正弦交流電的分析-電力牽引交流傳動系統(tǒng)

1、Unit4 Analysis of Sinusoidal Alternating Electricity 正弦交流電的分析正弦交流電的分析nR.M.S. (Effective) Values of Current and Voltage 電壓和電流的有效值電壓和電流的有效值nThe force between two current-carrying conductors is proportional to the square of the current in the conductors. The heat due to a current in a resistance over a

2、 period is also proportional to the square of that current.n兩載流導體之間的作用力與導體中的電流的平方成正比。某段時間兩載流導體之間的作用力與導體中的電流的平方成正比。某段時間內(nèi)電流通過一個電阻所產(chǎn)生的熱量也正比于電流的平方。內(nèi)電流通過一個電阻所產(chǎn)生的熱量也正比于電流的平方。New Words & Expressions:sinusoidal alternating electricity 正弦交流電正弦交流電effective values 有效值有效值r.m.s. values = root mean square val

3、ues 均方根值均方根值square平方平方nThis calls for knowledge of what is known as the root mean square (or effective) current defined as (Eq.1)nThe heat developed by a current i in a resistance r in time dt is (Eq.)n這便引出通常所說的均方根（或有效值）電流的概念，其定義如下：這便引出通常所說的均方根（或有效值）電流的概念，其定義如下：(Eq.1)n在在dt時間里電流時間里電流i通過電阻通過電阻r產(chǎn)生的熱量為產(chǎn)

4、生的熱量為(Eq.)nIt follows that the r.m.s. (effective) value of an alternating current is numerically equal to the magnitude of the steady direct current that would produce the same heating effect in the same resistance and over the same period of time.n句型句型It follows that 譯為譯為“由此得出由此得出”。賓語從句里面含有一個定。賓語從句

5、里面含有一個定語從句。語從句。n由此可得出，交流電的均方根（或有效）值等于在相同電阻、相同時由此可得出，交流電的均方根（或有效）值等于在相同電阻、相同時間內(nèi)產(chǎn)生相同熱量的恒穩(wěn)直流電的大小。間內(nèi)產(chǎn)生相同熱量的恒穩(wěn)直流電的大小。New Words & Expressions:steady direct current 恒穩(wěn)直流電恒穩(wěn)直流電nLet us establish the relationship between the r.m.s. and peak values of a sinusoidal current, I and ImnHence :(Eq.2)nThe r.m.s.

6、 (effective) values of e.m.f. and voltage are New Words & Expressions:peak values 峰值峰值nIn dealing with periodic voltages and currents, their r.m.s. (effective) value are usually meant, and the adjective “r.m.s.” or “effective” is simply implied.n在涉及交流電壓和電流時，通常指的值就是其均方根（有效）值，便在涉及交流電壓和電流時，通常指的值就是其

7、均方根（有效）值，便將限定詞將限定詞“均方根（有效）均方根（有效）”幾個字略去，并不明指。幾個字略去，并不明指。Representation of Sinusoidal Time Functions by Vectors andComplex Number 正弦時間函數(shù)的矢量和復數(shù)表示法正弦時間函數(shù)的矢量和復數(shù)表示法nA.C. circuit analysis can be greatly simplified if the sinusoidal quantities involved are represented by vectors or complex numbers.nLet the

8、re be a sinusoidal time function (current, voltage, magnetic flux and the like):n如果所涉及的正弦量用矢量和復數(shù)表示，便可大大地簡化交流電路的分如果所涉及的正弦量用矢量和復數(shù)表示，便可大大地簡化交流電路的分析。析。n設一正弦時間函數(shù)（電流、電壓、磁通等）設一正弦時間函數(shù)（電流、電壓、磁通等）New Words & Expressions:sinusoidal time function 正弦時間函數(shù)正弦時間函數(shù) vector 矢量矢量 complex number 復數(shù)復數(shù) sinusoidal quant

9、ity 正弦量正弦量 magnetic flux磁通磁通A.C. circuit=alternating current circuit 交流電路交流電路D.C. circuit=direct current circuit 直流電路直流電路nIt can be represented in vector form as follows. Using a right-hand set of Cartesian coordinates MON (Fig.1), we draw the vetor Vm to some convenient scale such that it represent

