《电工技能培训专题电路正弦稳态电路分析》由会员分享,可在线阅读,更多相关《电工技能培训专题电路正弦稳态电路分析(52页珍藏版)》请在金锄头文库上搜索。
1、JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jiang / Information School第九章第九章 正弦稳态电路的分析正弦稳态电路的分析o9.1 阻抗和导纳o9.2 电路的相量图o9.3 正弦稳态电路的分析o9.4 正弦稳态电路的功率o9.5 复功率o9.6 最大功率传输7/30/20241JiangSu University Of Science and Technology. Zhangjiagang Campus. Circu
2、it CourseLectured ByXuebin Jiang / Information School基本要求基本要求 o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;o熟练地运用相量法分析正弦电流电路;o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;o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7/30/20242JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit Co
3、urseLectured ByXuebin Jiang / Information School1. 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= = Uf fu .I = = If fi则:则:Zdef .U .I= =UIf fu- -f fi= = | Z |j jz| Z | = UI为阻抗的模,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7/30/20243JiangSu University Of Scien
4、ce and Technology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jiang / Information SchoolR2 + + 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 jzZ = = | Z |j jzZ = = R + + j X实部R称为电阻,虚部X称为电抗。Z+ +- - .U .IN0R = = |Z|cosj jzX = = |Z|sinj jzj jz| Z |RX|Z|、R、X构成的直角三角形称为阻抗三角形
5、。极坐标式:|Z| = =j jz = = arctgRX9.1 阻抗和导纳阻抗和导纳7/30/20244JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jiang / Information School(3) 单个元件的阻抗 R+ +- - .U .IN0LN0+ +- - .U .ICN0+ +- - .U .I说明 Z 可以是纯实数,也可以是纯虚数。 Z = = .U .I= = RZ = = .U .I= = jw wL = = j X
6、L纯电阻纯电感XL=L 称感性电抗, XL f !纯电容Z = = .U .I= =jw wC1= =w wC1- -j= = j XCXC = = - -w wC1称容性电抗, XC (1/f ) !9.1 阻抗和导纳阻抗和导纳7/30/20245JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jiang / Information School(4) RLC串联电路根据KVL和VCR的相量形式可得: .U= = w wL- -w wC1+ +
7、- -+ +- -Rjw wL+ +- - .UR .UL .UCjw wC1+ +- - .U .IN0= = R .I + + jw wL .I- - jw wC1 .I= = R + + jw wL- -w wC1 .Ij= = R + + j(XL+ +XC) .I .I= (= (R + + jX) )= = Z .IZ = = .I .U= = R + + j X = = | Z |j jzX = = XL + + XCj jz = = arctgRX9.1 阻抗和导纳阻抗和导纳7/30/20246JiangSu University Of Science and Technolo
8、gy. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jiang / Information School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 wC1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 .Uj jz9.1 阻抗和导纳阻抗和导纳7/30/20247JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin J
9、iang / Information School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 wC1时,有 X0 ,j jz0。 .I.UR.UC.ULj 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7/30/20248JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jiang / Information School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
10、电阻性。w wC1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7/30/20249JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jiang / Information School当R=0,X 0时,Z 为纯电感性;RLC 串联电路的电压 UR、UX、U 构
11、成电压三角形。满足:U =UR + UX22 .I.UR .Uj jz.UX|Z|XR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7/30/202410JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jiang / Information School2. 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 = 1Z .I= = |Y|j jY单位是SY = =f fi- -f fu .U= =
12、IU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,IU9.1 阻抗和导纳阻抗和导纳7/30/202411JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jiang / Information School9.1 阻抗和导纳阻抗和导纳G、B、|Y|、j jY 之间的关系为G= =|Y|cosj jYB= =|Y|sinj jY|
13、Y| = =G2 + + B2j jY = = arctgGBj jY| Y |GB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,等效导纳分别为 :Y+ +- - .U .IN07/30/202412JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jiang / Information School称为感性电纳;Y = .U .I纯电阻=R1= G纯电感Y = .U .I=jw wL1= jBLB
14、L = w wL1纯电容Y = .U .I= jw wC = jBCBC = w wC 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 .IN07/30/202413JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jiang / Information School(3) RLC并联电路 .I3jw wLjw wC1 .I2 .I1R .U+ +- - .I根据VCR和KCL的相量形式
15、可得: .I= = G .U .Ujw wL1+ + + jw wC .U .U .U = = Y .U= =w wL1+ + jw wC .U= = G + + j(BL+ + BC)= (= (G + + jB) )G - - j9.1 阻抗和导纳阻抗和导纳= = - -w wL1B = = BL + + BC+ +w wCj jY = = arctgGB7/30/202414JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jiang / I
16、nformation School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 .Ij jY.I3+ +.I2电流三角形电流三角形电流三角形电流三角形9.1 阻抗和导纳阻抗和导纳7/30/202415JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit CourseLectured
