Bridging Circuits and Fields Foundational Questions in Power Theory
... Three books have been written on the subject [15]- [17]. A fourth book will be available later in 2021 [18]. The available preview of [18] contains over one thousand references. ...
... A fourth book will be available later in 2021 [18]. The available preview of [18] contains over one thousand references. Attempts have been made to represent power phenomena in electric circuits directly in the time domain. ...
This paper proposes a systematic method for the identification of the load circuit parameters (say the R, L, and C elements) based only on the information of the instantaneous voltage and current measured at the point of common coupling (pcc). Geometric Algebra (GA) and concepts of differential geometry are used to produce a rigorous mathematical framework. The identification is formulated as a multidimensional geometrical problem that is solved conveniently by means of GA. Once the passive elements of the load have been identified, the active and reactive powers can be computed from first electromagnetic principles (Maxwell Equations). The theory is general and is verified with linear and nonlinear circuits. The paper shows single-phase circuits but the theory can be extended to three-phase circuits. The method is easy to program and has shown to be very robust for all tested cases. Because of its generality, the method presented will find applications beyond electric circuits.
... In the last few years, researchers from many different fields have taken this number system and used it in various fields of applied sciences. For some applications of hybrid numbers we refer the reader to [2,3]. There is no doubt that this number system will be studied by other applied science researchers in the near future. ...
In this paper, we introduce hybrid numbers with Fibonacci and Lucas hybrid number coefficients. We give the Binet formulas, generating functions, exponential generating functions for these numbers. Then we define an associate matrix for these numbers. In addition, using this matrix, we present two different versions of Cassini identitiy of these numbers.
Many properties of special numbers, such as sum formulas, symmetric properties, and their relationships with each other, have been studied in the literature with the help of the Binet formula and generating function. In this paper, higher-order generalized Fibonacci hybrid numbers with q-integer components are defined through the utilization of q-integers and higher-order generalized Fibonacci numbers. Several special cases of these newly established hybrid numbers are presented. The article explores the integration of q-calculus and hybrid numbers, resulting in the derivation of a Binet-like formula, novel identities, a generating function, a recurrence relation, an exponential generating function, and sum properties of hybrid numbers with quantum integer coefficients. Furthermore, new identities for these types of hybrids are obtained using two novel special matrices. To substantiate the findings, numerical examples are provided, generated with the assistance of Maple.
An electron model is developed based on a 4D sphere with a diameter of the Planck length. This model allows us to explain and calculate the intrinsic properties of the electron, such as its mass, charge, spin, etc., from the fundamental constants. Using this Planck sphere in four dimensions, we reach the conclusion that the electron particle has a size that is fixed by the Planck dimensions. The rotation of the Planck sphere generates the electron wave, the size of which depends on its wavelength. Our hypothesis is that the universe is composed of Planck spheres in four spatial dimensions, with two possible states: a rest state and rotational movement.
Geometric algebra has been presented in many different guises since its invention by William Kingdon Clifford shortly before his death in 1879. The guiding principle of this book is that it should be fully integrated into the foundations of mathematics, and in this regard nothing is more fundamental than the concept of number itself. Since I acquired this conviction as a graduate student at Arizona State University more than 50 years ago, much work has been done in applying geometric algebra to problems in mathematics, physics, engineering and computer science, in the gradual recognition of importance of geometric algebra and in the truth of this principle.
New-generation power networks, such as microgrids, are being affected by the proliferation of nonlinear electronic systems, resulting in harmonic disturbances both in voltage and current that affect the symmetry of the system. This paper presents a method based on the application of geometric algebra (GA) to the resolution of power flow in nonsinusoidal single-phase electrical systems for the correct determination of its components to achieve passive compensation of true quadrature current. It is demonstrated that traditional techniques based on the concepts of Budeanu, Fryze or IEEE1459 fail to determine the interaction between voltage and current and therefore, are not suitable for being used as a basis for the compensation of nonactive power components. An example is included that demonstrates the superiority of GA method and is compared to previous work where GA approaches and traditional methods have also been used.
