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1、第二章习题课第二章习题课(2-1b)2-1(b)试建立图所示电路的动态微分方试建立图所示电路的动态微分方程。程。uouiR1LR2Ci1=iL+icuL=LdiLdtuoiL=i2=R2uL=LR2duodtic=+CducdtCLR2d2uodt2duodt+uoR2CLR2d2uodt2duodti1=+Cuoi2=R2输入量为输入量为ui，输出量为，输出量为uo。ui=u1+uou1=i1R1ic=Cducdt=dtd(ui-uo)第1页/共43页习题课一习题课一 (2-2)求下列函数的拉氏变换求下列函数的拉氏变换。(1)f(t)=sin4t+cos4t解解:Lsinwt=ww2+s2s
2、w2+s2Lsin4t+cos4t=4s2+16ss2+16 =s+4s2+16+Lcoswt=第2页/共43页(2)f(t)=t3+e4t3!解解:Lt3+e4t=+=+3!s3+11s-4s41s-4(3)f(t)=tneat解解:Ltneat=n!(s-a)n+1(4)f(t)=(t-1)2e2t解解:L(t-1)2e2t=e-(s-2)2(s-2)3第3页/共43页2-3-1 函数的拉氏变换。函数的拉氏变换。F(s)=s+1(s+1)(s+3)解解：A1=(s+2)s+1(s+1)(s+3)s=-2=-1(s+1)(s+3)A2=(s+3)s+1s=-3=2F(s)=-2s+31s+2
3、f(t)=2e-3t-e-2t第4页/共43页F(s)=s(s+1)2(s+2)2-3-2 函数的拉氏变换。函数的拉氏变换。解解:f(t)=est +lim ests(s+1)2s=-2ddsss+2s -1=-2e-2t+lim(est+est)s -1 sts+22(s+2)2=-2e-2t-te-t+2e-t=(2-t)e-t-2e-2t第5页/共43页F(s)=2s2-5s+1s(s2+1)2-3-3 函数的拉氏变换。函数的拉氏变换。解解:F(s)(s2+1)s=+j=A1s+A2s=+jA1=1,A2=-5A3=F(s)s =1s=0 f(t)=1+cost-5sintF(s)=+1
4、ss2+1s-5s2+1第6页/共43页2-3-4 函数的拉氏变换。函数的拉氏变换。(4)F(s)=s+2s(s+1)2(s+3)解解:f(t)=est +ests+2(s+1)2(s+3)s=0s+2s(s+1)2s=-3+lim s -1d est s+2s(s+3)ds=+e-3t+lim +23112s -1(-s2-4s-6)est(s2+3)2(s+2)tests2+3s=+e-3t-e-t-e-t2311234t2第7页/共43页(2-4-1)求下列微分方程。求下列微分方程。d2y(t)dt2+5 +6y(t)=6，初始条件：初始条件：dy(t)dty(0)=y(0)=2。解解:
5、s2Y(s)-sY(0)-Y(0)+5sY(s)-5Y(0)+6Y(s)=1sA1=sY(s)s=0 y(t)=1+5e-2t-4e-3tA2=(s+2)Y(s)s=-2A3=(s+3)Y(s)s=-3A1=1,A2=5,A3=-4 Y(s)=6+2s2+12ss(s2+5s+6)第8页/共43页 (2-4-2)求下列微分方程。求下列微分方程。d3y(t)dt3 +4 +29 =29,d2y(t)dt2dy(t)dt初始条件初始条件:y(0)=0 ,y(0)=17 ,y(0)=-122 解解:第9页/共43页2-5-a 试画题试画题2-1图所示电路的动态结构图图所示电路的动态结构图,并并求传递
6、函数。求传递函数。CucR1R2uii1i2uoicC解解:ui=R1i1+uo ，i2=ic+i1UI(s)=R1I1(s)+UO(s)ducic=CdtI2(s)=IC(s)+I1(s)IC(s)=CsUC(s)即即:=I1(s)UI(s)-UO(s)R1UI(s)-UO(s)Cs=IC(s)第10页/共43页UO(s)UI(s)=1R1(sC)R21+1R1(sC)R2=R2+R1R2sCR1+R2+R1R2sC1R1sCR2UI(s)-UO(s)IC(s)I1(s)I2(s)1R1sC R2()UI(s)-UO(s)第11页/共43页2-5-b 试画出题试画出题2-1图所示的电路图所示
