Electrical properties of an arbitrary <i>m</i>×<i>n</i> rectangular network

Abstract

<sec>The development of natural science raises many complex new problems and requires people to find the basic method to resolve them. It was found that many problems could be resolved by building the resistor network model. In 1845, the German scientist Kirchhoff set up the node current law and the circuit voltage law.Since then the basic theory of electric circuit has been established. At present, three general theories for studying large-scale resistor networks have been developed, for example, In 2000 Cserti [<i>Am. J. Phys.</i> 2000, <b>68</b> , 896] set up the Green function technique to evaluate the resistance of infinite lattices. In 2004 Wu [<i>J. Phys. A: Math. Gen.</i> 2014, <b>37</b> , 6653] formulated a Laplacian matrix method and calculated the resistance of arbitrary finite and infinite lattices by using the eigenvalues and eigenvectors. In 2011 Tan [<i>Resistance Network Model</i> (Xi’an: Xidian University Press) 2011, pp16–216] proposed the recursion-transform (RT) method which depends on the one matrix along one directions and avoids the trouble of the Laplacian method that depends on two matrices along two directions. Among them, only two theories can calculate both finite and infinite networks. One is Wu's Laplacian matrix method and the other is Tan's RT method. However, there is only one way to compute a resistor network with arbitrary boundary, that is, the Tan's RT method.</sec><sec>Potential distribution problem in arbitrary rectangular circuit network has always been a problem of scientific research. In this paper, we develop the RT-I theory of resistor networks to calculate the arbitrary <i>m</i> × <i>n</i> circuit network model. We study the potential distribution and the equivalent resistance of a class of <i>m</i> × <i>n</i> rectangular network with an arbitrary boundary, a profound problem that has not been resolved so far, because previous research depends on the boundary conditions of rules or a zero-resistance boundary condition. Other methods, such as Green function technique and Laplacian method to calculate potential function are difficult and also impossible to study the resistor network with arbitrary boundary. Potential function problem is an important research subject in natural science and engineering technology, for example, the solution of Laplace's equation is one of research work. In this paper, we present an analytical expression of the node potential function of <i>m</i> × <i>n</i> rectangular resistor network with an arbitrary boundary, and also obtainan equivalent resistance formula between any two nodes, and the results in some special cases as well. In the comparative study of different results, a new mathematical identity and several interesting inferences are discovered.</sec>