17、 ByXuebin Jiang / Information School .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7/30/202416JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jiang / Information School3. 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
18、性无源元件的一端口N0+ +- - .UN0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 |= =1j jZ + + j jY = =09.1 阻抗和导纳阻抗和导纳7/30/202417JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit CourseLect
19、ured ByXuebin Jiang / Information School 若已知 Z=R+jX ,求等效的 Y=G+jB 。若已知 Y = G + jB则: .IR+ +- - .UjXjB .I+ +- - .UGY = =Z1= =R + + jX1= =(R + + jX) (R - - jX)(R - - jX)= =R2 + + X2R+ jR2 + + X2- -X= = G + + jBG = =|Z|2 RB = -= -|Z|2 X则:R = =|Y|2 GX = -= -|Y|2 B等效成 Z = R + jX9.1 阻抗和导纳阻抗和导纳7/30/202418Jia
20、ngSu University Of Science and Technology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jiang / Information School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Y10Y10 = =+ + jw wC= =10- -3 + + j3.1410- -3= =3.295410 - -372.33o S+ +- -R1 .I .U10jw wC1+ +- - .UsR2jw wL01 .I2 .I110W W0.5H10m m1k10
21、0Vw w = = 314rad/s则 Z1=10 + j157 R21Z10 = =Y101= = 303.45- -72.33o= = 92.11- - j289.13 W W9.1 阻抗和导纳阻抗和导纳7/30/202419JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jiang / Information SchoolZeq= = Z1+ +Z10- -52.30o= = (92.11+ +10)+ + j(157- -289.13)=
22、 = 102.11 - - j132.13= =166.99W W .I = =Zeq .US= =166.99- -52.30o100= = 0.652.30oA9.1 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 0.0031490o o - - 20.03o .I1= = jw wC .U10 = =0.5769.97oA .I2 = = .U10 R2= =0.182- -20.03oA7/30/202420JiangSu University Of Sci
23、ence and Technology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jiang / Information School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7/30/202421JiangSu University Of Science and Technology. Zhangjiag
24、ang Campus. Circuit CourseLectured ByXuebin Jiang / Information School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
25、7/30/202422JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jiang / Information School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 .U10jw wC1+ +- - .USR2jw wL01 .I2 .I1 .U10VCR .I1 .I2KCL .IVCR .I R1 jw wL .IKVL .US绘制时应根据已知条件,使图形大致符合比例。 .U10
26、 .U10 .I2 .I1 .I .I R1 jw wL .I .US9.2 电路的相量图电路的相量图7/30/202423JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jiang / Information School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KCLI = = = 14.14 A超前 .I 90o 由KVL知:+ +- - .I .Uab+ +-
27、- .URjXL .I2 .I1ab- -jXC100V10A10A .U 与 .I 同相。 .Uab .I = .I1+ .I2I12jXLI.+ + I22 .U = jXLI .+ Uab. .U .I .I1 .I2jXLI.jXLI.45o45o9.2 电路的相量图电路的相量图7/30/202424JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jiang / Information SchoolXLI = U = 100VXL =UI
28、= 7.07 Uab =U= 141.4VR =UabI2= 14.14 XC =UabI1= 14.14 若给定,还能进一步算出L和C。=14.141002I = = 14.14 A9.2 电路的相量图电路的相量图 .Uab .U .I .I1 .I2jXLI.jXLI.45o45o7/30/202425JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jiang / Information School本节的核心内容:将电阻电路的各种分析方法,
29、推广到正弦稳态电路中。电阻电路中的很多方法和定理,都以两类约束为基础,即: KCL、KVL和VCR。KCLi = 0I = 0.KVLu = 0VCRu = 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7/30/202426JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jiang / Information School推广时作如下变换
30、: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 .Iu .URZGY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7/30/202427JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jiang / Information School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
31、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7/30/202428JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jiang / Information School解题指导解题指导解:用观察法列结点电压方程+ +- - .IS5 .US
32、1Z1 .US3+ +- -Z2Z3Z4Z5(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- -Z2 .Il2 .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 = = .- -US3l3 .Il3 = =
33、 .- - IS59.3 正弦稳态电路的分析正弦稳态电路的分析7/30/202429JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jiang / Information School例2:求右图一端口的戴维宁等效电路。 ao .US1+ +- -Z1Z2 .I2 .IS3+ +- - .rI211 .Uoc+ +- - .Uoc= = - -r . I2+ .Uao . I2 = = Y2 .Uao解:求开路电压 .Uoc+ +- -Zeq11
34、 .Uoc= = - -rY2 .Uao+ + .Uao= = (1- -rY2) .Uao .Uao = =Y1 + + Y2 Y1 .US1 . - - IS3用结点法求出 .Uoc = =(1- -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7/30/202430JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jiang / Information School .
