Advantages of Geometric Algebra Over Complex Numbers in the Analysis of Networks With Nonsinusoidal Sources and Linear Loads
Abstract: Since 1892, the electrical engineering scientific community has been seeking a power theory for interpreting the power flow within electric networks under non-sinusoidal conditions. Although many power theories have been proposed regarding non-sinusoidal operation, an adequate solution is yet to be found. Using the framework based on complex algebra in non-sinusoidal circuit analysis (frequency domain), the verification of the energy conservation law is only possible in sinusoidal situations. In this case, reactive energy turns out to be proportional to the energy difference between the average electric and magnetic energies stored in the loads and its cancellation is mathematically trivial. However, in industrial architecture, apparent power definition of electric loads (non-sinusoidal conditions) is inconsistent with the energy conservation law. Up until now, in the classical complex algebra approach, this goal is only valid in the case of purely resistive loads. Thus, in this paper, a new circuit analysis approach using geometric algebra is used to develop the most general proof of energy conservation in industrial building loads. In terms of geometric objects, this powerful tool calculates the voltage, current, and apparent power in electrical systems in non-sinusoidal, linear/nonlinear situations. In contrast to the traditional method developed by Steinmetz, the suggested powerful tool extends the concept of phasor to multivector-phasors and is performed in a new Generalized Complex Geometric Algebra structure (CGn), where Gn is the Clifford algebra in n-dimensional real space and C is the complex vector space. To conclude, a numerical example illustrates the clear advantages of the approach suggested in this paper.