for i = 1, 2,…,N are called the poles of X(s)
Chapter 3 The Laplace TransformINC212 Signals and Systems : 2 / 2554
The Inverse ℒT using Partial-Fraction Expansion
Distinct Poles
NisXpsc
ps
c
ps
c
ps
csX
ipsii
N
N
,,2,1,)]()[(
)(2
2
1
1
0,)( 2121 tececectx tp
Ntptp N
Chapter 3 The Laplace TransformINC212 Signals and Systems : 2 / 2554
Example: Distinct Poles
3,1,0
)3)(1(0)(
)3)(1(34)(
34
2)(
321
23
23
ppp
ssssA
sssssssA
sss
ssX
0,6
1
2
1
3
2)( 3 teetx tt31
)(
)3()1(0)(
321
321
s
c
s
c
s
csX
s
c
s
c
s
csX
6
1
)1(
2)]()3[(
2
1
)3(
2)]()1[(
3
2
)3)(1(
2)]([
,,2,1,)]()[(
3
33
1
12
0
01
s
s
s
s
s
s
psii
ss
ssXsc
ss
ssXsc
ss
sssXc
NisXpsci
Chapter 3 The Laplace TransformINC212 Signals and Systems : 2 / 2554
Distinct Poles with 2 or More Poles Complex
N
N
ps
c
ps
c
ps
c
ps
csX
3
3
1
1
1
1)(
tpN
tpt
ttptp
tpN
tptptp
N
N
ececctectx
ctececec
ecececectx
3
11
311
311
1111
311
)cos(2)(
)cos(2
)(
Chapter 3 The Laplace TransformINC212 Signals and Systems : 2 / 2554
Example: Distinct Poles with 2 or More Poles Complex
1,1,1
)1)(1)(1(
243)(
243
12)(
321
23
23
2
pjpjp
sjsjs
ssssA
sss
sssX
0,4)87.126cos(5)( tetetx tt
111)(
)1()1()1()(
311
311
s
c
js
c
js
csX
s
c
js
c
js
csX
422
12)]()1[(
87.1263
4tan180
;2
54
4
9
22
3
)1)(1(
12)]()1[(
,,2,1,)]()[(
1
2
2
13
1
1
1
1
1
2
11
s
s
js
js
psii
ss
sssXsc
c
c
jc
sjs
sssXjsc
NisXpsci
Chapter 3 The Laplace TransformINC212 Signals and Systems : 2 / 2554
Repeated Poles
)(
)()(
sA
sBsX
0,)( 11111
121
tececetctecectx tp
Ntp
rtpr
rtptp Nr
N
N
r
rr
r
ps
c
ps
c
ps
c
ps
c
ps
csX
1
1
12
1
2
1
1
)()()(
1)]()[(
,,2,1,)]()[(
1 psr
r
psii
sXpsc
NrrisXpsci
1
1
)]()[(!2
1;2
)]()[(!1
1;1
12
2
2
11
ps
rr
ps
rr
sXpsds
dci
sXpsds
dci
1
)]()[(!
1
1,,2,1
1
ps
ri
i
ir sXpsds
d
ic
ri
Chapter 3 The Laplace TransformINC212 Signals and Systems : 2 / 2554
Example: Repeated Poles
23
15)(
3
ss
ssX
0,2)( 2 teteetx ttt
2)1(1)( 3
221
s
c
s
c
s
csX
1)1(
15)]()2[(
22
15)]()1[(
1)2(
9]
2
15[)]()1[(
!1
1
2223
11
22
12
11
21
s
s
ss
sss
s
ssXsc
s
ssXsc
ss
s
ds
dsXs
ds
dc
Chapter 3 The Laplace TransformINC212 Signals and Systems : 2 / 2554
Case when M ≥ N
)(
)()()(
sA
sRsQsX
)()();(
)();(
)()()(
)(
)()(),()()(
sVtvtv
stdt
dtq
tvtqtx
sA
sRsVsVsQsX
N
N
N
Chapter 3 The Laplace TransformINC212 Signals and Systems : 2 / 2554
011
1
011
1
)(
)(
)(
)()(
asasasasA
bsbsbsbsB
sA
sBsX
NN
NN
MM
MM
)(
)(
)(*)(
)()(sQ
sR
sQsA
sBsA
Example: Case when M ≥ N
0,6145.06145.20)(4)()(
0,6145.06145.20)(
4495.0
6145.0
4495.4
6145.20
24
1220)(
)(4)()(
4)(
)()()(24
12204
24
42)(
4495.04495.4
4495.04495.4
2
22
3
teettdt
dtx
teetv
ssss
ssV
ttdt
dtq
ssQ
sVsQsXss
ss
ss
sssX
tt
tt
Chapter 3 The Laplace TransformINC212 Signals and Systems : 2 / 2554
Transform of the I/O Differential Equation First-Order Case
)()()(
)(
)()(
)()0(
)(
sXsHsYas
bsH
sXas
bsY
sXas
b
as
ysY
H(s) Transfer Function (TF) of the system
)()0()()(
)()()0()(
)()()(
sbXysYas
sbXsaYyssY
tbxtaydt
tdy
Chapter 3 The Laplace TransformINC212 Signals and Systems : 2 / 2554
Transform of the I/O Differential Equation Example: First-Order Case
