The Laplace Transform

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The Laplace Transform
ECE 2221 Signals and Systems (Analysis)
Sem. II (03/04)

• Department of
• Electrical and Computer Engineering
• International Islamic University Malaysia

Lecture Chapter 5- Part 2
http//eng.iiu.edu.my/khaizura
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Laplace Transform Theorems

• Theorem 1 Linearity

Example Find the L.T for the following signal
using linearity theorem. x(t)cos 2pfot u(t)
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Theorem 2 Differentiation
Example Consider the circuit below
Find the current through the inductor as a
function of time
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Solution
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Theorem 3 Integration
Example Find the Laplace transform for the
current, i(t) as shown in the circuit below
Solution
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Theorem 4 Complex Frequency Shift (S-Shift)
Example Find the Laplace transform for the
following signal,
x(t)e-3t(cos2t 2.5sin2t) tgt0 Solution
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Theorem 5 Time-Delay
Example Find the Laplace transform for the
following signal, x(t)
u(t) - 2u(t-3) 2u(t-6) Solution
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Theorem 6 Convolution of Two Signals
Example Find the Laplace transform for the
following signal, Solution
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Theorem 8 Initial Value Theorem
Theorem 9 Final Value Theorem
Example Determine the initial and final values
of a signal x(t) whose Laplace transform is
Solution
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Another Example of Using the Final and Initial
Values Theorem

• Transfer function
• Poles at s 0, s -1 ? j2
• Zero at s -3/2

Initial Value
Final Value
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Inverse Laplace Transform

• Definition has integration in complex plane
• We will use lookup tables instead
• Table 5.3
• Many Laplace transform expressions are ratios of
  two polynomials, a.k.a. rational functions
• Convert complicated rational functions into
  simpler forms
• Apply partial fractions decomposition
• Use lookup tables

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Partial Fractions Example 1
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Partial Fractions Example 2
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Partial Fractions Example 3
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More Examples of How to find the Inverse Laplace
Transform

• Refer to Examples No. 5-9 to 5-14 (text book)

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The Double-Sided Laplace Transform

• Used for signals that are nonzero for tlt0
• Same as single-sided L.T except that the lower
  limit on the integral is -?
• The function of s denoting a double sided L.T
  does not correspond to a unique inverse laplace
  transform. Rather, both the function of s and the
  ROC of the L.T must be specified to define a
  unique inverse transform

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Example
Solution
Ims
Res
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