10、s the peak value Vm and makes the angle with the horizontal axis OM (positive values of are laid off counter-clockwise, and negative, clockwise).A makes angle with B: A與與B之間成之間成夾角夾角這個正弦時間函數(shù)可用如下的矢量形式表示。通過在笛卡爾坐標系這個正弦時間函數(shù)可用如下的矢量形式表示。通過在笛卡爾坐標系的右側的右側MON（如圖（如圖1所示）區(qū)域內(nèi)，取恰當?shù)谋壤嫵鍪噶克荆﹨^(qū)域內(nèi)，取恰當?shù)谋壤嫵鍪噶縑m，以便于代表該量的

11、幅值以便于代表該量的幅值Vm，并與橫坐標形成，并與橫坐標形成角（逆時針方向為正，角（逆時針方向為正，順時針方向為負）。順時針方向為負）。New Words & Expressions:clockwise 順時針方向順時針方向counter-clockwise 逆時針方向逆時針方向nNow we imagine that, starting at t=0, the vector Vm begins to rotate about the origin O counter-clockwise at a constant angular velocity equal to the angul

12、ar frequency . At time t, the vector makes the angle t+ with the axis OM. Its projection onto the vertical axis NN represents the instantaneous value of v to the scale chose.現(xiàn)在假設從現(xiàn)在假設從t=0開始，矢量開始，矢量Vm繞著原點繞著原點O以等于角頻率以等于角頻率的恒定角速度的恒定角速度逆時針旋轉。則逆時針旋轉。則t時刻矢量與橫坐標軸時刻矢量與橫坐標軸OM形成形成t+的夾角。它在縱軸的夾角。它在縱軸NN上的投影便表示在已

13、選用的比例尺下的瞬時值上的投影便表示在已選用的比例尺下的瞬時值v。New Words & Expressions:constant angular velocity 恒定角速度恒定角速度angular frequency 角頻率角頻率instantaneous value 瞬時值瞬時值nInstantaneous values of v, as projections of the vector on the vertical axis NN, can also be obtained by holding the vector Vm stationary and rotating t

14、he axis NN clockwise at the angular velocity , starting at time t=0. Now the rotating axis NN is called the time axis. 瞬時值瞬時值v（即矢量在縱坐標（即矢量在縱坐標NN上的投影）也能通過以下方法得到：上的投影）也能通過以下方法得到：即令矢量即令矢量Vm不動，將軸不動，將軸NN以角速度以角速度從從t=0開始順時針旋轉，此時開始順時針旋轉，此時旋轉的軸旋轉的軸NN稱為時間軸。稱為時間軸。New Words & Expressions:single-valued relat

15、ionship 單值關系（一一對應關系）單值關系（一一對應關系）vectors of voltages (e.m.f.s, currents, magnetic fluxes）電壓（電勢、電流、磁通）矢量電壓（電勢、電流、磁通）矢量nIn each case, there is a single-valued relationship between the instantaneous value of v and the vector Vm. Hence Vm may be termed the vector of the sinusoidal time function v. Likewi

16、se, there are vectors of voltages, e.m.f.s, currents, magnetic fluxes,etc.n兩種情況下，瞬時值兩種情況下，瞬時值v和矢量和矢量Vm之間都存在單值關系。因此，之間都存在單值關系。因此，Vm便便可稱為正弦時間函數(shù)可稱為正弦時間函數(shù)v的矢量。同理，還有電壓矢量、電勢矢量、電的矢量。同理，還有電壓矢量、電勢矢量、電流矢量、磁通矢量等。流矢量、磁通矢量等。n“True” vector quantities are denoted either by clarendon type, e.g. A, or by , while sin

17、usoidal ones are denoted by . Graphs of sinusoidal vectors, arranged in a proper relationship and to some convenient scale, are called vector diagrams.n真正的矢量是用粗體字真正的矢量是用粗體字A，或，或 表示，而正弦矢量則用表示，而正弦矢量則用 表示。按合表示。按合適的相對關系和某種恰當?shù)谋壤嫵龅恼蚁蛄康膱D解稱為矢量圖。適的相對關系和某種恰當?shù)谋壤嫵龅恼蚁蛄康膱D解稱為矢量圖。New Words & Expressions:e.g

18、. ,i:di: =exempli gratia 例如例如 vector diagrams 矢量圖矢量圖AAAANew Words & Expressions:real quantity 實量實量 imaginary quantity 虛量虛量 complex plane 復平面復平面complex number 復數(shù)復數(shù) absolute value 絕對值絕對值 modulus 模模 phase 相位相位 argument 相角相角 complex peak value 復數(shù)幅值復數(shù)幅值/峰值峰值Taking MM and NN as the axes of real and im

