第9章 正弦穩(wěn)態(tài)電路分析

1、JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jiang / Information SchoolTuesday, May 03, 20221o 9.1 阻抗和導(dǎo)納o 9.2 電路的相量圖o 9.3 正弦穩(wěn)態(tài)電路的分析o 9.4 正弦穩(wěn)態(tài)電路的功率o 9.5 復(fù)功率o 9.6 最大功率傳輸JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit Cour

2、seLectured ByXuebin Jiang / Information SchoolTuesday, May 03, 20222o 熟練掌握阻抗和導(dǎo)納的物理意義并了解它們之間等效變換的概念；o 熟練地運用相量法分析正弦電流電路；o 掌握正弦電流電路中的平均功率，無功功率、視在功率及功率因數(shù)的概念；o 掌握最大功率傳輸條件。 直流電路的分析 + 相量法基礎(chǔ) 正弦穩(wěn)態(tài)電路的分析方法，在第10、11、12章節(jié)中都要用到。本章與其它章節(jié)的聯(lián)系JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit Co

3、urseLectured ByXuebin Jiang / Information SchoolTuesday, May 03, 202231. 阻抗阻抗 Z (1) 定義 Fj jz就是該阻抗兩端的電壓與通過該阻抗電流的相位差j j ！ .I含線性無源元件的一端口N0+- .U設(shè)：設(shè)： .U= = Uf fu .I = = If fi則：則：Zdef .U .I= =UIf fu- -f fi= = | Z |j jz| Z | = UI為阻抗的模，也可以簡稱為阻抗。j jz = =f fu- -f fi為阻抗角。阻抗的單位與電阻相同。JiangSu University Of Scienc

4、e and Technology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jiang / Information SchoolTuesday, May 03, 20224R2 + + X2(2) 阻抗參數(shù)間的關(guān)系指數(shù)式：Z=| Z | e jj j z代數(shù)式：Z =| Z |cosj jz + j| Z |sinj jzZ = = | Z |j jzZ = = R + + j X實部R稱為電阻，虛部X稱為電抗。Z+ +- - .U .IN0R = = |Z|cosj jzX = = |Z|sinj jzj jz| Z |RX|

5、Z|、R、X構(gòu)成的直角三角形稱為阻抗三角形。極坐標式：|Z| = =j jz = = arctgRXJiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jiang / Information SchoolTuesday, May 03, 20225(3) 單個元件的阻抗 R+ +- - .U .IN0LN0+ +- - .U .ICN0+ +- - .U .I說明 Z 可以是純實數(shù)，也可以是純虛數(shù)。 Z = = .U .I= = RZ = = .U

6、.I= = jw wL = = j XL純電阻純電感XL=L 稱感性電抗， XL f ！純電容Z = = .U .I= =jw wC1= =w wC1- -j= = j XCXC = = - -w wC1稱容性電抗， XC (1/f ) ！JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jiang / Information SchoolTuesday, May 03, 20226+ + jw wL(4) RLC串聯(lián)電路根據(jù)KVL和VCR的相量形

7、式可得： .U= = w wL- -w wC1+ +- -+ +- -Rjw wL+ +- - .UR .UL .UCjw wC1+ +- - .U .IN0= = R .I .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 = = arctgRXJiangSu University Of Science and Technology. Zha

8、ngjiagang Campus. Circuit CourseLectured ByXuebin Jiang / Information SchoolTuesday, May 03, 20227當 w wL結(jié)論：表現(xiàn)為電壓超前電流，Z 呈感性，稱電路為感性電路。w wC1時，有 X0 ，j jz0以電流為參考相量相量圖以電流為參考相量相量圖 .I.UR.UC.UL .Uj jzJiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jiang / In

9、formation SchoolTuesday, May 03, 20228當 w wL表現(xiàn)為電壓滯后電流，Z 呈容性，稱電路為容性電路。w wC1時，有 X0 ，j jz0。 .I.UR.UC.ULj jz結(jié)論：以電流為參考相量相量圖以電流為參考相量相量圖 JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jiang / Information SchoolTuesday, May 03, 20229當 w wL =表現(xiàn)為電壓與電流同相位，電路發(fā)

