In this paper, the generalized analysis of the first Butterworth filter based on two passive elements is introduced in the fractional-order sense. The fractional-order condition of the Butterworth circuit is presented for the first time where it will lead us back to the known condition of the integer-order circuit when the two fractional-orders equal one. Therefore, the conventional behavior of the integer-order circuit is a narrow subset of the fractional-order ones. The circuit is studied under same and different order cases, and verified through their numerical simulations. Stability analysis is also introduced showing the poles location in the fractional-order versus integer order cases. © 2011 IEEE.
Impedance matching through a single passive fractional element
For the first time, a generalized admittance Smith chart theory is introduced to represent fractional order circuit elements. The principles of fractional order matching circuits are described. We show that for fractional order ? < 1, a single parallel fractional element can match a wider range of load impedances as compared to its series counterpart. Several matching examples demonstrate the versatility of fractional order series and parallel element matching as compared to the conventional approach. © 2012 IEEE.
General procedure for two integrator loops fractional order oscillators with controlled phase difference
This paper studies the fractional order two integrator loop based sinusoidal oscillators with two fractional order elements of different orders. Two general cases have been discussed and closed forms for the oscillation frequency and oscillation condition are driven. In addition, the effect of the fractional orders on the phase difference between the two oscillatory outputs is also presented. Design procedure for the two general cases is illustrated with numerical examples and validated through circuit simulations for three examples of oscillators based on two integrator loops. © 2013 IEEE.
CCII based KHN fractional order filter
This work aims to generalize the analysis of the fractional order filter to work for the low-pass, band-pass and high-pass responses. So, general expression for the maximum and minimum frequency points and the half power frequency points will be derived. In addition, the effect of the transfer function parameters on the filter poles and hence the stability is introduced. Besides, the effect of the fractional orders on the frequency response will be presented. Finally, to verify the numerical analysis and the proposed design procedure, circuit simulation will be used. © 2013 IEEE.
Current feedback operational amplifier (CFOA) based fractional order oscillators
This paper presents a study of fractional order oscillators based on current feedback operational amplifiers (CFOA). Two general cases have been discussed for the oscillation frequency and condition with the use of two fractional order elements of different orders. Design procedure for the two general cases is illustrated with numerical discussions. Circuit simulations for some special cases are presented to validate the theoretical findings. The simulations have been done using Ad844 commercial CFOA model © 2014 IEEE.
Fractional order oscillators with single non-zero transmission matrix element
This paper presents a study of fractional order oscillator design based on a matrix. The presented oscillator consists of a general two port network and three impedances. Only two port with single element in its transmission matrix is discussed which gives four possible networks. Different combinations for one element have been investigated. The impedances associated with the studied networks are series or parallel connection of resistors in addition to fractional order capacitors. The characteristic equation, oscillation frequency and condition for each combination are introduced. Numerical discussions of the presented oscillators with Spice simulations are presented to validate the theoretical findings. © 2015 IEEE.
Fractional-order inverting and non-inverting filters based on CFOA
This paper introduces a study to generalize the design of a continuous time filters into the fractional order domain. The study involves inverting and non-inverting filters based on CFOA where three responses are extracted which are high-pass, band-pass and low-pass responses. The proposed study introduces the generalized formulas for the transfer function of each response with different fractional orders. The fractional-order filters enhance the design flexibility and controllability due to the extra degree of freedom provided by the fractional order parameters. The general fundamentals of these filters are presented by calculating the cutoff frequency equation. Different numerical solutions for the generalized fractional order filters are introduced. Stability discussion is presented for different fractional order cases. Spice simulations results are introduced to validate the theoretical findings. © 2016 IEEE.
Fractional-order oscillator based on single CCII
This paper presents a generalization of well-known phase shift oscillator based on single CCII into the fractional order domain. The general state matrix, characteristic equation and design equations are presented. The general oscillation frequency, condition and the phase difference between the oscillatory outputs are introduced in terms of the fractional order parameters. These parameters add extra degrees of freedom which in turn increase the design flexibility and controllability. Numerical discussion of five special cases is investigated including the integer case. Spice simulations and experimental results are introduced to validate the theoretical findings with stability discussion. © 2016 IEEE.
Low pass filter design based on fractional power chebyshev polynomial
This paper introduces the design procedure for the low pass filter based on Chebyschev polynomials of fractional power of any order. The filter order is considered in intervals of width two. Only the first two intervals are considered along with their pole locus produced by varying the filter order and the magnitude response. A general formula for constructing the filter from its s-plane poles is suggested. Numerical analysis and circuit simulations using MATLAB and Advanced Design System (ADS) based on the proposed design procedure are presented. Good matching between the circuit simulation and the numerical analysis is obtained which proves the reliability of the proposed design procedure. © 2015 IEEE.
The generalized exponential function and fractional trigonometric identities
In this work, we recall the generalized exponential function in the fractional-order domain which enables defining generalized cosine and sine functions. We then re-visit some important trigonometric identities and generalize them from the narrow integer-order subset to the more general fractional-order domain. Generalized hyperbolic function relations are also given. © 2011 IEEE.