Advertisement

Circuits, Systems, and Signal Processing

, Volume 35, Issue 9, pp 3086–3112 | Cite as

Fractional Order Oscillator Design Based on Two-Port Network

  • Lobna A. Said
  • Ahmed G. Radwan
  • Ahmed H. Madian
  • Ahmed M. Soliman
Article

Abstract

In this paper, a general analysis of the generation for all possible fractional order oscillators based on two-port network is presented. Three different two-port network classifications are used with three external single impedances, where two are fractional order capacitors and a resistor. Three possible impedance combinations for each classification are investigated, which give nine possible oscillators. The characteristic equation, oscillation frequency and condition for each presented topology are derived in terms of the transmission matrix elements and the fractional order parameters \(\alpha \) and \(\beta \). Mapping between some cases is also illustrated based on similarity in the characteristic equation. The use of fractional order elements \(\alpha \) and \(\beta \) adds extra degrees of freedom, which increases the design flexibility and frequency band, and provides extra constraints on the phase difference. Study of four different active elements, such as voltage-controlled current source, gyrator, op-amp-based network, and second-generation current-conveyor-based network, serve as a two-port network is presented. The general analytical formulas of the oscillation frequency and condition as well as the phase difference between the two oscillatory outputs are derived and summarized in tables for each designed oscillator network. A comparison between fractional order oscillators with their integer order counterparts is also illustrated where some designs cannot work in the integer case. Numerical Spice simulations and experimental results are given to validate the presented analysis.

