正弦交流電的分析課堂PPT

1、.1unit4 analysis of sinusoidal alternating electricity 正弦交流電的分析.2nr.m.s. (effective) values of current and voltage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 period is als

2、o proportional to the square of that current.n兩載流導(dǎo)體之間的作用力與導(dǎo)體中的電流的平方成正比兩載流導(dǎo)體之間的作用力與導(dǎo)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。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。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 values 均方根值均方根值 squa

3、re平方平方.3nthis 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：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)生的熱量為(eq.).4ni

4、t 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。賓語從句里面含有一個定語從句。語從句里面含有一個定語

5、從句。n由此可得出，交流電的均方根（或有效）值等于由此可得出，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。流電的大小。new words & expressions:steady direct current 恒穩(wěn)直流電恒穩(wěn)直流電.5nlet 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. (effective) v

6、alues of e.m.f. and voltage are new words & expressions:peak values 峰值峰值.6nin 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，同時將限定詞均方根（有效）值，同時

7、將限定詞“均方根（均方根（有效）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 and complex number正弦時間函數(shù)的矢量和復(fù)數(shù)表示法正弦時間函數(shù)的矢量和復(fù)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 there be a sinuso

8、idal time function (current, voltage, magnetic flux and the like):n如果所涉及的正弦量用矢量和復(fù)數(shù)表示，便可大大地簡化交流電如果所涉及的正弦量用矢量和復(fù)數(shù)表示，便可大大地簡化交流電路的分析。路的分析。n設(shè)一正弦時間函數(shù)（電流、電壓、磁通等）設(shè)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(fù)數(shù)復(fù)數(shù) sinusoidal quantity 正弦量正弦量 magnetic

9、 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.8nit 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 represents the peak value

10、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之間之間成成夾角夾角這個正弦時間函數(shù)可用如下的矢量形這個正弦時間函數(shù)可用如下的矢量形式表示。通過在笛卡爾坐標(biāo)系的右側(cè)式表示。通過在笛卡爾坐標(biāo)系的右側(cè)mon（如圖（如圖1所示）區(qū)域內(nèi)，取恰當(dāng)所示）區(qū)域內(nèi)，取恰當(dāng)?shù)谋壤嫵鍪噶康谋壤嫵鍪噶縱m，以便于代表該量，以便于代表該量的幅值的幅值vm，并

11、與橫坐標(biāo)形成，并與橫坐標(biāo)形成角（逆角（逆時針方向?yàn)檎?，順時針方向?yàn)樨?fù)）。時針方向?yàn)檎槙r針方向?yàn)樨?fù)）。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 angular frequency . at ti

12、me 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(shè)從現(xiàn)在假設(shè)從t=0開始，矢量開始，矢量vm繞著繞著原點(diǎn)原點(diǎn)o以等于角頻率以等于角頻率的恒定角速的恒定角速度逆時針旋轉(zhuǎn)。則度逆時針旋轉(zhuǎn)。則t時刻矢量與橫時刻矢量與橫坐標(biāo)軸坐標(biāo)軸om形成形成t+的夾角。它在的夾角。它在縱軸縱軸nn上的投影便表示在已選用上的投影便表示在已選用的比例尺下的瞬

13、時值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.10ninstantaneous 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 the axis nn clockwise

14、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(biāo)（即矢量在縱坐標(biāo)nn上上的投影）也能通過以下方法得到：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以角以角速度速度從從t=0開始順時針旋轉(zhuǎn)，此開始順時針旋轉(zhuǎn)，此時旋轉(zhuǎn)的軸時旋轉(zhuǎn)的軸nn稱為時間軸。稱為時間軸。.11nin each case, there is a single-valued relationship between the

15、instantaneous value of v and the vector vm. hence vm may be termed the vector of the sinusoidal time function v. likewise, 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(guān)系。因之間都存在單值關(guān)系。因此，此，vm便可稱為正弦時間函數(shù)便可稱為正弦時間函數(shù)v的矢量。同理，還有電壓的矢量。同理，還有電壓矢量、電勢矢量、

16、電流矢量、磁通矢量等。矢量、電勢矢量、電流矢量、磁通矢量等。new words & expressions:single-valued relationship 單值關(guān)系（一一對應(yīng)關(guān)系）單值關(guān)系（一一對應(yīng)關(guān)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.12n“true” vector quantities are denoted either by clarendon type, e.g. a, or by a, while sinusoidal ones are d

17、enoted by a. 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a表示，而正弦表示，而正弦 矢矢量則用量則用a表示。按合適的相對關(guān)系和某種方便的表示。按合適的相對關(guān)系和某種方便的比例畫出的正弦向量的圖解稱為矢量圖。比例畫出的正弦向量的圖解稱為矢量圖。new words & expressions:e.g. ,i:di: =exempli gr

18、atia 例如例如 vector diagrams 矢量圖矢量圖ntaking mm and nn as the axes of real and imaginary 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

19、number is called the complex peak value of a given sinusoidal quantity.n在一復(fù)數(shù)平面內(nèi)，取在一復(fù)數(shù)平面內(nèi)，取mm和和nn分別為實(shí)數(shù)軸和虛數(shù)軸，矢分別為實(shí)數(shù)軸和虛數(shù)軸，矢量量vm可用一復(fù)數(shù)來表示，該復(fù)數(shù)的絕對值（即模）等于可用一復(fù)數(shù)來表示，該復(fù)數(shù)的絕對值（即模）等于vm，其相位角等于，其相位角等于。此復(fù)數(shù)稱為某一已知正弦量的復(fù)數(shù)。此復(fù)數(shù)稱為某一已知正弦量的復(fù)數(shù)峰值。峰值。new words & expressions:real quantity 實(shí)量實(shí)量 imaginary quantity 虛量虛量 complex pla

