Laplace Transform (1) Hany Ferdinando Dept. of Electrical Eng. Petra Christian University.
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Transcript of Laplace Transform (1) Hany Ferdinando Dept. of Electrical Eng. Petra Christian University.
Laplace Transform (1) - Hany Ferdinando 2
Overview
Introduction Laplace Transform Convergence of Laplace Transform Properties of Laplace Transform Using table Inverse of Laplace Transform
Laplace Transform (1) - Hany Ferdinando 3
Introduction
It was discovered by Pierre-Simon Laplace, French Mathematician (1749-1827)
Laplace Transform (1) - Hany Ferdinando 4
Introduction
It transforms signal/system from time-domain to s-domain for continuous-time LTI system
It is analogous to Z Transform in discrete-time LTI system
It is similar to Fourier Transform, but ‘j’ is substituted by s
Laplace Transform (1) - Hany Ferdinando 5
Introduction
Laplace Transform is continuous sum of exponential function of the form est, where s = + j is complex frequency
Therefore, Fourier can be viewed as a special case in which s = j
Laplace Transform (1) - Hany Ferdinando 7
Laplace Transform
For h(t) = e-at, find H(s) What is your assumption in finishing the
integration? If you do not have that assumption, then
what you can do? Is it important to have that assumption?
Laplace Transform (1) - Hany Ferdinando 8
Convergence…
The two-sided Laplace Transform exists if
dtetfsF st)()( is finite
Therefore,
dtetfdtetf tst )()( is finite
Laplace Transform (1) - Hany Ferdinando 9
Convergence…
Suppose there exists a real positive number R so that for some real and we know that
f(t) < R et for t > 0, and
f(t) < R et for t > 0
Laplace Transform (1) - Hany Ferdinando 10
Convergence…
0
)(
0
)(
0
)0
)
11)(
(()(
tt
tt
eeRsF
dteRdteRsF
Laplace Transform (1) - Hany Ferdinando 11
Convergence…
How did you make your assumption in order to solve the equation?
Can you solve it without that assumption?
The negative portion converges for < while the positive one converges for >
Laplace Transform (1) - Hany Ferdinando 16
Properties
Linearity Scaling
Time shift Frequency shift
(s)bF(s)aF(t)bf(t)af 2121
a
sF
a
1f(at)
τsF(s)eτ)f(t a)F(sf(t)e at
Laplace Transform (1) - Hany Ferdinando 17
Properties
Time convolution Frequency convolution
Time differentiation
jc
jc
2121 u)du(s(u)FF2ππ
1(t)(t)ff
sF(s)dt
df(t)
(s)(s)FF(t)f*(t)f 2121
Laplace Transform (1) - Hany Ferdinando 18
Properties
Time integration
Frequency differentation
βσα,0)max(,s
F(s)f(u)du
t
β,0)min(σα,s
F(s)f(u)du
t
n
nn
ds
F(s)df(t)t)(
Laplace Transform (1) - Hany Ferdinando 19
Properties
One-sided time differentiation
One-sided time integration
)0(...)0()0()()( )1()1(21 nnnn
n
n
ffsfssFsdt
tfd
t
s
f
s
sFduuf
)0()()(
)1(
Laplace Transform (1) - Hany Ferdinando 20
Using Standard Table
Use table from books both for transform and for its inverse
No RoC is needed Find the general form of the equation Properties of Laplace transform are
helpful You use that table also to find the
inverse
Laplace Transform (1) - Hany Ferdinando 21
Exercise
1)sin(2tf(t)
0.5ss
1F(s)
2
6ss
sF(s)
2
2)cos(5tef(t) 3t
Laplace Transform (1) - Hany Ferdinando 22
Next…
Signals and Linear Systems by Alan V. Oppenheim, chapter 9, p 603-616
Signals and System by Robert A. Gabel, chapter 6, p 373-394
The Laplace Transform is already discussed. It transforms continuous-time LTI system from
time-domain to s-domain. There are two types, one-sided (unilateral) and two-sided
Next, we will study the application of Laplace Transform in Electrical Engineering. Read the Electric Circuit handout to prepare yourself!