RLC electrical circuit of non-integer order

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1 ent. Eur. J. Phys. (0) DOI: 0.478/s entral European Journal of Physics L electrical circuit of non-integer order esearch Article Francisco Gómez, Juan osales, Manuel Guía Departamento de Ingeniería Eléctrica, División de Ingenierías ampus Irapuato-Salamanca, Universidad de Guanajuato arretera Salamanca-Valle de Santiago, km km, omunidad de Palo Blanco, 36885, Salamanca Guanajuato, México eceived 3 January 03; accepted 08 June 03 Abstract: In this work a fractional differential equation for the electrical L circuit is studied. The order of the derivative being considered is 0 < γ. To keep the dimensionality of the physical quantities, L and an auxiliary parameter σ is introduced. This parameter characterizes the existence of fractional components in the system. It is shown that there is a relation between γ and σ through the physical parameters L of the circuit. Due to this relation, the analytical solution is given in terms of the Mittag-Leffler function depending on the order γ of the fractional differential equation. PAS (008): 45.0.Hj; Bv; 84.3.Ff; 84.3.Tt Keywords: fractional calculus aputo derivative electrical circuits Mittag-Leffler function Versita sp. z o.o.. Introduction Although the application of Fractional alculus (F) has attracted interest of researches in recent decades, it has a long history when the derivative of order 0.5 has been described by Leibniz in a letter to L Hospital in 695. F, involving derivatives and integrals of non-integer order, is the natural generalization of the classical calculus [] [5]. Many physical phenomena have intrinsic fractional order description and so F is necessary in order to explain them. In many applications F provide more accurate models of the physical systems than ordinary calculus do. Since its success in description of anomalous diffusion [6] [9] non-integer order calculus both in one and multidimensional space, it has become an important tool in many areas of physics, mechanics, chemistry, engineer- jgomez@ugto.mx ing, finances and bioengineering [0] [6]. Fundamental physical considerations in favor of the use of models based on derivatives of non-integer order are given in [7] [9]. The Lagrangian and Hamilton formulation of dynamics and electromagnetic field in view of fractional calculus has been reported in [0]-[5]. Modeling as fractional order proves to be useful particulary for systems where memory or hereditary properties play a significant role. This is due to the fact that an integer order derivative at a given instant is a local operator which considers the nature of the function only at that instant and its neighborhood, whereas a fractional derivative takes into account the past history of the function from some earlier point in time, called lower terminal up to the instant at which the derivative is to be computed. Fractional derivatives provide an excellent instrument for the description of memory and hereditary properties of various materials and processes [6]. Another large field which requires the use of F is the theory of fractals [7] [30]. The development of the theory of fractals has 36 Download Date /4/9 : AM

2 L electrical circuit of non-integer order opened further perspective for the theory of fractional derivatives, especially in modeling dynamical processes in self-similar and porous structures. Fractional-order models have been already used for modeling of electrical circuits (such as domino ladders, tree structures, etc.) and elements (coils, memristor, etc.). The review of such models can be found in [3]-[33]. It has been suggested a fractional differential equation that combines the simple harmonic oscillations of an L circuit with the discharging of an circuit. The behavior of this new hybrid circuit without