Juraj Valsa

RC models of a constant phase element

SUMMARY The paper describes models of a constant-phase element consisting of passive R and C components. The models offer any input impedance argument (phase) between −90° and 0° over a selectable frequency band covering several decades. The design procedure makes it possible to choose values of average phase, phase ripple, frequency bandwidth, and total number of R and C elements. The model can cover three frequency decades with as few as five resistors and five capacitors. The models can be used for practical realization of fractional analog differentiators and integrators, fractional oscillators, chaotic networks or for analog simulation of fractional control systems. Copyright

Approximate formulae for numerical inversion of Laplace transforms

Most methods for the numerical calculation of inverse Laplace transformations f(t) = L−1[F(s)] have serious limitations concerning the class of functions F(s) that can be inverted or the achievable accuracy. The procedures described in the paper can be used to invert rational as well as irrational or transcendental functions of the complex variable s. The required accuracy of the results can be enhanced without changing the algorithm, only at the cost of a longer computation time. The described methods were verified with many examples including transients in lumped/distributed systems with sections of lossy multiconductor transmission lines or with distributed RC elements.

Analogue Realization of Fractional-Order Dynamical Systems

As it results from many research works, the majority of real dynamical objects are fractional-order systems, although in some types of systems the order is very close to integer order. Application of fractional-order models is more adequate for the description and analysis of real dynamical systems than integer-order models, because their total entropy is greater than in integer-order models with the same number of parameters. A great deal of modern methods for investigation, monitoring and control of the dynamical processes in different areas utilize approaches based upon modeling of these processes using not only mathematical models, but also physical models. This paper is devoted to the design and analogue electronic realization of the fractional-order model of a fractional-order system, e.g., of the controlled object and/or controller, whose mathematical model is a fractional-order differential equation. The electronic realization is based on fractional-order differentiator and integrator where operational amplifiers are connected with appropriate impedance, with so called Fractional Order Element or Constant Phase Element. Presented network model approximates quite well the properties of the ideal fractional-order system compared with e.g., domino ladder networks. Along with the mathematical description,

Conceptual design of a selectable fractional-order differentiator for industrial applications

In the past decade, researchers working on fractional-order systems modeling and control have been considering working on the design and development of analog and digital fractional-order differentiators, i.e. circuits that can perform non-integer-order differentiation. It has been one of the major research areas under such field due to proven advantages over its integer-order counterparts. In particular, traditional integer-order proportional-integral-derivative (PID) controllers seem to be outperformed by fractional-order PID (FOPID or PIλDμ) controllers. Many researches have emerged presenting the possibility of designing analog and digital fractional-order differentiators, but only restricted to a fixed order. In this paper, we present the conceptual design of a variable fractional-order differentiator in which the order can be selected from 0 to 1 with an increment of 0.05. The analog conceptual design utilizes operational amplifiers and resistor-capacitor ladders as main components, while a generic microcontroller is introduced for switching purposes. Simulation results through Matlab and LTSpiceIV show that the designed resistor-capacitor ladders can perform as analog fractional-order differentiation.

Optimization of perfectly matched layer for Laplace's equation

Parameters of the perfectly matched layer (PML) of circular boundary are rigorously calculated in the paper. The PML consists of a single, double, or triple layer of elements, whose artificial parameters are calculated by minimizing an error function of potential difference between the nodal potentials of the PML and of the original infinite grid.

Optimization of perfectly matched layer for 2D Poisson's equation with antisymmetrical or symmetrical boundary conditions

Parameters of the perfectly matched layer (PML) for 2D magnetic field in a region bounded by circular boundary are rigorously calculated for the case of symmetrical or antisymmetrical boundary conditions. The PML consists of a single or double layer of elements, whose artificial parameters are calculated by minimizing an error function of potential difference between the nodal potentials of the PML and of the original grid expanding to infinity.

Evaluation of transient field in unbounded dissipative media

A new finite element method (FEM) approach is used to evaluate transient electric field based on a surge wave in a dissipative open region. Instead of the solution of the basic equations in the time domain, the Laplace transform inversion is used to model planar and axisymmetrical transient fields. The main advantage of this approach consists in the possibility of modeling unbounded regions. Current‐driven electrodes, including electrodes with floating potential, are taken into consideration.