On the behavior of distributed networks of fractional-order capacitors

Abstract 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>z</mml:mi> <mml:mi>c</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>s</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>C</mml:mi> <mml:mrow> <mml:mi>α</mml:mi> </mml:mrow> </mml:msub> <mml:msup> <mml:mi>s</mml:mi> <mml:mrow> <mml:mi>α</mml:mi> </mml:mrow> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>=</mml:mo> <mml:mi>j</mml:mi> <mml:mi>ω</mml:mi> </mml:mrow> </mml:math> , <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>C</mml:mi> <mml:mrow> <mml:mi>α</mml:mi> </mml:mrow> </mml:msub> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> , <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>&lt;</mml:mo> <mml:mi>α</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> ), and its current–voltage relationship is given by the Caputo fractional-order derivative <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>i</mml:mi> <mml:mi>c</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>α</mml:mi> </mml:msub> <mml:msub> <mml:mrow/> <mml:mn>0</mml:mn> </mml:msub> <mml:msubsup> <mml:mtext>D</mml:mtext> <mml:mi>t</mml:mi> <mml:mi>α</mml:mi> </mml:msubsup> <mml:msub> <mml:mi>v</mml:mi> <mml:mi>c</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> . 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 distributed network of CPEs with multi-valued pseudocapacitances and orders is becoming necessary to investigate. In this study, we examine the behavior of such networks of CPEs, each of which is described by the Caputo fractional current–voltage equation, but with a non-negative, time-invariant weight function <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>ϕ</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>α</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> for the order α . The frequency-domain behavior as well as the time-domain response to a step current excitation are derived for two special cases of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>ϕ</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>α</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> including the case of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>ϕ</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>α</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:munder> <mml:mo>∑</mml:mo> <mml:mi>i</mml:mi> </mml:munder> <mml:msub> <mml:mi>C</mml:mi> <mml:mrow> <mml:msub> <mml:mi>α</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> </mml:msub> <mml:mstyle scriptlevel="0"/> <mml:mi>δ</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>α</mml:mi> <mml:mo>−</mml:mo> <mml:msub> <mml:mi>α</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> corresponding to a (discrete) network of parallel-connected CPEs of different fractional orders α i , and different pseudocapacitances