Numerical Tests of Novel Finite Difference Operator for Solving 1D Boundary-Value Problems

Jaraczewski, Marcin; Sobczyk, Tadeusz

doi:10.1109/wzee48932.2019.8979825

Cited by 3 publications

(5 citation statements)

References 2 publications

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“…Those straight lines have to have strictly the same but opposite derivatives, so the magnetic potential curve has its maximum exactly in the middle of the winding, i.e., the magnetic field takes the zero value at that point. The method of images, presented in [25,26], can be used to solve the case of only one winding in the air zone. The interval of analysis is extended to −d < Figure 6.…”

Section: Calculations Of Magnetic Field In the Transformer's Air Windowmentioning

confidence: 99%

“…It allows us to find a modified discrete difference operator (DDO) of the second order D (2) , as presented in [25,26], binding the values of the second derivatives and the function itself at the chosen point set {α n } a = D (2) • a (11) where:…”

Section: Of 13mentioning

confidence: 99%

“…This paper presents an approach for determining inductances L σ n,m for symmetrically-located windings, when the Rogowski factor can be applied to correct the length of the magnetic path of stray flux in the air window. This is a modification of 1D analysis, which uses the discrete differential operator developed in [25,26]. Difficulties in formulating the boundary conditions for the case in which a magnetic field is generated by only one winding has been omitted.…”

Section: Introductionmentioning

confidence: 99%

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On Simplified Calculations of Leakage Inductances of Power Transformers

Sobczyk

Jaraczewski

2020

Energies

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This paper deals with the problem of the leakage inductance calculations in power transformers. Commonly, the leakage flux in the air zone is represented by short-circuit inductance, which determines the short-circuit voltage, which is a very important factor for power transformers. That inductance is a good representation of the typical power transformer windings, but it is insufficient for multi-winding ones. This paper presents simple formulae for self- and mutual leakage inductance calculations for an arbitrary pair of windings. It follows from a simple 1D approach to analyzing the stray field using a discrete differential operator, and it was verified by the finite element method (FEM) calculation results.

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Section: Calculations Of Magnetic Field In the Transformer's Air Windowmentioning

confidence: 99%

Section: Of 13mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On Simplified Calculations of Leakage Inductances of Power Transformers

Sobczyk

Jaraczewski

2020

Energies

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“…An alternative approach is presented by Jaraczewski and Sobczyk (2019), Sobczyk and Jaraczewski (2020-E) and Jaraczewski and Sobczyk (2020-C) for solving 1D boundary-value problems and by Jaraczewski and Sobczyk (2020-E) for solving two-dimensional (2D) boundary-value problems. That approach is based on discrete differential operators (DDOs) of periodic functions used as FDOs, so that approach can be classified into a group of FDMs.…”

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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“…These operators re-lated the values of the derivatives at each point to the values of the function, at all points distributed uniformly over the function domain. The same types of FDOs were adapted to solve the one-dimensional (1D) boundary-value problems of ordinary differential equations [17,18] if the solution in the function domain repeated outside, i.e. it is periodic.…”

Section: Introductionmentioning

confidence: 99%

Bulletin of the Polish Academy of Sciences: Technical Sciences

Sobczyk

2020

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This paper presents the concept of using algorithms for reducing the dimensions of finite-difference equations of two-dimensional (2D) problems, for second-order partial differential equations. Solutions are predicted as two-variable functions over the rectangular domain, which are periodic with respect to each variable and which repeat outside the domain. Novel finite-difference operators, of both the first and second orders, are developed for such functions. These operators relate the value of derivatives at each point to the values of the function at all points distributed uniformly over the function domain. A specific feature of the novel operators follows from the arrangement of the function values as well as the values of derivatives, which are rectangular matrices instead of vectors. This significantly reduces the dimensions of the finite-difference operators to the numbers of points in each direction of the 2D area. The finite-difference equations are created exemplary for elliptic equations. An original iterative algorithm is proposed for reducing the process of solving finite-difference equations to the multiplication of matrices.

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Numerical Tests of Novel Finite Difference Operator for Solving 1D Boundary-Value Problems

Cited by 3 publications

References 2 publications

On Simplified Calculations of Leakage Inductances of Power Transformers

On Simplified Calculations of Leakage Inductances of Power Transformers

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

Bulletin of the Polish Academy of Sciences: Technical Sciences

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