Geometric Algebra Framework Applied to Single-Phase Linear Circuits with Nonsinusoidal Voltages and Currents

Abstract

We apply a well known technique of theoretical physics, known as geometric algebra or Clifford algebra, to linear electrical circuits with nonsinusoidal voltages and currents. We rederive from the first principles the geometric algebra approach to the apparent power decomposition. The important new point consists of endowing the space of Fourier harmonics with a structure of a geometric algebra (it is enough to define the Clifford product of two periodic functions). We construct a set of commuting invariant imaginary units which are used to define impedance and admittance for any frequency.

1. Introduction

Recently, another abstract mathematical structure has been applied to the classical problems of the electrical engineering involving distorted or nonsinusoidal currents. This is geometric algebra (also known as the Clifford algebra), a quite popular and convenient tool in mathematical and theoretical physics (including electrodynamics) [1,2], with recent applications also in electrical engineering [3,4,5].

For the last dozen years, several authors have tried to give a description of distorted currents in terms of geometric algebra. It seems that the first attempt [6] was already made in a good direction but later developments were of different values and sometimes contained mistakes or too cumbersome developments [7,8,9,10]; see also [11,12]. Montoya and his collaborators [13] presented the most complete formulation of this approach in the broader context involving existing theories of electric power and earlier implementations of geometric algebra. In particular, they discussed Czarnecki’s current physical components approach [14,15] in this context. Thus, this approach can be seen as a far-reaching extension of the Fryze concepts in the electric power theory; compare with [16].

Our paper has two main goals. First, we will derive the geometric algebra formulation (in a form very close to the theory of Montoya [13]) from the first principles. We do not need to introduce any abstract objects (including complex numbers). It is sufficient to give more structure to the space of periodic functions. Namely, using the standard scalar product, in this space, we define and use the Clifford product uniquely associated with this scalar product [1,2]. Second, we present some new results within this approach. In particular, we show that the geometric algebra approach can be applied to periodic voltages and currents both in the time domain and the frequency domain. However, the most important result consists of constructing commuting imaginary units (corresponding to particular frequencies) and showing that they are invariant with respect to changes of bases or phases. We do not present applications of geometric algebra in technical practice, as relevant examples can be found in the current literature; see, e.g., [9,10,13,17].

2. Complex Numbers and Alternating Currents

Complex numbers have a long and deeply rooted tradition in the description of alternating current (AC) circuits dating back to the seminal paper by Charles Steinmetz [18]; see also [11] or [19] for more historical details. Sinusoidal functions, representing voltage and current, are represented by the so-called phasors (complex functions):

$$\begin{array}{c}u\left(t\right)=\sqrt{2}Ucos(\omega t+\alpha )⟷\mathbf{u}\left(t\right)=\sqrt{2}U{e}^{j(\omega t+\alpha )},\hfill \\ i\left(t\right)=\sqrt{2}Icos(\omega t+\beta )⟷\mathbf{i}\left(t\right)=\sqrt{2}I{e}^{j(\omega t+\beta )},\hfill \end{array}$$

(1)

where j is the imaginary unit ($j^2=-1$) and $\omega =2\pi f$. The factor $\sqrt{2}$ 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 (nonphysical) space is a simple form of Ohm’s 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 ($\mathbf{u}=Z\mathbf{i}$), provided that instead of the real resistance, one uses the complex impedance Z which (for series connections) combines resistance, capacitance, and inductance ($R,C$, and L) in a well-known way:

$$Z=R+j\left(\omega L-{\displaystyle \frac{1}{\omega C}}\right),Z=\left|Z\right|e^{j\phi },$$

(2)

where by $\phi$ we denote the phase shift between the current and voltage (i.e., $\phi =\alpha -\beta$).

The complex impedance was first introduced by Arthur Kenelly [20]. In our article, we prefer to use admittance (the reciprocal of impedance):

$$Y=Z^{-1}=G+jB,\left(Y=\left|Y\right|e^{-j\phi },\left|Y\right|={\displaystyle \frac{1}{\left|Z\right|}}\right),$$

(3)

where the real part G is known as the conductance and the imaginary part B is called susceptance. Thus,

$$\mathbf{i}=Y\mathbf{u}$$

(4)

and the current is computed as the real part of the product:

$$i\left(t\right)=\mathrm{Re}\left((G+jB)\sqrt{2}U{e}^{j(\omega t+\alpha )}\right)=\sqrt{2}U\left(Gcos(\omega t+\alpha )-Bsin(\omega t+\alpha )\right).$$

(5)

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).

In single-phase linear circuits with a sinusoidal load, the active power ($P=UIcos\phi$), reactive power ($Q=UIsin\phi$), and apparent power ($S=UI$) can be conveniently represented by the complex power $\mathbf{S}$:

$$\mathbf{S}=UI{e}^{j\phi }=\mathbf{U}{\mathbf{I}}^{*},$$

(6)

where U and I are RMS values of the voltage and current, the asterisk denotes complex conjugation, and phasors $\mathbf{U}$ and $\mathbf{I}$ do not contain the factor $e^{j\omega t}$, i.e., $\mathbf{U}=U{e}^{j\alpha }$ and $\mathbf{I}=I{e}^{j\beta }$ (see [21]). Note that $\mathbf{I}=Y\mathbf{U}$; hence,

$$\mathbf{S}=U{U}^{*}{Y}^{*}=U^2(G-jB).$$

(7)

The complex conjugation of $\mathbf{I}$ is usually attributed to Paul Janet (see [22], where an interesting description of the evolution of the concept of electrical power is presented). We point out that (6) is generally valid only in the case of sinusoidal currents and voltages.

3. Geometric Algebra of Fourier Harmonics

The main goal of this paper is to show that both complex numbers and Clifford numbers appear in a quite natural way in the description of alternating currents.

Alternating currents are described by periodic waveforms which can be endowed with a structure of multidimensional Euclidean space by defining an appropriate well-known scalar product (see Appendix A). In particular, the active power, in a general case, can be expressed as a scalar product of the voltage and current:

$$P=\langle \mathit{u}\mid \mathit{i}\rangle ={\displaystyle \frac{1}{T}}{\int }_0^{T}u\left(t\right)i\left(t\right)dt.$$

(8)

The crucial point for the approach presented in this paper consists of treating products of functions: we use only the Clifford multiplication (the Clifford product of two functions is completely different than the usual product of functions); for more details, see Appendix A. In this article, we intentionally mark periodic functions in bold italics, which means that they have to be considered as Clifford vectors wherever any product of these functions is involved. Here, it is sufficient to remark that the Clifford square is always equal to

$$\mathit{f}\mathit{f}=\langle \mathit{f}\mid \mathit{f}\rangle ={\left|\mathit{f}\right|}^2,$$

(9)

where $\left|\mathit{f}\right|$ is a Euclidean norm of the function f, and the Clifford product of orthogonal functions is antisymmetric:

$$\langle \mathit{f}\mid \mathit{g}\rangle =0⟹\mathit{f}\mathit{g}=-\mathit{g}\mathit{f}.$$

(10)

It is natural to ask about the meaning of the Clifford product $\mathit{u}\mathit{i}$. The answer is promising (see Section 4), because this product can be interpreted as the apparent power, an element of the Clifford algebra with many mutually orthogonal components. The first of these components is the scalar product, i.e., the active power. Before we explore these details, we will illustrate the main idea of our approach with a simple example (compare [11]).

Example 1. 

Consider a circuit with sinusoidal waveforms:

$$\mathit{u}=U\tilde{\mathit{c}},\mathit{i}=Icos\phi \tilde{\mathit{c}}+Isin\phi \tilde{\mathit{s}},$$

(11)

where $\tilde{\mathit{c}}=\sqrt{2}cos(\omega t+\alpha )$, $\tilde{\mathit{s}}=\sqrt{2}sin(\omega t+\alpha )$, and $U,I,\phi ,\alpha$ are constants. Let us compute

$$\mathit{u}\mathit{i}=UIcos\phi {\left(\tilde{\mathit{c}}\right)}^2+UIsin\phi \tilde{\mathit{c}}\tilde{\mathit{s}}.$$

(12)

The computation of ${\left(\tilde{\mathit{c}}\right)}^2$ is rather elementary:

$${\left(\tilde{\mathit{c}}\right)}^2={\displaystyle \frac{1}{T}}{\int }_0^{T}2{cos}^2(\omega t+\alpha )dt=1,$$

