Laplace Transform (1)

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24 Νοε 2013 (πριν από 3 χρόνια και 8 μήνες)

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Laplace Transform (1)

Definition of Bilateral Laplace
Transform

(b for bilateral or two
-
sided transform)

Let s=
σ
+j
ω

Consider the two sided Laplace transform as the
Fourier transform of
f(t)
e
-
σ
t
. That is the Fourier transform
of an
exponentially windowed
signal.

Note also that if you set the evaluate the Laplace
transform F(s) at s= j
ω
, you have the Fourier transform
(F(
ω
))

Unilateral Laplace Transform

(Implemented in
Mathematica
)

Difference Between the
Unilateral Laplace Transform
and Bilateral Laplace transform


Unilateral

transform is used when we
choose
t=0

as the time during which
significant event occurs, such as
switching in an electrical circuit.


The
bilateral

Laplace transform are
needed for negative time as well as for
positive time.

Laplace Transform Convergence


The Laplace transform does not converge to a finite value for all
signals and all values of s


The values of s for which Laplace transform converges is called
the
Region Of Convergence

(ROC)


Always include ROC in your solution!



Example:

a
s
a
s
a
s
e
e
a
s
e
a
s
j
s
note
e
a
s
dt
e
e
dt
e
t
f
s
F
t
u
e
t
f
t
j
t
a
t
a
j
t
a
s
st
at
st
at













































)
Re(
;
1
0
)
Re(
1
1
:
;
1
)
(
)
(
);
(
)
(
0
)
(
0
)
(
0
)
(
0
0






Remember: e^jw is
sinusoidal; Thus, only
the real part is
important!

0+ indicates greater
than zero values

Example of Unilateral Laplace

Bilateral Laplace

a
s
a
s
a
s
a
s
a
s
a
s
e
a
s
dt
e
e
dt
e
t
f
s
F
t
u
e
t
f
t
a
s
st
at
st
at





































)
Re(
;
1
)
Re(
;
1
0
)
Re(
;
1
1
)
(
)
(
);
(
)
(
0
)
(
0
Example


RCO may not always exist!

;
3
1
2
1
)
(
3
)
Re(
;
3
1
)
(
2
)
Re(
;
2
1
)
(
)
(
)
(
)
(
)
(
)
(
3
2
3
2
























s
s
s
F
s
s
t
u
e
s
s
t
u
e
dt
e
t
f
s
F
t
u
e
t
u
e
t
f
t
t
st
t
t
Note that there is no
common

ROC


Laplace Transform can not be applied!

Laplace Transform & Fourier
Transform


Laplace transform is more general
than Fourier Transform


Fourier Transform: F(
ω
). (t→

ω
)


Laplace Transform: F(s=
σ
+j
ω
) (t→

σ
+j
ω
, a
complex plane)


How is Laplace Transform Used

(Building block of a

negative feedback system)

This system becomes unstable if
β
H(s) is
-
1. If you
subsittuted

s by j
ω
, you can use Bode plot to evaluate the stability of

the negative feedback system.

Understand Stability of a system
using Fourier Transform (Bode Plot)

(
unstable
)

Understand Stability of a System
Using Laplace Transform

Look at the roots of Y(s)/X(s)

Laplace Transform


We use the following notations for Laplace
Transform pairs


Refer to the table!

Table 7.1

Table 7.1 (Cont.)

Laplace Transform Properties (1)

Laplace Transform Properties (2)

Model an Inductor in the S
-
Domain


To model an inductor in the S
-
domain, we need to
determine the S
-
domain equivalent of derivative
(next slide)

Differentiation Property

Model a Capacitor in the S
-
Domain

If initial voltage is 0, V=I/
sC

1/(
sC
) is what we call the impedance of a capacitor.

Integration Property (1)

Integration Property (2)

Application


i
=
CdV
/
dt

(assume initial voltage is 0)



Integrate
i
/C with respect to
t,
will get
you I/(
sC
), which is the voltage in
Laplace domain


V=
L
di
/
dt

(assume initial condition is 0)


Integrate V/L with respect to t, get you
V/(
sL
), which is current in Laplace
domain.

