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# ppt M3 Laplace Transform.pdf

23 Mar 2023                      1 sur 22
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### ppt M3 Laplace Transform.pdf

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
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