10、生了串聯(lián)諧振，Z 呈純電阻性。w wC1時，有 X = 0 ，j jz = 0。 .I.UR.UC.UL .U = =從相量圖可以看出，正弦交流RLC串聯(lián)電路中，會出現(xiàn)分電壓大于總電壓的現(xiàn)象。 以電流為參考相量相量圖以電流為參考相量相量圖 結(jié)論：JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jiang / Information SchoolTuesday, May 03, 202210當R=0，X 0時，Z 為純電感性；FRLC 串聯(lián)電路的電

11、壓 UR、UX、U 構(gòu)成電壓三角形。滿足：U =UR + UX22 .I.UR .Uj jz.UX|Z|XR當R=0，X0時，Z 為純電容性。結(jié)論：JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jiang / Information SchoolTuesday, May 03, 2022112. 導(dǎo)納導(dǎo)納 Y(1) 阻抗阻抗Z的倒數(shù)定義為導(dǎo)納的倒數(shù)定義為導(dǎo)納Y， 即：即：Y = 1Z .I= = |Y|j jY單位是SY = =f fi- -f

12、 fu .U= =IU也可以簡稱為導(dǎo)納。j jY = =f fi- -f fu稱為導(dǎo)納角。|Y| = = 導(dǎo)納的代數(shù)形式為： Y = G + j B實部G稱為電導(dǎo)，虛部B稱為電納。稱為導(dǎo)納模，IUJiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jiang / Information SchoolTuesday, May 03, 202212G、B、|Y|、j jY 之間的關(guān)系為G= =|Y|cosj jYB= =|Y|sinj jY|Y| = =

13、G2 + + B2j jY = = arctgGBj jY| Y |GB導(dǎo)納三角形導(dǎo)納三角形(2) 單個單個R、L、C 元件的導(dǎo)納元件的導(dǎo)納 當無源網(wǎng)絡(luò)內(nèi)為單個元件時，等效導(dǎo)納分別為 ：Y+ +- - .U .IN0JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jiang / Information SchoolTuesday, May 03, 202213稱為感性電納；Y = .U .I純電阻=R1= G純電感Y = .U .I=jw wL1

14、= jBLBL = w wL1純電容Y = .U .I= jw wC = jBCBC = w wC 稱為容性電納；Y 可以是純實數(shù)，也可以是純虛數(shù)。 稱為電導(dǎo)；Y+ +- - .U .IN0JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jiang / Information SchoolTuesday, May 03, 202214(3) RLC并聯(lián)電路 .I3jw wLjw wC1 .I2 .I1R .U+ +- - .I根據(jù)VCR和KCL的

15、相量形式可得： .I= = G .U .Ujw wL1+ + + jw wC .U .U .U = = Y .U= =w wL1+ + jw wC .U= = G + + j(BL+ + BC)= (= (G + + jB) )G - - j= = - -w wL1B = = BL + + BC+ +w wCj jY = = arctgGBJiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jiang / Information SchoolTues

16、day, May 03, 202215結(jié)論： 對于 RLC 并聯(lián)電路 B0或j jY 0，稱Y為感性； B0或j jY 0，稱Y為容性； B=0或j jY =0，Y為純電阻性； G=0，B0，Y為純電感性； G=0，B0，Y為純電容性。以電壓為參考相量相量圖以電壓為參考相量相量圖 .U.I1.I2.I3 .Ij jY.I3+ +.I2JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jiang / Information SchoolTuesday

17、, May 03, 202216 .U.I1.I2.I3 .I= =從相量圖可以看出，正弦交流RLC并聯(lián)電路中，會出現(xiàn)分電流大于總電流的現(xiàn)象。 B=0、j jY =0，時的相量圖JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jiang / Information SchoolTuesday, May 03, 2022173. 阻抗與導(dǎo)納的相互等效F一端口的阻抗和導(dǎo)納可以互換，等效互換的條件為： .I含線性無源元件的一端口N0+ +- - .UN

18、0的等效阻抗(導(dǎo)納)、輸入阻抗(導(dǎo)納)或驅(qū)動點阻抗(導(dǎo)納)，它們的實部和虛部都是外施正弦激勵的角頻率 的函數(shù)：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分開寫 | Z | Y |= =1j jZ + + j jY = =0JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jiang / Information SchoolTuesd

