chapter 12 introduction to the laplace transformsdyang/courses/circuits/ch12... · 2014-06-09 ·...
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1
Chapter 12 Introduction To The Laplace Transform 12.1 Definition of the Laplace Transform
12.2-3 The Step & Impulse Functions
12.4 Laplace Transform of specific functions
12.5 Operational Transforms
12.6 Applying the Laplace Transform
12.7 Inverse Transforms of Rational Functions
12.8 Poles and Zeros of F(s)
12.9 Initial- and Final-Value Theorems
2
Overview
Laplace transform is a technique that is
particularly useful in linear circuit analysis when:
1. Considering transient response (e.g. switching)
of circuits with multiple nodes and meshes.
2. The sources are more complicated than the
simple dc level jumps.
3. Introducing the concept of transfer function to
analyze frequency-dependent sinusoidal
steady-state response (Chapters 13, 14).
3
Key points
What is the definition of the Laplace transform?
What are the Laplace transforms of unit step,
impulse, exponential, and sinusoidal functions?
What are the Laplace transforms of the
derivative, integral, shift, and scaling of a
function?
How to perform partial fraction expansion for a
rational function F(s) and perform the inverse
Laplace transform?
4
Section 12.1
Definition of the Laplace
Transform
5
What is Laplace transform?
Transforming a real function f (t) of real variable t
to a complex function F(s) of complex variable s:
The integral will converge (1) over a portion of
the s-plane (e.g. Re(s) > 0), and (2) for most of the
functions except for those of little interest (e.g. t t).
F(s) is determined by f (t) only for t > 0-. Thus we
use it to predict the response after initial
conditions have been established.
.)()}({)(0
CdtetftfLsF st
kernel
6
Section 12.2, 12.3
The Step and Impulse
Functions
1. Definition of unit step function u(t)
2. Definition of impulse function d(t)
3. Laplace transforms of d(t) and d'(t)
7
u(t) can be approximated by the limit of a linear
ramp function:
The unit step function u(t)
.0for ,1
;0for ,0)(
t
ttu
8
Representation of time shift and reversal
)( atKu
)( taKu
Time reversal:
Time shift:
9
Example 12.1: A pulse of finite width (1)
y1 y2 y3
Q: Express the piecewise linear function f (t) as
superposition of 3 functions.
iii btaty )(
10
Example 12.1 (2)
For each interval, f (t) can be expressed as the
product of a linear function and a square pulse
(difference between two step functions).
For example, for 1 < t < 3, the corresponding
linear and square pulse functions are:
The entire function can be represented by:
).3()1()( and ,42)( 22 tututptty
).()()()()()()( 332211 tptytptytptytf
11
The impulse function d(t)
An idealized math representation of sharply
peaked stimulus:
d(t) has (1) zero duration, (2) infinite peak
amplitude, (3) unit area (strength).
.1)(
;otherwise ,0
;0 ,)(
dtt
tt
d
d
12
d(t) is the derivative of u(t)
)(lim)(0
tftu
)()(lim)(0
tutft
d
13
The sifting property of d-function
Sampling of f (t) at t = a (>0) can be formulated
by integral of f (t) times d(t-a):
It can be used in calculating the Laplace
transform of a d-function:
).()(lim)(
)()(lim)()(
0
00
afdtataf
dtattfdtattf
a
a
a
a
d
dd
.any for ,1)()( )0(
0CsedtettL sst
dd
14
The derivative of d-function
area = 1
total area = 0
one-sided
area = 1/
)(lim)(0
tft
d
)(lim)(0
tft
d
15
.2
)(lim
2
)(lim
2lim
11lim)(lim)(
2
0020
0 2
0
2000
ss
see
s
see
s
ee
dtedtedtetftL
ssssss
ststst
d
Laplace transform of d'(t)
)(lim)(0
tft
d
Even d'(t) has well defined Laplace transform!
16
Section 12.4
Laplace Transform of
Specific Functions
17
E.g. Unit step function
.0Re if ,1
1)()(
0
00
sss
e
dtedtetutuL
st
stst
Re
Im
18
E.g. Single-sided exponential function
(a > 0)
Re
Im
-a
.Re if ,1
)(0
)(
0
)(
0
asassa
e
dtedteeeL
tsa
tsastatat
19
E.g. Sinusoidal function
.11
2
1
2
1
2)(sinsin
220
)()(
00
sjsjsjdtee
j
dtej
eedtettL
tjstjs
sttjtj
st
20
List of Laplace transform pairs (1)
.!
