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Fractal Fract 2018, 2(4), 24; https://doi.org/10.3390/fractalfract2040024

Article
Power Laws in Fractionally Electronic Elements
Shanghai Key Laboratory of Multidimensional Information Processing, School of Information Science and Technology, East China Normal University, No. 500, Dong-Chuan Road, Shanghai 200241, China
Received: 28 August 2018 / Accepted: 21 September 2018 / Published: 26 September 2018

Abstract

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

1. Introduction

Let ic and uc be the current and voltage through and over a capacitor C, with the constant capacitance denoted by C again. Then, one says that Cf stands for a pseudo-capacitance in the sense that
i c = C f d α u c ( t ) d t α = C d u c ( t ) d t   for   0   <   α   <   1 ,
where d α u c ( t ) d t α = u c ( α ) ( t ) denotes the fractional derivative of order α of uc [1]. One calls Cf the pseudo-capacitance of a capacitor because its unit is Farad × s1−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 Lf, is in the sense that
u L = L f d β i L ( t ) d t β = L d i L ( t ) d t   for   0   < β <   1 ,
where uL and iL are the voltage and current over and through an inductor L with the constant inductance denoted again by L. The unit of Lf is Henry × s1−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.

2. Results

Denoted by X(ω) the Fourier transform of x(t). Then, one has, for α > 0,
F [ x ( α ) ( t ) ] = x ( α ) ( t ) e j ω t d t = ( j ω ) α X ( ω ) ,
where j = 1 . Consequently,
x ( α ) ( t ) = F 1 [ ( j ω ) α X ( ω ) ] = 1 2 π ( j ω ) α X ( ω ) e j ω t d ω ,
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
Cf = ()1−αC.
Proof. 
The Fourier transform of C f d α u c ( t ) d t α in Equation (1) is given by
F [ C f d α u c ( t ) d t α ] = ( j ω ) α C f U c ( ω ) ,
where Uc(ω) = F[uc(t)]. On the other hand, doing the Fourier transform of C d u c ( t ) d t in Equation (1) produces
F [ C d u c ( t ) d t ] = j ω C U c ( ω ) .
Thus, according to Equation (1) and from Equations (6) and (7), we have ()αCfUc(ω) = jωCUc(ω). Therefore, we have Cf = ()1−αC. Hence, Theorem 1 holds. ☐
Note 1. 
Cf reduces to C if α → 1. We use the symbol Cf to represent either fractional capacitance or fractional capacitor.
Corollary 1. 
Denote the capacitance ratio by
Rc = C/Cf.
Then, Rc follows the power law in the form
R c = R c ( f , α ) = ( j 2 π f ) α 1 .
Proof. 
From Equation (2.3), we have R c = C C f = ( j ω ) α 1 = ( j 2 π f ) α 1 . 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 Hertzα−1. Figure 1 shows the plots of |Rc(f, α)| = (2πf )α−1.
Theorem 2. 
The fractional inductance Lf may be in the form
Lf = ()1−βL.
Proof. 
The Fourier transform of L f d β i L ( t ) d t β in Equation (2) is in the form
F [ L f i L ( β ) ( t ) ] = ( j ω ) β L f I L ( ω ) ,
where IL(ω) = F[iL(t)]. On the other side, in Equation (2), we have
F [ L i L ( t ) d t ] = ( j ω ) L I L ( ω ) .
From Equation (2) and according to Equations (11) and (12), we have ( j ω ) β L f I L ( ω ) = j ω L I L ( ω ) . Thus, we have Lf = (jω)1βL. This completes the proof. ☐
Note 2. 
The fractional inductance Lf degenerates to L when β → 1. The symbol Lf stands for both fractional inductance and fractional inductor.
Corollary 2. 
Let Rl be the inductance ratio in the form
Rl = L/Lf.
Then, it follows the power law in the form
R l = R l ( f , β ) = ( j 2 π f ) β 1 .
Proof. 
From Equation (10), we have R l = L L f = ( j ω ) β 1 = ( j 2 π f ) β 1 . This completes the proof. ☐
Corollary 2 exhibits a power law of Rl in terms of frequency with respect to Lf. The unit of Rl is Hertzβ−1.

3. 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 Cf (Lf) obeys (j2πf )α−1 with the unit Hertzα−1. Specifically for a fractional capacitor, due to 0 < α < 1, the power law described by Corollary 1 reveals that Cf → ∞ 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.

Funding

This work was supported in part by the National Natural Science Foundation of China under the project grant numbers 61672238 and 61272402.

Conflicts of Interest

The author declares no conflicts of interest.

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Figure 1. The plots of |Rc(f, α)|. Solid line: α = 0.5. Dot line: α = 0.8.
Figure 1. The plots of |Rc(f, α)|. Solid line: α = 0.5. Dot line: α = 0.8.
Fractalfract 02 00024 g001

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