“…The specific resistors and capacitors are given in Table 1 when a frequency of ω 0 = (2π)10 krad/s is used. At this time, fractional-order capacitors are not available for either simulations or physical realizations and require the use of approximations; however, it is important to note the recent progress in realizing devices with fractional-order impedances [26][27][28]. For the simulations and experimental circuits in this work, the fractional-capacitors were approximated using the 5th order Foster-I topology given in Figure 12.…”
In this paper, fractional-order transfer functions to approximate the passband and stopband ripple characteristics of a second-order elliptic lowpass filter are designed and validated. The necessary coefficients for these transfer functions are determined through the application of a least squares fitting process. These fittings are applied to symmetrical and asymmetrical frequency ranges to evaluate how the selected approximated frequency band impacts the determined coefficients using this process and the transfer function magnitude characteristics. MATLAB simulations of ( 1 + α ) order lowpass magnitude responses are given as examples with fractional steps from α = 0.1 to α = 0.9 and compared to the second-order elliptic response. Further, MATLAB simulations of the ( 1 + α ) = 1.25 and 1.75 using all sets of coefficients are given as examples to highlight their differences. Finally, the fractional-order filter responses were validated using both SPICE simulations and experimental results using two operational amplifier topologies realized with approximated fractional-order capacitors for ( 1 + α ) = 1.2 and 1.8 order filters.
“…The specific resistors and capacitors are given in Table 1 when a frequency of ω 0 = (2π)10 krad/s is used. At this time, fractional-order capacitors are not available for either simulations or physical realizations and require the use of approximations; however, it is important to note the recent progress in realizing devices with fractional-order impedances [26][27][28]. For the simulations and experimental circuits in this work, the fractional-capacitors were approximated using the 5th order Foster-I topology given in Figure 12.…”
In this paper, fractional-order transfer functions to approximate the passband and stopband ripple characteristics of a second-order elliptic lowpass filter are designed and validated. The necessary coefficients for these transfer functions are determined through the application of a least squares fitting process. These fittings are applied to symmetrical and asymmetrical frequency ranges to evaluate how the selected approximated frequency band impacts the determined coefficients using this process and the transfer function magnitude characteristics. MATLAB simulations of ( 1 + α ) order lowpass magnitude responses are given as examples with fractional steps from α = 0.1 to α = 0.9 and compared to the second-order elliptic response. Further, MATLAB simulations of the ( 1 + α ) = 1.25 and 1.75 using all sets of coefficients are given as examples to highlight their differences. Finally, the fractional-order filter responses were validated using both SPICE simulations and experimental results using two operational amplifier topologies realized with approximated fractional-order capacitors for ( 1 + α ) = 1.2 and 1.8 order filters.
“…1 depicts a mind map for the state-of-art realizations and approximation methods of CPEs. It is noted that two-terminal device realizations are based on electrochemical material properties [9] such as ionic gel-Cu electrodes, graphene-polymer dielectrics or metal-polymer composites among others [12] , [13] , [14] , [15] , [16] . Fig.…”
Graphical abstract
(a) Equivalent circuit Schematic of separate multiple CPEs design. (b) and (c) Relative phase error for different fractional-orders sharing the same capacitive loading with
n
= 8.
“…Due to the absence of commercially available fractional-order capacitors [4,5], it is difficult to implement fractional-order controllers through the substitution of conventional capacitors by fractional-order capacitors. Therefore, the implementation of fractional-order controllers can be performed through the utilization of RC networks [6][7][8][9][10][11][12][13][14][15], which approximate the behavior of the fractional-order capacitors.…”
This paper deals with fractional-order controller implementations, which can be constructed by electronically controlled building blocks fully integrated on a single chip. The proposed DC motor controller structure offers 50% reduction of the active components count, compared to the corresponding already published counterpart. The proposed liquid level of the two-interacting-tank controller scheme is the first one in the literature offering the aforementioned features. Simulation results, based on the 0.35 μ m Austria Mikro Systeme technology process, confirm the correct operation of both proposed controllers.
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