Skip to main content
SpringerLink
Log in

Comparison of \((1+\alpha )\) Fractional-Order Transfer Functions to Approximate Lowpass Butterworth Magnitude Responses

  • Published:
Circuits, Systems, and Signal Processing Aims and scope Submit manuscript

Abstract

Three fractional-order transfer functions are analyzed for differences in realizing (\(1+\alpha \)) order lowpass filters approximating a traditional Butterworth magnitude response. These transfer functions are realized by replacing traditional capacitors with fractional-order capacitors (\(Z=1/s^{\alpha }C\) where \(0\le \alpha \le 1\)) in biquadratic filter topologies. This analysis examines the differences in least squares error, stability, \(-\)3 dB frequency, higher-order implementations, and parameter sensitivity to determine the most suitable (\(1+\alpha \)) order transfer function for the approximated Butterworth magnitude responses. Each fractional-order transfer function for \((1+\alpha )=1.5\) is realized using a Tow–Thomas biquad a verified using SPICE simulations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Similar content being viewed by others

References

  1. A. Acharya, S. Das, I. Pan, S. Das, Extending the concept of analog Butterworth filter for fractional order systems. Signal Process. 94, 409–420 (2013)

    Article  Google Scholar 

  2. P. Ahmadi, B. Maundy, A.S. Elwakil, L. Belostostski, High-quality factor asymmetric-slope band-pass filters: a fractional-order capacitor approach. IET Circuits Devices Syst. 6(3), 187–197 (2012)

    Article  Google Scholar 

  3. A.S. Ali, A.G. Radwan, A.M. Soliman, Fractional order Butterworth filter: active and passive realizations. IEEE J. Emerg. Sel. Top. Circuits Syst. 3(3), 346–354 (2013)

    Article  Google Scholar 

  4. A.M. Elshurafa, M.N. Almadhoun, K.N. Salama, H.N. Alshareef, Microscale electrostatic fractional capacitors using reduced graphene oxide percolated polymer composites. Appl. Phys. Lett. 102(23), 232901 (2013). doi:10.1063/1.4809817

    Article  Google Scholar 

  5. A.S. Elwakil, Fractional-order circuits and systems: an emerging interdisciplinary research area. IEEE Circuits Syst. Mag. 10(4), 40–50 (2010)

    Article  Google Scholar 

  6. T.J. Freeborn, B. Maundy, A.S. Elwakil, Field programmable analogue array implementations of fractional step filters. IET Circuits Devices Syst. 4(6), 514–524 (2010)

    Article  Google Scholar 

  7. T.J. Freeborn, B. Maundy, A.S. Elwakil, Fractional-step Tow–Thomas biquad filters. Nonlinear Theory Appl. IEICE 3(3), 357–374 (2012)

    Article  Google Scholar 

  8. T.J. Freeborn, B. Maundy, A.S. Elwakil, Approximated fractional-order Chebyshev lowpass filters. Math. Prob. Eng. (2015). doi:10.1155/2015/832468

  9. T. Haba, G. Ablart, T. Camps, F. Olivie, Influence of the electrical parameters on the input impedance of a fractal structure realised on silicon. Chaos Solitons Fract. 24(2), 479–490 (2005)

    Article  Google Scholar 

  10. T. Helie, Simulation of fractional-order low-pass filters. IEEE/ACM Trans. Audio Speech Lang. Process. 22(11), 1636–1647 (2014)

    Article  Google Scholar 

  11. B. Krishna, K. Reddy, Active and passive realization of fractance device of order 1/2. Act. Passive Electron. Compon. (2008). doi:10.1155/2008/369421

  12. M. Li, Approximating ideal filters by systems of fractional order. Comput. Math. Methods Med. (2012). doi:10.1155/2012/365054

  13. A. Marathe, B. Maundy, A.S. Elwakil, Design of fractional notch filter with asymmetric slopes and large values of notch magnitude, in 2013 Midwest Symposium on Circuits and Systems, pp. 388–391 (2013)

  14. B. Maundy, A.S. Elwakil, T.J. Freeborn, On the practical realization of higher-order filters with fractional stepping. Signal Process. 91(3), 484–491 (2011)

    Article  MATH  Google Scholar 

  15. C. Psychalinos, G. Tsirimolou, A.S. Elwakil, Switched-capacitor fractional-step Butterworth filter design. Circuits Syst. Signal Process. (2015). doi:10.1007/s00034-015-0110-9

  16. 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)

    Article  Google Scholar 

  17. A. Radwan, A. Elwakil, A. Soliman, On the generalization of second-order filters to the fractional-order domain. J. Circuits Syst. Comput. 18(2), 361–386 (2009)

    Article  Google Scholar 

  18. A. Radwan, A. Soliman, A. Elwakil, A. Sedeek, On the stability of linear systems with fractional-order elements. Chaos Solitons Fract. 40(5), 2317–2328 (2009)

    Article  MATH  Google Scholar 

  19. M. Sivarama 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)

    Article  Google Scholar 

  20. A. Soltan, A.G. Radwan, A.M. Soliman, CCII based fractional filters of different orders. J. Adv. Res. 5(2), 157–164 (2014)

    Article  Google Scholar 

  21. A. Soltan, A.G. Radwan, A.M. Soliman, Fractional order Sallen–Key and KHN filters: stability and poles allocation. Circuits Syst. Signal Process. 34(5), 1461–1480 (2015)

    Article  Google Scholar 

  22. M.C. Tripathy, K. Biswas, 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. 32(4), 1523–1536 (2013)

    Article  MathSciNet  Google Scholar 

  23. M.C. Tripathy, D. Mondal, K. Biswas, S. Sen, Experimental studies on realization of fractional inductors and fractional-order bandpass filters. Int. J. Circuits Theory Appl. 43(9), 1183–1196 (2015)

    Article  Google Scholar 

  24. G. Tsirimokou, C. Laoudias, C. Psychalinos, 0.5-V fractional-order companding filters. Int. J. Circuits Theory Appl. 43(9), 1105–1126 (2015)

    Article  Google Scholar 

  25. G. Tsirimokou, C. Psychalinos, A.S. Elwakil, Digitally programmed fractional-order chebyshev filters realizations using current-mirrors, in 2015 International Symposium on Circuits and Systems 2337–2340 (2015)

  26. G. Tsirimokou, C. Psychalinos, Ultra-low voltage fractional-order circuits using current mirrors. Int. J. Circuits Theory Appl. (2015). doi:10.1002/cta.2066

Download references

Author information

Authors and Affiliations

  1. Department of Electrical and Computer Engineering, University of Alabama, Tuscaloosa, AL, USA

    Todd J. Freeborn

Authors
  1. Todd J. Freeborn
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Todd J. Freeborn.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Freeborn, T.J. Comparison of \((1+\alpha )\) Fractional-Order Transfer Functions to Approximate Lowpass Butterworth Magnitude Responses. Circuits Syst Signal Process 35, 1983–2002 (2016). https://doi.org/10.1007/s00034-015-0226-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00034-015-0226-y

Keywords

Search

Navigation