Circuits, Systems, and Signal Processing

, Volume 38, Issue 4, pp 1520–1546 | Cite as

Practical Design of RC Approximants of Constant Phase Elements and Their Implementation in Fractional-Order PID Regulators Using CMOS Voltage Differencing Current Conveyors

  • Ondrej DomanskyEmail author
  • Roman Sotner
  • Lukas Langhammer
  • Jan Jerabek
  • Costas Psychalinos
  • Georgia Tsirimokou


This paper brings a practical and straightforward view on the design of circuit elements described by fractional-order dynamics known as the constant phase element (CPE) and their implementation in a novel structure of a PIαDβ (or PIλDμ in some literature) regulator based on fabricated CMOS voltage differencing current conveyors. Comparison of presented topology with known solutions indicates significant improvements regarding overall simplification, simpler electronic controllability of time constants, and having all passive elements in grounded form. Step-by-step design of the CPE as well as the PIαDβ regulator is supported by experiments with active devices fabricated using the C07 I2T100 0.7 μm CMOS process (ON Semiconductor). Laboratory tests in frequency and time domain confirm the correct operation of the designed application and the accuracy of the derived results in comparison with the theoretical expectations.


Constant phase element (CPE) Differentiator Fractional-order circuit Integrator Proportional branch Proportional integrational and differential controller (PID) Voltage differencing current conveyor (VDCC) 



Research described in this paper was financed by Czech Science Foundation under Grant No. 16-06175S and the National Sustainability Program under Grant LO1401. For the research, infrastructure of the SIX Research Center was used. This article is based upon work from COST Action CA15225, a network supported by COST (European Cooperation in Science and Technology).


