Geometric algebra framework applied to circuits with non-sinusoidal voltages and currents

: We apply a well known technique of theoretical physics, known as Geometric Algebra or Cli ﬀ ord algebra, to linear electrical circuits with non-sinusoidal voltages and currents. We rederive from the ﬁrst principles the Geometric Algebra approach to the apparent power decomposition. The important new point consists in a choice of a natural convenient basis in the Cli ﬀ ord vector space which simpliﬁes considerably the presentation. Thus we are able to derive a number of general results which are missing in the former papers. In particular, a natural correspondence with the Current Physical Components approach is shown.


Introduction
Complex numbers are with a long and deeply rooted tradition in description of alternating current (AC) circuits [1]. A sinusoidal function f , representing current or voltage, is represented by the so called phasor (complex function): , where j is the imaginary unit ( j 2 = −1). The factor √ 2 is needed if we prefer to describe currents and voltages in terms of root mean square (RMS) values rather than in terms of amplitudes. Taking the real part of the phasor we recover the original sinusoidal function. The main reason for using this abstract (non-physical) space is a simple form of the Ohm Law for AC currents. For sinusoidal currents represented in the phasor space Ohm's Law has exactly the same form as for the direct current (DC) circuit (U U U = ZI I I) provided that instead of the real resistance one uses the complex impedance Z which combines resistance, capacitance and inductance (R, C and L) in a well known way: where by ϕ we denoted the phase shift between the current and voltage. The complex impedance was first introduced by Kenelly [2]. In our article we prefer to use admittance (the reciprocal of impedance) where the real part G is known as the conductance and the imaginary part B is called susceptance. Thus and the current is computed as the real part of the product: Here and in the rest of this paper we confine ourselves to single phase linear circuits (the nonlinear systems and three-phase systems will be considered elsewhere).
Complex numbers yield a convenient interpretation of the active, reactive and apparent power in single phase linear circuits with a sinusoidal load: where U and I are RMS values of the voltage and current, respectively. For our further purposes another representation of the apparent power will be useful: Recently, another abstract mathematical structure has been applied to the classical problems of the electrical engineering. This is the Geometric Algebra (also known as the Clifford algebra), a quite popular and convenient tool in mathematical and theoretical physics (including electrodynamics) [3,4], which recent applications also in electrical engineering [5].
Since a dozen years, several authors try to make description of distoreted currents in terms of the Geometric Algebra. It seems that already the first attempt [6] was made in a good direction but later developments were of diffferent value and sometimes contain mistakes or too cumbersome developments [7][8][9][10]. As an example of a recent critique of the Geometric Algebra applications, see [11]. The recent paper by Montoya et al [12] seems to contain the most mature formulation of the problem.
Our paper has two main goals. First, we will derive from the first principles the Geometric Algebra formulation (in a form very close to the theory of Montoya [12]). The important new point consists in a choice of a natural convenient basis in the Clifford vector space. Second, we present a number of general results which are missing in the former papers (focusing on discussion of more or less representative examples). In particular, a natural correspondence with Czarnecki's current physical components approach [13,14] is shown.