20、ne 復(fù)平面復(fù)平面complex number 復(fù)數(shù)復(fù)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(fù)數(shù)幅值復(fù)數(shù)幅值/峰值峰值ngenerally, a complex vector may be expressed in the following waysnwhere極坐標(biāo)的、指數(shù)的、三角的、直角或代數(shù)的new words & expressions:complex vector 復(fù)矢量復(fù)矢量nwhen the vector vm rotates counter-clockwis

21、e 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(dāng)矢量當(dāng)矢量vm從從t=0開始以角速度開始以角速度逆時針旋轉(zhuǎn)時，便逆時針旋轉(zhuǎn)時，便被稱之為復(fù)數(shù)時間函數(shù)，并定義為被稱之為復(fù)數(shù)時間函數(shù)，并定義為(eq.)?，F(xiàn)在，?，F(xiàn)在，既然它是一復(fù)函數(shù)，則可用實(shí)部和虛

22、部來表示既然它是一復(fù)函數(shù)，則可用實(shí)部和虛部來表示:new words & expressions:complex time function 復(fù)數(shù)時間函數(shù)復(fù)數(shù)時間函數(shù) real part 實(shí)部實(shí)部 imaginary part虛部虛部.16nwhere the sine term is the imaginary part of the complex variable equal (less j) to the sinusoidal quantity v, ornwhere the symbol im indicates that only the imaginary part of the

23、 function in the square brackets is taken.n其中正弦項(xiàng)是復(fù)數(shù)變量（除去其中正弦項(xiàng)是復(fù)數(shù)變量（除去j）的虛部，等于正）的虛部，等于正弦量弦量v，即，即n式中符號式中符號im是指只計及方括號中復(fù)數(shù)的虛部。是指只計及方括號中復(fù)數(shù)的虛部。nthe instantaneous 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 o

24、nly taken. for this case, the instantaneous value of v is represented by a projection of the vector onto the real axis.余弦函數(shù)的瞬時值由下式給余弦函數(shù)的瞬時值由下式給出：出：式中符號式中符號re是指只計及方括是指只計及方括號中復(fù)數(shù)的實(shí)部。在這種情號中復(fù)數(shù)的實(shí)部。在這種情況下，瞬時值由矢量況下，瞬時值由矢量 在實(shí)軸上的投影表示。在實(shí)軸上的投影表示。.18nthe representation of sinusoidal functions in complex form is

25、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(fù)數(shù)形式的正弦函數(shù)的表達(dá)式是交流電路分復(fù)數(shù)形式的正弦函數(shù)的表達(dá)式是交流電路分析中復(fù)數(shù)法的基礎(chǔ)?，F(xiàn)在所用的復(fù)數(shù)法的形析中復(fù)數(shù)法的基礎(chǔ)?，F(xiàn)在所用的復(fù)數(shù)法的形式是由式是由heaviside和和steinmetz提出的。提出的。new words & expressions:complex

26、-number method 復(fù)數(shù)法復(fù)數(shù)法 method of complex numbers addition of sinusoidal time functions正弦時間函數(shù)的加法na.c. circuit analysis involves the addition of harmonic time functions having the same frequencies but different peak values and epoch angles. direct addition of such functions would call for unwieldy trig

27、onometric 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(zhuǎn)換間函數(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(zhuǎn)換。簡單的方法是采用。簡單的方法是采用argand圖（圖

28、解法）和復(fù)數(shù)法（解析法）圖（圖解法）和復(fù)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(zhuǎn)換三角轉(zhuǎn)換 analytical solution 解析法解析法.20nsuppose we are to find the sum of two harmonic functionsnand nfirst, consider the application of the argan

29、d 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：首先，考慮采用argand圖法（作圖法）。我們畫出矢量(eq.)和(eq.)并由平行四邊形法則求出合成矢量(eq.)。resultant vector 合成矢量 .21nnow assume that the vectors begin to rotate about the origin of coordinates, o, at t=0, doing so with a constant a

30、ngular velocity in the counter-clockwise direction. n現(xiàn)在假設(shè)矢量 在t=0時刻開始逆時針方向繞著坐標(biāo)原點(diǎn)o以恒定角速度旋轉(zhuǎn)。new words & expressions:origin of coordinates坐標(biāo)原點(diǎn)坐標(biāo)原點(diǎn)nat any instant of time, 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 v

31、ectors and ,or 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 desired sinusoidal time function v=v1+v2.在任一時刻，旋轉(zhuǎn)矢量在任一時刻，旋轉(zhuǎn)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(biāo)軸上的投影之和，或者瞬在同一坐標(biāo)軸上的投影之和，或者瞬時

32、值時值v1和和v2之和。換句話說，矢量之和。換句話說，矢量(eq.)在縱坐標(biāo)上的投影表示瞬時值在縱坐標(biāo)上的投影表示瞬時值之和之和(v1+v2)，矢量，矢量(eq.)表示所要求表示所要求的正弦時間函數(shù)的正弦時間函數(shù)(eq.)。non finding the length of vm and the angle from the argand diagram, we may substitute them in the expression .nnow consider the analytical method/solution. referring to the diagram of fig.

33、 2, we may writenin the rectangular (algebraic) form, these complex numbers arenon adding them together we obtainnwhere.24nsince ,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

34、 function. for convenience the epoch angle may 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所在的象限是很重要的。通過函數(shù)的實(shí)部和虛部的符號能所在的象限是很重要的。通過函數(shù)的實(shí)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示初相角 。new words & expressions:quadrant 象限象限.25nthe two methods are appli

35、cable 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。數(shù)的疊加。new words & expressions:be applicable to (適適)用于用于.26nin practical work, one is usually interested in the r.m.s. values and phase displacements of sinusoidal quantities. therefore the