Unit4 Analysis of Sinusoidal Alternating Electricity 正弦交流电的分析nR.M.S. (Effective) Values of Current and Voltage电压和电流的有效值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 also proportional to the square of that current.n两载流导体之间的作用力与导体中的电流的平方成正比。两载流导体之间的作用力与导体中的电流的平方成正比。某段时间内电流通过一个电阻所产生的热量也正比于电某段时间内电流通过一个电阻所产生的热量也正比于电流的平方。流的平方。NewWords&Expressions:sinusoidalalternatingelectricity正弦交流电正弦交流电effectivevalues有效值有效值r.m.s.values=rootmeansquarevalues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这便引出通常所说的均方根（或有效值）电流的这便引出通常所说的均方根（或有效值）电流的概念，其定义如下：概念，其定义如下：(Eq.1)n在在dt时间里电流时间里电流i通过电阻通过电阻r产生的热量为产生的热量为(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 译为译为“由此得出由此得出”。宾语。宾语从句里面含有一个定语从句。从句里面含有一个定语从句。n由此可得出，交流电的均方根（或有效）值等于由此可得出，交流电的均方根（或有效）值等于在相同电阻、相同时间内产生相同热量的恒稳直在相同电阻、相同时间内产生相同热量的恒稳直流电的大小。流电的大小。NewWords&Expressions:steadydirectcurrent恒稳直流电恒稳直流电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. (effective) values of e.m.f. and voltage are NewWords&Expressions:peakvalues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在涉及交流电压和电流时，通常指的值就是其在涉及交流电压和电流时，通常指的值就是其均方根（有效）值，同时将限定词均方根（有效）值，同时将限定词“均方根均方根（有效）（有效）”几个字略去，并不明指。几个字略去，并不明指。Representation of Sinusoidal Time Functions by Vectors and Complex Number正弦时间函数的矢量和复数表正弦时间函数的矢量和复数表示法示法nA.C. circuit analysis can be greatly simplified if the sinusoidal quantities involved are represented by vectors or complex numbers.nLet there be a sinusoidal time function (current, voltage, magnetic flux and the like):n如果所涉及的正弦量用矢量和复数表示，便可大大地简化交流电如果所涉及的正弦量用矢量和复数表示，便可大大地简化交流电路的分析。路的分析。n设一正弦时间函数（电流、电压、磁通等）设一正弦时间函数（电流、电压、磁通等）NewWords&Expressions:sinusoidaltimefunction正弦时间函数正弦时间函数vector矢量矢量complexnumber复数复数sinusoidalquantity正弦量正弦量magneticflux磁通磁通A.C. circuit=alternating current circuit 交流电路交流电路D.C. circuit=direct current circuit 直流电路直流电路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 represents 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与与B之间之间成成夹角夹角这个正弦时间函数可用如下的矢量形这个正弦时间函数可用如下的矢量形式表示。通过在笛卡尔坐标系的右侧式表示。通过在笛卡尔坐标系的右侧MON（如图（如图1所示）区域内，取恰当所示）区域内，取恰当的比例画出矢量的比例画出矢量Vm，以便于代表该量，以便于代表该量的幅值的幅值Vm，并与横坐标形成，并与横坐标形成角（逆角（逆时针方向为正，顺时针方向为负）。时针方向为正，顺时针方向为负）。NewWords&Expressions:clockwise顺时针方向顺时针方向counter-clockwise逆时针方向逆时针方向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 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.现在假设从现在假设从t=0开始，矢量开始，矢量Vm绕着绕着原点原点O以等于角频率以等于角频率的恒定角速的恒定角速度逆时针旋转。则度逆时针旋转。则t时刻矢量与横时刻矢量与横坐标轴坐标轴OM形成形成t+的夹角。它在的夹角。它在纵轴纵轴NN上的投影便表示在已选用上的投影便表示在已选用的比例尺下的瞬时值的比例尺下的瞬时值v。NewWords&Expressions:constant angularvelocity恒定角速度恒定角速度angularfrequency角频率角频率instantaneousvalue瞬时值瞬时值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 the axis NN clockwise at the angular velocity , starting at time t=0. Now the rotating axis NN is called the time axis. 瞬时值瞬时值v（即矢量在纵坐标（即矢量在纵坐标NN上上的投影）也能通过以下方法得到：的投影）也能通过以下方法得到：即令矢量即令矢量Vm不动，将轴不动，将轴NN以角以角速度速度从从t=0开始顺时针旋转，此开始顺时针旋转，此时旋转的轴时旋转的轴NN称为时间轴。