Next time

Example


Unilateral Version


Find F(s):







Find F(s):

a
s
a
s
a
s
dt
e
e
dt
e
t
f
s
F
a
t
u
e
t
f
st
at
st
at



















)
Re(
0
)
Re(
;
1
)
(
)
(
0
);
(
)
(
0
0
0
)
Re(
;
1
]
1
[lim
1
]
1
[lim
1
)
(
)
(
)
(
)
(
)
(
)
(
0
0
























s
s
e
s
e
s
dt
t
u
e
dt
e
t
f
s
F
t
u
t
f
t
j
t
st
t
st
st


s
s
F
t
t
f
e
dt
t
t
e
dt
e
t
f
s
F
t
t
t
f
st
st
st

















;
1
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
0
0
0
0
0




Find F(s):







Find F(s):

a
s
a
s
a
s
a
s
dt
e
e
dt
e
t
f
s
F
a
e
t
f
st
at
st
at





















)
Re(
0
)
Re(
;
1
]
1
0
[
1
)
(
)
(
0
;
)
(
0
0
Example



0
)
Re(
;
2
/
1
2
/
1
)
(
0
)
Re(
;
2
/
1
2
/
1
;
2
/
1
2
/
1
)
Re(
;
1
)
(
)
(
2
/
1
2
/
1
)
cos(
)
(
2
2
0


























s
b
s
s
jb
s
jb
s
s
F
s
jb
s
e
jb
s
e
a
s
a
s
e
dt
e
t
f
s
F
e
e
t
f
bt
t
f
jbt
jbt
at
st
jbt
jbt


0
)
Re(
;
2
/
1
2
/
1
)
(
0
)
Re(
;
2
/
1
2
/
1
;
2
/
1
2
/
1
)
Re(
;
1
)
(
)
(
2
/
1
2
/
1
)
sin(
)
(
2
2
0


























s
b
s
b
jb
s
j
jb
s
j
s
F
s
jb
s
j
e
jb
s
j
je
a
s
a
s
e
dt
e
t
f
s
F
je
je
t
f
bt
t
f
jbt
jbt
at
st
jbt
jbt
Example



0
)
Re(
;
)
(
)
(
1
)
(
1
2
1
)
(
)
Re(
;
1
)
(
)
(
2
/
1
2
/
1
)
cos(
)
(
2
2
0


































a
s
b
a
s
a
s
jb
a
s
jb
a
s
s
F
a
s
a
s
e
dt
e
t
f
s
F
e
e
e
e
t
f
bt
e
t
f
at
st
at
jbt
at
jbt
at
Extra Slides


Building the Case…

Applications of Laplace
Transform


Easier than solving differential equations


Used to describe system behavior


We assume LTI systems


Uses S
-
domain instead of frequency domain


Applications of Laplace Transforms/


Circuit analysis


Easier than solving differential equations


Provides the general solution to any arbitrary wave (not
just LRC)


Transient


Sinusoidal steady
-
state
-
response (
Phasors
)


Signal processing


Communications


Definitely useful for Interviews!




Example of Bilateral Version

a
s
a
s
a
s
a
s
a
s
a
s
e
a
s
dt
e
e
dt
e
t
f
s
F
t
u
e
t
f
t
a
s
st
at
st
at






































)
Re(
;
1
)
Re(
;
1
0
)
Re(
;
1
1
)
(
)
(
);
(
)
(
0
)
(
0
a
s
a
s
a
s
a
s
a
s
a
s
e
a
s
dt
e
e
dt
e
t
f
s
F
t
u
e
t
f
t
a
s
st
at
st
at








































)
Re(
;
1
)
Re(
;
1
0
)
Re(
;
1
1
)
(
)
(
);
(
)
(
0
)
(
0
Find F(s):

Find F(s):

Re(s)<a

a

S
-
plane

Note that Laplace can also be found for periodic functions


ROC

Remembe
r

These!