Skip to main content

Analog Realization of Electronically Tunable Fractional-Order Differ-Integrators


This paper presents simple structures of fractional-order differentiators and fractional-order integrators. For these structures, both current mode and voltage mode designs have been implemented for differential input signals. The analog realizations of these fractional circuits use a single active element, the Differential Voltage Current Conveyor Transconductance Amplifier (DVCCTA), and a capacitive fractor. The passive elements present in the capacitive fractor are arranged in parallel RC ladder topology with their consecutive values in geometric progression. In the proposed structures, the capacitive fractor and the resistances connected at different ports of DVCCTA are all grounded, hence, making the designs simpler and less sensitive to the parasitic effects. Apart from this, the other key features of these designs are—electronically tunable, high dynamic range, robust and accurate. Simulations of all the four models are performed using TSMC \(0.25\,\upmu \hbox {m}\) technology in MentorGraphics. The non-ideal analysis of DVCCTA device and Monte Carlo analysis have been studied to validate the robustness of these models. The results obtained are in compliance with their corresponding ideal responses. Comparisons with other active element-based structures have been presented to substantiate the work proposed in this paper.

This is a preview of subscription content, access via your institution.


  1. 1.

    Biolek, D.; Senani, R.; Biolkova, V.; Kolka, Z.: Active elements for analog signal processing: classification, review, and new proposals. Radioengineering 17, 15–32 (2008)

    Google Scholar 

  2. 2.

    Patranabis, D.; Ghosh, D.: Integrators and differentiators with current conveyors. IEEE Trans. Circuits Syst. 31, 567–9 (1984)

    Article  Google Scholar 

  3. 3.

    Lee, J.Y.; Tsao, H.W.: True RC integrators based on current conveyors with tunable time constants using active control and modified loop technique. IEEE Trans. Instrum. Meas. 41, 709–14 (1992)

    Article  Google Scholar 

  4. 4.

    Liu, S.I.; Hwang, Y.S.: Dual-input differentiators and integrators with tunable time constants using current conveyors. IEEE Trans. Instrum. Meas. 43(4), 650–4 (1994)

    Article  Google Scholar 

  5. 5.

    Kumar P.; Verma R.: Realization of a novel current mode fully differential PID (FDPID) controller. In: IEEE 5th India International Conference on Power Electronics (IICPE), Dec 2012. pp. 1–5 (2012)

  6. 6.

    Erdal, C.; Kuntman, H.; Kafali, S.: A current controlled conveyor based proportional-integral-derivative (PID) controller. IU J. Electr. Electron. Eng. 4(2), 1243–8 (2004)

    Google Scholar 

  7. 7.

    Yuce, E.; Minaei, S.: New CCII-based versatile structure for realizing PID controller and instrumentation amplifier. Microelectron. J. 41(5), 311–6 (2010)

    Article  Google Scholar 

  8. 8.

    Srisakultiew, S.; Siripruchyanun, M.: A synthesis of electronically controllable current-mode PI, PD and PID controllers employing CCCDBAs. Circuits Syst. 4(3), 287 (2013)

    Article  Google Scholar 

  9. 9.

    Lawanwisut S.; Srisakultiew S.; Siripruchyanun M.: A synthesis of low component count for current-mode PID, PI and PD controllers employing single CCTA and Grounded elements. In: 2015 IEEE 38th International Conference on Telecommunications and Signal Processing (TSP), July 2015. pp. 1–5 (2015)

  10. 10.

    Mahmoud, S.A.: Low voltage wide range CMOS differential voltage current conveyor and its applications. Contemp. Eng. Sci. 1(3), 105–26 (2008)

    MathSciNet  Google Scholar 

  11. 11.

    Tangsrirat, W.: Floating simulator with a single DVCCTA. Indian J. Eng. Mater. Sci. 20, 79–86 (2013)

    Google Scholar 

  12. 12.

    Nandi, R.; Das, S.; Venkateswaran, P.: Floating lossless immittance functions using DVCCTA. Int. J. Electron. Lett. 4(1), 117–26 (2016)

    Article  Google Scholar 

  13. 13.

    Chien, H.C.; Chen, C.Y.: CMOS realization of single-resistance-controlled and variable frequency dual-mode sinusoidal oscillators employing a single DVCCTA with all-grounded passive components. Microelectron. J. 45(2), 226–38 (2014)

    Article  Google Scholar 

  14. 14.

    Pandey, N.; Pandey, R.: Approach for third order quadrature oscillator realisation. IET Circuits Devices Syst. 9(3), 161–71 (2015)

    Article  Google Scholar 

  15. 15.

    Pandey, N.; Arora, S.; Takkar, R.; Pandey, R.: DVCCCTA-based implementation of mutually coupled circuit. ISRN Electron. 2012, 1–6 (2012)

    Google Scholar 

  16. 16.

