Employment of the Clifford Geometric Algebra for the Analysis of Non-Linear Circuits
Electric power; Harmonics; Non-linear circuits; Reactive power; Clifford algebra; Steady-state.
The first theory created for the analysis of power in electric circuits under nonsinusoidal steady-state conditions
was proposed by Budeanu in 1927. It is based on an extrapolation of the classical method used for sinusoidal
steady-state conditions.
Since it does not explain adequately some aspects of the behavior of non-active
power flow, it has allowed the proposition of several other theories for almost the last
100 years.
Among the most recent propositions is the one by Castro-Nuñez, which uses, in
order to model the non-active power and the multivector aspect of the electric power, a
mathematical tool called Clifford Algebra or Geometric Algebra. However, this proposal
still does not reach its goal since its results diverge in the time-domain.
In this work, a new approach for the transformation between the time and
Clifford domains is presented, which is capable of reproducing the results from the
analysis of the instantaneous power in the time-domain. Also, it is proposed, for the first
time, a rotating operator, different for each frequency present in the circuit. Therefore,
the time-domain is perfectly reproduced in the Clifford-domain, a characteristic that is
not present in the proposal of other authors.
In order to validate the proposed method, four single-phase circuits in nonsinusoidal steady-state conditions
were solved in the time-domain and in the Clifford domain. These circuits contain fundamental frequency and
third harmonic voltage sources together with RLC loads. Also, a circuit containing a fundamental frequency
voltage source, a linear load, and a non-linear load consisting of a single-phase fullwave rectifier, was solved.