CHAPTER 2 | LAPLACE TRANSFORM &
INVERSE LAPLACE
TARNSFORM
INTENDED LEARNING OUTCOMES:
At the end of this chapter, it is expected that the students
will be able to:
1. Define the Laplace transform and discuss
existence and basic properties.
2. Derive the formula in taking the Laplace
transform of fundamental functions.
3. Solve the Laplace transform of different
functions not limited to elementary, unit step
and piecewise functions.
4. Derive the formula in taking the inverse
Laplace transform of fundamental functions.
5. Solve the inverse Laplace of different
functions using the methods of linearity
property, completing the squares and
translations and partial fractions
6. Use Laplace transforms to solve initial value
problems of linear differential equation with
constant coefficient.
7. Solve actual problems involving Laplace and
Inverse Laplace transform in electric circuit
problems
01 | INTRODUCTION
The Laplace Transform takes a function of time and
transforms it to a function of a complex variable . Because
the transform is invertible, no information is lost and it is
reasonable to think of a function ( ) and its Laplace
transform ( ) as two views of the same phenomenon.
Each view has its uses and some features of the
phenomenon are easier to understand in one view or the
other.
We can use the Laplace transform to transform a linear
time invariant system from the time domain to the sdomain. This leads to the system function ( ) for the
system –this is the same system function used in the
Nyquist criterion for stability.
One important feature of the Laplace transform is that it
can transform analytic problems to algebraic problems. We
will see examples of this for differential equations.
Can we obtain the Laplace transforms of ALL functions
( )? No, but it can be shown that Laplace transforms exist
(for some ) for all “reasonable” functions ( ). The
functions must be piecewise continuous, for instance, and
not go to infinity too rapidly as → ∞. Precise conditions
for the existence of the transform can be stated but these
are not necessary in this module. It is sufficient for you to
know that transforms can be found for all the standard
functions that normally arise Precise conditions for the
existence of the transform can be stated but these are not
necessary in this module. It is sufficient for you to know that
transforms can be found for all the standard functions that
normally arise.
02.1 | Transforms of Simple Functions
Example 1:
Let ( ) = 1, ≥ 0. Find the ( ) or ℒ[1 .
Solution:
The Laplace Transform satisfies:
02 | DEFINITION OF THE LAPLACE TRANSFORM
Let ( ) be a function on [0, ∞). The Laplace Transform of
a function ( ) is the function ( ) defined by the integral,
ℒ[ ( ) = ( ) =
( )
(1)
The domain ( ) is the set of all values of for which this
integral converges. The Laplace Transform of ( ) is
denoted by both ( ) and ℒ.
The definition shows that the Laplace operator ℒ
transforms the function ( ), in the time or -domain, to
( ) in the frequency or -domain.
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ℒ[ ( ) = ( ) =
( )
Such integral is called an improper integral and, by
definition, is evaluated according to the rule
ℒ[1 =
( )
= lim
→
( )
Substituting ( ) = 1 in the improper integral formula
leads us with
ℒ[1 =
(1)
= lim
→
(1)
EE 402 – Advanced Mathematics for EE
And hence
lim
→
(1)
= lim −
→
Note:
1
lim [
! =−
1
lim "
→
−
( )
#
=0
→
Using this limit, we deduce that:
1
ℒ[1 = − [0 − 1
1
ℒ[1 = − (−1)
ℒ[1 =
1
,
provided that > 0
Example 2: Exponentials
Let ( ) = . when ≥ 0, where / is a constant. Find the
ℒ[ ( ) .
Example 3: Powers of 0
Let ( ) = . , where / is a constant. Find the ℒ[ ( ) .
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EE 402 – Advanced Mathematics for EE
Theorem 2: First Shifting Theorem, ; – Shifting
If ℒ[ ( ) = ( ), when > / then,
02.2 | Properties of Laplace Transform
Theorem 1: Linearity of the Laplace Transform
If / and 1 are constants, then
ℒ2/[ ( ) + 1[4( ) 5 = /2ℒ[ ( ) 5 + 12ℒ[4( ) 5
This shows that the Laplace transform is a linear operation;
that is, for any ( ) and 4( ) whose transforms exist and
any constants / and 1, the transform of /[ ( ) + 1[4( )
exists.
