4正弦交流电的分析

单击此处编辑母版标题样式,单击此处编辑母版文本样式,第二级,第三级,第四级,第五级,*,Unit4 Analysis of Sinusoidal Alternating Electricity,正弦交流电的分析,R.M.S. (Effective) Values of Current and Voltage,电压和电流的有效值,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.,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。,New Words & Expressions:,sinusoidal alternating electricity,正弦交流电,effective values,有效值,r.m.s. values = root mean square values,均方根值,square,平方,This calls for knowledge of what is known as the root mean square (or effective) current defined as (Eq.1),The heat developed by a current,i,in a resistance,r,in time,dt,is (Eq.),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：,(Eq.1),在,dt,时间里电流,i,通过电阻,r,产生的热量为,(Eq.),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.,句型,It follows that ,译为“由此得出,”。宾语从句里面含有一个定语从句。,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。,New Words & Expressions:,steady direct current,恒稳直流电,Let us establish the relationship between the r.m.s. and peak values of a sinusoidal current,I,and,Im,Hence,:(Eq.2),The r.m.s. (effective) values of e.m.f. and voltage are,New Words & Expressions:,peak values,峰值,In 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.,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,正弦时间函数的矢量和复数表示法,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):,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）,New Words & Expressions:,sinusoidal time function,正弦时间函数,vector,矢量,complex number,复数,sinusoidal quantity,正弦量,magnetic flux,磁通,A.C. circuit=alternating current circuit,交流电路,D.C. circuit=direct current circuit,直流电路,It can be represented in vector form as follows. Using a right-hand set of Cartesian coordinates MON (Fig.1), we draw the,vetor,V,m,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,B,之间成,夹角,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）。,New Words & Expressions:,clockwise,顺时针方向,counter-clockwise,逆时针方向,Now we imagine that, starting at t=0,the vector,V,m,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,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,。,New Words & Expressions:,constant,angular velocity,恒定角速度,angular frequency,角频率,instantaneous value,瞬时值,Instantaneous values of v, as projections of the vector on the vertical axis NN,can also be obtained by,holding the vector,V,m 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,V,m. Hence,V,m 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,V,m,之间都存在单值关系。因此，,V,m,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。,New Words & Expressions:,single-valued relationship,单值关系（一一对应关系）,vectors of voltages (e.m.f.s, currents, magnetic fluxes,）,电压（电势、电流、磁通）矢量,“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.,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。,New Words & Expressions:,e.g.,i:di: =exempli gratia,例如,vector diagrams,矢量图,Taking MM and NN as the axes of real and imaginary quantities, respectively, in a complex plane,the vector,V,m 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,V,m,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。,New Words & Expressions:,real quantity,实量,imaginary quantity,虚量,complex plane,复平面,complex number,复数,absolute value,绝对值,modulus,模,phase,相位,argument,相角,complex peak value,复数幅值,/,峰值,Generally, a complex vector may be expressed in the following ways,where,极坐标的、指数的、三角的、直角或代数的,New Words & Expressions:,complex vector,复矢量,When the vector,V,m 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,:,New Words & Expressions:,complex time function,复数时间函数,real part,实部,imaginary part,虚部,Where the sine term is the imaginary part of the complex variable equal (less j) to the sinusoidal quantity v, or,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,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。,New Words & Expressions:,complex-number method,复数法,method of complex numbers,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. Simple 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）,New Words & Expressions:,harmonic time function,谐振时间函数,peak value,幅,/,峰值,epoch angle,初相角,trigonometric transformations,三角转换,analytical solution,解析法,Suppose we are to find the sum of two harmonic functions,and,First, 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,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。,New Words & Expressions:,origin of coordinates,坐标原点,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 Vm,and the angle,from the Argand diagram, we may substitute them in the expression .,Now consider the analytical method/solution. Referring to the diagram of Fig. 2, we may write,In the rectangular (algebraic) form, these complex numbers are,On adding them together we obtain,Where,Since ,it is important to know the quadrant where,V,m,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 ，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,。,New Words & Expressions:,quadrant,象限,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。,New Words & Expressions:,be applicable to (,适,),用于,In 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.,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,Argand,图（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。,New Words & Expressions:,phase displacements,相位差,Also, 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.,此外，不用长度等于正弦量峰值的 旋转矢量，而是改变其比例尺，将矢量的长度看成有效值。,New Words & Expressions:,rotating vectors,旋转矢量,In 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.,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 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复数电流、复数电压等。,