,2
1
3
2
n
n
s
ntL
stL
Laplace
transform of
polynomial
functions:
astue
sttu
stu
t
sFtf
at
1)(lexponentia
1)(ramp
1)(step
1)(impulse
)()(Type
2
d
21
List of Laplace transform pairs (2)
Damping by exponential decay function causes
a shift along the real axis in the s-domain.
22
2
22
22
)()(sinsine damped
)(
1)(ramp damped
)(coscosine
)(sinsine
)()(Type
astute
astute
s
stut
stut
sFtf
at
at
22
Section 12.5
Operational Transforms
23
What are operational transforms?
Operational transforms indicate how the
mathematical operations performed on either f (t)
or F(s) are converted into the opposite domain.
Useful in calculating the Laplace transform of a
function g(t) derived by performing some math
operation on f (t) with known F(s).
24
First-order time derivative
).0()()()0(0
))(()(
)()(
0
00
0
fssFdtetfsf
dtsetfetf
dtetftfL
st
stst
st
E.g.
.1)0(1
)(
,1
)( ),()(
us
stL
stuLtut
d
d
initial condition
integration by
parts
if f ()e-s() = 0
25
Higher-order time derivatives
).0()0()()0()0()(
)0()()()(
).0()()( ),()(Let
2
fsfsFsffssFs
gssGtgLtfL
fssFsGtftg
nth-order derivative:
2nd-order derivative:
.)0()0()0(
)()(
)1(21
)(
nnn
nn
ffsfs
sFstfL
initial conditions
initial conditions
26
Time integral
.)( ,)( );()( ,)()(
where,)()()()(
0
00
0
0
s
etvetvtftudxxftu
dttvtudtedxxfdxxfL
stst
t
sttt
.)(
)(1
)()(
)()(
)()()()()(
)0(0
0
)(
0
0
0
0
00
0
s
sFsF
ss
edxxf
s
edxxf
dts
etf
s
edxxf
dttvtutvtudxxfL
ss
ststt
t
1 0 finite
The formula is valid only if the function is integrable.
27
Scaling
Intuitively, a larger value of a corresponds to a
narrower function in the time domain but a
broader function in the frequency domain. The
width of f (at) times the width of F(s/a) is a
constant independent of a.
.0 if ,)(
)()(
,let ;)()(
0
0
aa
asF
a
tdetfatfL
tatdteatfatfL
ats
st
28
Translation in the t and s domains
.0for ),()()( asFeatuatfL as
Translation in the time domain:
).()( asFetfL at
Translation in the frequency domain:
Both relations can be proven by change of
variable of integration.
29
Section 12.7
Inverse Transforms of
Rational Functions
1. Distinct real roots
2. Distinct complex roots
3. Repeated real roots
4. Repeated complex roots
5. Improper rational functions
30
Why only rational functions?
For linear, lumped-parameter circuits with
constant component parameters, the s-domain
expression for v(t), i(t) are always rational
functions, i.e. ratio of two polynomials:
General inverse Laplace transform:
...
..
)(
)()(
0
1
1
0
1
1
bsbsb
asasa
sD
sNsF
m
m
m
m
n
n
n
n
,)(2
1)()(
1
ir
ir
stdsesFi
sFLtf
involves with complex integral.
31
How to calculate?
If F(s) is a proper (m > n) rational function, the
inverse transform is calculated by (1) partial
fraction expansion, (2) individual inverse
transforms (4 types).
If F(s) is an improper (m n) rational function,
decompose F(s) as the summation of a
polynomial function and a proper rational
function, which are inverse transformed
individually.
32
Type I: D(s) has distinct real roots (1)
.68)6)(8(
)12)(5(96)( 321
s
K
s
K
s
K
sss
sssF
.120)6)(8(
)12)(5(96
)6)(8(
)12)(5(96
;68
)(
1
0
0
3210
Kss
ss
s
sK
s
sKKssF
s
s
s
.48)6)((
,72)8)((
63
82
s
s
ssFK
ssFK
33
Type I: D(s) has distinct real roots (2)
).( 4872120
148
172
1120
6
148
8
172
1120)(
,6
48
8
72120)(
68
61811
111
tuee
es
Les
Ls
L
sL
sL
sLtf
ssssF
tt
tt
The circuit is over-damped.