)(1
1
1
)0()( sX
RCs
RC
RCs
ysY
)(1
)(1)(
txRC
tyRCdt
tdy
0,1)0()(
1
11
1
)0()(
1
1
1
)0()(
)1()1(
teeyty
RCs
RC
sRCs
ysY
sRCs
RC
RCs
ysY
tRCtRC
Chapter 3 The Laplace TransformINC212 Signals and Systems : 2 / 2554
Transform of the I/O Differential Equation Second-Order Case
012
01
012
01 )(;)()(asas
bsbsHsX
asas
bsbsY
)(
)0()0()0()(
)()()()0()()0()0()(
)()(
)()()(
012
01
012
1
01012
01012
2
sXasas
bsb
asas
yaysysY
sXbssXbsYayssYaysysYs
txbdt
tdxbtya
dt
tdya
dt
tyd
If initial condition = 0 :
Chapter 3 The Laplace TransformINC212 Signals and Systems : 2 / 2554
Transform of the I/O Differential Equation Example: Second-Order Case
x(t) = u(t) so that X(s) = 1/s; initial cond. = 0
86
2)(
)(2)(8)(
6)(
2
2
2
sssH
txtydt
tdy
dt
tyd
0,25.05.025.0)(
4
25.0
2
5.025.0)(
1
86
2)()()(
42
2
teety
ssssY
ssssXsHsY
tt
Chapter 3 The Laplace TransformINC212 Signals and Systems : 2 / 2554
Transform of the I/O Differential Equation Example: Second-Order Case
x(t) = u(t) with the initial condition
0,75.15.225.0)(
4
75.1
2
5.225.0)(
)86(
28
1
86
2
86
8)(
42
2
2
22
teety
ssssY
sss
sssssss
ssY
tt
2)0(
1)0(
y
y
Chapter 3 The Laplace TransformINC212 Signals and Systems : 2 / 2554
Transform of the I/O Differential Equation Nth-Order Case
011
1
01
011
1
01
)(
)()()(
)()(
asasas
bsbsbsH
sXasasas
bsbsbsX
sA
sBsY
NN
N
MM
NN
N
MM
)0()0()0()(
)(;)(
)()(
)(
)(
)()(;
)()()(
1
011
1011
1
0
1
0
yaysysC
asasassAbsbsbsbsB
sXsA
sB
sA
sCsY
dt
txdb
dt
tyda
dt
tyd
NN
NMM
MM
M
ii
i
i
N
ii
i
iN
N
Chapter 3 The Laplace TransformINC212 Signals and Systems : 2 / 2554
Transform of the I/O Convolution Integral
)(
)()(
)()(
)()()(
0,)()()()()(0
sX
sYsH
sHth
sXsHsY
tdtxhtxthtyt
Chapter 3 The Laplace TransformINC212 Signals and Systems : 2 / 2554
Transform of the I/O Convolution Integral Example: Determining the TF
1
1)(
4)2(
2
1
32)(
0,2cos32)(
2
2
ssX
s
s
sssY
tteety tt
sss
ssss
sssssss
ss
s
ss
ss
sssH
84
162]4)2[(
)2)(1(]4)2][(3)1(2[4)2(
)2)(1(3
)1(21
14)2(
21
32
)(
23
2
2
2
2
2
Chapter 3 The Laplace TransformINC212 Signals and Systems : 2 / 2554
Transform of the I/O Convolution Integral Finite-Dimensional Systems
M
ii
i
i
N
ii
i
iN
N
MM
MM
NN
N
NN
N
MM
MM
dt
txdb
dt
tyda
dt
tyd
sXbsbsbsbsYasasas
asasas
bsbsbsbsH
0
1
0
011
1011
1
011
1
011
1
)()()(
)()()()(
)(
Chapter 3 The Laplace TransformINC212 Signals and Systems : 2 / 2554
Transform of the I/O Convolution Integral Poles and zeros of a Systems
)())((
)())(()(
)(
21
21
011
1
011
1
N
MM
NN
N
MM
MM
pspsps
zszszsbsH
asasas
bsbsbsbsH
zi : “zeros of H (s)” or “zeros of system”pi : “poles of H (s)” or “poles of system”N : “number of poles of system” or “order N of system”
Chapter 3 The Laplace TransformINC212 Signals and Systems : 2 / 2554
Transform of the I/O Convolution Integral Example: Third-Order System
jpjpp
jzjz
jsjss
jsjssH
sss
sssH
1,1,4
3and3
)1)(1)(4(
)3)(3(2)(
8106
20122)(
321
11
23
2
Chapter 3 The Laplace TransformINC212 Signals and Systems : 2 / 2554
Exercises Sketch the pole-zero plot and ROC for
these signals.
