电工技能培训专题-电路-正弦稳态电路分析

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1、JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit Course Lectured By Xuebin Jiang / Information School Thursday, April 8, 2021 1 第九章第九章正弦稳态电路的分析正弦稳态电路的分析 9.1 阻抗和导纳 9.2 电路的相量图 9.3 正弦稳态电路的分析 9.4 正弦稳态电路的功率 9.5 复功率 9.6 最大功率传输 JiangSu University Of Science and Technology. Zhangj

2、iagang Campus. Circuit Course Lectured By Xuebin Jiang / Information School Thursday, April 8, 2021 2 基本要求基本要求熟练掌握阻抗和导纳的物理意义并了解它们之间等效变换的概念；熟练地运用相量法分析正弦电流电路；掌握正弦电流电路中的平均功率，无功功率、视在功率及功率因数的概念；掌握最大功率传输条件。直流电路的分析 + 相量法基础正弦稳态电路的分析方法，在第10、11、12章节中都要用到。本章与其它章节的联系 JiangSu University Of Science and

3、Technology. Zhangjiagang Campus. Circuit Course Lectured By Xuebin Jiang / Information School Thursday, April 8, 2021 3 1. 阻抗阻抗 Z (1) 定义 j jz就是该阻抗两端的电压与通过该阻抗电流的相位差j j ！ . I 含线性无源元件的一端口N0 + - . U 设：设： . U = = U f fu . I = = I f fi 则：则：Z def . U . I = = U I f fu- -f fi = = | Z | j jz | Z | = U I 为

4、阻抗的模，也可以简称为阻抗。 j jz = =f fu- -f fi 为阻抗角。阻抗的单位与电阻相同。 9.1 阻抗和导纳阻抗和导纳 JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit Course Lectured By Xuebin Jiang / Information School Thursday, April 8, 2021 4 R2 + + X2 (2) 阻抗参数间的关系指数式：Z=| Z | e jj j z 代数式：Z =| Z |cosj jz + j| Z |sinj j

5、z Z = = | Z | j jz Z = = R + + j X 实部R称为电阻，虚部X称为电抗。 Z + + - - . U . I N0 R = = |Z|cosj jz X = = |Z|sinj jz j jz R X |Z|、R、X构成的直角三角形称为阻抗三角形。极坐标式： |Z| = = j jz = = arctg R X 9.1 阻抗和导纳阻抗和导纳 JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit Course Lectured By Xuebin Jiang / In

6、formation School Thursday, April 8, 2021 5 (3) 单个元件的阻抗 R + + - - . U . I N0 L N0 + + - - . U . I C N0 + + - - . U . I 说明 Z 可以是纯实数，也可以是纯虚数。 Z = = . U . I = = R Z = = . U . I = = jw wL = = j XL 纯电阻纯电感 XL=L 称感性电抗， XL f ！纯电容 Z = = . U . I = = jw wC 1 = = w wC 1 - -j = = j XC XC = = - - w wC 1 称容性电抗， X

7、C (1/f ) ！ 9.1 阻抗和导纳阻抗和导纳 JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit Course Lectured By Xuebin Jiang / Information School Thursday, April 8, 2021 6 (4) RLC串联电路根据KVL和VCR的相量形式可得： . U = = w wL- - w wC 1 + + - - + + - - R jw wL + + - - . UR . UL . UC jw wC 1 + + - - . U

8、 . I N0 = = R . I + + jw wL . I - - j w wC 1 . I = = R + + jw wL- - w wC 1 . I j = = R + + j(XL+ +XC) . I . I = (= (R + + jX) ) = = Z . I Z = = . I . U = = R + + j X = = | Z | j jz X = = XL + + XC j jz = = arctg R X 9.1 阻抗和导纳阻抗和导纳 JiangSu University Of Science and Technology. Zhangjiagang Campus. Ci

9、rcuit Course Lectured By Xuebin Jiang / Information School Thursday, April 8, 2021 7 当 w wL 结论：表现为电压超前电流， Z 呈感性，称电路为感性电路。 w wC 1 时，有 X0 ，j jz0 以电流为参考相量相量图以电流为参考相量相量图 . I . UR . UC . UL . U j jz 9.1 阻抗和导纳阻抗和导纳 JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit Course Lectur

