The Laplace transform is used to solve differential equations by converting them to algebraic equations that can be solved using algebra. It relates a function of time (f(t)) to a function of complex values (F(s)), allowing differential equations describing systems to be converted to an algebraic form by taking the Laplace transform. Key properties include linearity, scaling, time shifting, and how various standard functions like exponentials, sinusoids and polynomials transform. The inverse Laplace transform is then used to convert back from the function of complex values to the original time domain function to obtain the solution to the differential equation.

1. LAPLACE TRANSFORM

2. 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

3. 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

4. WHAT ARE LAPLACE
TRANSFORMS?
















j
j
st
1
0
st
ds
e
)
s
(
F
j
2
1
)}
s
(
F
{
L
)
t
(
f
dt
e
)
t
(
f
)}
t
(
f
{
L
)
s
(
F
• t is real, s is complex!
• Inverse requires complex analysis to solve
• Note “transform”: f(t)  F(s), where t is
integrated and s is variable
• Conversely F(s)  f(t), t is variable and s is
integrated

5. EVALUATING F(S) = L{F(T)}
• Hard Way – do the integral

 



 




















0
st
0 0
t
)
a
s
(
st
at
at
0
st
dt
)
t
sin(
e
)
s
(
F
t
sin
)
t
(
f
a
s
1
dt
e
dt
e
e
)
s
(
F
e
)
t
(
f
s
1
)
1
0
(
s
1
dt
e
)
s
(
F
1
)
t
(
f
let
let
let
Integrate by parts

6. EVALUATING F(S)=L{F(T)}- HARD
WAY
remember
 

 vdu
uv
udv
)
t
cos(
v
,
dt
)
t
sin(
dv
dt
se
du
,
e
u st
st





 


 



 











0
0 0
0
)
cos(
)
1
(
)
cos(
)
cos(
[
)
sin( ]
dt
t
e
s
e
dt
t
e
s
t
e
dt
t
e
st
st
st
st
st
)
t
sin(
v
,
dt
)
t
cos(
dv
dt
se
du
,
e
u st
st




 




















0
st
st
0
st
0
st
0
st
dt
)
t
sin(
e
s
)
0
(
e
dt
)
t
sin(
e
s
)
t
sin(
e
[
dt
)
t
cos(
e
]
2
0
st
0
st
2
0 0
st
2
st
s
1
1
dt
)
t
sin(
e
1
dt
)
t
sin(
e
)
s
1
(
dt
)
t
sin(
e
s
1
dt
)
t
sin(
se









 




 


let
let
Substituting, we get:

7. EVALUATING F(S) = L{F(T)}
• This is the easy way ...
• Recognize a few different transforms
• Learn a few different properties
• Do a little math

8. TABLE OF SELECTED LAPLACE
TRANSFORMS
1
1
)
(
)
sin(
)
(
1
)
(
)
cos(
)
(
1
)
(
)
(
1
)
(
1
)
(
2
2
















s
s
F
t
t
f
s
s
s
F
t
t
f
a
s
s
F
e
t
f
s
s
F
t
f
at

9. MORE TRANSFORMS
1
!
)
(
)
( 


 n
n
s
n
s
F
t
t
f
6
6
5
2
1
s
120
s
!
5
)
s
(
F
)
t
(
u
t
)
t
(
f
,
5
n
s
!
1
)
s
(
F
)
t
(
tu
)
t
(
f
,
1
n
s
1
s
!
0
)
s
(
F
)
t
(
u
)
t
(
f
,
0
n















10. PROPERTIES OF LAPLACE
TRANSFORMS
• Linearity
• Change of scale
• Time Shift
• “frequency” or s-plane shift
• Multiplication by tn
• Integration
• Differentiation

11. PROPERTIES: LINEARITY
)
s
(
F
c
)
s
(
F
c
)}
t
(
f
c
)
t
(
f
c
{
L 2
2
1
1
2
2
1
1 


Example :
Proof :
)
s
(
F
c
)
s
(
F
c
dt
e
)
t
(
f
c
dt
e
)
t
(
f
c
dt
e
)]
t
(
f
c
)
t
(
f
c
[
)}
t
(
f
c
)
t
(
f
c
{
L
2
2
1
1
0
st
2
2
0
st
1
1
st
2
2
0
1
1
2
2
1
1
















