電路理論第二冊時(shí)域與頻域分析第七章動態(tài)網(wǎng)絡(luò)的復(fù)頻域分析法

1、第七章域分析法第七章動態(tài)網(wǎng)絡(luò)的復(fù)頻域分析引言1、動態(tài)網(wǎng)絡(luò)的描述aadn1ydn2y adyayn dtn n2 dtn2+1dt+0dmx=bdm1xdmxdm2xdxbxm dtm+m2dtm2+ +b1 dt +0*y(0+)，y(1)(0+)，y(n1)(0+)(時(shí)域分析)XY對正弦穩(wěn)態(tài),x(t). ,y(t).XYd jdta (j)n+a(j)n1+(j)+ann110=bm(j)m+bm1(j)m1+b b 112、為什么要將拉普拉斯變換引入動態(tài)網(wǎng)絡(luò)分析？拉普拉斯變換拉普拉斯變換的定義0-f(t)=eStt 0-S= + j關(guān)于積分下限0例 KeStdt 1 KeSt = K0-1

2、(t)=Sdt0-S0=SeStdt10=S+(t)=0-0-(t)eStdt0-0-0t =10-et=et eStdt0-= (+S)t=1e(+S)t 10-edt0- =S+7-1 拉普拉斯變換拉普拉斯變換的基本性質(zhì)1、線性性質(zhì)設(shè)f1(t)=F1(S)f2(t)=F2(S) 1f1(t)+2f2(t)=1F1(S) +2F2(S) kcost = 0.5k(ejt+ejt)=0.5k(1+1)SjS+j=kSS2+22、微分性質(zhì)設(shè) f (t)=FS)df(t)dt=SF(S)f(0-)7-1 拉普拉斯變換7-1-2拉普拉斯變換的基本性質(zhì)3、積分性質(zhì)設(shè) f (t)=Ft t0-f(t)d

3、t= 1SF(S)+s(t)-iu+-CRL+s(t)-iu+-C i(t)=I(S) uS(t)=US(S)Ri+L +u)+ 1 t idt (t)dtCC0SRi (t)+L di +uC(0)S1 t0idt =t0(t)C(R+SL+ 1 )I(S) Li(0 ) + uC(0) =U (S)CSCSS7-1 拉普拉斯變換拉普拉斯變換的基本性質(zhì)(R+SL+ 1 )I(S) Li(0 ) + uC(0) =U (S)SCSSI(S)= SCUS(S)+SLCi(0)CuC(0)S2LC+SRC+1部分分式法求拉普拉斯反變換出發(fā)點(diǎn) ket k S+1k=ket集中參數(shù)電路中響應(yīng)變換式的特

4、點(diǎn) 變換式在一般情況下為S的實(shí)系數(shù)有理函數(shù)F (S)bmSm + bm1Sm1 + + b1S + b0F (S)=1=F2(S)anSn + an1Sn1 + + a1S + a07-1 拉普拉斯變換7-1-3部分分式法求拉普拉斯反變換F (S)=F1(S)F2(S)mbmSm + bm1Sm1 + anSn + an1Sn1 + + b1S + b0+ a1S + a0(Szi)i=1nH0實(shí)數(shù)常數(shù)(Spj)j=1ziF(S)的零點(diǎn)(1)nm(1)pjF(S)的極點(diǎn)F(S)可展開為部分分式之和(2)nmF(S)=Q(S)R(S)F2(S)例S3+1S2+2S+2=S 2+ 2S+5 S2+

5、2S+2其 中 1(S2)=(t)2(t)實(shí)數(shù)F(S)的極點(diǎn)單極點(diǎn)復(fù)數(shù)重極點(diǎn)實(shí)復(fù)1、F(S)只含實(shí)數(shù)單極點(diǎn)A1S p1 +A2 S + AkS pk +An S nf(t)= 1F(S)=Ake pktk=1問題歸結(jié)為求F(S)的極點(diǎn)和確定相應(yīng)的常數(shù)AkAn S 7-1 拉普拉斯變換An S 7-1-3部分分式法求拉普拉斯反變換A1S p1 +A2 S + AkS pk +S= pk例求S2+3S+5 的反變換S2+3S+5=A1+A2+A3(S+1)(S+2)(S+3)S+1S+2S+3A1=(S+1)F(S)=S2+3S+5 (S+2)(S+3)=1.5S= 1A A2=(S+2)F(S)

