第3讲：正弦电路C

1、 Lecturer:KangCollege of Electrial & Information Engineering XiHua University Electric Circuit Analysis电路分析电路分析 Lecturer:KangCollege of Electrial & Information Engineering XiHua University Electric Circuit Analysis电路分析电路分析u瞬时功率瞬时功率u平均功率平均功率u无功功率无功功率u视在功率视在功率u复功率复功率u功率因数提高功率因数提高u共轭匹配共轭匹配 Lect

2、urer:KangCollege of Electrial & Information Engineering XiHua University Electric Circuit Analysis电路分析电路分析)sin(2)( )sin(2)( iutItitUtu iu21- )2cos(cos )2cos()cos( )sin(2)sin(2 pptUItUItItUuipiuiuiuiu无无源源+ui_)cos()cos(21sinsinyxyxyx Lecturer:KangCollege of Electrial & Information Engineering

3、XiHua University Electric Circuit Analysis电路分析电路分析 tO iuP1=UIcos p p2=-UIcos(2 t +u+i) Lecturer:KangCollege of Electrial & Information Engineering XiHua University Electric Circuit Analysis电路分析电路分析iRu+- -)cos( )cos(cos )sin()sin( uiuuiu2t21UI2t20UItI2tU2uip iLu+- -)cos( )cos(cos )sin()sin( 22t2U

4、I222t22UItI2tU2uipuiuuiu Lecturer:KangCollege of Electrial & Information Engineering XiHua University Electric Circuit Analysis电路分析电路分析波形和能量转化如下：波形和能量转化如下：则：则：若若),sin( sin),sin( t2UIptI2i2tU2u t0 iupLiLu+- -iLu+- -iLu+- -iLu+- - Lecturer:KangCollege of Electrial & Information Engineering XiH

5、ua University Electric Circuit Analysis电路分析电路分析iCu+- -)cos( )cos()cos( )sin()sin( 22t2UI222t22UItI2tU2uipuiuuiu Lecturer:KangCollege of Electrial & Information Engineering XiHua University Electric Circuit Analysis电路分析电路分析 TtpTP0d1 =u- -i：功率因数角。功率因数角。为无源网络等效阻抗的为无源网络等效阻抗的阻抗角阻抗角cos ：功率因数。功率因数。 T0i

6、uttUIUIT1d)cos(cos UI cos UIPcos Lecturer:KangCollege of Electrial & Information Engineering XiHua University Electric Circuit Analysis电路分析电路分析 tO iuP=UIcos piRu+- -RIUIUItpT1P2T0R cosd Lecturer:KangCollege of Electrial & Information Engineering XiHua University Electric Circuit Analysis电路分析

7、电路分析0dt2u2t2UIT1tpT1PT0T0L )cos(d iLu+- -iCu+- -0dt2u2t2UIT1tpT1PT0T0L )cos(d Lecturer:KangCollege of Electrial & Information Engineering XiHua University Electric Circuit Analysis电路分析电路分析一般地一般地 , 有有 0cos 1X0, 0 , 感性感性X0, 0 , 容性容性cos 1, 纯电阻纯电阻0， 纯电抗纯电抗22coscosRIZIUIP Lecturer:KangCollege of Elec

8、trial & Information Engineering XiHua University Electric Circuit Analysis电路分析电路分析 sinUIQ sinsinZIUIQ2 Lecturer:KangCollege of Electrial & Information Engineering XiHua University Electric Circuit Analysis电路分析电路分析0Q00X0Q00X 容性网络，容性网络，感性网络，感性网络，, ?Q?Pcos)A30sin(3t22i)V,30sin(3t210u 求：无源网络例：例：

9、Var3217UIQW10UIP5060.sin,cos,.)cos(cos 解：解： Lecturer:KangCollege of Electrial & Information Engineering XiHua University Electric Circuit Analysis电路分析电路分析UIS cos SP Lecturer:KangCollege of Electrial & Information Engineering XiHua University Electric Circuit Analysis电路分析电路分析 Lecturer:KangCol

