UNIT_5_Analysis_of_Sinusoidal_Current_and_Voltage(兰州交通大学-电气工程专业英语).docx

UNIT 5Analysis of Sinusoidal Current and Voltage正弦交流电的分析R.M.S. (Effective) Values of Current and Voltage电流和电压的有效值The 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. This calls for knowledge of what is known as the root mean square (or effective) current defined as:I= (4.1)The heat developed by a current i in a resistance r in time dt is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流的平方。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：I= (4.1)在dt时间里电流i通过电阻r产生的热量为It 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. 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流电的大小。Let us establish the relationship between the R.M.S. and peak values of a sinusoidal current, I and ImI2=Hence I=Im/ (4.2)The R.M.S. (effective) values of e.m.f. and voltage areandIn dealing with periodic voltages and currents, their R.M.S. (effective) values are usually meant, and the adjective “r.m.s.” or “effective” is simply implied. 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）”几个字略去，并不明指。Representation of Sinusoidal Time Functions by Vectors and Complex Number A.C. circuit analysis can be greatly simplified if the sinusoidal quantities involved are represented by vectors or complex numbers.Let there be a sinusoidal time function (current, voltage, magnetic flux and the like): v=Vmsin(t+)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。设一正弦时间函数（电流、电压、磁通等） v=Vmsin(t+)It can be represented in vector form as follows. Using a right-hand set of Cartesian coordinates MON(Fig.5.1) (omitted), we draw the vector 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).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MON（如图1所示）区域内，取恰当的比例画出矢量Vm，以便于代表该量的幅值Vm，并与横坐标形成角（逆时针方向为正，顺时针方向为负）。Now we imagine that, starting at t=0, the vector 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 N represents the instantaneous value of v to the scale chose.现在假设从t=0开始，矢量Vm绕着原点O以等于角频率的恒定角速度逆时针旋转。则t时刻矢量与横坐标轴OM形成t+的夹角。它在纵轴NN上的投影便表示在已选用的比例尺下的瞬时值v。Instantaneous values of v, as projections of the vector on the vertical axis NN, can also be obtained by holding the vector 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v（即矢量在纵坐标NN上的投影）也能通过以下方法得到：即令矢量Vm不动，将轴NN以角速度从t=0开始顺时针旋转，此时旋转的轴NN称为时间轴。In each case, there is a single-valued relationship between the instantaneous value of v and the vector. Hence 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.两种情况下，瞬时值v和矢量Vm之间都存在单值关系。因此，Vm便可称为正弦时间函数v的矢量。同理，还有电压矢量、电势矢量、电流矢量、磁通矢量等。 “True” vector quantities are demoted either by clarendon type, e.g. A, or by, while sinusoidal ones are demoted by. Graphs of sinusoidal vectors, arranged in a proper relationship and to some convenient scale, are called vector diagrams.真正的矢量是用粗体字A，或A表示，而正弦矢量则用A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Taking MM and NN as the axes of real and imaginary quantities, respectively, in a complex plane, the vector 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.在一复数平面内，取MM和NN分别为实数轴和虚数轴，矢量Vm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。此复数称为某一已知正弦量的复数峰值。Generally, a complex vector may be expressed in the following ways:极坐标的、指数的、三角的、直角或代数的 (5.3) Where j=When the vector 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 parts:当矢量Vm从t=0开始以角速度逆时针旋转时，便被称之为复数时间函数，并定义为(Eq.)。现在，既然它是一复函数，则可用实部和虚部来表示:Where the sine term is the imaginary part of the complex variable equal (less j) to the sinusoidal quantity v, or (5.4)Where, the symbol Im indicates that only the imaginary part of the function in the square brackets is taken.其中正弦项是复数变量（除去j）的虚部，等于正弦量v，即式中符号Im是指只计及方括号中复数的虚部。The instantaneous value of a cosinusoidal function is given by (5.5)Where 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出：式中符号Re是指只计及方括号中复数的实部。在这种情况下，瞬时值由矢量在实轴上的投影表示The 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.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式是由Heaviside和Steinmetz提出的。Addition of Sinusoidal Time Functions正弦时间函数的加法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 such functions would call for unwieldy trigonometric transformations. Simpler approaches are provided by the Argand diagram (graphical solution) and by the method of complex numbers (analytical solution).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。简单的方法是采用Argand图（图解法）和复数法（解析法）Suppose we are to find the sum of two harmonic functions:andFirst, 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.)Now 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.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。At 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(Eq.)在纵轴NN上的投影等于矢量(Eq.)和(Eq.)在同一坐标轴上的投影之和，或者瞬时值v1和v2之和。换句话说，矢量(Eq.)在纵坐标上的投影表示瞬时值之和(v1+v2)，矢量(Eq.)表示所要求的正弦时间函数(Eq.)。On finding the length of Vmand the angle from the Argand diagram, we may substitute them in the expression Now consider the analytical method. Referring to the diagram of Fig 4.2(omitted), we may write In the rectangular (algebraic) form, these complex numbers are On adding them together we obtainWhere Since tan=tan(), it is important to know the quadrant where 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.由于tan=tan()，在我们确定之前，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 。The two methods are applicable to the addition of any number of sinusoidal functions of the same frequency.这两种方法可用于任何数目的同频率正弦函数的叠加。In practical work, one is usually interested in the r.m.s. values and phase displacements of sinusoidal quantities. Therefore the Argand diagram is simplifie