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.

Figure 1

Relevant variables for the computation of magnetic and electric fields for two long parallel wires. Only the current on the top cable has been considered. The orientation for $\mathit{e}$ and $\mathit{H}$ follows positive values of the fields.

Figure 2

Generic multi-phase circuit with m wires. The power flows through every power channel ${\alpha }_k$ delimited by wires k and $k+1$.

Figure 3

Magnetic and electric field in a 2D circuit with unbalanced three-phase load and symmetrical supply.

Figure 4

Power flow in channels ${\alpha }_1$ and ${\alpha }_2$, total instantaneous power and active power for the circuit in Fig. 3.

Figure 5

Steinmetz compensator. An inductor and a capacitor are added to the resistor in Fig. 3 to restore the symmetry in the currents.

Figure 6

Generic Four-wire three-phase circuit. Three different power channels are formed (green, red and blue).

Figure 7

4-wire unbalanced 3-phase circuit with resistor between line a and neutral n.

Figure 8

4-wire unbalanced 3-phase circuit with resistor between line a and neutral n and line b in between.

Figure 9

4-wire unbalanced 3-phase circuit with resistor between line a and neutral n with 2 phases in between.