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Generalized Vandermonde matrices and determinants in electromagnetic compatibility
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. (Engineering mathematics)ORCID iD: 0000-0003-3204-617X
2017 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

Matrices whose rows (or columns) consists of monomials of sequential powers are called Vandermonde matrices and can be used to describe several useful concepts and have properties that can be helpful for solving many kinds of problems. In this thesis we will discuss this matrix and some of its properties as well as a generalization of it and how it can be applied to curve fitting discharge current for the purpose of ensuring electromagnetic compatibility.

In the first chapter the basic theory for later chapters is introduced. This includes the Vandermonde matrix and some of its properties, history, applications and generalizations, interpolation and regression problems, optimal experiment design and modelling of electrostatic discharge currents with the purpose to ensure electromagnetic compatibility.

The second chapter focuses on finding the extreme points for the determinant for the Vandermonde matrix on various surfaces including spheres, ellipsoids, cylinders and tori. The extreme points are analysed in three dimensions or more.

The third chapter discusses fitting a particular model called the p-peaked Analytically Extended Function (AEF) to data taken either from a standard for electromagnetic compatibility or experimental measurements. More specifically the AEF will be fitted to discharge currents from the IEC 62305-1 and IEC 61000-4-2 standards for lightning protection and electrostatic discharge immunity as well as some experimentally measured data of similar phenomena.

Place, publisher, year, edition, pages
Västerås: Mälardalen University , 2017.
Series
Mälardalen University Press Licentiate Theses, ISSN 1651-9256 ; 253
National Category
Mathematics Mathematical Analysis Computational Mathematics Probability Theory and Statistics Algebra and Logic
Research subject
Mathematics/Applied Mathematics
Identifiers
URN: urn:nbn:se:mdh:diva-34864ISBN: 978-91-7485-312-4 (print)OAI: oai:DiVA.org:mdh-34864DiVA: diva2:1073960
Presentation
2017-03-23, Kappa, Västerås, 13:15 (English)
Opponent
Supervisors
Available from: 2017-02-14 Created: 2017-02-14 Last updated: 2017-03-14Bibliographically approved
List of papers
1. Extreme points of the Vandermonde determinant on the sphere and some limits involving the generalized Vandermonde determinant
Open this publication in new window or tab >>Extreme points of the Vandermonde determinant on the sphere and some limits involving the generalized Vandermonde determinant
2013 (English)Report (Other academic)
Abstract [en]

The values of the determinant of Vandermonde matrices with real elements are analyzed both visually and analytically over the unit sphere in various dimensions. For three dimensions some generalized Vandermonde matrices are analyzed visually. The extreme points of the ordinary Vandermonde determinant on finite-dimensional unit spheres are given as the roots of rescaled Hermite polynomials and a recursion relation is provided for the polynomial coefficients. Analytical expressions for these roots are also given for dimension three to seven. A transformation of the optimization problem is provided and some relations between the ordinary and generalized Vandermonde matrices involving limits are discussed.

Place, publisher, year, edition, pages
Mälardalen University, 2013. 28 p.
Keyword
Vandermonde matrix, Determinants, Extreme points, Unit sphere, Generalized Vandermonde matrix
National Category
Mathematical Analysis
Research subject
Mathematics/Applied Mathematics
Identifiers
urn:nbn:se:mdh:diva-23914 (URN)
Funder
Swedish Research Council
Available from: 2013-12-27 Created: 2013-12-21 Last updated: 2017-02-14Bibliographically approved
2. Optimization of the determinant of the Vandermonde matrix and related matrices
Open this publication in new window or tab >>Optimization of the determinant of the Vandermonde matrix and related matrices
2014 (English)In: 10TH INTERNATIONAL CONFERENCE ON MATHEMATICAL PROBLEMS IN ENGINEERING, AEROSPACE AND SCIENCES: ICNPAA 2014 Conference date: 15–18 July 2014 Location: Narvik, Norway ISBN: 978-0-7354-1276-7 Editor: Seenith Sivasundaram Volume number: 1637 Published: 10 december 2014 / [ed] Seenith Sivasundaram, American Institute of Physics (AIP), 2014, 627-636 p.Conference paper, (Refereed)
Abstract [en]

