AN INVESTIGATION INTO A CLASS OF FRACTIONAL ORDER ANALOG CIRCUITS

Abstract

This thesis has dealt with the realization of voltage-mode/current-mode fractional order analog circuits viz. fractional order filters, fractional order inverse filters and fractional order sinusoidal oscillators using different active building blocks such as op-amp, CFOAs and OTRA. In the following, we present a summary of the main contributions of this thesis. 6.1. Summary In Chapter 1of the thesis, a preliminary introduction of fractional order analog circuits, the objectives of the research work and the organization of the thesis has been presented. Chapter 2 dealt with different types of fractional order elements (FOEs) available in open literature, viz. single component based and multicomponent-based FOEs. We have presented the details of some of the important works dealing with the realizations of the FOEs using single, as well as multi-component based realizations of the FOEs. We have also presented the details of the methodology and design of the fractional order capacitors, which we have used for the realization of various fractional order analog circuits in this thesis. The methods proposed by Valsa, Dvorak and Friedl and Oustaloup, Levron, Mathieu and Nanot, for implementation of fractional order capacitors have been explained in detail for finding the component values in the R-C ladder network. Using Valsa, Dvorak and Friedl’s method,

245 fractional order capacitors of values 0.382μF/sec(α-1)

, 0.0955μF/sec(α-1) and 1nF/sec(α-1) for different values of α have been designed, whereas fractional order capacitors of value 10nF/sec(α-1) for α = 0.9 and α = 0.8 have been designed using Oustaloup, Levron, Mathieu and Nanot’s method. To verify the performance of these fractional order capacitors, the magnitude and phase responses have also been presented. Experimentally obtained frequency response characteristics of a fractional order capacitor have also been presented. In chapter 3, a brief literature overview of fractional order filters, designed using various active building blocks has been presented. This Chapter is also concerned with the design of CFOA-based fractional order filters in (i) voltage-mode and (ii) current-mode. Two different structures of voltage-mode fractional order filters have been presented in which the fractional order low pass filter, fractional order band pass filter and fractional order high pass filter can be realized by selecting the branch admittances appropriately. Both the presented structures employ only a single CFOA and five passive components (three/two conventional resistors and two/three fractional order capacitors). The proposed current-mode fractional order filter structure, on the other hand, employs only a single CFOA, two resistors and two fractional order capacitors. The various output responses viz, FOLP, FOBP and FOHP can be obtained by appropriate selection of the branch admittances The workability of all the proposed circuits has been verified by theoretical calculations, MATLAB and PSPICE simulations for different values of ‘α’ (0 < α < 1). Experimental results for an exemplary single CFOA based fractional order filter operating in voltage mode (for α = β = 0.7) have also been presented.

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In chapter 4, a brief literature overview of conventional inverse filters, designed using various active building blocks has been presented. Prior to the commencement of this research work, no analog inverse active filter in fractional order domain was reported in open literature. In view of this, we have presented several structures of fractional order inverse active filters, designed with various active building blocks viz. op-amp, CFOAs and OTRA, passive resistors and simulated fractional order capacitors. Three circuits of op-amp-based fractional order multifunction inverse active filters, two circuits of CFOA-based multifunction inverse filters and one circuit of OTRA-based fractional order inverse filter have been presented. From the same topology, in all the presented configurations, FOILP, FOIBP and FOIHP filter responses can be obtained by appropriately selecting the branch admittances. The performance of all presented circuits has been verified by theoretical calculations, MATLAB and PSPICE simulations for various values of ‘α’ (0 < α < 1). One of the op-amp-based fractional order inverse filter circuit (minimal realization of fractional order inverse filter) has also been verified experimentally for α = 0.7. The sensitivity and stability analysis of fractional order inverse active filters have also been described in this chapter. In chapter 5, we have presented an extensive literature review of fractional order sinusoidal oscillators, which were designed by merely replacing the integer order capacitors by fractional order capacitors or designed ab-initio using various active building blocks. Two new configurations of fractional order sinusoidal oscillators have been implemented using single active device and other passive components viz. passive resistors and fractional order capacitors. One of the structures of FOSO is realized with a single OTRA, four resistors and two fractional

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order capacitors. Nine different cases of OTRA-based fractional order oscillators for different combinations of α and β were simulated in PSPICE. The other structure was implemented using a single CFOA, three resistors and two fractional order capacitors. Nine different cases of CFOA-based fractional order oscillators for different combinations of α and β were simulated in PSPICE. For both the structures of FOSOs, PSPICE simulations were carried out to show the transient response and frequency spectra of the proposed FOSOs. For one case, (α = β = 0.8) of CFOA based FOSO, experimental results have also been presented. The stability analysis has also been carried out for the presented fractional order sinusoidal oscillators. 6.2. Future Scope Fractional order circuits and systems is an emerging interdisciplinary area of science and engineering. Several interesting developments have taken place in the area of fractional order analog active circuits during the last two decades resulting in appearance of various types of analog circuit implementation of fractional order filters, fractional order sinusoidal oscillators and fractional order PID controllers with different properties, which were not reported in their integer order counterparts, in open literature. When this research work was initiated, there were no fractional order inverse filter structure available in the open literature. We have proposed a new configuration of fractional order inverse filters in 2018 and proposed several other inverse active filters in fractional order domain. Apart from the work reported in this thesis, in the following, we suggest few other directions in which the work presented in this thesis can be extended:

1. The design and development of a compact fractance device, in which the order of

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the FOE can be changed smoothly, by means of some electronic control (voltage /current), will be helpful in exploiting the real potential of the fractional order active circuits. This is still an open problem which can be taken up for further investigation. 2. Various approximation methods have been used for simulation of the FOE in general. A systematic evaluation of the performance of the realized FOE for a specific application will help in benchmarking of the different approximation methods for different applications, thus, will help in the development of commercially available FOEs, and therefore, is an interesting problem open for future research.

3. The inverse analog filters presented in this thesis have been realized using an approach wherein we have realized the transfer functions with the help of fractional order capacitor(s). These transfer functions could also be realized using any one of the standard methods of approximations of the fractional order Laplacian operator and the performance may be compared in terms of the tunability properties and other specifications. 4. The development of design tables and related softwares for the design of fractional order active filters for different values of the fractional order parameters and design specifications is an interesting problem which may be taken up for further research.

5. New structures of multiphase sinusoidal oscillators realized with different ABBs may be developed.

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6. Analog circuit implementation of multi frequency fractional order oscillators is yet another area in which the work presented in this thesis may be extended. Finally, it must be concluded that there is no dearth of new possibilities, problems and ideas to pursue, as far as the design of fractional order analog circuits is concerned and this is, undoubtedly, a very exciting area of research.