35、 I2 = =Z1+ +Z2Z1 . I . I = =Z1Z1+ +Z2 . I2= =(1+ +Y1Z2 ) . U = = - -r . I2+ + Z2 . I2= = (Z2- -r) . I2Zeq = = . U . I= =Z2- -r1+ +Y1Z2aoZ1Z2 .I2+ +- - .rI211 .U+ +- - .I再求等效阻抗 . I2 .Uoc+ +- -Zeq119.3 正弦稳态电路的分析正弦稳态电路的分析7/30/202431JiangSu University Of Science and Technology. Zhangjiagang Campus. Circ
36、uit CourseLectured ByXuebin Jiang / Information SchoolI1 =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 .USIC = 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 .US9.662+ 2.592= 10 AR1 .Ijw wC1+ +- - .USjw wL1 .I1 .ICAA1为参考相量始终构成封闭三角形。当与同相时最小。 .IC
37、.US .I1 .IC .I .US .I1 .IC .I9.3 正弦稳态电路的分析正弦稳态电路的分析7/30/202432JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jiang / Information School解法2:电路的输入导纳为Y = jC+|Z1|2R1- j|Z1|2L1R1 .Ijw wC1+ +- - .USjw wL1 .I1 .ICAA1调C,只改变ImY。当ImY =0时,| Y |最小,I = | Y |U 也
38、最小。电路呈纯电阻性, .US 与 .I 同相。设 .US = 380 0oV则 .I = 2.590oA .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7/30/202433JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jiang / Information School实践中,可以用这种电路测量一个电感
39、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= 113.7mH35.71w wR1 .Ijw wC1+ +- - .USjw wL1 .I1 .ICAA1 .I1= 10-70o AZ1 = .Us .I1 = 3870o= 13 + j 35.71 W W9.3 正弦稳态电路的分析正弦稳态电路的分析7/30/202434JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jiang / Information Sch
40、ool为分析方便,以电流为参考正弦量。即把计时起点选在fi = 0的时刻:电压与电流之间的相位差j j = = f fu - -f fi= = f fu1. 瞬时功率瞬时功率pp = = u i = =只含无只含无源元件源元件的一端的一端口口 (N)+-uii = = I cosw wt2u = = U cos(w wt+ +j j)2I cosw wt2U cos(w wt+ +j j)2w wtou ,i j jiu因 u、i采用关联的参考方向,故一端口吸收的瞬时功率为:9.4 正弦稳态电路的功率正弦稳态电路的功率7/30/202435JiangSu University Of Scien
41、ce and Technology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jiang / Information School第一项为恒定量。第二项仍为正弦量,但频率是电压或电流的两倍。p = = u i = =I cosw wt2U cos(w wt+ +j j)2= = UIcosj j+ + UIcos(2w wt+ +j j)UI cosj jw wtou ,i , pj jiu pUIcos(2w wt+ +j j)瞬时功率式还可以改写成p = =UIcosj j (1+(1+cos2w wt) )- -UIsin
42、j j sin2w wt第1项始终0为不可逆部分。第2项为两倍电压或电流频率的正弦量,是瞬时功率的可逆部分。9.4 正弦稳态电路的功率正弦稳态电路的功率7/30/202436JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jiang / Information School不可逆部分是一端口内部所有电阻消耗的功率。p = =UIcosj j (1+ (1+ cos2w wt) )- - UIsinj j sin2w wt只含无只含无源元件源元件的
43、一端的一端口口 (N)+-uip0,表示电路吸收功率,不可逆部分不可逆部分可逆部分可逆部分可逆部分正负交替,说明一端口与电源之间有能量交换情况。p0,表示电路发出功率。 因为没有必要研究电路中每时每刻的功率情况,而且瞬时功率也不便于测量。所以瞬时功率的实际意义不大。9.4 正弦稳态电路的功率正弦稳态电路的功率7/30/202437JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jiang / Information School2. 有功功率P
