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. Copyright © 2013 John Wiley & Sons, Ltd.