19、ay, May 03, 202218 若已知 Z=R+jX ，求等效的 Y=G+jB 。 若已知 Y = G + jB則： .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則：R = =|Y|2 GX = -= -|Y|2 B等效成 Z = R + jXJiangSu University Of Science and Technology. Zha

20、ngjiagang Campus. Circuit CourseLectured ByXuebin Jiang / Information SchoolTuesday, May 03, 202219例：電路如圖，求各支路電流和解：設(shè) 串聯(lián)支路阻抗為Z1 .U10 。Z1并聯(lián)支路導(dǎo)納為Y10Y10 = =+ + jw wC= =10- -3 + + j3.1410- -3= =3.295410 - -372.33o S+ +- -R1 .I .U10jw wC1+ +- - .UsR2jw wL01 .I2 .I110W W0.5H10m m1k100Vw w = = 314rad/s則 Z1=

21、10 + j157 R21Z10 = =Y101= = 303.45- -72.33o= = 92.11- - j289.13 W WJiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jiang / Information SchoolTuesday, May 03, 202220Zeq= = Z1+ +Z10- -52.30o= = (92.11+ +10) + + j(157- -289.13)= = 102.11 - - j132.13= =

22、166.99W W .I = =Zeq .US= =166.99- -52.30o100= = 0.652.30oA- -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- -20.03oAJiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit

23、 CourseLectured ByXuebin Jiang / Information SchoolTuesday, May 03, 202221F 相量作為一個復(fù)數(shù)，可以用復(fù)平面上的有向線段來表示。F 按照大小和相位關(guān)系，用初始位置的有向線段畫出的若干個相量的圖形，稱為相量圖。F 因相量圖能直觀地反映各相量之間的關(guān)系，所以借助于相量圖對電路進行輔助分析和計算，有時能起到“事半功倍”的效果。JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jia

24、ng / Information SchoolTuesday, May 03, 202222相量圖的定性畫法選一參考相量，通常是某一并聯(lián)部分的電壓，習慣上把它畫在水平方向。由VCR確定并聯(lián)支路電流的相量由KCL確定結(jié)點電流相量；對串聯(lián)部分，以電流相量為參考由VCR確定有關(guān)電壓相量由KVL確定回路上各電壓相量。繪制時，可以用平移求和法則，使各相量（有關(guān)結(jié)點電流相量、回路電壓相量等）構(gòu)成若干個封閉的多邊形。也可以使各相量都從原點向外輻射，用平行四邊形法則求和。 一般是根據(jù)需要，結(jié)合上述兩種方式，畫成便于分析計算的形狀。JiangSu University Of Science and Techno

25、logy. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jiang / Information SchoolTuesday, May 03, 202223當需要借助相量圖進行分析計算時，右圖選并聯(lián)部分電壓為參考相量比較方便。定性繪制過程：+ +- -R1 .I .U10jw wC1+ +- - .USR2jw wL01 .I2 .I1 .U10VCR .I1 .I2KCL .IVCR .I R1 jw wL .IKVL .US繪制時應(yīng)根據(jù)已知條件，使圖形大致符合比例。 .U10 .U10 .I2 .I1 .I .I R1 jw wL

26、 .I .USJiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jiang / Information SchoolTuesday, May 03, 202224例題：I、R、XC、XL 。求：解：選為參考相量 .Uab .I1 超前 .Uab 90o .I2 與 .Uab 同相由KCLI = = = 14.14 A超前 .I 90o 由KVL知：+ +- - .I .Uab+ +- - .URjXL .I2 .I1ab- -jXC100V10A1

27、0A .U 與 .I 同相。 .Uab .I = .I1+ .I2I12jXLI.+ + I22 .U = jXLI .+ Uab. .U .I .I1 .I2jXLI.jXLI.45o45oJiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jiang / Information SchoolTuesday, May 03, 202225XLI = U = 100VXL =UI= 7.07 Uab =U= 141.4VR =UabI2= 14.14

28、 XC =UabI1= 14.14 若給定，還能進一步算出L和C。=14.141002I = = 14.14 A .Uab .U .I .I1 .I2jXLI.jXLI.45o45oJiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jiang / Information SchoolTuesday, May 03, 202226本節(jié)的核心內(nèi)容：將電阻電路的各種分析方法，推廣到正弦穩(wěn)態(tài)電路中。電阻電路中的很多方法和定理，都以兩類約束為基礎(chǔ)，即： KC

