Laplace Transform
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Transcript Laplace Transform
Laplace
Transform
Prepared By :
Akshay Gandhi
Kalpesh kale
Jatin Patel
Prashant Dhobi
Azad Hudani
: 130460119029
: 130460119038
: 130460119036
: 130460119026
: 130460119031
The French Newton
Pierre-Simon Laplace
Developed mathematics in
astronomy, physics, and
statistics
Began work in calculus which
led to the Laplace Transform
Focused later on celestial
mechanics
One of the first scientists to
suggest the existence of black
holes
Why use Laplace
Transforms?
Find solution to differential
equation using algebra
Relationship to Fourier Transform
allows easy way to characterize
systems
No need for convolution of input
and differential equation solution
Useful with multiple processes in
system
How to use Laplace
Find differential equations that
describe system
Obtain Laplace transform
Perform algebra to solve for output
or variable of interest
Apply inverse transform to find
solution
What are Laplace
transforms?
F ( s) L{ f (t )} f (t )e st dt
0
t is real, s is complex!
II.
Note “transform”: f(t) F(s), where t is
integrated and s is variable
III. Conversely F(s) f(t), t is variable and s is
integrated
IV. Assumes f(t) = 0 for all t < 0
I.
Laplace Transform Theory
•General Theory
•Example
Laplace Transform for ODEs
•Equation with initial conditions
•Laplace transform is linear
•Apply derivative formula
Table of selected Laplace
Transforms
1
f ( t ) u ( t ) F(s)
s
1
f ( t ) e u ( t ) F(s)
sa
at
s
f ( t ) cos( t )u ( t ) F(s) 2
s 1
1
f ( t ) sin( t )u ( t ) F(s) 2
s 1
More transforms
n!
f ( t ) t u ( t ) F(s) n 1
s
n
0! 1
n 0, f ( t ) u ( t ) F(s) 1
s s
1!
n 1, f ( t ) tu ( t ) F(s) 2
s
5! 120
n 5, f ( t ) t 5 u ( t ) F(s) 6 6
s
s
f ( t ) ( t ) F(s) 1
Note on step functions in
Laplace
Unit step function definition:
u ( t ) 1, t 0
u ( t ) 0, t 0
Used in conjunction with f(t)
f(t)u(t) because of Laplace integral
limits:
L{f ( t )} f ( t )e dt
0
st
Properties of Laplace
Transforms
Linearity
Scaling in time
Time shift
“frequency” or s-plane shift
Multiplication by tn
Integration
Differentiation
Properties: Linearity
L{c1f1 (t ) c2f 2 (t )} c1F1 (s) c2 F2 (s)
Example : L{sinh( t )}
Proof :
1 t 1 t
y{ e e }
2
2
1
1
L{e t } L{e t }
2
2
1 1
1
(
)
2 s 1 s 1
1 (s 1) (s 1)
1
(
)
2
s2 1
s2 1
L{c1f1 ( t ) c 2 f 2 ( t )}
st
[
c
f
(
t
)
c
f
(
t
)]
e
dt
2 2
11
0
0
0
c1 f1 ( t )e st dt c 2 f 2 ( t )e st dt
c1F1 (s) c 2 F2 (s)
Properties: Scaling in Time
1 s
L{f (at )} F( )
a a
Example : L{sin( t )}
1
1
(
1)
2
s
( )
1
2
( 2
)
2
s
s 2 2
Proof :
L{f (at )}
st
f
(
at
)
e
dt
0
let
u at , t
a
u
1
, dt du
a
a
s
( ) u
1
f (u )e a du
a0
1 s
F( )
a a
Properties: Time Shift
L{f ( t t 0 )u ( t t 0 )} e
a ( t 10)
u ( t 10)}
Example : L{e
e 10s
sa
Proof :
st 0
F(s)
L{f ( t t 0 )u ( t t 0 )}
st
f
(
t
t
)
u
(
t
t
)
e
dt
0
0
0
st
f
(
t
t
)
e
dt
0
t0
let
u t t0, t u t0
t0
s ( u t 0 )
f
(
u
)
e
du
0
e
st 0
st 0
su
f
(
u
)
e
du
e
F(s)
0
Properties: S-plane
(frequency) shift
at
L{e f (t )} F(s a )
Example : L{e at sin( t )}
(s a ) 2 2
Proof :
L{e at f ( t )}
at
st
e
f
(
t
)
e
dt
0
(s a ) t
f
(
t
)
e
dt
0
F(s a )
Properties: Multiplication
by tn n
n
n d
L{t f ( t )} (1)
F
(
s
)
n
ds
Example :
Proof :
L{t n u ( t )}
(1) n
n!
s n 1
L{t n f ( t )} t n f ( t )e st dt
0
n
d 1
( )
n
ds s
n st
f
(
t
)
t
e dt
0
n
(1) n f ( t ) n e st dt
s
0
n
n
st
n
(1)
f ( t )e dt (1)
F(s)
n
n
s 0
s
n
The “D” Operator
1.
2.
Differentiation shorthand
Integration
t
if
g( t ) f ( t )dt
then
Dg ( t ) f ( t )
df ( t )
Df ( t )
dt
d2
2
D f (t) 2 f (t)
dt
shorthand
t
if g( t ) f ( t )dt
a
1
then g( t ) D a f ( t )
Difference in
f (0 ), f (0 ) & f (0)
The values are only different if f(t)
is not continuous @ t=0
Example of discontinuous function:
u(t)
f (0 ) lim u ( t ) 0
t 0
f (0 ) lim u ( t ) 1
t 0
f (0) u (0) 1
Properties: Nth order
derivatives
2
L{D f ( t )} ?
let
g( t ) Df ( t ), g(0) Df (0) f ' (0)
L{D 2 g( t )} sG (s) g(0) sL{Df ( t )} f ' (0)
s(sF(s) f (0)) f ' (0) s 2 F(s) sF(0) f ' (0)
L{Dn f (t )} s n F(s) s( n 1) f (0) s( n 2) f ' (0) sf ( n 2)' (0) f ( n 1)' (0)
NOTE: to take L{D n f ( t )}
you need the value @ t=0 for
Dn 1f (t ), Dn 2f (t ),...Df (t ), f (t ) called initial conditions!
We will use this to solve differential equations!
Real-Life Applications
Semiconductor
mobility
Call completion in
wireless networks
Vehicle vibrations on
compressed rails
Behavior of magnetic
and electric fields
above the atmosphere
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