The calculation of power flow in power systems with the presence of harmonics has been properly studied in the scientific literature. However, power flow calculation considering interharmonic components is still an open question. Traditional methods based on the IEEE1459 standard have proven to be valid and accurate only for linear and sinusoidal systems, but have been criticized for non-linear and non-sinusoidal systems because they are not able to explain correctly the current and voltage interactions beyond the active power. This paper proposes the use of a novel mathematical framework called geometric algebra (GA) to study the power flow considering the interaction of current and voltage harmonics and interharmonics. The use of GA enables the precise determination of the direction and magnitude of the total and single active power flow for each component, as well as other power elements related to the non-active power due to cross interaction. Moreover, this paper makes a novel contribution to the definition of interharmonics in geometric algebra space that has not been done before. To test the validity of the method, both linear and non-linear circuits are proposed and solved by applying voltages and currents with harmonic and interharmonic components. The results obtained show that power flow can be analyzed under the prism of the principle of energy conservation (PoCoE) in a way that allows a better understanding of the power spectrum due to the interaction of harmonics and interharmonics of voltage and current.
A new approach for the definition of non-active power in electrical systems is presented in this paper. Through the use of geometric algebra, it is possible to define a new term called geometric non-active power, which is applicable to both sinusoidal and non-sinusoidal systems, and to both linear and nonlinear loads. The classic definitions of distortion and reactive power are compared and discussed in our proposal. We verify how geometric non-active power can appear in both purely resistive and purely reactive systems. The superiority of geometric algebra is revealed through several examples of electrical circuits previously analysed in specialised literature. Furthermore, a new geometrical current decomposition is proposed, for the first time, to provide a greater physical sense to existing geometric power. The results obtained confirm that classic concepts based on apparent power S are based on a lack of physical meaning, which is why geometric algebra theory should be adopted instead.
This paper is discussing how and where imaginary numbers came to be and how their extension to our classic number line helped mathematics to grow even faster. We talk about the beginning of imaginary numbers and the set of rules that come with them. We show how an error that occurred in an equation started the discovery of these. These numbers also help us achieve a better perspective towards the parabolas we see every day. At the end, you can see how these new numbers found the perfect place on the number line and fit in well with different categories we all know.
Building on the work in [1], this paper shows how Conformal Geometric Algebra (CGA) can be used to model an arbitrary two-port network as a rotation in four dimensional space, known as a spinor. This spinor model plays the role of the wave-cascading matrix in conventional network theory. Techniques to translate two-port scattering matrix in and out of spinor form are given. Once the translation is laid out, geometric interpretations are given to the physical properties of reciprocity, loss, and symmetry and some mathematical groups are identified. Methods to decompose a network into various sub-networks, are given. Since rotations in four dimensional minkowski space are Lorentz transformations, our model opens up up the field of network theory to physicists familiar with relativity, and vice versa. The major drawback to the approach is that Geometric Algebra is relatively unknown. However, it is precisely the Geometric Algebra which provides the insight and universality of the model.
Non-linear loads in circuits cause the appearance of harmonic disturbances both in voltage and current. In order to minimize the effects of these disturbances and, therefore, to control the flow of electricity between the source and the load, passive or active filters are often used. Nevertheless, determining the type of filter and the characteristics of their elements is not a trivial task. In fact, the development of algorithms for calculating the parameters of filters is still an open question. This paper analyzes the use of genetic algorithms to maximize the power factor compensation in non-sinusoidal circuits using passive filters, while concepts of geometric algebra theory are used to represent the flow of power in the circuits. According to the results obtained in different case studies, it can be concluded that the genetic algorithm obtains high quality solutions that could be generalized to similar problems of any dimension.
The set of equations known today as Maxwell's equations along with a few constitutive equations lie at the heart of classical electromagnetism. A common misconception held by many is that Maxwell's equations are essential, and that classical electromagnetic theory is settled science and is no longer an active field of investigations. We will review the four Maxwell's equations and related equations, their supporting experimental evidence, the field concept, and the Lorentz and Ritz force laws. We will give a brief outline of two approaches to classical electromagnetism which bypass Maxwell's equations, the propagated potential approach and the direct action approach which bypasses even the field concept. We will also give a few indications of some current research being done in comparing the fitness of the Lorentz and Ritz force laws, and point out some area of research yet to be explored concerning the direct action approach to classical electromagnetism using the Ritz force formula.
