402

THE LAPLACE TRANSFORM METHOD

[CHAP. 16

But est in the integrand approaches zero as s ! 1. Thus, lim fsFðsÞ  f ð0þ Þg ¼ 0

s!1

Since f ð0þ Þ is a constant, we may write f ð0þ Þ ¼ lim fsFðsÞg s!1

which is the statement of the initial-value theorem. EXAMPLE 16.3 In Example 16.1,  lim fsIðsÞg ¼ lim 10 

s!1

s!1

 8s ¼ 10  8 ¼ 2 s þ 2000

þ

which is indeed the initial current, ið0 Þ ¼ 2 A. The ﬁnal-value theorem is also developed from the direct Laplace transform of the derivative, but now the limit is taken as s ! 0 (through real values).   ð1 df ðtÞ df ðtÞ st lim l e dt ¼ limfsFðsÞ  f ð0þ Þg ¼ lim s!0 s!0 0 s!0 dt dt ð1 ð1 df ðtÞ st lim But df ðtÞ ¼ f ð1Þ  f ð0þ Þ e dt ¼ s!0 0 dt 0 and f ð0þ Þ is a constant. Therefore, f ð1Þ  f ð0þ Þ ¼ f ð0þ Þ þ limfsFðsÞg s!0

f ð1Þ ¼ limfsFðsÞg

or

s!0

This is the statement of the ﬁnal-value theorem. The theorem may be applied only when all poles of sFðsÞ have negative real parts. This excludes the transforms of such functions as et and cos t, which become inﬁnite or indeterminate as t ! 1.

16.6

PARTIAL-FRACTIONS EXPANSIONS

The unknown quantity in a problem in circuit analysis can be either a current iðtÞ or a voltage vðtÞ. In the s-domain, it is IðsÞ or VðsÞ; for the circuits considered in this book, this will be a rational function of the form RðsÞ ¼

PðsÞ QðsÞ

where the polynomial QðsÞ is of higher degree than PðsÞ. Furthermore, RðsÞ is real for real values of s, so that any nonreal poles of RðsÞ, that is, nonreal roots of QðsÞ ¼ 0, must occur in complex conjugate pairs. In a partial-fractions expansion, the function RðsÞ is broken down into a sum of simpler rational functions, its so-called principal parts, with each pole of RðsÞ contributing a principal part. Case 1: s ¼ a is a simple pole. principal part of RðsÞ is

When s ¼ a is a nonrepeated root of QðsÞ ¼ 0, the corresponding

A sa

where

A ¼ limfðs  aÞRðsÞg s!a



If a is real, so will be A; if a is complex, then a is also a simple pole and the numerator of its principal part is A . Notice that if a ¼ 0, A is the ﬁnal value of rðtÞ Case 2: s ¼ b is a double pole. When s ¼ b is a double root of QðsÞ ¼ 0, the corresponding principal part of RðsÞ is

Mahmood_Nahvi_eBook_Schaum_s_Outlines_Theory_An
Mahmood_Nahvi_eBook_Schaum_s_Outlines_Theory_An