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Calculation of the general impedance between adjacent nodes of infinite uniform N-dimensional resistive, inductive, or capacitive lattices Introduction

All undergraduate Electrical Engineering (EE) students study basic electric circuit theory where they get introduced to the fundamental concepts and principles of electric circuits (e.g., Ohm’s law, Kirchhoff’s circuit laws, superposition, symmetry, and the concept of equivalent circuits). To help them better understand and gain the ability to apply these concepts and principles, the students are exposed to numerous and relatively straightforward electric circuit problems involving impedance networks comprised of a finite number of circuit elements. In addition, some more complex examples of electric circuits with a special configuration are used to provoke student’s motivation and curiosity and to engage them in the subject matter. One such well-known problem is finding the effective resistance seen between opposite corners of a cubic resistor network1 comprised of twelve equal resistors each of value R resulting in an effective resistance given by Reff = 5R/6. Another well-known problem is finding the effective resistance of the one-dimensional, semi-infinite resistive-ladder network1 comprised of equal resistors each of value R which is found to be Reff = R(1+50.5)/2. Although two-dimensional (2D) network problems are generally considered to be more difficult and challenging for undergraduate EE education, there are exceptions to this view. For example, Aitchison2 considered the problem of finding the effective resistance between any two adjacent nodes of the infinite uniform 2D Liebman resistive mesh (i.e., the infinite 2D square resistive lattice comprised of identical resistors each of value R) and, using a simple solution based on superposition and symmetry, he found it to be Reff = R/2. Aitchison pointed out that, in general, undergraduate EE students found this class of problems with such solutions to be more instructive and satisfying, rather than exclusively analyzing simpler, standard circuit problems. Similarly, again using superposition and symmetry, Bartis3 calculated the effective resistance between any two adjacent nodes of three other cases of infinite 2D resistive lattices including the triangular, Honeycomb and Kagomé lattices for which the effective resistances were found to be R/3, 2R/3, and R/2, respectively. Bartis provided these problems as counter-examples to those who believe that such complex resistive network problems can not be treated at the elementary level. He also elaborated that these types of problems help teach the students how to approach more complex network problems and tackle their solutions. Indeed, these more complex infinite resistive lattice problems can serve as excellent pedagogical vehicles for teaching and motivating EE students to appreciate the power of superposition and symmetry in electrical circuit analysis. (Note that this class of 2D infinite resistive lattice problems was subsequently extended to the much more difficult calculation of the effective resistance between any two arbitrary, non- adjacent nodes using more advanced mathematical techniques including finite difference equations, random walks, and the lattice Green’s function4-9. However, the authors consider this class of problems 4-9 too advanced and, therefore, well beyond the reach of undergraduate EE students.) Recently, the authors observed that the straightforward analysis of the Aitchison and Bartis class of 2D infinite resistive lattice circuit problems2,3, above, can be extended to the most general case of calculating the effective impedance between any two adjacent nodes of any infinite uniform N-dimensional resistive, inductive, or capacitive lattice (where N = 1, 2, or 3),

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Osterberg, P., & Inan, A. (2009, June), Calculation Of The General Impedance Between Adjacent Nodes Of Infinite Uniform N Dimensional Resistive, Inductive, Or Capacitive Lattices Paper presented at 2009 Annual Conference & Exposition, Austin, Texas. 10.18260/1-2--5872