We consider the problem of two-point resistance in a resistor network previously studied by one of us [F. Y. Wu, J. Phys. A {\bf 37}, 6653 (2004)]. By formulating the problem differently, we obtain a new expression for the two-point resistance between two arbitrary nodes which is simpler and can be easier to use in practice. We apply the new formulation to the cobweb resistor network to obtain the resistance between two nodes in the network. Particularly, our results prove a recently proposed conjecture on the resistance between the center node and a node on the network boundary. Our analysis also solves the spanning tree problem on the cobweb network.

The two-point resistance of a resistor network: A new formulation and application to the cobweb network

We consider the problem of two-point resistance in a resistor network previously studied by one of us (Wu 2004 J. Phys. A: Math. Gen. 37 6653). By formulating the problem differently, we obtain a new expression for the two-point resistance between two arbitrary nodes which is simpler and can be easier to use in practice. We apply the new formulation to the cobweb resistor network to obtain the resistance between two nodes in the network. Particularly, our results prove a recently proposed conjecture on the resistance between the center node and a node on the network boundary. Our analysis also solves the spanning tree problem on the cobweb network.

https://iopscience.iop.org/article/10.1088/1751-8113/47/3/035003/pdf

The two-point resistance of a resistor network: a new formulation and application to the cobweb network

The resistance between two nodes in general position on a fan network with $n$ radial lines and $m$ transverse lines is determined. Also a similar result of Izmailian, Kenna, and Wu [J. Phys. A: Math. Theor. 47, 035003 (2014)] for an $m\ifmmode\times\else\texttimes\fi{}n$ cobweb network is reproduced, but the method used here is significantly different. It avoids the use of the Kirchhoff matrix, requires the solution of just one instead of two eigenvalue problems, and results directly in only a single summation. Further, the current distribution is given explicitly as a byproduct of the method. The method is the same as that used by Tan, Zhou, and Yang [J. Phys. A: Math. Theor. 46, 195202 (2013)] to find the cobweb resistance between center and perimeter for $1\ensuremath{\le}m\ensuremath{\le}3$ and general $n$. Proof of their conjecture for general $m$ is discussed.

Resistance between two nodes in general position on an m×n fan network.

We develop a general recursion-transform (R-T) method for a two-dimensional resistor network with a zero resistor boundary. As applications of the R-T method, we consider a significant example to illuminate the usefulness for calculating resistance of a rectangular $m\ifmmode\times\else\texttimes\fi{}n$ resistor network with a null resistor and three arbitrary boundaries, a problem never solved before, since Green's function techniques and Laplacian matrix approaches are invalid in this case. Looking for the exact calculation of the resistance of a binary resistor network is important but difficult in the case of an arbitrary boundary since the boundary is like a wall or trap which affects the behavior of finite network. In this paper we obtain several general formulas of resistance between any two nodes in a nonregular $m\ifmmode\times\else\texttimes\fi{}n$ resistor network in both finite and infinite cases. In particular, 12 special cases are given by reducing one of the general formulas to understand its applications and meanings, and an integral identity is found when we compare the equivalent resistance of two different structures of the same problem in a resistor network.

Recursion-transform method for computing resistance of the complex resistor network with three arbitrary boundaries.

A basic theorem of equivalent resistance between two arbitrary nodes in an m×n cobweb network in both finite and infinite conditions is discovered, and two conjectures on the equivalent resistance are proved in terms of the basic theorem. We built a tridiagonal matrix equation by means of network analysis and made a diagonalization method of matrix transformation and work out its explicit expressions. The new formulae obtained here can be effectively applied in complex impedance network, especially the formulation leads to the occurrence of resonances and a series of novel results in RLC denote resistor, inductance and capacitance network. These curious results suggest the possibility of practical applications to resonant circuits. Copyright © 2014 John Wiley & Sons, Ltd.

Theory on resistance of m×n cobweb network and its application

Summary
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. Copyright © 2017 John Wiley & Sons, Ltd.

https://scholar.ptuk.edu.ps/bitstream/123456789/616/1/37-%20Resistance%20Formulae%20of%20a%20multipurpose%20n-%20step%20network%20and%20its%20application%20in%20LC%20network.pdf

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

Summary

This paper deals with the equivalent resistance for the m × n resistor network in both finite and infinite cases. Firstly, we build a difference equation driven by a tridiagonal matrix to model the network; then by performing the diagonalizing transformation on the driving matrix, and using the auxiliary function tz(x,n), we derive two formulae of the equivalent resistance between two corner nodes on a common edge of the network. By comparing two different formulae, we also obtain a new trigonometric identity here. Our framework can be effectively applied in complex impedance networks. As in applications in the LC network, we find that our formulation leads to the occurrence of resonances at frequencies associated with (n + 1)ϕt = kπ. This somewhat curious result suggests the possibility of practical applications of our formulae to resonant circuits. At the end of the paper, two other formulae of an m × n resistor network are proposed. Copyright © 2014 John Wiley & Sons, Ltd.

Formulae of resistance between two corner nodes on a common edge of the m × n rectangular network

We investigate a multifunctional n-step honeycomb network which has not been studied before. By adjusting the circuit parameters, such a network can be transformed into several different networks with a variety of functions, such as a regular ladder network and a triangular network. We derive two new formulae for equivalent resistance in the resistor network and equivalent impedance in the LC network, which are in the fractional-order domain. First, we simplify the complex network into a simple equivalent model. Second, using Kirchhoff’s laws, we establish a fractional difference equation. Third, we construct an equivalent transformation method to obtain a general solution for the nonlinear differential equation. In practical applications, several interesting special results are obtained. In particular, an n-step impedance LC network is discussed and many new characteristics of complex impedance have been found.

A fractional-order multifunctional n -step honeycomb RLC circuit network

A classic problem in electric circuit theory studied by numerous authors over 160 years is the computation of the resistance between two nodes in a resistor network, yet some basic problem in m×n cobweb network is still not solved ideally. The equivalent resistance and capacitance of 4×n cobweb network are investigated in this paper. We built a quaternion matrix equation and proposed the method of matrix transformations in terms of the network analysis. We proposed a brief equivalent resistance formula and find that the equivalent resistance is expressed by coskπ/9 in a series of strict calculation. Meanwhile, an equivalent resistance of infinite networks is gained. Using the inverse mapping relation between capacitance parameters and resistance parameters, the equivalent capacitance formula is also given for the 4×n capacitance cobweb network. By analyzing and comparing the equivalent resistances of the 1×n, 2×n, 3×n and 4×n cobweb networks, two conjectures on the equivalent resistance and capacitance of the m×n cobweb network are proposed. Copyright © 2013 John Wiley & Sons, Ltd.

Zhi-Zhong Tan

Papers

Theory on resistance of m×n cobweb network and its application

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

Formulae of resistance between two corner nodes on a common edge of the m × n rectangular network

A fractional-order multifunctional n -step honeycomb RLC circuit network

Resistance and capacitance of 4×n cobweb network and two conjectures