An Interpolation in Polygonal Networks of Resistors

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.


Introduction
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] .
Fibonacci numbers appear also in a polygon of resistors where equal resistors are connected along sides and spokes 8 . This can be understood since such a polygon of resistors can be deformed to a ladder network [9][10] . 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.
Their expressions depend on the parity of the number of the sides of the polygons and are expressed in terms of the Chebyshev polynomials of the second kind or equivalently in terms of the Fibonacci polynomials. Finally we find a polygon of resistors interpolating the ones with odd numbers of sides.

Polygon of Resistors
while the resistors on the spokes have 2r p .  Following Sidhu 9 and Pareta-Lopez 10 we transform the polygon of resistors in Figure 1 or Figure 2 into a ladder network of resistors in Figure 3. In the ladder network equivalents series resistors have resistance r s while parallel resistors have resistance r p except the rightmost one with ρr p for a continuous parameter ρ in Figure 3 where ρ = 1 for odd n m = + 2 1 and ρ = 2 for even n m = 2 . We see that the nodal potential V i at the node i in Figure 1 is equal to that in Figure 3 with ρ = 1 and that the input impedance R m between nodes ⊕ = m and  in Figure 1 is equal to that in the ladder network shown in Figure 3 with ρ = 1 . The same is true for Figure 2 and Figure 3 with ρ = 2 for even n case. We apply a d.c. voltage source to the network so that which can be written in matrix form as where T is a transfer matrix given by with x r r = s p / (see Trzaska 11 ). The characteristic equation for T is then we find the recurrence relation for q i q x q q q q Vol 11 (25) | July 2018 | www.indjst.org Iterating equation (3) gives from which we find the recurrence relation for V i Since we see V V r r r 1 0 we have the attenuation (or transfer ratio) at the node i where equation (8) has been used. The first equality in equation (15) in the case of ρ = 1 was obtained by Trzaska. 11 Letting and obtain the result In view of equation (14), we see that equation (17) holds also at i = 0 .
As shown in Figure 3 the ladder equivalent of the n-sided polygon with n m = + 2 1or n m = 2 includes the ( 2 1 i+ ) -or 2i -sided polygon equivalent for 0 < < i m, respectively. In order to evaluate the input impedance R i , we use the proportionality where we have used the operator || defined by r r r r r r for the parallel combination of resistors. Solving for R i yields which becomes, due to equations (15) and (8), which can also be read off from a result by Hong and Choi. 12 The resulting expressions for A i , V i /  and R r i / p depend on the value of ρ .

Odd n Case
For n m = + 2 1we have ρ = 1 . Then we find the attenuation

so that
But due to equation (11) they are written in terms of Chebyshev polynomials of the second kind as follows (26) as was obtained by Mowery 13 and Trzaska. 11 It was also proved that 3,4 Hence A i ,V i /  , and R r i / p can be expressed in terms of the Fibonacci polynomials:

Even n Case
Using the Fibonacci polynomials, they are given by R r

Interpolation Between Odd n Cases
The expressions for even n cases have different forms from those for odd n cases. We can find expressions interpolating odd n cases by choosing ρ=0 with the corresponding polygon of resistors depicted in Figure 4. ThenV 0 0 = and equation (15) cannot be used. But from equation (13) we have In terms of the Fibonacci polynomials, they are expressed as which interpolate the results given by equation (29) for the polygons with 2 1 i+ and 2 1 i− sides.

Concluding Remarks
We (40) Finally we note that physical quantities in a ladder network with inhomogeneous resistances, for example, with exponentially varying resistances along the ladder can be calculated and will be an interesting system to be analyzed.