  1. 1.
    A. Adhikary, M. Khanra, S. Sen, K. Biswas, Realization of carbon nanotube based electrochemical fractor, in Proceedings of IEEE International Symposium on Circuits and Systems (ISCAS) (2015), pp. 2329–2332Google Scholar
  2. 2.
    C.L. Alexander, B. Tribollet, M.E. Orazem, Contribution of surface distributions to constant-phase-element (CPE) Behavior: 1. Influence of roughness. Electrochim. Acta 173(10), 416–424 (2015)CrossRefGoogle Scholar
  3. 3.
    J. Ashraf, M.S. Alam, D. Rathee, A new proportional-integral-derivative (PID) controller realization by using current conveyors. Int. J. Electron. Eng. 3(2), 237–240 (2011)Google Scholar
  4. 4.
    M. Axtell, M.E. Bise, Fractional calculus application in control systems, in Proceedings of IEEE Conference on Aerospace and Electronics (1990), pp. 563–566Google Scholar
  5. 5.
    S. Bennett, Development of the PID controller. IEEE Control Syst. 13(6), 58–62 (1993)CrossRefGoogle Scholar
  6. 6.
    D. Biolek, R. Senani, V. Biolkova, Z. Kolka, Active elements for analog signal processing: classification, review, and new proposal. Radioengineering 17(4), 15–32 (2008)Google Scholar
  7. 7.
    G.J. Brug, A.L.G. Eeden, M. Sluyters-Rehbach, J.H. Sluyters, The analysis of electrode impedances complicated by the presence of a constant phase element. J. Electroanal. Chem. Interfacial Electrochem. 176(1), 275–295 (1984)CrossRefGoogle Scholar
  8. 8.
    A. Charef, Analogue realisation of fractional-order integrator, differentiator and fractional PI/spl lambda/D/spl mu/controller. IEE Proc. Control Theory Appl. 153(6), 714–720 (2006)CrossRefGoogle Scholar
  9. 9.
    L.A. Christopher, B. Tribollet, M.E. Orazem, Contribution of surface distributions to constant-phase-element (CPE) behavior: 2. Capacitance. Electrochim. Acta 188(10), 566–573 (2016)Google Scholar
  10. 10.
    I. Dimeas, I. Petras, C. Psychalinos, New analog implementation technique for fractional-order controlled: a dc motor control. AEU—Int. J. Electron. Commun. 78(8), 192–200 (2017)Google Scholar
  11. 11.
    A.M. Elshurafa, M.N. Almadhoun, H.K. Salama, H.N. Alshareef, Microscale electrostatic fractional capacitors using reduced graphene oxide percolated polymer composites. Appl. Phys. Lett. 102(23), 232901–232904 (2013)CrossRefGoogle Scholar
  12. 12.
    A. Elwakil, Fractional-order circuits and systems: an emerging interdisciplinary research area. IEEE Circuits Syst. Mag. 10(4), 40–50 (2010)CrossRefGoogle Scholar
  13. 13.
    C. Erdal, H. Kuntman, S.A. Kafali, A current controlled conveyor based proportional-integral-derivative (PID) controller. J. Electr. Electron. Eng. 4(2), 1243–1248 (2004)Google Scholar
  14. 14.
    C. Erdal, A. Toker, C. Acar, Ota-C based proportional-integral-derivative (PID) controller and calculating optimum parameter tolerances. J. Appl. Sci. 9(2), 189–198 (2001)Google Scholar
  15. 15.
    T. Freeborn, B. Maundy, A. Elwakil, Approximated fractional order Chebyshev lowpass filters. Math. Probl. Eng. 2015 (2015).
  16. 16.
    T. Freeborn, Comparison of (1 + α) fractional-order transfer functions to approximate lowpass butterworth magnitude responses. Circuits Syst. Signal Process. 35(6), 1983–2002 (2016)MathSciNetCrossRefGoogle Scholar
  17. 17.
    J. Jerabek, R. Sotner, N. Herencsar, K. Vrba, T. Dostal, Behavioral model for emulation of ZC-CG-VDCC. IEICE Electron. Express 13(18), 1–6 (2016)CrossRefGoogle Scholar
  18. 18.
    J.B. Jorcin, M.E. Orazem, N. Pebere, B. Tribollet, CPE analysis by local electrochemical impedance spectroscopy. Electrochim. Acta 51(8–9), 1473–1479 (2006)CrossRefGoogle Scholar
  19. 19.
    A.U. Keskina, Design of a PID controller circuit employing CDBAs. Int. J. Electr. Eng. Educ. 43(1), 48–56 (2001)CrossRefGoogle Scholar
  20. 20.
    J. Kittel, N. Celati, M. Keddam, H. Takenouti, New methods for the study of organic coatings by EIS: new insights into attached and free films. Prog. Org. Coat. 41(1–3), 93–98 (2001)CrossRefGoogle Scholar
  21. 21.
    M. Krishna, S. Das, K. Biswas, B. Goswami, Fabrication of a fractional order capacitor with desired specifications: a study on process identification and characterization. IEEE Trans. Electron Devices 58(11), 4067–4073 (2011)CrossRefGoogle Scholar
  22. 22.
    K.S. Miller, B. Ross, An introduction to the fractional calculus and fractional differential equations (Willey, New York, 1993)zbMATHGoogle Scholar
  23. 23.
    C. Muniz-Montero, L.V. Garcia-Jimenez, L.A. Sanchez-Gaspariano, C. Sanchez-Lopez, V.R. Gonzalez-Diaz, E. Tlelo-Cuautle, New alternatives for analog implementation of fractional-order integrators, differentiators and PID controllers based on integer order integrators. Nonlinear Dyn. 90(1), 241–256 (2017)MathSciNetCrossRefGoogle Scholar
  24. 24.
    M.D. Ortigueira, Introduction to fractional signal processing. Part 1: Continuous-time systems. IEEE Proc. Vis. Image Signal Process. 147(1), 62–70 (2000)CrossRefGoogle Scholar
