Geometric algebra: a multivectorial proof of Tellegen's theorem in multiterminal networks

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Published in IET Circuits, Devices & SystemsReceived on 20th July 2007Revised on 4th December 2007doi: 10.1049/iet-cds:20070245

ISSN 1751-858X

Geometric algebra: a multivectorial proof ofTellegen’s theorem in multiterminal networksM. Castilla1 J.C. Bravo1 M. Ordonez21Electrical Engineering Department, University of Sevilla, Escuela Universitaria Politecnica, Virgen de Africa, 7, Sevilla 41011,Spain2Applied Mathematics Department, University of Sevilla, Escuela Universitaria Politecnica, Virgen de Africa, 7, Sevilla 41011,SpainE-mail: [email protected]

Abstract: A generalised and multivectorial proof of Tellegen’s theorem in multiterminal systems is presentedusing a new power multivector concept defined in the frequency domain. This approach permits innonsinusoidal/linear and nonlinear situations formulating Tellegen’s theorem in a novel complex-multivectorrepresentation, similar to Steinmetz’s phasor model, based on complex numbers and limited to the purelysinusoidal case. In this sense, a suitable notation of voltage and current complex-vectors, associated to theelements and nodes of the network, is defined for easy development to Kirchhoff’s laws in this environment.A numerical example illustrates the clear advantages of the suggested proof.

List of symbols (nomenclature)V n linear space over real numbers

si1, i2, . . . , ikbasis of the Clifford algebra

Xi1, i2, . . . , ikcomplex numbers [ C or Cliffor’s coefficients

A0 complex-scalar

Ap complex-vector

Apq complex-bivector

<(a, a0) associated rotation into multivectorial planes

g geometric product with associated rotation

X complex-vector

X complex number

jXpj, Xp magnitude of pth complex-vector

kXk norm

(X)† (21)k(k21)/2X (reverse element)

a . b inner product

a ^ b outer product or bivector

Up pth voltage complex-vector

Ip pth current complex-vector

S multivectorial apparent power

D distortion power bivector

D magnitude of distortion power bivector

Circuits Devices Syst., 2008, Vol. 2, No. 4, pp. 383–39010.1049/iet-cds:20070245

s0 plane scalar plane

spq plane bivectorial plane

R real numbers

C complex vector space

Cln Clifford algebra in n-dimensional real space

ap phase angle of pth voltage complex-vector

aq phase angle of qth current complex-vector

wq phase angle between qth voltage complex-vectorangle and qth current complex-vector angle.

� conjugated operation

1 IntroductionGeometric algebra (GA) is a mathematical frameworkdeveloped over Clifford algebras [1–3] that unifiescoherently a set of concepts and algebraic operations. Thisalgebra is based on the concept of real vector space anddefines a geometric product for vectors that satisfies theassociative, distributive and contraction rules. All of theserules should be familiar from ordinary scalar algebra. Thefundamental difference is the absence of a commutativerule. The contraction rule is related to GA anddistinguishes it from all other associative algebras.

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The application of a GA to the circuit and power theoryanalysis [4] has a very short history and few advanceshave been made in this area. The GA introduces a productnamed ‘geometric’ product whose properties are extraordinarilyuseful to generalise and to interpret power theory in electricsystems.

On the other hand, a distinctive feature of electric circuits inactual communication equipment is the presence ofmultiterminal elements. In this sense, the purpose of thispaper is to show that the conservation of the multivectorialapparent power in multiterminal systems is the fundamentalkey to prove Tellegen’s theorem [5, 6]. In order to obtain thisgoal it is important to know the multidimensional character ofpower theory. This new aspect is better comprehended if avector space, defined as generalised complex GA (GCGA), isobtained for the representation of voltage and currentperiodical nonsinusoidal signals. This algebra is based on anew geometric product of the electrical quantities characterisedexclusively by the properties of magnitude, direction and sense.Our work is concerned with a representation of voltage andcurrent complex-vectors and power components as complex-bivectors of similar form to the complex representation validonly under sinusoidal conditions.