ℒ[
=/
Proof:
( )=
= /2ℒ[ ( ) 5 + 12ℒ[4( ) 5
9
cosh / = (
:
.
+
.
9
) and sinh / = (
:
.
−
.
)
.)
( )
( )
=
<.
( )
=
.
( )
.
( )
Example:
1. ℒ[ . cos =
2. ℒ[
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.)
(
( − /) = ℒ[
Application of Theorem 1:
1:
Find the Laplace transform of cosh / and sinh / .
Solution:
Recall from Chapter 1,
(
=
4( )
+1
( )
( − /) =
2/[ ( ) + 1[4( ) 5
( )
( ) = ( − /)
In words, the substitution − / for in the transform
corresponds to the multiplication of the original function
by . .
PROOF
ℒ2/[ ( ) + 1[4( ) 5 =
.
.
sin =
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Some Functions >(0) and Their Laplace Transforms
( )
1
1
ℒ[ ( )
1
1
2
:
?!
:
3
4
B
,? ≥ 0
5
.
,/ > 0
.
6
7
sin =
8
cos =
9
sinh /
10
cosh /
11
.
sin =
12
.
cos =
A
?!
B<9
Γ(/ + 1)
.<9
1
−/
=
: + =:
+ =:
/
: − /:
:
− /:
=
( − /): − = :
−/
( − /): − = :
:
Exercise:
Find the Laplace Transform of the following:
a. ( ) = 5 − 2
b. ( ) = 3 − 6 :
c. ( ) = : :
d. ( ) = H sin 3
e. ( ) = cos :
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EE 402 – Advanced Mathematics for EE
Theorem 3: Second Shifting of Laplace Transform
( − /) > /
If ℒ[ ( ) = ( ), and 4( ) = I
0
</
Theorem 4: Multiplication by Power of 0
If ℒ[ ( ) = ( ), then
then,
ℒ[4( ) =
Proof:
4( ) = I
0
( − /)
ℒ[4( ) =
>/
</
B
( ) = (−1) M
ℒ[
B
( ) = (−1)B
.
(0)
Proof:
+
.
B
(B)
[ ( )N
( )
( ) = ℒ[ ( )
( − /)
( )=
( )
=
( )
Differentiate both sides with respect to ,
( − /)
.
B
Where ? = 1,2,3, …
4( )
ℒ[4( ) =
ℒ[4( ) =
( )
.
ℒ[
Let,
K = −/
= K+/
= K
Recall from Leibniz Rule of differentiation under integral sign,
=
P
P
( )
=
P
P
( )
=
(−
) ( )
when = /, K = 0
when = ∞, K = ∞
ℒ[4( ) =
ℒ[4( ) =
L
ℒ[4( ) =
ℒ[4( ) =
L
.
ℒ[4( ) =
.
ℒ[4( ) =
.
ℒ[4( ) =
(K) K
(L<.)
(K) K
.
.
L
(K) K
= −ℒ[ ( )
Thus,
(K) K
ℒ[ ( ) = −
Assuming the theorem is true for ? = U,
ℒ[ ( − /)
( )
ℒ[
Example 1:
Find the Laplace transform of:
( − 1):
4( ) = M
0
= − Q ( ) → Equation 1
which proves the theorem for ? = 1.
ℒ[ (K)
.
