一阶电路的阶跃响应

6.5一阶电路的阶跃响应1.单位阶跃函数定义0)(10)(0)(tttt(t)01单位阶跃函数的延迟)(1)(0)(000ttttttt(t-t0)t001t=0合闸i(t)=Is)(tIsK)(tiu(t))(tISKEu(t)u(t))(tE（1）在电路中模拟开关的动作t=0合闸u(t)=E)(t单位阶跃函数的作用（2）延迟一个函数tf(t)0)(sintttf(t)0)()sin(00ttttt0（3）起始一个函数tf(t)0t0)()sin(tt)()sin(0ttt用单位阶跃函数表示复杂的信号例1)()()(0ttttf(t)tf(t)101t0tf(t)0t0-(t-t0))4()3()1(2)(ttttf例21t1f(t)0243)1()]1()([)(tttttf例41t1f(t)0)1()1()(tttt)1()1(tt)(tt)4()3()1()()(tttttf例31t1f(t)0243)()()1(ttu例5t1u(t)02已知电压u(t)的波形如图，试画出下列电压的波形。)1()2()4(ttu)1()1()3(ttu)()1()2(ttut1u(t)0－22t1u(t)0－11t1u(t)01u(t)t1021iC+–uCRuC(0－)=0)(t)()1()(tetuRCtC)(1)(teRtiRCttuc1注意)(teiRCt和0teiRCt的区别t01it0R1i2.一阶电路的阶跃响应激励为单位阶跃函数时，电路中产生的零状态响应。阶跃响应tiC0激励在t=t0时加入，则响应从t=t0开始。iC(t-t0)C+–uCR+-t-t0RCCeRi1(t-t0)R1t0注意RCeR1－t(t-t0)不要写为)5.0(10)(10ttuS求图示电路中电流iC(t）。10k10kus+-ic100FuC(0－)=00.510t(s)us(V)0例+-ic100FuC(0－)=05kSU5.0等效)(5t5k+-ic100F叠加)5.0(5t5k+-ic100Fs5.01051010036RCmA)(51dd2CtetuCitCSU5.05k+-ic100F)()1()(2ttetuC阶跃响应为：由齐次性和叠加性得实际响应为：)]5.0(51)(51[5)5.0(22teteittCmA)5.0()()5.0(22tetett分段表示为s)0.5(mA0.632-s)5.0(0mA)(5)0.-2(-2tetetittt(s)i(mA)01-0.6320.5波形0.368)5.0()5.0()5.0()()5.0(2222teteteteittttC)5.0()5.0()]5.0()([)5.0(222tetettettt)5.0()5.0()]5.0()([)5.0(2)5.0(212teteettettt)5.0(632.0)]5.0()([)5.0(22tettett10)()(lim0ttp/21/tp(t)-/2)]2()2([1)(tttpt(t)(1)06.5一阶电路的冲激响应1.单位冲激函数定义)0(0)(tt1d)(tt单位脉冲函数的极限单位冲激函数的延迟1d)()(0)(000tttttttt(t-t0)t00（1）单位冲激函数的性质（1）冲激函数对时间的积分等于阶跃函数。)(0100d)(tttttt)()(tdttd2.冲激函数的筛分性)0(d)()0(d)()(fttftttf)(d)()(00tfttttf同理有：d)6()(sintttt02.162166sin例t(t)(1)0f(t)f(0)*f(t)在t0处连续f(0)(t))0(1)0(CcuCu)(tRudtduCccuc不是冲激函数,否则KCL不成立分二个时间段来考虑冲激响应。电容充电，方程为：(1).t在0－→0+间电容中的冲激电流使电容电压发生跃变例1.2.一阶电路的冲激响应激励为单位冲激函数时，电路中产生的零状态响应。冲激响应uC(0－)=0iCR（t）C+-uC1d)(dd000000tttRutdtduCcc01)]0()0([ccuuC(2).t0+为零输入响应（RC放电）Cuc1)0(icRC+uc－01teCuRCtc01teRCRuiRCtccuCt0C1iCt(1)RC1)(1)()(1teRCtiteCuRCtcRCtc)(tdtdiLRiLLiL不可能是冲激函数1)(000000dttdtdtdiLdtRiLL1)0()0(LLiiL)0(1)0(LLiLi(1).t在0－0+间方程为：例20)0(LiL+-iLR)(t+-uL分二个时间段来考虑冲激响应。