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Theory of Laplace and Fourier Transform with Its Applications

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Theory of Laplace and Fourier Transform with Its Applications International Journal of Emerging Technologies in Engineering Research (IJETER) Volume 4, Issue 6, June (2016) www.ijeter.everscience.org Theory of Laplace and Fourier Transform With Its Applications J.R.Sontakke Department Of Mathematics, Datta Meghe Institute Of Engineering Technology And Research (Wardha) M.S. Abstract – The concept of Laplace Transformation and Fourier range of frequencies.[8][5]. This is done by converting Transformation play a vital role in various areas of science and infinite Fourier series 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 frequency equations, time and frequency domains for non-periodic wave domain representation of non-periodic function, 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 signal design and solution accomplished by means of Fourier integral[7][5]. to related problems. 1 it it Index Terms – Laplace Transform & Fourier Transform. f (t) e f ( )e d d 2 1. INTRODUCTION Laplace transform is an integral transform 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 ()eit d physics , electrical engineering, control engineering, optics, 2 mathematics , signal processing. 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. Linearity 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 derivative 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 signals into its 2.4 Mutipication by t constituent frequencies and its oscillatory functions. It represent signal in frequency domain 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 multiplication 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 Periodic function L A function f(t) is said to be periodic if and only if there exists R C a positive real number 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 Transfer function 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, control theory , 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 differential equation 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.
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