Electric Circuits, 9th Edition P49. 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. | 456 Introduction to the Laplace Transform Observe that the right-hand side of Eq. may be written as r rdf n f df lim ei dt - -e dt . s- oo j r dt J r dt As 5 oo df dt e st 0 hence the second integral vanishes in the limit. The first integral reduces to 0 - O which is independent of 5. Thus the right-hand side of Eq. becomes df lim A7 z 0 - 0 . s Jo dt Because f 0 is independent of s the left-hand side of Eq. may be written lim F s - 0 lim sF s - 0 s OC 5 oc From Eqs. and Jim sF s 0 Jjm f t which completes the proof of the initial-value theorem. The proof of the final-value theorem also starts with Eq. . Here we take the limit as .v 0 pOO r 12-99 Jo- at The integration is with respect to t and the limit operation is with respect to 5 so the right-hand side of Eq. reduces to f df _ f df lim - -e dt - -dt. Jo- dt J J dt v Because the upper limit on the integral is infinite this integral may also be written as a limit process f df f df dt lim rdy Jo- dt i Jo-dy where we use y as the symbol of integration to avoid confusion with the upper limit on the integral. Carrying out the integration process yields lim f t - 0 lim f t - 0 . Substituting Eq. into Eq. gives limfs s - O lim f t - O . y- 0 Because 0 cancels Eq. reduces to the final-value theorem namely lim sF s lim i . y - oc The final-value theorem is useful only if 00 condition is true only if all the poles of F s except for a simple pole at the origin lie in the left half of the 5 plane. Initial- and Final-Value Theorems 457 The Application of Initial- and Final-Value Theorems To illustrate the application of the initial- and final-value theorems we apply them to a function we used to illustrate partial fraction expansions. Consider the transform pair given by Eq. . The initial-value theorem gives 100 l 3 5 1 lim sF s lim -z------------------------------z 0 6 5 l 6 5 25 52 lim f t -12 .
