Lubomir Brancik

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.

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.

Numerical Inversion of Two-Dimensional Laplace Transforms Based on Partial Inversions

The paper deals with a technique for numerical inversion of two-dimensional Laplace transforms based on the FFT & IFFT in conjunction with a quotient-difference algorithm of Rutishauser. In contrast to the existing FFT-based technique the presented one is based on a repeated application of one-dimensional partial Laplace transform inversions. The method promises a generalization towards multidimensional numerical inverse Laplace transforms because it establishes more effective unified algorithmic approach to a computation. The method was programmed using Matlab language and analyzed as for its accuracy.

An improvement of FFT-based numerical inversion of two-dimensional Laplace transforms by means of /spl epsiv/-algorithm

The paper describes a fast computational method for the numerical inversion of two-dimensional Laplace transforms (2D-NILT). The numerical procedure is basically that based on FFT algorithms (Brancik and Valsa, 1998), but it is improved significantly using an /spl epsiv/-algorithm to give precision to the result. Compared with the original method, the procedure under consideration enables the error to be decreased by a few orders. The method has been programmed and verified using the universal mathematical language Matlab.

Numerical Inverse Laplace Transforms for Electrical Engineering Simulation

Numerical inverse Laplace transform (NILT) methods are widely used in various scientific areas, especially for a solution of respective differential equations. In field of an electrical engineering many various approaches have been considered so far, but mostly for a single variable (1D NILT), see at least (Brancik, 1999, 2007b; Cohen, 2007; Valsa & Brancik, 1998; Wu at al., 2001) from plenty of papers. Much less attention was paid to multidimensional variable (nD NILT) methods, see e.g. (Hwang at al., 1983; Singhal at al., 1975), useful rather for more complicated electromagnetic systems. The 2D NILT methods, see e.g. (Brancik, 2005, 2007a, 2007b; Hwang & Lu, 1999), can be applied for a transmission line analysis, or nD NILT methods, n ≥ 2, for a nonlinear circuits analysis, if relevant Laplace transforms are developed through a Volterra series expansion, see e.g. (Brancik, 2010a, 2010b, Karmakar, 1980; Schetzen, 2006), to highlight at least a few applications. This paper is focused on the class of NILT methods based on complex Fourier series approximation, their error analysis, their effective algorithms development in a Matlab language, and after all, on their selected applications in field of electrical engineering to show practical usefulness of the algorithms.

Matlab based time-domain simulation of multiconductor transmission line systems

The paper deals with a technique of the time-domain domain simulation of transient phenomena in linear distributed systems using Matlab language. The distributed parts are formed by multiconductor transmission lines (MTL) which can generally be nonuniform, frequency-dependent, and under nonzero initial conditions. In principle the solution is formulated in terms of a modified nodal admittance (MNA) equation method in the frequency domain. Then an improved fast method of numerical inversion of Laplace transforms (NILT) in vector or matrix form is applied to gain a solution in the time domain. This NILT method is based on the FFT in conjunction with the quotient-difference algorithm of Rutihauser to ensure both high speed of computation and sufficient accuracy of 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.

Simulation of multiconductor transmission line circuits combining 1D and 2D Laplace transformations

The paper presents an innovative way for a simulation of multiconductor transmission line (MTL) circuits which combines the Laplace transformations in one and in two variables. This approach enables computation of both nodal voltages (branch currents) in lumped-element parts of a circuit and voltage (current) waves distributions along MTL wires effectively. The first one is performed by the numerical inversion of 1D Laplace transforms when the solution in frequency s-domain is formulated using a modified nodal admittance equation method (MNA). The second one is done by the numerical inversion of 2D Laplace transforms when the solution appertaining to distributed parts of the circuit is formulated in the (q,s)-domain. Both NILT (numerical inversion of Laplace transforms) methods utilize the FFT in conjunction with quotient-difference algorithm to provide the high speed calculation and precision of results.

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.