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

Abstract

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.

Two-point resistance of a resistor network embedded on a globe

“…Early discussions along this line are due to Tan et al [6,[8][9][10]. A similar analysis for a fan network has been given recently in [11].…”

“…An alternative direct approach of computing resistances had been developed by Tan et al [6,[8][9][10] that, when applied to the cobweb and globe networks, gives the result in terms of a single summation. This offers a direct and somewhat simpler approach.…”

“…Studies of the resistance problem have also been carried out independently by Tan et al along a route that we shall refer to as the method of direct evaluation [6,[8][9][10]. The direct method is useful in cases when there exists a special node such as a pole of the globe or the center of the cobweb, with all other nodes connected to it equally along longitudes of a globe or the radii of a cobweb.…”

“…One thus circumvents the need of diagonalizing a nonregular Laplacian matrix. The direct method of computing resistances had been applied successively to the cobweb network for fixed values of m up to m = 4 [6,8,9]. It has also been used recently to compute the resistances in a fan network [11].…”

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

“…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 . Now the research on resistor network model is no longer confined to the circuit field; it has expanded into the basic model in the application of various disciplines . Human beings have solved many abstract and complex scientific problems through modeling resistor network since the German scientist Kirchhoff (1824–1887) founded the node current law and the circuit voltage law in 1845.…”