FRACTIONAL ORDER CURRENT MODE CIRCUITS

Abstract

Fractional calculus, i.e., fractional-order differentiation or integration, is a part of mathematics dealing with derivatives of arbitrary order. The fractional calculus is more than 300 years old topic and gaining research interest in recent past. It has become a powerful and widely used tool to demonstrate the characteristics of many systems in the real world. The fractional order dynamic system offers extra degree of freedom to control the phenomena of system. Fractional approach has been used in modeling of various physical processes such as anomalous diffusion, flow of fluid in porous media, heat conduction in a semi infinite slab, voltage-current relation in a semi-infinite transmission line, the charging and discharging of lossy capacitors etc. The fractional-order circuits and systems incorporate fractional calculus concepts and have immense potential in the field of signal processing, control systems, biomedical instrumentation, and many more. Thus the aim of this thesis is to generalize the narrow integer-order circuits to more general fractional-order counterparts. In this work design of current mode circuits has been investigated from fractional-order perspective and is briefly presented below. The capacitance scaling in integer and fractional order capacitors is addressed first and a novel Current Feedback Operational Amplifier based capacitance multiplier is proposed. It provides high multiplication factor with low component spread. This circuit is generalized in fractional domain along with three other xviii

capacitance multipliers. An application based on parallel RLC resonator is included to show its usefulness. The concept of Operational Transconductance Amplifier based impedance inverted is used to present a novel Inverted Impedance Multiplier Circuit which is further generalized to fractional domain. The proposal is illustrated through implementation of fractional higher order filter. Further two topologies for electronically tunable fractional order filters based on Operational Transconductance Amplifier are presented. The first is multi input single output structure which provides all pass and low pass responses. The second topology provides low pass and band pass responses simultaneously. The next objective of this work is proposition of higher fractional order filters which are realized by cascading filters of order (1+α) with higher integer order filters. The proposed filters are designed by approximating the fractional Laplacian operator by an appropriate integer order transfer function. Subsequently, functional block diagram approach is used for Current Feedback Operational Amplifier based realization of low and high filters of order (1+α). The concept is illustrated through Current Feedback Operational Amplifier based low pass filter of order (5+α) as obtained by cascading low pass filter of order (1+α) with proposed leapfrog realization of 4th order low pass filter. The work is extended to Current Feedback Operational Amplifier based high pass filter of order (5+α). xix

The functionality of all the propositions is verified through SPICE simulations. The Current Feedback Operational Amplifier based circuits are simulated using its macro model whereas 0.18 μm TSMC CMOS process parameters are used for Operational Transconductance Amplifier based circuits. Some of the circuits are also verified experimentally. Mathematical formulation for sensitivity of filters is included and examined through MATLAB simulations. The stability of proposed structures is also investigated.