Complex-multivectors are introduced as a particular case ofquaternion-multivectors [2]. To define these new elements,we start from an n-dimensional linear space over the realnumbers, Vn, whose elements are named vectors. If {s1, s2,s3, . . . , sn} is an orthonormal basis of V n (n is equal to thenumber of harmonic components in periodic nonsinusoidalsignals), the unit element of this algebra is denoted by s0.The vector basis for the Clifford algebra {Cln} is generated by{si1,...,ik

¼ si1^ � � � ^ sik

=is [ {1, . . . , n} 8s:1, . . . , k} andspans the CCln vector space defined by

CCln ¼X

1�i1,...,ik�n

�Ci1i2,...,iksi1i2,...,ik

( )(1)

where ^ denotes the Grassman product. Each coefficient of abasic vector sj replaces one of the orthonormal functions inthe Fourier decomposition. The elements in this GA aretermed multivectors [1, 2] and have the expanded form

1|{z}scalar

, sk|{z}vectors(k:1,...,n)

, sk ^ sh|fflfflfflffl{zfflfflfflffl}bivectors(k,h:1,...,n;k=h)

, . . . , s1s2s3, . . . , sn|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}pseudoscalar

8>>><>>>:

9>>>=>>>;

In our work, we have following convention in referring to thehighest grade in a GA as the pseudoscalar ( J ). The two mostimportant properties of J are that it squares to 21 and definesan orientation.

Due to the difficulty of developing a multivectorial apparentpower equation in the Clifford algebra, we will define a newGA GCGA that is a generalisation of the Clifford algebra

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[2]. With this structure it is possible to obtain a generalisedpower multivector model in nonsinusoidal situations validfor linear and nonlinear loads. In general, this tool offersmany clear advantages over the use of more classicaltechniques, and these are summarised below.

† It provides a clear definition of the concept of complex-vector, applied to algebraic and geometric representation ofvoltage and current signals in frequency domain analysis.

† It is valid to geometrical interpretation and representation ofpower theory and the new features of electrical powerdecomposition into active, reactive and distortion powers withmagnitude, direction and sense. This interpretation is based onthe association of complex planes (C) to each multivectorialelement of Clifford basis (Cln). Apparent power decompositionin vectors and bivectors permits a vectorial sum of powercomponents into each plane. This possibility is the basicconcept for a multivectorial proof of Tellegen’s theorem.

† It is fundamental to define the distortion power inmultivectorial form. In this sense, the apparent powermultivector can be conceived as the generalisation of thecomplex power in sinusoidal situations.

† GCGA is operationally easy by using matrix calculus.

† GCGA explains clearly the reversibility frequencydomain–time domain via Clifford Fourier transform [7].

† The new structure permits the possibility of extending topower phenomena in poly-phase systems.

† Finally, GCGA provides a unified language for powertheory analysis in nonsinusoidal situations that is conceptuallyand computationally superior to more classical techniques.

2 Mathematical foundationsWe define as C the complex-vector space and Cln the Cliffordalgebra on n-dimensional real space. According to (1), in thecomplex-vectors case, we obtain the vector subspace

[CCln]1 ¼Pn

p¼1�Cpsp, where the coefficients �Cp [ C and

the basis vector sp [ Cln. The generic element �Cpsp is a pth

complex-vector, and it can be represented by ap þ jbp

� �sp,

where j is the imaginary unity. In the complex-bivectors case,we obtain the vector subspace CCln

� �2¼P

p=q�Cpqspq. The

generic element �Cpqspq, is a pqth complex-bivector, and it

can be represented by apq þ jbpq

� �spq. In the most general

form, complex-multivectors, we obtain the vector subspace

CCln� �

k¼P

ij=isj,s:1,...,k


. Therefore CCln in (1),

also can be represented as

CCln ¼ C|{z}complexscalar

� [CCln]1|fflfflfflffl{zfflfflfflffl}complexvectors

� [CCln]2|fflfflfflffl{zfflfflfflffl}complexbivectors

�� [CCln]n|fflfflfflffl{zfflfflfflffl}complexpseudoscalar

(2)

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For our goal, complex-vector elements ~Ap ¼�Cpsp ¼ Ape japsp

are used to represent voltage and current harmonics and

complex-bivector elements ~Apq ¼�Cpqspq ¼ Apqe japqspq are

used to analyse power theory. The corresponding complex-

vector and complex-bivector are given by ~AVector ¼P

p~Ap and

~ABivector ¼P

p,q~Apq, respectively. More complete information

can be seen in [1–3] and in Sections 10.1–10.3.

