第13讲任意激励下动态电路的求解1阶跃函数与阶跃响应重点2冲激函数与冲激响应3利用卷积积分求解任意激励作用下的动态电路PrinciplesofElectricCircuitsLecture13TsinghuaUniversity20051阶跃函数与阶跃响应（1）单位阶跃函数(Unitstep)ε(t)ε(t)1⎧0(t<0)1).定义ε(t)=⎨⎩1(t>0)0t可用ε(t)来表示电压/电流的突然变化。t=0时开关闭合u(t)=USε(t)Ku(t)USUSε(t)u(t)i(t)Kt=0时开关闭合,i(t)=Iε(t)IssPrinciplesofElectricCircuitsLecture13TsinghuaUniversity200512).单位阶跃函数的延时ε(t-t0)1⎧0(t<t0)ε(t−t0)=⎨t⎩1(t>t0)0t03).用单位阶跃函数来表示比较复杂的信号例1f(t)ε(t)1f(t)=ε(t)−ε(t−t0)0t0t-ε(t-t0)tε(t)例2f(t)f(t)=t[ε(t)−ε(t−1)]+ε(t−1)1t[]ε()t−ε(t−1)01tft=tε(t)−(t−1)ε(t−1)−(t−1)ε(t−1)()（2）单位阶跃响应单位阶跃函数引起的零状态响应Riu1c+ε(t)CuC–t1-iuC(0)=0Rt−RCuC(t)=(1−e)ε(t)0tti1−i(t)=eRCε(t)1Re−tε(t)今后要注意规范表达！0t−t−t思考i=eε(t)和i=et≥0有什么区别？PrinciplesofElectricCircuitsLecture13TsinghuaUniversity20052RiCu(0-)=0++Cε(t-t0)CuC求u.-–Ct-tt01−i=eRCε(t-t)CR0注意f(t)t−1RCf(t)ε(t)eε(t-t0)错误.Rf(t-t0)ε(t-t0)f(t)ε(t-t0)1iCt−tt1−0ReRCε()t−t0t0R00tt0PrinciplesofElectricCircuitsLecture13TsinghuaUniversity2005复杂一点的例子求i(t).C10k10k+ic+ic10ε(t)10ku10k100µFs--100µFu(0-)=0-CuC(0)=0叠加定理10kus(V)看作由阶跃函数构成+ic10−10ε(t−0.5)10k-100µF00.5t(s)-uC(0)=0uS=10ε(t)−10ε(t−0.5)PrinciplesofElectricCircuitsLecture13TsinghuaUniversity20053uS=10ε(t)−10ε(t−0.5)10k5k+ic+i10ε(t)10k等效c100µF5ε(t)100µF----uC(0)=0uC(0)=0+iC(0)=1mAiC(∞)=0−63−2tτ=100×10×5×10=0.5siC=eε(t)mA10k+i−2(t−0.5)ciC=−eε(t−0.5)mA−10ε(t−0.5)10k100µF--uC(0)=0−2t−2(t−0.5)iC=eε(t)−eε(t−0.5)mA还可以用二次换路和卷积积分来求解。PrinciplesofElectricCircuitsLecture13TsinghuaUniversity20052单位冲激函数与冲激响应（1）单位冲激(Unitimpulse)函数δ(t)1).单位脉冲(Unitpulse)函数p(t)p(t)1p(t)=[ε(t)−ε(t−∆)]1/∆∆∞p(t)dt=1∫−∞0∆tPrinciplesofElectricCircuitsLecture13TsinghuaUniversity200542).单位冲激函数δ(t)∞p(t)dt=1∫−∞p(t)1pt()=[εε()t−−∆(t)]1/∆∆1∆→0→∞∆0∆tlimp(t)=δ(t)∆→0定义:⎧0(t<0)δ(t)δ(t)=⎨⎩0(t>0)(1)∞δ(t)dt=10t∫−∞PrinciplesofElectricCircuitsLecture13TsinghuaUniversity20053).单位冲激函数的延时δ(t-t0)δ(t-t0