[EC 211]
ELECTRICAL NETWORKS ANALYSIS
LECTURE#04
The Laplace Transform
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Lecture#04 - Layout
Introduction
Definition of the Laplace Transform
Properties of the Laplace Transform
The Inverse Laplace Transform
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Introduction
In this lecture, we discuss one of the techniques
for analyzing circuits with a wide variety of
inputs and responses.
Such circuits are modeled by differential
equations whose solutions describe the total
response behavior of the circuits.
Mathematical methods have been devised to
systematically determine the solutions of
differential equations.
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Introduction
Laplace transformation involves turning
differential equations into algebraic equation.
When using phasors for the analysis of circuits,
we transform the circuit from the time domain to
the frequency or phasor domain.
Once we obtain the phasor result, we transform
it back to the time domain.
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Introduction
The Laplace transform method follows the same
process.
We use Laplace transformation to transform the
circuit from the time domain to the frequency
domain, obtain the solution, and apply the
inverse Laplace transform to the result to
transform it back to the time domain.
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Introduction
The Laplace transform is significant for a
number of reasons:
1. It can be applied to a wider variety of inputs
than phasor analysis.
2. It provides an easy way to solve circuit problems
involving initial conditions, because it allows us
to work with algebraic equations instead of
differential equations.
3. It is capable of providing us, in one single
operation, the total response of the circuit
comprising both the natural and forced
responses.
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Introduction
We begin with the definition of the Laplace
transform which gives rise to most essential
properties.
By examining these properties, we shall see
how and why the method works.
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Definition of the Laplace Transform
The Laplace transform is an integral
transformation of a function f(t) from the
time domain into the complex frequency
domain, giving F(s).
Therefore, given a function f(t), its Laplace
transform, denoted by F(s) or 𝓛[f(t)], is
defined by;
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Definition of the Laplace Transform
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Definition of the Laplace Transform
Note that a function f(t) may not have a
Laplace transform.
In order for f(t) to have a Laplace transform,
the integral in the Laplace transform
equation must converge to a finite value.
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Examples
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Properties of the Laplace Transform
The properties of the Laplace transform help us to
obtain transform pairs without directly using
Laplace transform equation.
These properties include;
1. Linearity
2. Scaling
3. Time Shift
4. Frequency Shift
5. Time Differentiation
6. Time Integration
7. Frequency Differentiation
8. Time Periodicity
9. Initial and Final Values
10. Convolution.
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Properties of the Laplace Transform
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Properties of the Laplace Transform
Example-01
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Properties of the Laplace Transform
Example-02
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Properties of the Laplace Transform
Example-03
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Inverse Laplace Transform
The inverse Laplace transform is given by:
The functions f(t) and F(s) are regarded as a
Laplace transform pair, where f(t)⇔F(s)
meaning that there is a one-to-one
correspondence between f(t) and F(s).
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Inverse Laplace Transform
Given F(s), how do we transform it back to
the time domain and obtain the
corresponding f(t)?
There are mainly two ways;
1. Use the inverse Laplace transform equation
(cumbersome!!)
2. Use look-up tables that shows Laplace
transform pairs.
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Inverse Laplace Transform
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Inverse Laplace Transform
Steps to find the Inverse Laplace Transform:
1. Decompose F(s) into simple terms using
partial fraction expansion.
2. Find the inverse of each term by
matching entries in look-up tables
showing Laplace transform pairs (shown
in next slide).
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Inverse Laplace Transform
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Inverse Laplace Transform
Example-01
Find the inverse Laplace transform of:
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Inverse Laplace Transform
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Inverse Laplace Transform
Example-02
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