29、L、KVL和VCR。KCLi = 0I = 0.KVLu = 0VCRu = Ri在引入相量和復(fù)阻抗的概念以后，兩類約束的相量表達式與時域表達式具有相同的形式：U = 0.U = Z I.JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jiang / Information SchoolTuesday, May 03, 202227F推廣時作如下變換： 以兩類約束為基礎(chǔ)的各種計算方法和定理必然也具有相同的形式。 所以電阻電路的各種分析方法和定理就

30、能推廣到正弦穩(wěn)態(tài)電路中來。 例如：Req的定義與求法，可以推廣成 Zeq的定義與求法；i .Iu .URZGY電阻電路 正弦穩(wěn)態(tài)電路JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jiang / Information SchoolTuesday, May 03, 202228F 相量法把描述動態(tài)電路的微分方程變?yōu)閺?fù)數(shù)的代數(shù)方程。與求微分方程的特解(正弦穩(wěn)態(tài)解)相比，使計算簡化，書寫方便，物理概念也更加突出。F 由于描述的物理過程不同，所以方程為

31、相量形式，計算為復(fù)數(shù)運算。F 因為 p = ui 是非正弦量，所以，功率的計算要單獨考慮(后述)。推廣后的差別JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jiang / Information SchoolTuesday, May 03, 202229解題指導(dǎo)解題指導(dǎo)解：用觀察法列結(jié)點電壓方程+ +- - .IS5 .US1Z1 .US3+ +- -Z2Z3Z4Z5(Y1+ +Y2+ +Y3) .Un1- - Y3 .Un2= = Y1 .U

32、S1+ + Y3 .US3- - Y3 .Un1+ + (Y3+ +Y4) .Un2= = - -Y3 .US3 .+ + IS5 .Il1 .Il2 .Il3用觀察法列回路電流方程l1(Z1+ +Z2) .Il1- -Z2 .Il2 .US1+ +- - .U例1：激勵均為同頻率正弦量。試列出該電路的結(jié)點電壓方程和回路電流方程。= =l2- -Z2 .Il1+ + (Z2+ +Z3 + +Z4) .Il2 - -Z4 .Il3 = = .- -US3l3 .Il3 = = .- - IS5JiangSu University Of Science and Technology. Zhangj

33、iagang Campus. Circuit CourseLectured ByXuebin Jiang / Information SchoolTuesday, May 03, 202230例2：求右圖一端口的戴維寧等效電路。 ao .US1+ +- -Z1Z2 .I2 .IS3+ +- - .rI211 .Uoc+ +- - .Uoc= = - -r . I2+ .Uao . I2 = = Y2 .Uao解：求開路電壓 .Uoc+ +- -Zeq11 .Uoc= = - -rY2 .Uao+ + .Uao= = (1- -rY2) .Uao .Uao = =Y1 + + Y2 Y1 .US

34、1 . - - IS3用結(jié)點法求出 .Uoc = =(1- -rY2)(Y1 .US1 . - - IS3)Y1 + + Y2 代入上式得JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jiang / Information SchoolTuesday, May 03, 202231 . I2 = =Z1+ +Z2Z1 . I . I = =Z1Z1+ +Z2 . I2= =(1+ +Y1Z2 ) . U = = - -r . I2+ + Z2

35、. I2= = (Z2- -r) . I2Zeq = = . U . I= =Z2- -r1+ +Y1Z2aoZ1Z2 .I2+ +- - .rI211 .U+ +- - .I再求等效阻抗 . I2 .Uoc+ +- -Zeq11JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jiang / Information SchoolTuesday, May 03, 202232I1 =例3：US =380V，f =50Hz，C為可變電容，當C=80.

