 # Application of laplace transform pdf

It only takes a minute to sign up. I am curious to know what kind of applications the Laplace transform has.

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Yes, I know people will reference Wikipedia, and other online sites which discuss the Laplace transform at length. However, all the applications are very one-dimensional.

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For example, even looking at Wikipedia most the "applications" are towards solving differential equations. Furthermore, I have been searching for many books, engineering books, physics books, math books, ect.

All of those books use the Laplace transform only as a means to solve differential equations. I never see any other applications. To add further to my question, I heard it said, each time the Laplace transform is introduced, of how valuable it is to electrical engineering. In fact, I said so myself, but looking at books, I again only find the applications of the transform to solving differential equations.

## The Laplace Transform and Its Application to Circuit Problems

Nothing really beyond that. This is what I mean by "one-dimensional applications". Yes, the Laplace transform has "applications", but it really seems that the only application is solving differential equations and nothing beyond that.

Though, that is not entirely true, there is one more application of the Laplace transform which is not usually mentioned. And that is the moment generating function from probability theory. After all that is the original motivation of Laplace to create that transform in the first place. Unfortuantely, moment generating functions are not of superior importance to probability theory to the best of my knowledgeand so the the only "big" applications of this transform appears to be only to the solution of differential equations both ordinary and partial.

Contrast this with the Fourier transform. The Fourier transform can be used to also solve differential equations, in fact, more so. The Fourier transform can be used for sampling, imaging, processing, ect. And even in probability theory the Fourier transform is the characteristic function which is far more fundamental than the moment generating function.

The Fourier transform is certaintly a huge powerful tool with vast applications all across mathematics, physics, and engineering. There are books, across all fields, all devoted to the different applications of this transform.

Advertisement Hide. Introduction to the Theory and Application of the Laplace Transformation. Pages Examples of Laplace Integrals.

Precise Definition of Integration. The Half-Plane of Convergence. The Laplace Integral as a Transformation. The Unique Inverse of the Laplace Transformation. The Laplace Transform as an Analytic Function.

The Mapping of a Linear Substitution of the Variable. The Mapping of Integration. The Mapping of Differentiation. The Mapping of the Convolution. Applications of the Convolution Theorem: Integral Relations. The Laplace Transformation of Distributions.

Rules of Mapping for the L -Transformation of Distributions. The Normal System in the Space of Distributions. The Behaviour of the Laplace Transform near Infinity. The Fourier Transformation. The Image of the Product.This paper presents an overview of the Laplace transform along with its application to basic circuit analysis. There is a focus on systems which other analytical methods have difficulty solving. The concept of Laplace Transformation plays a vital role in diverse areas of science and technology such as electric circuit analysis, communication engineeringcontrol engineering, linear system analysis, statistics, optics, quantum physics, etc.

The Laplace Transform is an integral transform method which is particularly useful in solving linear ordinary differential equations.

It finds very wide applications in various areas of physics, optics, electrical engineering, control engineering, mathematics, signal processing and probability theory. The Laplace transform-is an important concept from the branch of mathematics called functional analysis. It is a powerful technique for analyzing linear time-invariant systems such as electrical circuits, harmonic oscillators, mechanical systems, control theory and optical devices using algebraic methods.

Given a simple mathematical or functional description of an input or output to a system, the Laplace transform provides an alternative functional description that often simplifies the process of analyzing the behavior of the system, or in synthesizing a new system based on a set of specifications. The analysis of electrical circuits and solution of linear differential equations is simplified by use of Laplace transform.

In actual Physics systems the Laplace transform can be interpreted as a transformation from the time domain, where input and output are functions of time to the frequency in the domain, where input and output are functions of complex angular frequency. The basic process of analyzing a system using Laplace transform involves conversion of the system transfer function or differential equation into s —domain, using s —domain to convert input functions, finding an output function by algebraically combing the input and transfer functions, using partial functions to reduce the output function to simpler components and conversion of output equation back to time domain.

Furthermore, we use the Laplace transform 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. The major purpose of this work is to explore the procedure for solving a complete circuit problem by transform technique. This paper provides the reader with a solid foundation in the fundamentals of Laplace transform and circuit theory, and gain an understanding of some of the very important and basic applications of these fundamentals to electric circuit and solution to related problems.