Keywords

Fractional-order Two-port network Oscillator Gyrator CCII 

References

  1. 1.
    G.C. Alexander, M. Sadiku, Fundamentals of Electric Circuits (McGraw-Hill International Edition, Singapore, 2000)Google Scholar
  2. 2.
    G.E. Carlson, C.A. Halijak, Approximation of fractional capacitors (1/s)1/n by a regular Newton process. IEEE Trans. Circ. Theor. CT–11(2), 210–213 (1964)CrossRefGoogle Scholar
  3. 3.
    H. Chen, Robust stabilization for a class of dynamic feedback uncertain nonholonomic mobile robots with input saturation. Int. J. Control Autom. Syst. 12(6), 1216–1224 (2014)CrossRefGoogle Scholar
  4. 4.
    A. Dumlu, K. Erenturk, Trajectory tracking control for a 3-DOF parallel manipulator using fractional-order control. IEEE Trans. Ind. Electron. 61(7), 3417–3426 (2014)CrossRefGoogle Scholar
  5. 5.
    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)CrossRefGoogle Scholar
  6. 6.
    A.S. Elwakil, Design of non-balanced cross-coupled oscillators with no matching requirements. IET Circ. Devices Sys. 4(5), 365–373 (2010)CrossRefGoogle Scholar
  7. 7.
    A.S. Elwakil, M.A. Al-Radhawi, All possible second-order four-impedance two-stage Colpitts oscillators. IET Circ. Devices Sys. 5, 196–202 (2011)CrossRefGoogle Scholar
  8. 8.
    A.S. Elwakil, On the two-port network classification of Colpitts oscillators. IET Circ. Devices Sys. 3, 223–232 (2009)CrossRefGoogle Scholar
  9. 9.
    A.S. Elwakil, B.J. Maundy, Single transistor active filters: What is possible and what is not. Mathemat. IEEE Trans. Circ. Syst. 61(9), 2517–2524 (2014)Google Scholar
  10. 10.
    M. Faryad, Q.A. Naqvi, Fractional rectangular waveguide. Prog. Electromagn. Res. 75, 384–396 (2007)CrossRefGoogle Scholar
  11. 11.
    R. Gorenflo, F. Mainardi, Fractional calculus: integral and differential equations of fractional irder, in Fractal and Fractional Calculus in Continuum Mechanics (Springer, Berlin, 1997)Google Scholar
  12. 12.
    L.J. Guo, Chaotic dynamics and synchronization of fractional-order Genesio–Tesi systems. Chinese Phys. 14(8), 1517–1521 (2005)CrossRefGoogle Scholar
  13. 13.
    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 Fractals 24(2), 479–490 (2005)CrossRefGoogle Scholar
  14. 14.
    R. Hilfer, Applications of Fractional Calculus in Physics (World Scientific Publishers, Singapore, 2000)CrossRefzbMATHGoogle Scholar
  15. 15.
    I.S. Jesus, T.J.A. Machado, B.J. Cunha, Fractional electrical impedances in botanical elements. J. Vib. Control 14(9), 1389–1402 (2008)CrossRefzbMATHGoogle Scholar
  16. 16.
    J. Kim, W.-S. Ohm, N.-C. Park,Two-Port Network Model of Fluid-Loaded SAW Delay-Line Sensors. ASME (2011)Google Scholar
  17. 17.
    M.N. Kuperman, D.H. Zanette, Synchronization of multi-phase oscillators: an Axelrod-inspired model. Eur. Phys. J. B 70(2), 243–248 (2009)CrossRefGoogle Scholar
  18. 18.
    H. Li, Y. Luo, Y.Q. Chen, A fractional-order proportional and derivative (FOPD) motion controller: tuning rule and experiments. IEEE Trans. Control Syst. Tech. 18(2), 516–520 (2010)CrossRefGoogle Scholar
  19. 19.
    C. Ma, Y. Hori, Backlash vibration suppression control of torsional system by novel fractional order P \(ID^{\text{ k }}\) controller. IEEJ Trans. Ind. Applicat. 124–D(3), 312–317 (2004)CrossRefGoogle Scholar
  20. 20.
    J.J. Meng, J.T. Yun, Application of two-port network in Bilateral control for a Haptic interface with force-position compensation. Adv. Mater. Res. 199–200, 1211–1216 (2011)CrossRefGoogle Scholar
  21. 21.
    F. Miguel, M. Lima, J. Machado, M. Crisostomo, Experimental signal analysis of robot impacts in a fractional calculus perspective. J. Adv. Comput. Intell. Intell. Inf 11(9), 1079–1085 (2007)Google Scholar
  22. 22.
    K.S. Miller, Derivatives of noninteger order. Mathemat. Mag. 68(3), 183–192 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    K. Moaddy, A.G. Radwan, K.N. Salama, S. Momani, I. Hashim, The fractional-order modeling and synchronization of electrically coupled neurons system. Comput. Math. Appl. 34(10), 3329–3339 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    M. Nakagava, K. Sorimachi, Basic characteristics of a fractance device. IEICE Trans. Fund. Electr. E75–A(12), 1814–1818 (1992)Google Scholar
  25. 25.
    I. Podlubny, Fractional Differential Equations (Academic Press, San Diego, 1999)zbMATHGoogle Scholar
  26. 26.
    A.G. Radwan, A. Shamim, K.N. Salama, Theory of fractional order elements based impedance matching networks. IEEE Microw. Wireless Comp. Lett. 21(3), 120–122 (2011)CrossRefGoogle Scholar
  27. 27.
    A.G. Radwan, A.M. Soliman, A.S. Elwakil, Design equations for fractional-order sinusoidal oscillators: four practical circuits examples. Int. J. Circ. Theor. Appl. 36(4), 473–492 (2008)CrossRefzbMATHGoogle Scholar
  28. 28.
    A.G. Radwan, A.S. Elwakil, A.M. Soliman, Fractional-order sinusoidal oscillators: design procedure and practical examples. IEEE Trans. Circ. Syst. I 55(7), 2051–2063 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    A.G. Radwan, K.N. Salama, Passive and active elements using fractional \(L_{\beta } C_{\alpha }\) circuit. IEEE Trans. Circ. Syst. 58, 2388–2397 (2011)MathSciNetCrossRefGoogle Scholar
  30. 30.
    A.G. Radwan, A.M. Soliman, A.S. Elwakil, A. Sedeek, On the stability of linear systems with fractional-order elements. Chaos Solitons Fractals 40, 2317–2328 (2009)CrossRefzbMATHGoogle Scholar
  31. 31.
    R.S. Raven, Requirements on master oscillators for coherent radar. Proc. IEEE 54(2), 237–243 (1966)CrossRefGoogle Scholar
  32. 32.
    B. Ross, Fractional calculus. Mathemat. Mag. 50(3), 115–122 (1977)CrossRefzbMATHGoogle Scholar
  33. 33.
    K.N. Salama, A.M. Soliman, Novel oscillators using the operational transresistance amplifier. Microelectr. J 31, 39–47 (2000)CrossRefGoogle Scholar
  34. 34.
    A.M. Soliman, Two integrator loop quadrature oscillators: a review. J. Adv. Res. 4(1), 1–11 (2013)CrossRefGoogle Scholar
  35. 35.
    A.M. Soliman, Applications of current feedback operational amplifiers. Analog Integr. Circ. Signal Process. 11, 265–302 (1996)CrossRefGoogle Scholar
  36. 36.
    K.R. Sturley, in Radio Receiver Design (Chapman and hall, London, 1943)Google Scholar
  37. 37.
    M. Sugi, Y. Hirano, Y.F. Miura, K. Saito, Simulation of fractal immittance by analog circuits: an approach to the optimized circuits. Commun. Comput. Sci E82–A(8), 1627–1635 (1999)Google Scholar
  38. 38.
    M.C. Tsai, D.W. Gu, Robust and Optimal Control: A Two-Port Framework Approach, Advances in Industrial Control (Springer, London, 2014)CrossRefzbMATHGoogle Scholar
  39. 39.
    J. Williams, Practical circuitry for measurement and control problems. Linear Tech. Corp. AN 61, 1–40 (1994)Google Scholar
  40. 40.
    Y. Xu, A. Srivastava, in A Two Port Network Model of CNT-FET for RF Characterization, MWCAS (2007)Google Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Faculty of IETGerman University in Cairo (GUC)CairoEgypt
  2. 2.Engineering Mathematics and Physics DepartmentCairo UniversityGizaEgypt
  3. 3.Nanoelectronics Integrated Systems Center (NISC)Nile UniversityCairoEgypt
  4. 4.Radiation Engineering DepartmentNCRRT, Egyptian Atomic Energy, AuthorityCairoEgypt
  5. 5.Electronics and Communication Engineering DepartmentCairo UniversityGizaEgypt

Personalised recommendations