13动态电路的复频域分析

1、第十三章网络的复频域分析法拉氏变换运算法 网络函数相量分析法：频域(相量)时域时域复频域分析法：复频域时域时域拉氏变换拉氏逆变换解高阶电路变换过程中已经以某种形式计入原微分方程的初始条件。优点：13.1变换变换的定义13.1.1在数学中，F (s) 正变换定义为：f (t)estdt0式中，s=+j为复数变量，通常称为复频率。f(t)：原函数；F(S)：f(t)在s域中的象函数。逆变换定义为： 1 2j jf (t) stF (s)e dt j典型函数的拉氏变换例1：求F (s) L冲击函数(t)的象函数。 (t)est (t)f (t)estdtdt000 (t)dt例2：求 10阶跃函数的

2、象函数。st1sF (s) L1(t)f (t)estdt1(t)edt001s est dtest 00例3：求指数函数e-at的象函数。 1F (s) Leat f (t)estdtes a0013.1.2 拉氏变换的基本性质1 线性性质：K1 f 1 ( t )+ K2 f 2 ( t )= K1F1 (s)+ K2F2 (s)例1：求cost的象函数。e jte jte jte jtsin t 由公式得：cost 2 j21212112Lcost jtjtjtjt LeeLeLe2 s 11 11 2s22 s j2 s j Lsin t 2s2LL d2微分性质：f (t) sF (

3、s) f (0)dt d dtd f (t)dtLf (t)ststeedf (t)dt00f (t) s estf (t)dest f (0) sF (s)00微分定理说明：时域中的求导运算，对应复频域中乘以s的运算，并以f(0-)计入原始条件。d 2ddd Lf (t) sLf (t) f (0)Lf (t)dtdtdtdt 2 s2F (s) sf (0) f (0) ssF (s) f (0) f (0) d n f n 1n 2(n 2)(n1)n S F (S ) Sf (0 ) Sf (0 ) Sf(0 ) f(0 )ndt3积分性质：f 1(0)1tf (t)dt F (s)

4、sLs0(0) 1ff (t)dt d 是函数f(t)的积分式在t=0-时的值。tf (t)dt f (t)dt d dttf (t)dt F (s)Lttf (t)dt F (s)sLf (t)dt00f (t)dtf 1(0)11tf (t)dt F (s) sF (s) sLssf (t)dt 1 F (s)stL0例：L1(t) 1s1(t)dt t1(t)t0Lt1(t) L1(s) 1s2s4时域延迟定理 stL f (t t0 )1(t t0 ) e0 F (s)求f (t) e(t3)1(t 3)的象函数1f (t) e(t3)1(t 3) e3ss 113.1.3拉氏反变换：

5、sm1b bsmbF1 (s) m1F (s) m0sm1a asnF (s)a2m1n0F1 (s) K F10 (s)F (s) (n m)F2 (s)F2 (s)1单根的拉氏逆变换F1 (s) F1 (s)F (s) (s s1)(s s2 )(s sn )F2 (s)KnK1K2(s s1 )(s s2 )(s sn )KnK2(s s )F (s) K (s s )111(s s )(s s)2nK1 (s s )F (s)Ki (s si )F (s)(i 1,n)1s sssi12例1：求F (s) 的原函数。2 2s 5s22F (s) 2s 5s2(s 1)2 22f (t)

6、 L1F (s) et sin 2t(t 0)K12K2F (s) 2s 5s (1 j2)s (1 j2)s22 j 112K s (1 j2)K2 j1 2s 5s22s( 1 j 2)j 1j 1f (t) L1F (s) L12 2s (1 j2)s (1 j2)e j 2t e j 2t j 1 e( 1 j 2)t j 1 e( 1 j 2)t te2sin 2t2(t 0)2 j ets 3例2：求F (s) 的原函数(s 1)(s2 2s 5)F (s) s 3 as b 0.5(s 1)(s2 2s 5) 2s 5s 1s2 (a 0.5)s2 (a b 1)s b 2.5(