6、=AS2+3S+5 S2+3S+5= 3S= 2= 2.53S= 37-1 拉普拉斯變換部分分式法求拉普拉斯反變換1、F(S)只含實(shí)數(shù)單極點(diǎn)S2+3S+5=1.532.5(S+1)(S+2)(S+3)S+1S+2S+3f(t)=1F(S)=1.5et3e2t+2.5e3tt 02、F(S)除含實(shí)數(shù)單極點(diǎn)外，還含有復(fù)數(shù)單極點(diǎn)復(fù)數(shù)極點(diǎn)是共軛形式成對出現(xiàn)的F(S)=A1S(+j)A2+S(j) + 與復(fù)數(shù)極點(diǎn)對應(yīng)的兩個(gè)常數(shù)也互為共軛復(fù)數(shù)A2=1令A(yù)1 ej則A1 ej拉普拉斯變換7-1-3部分分式法求拉普拉斯反變換2、F(S)除含實(shí)數(shù)單極點(diǎn)外，還含有復(fù)數(shù)單極點(diǎn)F(S)=A1+S(+j)+A2S(j)

7、 + 令A(yù)1 ej則A1 ejf(t)= A1eje(+j)t +A1 eje(j)t + = A1et ej(t + ) + ej(t + )+ =2 A1 et cos(t+)+ 注意A1是虛部為正的極點(diǎn)對應(yīng)的那個(gè)常數(shù)例求S2+3S+7 的反變換A11A3F(S)= S + S (2j2) +S +1S2+3S+7S (2j2)(S+1)A1=S2+3S+7S (2j2)(S+1)=0.25ej90S= 2+j2A3=S2+3S+7 (S+2)2+4=1S= 1 f(t)=1F(S)=0.5e2tcos(2t+90)+ett 0方程*S域代數(shù)方程（條件含在其中）方程*（時(shí)域） 初始條件y(

8、t)1（復(fù)頻域）Y(S)第七章動態(tài)網(wǎng)絡(luò)的復(fù)頻域分析運(yùn)算法時(shí)域電路（討論電路基本定律，元件特性方程的復(fù)頻域形式）（頻域電路運(yùn)算模型）微分方程)代數(shù)方程獲得復(fù)頻域代數(shù)方程的途徑時(shí)域電路（討論電路基本定律，元件特性方程的復(fù)頻域形式）（頻域電路運(yùn)算模型）微分方程)代數(shù)方程運(yùn)算法KCL與KVL的運(yùn)算形式0-1、KCL0-ik(t)=0i1i2i3Ik(S)=0i1i2i3ik(t)= ik(t)eStdt=Ik(S)（運(yùn)算電流） I1(S) +I2(S) I3(S)=0I1(S)I2(S)I3(S)2、KVLI1(S)I2(S)I3(S)Uk(S)=0I(S)+RU(S)-電路元件的運(yùn)算模型I(S)+R

9、U(S)-1、線性時(shí)不變電阻元件+u(t)-i(t)R+u(t)-I(S)SLLi(0-)+2、線性時(shí)不變電感元件I(S)SLLi(0-)+i(t)+Lu(t)-+U(S)-u(t)=Ldi(t)dt微分性質(zhì)U(S)=SLI(S)Li(0-)i(0-) S 1 i(0-) SI(S)= 1U(S)+i(0-)I(S)SLSLS+U(S)-7-2-2電路元件的運(yùn)算模型3、線性時(shí)不變電容元件U(S)= 1I(S)+u(0-)I(S)=SCU(S)Cu(0-)I(S)SCcu(0-)+U(S)SCSI(S)SCcu(0-)+U(S)I(S)SC+u(0-)/S 1 I(S)SC+u(0-)/S 1

10、+U(S)4、線性時(shí)不變耦合電感元件u di1 +di2U1(S)=SL11(S) SMI2(S)L1i1(0)+11 dtdt+ Mi2(0)u =di1 di2U2(S)=SL22(S) SMI1(S)L2i2(0)+2+2 dt Mi1(0)I(S)SL-Li(0-)+ 1 I(S)SC+u(0-)/S+U(S)7-2-2電路元件的運(yùn)算模型I(S)SL-Li(0-)+ 1 I(S)SC+u(0-)/S+U(S)+U(S)-討論:1）初具電源(附加電源)由uC(0-)、iL(0-)提供，參考方向，UL(S),UC(S)等的計(jì)算2）考慮零狀態(tài)情況運(yùn)算阻抗與運(yùn)算導(dǎo)納U(S)=RI(S)I(S)