7、的电路的动态结构图的动态结构图,并并求传递函数。求传递函数。uouiR1LR2C解：解：ui=R1I1+ucuc=uo+uLuL=LdiLdtiL=uoR2i1=iL+icic=CducdtUi(s)=R1I1(s)+UC(s)UC(s)=UO(s)+UL(s)UL(s)=sLIL(s)I1(s)=IL(s)+IC(s)第12页/共43页1R1CssLR2I1UOUiIC-UC=UO+ULILULI2(s)=UO(s)R2IC(s)=CsUC(s)I1(s)=UO(s)R2I1(s)=UI(s)+UC(S)R1即：即：IL(s)=I1(s)-IC(s)IC(s)=UC(s)Cs第13页/共43
8、页解解:电路等效为电路等效为:2-6-a 用运算放大器组成的有源电网络如图用运算放大器组成的有源电网络如图所示所示,试采用复数阻抗法写出它们的传递函数。试采用复数阻抗法写出它们的传递函数。UO=R3SCR2R21UIR1UOR3SCR2R21SC1=第14页/共43页R1+R3+R2R3CS=R1(R2SC+1)R2R3=(+)R1(R2SC+1)R1R1R2=(+R3)(R2SC+1)1=R21R3R2SCR1C(S)=UO(S)UI(S)CR1R2R3uiuoCR1R2R3uiuo第15页/共43页CR1R2R3uiuoR4R52-6-b 用运算放大器组成的有源电网络如用运算放大器组成的有
9、源电网络如力所示力所示,试采用复数阻抗法写出它们的传试采用复数阻抗法写出它们的传递函数。递函数。第16页/共43页=R5R4+R5UO(R3SC+1)R2R3SC+R2+R3UOUI=(R2R3SC+R2+R3)(R4+R5)R1(R3SC+1)R5=(R4+R5)(R2+R3)(SC+1)R2R3R2+R3R1R5(R3SC+1)UIR1=R5R4+R5UOR2R3SCSCR3SC1R5R4+R5UOR2R3R3SC 1第17页/共43页c(t)t0TK(t)(t)2-8 设有一个初始条件为零的系统，系设有一个初始条件为零的系统，系统的输入、输出曲线如图，求统的输入、输出曲线如图，求G(s)
10、。c(t)t0TK(t)(t)c(t)=KTt-(t-T)KT C(s)=K(1-e )Ts2-TSC(s)=G(S)第二章习题课第二章习题课(2-8)解解:第18页/共43页2-9 若系统在单位阶跃输入作用时，已若系统在单位阶跃输入作用时，已知初始条件为零的条件下系统的输出响知初始条件为零的条件下系统的输出响应，求系统的传递函数和脉冲响应。应，求系统的传递函数和脉冲响应。r(t)=I(t)c(t)=1-e +e-2t-t解解:R(s)=1sG(S)=C(s)/R(s)1s+21s-C(s)=1s+1+=s(s+1)(s+2)(s2+4s+2)=(s+1)(s+2)(s2+4s+2)C(s)=
11、(s+1)(s+2)(s2+4s+2)脉冲响应脉冲响应:2s+2=1+1s+1-c(t)=(t t)+2e +e-2t-t第二章习题课第二章习题课(2-)第19页/共43页2-10 已知系统的微分方程组的拉氏变换已知系统的微分方程组的拉氏变换式，试画出系统的动态结构图并求传递式，试画出系统的动态结构图并求传递函数。函数。解解:X1(s)=R(s)G1(s)-G1(s)G7(s)-G8(s)C(s)X2(s)=G2(s)X1(s)-G6(s)X3(s)X3(s)=G3(s)X2(s)-C(s)G5(s)C(s)=G4(s)X3(s)G1G2G3G5-C(s)-R(s)G4G6G8G7X1(s)=