1
1
)
1
)
1
(
)
1
(
(
2
1
)
1
1
1
1
(
2
1
}
{
2
1
}
{
2
1
}
2
1
2
1
{
)}
{sinh(
2
2

















s
s
s
s
s
s
e
L
e
L
e
e
L
t
L
t
t
t
t

12. )
a
s
(
F
a
1
)}
at
(
f
{
L 
Example :
2
2
2
2
2
2
s
)
s
(
1
)
1
)
s
(
1
(
1
)}
t
{sin(
L













Proof :
)
a
s
(
F
a
1
du
e
)
u
(
f
a
1
du
a
1
dt
,
a
u
t
,
at
u
dt
e
)
at
(
f
)}
at
(
f
{
L
a
0
u
)
a
s
(
0
st












let
PROPERTIES: CHANGE OF SCALE

13. PROPERTIES: TIME SHIFT
)
s
(
F
e
)}
t
t
(
u
)
t
t
(
f
{
L 0
st
0
0




Example :
a
s
e
t
u
e
L
s
t
a






10
)
10
(
)}
10
(
{
Proof :
)
s
(
F
e
du
e
)
u
(
f
e
du
e
)
u
(
f
t
u
t
,
t
t
u
dt
e
)
t
t
(
f
dt
e
)
t
t
(
u
)
t
t
(
f
)}
t
t
(
u
)
t
t
(
f
{
L
0
0
0
0
0
st
0
su
st
t
0
)
t
u
(
s
0
0
t
st
0
0
st
0
0
0
0






























let

14. PROPERTIES: S-PLANE
(FREQUENCY) SHIFT
)
a
s
(
F
)}
t
(
f
e
{
L at



Example :
2
2
at
)
a
s
(
)}
t
sin(
e
{
L







Proof :
)
a
s
(
F
dt
e
)
t
(
f
dt
e
)
t
(
f
e
)}
t
(
f
e
{
L
0
t
)
a
s
(
0
st
at
at














15. PROPERTIES:
MULTIPLICATION BY tn
)
s
(
F
ds
d
)
1
(
)}
t
(
f
t
{
L n
n
n
n


Example :
1
n
n
n
n
n
s
!
n
)
s
1
(
ds
d
)
1
(
)}
t
(
u
t
{
L




Proof :
)
s
(
F
s
)
1
(
dt
e
)
t
(
f
s
)
1
(
dt
e
s
)
t
(
f
)
1
(
dt
e
t
)
t
(
f
dt
e
)
t
(
f
t
)}
t
(
f
t
{
L
n
n
n
0
st
n
n
n
0
st
n
n
n
0
st
n
0
st
n
n



























16. THE “D” OPERATOR
1. Differentiation shorthand
2. Integration shorthand
)
t
(
f
dt
d
)
t
(
f
D
dt
)
t
(
df
)
t
(
Df
2
2
2


)
t
(
f
)
t
(
Dg
dt
)
t
(
f
)
t
(
g
t

 

)
t
(
f
D
)
t
(
g
dt
)
t
(
f
)
t
(
g
1
a
t
a


 
if
then then
if

17. PROPERTIES: INTEGRALS
s
s
F
t
f
D
L
)
(
)}
(
{ 1
0 

Example :
)}
t
{sin(
L
1
s
1
)
1
s
s
)(
s
1
(
)}
t
cos(
D
{
L
2
2
1
0





Proof :
let
st
st
0
st
1
0
e
s
1
v
,
dt
e
dv
dt
)
t
(
f
du
),
t
(
g
u
dt
e
)
t
(
g
)}
t
{sin(
L
)
t
(
f
D
)
t
(
g



















 


t
0
st
0
st
dt
)
t
(
f
)
t
(
g
s
)
s
(
F
dt
e
)
t
(
f
s
1
]
e
)
t
(
g
s
1
[








0
)
(
)
( dt
e
t
f
t st
If t=0, g(t)=0
for so
slower than





0
)
(
)
( t
g
dt
t
f 0

st
e

18. PROPERTIES: DERIVATIVES
(THIS IS THE BIG ONE)
)
0
(
f
)
s
(
sF
)}
t
(
Df
{
L 


Example :
)}
t
sin(
{
L
1
s
1
1
s
)
1
s
(
s
1
1
s
s
)
0
(
f
1
s
s
)}
t
cos(
D
{
L
2
2
2
2
2
2
2
2















