419附录A拉普拉斯变换及反变换1.表A-1拉氏变换的基本性质齐次性)()]([saFtafL=1线性定理叠加性)()()]()([2121sFsFtftfL±=±一般形式=−=][′−ٛ−=−=−−−−=−∑11)1()1(1222)()()0()()(0)0()(])([)0()(])([kkkknkknnnndttfdtffssFsdttfdLfsfsFsdttfdLfssFdttdfLM)(2微分定理初始条件为0时)(])([sFsdttfdLnnn=一般形式}}∑∫∫∫∫∫∫∫∫∫∫∫==+−===+=++=+=nktnnknnnntttdttfsssFdttfLsdttfsdttfssFdttfLsdttfssFdttfL1010220220]))(([1)(])()([]))(([])([)(]))(([])([)(])([个共个共LLM3积分定理初始条件为0时}nnnssFdttfL)(]))(([=∫∫个共L4延迟定理(或称t域平移定理))()](1)([sFeTtTtfLTs−=−−5衰减定理(或称s域平移定理))(])([asFetfLat+=−6终值定理)(lim)(lim0ssFtfst→∞→=7初值定理)(lim)(lim0ssFtfst∞→→=8卷积定理)()(])()([])()([21021021sFsFdtftfLdftfLtt=−=−∫∫τττττ4202.表A-2常用函数的拉氏变换和z变换表序号拉氏变换E(s)时间函数e(t)Z变换E(z)11δ(t)12Tse−−11∑∞=−=0)()(nTnTttδδ1−zz3s1)(1t1−zz421st2)1(−zTz531s22t32)1(2)1(−+zzzT611+ns!ntn)(!)1(lim0aTnnnaezzan−→−∂∂−7as+1ate−aTezz−−82)(1as+atte−2)(aTaTezTze−−−9)(assa+ate−−1))(1()1(aTaTezzze−−−−−10))((bsasab++−btatee−−−bTaTezzezz−−−−−1122ωω+stωsin1cos2sin2+−TzzTzωω1222ω+sstωcos1cos2)cos(2+−−TzzTzzωω1322)(ωω++asteatωsin−aTaTaTeTzezTze22cos2sin−−−+−ωω1422)(ω+++asasteatωcos−aTaTaTeTzezTzez222cos2cos−−−+−−ωω15aTsln)/1(1−Tta/azz−4213.用查表法进行拉氏反变换用查表法进行拉氏反变换的关键在于将变换式进行部分分式展开,然后逐项查表进行反变换。设)(sF是s的有理真分式01110111)()()(asasasabsbsbsbsAsBsFnnnnmmmm++++++++==−−−−LL(mn)式中系数nnaaaa,,...,,110−,mmbbbb,,,110−L都是实常数;nm,是正整数。按代数定理可将)(sF展开为部分分式。分以下两种情况讨论。①0)(=sA无重根这时,F(s)可展开为n个简单的部分分式之和的形式。∑=−=−++−++−+−=niiinniisscsscsscsscsscsF12211)(LL(F-1)式中,nsss,,,21L是特征方程A(s)=0的根。ic为待定常数,称为F(s)在is处的留数,可按下式计算:)()(limsFsscissii−=→(F-2)或issisAsBc=′=)()((F-3)式中,)(sA′为)(sA对s的一阶导数。根据拉氏变换的性质,从式(F-1)可求得原函数[]⎥⎦⎤⎢⎣⎡−==∑=−−niiisscLsFLtf111)()(=tsniiiec−=∑1(F-4)②0)(=sA有重根设0)(=sA有r重根1s,F(s)可写为())()()()(11nrrsssssssBsF−−−=+L=nniirrrrrrsscsscsscsscsscssc−++−++−+−++−+−++−−LLL11111111)()()(式中,1s为F(s)的r重根,1+rs,…,ns为F(s)的n-r个单根;422其中,1+rc,…,nc仍按式(F-2)或(F-3)计算,rc,1−rc,…,1c则按下式计算:)()(lim11sFsscrssr−=→)]()([lim111sFssdsdcrssr−=→−M)()(lim!11)()(1sFssdsdjcrjjssjr−=→−(F-5)M)()(lim)!1(11)1()1(11sFssdsdrcrrrss−−=−−→原函数)(tf为[])()(1sFLtf−=⎥⎦⎤⎢⎣⎡−++−++−+−++−+−=++−−−nniirrrrrrsscsscsscsscsscsscLLLL111111111)()()(tsnriitsrrrriecectctrctrc∑+=−−−+⎥⎦⎤⎢⎣⎡+++−+−=1122111)!2()!1(L(F-6)