Fractional Order Butterworth Filter: Active and Passive Realizations

This paper presents a general procedure to obtain Butterworth filter specifications in the fractional-order domain where an infinite number of relationships could be obtained due to the extra independent fractional-order parameters which increase the filter degrees-of-freedom. The necessary and sufficient condition for achieving fractional-order Butterworth filter with a specific cutoff frequency is derived as a function of the orders in addition to the transfer function parameters. The effect of equal-orders on the filter bandwidth is discussed showing how the integer-order case is considered as a special case from the proposed procedure. Several passive and active filters are studied to validate the concept such as Kerwin-Huelsman-Newcomb and Sallen-Key filters through numerical and Advanced Design System (ADS) simulations. Moreover, these circuits are tested experimentally using discrete components to model the fractional order capacitor showing great matching with the numerical and circuit simulations.

[1]  R. Magin.  Fractional Calculus in Bioengineering , 2006 .

[2]  Isabel S. Jesus,et al.  Fractional Electrical Impedances in Botanical Elements , 2008 .

[3]  Khaled N. Salama,et al.  Passive and Active Elements Using Fractional Circuit , 2011 .

[4]  Ahmed S. Elwakil,et al.  On the Generalization of Second-Order Filters to the fractional-Order Domain , 2009, J. Circuits Syst. Comput..

[5]  Ahmed M. Soliman,et al.  Fractional order filter with two fractional elements of dependant orders , 2012, Microelectron. J..

[6]  I. Podlubny.  Fractional differential equations , 1998 .

[7]  M. Nakagawa,et al.  Basic Characteristics of a Fractance Device , 1992 .

[8]  Steve Winder.  Time and frequency response , 2002 .

[9]  Ahmed Gomaa Radwan,et al.  Stability and non-standard finite difference method of the generalized Chua's circuit , 2011, Comput. Math. Appl..

[10]  Kazuhiro Saito,et al.  Simulation of Power-Law Relaxations by Analog Circuits : Fractal Distribution of Relaxation Times and Non-integer Exponents , 1993 .

[11]  Ahmed S. Elwakil,et al.  First-Order Filters Generalized to the fractional Domain , 2008, J. Circuits Syst. Comput..

[12]  Todd C Doehring,et al.  Fractional order viscoelasticity of the aortic valve cusp: an alternative to quasilinear viscoelasticity. , 2005, Journal of biomechanical engineering.

[13]  B. T. Krishna,et al.  Active and Passive Realization of Fractance Device of Order 1/2 , 2008 .

[14]  Les Thede,et al.  Practical Analog And Digital Filter Design , 2004 .

[15]  A. Elwakil,et al.  On the stability of linear systems with fractional-order elements , 2009 .

[16]  Khaled N. Salama,et al.  The fractional-order modeling and synchronization of electrically coupled neuron systems , 2012, Comput. Math. Appl..

[17]  Muhammad Faryad,et al.  Fractional Rectangular Waveguide , 2007 .

[18]  A. G. Radwan,et al.  Butterworth passive filter in the fractional-order , 2011, ICM 2011 Proceeding.

[19]  Yangquan Chen,et al.  A Fractional Order Proportional and Derivative (FOPD) Motion Controller: Tuning Rule and Experiments , 2010, IEEE Transactions on Control Systems Technology.

[20]  K. Salama,et al.  Theory of Fractional Order Elements Based Impedance Matching Networks , 2011, IEEE Microwave and Wireless Components Letters.

[21]  B. T. Krishna.  Studies on fractional order differentiators and integrators: A survey , 2011, Signal Process..

[22]  Yoshiaki Hirano,et al.  Simulation of Fractal Immittance by Analog Circuits: An Approach to the Optimized Circuits , 1999 .

[23]  Calvin Coopmans,et al.  Purely Analog Fractional Order PI Control Using Discrete Fractional Capacitors (Fractors): Synthesis and Experiments , 2009 .

[24]  Steve Winder,et al.  Analog and digital filter design , 2002 .

[25]  Khaled N. Salama,et al.  Passive and Active Elements Using Fractional ${\rm L}_{\beta} {\rm C}_{\alpha}$ Circuit , 2011, IEEE Transactions on Circuits and Systems I: Regular Papers.

[26]  Adel S. Sedra,et al.  Filter Theory and Design: Active and Passive , 1977 .

[27]  Ahmed S. Elwakil,et al.  Field programmable analogue array implementation of fractional step filters , 2010, IET Circuits Devices Syst..

[28]  Ahmed S. Elwakil,et al.  Fractional-order sinusoidal oscillators: Design procedure and practical examples , 2008, IEEE Transactions on Circuits and Systems I: Regular Papers.

[29]  K. Salama,et al.  Fractional Smith Chart Theory , 2011, IEEE Microwave and Wireless Components Letters.

[30]  K. Miller,et al.  An Introduction to the Fractional Calculus and Fractional Differential Equations , 1993 .

[31]  K. Biswas,et al.  Performance study of fractional order integrator using single-component fractional order element , 2011, IET Circuits Devices Syst..