Beginning to Count the Number of Equivalent Resistances

Sameen Ahmed Khan lowast

doi:10.17485/ijst/2016/v9i44/88086

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Beginning to Count the Number of Equivalent Resistances

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Indian Journal of Science and Technology

DOI: 10.17485/ijst/2016/v9i44/88086

Year: 2016, Volume: 9, Issue: 44, Pages: 1-7

Original Article

Beginning to Count the Number of Equivalent Resistances

Sameen Ahmed Khan^∗

*Author for correspondence
Sameen Ahmed Khan Department of Mathematics and Sciences, College of Arts and Applied Sciences, Dhofar University, Salalah, Onam; [email protected]

This work is licensed under a Creative Commons Attribution 4.0 International License.

Abstract

Given a bunch of equal resistors, what are all the values of the equivalent resistances which one can obtain? Is there a formula? In this note, we address such questions analytically and establish strict lower and upper bounds of the cardinality (or the order) of the set of equivalent (or net) resistances using series and parallel connections. Our results are obtained using the Farey sequences and Fibonacci numbers. For n equal resistors, the order of the set is bounded from below by (0.25)(2.414)n and from above by (0.318)(2.618)n .

Keywords: Bridge, Fibonacci Numbers, Haros-Farey Sequences, Non-Planar Circuits, Parallel, Resistor Networks, Series .