34
Type II: D(s) has distinct complex roots (1)
.86)43)((
,86)43)(6(
)3(100)43)((
,12)6)((
,43436
)(
2433
43
432
61
321
KjjssFK
jjss
sjssFK
ssFK
js
K
js
K
s
KsF
js
js
js
s
.43 ,6 :)( of roots ,)256)(6(
)3(100)(
2jssD
sss
ssF
Conjugate roots must have conjugate coefficients.
35
Type II: D(s) has distinct complex roots (2)
).()534cos(2012
)()86(Re212
)()86()86(12
)()86()86(12
43
86
43
86
6
12)(
,43
86
43
86
6
12)(
36
436
4436
)43()43(6
111
tutee
tuejee
tuejejee
tuejeje
js
jL
js
jL
sLtf
js
j
js
j
ssF
tt
tjtt
tjtjtt
tjtjt
5310
The circuit shows over-damped (or 1st-order)
and under-damped characteristics.
36
highest order
Type III: D(s) has repeated real roots (1)
.5)5()5()5(
)25(100)( 4
2
3
3
21
3
s
K
s
K
s
K
s
K
ss
ssF
.400)5(
)20(100)5(
;)5()5()5(
)25(100)5)(()(
2
2
432
3
1
3
KG
sKsKKs
sK
s
sssFsG
37
Type III: D(s) has repeated real roots (2)
.5)5()5()5(
)25(100)( 4
2
3
3
21
3
s
K
s
K
s
K
s
K
ss
ssF
.10025
2500)5(
);5(2)5()5(3
2500)25(100)(
;)5()5()5()25(100
)(
3
432
32
1
22
2
432
3
1
KG
sKKs
sssK
ss
sssG
sKsKKs
sK
s
ssG
38
Type III: D(s) has repeated real roots (3)
.5)5()5()5(
)25(100)( 4
2
3
3
21
3
s
K
s
K
s
K
s
K
ss
ssF
.20 ,240125
5000)5(
;2)255)(5(25000
)(
);5(2)52()5(2500
)(
44
43
2
13
432
2
12
KKG
Ks
sssK
ssG
sKKs
ssK
ssG
39
Type III: D(s) has repeated real roots (4)
.5
20
)5(
100
)5(
40020)(
23
sssssF
).(20!1
100!2
40020
5
20
)5(
100
)5(
40020)(
5552
1
2
1
3
11
tueet
et
sL
sL
sL
sLtf
ttt
The circuit shows critically-damped behavior.
40
Type IV: D(s) has repeated complex roots (1)
.43)43(43)43(
)43()43(
768
)256(
768)(
*
2
2
*
12
2
1
2222
js
K
js
K
js
K
js
K
jsjssssF
.3)43(.12)8(
768)43(
;43
)43(
)43(
)43()43(
)43(
768)43)(()(
212
2*
22
2*
121
2
2
jjGKKj
jG
js
jsK
js
jsKjsKK
jsjssFsG
highest order
41
Type IV: D(s) has repeated complex roots (2)
43
3
)43(
12
43
3
)43(
12)(
22 js
j
jsjs
j
jssF
).( 4sin64cos24
)( )904cos(64cos24
)( ..3.12
..43
903..
)43(
12)(
33
33
)904(343
1
2
1
tutette
tutette
tucceeccete
ccjs
Lccjs
Ltf
tt
tt
tjttjt
The circuit shows critically- and under-damped
characteristics.
42
Useful transformation pairs
)()cos(2)()(
)()cos(2
)()(
1
)(1
)()(
2
*
2
*
2
tutteKjs
K
js
K
tuteKjs
K
js
K
tuteas
tueas
tfsF
K
t
K
t
at
at
43
Inverse transform of improper rational functions
( ) .5
50
4
20104
204
3002006613)(
2
2
234
ssss
ss
sssssF
).(5020
)(10)(4)()(
54 tuee
ttttf
tt
ddd
44
Section 12.8
Poles and Zeros of F(s)
45
Definition
F(s) can be expressed as the ratio of two
factored polynomials N(s)/D(s).
The roots of the denominator D(s) are called
poles and are plotted as Xs on the complex s-
plane.
The roots of the numerator N(s) are called zeros
and are plotted as os on the complex s-plane.
46
Example
.)]86()][86()[10(
)]43()][43()[5(
)(
)()(
jsjsss
jsjss
sD
sNsF
47
Key points
What is the definition of the Laplace transform?
What are the Laplace transforms of unit step,
impulse, exponential, and sinusoidal functions?
What are the Laplace transforms of the
derivative, integral, shift, and scaling of a
function?
How to perform partial fraction expansion for a
rational function F(s) and perform the inverse
Laplace transform?
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