称为时间轴。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. Likewise, there are vectors of voltages, e.m.f.s, currents, magnetic fluxes,etc.n两种情况下，瞬时值两种情况下，瞬时值v和矢量和矢量Vm之间都存在单值关系。因之间都存在单值关系。因此，此，Vm便可称为正弦时间函数便可称为正弦时间函数v的矢量。同理，还有电压的矢量。同理，还有电压矢量、电势矢量、电流矢量、磁通矢量等。矢量、电势矢量、电流矢量、磁通矢量等。NewWords&Expressions:single-valuedrelationship单值关系（一一对应关系）单值关系（一一对应关系）vectorsofvoltages(e.m.f.s,currents,magneticfluxes）电压（电势、电流、磁通）矢量电压（电势、电流、磁通）矢量n“True” vector quantities are denoted either by clarendon type, e.g. A, or by A, while sinusoidal ones are denoted by A. Graphs of sinusoidal vectors, arranged in a proper relationship and to some convenient scale, are called vector diagrams.n真正的矢量是用粗体字真正的矢量是用粗体字A，或，或A表示，而正弦表示，而正弦 矢矢量则用量则用A表示。按合适的相对关系和某种方便的表示。按合适的相对关系和某种方便的比例画出的正弦向量的图解称为矢量图。比例画出的正弦向量的图解称为矢量图。NewWords&Expressions:e.g.,i:di:=exempligratia例如例如vectordiagrams矢量图矢量图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 number is called the complex peak value of a given sinusoidal quantity.n在一复数平面内，取在一复数平面内，取MM和和NN分别为实数轴和虚数轴，矢分别为实数轴和虚数轴，矢量量Vm可用一复数来表示，该复数的绝对值（即模）等于可用一复数来表示，该复数的绝对值（即模）等于Vm，其相位角等于，其相位角等于。此复数称为某一已知正弦量的复数峰值。此复数称为某一已知正弦量的复数峰值。NewWords&Expressions:realquantity实量实量imaginaryquantity虚量虚量complexplane复平面复平面complexnumber复数复数absolutevalue绝对值绝对值modulus模模phase相位相位argument相角相角complexpeakvalue复数幅值复数幅值/峰值峰值nGenerally, a complex vector may be expressed in the following waysnwhere极坐标的、指数的、三角的、直角或代数的NewWords&Expressions:complexvector复矢量复矢量nWhen the 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当矢量当矢量Vm从从t=0开始以角速度开始以角速度逆时针旋转时，便逆时针旋转时，便被称之为复数时间函数，并定义为被称之为复数时间函数，并定义为(Eq.)。现在，既。现在，既然它是一复函数，则可用实部和虚部来表示然它是一复函数，则可用实部和虚部来表示:NewWords&Expressions:complextimefunction复数时间函数复数时间函数realpart实部实部imaginarypart虚部虚部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 that only the imaginary part of the function in the square brackets is taken.n其中正弦项是复数变量（除去其中正弦项是复数变量（除去j）的虚部，等于正）的虚部，等于正弦量弦量v，即，即n式中符号式中符号Im是指只计及方括号中复数的虚部。是指只计及方括号中复数的虚部。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 only taken. For this case, the instantaneous value of v is represented by a projection of the vector onto the real axis.余弦函数的瞬时值由下式给余弦函数的瞬时值由下式给出：出：式中符号式中符号Re是指只计及方括是指只计及方括号中复数的实部。在这种情号中复数的实部。在这种情况下，瞬时值由矢量况下，瞬时值由矢量 在实轴上的投影表示。在实轴上的投影表示。nThe representation of sinusoidal 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复数形式的正弦函数的表达式是交流电路分复数形式的正弦函数的表达式是交流电路分析中复数法的基础。现在所用的复数法的形析中复数法的基础。现在所用的复数法的形式是由式是由Heaviside和和Steinmetz提出的。提出的。NewWords&Expressions:complex-numbermethod复数法复数法method of complex numbers Addition of Sinusoidal Time Functions正弦时间函数的加法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 trigonometric transformations. Simple approaches are provided by the Argand diagram (graphical solution) and by the method of complex numbers (analytical solution).n交流电路的分析包括对有相同频率、不同幅值和初相角的谐振时交流电路的分析包括对有相同频率、不同幅值和初相角的谐振时间函数的加法。这些函数的直接相加将要求用到繁杂的三角转换。间函数的加法。