sources has been analyzed [34]. In the work [35] 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. ecently, in [36] has been proposed a systematic way to construct fractional differential equations for the physical systems. In particular, the systems mass-spring and spring-damper has been analyzed. Such a way consists in to analyze the dimensionality of the ordinary derivative operator and try to bring it to a fractional derivative operator consistently. In the present work we obtain a solution for the fractional L circuit which is the generalization of the solution obtained in [36].. Overview on fractional calculus To analyze the dynamical behavior of a fractional system it is necessary to use an appropriate definition of fractional derivative. In fact, the definition of the fractional order derivative is not unique and there are several definitions, including: Grünwald-Letnikov, iemann-liouville, Weyl, iesz and the aputo representation. In the aputo case, the derivative of a constant is zero and we can define, properly, the initial conditions for the fractional differential equations which can be handled by using an analogy with the classical case (ordinary derivative). aputo derivative implies a memory effect by means of a convolution between the integer order derivative and a power of time. For this reason, in this paper we prefer to use the aputo fractional derivative [] [4]: d γ dt γ f(t) = 0 D γ t f(t) = t Γ(n γ) 0 f (n) (τ) dτ () (t τ) γ n+ where γ is the order of the fractional derivative, n < γ n N =,,..., and f (n) (τ) = dn f(τ) dτ n, Γ(x) = 0 e t t x dt, () are the ordinary derivative and the Gamma function, respectively. The Laplace transform of the aputo fractional derivative is given by [4] d γ f(x) L dx γ n = s γ F(s) s γ m f (m) (0), n < γ < n. (3) The aputo derivative operator satisfies the following relations 0 D γ t [f(t) + g(t)] = 0 D γ t f(t) + 0 D γ t g(t), 0 D γ t = 0, where is constant. (4) For example, in the case f(t) = t k, where k is arbitrary number and 0 < γ we have the following expression for the fractional derivative 0 D γ t t k kγ(k) = Γ(k + γ) tk γ, (0 < γ ) (5) where Γ(k) and Γ(k + γ) are the Gamma functions. If γ = the expression (5) yields the ordinary derivative 0 Dt t k = dtk dt = ktk. (6) During the recent years the Mittag-Leffler function has caused extensive interest among physicist due to its role played in describing realistic physical systems with memory and delay. The Mittag-Leffler function is defined by the series expansion as E a (t) = t m, (a > 0), (7) Γ(am + ) where Γ( ) is the Gamma function. When a =, from (7) we have E (t) = t m Γ(m + ) = t m m! = et. (8) Therefore, the Mittag-Leffler function is a generalization of the exponential function. 36 Download Date /4/9 : AM

3 Francisco Gómez, Juan osales, Manuel Guía 3. Fractional L circuit An oscillating circuit in series, in general, is an electrical circuit consisting of three kinds of circuit elements: a resistor with a resistance measured in ohms, an inductor with an inductance L measured in henries, and a capacitor with capacitance measured in farads. The change with respect to time of the electric charge q(t) in the shell of the capacitor is described by the homogeneous differential equation L d q(t) dt + dq(t) dt + q(t) = 0. (9) q(t) The term,, is very important because its lack in (9) implies that we have not an oscillating circuit. The main goal of this work is the study of the differential equation (9) from the point of view of the fractional calculus. Unlike other works [34]-[38], in which the pass from ordinary derivative to fractional one is direct, in [36] a systematic way to construct fractional differential equations for the physical systems has been proposed. It was proposed the transition of the ordinary derivative operator to the fractional operator as follows: d dt d γ σ γ dt, (0) γ where the auxiliary parameter σ has dimension of seconds and γ is an arbitrary parameter representing the order of the fractional time derivative, and in the case γ = it becomes the usual derivative. The expression (0) is a time derivative in the usual sense, because its dimension is s. The parameter σ (auxiliary parameter) represents the fractional time components in the system, components that show an intermediate behavior between a system conservative and dissipative. The physical and geometrical interpretation of the fractional operators is given in [39]-[40]. Using (0), the fractional differential equation corresponding to (9) is given by L d γ q σ ( γ) dt + d γ q γ σ γ dt + q(t) = 0, 0 < γ. () γ where the fractional derivative in () is the aputo derivative (). The solution of the equation () may be obtained applying direct and inverse Laplace transform. For underdamped case, the solution is given by γ σ q γl (t) = q 0 E γ tγ L [ E γ L ] σ ( γ) t γ, 4L () with < L/, and q γl (0) = q 0. In () ω0 = is L the undamped natural frequency expressed in radians per second, and α = is the damping factor expressed in L nepers per second. In the case γ =, from (), we have the well-known result q L (t) = q 0 e L t cos ( ) L 4L t, (3) showing an ordinary underdamped system characterized by a constant of time τ = L/ and undamped natural frequency ω 0. From (), we see that there is a relation between γ and σ given by ( γ = L ) / σ, 0 < σ 4L ) /. L 4L (4) Then, the solution () for the underdamped case < L/ or α < ω 0 takes the form where q γl ( t) = q 0 E γ L L E γ γ ( γ) t γ, ( 4L γ ( γ) t γ (5) ( t = L ) / t. (6) 4L Due to the condition < L/ we can choose, as an example = L 5, 0 <. (7) L L 4L L 4L So, the solution (5) takes its final form q γl ( t) = q 0 E γ 5 γ( γ) t γ E γ γ ( γ) t γ. (8) Figure shows the plot of the solution (8) for different values of γ. In the overdamped case α > ω 0 or > L/ the solution of the equation () has the form q γl (t) = q 0 E γ [ E γ 4L L σ ( γ) t γ L ] / σ ( γ) t γ. (9) 363 Download Date /4/9 : AM

4 L electrical circuit of non-integer order q γl ( t)/q γ = t Then, the solution (9) can be written in its final form q γl ( t) = q 0 E γ γ ( γ) t γ E γ γ ( γ) t γ. (5) Figure shows the plot of the solution (5) for different values of γ. q γl ( t)/ q γ = Figure. Solution of (8) for different values of γ In the particular case γ =, we have q L (t) = q 0 e L (+ 4L )t, (0) where q L (0) = q 0 is the charge on the capacitor in t = 0. The solution (0) represents the change of charge q(t) on the capacitor and has aperiodic character. This is a well-known result t Taking into account the relation between γ and σ Figure. Solution of (5), for different values of γ. ( γ = 4L ) / σ, 0 < σ L the solution (9) takes the form where q γl ( t) = q 0 E γ L 4L L E γ γ ( γ) t γ ( 4L L γ ( γ) t γ ) /, () () ( t = 4L ) / t. (3) L If the condition > L/ is fulfilled, we have the following range of values =, < L 4L L L 4L L <. (4) 4. onclusion We have presented an analysis of the electrical L circuit in the time domain, from the point of view of fractional calculus. We also found out that there is a relation between γ and σ through the physical parameters, depending on the system in studies, see (4, ). Due to these relations, we obtained the solutions (8) and (5) in terms of the Mittag-Leffler function depending on the parameter γ. The classical cases are recovered by taking the limit when γ =. We hope to study some other aspect of the fractional modified electrical circuits in future. We also hope that it can give some new insights about some promising topics for future research such as fractional filters (fractional analogical filters of second order) and communication theory. 364 Download Date /4/9 : AM

