Edita Kolarova

Statistical Estimates of Stochastic Solutions of RL Electrical Circuits

This paper shows an application of the Ito stochastic calculus to the problem of modelling inductor-resistor electrical circuits. The deterministic model of the circuit is replaced by a stochastic model by adding a noise term in both the source and the resistance. The analytic solution of the resulting stochastic differential equations is obtained using the multidimensional Ito formula. Some statistical properties of the stochastic solutions are presented. The programming language C#, a part of the new MS .NET platform, is used to generate numerical solutions and their graphical representations.

Simulation of Higher-Order Electrical Circuits with Stochastic Parameters via SDEs

The paper deals with a technique for the simulation of higher-order electrical circuits with parameters varying randomly. The principle consists in the utilization of the theory of sto ...

Stochastic differential equations approach in the analysis of MTLs with randomly varied parameters

The paper deals with a technique for the analysis of multiconductor transmission lines (MTL) with randomly varied parameters, that is based on the theory of stochastic differential equations (SDE). Sets of stochastic trajectories are computed as voltage or current responses, accompanied by relevant sample means and confidence intervals. The MTL model is based on a cascade connection of generalized RLGC networks, terminating circuits are replaced by their generalized Thévenin equivalents. To develop model equations a state-variable method is applied, and then a corresponding vector SDE is formulated. Finally, a stochastic implicit Euler numerical technique is used for the numerical solution being consistent with Itô stochastic calculus. All the computation were done in the MATLAB® language and deterministic responses are also stated via a numerical inverse Laplace transforms (NILT) procedure to verify the results.

Simulation of random effects in transmission line models via stochastic differential equations

The paper deals with a method for simulation of transmission line (TL) models with randomly varied parameters, based on the theory of stochastic differential equations (SDE). The random changes of both excitation sources and TL model parameters can be considered. Voltage and/or current responses are represented in the form of the sample means and proper confidence intervals to provide reliable estimates. The TL models are based on a cascade connection of RLGC networks enabling to model nonuniform TLs in general. To develop model equations a state-variable method is used, and afterwards a corresponding vector SDE is formulated. A stochastic implicit Euler numerical scheme is used while using the MATLAB® language environment for all the computations. To verify the results the deterministic responses are also computed by the help of a numerical inversion of Laplace transforms procedure.

Vector linear stochastic differential equations and their applications to electrical networks

In this paper we present an application of the Itô stochastic calculus to the problem of modelling RLC electrical circuits. The deterministic model of the circuit is replaced by a stochastic model by adding a noise term to various parameters of the circuit. The analytic solutions of the resulting stochastic integral equations are found using the multidimensional Itô formula. For the numerical simulations in the examples we used Matlab.

SDE-based variance simulation in transmission line models with random excitations

This paper is focused on the development of a method to determine variances of the stochastic responses in transmission line models with random excitations. It is based on the theory of stochastic differential equations (SDE), namely a system of vector linear SDEs with additive noise is formulated to be prepared to simulate respective stochastic trajectories. One possibility to get the moments of the stochastic process is to determine sample statistics from a set of individual trajectories. This is, however, rather time consuming, and its benefit when compared to common methods is not so obvious. In this paper differential equations for getting directly the first two stochastic response moments of transmission line models under randomly varying excitations are formulated and solved. A verification is performed via the above stated approach, while the Students t-distribution can directly be used to determine respective confidence intervals in case of an additive noise. All simulations have been performed in a Matlab language environment.

