ABSTRACT
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 engineering, control engineering, linear system analysis, statistics, optics, quantum physics, etc.
CHAPTER ONE
1.0 INTRODUCTION
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.
AIMS/OBJECTIVE
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
SCOPE OF STUDY
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. 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.
BACKGROUND OF STUDY
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 (1749-1827) 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. Most of electrical engineering was invented by 1925, reduced to practice by 1935 and mathematically analyzed and scientifically understood by 1945.
The Laplace transform is denoted byL{f(t) }has it function f(t) with t (t> 0) that transforms it to a function f(s) with a real argument s. In order for any function of time f(t) to be Laplace transformable it must satisfy the following Dirichlet conditions;
f(t)Must be piecewise continuous which means that it must be single valued but can have a finite number of finite isolated discontinuities for t>0.
f(t)Must be of exponential order, which means f(t)must remain less than
∫▒e^(〖-a〗_0 t) Ast approaches ∞ where s is a positive constant and a_0is a real positive number.
If there is any function f(t)that satisfies the Dirichlet conditions then
F(s)=∫_0^∞▒〖f(t) e^(-st) dt〗
Written as L{f(t) } is called the Laplace transform of f(t)Here s can be either a real variable or a complex quantity. 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. One way to circumvent this problem is to apply LAPLACE TRANSFORMATION. 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.
The Laplace transform is significant for a number of reasons.
First, it can be applied to a wider variety of inputs than other methods of analysis.
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.
The Laplace Transform
The Laplace transform of a function f(t)defined for all real numbers t≥0 is the function F(s), defined by:
F(s)=L{f(t) }=∫_(0^-)^∞▒e^(-st) (1.0)
Where:
F(s) Indicates the Laplace transform of the function f(t)on condition that;
f(t)=0,
t
s= Complex variable known as Laplace variable given by s=σ+jω
L= Laplacian transform operator
Since the arguments of the exponent in (1.0) must be dimensionless, it follows thats has the dimensions of frequency and units of inverse seconds (s^(-1) ).
In (1.0), the lower limit of 0^-is short notation to means lim┬(ϵ→0)– ϵ and assures the inclusion of the entire Dirac delta function δ(t)at 0 if there is such an impulse in
f (t) at 0.
The Laplace transform is an integral transformation of a function f(t) from the time domain into the complex frequency domain, giving F(s).
We assume in (1.0) that f(t) is ignored for t