Application of the eigenvalue theory and the Mohr's circle in electrical network analysis

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.