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International Journal of Emerging Technologies in Research (IJETER) Volume 4, Issue 6, June (2016) www.ijeter.everscience.org

Theory of Laplace and With Its Applications

J.R.Sontakke Department Of , Datta Meghe Institute Of Engineering Technology And Research (Wardha) M.S.

Abstract – The concept of Laplace Transformation and Fourier range of .[8][5]. This is done by converting Transformation play a vital role in various areas of science and infinite in terms of series and cosines into a engineering. In solving problems relating to these fields, one double infinite series involving complex exponentials. On usually encounters problems on time invariants, differential other hands, Fourier transform basically involve equations, time and frequency domains for non-periodic wave domain representation of non-periodic , in which such forms. This paper will discuss the fundamentals of Laplace transform and Fourier transform and basic applications of these representation is valid over the entire time domain and fundamentals to electric circuits and design and solution accomplished by means of Fourier [7][5]. to related problems.   1 it  it  Index Terms – Laplace Transform & Fourier Transform. f (t)  e  f ( )e d d 2   1. INTRODUCTION    Laplace transform is an method which is It is reduces to particularly useful in solving linear ordinary differential 1  equation. It finds very wide applications in various areas of f (t)  f ()eit d , , , optics, 2  mathematics , . The laplace transform can be  interpreted as a transformation from the time to the frequence 2. PROPERTIES OF LAPLACE TRANSFORM domain where inputs and outputs are function of time f(t) to Some of the very important properties of Laplace transforms be laplace transformable, it must satisfy the following which will be used in its applications to be discussed later on Dirichlet conditions : are described as follows: [3] 1) f(t) must be piecewise continuous which means that it 2.1. must be single valued but can have a finite number of finite isolated discontinuities for t > 0. If Laplace Transform of function f(t),g(t) and h(t) be ̅̅̅̅̅̅̅ ̅̅̅(̅̅̅) ̅ 2) f(t) must be exponential order which means that f(t) must 푓(푠), 푔 푠 푎푛푑 ℎ(푆) respectively , then remain less than 푠푒−푎0푡 as t approaches ∞ where S is a L{푎푓(푡) + 푏푔(푡) + ℎ(푡)} = 푎푓(̅ 푠) + 푏푔̅ (푠) + 푐ℎ̅ (푠) Where positive constant and 푎0 is a real positive number. a, b, c are constant. If there is any function f(t) that satisfies the Dirichlet 2.2. Differentiation Conditions then, If L{푓(푡)} = 푓(̅ 푠), then 퐿{푓푛(푡)} = 푠푛푓(̅ 푠) − 푠푛−1푓(0) − ∞ ( ) −푠푡 푛−2 ′ 푛−1 퐹 푠 = ∫0 푓(푡)푒 푑푡 written as L(f(t)) is called the laplace 푠 푓 (0) − − 푓 (0) transformation of f(t). Here, s can be either a real variable or a Where n denote nth complex quantity. 2.3 Integral ∞ −푠푡 | −푠푡| The integral ∫0 푓(푡)푒 푑푡 converges if ∫ 푓(푡)푒 푑푡 < 푡 푓̅(푠) ∞ , 푠 = 휎 + 푗푤 [1][2] If L{푓(푡)} = 푓(̅ 푠), then 퐿 {∫ 푓(푢)푑푢} = 0 푠 Fourier transform is used for decomposing into its 2.4 Mutipication by t constituent frequencies and its oscillatory functions. It represent signal in and transforms one If L{푓(푡)} = 푓(̅ 푠) , then complex value function of a real variable into another. 푑 (i) 퐿{푡푓(푡)} = − 푓(̅ 푠) Fourier transform entails representation of a non-periodic 푑푠 dn function not as a sum but as an integral over a continuous (ii) 퐿{푡푛푓(푡)} = (−1)n f(̅ s) dsn

ISSN: 2454-6410 ©EverScience Publications 103

International Journal of Emerging Technologies in Engineering Research (IJETER) Volume 4, Issue 6, June (2016) www.ijeter.everscience.org

