# The Time Dimensional Measurability Aware FDE Based Analysis of Active Circuit in The Fractional Domain

## Main Article Content

## Abstract

In this research, the analysis of the active circuit in the fractional domain has been performed by using the fractional differential equation approach where the measurability in time dimension of the derivative term has also been concerned unlike the previous work. The OTA-C filter has been adopted as the candidate active circuit due to its compactness and renown. The derivative term of the fractional differential equation which includes the fractional time component parameters for obtaining such time dimensional measurability, has been interpreted in Caputo sense and the analytical solution of such equation has been determined with the aid of Laplace transformation. With the obtained solution, the time dimensional measurability aware fractional derivative based circuit responses to various inputs have been determined, the circuit fractional time constant and other crucial time parameters has been determined and the temporal behaviour of the circuit has been analysed in the fractional domain. The loci of the pole on W-plane has been determined and the stability analysis has also been presented. Moreover, we also mathematically proof that the OTA-C filter in the fractional domain can be electronically realized with time dimensional measurability awareness by using the state of the art fractional capacitor.

## Article Details

*ECTI-CIT*, vol. 13, no. 1, pp. 59-70, Jun. 2019.

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

## References

[2] 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.

[3] 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.

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

[5] 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.

[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] M. F. M. Lima, J. A. T. Machado, and M. Criśostomo, “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.

[8] J. Rosario, D. Dumur, and J. T. Machado, “Analysis of Fractional-order Robot Axis Dynamics,” IFAC Proceedings, Vol. 39, No. 11, pp. 367-372, 2006.

[9] 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.

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

[11] 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.

[12] 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.

[13] 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.

[14] 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.

[15] 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.

[16] 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.

[17] J.F. Gómez-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

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

[19] 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

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

[21] M. Hasan and Y. Sun, “Oscillation-Based Test Structure and Method for OTA-C Filters”, Proceedings of the 13th IEEE International Conference on Electronics, Circuits and Systems (ICECS'06), pp. 98-101, 2006

[22] D. Csipkes, G. Csipke, P. Farago, H. Fernandez-Canqueand and S. Hintea, “An OTA-C field programmable analog array for multi-mode filtering applications,” Proceedings of the 13th International Conference on Optimization of Electrical and Electronic Equipment (OPTIM’12), pp. 1176-1182, 2012

[23] P.V. Anada Mohan, VLSI analog filters: Active RC, OTA-C and SC. Springer, Newyork, 2013.

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

[25] 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.

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

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

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

[29] M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover, New York, 1972.

[30] E. Ziedler, Applied Functional Analysis, Springer, New York, 1995.

[31] 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.

[32] 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

[33] 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.

[34] 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.

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

[36] 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.

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

[38] 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.

[39] 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.

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