The analysis of nonlinear fractional-order circuits is a challenging problem. This is due to the lack of
nonlinear circuit theorems and designs particularly in the presence of memristive elements. The response
of a series connection of a simple resistor with fractional order capacitor and its analytical formulation in
both charging and discharging phases is considered. The numerical simulation of fractional order HP
memristor in series with a fractional order capacitor is also discussed. It is a demonstration of a simple
nonlinear fractional-order memristive circuit in both charging and discharging cases. Furthermore, this
paper introduces an approach to approximate nonlinear fractional-order memrisitve circuits by linear
circuits using a minimax optimization technique. Hence, the new circuit can be analyzed using the con-
ventional linear circuit theorems. The charging and discharging of a series fractional-order memristor with
a fractional-order capacitor are discussed numerically. The effect of fractional-order parameters and
memristor polarity are also investigated. Using a suitable optimization technique, an accurate approx-
imation by a circuit that include a resistor and a fractional-capacitor is obtained for both charging and
discharging cases. A great matching was observed between the frequency responses of the fractional-order
nonlinear low pass filter based on fractional-order memristor and fractional-order capacitor and that of the
optimized linear fractional order case. Similar matching is observed for the nonlinear and optimized cases
when a periodic triangular waveform is applied using Fourier series expansion. |