10、ed By Xuebin Jiang / Information School Thursday, April 8, 2021 8 当 w wL 表现为电压滞后电流，Z 呈容性，称电路为容性电路。 w wC 1 时，有 X0 ，j jz0。 . I . UR . UC . UL j jz 结论：以电流为参考相量相量图以电流为参考相量相量图 9.1 阻抗和导纳阻抗和导纳 JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit Course Lectured By Xuebin Jiang / In

11、formation School Thursday, April 8, 2021 9 当 w wL = 表现为电压与电流同相位，电路发生了串联谐振，Z 呈纯电阻性。 w wC 1 时，有 X = 0 ，j jz = 0。 . I . UR . UC . UL . U = = 从相量图可以看出，正弦交流RLC串联电路中，会出现分电压大于总电压的现象。以电流为参考相量相量图以电流为参考相量相量图结论： 9.1 阻抗和导纳阻抗和导纳 JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit Cou

12、rse Lectured By Xuebin Jiang / Information School Thursday, April 8, 2021 10 当R=0，X 0时，Z 为纯电感性； RLC 串联电路的电压 UR、 UX、U 构成电压三角形。满足： U = UR + UX 2 2 . I . UR . U j jz . UX |Z| X R 当R=0，X0时，Z 为纯电容性。结论： 9.1 阻抗和导纳阻抗和导纳 JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit Course Lect

13、ured By Xuebin Jiang / Information School Thursday, April 8, 2021 11 2. 导纳导纳 Y (1) 阻抗阻抗Z的倒数定义为导纳的倒数定义为导纳Y，即：即：Y = 1 Z . I = = |Y| j jY 单位是S Y = = f fi- -f fu . U = = I U 也可以简称为导纳。 j jY = =f fi- -f fu 称为导纳角。 |Y| = = 导纳的代数形式为： Y = G + j B 实部G称为电导，虚部B称为电纳。称为导纳模， I U 9.1 阻抗和导纳阻抗和导纳 JiangSu University

14、Of Science and Technology. Zhangjiagang Campus. Circuit Course Lectured By Xuebin Jiang / Information School Thursday, April 8, 2021 12 9.1 阻抗和导纳阻抗和导纳 G、B、|Y|、j jY 之间的关系为 G= =|Y|cosj jY B= =|Y|sinj jY |Y| = = G2 + + B2 j jY = = arctg G B j jY G B 导纳三角形导纳三角形 (2) 单个单个R、L、C 元件的导纳元件的导纳当无源网络内为单个元件时，等效导

15、纳分别为： Y + + - - . U . I N0 JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit Course Lectured By Xuebin Jiang / Information School Thursday, April 8, 2021 13 称为感性电纳； Y = . U . I 纯电阻 = R 1 = G 纯电感 Y = . U . I = jw wL 1 = jBL BL = w wL 1 纯电容 Y = . U . I = jw wC = jBC BC = w wC

16、称为容性电纳； Y 可以是纯实数，也可以是纯虚数。称为电导； 9.1 阻抗和导纳阻抗和导纳 Y + + - - . U . I N0 JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit Course Lectured By Xuebin Jiang / Information School Thursday, April 8, 2021 14 (3) RLC并联电路 . I3 jw wL jw wC 1 . I2 . I1 R . U + + - - . I 根据VCR和KCL的相量形式可

17、得： . I = = G . U . U jw wL 1 + + + + jw wC . U . U . U = = Y . U = = w wL 1 + + jw wC . U = = G + + j(BL+ + BC) = (= (G + + jB) ) G - - j 9.1 阻抗和导纳阻抗和导纳 = = - - w wL 1 B = = BL + + BC + +w wC j jY = = arctg G B JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit Course Lecture

18、d By Xuebin Jiang / Information School Thursday, April 8, 2021 15 结论：对于 RLC 并联电路 B0或j jY 0，称Y为感性； B0或j jY 0，称Y为容性； B=0或j jY =0，Y为纯电阻性； G=0，B0，Y为纯电感性； G=0，B0，Y为纯电容性。以电压为参考相量相量图以电压为参考相量相量图 . U . I1 . I2 . I3 . I j jY . I3 + + . I2 电流三角形电流三角形 9.1 阻抗和导纳阻抗和导纳 JiangSu University Of Science and Technolog