11、=GU(S)U(S)=SLI(S)I(S)= 1 U(S) SLU(S)= 1 I(S)SCI(S)=SCU(S)1U=RIU=jLIU=jCI 1I=GUI= jLUI=jCU運(yùn)算電路，電阻性網(wǎng)絡(luò)各種解法的適用性電路基本定律、元件特性的描述ik(t)=0uk(t)=0u (t)=Ri(t)Ik(S)=0Uk(S)=0U(S)=RI(S) U(S)=SLI(S)Li(0-)u (t)、(t)U(S)= 1SCI(S)+u(0-)SSSUS(S)、IS(S)uk(t)、ik(t)時(shí)域電路R、L、C等元件時(shí)域電路Uk(S)、Ik(S)運(yùn)算電路(頻域電路)運(yùn)算阻抗(或?qū)Ъ{)和初具電源運(yùn)算電路(頻域電

12、路)7-2-3運(yùn)算電路，電阻性網(wǎng)絡(luò)各種解法的適用性例1求圖示電路的沖激響應(yīng)+1+1F1+u+1F1+u1F1SU(S)11S1時(shí)域分析的困難節(jié)點(diǎn)方程(2S+1)U(S) =S2S+1U(S)=S2S+111=224(S+1/2)=2 1U(S)=1(t) 1 t 4e2 1(t)47-2-3運(yùn)算電路，電阻性網(wǎng)絡(luò)各種解法的適用性例2(見教材7-9)0.4H+100iL100+50v1000.4H+100iL100+50v100Fuck+50 0.1 +0.4SIL(S)+ 25SS 104/SiL(0-)=0.25AuC(0-)=25vSIL(S)=S50+0.125IL(S)A S+125j9

13、6.8*+S+125+j96.8+100+0.4s+104/SLI(S)= 0.25S+62.5 S2+250S+25000LSA=0.25S+62.5 S|S= 125+j96.8=0.25S+62.5 =0.204 52.2+ 0.20452.2S+125+j96.87-2-3運(yùn)算電路，電阻性網(wǎng)絡(luò)各種解法的適用性+ 0.20452.2S+125+j96.80.4H100iL+100+50v100FuckIL(S) =0.4H100iL+100+50v100Fuck0.20452.2 S+125j96.8i(0-)=0.25A(0-)=25viL(t)=0.408e125tcos(96.8t

14、52.2)(t0)LC例3圖示電路在開關(guān)閉合前處于穩(wěn)態(tài)，t=0時(shí)將開關(guān)閉合求開關(guān)閉合后uC(t)和iL(t)的變化規(guī)律。iL25H+40V-+uC-2030500.01FiL25H+40V-+uC-203050iL(0-)=40 =0.8AuC(0-)= 0.820 =16 V0.01FiL(0-)= 40 50=0.8 A16+ILiL25H+40V-+uC-20iL25H+40V-+uC-203020+ 40 -uC(0-)= 0.820 =16 VS-UC20S100S- 11U(S)= 16S2+80S+16020 25S)UCCS(S+1)(S+4)40+20I(S)=20 + 40

15、/SUC=0.16 +S25SL25S32(S)= 20S2+124S+200(S +5S +4S)UCL25S(S+1)(S+4)= 16S2+80S +160U(S)=16S2+80S+160I(S)= 20S2+124S+200CS(S+1)(S+4)L25S(S+1)(S+4)UC=A1A216S2+80S +160 (S+1)(S+4)+SS+1 16S2+80S +160 (S+1)(S+4)+A3 S+4A1=UCS=0=S=0=40A2=(S+1)UCS= 1=S= = 32A3=(S+4)UCS= 4=16S2+80S +16016S2+80S +160 S(S+4)S(S+

16、1)16S2+80S +160 S(S+4)S= 4= 8 uC(t)=4032et+8e4tt 0iL(t)=21.28et+0.08e4tt 0 9+S 6+S 1 2SI(S) 1 3S 6+S 10 9+S 6+S 1 2SI(S) 1 3S 6+S 10 1 3S 1 2S+S15 + 9S+u1 + u2 2F3Fi153F2F 9+S 6+S 1 2S 6+S 1 3S 1 3SU+ 9 9+S 6+S 1 2S 6+S 1 3S 1 3SU+ 90C + S 1 2S15+S5u1(0-)= 515=9Vu2(0-)=6V解法一、應(yīng)用戴維南定理U(S)= + 6= 3OCSSS

