Passive and Active Elements Using Fractional ${\rm L}_{\beta} {\rm C}_{\alpha}$ Circuit

This paper introduces a qualitative revision of the traditional LC tank circuit in the fractional domain. The paper can be divided into six major parts, aiming in turn to establish the various conditions under which LβCα impedance may act as a resistor, negative resistor, or a positive or negative pure imaginary inductor or capacitor, in accordance to new frequency definitions; illustrate the process by which the phase response chooses the shortest path from initial to final phase, and use this illustration to verify the cases discussed in part one; develop the generalized parameters for the bandpass filter response of the LβCα circuit, such as the resonance frequency and quality factor versus α- β plane; discuss sensitivity analyses with respect to the fractional orders, as well as the time domain analyses for the impulse and step responses with their analytical formulas; and lastly, to propose some possible applications for this generalized circuit. Mathematical and PSpice simulation results are included to validate the discussion.

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

[2]  T. Hartley,et al.  Generalized functions for the fractional calculus. , 2008, Critical reviews in biomedical engineering.

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

[4]  YangQuan Chen,et al.  A Physical experimental study of variable-order fractional integrator and differentiator , 2011 .

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

[6]  K. Moore,et al.  Discretization schemes for fractional-order differentiators and integrators , 2002 .

[7]  C. Halijak,et al.  Approximation of Fractional Capacitors (1/s)^(1/n) by a Regular Newton Process , 1964 .

[8]  Alessandro Lo Schiavo,et al.  On the Theory of Quadrature Oscillations Obtained Through Parallel $LC$ VCOs , 2010, IEEE Transactions on Circuits and Systems I: Regular Papers.

[9]  R. Hilfer,et al.  NUMERICAL RESULTS FOR THE GENERALIZED MITTAG-LEER FUNCTION , 2005 .

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

[11]  O. Agrawal,et al.  Advances in Fractional Calculus , 2007 .

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

[13]  Karabi Biswas,et al.  Realization of a Constant Phase Element and Its Performance Study in a Differentiator Circuit , 2006, IEEE Transactions on Circuits and Systems II: Express Briefs.

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

[15]  Yangquan Chen,et al.  Robust stability check of fractional order linear time invariant systems with interval uncertainties , 2005, IEEE International Conference Mechatronics and Automation, 2005.

[16]  B. Maundy,et al.  On a multivibrator that employs a fractional capacitor , 2009 .

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

[18]  Tom T Hartley,et al.  R-function relationships for application in the fractional calculus. , 2008, Critical reviews in biomedical engineering.

[19]  O. Agrawal,et al.  Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering , 2007 .

[20]  Luigi Fortuna,et al.  Non-integer order integration by using neural networks , 2001, ISCAS 2001. The 2001 IEEE International Symposium on Circuits and Systems (Cat. No.01CH37196).

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

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

[23]  Georges L. Loum,et al.  An analytical expression for the input impedance of a fractal tree obtained by a microelectronical process and experimental measurements of its non-integral dimension , 2007 .

[24]  O. Marichev,et al.  Fractional Integrals and Derivatives: Theory and Applications , 1993 .

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

[26]  YangQuan Chen,et al.  ANALOGUE FRACTIONAL-ORDER GENERALIZED MEMRISTIVE DEVICES , 2009 .

[27]  Gary W. Bohannan,et al.  Analog Realization of a Fractional Control Element-Revisited , 2002 .

[28]  David Murphy,et al.  Phase Noise in LC Oscillators: A Phasor-Based Analysis of a General Result and of Loaded $Q$ , 2010, IEEE Transactions on Circuits and Systems I: Regular Papers.

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

[30]  Olivier Lavialle,et al.  The CRONE toolbox for Matlab: fractional path planning design in robotics , 2001, Proceedings 10th IEEE International Workshop on Robot and Human Interactive Communication. ROMAN 2001 (Cat. No.01TH8591).

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

[32]  Chu-Sun Yen,et al.  Time-domain skin-effect model for transient analysis of lossy transmission lines , 1982, Proceedings of the IEEE.

[33]  Robin De Keyser,et al.  Relations Between Fractional-Order Model Parameters and Lung Pathology in Chronic Obstructive Pulmonary Disease , 2009, IEEE Transactions on Biomedical Engineering.

[34]  Hon Keung Kwan,et al.  FIR, Allpass, and IIR Variable Fractional Delay Digital Filter Design , 2009, IEEE Transactions on Circuits and Systems I: Regular Papers.

[35]  R. Martin,et al.  Modeling electrochemical double layer capacitor, from classical to fractional impedance , 2008, MELECON 2008 - The 14th IEEE Mediterranean Electrotechnical Conference.

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

[37]  Chien-Cheng Tseng,et al.  Design of Fractional Order Digital Differentiator Using Radial Basis Function , 2010, IEEE Transactions on Circuits and Systems I: Regular Papers.