19、y. Zhangjiagang Campus. Circuit Course Lectured By Xuebin Jiang / Information School Thursday, April 8, 2021 16 . U . I1 . I2 . I3 . I = = 从相量图可以看出，正弦交流RLC并联电路中，会出现分电流大于总电流的现象。 B=0、j jY =0，时的相量图 9.1 阻抗和导纳阻抗和导纳 JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit Course Lectur

20、ed By Xuebin Jiang / Information School Thursday, April 8, 2021 17 3. 阻抗与导纳的相互等效一端口的阻抗和导纳可以互换，等效互换的条件为： . I 含线性无源元件的一端口N0 + + - - . U N0的等效阻抗(导纳)、输入阻抗(导纳)或驱动点阻抗(导纳)，它们的实部和虚部都是外施正弦激励的角频率的函数： Z(jw w) = = R(w w) + + jX(w w) Y(jw w) = = G(w w) + + jB(w w) Z(jw w) Y(jw w) = =1 分开写 | Z | Y |= =1 j

21、 jZ + + j jY = =0 9.1 阻抗和导纳阻抗和导纳 JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit Course Lectured By Xuebin Jiang / Information School Thursday, April 8, 2021 18 若已知 Z=R+jX ，求等效的 Y=G+jB 。若已知 Y = G + jB 则： . I R + + - - . U jX jB . I + + - - . U G Y = = Z 1 = = R + + jX 1 =

22、 = (R + + jX) (R - - jX) (R - - jX) = = R2 + + X2 R + j R2 + + X2 - -X = = G + + jB G = = |Z|2 R B = = - - |Z|2 X 则： R = = |Y|2 G X = = - - |Y|2 B 等效成 Z = R + jX 9.1 阻抗和导纳阻抗和导纳 JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit Course Lectured By Xuebin Jiang / Information Sc

23、hool Thursday, April 8, 2021 19 例：电路如图，求各支路电流和解：设串联支路阻抗为Z1 . U10 。 Z1 并联支路导纳为Y10 Y10 = = + + jw wC = =10- -3 + + j3.1410- -3 = =3.295410 - -3 72.33o S + + - - R1 . I . U10 jw wC 1 + + - - . Us R2 jw wL 0 1 . I2 . I1 10W W 0.5H 10m m 1k 100V w w = = 314rad/s 则 Z1=10 + j157 R2 1 Z10 = = Y10 1 = = 3

24、03.45 - -72.33o = = 92.11- - j289.13 W W 9.1 阻抗和导纳阻抗和导纳 JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit Course Lectured By Xuebin Jiang / Information School Thursday, April 8, 2021 20 Zeq= = Z1+ +Z10 - -52.30o = = (92.11+ +10) + + j(157- -289.13) = = 102.11 - - j132.13 = =

25、166.99 W W . I = = Zeq . US = = 166.99 - -52.30o 100 = = 0.6 52.30o A 9.1 阻抗和导纳阻抗和导纳 - -72.33o + + 52.30o . U10 = = Z10 . I = =303.45 0.6 =182.07 - -20.03o V = =182.07 0.00314 90o o - - 20.03o . I1= = jw wC . U10 = =0.57 69.97o A . I2 = = . U10 R2 = =0.182 - -20.03o A JiangSu University Of Science a

26、nd Technology. Zhangjiagang Campus. Circuit Course Lectured By Xuebin Jiang / Information School Thursday, April 8, 2021 21 相量作为一个复数，可以用复平面上的有向线段来表示。按照大小和相位关系，用初始位置的有向线段画出的若干个相量的图形，称为相量图。因相量图能直观地反映各相量之间的关系，所以借助于相量图对电路进行辅助分析和计算，有时能起到“事半功倍” 的效果。 9.2 电路的相量图电路的相量图 JiangSu University Of Science and T

27、echnology. Zhangjiagang Campus. Circuit Course Lectured By Xuebin Jiang / Information School Thursday, April 8, 2021 22 相量图的定性画法选一参考相量，通常是某一并联部分的电压，习惯上把它画在水平方向。由VCR确定并联支路电流的相量由KCL确定结点电流相量；对串联部分，以电流相量为参考由VCR确定有关电压相量由KVL确定回路上各电压相量。绘制时，可以用平移求和法则，使各相量（有关结点电流相量、回路电压相量等）构成若干个封闭的多边形。也可以使各相量都