)[ ( )
(
=−
V
V
( ) = (−1)V
V
= (−1)V
V(
) → Equation 2
Differentiating both sides with respect to ,
[
>1
<1
−
(
)"
P
P
[
(−
)[
(
(V<9)
)"
( )#
V
V
( )
V
= (−1)V
( )
= (−1)V W
( )
(V<9)
= (−1)V W
( )#
(V<9)
(V<9)
(V<9)
V
(V<9)
X = (−1)V
X = (−1)V
(V<9)
(V<9)
= (−1)V W
= (−1)(V<9) W
V
W
"
X = (−1)V
(V<9)
(V<9)
V(
"
)
(V<9)
"
X = (−1)V
X = (−1)(V<9)
[
( )#
(V<9)
"
(V<9)
( )#
(V<9)
( )#
( )# → Equation 3
This shows that the theorem is true for ? = U from Equation 2 and for ? = U + 1 from Equation 3. From
Equation 1, the theorem is true for ? = 1. Hence, it is true for ? = 1 + 1 = 2 and ? = 2 + 1 = 3, and so on, thus,
for all positive integer values of ?. Therefore,
where ? = 1,2,3, …
ℒ[
B
( ) = (−1)B
(B)
( )
Example 1:
Find the Laplace transform of
( ) = cos 5
Example 2:
Find the Laplace transform of:
( − 2)A
4( ) = M
0
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>2
<2
EE 402 – Advanced Mathematics for EE
Example 2:
Find the Laplace transform of
( )=
: Y
02 | DEFINITION OF THE INVERSE LAPLACE TRANSFORM
From ℒ[ ( ) = ( ), the value of ( ) is called the inverse
Laplace transform of ( ). In symbol,
ℒ
where ℒ
9
9[
( ) = ( )
is called the inverse Laplace transform operator.
02.1 | Theorem on the Inverse Laplace Transform
Theorem 1: If / and 1 are constants, then
ℒ
9 [/
( ) + 1 ( ) = /ℒ
9[
( ) + 1ℒ
9[
( )
Theorem 2:
9[
ℒ
( ) =
.
9[
ℒ
( − /)
Table of Inverse Laplace Transform
( )
ℒ 9[ ( )
1
1
1
−/
2
;
(? − 1)!
? = 1,2,3, …
1
( − /)B
5
:
6
:
.
1
+ =:
1
sin =
=
cos =
1
sinh /
/
cosh /
− /:
=
( − /): − = :
−/
( − /): − = :
:
11
(
:
12
(
:
B 9
(? − 1)!
+ =:
/
: − /:
7
10
B 9
B
4
9
.
1
3
8
1
1
=
/
.
sin =
cos =
1
sin =
2=
+ = : ):
1
+ = : ):
1
(sin = − = cos = )
2= A
Example 1:
Find the inverse Laplace transform of
( )=
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6
−
1
4
+
−8
−3
EE 402 – Advanced Mathematics for EE
Example:
Find
Example 2:
Find the Laplace Transform of
( )=
ℒ
19
1
7
−
+ H
+2 3 −5
9[
( ) =ℒ
9
7 −1
!
( + 1)( + 2)( − 3)
02.2 | Solution of the Inverse Laplace Transform using
Partial Fraction
Many transforms that one encounters are of form
( )=
_( )
`( )
Where _ and ` are polynomials in terms of
deg2_5. To evaluate ℒ
9[
with deg2`5 >
( ) , we can write ( ) =
terms of Partial Fractions.
b( )
c( )
in
Partial Fractions is a method to reduce a complex rational
function into a sum of much simpler terms.
1. Non
Non--repeated Linear Factors
If `( ) = ( − d9 )( − d: )( − dA ) … ( − dB ) and de ≠ dg
for h ≠ i
_( )
j9
j:
jA
jB
=
+
+
+ ⋯+
( − dB )
`( ) ( − d9 ) ( − d: ) ( − dA )
Then we solve for the values of j9 , j: , jA , … jB
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EE 402 – Advanced Mathematics for EE
2. Repeated Linear Factors
If `( ) contains a factor of the form ( − d)l then you
must have the following terms
3. Non
Non--repeated Quadratic Factors
If `( ) contains a factor of the form (/
then you must have the following terms
_( )
j9
j:
jA
jl
=
+
+
+⋯+
:
A
( − d)
( − d)l
`( ) ( − d) ( − d)
Then we solve for the values of j9 , j: , jA , … jl
Example:
Find
ℒ
9
W
+9 +2
X
( − 1): ( + 3)
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:
:
− 1 + m)
_( )
j +n
= :
`( ) / − 1 + m
Then we solve for the values of j and n.
Example:
Find
ℒ
9
W
3 : − 28
X
( − 4)( : + 4)
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4. Repeated Quadratic Factors
If `( ) contains a factor of the form (/
then you must have the following terms
:
− 1 + m)V
_( )
j9 + n9
j: + n:
=
+
+⋯
:
:
`( ) (/ − 1 + m) (/ − 1 + m):
jV + nV
+
(/ : − 1 + m)V
Then we solve for the values of jV ′ and nV ′ .
