Analog Realization of Fractional-Order Capacitor and Inductor via the Caputo–Fabrizio Derivative
Manjie Ran, Xiaozhong Liao, Da Lin, and Ruocen Yang
Automation Department, Beijing Institute of Technology
No.5 South Street, Zhongguancun, Haidian District, Beijing 100081, China
Capacitors and inductors have been proven to exhibit fractional-order characteristics. Therefore, the establishment of fractional-order models for circuits containing such components is of great significance in practical circuit analysis. This study establishes the impedance models of fractional-order capacitors and inductors based on the Caputo–Fabrizio derivative and performs the analog realization of fractional-order electronic components. The mathematical models of fractional RC, RL, and RLC electrical circuits are deduced and verified via a comparison between the numerical simulation and the corresponding circuit simulation. The electrical characteristics of the fractional circuits are analyzed. This study not only enriches the models of fractional capacitors and inductors, but can also be applied to the description of circuit characteristics to obtain more accurate results.
-  I. S. Jesus and J. A. Tenreiro Machado, “Development of Fractional Order Capacitors Based on Electrolyte Processes,” Nonlinear Dyn., Vol.56, No.1-2, pp. 45-55, 2009.
-  A. Allagui, T. J. Freeborn, A. S. Elwakil, M. E. Fouda, B. J. Maundy, A. G. Radwan, Z. Said, and M. A. Abdelkareem, “Review of Fractional-Order Electrical Characterization of Supercapacitors,” J. Power Sources, Vol.400, pp. 457-467, 2018.
-  A. A. Raorane, M. D. Patil, and V. A. Vyawahare, “Analysis of Full-Wave Controlled Rectifier with Lossy Inductive Load Using Fractional-Order Models,” Proc. of the 2015 Int. Conf. on Industrial Instrumentation and Control (ICIC), pp. 750-755, 2015.
-  J. F. Gómez Aguilar, “Behavior Characteristics of a Cap-Resistor, Memcapacitor, and a Memristor from the Response Obtained of RC and RL Electrical Circuits Described by Fractional Differential Equations,” Turk. J. Elec. Eng. & Comp. Sci., Vol.24, No.3, pp. 1421-1433, 2016.
-  I. Petráš, “A Note on the Fractional-Order Chua’s System,” Chaos Solitons Fractals, Vol.38, No.1, pp. 140-147, 2008.
-  A. Hajiloo and W.-F. Xie, “Multi-Objective Optimal Fuzzy Fractional-Order PID Controller Design,” J. Adv. Comput. Intell. Intell. Inform., Vol.18, No.3, pp. 262-270, 2014.
-  F. Liu, Z. Zhang, X. Wang, and F. Sun, “Stability and Synchronization Control of Fractional-Order Gene Regulatory Network System with Delay,” J. Adv. Comput. Intell. Intell. Inform., Vol.21, No.1, pp. 148-152, 2017.
-  P. Mu, L. Wang, and C. Liu, “A Control Parameterization Method to Solve the Fractional-Order Optimal Control Problem,” J. Optim. Theory Appl., Vol.187, No.1, pp. 234-247, 2020.
-  I. Petráš and Y. Chen, “Fractional-Order Circuit Elements with Memory,” Proc. of the 13th Int. Carpathian Control Conf. (ICCC), pp. 552-558, 2012.
-  A. Adhikary, P. Sen, S. Sen, and K. Biswas, “Design and Performance Study of Dynamic Fractors in Any of the Four Quadrants,” Circuits Syst. Signal Process., Vol.35, No.6, pp. 1909-1932, 2016.
-  Z. Zhang, T. Ushio, J. Zhang, C. Ding, and F. Liu, “Bifurcation Analysis of a Class Fractional-Oder Nonlinear Chua’s Circuit System,” J. Adv. Comput. Intell. Intell. Inform., Vol.24, No.4, pp. 549-556, 2020.
-  A. G. Radwan and K. N. Salama, “Passive and Active Elements Using Fractional LβCα Circuit,” IEEE Trans. Circuits Syst. I, Reg. Papers, Vol.58, No.10, pp. 2388-2397, 2011.
-  M. C. Tripathy, K. Biswas, and S. Sen, “A Design Example of a Fractional-Order Kerwin–Huelsman–Newcomb Biquad Filter with Two Fractional Capacitors of Different Order,” Circuits Syst. Signal Process., Vol.32, No.4, pp. 1523-1536, 2013.
-  R. Martínez, Y. Bolea, A. Grau, and H. Martínez, “Fractional DC/DC Converter in Solar-Powered Electrical Generation Systems,” Proc. of the 14th IEEE Int. Conf. on Emerging Technologies & Factory Automation (ETFA’09), pp. 1475-1480, 2009.
-  G. Teodoro, J. A. Tenreiro Machado, and E. C. de Oliveira, “A Review of Definitions of Fractional Derivatives and Other Operators,” J. Comput. Phys., Vol.388, pp. 195-208, 2019.
-  A. Atangana and A. Secer, “A Note on Fractional Order Derivatives and Table of Fractional Derivatives of Some Special Functions,” Abstr. Appl. Anal., Vol.2013, Article No.279681, 2013.
-  M. Caputo and M. Fabrizio, “A New Definition of Fractional Derivative Without Singular Kernel,” Progr. Fract. Differ. Appl., Vol.1, No.2, pp. 73-85, 2015.
-  A. Atangana and R. T. Alqahtani, “Numerical Approximation of the Space-Time Caputo-Fabrizio Fractional Derivative and Application to Groundwater Pollution Equation,” Adv. Differ. Equ., Vol.2016, No.1, Article No.156, 2016.
-  A. Alshabanat, M. Jleli, S. Kumar, and B. Samet, “Generalization of Caputo-Fabrizio Fractional Derivative and Applications to Electrical Circuits,” Front. Phys., Vol.8, Article No.64, 2020.
-  J. J. Rosales García, J. D. Filoteo, and A. González, “A Comparative Analysis of the RC Circuit with Local and Non-Local Fractional Derivatives,” Rev. Mex. Fis., Vol.64, No.6, pp. 647-654, 2018.
-  K. A. Abro, A. A. Memon, and M. A. Uqaili, “A Comparative Mathematical Analysis of RL and RC Electrical Circuits via Atangana-Baleanu and Caputo-Fabrizio Fractional Derivatives,” Eur. Phys. J. Plus, Vol.133, No.3, Article No.133, 2018.
-  A. Alsaedi, J. J. Nieto, and V. Venktesh, “Fractional Electrical Circuits,” Adv. Mech. Eng., Vol.7, No.12, 7pp., 2015.
-  K. A. Abro, A. A. Memon, and A. A. Memon, “Functionality of Circuit via Modern Fractional Differentiations,” Analog Integr. Circuits Signal Process., Vol.99, No.1, pp. 11-21, 2019.
This article is published under a Creative Commons Attribution-NoDerivatives 4.0 International License.