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Generalized Vandermonde matrices and determinants in electromagnetic compatibility
Mälardalens högskola, Akademin för utbildning, kultur och kommunikation, Utbildningsvetenskap och Matematik. (MAM)ORCID-id: 0000-0003-3204-617X
2017 (Engelska)Licentiatavhandling, sammanläggning (Övrigt vetenskapligt)
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.

Ort, förlag, år, upplaga, sidor
Västerås: Mälardalen University , 2017.
Serie
Mälardalen University Press Licentiate Theses, ISSN 1651-9256 ; 253
Nationell ämneskategori
Matematik Matematisk analys Beräkningsmatematik Sannolikhetsteori och statistik Algebra och logik
Forskningsämne
matematik/tillämpad matematik
Identifikatorer
URN: urn:nbn:se:mdh:diva-34864ISBN: 978-91-7485-312-4 (tryckt)OAI: oai:DiVA.org:mdh-34864DiVA, id: diva2:1073960
Presentation
2017-03-23, Kappa, Västerås, 13:15 (Engelska)
Opponent
Handledare
Tillgänglig från: 2017-02-14 Skapad: 2017-02-14 Senast uppdaterad: 2017-09-28Bibliografiskt granskad
Delarbeten
1. Extreme points of the Vandermonde determinant on the sphere and some limits involving the generalized Vandermonde determinant
Öppna denna publikation i ny flik eller fönster >>Extreme points of the Vandermonde determinant on the sphere and some limits involving the generalized Vandermonde determinant
2013 (Engelska)Rapport (Övrigt vetenskapligt)
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.

Ort, förlag, år, upplaga, sidor
Mälardalen University, 2013. s. 28
Nyckelord
Vandermonde matrix, Determinants, Extreme points, Unit sphere, Generalized Vandermonde matrix
Nationell ämneskategori
Matematisk analys
Forskningsämne
matematik/tillämpad matematik
Identifikatorer
urn:nbn:se:mdh:diva-23914 (URN)
Forskningsfinansiär
Vetenskapsrådet
Tillgänglig från: 2013-12-27 Skapad: 2013-12-21 Senast uppdaterad: 2019-06-24Bibliografiskt granskad
2. Optimization of the determinant of the Vandermonde matrix and related matrices
Öppna denna publikation i ny flik eller fönster >>Optimization of the determinant of the Vandermonde matrix and related matrices
2014 (Engelska)Ingår i: 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, Vol. 1637, s. 627-636Konferensbidrag, Publicerat paper (Refereegranskat)
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.

Ort, förlag, år, upplaga, sidor
American Institute of Physics (AIP), 2014
Nationell ämneskategori
Matematik Beräkningsmatematik
Forskningsämne
matematik/tillämpad matematik
Identifikatorer
urn:nbn:se:mdh:diva-27255 (URN)10.1063/1.4904633 (DOI)000347812200076 ()2-s2.0-85031861766 (Scopus ID)978-0-7354-1276-7 (ISBN)
Konferens
10TH INTERNATIONAL CONFERENCE ON MATHEMATICAL PROBLEMS IN ENGINEERING, AEROSPACE AND SCIENCES: ICNPAA 2014
Tillgänglig från: 2014-12-31 Skapad: 2014-12-31 Senast uppdaterad: 2018-12-17Bibliografiskt granskad
3. Optimization of the determinant of the Vandermonde matrix on the sphere and related surfaces
Öppna denna publikation i ny flik eller fönster >>Optimization of the determinant of the Vandermonde matrix on the sphere and related surfaces
2015 (Engelska)Ingår i: 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, s. 637-648Konferensbidrag, Publicerat paper (Refereegranskat)
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.

Ort, förlag, år, upplaga, sidor
ISAST: International Society for the Advancement of Science and Technology, 2015
Nyckelord
Vandermonde determinant, optimization, Gr¨obner basis, orthogonal polynomials, ellipsoid, optimal experiment design, homogeneous polynomials
Nationell ämneskategori
Matematisk analys
Forskningsämne
matematik/tillämpad matematik
Identifikatorer
urn:nbn:se:mdh:diva-29938 (URN)978-618-5180-05-8 (ISBN)
Konferens
16th Applied Stochastic Models and Data Analysis International Conference (ASMDA2015) with Demographics 2015 Workshop, 30 June – 4 July 2015, University of Piraeus, Greece
Tillgänglig från: 2015-12-15 Skapad: 2015-12-15 Senast uppdaterad: 2018-03-05Bibliografiskt granskad
4. On Some Properties of the Multi-Peaked Analytically Extended Function for Approximation of Lightning Discharge Currents
Öppna denna publikation i ny flik eller fönster >>On Some Properties of the Multi-Peaked Analytically Extended Function for Approximation of Lightning Discharge Currents
2016 (Engelska)Ingår i: 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, s. 151-172Kapitel i bok, del av antologi (Refereegranskat)
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.

Ort, förlag, år, upplaga, sidor
Heidelberg: Springer, 2016
Serie
Springer Proceedings in Mathematics & Statistics, ISSN 2194-1009 ; 178
Nationell ämneskategori
Beräkningsmatematik Annan elektroteknik och elektronik
Forskningsämne
matematik/tillämpad matematik
Identifikatorer
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)
Tillgänglig från: 2016-09-08 Skapad: 2016-09-08 Senast uppdaterad: 2019-06-24Bibliografiskt granskad
5. Estimation of Parameters for the Multi-peaked AEF Current Functions
Öppna denna publikation i ny flik eller fönster >>Estimation of Parameters for the Multi-peaked AEF Current Functions
2017 (Engelska)Ingår i: Methodology and Computing in Applied Probability, ISSN 1387-5841, E-ISSN 1573-7713, Vol. 19, nr 4, s. 1107-1121Artikel i tidskrift (Refereegranskat) Published
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.

Nyckelord
Analytically extended function, Electromagnetic compatibility, Lightning discharge, Marquardt least-squares method
Nationell ämneskategori
Beräkningsmatematik Annan elektroteknik och elektronik
Forskningsämne
matematik/tillämpad matematik
Identifikatorer
urn:nbn:se:mdh:diva-33096 (URN)10.1007/s11009-016-9501-z (DOI)000413792200007 ()2-s2.0-84973100963 (Scopus ID)
Konferens
15th Applied Stochastic Models and Data Analysis International Conference (ASMDA), Univ Piraeus, Piraeus, GREECE, JUN 30-JUL 04, 2015
Tillgänglig från: 2016-09-08 Skapad: 2016-09-08 Senast uppdaterad: 2019-06-24Bibliografiskt granskad
6. Electrostatic Discharge Currents Representation using the Multi-Peaked Analytically Extended Function by Interpolation on a D-Optimal Design
Öppna denna publikation i ny flik eller fönster >>Electrostatic Discharge Currents Representation using the Multi-Peaked Analytically Extended Function by Interpolation on a D-Optimal Design
(Engelska)Manuskript (preprint) (Övrigt vetenskapligt)
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.

Nationell ämneskategori
Beräkningsmatematik
Forskningsämne
matematik/tillämpad matematik
Identifikatorer
urn:nbn:se:mdh:diva-34861 (URN)
Tillgänglig från: 2017-02-13 Skapad: 2017-02-13 Senast uppdaterad: 2017-02-14Bibliografiskt granskad

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