A new semiclassical model of the electron with helical solenoid geometry is presented. This new model is an extension of both the Parson Ring Model and the Hestenes Zitterbewegung Model. This model interprets the Zitterbewegung as a real motion that generates the electron’s rotation (spin) and its magnetic moment. In this new model, the g-factor appears as a consequence of the electron’s geometry while the quantum of magnetic flux and the quantum Hall resistance are obtained as model parameters. The Helical Solenoid Electron Model necessarily implies that the electron has a toroidal moment, a feature that is not predicted by Quantum Mechanics. The predicted toroidal moment can be tested experimentally to validate or discard this proposed model.
A charged particle immersed in the random fluctuating zeropoint field is considered as an oscillator with oscillations at random directions and the stochastic average of all such oscillations may be considered as local complex rotation in complex vector space. The average internal oscillations or rotations of the charged particle reveal the particle extended structure with separated centre of charge and centre of mass. The aim of this short review article is to give an account of the extended particle structure in complex vector space and to study the origin of relativistic effects due to a charged particle motion in the presence of zeropoint field.
We present a particular geometric algebra together with such an embedding of two–dimensional Euclidean space that the algebra elements may be in the most efficient way interpreted as arbitrary conic sections. Consequently, in this setting we provide full description of the conic sections analysis, classification and their transformations. Examples that show the functionality and consistency are provided in Maple together with the source code. © 2018, Springer International Publishing AG, part of Springer Nature.
An extension of Maxwell equations to include root-mass terms. Expressing the Maxwell equations using a fully relativistic algebra designed to parallel the underlying nature of space and time as closely as possible leads to the possibilty of describing both light and material "particles" within the same framework.This, the first conference paper in the series, uses an earlier version of the algebra with space considered before time rather than time before space as in later work.
In this study, we define a new non-commutative number system called hybrid numbers. This number system can be accepted as a generalization of the complex , hyperbolic and dual number systems. A hybrid number is a number created with any combination of the complex, hyperbolic and dual numbers satisfying the relation Because these numbers are a composition of dual, complex and hyperbolic numbers, we think that it would be better to call them hybrid numbers instead of the generalized complex numbers. In this paper, we give some algebraic and geometric properties of this number set with some classifications. In addition, we examined the roots of a hybrid number according to its type and character.
In this paper we present the application of a projective geometry tool known as Conformal Geometric Algebra (CGA) to transmission line theory. Explicit relationships between the Smith Chart, Riemann Sphere, and CGA are developed to illustrate the evolution of projective geometry in transmission line theory. By using CGA, fundamental network operations such as adding impedance, admittance, and changing lines impedance can be implemented with rotations, and are shown to form a group. Additionally, the transformations relating different circuit representations such as impedance, admittance, and reflection coefficient are also related by rotations. Thus, the majority of relationships in transmission line theory are linearized. Conventional transmission line formulas are replaced with an operator-based framework. Use of the framework is demonstrated by analyzing the distributed element model and solving some impedance matching problems.
The Editor of the Mathematical Gazette has kindly allowed me to see an advance proof of Prof. Carslaw’s paper, and I take the opportunity of making some comments both on that paper and on Dr. Jeffreys’ Tract .
The publication in 1890 of the two-volume Scientific Papers of James Clerk Maxwell, edited by W. D. Niven, was one of the two objects of a committee formed 'for the purpose of securing a fitting memorial of him' (the other object being the commissioning of a marble bust for the Cavendish Laboratory). Before his death in 1879 at the age of 48, Clerk Maxwell had made major contributions to many areas of theoretical physics and mathematics, not least his discoveries in the fields of electromagnetism and of the kinetic theory of gases, which have been regarded as laying the foundations of all modern physics. He is generally considered the third most important physicist of all time, after Newton and Einstein. These collected shorter works, beginning with a paper written at the age of 15, show the wide range of Clerk Maxwell's interests across mathematics, physics and chemistry.
In Order to fully appreciate the study of vectors, one should have some idea of how they came about. This paper is devoted to the men responsible for the development of vectors. Since not all the men responsible for vector development could be discussed here, I have taken a few of the important men in the field and searched for information about them.
Cambridge Core - General and Classical Physics - Electricity and Magnetism - by Edward M. Purcell
This chapter is devoted to a closer study of the rotor group , and as a byproduct of its primacy (in the sense given to this expression in the last chapter), also of the other spinorial and orthogonal groups.