(13)

where we take into account $\omega T=2\pi$ (similarly, one can show that ${\left(\tilde{\mathit{c}}\right)}^2=1$ and $\langle \tilde{\mathit{c}}\mid \tilde{\mathit{s}}\rangle =0$, hence $\tilde{\mathit{s}}\tilde{\mathit{c}}=-\tilde{\mathit{c}}\tilde{\mathit{s}}$). The second term, $\tilde{\mathit{c}}\tilde{\mathit{s}}$, cannot be simplified in a straightforward way, but the following observations are crucial and sufficient for its interpretation:

$$\begin{array}{c}{\left(\tilde{\mathit{c}}\tilde{\mathit{s}}\right)}^2=\tilde{\mathit{c}}\tilde{\mathit{s}}\tilde{\mathit{c}}\tilde{\mathit{s}}=-\tilde{\mathit{c}}\tilde{\mathit{s}}\tilde{\mathit{s}}\tilde{\mathit{c}}=-{\left(\tilde{\mathit{c}}\right)}^2{\left(\tilde{\mathit{s}}\right)}^2=-1,\hfill \\ {\displaystyle \frac{d}{dt}}\left(\tilde{\mathit{c}}\tilde{\mathit{s}}\right)=-\omega \tilde{\mathit{s}}\tilde{\mathit{s}}+\omega \tilde{\mathit{c}}\tilde{\mathit{c}}=0,\hfill \end{array}$$

(14)

where we take into account the Leibnitz rule, valid also in the case of the Clifford product.

Thus, $\tilde{\mathit{c}}\tilde{\mathit{s}}$ is a constant element squared to $-1$. Therefore, it can be easily interpreted as an imaginary unit, and the formula (12) becomes identical to the complex power (6). By the way, we obtained this formula without any complex conjugation, which is not bad because the conjugation in the formula (6) seems to be somewhat artificial, as discussed in [11].

Throughout this paper, we consider nonsinusoidal voltages and currents in the form of finite sum of Fourier harmonics:

$$\begin{array}{c}{\displaystyle u\left(t\right)=\sum _{k=1}^N{U}_k{\tilde{\mathit{c}}}_k,}\hfill \\ {\displaystyle i\left(t\right)=\sum _{k=1}^N\left(I_{k}cos{\phi }_k{\tilde{\mathit{c}}}_k+I_{k}sin{\phi }_k{\tilde{\mathit{s}}}_k\right),}\hfill \end{array}$$

(15)

where ${\phi }_k$ are currents’ phase shifts and

$${\tilde{\mathit{c}}}_k=\sqrt{2}cos(k\omega t+{\alpha }_k),{\tilde{\mathit{s}}}_k=\sqrt{2}sin(k\omega t+{\alpha }_k).$$

(16)

Taking $N=\infty$, we can also apply the geometric algebra approach in the general case of any periodic voltages and currents.

4. Geometric Power

Following Menti et al. [6] and Montoya et al. [13], we define geometric (Clifford) power as the Clifford product of $\mathit{u}$ and $\mathit{i}$.

Definition 1. 

Geometric power is defined as Clifford product of the voltage and the current (considered as Clifford vectors):

$$\mathit{M}=\mathit{u}\mathit{i}=\langle \mathit{u}\mid \mathit{i}\rangle +\mathit{u}\wedge \mathit{i},$$

(17)

where the wedge (skew) product is defined by $\mathit{u}\wedge \mathit{i}:={\displaystyle \frac{1}{2}}\left(\mathit{u}\wedge \mathit{i}-\mathit{i}\wedge \mathit{u}\right)$.

Assuming $\mathit{u}$ and $\mathit{i}$ in the form (15), we obtain

$$\mathit{M}=\sum _{k=1}^N{U}_k{I}_{k}cos{\phi }_k+\sum _{k<m}^N(U_k{I}_{m}cos{\phi }_m-U_m{I}_{k}cos{\phi }_k){\mathit{c}}_k{\mathit{c}}_m+\sum _{k,m=1}^N{U}_k{I}_{m}sin{\phi }_m{\mathit{c}}_k{\mathit{s}}_m.$$

(18)

It is worth noting that the above formula has a simplified form (as compared to earlier papers) due to the convenient choice of the basis ${\tilde{\mathit{c}}}_k,{\tilde{\mathit{s}}}_k$ ($k=1,\dots ,N$). It is a sum of orthogonal components. One can easily verify that the number of the orthogonal components (denoted by $C_N$) is given by

$$C_N=1+{\displaystyle \frac{1}{2}}N(N-1)+N^2.$$

(19)

The so-called spinor norm of $\mathit{M}$ (the Clifford product of $\mathit{M}$ and its reversion, denoted by ${\left|\mathit{M}\right|}^2$), is equal to the apparent power. Indeed,

$${\left|\mathit{M}\right|}^2\equiv \mathit{u}\mathit{i}\mathit{i}\mathit{u}={\mathit{u}}^2{\mathit{i}}^2={\left|\mathit{u}\right|}^2{\left|\mathit{i}\right|}^2.$$

