Power Laws in Fractionally Electronic Elements

The highlight presented in this short article is about the power laws with respect to fractional capacitance and fractional inductance in terms of frequency.


Introduction
Let i c and u c be the current and voltage through and over a capacitor C, with the constant capacitance denoted by C again.Then, one says that C f stands for a pseudo-capacitance in the sense that where c (t) denotes the fractional derivative of order α of u c [1].One calls C f the pseudo-capacitance of a capacitor because its unit is Farad × s 1−α instead of Farad [1].In this article, we call it fractional capacitance of order α of a capacitor.Similarly, the fractional inductance of order β, denoted by L f , is in the sense that where u L and i L are the voltage and current over and through an inductor L with the constant inductance denoted again by L. The unit of L f is Henry × s 1−β .It is also called the pseudo-inductance [1,2].Fractional elements, including a fractional capacitor and a fractional inductor, attract research interests in engineering.The literature about their analysis and applications is rich, see References [1][2][3][4][5][6][7][8][9][10], referring [11][12][13][14] to some recent work on fractional calculus.However, reports about power laws that fractional elements follow are rarely seen.This short article aims at expounding the power laws that fractional elements follow.
In the rest of this article, we present the results in Section 2, which is followed by concluding remarks.
Theorem 1.The fractional capacitance C f may be expressed by Proof.The Fourier transform of C f dt α in Equation ( 1) is given by where On the other hand, doing the Fourier transform of Thus, according to Equation (1) and from Equations ( 6) and ( 7), we have (jω We use the symbol C f to represent either fractional capacitance or fractional capacitor.

Corollary 1. Denote the capacitance ratio by
Then, Rc follows the power law in the form Proof.From Equation (2.3), we have Corollary 1 suggests a power law of Rc in terms of frequency with respect to the fractional capacitor C f .The unit of Rc is Hertz α−1 .Figure 1 shows the plots of |Rc(f, see Miller and Ross [15], Uchaikin [16], Section 4.5.3, and Lavoie [17], p. 246.Following Miller and Ross [15], Raina and Koul [18], we explain our research in the domain of generalized functions.Thus, any function considered in this article is differentiable of any times and its Fourier transform exists (Gelfand and Vilenkin [19]).

Theorem 1. The fractional capacitance Cf may be expressed by
Proof.The Fourier transform of 1) is given by where

Corollary 1. Denote the capacitance ratio by
Then, Rc follows the power law in the form Proof.From Equation (2.3), we have .
The proof completes.□ Corollary 1 suggests a power law of Rc in terms of frequency with respect to the fractional capacitor Cf.The unit of Rc is Hertz α−1 .Figure 1 shows the plots of |Rc(f, α)| = (2πf ) α−1 .Theorem 2. The fractional inductance L f may be in the form Proof.The Fourier transform of L f dt β in Equation ( 2) is in the form where On the other side, in Equation ( 2), we have From Equation ( 2) and according to Equations ( 11) and ( 12), we have (jω) β L f I L (ω) = jωLI L (ω).Thus, we have L f = (jω) 1−β L. This completes the proof.Note 2. The fractional inductance L f degenerates to L when β → 1.The symbol L f stands for both fractional inductance and fractional inductor.

Corollary 2. Let Rl be the inductance ratio in the form
Then, it follows the power law in the form Proof.From Equation (10), we have Rl = L L f = (jω) β−1 = (j2π f ) β−1 .This completes the proof.
Corollary 2 exhibits a power law of Rl in terms of frequency with respect to L f .The unit of Rl is Hertz β−1 .

Concluding Remarks
We have presented Theorems 1 and 2 to express the fractional capacitance and fractional inductance, respectively.In addition, power laws in terms of frequency with respect to fractional capacitance and fractional inductance have been given in Corollaries 1 and 2. To be precise, for a fractional capacitor (inductor) of order α, the ratio of C (L) to C f (L f ) obeys (j2πf ) α−1 with the unit Hertz α−1 .Specifically for a fractional capacitor, due to 0 < α < 1, the power law described by Corollary 1 reveals that C f → ∞ when f → 0. Note that a key property of a supercapacitor or an ultracapacitor utilized in batteries is that it has an infinitely large capacitance for f → 0 [20][21][22].Therefore, the power law presented in Corollary 1 provides a new explanation about that as an application in the case of supercapacitors.