Resistance formulae of a multipurpose n‐step network and its application in LC network

Abstract

We consider a multipurpose n-step network with cross resistors that is a profound problem that has not been resolved before. This network contains a number of different types of resistor network model. This problem is resolved by three steps: First of all, we simplify a complex graphics into a simple equivalent model; next, we use Kirchhoff's laws to analyse the network and establish a nonlinear difference equation; and finally, we construct the method of equivalent transformation to obtain the general solution of the nonlinear difference equation. In this paper, we created a new concept of negative resistance for the needs of the equivalent conversion and obtain two general resistance formulae of a multipurpose ladder network of cross resistors. As applications, several interesting specific results are produced. In particular, an n-step impedance LC network is discussed. Our method and the results are suitable for the research of complex impedance network as well.

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

“…In particular, we provide three special results when taking x x x 1 1 = = and x 0, 1 = x n. 2 = It is interesting that we established a new method to study the equivalent resistance of a 3D network, which is a very clever design that applies the results of a 2×n resistor network to a ,×n circuit network. It is necessary for us to explain that figure 2 is a new model because there are two different resistor elements on the horizontal axis, which has not been resolved before, and the general results of equations (44) and (45) are discovered for the first time.…”

The wave concept iterative process (WCIP) method for electrical circuit network with triangular and hexagonal topology

“…Over the past few decades, the rectangular network topology has been published in some papers with the superposition current distribution, the random walks, the lattice Green function,() the Laplacian matrix,() the analytical solution obtained by recursion or equivalent transform methods,() and so on.…”

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

“…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 10 – 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 superiority of the RT method is that studying resistor network just need one matrix along one directions, which, avoids two Laplacian matries 13 , 21 , the solution required is just one instead of two eigenvalues, and results given by the RT method is in a single sum. In fact, the RT method have been developed and used to study various types of resistance networks 24 – 37 . Such as, ref.…”

“…35 researched the equivalent resistance and impedance of the cylindrical network. Refs 36 , 37 researched the complex impedance of the two networks. Recently, ref.…”