Taking into account the definitions developed in (29) and(30), for ~A ¼

Pp Ape japsp and ~B ¼

PBqe j(aq�wq)sq

complex-vectors, the generalised complex geometric productcan be defined

~Ag ~B ¼X

p

~Ap

�� j ~Bpjejwp þ

Xp,q

e j(ap�aq) ~Ap

�� ~Bq

�� e jwqspq

þXq,p

e j(aq�ap) ~Aq

�� ~Bp

�� e jwpsqp

¼X

p

~Ap

�� ~Bp

�� e jwp þXp,q

e j(ap�aq) ~Ap

�� ~Bq

�� e jwq

n

�<ap,aqe j(aq�ap) ~Aq

�� ~Bp

�� e jwp

ospq

(3)

where

<ap ,aq¼ e�2j(aq�ap) (4)

is an application into complex planes (30) and the angles ap

and aq � wq identify the phase of the pth and qth harmonics,respectively, in the linear case.

If sp = sq, the basic operations as sum, product, inverseand quotient of complex-vectors are given by:

† Sum

~Ap þ~Bq ¼ ap þ jbp

� �sp þ cq þ jdq

� �sq (5)

† Product

~Apg ~Bq ¼ Ape jap Bqe j(bq)spsq (6)

† Inverse

~Bq

� ��1

¼1

~Bq

¼1

Bq

e�j(bq)sq (7)

† Quotient

~Ap

~Bq

¼Ap

Bq

e j(ap�bq)spq ¼~Ap

~Bq

� ��1

(8)

If sp ¼ sq, the equality of two ~Ap and ~Bq complex-vectors, aswell as the addition, the product and the quotient rules, arecompletely similar to those for common complex numbers.


If we restricted C to the real numbers, we obtain the classicClifford algebra.

The apparent complexity of the mathematical structure(CCln, g) is justified by the operative facility and intuitivegeometrical interpretation of power theory.

3 Multivectorial apparent powerSuppose a nonsinusoidal voltage

u(t) ¼ffiffiffi2p X

p[L<N

Up sin pvt þ ap

� �

applied to a nonlinear load, Fig. 1, where p is the harmonicorder of u(t). The resulting current has an instantaneousvalue given by

i(t) ¼ffiffiffi2p X

q[N <M

Iq sin qvt þ bq

� �

where q is the harmonic order of i(t), in linear conditions,bq ¼ aq � wq and wq is the impedance phase angle. It isassumed that a group of voltage harmonics N exist thathave corresponding current harmonics of the samefrequencies, that components L of the supply voltage existwithout corresponding current and that components M ofcurrent exist without corresponding voltages. The capitalUp and Iq represent rms values of up(t) and iq(t).~U ¼

Pp[L<N

~U p and ~I ¼P

q[N <M~I q are the complex-

vector voltage and current, respectively.

For linear and nonlinear load the multivectorial apparentpower can be deduced from (3) by

~S ¼ ~U g ~I�¼

Xp[N <Lq[N <M

~U pg ~I�

q ¼X

p[N <Lq[N <M

~U p �~I�

q þ~U p ^ ~I

�

q

� �

¼Xp¼q

j ~U pjj~I pj cosfp þ j

Xp¼q

j ~U pjj~I pj sinfp

þXp,q

(j ~U pjj~I qje

jfq � j ~U qjj~I pje

jfp )e j(ap�aq)spq

¼ C0,j0 þ C

0,j2 ¼

~P þ j ~Qþ ~D (9)

where ~U ¼P

p[N <L j~U pje

japsp and ~I�¼P

q[N <M j~I qj

e j(aq�wq )sq are voltage and conjugated current complex-vectors, respectively, C

0,j0 ¼

~P þ j ~Q, C0,j2 ¼

~D, and ~P, ~Qand ~D are the active, reactive and distortion power

Figure 1 Generic nonlinear circuit

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complex-bivectors correspondingly. The squared value j ~Sj2

may be represented, (36), as

~S�� 2¼ j ~U g ~I

�j2¼

Xp[N <Lq[N <M

j ~U pj2X

p[N <Lq[N <M

j~I qj2¼ j ~U j2j~I j2 (10)

The conventional concept of apparent power S ¼ j ~Sj, is simplya consequence of multivectorial apparent power, ~S. This isthe fundamental and original characteristic of ourformulation. The S value results as the magnitude ofmultivector ~S (9) and not as a simple product of the voltageand current rms values. It should be noted that the suggestedpower multivector ~S is decomposed into complex-scalar andcomplex-bivectors with magnitude, direction and sense.