这些函数的直接相加将要求用到繁杂的三角转换。简单的方法是采用简单的方法是采用Argand图（图解法）和复数法（解析法）图（图解法）和复数法（解析法）NewWords&Expressions:harmonictimefunction谐谐振振时时间间函函数数peakvalue幅幅/峰峰值值epochangle初初相相角角trigonometrictransformations三角转换三角转换analyticalsolution解析法解析法nSuppose we are to find the sum of two harmonic functionsnand nFirst, consider the application of the Argand diagram (graphical solution). We lay off the vectors and find the resultant vector .假如我们要求两个谐振函数的和：首先，考虑采用Argand图法（作图法）。我们画出矢量(Eq.)和(Eq.)并由平行四边形法则求出合成矢量(Eq.)。resultant vector 合成矢量 nNow assume that 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现在假设矢量 在t=0时刻开始逆时针方向绕着坐标原点O以恒定角速度旋转。NewWords&Expressions:originofcoordinates坐标原点坐标原点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 vectors 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.在任一时刻，旋转矢量在任一时刻，旋转矢量(Eq.)在纵轴在纵轴NN上的投影等于矢量上的投影等于矢量(Eq.)和和(Eq.)在同一坐标轴上的投影之和，或者瞬在同一坐标轴上的投影之和，或者瞬时值时值v1和和v2之和。换句话说，矢量之和。换句话说，矢量(Eq.)在纵坐标上的投影表示瞬时值在纵坐标上的投影表示瞬时值之和之和(v1+v2)，矢量，矢量(Eq.)表示所要求表示所要求的正弦时间函数的正弦时间函数(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. 2, we may writenIn the rectangular (algebraic) form, these complex numbers arenOn adding them together we obtainnWherenSince ,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 may be expressed in degrees rather than in radians.n由于由于 ，在我们确定，在我们确定之前，知道之前，知道Vm所在的象限是很重要的。通过函数的实部和虚部的符号能所在的象限是很重要的。通过函数的实部和虚部的符号能很容易地确定象限。为方便起见，用角度而不用弧度来表很容易地确定象限。为方便起见，用角度而不用弧度来表示初相角示初相角 。NewWords&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这两种方法可用于任何数目的同频率正弦函数这两种方法可用于任何数目的同频率正弦函数的叠加。的叠加。NewWords&Expressions:beapplicableto(适适)用于用于nIn practical work, one is usually interested in the r.m.s. values and phase displacements of sinusoidal quantities. Therefore the Argand diagram is simplified by omitting the axes (whose position is immaterial), while the phase displacement between the vectors is faithfully reproduced. n在实际中，人们通常对正弦量的有效值和相位差在实际中，人们通常对正弦量的有效值和相位差感兴趣。于是可通过省略坐标轴来简化感兴趣。于是可通过省略坐标轴来简化Argand图图（坐标轴的位置是不重要的），但矢量之间的相（坐标轴的位置是不重要的），但矢量之间的相位差并没有改变。位差并没有改变。NewWords&Expressions:phasedisplacements相位差相位差nAlso, instead of rotating vectors of length equal to the peak values of sinusoidal quantities, the scale is changed and the vector lengths are treated as r.m.s. values.n此外，不用长度等于正弦量峰值的此外，不用长度等于正弦量峰值的 旋转矢量，旋转矢量，而是改变其比例尺，将矢量的长度看成有效值。而是改变其比例尺，将矢量的长度看成有效值。NewWords&Expressions:rotatingvectors旋转矢量旋转矢量nIn analytical treatment, it is usual to arrange the sinusoidal quantities so that the epoch angle of any one becomes zero. Likewise, instead of complex peak values, the respective complex r.m.s. values, obtained by division of complex peak values by ,are used. For brevity, complex r.m.s. values are called simply a complex current, a complex voltage, etc.n在分析中，常将正弦量的初相角设定为零。同理，在分析中，常将正弦量的初相角设定为零。同理，系数采用复数峰值除以系数采用复数峰值除以 后得到的有效值，而不后得到的有效值，而不是复数峰值。为简单起见，复数有效值简单地称为是复数峰值。为简单起见，复数有效值简单地称为复数电流、复数电压等。复数电流、复数电压等。