This document discusses the Laplace transform and its applications. It begins with background on Pierre-Simon Laplace who developed calculus and the Laplace transform. It then defines the Laplace transform mathematically and provides examples of calculating transforms of basic functions. The document outlines several properties of the Laplace transform and discusses applications in various domains including physics, electric circuits, control engineering, signal processing, and more. It concludes by noting some limitations of the Laplace transform.

1. VISVESVARAYA TECHNOLOGICAL UNIVERSITY
BELAGAVI
HKBK COLLEGE OF ENGINEERING
SUB-TRANSFORMS CALCULUS, FOURIER SERIES AND
NUMERICAL TECHNIQUES (21MAT31)
Under the Guidance of
PROF. SNEHA SRINIVAS
Department of Engineering Mathematics
HKBK COLLEGE OF ENGINEERING
2022-2023
HKBK COLLEGE OF ENGINEERING
22/1, Nagawara, Bengaluru – 560045.
E-mail: info@hkbk.edu.in, URL: www.hkbk.edu.in
TOPIC- LAPLACE TRANSFORM
SUBMITTED BY
OSMAN GONI
DEPARTMENT OF
ELECTRONICS AND COMMUNICATION ENGINEERING
HKBK COLLEGE OF ENGINEERING

2. THE FRENCH NEWTON
PIERRE-SIMON LAPLACE
 Developed mathematics in astronomy, physics, and statistics
 Began work in calculus which led to the Laplace Transform
 Focused later on celestial mechanics
 One of the first scientists to suggest the existence of black
holes

3. DEFINITION OF LAPLACE TRANSFORM ?

4. MATHEMATICAL OPERATOR OF
LAPLACE TRANSFORM
 A LAPLACE TRANSFORM OF FUNCTION F (T) IN A TIME
DOMAIN, WHERE T IS THE REAL NUMBER GREATER THAN OR
EQUAL TO ZERO, IS GIVEN AS F(S), WHERE THERE
S IS THE COMPLEX NUMBER IN FREQUENCY DOMAIN .I.E. S = Σ+JΩ
THE ABOVE EQUATION IS CONSIDERED AS
UNILATERAL LAPLACE TRANSFORM EQUATION.
 WHEN THE LIMITS ARE EXTENDED TO THE ENTIRE REAL AXIS
THEN THE BILATERAL LAPLACE TRANSFORM CAN BE
DEFINED AS-

5. EXAMPLE-

6. SOME BASIC FUNCTION

7. HOW TO CALCULATE LAPLACE TRANSFORM?

8. PROPERTIES OF LAPLACE TRANSFORM:
LINEARITY
TIME SHIFTING
DIFFERENTIATION IN S-DOMAIN

9. PROPERTIES OF LAPLACE TRANSFORM:
SHIFT IN S-DOMAIN
TIME-REVERSAL
CONVOLUTION IN TIME

10.  APPLICATIONS OF LAPLACE TRANSFORMS THIS SECTION DESCRIBES THE
APPLICATIONS OF LAPLACE TRANSFORMS IN THE AREAS OF SCIENCE AND
ENGINEERING.
.
APPLICATIONS OF LAPLACE
TRANSFORMS
 AT FIRST, SIMPLE APPLICATION IN THE AREA OF PHYSICS AND ELECTRIC
CIRCUIT THEORY IS PRESENTED WHICH WILL BE FOLLOWED BY A MORE
COMPLEX APPLICATION TO POWER SYSTEM WHICH INCLUDES THE
DESCRIPTION OF LOAD FREQUENCY CONTROL (LFC) FOR TRANSIENT
STABILITY STUDIES

11. A. APPLICATION IN PHYSICS
A very simple application of Laplace transform in the area of
physics could be to find out the harmonic vibration of a beam
which is supported at its two ends.
Let us consider a beam of length l and uniform cross section
parallel to the yz plane so that the normal deflection w(x,t) is
measured downward if the axis of the beam is towards x axis.
EId4w/dx4 − mω2w = 0

12. B. APPLICATION IN ELECTRIC CIRCUIT THEORY
The Laplace transform can be applied to solve the switching
transient phenomenon in the series or parallel RL,RC or RLC
circuits. Let us consider a series RLC circuit as shown in Fig to
which a d.c. voltage Vo is suddenly applied.
Now, applying Kirchhoff’s Voltage Law (KVL) to the circuit, we
have,
Ri + Ldi/dt + 1/C R idt = Vo

13. ORDINARY DIFFERENTIAL EQUATION CAN BE EASILY SOLVED
BY THE LAPLACE TRANSFORM METHOD WITHOUT FINDING
THE GENERAL SOLUTION AND THE ARBITRARY CONSTANTS.
THE METHOD IS ILLUSTRATED BY FOLLOWING EXAMPLE,
C. Laplace Transform to solve Differential Equation:
Differential equation is

14. The following example is based on concepts from nuclear physics. Consider
the following first order linear differential equation
Where represents the number of un decayed atoms remaining in a sample of a
radioactive isotope at time and is the decay constant.
D. Laplace Transform in Nuclear Physics:
𝑑𝑁
𝑑𝑡
+ λ𝑁=0

15. E. LAPLACE TRANSFORM IN CONTROL ENGINEERING.
In Mechanical engineering field Laplace Transform is widely used to solve
differential equations occurring in mathematical modeling of mechanical system
to find transfer function of that particular system. Following example describes
how to use Laplace Transform to find transfer function.
Example: The tank shown in figure is initially empty . A constant rate of flow is
added for The rate at which flow leaves the tank is The cross sectional area of
the tank is . Determine the differential equation for the head Identify the time
constant and find the transfer function of system.

16. F. Application of Laplace Transform In Signal
Processing
Laplace transforms are frequently opted for signal
processing. Along with the Fourier transform, the Laplace
transform is used to study signals in the frequency domain.
When there are small frequencies in the signal in the
frequency domain then one can expect the signal to be smooth
in the time domain. Filtering of a signal is usually done in the
frequency domain for which Laplace acts as an important tool
for converting a signal from time domain to frequency
domain.

17. G. REAL-LIFE APPLICATIONS
 Semiconductor mobility
 Call completion in wireless networks
 Vehicle vibrations on compressed rails
 Behavior of magnetic and electric fields
above the atmosphere

18. CHARACTERIZATION OF LINEAR TIME-INVARIANT SYSTEMS
USING LAPLACE TRANSFORM
For a casual system ROC associated with the system, the
function is the right half plane. A system is anti-casual if its
impulse response h(t) =0 for t > 0.
If ROC of the system functions H(s) includes the jω axis then
the L.T.I. the system is called a stable system. If a casual
system with rational system functions H(s) have negative real
parts for all of its poles then the system is stable.

19. LIMITATION OF LAPLACE TRANSFORM
Only be used to solve differential equations with known
constants. An equation without the known constants, then this
method is useless.

20. Conclusion
The paper presented the application of Laplace transform
in different areas of physics and electrical power
engineering.
 Besides these, Laplace transform is a very effective
mathematical tool to simplify very complex problems in
the area of stability and control.

21. REFERENCE
 https://www.elprocus.com/what-is-laplace-transform-formula-
properties-conditions-and-applications/
 http://sces.phys.utk.edu/~moreo/mm08/sarina.pdf
 https://www.sjsu.edu/me/docs/hsuChapter%206%20Laplace%20tran
sform.pdf

22. Design by Goni