4 Multivectorial form ofTellegen’s theoremFor a z-element, the associated pth harmonic voltage ~U p ¼

j ~U pjejapsp and the qth harmonic current ~I q ¼

j~I qjej(aq�wq)sq are complex-vectors. The pqth component of

the multivectorial apparent power, delivered in this element, isgiven by

~Sz

pq ¼~U

z

pg(~Iz

q)� (11)

Equation (11) can also be expressed through potentials of itsnodes. In this sense, we consider the two-terminal z-element(Fig. 2) with nodes 1 and 2. It is possible to designate

two currents ~I1z

q ¼~I

z

q and ~I2z

q ¼ �~I

z

q which have the same

value but opposite senses [8].

Thus, (11) can be transformed as follows

~Sz

pq ¼~U

z

pg(~Iz

q)� ¼ ~U1

p g(~I1z

q )� þ ~U2

p g(~I2z

q )� (12)

where ~Uz

p ¼~U

1

p �~U

2

p is the pth voltage harmonic of the z-element, ~U

n

p is the n-node voltage harmonic, ~Inz

q is the qthcurrent harmonic in the zth element that leaves the n-nodeand (~I

nz

q )� is the conjugate of ~Inz

q .

The definition of pqth multivectorial apparent power of thez-element (12), is easily generalised to any multiterminalnetwork (Fig. 3) with nþ 1 nodes, n ¼ 1, . . . , N andz ¼ 1, . . . , Z elements for each pqth multivectorialcomponent is given by

~Spq ¼XN

n¼1

XZ

z¼1

~Un

p g(~Inz

q )� 8p � q (13)

Figure 2 z-element

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In (13), the summation extends to n nodes that agree withexternal terminals of the zth network element and also to zelements which are connected to the nth node.

One of the most important properties of this powermultivector concept (13) is the validity of

~Spq ¼XN

n¼1

XZ

z¼1

~Un

p g(~Inz

q )� ¼ 0 8p � q and

~S ¼X

pq

~Spq ¼ 0 8p � q

(14)

if Kirchhoff’s first law is fulfilled,Pz

z¼1 (~Inz

q )� ¼ 0. Then, (14)states the generalised multivectorial Tellegen’s theorem.Therefore the total multivectorial apparent power transferredby the sources must equal that delivered by the load. In thiscontext, we have used the GCGA structure to introduce anew reformulation of Tellegen’s theorem for multiterminalcircuits (14). This new version of the theorem is useful and isapplicable to linear and nonlinear passive and active networks.

5 Geometric interpretationIn the CCln algebraic structure, voltages and currents must beconsidered as complex-vectors and the multivectorial apparentpower (13) is a complex-bivector with three differentcomponents: active, reactive and distortion powers. Theactive and reactive powers result in real and imaginary parts,respectively, into the s0-plane in which all products of thesame frequency are contained (p ¼ q). On the other hand,the distortion power is a formal sum of complex-bivectors(all spq-planes), involving cross-frequency products (p = q).

For a pair of generic indexes (p ¼ q) like-frequencyproducts are represented into the s0-plane (Fig. 4). Thesum of these products is a complex-vector ( ~A0) and their

components are the active power ~P ¼P

p~Pp and the

reactive power ~Q ¼P

p~Qp.

Figure 3 zth n-terminal component

Figure 4 Active P and reactive Q powers


gdt

afdma

Tt

6LzTtcnt

K

Fnts

F

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If p = q and p, q [ N , bivectors from cross-frequencyeometric products are represented in Fig. 5. Theifference of these products is a ~Dpq complex-bivector intohe spq-plane and is associated to linear distortion power.

On the other hand, if p = q, p [ L < N , q [ Mnd p [ L, q [ N < M , bivectors involving cross-requency geometric products are associated to nonlinearistortion power. From geometrical interpretation, theultivectorial apparent power squared value may be

nalytically decomposed, (36), by

~S�� 2 ¼X

p¼q

~A0

�� 2þXp=q

~D2

pq

�� ¼ Xp[N <Lq[N <M

p=q

~U pg ~I�

q

�� 2

¼X

p[N <L

~U p

�� 2 Xq[N <M

~I q

�� 2 (15)

he above equation is identical to the classic squared value ofhe apparent power.

