Further Generalization and Approximation of Fractional-Order Filters and Their Inverse Functions of the Second-Order Limiting Form

This paper proposes a further generalization of the fractional-order filters whose limiting form is that of the second-order filter. This new filter class can also be regarded as a superset of the recently reported power-law filters. An optimal approach incorporating constraints that restricts the real part of the roots of the numerator and denominator polynomials of the proposed rational approximant to negative values is formulated. Consequently, stable inverse filter characteristics can also be achieved using the suggested method. Accuracy of the proposed low-pass, high-pass, band-pass, and band-stop filters for various combinations of design parameters is evaluated using the absolute relative magnitude/phase error metrics. Current feedback operational amplifier-based circuit simulations validate the efficacy of the four types of designed filters and their inverse functions. Experimental results for the frequency and time-domain performances of the proposed fractional-order band-pass filter and its inverse counterpart are also presented

On the Design of Power Law Filters and Their Inverse Counterparts

This paper presents the optimal modeling of Power Law Filters (PLFs) with the low-pass (LP), high-pass (HP), band-pass (BP), and band-stop (BS) responses by means of rational approximants. The optimization is performed for three different objective functions and second-order filter mother functions. The formulated design constraints help avoid placement of the zeros and poles on the right-half s-plane, thus, yielding stable PLF and inverse PLF (IPLF) models. The performances of the approximants exhibiting the fractional-step magnitude and phase responses are evaluated using various statistical indices. At the cost of higher computational complexity, the proposed approach achieved improved accuracy with guaranteed stability when compared to the published literature. The four types of optimal PLFs and IPLFs with an exponent alpha of 0.5 are implemented using the follow-the-leader feedback topology employing AD844AN current feedback operational amplifiers. The experimental results demonstrate that the Total Harmonic Distortion achieved for all the practical PLF and IPLF circuits was equal or lower than 0.21%, whereas the Spurious-Free Dynamic Range also exceeded 57.23 and 54.72 dBc, respectively

Adjustable Multiphase Sinusoidal Oscillator with Fractional–Order Elements

This paper presents a fractional–order multiphase oscillator with adjustable FO (Frequency of Oscillation). The oscillator utilizes the following components: CCTA (Current Conveyor Transconductance Amplifier), OTA (Operational Transconductance Amplifier), auxiliary OTRA (Operational Transresistance Amplifier), two FOC (Fractional–Order Capacitors) and auxiliary resistors. The proposed structure provides three voltage outputs and their inverted variants for three selected values of alpha = 0.3, 0.5, and 0.8 of FOC. The FO is tuned by changing the values of the transconductances of the OTA and CCTA. The features of the presented circuit were verified by PSpice simulations with behavioural models of the active elements. Analysis of the dependence of the values of the transconductances on values of alpha and frequency is provided. A THD (Total Harmonic Distortion) and parasitic analysis of the circuit was also made

Electronically Adjustable Emulator of the Fractional-Order Capacitor

This paper presents a design of the controllable emulator of the FOC (Fractional-Order Capacitor) and its application. The circuit is based on 5th-order RC topology (type Foster I) where the passive elements in the topology are replaced by electronically adjustable components. The proposed emulator is based on OTA (Operational Transconductance Amplifier) and VDCC (Voltage Differencing Current Conveyor). The electronically controllable resistors are implemented by OTAs. The electronically tunable capacitors are implemented using capacitance multipliers, which employ VDCCs. The proposed structure provides the electronic control of the order and electronic shifting of the frequency band of the approximation validity. The proposed FOC emulator is also used for fractional-order filter design. The proposed circuits are verified using PSpice simulations

A Fractional-Order Transitional Butterworth-Butterworth Filter and Its Experimental Validation

This paper introduces the generalization of the classical Transitional Butterworth-Butterworth Filter (TBBF) to the Fractional-Order (FO) domain. Stable rational approximants of the FO-TBBF are optimally realized. Several design examples demonstrate the robustness and modeling efficacy of the proposed method. Practical circuit implementation using the current feedback operational amplifier employed as an active element is presented. Experimental results endorse good agreement (R2= 0.999968) with the theoretical magnitude-frequency characteristic