Abstract
This paper provides an analytic solution of electrical circuit described by a fractional differential equation of the order . We use the Laplace transform of the fractional derivative in the Caputo sense. Some special cases for the different source terms have also been discussed.
1. Introduction
Fractional calculus, involving derivatives of integrals of noninteger order, is the natural generalization of the classical calculus, which during recent years became a powerful and widely used tool for better modeling and control of processes in many areas of science and engineering [1–3]. Many physical phenomena have been discussed by fractional calculus approach [4]. In many applications, fractional calculus provides more accurate models of the physical systems than ordinary calculus does. Fundamental physical considerations in favor of the use of models based on derivatives of noninteger order are given in [5]. Fractional derivatives provide an excellent instrument for the description of memory and hereditary properties of various materials and processes [6]. This is the main advantage of fractional calculus in comparison with the classical integer-order models, in which such effects are in fact neglected. Fractional order models have been already used for modeling of electrical circuits (such as domino ladders and tree structures) and elements (coils, memristor, etc.). The review of such models can be found in [7].
Recently, a fractional differential equation has been suggested that combines the simple harmonic oscillations of an LC circuit with the discharging of an RC circuit. The behavior of this new hybrid circuit without sources has been analyzed [8]. In the work of [9], the simple current source-wire circuit has been studied fractionally using direct and alternating current source. It was shown that the wire acquires an inducting behavior as the current is initiated in it and gradually recovers its resisting behavior. Recently, Guia et al. [10] have analyzed time delay, rise time, and settling time of an RC circuit. In this paper, in the framework of fractional calculus, we are interested in the solution of an circuit for different source terms.
2. Preliminary
The function of the complex variable defined by is called the Laplace transform of the function [3]. For the existence of the integral (1), the function must be piecewise continuous and of exponential order .
The original can be restored from the Laplace transform with the help of the inverse Laplace transform [3] where lies in the right half plane of the absolute convergence of the Laplace integral (1). In this work, we use the Caputo definition of the fractional derivative [3] where is the order of the fractional derivative and , , and is the Euler Gamma function. We consider the case ; then ; that is, in the integrand (3), there is only first derivative. The Caputo definition of the fractional derivative is very useful in the time domain studies, because the initial conditions for the fractional order differential equations with the Caputo derivatives can be given in the same manner as for the ordinary differential equations with a known physical interpretation.
The formula for the Laplace transform of the Caputo fractional derivative (3) has the form [3] where is the ordinary derivative. The inverse Laplace transform requires the introduction of the Mittag-Leffler function [3], which is defined as where is the Gamma function. When , from (5), we have Therefore, the Mittag-Leffler function includes the exponential function as a special case.
3. Formulation of Fractional Differential Equation Models for Flow of Electricity in Resistance-Inductance Circuit
The differential equation for the circuit shown in Figure 1 is given by where is the current and is the inductance.
The solution of circuit is reported by Kreyszig [11] as In this paper, we develop the resistance-inductance circuit model in the form of fractional differential equation as If , then (9) reduces to its original form .
Since , it means that is very, very small.
Throughout the paper, we consider and . Therefore, we are not interested to consider the dimensionless variables.
Solution of (9) Using the Initial Condition . Applying the Laplace transform on (9) using the initial condition , we have Now, applying the inverse Laplace transform and convolution, we have Here, we obtain the solution of different cases of the resistance-inductance circuit model (9) for different source terms.
Case 1 (when no electromotive force is applied (no source term), i.e., ). In this case, (9) becomes and the initial condition is
Solution. Rewrite (12a) as Taking the Laplace transform on both sides and using (12b), we get Taking the inverse Laplace transform on both sides, we obtain
Case 2 (when constant electromotive force is applied, i.e., ). In this case, (9) becomes and the initial condition is Rewrite (16a) as Taking the Laplace transform on both sides and using (16b), we get Taking the inverse Laplace transform on both sides, we have on applying the convolution theorem and using [12, page 109], we get and from (15), we have Using (19), (20), and (21), we get
In Figure 2, we observe the interesting behavior of current by using fractional calculus approach for different values of . When , the current decreases very sharply and it moves towards stability as time increases. On increasing the value of , the current increases for the specific time and afterwards attains its stability. Finally, when , then current shows its natural behavior. This exhibits the behavior of current for different values of with respect to time () before it attends the natural behavior.
| (a) |
| (b) |
Case 3 (when electromotive force is in terms of unit step function, i.e., ). In this case, (9) becomes
where is a unit step function.
The initial condition is
Solution. Rewrite (23a) as Taking the Laplace transform on both sides and using (23b), we get Taking the inverse Laplace transform, we obtain on applying the convolution theorem and using the result = [12, page 109], we get and using (15), (26), and (27), we obtain
Case 4 (when periodic electromotive force is applied, i.e., ). In this case, (9) becomes The initial condition is
Solution. Rewrite (29a) as Taking the Laplace transform on both sides and using (29b), we get Taking the inverse Laplace transform, we obtain on applying the convolution theorem, this yields Using (15), (32), and (33), we obtain
Case 5 (when periodic electromotive force is applied, i.e., ). In this case, (9) becomes The initial condition is
Solution. Rewrite (35) as Taking the Laplace transform on both sides and using (36), we get Taking the inverse Laplace transform, we obtain on applying the convolution theorem, this gives Further, using (15), (39), and (40) we get
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors are grateful to the anonymous referees for their careful reading and helpful comments to improve the presentation of the paper. The third author is thankful to University Grants Commission, New Delhi, India, for awarding Dr. D. S. Kothari Postdoctoral Fellowship (Award no. File no. F.4-2/2006 (BSR)/13-803/2012 (BSR) dated 09/11/2012).