The Laplace Transform Method 16.1

INTRODUCTION

The relation between the response yðtÞ and excitation xðtÞ in RLC circuits is a linear diﬀerential equation of the form an yðnÞ þ    þ aj yð jÞ þ    þ a1 yð1Þ þ a0 y ¼ bm xðmÞ þ    þ bi xðiÞ þ    þ b1 xð1Þ þ b0 x ð jÞ

ð1Þ

ðiÞ

where y and x are the jth and ith time derivatives of yðtÞ and xðtÞ, respectively. If the values of the circuit elements are constant, the corresponding coeﬃcients aj and bi of the diﬀerential equation will also be constants. In Chapters 7 and 8 we solved the diﬀerential equation by ﬁnding the natural and forced responses. We employed the complex exponential function xðtÞ ¼ Xest to extend the solution to the complex frequency s-domain. The Laplace transform method described in this chapter may be viewed as generalizing the concept of the s-domain to a mathematical formulation which would include not only exponential excitations but also excitations of many other forms. Through the Laplace transform we represent a large class of excitations as an inﬁnite collection of complex exponentials and use superposition to derive the total response.

16.2

THE LAPLACE TRANSFORM

Let f ðtÞ be a time function which is zero for t  0 and which is (subject to some mild conditions) arbitrarily deﬁned for t > 0. Then the direct Laplace transform of f ðtÞ, denoted l½ f ðtÞ, is deﬁned by ð1 l½ f ðtÞ ¼ FðsÞ ¼ f ðtÞest dt ð2Þ 0þ

Thus, the operation l½  transforms f ðtÞ, which is in the time domain, into FðsÞ, which is in the complex frequency domain, or simply the s-domain, where s is the complex variable  þ j!. While it appears that the integration could prove diﬃcult, it will soon be apparent that application of the Laplace transform method utilizes tables which cover all functions likely to be encountered in elementary circuit theory. There is a uniqueness in the transform pairs; that is, if f1 ðtÞ and f2 ðtÞ have the same s-domain image FðsÞ, then f1 ðtÞ ¼ f2 ðtÞ. This permits going back in the other direction, from the s-domain to the time 398 Copyright 2003, 1997, 1986, 1965 by The McGraw-Hill Companies, Inc. Click Here for Terms of Use.

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