Generalized modeling and character analyzing of composite fractional-order memristors in series connection

Memristor is a type of memory device representing the relation between charge q and flux phi. Recently, memristor, as well as other memristive elements and their corresponding fractional elements, have become very attractive in many applications, due to their unique behavior which cannot be obtained by using conventional elements. However, there are few studies about composite behaviors or characteristics of multiple memristive elements connected in series or parallel, especially for the fractional-order memristors. This paper focuses on analyzing the composite characteristics of two fractional-order memristors connected in series as well as concerning the window function. Under the applied sinusoidal current source, two charge-controlled memristors connected in series are adopted to theoretically demonstrate the variation of memristance and v-i curves. The two fractional order alpha and beta are interpreted as two phase offset which makes the analysis much convenient. The obtained formulas of instantaneous memristance of composite memristor are derived and some influential factors are analyzed in detail. The further analyzing demonstrates that the composite memristor circuits behave as a new memristor with higher complexity and some typical phenomena are found especially for the pinched hysteresis loops in v-i plane.