Todd J. Freeborn

A Survey of Fractional-Order Circuit Models for Biology and Biomedicine

In this survey, we present many of the fractional- order circuit models used in biomedicine and biology to fit experimentally collected impedance data. An overview of the different methods used to extract the impedance parameters from collected datasets are also presented. Applications of fractional order circuit models for modelling human tissue, plant physiology, respiratory systems, and tissue-electrode interfaces are presented to highlight the significance of these models and their potential for further research. This survey is of a tutorial nature intended as an introduction to fractional-order circuit models and to consolidate the many models reported across literature.

Field programmable analogue array implementation of fractional step filters

In this study, the authors propose the use of field programmable analogue array hardware to implement an approximated fractional step transfer function of order (n+α) where n is an integer and 0 < α < 1. The authors show how these filters can be designed using an integer order transfer function approximation of the fractional order Laplacian operator sα. First and fourth-order low- and high-pass filters with fractional steps from 0.1 to 0.9, that is of order 1.1–1.9 and 4.1–4.9, respectively, are given as examples. MATLAB simulations and experimental results of the filters verify the implementation and operation of the fractional step filters.

Measurement of Supercapacitor Fractional-Order Model Parameters From Voltage-Excited Step Response

In this paper, we propose using a numerically solved least squares fitting process to estimate the impedance parameters of a fractional order model of supercapacitors from their voltage excited step response, without requiring direct measurement of the impedance or frequency response. Experimentally estimated parameters from low capacity supercapacitors of 0.33, 1, and 1.5 F in the time range 0.2-30 s and high capacity supercapacitors of 1500 and 3000 F in the time range 0.2-90 s verify the proposed time domain method showing less than 3% relative error between the simulated response (using the extracted fractional parameters) and the experimental step response in these time ranges. An application of employing supercapacitors in a multivibrator circuit is presented to highlight their fractional time-domain behavior.

On the practical realization of higher-order filters with fractional stepping

We propose the use of a compact integer-order transfer function approximation of the fractional-order Laplacian operator s^@a to realize fractional-step filters. Lowpass and bandpass filters of orders (n+@a) and 2(n+@a), where n is an integer and 0 5.9) is given as an example with its characteristics compared to 5th- and 6th-order Butterworth filters. Spice simulations and experimental results are shown.

Fractional-order models of supercapacitors, batteries and fuel cells: a survey

This paper surveys fractional-order electric circuit models that have been reported in the literature to best fit experimentally collected impedance data from energy storage and generation elements, including super-capacitors, batteries, and fuel cells. In all surveyed models, the employment of fractional-order capacitors, also known as constant phase elements, is imperative not only to the accuracy of the model but to reflect the physical electrochemical properties of the device.

Approximated Fractional Order Chebyshev Lowpass Filters

We propose the use of nonlinear least squares optimization to approximate the passband ripple characteristics of traditional Chebyshev lowpass filters with fractional order steps in the stopband. MATLAB simulations of , , and order lowpass filters with fractional steps from  = 0.1 to  = 0.9 are given as examples. SPICE simulations of 1.2, 1.5, and 1.8 order lowpass filters using approximated fractional order capacitors in a Tow-Thomas biquad circuit validate the implementation of these filter circuits.

Reevaluation of Performance of Electric Double-layer Capacitors from Constant-current Charge/Discharge and Cyclic Voltammetry

The electric characteristics of electric-double layer capacitors (EDLCs) are determined by their capacitance which is usually measured in the time domain from constant-current charging/discharging and cyclic voltammetry tests, and from the frequency domain using nonlinear least-squares fitting of spectral impedance. The time-voltage and current-voltage profiles from the first two techniques are commonly treated by assuming ideal SsC behavior in spite of the nonlinear response of the device, which in turn provides inaccurate values for its characteristic metrics. In this paper we revisit the calculation of capacitance, power and energy of EDLCs from the time domain constant-current step response and linear voltage waveform, under the assumption that the device behaves as an equivalent fractional-order circuit consisting of a resistance Rs in series with a constant phase element (CPE(Q, α), with Q being a pseudocapacitance and α a dispersion coefficient). In particular, we show with the derived (Rs, Q, α)-based expressions, that the corresponding nonlinear effects in voltage-time and current-voltage can be encompassed through nonlinear terms function of the coefficient α, which is not possible with the classical RsC model. We validate our formulae with the experimental measurements of different EDLCs.

Towards the realization of fractional step filters

In this paper we propose a fractional lowpass transfer function of the order (n + α) where n is an integer and 0 < α < 1. We show how this filter can be designed using an integer-order transfer function approximation of the fractional-order Laplacian operator s^α. A 1^st order lowpass filter with fractional steps from 0.1 to 0.9, that is of order 1.1 to 1.9 is given as an example with its characteristics compared to 1^st and 2^nd order Butterworth filters. PSPICE simulations and experimental results of a prototype filter verify the operation of the fractional step filter.

Emulation of current excited fractional-order capacitors and inductors using OTA topologies

A novel topology suitable for emulating fractional-order capacitors and inductors using current excitation is achieved using a fractional-order differentiator/integrator block and appropriately configured Operational Transconductance Amplifiers. The scheme is capable of emulating both fractional-order capacitors and fractional-order inductors without any modifications to its structure. This implementation allows for electronic tuning of the order, capacitance/inductance, and bandwidth of operation by modification of only the bias current. Post-layout simulation results of the impedance magnitude and phase confirm the correct emulated behavior of both fractional-order capacitors and inductors. Two examples highlight the applications of this topology in i) realizing a fractional-order bandpass filter and ii) emulating a Cole-impedance model for biological applications. For both examples the characteristics of each circuit are validated in simulation.

Approximated Fractional-Order Inverse Chebyshev Lowpass Filters

In this paper we use a least-squares fitting routine to approximate the stopband ripple characteristics of fractional-order inverse Chebyshev lowpass filters which have fractional-order zeros and poles. MATLAB simulations of