7、s2 2s 5)(s 1) 0.5s 0.5 0.5s 0.5 0.50.5 2s 5s 1(s 1)2 22s 1s2 0.5(s 1)2 10.5(s 1)2 222(s 1)2 22 s 1 0.5et0.5et sin 2t 0.5et(t 0)cos 2t 0.5et 0.5et cos(2t )42重根的拉氏逆变换F1 (s)F (s) (s s )3 (s s )(s s)12nn K13K12K11Ki(s s )(s s )2(s s )3(s s )i2111in i2K13K12Ki(s s )3 F (s) K (s s )3(1)1111(s s )(s s )2(s

8、 s )11iK (s s )3 F (s)111ss1对(1)式两边求导得:n d ds d ds1Ki1(s K(s s )33s )F (s)2(ss )K111312(s s )i2idd 2(s 3Ks )F (s)(s s1) F (s)3K13121ds22 dss s1s s12s 3例3：求F (s) 的原函数。(s 2)3(s 1)2s 3K13K12K11K2F (s) (s 2)3 (s 1)s 2(s 2)2(s 2)3s 1K (s 2)3 F (s) 111s 2d 2s 31d (s 2)3 F (s)s2s 2 1K12s 1(s 1)2dsdss2d 21d

9、s2(s 2) F (s)3K13(s 1)3 1s2s2K (s 1)3 F (s) 12s11111f (t) L1s 2(s 2)2(s 2)3s 1 e2t te2t 1 t 2e2t et(t 0)213.2动态电路的复频域分析法13.2.1KCL与KVL的复频域模型 i 0 u 0L I (s) 0U (s) 0KCLKVLRiu (0 ) 0i (0 ) 0+uCLCu iR L di 1tidt-dtC013.2.21、电路元件的复频域模型电阻元件I(S)i(t)RR-+U(S)+u(t)2、电感元件Li(0-)-i(t)LSLI(S)+-+u(t)-+U(S)u(t)=L d

10、i(t)微分性质U(S)=SLI(S) Li(0-) 1 dtI(S)SLI(S)= 1U(S)+i(0-) i(0-) SSLS-+U(S)13.2.2电路元件的复频域模型3、 电容元件U(S)= 1 I(S)+u(0-)I(S)=SCU(S) Cu(0-)SC 1 SCSSCI(S)u(0-)/SI(S)+cu(0-)+U(S)+U(S)4、线性时不变耦合电感元件didiU1(S)=SL1I1(S) + SMI2(S) L 1i1(0 )1+2u =LM11 dt Mi (0 )dt+2didi2U2(S)=SL2I2(S) + SMI1(S) L 2i2(0 )1+u =M+Ldt2 d

11、t2 Mi (0 )+113.2.2电路元件的复频域模型 1 SCLi(0-)u(0-)/S-SLI(S)I(S)+-+U(S)+U(S):1）初具电源(附加电源)由uC(0-)、iL(0-)提供，参考方向，UL(S), UC(S)等的计算2）考虑零状态情况运算阻抗与运算导纳U(S)= 1 I(S)U(S)=RI(S)I(S)=GU(S)U(S)=SLI(S)SCI(S)=SCU(S)I(S)= 1 U(S)SL 1 U=jLIU= jC II=jCUU=RII=GU 1 I= jLU13.2.3动态电路的复频域分析法1 运算电路模型RiI (S)1i21I2(S)RLRRL(t)C1/SCE

12、/SLSLuc (0 ) 0iL (0 ) 0运算电路时域电路电压、电流用象函数形式元件用运算阻抗或运算导纳电容电压和电感电流初始值用附加电源表示例2050V+-IL(s)201/2siL50.5s-0.5H+ u-c102.5+25/s-2FUC(s)+105t=0时打开开关t 0复频域模型时域电路uC(0-)=25ViL(0-)=5A总结：变换法分析电路步骤：由.换路前电路计算uc(0-) , iL(0-) 。画运算电路模型应用电路分析方法求象函数。反变换求原函数。0.1例1：30已知：uc (0 ) 100Vt = 0时闭合s,求iL，uL.。-iLuc100010+200例1：30解：