Computational Methods for Electric Power Systems introduces computational methods that form the basis of many analytical studies in power systems. The book provides the background for a number of widely used algorithms that underlie several commercial software packages, linking concepts to power system applications. By understanding the theory behind many of the algorithms, the reader can make better use of the software and make more informed decisions (e.g., choice of integration method and step size in simulation packages). This Third Edition contains new material on preconditioners for linear iterative methods, Broyden’s method, and Jacobian-free Newton–Krylov methods. It includes additional problems and examples, as well as updated examples on sparse lower-upper (LU) factorization. It also adds coverage of the eigensystem realization algorithm and the double-shift method for computing complex eigenvalues.
The association of the geometrical structures with the electrical network values that has been formerly handled just as an abstract set of equations is clarified and made understandable by introducing a geometrical viewpoint that corresponds to the necessity of reducing the great number of output data generated by the computer to a small number of pertinent, synthetic and intuitive information.
Band 3 des 3teiligen Werkes Elektrische Energieversorgung erscheint nun in der 2. Auflage. Die behandelten Technologien und Verfahren wurden umfassend aktualisiert.
Der dritte Band behandelt die Themen Regelung und Stabilität des Energieversorgungsnetzes, Netzplanung, Betriebsplanung und -führung, Netzleittechnik sowie Leistungselektronische Netzsteuerung (FACTS) und Hochspannungsgleichstromübertragung.
Die drei Bände der Elektrischen Energieversorgung zeichnen sich durch die Synthese von theoretischer Fundierung und unmittelbarem Praxisbezug aus und unterstützen das Verständnis und den Lernerfolg mit Übungsaufgaben, Modellbeispielen und Simulationen. Die Autoren schöpfen inhaltlich aus ihrer langjährigen Erfahrung auf dem Gebiet der Energieversorgung sowie didaktisch aus ihrer Lehrtätigkeit als Professoren.
Der Inhalt
Modellierung und Simulation.- Drehzahl- und Frequenzleistungsregelung.- Synchronisierung und Polradwinkelstabilität.- Spannungsregelung und Spannungsstabilität.- Versorgungsqualität.- Netzplanung und Netzberechnung.- Flexible AC Transmission Systems.- Hochspannungsgleichstromübertragung.- Betriebsplanung.- Leit- und Informationstechnik.
Die Zielgruppen
Das Werk richtet sich an Studierende der Elektrotechnik und Energietechnik sowie an Ingenieure als Nachschlagewerk für die Praxis.
Die Autoren
Valentin Crastan ist emeritierter Professor. Er schöpft inhaltlich aus seiner langjährigen Erfahrung auf dem Gebiet der Energieversorgung. Nach praktischer und leitender Tätigkeit in Unternehmen der Energietechnik (Brown Boveri/ABB, Baden, Schweizerische Elektrizitäts- und Verkehrsgesellschaft, Basel) ist er als Professor für Energiesysteme und Regelungstechnik an der Berner Fachhochschule, Hochschule für Technik und Informatik, Biel, berufen worden, wo er zudem neun Jahre lang auch als Dekan des Fachbereichs Elektrotechnik/Kommunikationstechnik amtete.
Dirk Westermann ist Professor für Elektrische Energieversorgung. Nach vieljähriger Tätigkeit in der Industrie bei ABB in Zürich wechselte er an die Technische Universität Ilmenau. Dort leitet er seit vielen Jahren das Institut für Elektrische Energie- und Steuerungstechnik und forscht und lehrt auf dem Gebiet des Designs, der Analyse und des Betriebs elektrischer Energiesysteme.
An important new resource for the international utility market Over the past two decades, static reactive power compensators have evolved into a mature technology and become an integral part of modern electrical power systems. They are one of the key devices in flexible AC transmission systems (FACTS). Coordination of static compensators with other controllable FACTS devices promises not only tremendously enhanced power system controllability, but also the extension of power transfer capability of existing transmission corridors to near their thermal capacities, thus delaying or even curtailing the need to invest in new transmission facilities. Offering both an in-depth presentation of theoretical concepts and practical applications pertaining to these power compensators, Thyristor-Based FACTS Controllers for Electrical Transmission Systems fills the need for an appropriate text on this emerging technology. Replete with examples and case studies on control design and performance, the book provides an important resource for both students and engineers working in the field. © 2002 by the Institute of Electrical and Electronics Engineers, Inc. All rights reserved.