Various techniques for interpolation of data, moment matching in stochastic applications and various methods in numerical analysis can be described using Vandermonde matrices. For this reason the properties of the determinant of the Vandermonde matrix and related matrices are interesting. Here the extreme points of the Vandermonde determinant, and related determinants, on some simple surfaces such as the unit sphere are analyzed, both numerically and analytically. Some results are also visualized in various dimensions. The extreme points of the Vandermonde determinant are also related to the roots of certain orthogonal polynomials such as the Hermite polynomials.

Place, publisher, year, edition, pages
American Institute of Physics (AIP), 2014
National Category
Mathematics Computational Mathematics
Research subject
Mathematics/Applied Mathematics
Identifiers
urn:nbn:se:mdh:diva-27255 (URN)10.1063/1.4904633 (DOI)000347812200076 ()978-0-7354-1276-7 (ISBN)
Conference
10TH INTERNATIONAL CONFERENCE ON MATHEMATICAL PROBLEMS IN ENGINEERING, AEROSPACE AND SCIENCES: ICNPAA 2014
Available from: 2014-12-31 Created: 2014-12-31 Last updated: 2017-02-14Bibliographically approved
3. Optimization of the determinant of the Vandermonde matrix on the sphere and related surfaces
Open this publication in new window or tab >>Optimization of the determinant of the Vandermonde matrix on the sphere and related surfaces
2015 (English)In: ASMDA 2015 Proceedings: 16th Applied Stochastic Models and Data Analysis International Conference with 4th Demographics 2015 Workshop / [ed] Christos H Skiadas, ISAST: International Society for the Advancement of Science and Technology , 2015, 637-648 p.Conference paper, (Refereed)
Abstract [en]

The value of the Vandermonde determinant is optimized over various surfaces, including the sphere, ellipsoid and torus. Lagrange multipliers are used to find a system of polynomial equations which give the local extreme points in its solutions. Using Gröbner basis and other techniques the extreme points are given either explicitly or as roots of polynomials in one variable. The behavior of the Vandermonde determinant is also presented visually in some interesting cases.

Place, publisher, year, edition, pages
ISAST: International Society for the Advancement of Science and Technology, 2015
Keyword
Vandermonde determinant, optimization, Gr¨obner basis, orthogonal polynomials, ellipsoid, optimal experiment design, homogeneous polynomials
National Category
Mathematical Analysis
Research subject
Mathematics/Applied Mathematics
Identifiers
urn:nbn:se:mdh:diva-29938 (URN)www.asmda.es/images/1_L-MOH_ASMDA2015_Proceedings.pdf (DOI)978-618-5180-05-8 (ISBN)
Conference
16th Applied Stochastic Models and Data Analysis International Conference (ASMDA2015) with Demographics 2015 Workshop, 30 June – 4 July 2015, University of Piraeus, Greece
Available from: 2015-12-15 Created: 2015-12-15 Last updated: 2017-02-14Bibliographically approved
4. On Some Properties of the Multi-Peaked Analytically Extended Function for Approximation of Lightning Discharge Currents
Open this publication in new window or tab >>On Some Properties of the Multi-Peaked Analytically Extended Function for Approximation of Lightning Discharge Currents
2016 (English)In: Engineering Mathematics I: Electromagnetics, Fluid Mechanics, Material Physics and Financial Engineering, Series: Springer Proceedings in Mathematics & Statistics / [ed] Sergei Silvestrov, Milica Rančić, Heidelberg: Springer, 2016, 151-172 p.Chapter in book (Refereed)
Abstract [en]

