# Time Dimensional Consistency Aware Analysis of Voltage Mode and Current Mode Active Fractional Circuits

## Main Article Content

## Abstract

In this research, the analysis of the active fractional circuits has been performed by using the fractional differential equation approach. Both voltage and current mode circuits have been taken into account. The fractional time component parameters have been included in the derivative terms within the fractional differential equations. This is because the consistency in time dimension between the fractional derivative and the conventional one which is also related to the physical measurability, is concerned. The fractional derivatives have been interpreted in Caputo sense. The resulting analytical solutions of the time dimensional consistency aware fractional differential equations have been determined. We have found that the dimensional consistency between both sides of the equations of the solutions which cannot be achieved in the previous works, can be obtained. By applying different source terms to the obtained analytical solutions, the response of both voltage and current mode circuits have been determined and the behaviours of the circuits have been analysed. The fractional time constant and pole locations in the F-plane of these circuits have been determined. Their dynamic behaviours, stabilities have been analysed. Moreover, the discussion on circuit realizations with fractional capacitor has also been made.

## Article Details

How to Cite

[1]

R. Banchuin and R. Chaisrichaoren, “Time Dimensional Consistency Aware Analysis of Voltage Mode and Current Mode Active Fractional Circuits”,

*ECTI-CIT*, vol. 13, no. 1, pp. 81-93, Oct. 2019.
Issue

Section

Review Article

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

## References

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[25] G.-A. J. Francisco1, R.-G. Juan3, G.-C. Manuel and R.-H. J. Roberto, “Fractional RC and LC Electrical Circuits,” Ingeniería, Investigación y Tecnología, Vol. 15, No. 2, , pp. 311–319, 2014

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[29] I. Podlubny, Fractional Differential Equations, Vol. 198 of Mathematics in Science and Engineering, Academic Press, New York, 1999.

[30] A. Atangana and A. Secer, “A Note on Fractional Order Derivatives and Table of Fractional Derivatives of Some Special Functions,” Abstract and Applied Analysis, Vol. 2013, pp. 1-8, 2013.

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[32] A. M. Mathai and H. J. Haubold, Special Functions for Applied Scientists, Springer, New York, 2010.

[33] T. Deliyannis, Y. Sun, and J.K. Fidler, Continuous – Time Active Filter Design, CRC Press, Florida,1999

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[35] B. Dwork, Generalized Hypergeometric Functions, Clarendon Press, Oxford,1990.

[36] M.S. Semary, A.G. Radwan and H.N. Hassan, “Fundamentals of Fractional-order LTI Circuits and Systems: Number of Poles, Stability, Time and Frequency Responses”, International Journal of Circuit Theory and Applications, Vol. 44, No. 12, pp. 2114-2133, 2016.

[37] A. Agambayev, S. Patole, M. Farhat, A. Elwakil, H. Bagci and K. N. Salama, “Ferroelectric Fractional-Order Capacitors,” ChemElectroChem, Vol. 4, No. 11, pp. 2807–2813, 2017.

[38] D. A. John, S. Banerjee, G. W. Bohannan and K. Biswas, “Solid-state Fractional Capacitor Using MWCNT-epoxy Nanocomposite,” Applied Physics Letters, Vol. 110, No. 16, pp. 163504-1-163504-5, 2017.

[39] C. Halijak, “An RC Impedance Approximant to (1/s)1/2, IEEE Transactions on Circuit Theory, Vol. 11, No. 4, pp. 494–495, 1964.

[40] J. Valsa and J. Vlach, RC Models of A Constant Phase Element, International Journal of Circuit Theory and Applications. Vol. 41, No. 1, pp. 59–67, 2013.

[41] R. Banchuin, “Novel Expressions for Time Domain Responses of Fractance Device,” Cogent Engineering, Vol. 4, No. 1, pp. 1-28, 2017.

[42] T. J. Freeborn, B, Maundy and A.S. Elwakil, “Measurement of Supercapacitor Fractional-order Model Parameters From Voltage-excited Step Response,” IEEE Journal on Emerging and Selected Topics in Circuits and Systems, Vol. 3, No. 3, pp. 367-376, 2013.

[43] S. Das, K. Biswas and B. Goswami, “Study of The Parameters of A Fractional Order Capacitor,” Proceedings of INDICON 2015 Annual IEEE India Conference, pp. 1-5, 2015.

[2] M. F. M. Lima, J. A. T. Machado, and M. Cris´ostomo, “Experimental Signal Analysis of Robot Impacts in A Fractional Calculus Perspective,” Journal of Advanced Computational Intelligence and Intelligent Informatics, Vol. 11, No. 9, pp. 1079–1085, 2007.