(20)

By the standard Pythagorean-like rule, it can be written as a sum of squares of the orthogonal components from the formula (18):

$${\left|\mathit{M}\right|}^2={\left(\sum _{k=1}^N{U}_k{I}_{k}cos{\phi }_k\right)}^2+\sum _{k<m}^N{(U_k{I}_{m}cos{\phi }_m-U_m{I}_{k}cos{\phi }_k)}^2+\left(\sum _{k=1}^N{U}_k^2\right)\left(\sum _{m=1}^N{I}_m^2{sin}^2{\phi }_m\right).$$

(21)

5. Geometric Power in Terms of Clifford Admittancies

In the last part of our paper, we focus on the linear case when for all frequencies corresponding admittances are given.

Definition 2. 

The Clifford admittance ${\mathit{y}}_k$, corresponding to the k-th Fourier harmonics, is defined as

$${\mathit{y}}_k=G_k+{\mathit{j}}_k{B}_k,$$

(22)

where ${\mathit{j}}_k$ is a Clifford product of ${\tilde{\mathit{c}}}_k$ and ${\tilde{\mathit{s}}}_k$:

$${\mathit{j}}_k={\tilde{\mathit{c}}}_k{\tilde{\mathit{s}}}_k.$$

(23)

One can easily obtain the following properties of ${\mathit{j}}_k$:

$${\mathit{j}}_k^2=-1,{\mathit{j}}_{\mu }{\mathit{j}}_{\nu }={\mathit{j}}_{\nu }{\mathit{j}}_{\mu },{\displaystyle \frac{d}{dt}}{\mathit{j}}_k=0;$$

(24)

see (A8) and (A9). Therefore, these entities can be interpreted as commuting constant imaginary units (more details are given in Appendix A).

The formula (23) apparently depends on the initial choice of the basis elements. For instance, we could take ${\mathit{c}}_k=\sqrt{2}cos\left(k\omega t\right)$ and ${\mathit{s}}_k=\sqrt{2}sin\left(k\omega t\right)$ instead of ${\tilde{\mathit{c}}}_k$ and ${\tilde{\mathit{s}}}_k$. Fortunately, the following theorem shows that the imaginary units do not depend on the parameter $\alpha$ entering formulas (16).

Theorem 1. 

Clifford imaginary units depend only on the frequency (i.e., on k) and do not depend on the basis chosen (provided that it is also orthonormal and with the same orientation). In other words, ${\tilde{\mathit{c}}}_k{\tilde{\mathit{s}}}_k={\mathit{c}}_k{\mathit{s}}_k$.

Indeed, the transformation between the bases has the following form:

$$\begin{array}{c}{\displaystyle {\tilde{\mathit{c}}}_k={\mathit{c}}_{k}cos{\alpha }_k-{\mathit{s}}_{k}sin{\alpha }_k=(cos{\alpha }_k+sin{\alpha }_k{\mathit{c}}_k{\mathit{s}}_k){\mathit{c}}_k=e^{{\alpha }_k{\mathit{j}}_k}{\mathit{c}}_k,}\hfill \\ {\displaystyle {\tilde{\mathit{s}}}_k={\mathit{c}}_{k}sin{\alpha }_k+{\mathit{s}}_{k}cos{\alpha }_k=(cos{\alpha }_k+sin{\alpha }_k{\mathit{c}}_k{\mathit{s}}_k){\mathit{s}}_k=e^{{\alpha }_k{\mathit{j}}_k}{\mathit{s}}_k.}\hfill \end{array}$$

(25)

Then, we easily compute the following:

$${\tilde{\mathit{c}}}_k{\tilde{\mathit{s}}}_k={\mathit{c}}_k{\mathit{s}}_k{cos}^2{\alpha }_k-{\mathit{s}}_k{\mathit{c}}_k{sin}^2{\alpha }_k+({\mathit{c}}_k{\mathit{c}}_k-{\mathit{s}}_k{\mathit{s}}_k)cos{\alpha }_{k}sin{\alpha }_k={\mathit{c}}_k{\mathit{s}}_k$$

(26)

which ends the proof.