10、lege of Electrial & Information Engineering XiHua University Electric Circuit Analysis电路分析电路分析22QPS SPQ ZRX UURUXRX+_+_+_URUXU Lecturer:KangCollege of Electrial & Information Engineering XiHua University Electric Circuit Analysis电路分析电路分析uiR+- -PR =UIcos =UIcos0 =UI=I2R=U2/RQR =UIsin =UIsin0

11、=0iuL+- -PL=UIcos =UIcos90 =0QL =UIsin =UIsin90 =UIiuC+- -PC=UIcos =Uicos(- -90 )=0QC =UIsin =UIsin (- -90 )= - -UI Lecturer:KangCollege of Electrial & Information Engineering XiHua University Electric Circuit Analysis电路分析电路分析 SSjQPSUIQUIPUISS sincos PQQPS22arctan, Lecturer:KangCollege of Electr

12、ial & Information Engineering XiHua University Electric Circuit Analysis电路分析电路分析* )(IUIUUIUISiuiu 互为共轭复数互为共轭复数,*jbaIIjbaIIii 222jXIRIZIIZISIUSZIUjXRZ *)(222IZSXIQRIP , Lecturer:KangCollege of Electrial & Information Engineering XiHua University Electric Circuit Analysis电路分析电路分析222jBUGUUYYUUY

13、UUSIUSYUIjBGY *)(222UYSBUQGUP , x22xxxxxxxSQPSSPjQPQjPjQPSQQPP)(, Lecturer:KangCollege of Electrial & Information Engineering XiHua University Electric Circuit Analysis电路分析电路分析 00 11bkkbkkQP )j(0SQPb1kkb1kkk 路路复复功功率率求求各各支支路路复复功功率率和和全全电电 1Z1jZ1jZ2jZVe2U4321j10,Z3Z2Z41U1I2I3I+_Z1 Lecturer:KangColl

14、ege of Electrial & Information Engineering XiHua University Electric Circuit Analysis电路分析电路分析 45j4324321e2ZZZZZZZZ)(Ae2e2e2ZUI45j45j0j11 A1352IZZZZIA02IZZZZZI1432231432432 VA2IZSVA2jIZSVA4jIZSVA4jIZSVA4522IUS2344233322222111 ,*Z3Z2Z41U1I2I3I+_Z1 Lecturer:KangCollege of Electrial & Informatio

15、n Engineering XiHua University Electric Circuit Analysis电路分析电路分析SSQPNetA30t10titV75tu)sin()(sin)(、的的求求 uNeti+- -VA30350IUS30210I0275U * Var175S1QW1303SP866030VA350S2 .)cos( Lecturer:KangCollege of Electrial & Information Engineering XiHua University Electric Circuit Analysis电路分析电路分析A68. 58 . 022

16、01000cos DDD UPI+_DCUICIDI解：解：oDD. ,0.8(cos 836 感性，滞后）感性，滞后）V0220Uo 设设082C0220I836685IC.jj , . ooD A316734331544IIICoD.j. 96. 0)3 .16cos(cos o Lecturer:KangCollege of Electrial & Information Engineering XiHua University Electric Circuit Analysis电路分析电路分析（六）（六） 功率因数提高功率因数提高所以：设备容量所以：设备容量 S ( (额定额定

17、) )向负载送多向负载送多少有功，要由负载的阻抗角决定。少有功，要由负载的阻抗角决定。P=UIcos =Scos S75kVA负载负载cos =1, P=S=75kWcos =0.7, P=0.7S=52.5kW一般用户：一般用户： 异步电机异步电机 空载空载cos =0.20.3 满载满载cos =0.70.85日光灯日光灯 cos =0.450.6 (1) (1) 发电设备不能充分利用发电设备不能充分利用(2) (2) 线路上电流大，线路损耗大线路上电流大，线路损耗大 Lecturer:KangCollege of Electrial & Information Engineeri

18、ng XiHua University Electric Circuit Analysis电路分析电路分析分析：分析：CIULI 1I 2LRCUILICI+_并联电容后，原负载的电压和电流不变，吸收并联电容后，原负载的电压和电流不变，吸收的有功功率和无功功率不变，即：负载的工作的有功功率和无功功率不变，即：负载的工作状态不变。但电路端口的功率因数提高了。状态不变。但电路端口的功率因数提高了。特点：特点： Lecturer:KangCollege of Electrial & Information Engineering XiHua University Electric Circu

