Results of analysis of transient states in a series circuit of the class \({RL}_{\beta }{C}_{\alpha }\), supplied by an ideal voltage source, have been described in the paper. This circuit consists of a coil \({L}_{\beta }\) and a supercapacitor \({C}_{\alpha }\) described by fractional-order differential equations. A method for determining the current and voltage waveforms in the analyzed circuit, based on the decomposition of rational functions into partial fractions, has been described. This method allows to determine transient waveform shapes in the system for any kind of voltage excitation. Two cases of the problem solutions have been considered. The first case concerns a situation where poles of rational functions are real, and the second where rational functions have complex poles. Effective relations enabling the determination of transient waveforms in a closed form have been given. Analytical formulae describing transient state waveforms in the system for different types of voltage excitations: constant, monoharmonic, periodic and arbitrary being an element of a Hilbert space, have been determined, too. The obtained results have been illustrated by an example.

The article introduces a fractional-order electric arc furnace (EAF) model. It is based on a power balance equation which, as has been shown, can be represented by a Hammerstein-Wiener model. Such a model includes a block described by a linear integer-order differential equation, in this case the order is equal to one, which can be extended to a fractional-order equation. The introduced fractional order is either constant or variable in time. The variable coefficients of the model have been identified by a genetic algorithm The proposed model has been tested using measurement data from two different EAFs and its accuracy has been compared with the classical integer-order model based on the RMS error between the model output and measurement waveforms. The results show that the fractional-order EAF model enables error reduction and consequently reflects more accurately the phenomena taking place in the furnace.

The paper presents the results of research on new electroconductive polymer materials, based on polypyrrole, for the different supercapacitor constructions, i.e., the symmetric and asymmetric constructions. All the supercapacitors considered contain ITO collectors. Measurements of the complex impedance frequency characteristics were performed for these elements using the electrochemical impedance spectroscopy (EIS) method. Selected fractional-order models, known from the literature, have been used to model the impedance of these elements. The Particle Swarm Optimization (PSO) algorithm was used to estimate the model parameters. Selected estimation results, their comparison, and conclusions are also presented in the paper. The type of active electrolyte component has the greatest impact on the shape of the impedance frequency characteristics. In most cases, the highest capacitance values and the smallest resistance values were obtained for asymmetric supercapacitors.

The paper presents the results of research on new electroconductive polymer materials, based on polypyrrole, for the construction of supercapacitors. Measurements of the complex impedance frequency characteristics were performed for these elements using electrochemical impedance spectroscopy (EIS) method. Selected fractional-order models, known from literature, have been used to model the impedance of these elements. The Particle Swarm Optimization (PSO) algorithm has been used to estimate parameters of these models.