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

Abstract

We consider the problem of two-point resistance in an (m-1) × n resistor network embedded on a globe, a geometry topologically equivalent to an m × n cobweb with its boundary collapsed into one single point. We deduce a concise formula for the resistance between any two nodes on the globe using a method of direct summation pioneered by one of us [Z.-Z. Tan, L. Zhou, and J. H. Yang, J. Phys. A: Math. Theor. 46, 195202 (2013)]. This method is contrasted with the Laplacian matrix approach formulated also by one of us [F. Y. Wu, J. Phys. A: Math. Gen. 37, 6653 (2004)], which is difficult to apply to the geometry of a globe. Our analysis gives the result in the form of a single summation.

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

“…At the end of this paper, we propose a profound problem based on the previous research in the article : how to derive the resistance of an m × n network of cross resistors as shown in Figure . We look forward to the solution of the problem.…”

“…The computation of the two‐point resistance in networks is a classical problem in circuit theory and graph theory, which has been researched for more than 170 years . The computation of resistances is relevant to a wide range of problems ranging from science and technology to engineering . However, it is usually very difficult to obtain the exact resistance in complex resistor networks ; the construction of and research on the models of complex networks therefore make sense for theories and applications.…”

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

“…where t i is given in equation (5). Substituting equations (32) and (33) into equation 31   ) can be obtained after some algebra and reduction…”

Recursion-Transform method to a non-regular m×n cobweb with an arbitrary longitude

“…After some improvements, the Laplacian approach has been extended to the complex impedance network 12 and to the resistor network with zero resistor boundary 13 14 15 ; The fourth method is that Tan created the Recursion-Transform ( R-T ) method 16 (one may refer Ref. 17 , 18 , 19 , 20 , 21 ), which compute the equivalent resistance relies on just one matrix along one direction, and the explicit resistance is expressed by a single summation 17 18 19 20 21 . The advantage of the R-T method is different from the Laplacian matrix method is that it only dependent on a matrix not two matrices.…”

“…For the arbitrary resistor networks of various topologies, various new results of the resistor networks have been obtained by applying the R-T method. For example, a cobweb model 17 , a globe network 18 , a fan network 19 , a cobweb network with 2 r boundary 20 , a hammock network 15 , especially, very recently this method was further generalized in Ref. 21 , which can applied to the resistor network with an arbitrary boundary.…”