13、(1)i (0 ) 5 A0.1HLuc (0 ) 100V(2)画运算电路SL 0.1S-i Luc101000F+200V0.5V11S 1000 106300.1sSCI (S)I (S)L21000/S100010S200/S V100/S V0.5 V300.1sIL(S)I2(S)1000/SI1 (s)10100/S V200/SVI2 (s)I ( S )(40 0.1S ) 10I ( S ) 200 0.512S100(3)回路法1000 - 10I1( S ) (10 )I2( S ) SS5( S 2 700S 40000)I1 ( S ) S( S 200)25( S

14、 2 700S 40000)I1 ( S ) S( S 200)25( S 2 700S 40000) lims 5i(0 ) lim SF ( S )s22 400S 200S5( S 2 700S 40000)i() lim SF ( S ) 5lims022 400S 200s0S(4)反变换求原函数F2 ( S ) 0有3个根S1 0，S2 S3 200I ( S ) K1 K21S 200K221( S 200)2SI ( S ) K1 K21S 200K221( S 200)2S2 5( S 700S 40000)K F ( S )S5S 01S 0S 2 400S 2002 F

15、( S )( S 200)2 1500K22S 200 d ( S 200)2 F ( S )0KS 20021ds5 0( S 200)1500I ( S ) 1( S 200)2Si1(t ) (5 1500te200t ) (t ) A0.5 V300.1sIL(S)I2(S)1000/SUL(S)10200/SV100/SVU L ( S ) ? I1 ( S ) SL求UL(S) 30000150S 200U ( S ) I ( S )SL 0.5 ( S 200)2L1ete200tVtuL (t ) is (t ), uc (0 ) 0例2：求冲激响应+ucI (s)issU (

16、S)Rc1/SCCRIs (s) 1R1RU (s) I(s)RC ( S 1 / RC )CsR 1 / SCSC RSC 1 1RSC1I ( S ) U( S )CCRSC 1RSC 1RSC 1SCu 1 et / RC (t 0)1et / RC (t 0)i (t ) ccCRC例3 求图示电路的冲激响应+ 11F+S1S1(t)U(S)u111F时域分析的节点方程(2S+1)U(S) =SU(S)= S = 112S+124(S+1/2) t 211u(t)= U(S)=1(t) e1(t)24例4(见13-9) 0.1 +1000.4SIL(S)0.4H100100 iL+50

17、S 25S+uc100F50vk104/SuC(0-)=25viL(0-)=0.25AA* S+125+j96.850(S) =A+0.1 25I+IL(S)= SSS+125 j96.8L100+0.4s+104/SA= 0.25S+62.5 | 0.25S+62.5IL(S)=S2+250SS= 125+j96.8S+125+j96.8=0.204 52.2 = 0.25S+62.5(S+125)2(S) = 0.204 52.2 + 0.20452.2 0.4H100I100 iS+125 j96.8LS+125+j96.8L+uc100F50vkiL(t)=0.408e125t cos

18、(96.8t 52.2 )(t0)u (0-)=25vi (0-)=0.25ACL例5图示电路在开关闭合前处于稳态，t=0时将开关闭合，求开关闭合后uC(t)和iL(t)的变化规律。+ 40V-iL25H 40 50i (0-)=0.8 AL+uC-20300.01FuC(0-)= 0.820 =16 V+ 40V-iL25H 40 50i (0-)=0.8 A+LuC-20300.01FuC(0-)= 0.820 =16 VIL-+-2016 + 4025SS -UC-20S100S(S)= 16S2+80S +160 1 1U(0.01S+20 + 25S )UCCS(S+1)(S+4)2

19、0 + 40/S U C40 +20I (S)=0.16 + SL25S25S(S)= 20S2+124S +200I(S +5S +4S)UC32L25S(S+1)(S+4)= 16S2+80S +160(S)= 16S2+80S +160(S)= 20S2+124S +200IUS(S+1)(S+4)L25S(S+1)(S+4)CA2 S+1A3 S+4A1 SU =+C= 16S2+80S +160A =U (S)S=401CS=0(S+1)(S+4)S=0= 16S2+80S +160A =(S+1)U (S)= 322CS(S+4)S= 1S= 1= 16S2+80S +160= 8

