# Power Laws in Fractionally Electronic Elements

## Abstract

**:**

## 1. Introduction

_{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

_{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

_{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].

## 2. Results

**Theorem 1.**

_{f}may be expressed by

_{f}= (jω)

^{1−α}C.

**Proof.**

_{c}(ω) = F[u

_{c}(t)]. On the other hand, doing the Fourier transform of $C\frac{d{u}_{c}(t)}{dt}$ in Equation (1) produces

^{α}C

_{f}U

_{c}(ω) = jωCU

_{c}(ω). Therefore, we have C

_{f}= (jω)

^{1−α}C. Hence, Theorem 1 holds. ☐

**Note 1.**

_{f}reduces to C if α → 1. We use the symbol C

_{f}to represent either fractional capacitance or fractional capacitor.

**Corollary 1.**

_{f}.

**Proof.**

_{f}. The unit of Rc is Hertz

^{α}

^{−1}. Figure 1 shows the plots of |Rc(f, α)| = (2πf )

^{α}

^{−1}.

**Theorem 2.**

_{f}may be in the form

_{f}= (jω)

^{1−β}L.

**Proof.**

_{L}(ω) = F[i

_{L}(t)]. On the other side, in Equation (2), we have

_{f}= (jω)

^{1}

^{−β}L. This completes the proof. ☐

**Note 2.**

_{f}degenerates to L when β → 1. The symbol L

_{f}stands for both fractional inductance and fractional inductor.

**Corollary 2.**

_{f}.

**Proof.**

_{f}. The unit of Rl is Hertz

^{β}

^{−1}.

## 3. Concluding Remarks

_{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.

## Funding

## Conflicts of Interest

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**MDPI and ACS Style**

Li, M.
Power Laws in Fractionally Electronic Elements. *Fractal Fract.* **2018**, *2*, 24.
https://doi.org/10.3390/fractalfract2040024

**AMA Style**

Li M.
Power Laws in Fractionally Electronic Elements. *Fractal and Fractional*. 2018; 2(4):24.
https://doi.org/10.3390/fractalfract2040024

**Chicago/Turabian Style**

Li, Ming.
2018. "Power Laws in Fractionally Electronic Elements" *Fractal and Fractional* 2, no. 4: 24.
https://doi.org/10.3390/fractalfract2040024