Chapter 3 The Laplace TransformINC212 Signals and Systems : 2 / 2554
)()( 8 tuetx t
)()20cos()( 3 tutetx t
)()()( 52 tuetuetx tt
Exercises Using the time-shifting property, find
the LT of these signals.
Chapter 3 The Laplace TransformINC212 Signals and Systems : 2 / 2554
)1()()( tututx
)2(3)( )2(3 tuetx t
)2(3)( 3 tuetx t
)1())1(sin(5)( tuttx
Exercises Find the inverse LT of these functions.
Chapter 3 The Laplace TransformINC212 Signals and Systems : 2 / 2554
)8(
24)(
sssX
34
20)(
2
sssX
44)(
2
ss
ssX
44)(
2
2
ss
ssX
736
5)(
2
sssX
132
2)(
2
ss
ssX
Chapter 3 The Laplace TransformINC212 Signals and Systems : 2 / 2554
Direct Construction of the TF RLC Circuits
)0()()(
)()(
LisLsIsVdt
tdiLtv
)0(1
)(1
)(
)(1
)0()(
)(1)(
vs
sICs
sV
sIC
vssV
tiCdt
tdv
)()(
)()(
sRIsV
tRitv
Chapter 3 The Laplace TransformINC212 Signals and Systems : 2 / 2554
Direct Construction of the TF Series and Parallel Connection
)()()(
)()(
)()()(
)()(
21
12
21
21
sIsZsZ
sZsI
sIsZsZ
sZsI
)()()(
)()(
)()()(
)()(
21
22
21
11
sVsZsZ
sZsV
sVsZsZ
sZsV
Chapter 3 The Laplace TransformINC212 Signals and Systems : 2 / 2554
Direct Construction of the TF Example: Series RLC Circuit
)1()(
1)(
)()1()(
1
)()1(
1)(
2
2
LCsLRs
LCsH
sXLCsLRs
LC
sXCsRLs
CssVc
)1()(
)()(
)()1()(
)(
)()1(
)(
2
2
LCsLRs
sLRsH
sXLCsLRs
sLR
sXCsRLs
RsVR
Output = VR(s)Output = VC(s)
Chapter 3 The Laplace TransformINC212 Signals and Systems : 2 / 2554
Direct Construction of the TF Interconnections of Integrators
Chapter 3 The Laplace TransformINC212 Signals and Systems : 2 / 2554
Direct Construction of the TF Example:
127
178
)4)(3(
178)(
)()4)(3(
178
)()()4)(3(
5)()()(
)()4)(3(
5
)(14
1
3
1
)]()([3
1)(
)()(3)()(
)(4
1)(
)()(4)(
2
22
2
2
12
212
1
11
ss
ss
ss
sssH
sXss
ss
sXsXss
ssXsQsY
sXss
s
sXss
sXsQs
sQ
sXsQsQssQ
sXs
sQ
sXsQssQ
Chapter 3 The Laplace TransformINC212 Signals and Systems : 2 / 2554
TF of Block Diagrams Parallel Interconnection
)()()(
)())()((
)()()()()(
)()()(
)()()(
)()()(
21
21
21
22
11
21
sHsHsH
sXsHsH
sXsHsXsHsY
sXsHsY
sXsHsY
sYsYsY
Chapter 3 The Laplace TransformINC212 Signals and Systems : 2 / 2554
TF of Block Diagrams Series Connection
)()()()()(
)()()()()(
)()()(
)()()(
2112
122
122
11
sHsHsHsHsH
sXsHsHsYsY
sYsHsY
sXsHsY
Chapter 3 The Laplace TransformINC212 Signals and Systems : 2 / 2554
TF of Block Diagrams Feedback Connection
)()(1
)()(
)()()(1
)()(
)]()()()[()(
)()()(
)()()(
)()()(
21
1
21
1
21
2
21
11
sHsH
sHsH
sXsHsH
sHsY
sYsHsXsHsY
sYsHsX
sYsXsX
sXsHsY
)()(1
)()(
21
1
sHsH
sHsH
Chapter 3 The Laplace TransformINC212 Signals and Systems : 2 / 2554
Chapter 3 The Laplace TransformINC212 Signals and Systems : 2 / 2554
