Potential formula of the nonregular m × n fan network and its application

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.

Potential formula of an m × n globe network and its application

“…In recent years, Tan 23 – 25 proposed the Recursion-Transform (RT) method which, depends on the one matrix along one directions, avoids the trouble of the Laplacian method depends on two matrices along two directions. Researchers have resolved a lot of complex resistor network by the RT technique 26 – 37 , and the results expressed by fractional-order are in the form of a single summation. As a summary, the RT method is expressed by two forms, one form is expressed by current parameters 23 – 25 , which is simply called the RT-I method; another form is expressed by potential parameters 36 , 37 , which is simply called the RT-V method.…”

“…Researchers have resolved a lot of complex resistor network by the RT technique 26 – 37 , and the results expressed by fractional-order are in the form of a single summation. As a summary, the RT method is expressed by two forms, one form is expressed by current parameters 23 – 25 , which is simply called the RT-I method; another form is expressed by potential parameters 36 , 37 , which is simply called the RT-V method. Very recently, the researchers have also made new progress in the multi-functional N-order resistance network 38 , 39 , and the result derived by the RT method has also been applied to the impedance network 40 .…”

“…According to research information obtained from the above literature, we know the computation of equivalent resistance has made great progress 2 – 40 , but the calculation of the nodal potential has been a difficult problem and has not been resolved until a recent research of literature 36 , 37 . Ref.…”

“… 36 gives the precise potential formulae of the fan and cobweb networks by the RT-V method for the first time, next ref. 37 studies the potential formula of nonregular fan network by the RT-V method. However, due to the diversity of network types, there are still many potential of the resistor networks needs to be calculated.…”

Electrical characteristics of the 2 × n and □ × n circuit network

“…, when we just calculate the equivalent resistance between two nodes of A x 1 and A x 2 or A x 1 and C ,x 2 the 3D ,×n circuit network in figure 2 can be equivalent to the structure of the 2×n plane network model shown in figure 1.Thus, formulae (1) and (2) can be applied to the 3D ,×n circuit network in figure 2 if we change the corresponding resistor elements in the different networks, According to the transformation between figures 1 and 2, and using (42), we have the relations equations(42) and(43) into equations (6)-(8), we have the equivalent resistance between two nodes of A C ,…”