According to experimental results for lightning discharge currents, they are classified in the IEC 62305 Standard into waveshapes representing the first positive, first and subsequent negative strokes, and long-strokes. These waveshapes, especially shot-term pulses, are approximated with a few mathematical functions in literature, in order to be used in lightning discharge models for calculations of electromagnetic field and lightning induced effects. An analytically extended function (AEF) is presented in this paper and used for lightning currents modeling. The basic properties of this function with a finite number of peaks are examined. A general framework for estimating the parameters of the AEF using the Marquardt least-squares method (MLSM) for a waveform with an arbitrary (finite) number of peaks as well as for the given charge transfer and specific energy is described. This framework is used to find parameters for some common single-peak wave-forms and some advantages and disadvantages of the approach are also discussed.

Place, publisher, year, edition, pages
Heidelberg: Springer, 2016
Series
Springer Proceedings in Mathematics & Statistics, ISSN 2194-1009 ; 178
National Category
Computational Mathematics Other Electrical Engineering, Electronic Engineering, Information Engineering
Research subject
Mathematics/Applied Mathematics
Identifiers
urn:nbn:se:mdh:diva-33097 (URN)10.1007/978-3-319-42082-0_10 (DOI)2-s2.0-85015267484 (Scopus ID)978-3-319-42082-0 (ISBN)978-3-319-42081-3 (ISBN)
Available from: 2016-09-08 Created: 2016-09-08 Last updated: 2017-03-30Bibliographically approved
5. Estimation of Parameters for the Multi-peaked AEF Current Functions
Open this publication in new window or tab >>Estimation of Parameters for the Multi-peaked AEF Current Functions
(English)In: Methodology and Computing in Applied Probability, ISSN 1387-5841, E-ISSN 1573-7713Article in journal (Refereed) Epub ahead of print
Abstract [en]

An examination of how the analytically extended function (AEF) can be used to approximate multi-peaked lightning current waveforms, is presented in the paper. A general framework for estimating the parameters of the AEF using the Marquardt least-squares method (MLSM) for a waveform with an arbitrary (finite) number of peaks is presented. This framework is used to find parameters for some common waveforms with a single peak, such as Standard IEC 62305 lightning currents. Illustration of fitting a p-peak AEF to recorded lightning current data is also presented.

Keyword
Analytically extended function, Electromagnetic compatibility, Lightning discharge, Marquardt least-squares method
National Category
Computational Mathematics Other Electrical Engineering, Electronic Engineering, Information Engineering
Research subject
Mathematics/Applied Mathematics
Identifiers
urn:nbn:se:mdh:diva-33096 (URN)10.1007/s11009-016-9501-z (DOI)2-s2.0-84973100963 (Scopus ID)
Available from: 2016-09-08 Created: 2016-09-08 Last updated: 2017-02-14Bibliographically approved
6. Electrostatic Discharge Currents Representation using the Multi-Peaked Analytically Extended Function by Interpolation on a D-Optimal Design
Open this publication in new window or tab >>Electrostatic Discharge Currents Representation using the Multi-Peaked Analytically Extended Function by Interpolation on a D-Optimal Design
(English)Manuscript (preprint) (Other academic)
Abstract [en]

Multi-peaked analytically extended function (AEF), previously applied by the authors to modelling of lightning discharge currents, is used in this paper for representation of the electrostatic discharge (ESD) currents. The fitting to data is achieved by interpolation of certain data points. In order to minimize unstable behaviour, the exponents of the AEF are chosen from a certain arithmetic sequence and the interpolated points are chosen according to a D-optimal design. ESD currents' modelling is illustrated through two examples: one corresponding to an approximation of the IEC Standard 61000-4-2 waveshape, and the other to representation of some measured ESD current.

National Category
Computational Mathematics
Research subject
Mathematics/Applied Mathematics
Identifiers
urn:nbn:se:mdh:diva-34861 (URN)
Available from: 2017-02-13 Created: 2017-02-13 Last updated: 2017-02-14Bibliographically approved

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