清华考研-电路原理课件-第9章--阶跃响应、冲激响应和卷积积分的应用

清华大学电路原理电子课件江辑光版参考教材：《电路原理》（第2版）清华大学出版社，2007年3月江辑光刘秀成《电路原理》清华大学出版社，2007年3月于歆杰朱桂萍陆文娟《电路》（第5版）高等教育出版社，2006年5月邱关源罗先觉第第第第9999章章章章阶跃响应、冲激响应阶跃响应、冲激响应阶跃响应、冲激响应阶跃响应、冲激响应和卷积积分的应用和卷积积分的应用和卷积积分的应用和卷积积分的应用第第第第第第第第99999999章章章章章章章章阶跃响应、冲激响应阶跃响应、冲激响应阶跃响应、冲激响应阶跃响应、冲激响应阶跃响应、冲激响应阶跃响应、冲激响应阶跃响应、冲激响应阶跃响应、冲激响应和卷积积分的应用和卷积积分的应用和卷积积分的应用和卷积积分的应用和卷积积分的应用和卷积积分的应用和卷积积分的应用和卷积积分的应用9.19.19.19.1阶跃函数和冲激函数9.19.19.19.1阶跃函数和冲激函数本章重点本章重点9.49.49.49.4电路在任意激励作用下的零状态响应————————卷积积分9.49.49.49.4电路在任意激励作用下的零状态响应————————卷积积分9.59.59.59.5电容电压和电感电流的跃变9.59.59.59.5电容电压和电感电流的跃变9.29.29.29.2阶跃响应9.29.29.29.2阶跃响应9.39.39.39.3冲激响应9.39.39.39.3冲激响应••••阶跃响应和冲激响应����本章重点本章重点本章重点本章重点••••阶跃函数和冲激函数••••卷积积分返回目录返回目录返回目录返回目录••••电容电压和电感电流的跃变9.1阶跃函数和冲激函数一、单位阶跃函数（unitstepfunctionunitstepfunctionunitstepfunctionunitstepfunction）1.1.1.1.定义ttttεεεε((((tttt))))11110000()()()()ttttεεεε用可描述开关的动作。++++––––uuuuCCCCUUUUSSSSεεεε((((tttt))))RRRRCCCCdefdefdefdef0(0)0(0)0(0)0(0)()()()()1(0)1(0)1(0)1(0)ttttttttttttεεεε⎧⎧⎧⎧====⎨⎨⎨⎨⎩⎩⎩⎩defdefdefdefSSSSSSSS0(0)0(0)0(0)0(0)()()()()(0)(0)(0)(0)ttttUtUtUtUtUtUtUtUtεεεε⎧⎧⎧⎧====⎨⎨⎨⎨⎩⎩⎩⎩UUUUSSSSSSSS++++––––uuuuCCCCRRRRCCCC开关在tttt=0=0=0=0时闭合2.2.2.2.延迟的单位阶跃函数ttttεεεε((((t-tt-tt-tt-t0000))))tttt00000000defdefdefdef0000000000000()0()0()0()()()()()1()1()1()1()ttttttttttttttttttttttttεεεε⎧⎧⎧⎧−=−=−=−=⎨⎨⎨⎨⎩⎩⎩⎩3.3.3.3.由单位阶跃函数可组成复杂的信号UUUUSSSSSSSS++++––––uuuuCCCCRRRRCCCC开关在tttt====tttt0000时闭合0000()()()()()()()()()()()()fttttfttttfttttfttttεεεεεεεε=−−=−−=−−=−−tttt0000tttt-εεεε((((tttt-tttt0000))))εεεε((((tttt))))0000ffff((((tttt))))1111解所示矩形脉冲可分解为阶跃函数和延迟阶跃函数相加。例1111⎩⎩⎩⎩⎨⎨⎨⎨⎧⎧⎧⎧====)))),,,,0000((((0000))))0000((((1111))))((((00000000ttttttttttttttttttttttttffff1111tttt0000ttttffff((((tttt))))0000试用阶跃函数表示上图所示的矩形脉冲。()[()(1)](1)()[()(1)](1)()[()(1)](1)()[()(1)](1)ftttttftttttftttttftttttεεεεεεεεεεεε=−−+−=−−+−=−−+−=−−+−11111111tttt000000001111tttt1111ffff((((tttt))))例2222试用阶跃函数表示图示的波形。解ffff((((tttt))))分成两段表示。1111tttt111100001111tttt1111++++(0(0(0(0tttt1)1)1)1)()[()(1)]()[()(1)]()[()(1)]()[()(1)]fttttfttttfttttfttttεεεεεεεε=−−=−−=−−=−−(1(1(1(1tttt))))()(1)()(1)()(1)()(1)fttfttfttfttεεεε=−=−=−=−则二、单位冲激函数（unitpulsefunctionunitpulsefunctionunitpulsefunctionunitpulsefunction）1.1.1.1.单位脉冲函数1111()[()()]()[()()]()[()()]()[()()]ptttptttptttptttεε∆εε∆εε∆εε∆∆∆∆∆=−−=−−=−−=−−0000lim()()lim()()lim()()lim()()pttpttpttptt∆∆∆∆δδδδ→→→→====令11110000∆∆∆∆∆∆∆∆→→∞→→∞→→∞→→∞面积不变∆∆∆∆1/1/1/1/∆∆∆∆ttttpppp((((tttt))))0000∆∆∆∆减小，脉冲变窄，面积不变。