    Lee, C.N.: Independently tunable mixed-mode universal biquad filter with versatile input/output functions. AEU Int. J. Electron. Commun. 70(8), 1006–19 (2016)

    Article  Google Scholar 

  17. 17.

    Khateb, F.; Kubánek, D.; Tsirimokou, G.; Psychalinos, C.: Fractional-order filters based on low-voltage DDCCs. Microelectron. J. 50, 50–9 (2016)

    Article  Google Scholar 

  18. 18.

    Phatsornsiri, P.; Kumngern, M.; Lamun, P.: A voltage-mode universal biquadratic filter using DDCCTA. J. Circuits Syst. Comput. 25(05), 1650034–57 (2016)

    Article  Google Scholar 

  19. 19.

    Tangsrirat, W.; Channumsin, O.; Pukkalanun, T.: Resistorless realization of electronically tunable voltage-mode SIFO-type universal filter. Microelectron. J. 44(3), 210–5 (2013)

    Article  Google Scholar 

  20. 20.

    Chen, H.P.; Hwang, Y.S.; Ku, Y.T.; Lin, T.J.: Voltage-mode biquadratic filters using single DDCCTA. AEU Int. J. Electron. Commun. 70(10), 1403–11 (2016)

    Article  Google Scholar 

  21. 21.

    Kuntman H.H.; Uygur A.: New possibilities and trends in circuit design for analog signal processing. In: 2012 IEEE International Conference on Applied Electronics (AE), Sept 2012. pp. 1–9 (2012)

  22. 22.

    Nezha M.; Massinissa T.; Jean-Claude T.: Physical interpretation and initialization of the fractional integrator. In: 2014 IEEE International Conference on Fractional Differentiation and Its Applications (ICFDA) June 2014. pp. 1–6 (2014)

  23. 23.

    Petras, I.; Sierociuk, D.; Podlubny, I.: Identification of parameters of a half-order system. IEEE Trans. Signal Process. 60(10), 5561–6 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  24. 24.

    Sierociuk, D.; Podlubny, I.; Petras, I.: Experimental evidence of variable-order behavior of ladders and nested ladders. IEEE Trans. Control Syst. Technol. 21(2), 459–66 (2013)

    Article  Google Scholar 

  25. 25.

    Mondal, D.; Biswas, K.: Performance study of fractional order integrator using single-component fractional order element. IET Circuits Devices Syst. 5(4), 334–42 (2011)

    Article  Google Scholar 

  26. 26.

    Radwan, A.G.; Elwakil, A.S.; Soliman, A.M.: On the generalization of second-order filters to the fractional-order domain. J. Circuits Syst. Comput. 18(02), 361–86 (2009)

    Article  Google Scholar 

  27. 27.

    Podlubny, I.; Petraš, I.; Vinagre, B.M.; O’leary, P.; Dorčák, L’.: Analogue realizations of fractional-order controllers. Nonlinear Dyn. 29(1), 281–96 (2002)

    MathSciNet  Article  MATH  Google Scholar 

  28. 28.

    Charef, A.: Analogue realisation of fractional-order integrator, differentiator and fractional PI\(^{\lambda }\)D\(^{\mu }\) controller. IEE Proc. Control Theory Appl. 153(6), 714–20 (2006)

    Article  Google Scholar 

  29. 29.

    Djouambi A.; Charef A.; Voda A.: Numerical simulation and identification of fractional systems using digital adjustable fractional order integrator. In: 2013 IEEE European Conference on Control (ECC) July 2013. pp. 2615–2620 (2013)

  30. 30.

    Adhikary, A.; Khanra, M.; Pal, J.; Biswas, K.: Realization of fractional order elements. INAE Lett. 2, 41–47 (2017)

    Article  Google Scholar 

  31. 31.

    Khanra M.; Pal J.; Biswas K.: Rational approximation and analog realization of fractional order differentiator. In: 2011 IEEE International Conference on Process Automation, Control and Computing (PACC), July 2011. pp. 1–6 (2011)

  32. 32.

    Valsa, J.; Vlach, J.: RC models of a constant phase element. Int. J. Circuit Theory Appl. 41(1), 59–67 (2013)

    Google Scholar 

  33. 33.

    Gonzalez E.A.; Petráš I.: Advances in fractional calculus: control and signal processing applications. In: 2015 IEEE 16th International Conference on Carpathian Control (ICCC), May 2015. pp. 147–152 (2015)

  34. 34.

    Abulencia G.L.; Abad AC.: Analog realization of a low-voltage two-order selectable fractional-order differentiator in a 0.35 um CMOS technology. In: 2015 IEEE International Conference on Humanoid, Nanotechnology, Information Technology, Communication and Control, Environment and Management (HNICEM), Dec 2015. pp. 1–6 (2015)

  35. 35.

