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Introduction to the Laplace Transform

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Introduction to the Laplace Transform

The important thing about a problem is not its solution, but the strength we gain in finding the solution.

—Anonymous

Enhancing Your Skills and Your Career

ABET EC 2000 criteria (3.h), “the broad education necessary to understand the impact of engineering solutions in a global and societal context.”

As a student, you must mak e sure you acquire “the broad education necessary to understand the impact of engineering solutions in a global and societal context.” To some extent, if you are already enrolled in an ABET-accredited engineering program, then some of the courses you are required to take must meet this criteria. My recommendation is that even if you are in such a program, you look at all the elective courses you take to make sure that you e xpand your awareness of global issues and societal concerns. The engineers of the future must fully understand that they and their activities affect all of us in one way or another.

ABET EC 2000 criteria (3.i), “need for, and an ability to engage in life-long learning.”

You must be fully aware of and recognize the “need for, and an ability to engage in life-long learning. ” It almost seems absurd that this need and ability must be stated. Yet, you w ould be surprised at ho w m any engineers do not really understand this concept. The only way to be really able to keep up with the explosion in technology we are facing now and will be f acing in the future is through constant learning. This learning must include nontechnical issues as well as the latest technology in your field.

The best w ay to k eep up with the state of the art in your field is through your colleagues and association with indi viduals you meet through your technical organization or organizations (especially IEEE). Reading state-of-the-art technical articles is the next best way to stay current.

Photo by Charles Alexander

Pierre Simon Laplace (1749–1827), a French astronomer and mathematician, first presented the transform that bears his name and its applications to differential equations in 1779.

Born of humble origins in Beaumont-en-Auge, Normandy, France, Laplace became a professor of mathematics at the age of 20. His math ematical abilities inspired the f amous mathematician Simeon Poisson, who called Laplace the Isaac Newton of France. He made important contributions in potential theory, probability theory, astronomy, and celestial mechanics. He w as widely kno wn for his w ork, Traite de Mecanique Celeste (Celestial Mechanics), which supplemented the work of Newton on astronomy . The Laplace transform, the subject of this chapter , is named after him.

Learning Objectives

By using the information and exercises in this chapter you will be able to:

    1. Understand the Laplace transform, its importance in circuit analysis, and how to determine the Laplace transform of functions common to circuit analysis.
    1. Understand the properties of the Laplace transform.
    1. Understand the inverse Laplace transform and how to determine its given functions in the s-domain.
    1. Understand the convolution integral and how to use it in the time domain and its equivalence in the s-domain.

15.1 Introduction

Our goal in this and the follo wing chapters is to develop techniques for analyzing circuits with a wide variety of inputs and responses. Such circuits are modeled by differential equations whose solutions describe the total response behavior of the circuits. Mathematical methods have been devised to systematically determine the solutions of dif ferential equa tions. We now introduce the po werful method of Laplace transformation, which involves turning differential equations into algebraic equations, thus greatly facilitating the solution process.

The idea of transformation should be f amiliar by now. When using phasors for the analysis of circuits, we transform the circuit from the time domain to the frequency or phasor domain. Once we obtain the phasor result, we transform it back to the time domain. The Laplace transform method follows the same process: We use the Laplace transformation to transform the circuit from the time domain to the frequenc y domain, obtain the solution, and apply the inverse Laplace transform to the result to transform it back to the time domain.

The Laplace transform is significant for a number of reasons. First, it can be applied to a wider variety of inputs than phasor analysis. Second, it provides an easy way to solve circuit problems involving initial conditions, because it allo ws us to w ork 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 responses.

We begin with the definition of the Laplace transform which gives rise to its most essential properties. By e xamining these properties, we shall see ho w and wh y the method w orks. This also helps us to better appreciate the idea of mathematical transformations. We also consider some properties of the Laplace transform that are very helpful in circuit analysis. We then consider the inverse Laplace transform, transfer functions, and convolution. In this chapter , we will focus on the mechanics of the Laplace transformation. In Chapter 16 we will e xamine how the Laplace transform is applied in circuit analysis, netw ork stability, and network synthesis.