19、it Analysis电路分析电路分析21sinsin IIILC 不同的不同的C容量的补偿效果不同：容量的补偿效果不同：全补偿全补偿不要求不要求(电容设备投资增加电容设备投资增加,经济效果不明显经济效果不明显)欠补偿欠补偿达不到要求的功率因素达不到要求的功率因素过补偿过补偿使功率因数又由高变低使功率因数又由高变低(电路性质不同电路性质不同)CIULI 1I 2代入得代入得将将 cos , cos 12 UPIUPIL )tgtg( 21UPCUIC)tgtg(212 UPC Lecturer:KangCollege of Electrial & Information Enginee

20、ring XiHua University Electric Circuit Analysis电路分析电路分析 1 2PQCQLQ)tgtg( )tgtg( 2122C21CLUPCCUQPQQQ Lecturer:KangCollege of Electrial & Information Engineering XiHua University Electric Circuit Analysis电路分析电路分析已知：已知：f=50Hz, U=220V, P=10kW, cos 1=0.6，要使功率，要使功率因数提高到因数提高到0.9 , 求并联电容求并联电容C，并联前后电路的总电流

21、并联前后电路的总电流各为多大？各为多大？o1113.53 6 . 0cos 例：例：解：解：o2284.25 9 . 0cos F 557 )84.25tg13.53tg(2203141010 )tgtg(23212 UPCLRCUILICI+_AUPIIL8 .756 . 02201010cos31 未并电容时：未并电容时：并联电容后：并联电容后：AUPI5 .509 . 02201010cos32 Lecturer:KangCollege of Electrial & Information Engineering XiHua University Electric Circuit

22、 Analysis电路分析电路分析o2219.18 95. 0cos 解：解：o1184.25 9 . 0cos F 103 )8.191tg5.842tg(2203141010 )tgtg(23212 UPCAI8 .4795. 022010103 功率因数提高后线路上总电流减少，但继续提高功率因数所需功率因数提高后线路上总电流减少，但继续提高功率因数所需电容成本很大，总电流减小却不明显。因此一般将提高到电容成本很大，总电流减小却不明显。因此一般将提高到0.9即可。即可。 Lecturer:KangCollege of Electrial & Information Engineer

23、ing XiHua University Electric Circuit Analysis电路分析电路分析（七）共轭匹配（七）共轭匹配S UZLZiI+- -Zi= Ri + jXi， ZL= RL + jXL2Li2LiSLiS)()( , XXRRUIZZUI 2Li2Li2SL2L)()( XXRRURIRP 有功功率有功功率戴维南等效戴维南等效 Lecturer:KangCollege of Electrial & Information Engineering XiHua University Electric Circuit Analysis电路分析电路分析(1) ZL=

24、 RL + jXL可任意改变可任意改变 (a) 先先设设RL不变，不变，XL改变改变显然，当显然，当Xi + XL=0，即，即XL =- -Xi时，时，P获得获得最大最大值值(b) 再讨论再讨论RL改变时，改变时，P的最大值的最大值当当 RL= Ri 时，时，P获得最大值获得最大值i2Smax4RUP 2Li2Li2SL)()(XXRRURP ZL= Zi*RL= RiXL =- -Xi2Li2SL)(RRURP Lecturer:KangCollege of Electrial & Information Engineering XiHua University Electric C

25、ircuit Analysis电路分析电路分析(2) 若若ZL= RL为纯电阻为纯电阻负载获得的功率为：负载获得的功率为：2i2LiSLiS)( , XRRUIRZUI 电路中的电流为：电路中的电流为：2i2Li2SL)(XRRURP iiiLLZXRRdRdP 22 0 获得最大功率条件：获得最大功率条件：令令 Lecturer:KangCollege of Electrial & Information Engineering XiHua University Electric Circuit Analysis电路分析电路分析求：（求：（1）RL=5 时其消耗的功率；时其消耗的功率

