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Laplace Transformation

The document provides an overview of the Laplace Transform, a mathematical technique that converts functions of time into functions of a complex variable, facilitating the analysis of differential equations. It outlines key properties, applications in engineering, physics, and mathematics, as well as advantages and limitations of the Laplace Transform. Specific applications in electronics engineering include circuit analysis, transfer functions, filter design, and signal processing.

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ratnamukherjee975
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28 views9 pages

Laplace Transformation

The document provides an overview of the Laplace Transform, a mathematical technique that converts functions of time into functions of a complex variable, facilitating the analysis of differential equations. It outlines key properties, applications in engineering, physics, and mathematics, as well as advantages and limitations of the Laplace Transform. Specific applications in electronics engineering include circuit analysis, transfer functions, filter design, and signal processing.

Uploaded by

ratnamukherjee975
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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LAPLACE

TRANSFORMATION
AND ITS APPLICATION
ALIPURDUAR GOVERNMENT ENGINEERING
& MANAGEMENT COLLEGE
Name : Sarup Mukherjee
Department : Electronics & Communication Engineering
Subject : Network Theory
Subject Code : EC304
Roll No. :
Registration No. :
Topic : C1
Session : 2024-2025
Introduction to Laplace
Transform
Definition: The Laplace Transform is a mathematical
technique used to transform a function of time (t) into a
function of a complex variable (s).
AGENDA
∞ −𝑠𝑡
Formula: ℒ 𝑓 𝑡 = 0
𝑒 𝑓 𝑡 𝑑𝑡

Key Idea: Converts differential equations into algebraic


equations.
Properties of Laplace Transform
Linearity: ℒ 𝑎𝑓 𝑡 + 𝑏𝑔(𝑡) = 𝑎ℒ f t + bℒ 𝑔 𝑡

First Derivative: ℒ 𝑓′ 𝑡 = 𝑠ℒ f t − 𝑓(0)

Shifting: ℒ 𝑒 𝑎𝑡 𝑓(𝑡) = 𝐹(𝑠 − 𝑎)


Inverse Laplace Transform
DEFINITION: CONVERTS THE
LAPLACE DOMAIN FUNCTION
BACK TO THE TIME DOMAIN.

FORMULA: ℒ −𝟏 𝑭 𝒔 = 𝒇(𝒕)
𝟏
EXAMPLE: ℒ −𝟏 =𝒕
𝒔𝟐
Applications of Laplace Transform
1. Engineering:
• Electrical circuits (RLC analysis).
• Control systems (stability analysis).

2. Physics:
• Heat conduction problems.
• Vibrations and oscillations.

3. Mathematics:
• Solving differential equations.
Laplace Transform in Electronics
Engineering
1. CIRCUIT ANALYSIS
• SIMPLIFIES THE ANALYSIS OF RLC CIRCUITS BY CONVERTING DIFFERENTIAL EQUATIONS INTO
ALGEBRAIC EQUATIONS IN THE S-DOMAIN.
• EXAMPLE: 𝑉(𝑠) = 𝐼(𝑠)𝑍(𝑠), WHERE 𝑍(𝑠) REPRESENTS IMPEDANCE IN THE LAPLACE DOMAIN.
2. TRANSFER FUNCTIONS
• USED TO DERIVE THE RELATIONSHIP BETWEEN INPUT AND OUTPUT IN SYSTEMS.
𝑌(𝑠)
• EXAMPLE: 𝐻(𝑠) = 𝑋(𝑠) WHERE 𝐻(𝑠) IS THE SYSTEM'S TRANSFER FUNCTION.
3. FILTER DESIGN
• HELPS IN DESIGNING AND ANALYZING FILTERS (LOW-PASS, HIGH-PASS, BAND-PASS) USING POLE-ZERO
PLOTS.
• USED TO DETERMINE FREQUENCY RESPONSE.
4. SIGNAL PROCESSING
• USEFUL FOR ANALYZING AND PROCESSING SIGNALS (AMPLIFICATION, NOISE FILTERING).
• CONVERTS TIME-DOMAIN SIGNALS TO FREQUENCY-DOMAIN FOR EASIER MANIPULATION.
Advantages of Laplace Transform

• Simplifies complex differential


equations.

• Provides a unified approach for solving


linear systems.

• Useful in both theoretical and practical


problems.
Limitations
• Not suitable for nonlinear systems.

• Requires the function to be piecewise


continuous and of exponential order.
THANK
YOU

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