20、A =(S+4)U (S)S= 43CS(S+1)S= 4t 0t 0uC(t)=40 32e t + 8e 4tiL(t)=2 1.28e t +0.08e 4t13.3网络函数(networkfunctions)13.3.1 网络函数的定义性时不变零状态电路中，网络函数 H(S) 定义为响应 R(S)与输入激励 E(S)之比。即：S域中的网络函数=零状态响应象函数/激励象函数R(S)=Lr(t)，E(S)=L e(t)，网络函数为：R(S ) E(S )H (S )网络函数有六种类型：I1(S)I2(S)+U2(S)U1(S)NZ0I2(S)+U2(S)I1(S)+U1(S)N0ZI1(S

21、)I2(S)+U2(S)U1(S)NZ0I1(S )H (S ) =Y11(S)Driving-poadmittanceU1(S )策动点导纳型H (S ) I2 (S ) =Y21(s)Transform admittanceU1 (S )转移导纳型H (S ) U2 (S ) =HU(S)Voltage gain转移电压比U1(S )I2(S)+U2(S)I1(S)+U1(S)N0ZH (S ) U1(S )=Z(S)Driving-poimpedance11I (S )1策动点阻抗型H (S ) U2 (S ) =Z (s)Transformimpedance21I (S )1转移阻抗型

22、I 2 ( S )H ( S ) 转 移 电 流 比 型=H ( S )C urrent gainII1 ( S ) 1 , R 1 , L 1 H , C 1F ，电路初始状态为零。求网络函数例： R12224H (S ) U1(S ) , H(S ) I1(S )R1R2I2(S)I1(S)12U (S )U (S )+U1(S)U(S)LC1解：(R1+SL)I1(S)SLI2(S)=U(S)1SLI (S ) (R SL (S ) 0)I122SCS 2S 2 2S 4I1(S ) 2U (S ) ,S 2S 2I2 (S ) 2SU (S ) 2S 21I (S )2(S ) I1(

23、S ) S 2S 4 H (S ) U1(S ) 2SSC, H12S 2S 2 2S 2 2S 2U (S )U (S )U (S )213.3.2网络函数H(S)与冲击响应h(t)之间的关系： H (S ) R(S ) , R(S ) H (S )E(S ) .E(S )当输入 e(t)=(t)时H (S ) R(S ) h(t) h(t) .E(S)= 1 and (t)E(S )或h(t) L1H (S )网络函数等于 对应冲激响应的拉氏变换。1例：R =8，R =4，L=2H， C F ，求冲激响应。1216LR1H (S ) h(t)+ (t)(t)RCh(t)21 R12SC11

24、(R SL)SL R22SC1SCR1R SL 2SC222(S 2)2 (22)2 h(t) L1H (S ) (2e2t2t) I (t)sin 2零点、极点与零极点图：bm Sm bm1S m1 b1S b0P(S )H (S ) Q(S )Sn1 a S aSnaan 1n10m(S zi )H (S ) H (0) (S z1 )(S z2 )(S zm ) H (0) i1(S p1)(S p2 )(S pn )n(S p j )j1H (0) bm是实系数。an当S等于 z1, z2, ., zm时， 网络函数等于零。 所以Zi称为网络函数的零点。当S等于 p1p, ., pn时

25、， 网络函数趋于无穷。 所以pj称为网络函数的极点。上以s的实部为横轴、虚部j 为纵轴，在其上标出极零点的位置。用符号 表示零点，用符号 表示极点。称该图为网络函数的零、极点图。S 2 (S 3)例： H (S ) (S 1)(S 2 j)(S 2 j)j1解：该网络函数有三个零点，三个极点。Z1=Z2=0，Z3= 3，P1= 1，P2= 2+j，P3= 2j31113.3.3 网络函数的极点与网络的稳定性的关系0H (S ) (S )2 20 j0p12h(t) e0t1h(t) etH (S ) S 若网络函数的极点在s平面的左半开平面，则当t趋于无穷时，冲击响应将趋于零，对应的网络是渐近稳定的。距离虚轴越远，稳定性越好。1h(t) etH (S ) S a0h(t) et sin t, p a jH (S ) 1200(S a)2 202 若网络函数含有在s平面的右半开平面的极点，则当t趋于