Conformal groups play a key role in geometry and spin structures. This book provides a self-contained overview of this important area of mathematical physics, beginning with its origins in the works of Cartan and Chevalley and progressing to recent research in spinors and conformal geometry.
Key topics and features:
* Focuses initially on the basics of Clifford algebras
* Studies the spaces of spinors for some even Clifford algebras
* Examines conformal spin geometry, beginning with an elementary study of the conformal group of the Euclidean plane
* Treats covering groups of the conformal group of a regular pseudo-Euclidean space, including a section on the complex conformal group
* Introduces conformal flat geometry and conformal spinoriality groups, followed by a systematic development of riemannian or pseudo-riemannian manifolds having a conformal spin structure
* Discusses links between classical spin structures and conformal spin structures in the context of conformal connections
* Examines pseudo-unitary spin structures and pseudo-unitary conformal spin structures using the Clifford algebra associated with the classical pseudo-unitary space
* Ample exercises with many hints for solutions
* Comprehensive bibliography and index
This text is suitable for a course in mathematical physics at the advanced undergraduate and graduate levels. It will also benefit researchers as a reference text.
Baffled by maths? Then don't give up hope.
John Vince will show you how to understand many of the mathematical ideas used in computer animation, virtual reality, CAD, and other areas of computer graphics.
In thirteen chapters you will rediscover - and hopefully discover for the first time a new way of understanding - the mathematical techniques required to solve problems and design computer programs for computer graphic applications. Each chapter explores a specific mathematical topic and takes you forward into more advanced areas until you are able to understand 3D curves and surface patches, and solve problems using vectors.
After reading the book, you should be able to refer to more challenging books with confidence and develop a greater insight into the design of computer graphics software.
Get to grips with mathematics fast ...
• Numbers
• Algebra
• Trigonometry
• Coordinate geometry
• Transforms
• Vectors
• Curves and surfaces
• Barycentric coordinates
• Analytic geometry
Mathematics for Computer Graphics, Second Edition
The book you will read once, and refer to over and over again!
Dieses Buch vermittelt in verständlicher, leicht lesbarer Weise einen vollständigen Überblick über die hohe Komplexität moderner Stromversorgung.
Beginnend mit der effizienten Umwandlung der Primärenergieressourcen der Erde in kohlebefeuerten Kraftwerken, Kernkraftwerken und der Nutzung erneuerbarer Energien behandelt das Buch das gesamte Spektrum der Erzeugung, Übertragung und Verteilung elektrischer Energie und der hierzu erforderlichen Einrichtungen, einschließlich der Informationssysteme der Kraftwerke, Netz- und Schaltanlagenleittechnik bis hin zum Endverbraucher. Besondere Beachtung finden die aktuelle Deregulierung und Liberalisierung der Strommärkte in Europa.
Dieses Buch ist nicht für versierte Fachleute geschrieben, sondern vorrangig für solche, die es werden wollen. Es wendet sich an Studierende der Elektrotechnik, des Maschinenbaus und des Wirtschaftsingenieurwesens, ferner an alle Ingenieure und Fachleute anderer Disziplinen, die mit Elektroenergiesystemen bzw. der öffentlichen und industriellen Stromversorgung im engeren und weiteren Sinn befasst sind. In zahlreichen Passagen spricht dieses Buch auch politische Entscheidungsträger an sowie all jene, für die der Strom aus der Steckdose kommt.
This is a textbook on classical mechanics at the intermediate level, but its main purpose is to serve as an introduction to a new mathematical language for physics called geometric algebra. Mechanics is most commonly formulated today in terms of the vector algebra developed by the American physicist J. Willard Gibbs, but for some applications of mechanics the algebra of complex numbers is more efficient than vector algebra, while in other applica tions matrix algebra works better. Geometric algebra integrates all these algebraic systems into a coherent mathematical language which not only retains the advantages of each special algebra but possesses powerful new capabilities. This book covers the fairly standard material for a course on the mechanics of particles and rigid bodies. However, it will be seen that geometric algebra brings new insights into the treatment of nearly every topic and produces simplifications that move the subject quickly to advanced levels. That has made it possible in this book to carry the treatment of two major topics in mechanics well beyond the level of other textbooks. A few words are in order about the unique treatment of these two topics, namely, rotational dynamics and celestial mechanics.