5 Francisco Gómez, Juan osales, Manuel Guía Acknowledgments The authors acknowledge fruitful discussions with Prof. V.I. Tkach, Prof. D. Baleanu, I. Lyanzuridi and J. Martínez. This research was supported by ONAYT and POMEP under the Grant: Fortalecimiento de As., 0, UGTO- A-7. eferences [] K.B. Oldham, J. Spanier, The Fractional alculus (Academic Press, New York, 974) [] K.S. Miller, B. oss, An Introduction to the Fractional alculus and Fractional Differential Equations (Wiley, New York, 993) [3] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives, Theory and Applications (Gordon and Breach Science Publishers, Langhorne, 993) [4] I. Podlubny, Fractional Differential Equations (Academic Press, New York, 999) [5] D. Baleanu, K. Diethelm, E. Scalas, J.J. Trujillo, Fractional alculus Models and Numerical Methods. Series on omplexity, Nonlinearity and haos (World Scientific, Singapore, 0) [6] W. Wyss, J. Math. Phys. 7, 78 (986) [7]. Hilfer, J. Phys. hem. B 04, 394 (000) [8]. Metzler, J. Klafter, Phys. ep. 339, (000) [9]. Metzler, J. Klafter, J. Phys. A 37, 6 (004) [0] O.P. Agrawal, J.A. Tenreiro-Machado, I. Sabatier, Fractional Derivatives and Their Applications: Nonlinear Dynamics (Springer-Verlag, Berlin, 004) []. Hilfer, Applications of Fractional alculus in Physics (World Scientific, Singapore, 000) [] B.J. West, M. Bologna, P. Grigolini, Physics of Fractional Operators (Springer-Verlag, Berlin, 003) [3] F. Gómez-Aguilar, J. osales-garcía, M. Guía- alderón, J. Bernal-Alvarado, UNAM. XIII, 3 (0) [4] J.J. osales, M. Guía, J.F. Gómez, V.I. Tkach, Discontinuity, Nonlinearity and omplexity 4, 35 (0) [5] F. Gómez, J. Bernal, J. osales, T. órdova, Journal of Electrical Bioimpedance 3, (0) [6].L. Magin, Fractional calculus in Bioengineering (Begell House Publisher, oddin, 006) [7] M. aputo, F. Mainardi, Pure Appl. Geophys. 9, 34 (97) [8] S. Westerlund, ausality, University of Kalmar, ep. No (994) [9] D. Baleanu, Z.B. Günvenc, J.A. Tenreiro Machado, New Trends in Nanotechnology and Fractional alculus Applications (Springer, London, 00) [0] F. iewe. Phys. ev. E 53, 4098 (996) [] A.E. Herzallah Mohamed, I. Muslih Sami, D. Baleanu, M. abei Eqab, Nonlinear Dynam. 66, 4 (0) [] K. Golmankhaneh Alireza, M. Yengejeh Ali, D. Baleanu, Int. J. Theor Phys. 5, 9 (0) [3] K. Golmankhaneh Alireza, L. Lambert, Investigations in Dynamics: With Focus on Fractional Dynamics (Academic Publishing, Germany, 0) [4] I. Muslih Sami, M. Saddallah, D. Baleanu, E. abei, omanian Journal of Physics. 55, 7 (00) [5] D. Baleanu, I. Muslih Sami, M. abei Eqab, Nonlinear Dynam. 53, (008) [6] V. Uchaikin, Fractional Derivatives for Physicists and Engineers (Springer, London, 03) [7] B. Mandelbrot. The Fractal Geometry of Nature (Freeman, San Francisco, A, 98) [8] A.E. Herzallah Mohamed, D. Baleanu, omputers Mathematics with Applications 64, 0 (0) [9] S. Bhalekar, V. Daftardar-Gejji, D. Baleanu, omputers Mathematics with Applications 64, 0, (0) [30] A. azminia, D. Baleanu, Proceeding of the omanian Academy Series A-Mathematics Physics Technical Sciences 3, 4 (0) [3] I. Petrás, I. Podlubny, P. O Leary, L. Dorcák, B.M. Vinagre, Analoge ealization of Fractional-Order ontrollers, Tu Kosice: BEG, Faculty, Slovakia, (00) [3] I. Petrás, IEEE Transactions on ircuits and Systems- II: Express Briefs 57, (00) [33] I. Petrás, Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation (Springer and Beijing: HEP, London, 0) [34] A.A. ousan et al., Fract. alc. App. Anal. 9, (006) [35] A. Obeidat, M. Gharibeh, M. Al-Ali, A. ousan, Fract. alc. App. Anal. 4, (0) [36] J.F. Gómez-Aguilar, J.J. osales-garcía, J.J. Bernal- Alvarado, T. órdova-fraga,. Guzmán-abrera, ev. Mex. Fís. 58, 348 (0) [37] Ya.E. yabov, A. Puzenko, Phys. ev. B 66, 840 (00) [38] A.A. Stanislavsky, Acta Physica Polonica B 37, 39 (006) [39] I. Podlubny, Fract. alc. App. Anal. 5, 4 (00) [40] M. Moshre-Torbati, J.K. Hammond, J. Franklin Inst. 335B, 6 (998) 365 Download Date /4/9 : AM