Proof :
)
s
(
sF
)
0
(
f
dt
e
)
t
(
f
s
)]
t
(
f
e
[
)
t
(
f
v
,
dt
)
t
(
f
dt
d
dv
se
du
,
e
u
dt
e
)
t
(
f
dt
d
)}
t
(
Df
{
L
0
st
0
st
st
st
0
st





















let

19. DIFFERENCE IN
• The values are only different if f(t) is not
continuous at t=0
• Example of discontinuous function: u(t)
)
0
(
f
&
)
0
(
f
),
0
(
f 

1
)
0
(
u
)
0
(
f
1
)
t
(
u
lim
)
0
(
f
0
)
t
(
u
lim
)
0
(
f
0
t
0
t













20. ?
)}
t
(
f
D
{
L 2

)
0
(
'
f
)
0
(
sF
)
s
(
F
s
)
0
(
'
f
))
0
(
f
)
s
(
sF
(
s
)
0
(
'
f
)}
t
(
Df
{
sL
)
0
(
g
)
s
(
sG
)}
t
(
g
D
{
L
)
0
(
'
f
)
0
(
Df
)
0
(
g
),
t
(
Df
)
t
(
g
2
2














let
)
0
(
f
)
0
(
sf
)
0
(
'
f
s
)
0
(
f
s
)
s
(
F
s
)}
t
(
f
D
{
L )'
1
n
(
)'
2
n
(
)
2
n
(
)
1
n
(
n
n 








 
NOTE: to take
you need the value of t=0 for
called initial conditions!
We will use this to solve differential equations!



)
t
(
f
),
t
(
Df
),...
t
(
f
D
),
t
(
f
D 2
n
1
n
)}
t
(
f
D
{
L n
PROPERTIES: NTH ORDER
DERIVATIVES

21. PROPERTIES: NTH ORDER
DERIVATIVES
)
0
(
f
)
s
(
sF
)}
t
(
Df
{
L 

)}
t
(
f
D
{
L 2
)
0
(
f
)
s
(
sF
)}
t
(
Df
{
L
)}
t
(
g
{
L
)
s
(
G
)
0
(
'
f
)
0
(
g
)
t
(
Df
)
t
(
g
)
0
(
g
)
s
(
sG
)}
t
(
Dg
{
L
)
t
(
f
D
)
t
(
Dg
and
)
t
(
Df
)
t
(
g 2











)
0
(
'
f
)
0
(
sf
)
s
(
F
s
)
0
(
'
f
)]
0
(
f
)
s
(
sF
[
s
)
0
(
g
)
s
(
sG
)}
t
(
Dg
{
L 2









.
etc
),
t
(
f
D
),
t
(
f
D 4
3
Start with
Now apply again
let
then
remember
Can repeat for
)
0
(
f
)
0
(
sf
)
0
(
'
f
s
)
0
(
f
s
)
s
(
F
s
)}
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22. REAL WORLD APPLICATIONS OF
LAPLACE TRANSFORM
• A simple Laplace Transform is conducted while sending
signals over any two-way communication medium
(FM/AM stereo, 2-way radio sets, cellular phones).
When information is sent over medium such as cellular
phones, they are first converted into time-varying wave,
and then it is super-imposed on the medium. In this
way, the information propagates. Now, at the receiving
end, to decipher the information being sent, medium
wave’s time functions are converted to frequency
functions.

23. MORE EXAMPLE
• Laplace transform is crucial for the study of
control systems, hence they are used for the
analysis of HVAC (Heating, Ventilation and Air
Conditioning) control systems, which are used in
all modern buildings and constructions

24. ENGINEERING APPLICATIONS OF
LAPLACE TRANSFORM
• System Modeling
• Analysis of Electrical Circuits
• Analysis of Electronic Circuits
• Digital Signal Processing
• Nuclear Physics

25. THANKS TO ALL
