Applications of the Laplace Transform

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Applications of the Laplace Transform
ECE 2221/MCT 2210 Signals and Systems
(Analysis) Sem. I (03/04)

• Br. Shahrul Naim Sidek
• Department of Mechatronics Engineering
• International Islamic University Malaysia

Lecture Chapter 6
http//eng.iiu.edu.my/snaim
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Topic covered

• Analysis of Electrical Networks
• Concept of Transfer function
• Stability via Routh Test
• Bode plot to display Frequency Response

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Analysis of Electrical Networks Passive Circuit
Elements

• Laplace transforms with zero-valued initial
  conditions
• Capacitor

• Inductor
• Resistor

Transfer Function
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Ex First-Order RC Lowpass Filter
R


x(t)
y(t)
C
i(t)
Time domain
R


X(s)
Y(s)
I(s)
Laplace domain
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Passive Circuit Elements

• Laplace transforms with non-zero initial
  conditions
• Capacitor

• Inductor
• Resistor ??

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Initial and Final Values

• Values of f(t) as t ? 0 and t ? ? may be computed
  from its Laplace transform F(s)
• Initial value theorem
• If f(t) and its derivative df/dt have Laplace
  transforms, then provided
  that the limit on the right-
• hand side of the equation exists.
• Final value theorem
• If both f(t) and df/dt have Laplace transforms,
  then provided that s
  F(s) has no poles in the
• RHP or on the imaginary axis.

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Final and Initial Values Example

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

Initial Value
Final Value
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Transfer Function

• Provides a complete behaviors of a specified LTI
  system.
• T.F. is independent of the particular input and
  is a property of the circuit only.
• T.F. is obtained for the case of zero initial
  conditions.
• T.F. for LTI circuit is a rational function of s
• Frequency response is H(s) and let s jw

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Component of response
The Laplace transform of differential equation
with initial condition is given
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Component of response