Discrete differential operators for periodic and two-periodic time functions

Sobczyk, Tadeusz; Radzik, M.; Radwan-Pragłowska, Natalia

doi:10.1108/compel-03-2018-0123

Cited by 8 publications

(9 citation statements)

References 23 publications

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“…It should be noticed that the F matrix becomes Hermitian when choosing the uniform set of points, for Δ𝑥 = 2𝜋/(2𝑅 + 1) [15]. Its invers matrix has the form:…”

Section: First-order Discrete Operatormentioning

confidence: 99%

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Archives of Electrical Engineering

Sobczyk

2022

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This paper presents novel discrete differential operators for periodic functions of one-and two-variables, which relate the values of the derivatives to the values of the function itself for a set of arbitrarily chosen points over the function's area. It is very characteristic, that the values of the derivatives at each point depend on the function values at all points in that area. Such operators allow one to easily create finite-difference equations for boundaryvalue problems. The operators are addressed especially to nonlinear differential equations.

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“…It should be noticed that the F matrix becomes Hermitian when choosing the uniform set of points, for Δ𝑥 = 2𝜋/(2𝑅 + 1) [15]. Its invers matrix has the form:…”

Section: First-order Discrete Operatormentioning

confidence: 99%

“…The FDOs of periodic and two-periodic time functions are presented in [14] and [15] and have been successfully tested in [16,17] for steady-state analysis of electromagnetic circuits. The same methodology has been used in [18] to develop FDOs to solve boundary-value problems for Ordinary Differential Equations (ODEs).…”

Section: Introductionmentioning

confidence: 99%

Archives of Electrical Engineering

Sobczyk

2022

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“…Choosing (2 R + 1) points uniformly located along the period, matrix D (1) presented in Sobczyk et al (2019) can be determined. This matrix relates vectors containing the values of the function x and its first derivative x’ : …”

Section: Algorithms Applying Ddomentioning

confidence: 99%

“…The idea of an alternative approach for determining a periodic steady-state solution was presented by Sobczyk and Radzik (2017, 2019) using the so-called first-order discrete differential operator (DDO) developed by Sobczyk et al (2019). This operator relates the first-derivative values of a periodic function to values of that function in a set of time instants that are uniformly distributed over the period.…”

Section: Introductionmentioning

confidence: 99%

Direct steady-state solutions for circuit models of nonlinear electromagnetic devices

Sobczyk

Radzik

Tulicki

2021

COMPEL

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Purpose This paper aims to omit the difficulties of directly finding the periodic steady-state solutions for electromagnetic devices described by circuit models. Design/methodology/approach Determine the discrete integral operator of periodic functions and develop an iterative algorithm determining steady-state solutions by a multiplication of matrices only. Findings An alternative method to creating finite-difference relations directly determining steady-state solutions in the time domain. Research limitations/implications Reduction of software and hardware requirements for determining steady-states of electromagnetic. Practical implications A unified approach for directly finding steady-state solutions for ordinary nonlinear differential equations presented in the normal form. Originality/value Eliminate the necessity of solving high-order finite-difference equations for steady-state analysis of electromagnetic devices described by circuit models.

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“…Initially, DDOs were developed to solve, directly in time, steady-state solutions of nonlinear electromagnetic circuits described by ordinary differential equations (Sobczyk et al , 2019) and were then adapted to 1D boundary condition problems. Discrete partial differential operators (DPDOs) for 2D problems were proposed by Sobczyk (2019).…”

Section: Introductionmentioning

confidence: 99%

Solving 2D boundary-value problems using discrete partial differential operators

Jaraczewski

Sobczyk

2021

COMPEL

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Purpose Discrete differential operators of periodic base functions have been examined to solve boundary-value problems. This paper aims to identify the difficulties of using those operators to solve ordinary linear and nonlinear differential equations with Dirichlet and Neumann boundary conditions. Design/methodology/approach This paper presents a promising approach for solving two-dimensional (2D) boundary problems of elliptic differential equations. To create finite differential equations, specially developed discrete partial differential operators are used to replace the partial derivatives in the differential equations. These operators relate the value of the partial derivatives at each point to the value of the function at all points evenly distributed over the area where the solution is being sought. Exemplary 2D elliptic equations are solved for two types of boundary conditions: the Dirichlet and the Neumann. Findings An alternative method has been proposed to create finite-difference equations and an effective method to determine the leakage flux in the transformer window. Research limitations/implications The proposed approach can be classified as an extension of the finite-difference method based on the new formulas approximating the derivatives. This method can be extended to the 3D or time-periodic 2D cases. Practical implications This paper presents a methodology for calculations of the self- and mutual-leakage inductances for windings arbitrarily located in the transformer window, which is needed for special transformers or in any case of the internal asymmetry of windings. Originality/value The presented methodology allows us to obtain the magnetic vector potential distribution in the transformer window only, for example, to omit the magnetic core of the transformer from calculations.

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Discrete differential operators for periodic and two-periodic time functions

Cited by 8 publications

References 23 publications

Archives of Electrical Engineering

Archives of Electrical Engineering

Direct steady-state solutions for circuit models of nonlinear electromagnetic devices

Solving 2D boundary-value problems using discrete partial differential operators

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