12、R(s)-C(s)G7(s)-G8(s)G1(s)C(s)G7(s)-G8(s)G6(s)X3(s)X1(s)X2(s)C(s)G5(s)X3(s)G1G2G3G5-C(s)-R(s)G4G2G6G8G7G1G2G5-C(s)-R(s)G7-G81+G3G2G6G3G4-C(s)R(s)G7-G81+G3G2G6+G3G4G5G1G2G3G41+G3G2G6+G3G4G5+G1G2G3G4(G7-G8)G1G2G3G4R(s)C(s)=第二章习题课第二章习题课(2-10)第20页/共43页解解:2-11(a)G1(s)G2(s)G3(s)H1(s)_+R(s)C(s)H2(s)G1(s)G2(
13、s)H1(s)_+R(s)C(s)H2(s)G3(s)求系统的求系统的传递函数传递函数1+G2H1G2 G1+G31+G1H21+G2H1G2 1+G2H1G2=1+G2H1+G1G2H2G2 R(s)C(s)=1+G2H1+G1G2H2G2G1+G2G3G1(s)G2(s)G3(s)H1(s)_+R(s)C(s)G 1(s)H2(s)第二章习题课第二章习题课(2-11a)第21页/共43页2-11(a)G1(s)G2(s)G3(s)H1(s)_+R(s)C(s)H2(s)求系统的求系统的传递函数传递函数解解:L1L1=-G2H1L2L2=-G1G2H1P1=G1G2P2=G3G21=12=1
14、R(s)C(s)=nk=1Pkk=1+G2H1+G1G2H21+G2H1+G1G2H2G2G1+G2G3=第二章习题课第二章习题课(2-11a)第22页/共43页解解:2-11(b)G1(s)G2(s)G3(s)G4(s)_+R(s)C(s)H(s)求系统的求系统的传递函数传递函数G1(s)G2(s)G3(s)G4H_+R(s)C(s)H(s)1+G4G1HG1 G2(s)G3(s)_+R(s)C(s)H(s)1+G4HG1G1 G2G3_+R(s)C(s)1+G4HG1G1 HG1 1+G4HG1G1+G3(1+HG1G4)1+G4HG1G2(1+HG1G4)1+G4G1H+G1G2HR(s
15、)C(s)=1+G1G2H+G1G4HG1G2+G2G3+G1G2G3G4 H第二章习题课第二章习题课(2-11b)第23页/共43页解解:2-11(b)G1(s)G2(s)G3(s)G4(s)_+R(s)C(s)H(s)求系统的求系统的传递函数传递函数R(s)C(s)=1+G1G2H+G1G4HG1G2+G2G3+G1G2G3G4 HL1L1=-G1G2HL1=-G1G4HL2P1=G1G21=1P2=G3G2=1+G4G2H+G1G2H2=1+G1G4H第二章习题课第二章习题课(2-11b)第24页/共43页H1_+G1+C(s)R(s)G3G22-11c 求系统的闭环传递函数求系统的闭环
16、传递函数。解解:H1_+G1+C(s)R(s)G3G2H1R(s)C(s)1+G1G2+G1H1G3H1G1G2(1 G3H1)=_G1C(s)R(s)G2H1+G21-G3H11第二章习题课第二章习题课(2-11c)第25页/共43页H_G1+C(s)R(s)G22-11d 求系统的闭环传递函数求系统的闭环传递函数。解解:(1)_G1+C(s)R(s)G2HG21+G2H1(G1+G2)R(s)C(s)=(2)L1L1=-G2HP1=G11=1P2=G22=1第二章习题课第二章习题课(2-11d)第26页/共43页-_G1+C(s)R(s)G2G3G42-11e 求系统的闭环传递函数求系统的
17、闭环传递函数。解解:(1)_C(s)R(s)G1+G2G3-G4C(s)=R(s)1+(G1+G2)(G3-G4)(G1+G2)1+G1G3+G2G3G1G4-G2G4=(G1+G2)第二章习题课第二章习题课(2-11e)L1L2L3L4L2=G1G4L3=-G2G3L4=G2G4(2)L1=-G1G3P1=G11=1P2=G22=11+G1G3+G2G3G1G4-G2G4=(G1+G2)C(s)R(s)第27页/共43页_G1+C(s)R(s)G22-11f 求系统的闭环传递函数求系统的闭环传递函数。_C(s)R(s)G11-G2G2C(s)=R(s)1+1-G2G1G1G21+G1G2G2
