look exactly like those calculated by hand. However, the result will be mathematically
equivalent. While this is acceptable for actual calculations, it can be a hindrance when
using the software to check your work. In these cases, you can at least check your
intermediate partial fraction decomposition, as well. For example, if we use the
from
the Case 4 example, one will notice that the result is the same as our own expansion:
>
convert( (10*s*(s-3))/((s+5)*(s^2+4*s+5)^2), parfrac, s,
complex );
Once again, it is strongly recommended that you acquire and become familiar with Maple;
not only is it an immense time-saver when doing real circuits work, but it can also be used
to save time by checking your work.
48
e
(
)
6
t
72
e
(
)
8
t
120
5.000000000
7.500000000
I
(
)
s
2.000000000
1.000000000
I
2
2.000000000
11.00000000
I
s
2.
1.
I
4.000000000
s
5.
5.000000000
7.500000000
I
(
)
s
2.
1.
I
2
2.000000000
11.00000000
I
s
2.000000000
1.000000000
I
Tip:
Maple and Inverse Laplace

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42
Lecture Slides
: None
Current Topics
: Inverse Laplace Transform, Circuit Elements, and Initial Conditions Problems
Next Topics
: Final Value Problems, Introduction to Op-Amps and Filters
Laplace Transform and Differential Equations
The original (historical) application of the Laplace Transform was to provide a simpler method
of solving differential equations. Circuits can be represented and solved using differential
equations.
Example 1
–
Find the Laplace Transform of the following time-domain function:
Find a simple Laplace transform from the table, such as
:
(1)
Using the Differentiation property of the Laplace transform:
(2)
Class Notes (and Review)

43
Example 2
–
Find the Laplace Transform of the following differential equation:
̈ ̇ ̇
Relevant Laplace Transform properties:
(1)
Final transform after applying the above properties:
(2)
Inverse Laplace Transform
–
Partial Fractions
General procedure for finding Inverse Laplace Transform:
Use table of common transforms and properties to reverse Laplace process
Apply Linearity in particular to reduce complex problems to simple ones

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Often we need to make use of partial fraction expansion to determine the inverse Laplace
transform of a particular function. The procedure depends on the type and nature of poles.
Cases are as follows:
(A) All real poles, no repeated poles
(B) All real poles, with one or more repeated poles
(C) At least one pair of complex poles, but no repeated pairs
(D) At least one pair of complex poles, with one or more repeated pairs
Example 1
–
Case A
Step 1
–
Write partial fraction terms.
(1)
Step 2
–
For each pole, evaluate the unknown coefficients.
(2)