New mathematical model based on geometric algebra for physical power flow in theoretical two-dimensional multi-phase power circuits

Abstract

This study proposes an explanation for the physical power flow in planar circuits by analogy to theoretical two-dimensional circuits using a new mathematical model based on Geometric Algebra (GA) and 2D Maxwell’s equations. In contrast with traditional 3D physics in the observable real world, the magnetic field can be defined as a bivector instead of an axial vector allowing to obtain the Poynting Vector directly in a 2D flat world, where physical variables of planar circuits can be obtained. This approach is presented here for the first time to the best of the author’s knowledge. Previous investigations have focused on simplifications and symmetries of real 3D circuits studied mainly in the phasor and frequency domain. In this work, the electromagnetic power flow phenomenon is analyzed on a completely 2D time-domain basis and derived directly from the undisputed Maxwell equations, formulated in two dimensions. Several cases of special interest in AC multi-phase circuits are presented using the proposed technique, bringing a new simplified approach to the measurement of power flow exchange between the source and the load. It suggests a new way to understand energy propagation from a purely physical point of view.

Geometric Foundations of Power Theory for Multiphase AC Systems in the Frequency Domain

“…Geometric algebra (GA) has been revealed as a universal and unifying framework that includes a large number of traditional tools such as complex numbers, quaternions, matrices, differential forms, tensors, etc [14]. Recently, its versatility and capability have been demonstrated in various fields of science and engineering such as quantum physics, mechanics, robotics, computer vision, and also electrical engineering [15], [16], [17]. The use of GA makes it possible to solve electrical circuits in both frequency and time domain [18], [19] with a very similar approach.…”

“…To obtain the sequence shifted vectors, it is necessary to consider the expressions given in Eqs. ( 13) through (16).…”

Geometric Foundations of Power Theory for Multiphase AC Systems in the Frequency Domain

“…Geometric algebra (GA) has been revealed as a universal and unifying framework that includes a large number of traditional tools such as complex numbers, quaternions, matrices, differential forms, tensors, etc [14]. Recently, its versatility and capability have been demonstrated in various fields of science and engineering such as quantum physics, mechanics, robotics, computer vision, and also electrical engineering [15], [16], [17]. The use of GA makes it possible to solve electrical circuits in both frequency and time domain [18], [19] with a very similar approach.…”

“…To obtain the sequence shifted vectors, it is necessary to consider the expressions given in Eqs. ( 13) through (16).…”

Reduced Reference Frame Transformation Assessment in Unbalanced Three-Phase Four-Wire Systems

“…Of special interest are those based on Singular Value Decomposition proposed by V. Choqueuse in [9] or those based on the Frenet-Serret frame proposed by P. Granjon [10] and more recently F. Milano [11]. Others authors use tensor algebra [12], while others apply the geometric or Clifford algebra to analyze three-phase four wire systems [13], [14] or multiphase power systems in time-domain [15], [16].…”