18、G1(1 G2)=第二章习题课第二章习题课(2-11f)解解:(1)(2)L1L1=-G1G2L2L2=G2P1=G11=1-G2=1+G1G2-G2C(s)R(s)1+G1G2G2G1(1 G2)=第28页/共43页2-12(a)R(s)G1(s)G2(s)H2(s)_+C(s)H3(s)H1(s)_+D(s)解解:求求:D(s)C(s)R(s)C(s)D(s)=01-G2H2G2 G(s)=1-G2H2G1G2 C(s)=R(s)1+1-G2H2G1G2H31-G2H2G1G2 1-G2H2+G1G2H3G2G1=R(s)=0结构图变结构图变 换成：换成：G2(s)H2(s)_+C(s)G
19、1H3G1H1_D(s)1-G2H2G2 1-G1H1C(s)=D(s)1+1-G2H2G21-G2H2G2 G1H3(1-G1H1)1-G2H2+G1G2H3G2(1-G1H1)=第二章习题课第二章习题课(2-12a)第29页/共43页2-12(b)求求:D(s)C(s)R(s)C(s)R(s)Gn+D(s)解解:D(s)=0G(s)=1+G1G2HG1G2 G1G2H_C(s)C(s)=R(s)1+1+G1G2HG1G2 1+G1G2HG1G2 1+G1G2H+G1G2G1G2=R(s)=0Gn+D(s)结构图变结构图变 换成：换成：G1G2H-C(s)Gn+D(s)G1G2H-C(s)G
20、n/G1+D(s)1+G1G2HG1G2-C(s)+D(s)1+G1G2HG2Gn 1+G1G2HG2G1系统的传递函数系统的传递函数:)C(s)=D(s)1+1(1+1+G1G2HG1G2 1+G1G2HGnG2 1+G1G2+G1G2H=1+GnG2+G1G2H第二章习题课第二章习题课(2-12b)第30页/共43页2-13(a)求求:R(s)E(s)R(s)C(s)C(s)E(s)G1G2G3_+R(s)解解:L1L1=-G2L2L2=-G1G2G3P1=G2G3P2=G1G2G3R(s)C(s)=1+G2+G1G2G3G2G3+G1G2G31=12=1E(s)结构图变结构图变 换成：换
21、成：G1G2+-E(s)G3-R(s)G1+-E(s)R(s)1+G2G3G2-E(s)-R(s)1+G2G2G3G1G2G31+G2系统的传递函数系统的传递函数:)E(s)=R(s)1+1(1-G1G2G31+G2 1+G2G2G31+G2+G1G2G3=1+G2-G2G3第二章习题课第二章习题课(2-13a)第31页/共43页R(s)G4(s)+C(s)E(s)G1(s)G2(s)-+G3(s)第二章习题课第二章习题课(2-14)D(s)X(s)2-14 求求:R(s)C(s)解解:D(s)=0结构图结构图变换为变换为 R(s)G4(s)+C(s)G1(s)G2(s)-+G3(s)G3(s
22、)(G1+G2)(G3+G4)1+(G1+G2)G3G1+G2C(s)R(s)=1+G3(G1+G2)(G1+G2)(G3+G4)第32页/共43页D(s)第二章习题课第二章习题课(2-14)R(s)+-E(s)G3G2G1E(s)R(s)=1+G3(G1+G2)1R(s)G4(s)+C(s)E(s)G1(s)G2(s)-+G3(s)X(s)求求:R(s)E(s)2-14解解:D(s)=0结构图变换为结构图变换为 G3(G1+G2)第33页/共43页D(s)C(s)D(s)R(s)C(s)G4(s)+E(s)G1(s)G2(s)-+G3(s)X(s)求求:2-14解解:R(s)=0D(s)C(
23、s)=1E(s)X(s)=G2(s)E(s)X(s)第二章习题课第二章习题课(2-14)第34页/共43页C1(s)R1(s)第二章习题课第二章习题课(2-15)求求:2-15+G1G2G3C1(s)R1(s)+-H2H1G4G5-G6C2(s)R2(s)解解:结构图结构图变换为变换为+G1G2G3C1(s)R1(s)-H2H1G4G5-1+G4G4G5H1H21+G4G4G5H1H21+G1G1G21+G4+G1G4G5H1H2-G1G2-G1G2G4G1G2G3(1+G4)=C1(s)R1(s)1+G4+G1G4G5H1H2G1G2(1+G4)=第35页/共43页C2(s)R2(s)G3C