[3] L. Debnath, “Recent Applications of Fractional Calculus to Science and Engineering,” International Journal of Mathematics and Mathematical Sciences, Vol. 2003, No. 54, pp. 3413–3442, 2003.

[4] L. Sommacal, P.Melchior, A. Oustaloup, J.-M. Cabelguen, and A. J. Ijspeert, “Fractional Multi-Models of The Frog Gastrocnemius Muscle,” Journal of Vibration and Control, Vol. 14, No. 9-10, pp. 1415–1430, 2008.

[5] R. L. Magin and M. Ovadia, “Modeling The Cardiac Tissue Electrode Interface Using Fractional Calculus,” Journal of Vibration and Control, Vol. 14, No. 9-10, pp. 1431–1442, 2008.

[6] Y. Pu, X. Yuan, K. Liao, et al., “A Recursive Two-circuits Series Analog Fractance Circuit For Any Order Fractional Calculus,” Proceedings of SPIE ICO20 Optical Information Processing, pp. 509–519, 2006.

[7] B. T. Krishna and K. V. V. S. Reddy, “Active and Passive Realization of Fractance Device of Order 1/2,” Active and Passive Electronic Components, Vol. 2008, pp. 1-5, 2008.

[8] Z.-Z. Yang and J.-L. Zhou, “An Improved Design for The IIR-type Digital Fractional Order Differential Filter,” Proceedings of the International Seminar on Future BioMedical Information Engineering (FBIE ’08), pp. 473–476, 2008.

[9] R. Panda and M. Dash, “Fractional Generalized Splines and Signal Processing,” Signal Processing, Vol. 86, No. 9, pp. 2340–2350, 2006.

[10] J. Cervera and A. Ba˜ nos, “Automatic Loop Shaping in QFT Using CRONE Structures,” Journal of Vibration and Control, Vol. 14, No. 9-10, pp. 1513–1529, 2008.

[11] G.W. Bohannan, “Analog Fractional Order Controller in Temperature and Motor Control Applications,” Journal of Vibration and Control, Vol. 14, No. 9-10, pp. 1487–1498, 2008.

[12] R. Caponetto, G. Dongola, L. Fortuna, I. Petras, Fractional Order System-Modeling and Control Applications,World Scientific Publishing, Singapore, 2010

[13] A.G. Radwan, A.M. Soliman, A.S. Elwakil and A. Sedeek, “On The Stability of Linear Systems with Fractional Order Elements,” Chaos, Solitons and Fractals, Vol. 40, No 5. pp. 2317–2328, 2009

[14] A.G. Radwan, A.M. Soliman and A.S. Elwakil, “First-Order Filters Generalized to The Fractional Domain,” Journal of Circuits, Systems, and Computers, Vol. 17, No. 1,pp. 55–66, 2008.

[15] A.G. Radwan, A.S. Elwakil and A.M. Soliman, “On The Generalization of Second Order Filters to The Fractional Order Domain,” Journal of Circuits, Systems, and Computers , Vol.18, No. 2, pp. 361–386, 2009.

[16] T.J. Freeborn, B. Maundy and A.S. Elwakil, “Field Programmable Analogue Array Implementation of Fractional Step Filters,” IET circuits, devices & systems, Vol. 4, No. 6, pp. 514–524, 2010

[17] A.G. Radwan, A.S. Elwakil and A.M. Soliman, “Fractional–order Sinusoidal Oscillator: Design Procedure and Practical Examples,” IEEE Transactions on Circuits and Systems I: Regular Papers, Vol. 55, No.7, pp. 2051–2063, 2008.

[18] D. Mondal and K. Biswas, “Performance Study of Fractional Order Integrator Using Single-component Fractional Order Element,” IET circuits, devices & systems, Vol.5, No. 4, pp. 334–342, 2011.

[19] M. Guia, F. Gomez, and J. Rosales, “Analysis on The Time and Frequency Domain for The RC Electric Circuit of Fractional Order,” Central European Journal of Physics, Vol. 11, No. 10, pp. 1366–1371, 2013

[20] A. A. Rousan, N. Y. Ayoub, F. Y. Alzoubi et al., “A Fractional LC−RC Circuit,” Fractional Calculus & Applied Analysis, Vol. 9, No. 1, pp. 33–41, 2006.

[21] P. V. Shah, A. D. Patel, I. A. Salehbhai, and A. K. Shukla, “Analytic Solution for the Electric Circuit Model in Fractional Order,” Abstract and Applied Analysis, Vol. 2014, pp. 1-5, 2014.

