chapter13 拉普拉斯变换分析，1：基础.pptVIP

Partial Fraction Expansions: Distinct Poles Our focus will center on proper3 rational functions, where and are the zeros of the denominator polynomial and are called the finite poles of F(s). For the most part, rational functions are sufficient for the study of basic circuits. There are three cases of partial fraction expansions to consider: 1.Case of distinct poles: 2.Case of repeated poles: 3.Case of complex poles. for all for at least one Although case 3 is a subcategory of case 1 or case 2, or both, its attributes warrant special recognition. If F(s) is a proper rational function with distinct (equivalently, simple) poles then where And where the residue Ai of pole pi is Indeed, by inspection, SOLUTION Example13.11 Find f(t) when By linearity, Hence, -1 -1 Example13.12 the numerator and denomanator coincide Find f(t) when SOLUTION Partial Fraction Expansions: Repeated Poles Proper rational functions with repeated roots have a more intricate partial fraction expansion, and calculation of the residues often proves cumbersome. For example, suppose where are unknown constants. The formulas for computing the of equation 13.31 are And, in general Of these expressions, only the first looks like the case with distinct roots; the other require derivatives of . Example13.13 repeated poles The goal here is to illustrate the computation of f(t) when The two easiest constants to find are A2 and B2 and According to equation, Consequently, Similarly, so Which again yields B1 =3. Hence Since were known, a simple trick allows a more direct computation Of : merely evaluate equation 13.33 at Partial Fraction Expansions: Distinct Complex Poles Consider a rational function having a pair of distinct complex poles, as in The following equation: It is possible to write the partial fraction expansion of F(s) as For appropriate polynomials n1(t) and d(s). The res