To apply basic circuit analysis method transform domain circuit to obtain desired response function, write the differential equation in the time domain and solve for a desired variable using transform method. The aim of this monograph as stated earlier is to supply a treatment of the Laplace transform and its applications to circuit analysis. This project intends to study the behavior of circuit.

### Introduction to the Theory and Application of the Laplace Transformation

How does it respond to a given input? How do the interconnected elements and devices in the circuit interact? The study is about creating and solving equations for 1 the inputs, 2 the transmission or intermediate processing function and 3 the output of electrical circuit. It concentrates upon transient analysis and the solution of circuit equations with differential and integral terms using the Laplace transform.

Accordingly, the main body of this thesis is divided into three parts. Part one provides an elementary discussion of the unilateral Laplace transform in addition to derivation of the basic properties of this transform.

Part two will consider some properties of the Laplace transform that are very helpful in circuit analysis. A brief discussion of the Heaviside function, the Delta function, Periodic functions and the inverse Laplace transform. Finally, the third part will outline with proper examples how the Laplace transform is applied to circuit analysis. Furthermore, discuss solutions to few problems related to circuit analysis.

The Laplace transform was discovered originally by Leonhard Euler, the eighteenth-century Swiss mathematician but the technique is named in the honor of Pierre-Simon Laplace a French mathematician and astronomer who used the transform in his work on probability theory and developed the transform as a technique for solving complicated differential equation.

Although the Laplace transform is often taught simply as a method of solving electrical circuit, differential equations, its use and influence is much wider than that in the field of electronics and communication. The use of Laplace transform has produced a literature and a tradition that is the foundation of transient analysis. The transform itself did not become popular until Oliver Heaviside a famous electrical engineer began using a variation of it to solve electrical circuit.

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Most of electrical engineering was invented byreduced to practice by and mathematically analyzed and scientifically understood by In order for any function of time f t to be Laplace transformable it must satisfy the following Dirichlet conditions. In analyzing a system, one usually recounts Time—Invariant, Linear Differential Equations of second or high orders. Generally, it is difficult to obtain solutions of these equations in closed form via the solution methods in ordinary differential equations.

Laplace transforms converts a differential equation into an algebraic equation in terms of the transform function of the unknown quantity intended. Transform domain equivalent circuit are developed for representing the voltage current relationship of all circuit components the use of these equivalent circuit permits the application of basic algebraic circuit analysis schemes to be applied directly to complex circuit.

Second, 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. Third, the Laplace transform is capable of providing us, in one single operation, the total response of the circuit comprising both the natural and forced response. Since the arguments of the exponent in 1.To browse Academia.

Skip to main content. Log In Sign Up. IOSR Journals. Volume 9, Issue 2 Nov. Dhunde1 and G. Through this method the boundary value problem is solved without converting it into Ordinary Differential equation, therefore no need to find complete solution of Ordinary Differential equation.

This is the biggest advantage of this method. The problem related to partial differential equation commonly can be solved by using a special Integral transform thus many authors solved the boundary value problems by using single Laplace Transform . Eltayeb and Kilicman  has worked on the non-homogeneous wave equation with variable coefficients is solved by applying the Double Laplace Transform. In [4 ], R. They also considered two boundary value problems.

The first was related to heat transfer for cooling off a very thin semi-infinite homogeneous plate into the surrounding medium solved by using double Laplace Transforms, the second, was heat equation for the semi-infinite slab where the sides of the slab are maintained at prescribed temperature. The scheme is tested through three different examples which are being referred from [7, 8]. Here p, s are complex numbers.

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References:  D. Dahiya, M. Modelling, Vol. Aghili, A. Dhunde, N. Related Papers.A word in response to the corona virus crisis: Your print orders will be fulfilled, even in these challenging times. In anglo-american literature there exist numerous books, devoted to the application of the Laplace transformation in technical domains such as electrotechnics, mechanics etc.

By contrast, the present book intends principally to develop those parts of the theory of the Laplace transformation, which are needed by mathematicians, physicists a,nd engineers in their daily routine work, but in complete generality and with detailed, exact proofs. For those who are interested only in particular details, all results are specified in "Theorems" with explicitly formulated assumptions and assertions. Chapters treat the question of convergence and the mapping properties of the Laplace transformation. The Fourier Transformation. The Image of the Product. The Holomorphy of the Represented Function.