2.5 Division by t inputs and output function of complex angular frequency in

푓(푡) ∞ radians per unit time). [5] If L{푓(푡)} = 푓(̅ 푠) , then 퐿 { } = ∫ 풇̅(풔)ds 푡 푠 3.1 Application in electric circuit theory 2.6 Time delay The Laplace transform can be applied to solve the switching The substitution of 푡 − 휆 for the variable t in the transform transient phenomenon in the series or parallel RL,RC or RLC Lf(t) corresponds to the of the function F(s) by circuits [6]. A simple example of showing this application 푒−휆푠 that is follows next. 퐿(푓(푡 − 휆)) = 푒−휆푠퐹(푠) . [2] Let us consider a series RLC circuit as shown in Fig 1. to which a d.c. voltage Vo is suddenly applied. 2.7 L A function f(t) is said to be periodic if and only if there exists R C a positive T such that for all values of t, 푓(푡) = 푓(푡 + 푇) = 푓(푡 + 2푡) = −−= 푓(푡 + 푛푇) The positive real number T is called the period of the function f(t) and it is the least value of variable t by which it may be V increased or decreased without affecting the value of the 0 function f(t).

[ ] 1 푇 −푠푡 ( ) 퐿 푓(푡) = −푠푇 ∫0 푒 푓 푡 푑푡. 1−푒 Fig.1 : Series RLC circuit 2.8 Unit step function Now, applying Kirchhoff’'s Voltage Law (KVL) to the circuit, The unit step function can be said to be the force of we have, magnitude 1 which is applied to a system at time 푡 ≥ 0 and it 푑푖 1 푅푖 + 퐿 + ∫ 푖 푑푡 = 푉 (3) is defined as 푑푡 퐶 0 0 , 푡 < 0 푈(푡) = { Differentiating both sides, 1 , 푡 ≥ 0 푑2푖 1 푑푖 퐿 + 푖 + 푅 = 0 푑푡2 퐶 푑푡 1 푑2푖 푅 푑푖 1 L[U(t)] = , s > 0 퐿 + ( ) + ( ) 푖 = 0 (4) s 푑푡2 퐿 푑푡 퐿퐶 2.9 Now , Applying laplace transform to this equation, let us assume that the solution of this equation is Mathematically it is defined as the ratio of Laplace transform of output (response) of the system to the Laplace transform of 푖(푡) = 퐾푒푠푡 where K and s are constants which may be real , input ( driving function) , under the assumption that all initial imaginary or complex. conditions are zero.[4] Now, from eqn (4), If T(s) is the transfer function of the system then , 2 푠푡 푠푡 1 푠푡 퐿푎푝푙푎푐푒 푡푟푎푛푠푓표푟푚 표푓 표푢푡푝푢푡 퐶(푠) 퐿퐾푠 푒 + 푅퐾푒 + 퐾푒 = 0 푇(푠) = = 퐶 퐿푎푝푙푎푐푒 푡푟푎푛푠푓표푟푚 표푓 푖푛푝푢푡 푅(푠) Which on simplification gives,

3. APPLICATIONS OF LAPLACE TRANSFORM 푅 1 Or 푠2 + ( ) 푠 + = 0 In engineering and science, the Laplace transform is used for 퐿 퐿퐶 solving problems of time invariant systems such as electrical The roots of the equation would be circuits, harmonics, oscillations, mechanical system, , optical devices ,signal analysis and design , system 푅 푅2 1 푠 , 푠 = ± √( ) − ( ) analysis, and solving differential equations. The Laplace in its 1 2 2퐿 4퐿2 퐿퐶 analysis transforms the time domain in which outputs and inputs are function of time to the frequency domain (the The general solution of the is thus,

푠1푡 푠2푡 푖(푡) = 퐾1푒 + 퐾2푒

ISSN: 2454-6410 ©EverScience Publications 104

International Journal of Emerging Technologies in Engineering Research (IJETER) Volume 4, Issue 6, June (2016) www.ijeter.everscience.org

where K1 and K2 are determined from the initial conditions. Solution, Now , if we define, Draw the Laplace domain network of the given system 푅 훼 = 퐷푎푚푝푖푛푔 퐶표푒푓푓푖푐푖푒푛푡 = R 2퐿 1 1 and natural frequency , 휔 = which is also 푛 √퐿퐶 known as undamped natural frequency or resonant frequency. R2 1 2 2 Thus roots are : 푠1, 푠2 = −훼 ± √훼 − 휔푛 SC 1 Eo(s) Ei(s) The final form of solution depends on whether 1 푅2 1 푅2 1 푅2 1 SC ( ) > ; ( ) = and ( ) < 2 4퐿2 퐿퐶 4퐿2 퐿퐶 4퐿2 퐿퐶 The three cases can be analysed based on the initial Fig. : 3 conditions of the circuit which are : overdamped case if Eo(s)

