Abstract
Potential formula of an arbitrary resistor network has been an unsolved problem for hundreds of years, which is an interdisciplinary problem that involves many areas of natural science. A new progress has been made in this paper, which discovered the potential formula of a nonregular m × n fan network with two arbitrary boundaries by the Recursion-Transform method with potential parameters (simply call RT-V). The nonregular m × n fan network is a multipurpose network contains several different types of network model such as the interesting snail network and hart network. In the meantime, we discussed the semi-infinite fan network and a series of novel and special conclusions are produced, the effective resistance is educed naturally. The discovery of potential formulae of resistor network provides new theoretical tools and techniques for related scientific research.
Highlights
Modelling resistor network to study scientific problem is an important idea, the initial progress of circuit theory dates back to 1845, a German scientist Kirchhoff who proposed the node current law and the circuit voltage law1
We revisit the research history of the network model, it is found that it is usually very difficult to obtain the explicit resistance and potential formulae of the complex networks because the boundary condition is like a trap or wall, which affects the electrical characteristics of the finite network10–37
As this reason that researchers have found several different effective methods to evaluate the effective resistance of resistor network with different structure, but the potential formula of the complex resistor network has always been an unsolved problem for hundreds of years
Summary
Modelling resistor network to study scientific problem is an important idea, the initial progress of circuit theory dates back to 1845, a German scientist Kirchhoff who proposed the node current law and the circuit voltage law. We revisit the research history of the network model, it is found that it is usually very difficult to obtain the explicit resistance and potential formulae of the complex networks because the boundary condition is like a trap or wall, which affects the electrical characteristics (current, resistance, potential) of the finite network. We revisit the research history of the network model, it is found that it is usually very difficult to obtain the explicit resistance and potential formulae of the complex networks because the boundary condition is like a trap or wall, which affects the electrical characteristics (current, resistance, potential) of the finite network10–37 As this reason that researchers have found several different effective methods to evaluate the effective resistance of resistor network with different structure, but the potential formula of the complex resistor network has always been an unsolved problem for hundreds of years. The Laplacian method is difficult to study the resistor network with arbitrary elements because it is bound by the explicit solution of the matrix with different parameters
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