Applications of the Carathéodory’s Inequality for Driving Point Impedance Functions

Abstract

In this study, the Carathéodory's Inequality, which is a highly popular topic of complex analysis theory, has been applied to electrical engineering to obtain novel driving point impedance functions. In electrical engineering, driving point impedance functions correspond to positive real functions and they are used for representation of the spectral characteristics of a particular circuit. Accordingly, boundary version of the Carathéodory's inequality has been considered here assuming that the driving point empedance function, 𝑍(𝑠) has a fractional function structure with 0 < ℜ𝑍(𝑠) ≤ 𝐴 for ℜ𝑠 ≥ 0 and it is analytic in the right half plane. At the end of the analyses, new driving point impedance functions have been obtained and they have been presented with their spectral characteristics. According to simulation results, it is possible to say that the frequency responses of the obtained generic driving point impedance functions have spiky filter structures where the number of the spikes in the frequency response of these filters depend on a pre-defined parameter, 𝑛.