^{1}

This paper describes the use of integer and fractional electrical elements, for modelling two electrochemical systems. A first type of system consists of botanical elements and a second type is implemented by electrolyte processes with fractal electrodes. Experimental results are analyzed in the frequency domain, and the pros and cons of adopting fractional-order electrical components for modelling these systems are compared.

Fractional calculus (FC) is a generalization of the integration and differentiation to a noninteger order. The fundamental operator is

Recent studies brought FC into attention revealing that many physical phenomena can be modelled by fractional differential equations [

Capacitors are one of the crucial elements in integrated circuits and are used extensively in many electronic systems [

Bearing these ideas in mind, this paper analyzes the fractional modelling of several electrical devices and is organized as follows. Section

In an electrical circuit, the voltage

Consequently, in complex form, the electrical impedance

In the case of a CPE, we have the model:

It is well known that, in electrochemical systems with diffusion, the impedance is modelled by the so-called Warburg element [

Based on these concepts, and in the previous works developed by the authors [

This different behavior, for low and for high frequencies, makes difficult the modelling of these systems in all frequency range. This fact motivated the study of both systems with different type of

Table

Elementar circuits of integer and fractional order.

Circuit | Circuit | |
---|---|---|

In this line of thought, in the following two sections, the impedances of botanical and fractal electrolyte systems are analyzed. In both cases, a large number of measurements were performed in order to understand and minimize the effect of nonlinearities, initial conditions, and experimental and instrumentation limitations.

The structures of fruits and vegetables have cells that are sensitive to heat, pressure, and other stimuli. These systems constitute electrical circuits exhibiting a complex behavior. Bearing these facts in mind, in our work, we study the electrical impedance of the

We apply sinusoidal excitation signals

We start by analyzing the impedance for an amplitude of input signal of

Polar and Nichols diagrams of the impedance

For the approximate modelling of the results presented in Figure

In the botanical system are applied the circuits

The resulting numerical values of

Comparison of circuit parameters for the circuits

Circuit | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

1 | 2487.00 | 4.23 | ||||||||

2 | 2271.00 | 0.84 | 1.97 | |||||||

3 | 116.00 | 2309.00 | 0.57 | |||||||

4 | 55.70 | 3198.00 | 0.68 | 0.06 | ||||||

5 | 115.00 | 2325.00 | 0.57 | |||||||

6 | 61.50 | 3274.00 | 0.68 | 0.06 | ||||||

13 | 137.90 | 0.10 | 2384.80 | 0.63 | ||||||

14 | 18.50 | 0.0002 | 4998.00 | 0.52 | 0.46 | |||||

15 | 18.80 | 0.0002 | 4968.90 | 0.52 | 0.46 | |||||

16 | 17.50 | 0.0002 | 0.52 | 0.001 | 0.46 | |||||

17 | 62.60 | 2976.90 | 759.00 | 0.62 | 0.64 | 0.18 | ||||

18 | 662.40 | 79.30 | 0.31 | 0.65 | 2.67 |

Comparison of circuit parameters for the circuits

Circuit | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

1 | 1472.10 | 15.70 | ||||||||

2 | 2889.60 | 0.59 | 1.58 | |||||||

3 | 104.50 | 2309.10 | 1.91 | |||||||

4 | 67.50 | 3198.00 | 0.68 | 0.19 | ||||||

5 | 103.60 | 2325.00 | 1.91 | |||||||

6 | 64.30 | 2691.90 | 0.69 | 0.14 | ||||||

13 | 104.90 | 0.10 | 2384.80 | 1.91 | ||||||

14 | 35.90 | 0.0002 | 3019.00 | 0.55 | 0.83 | |||||

15 | 34.60 | 0.0002 | 4968.90 | 0.53 | 1.28 | |||||

16 | 43.40 | 0.0002 | 0.58 | 1.00 | 0.62 | |||||

17 | 59.10 | 2189.70 | 630.90 | 0.62 | 0.85 | 0.16 | ||||

18 | 0.10 | 79.30 | 0.35 | 0.66 | 3.78 |

Polar and Nichols diagrams of

In a second experiment, we organized similar studies for a Kiwi. In this case, the constant adaptation resistance is