26、； （2）RL=?能获得最大功率，并求最大功率；能获得最大功率，并求最大功率； （3）在）在RL两端并联一电容，问两端并联一电容，问RL和和C为多大时能与内为多大时能与内阻抗最佳匹配，并求最大功率。阻抗最佳匹配，并求最大功率。AjI)6 .26(89. 0555010 (1) oo 例：例：解：解： 55105010565jjjXRZLi+_U100o V50 H HRL5 I =105rad/sWRIPLL4589. 022 获最大功率获最大功率当当 07. 755 )2(2222 iiLXRRAjI)5 .22(766. 007. 755010 oo WRIPLL15. 407. 7766

27、. 022 Lecturer:KangCollege of Electrial & Information Engineering XiHua University Electric Circuit Analysis电路分析电路分析+_U100o V50 H HRL5 I =105rad/sCCjRYL 1 (3) 222L)(1)(111 LLLLLLCRCRjCRRCRjRYZ 5)(15)(1 222LLLLCRCRCRR 当当获最大功率获最大功率 110 FCRL AI110010 o WRIPi5512max Lecturer:KangCollege of Electrial

28、 & Information Engineering XiHua University Electric Circuit Analysis电路分析电路分析电路如图，求电路如图，求ZL=?时能获得最大功率，并求最大功率时能获得最大功率，并求最大功率.例：例： I190o AZLj30 30 - -j30 S UZLZiI+- -解：解： 4515)30/30(30jjjZi045260)30/30(4 jjUS 4515 *jZZiL当当WP120154)260( 2max 有有 Lecturer:KangCollege of Electrial & Information En

29、gineering XiHua University Electric Circuit Analysis电路分析电路分析 Lecturer:KangCollege of Electrial & Information Engineering XiHua University Electric Circuit Analysis电路分析电路分析i1u1+Mu2+i2L1L2 Lecturer:KangCollege of Electrial & Information Engineering XiHua University Electric Circuit Analysis电路分

30、析电路分析+ui NedtdiLudtdiLeiL , Lecturer:KangCollege of Electrial & Information Engineering XiHua University Electric Circuit Analysis电路分析电路分析 21L2L1i1magnetic linkage) Lecturer:KangCollege of Electrial & Information Engineering XiHua University Electric Circuit Analysis电路分析电路分析 22L2L1i2 Lecture

31、r:KangCollege of Electrial & Information Engineering XiHua University Electric Circuit Analysis电路分析电路分析 2L2L1i2 i1 Lecturer:KangCollege of Electrial & Information Engineering XiHua University Electric Circuit Analysis电路分析电路分析注解：注解： 11121 212 22 1 MiiL211 2 MiiL122 Lecturer:KangCollege of Ele

32、ctrial & Information Engineering XiHua University Electric Circuit Analysis电路分析电路分析(coupling coefficient) 121def LLMk1iLiLMiMiLLMLLMk2211211222112121221 )(注解：注解： Lecturer:KangCollege of Electrial & Information Engineering XiHua University Electric Circuit Analysis电路分析电路分析12112111uudtdiMdtdiL

33、dtdu 1 21221222uudtdiMdtdiLdtdu 2 注解：注解：普通电感引起普通电感引起自感电压自感电压 - 21 dtdiLudtdiLu222111耦合电感引起耦合电感引起互感电压互感电压 - dtdiMudtdiMu121212 Lecturer:KangCollege of Electrial & Information Engineering XiHua University Electric Circuit Analysis电路分析电路分析注解：注解： Lecturer:KangCollege of Electrial & Information E

34、ngineering XiHua University Electric Circuit Analysis电路分析电路分析dtdiMdtdiMtiLuuuu3b2a113112111 dd* i1 11N1N2+u11+u21+u31N3 22i2i3 33+u12+u22+u32+u13+u23+u33dtdiMdtdiLtiMuuuu3c221a3222212 dd0MMM0MMM0MMM2323c1313b1212a ,dtdiLdtdiMtiMuuuu332c1b3332313 dd Lecturer:KangCollege of Electrial & Information