17、 9+S 6 9+S 6+S 1 2SI(S) 1 3S 6+S 10 1 3S 1 2S+S15 + 9S 1 2S 1 3S 1 3S 1 2SU(S)= 3 1 2S 1 3S 1 3S 1 2S 1 1OCSZ0(S)=2 2S3S=2 1+ 15S+3S+3S25S 32S3S1510I(S)=S2= 50S+2= 3110+ 5S10(S+ 25 )i(t)= 0.3e0.04t7-2-3運(yùn)算電路，電阻性網(wǎng)絡(luò)各種解法的適用性解法二：節(jié)點(diǎn)分析U3(S)= 15+9S 1 2S13S1I(S)S+ 6 6+S 210+9S 1 2S13S1I(S)S+ 6 6+S 2109+ S 1

18、3S1 2S315+SSSS= 18+18S0.1U1+(5S+0.1)U23S 15= 18+18(S)=U150S+7. 5(S)=U1S(25S+1)225S+7.5(S)=U2(S)= S(25S+1)(S)=I(S)=0.1U1(S) U2 7.5 (25S+1) 0.325=(S+ 1 )25=i(t)= 1I(S)=0.3e0.04t(t0)零狀態(tài)響應(yīng)的拉氏變換網(wǎng)絡(luò)函數(shù)零狀態(tài)響應(yīng)的拉氏變換7-2運(yùn)算法7-3-1定義與分類7-2運(yùn)算法網(wǎng)絡(luò)函數(shù)H(S)=輸入的拉氏變換討論:(1)是網(wǎng)絡(luò)函數(shù)的一般定義對正弦穩(wěn)態(tài)電路中定義的網(wǎng)絡(luò)函數(shù)如何理解 (2)討論網(wǎng)絡(luò)函數(shù)的意義H(S)與h(t)的關(guān)

19、系 與對應(yīng)正弦穩(wěn)態(tài)響應(yīng)的關(guān)系H(S)H(j)與對應(yīng)正弦穩(wěn)態(tài)響應(yīng)的關(guān)系H(S)H(j)R 1.+ UR -jLR 1.UR(S)SL-+-jC-+ .+R1 U(S)SC-SCH=.RU =R1H (S)=UR(S)U(S)=R1R+jL+jCR+SL+ SC網(wǎng)絡(luò)函數(shù)定義與分類例N0為線性定常松弛網(wǎng)絡(luò)。在端口1施加電流源iS(t)=1(t)A時(shí)，端口2電壓的零狀態(tài)響應(yīng)為u2(t)=2 1(t) (1e2t )V 。如果將電流源波形改為iS(t)=10sin2tA，求穩(wěn)態(tài)響應(yīng)u2(t)。iS(t)u (t)12R2N012R2N02( 11)+u2(t)H(S)=SS+2=4iS(t)4 1S+2

20、SU2=H(j2)IS=2+j210=14.1 45u2(t)=14.1sin(2t45)網(wǎng)絡(luò)函數(shù)定義與分類2、網(wǎng)絡(luò)函數(shù)的分類I2(S)+U1(S)I1(S)N0N+U2(S)Z2(S)策（驅(qū)）動點(diǎn)函數(shù)輸入和輸出在同一端口（或支路）Z(S)=U1(S)Y(S)=I1(S)Z(S)=1/ (S)inI1(S)inU1(S)inin(分別具有阻抗導(dǎo)納的量綱)網(wǎng)絡(luò)函數(shù)定義與分類2、網(wǎng)絡(luò)函數(shù)的分類I2(S)+U1(S)I1(S)N0N+U2(S)Z2(S)傳輸(傳遞、轉(zhuǎn)移)函數(shù)輸入和輸出不在同一端口（或支路）(S)=I1(S)(轉(zhuǎn)移阻抗)21(S)=I2(S)U1(S)(轉(zhuǎn)移導(dǎo)納)H(S)= U2(S