Polar and Nichols diagrams of the impedance

In this case, for modelling

Comparison of circuit parameters for

Circuit | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

1 | 178.00 | 2.48 | ||||||||

2 | 222.00 | 0.69 | 0.35 | |||||||

3 | 56.00 | 180.40 | 0.60 | |||||||

4 | 27.30 | 250.70 | 0.54 | 0.09 | ||||||

5 | 56.30 | 180.50 | 0.61 | |||||||

6 | 34.60 | 257.10 | 2887 | 0.63 | 0.13 | |||||

13 | 60.20 | 192.9 | 0.62 | |||||||

14 | 19.90 | 0.009 | 284.10 | 0.45 | 0.11 | |||||

15 | 20.10 | 0.0004 | 283.60 | 11.30 | 0.45 | 0.12 | ||||

16 | 30.20 | 0.0013 | 0.46 | 0.994 | 0.001 | 0.19 | ||||

17 | 28.10 | 249.30 | 755.10 | 0.55 | 0.86 | 1.64 | 0.09 | |||

18 | 16.80 | 30.20 | 86.50 | 0.13 | 0.0069 | 0.49 | 0.68 |

Comparison of circuit parameters for

Circuit | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

1 | 142.80 | 9.63 | ||||||||

2 | 217.80 | 0.59 | 1.48 | |||||||

3 | 39.70 | 158.40 | 3.13 | |||||||

4 | 14.20 | 241.30 | 0.54 | 0.21 | ||||||

5 | 39.60 | 158.20 | 3.13 | |||||||

6 | 21.80 | 242.00 | 2887 | 0.63 | 0.44 | |||||

13 | 39.70 | 169.9 | 3.13 | |||||||

14 | 4.30 | 0.009 | 279.00 | 0.43 | 0.32 | |||||

15 | 3.50 | 0.0004 | 280.90 | 11.30 | 0.43 | 0.33 | ||||

16 | 8.50 | 0.0013 | 0.49 | 1.00 | 0.19 | |||||

17 | 16.60 | 239.30 | 755.10 | 0.55 | 0.86 | 1.64 | 0.25 | |||

18 | 0.10 | 30.20 | 86.50 | 0.13 | 0.0069 | 0.49 | 2.85 |

We verify that a significant decreasing of the error

Polar and Nichols diagrams of

Fractals can be found both in nature and abstract objects. The impact of the fractal structures and geometries, is presently recognized in engineering, physics, chemistry, economy, mathematics, art, and medicine [

The concept of fractal is associated with Benoit Mandelbrot, that led to a new perception of the geometry of the nature [

A geometric important index consists in the fractal dimension

In this work, we adopted the classical fractal Carpet of Sierpinski and the Triangle of Sierpinski with

The simplest capacitors are constituted by two parallel electrodes separated by a layer of insulating dielectric. There are several factors susceptive of influencing the characteristics of a capacitor [^{2}.

We apply sinusoidal excitation signals

For the first experiment with fractal structures, we consider two identical single-face electrodes. The voltage, the adaptation resistance ^{−1} and two single-face copper electrodes with the Carpet of Sierpinski printout.