36、95F時，表A讀數(shù)最小為2.59A。求：表A1的讀數(shù)。解法1：借助相量圖求解選 .USIC = 2fCUS= 9.66A .I = .I1+ .IC調(diào)C，IC變。但： .I1 不變， .IC 始終與 .US 正交。 .US .I1 .I .I .US9.662+ 2.592= 10 AR1 .Ijw wC1+ +- - .USjw wL1 .I1 .ICAA1為參考相量始終構(gòu)成封閉三角形。當與同相時最小。 .IC .US .I1 .IC .I .US .I1 .IC .IJiangSu University Of Science and Technology. Zhangjiagang Cam

37、pus. Circuit CourseLectured ByXuebin Jiang / Information SchoolTuesday, May 03, 202233解法2：電路的輸入導(dǎo)納為Y = jC+|Z1|2R1- j|Z1|2L1R1 .Ijw wC1+ +- - .USjw wL1 .I1 .ICAA1F調(diào)C，只改變ImY。當ImY =0時，| Y |最小，I = | Y |U 也最小。電路呈純電阻性， .US 與 .I 同相。設(shè) .US = 380 0oV則 .I = 2.590oA .IC = jC= j9.66A .US .I1= .I - .IC = 2.59 - j9

38、.66= 10-70o A表A1的讀數(shù)為10 A。JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jiang / Information SchoolTuesday, May 03, 202234實踐中，可以用這種電路測量一個電感線圈的參數(shù)。由以上(測得的)數(shù)據(jù)算出電感線圈電感線圈R1=13 W W，L1= 113.7mH35.71w wR1 .Ijw wC1+ +- - .USjw wL1 .I1 .ICAA1 .I1= 10-70o AZ1

39、= .Us .I1 = 3870o= 13 + j 35.71 W WJiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jiang / Information SchoolTuesday, May 03, 202235為分析方便，以電流為參考正弦量。即把計時起點選在fi = 0的時刻：電壓與電流之間的相位差j j = = f fu - -f fi= = f fu1. 瞬時功率瞬時功率pp = = u i = =只含無只含無源元件源元件的一端的一端口

40、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采用關(guān)聯(lián)的參考方向，故一端口吸收的瞬時功率為：JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jiang / Information SchoolTuesday, May 03, 202236第一項為恒定量。第二項仍為正弦量，但頻率是電壓或電流的兩倍。p

41、= = 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)瞬時功率式還可以改寫成p = =UIcosj j (1+(1+cos2w wt) )- -UIsinj j sin2w wt第1項始終0為不可逆部分。第2項為兩倍電壓或電流頻率的正弦量，是瞬時功率的可逆部分。JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit C

42、ourseLectured ByXuebin Jiang / Information SchoolTuesday, May 03, 202237F 不可逆部分是一端口內(nèi)部所有電阻消耗的功率。p = =UIcosj j (1+ (1+ cos2w wt) )- - UIsinj j sin2w wt只含無只含無源元件源元件的一端的一端口口 (N)+-uip0，表示電路吸收功率，不可逆部分不可逆部分可逆部分可逆部分F 可逆部分正負交替，說明一端口與電源之間有能量交換情況。p0，表示電路發(fā)出功率。 因為沒有必要研究電路中每時每刻的功率情況，而且瞬時功率也不便于測量。所以瞬時功率的實際意義不大。Jia

43、ngSu University Of Science and Technology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jiang / Information SchoolTuesday, May 03, 2022382. 有功功率P 和 功率因數(shù)cosj (或用l表示)F 為便于測量，通常采用平均功率P的概念。P為瞬時功率在一個周期內(nèi)的平均值，即：P =P = UIcosj jFP 不僅與電壓和電流有效值的乘積有關(guān)，而且還與只含無只含無源元件源元件的一端的一端口口 (N)+-ui積分結(jié)果是T10Tp dt一般有 0|co

44、sj j | 1。cosj j =1，表示一端口的等效cosj j =0，表示一P實際上是電阻消耗的功率，亦稱為有功功率。 阻抗為純電阻，P = UI 達到最大； 端口的等效阻抗為純電抗，P = 0。它們之間的相位差有關(guān)！cosj j 稱為功率因數(shù)。JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jiang / Information SchoolTuesday, May 03, 2022393. 無功功率Q 和視在功率 S由于Q并非一端口實際消

45、耗的功率，故稱無功功率。Q與瞬時功率的可逆部分有關(guān)，反映了一端口與外電路之間進行能量交換的規(guī)模(最大值)。為便于區(qū)分，Q的單位用Var(乏)。視在功率反映含源一端口的做功能力。也稱表觀功率。發(fā)電機、變壓器等許多電力設(shè)備的容量就用S表示。S的單位是VA(伏安)。QdefUIsinj jSdefU I 無源無源一端一端口口N0+-ui有源有源一端一端口口NSJiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jiang / Information Sch