44、和 功率因数cosj (或用l表示)为便于测量,通常采用平均功率P的概念。P为瞬时功率在一个周期内的平均值,即:P =P = UIcosj jP 不仅与电压和电流有效值的乘积有关,而且还与只含无只含无源元件源元件的一端的一端口口 (N)+-ui积分结果是T10Tp dt一般有 0|cosj j | 1。cosj j =1,表示一端口的等效cosj j =0,表示一P实际上是电阻消耗的功率,亦称为有功功率。 阻抗为纯电阻,P = UI 达到最大; 端口的等效阻抗为纯电抗,P = 0。它们之间的相位差有关!cosj j 称为功率因数。9.4 正弦稳态电路的功率正弦稳态电路的功率7/30/20243
45、8JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jiang / Information School3. 无功功率Q 和视在功率 S由于Q并非一端口实际消耗的功率,故称无功功率。Q与瞬时功率的可逆部分有关,反映了一端口与外电路之间进行能量交换的规模(最大值)。为便于区分,Q的单位用Var(乏)。视在功率反映含源一端口的做功能力。也称表观功率。发电机、变压器等许多电力设备的容量就用S表示。S的单位是VA(伏安)。QdefUIsinj jSdefU
46、 I 无源无源一端一端口口N0+-ui有源有源一端一端口口NS9.4 正弦稳态电路的功率正弦稳态电路的功率7/30/202439JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jiang / Information School4. S、P、Q、j 之间的关系通过上述分析可知:S = UIP = UIcosj j = Scosj j Q = UIsinj j = Ssinj j功率三角形功率三角形S = =P2 + Q2j j = = arctg
47、PQj jSPQRX|Z|UURUX9.4 正弦稳态电路的功率正弦稳态电路的功率7/30/202440JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jiang / Information School5. 单一R、L、C元件时的功率情况(1) R :u与i 同相,j j =0QR= =UIsinj j = = 0i = = I cosw wt2u = = I cosw wt2w wtou ,i , piup= =ui= =UI(1+ +cos2
48、w wt)0总有 p0,说明R一直在吸收功率。PR= =UIcosj j= =UI = = I2R= =U2 GUIR+ +- -ui9.4 正弦稳态电路的功率正弦稳态电路的功率7/30/202441JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jiang / Information School(2) L:u 超前 i 90o , 即j j =90ow wtou , i , pQ=UIi = = I cosw wt2u = = I cos(w
49、 wt+ +9090o o) )2p= =ui= = - -UIsin2w wtP= =UIcosj j = = 0,不耗能。 p交替变化,说明L对外有能量交换,其规模为: L+ +- -uiQL= =UIsin90o = =UI= = w wLI I= =w wLU2= = I2w wL工程上认为,L吸收无功功率。 放 吸 放 吸= = I2XL9.4 正弦稳态电路的功率正弦稳态电路的功率7/30/202442JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit CourseLectured By
50、Xuebin Jiang / Information School(3) C:u 滞后i 90o,即j j = -90o= = - -= = - -U2w wCi = = I cosw wt2u = = I cos(w wt- -9090o o) )2p= =ui= = UIsin2w wtP=UIcosj j = 0,不耗能。 p交替变化,说明C对外有能量交换,其规模为: QC= =UIsin(- -90o) = -= -UIw wC1I2工程上认为,C放出无功功率(以示与L的区别)。 C+ +- -uiw wtou , i , p吸 放 吸 放Q=UI= = I2XC9.4 正弦稳态电路的
51、功率正弦稳态电路的功率7/30/202443JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jiang / Information School例题:电感参数R、L的测量电路如图。根据测出的数据求R、L。解: 功率表的读数就是w wL = =502 - - 302= = 40W W L = =w w40= =31440= = 127mH WAV*+ +- - .US .I50V1A30Wf= =50HzRL电感线圈电阻 R 消耗的或者P= = U
52、I cosj j = =30 Wcosj j = =PUI=0.6=0.6j j = 53.1= 53.1o oZ = =5053.153.1o o= =30+ +j40 W W得 R=30W W,w wL= 40。 有功功率。由P= I2R =30 W得 R=30电压有效值与电流有效值之比即为电感线圈的阻抗(模)。|Z|=UI= 509.4 正弦稳态电路的功率正弦稳态电路的功率7/30/202444JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin
53、 Jiang / Information School正弦电流电路的有功功率、无功功率和视在功率1. 定义三者之间的关系可以通过“复功率”表述。 .U= =Uf fu .I = = If fi,则:则:del .U .I*= = UIf fu- -f fi= = Sj j把 P、Q、S 联系在一起,它的实部是平均功率;虚部是无功功率,模是视在功率;辐角是功率因数角。SS设一端口的= =UIcosj j + jUIsinj j = = P+ +jQ9.5 复功率复功率7/30/202445JiangSu University Of Science and Technology. Zhangjia