Proofet us consider a multiterminal network (Fig. 6a) with-elements (z ¼ 1, . . ., Z) and (nþ 1)-nodes (n ¼ 1, . . . , N ).he jth generic node is represented in (Fig. 6b). In Fig. 6a,

he source network is represented by a simple staronnection of voltages. We consider that the nodes of aetwork are the external terminals of each component orhey are a union of nodes of some zth components.

Thus, for each nth node and qth harmonic, the equation ofirchhoff’s first law can be written down in the following form

XZ

z¼1

(~Inz

q )� ¼ 0 (16)

or a generic qth harmonic, the summation extends to the zumbers of those network components which are connectedo the nth node. If we set up (16) for all nodes, we obtain aet of equations for the system of Fig. 6a

n ¼ 1) (~I11

q )� þ � � � þ (~I1k

q )� þ � � � þ (~I1Z

q )� ¼ 0

n ¼ j ) (~I qj1)� þ � � � þ (~I q

jk)� þ � � � þ (~I qjZ)� ¼ 0

n ¼ N ) (~IN 1

q )� þ � � � þ (~INk

q )� þ � � � þ (~INZ

q )� ¼ 0

(17)

igure 5 Dpq linear distortion multivector

ircuits Devices Syst., 2008, Vol. 2, No. 4, pp. 383–3900.1049/iet-cds:20070245

If we multiplied (17) by the pth voltage harmonic complex-

vector in the nth node, ~Un

p , these equations are still fulfilled

in the general form

XZ

z¼1

~Un

p g ~Inz

q

� ��¼ 0 8p � q (18)

where the summation is executed under all z-components,which have connections with the nth node. The set (18),made for all nth nodes, again contains each qth current ~I

nz

q

complex-vector

~U1

pg (~I11

q )� þ � � � þ (~I1k

q )� þ � � � þ (~I1Z

q )�h iþ � � � þ ~U

j

p g (~Ij1

q )� þ � � � þ (~Ijk

q )� þ � � � þ (~IjZ

q )�h i

þ � � � þ ~UN

p g (~IN 1

q )� þ � � � þ (~INk

q )�h

þ � � � þ (~INZ

q )�i¼ 0 8p � q (19)

The above equation can also be expressed from distributiveproperty by

~U1

p g(~I11

q )� þ � � � þ ~Uj

pg(~Ij1

q )� þ � � � þ ~UN

p g(~IN 1

q )�|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}Sz¼1

pq

þ � � �

Figure 6 n-terminal source-load network and jth node

a n-terminal source-load networkb jth node

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þ ~U1

p g(~I1k

q )� þ � � � þ ~Uj

p g(~Ijk

q )� þ � � � þ ~UN

p g(~INk

q )�|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}Sz¼k

pq

þ � � �

þ ~U1

p g(~I1Z

q )� þ � � � þ ~Uj

p g(~IjZ

q )� þ � � � þ ~UN

p g(~INZ

q )�|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}Sz¼Z

pq

¼ 0

(20)

In accordance with (20), (14) may be rewritten

~Spq ¼XZ

z¼1

~Sz

pq ¼XZ

z¼1

XN

n¼1

~Un

p g(~Inz

q )�" #

¼ 0 8p � q (21)

and taking into account (13) and (21), the global multivectorapparent power is

~S ¼X

pq

~Spq ¼ 0 8p � q (22)

The above equation represents a multivectorial proof ofmultivectorial Tellegen’s theorem in electric networks withmultiterminals components in nonsinusoidal situations.From (21) it is obvious that the same results can beobtained for any other networks satisfying the first andsecond Kirchhoff’s laws.

7 Numerical exampleA star connection of three voltage sources, whose complex-vectors ~U

n¼P

p[{1,2}~U

n

p , are given by

~U1¼ ~U

1

1 þ~U

1

2 ¼ 10e j30s1 þ 5e j0s2 (V )

~U2¼ ~U

2

1 þ~U

2

2 ¼ 5e j0s1 þ 5e j30s2 (V )

~U3¼ ~U

3

1 þ~U

3

2 ¼ 10e j60s1 þ 5e�j30s2 (V )

(23)

is applied to linear load components, (z ¼ 1, . . . , 5) (Fig. 7).

