Abstract
Objectives: Ladder networks of resistors have been discussed extensively. This paper considers polygons of resistors where the resistors on sides are different from those on spokes. The objective is to find how their physical quantities depend on the parity of the number of the sides. Methods: We calculate attenuations, nodal potentials, and input impedances when a voltage source is connected between a node and the center. We introduce a continuous parameter ρ in equivalent ladder networks where ρ =1 and ρ = 2 correspond to odd and even numbers of sides, respectively. Findings: Attenuations, nodal potentials, and input impedances are expressed in terms of the Chebyshev polynomials of the second kind or the Fibonacci polynomials. The results depend on the parity of the number of sides. The case ρ = 0 interpolates the case with the odd numbers of sides. Application: The method presented in this document can be applicable to networks with inhomogeneous resistances around the sides. Keywords: Chebychev Polynomials, Electric Circuit, Fibonacci polynomials, Interpolation, Polygonal Network
Highlights
Ladder networks consist of passive elements like resistors, capacitors, and inductors and have applications in filters and transmission lines
Attenuations, nodal potentials, and input impedances are expressed in terms of the Chebyshev polynomials of the second kind or the Fibonacci polynomials
It is well known that the Fibonacci numbers appear in a ladder network of equal resistors[1]
Summary
Ladder networks consist of passive elements like resistors, capacitors, and inductors and have applications in filters and transmission lines. It is well known that the Fibonacci numbers appear in a ladder network of equal resistors[1] Physical quantities such as input impedances (or equivalent resistances), attenuations, and nodal potentials in a ladder of resistors which is homogeneous along the ladder, that is, has identical series and identical parallel (shunt) resistors, respectively, have been calculated. They are expressed in terms of Morgan-Voyce polynomials which have been studied extensively[2,3,4,5,6,7]. The purpose of this article is to determine physical quantities of the polygons where the resistors on the sides are different from those on the spokes
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