Z1 훼 > 휔푛 Critically damped case if 훼 = 휔푛 and under damped case if 훼 < 휔푛. [2]

3.2 Application in control system Z2

Ei(s) The control system can be classified as electrical, mechanical, Eo(s) hydraulic, thermal and so on. All system can be described by I(s) integro-differential equations of various orders. While the output of such systems for any input can be obtained by solving such integro-differential equations. Mathematically, it is very difficult to solve such equations in time domain. The Fig. : 4 laplace transform of such integro-differential equations 1 푅1× 1 푠퐶1 푅1 converts them into simple algebraic equations. All the 푍1 = 푅1 ∥ = 1 = 푠퐶1 푅1+ 1+푠푅1퐶1 complicated computations then can be easily transformed 푠퐶1 equations are known as equations in frequency domain. 1 1+푠푅2퐶2 푍2 = 푅2 + = Then by eliminating unwanted variable, the required variable 푠퐶2 푠퐶2 퐸 (푠)−퐸 (푠) in s domain can be obtained. Then by using technique of 퐼(푠) = 푖 0 ------(1) laplace inverse , time domain function for the required 푍1 variable can be obtained. Hence making the computations 퐸0(푠) = 퐼(푠)푍2 ------(2) easy by converting the integro-differential equations into 퐸 (푠) − 퐸 (푠) algebraic is the main essence of the laplace transform.[4] ∴ 퐸 (푠) = 푖 0 × 푍 0 푍 2 Example 1 Find the transfer function of the given circuit.[4] 1 푍2 푍2 ∴ 퐸0(푠) [1 + ] = 퐸푖(푠) R1 푍1 푍1

1+푠푅2퐶2 퐸0(푠) 푍2 푠퐶2 ∴ = = 푅 (1+푠푅 퐶 ) 퐸푖(푠) 푍1 + 푍2 1 + 2 2 1+푠푅1퐶1 푠퐶2 R2 C 1 퐸 (푠) ei 0 eo ∴ 퐸푖(푠) (1 + 푠푅2퐶2)(1 + 푠푅1퐶1) C2 = 푠2푅 푅 퐶 퐶 + 푠[푅 퐶 + 푅 퐶 + 푅 퐶 ] + 1 1 2 1 2 1 1 2 2 1 2 4. PROPERTIES OF FOURIER TRANSFORM Fig.: 2 The properties of Fourier transform are as follows [3]

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International Journal of Emerging Technologies in Engineering Research (IJETER) Volume 4, Issue 6, June (2016) www.ijeter.everscience.org

4.1 Linearity 6. CONCLUSION F (s) F (s) In this paper the applications of laplace transform in areas of If 1 and 2 are Fourier transform of electrical power engineering and control system are given. f1(x) and f2 (x) respectively then Also the Laplace transform resolves a function into its moments whereas the Fourier transform resolves functions or Faf1(x) bf 2 (x) aF1(s) bF2 (s) signal into its mode of vibration. 4.2 Change of scale property REFERENCES [1] A.D. Poularikas, The Transform and Applications Handbook (McGraw If F(s) is the complex Fourier Transform of f(x) then Hill,2000), 2 nd ed. [2] Sarina Adhikari, Laplace Transform And Its Applications , Department 1  s  of Electrical Engineering and Computer Science, University of Ff (ax) F  Tennessee. a  a  [3] H.K.Dass , Advanced Engineering Mathematics , S.Chand and company ltd New Delhi,2009. 4.3 Shifting property [4] U.A.Bakshi, V.U.Bakshi , Control System Engineering. [5] M.C.Anumaka, Analysis and applications of laplace /fourier transforms If is the complex Fourier Transform of f(x) then in electric circuit, IJRRAS Volume 12,Issue2 August 2012. [6] M.N.S. Charles K. M.N.S. Charles K. Alexander, Fundamentals of Ff (x  a) eisaF(s) Electric Circuits (McGraw Hill,2006), 2 nd ed. [7] H.F. Davis, Fourier series and orthogonal Functions New York ,1963 5. APPLICATION OF FOURIER TRANSFORM [8] K.A.Stroud “ Advanced Engineering Mathematics” Mac Williams 2003 ISBN 1-4039-0312-3 Fourier transform can be used in communications, analysis, statistics, quantum physics,optics, solution of partial differential equations and antennas, etc.[5]

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