最全拉氏变换计算公式

PAGE\*MERGEFORMAT#最全拉氏变换计算公式1.拉氏变换的基本性质1线性定理齐次性L[af(t)]=aF(s)叠加性L[f(t)土f(t)]二F(s)土F(s)1212L[f(t)]-sF(s)-f(0)dtL[]=s2F(s)-sf(0)-f(0)dt22微分定理一般形式ML[]-SnF(s)-Xsn-kf(k-1)(0)dtnk-1f(t)dk-1/(t)f(k-1)(t)-,dtk-1初始条件为0时dnf(t)L[八丿]-snF(s)dtnL[ff(t)dt]-加+一般形式ssL山f(t)(dt)2]-F(s)++山f(t)(dt巴-0s2s2s3积分定理ML[Ljf(t)(dt)n]-F(s)+X1[Ljf(t)(dt)n]snsn-k+1t-0k1初始条件为0时L[JaJf(t)(dt)n]-加sn4延迟定理（或称t域平移定理）L[f(t—T)1(t—T)]-e-tsF(s)5衰减定理（或称s域平移定理）L[f(t)e-at]-F(s+a)6终值定理limf(t)-limsF(s)tSsT07初值定理limf(t)-limsF(s)tt0ss8卷积定理L[jtf(t-T)f(t)dT]-L[jtf(t)f(t-T)dT]-F(s)F(s)012012121PAGE\*MERGEFORMAT#2.常用函数的拉氏变换和z变换拉氏变换E(s)时间函数e(t)Z变换E(z)6(t)1—e-Ts5(t)=天5(t—nT)T1(t)Tz(z-1)2t22T2z(z+1)2(z—1)3tnsn+ln!Hm(-1)nQna—0(—z—)n!Qanz—e—aTz一e—aT(s+a)2te—atTze—aT(z—e—aT)2as(s+a)1—e—at(1一e-aT)z(z—1)(z—e—aT)10(s+a)(s+b)e—at—e—bt11sinotzsin①Tz2—2zcosoT+112s2+o2cosotz(z-cosoT)z2—2zcosoT+113(s+a)2+o2e—atsinotze—aTsinoTz2—2ze—aTcosoT+e—2aT14s+a(s+a)2+o2e—atcosotz2—ze—aTcosoTz2—2ze—aTcosoT+e—2aT15s—(1/T)lnaat/t11PAGE\*MERGEFORMAT#3．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F(s)是s的有理真分式F(s)=A(s)B(s)bsm+bsm一1+A+bs+bmm—110asn+asn—1+A+as+ann一11m,n是正整数。按代数定理可将F(s)展式中系数a,aa,a,b,b,Ab,b都是实常数;01n一1n01m一1m开为部分分式。分以下两种情况讨论。①A(s)二0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。式中，cccHYPERLINK\l"bookmark25"\o"CurrentDocument"F(s)=1—+2—+A+i—s一ss一ss一s12is,s,A,s1c+A+n——s一sn是特征方程A(s)=O的根。nnc=2—s一si=1ic为待定常数，i称为F(s)在s处的留数，可按下式计i算：c=lim(s一s)F(s)iific=如iA'(s)s=si式中，A'(s)为A(s)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f(t)二L-1If(s)LL-1£c―is-s1-i=1i」n=乙ce-呼ii=1A(s)二0有重根设A(s)二0有r重根s,1F(s)可写为B(s)F(s)=(s一s)r(s一s)A(s一s)1r+1n式中,ccr+r—1(s一s)r(s一s)r-1+Acc+1+r+1+A(s一s)s一s1r+1c+i+A+s一sicns一sns为F(s)的r重根，s,1r+1为F(s)的n-r个单根;PAGE\*MERGEFORMAT#其中，c，…，c仍按式(F-2)或(F-3)计算，c,c,r+1nrr-1c1则按下式计算:c=lim(s-s)rF(s)r1sTs[c=lim—[(s-s)rF(s)]r-1ds1sTs1cr-j原函数f(t)为f(t)=L-i[F(s)]=L-icr—(s-s)r1=却暑js-s1)rF(s)(F-5)d(r-1)-lim(r-1)!$6]ds(r-1)(s-s)rF(s)1cc+T+A+1—(s-s)r-1(s-s)11c++As-sr+1c+i+As-sic+n—s-sncrtr-1(r-1)!+Cr-1tr-2+A+C(r-2)!t+c21es1tn+5cestii=r+1F-6)