54、gang Campus. Circuit CourseLectured ByXuebin Jiang / Information School2. 复功率也可表示为 它作为一个复数,只用于本身不代表正弦量, .U .I* .I ) .I* = (= (Z = (= (R + + jX) ) I2 = = RI2 + + jXI2 = = ZI2P + + jQ或 .U .I* S = =S = =S辅助计算功率。 电路也满足复功率守恒 S = 0= 0 P = 0= 0Q = 0= 0 .U= =( (Y .U )* = = YU29.5 复功率复功率7/30/202446JiangSu Un
55、iversity Of Science and Technology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jiang / Information SchoolP239 例9-10 :已知 U=380V,f =50Hz,cosj j1=0.6,P1=20kW欲使cosj j=0.9,求补偿电容C。Rjw wLjw wC1+ +- - .I .U .I1 .ICQ1= P1tg j j1j j1 1S1Q1P1= 20tg53.13o= = 26.67 kvarRL支路的复功率为S1= =20+ +j26.67 kVA并联C后
56、 不会改变 S1; 不会改变 P1。解法1:并联C前cosj j1= 0.6j j1=53.13oRjw wLjw wC1+ +- - .I .U .I1 .IC根据功率三角形得= =P1+ + jQ19.5 复功率复功率7/30/202447JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jiang / Information School若 C的复功率为Rjw wLjw wC1+ +- - .I .U .I1 .IC= P1+ j(Q1+ Q
57、C)因 要求把cosj j 提高到 0.9故 j j =25.84oSC = 0 + jQC则 S = S1+ SC= =9.69 kvar因此补偿后总无功功率需要C补偿的无功功率为QC = =9.69 - - Q1QC = -16.98 或 -36.36 kvarQ = = Q1+ + QC= = P1tgj j= =9.69- -26.67 kvar由QC = -w wCU2 得C =QC-w wU2=19.681033143802= 374.49 mF取较小(绝对)值,节省成本。9.5 复功率复功率7/30/202448JiangSu University Of Science and
58、Technology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jiang / Information School解法2:借助相量图。需要的补偿电流 IC 应为IC =w wCU 代入上式并整理得 .UI1= = P1Ucosj jZ1I = = P1Ucosj jZC = = w wU2 P1(tgj jZ1 - - tgj jZ) )代入已知数据可求得 C=375mFRjw wLjw wC1+ +- - .I .U .I1 .IC.I1j jZ1j jZ .I接C前I1的无功电流分量为I1sinj jZ1接C后I1的无功电
59、流分量变为Isinj jZIC = I1sinj jZ1 - Isinj jZ将.ICIcosj jZ = = I1cosj jZ19.5 复功率复功率解法3:参考教材7/30/202449JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jiang / Information School提高功率因数的意义接C前,I = I1提高了电源设备的利用率;降低了传输线路上的损耗。I1= P1U cosj1=201033800.6= 87.72AI =
60、= P1Ucosj j= =201033800.9= 58.48A= 58.48A接C后P1不变,I1仍为87.72A。但电源提供的电流 I 减小了:Rjw wLjw wC1+ +- - .I .U .I1 .ICRjw wLjw wC1+ +- - .I .U .I1 .IC9.5 复功率复功率7/30/202450JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jiang / Information SchoolNSZ+ +- - .U .I
61、 .IZ+ +- - .UZeq+ +- - .UocZ=?能获得最大功率?Pmax=?设:Z=R + jX,Zeq=Req+ jXeq则负载吸收的功率为:P= = I2R = =|Z+ +Zeq|Uoc2R2= = (R + + Req)2+ + (X + + Xeq)2R Uoc29.6 最大功率传输最大功率传输7/30/202451JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jiang / Information School9.6 最大功率传输最大功率传输若Z无约束,则若为诺顿等效电路,则条件是 Y=Yeq*。X +Xeq=0时可获得最大功率。dPdR=0解得:Z=Req- jXeq =Z*eqPmax=4ReqUoc2共轭匹配 .IZ+ +- - .UZeq+ +- - .Uoc7/30/202452