特征值理论及莫尔圆在电网络分析中的应用

Application of the Eigenvalue Theory and the Mohr’s Circle in Electrical Network Analysis

电网络分析是电工电子学中的一个基本问题,人们对此进行了较多的研究,并给出了许多行之有效的解法。但以往的工作主要是考虑如何将规模较大的问题化为规模较小的问题,以及如何利用系数方程的稀疏性来节约计算机时等,而在如何使用特征值理论及几何直观意义对电网络状态量的总体性的内在联系进行分析方面,却未得到应用的重视。所以根据直流网络电导矩阵的正定性,提出用一个超椭球面将其电流向量、电压向量和功率三者联系起来,从而发现电网络的稳定状态与弹性力学中所考虑的一点的应力应变状态有着相似的关系,而网络的最大和最小功率可由上述超椭球面最长和最短主轴直接确定。由此得到了一种确定网络最大和最小功率的作图法———莫尔圆法。结果表明,该作图法可被方便地应用于三相交流电路。为用几何方法分析与电网络相关的一类问题提供了启示。

Electric network analysis is a basic problem in electrical engineering and electronics field. A lot of research work has been done and many effective solutions for it have been presented. These methods mainly take consideration on how to change a problem of large scale to problem of smaller scale, how to reduce the computer time of calculation according to the sparse characteristic of the coefficient matrix. But it has not attracted the deserved attention in how to analyze the general relation of the state variables of the electric network with the eigenvalue theory and the geometrical intuition meaning. According to the positive characteristic of the conductance matrix of a direct current network, this paper proposed a super elliptic surface with which the current vector, the voltage vector and the power of the network were related. Together it has been found that there existed a similarity between the stable state of an electric network and the stress state or the strain state considered in the elastic mechanics. The maximum power and the minimum power of the direct current network could be determined by the longest axle and the shortest axle of super elliptic surface. Thus a illustrating method——the method of using Mohr’s circle was proposed for this purpose. It was showed that the illustrating method could be used to a three-phase alternating current network. This paper gave an inspiration of using geometrical methods to analyze the kind of problem of electric networks.