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1、7.5 The Zero-State Response,If all of the initial conditions have zero value, then the circuit is said to be in the zero-state, and the solution to nonzero inputs for the circuit is known as the zero-state response. (零状态响应),Zero-State Response Of An RC Circuit,t=0：uC(0)=0,The particular solution：,It
2、 is usually called the forced response or the steady-state response.,uC ()=US ，uC p = uC ()=US,The homogeneous solution：,The characteristic equation： RCS+1=0,The characteristic value：,The homogeneous solution is usually called the natural or transient response.,uC (0+)=uC (0)=0=US +A A= US,The gener
3、al form of the step response of RC circuits：,1 zero-state response of DC input:,（2）Discussion,7.6 Step Response Circuit,When the dc source is suddenly applied, the signal of the source can be modeled as a step function, the response is known as a step response.,The step response of a circuit is its
4、behavior when the excitation is the step function, which may be a voltage or a current source.,7.6.1 Step Response of RC Circuits,Us1(t),t=0：uC(0)=U0,Its solution includes two parts : the particular solution and the homogeneous solution.,Complete response: external source and initial condition of th
5、e storage element,The particular solution：,It is usually called the forced response or the steady-state response.,uC ()=US ，uC p = uC ()=US,The homogeneous solution：,The characteristic equation： RCS+1=0,The characteristic value：,The homogeneous solution is usually called the natural or transient res
6、ponse.,uC (0+)=uC (0)=U0=US +A A= U0 US,The general form of the step response of RC circuits：,Complete Response,uc(0+)=U0,There are two ways to excite the circuit: By initial conditions of the storage elements in the circuits. By independent sources.,1、Two ways to determine complete response:,（1）to
7、linear circuit:,Complete response,Zero_state response,Zero_input response,Complete response = zero-input response + zero-state response.,Zero_input response is a special complete response when y( )=0;,Zero_state response is a special complete response when y( 0 )=0;,y(t) = yh(t) + yp(t),(2) To linea
8、r circuit, complete response=natural response+forced response,Thus, to find the step response requires three things: The initial value y(0+); The final value y( ); The time constant ;,（2） 与输入无关，归结为求由电容元件或电感元件观 察的入端电阻R,关于uc(0-)，iL(0-) 和 uc(0+)，iL(0+),（1） y() 归结为求解电阻网络（电容元件相当于开路， 电感元件相当于短路）,Three fact
9、ors:,Switch s is closed at t=0, uc(0-)=2V, calculate the response uc.,+ u -,iC,Example:,Method 1:,Reduce the circuit at first.,Method 2:,L=L1+L2=3mH,Drill 1. Calculate Uc.,uC(0)=150V,uC(0+)=150V,=0.3 5000/150=10S,uC=100 + 50e0.1t (t 0),uC()=35000/150=100V,Drill 2. Calculate Ul.,iL(0)=5mA,iL(0+)=5mA,
10、iL()=5+5=10mA,uL(0+)=5V,uL()=0,=103S,iL=10 5e1000t mA,uL=5e1000t V,Drill 3. Calculate Uc, il, ik.,uC(0)=20V,iL(0)=1A,uC(0+)=20V,iL(0+)=1A,i (0+)=20/15=4/3A,uC()=1.215=18V,i()=30/25=1.2A,iL()=0,C=3106 150/25=18 106 S,L=2103/5=1/2500 S,iL=e2500t A,solution：,uC( )=US=6V， uC (0+)=uC (0)=0， 1=C(R1+R2)=0.
11、52=1s,t=2s：uC (2)=66e2=5.19 V,uC ( )=6V， uC (2+)=uC (2)=5.19 V， 2=R1C=0.5s,+ U* -,Fig.a,Fig.b,N is a linear resistive circuit, in Fig.a, c=1F, Uc(0)=0, Us=U0, u=(0.75-e-2t)V; In Fig.b, L=1H, iL(0 )=0, calculate u*.,Special Cases:,已知：,解：此题为初始值跃变情况,电流发生跃变,回路磁链守恒,电容电压初值一定会发生跃变。,合k后,已知图中E=1V , R=1 , C1=
12、0.25F , C2=0.5F 。 求： uC1 、 uC2 、 iC1 和 iC2 并画出波形。,节点电荷守恒,q(0+)= q(0-),uC() = 1V, = R (C1+C2),7.7 First_order OP AMP Circuits,An op amp circuit containing a storage element will exhibit first_order behavior.,Examples: Differentiators and integrators,Analyze first_order op amp circuits,Classical way:
13、write the circuit equation in the first_order differential form at first, get the solution. Use the general form of the complete response of the first_order circuits.,Drill 1: For the op amp circuit , find vo(t) for t0, given that v(0)=3V.,Solution: vo(0+)=12V, Vo()=0V. Req=20K =ReqC=0.1s,7.7 First_order OP AMP Circuits,Drill 2: Determine v(t) and vo(t).,Drill 3: For the op amp circuit , find vo(t) for t0, given that vi=21(t)V.,1. Sinusoidal response:,i (0-)=0,calculate：i (t),强制分量(稳态分量),自由分量(暂态分量),Poticular solution:,
