Lubomir Brancik

In this paper we introduce stochastic ordinary and stochastic partial differential equations and apply the theory to the model of transmission line. First we consider transmission line with external sources affected by randomness. Here the theory of ordinary stochastic differential equations is applied. We illustrate the solution by an example, where we simulate the stochastic responses of the system. While considering stochastic effects inside of the transmission line, we have to use the stochastic partial differential equation model. Here we present the theoretical background of the situation.

This study proposes a new approach for the optimization of phase and magnitude responses of fractional-order capacitive and inductive elements based on the mixed integer-order genetic algorithm (GA), over a bandwidth of four decade, and operating up to 1 GHz with low phase error of approximately ±1°. It provides a phase optimization in the desired bandwidth with minimal branch number, and avoids use of negative component values, and any complex mathematical analysis. Standardized, IEC 60063 compliant commercially available passive component values are used; hence, no correction on passive elements is required. To the best knowledge of the authors, this approach is proposed for the first time in the literature. As validation, we present numerical simulations using MATLAB® and experimental measurement results, in particular the Foster-II and Valsa structures with five branches for precise and/or high-frequency applications. Indeed, the results demonstrate excellent performance and significant improvements over the Oustaloup approximation, the Valsa recursive algorithm, and the continued fraction expansion as well as the adaptability of the GA-based design with five different types of distributed RC/RL network.

In this paper we deal with stochastic differential equations, that describe systems effected by coloured noise. In electrical systems this can be the case, when e.g. transmission line is modelled by means of proper higher-order ladder network. We define the mathematical representation of the coloured noise as a solution of the Langevin equation and formulate the corresponding Itô type stochastic differential equation. Applying this theory we derive the stochastic model of the network and find sets of individual stochastic trajectories numerically via a stochastic version of the backward Euler scheme. Afterwards respective confi-dence intervals are computed statistically while utilizing Student’s t distribution. The theoretical results are illustrated by an example of a higher-order ladder network. Numerical simulations in the example are carried out using Matlab.

This paper deals with CMOS fractional-order inductance (FoL) simulator design and its utilization in 2.75 ^th order Colpitts oscillator providing high frequency of oscillation. The proposed floating FoL is composed of two unity-gain current followers (CF±s), two inverting voltage buffers, a transconductor, and a fractional-order capacitor (FoC) of order 0.75, while the input intrinsic resistance of CF± is used as design parameter instead of passive resistor. The resulting equivalent inductance value of the FoL can be adjusted via order of FoC, which was emulated via 5 ^th -order Foster II RC network and values optimized using modified least squares quadratic method. In frequency range 138 kHz - 2.45 MHz the L _y shows ±5 degree phase angle deviation. Theoretical results are verified by SPICE simulations using TSMC 0.18 μm level-7 LO EPI SCN018 CMOS process parameters with ±1 V supply voltages.

The paper presents a computationally efficient method for modeling and simulating distributed systems with lossy transmission line (TL) including multiconductor ones, by a less conventional method. The method is devised based on 1D and 2D Laplace transforms, which facilitates the possibility of incorporating fractional-order elements and frequency-dependent parameters. This process is made possible due to the development of effective numerical inverse Laplace transforms (NILTs) of one and two variables, 1D NILT and 2D NILT. In the paper, it is shown that in high frequency operating systems, the frequency dependencies of the system ought to be included in the model. Additionally, it is shown that incorporating fractional-order elements in the modeling of the distributed parameter systems compensates for losses along the wires, provides higher degrees of flexibility for optimization and produces more accurate and authentic modelling of such systems. The simulations are performed in the Matlab environment and are effectively algorithmized.

Numerical inverse Laplace transform (NILT) methods are potential methods for time domain simulations, for instance the analysis of the transient phenomena in systems with lumped and/or distributed parameters. This paper proposes a numerical inverse Laplace transform method based originally on hyperbolic relations. The method is further enhanced by properly adapting several convergence acceleration techniques, namely, the epsilon algorithm of Wynn, the quotient-difference algorithm of Rutishauser and the Euler transform. The resulting accelerated models are compared as for their accuracy and computational efficiency. Moreover, an expansion to two dimensions is presented for the first time in the context of the accelerated hyperbolic NILT method, followed by the error analysis. The expansion is done by repeated application of one-dimensional partial numerical inverse Laplace transforms. A detailed static error analysis of the resulting 2D NILT is performed to prove the effectivness of the method. The work is followed by a practical application of the 2D NILT method to simulate voltage/current distributions along a transmission line. The method and application are programmed using the Matlab language.

The paper describes two approaches to a numerical simulation of waves propagation on multiconductor transmission lines (MTL) in case the additional losses at high frequencies (HF), like is a skin effect, are to be taken into account. Such numerical simulations can be required at solving signal integrity issues in current high-speed electronic systems where the interconnect structures, often modelled as general MTLs, play an important role. Both the Laplace-domain and time-domain approaches are discussed, corresponding to telegraph equations with a half-order derivative term. A simple example is simulated in Matlab, based on Laplace domain and a numerical inverse Laplace transform (NILT). Basic starting points for a direct time-domain solution are also briefly sketched, as well as utilization of a modified nodal analysis (MNA) method enabling to cover the MTL systems with more complex load circuits.