24、1(s)R1(s)+G1G2-H2H1G4G5G6-C2(s)R2(s)求求:2-14解解:结构图结构图变换为变换为+G4G5G6C2(s)R2(s)-H1H2G2G1-第二章习题课第二章习题课(2-15)1+G4G4G5G1H1H21-G1G21+G4+G1G4G5H1H2-G1G2-G1G2G4G4G5G6(1-G1G2)=1+1+G4G6G4G5G1H1H21-G2G21+G4G4G5C2(s)R2(s)第36页/共43页C2(s)G4G5G6-+G1G2-H2H1R1(s)G3C1(s)R2(s)求求:2-14解解:结构图变换为结构图变换为 G4G5G6-+-G1G2H2H1R1(s)
25、C2(s)第二章习题课第二章习题课(2-15)C2(s)R1(s)1+G4H2G4G5G11-G1G21+G4+G1G4G5H1H2-G1G2-G1G2G4G1G4G5G6H2=1+G6H1C2(s)R1(s)G11-G1G21+G4H2G4G5G11-G1G21+G4H2G4G5第37页/共43页C2(s)G6R1(s)G4G5-+G1G2-H2H1G3C1(s)R2(s)求求:2-14解解:结构图变换为结构图变换为 第二章习题课第二章习题课(2-15)G1G2G3+G5G4-H1R2(s)C1(s)H2-C1(s)R2(s)G1G2G3+G5G4-H1R2(s)C1(s)H2-G21+G4
26、G4G51-G1G2-G1H11+G4+G1G4G5H1H2-G1G2-G1G2G4-G1G2G3G4G5H1=1+1+G4G4G5G1H11-G2G21+G4G4G5C1(s)R2(s)H2-G1H11-G2G2G2G3第38页/共43页G3C1(s)R1(s)+G1G2+-H2H1G4G5-G6C2(s)R2(s)C1(s)R1(s)求求2-15第二章习题课第二章习题课(2-15)解解:L1=G1G2L3=-G4L2=-G1G4G5H1H2P1=G1G2G3=1-G1G2+G1G4G35H1H2+G4-G1G2G41=1+G41+G4+G1G4G5H1H2-G1G2-G1G2G4G1G2G
27、3(1+G4)=C1(s)R1(s)第39页/共43页G3C1(s)R1(s)+G1G2+-H2H1G4G5-G6C2(s)R2(s)C2(s)R2(s)求求2-15解解:L1=G1G2L3=-G4L2=-G1G4G5H1H2P1=G4G5G6=1-G1G2+G1G4G5H1H2+G4-G1G2G41=1-G1G21+G4+G1G4G5H1H2-G1G2-G1G2G4G4G5G6(1-G1G2)=C2(s)R2(s)第二章习题课第二章习题课(2-15)第40页/共43页G3C1(s)R1(s)+G1G2+-H2H1G4G5-G6C2(s)R2(s)C1(s)R2(s)求求2-15解解:L1=G
28、1G2L3=-G4L2=-G1G4G5H1H2=1-G1G2+G1G4G5H1H2+G4-G1G2G41=1P1=-G1G2G3G4G5H11+G4+G1G4G5H1H2-G1G2-G1G2G4-G1G2G3G4G5H1=C1(s)R2(s)第二章习题课第二章习题课(2-15)第41页/共43页G3C1(s)R1(s)+G1G2+-H2H1G4G5-G6C2(s)R2(s)C2(s)R1(s)求求2-15解解:L1=G1G2L3=-G4L2=-G1G4G5H1H2=1-G1G2+G1G4G5H1H2+G4-G1G2G41=1P1=G1G4G5G6H21+G4+G1G4G5H1H2-G1G2-G1G2G4G1G4G5G6H2=C2(s)R1(s)第二章习题课第二章习题课(2-15)返回返回第42页/共43页感谢您的观看！第43页/共43页