This book is a first year graduate text on electromagnetic fields and waves. At the same time it serves as a useful reference for researchers and engineers in the areas of microwaves and optoelectronics.
Following the presentation of the physical and mathematical foundations of electromagnetic theory, the book discusses the field analysis of electromagnetic waves confined in material boundaries, or so-called guided waves, electromagnetic waves in open space, scalar diffraction theory and active devices. The theories and methods presented in the book are foundations of wireless engineering, microwave and millimeter wave techniques, optoelectronics and optical fiber transmission.
Das Lehrbuch behandelt die nichtlineare Netzwerktheorie, ausgehend von einem geometrischen Standpunkt. Die zur Geometrisierung der Theorie verwendeten mathematischen Konzepte werden in einleitenden Kapiteln dargestellt. Der Leser findet eine präzise Beschreibung in einer Sprache, die die Theorie transparent und leicht merkbar macht. Nach Definition von System- und Netzwerkbegriff wird auf die Modellbildung eingegangen und das Modell der Realisierung vergleichend gegenüberstellt. Neben klassischen Beiträgen zur zeitvarianten Theorie findet man auch störungstheoretische Methoden. Das Buch stellt die Theorie nichtlinearer Netzwerke in neuer Konzeption dar, die den Leser an die neuere Entwicklung heranführt. Zahlreiche Beispiele illustrieren das primär für Elektrotechnik-Studenten höherer Semester an technischen Hochschulen bestimmte Werk.
Die Energieform elektrische Energie gewinnt immer mehr an Be deutung. Aufgrund langjähriger Erfahrung rechnet man mit einer Ver dopplung des Bedarfs an elektrischer Energie in zehn Jahren. Roh energie wird heute in Kraftwerken mit Leistungen bis zu einigen 1000 MW in elektrische Energie umgewandelt, die einer Vielzahl von Verbrauchern mit zum Teil sehr kleinen Leistungen möglichst unter brechungslos zugeführt werden muß. So sind aus den ursprünglichen Einzelleitungen und eng begrenzten Ortsnetzen ganze Länder über spannende übertragungsleitungen und mehrfach überlagerte Ver teilernetze entstanden. Die Übertragung elektrischer Energie hat sich zu einem Hauptgebiet der Starkstromtechnik entwickelt. Voraussetzung für eine fruchtbare Tätigkeit auf diesem Gebiet ist die Beherrschung der Grundlagen der Übertragung elektrischer Energie. Da ein zusammenfassendes Lehrbuch hierüber seit Jahren fehlt, ist das vorliegende Buch aufgrund meiner langjährigen Tätigkeit in verschie denen Energie-Versorgungsunternehmen und meiner Vorlesungen über elektrische Anlagen an der Rhein.-Westf. Technischen Hochschule Aachen entstanden. Es handelt sich um ein ausgesprochenes Lehrbuch, dessen Stoff jedem Elektroingenieur und insbesondere den Studenten der Starkstromtechnik bekannt sein müßte. Es enthält den Stoff meiner Vor lesungen, die ich auf dem Gebiet der Energieübertragung für Studierende des sechsten bis achten Semesters halte. Aus den zu diesen Vorlesungen gehörenden übungen sind fast jedem Kapitel einige Aufgaben mit aus geführtem Rechengang zur Anwendung der Theorie auf praktische Bei spiele angefügt.
Around 1850, the idea originated that electromagnetic forces between moving charges in circuits are propagated with the velocity of light. After such a speculation by C. F. Gauss in 1845, B. Riemann, in 1858, suggested the inhomogeneous wave equation in 3-dimensional space for the modeling of this propagation. He found a particular solution replacing Coulomb’s potential, now called the retarded potential. His attempt failed to derive from this solution Weber’s action-at-a-distance potential. Riemann withdrew his pertinent paper before it became printed. After a description of some aspects of research by Gauss, Weber and Riemann, a likely reason for Riemann’s withdrawal is specified differing from recent suggestions by historians of mathematics.