  25. 25.
    J. Petrzela, A note on fractional-order two-terminal devices in filtering applications, in Proceedings of 24th International Conference Radioelektronika (2014), pp. 1–4Google Scholar
  26. 26.
    J. Petrzela, Arbitrary phase shifters with decreasing phase, in Proceedings of 38th International Conference on Telecommunications and Signal Processing (TSP) (2015), pp. 682–686Google Scholar
  27. 27.
    J. Petrzela, Arbitrary phase shifters with increasing phase, In Proceedings of 38th International Conference on Telecommunications and Signal Processing (TSP) (2015), pp. 319–324Google Scholar
  28. 28.
    J. Petrzela, Matrix pencil design approach towards fractional-order PI, PD and PID regulators, in Proceedings of 27th International Conference Radioelektronika (2017), pp. 1–4Google Scholar
  29. 29.
    J. Petrzela, New network structures of reconfigurable fractional-order PID regulators with DVCC, in Proceedings of 2017 24th International Conference “Mixed Design of Integrated Circuits and Systems (MIXDES) (2017), pp. 527–531Google Scholar
  30. 30.
    I. Podlubny, L. Dorcak, I. Kostial, On fractional derivatives, fractional-order dynamic systems and PIλDμ-controllers, in Proceedings of the 36th IEEE Conference on Decision and Control (1997), pp. 4985–4990Google Scholar
  31. 31.
    I. Podlubny, B. Vinagre, P. O’leary, L. Dorcak, Analogue realizations of fractional-order controllers. Nonlinear Dyn. 29(1–4), 281–296 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    I. Podlubny, Fractional-Order Systems and Fractional-Order Controllers, UEF-03-94, Inst. Exp. Phys, Slovak Acad. Sci., Kosice, 1994. Accessed 26 Sept 2018
  33. 33.
    I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications (Academic Press, San Diego, 1999)zbMATHGoogle Scholar
  34. 34.
    A.G. Radwan, A.M. Soliman, A.S. Elwakil, First-order filters generalized to the fractional domain. J. Circuits Syst. Comput. 17(1), 55–66 (2008)CrossRefGoogle Scholar
  35. 35.
    V. Silaruam, A. Lorsawatsiri, C. Wongtaychatham, Novel resistorless mixed-mode PID controller with improved low-frequency performance. Radioengineering 22(3), 932–940 (2013)Google Scholar
  36. 36.
    R. Sotner, J. Jerabek, N. Herencsar, R. Prokop, K. Vrba, T. Dostal, Resistor-less first-order filter design with electronical reconfiguration of its transfer function, in Proceedings of 24th Int. Conference Radioelektronika (2014), pp. 1–4Google Scholar
  37. 37.
    R. Sotner, J. Jerabek, J. Petrzela, O. Domansky, G. Tsirimokou, C. Psychalinos, Synthesis and design of constant phase elements based on the multiplication of electronically controllable bilinear immittances in practice. AEU—Int. J. Electron. Commun. 78(8), 98–113 (2017)Google Scholar
  38. 38.
    R. Sotner, J. Jerabek, R. Prokop, V. Kledrowetz, J. Polak, L. Fujcik, T. Dostal, Practically implemented electronically controlled CMOS voltage differencing current conveyor, in Proceedings of 2016 IEEE 59th International Midwest Symposium on Circuits and Systems (MWSCAS) (2016), pp. 667–670Google Scholar
  39. 39.
    R. Sotner, J. Jerabek, R. Prokop, V. Kledrowetz, Simple CMOS voltage differencing current conveyor-based electronically tuneable quadrature oscillator. Electron. Lett. 52(12), 1016–1018 (2016)CrossRefGoogle Scholar
  40. 40.
    A. Sylvain, M. Marco, M.E. Orazem, N. Pebere, B. Tribollet, V. Vivier, Constant-phase-element behavior caused by inhomogeneous water uptake in anti-corrosion coatings. Electrochim. Acta 87(1), 693–700 (2013)Google Scholar
  41. 41.
    G. Tsirimokou, C. Psychalinos, A.S. Elwakil, K.N. Salama, Experimental verification of on-chip CMOS fractional-order capacitor emulators. Electron. Lett. 52(15), 1298–1300 (2016)CrossRefGoogle Scholar
  42. 42.
    P. Ushakov, A. Shadrin, A. Kubanek, J. Koton, Passive fractional-order components based on resistive-capacitive circuits with distributed parameters, in Proceedings of 39th International Conference on Telecommunications and Signal Processing (TSP) (2016), pp. 638–462Google Scholar
  43. 43.
    J. Valsa, P. Dvorak, M. Friedl, Network model of the CPE. Radioengineering 20(3), 619–626 (2011)Google Scholar
  44. 44.
    J. Valsa, J. Vlach, RC models of a constant phase element. Int. J. Circuit Theory Appl. 41(1), 59–67 (2013)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Ondrej Domansky
    • 1
    Email author
  • Roman Sotner
    • 1
  • Lukas Langhammer
    • 1
  • Jan Jerabek
    • 2
  • Costas Psychalinos
    • 3
  • Georgia Tsirimokou
    • 3
  1. 1.Department of Radio Electronics, Faculty of Electrical Engineering and CommunicationBrno University of Technology (BUT)BrnoCzech Republic
  2. 2.Department of Telecommunications, Faculty of Electrical Engineering and CommunicationBrno University of Technology (BUT)BrnoCzech Republic
  3. 3.Department of PhysicsUniversity of PatrasRio PatrasGreece

Personalised recommendations