The resulting polar and Nichols diagrams of the electrical impedance

Comparison of circuit parameters for

Circuit | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

7 | 17.00 | 0.0027 | 2.03 | |||||||

8 | 14.20 | 0.037 | 0.60 | 0.20 | ||||||

9 | 17.20 | 0.0027 | 2.03 | |||||||

10 | 14.30 | 0.035 | 0.61 | 0.20 | ||||||

11 | 14.90 | 41.20 | 0.0033 | 0.58 | ||||||

12 | 10.20 | 4.10 | 0.038 | 0.60 | 0.18 | |||||

13 | 15.00 | 70.6 | 0.0026 | 0.001 | 0.35 | |||||

14 | 14.10 | 0.0001 | 0.040 | 0.59 | 0.20 | |||||

15 | 13.90 | 20.90 | 0.045 | 0.56 | 0.27 | |||||

16 | 14.10 | 3.50 | 0.057 | 0.52 | 0.86 | 0.31 | ||||

17 | 9.80 | 782.40 | 4.30 | 0.019 | 0.68 | 0.79 | 0.26 | |||

18 | 11.90 | 2.30 | 0.042 | 0.59 | 0.76 | 0.20 |

Comparison of circuit parameters for

Circuit | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

7 | 3.36 | |||||||||

8 | 0.032 | 0.62 | 0.23 | |||||||

9 | 2.42 | |||||||||

10 | 0.032 | 0.62 | 0.23 | |||||||

11 | 0.0034 | 0.55 | ||||||||

12 | 0.031 | 0.61 | 0.28 | |||||||

13 | 0.0027 | 0.001 | 0.37 | |||||||

14 | 0.032 | 0.62 | 0.23 | |||||||

15 | 20.90 | 0.041 | 0.58 | 0.28 | ||||||

16 | 0.031 | 0.60 | 0.006 | 0.63 | 0.28 | |||||

17 | 3.90 | 0.026 | 0.66 | 0.60 | 0.43 | |||||

18 | 2.30 | 0.043 | 0.58 | 0.009 | 1.30 | 0.20 |

Polar and Nichols diagrams of the impedance

We can analyze the approximation errors

Polar and Nichols diagrams of

In order to study the influence of the fractal printed in the surface of the electrode, we adopted another fractal, namely, the Triangle of Sierpinski.

In this case the voltage,

The resulting polar and Nichols diagrams of the electrical impedance

Comparison of circuit parameters for

Circuit | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

7 | 7.40 | 7.81 | ||||||||

8 | 5.40 | 0.072 | 0.52 | 1.48 | ||||||

9 | 7.20 | 7.81 | ||||||||

10 | 5.40 | 0.072 | 0.52 | 1.48 | ||||||

11 | 7.00 | 267.70 | 2.07 | |||||||

12 | 4.20 | 1.50 | 0.069 | 0.53 | 1.48 | |||||

13 | 6.80 | 1.07 | ||||||||

14 | 5.70 | 0.0002 | 987.00 | 0.039 | 0.61 | 0.60 | ||||

15 | 5.60 | 29.3 | 0.063 | 0.60 | 0.56 | |||||

16 | 5.80 | 0.0002 | 0.051 | 0.56 | 0.0009 | 0.87 | 0.64 | |||

17 | 5.60 | 337.30 | 30.90 | 0.031 | 0.64 | 0.0001 | 1.70 | 0.13 | ||

18 | 3.30 | 2.30 | 0.046 | 0.59 | 0.257 | 0.59 | 0.66 |

Comparison of circuit parameters for

Circuit | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

7 | 7.20 | 6.35 | ||||||||

8 | 5.70 | 0.072 | 0.52 | 1.45 | ||||||

9 | 7.20 | 6.35 | ||||||||

10 | 5.70 | 0.072 | 0.52 | 1.45 | ||||||

11 | 6.80 | 309.70 | 1.18 | |||||||

12 | 4.20 | 1.50 | 0.068 | 0.53 | 1.45 | |||||

13 | 6.70 | 1.05 | ||||||||

14 | 6.00 | 0.0002 | 420.40 | 0.013 | 0.75 | 0.59 | ||||

15 | 5.30 | 29.3 | 0.072 | 0.52 | 1.25 | |||||

16 | 6.00 | 0.0002 | 0.997 | 0.0005 | 0.012 | 0.76 | 0.59 | |||

17 | 5.80 | 363.90 | 53.20 | 0.035 | 0.63 | 0.0001 | 1.60 | 0.34 | ||

18 | 3.40 | 2.00 | 0.072 | 0.52 | 1.476 | 0.57 | 1.38 |

Polar and Nichols diagrams of the impedance

Polar and Nichols diagrams of

In conclusion, we verify that, in general, the adoption of fractional electrical elements leads to modelling circuits well adapted to the experimental data and that this direction of research should be further explored in other complex systems.

FC is a mathematical tool applied in scientific areas such as electricity, magnetism, fluid dynamics, and biology. In this paper, FC concepts were applied to the analysis of electrical fractional impedances, in botanical elements and in electrical capacitors with fractal characteristics. The introduction of the CPE element in the electric circuits led us to conclude that, for the same number of elements, we have a better approximation model and consequently a decrease in the error value. The different configurations of the polar and Nichols diagrams of the systems studied led us to modelling the systems through electrical circuit with different configurations (series and parallel) and the combination of integer and fractional-order elements in the circuits.