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Electric Circuits, 9th Edition P47

Electric Circuits, 9th Edition P47. Designed for use in a one or two-semester Introductory Circuit Analysis or Circuit Theory Course taught in Electrical or Computer Engineering Departments. Electric Circuits 9/e is the most widely used introductory circuits textbook of the past 25 years. As this book has evolved over the years to meet the changing learning styles of students, importantly, the underlying teaching approaches and philosophies remain unchanged. | 436 Introduction to the Laplace Transform 12.4 Functional Transforms A functional transform is simply the Laplace transform of a specified function of t. Because we are limiting our introduction to the unilateral or one-sided Laplace transform we define all functions to be zero for t 0 . We derived one functional transform pair in Section 12.3 where we showed that the Laplace transform of the unit impulse function equals 1 see Eq. 12.14 . A second illustration is the unit step function of Fig. 12.13 a where Figure 12.14 A decaying exponential function. Figure 12.15 A sinusoidal function for t 0. 12.18 Equation 12.18 shows that the Laplace transform of the unit step function is 1 s. Tile Laplace transform of the decaying exponential function shown in Fig. 12.14 is e at 12.19 In deriving Eqs. 12.18 and 12.19 we used the fact that integration across the discontinuity at the origin is zero. A third illustration of finding a functional transform is the sinusoidal function shown in Fig. 12.15. The expression for f t for t 0 is sino z hence the Laplace transform is Z.CC .T sin art sin a t e sl dt dt 00 -----------z------------dt - 1 1 2 V - 1 5 j t i s2 a 2 12.20 Table 12.1 gives an abbreviated list of Laplace transform pairs. It includes the functions of most interest in an introductory course on circuit applications. 12.5 Operational Transforms 437 TABLE 12.1 An Abbreviated List of Laplace Transform Pairs Type 0 t 0- F s impulse 0 r 1 step 0 1 ramp t 1 5 exponential e at 1 5 a sine sin art s2 i 2 cosine COS ü t s s2 i 2 damped ramp to al 1 te 5 a 2 damped sine e al sin art s a 2 it 2 damped cosine e l cos a t s a s a 2 a 2 ASSESSMENT PROBLEM Objective 1 Be able to calculate the Laplace transform of a function using the definition of Laplace transform 12.1 Use the defining integral to a find the Laplace transform of cosh ßt b find the Laplace transform of sinh ßt. Answer a s s2 ß2 b ß s2 - 32 . NOTE Also try Chapter Problem 12.17. 12.5 Operational Transforms Operational