19、aginary quantities, respectively, in a complex plane, the vector Vm can be represented by a complex number whose absolute value (or modulus) is equal to Vm, and whose phase (or argument) is equal to the angle . This complex number is called the complex peak value of a given sinusoidal quantity.在一復數(shù)平

20、面內(nèi)，取在一復數(shù)平面內(nèi)，取MM和和NN分別為實數(shù)軸和虛數(shù)軸，矢量分別為實數(shù)軸和虛數(shù)軸，矢量Vm可用可用一復數(shù)來表示，該復數(shù)的絕對值（即模）等于一復數(shù)來表示，該復數(shù)的絕對值（即模）等于Vm，其相位角等于，其相位角等于。此。此復數(shù)稱為某一已知正弦量的復數(shù)峰值。復數(shù)稱為某一已知正弦量的復數(shù)峰值。nGenerally, a complex vector may be expressed in the following waysnwhere通常，通常，復向量復向量可表示為如下幾種形式：極坐標的、指數(shù)的、三角的、直可表示為如下幾種形式：極坐標的、指數(shù)的、三角的、直角或代數(shù)的角或代數(shù)的nWhen the

21、vector Vm rotates counter-clockwise at angular velocity , starting at t=0, it is said to be a complex time function, defined so that . Now, since this is a complex function it can be expressed in terms of its real and imaginary partsn當矢量當矢量Vm從從t=0開始以角速度開始以角速度逆時針旋轉時，便被稱之為復數(shù)時間函逆時針旋轉時，便被稱之為復數(shù)時間函數(shù)，并定義為數(shù)

22、，并定義為(Eq.)。現(xiàn)在，既然它是一復函數(shù)，則可用實部和虛部來?，F(xiàn)在，既然它是一復函數(shù)，則可用實部和虛部來表示表示:New Words & Expressions:complex time function 復數(shù)時間函數(shù)復數(shù)時間函數(shù) real part 實部實部 imaginary part虛部虛部nWhere the sine term is the imaginary part of the complex variable equal (less j) to the sinusoidal quantity v, ornWhere the symbol Im indicates

23、that only the imaginary part of the function in the square brackets is taken.n其中正弦項是復數(shù)變量（除去其中正弦項是復數(shù)變量（除去j）的虛部，等于正弦量）的虛部，等于正弦量v，即，即n式中符號式中符號Im是指只計及方括號中復數(shù)的虛部。是指只計及方括號中復數(shù)的虛部。余弦函數(shù)的瞬時值由下式給出：余弦函數(shù)的瞬時值由下式給出：式中符號式中符號Re是指只計及方括號中復數(shù)的實部。在這種情況下，瞬時值由是指只計及方括號中復數(shù)的實部。在這種情況下，瞬時值由矢量矢量 在實軸上的投影表示。在實軸上的投影表示。nThe instantan

24、eous value of a cosinusoidal function is given bynWhere the symbol Re indicates that the real part of the complex variable in the square brackets is only taken. For this case, the instantaneous value of v is represented by a projection of the vector onto the real axis.nThe representation of sinusoid

25、al functions in complex form is the basis of the complex-number method of A.C. circuit analysis. In its present form, the method of complex numbers was introduced by Heaviside and Steinmetz.n復數(shù)形式的正弦函數(shù)的表達式是交流電路分析中復數(shù)法的基礎?，F(xiàn)在復數(shù)形式的正弦函數(shù)的表達式是交流電路分析中復數(shù)法的基礎?，F(xiàn)在所用的復數(shù)法的形式是由所用的復數(shù)法的形式是由Heaviside和和Steinmetz提出的。提出的

26、。New Words & Expressions:complex-number method 復數(shù)法復數(shù)法method of complex numbers Addition of Sinusoidal Time Functions 正弦時間函數(shù)的加法正弦時間函數(shù)的加法A.C. circuit analysis involves the addition of harmonic time functions having the same frequencies but different peak values and epoch angles. Direct addition of

27、 such functions would call for unwieldy trigonometric transformations. Simple approaches are provided by the Argand diagram (graphical solution) and by the method of complex numbers (analytical solution).n交流電路的分析包括對有相同頻率、不同幅值和初相角的諧振時間函交流電路的分析包括對有相同頻率、不同幅值和初相角的諧振時間函數(shù)的加法。這些函數(shù)的直接相加將要求用到繁雜的三角轉換。簡單的方數(shù)的加法