28、从原点向外辐射，用平行四边形法则求和。一般是根据需要，结合上述两种方式，画成便于分析计算的形状。 9.2 电路的相量图电路的相量图 JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit Course Lectured By Xuebin Jiang / Information School Thursday, April 8, 2021 23 当需要借助相量图进行分析计算时，右图选并联部分电压为参考相量比较方便。定性绘制过程： + + - - R1 . I . U10 jw wC 1

29、 + + - - . US R2 jw wL 0 1 . I2 . I1 . U10 VCR . I1 . I2 KCL . I VCR . I R1 jw wL . I KVL . US 绘制时应根据已知条件，使图形大致符合比例。 . U10 . U10 . I2 . I1 . I . I R1 jw wL . I . US 9.2 电路的相量图电路的相量图 JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit Course Lectured By Xuebin Jiang / Informat

30、ion School Thursday, April 8, 2021 24 例题： I、R、XC、XL 。求：解：选为参考相量 . Uab . I1 超前 . Uab 90o . I2 与 . Uab 同相由KCL I = = = = 14.14 A 超前 . I 90o 由KVL知： + + - - . I . Uab + + - - . U R jXL . I2 . I1 a b - -jXC 100V 10A 10A . U 与 . I 同相。 . Uab . I = . I1+ . I2 I1 2 jXLI . + + I2 2 . U = jXLI . + Uab . . U

31、 . I . I1 . I2 jXLI . jXLI . 45o 45o 9.2 电路的相量图电路的相量图 JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit Course Lectured By Xuebin Jiang / Information School Thursday, April 8, 2021 25 XLI = U = 100V XL = U I = 7.07 Uab = U = 141.4V R = Uab I2 = 14.14 XC = Uab I1 = 14.14 若给定，

32、还能进一步算出L和C。 = 14.14 100 2 I = = 14.14 A 9.2 电路的相量图电路的相量图 . Uab . U . I . I1 . I2 jXLI . jXLI . 45o 45o JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit Course Lectured By Xuebin Jiang / Information School Thursday, April 8, 2021 26 本节的核心内容：将电阻电路的各种分析方法，推广到正弦稳态电路中。电阻电路中的很多

33、方法和定理，都以两类约束为基础，即： KCL、KVL和VCR。 KCL i = 0 I = 0 . KVL u = 0 VCR u = Ri 在引入相量和复阻抗的概念以后，两类约束的相量表达式与时域表达式具有相同的形式： U = 0 . . U = Z I . 9.3 正弦稳态电路的分析正弦稳态电路的分析 JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit Course Lectured By Xuebin Jiang / Information School Thursday, April 8

34、, 2021 27 推广时作如下变换：以两类约束为基础的各种计算方法和定理必然也具有相同的形式。所以电阻电路的各种分析方法和定理就能推广到正弦稳态电路中来。例如：Req的定义与求法，可以推广成 Zeq的定义与求法； i . I u . U R Z G Y 电阻电路正弦稳态电路 9.3 正弦稳态电路的分析正弦稳态电路的分析 JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit Course Lectured By Xuebin Jiang / Information School Thu

35、rsday, April 8, 2021 28 相量法把描述动态电路的微分方程变为复数的代数方程。与求微分方程的特解(正弦稳态解)相比，使计算简化，书写方便，物理概念也更加突出。由于描述的物理过程不同，所以方程为相量形式，计算为复数运算。因为 p = ui 是非正弦量，所以，功率的计算要单独考虑 (后述)。推广后的差别 9.3 正弦稳态电路的分析正弦稳态电路的分析 JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit Course Lectured By Xuebin Jiang / I

36、nformation School Thursday, April 8, 2021 29 解题指导解题指导解：用观察法列结点电压方程 + + - - . IS5 . US1 Z1 . US3 + + - - Z2 Z3 Z4 Z5 (Y1+ +Y2+ +Y3) . Un1 - - Y3 . Un2 = = Y1 . US1 + + Y3 . US3 - - Y3 . Un1 + + (Y3+ +Y4) . Un2 = = - -Y3 . US3 . + + IS5 . Il1 . Il2 . Il3 用观察法列回路电流方程 l1 (Z1+ +Z2) . Il1 - -Z 2 . Il2 .