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Ordinary Differential Equations with Polynomial Coefficients. JavaScript is currently disabled, this site works much better if you enable JavaScript in your browser. Mathematics Probability Theory and Stochastic Processes. Free Preview. Buy eBook. Buy Softcover. FAQ Policy. About this book In anglo-american literature there exist numerous books, devoted to the application of the Laplace transformation in technical domains such as electrotechnics, mechanics etc.

Show all. Examples of Laplace Integrals.

4. Laplace Transforms - Problem#1 - Complete Concept

Integral Equations Pages Doetsch, Gustav. Show next xx.To browse Academia. Internal boundaries are viewed as circles with infinite radii and act as a known limiting case for finite radii internal boundaries.

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The time it takes pressure transient to reach the internal circular boundary and the permeability of the reservoir formation bounded by an internal discontinuity is estimated using generalized type curves. Following the earlier concepts of finding the Reservoir with constant pressure internal circular boundaries slopes of the semi-log plots, derivative type curves are plots occurs naturally in oil fields as gas caps and in geothermal of the derivative of the semi-log curve plotted on a log-log fields as steam or non-condensable gas caps.

Mangold et al. Derivative curves allowed for determination of the studied the effects of a thermal discontinuity on well reservoir parameters like skin, permeability, e. These boundaries the identification of well and reservoir flow behavior . The can also be induced artificially during steam flooding, in- situ thrust of this research is to develop a pressure transient combustion, immiscible gas drive, aquifer gas storage and analysis method for a drawdown constant rate test for a well growth of steam or gas bubbles below the bubble point near an internal circular boundary.

This reservoir limit test pressure. A stimulation program, such as acidizing, can also may be analyzed to determine permeability of the reservoir result in a permeability discontinuity , . Wattenburger and section within the discontinuity and the transient time it takes Ramey as reported in , ,  treated a finite to reach the circular discontinuity. Also, this research is aimed thickness skin region as a composite system. In any of these at using Laplace transform to further simplify the differential cases, testing a well completed in the liquid zone, exterior to diffusivity equation describing fluid flow through a reservoir the circular discontinuity, can provide estimates of system with an internal circular boundary, developing a permeability of the reservoir section within this internal sub- general analytic solution in Laplace space to determine region, time taken for pressure transient to reach it and the wellbore pressure, pressure transient distribution and their distance to it using, diffusivity equation, numerical Laplace time rate of change in the considered reservoir system and transform and a generalized semi-log type curve.

In recent finally to determine the effects of internal reservoir limits on years, the numerical Laplace transformation of initial the pressure response of a well, permeability of the reservoir boundary value problems has proven to be useful for well test section within the discontinuity and the transient time to the analysis applications.

However, the success of this approach internal circular boundary.The Laplace Transform has many applications. Two of the most important are the solution of differential equations and convolution. These are discussed below. The Laplace Transform can greatly simplify the solution of problems involving differential equations. Two examples are given below, one for a mechanical system and one for an electrical system. Free Body Diagram. This is equivalent to the original equation with output e o t and input i a t.

The convolution integral is very important in the study of systems. A detailed description is available here. In short, convolution can be used to calculate the zero state response i. Given a system impulse response, h tand the input, f tthe output, y t is the convolution of h t and f t :.

However, this integral can be quite hard to calculate in this form, but is quite easy if using the Laplace Transform. Note: This problem is solved elsewhere in the time domain using the convolution integral.

If you examine both techniques, you can see that the Laplace domain solution is much easier. Solution : To evaluate the convolution integral we will use the convolution property of the Laplace Transform:. We need the Laplace Transforms of f t and h tbut we can look them up in the tables :. We can look up both of these terms in the tables. Aside: Origin of the First Order Differential Equation The differential equation with input f t and output y t can represent many different systems. Take the Laplace Transform of the differential equation using the derivative property and, perhaps, others as necessary. Put initial conditions into the resulting equation. Solve for the output variable.

Get result from Laplace Transform tables. If the result is in a form that is not in the tables, you'll need to use the Inverse Laplace Transform.