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M, The Power Definition in Geometric Algebra that Unveils the Shortcomings of the Nonsinusoidal Apparent Power S

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Abstract

The circuit analysis approach based on geometric algebra and \(\varvec{M}\), the power definition based on the geometric product between the voltage and the current multivectors, are used here to demonstrate the shortcomings of the traditional definition of the non-sinusoidal apparent power S. The shortcomings of S are illustrated in three ways. Firstly, by showing an example of how the norm of \(\varvec{M}\) contains S. Secondly, through six experiments that involve compliance with: Kirchhoff’s circuit laws, Tellegen’s theorem, the principle of conservation of energy, the equivalency of two terminal networks and the concept of reactive power compensation. Lastly, by showing how the use of S leads the current’s physical component power theory astray. The experiments show contradictions between the aforementioned circuit theory fundamentals and the results attained with S but a compelling harmony with the results attained with \(\varvec{M}\). The evidence reveals two unprecedented discoveries: (1) that mathematical models aimed at explaining energy flow in non-sinusoidal circuits shouldn’t be based on the decomposition of S—as traditionally done— and, (2) the inappropriateness of extrapolating definitions from sinusoidal conditions to non-sinusoidal settings.

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Acknowledgements

The authors express their gratitude to El Roi. Dr. Jaime Castro Núñez, Valery Castro-Londoño and Daniel Castro-Londoño are also thanked for their unconditional support.

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Authors and Affiliations

  1. Alberta Electric System Operator (AESO), 25300, 330-5th Ave, Calgary, T2P 0L4, Canada

    Milton Castro-Núñez

  2. Department of Electrical and Electronic Engineering, Universidad Nacional de Colombia, Cra. 30 No. 45-03, Bogotá, Colombia

    Milton Castro-Núñez

  3. Department of Electrical and Electronic Engineering, Universidad Nacional de Colombia, Cra. 27 No. 64-60, Manizales, Colombia

    Deysy Londoño-Monsalve

  4. Department of Mathematics, Universidad de Córdoba, Cra. 6 No. 77-305, Montería, Colombia

    Róbinson Castro-Puche

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  1. Milton Castro-Núñez
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This article is part of the ENGAGE 2019 Topical Collection on Geometric Algebra for Computing, Graphics and Engineering edited by Linwang Yuan (EiC), Werner Benger, Dietmar Hildenbrand, and Eckhard Hitzer.

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Castro-Núñez, M., Londoño-Monsalve, D. & Castro-Puche, R. M, The Power Definition in Geometric Algebra that Unveils the Shortcomings of the Nonsinusoidal Apparent Power S. Adv. Appl. Clifford Algebras 32, 18 (2022). https://doi.org/10.1007/s00006-022-01200-8

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