On the behavior of distributed networks of fractional-order capacitors

Fractional calculus has seen tremendous applications in the past couple of decades particularly in the electrochemistry field for the modeling of anomalous dynamics at electrode/electrolyte interfaces. At the core of this is the fractional-order capacitor (also known as the constant phase element (CPE)), which is a non-ideal capacitive model whose impedance is zc(s) = 1/(Cα sα) (s=j ω, Cα>0, 0<α<1), ant its current-voltage relationship is given by the Caputo fractional-order derivative ic(t) = Cα\, ₀Dₜα vc(t). The CPE has been successfully used for the modeling of impedance spectroscopy data and time-domain response of a wide range of electrochemical systems. However, a CPE of single-valued parameters can be limited in terms of modeling capability, thus a more general approach of a distributed network of CPE with multi-valued pseudocapacitances and orders is becoming necessary to investigate. In this study, we examine the behavior such a networks of CPEs, each of which is described by the Caputo fractional current-voltage equation, but with a non-negative, time-invariant weight function φ(α) for the order α. The frequency-domain behavior of the distributed-order network as well as its time-domain response to a step current excitation are derived for two special cases of φ(α) including the case of φ(α) = ∑ᵢ Cαᵢ\, δ(α-αᵢ) corresponding to a (discrete) network of parallel-connected CPEs of different fractional orders αᵢ, and different pseudocapacitances Cαᵢ, and the case of φ(α) following a (continuous) uniform distribution. Experimental validation of the derived expressions are done on a reconfigurable field programmable analog array (FPAA) platform.