We investigate the equivalent resistance of a 3 × n cobweb network. The difference equations of the model are constructed by network analysis and their general solution is obtained by matrix transformations. It is found that the equivalent resistance can be expressed by the trigonometric function of kπ/7, which decreases with the increase of the order n. By analyzing and comparing the equivalent resistances of the 1 × n, 2 × n and 3 × n cobweb networks, a conjecture on the equivalent resistance of the m × n cobweb network is proposed.

SUMMARYA 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 cos(kπ/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.

Searching for the explicit solutions of the potential function in an arbitrary resistor network is important but difficult in physics. We investigate the problem of potential formula in an arbitrary m × n globe network of resistors, which has not been resolved before (the previous study only calculated the resistance). In this paper, an exact potential formula of an arbitrary m × n globe network is discovered by means of the Recursion-Transform method with current parameters (RT-I). The key process of RT method is to set up matrix equation and to transform two-dimensional matrix equation into one-dimensional matrix equation. In order to facilitate practical application, we deduced a series of interesting results of potential by means of the general formula, and the effective resistance between two nodes in the m × n globe network is derived naturally by making use of potential formula.

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.

We consider the problem of electrical properties of an m × n cylindrical network with two arbitrary boundaries, which contains multiple topological network models such as the regular cylindrical network, cobweb network, globe network, and so on. We deduce three new and concise analytical formulae of potential and equivalent resistance for the complex network of cylinders by using the RT-V method (a recursion-transform method based on node potentials). To illustrate the multiplicity of the results we give a series of special cases. Interestingly, the results obtained from the resistance formulas of cobweb network and globe network obtained are different from the results of previous studies, which indicates that our research work creates new research ideas and techniques. As a byproduct of the study, a new mathematical identity is discovered in the comparative study.

The unified processing and research of multiple network models are implemented, and a new theoretical advance has been made, which sets up two new theorems on evaluating the exact electrical characteristics (potential and resistance) of the complex m × n resistor networks by the recursion-transform method with potential parameters, and applies to a variety of different types of lattice structure with arbitrary boundaries such as the nonregular m × n rectangular networks and the nonregular m × n cylindrical networks. Our research gives the analytical solutions of electrical characteristics of the complex networks (finite, semi-infinite and infinite), which has not been solved before. As applications of the theorems, a series of analytical solutions of potential and resistance of the complex resistor networks are discovered.

SUMMARYA 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.

SUMMARYThis 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)ϕ t = 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.