Variations of vertex potentials by short-circuiting two vertices in resistance networks
| Autor: | Kimio Sato |
|---|---|
| Rok vydání: | 1990 |
| Předmět: | |
| Zdroj: | Electronics and Communications in Japan (Part III: Fundamental Electronic Science). 73:31-37 |
| ISSN: | 1520-6440 1042-0967 |
| DOI: | 10.1002/ecjc.4430730404 |
| Popis: | In the resistance network N, the potential of each vertex determined in reference to one vertex of the input vertex-pair, where a unit current source is placed, can be expressed by using the topological formula as the ratio of the conditional 2-tree conductance product sum to the tree conductance product sum of N. It is a well-known empirical law that the potential of the input vertex (equi-valently, the input resistance) does not increase if any two vertices of N are short-circuited. However, no rigorous and simple algebraic proof including the potential variation of the input vertex is available, and such studies on an arbitrary vertex as its generalization have not been carried out. In this paper, the variations of vertex potentials by short-circuiting two vertices of N are formulated by using the topological process. If the formula obtained here is used, a necessary and sufficient condition can be provided for increase (or decrease) of any vertex potentials by short-circuiting two vertices of N besides a proof of the fore-mentioned empirical law. In addition, the following can readily be verified. If the potentials of two vertices of N are equal, the potentials of any vertices do not change by short-circuiting the two vertices. Even if a specific vertex potential does not change after short-circuiting two vertices, the potentials of these two vertices are not necessarily equal. |
| Databáze: | OpenAIRE |
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