37、US1 + + - - . U 例1：激励均为同频率正弦量。试列出该电路的结点电压方程和回路电流方程。 = = l2 - -Z2 . Il1 + + (Z2+ +Z3 + +Z4) . Il2 - -Z4 . Il3 = = . - -US3 l3 . Il3 = = . - - IS5 9.3 正弦稳态电路的分析正弦稳态电路的分析 JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit Course Lectured By Xuebin Jiang / Information School Th

38、ursday, April 8, 2021 30 例2：求右图一端口的戴维宁等效电路。 a o . US1 + + - - Z1 Z2 . I2 . IS3 + + - - . rI2 1 1 . Uoc + + - - . Uoc = = - -r . I2 + . Uao . I2 = = Y2 . Uao 解：求开路电压 . Uoc + + - - Zeq 1 1 . Uoc = = - -rY2 . Uao + + . Uao = = (1- -rY2) . Uao . Uao = = Y1 + + Y2 Y1 . US1 . - - IS3 用结点法求出 . Uoc = = (1-

39、 -rY2) (Y1 . US1 . - - IS3) Y1 + + Y2 代入上式得 9.3 正弦稳态电路的分析正弦稳态电路的分析 JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit Course Lectured By Xuebin Jiang / Information School Thursday, April 8, 2021 31 . I2 = = Z1+ +Z2 Z1 . I . I = = Z1 Z1+ +Z2 . I2 = =(1+ +Y1Z2 ) . U = = - -r .

40、I2 + + Z2 . I2 = = (Z2- -r) . I2 Zeq = = . U . I = = Z2- -r 1+ +Y1Z2 a o Z1 Z2 . I2 + + - - . rI2 1 1 . U + + - - . I 再求等效阻抗 . I2 . Uoc + + - - Zeq 1 1 9.3 正弦稳态电路的分析正弦稳态电路的分析 JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit Course Lectured By Xuebin Jiang / Information Scho

41、ol Thursday, April 8, 2021 32 I1 = 例3：US =380V，f =50Hz，C为可变电容，当C=80.95F时，表A读数最小为2.59A。求：表A1的读数。解法1：借助相量图求解选 . US IC = 2fCUS = 9.66A . I = . I1+ . IC 调C，IC变。但： . I1 不变， . IC 始终与 . US 正交。 . US . I1 . I . I . US 9.662 + 2.592 = 10 A R1 . I jw wC 1 + + - - . US jw wL1 . I1 . IC A A1 为参考相量始终构成封闭三角形。

42、当与同相时最小。 . IC . US . I1 . IC . I . US . I1 . IC . I 9.3 正弦稳态电路的分析正弦稳态电路的分析 JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit Course Lectured By Xuebin Jiang / Information School Thursday, April 8, 2021 33 解法2：电路的输入导纳为 Y = jC+ |Z1|2 R1 - j |Z1|2 L1 R1 . I jw wC 1 + + - - .

43、 US jw wL1 . I1 . IC A A1 调C，只改变ImY。当ImY =0时，| Y |最小， I = | Y |U 也最小。电路呈纯电阻性， . US 与 . I 同相。设 . US = 380 0o V 则 . I = 2.59 0o A . IC = jC = j9.66A . US . I1= . I - . IC = 2.59 - j9.66 = 10 -70o A 表A1的读数为10 A。 9.3 正弦稳态电路的分析正弦稳态电路的分析 JiangSu University Of Science and Technology. Zhangjiagang Campus

44、. Circuit Course Lectured By Xuebin Jiang / Information School Thursday, April 8, 2021 34 实践中，可以用这种电路测量一个电感线圈的参数。由以上(测得的)数据算出电感线圈电感线圈 R1=13 W W， L1= = 113.7mH 35.71 w w R1 . I jw wC 1 + + - - . US jw wL1 . I1 . IC A A1 . I1= 10 -70o A Z1 = . Us . I1 = 38 70o = 13 + j 35.71 W W 9.3 正弦稳态电路的分析正弦稳态电路的分析 JiangSu University Of Science and Technology. Zhangjiag