46、oolTuesday, May 03, 2022404. S、P、Q、j 之間的關(guān)系通過上述分析可知：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 = = arctgPQj jSPQRX|Z|UURUXJiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jiang / Information SchoolTuesday, May 03, 202

47、2415. 單一R、L、C元件時的功率情況(1) R ：u與i 同相，j j =0QR= =UIsinj j = = 0i = = I cosw wt2u = = Ucosw wt2w wtou ,i , piup= =ui= =UI(1+ +cos2w wt)0總有 p0，說明R一直在吸收功率。PR= =UIcosj j= =UI = = I2R= =U2 GUIR+ +- -uiJiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jiang /

48、Information SchoolTuesday, May 03, 202242(2) L：u 超前 i 90o , 即j j =90ow wtou , i , pQ=UIi = = I cosw wt2u = = Ucos(w wt+ +9090o o) )2p= =ui= = - -UIsin2w wtP= =UIcosj j = = 0，不耗能。 p交替變化，說明L對外有能量交換，其規(guī)模為： L+ +- -uiQL= =UIsin90o = =UI= = w wLI I= =w wLU2= = I2w wL工程上認為，L吸收無功功率。 放 吸 放 吸= = I2XLJiangSu Un

49、iversity Of Science and Technology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jiang / Information SchoolTuesday, May 03, 202243(3) C：u 滯后i 90o，即j j = -90o= = - -= = - -U2w wCi = = I cosw wt2u = = Ucos(w wt- -9090o o) )2p= =ui= = UIsin2w wtP=UIcosj j = 0，不耗能。 p交替變化，說明C對外有能量交換，其規(guī)模為： QC= =UI

50、sin(- -90o) = -= -UIw wC1I2工程上認為，C放出無功功率(以示與L的區(qū)別)。 C+ +- -uiw wtou , i , p吸 放 吸 放Q=UI= = I2XCJiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jiang / Information SchoolTuesday, May 03, 202244例題：電感參數(shù)R、L的測量電路如圖。根據(jù)測出的數(shù)據(jù)求R、L。解： 功率表的讀數(shù)就是w wL = =502 - - 30

51、2= = 40W W L = =w w40= =31440= = 127mH WAV*+ +- - .US .I50V1A30Wf= =50HzRL電感線圈電阻 R 消耗的或者P= = UI 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電壓有效值與電流有效值之比即為電感線圈的阻抗(模)。|Z|=UI= 50JiangSu University Of Science and Tec

52、hnology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jiang / Information SchoolTuesday, May 03, 202245F 正弦電流電路的有功功率、無功功率和視在功率1. 定義三者之間的關(guān)系可以通過“復(fù)功率”表述。 .U= =Uf fu .I = = If fi，則：則：del .U .I*= = UIf fu- -f fi= = Sj j把 P、Q、S 聯(lián)系在一起，它的實部是平均功率；虛部是無功功率，模是視在功率；輻角是功率因數(shù)角。SS設(shè)一端口的= =UIcosj j + jUIsinj j

53、 = = P+ +jQJiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jiang / Information SchoolTuesday, May 03, 2022462. 復(fù)功率也可表示為 它作為一個復(fù)數(shù)，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輔助計算功率。 電路也

54、滿足復(fù)功率守恒 S = 0= 0 P = 0= 0Q = 0= 0 .U= =( (Y .U )* = = YU2JiangSu University Of Science and Technology. Zhangjiagang Campus. Circuit CourseLectured ByXuebin Jiang / Information SchoolTuesday, May 03, 202247P239 例9-10 ：已知 U=380V，f =50Hz，cosj j1=0.6，P1=20kW欲使cosj j=0.9，求補償電容C。Rjw wLjw wC1+ +- - .I .U .I1 .ICQ1= P1tg j j1j j1 1S1Q1P1= 20tg53.13o= = 26.67 kvarRL支路的復(fù)功率為S1= =20+ +j26.67 kVA并聯(lián)C后 不會改變 S1； 不會改變 P1。解法1：并聯(lián)C前cosj j1= 0.6j j1=53.13oRjw wLjw wC1+ +- - .I .U .I1 .IC根據(jù)功率三角形得= =P1+ +