Example:
Find
ℒ
9
W
(
:
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− 10
X
− 2 + 1):
:
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03 | SOLVING INITIAL VALUE PROBLEMS (IVP) UISNG
LAPLACE TARNSFORM
03.1 | Theorem 1: Laplace Transform of Derivatives
Suppose that
is continuous and ′ is a piecewise
continuous on any interval 0 ≤ ≤ j. Suppose that and
′ are of exponential order with q (e) ( )q ≤ r . for some
constant r and / and h = 0,1. Then ℒ[ ’( ) exist for >
/, and satisfies
ℒ[ ’( ) = ℒ[ ( ) − (0).
03.2 | Theorem 2: Laplace Transform of Derivative >(u) of
Any Order u
Suppose that , ′, ′′, … , (B 9) are continuous and that
(B)
is piecewise continuous on any interval 0 ≤ ≤ j.
Suppose that , ′, ′′, … , (B) are of exponential order with
q (e) ( )q ≤ r . for some constant r and / and 0 ≤ h ≤ ?.
Then Then ℒ"
ℒ"
(B)
(B)
( )# =
( )# exist for
B
ℒ[ ( ) −
−
> /, and satisfies
B 9
(B 9)
(0).
(0) − ⋯ −
(B :)
(0)
Such that for a 2nd order differential equation we have
ℒ[ ′′( ) =
:
ℒ[ ( ) −
(0) − (0)
and for a 3rd order differential equation we have
ℒ[ ′′′( ) =
A
ℒ[ ( ) −
:
(0) −
Q
(0) − ′′(0)
Example 1:
Consider the initial value problem below.
v QQ − v Q = 2
v(0) = 1, v Q (0) = −2
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EE 402 – Advanced Mathematics for EE
Example 2:
Solve the initial value problem below.
v QQ − 2v Q + 5v = −8 w
v(0) = 2, v Q (0) = 12
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EE 402 – Advanced Mathematics for EE
04 | APPLICATION OF AND LAPLACE TRANSFORM AND
INVERSE LAPLACE IN SOLVING ELECTRICAL
ENGINEERING PROBLEMS
The Laplace transform is used frequently in engineering
and physics; the output of a linear dynamic system can be
calculated by convolving its unit impulse response with the
input signal. Performing this calculation in Laplace space
turns the convolution into a multiplication; the latter being
easier to solve because of its algebraic from.
04.1 | Laplace Transform in Circuit Analysis
The Laplace transform can be applied to solve the
switching transient phenomenon in the series or parallel
RL, RC or RLC circuits.
How can we use the Laplace transform to solve problems?
1. Write the set of differential equations in the time
domain that describe the relationship between
voltage and current for the circuit.
2. Use KVL, KCL and the laws governing voltage and
current of resistor, inductors (and coupled coils) and
capacitors.
3. Laplace transform the equations to eliminate the
integrals and derivates and solve these equations
for x( ) and y( ).
4. Inverse Laplace transform to get z( ) and h( ).
Laplace Transform – Resistors
Example:
1. Given the differential equation:
{
h
+ |h = 100 sin 377
where { = 0.1 henry, | = 10 ohms and h(0) = 0. Determine
the current at = 0.01 second.
2. A series RC circuit with | = 30 ohms and } = 250 µF is
connected across an AC voltage source of z =
100 sin 100 through a switch. The switch is closed at =
0. Determine the current when = 0.01 second. Assume
at = 0, the capacitor is uncharged. Use the differential
equation:
z
h 1
=| + h
}
3. A series circuit has | = 10 ohms, { = 0.1 henry and } =
0.0001 farad. An AC voltage z = 100 sin 377 is applied
across the series circuit and the applicable differential
equation is:
z
={
:
h
:
+|
h
+
1
h
}
Solve for the particular solution (without the
complementary solution) and determine the amplitude of
the resulting sinusoidal current h( ).
Laplace Transform – Inductor
Laplace Transform – Capacitor
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EE 402 – Advanced Mathematics for EE