This paper introduces the fundamentals of the conventional LC low-pass filter circuit in the fractional domain. First, we study the new fundamentals of fractional-order LC low-pass filter circuit including the pure real angular frequency, the pure imaginary angular frequency and the short circuit angular frequency. Moreover, sensitivity analysis of the impedance characteristics and phase characteristics of the LC low-pass filter circuit with respect to the system variables is studied in detail, which shows the greater flexibility of the fractional-order filter circuit in designs. Furthermore, from the filtering property perspective, we systematically investigate the effects of the system variables (LC, frequency f and fractional orders) on the amplitude-frequency characteristics and phase-frequency characteristics. In addition, the detailed analyses of the cut-off frequency and filter factor are presented. Numerical experimental results are presented to verify the theoretical results introduced in this paper.
J. Sabatier, O. P. Agrawal and J. A. Tenreiro Machado, “Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering,” Berlin: Springer-Verlag New York Inc., 2007.
T. C. Doehring, A. H. Freed, E. O. Carew and I. Vesely, “Fractional order viscoelasticity of the aortic valve cusp: An alternative to quasilinear viscoelasticity,” J. Biomech. Eng.-Trans. ASME, vol. 127, no. 4, pp. 700-708, Aug. 2005.
A. G. Radwan, A. M. Soliman, A. S. Elwakil and A. Sedeek, “On the stability of linear systems with fractional-order elements,” Chaos. Solition. Fract., vol. 40, no. 5, pp. 2317-2328, Jun. 2009.
W. B. Zhang, J. A. Fang and Y. Tang, “Robust stability for genetic regulatory networks with linear fractional uncertainties,” Commun. Nonl. Sci. Numer. Simul., vol. 17, no. 4, pp. 1753-1765, Apr. 2012.
Y. Tang, H. J. Gao and J. Kurths, “Distributed robust synchronization of dynamical networks with stochastic coupling,” IEEE T. CAS-I, vol. 61, no. 5, pp. 1508-1519, May. 2014.
L. Tan, J. Jiang and L. M. Wang, “Multirate Processing Technique for Obtaining Integer and FractionalOrder Derivatives of Low-Frequency Signals,” IEEE Trans. Instrum. Meas., vol. 63, no. 4, pp. 904-912, Apr. 2014.
H. Li, Y. Luo and Y. Q. Chen, “A fractional-order proportional and derivative (FOPD) motion controller:tuning rule and experiments,” IEEE Trans. Control Syst. Technol., vol. 18, no. 2, pp. 516-520, Mar. 2010.
A. G. Radwan, A. S. Elwakil and A. M. Soliman, “Fractional-order sinusoidal oscillators: Design procedure and practical examples,” IEEE T. Circ. Syst., vol. 55, no. 7, pp. 2051-2063, Aug. 2008.
U. J. Seo, Y. D. Chun, J. H. Choi, S. U. Chung and P. W. Han, “General characteristic of fractional slot double layer concentrated winding synchronous machine,” J. Electr. Eng. Techn. vol. 8, no. 3, pp. 282-287, Mar. 2013.
P. Ahmadi, B. Maundy, A. S. Elwakil and L. Belostostski, “High-quality factor asymmetric-slope band-pass filters: a fractional-order capacitor approach,” IET Circ. Dev. Syst., vol. 6, no. 3, pp. 187-197, May. 2012.
S. M. Shah, R. Samar, M. A. Z. Raja and J. A. Chambers, “Fractional normalised filtered-error least mean squares algorithm for application in active noise control systems,” Electron. Lett., vol. 50, no. 14, pp. 973-U9874, Jul. 2014.
M. Tabatabaei and M. Haeri, “Sensitivity analysis of CRA based controllers in fractional order systems,” Signal Process., vol. 92, no. 9, pp. 2040-2055, Sep. 2012.
G. W. Bak and A. K. Jonscher, “Low-frequency dispersion in hopping electronic systems,” Journal of Materials Science, vol. 34, no. 22, pp. 5505-5508, Nov. 1999.
S. Westerlund and L. Ekstam, “Capacitor theory,” IEEE T. Dielectr. Electr. Insulat., vol. 1, no. 5, pp. 826-839, Oct. 1994.
I. Petrá, “A note on the fractional-order Chua’s system,” Chaos Solitons Fractals, vol. 38, no. 1, pp. 140-147, Oct. 2008.
S. Westerlund, “Dead Matter Has Memory,” Kalmar, Sweden: Causal Consulting, 2002.
I. S. Jesus, J. A. Tenreiro Machado, “Development of fractional order capacitors based on electrolyte processes,” Nonlinear Dyn., vol. 68, no. 1-2, pp. 107-115, Apr. 2012.
M. Sivarama Krishna, S. Das, K. Biswas and B. Goswami, “Fabrication of a fractional order capacitor with desired specifications: A study on process identification and characterization,” IEEE Trans. Electron. Devices, vol. 58, no. 11, pp. 4067-4073, Nov. 2011.
T. Haba, G. Ablart, T. Camps and F. Olivie, “Influence of the electrical parameters on the input impedance of a fractal structure realised on silicon,” Chaos, Solitons Fractals, vol. 24, no. 2, pp. 479-490, Apr. 2005.
A. M. Elshurafa, M. N. Almadhoun, K. N. Salama and H. N. Alshareef, “Microscale electrostatic fractional capacitors using reduced graphene oxide percolated polymer composites,” Appl. Phys. Lett., vol. 102, no. 23, Jun. 2013.
J.A.T. Machado and A.M.S.F. Galhano, “Fractional order inductive phenomena based on the skin effect,” Nonlinear Dyn., vol. 68, no. 1-2, pp. 107-115, Apr. 2012.
Salim; M. T. ,Benchouia and A. Golea, “Harmonic Current Compensation based on Three-phase Threelevel Shunt Active Filter using Fuzzy Logic Bandstop-to-Bandpass Switchable Filter,” IEEE Trans. Microw. Theory Tech., vol. 61, no. 3, pp. 1114-1123, Mar. 2013.
J. H. Park, S. J. Cheon and J. Y. Park, “A Compact LTCC Dual-Band WLAN Filter using Two Notch Resonators,” J. Electr. Eng. Technol., vol. 8, no. 1, pp. 168-175, Jan. 2013.
S. Kahng, B. Lee and T. Park, “Compact UHF 9thorder bandpass filter with sharp skirt by cascadedtriplet CRLH-Zor,” J. Electr. Eng. Technol., vol. 8, no. 5, pp. 1152-1156, Sep. 2013.
J. D. Ruiz and J. Hinojosa, “Shunt series LC circuit for compact coplanar waveguide notch filter design,” IET Microw Antenna P., vol. 8, no. 2, pp. 125-129, Jan. 2014.
S. J. Cheon, J. H. Park and J. Y. Park, “Highly Miniaturized and Performed UWB Bandpass Filter Embedded into PCB with SrTiO3 Composite Layer,” J. Electr. Eng. Technol. vol. 7, no. 4, pp. 528-588, Jul. 2012.
A. G. Radwan and M. E. Fouda, “Optimization of Fractional-Order RLC Filters,” Circuits Syst. Signal Process., vol. 32, no. 5, pp. 2097-2118, Oct. 2013.
A. S. Ali, A. G. Radwan and A. M. Soliman, “Fractional Order Butterworth Filter: Active and Passive Realizations,” IEEE Journal on emerging and selected topics in circuits and systems, vol. 3, no. 3, pp. 346-354, Sep. 2013.
I. Podlubny, “Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and some of their Applications,” vol.198, Academic Press, San Diego, Calif, USA, 1999.