35、 Engineering XiHua University Electric Circuit Analysis电路分析电路分析 i1122*112233 例：判断下列耦合电感的同名端例：判断下列耦合电感的同名端 Lecturer:KangCollege of Electrial & Information Engineering XiHua University Electric Circuit Analysis电路分析电路分析tiMudd121 tiMudd121 i1*u21+Mi1*u21+M注解：注解： Lecturer:KangCollege of Electrial &am

36、p; Information Engineering XiHua University Electric Circuit Analysis电路分析电路分析tiMtiLudddd2111 tiLtiMu2212dddd i1*L1L2+_u1+_u2i2MtiMtiLudddd2111 tiLtiMudddd2212 i1*L1L2+_u1+_u2i2Mi1*L1L2+_u1+_u2i2Mi1*L1L2+_u1+_u2i2M例：例：tiMtiLudddd2111 tiLtiMu2212dddd tiMtiLu2111dddd tiLtiMu2212dddd Lecturer:KangColleg

37、e of Electrial & Information Engineering XiHua University Electric Circuit Analysis电路分析电路分析例：例：i1*L1L2+_u2MR1R2+_u21010i1/At/s)()(H,1,H2,H5,10 2211tutuMLLR和和求求已知已知 tstVstVtiMtu2 021 1010 10dd)(12解：解： tstVtstVttiLiRtu2 021 150 10010 50 100dd)(111 Lecturer:KangCollege of Electrial & Informatio

38、n Engineering XiHua University Electric Circuit Analysis电路分析电路分析I1U1+jMU2+I2jL1jL2 2111IMjILjU 2212ILjIMjU Lecturer:KangCollege of Electrial & Information Engineering XiHua University Electric Circuit Analysis电路分析电路分析IM*L1L2+UILeq+Ueq2121LI jM2LLI jIMjILjIMjILjU )( )( )(M2LLL21eq Lecturer:KangCo

39、llege of Electrial & Information Engineering XiHua University Electric Circuit Analysis电路分析电路分析ILeq+Ueq2121LI jM2LLI jIMjILjIMjILjU )( )( )(M2LLL21eq IM*L1L2+U*)(21LL21M 0M2LLL21 Lecturer:KangCollege of Electrial & Information Engineering XiHua University Electric Circuit Analysis电路分析电路分析I*L

40、1L2+_UI2MI1ILeq+UUMLLjM2LLIIIIUILjIMjUIMjILj2212121221211)( M2LLMLLLqLjIUZ21221eqe 1LLMkLLM0L2121eq , Lecturer:KangCollege of Electrial & Information Engineering XiHua University Electric Circuit Analysis电路分析电路分析I*L1L2+_UI2MI1ILeq+UUMLLjM2LLIIIIUILjIMjUIMjILj2212121221211)( M2LLMLLLLjIUZ21221eqe

41、q Lecturer:KangCollege of Electrial & Information Engineering XiHua University Electric Circuit Analysis电路分析电路分析*j L1 I1 I2 I123j L2j M21113 jj IMILU12223 jj IMILU 21 III j)(j11IMIML j)(j22IMIML j (L1-M) I1 I2 I123j Mj (L2-M) Lecturer:KangCollege of Electrial & Information Engineering XiHua

42、University Electric Circuit Analysis电路分析电路分析*j L1 I1 I2 I123j L2j Mj (L1M) I1 I2 I123j Mj (L2M)21113 jj IMILU 12223 jj IMILU 21 III j)(j11IMIML j)(j22IMIML Lecturer:KangCollege of Electrial & Information Engineering XiHua University Electric Circuit Analysis电路分析电路分析*Mi2i1L1L2ui+*Mi2i1L1L2u+u+j (

43、L1M) I1 I2 Ij Mj (L2M)j (L1M)1 I2 Ij Mj (L2M) Lecturer:KangCollege of Electrial & Information Engineering XiHua University Electric Circuit Analysis电路分析电路分析abL 求等效电感求等效电感M=4H6H2H3H5HabM=1H4H3H2H1Hab3HLab=6H Lecturer:KangCollege of Electrial & Information Engineering XiHua University Electric

