The Laplace Transform Method

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16.1 INTRODUCTION The relation between the response y t and excitation x t in RLC circuits is a linear di erential equation of the form an y n aj y j a1 y 1 a0 y bm x m bi x i b1 x 1 b0 x

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where y and x are the jth and ith time derivatives of y t and x t , respectively. If the values of the circuit elements are constant, the corresponding coe cients aj and bi of the di erential equation will also be constants. In s 7 and 8 we solved the di erential equation by nding the natural and forced responses. We employed the complex exponential function x t 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 in nite collection of complex exponentials and use superposition to derive the total response.

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THE LAPLACE TRANSFORM

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Let f t be a time function which is zero for t 0 and which is (subject to some mild conditions) arbitrarily de ned for t > 0. Then the direct Laplace transform of f t , denoted l f t , is de ned by 1 l f t F s f t e st dt 2

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Thus, the operation l transforms f t , which is in the time domain, into F s , which is in the complex frequency domain, or simply the s-domain, where s is the complex variable j!. While it appears that the integration could prove di cult, 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 t and f2 t have the same s-domain image F s , then f1 t f2 t . This permits going back in the other direction, from the s-domain to the time 398

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THE LAPLACE TRANSFORM METHOD

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domain, a process called the inverse Laplace transform, l 1 F s f t . The inverse Laplace transform can also be expressed as an integral, the complex inversion integral: 1 0 j1 l 1 F s f t F s est ds 3 2j 0 j1 In (3) the path of integration is a straight line parallel to the j!-axis, such that all the poles of F s lie to the left of the line. Here again, the integration need not actually be performed unless it is a question of adding to existing tables of transform pairs. It should be remarked that taking the direct Laplace transform of a physical quantity introduces an extra time unit in the result. For instance, if i t is a current in A, then I s has the units A s (or C). Because the extra unit s will be removed in taking the inverse Laplace transform, we shall generally omit to cite units in the s-domain, shall still call I s a current, indicate it by an arrow, and so on.

SELECTED LAPLACE TRANSFORMS The Laplace transform of the unit step function is easily obtained: 1 1 1 l u t 1 e st dt e st 1 0 s s 0

From the linearity of the Laplace transform, it follows that v t Vu t in the time domain has the sdomain image V s V=s. The exponential decay function, which appeared often in the transients of 7, is another time function which is readily transformed. 1 A a s t 1 A e l Ae at Ae at e st dt 0 A s s a 0 or, inversely, l