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 ${\mathit{u}}_k=U_k{\tilde{\mathit{c}}}_k$ 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:

$${\mathit{i}}_k={\mathit{y}}_k{\mathit{u}}_k.$$

(27)

The proof is straightforward:

$${\mathit{i}}_k={\mathit{y}}_k{\mathit{u}}_k=(G_k+{\tilde{\mathit{c}}}_k{\tilde{\mathit{s}}}_k{B}_k)U_k{\tilde{\mathit{c}}}_k=U_k\left(G_k{\tilde{\mathit{c}}}_k-B_k{\tilde{\mathit{s}}}_k\right).$$

(28)

The corresponding calculation using the usual complex phasors (compare (5) is a little bit more complicated, but, obviously, leads to the same result:

$${\mathit{I}}_k=\mathrm{Re}\left(Y_k{\mathit{U}}_k\right)=U_k\left(G_k\sqrt{2}cos({\omega }_{k}t+{\alpha }_k)-B_k\sqrt{2}sin({\omega }_{k}t+{\alpha }_k)\right).$$

(29)

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 ${\mathit{j}}_1,\dots ,{\mathit{j}}_N$ instead of the single complex imaginary unit $\mathit{j}$.

The distorted (i.e., nonsinusoidal) case is treated in an analogous way due to the linearity of the problem:

$$\mathit{i}=\sum _{k=1}^N{\mathit{y}}_k{\mathit{u}}_k=\sum _{k=1}^N\left(G_k+{\tilde{\mathit{c}}}_k{\tilde{\mathit{s}}}_k{B}_k\right){\tilde{\mathit{c}}}_k{U}_k=\sum _{k=1}^N\left(G_k{U}_k{\tilde{\mathit{c}}}_k-B_k{U}_k{\tilde{\mathit{s}}}_k\right).$$

(30)

The geometric power can be expressed in terms of the admittances as follows:

$$\mathit{M}=\mathit{u}\mathit{i}=\langle \mathit{u}\mid \mathit{i}\rangle +\mathit{u}\wedge \mathit{i}=\sum _{k=1}^N{G}_k{U}_k^2+\sum _{k,j=1}^N{U}_j{U}_k\left(G_k{\tilde{\mathit{c}}}_k\wedge {\tilde{\mathit{c}}}_j+B_k{\tilde{\mathit{c}}}_j\wedge {\tilde{\mathit{s}}}_k\right).$$

(31)

Below, we present and discuss this formula for the low-dimensional special cases.

The case $N=1$.

The geometric power is computed easily as a sum of two terms:

$$\mathit{M}=\mathit{u}\mathit{i}=G{U}^2+B{U}^2{\tilde{\mathit{c}}}_1\wedge {\tilde{\mathit{s}}}_1=G{U}^2+{\mathit{j}}_{1}B{U}^2,$$

(32)

where we take into account, here and below, ${\tilde{\mathit{c}}}_1\wedge {\tilde{\mathit{s}}}_1={\tilde{\mathit{c}}}_1{\tilde{\mathit{s}}}_1$ (because ${\tilde{\mathit{c}}}_1$ and ${\tilde{\mathit{s}}}_1$ anticommute).

This result is almost the same as the result obtained with the complex phasor approach (7) except for the sign of the imaginary part. It means that $\mathit{M}$ corresponds to ${\mathit{S}}^{*}$ rather than $\mathit{S}$. This is a minor problem (the sign of the imaginary part has no deeper meaning) but we could reconcile both notations by changing the definition of a geometric power from $\mathit{M}$ to $\mathit{S}=\mathit{i}\mathit{u}$.

The case $N=2$.

The geometric power is decomposed into the sum of $C_2=6$ terms (compare (19)):

$$\begin{array}{c}\mathit{M}=\mathit{u}\mathit{i}=G_1{U}_1^2+G_2{U}_2^2+(G_1-G_2)U_1{U}_2{\tilde{\mathit{c}}}_1\wedge {\tilde{\mathit{c}}}_2\hfill \\ +B_1{U}_1^2{\tilde{\mathit{c}}}_1\wedge {\tilde{\mathit{s}}}_1+B_2{U}_2^2{\tilde{\mathit{c}}}_2\wedge {\tilde{\mathit{s}}}_2+B_1{U}_1{U}_2{\tilde{\mathit{c}}}_2\wedge {\tilde{\mathit{s}}}_1+B_2{U}_1{U}_2{\tilde{\mathit{c}}}_1\wedge {\tilde{\mathit{s}}}_2,\hfill \end{array}$$

(33)

where we take into account ${\tilde{\mathit{c}}}_2\wedge {\tilde{\mathit{c}}}_1=-{\tilde{\mathit{c}}}_1\wedge {\tilde{\mathit{c}}}_2$.