44、 Circuit Analysis电路分析电路分析j2 I1I j1+s Uj3 I1I j2+s UV30t210us)sin( ,SQPii1 182482111j202j3j10Z. )/()(A825850ZUIS .A7251I2j1j201j20I1 .A825t2850i).sin(. A725t2i1).sin( jQP483j757182458IUSS .*9101824VA58SVar483QW757P.cos.,.,. Lecturer:KangCollege of Electrial & Information Engineering XiHua Univers

45、ity Electric Circuit Analysis电路分析电路分析*j L11 I2 Ij L2j M+S UR1R2Z=R+jX Lecturer:KangCollege of Electrial & Information Engineering XiHua University Electric Circuit Analysis电路分析电路分析1 j) j( UIMILR2111 令令 Z11=R1+j L1 Z22=(R2+R)+j( L2+X) ZM= j M1 UIZIZ2M1110IZIZ2221M )( )(11 22211222111YMZUZMZUI 112

46、222YMZUYI)(Z-1M11 0)(2221 IZLjRIMjj L11 I2 Ij L2+1 UZ=R+jX Lecturer:KangCollege of Electrial & Information Engineering XiHua University Electric Circuit Analysis电路分析电路分析r1111ZZIUZ 1in 输入阻抗：输入阻抗：称称为为反反射射阻阻抗抗阻阻抗抗表表示示副副边边对对原原边边的的等等效效其其中中，,)( 1rr1r1222jXRYMZ 反射电阻，正值反射电阻，正值,)()Re( 1r222222222222RXRMY

47、MR 反性质反性质与与反射电抗反射电抗22222222222222MZXXRMYMIX,)()( 1r 1 I+1 UZ11r1Z11in , ,ZZ0I2 即副边开路即副边开路当当 )( 1 222111YMZUI Lecturer:KangCollege of Electrial & Information Engineering XiHua University Electric Circuit Analysis电路分析电路分析111M1M11ocUZZUZYU 其其中中，开开路路电电压压 2I +oc UZ22r2Zr222oc112222ZZUYMZUYI )(Z-1M11

48、222222jXRLLjRRZ )() 22 （副副边边阻阻抗抗112YMZ)( 2r 原原边边对对副副边边的的等等效效阻阻抗抗 Lecturer:KangCollege of Electrial & Information Engineering XiHua University Electric Circuit Analysis电路分析电路分析10j1010Z4ZMZX2222r1 j 8 . 9 j2 . 010200)1010(41010104 jjjjZX此时负载获得的功率：此时负载获得的功率： W10R101020PPr12Zr1 ）（ W , *S10R4UPZZ211r

49、1 此时，是最佳阻抗匹配：此时，是最佳阻抗匹配：* *j10 2 Ij10 j2+S U10 ZX+S U10+j10 Z1r=10j10 Lecturer:KangCollege of Electrial & Information Engineering XiHua University Electric Circuit Analysis电路分析电路分析L1=3.6H , L2=0.06H , M=0.465H , R1=20 , R2=0.08 , RL=42 , 314 314rad/s,V 0115o sU. , :21II求求应用原边等效电路应用原边等效电路4113020L

50、RZ1111.jj .j8518j0842LRRZ2L222 8 .188-j42212434621241146146ZXZ2222Mr1).(.o1 I+S UZ11222)(ZM 例例2：*j L11 I2 Ij L2j M+S UR1R2RL解解：(1) Lecturer:KangCollege of Electrial & Information Engineering XiHua University Electric Circuit Analysis电路分析电路分析A)9 .64(111. 08 .1884224 .1130200115o11S1 jjZZUIl应用副边等效

51、电路应用副边等效电路VjjLjRUMjIMjUSOC085.144 .1130200115146 111 85.18906 .1130213164 .113020146)(2112jjZM解解 (2)2 I+oc UZ22112)(ZMAjjUIOC0353. 008.42085.1485.1808.425 .182 Lecturer:KangCollege of Electrial & Information Engineering XiHua University Electric Circuit Analysis电路分析电路分析 Lecturer:KangCollege of Electrial & Information Engineering Xi