Geometric Algebra
Throughout this paper we consider non-sinusoidal currents in the form of finite sum of Fourier harmonics: Taking into account the possibility of currents' phase-shifts we have to consider 2N dimensional vector space spanned bỹ The voltage is a linear combination ofc c c k and the current, in general, can be a combination of all basis vectors. The crucial point for the approach presented in this paper consists in treating products of functions: we use only the Clifford product (the Clifford product od two functions is completely different than the usual product of functions), for more details see the Appendix.
In this article, we intentionally mark periodic functions in bold, which means they have to be considered Clifford vectors wherever any product of these functions is involved. Definition 1. The Clifford admittance y y y k is defined as where j j j k is a Clifford product ofc c c k ands s s k : j j j k =c c c ks s s k .
The properties of j j j k are very nice (for more details see the Appendix): j j j 2 k = −1 , j j j µ j j j ν = j j j ν j j j µ .
These entities can be interpreted as commuting imaginary units related to every harmonics. First of all, we are going to show that for sinusoidal currents our Clifford algebra approach yields the same results as the standard approach using complex phasors.
Theorem 2. Given a harmonic voltage u u u k = U kc c c k and a load with conductance G k and susceptance B k , the current can be computed as the Clifford product of the voltage and the Clifford admittance: The proof is straightforward: The corresponding calculation using the usual complex phasors (compare (4) is a little bit more complicated, but, obviously, leads to the same result: The important corollary is that a kind of a phasor-like structure is automatically built into the Clifford algebra structure and there is no need for a complexification of the Clifford algebra (as done, for instance, in [10]). In fact we already have N commuting imaginary units j j j 1 , . . . , j j j N instead of the single complex imaginary unit j j j.
The distorted (i.e., nonsinusoidal) case is treated in an analogous way due to the linearity of the problem: Following Menti et al. [6] and Montoya et al. [12], we define geometric (Clifford) power as the Clifford product of u u u and i i i.
U j U k (G kc c c k ∧c c c j + B kc c c j ∧s s s k ) (16) It is worth to be noted that the above formula has a simplified form (as compared to earlier papers) due to the convenient choice of the basisc c c k ,s s s k (k = 1, . . . , N). It is a sum of orthogonal components. One can easily verify that the number of the orthogonal components C N is given by The  Indeed, the transformation between the bases has the following form: c c c k = c c c k cos α k − s s s k sin α k = (cos α k + sin α k c c c k s s s k )c c c k = e α k j j j k c c c k , s s s k = c c c k sin α k + s s s k cos α k = (cos α k + sin α k c c c k s s s k )s s s k = e α k j j j k s s s k .
Then, we easily compute: c c c ks s s k = c c c k s s s k cos 2 α k − s s s k c c c k sin 2 α k + (c c c k c c c k − s s s k s s s k ) cos α k sin α k = c c c k s s s k which ends the proof.

Low dimensional special cases
3.1. The case N = 1.
The geometric power is computed easily as a sum of two terms: where we took into account, here and below,c c c 1 ∧s s s 1 =c c c 1s s s 1 (becausec c c 1 ands s s 1 anticommute). This result is, in principle, the same as obtained with the complex phasor approach (6). The difference in sign is either question of a convention or redefinition of the imaginary unit ( j → − j).
The geometric power is decomposed into the sum of C 2 = 6 terms (compare (17)): where we took into accountc c c 2 ∧c c c 1 = −c c c 1 ∧c c c 2 . The scalar component (G 1 U 2 1 + G 2 U 2 2 ) is easily recognized as the active power. The next term, byc c c 1 ∧c c c 2 , can be identified with the scattered power (G 1 − G 2 )U 1 U 2 . The remaining terms are related to the non-active power. The square of the apparent power can be decomposed as follows: 3.3. The case N = 3.
In this case we have C 3 = 13 components: c c 3 ∧s s s 3 The scalar component (G 1 U 2 1 + G 2 U 2 2 + G 3 U 2 3 ) is, as always, the active power. The next three components form a vector sum of the orthogonal components of the scattered power. The square of its value is given by: We can see that the geometric (or Clifford) power is a sum of orthogonal componenets which can be interpreted within the standard CPC (curents' physical components) theory [13,14]. In fact, one can also make one to one correspondence on the level of currents. The active current is a Clifford vector parallel to the voltage. The scattered current is a component belonging to the subspace spanned byc c c 1 , . . . ,c c c N . The rest of the current is interpreted as non-active current.

Conclusions
In this paper we rederived the Geometric Algebra approach in application to the apparent power theory. The results are satisfactory and fully general. We would like to argue that the approach based on using the Clifford algebra is not more difficult than the standard complex phasor approach in the sinusoidal case but can be easily used also in the distorted case. Thus Clifford numbers can naturally replace complex numbers in the nonsinusoidal case.