[22] R. Banchuin and R. Chaisricharoen. “The analysis of active circuit in fractional domain,” Proceedings of the

2018 ECTI Northern Section Conference on Electrical Electronics, Computer and Telecommunications Engineering (ECTI-NCON 2018), pp. 25-28, 2018.

[23] R. Banchuin and R. Chaisricharoen.“Time Domain FDE Based Analysis of Active Fractional Circuit,” Proceedings of the 2018 International Conference on Digital Arts, Media and Technology (ICDAMT 2018), pp. 25-28, 2018.

[24] J.F. G´omez-Aguilara, J.J. Rosales-Garc´ıab, J.J. Bernal-Alvaradoa, T. C´ordova-Fragaa and R. Guzm´an-Cabrerab, “Fractional Mechanical Oscillators,” Revista Mexicana de F´ısica, Vol. 58, No. 8, pp. 348–352, 2012

[25] G.-A. J. Francisco1, R.-G. Juan3, G.-C. Manuel and R.-H. J. Roberto, “Fractional RC and LC Electrical Circuits,” Ingeniería, Investigación y Tecnología, Vol. 15, No. 2, , pp. 311–319, 2014

[26] F. Gómez, J. Rosales and M. Guía, “RLC Electrical Circuit of Non-integer Order,” Central European Journal of Physics, Vol.11, No. 10, pp 1361-1365

[27] A. Fabre, “Insensitive Voltage-mode and Current-mode Filters from Commercially Available Transimpedance Opamps,” IEE Proceedings G (Circuits, Devices and Systems), Vol. 140, No. 5, pp. 319–321, 1993.

[28] J. Mahattanakul and C. Toumazou, “Current-mode Versus Voltage-mode Gm–C Biquad Filters: What the theory says,” IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing, Vol. 45, No. 2, pp. 173–186, 1998.

[29] I. Podlubny, Fractional Differential Equations, Vol. 198 of Mathematics in Science and Engineering, Academic Press, New York, 1999.

[30] A. Atangana and A. Secer, “A Note on Fractional Order Derivatives and Table of Fractional Derivatives of Some Special Functions,” Abstract and Applied Analysis, Vol. 2013, pp. 1-8, 2013.

[31] E. Kreyszig, Advanced Engineering Mathematics, John Wiley and Sons, Inc., New York, 1999.

[32] A. M. Mathai and H. J. Haubold, Special Functions for Applied Scientists, Springer, New York, 2010.

[33] T. Deliyannis, Y. Sun, and J.K. Fidler, Continuous – Time Active Filter Design, CRC Press, Florida,1999

[34] E. W. Weisstein, “Regularized Hypergeometric Function,” MathWorld-A Wolfram Web Resource.

[35] B. Dwork, Generalized Hypergeometric Functions, Clarendon Press, Oxford,1990.

[36] M.S. Semary, A.G. Radwan and H.N. Hassan, “Fundamentals of Fractional-order LTI Circuits and Systems: Number of Poles, Stability, Time and Frequency Responses”, International Journal of Circuit Theory and Applications, Vol. 44, No. 12, pp. 2114-2133, 2016.

[37] A. Agambayev, S. Patole, M. Farhat, A. Elwakil, H. Bagci and K. N. Salama, “Ferroelectric Fractional-Order Capacitors,” ChemElectroChem, Vol. 4, No. 11, pp. 2807–2813, 2017.

[38] D. A. John, S. Banerjee, G. W. Bohannan and K. Biswas, “Solid-state Fractional Capacitor Using MWCNT-epoxy Nanocomposite,” Applied Physics Letters, Vol. 110, No. 16, pp. 163504-1-163504-5, 2017.

[39] C. Halijak, “An RC Impedance Approximant to (1/s)1/2, IEEE Transactions on Circuit Theory, Vol. 11, No. 4, pp. 494–495, 1964.

[40] J. Valsa and J. Vlach, RC Models of A Constant Phase Element, International Journal of Circuit Theory and Applications. Vol. 41, No. 1, pp. 59–67, 2013.

[41] R. Banchuin, “Novel Expressions for Time Domain Responses of Fractance Device,” Cogent Engineering, Vol. 4, No. 1, pp. 1-28, 2017.

[42] T. J. Freeborn, B, Maundy and A.S. Elwakil, “Measurement of Supercapacitor Fractional-order Model Parameters From Voltage-excited Step Response,” IEEE Journal on Emerging and Selected Topics in Circuits and Systems, Vol. 3, No. 3, pp. 367-376, 2013.

[43] S. Das, K. Biswas and B. Goswami, “Study of The Parameters of A Fractional Order Capacitor,” Proceedings of INDICON 2015 Annual IEEE India Conference, pp. 1-5, 2015.