    Gonzalez, E.; Dorčák, L’.; Monje, C.; Valsa, J.; Caluyo, F.; Petráš, I.: Conceptual design of a selectable fractional-order differentiator for industrial applications. Fract. Calc. Appl. Anal. 17(3), 697–716 (2014)

    Article  MATH  Google Scholar 

  36. 36.

    Krishna, B.T.: Studies on fractional order differentiators and integrators: a survey. Signal Process. 91(3), 386–426 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  37. 37.

    Hollmann, L.J.; Stevenson, R.L.: Pole-zero placement algorithm for the design of digital filters with fractional-order rolloff. Signal Process. 107, 218–29 (2015)

    Article  Google Scholar 

  38. 38.

    Dhabale, A.S.; Dive, R.; Aware, M.V.; Das, S.: A new method for getting rational approximation for fractional order differintegrals. Asian J. Control 17(6), 2143–52 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  39. 39.

    Tsirimokou, G.; Koumousi, S.; Psychalinos, C.: Design of fractional-order filters using current feedback operational amplifiers. J. Eng. Sci. Technol. Rev. 9, 77–81 (2016)

    Article  Google Scholar 

  40. 40.

    Verma, R.; Pandey, N.; Pandey, R.: Electronically tunable fractional order filter. Arab. J. Sci. Eng. 42, 3409–3422 (2017)

    Article  Google Scholar 

  41. 41.

    Psychalinos, C.; Elwakil, A.; Maundy, B.; Allagui, A.: Analysis and realization of a switched fractional-order-capacitor integrator. Int. J. Circuit Theory Appl. 44(11), 2035–40 (2016)

    Article  Google Scholar 

  42. 42.

    Caponetto, R.; Dongola, G.; Maione, G.; Pisano, A.: Integrated technology fractional order proportional-integral-derivative design. J. Vib. Control 20(7), 1066–75 (2014)

    MathSciNet  Article  Google Scholar 

  43. 43.

    Tsirimokou, G.; Psychalinos, C.: Ultra-low voltage fractional-order circuits using current mirrors. Int. J. Circuit Theory Appl. 44(1), 109–26 (2016)

    Article  Google Scholar 

  44. 44.

    Vastarouchas, C.; Tsirimokou, G.; Freeborn, T.J.; Psychalinos, C.: Emulation of an electrical-analogue of a fractional-order human respiratory mechanical impedance model using OTA topologies. AEU Int. J. Electron. Commun. 78, 201–8 (2017)

    Article  Google Scholar 

  45. 45.

    Dimeas I.; Tsirimokou G.; Psychalinos C.; Elwakil A.: Realization of fractional-order capacitor and inductor emulators using current feedback operational amplifiers. In: 2015 International Symposium on Nonlinear Theory and its Application (NOLTA), Dec 2015. pp. 237–240 (2015)

  46. 46.

    Sheng, H.; Sun, H.G.; Coopmans, C.; Chen, Y.Q.; Bohannan, G.W.: A physical experimental study of variable-order fractional integrator and differentiator. Eur. Phys. J. Spec. Top. 193(1), 93–104 (2011)

    Article  Google Scholar 

  47. 47.

    Biswas, K.; Sen, S.; Dutta, P.K.: Realization of a constant phase element and its performance study in a differentiator circuit. IEEE Trans. Circuits Syst. II Expr. Br. 53(9), 802–6 (2006)

    Article  Google Scholar 

  48. 48.

    Sarafraz, M.S.; Tavazoei, M.S.: Realizability of fractional-order impedances by passive electrical networks composed of a fractional capacitor and RLC components. IEEE Trans. Circuits Syst. I Regul. Pap. 62(12), 2829–35 (2015)

    MathSciNet  Article  Google Scholar 

  49. 49.

    Pu, Y.F.: Measurement units and physical dimensions of fractance-part I: Position of purely ideal fractor in Chua’s axiomatic circuit element system and fractional-order reactance of fractor in its natural implementation. IEEE Access 4, 3379–97 (2016)

    Article  Google Scholar 

  50. 50.

    Yifei P.; Xiao Y.; Ke L.; Jiliu Z.; Ni Z.; Yi Z.; Xiaoxian P.: Structuring analog fractance circuit for 1/2 order fractional calculus. In: 2005 IEEE 6th International Conference on ASICON, 2005. pp. 1136–1139 (2005)

Download references

Author information



Corresponding author

Correspondence to Divya Goyal.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Goyal, D., Varshney, P. Analog Realization of Electronically Tunable Fractional-Order Differ-Integrators. Arab J Sci Eng 44, 1933–1948 (2019).

Download citation


  • Integrator
  • Differentiator
  • Differential input
  • Fractional
  • Fractor
  • Current conveyor