28、。這些函數(shù)的直接相加將要求用到繁雜的三角轉換。簡單的方法是采用法是采用Argand圖（圖解法）和復數(shù)法（解析法）圖（圖解法）和復數(shù)法（解析法）New Words & Expressions:harmonic time function 諧振時間函數(shù)諧振時間函數(shù) peak value 幅幅/峰值峰值 epoch angle 初相角初相角 trigonometric transformations 三角轉換三角轉換 analytical solution 解析法解析法nSuppose we are to find the sum of two harmonic functionsnand

29、nFirst, consider the application of the Argand diagram (graphical solution). We lay off the vectors and find the resultant vector .假如我們要求兩個諧振函數(shù)的和：假如我們要求兩個諧振函數(shù)的和：首先，考慮采用首先，考慮采用Argand圖法（作圖法）。我們畫出矢量圖法（作圖法）。我們畫出矢量(Eq.)和和(Eq.)并由平并由平行四邊形法則求出合成矢量行四邊形法則求出合成矢量(Eq.)。resultant vector 合成矢量合成矢量 nNow assume that

30、the vectors begin to rotate about the origin of coordinates, O, at t=0, doing so with a constant angular velocity in the counter-clockwise direction. n現(xiàn)在假設矢量現(xiàn)在假設矢量 在在t=0時刻開始逆時針方向繞著坐標原點時刻開始逆時針方向繞著坐標原點O以以恒定角速度恒定角速度旋轉。旋轉。New Words & Expressions:origin of coordinates 坐標原點坐標原點At any instant of time,

31、a projection of the rotating vector onto the vertical axis NN is equal to the sum of projections onto the same axis of the rotating vectors andor the instantaneous values v1 and v2. in other words, the projection of onto the vertical axis represents the sum (v1+v2),and the vector represents the desi

32、red sinusoidal time function v=v1+v2.在任一時刻，旋轉矢量在任一時刻，旋轉矢量(Eq.)在縱軸在縱軸NN上的投影等于矢量上的投影等于矢量(Eq.)和和(Eq.)在同一坐標軸上的投影之和，或者瞬時值在同一坐標軸上的投影之和，或者瞬時值v1和和v2之和。換句話說，矢之和。換句話說，矢量量(Eq.)在縱坐標上的投影表示瞬時值之和在縱坐標上的投影表示瞬時值之和(v1+v2)，矢量，矢量(Eq.)表示所要表示所要求的正弦時間函數(shù)求的正弦時間函數(shù)(Eq.)。On finding the length of Vm and the angle from the Argand

33、 diagram, we may substitute them in the expression .Now consider the analytical method/solution. Referring to the diagram of Fig. 2, we may write從阿爾岡圖中求出從阿爾岡圖中求出Vm的長度和角度的長度和角度后，可將其表示為：后，可將其表示為：下面討論解析法。對照圖下面討論解析法。對照圖2，我們可寫為：，我們可寫為：In the rectangular (algebraic) form, these complex numbers areOn addin

34、g them together we obtainWhere用直角（代數(shù)）形式，這些復數(shù)是用直角（代數(shù)）形式，這些復數(shù)是將其相加，得將其相加，得式中式中nSince , it is important to know the quadrant where Vm occurs, before we can determine . The quadrant can be readily identified by the signs of the real and imaginary parts of the function. For convenience the epoch angle ma

35、y be expressed in degrees rather than in radians.n由于由于 ，在我們確定，在我們確定之前，知道之前，知道Vm所在的象限是所在的象限是很重要的。通過函數(shù)的實部和虛部的符號能很容易地確定象限。為方很重要的。通過函數(shù)的實部和虛部的符號能很容易地確定象限。為方便起見，用角度而不用弧度來表示初相角便起見，用角度而不用弧度來表示初相角 。New Words & Expressions:quadrant 象限象限nThe two methods are applicable to the addition of any number of sinusoidal functions of the same frequency.n這兩種方法可用于任何數(shù)目的同頻率正弦函數(shù)的疊加。這兩種方法可用于任何數(shù)目的同頻率正弦函數(shù)的疊加。New Words & Expressions:be applicable to (適適)用于用于nIn practical work, one is usually interested in the r.m.s. values and phase displacements of sinusoidal quantiti