0电感上的冲激电压使电感电流发生跃变(2).t0+RL放电RL01teLitL0teLRRiutLL)(1teLitL)()(teLRtutLLiL1)0(RuLiL+－tiL0L1tuL)(tLR零状态R(t))(te单位阶跃响应和单位冲激响应关系单位阶跃响应单位冲激响应h(t)s(t)单位冲激(t)单位阶跃(t)dttdt)()()()(tsdtdth零状态h(t))(t零状态s(t))(t证明：1f(t)t)(1)(1)(tttf)(1ts)(1ts)]()([1lim)(0tststh)(tsdtd1s(t)定义在（-，）整个时间轴注)()(ttiS先求单位阶跃响应,令：)()1()(teRtuRCtCiCRisC例1+-uCuC(0+)=0uC()=R=RC0)0(cu已知：求：is(t)为单位冲激时电路响应uC(t)和iC(t)iC(0+)=1iC()=0)(teiRCtc)()1(teRdtduRCtC)()1(teRRCt)(1teCRCt)(1teCRCt)()0()()(tfttf0再求单位冲激响应,令：)()(ttiS)]([ddtetiRCtc)(1)(teRCteRCtRCt)(1)(teRCtRCtuCRt0iC1t0uCt0C1iCt(1)RC1冲激响应阶跃响应3.电容电压或电感电流初值的跃变(1)在冲激激励下，电容电压或电感电流初值的跃变tiCuutCccd1)0()0(0CAuC)0(iCC)(tA例1+-uCtucCAuC(0-)uC(0+)0tic)(tA0(2).换路后电路有纯电容(或纯电容和电压源）构成的回路。tuLiitLLLd1)0()0(0LAiL)0(tuL)(tAuL+-iL)(tA例2+-tiLLAiL(0-)iL(0+)0合闸后由KVLuc(0+)=E)(tEuc)(tcEic0)0(cuic不是冲激函数,uc不会跳变CEteREdtiqRCtc0-0d)(00tCEiRcEiCCk(t=0)例3tic0uctE0ict(CE)0RRCtceREi-已知：E=1V,R=1,C1=0.25F,C2=0.5F,t=0时合k。求：uC1，uC2。解V1)0(1EuC0)0(2Cu电容电压初值发生跃变。)0()0()0(21CCCuuu合k前合k后tuCtuCiCCdddd2211ttuCtuCtiCCd)dddd(d00221100)]0()0([)]0()0([0222111CCCCuuCuuC例4ERC1C2+-uc1+-uc2k(t=0)iiC1iC2i为有限值0（1）确定初值节点电荷守恒V315.025.0125.0)0(211CCECuC可解得)0()0()0()0(22112211CCCCuCuCuCuC)0()()0()0(212211CCCuCCuCuCq(0+)=q(0－)uC()=4V0321)131(1)(3434teetuttC=R(C1+C2)s43iiC1iC2（2）确定时间常数ut0)(92)(61)](98)()321()([41dd3434111tettetttuCittC0321)131(1)(3434teetuttC0)0(1)0(021CCuut)()321()()(341tettutC)()321()(342tetutC1/3uC21uC1tuCiCdd222)()321()(342tetutCit-1/62/9iC14/91/6iC2iiC1iC2)(94)(6134tett)](98)()321[(2134tetti无冲激冲激电流由C1流向C2根据物理概念求电容电流00321311Cu转移的电荷q1=0.25(-2/3)=-1/63/103/12Cu转移的电荷q2=0.51/3=1/6t0+ttCetei3434192d)321(d41ttCetei3434294d)321(d21)(92)(61341tetitC)(94)(61342tetitC-1/6(t)冲激电流1/6(t)冲激电流3.换路后电路有纯电感(或纯电感和电流源）构成的割集k例5+-10V20.3H0.1Hi1i23+u1-+u2-A5)0(1i0)0(2i)0()0(21ii而电感电流发生跃变已知如图求：i1，i2和u1,u2。解)0(i10dd1.03dd3.022211tiitiit0+电流方程为0)]0()0([1.0)]0()0([3.02211iiii)0(1.0)0(3.0)0(1.03.021iii）（即：(0+)=(0－)回路磁链守恒A75.31.03.053.0)0(ii()=2A075.12)275.3(2)(5.125.12teetitt=(0.3+0.1)/(2+3)＝0.08s0)0(5)0(021iit)()75.12()(5)(5.121tettit)()75.12()(5.122tetitit0)(5625.6)(375.05.12tett3.75)](875.21)(75.3)(5[3.05.12tettttiudd3.0115i2i12)()75.12()(5.122tetittiudd1.022)(1875.2)(375.05.12tett)](875.21)(75.3[1.05.12tettutu10.375-2.1875-6.5625u2-0.375根据物理概念求电压0025.1575.31i转移的磁链1=