# A New Look at the Capacitor Theory

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## Abstract

**:**

## 1. Introduction

#### Remarks

- Our working domain is always $\mathbb{R}$.
- We use the bilateral Laplace transform (LT):$$\mathcal{L}\left(\right)open="["\; close="]">f\left(t\right)$$
- The Fourier transform is obtained from the LT through the substitution $s=j\omega $ with $\omega \in \mathbb{R}$ and $j=\sqrt{-1}$.
- The inverse LT is given by the Bromwich integral$$f\left(t\right)={\mathcal{L}}^{-1}\left(\right)open="["\; close="]">F\left(s\right)$$
- Current properties of the Dirac delta distribution, $\delta \left(t\right)$, and its derivatives will be used.
- The order of the fractional derivative is assumed to be any real number.
- The multi-valued expression ${s}^{\alpha}$ is used. To obtain a function we will fix for a branch-cut line the negative real half axis and select the first Riemann surface.
- It is very common to add the prefix pseudo” to the “fractionalisation” of classic entities, as is also the case for "capacitance", which appears as “pseudo-capacitance”. We do not find any particular reason to do so [1].

## 2. Fractional Devices and Derivatives

#### 2.1. The Differintegrator

**Remark**

**1.**

#### 2.2. Suitable Fractional Derivatives

**Example**

**1.**

## 3. On the Capacitor

#### 3.1. Classic: $q\left(t\right)=c\left(t\right)v\left(t\right)$

#### 3.2. Fractional: $\frac{{d}^{1-\alpha}q\left(t\right)}{d{t}^{1-\alpha}}=c\left(t\right)v\left(t\right)$

#### 3.3. A Strange Result

## 4. On the Fractional Inductor

**Remark**

**2.**

## 5. Responses of Fractional Ideal Capacitor

#### 5.1. Formulation

- 1.
- Voltage$$i\left(t\right)={\displaystyle \frac{{d}^{\alpha}\left(\right)open="["\; close="]">c\left(t\right)v\left(t\right)}{}d{t}^{\alpha}}$$
- 2.
- Current$$v\left(t\right)={\displaystyle \frac{1}{c\left(t\right)}}{\displaystyle \frac{{d}^{-\alpha}}{d{t}^{-\alpha}}}i\left(t\right),$$

**Remark**

**3.**

#### 5.2. The Voltage Input Case

**Remark**

**4.**

**Example**

**2.**

**Example**

**3.**

**Example**

**4.**

**Remark**

**5.**

**Example**

**5.**

#### 5.3. The Current Input Case

- 1.
- The use of the fractional anti-derivative ($\alpha <0$) that is defined by a regular integral:$$\begin{array}{cc}\hfill v\left(t\right)=& \frac{1}{\Gamma \left(\alpha \right)c\left(t\right)}{\int}_{0}^{\infty}{\tau}^{\alpha -1}i(t-\tau )d\tau \hfill \\ \hfill =& \frac{1}{\Gamma \left(\alpha \right)c\left(t\right)}{\int}_{-\infty}^{t}i\left(\tau \right){(t-\tau )}^{\alpha -1}d\tau ;\hfill \end{array}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.166667em}{0ex}}t\in \mathbb{R}.$$
- 2.
- The involvement of the function $\frac{1}{c\left(t\right)}$ does not add complexity to the situation.

**Example**

**6.**

**Example**

**7.**

## 6. Power and Energy

- 1.
- $\alpha =1$The integrand degenerates into a $\delta \left(t\right)$ and the integral gives 1. Thus, the energy is $\mathcal{E}={C}_{0}$.
- 2.
- $\alpha <1$$$\mathcal{E}={C}_{0}{\left(\right)}_{\frac{{t}^{1-\alpha}}{\Gamma (2-\alpha )}}^{}0\infty $$

- $t\in \mathbb{R}$;
- ${\omega}_{m}=-{\omega}_{-m},\phantom{\rule{0.277778em}{0ex}}m\in \mathbb{Z}$;
- ${V}_{-m}={V}_{m}^{\ast},\phantom{\rule{0.277778em}{0ex}}m\in \mathbb{Z}$;
- ${\sum}_{m=-\infty}^{\infty}{\left|{V}_{m}\right|}^{2}<\infty .$

## 7. Variable Order Capacitors and Inductors

## 8. Tempered Fractors

## 9. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**MDPI and ACS Style**

Ortigueira, M.D.; Martynyuk, V.; Kosenkov, V.; Batista, A.G.
A New Look at the Capacitor Theory. *Fractal Fract.* **2023**, *7*, 86.
https://doi.org/10.3390/fractalfract7010086

**AMA Style**

Ortigueira MD, Martynyuk V, Kosenkov V, Batista AG.
A New Look at the Capacitor Theory. *Fractal and Fractional*. 2023; 7(1):86.
https://doi.org/10.3390/fractalfract7010086

**Chicago/Turabian Style**

Ortigueira, Manuel Duarte, Valeriy Martynyuk, Volodymyr Kosenkov, and Arnaldo Guimarães Batista.
2023. "A New Look at the Capacitor Theory" *Fractal and Fractional* 7, no. 1: 86.
https://doi.org/10.3390/fractalfract7010086