Confidence intervals for RLCG cell influenced by coloured noise

Purpose The purpose of this paper is to determine confidence intervals for the stochastic solutions in RLCG cells with a potential source influenced by coloured noise. Design/methodology/approach The deterministic model of the basic RLCG cell leads to an ordinary differential equation. In this paper, a stochastic model is formulated and the corresponding stochastic differential equation is analysed using the Ito stochastic calculus. Findings Equations for the first and the second moment of the stochastic solution of the coloured noise-affected RLCG cell are obtained, and the corresponding confidence intervals are determined. The moment equations lead to ordinary differential equations, which are solved numerically by an implicit Euler scheme, which turns out to be very effective. For comparison, the confidence intervals are computed statistically by an implementation of the Euler scheme using stochastic differential equations. Practical implications/implications The theoretical results are illustrated by examples. Numerical simulations in the examples are carried out using Matlab. A possible generalization for transmission line models is indicated. Originality/value The Ito-type stochastic differential equation describing the coloured noise RLCG cell is formulated, and equations for the respective moments are derived. Owing to this original approach, the confidence intervals can be found more effectively by solving a system of ordinary differential equations rather than by using statistical methods.

Multiconductor transmission line models excited from multiple stochastic sources

The paper deals with a technique for simulation of statistical characteristics of random responses in multiconductor transmission line (MTL) models excited from multiple stochastic sources. The method follows the theory of stochastic differential equations (SDE), specially a vector linear SDE with additive noise is developed for the solution. The MTLs model is prepared for testing number of situations at stochastic and/or deterministic excitations in its arbitrary nodes. In such a way one can cover effects of possible stochastic disturbances along the MTL wires. The MTL model is formed by generalized II networks in cascade, while the state-variable method is used to derive its mathematical description. The excitations are permitted through generalized Thévenin equivalents of external circuits. To get the characteristics of the stochastic responses, a set of trajectories is statistically processed, while a weak stochastic backward Euler scheme, consistent with the Itô stochastic calculus, is applied.

Variance assessment at transmission lines with randomly varying parameters via SDE theory

The paper addresses a method for the assessment of variances of responses at transmission lines (TL) with randomly varying primary parameters, based on the theory of stochastic differential equations (SDE). In contrast to previous works where stochastic trajectories were statistically processed to get sample means and sample standard deviations, herein the statistical characteristics are solved directly through ordinary differential equations (ODE) prepared. The TL model is formed by cascade connection of the RLGC-based Π-sections enabling to model also nonuniform TLs if necessary. To develop the model equations a state-variable method is applied, and then a corresponding vector SDE is formulated. To verify the novel approach the SDEs are solved to obtain batches of individual stochastic trajectories, and then sample statistics and corresponding confidence intervals are computed. A stochastic implicit Euler numerical scheme is used for this purpose while utilizing the Matlab language.

Evaluation of variances in hybrid MTL systems with stochastic parameters via SDAE approach

Summary form only given. The application of stochastic differential equation (SDE) approach find its usage in various areas of the engineering when stochastic changes in physical systems are to be considered [1]. In the electrical engineering this theory can very be useful at the solution of systems with distributed parameters, particulary containing multiconductor transmission lines (MTL), which are often exploited in high-speed circuits for the data transmission [2]. In the paper, an attention will be paid to hybrid (lumped-distributed) systems with the MTLs as their distributed parts, whereas parameters of the system can vary randomly. In case of the MTL itself, with rather simple terminating elements, just the SDE theory would be applicable when taking into account some proper numerical technique. Usually, evaluation of the dispersion of responses of the system is needed which can be resolved via confidence intervals determination. To do it, either the solution of the SDE and subsequent statistical processing is applied or the direct solution of variances via Lyapunov-like equations can be performed. In hybrid systems, however, due to lumped-parameter parts included, a non-differential (algebraic) part is generally present in respective mathematical model. Therefore, a stochastic differential-algebraic equation (SDAE) approach has to be considered. The SDAE solution is more complicated in general while a proper numerical technique has to be chosen [3]. Utilization of approach in [3, 4] leads to necessity to repeatedly solve the SDAEs and to process the results statistically. In this paper a way of direct solution of variances and confidence intervals will be elaborated to be able to avoid the SDAE solution itself, namely by formulating and numerically solving Lyapunov-like equations adapted for hybrid MTL systems. All the computations will be done by using the Matlab language, with some examples provided.