defdefdefdef1111(0)(0)(0)(0)()()()()0(0,)0(0,)0(0,)0(0,)ttttptptptpttttttttt∆∆∆∆∆∆∆∆∆∆∆∆⎧⎧⎧⎧⎪⎪⎪⎪====⎨⎨⎨⎨⎪⎪⎪⎪⎩⎩⎩⎩∆∆∆∆////22222/2/2/2/∆∆∆∆2.2.2.2.单位冲激函数的定义∫∫∫∫++++−−−−====000000001111dddd))))((((δδδδttttttttttttδδδδ((((tttt))))0000符号kkkkδδδδ(t(t(t(t))))脉冲强度为kkkk的冲激函数ttttkkkkδδδδ((((tttt))))0000()d()d()d()dkttkkttkkttkkttkδδδδ∞∞∞∞−∞−∞−∞−∞====∫∫∫∫0(0)0(0)0(0)0(0)()()()()0(0)0(0)0(0)0(0)ttttktktktktttttδδδδ⎧⎧⎧⎧====⎨⎨⎨⎨⎩⎩⎩⎩0(0)0(0)0(0)0(0)()()()()0(0)0(0)0(0)0(0)ttttttttttttδδδδ⎧⎧⎧⎧====⎨⎨⎨⎨⎩⎩⎩⎩()d1()d1()d1()d1ttttttttδδδδ∞∞∞∞−∞−∞−∞−∞====∫∫∫∫SSSS0(0)0(0)0(0)0(0)(0)(0)(0)(0)()()()()ttttUUUUuttuttuttuttEtEtEtEtττττττττττττ⎧⎧⎧⎧⎪⎪⎪⎪⎪⎪⎪⎪====⎨⎨⎨⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩⎩⎩⎩SSSSuuuuuuuuCCCC====ttttuuuuCCCCiiiiCCCCCCCCdddddddd====uuuuSSSSttttUUUUττττ0000例讨论电路中uuuuCCCC，iiiiCCCC的变化情况解++++----CCCC++++-iiiiCCCCuuuuSSSSuuuuCCCCττττ[()()][()()][()()][()()]CCCCCUCUCUCUittittittittεετεετεετεετττττ=−−=−−=−−=−−ddddCCCCitqCUitqCUitqCUitqCU+∞+∞+∞+∞−∞−∞−∞−∞========∫∫∫∫ττττ当ττττ→→→→0000，uuuuCCCC→→→→UUUUεεεε((((tttt))))iiiiCCCC→→→→CUCUCUCUδδδδ((((tttt))))uuuuCCCCttttUUUU0000iiiiCCCCttttCUCUCUCUδδδδ((((tttt))))0000qqqq不变。iiiiCCCCttttττττ0000CUCUCUCUττττuuuuCCCCttttUUUUττττ0000，qqqq不变当ττττ则iiii====CCCCUUUUSSSSδδδδ((((tttt))))tttt=0=0=0=0时合SSSS))))0000(((())))0000((((−−−−++++≠≠≠≠CCCCCCCCuuuuuuuu0000))))0000((((====−−−−CCCCuuuuξξξξξξξξdddd))))((((1111))))0000(((())))0000((((00000000∫∫∫∫++++−−−−++++====−−−−++++iiiiCCCCuuuuuuuuCCCCCCCC====UUUUSSSS000000000000()0()()0()()0()()0()()d1()d1()d1()d1ttttttttttttttttttttttttttttδδδδδδδδ∞∞∞∞−∞−∞−∞−∞−=≠−=≠−=≠−=≠⎧⎧⎧⎧⎪⎪⎪⎪⎨⎨⎨⎨−=−=−=−=⎪⎪⎪⎪⎩⎩⎩⎩∫∫∫∫ttttδδδδ((((t-tt-tt-tt-t0000))))tttt000000003.3.3.3.延迟单位冲激函数δδδδ((((t-tt-tt-tt-t0000))))SSSS++++––––uuuuCCCCUUUUSSSSCCCCiiii特例4.4.4.4.δδδδ函数的筛分性质()()d()()d()()d()()dftttftttftttftttδδδδ∞∞∞∞−∞−∞−∞−∞∫∫∫∫00000000()()d()()()d()()()d()()()d()fttttftfttttftfttttftfttttftδδδδ∞∞∞∞−∞−∞−∞−∞−=−=−=−=∫∫∫∫同理有ffff(0)(0)(0)(0)δδδδ((((tttt))))(0)()d(0)(0)()d(0)(0)()d(0)(0)()d(0)fttffttffttffttfδδδδ∞∞∞∞−∞−∞−∞−∞========∫∫∫∫条件：ffff((((tttt))))在tttt0000处连续。设函数ffff((((tttt))))在tttt=0=0=0=0处连续，则三、εεεε((((tttt))))和δδδδ((((tttt))))的关系0(0)0(0)0(0)0(0)()d()d()d()d1(0)1(0)1(0)1(0)ttttttttttttttttttttδδδδ−∞−∞−∞−∞⎧⎧⎧⎧====⎨⎨⎨⎨⎩⎩⎩⎩∫∫∫∫=εεεε((((tttt))))d()d()d()d()()()()()ddddttttttttttttεεεεδδδδ====ttttεεεε((((tttt))))00001111ttttδδδδ((((tttt))))(1)(1)(1)(1)0000返回目录返回目录返回目录返回目录阶跃响应（stepresponsestepresponsestepresponsestepresponse）：阶跃函数激励下电路中产生的零状态响应。9.2阶跃响应单位阶跃响应（unitstepresponseunitstepresponseunitstepresponseunitstepresponse）：单位阶跃函数激励下电路中产生的零状态响应。阶跃响应的求解：阶跃激