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1

Equivalent diagrams of fractional order elements

Różowicz Sebastian, Włodarczyk Maciej, Zawadzki Andrzej

Archives of Control Sciences

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2023

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Vol. 33, No. 4

801--827

EN

This paper presents equivalent impedance and operator admittance systems for fractional order elements. Presented models of fractional order elements of the type: sαLα and 1/sαCα, (0 α 1) were obtained using the Laplace transform based on the expansion of the factor sign to an infinite fraction with varying degrees of accuracy - the continued fraction expansion method (CFE). Then circuit synthesis methods were applied. As a result, equivalent circuit diagrams of fractional order elements were obtained. The obtained equivalent schemes consist both of classical RLC elements, as well as active elements built based on operational amplifiers. Numerical experiments were conducted for the constructed models, presenting responses to selected input signals.

2

Analysis of Caputo-Fabrizio operator application for synthesis of fractional order PID-controller

Lozynskyy Andriy, Lozynskyy Orest., Kasha Lidiya, Holovach Ihor

Przegląd Elektrotechniczny

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2021

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R. 97, nr 6

66--71

EN

Using the representation of Caputo and Fabrizio, the influence of substitution in the linear model of a two-mass system of integer derivatives on fractional order derivatives is shown in this paper. So, the change of parameters of PID-controller of fractional order in comparison with classical PID-controller is analyzed. The influence of the use of PID-controller of fractional order on the transient characteristics of the system is demonstrated in the results of this paper.

PL

Korzystając z reprezentacji Caputo i Fabrizio, ukazano wpływ zastąpienia pochodnych całkowitego rzędu w liniowym, modelu układu dwóch mas, pochodnymi ułamkowego rzędu. Przeanalizowano zmianę parametrów regulatora PID ułamkowego rzędu względem klasycznego regulatora PID. Zademonstrowano wyniki przedstawiające wpływ wykorzystania regulatora PID ułamkowego rzędu na charakterystykę przejściową układu.

3

A quadrature-free Legendre polynomial approach for the fast modelling guided circumferential wave in anisotropic fractional order viscoelastic hollow cylinders

Zhang X., Liang S., Shao S., Yu J.

Archives of Mechanics

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2021

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Vol. 73, nr 2

121--152

EN

Compared to the traditional integer order viscoelastic model, a fractional order derivative viscoelastic model is shown to be advantageous. The characteristics of guided circumferential waves in an anisotropic fractional order Kelvin–Voigt viscoelastic hollow cylinder are investigated by a quadrature-free Legendre polynomial approach combining the Weyl definition of fractional order derivatives. The presented approach can obtain dispersion solutions in a stable manner from an eigenvalue/eigenvector problem for the calculation of wavenumbers and displacement profiles of viscoelastic guided wave, which avoids a lot of numerical integration calculation in a traditional polynomial method and greatly improves the computational efficiency. Comparisons with the related studies are conducted to validate the correctness of the presented approach. The full three dimensional spectrum of an anisotropic fractional Kelvin–Voigt hollow cylinder is plotted. The influence of fractional order and material parameters on the phase velocity dispersion and attenuation curves of guided circumferential wave is discussed in detail. Moreover, the difference of the phase velocity dispersion and attenuation characteristics between the Kelvin–Voigt and hysteretic viscoelastic models is also illustrated. The presented approach along with the observed wave features should be particularly useful in non-destructive evaluations using waves in viscoelastic waveguides.

4

Fractional models of selected combustion engine ignition systems

Włodarczyk M., Zawadzki A., Różowicz S.

Przegląd Elektrotechniczny

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2016

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R. 92, nr 4

30--33

EN

This paper attempts to introduce a quasi-inductive element into ignition system and describes it by fractional order equation. Two typical systems have been studied and numerical analysis has been conducted.

PL

W pracy podjęto próbę wprowadzenia elementów quasi-indukcyjnych, aby opisać równaniem ułamkowego rzędu układy zapłonowe. Dwa typowe systemy zostały zbadane, po czym przeprowadzono dla nich analizę numeryczną.

5

Fractional-order derivative and time-dependent viscoelastic behaviour of rocks and minerals

Kawada Y., Yajima T., Nagahama H.

Acta Geophysica

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2013

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Vol. 61, no. 6

1690-1702

EN

A general constitutive equation for viscoelastic behaviour of rocks and minerals with fractional-order derivative is investigated. This constitutive law is derived based on differential geometry and thermodynamics of rheology, and the fractional order of derivative represents the degree of time delay. Analyzing some laboratory experimental data of high temperature deformation of rocks and minerals such as halite, marble and orthopyroxene, we propose how to determine the orders of fractional derivative for viscoelastic behaviours of rocks and minerals. The order is related to the exponents for the temporal scaling in the relaxation modulus and the stress power-law of strain rate, i.e., the non-Newtonian flow law, and considered as an indicator representing the macroscopic behaviour and microscopic dynamics of rocks.

6

Modelowanie procesów ładowania i rozładowania superkondensatora

Zawadzki A., Włodarczyk M.

Pomiary Automatyka Kontrola

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2010

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R. 56, nr 12

1413-1415

PL

W artykule przedstawiono zastosowanie rachunku różniczkowego ułamkowego rzędu do analizy stanu nieustalonego: ładowania i rozładowania superkondensatora w układzie RC. Obiektem badań był superkondensator wykorzystywany w elektronice, trakcji i różnych gałęziach przemysłu jako dodatkowe źródło o dużej pojemności pełniące funkcję zaawansowanego magazynu energii. Pokazano wyniki badań eksperymentalnych i symulacyjnych. Badania eksperymentalne przeprowadzono w różnych temperaturach pracy w celu analizy wpływu środowiska na parametry superkondensatorów. W badaniach symulacyjnych wykonano obliczenia dla klasycznego układu RC oraz dokonano wyboru rzędu pochodnej ułamkowej, dla której otrzymuje się najlepsze przybliżenia w stosunku do danych pomiarowych.

EN

In the paper there is presented use of differential calculus of fractional grade for analysis of transient state of charging and discharging supercapacitor in the RC system. The research object was a supercapacitor used in electronic, traction and other industry branches as an additional source of large capacity functioning as the energy store. The results of experimental and simulation research are given. In order to analyse the environmental influence on supercapacitor parameters, experimental investigations were carried out in different work temperatures for analysis. In simulation research there were carried out calculations for the classic RC system and there was determined the order of fractional derivative which gives the best closeness to the measurement data. In the figures placed in the paper the simulation and experiments results are shown. From the performed investigations it follows that the classical theory gives better approximation of charging and uncharging current, altough for time instants, where the current changes more slowly, fractional derivatives give better approximations.