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Elevar la calidad del servicio que la revista presta a los autores. Asegurar la eficacia y la mejora continua del servicio. In this paper we propose a fractional differential equation for the electrical RC and LC circuit in terms of the fractional time derivatives of the Caputo type. To keep the dimensionality of the physical parameters R, L, C the new parameter a is introduced. This parameter characterizes the existence of fractional structures in the system.
The numeric Laplace transform method was used for the simulation of the equations results. An analysis in the frequency domain of an RC circuit shows the application and use of fractional order differential equations.
Although the mathematical foundation of Fractional Calculus FC was established over years ago, there remains a subject quite new to mathematicians. FC, involving derivatives and integrals of non-integer order, is the natural generalization of the classical calculus Oldham and Spanier, ; Miller and Ross, ; Samko et al , ; Podlubny et al , ; Uchaikin, In many applications FC provides more accurate models of the physical systems than ordinary calculus do.
Since its success describing anomalous diffusion Wyss et al , ; Hilfer, , Meteler and Klafter, ; Agrawal et al. Fundamental physical considerations in favor of the use of models based on derivatives of non-integer order are given in Westerlund , Veliev and Baleanu et al.
The advantage of using fractional order systems compared with systems of integer order is that the former has infinite memory, while others have finite memory. This is the main advantage of FC in comparison with the classical integer-order models, in which such effects are in fact neglected.
To analyze the dynamical behavior of a fractional system it is necessary to use an appropriate definition of the fractional derivative. In the Caputo case, the derivative of a constant is zero and we can properly define the initial conditions for the fractional differential equations so that they can be handled analogously to the classical integer case.
Working with this definition is important because of the ability to be implemented numerically Podlubny, The Caputo derivative has the following property: if f t is a constant, then its derivative is zero.
This does not happen with other representations. Another very important feature in the form of Caputo fractional derivative is that its Laplace transform is:. From equation 2 , we can see that the representation of the Caputo derivative in Laplace domain using the initial conditions f k 0 where k is integer.
If the initial conditions are zero, this reduces to:. The Laplace transform is a useful tool for analyzing linear systems because it simplifies the problem of dealing with differential equations in the time domain by converting them into algebraic equations within the frequency domain. The numerical Laplace transform NLT is essentially a modified discrete Fourier transform DFT through a windowing function Gibbs phenomenon and a stability factor aliasing , Proakis and Manolakis, Development of the NLT and its application to the analysis of systems has been well documented over the past 40 years Ramirez et al.
When using discrete techniques in the frequency domain, computation time becomes an important factor since it requires a certain amount of time to transform the data from the frequency domain to the time domain or vice versa. However, by using the fast Fourier transform FFT the time necessary for computation is greatly reduced and as a result the techniques of analysis in the frequency domain become an attractive option.
Sheng investigated the validity of applying numerical inverse Laplace transform algorithms in fractional calculus Sheng y Chen, In a paper b , an overview of a methodology based on the NLT and applied to the analysis of electromagnetic transient phenomena in power systems described by differential equations of fractional order, a Newton-type methodology to calculate either the transient or the periodic steady state is used and the definition of Caputo fractional derivative is applied.
To electrical networks including nonlinear reactors and electronic devices. In the present work we are interested in the study of a simple electrical circuit consisting of a resistor, an inductor and a capacitor, in the framework of the fractional derivative applying the method of the NLT for the simulations.
Using the Kirchhoff voltage law and the circuit of Figure 1 , we have: where E t is the source voltage, V e t is the voltage in the fractal element and V C t is the voltage in the capacitor.
Circuit with a fractal element and a capacitor. Such a way consists in analyzing the dimensionality of the ordinary derivative operator and trying to bring it to a fractional derivative operator consistently. Using the expression 8 , the fractional differential equation for a circuit of a capacitor and a resistor has the form with.
In this case the empiric relationship is given by the expression thus, the magnitude characterizes the existence of fractional structures in the system. The voltage in the capacitor is. The current can be obtained applying inverse numeric Laplace transform in 14 and differentiating with respect to time. Using the expression 16 , the fractional differential equation for a circuit containing a capacitor and an inductor has the form where.
In this case the empiric relationship is given by the expression thus, the magnitude characterize the existence of fractional structures in the system. The current can be obtained applying inverse numeric Laplace transform in 21 and differentiating with respect to time. The electrical impedance spectroscopy technique applies a potential difference between the two electrodes by passing a low power alternating current through the sample and this is compared with the voltage and current detected to the output.
The fidelity of the frequency sweep for these tests was important since it shows the characteristic spectrum of the sample, which is necessary for comparing with the electrical parameters of an equivalent circuit. The frequency range used was from 10 Hz to kHz. A voltage of 25 mV was applied across the RC circuit.
To determine the equivalent equation, the impedance is determined, which is given by the following formula in the complex frequency domain. Applying Kirchhoff laws to the circuit of Figure 1 , we have:. Before applying the Laplace transform of 24 and 25 must make the following considerations. In applying the definition of fractional derivative Podlubny, , we have:. Substituting 28 and 29 into 25 , we obtain:.
Applying Laplace transform to 30 and 31 is obtained:. Finally from 32 and 33 we have:. This value was obtained by a least-squares fit. From the Figure 7 , it can be seen that the fractional differential equation 35 obtained describes best the measurement of electrical impedance spectroscopy.
The Figures 2 and 4 show the charge and current in the RC circuit, respectively. From Figure 3 can observe that as y increases from 0. The fractional differential equation for the RC and LC circuits has been proposed. The relevant aspect of this work is the way to introduce the fractional derivatives operators, providing a systematic way to construct the fractional differential equations of any physical system keeping the dimensionality of the physical parameters.
Also, the concept of fractional time constant has been introduced. From the description of the fractional differential equation models can be noted that the representation of the Cole model is generated to solve the RC circuit in the formalism of fractional calculus.
The simulations obtained from the fractional representation showing a better description than those obtained by the equations of integer order. With the approach presented here, it will be possible to have a better study of the transient effects in the electrical systems. Also, the electrical circuit will provide a robust framework for studying the bioelectrical response to transient stimuli, when an equivalent electrical circuit has been used.
His interests are electromagnetic effects in biological systems, numerical methods applied to engineering, fractional calculus and its application to electrical circuit theory.. His interests are fractional calculus and circuit theory.. His interests are numerical methods applied to engineering and fractional calculus.. His interests are fractional calculus and its application to electrical circuit theory..
ISSN: Descargar PDF. Under a Creative Commons license. Texto completo. Introduction Although the mathematical foundation of Fractional Calculus FC was established over years ago, there remains a subject quite new to mathematicians.
For these reasons, in this paper we prefer to use the Caputo fractional derivative defined by:. Figure 1. Figure 2. Figure 3. Figure 4. Figure 5. Figure 6. Figure 7. Agrawal, J. Tenreiro-Machado, I. Fractional Derivatives and Their Applications. Nonlinear Dynamics,. Baleanu, Z. Caputo, F. Pure and Applied Geophysics, 91 , pp.
Cole, R. Bernal-Alvarado, T. Rosales, J. Bernal, M. Prespacetime Journal, 3 , pp. Bernal, J. Rosales, T. Journal of Electrical Bioimpedance, 3 ,. B, , pp. Meteler, J. Miller, B. Implementation of the Numerical Laplace Transform: a Review.
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