21、)V(轉(zhuǎn)移電壓比)(S)=I2(S)(轉(zhuǎn)移電流比)I1(S)網(wǎng)絡(luò)函數(shù)的確定例求如圖所示電路的傳遞函數(shù)應(yīng)的沖激響應(yīng)。U0(S)Ui(S) 1SC，并求對12SCR2R( 1 +2SC)U1R2R2R2U1 = SCU2R212R12Rui 1CRC-uR(2SC + 1R)U2 SCU0 =0Ui(S)2SC+0U0(S)H(S)=U0(S)=1Ui(S)4R2C2S2h(t)=1H(S)=14R2C2t1(t)7-3-2網(wǎng)絡(luò)函數(shù)的確定1I1(S)+U1(S)2N0+N0U2(S)3I2(S)Z2(S)Yn(S)E(S) =InS(S)InS(S)=I1(S) 0TE1(S)=11II1(S)E

22、2(S)=12I1(S)IE3(S)=13I1(S)IZ= 11 = 1Z= 12 -13Y=12 -13 inYin212111Z2H= 12 -13H = 12 - 13V11IZ2Z= 11 = 1Z= 12 -13Y=12 -13inYin212111Z2H= 12 -13H = 12 - 13V11IZ2網(wǎng)絡(luò)函數(shù)是復(fù)頻率變量S的實(shí)系數(shù)有理函數(shù)H(S)=F1(S)bmSm +bm-1Sm1+b1S+b0F2(S)=a Sn + aSn1 + +a S+a=H0nm(Szi )mi=1nn-11n(Sp )jj=1j網(wǎng)絡(luò)函數(shù)的零點(diǎn)和極點(diǎn)7-3 網(wǎng)絡(luò)函數(shù)7-3-5極點(diǎn)和網(wǎng)絡(luò)的穩(wěn)定性1、網(wǎng)絡(luò)

23、穩(wěn)定性的概念adny +dn1y+ +ady+ a ym dtnn-1dtn11 dt0=bdmx +dm1x+ +bdx + b xmdtm1 dt0考慮零輸入響應(yīng)若微分方程對應(yīng)的特征方程無重根，則ny(t)=kiei=1Sit7-3 網(wǎng)絡(luò)函數(shù)7-3-5極點(diǎn)和網(wǎng)絡(luò)的穩(wěn)定性1、網(wǎng)絡(luò)穩(wěn)定性的概念ny(t)=kiei=1Sit若微分方程對應(yīng)的特征方程包含p階重根Sp，則npi=1+(kp1+kp2t+ kpptp1)eSpt 網(wǎng)絡(luò)的固有頻率 1)如果全部固有頻率位于復(fù)平面的開左半平面,即eSi0則ty(t)=07-3 網(wǎng)絡(luò)函數(shù)m7-3-5極點(diǎn)和網(wǎng)絡(luò)的穩(wěn)定性m(Szi )H(S)=H0 i=1nj(

24、Sp )jj=11=npjt沖激響應(yīng)h(t)= H(S)kjej=1 特定初始條件下的零輸入響應(yīng)反之，網(wǎng)絡(luò)函數(shù)的極點(diǎn)不一定包含了網(wǎng)絡(luò)變量的 全部固有頻率。一個(gè)網(wǎng)絡(luò)穩(wěn)定的必要條件是，該網(wǎng)絡(luò)中任一 或是虛數(shù)軸上的單極點(diǎn)。11HiS+uC1F1網(wǎng)絡(luò)函數(shù)11HiS+uC1F1例求網(wǎng)絡(luò)函數(shù)H(S)=UC(S)/IS(S)；和iL(0-)，求零輸入響應(yīng)uC；討論網(wǎng)絡(luò)函數(shù)極點(diǎn)與對應(yīng)網(wǎng)絡(luò)變量固有頻率的關(guān)系。UC(S)IS(S)(1)H(S)=UC(S)IS(S)1S=S+11SS+2+ 1SSIS(S)+UC(S)1/S1=S+1=S2+2S+1 1S+11是H(S)的一個(gè)極點(diǎn)，也是uC的一個(gè)固有頻率。1+SuC(0-)iL(0-)S1/S17-3 網(wǎng)絡(luò)函數(shù)1+SuC(0-)iL(0-)S1/S111HiS+uC1F1UCu (0-)uc(0-) +i(0-)(S)=cL 1SCSS+2+ 1SS= SuC(0-)+2uC(0-)iL(0-) S2+2S+1= uC(0-)iL(0-)(S+1)2+ uC(0-)S+111HiS+uC1F17-3 網(wǎng)絡(luò)函數(shù)11HiS+uC1F1UC(S)(S+1)2+ uC(0-)S+1uC(t)=uC(0-)et +uC(0-)iL(0