The scalar component ($G_1{U}_1^2+G_2{U}_2^2$) is easily recognized as the active power. The next term, by ${\tilde{\mathit{c}}}_1\wedge {\tilde{\mathit{c}}}_2$, can be identified with the scattered power $(G_1-G_2)U_1{U}_2$. The remaining terms are related to the nonactive power. The square of the apparent power can be decomposed as follows:

$$\begin{array}{c}{\parallel \mathit{u}\parallel }^2{\parallel \mathit{i}\parallel }^2={(G_1{U}_1^2+G_2{U}_2^2)}^2+{\left((G_1-G_2)U_1{U}_2\right)}^2\hfill \\ +{\left(B_1{U}_1^2\right)}^2+{\left(B_2{U}_2^2\right)}^2+{\left(B_1{U}_1{U}_2\right)}^2+{\left(B_2{U}_1{U}_2\right)}^2.\hfill \end{array}$$

(34)

The case $N=3$.

In this case, we have $C_3=13$ components:

$$\begin{array}{c}\mathit{M}=\mathit{u}\mathit{i}=G_1{U}_1^2+G_2{U}_2^2+G_3{U}_3^2\hfill \\ +(G_1-G_2)U_1{U}_2{\tilde{\mathit{c}}}_1\wedge {\tilde{\mathit{c}}}_2+(G_1-G_3)U_1{U}_3{\tilde{\mathit{c}}}_1\wedge {\tilde{\mathit{c}}}_3+(G_2-G_3)U_2{U}_3{\tilde{\mathit{c}}}_2\wedge {\tilde{\mathit{c}}}_3\hfill \\ +B_1{U}_1^2{\tilde{\mathit{c}}}_1\wedge {\tilde{\mathit{s}}}_1+B_2{U}_2^2{\tilde{\mathit{c}}}_2\wedge {\tilde{\mathit{s}}}_2+B_3{U}_3^2{\tilde{\mathit{c}}}_3\wedge {\tilde{\mathit{s}}}_3\hfill \\ +B_1{U}_1{U}_2{\tilde{\mathit{c}}}_2\wedge {\tilde{\mathit{s}}}_1+B_2{U}_2{U}_1{\tilde{\mathit{c}}}_1\wedge {\tilde{\mathit{s}}}_2+B_1{U}_1{U}_3{\tilde{\mathit{c}}}_3\wedge {\tilde{\mathit{s}}}_1\hfill \\ +B_3{U}_3{U}_1{\tilde{\mathit{c}}}_1\wedge {\tilde{\mathit{s}}}_3+B_2{U}_2{U}_3{\tilde{\mathit{c}}}_3\wedge {\tilde{\mathit{s}}}_2+B_3{U}_3{U}_2{\tilde{\mathit{c}}}_2\wedge {\tilde{\mathit{s}}}_3.\hfill \end{array}$$

(35)

The scalar component ($G_1{U}_1^2+G_2{U}_2^2+G_3{U}_3^2$) 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

$${\left(D_s\right)}^2={(G_1-G_2)}^2{U}_1^2{U}_2^2+{(G_1-G_3)}^2{U}_1^2{U}_3^2+{(G_2-G_3)}^2{U}_2^2{U}_3^2.$$

(36)

We can see that the geometric (or Clifford) power is a sum of orthogonal components which can be grouped into three parts which can be interpreted as the currents’ physical component [14,15]. In fact, a similar correspondence takes place also at the level of currents. The active current ${\mathit{i}}_a$ is a Clifford vector parallel to the voltage. The scattered current ${\mathit{i}}_s$ is a component orthogonal to ${\mathit{i}}_a$ and belonging to the subspace spanned by ${\tilde{\mathit{c}}}_1,\dots ,{\tilde{\mathit{c}}}_N$. The rest of the current (the component spanned by ${\tilde{\mathit{s}}}_1,\dots ,{\tilde{\mathit{s}}}_N$) is interpreted as reactive current ${\mathit{i}}_r$.

6. Conclusions

In this paper, we rederived the geometric algebra approach in application to the apparent power theory, focusing on the case of linear electrical circuits with nonsinusoidal voltages and currents. The key idea consists of endowing the space of Fourier harmonics with a structure of geometric algebra. The results are satisfactory. 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 implemented also in the distorted case. Thus, Clifford numbers can naturally replace complex numbers in the nonsinusoidal case. It turns out that we need more imaginary units (corresponding to different frequencies) and they are constructed in a natural way (see Definition 2). Theorem 1 shows the invariance of the imaginary units with respect to changes of bases and phases.