The effect of non-linear capacitances in the localization properties of aperiodic dual electric transmission lines

Abstract

This study investigates the localization properties of dual electric transmission lines with non-linear capacitances. The VC,n voltage across each capacitor is selected as a non-linear function of the electric charge qn, i.e., VC,n = qn(1/Cn -en|qn|2) where Cn is the linear part of the capacitance and en the amplitude of the non-linear term. We follow a binary distribution of values of en, according to the Thue-Morse m-tupling sequence. The localization behavior of this non-linear case indicates that the case m = 2 does not belong to the m ≥ 3, family because when m changes from m = 2 to m = 3, the number of extended states diminishes dramatically. This proves the topological difference of the m = 2 and m = 3 families. However, by increasing m values, localization behavior of the m-tupling family resembles that of the m = 2, case because the system begins to regain its extended states. The exact same result was obtained recently in the study of linear direct transmission lines with m-tupling distribution of inductances. Consequently, we state that the localization behavior of the m-tupling family as a function of the m value is independent of both the linear and the non-linear system under study, but independent of the kind of transmission line (dual or direct). This is curious behavior of the m-tupling family and thus deserves more scholarly attention.