Geometric Algebra for Electrical and Electronic Engineers

Chappell, James M.; Drake, Samuel Picton; Seidel, Cameron L.; Gunn, Lachlan J.; Iqbal, Azhar; Allison, Andrew; Abbott, Derek

doi:10.1109/jproc.2014.2339299

Cited by 58 publications

(58 citation statements)

References 47 publications

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“…We shall give a presentation of the Maxwell equations in the geometric algebra G 3 = Cℓ 3,0 . The exposition follows closely the presentation of Chappell et al [4]. In this presentation we assume c = µ 0 = ǫ 0 = 1.…”

Section: Geometric Calculus Functionality In Cliffordmentioning

confidence: 80%

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Sparse Representations of Clifford and Tensor Algebras in Maxima

Prodanov¹,

Toth

2016

Adv. Appl. Clifford Algebras

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Clifford algebras have broad applications in science and engineering. The use of Clifford algebras can be further promoted in these fields by availability of computational tools that automate tedious routine calculations. We offer an extensive demonstration of the applications of Clifford algebras in electromagnetism using the geometric algebra G 3 ≡ Cℓ3,0 as a computational model in the Maxima computer algebra system. We compare the geometric algebra-based approach with conventional symbolic tensor calculations supported by Maxima, based on the itensor package. The Clifford algebra functionality of Maxima is distributed as two new packages called clifford -for basic simplification of Clifford products, outer products, scalar products and inverses; and cliffordan -for applications of geometric calculus. 1 2 DIMITER PRODANOV 1 AND VIKTOR T. TOTH 2for Computer Science during the years 1969-1972. Maxima is written entirely in Lisp and is distributed under open source license 1 . Maxima has its own programming language, which is particularly well-suited for handling symbolical mathematical expressions or mixed numerical-symbolical expressions. In addition, the system supports 64-bit precision floating point and arbitrary precision arithmetics. Maxima programs can be automatically translated and compiled to Lisp within the program environment itself. Third-party Lisp programs can be also loaded and accessed from within the system. The system also offers the possibility of running batch unit tests.Maxima supports several primitive data types [11]: numbers (rational, float and arbitrary precision); strings and symbols. In addition there are compound data types, such as lists, arrays, matrices and structs. There are also special symbolic constants, such as the Boolean constants true and false or the complex imaginary unit %i.Several types of operators can be defined in Maxima. An operator is a defined symbol that may be unary prefix, unary postfix, binary infix, n-ary matchfix, or nofix types. For example, the inner and outer products defined in clifford are of the binary infix type. The scripting language allows for defining new operators with specified precedence or redefining the precedence of existing operators. Maxima distinguishes between two forms of applications of operators -forms which are nouns and forms which are verbs. The difference is that the verb form of an operator evaluates its arguments and produces an output result, while the noun form appears as an inert symbol in an expression, without being executed. A verb form can be mutated into a noun form and vice-versa. This allows for context-dependent evaluation, which is especially suited for symbolic processing.Listing 1: Dot products and exponent simplification rules in clifford.if get('clifford,'version)=false then ( tellsimp(aa[kk].

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Section: Geometric Calculus Functionality In Cliffordmentioning

confidence: 80%

“…where i is the pseudoscalar of the algebra. This field object has the required transformation properties under reflection [4]. In the conventional Heaviside-Gibbs notation, the fields are represented by the vectors E and B, which change sign under reflection of the coordinate system.…”

Section: Geometric Calculus Functionality In Cliffordmentioning

confidence: 99%

Sparse Representations of Clifford and Tensor Algebras in Maxima

Prodanov¹,

Toth

2016

Adv. Appl. Clifford Algebras

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“…The multivector naturally describes algebraically the four geometric elements of three-dimensional space, that of points, lines, areas and volumes, as shown in Equation (6). These four quantities also describe the physical quantities referred to as scalar, vectors, pseudovectors and pseudoscalars and found to be a natural language to describe physical theories in three dimensions [24,25]. We thus consider that the four geometrical quantities of physical space are the fundamental basis upon which we abstract such local concepts as time and space [26,27].…”

Section: Resultsmentioning

confidence: 99%

Time As a Geometric Property of Space

et al. 2016

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The proper description of time remains a key unsolved problem in science. Newton conceived of time as absolute and universal which "flows equably without relation to anything external." In the nineteenth century, the four-dimensional algebraic structure of the quaternions developed by Hamilton, inspired him to suggest that he could provide a unified representation of space and time. With the publishing of Einstein's theory of special relativity these ideas then lead to the generally accepted Minkowski spacetime formulation of 1908. Minkowski, though, rejected the formalism of quaternions suggested by Hamilton and adopted an approach using four-vectors. The Minkowski framework is indeed found to provide a versatile formalism for describing the relationship between space and time in accordance with Einstein's relativistic principles, but nevertheless fails to provide more fundamental insights into the nature of time itself. In order to answer this question we begin by exploring the geometric properties of three-dimensional space that we model using Clifford geometric algebra, which is found to contain sufficient complexity to provide a natural description of spacetime. This description using Clifford algebra is found to provide a natural alternative to the Minkowski formulation as well as providing new insights into the nature of time. Our main result is that time is the scalar component of a Clifford space and can be viewed as an intrinsic geometric property of three-dimensional space without the need for the specific addition of a fourth dimension.

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“…It is based on the notion of an invertible product of vectors that captures the geometric relationship between two vectors, i.e., their relative magnitudes and the angle between them [23]. Some investigations have defined the properties of geometric algebra [24,25] applied to physics and engineering. Traditional concepts such as vector, spinor, complex numbers, or quaternions are naturally explained as members of subspaces in geometric algebra.…”

Section: Basic Definitions Of Geometric Algebramentioning

confidence: 99%

Quadrature Current Compensation in Non-Sinusoidal Circuits Using Geometric Algebra and Evolutionary Algorithms

2019

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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.

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Geometric Algebra for Electrical and Electronic Engineers

Abstract: This tutorial paper provides a short introduction to geometric algebra, starting with its history and then presenting its benefits and exploring its applications.

Cited by 58 publications

References 47 publications

Sparse Representations of Clifford and Tensor Algebras in Maxima

Sparse Representations of Clifford and Tensor Algebras in Maxima

Time As a Geometric Property of Space

Quadrature Current Compensation in Non-Sinusoidal Circuits Using Geometric Algebra and Evolutionary Algorithms

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