The resulting nodal current complex-vector ~Inz¼P

q[ 1,2f g~I

nz

q is given by

~I11¼ ~I

11

1 þ~I

11

2 ¼ 3:20e j62:1W

s1 þ 2:57e�j169:5s2

~I22¼ ~I

22

1 þ~I

22

2 ¼ 3:43e j151W

s1 þ 3:37e�j21:5s2

~I33¼ ~I

33

1 þ~I

33

2 ¼ 4:74e�j71:5W

s1 þ 1:80e j109:7s2

~I14¼ ~I

14

1 þ~I

14

2 ¼ 1:05e�j71:7W

s1 þ 2e j18:4s2

~I24¼ �~I

22

~I34¼ ~I

34

1 þ~I

34

2 ¼ 4:26e j141:4W

s1 þ 2:24e�j56:6s2

~I15¼ ~I

15

1 þ~I

15

2 ¼ 2:59e�j135W

s1 þ 0:65e�j15s2

~I35¼ �~I

15(24)

Proceeding from proposed notations and applying (16) and(21) for n ¼ 1, 2, 3, z ¼ 1,2, . . . , 5, we obtain a set of

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equations for the system of Fig. 7

~U1

p g ~I11

q

� ��|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}

~S1pq

þ ~U2

p g ~I22

q


~S2pq

þ ~U3

p g ~I33

q


~S3pq

þ ~U1

p g ~I14

q

� ��þ ~U

2

p g ~I24

q

� ��þ ~U

3

p g ~I34

q

� ��|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

~S4pq

þ ~U1

p g ~I15

q

� ��þ ~U

3

p g ~I35

q

� ��|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

~S5pq

¼ 0 (25)

where

~Sz

pq ¼�S

z

11 þ�S

z

22

s0|fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}

~Az0

þ �Sz

12s12|fflfflffl{zfflfflffl}~D12

¼ Pzþ jQz

s0 þ

�Dz

12s12 (26)

and

~S1

pq ¼ (14:5� j14:7)s0 þ (�40:2� j8)s12

~S2

pq ¼ (�4:5þ j4:9)s0 þ (32:8þ j5:9)s12

~S3

pq ¼ (�38:3þ j29:6)s0 þ (29:4þ j1:9)s12

~S4

pq ¼ (28:3� j34:9)s0 þ (�40:8þ j11)s12

~S5

pq ¼ ( j15:1)s0 þ (18:8� j10:8)s12

(27)

Combination of (21), (22) and (27), gives the total apparentpower multivector ~S for all network components

~Spq ¼X

p[ 1,2f gq[ 1,2f g

~S1

pq þ~S

2

pq þ~S

3

pq þ~S

4

pq þ~S

5

pq ¼ 0

~S ¼Xp,q

~Spq ¼ 0

(28)

Figure 7 Three terminals network


IETdoi:

www.ietdl.org

Geometric interpretation of (28) is represented in Figs. 8and 9.

When p ¼ q and p, q [ 1, 2f g we represent in thes0-plane (Fig. 8) the conservation of geometric productsexpressed in (25) involving like frequency ( ~A0 complex-vector).

If p = q, we represent in the s12-plane (Fig. 9) theconservation of geometric products expressed in (25)involving cross-frequency ( ~D12 distortion complex-bivector).

8 ConclusionThe work in this paper has used an n-dimensional lineardecomposition of periodic nonsinusoidal voltage andcurrent waveforms to show the multivectorial character ofapparent power. This new power multivector permits avectorial sum of all their components into complex planes,(C ), defined for each multivectorial element of Cliffordbasis (Cln). In classical approaches this goal is notpossible. In this sense, a multivectorial proof of Tellegen’stheorem is presented, based on the conservation law ofmultivectorial apparent power in each plane. Therefore wecan conclude that the multivectorial proof of Tellegen’stheorem, in multiterminal networks under nonsinusoidalconditions, is a generalisation of that demonstration basedon the complex power under sinusoidal situations. Like thewell-known complex power, the power multivector conceptcarries all power information and obeys the usualconservation laws.

Figure 8 A0z vectors)

PzA0

z ¼ 0

Figure 9 D12z bivector )

PzD12

z ¼ 0


9 References

[1] HESTENES D., SOBCZYK G.: ‘Clifford algebra to geometriccalculus: a unified language for mathematics and physics’(Kluwer Academia, 1986)

[2] DORST L., DORAN C., LASENBY J.: ‘Applications of geometricalalgebra in computer science and engineering’ (Birkhauser,2002)

[3] ABLAMOWICZ R., SOBCZYK G.: ‘Lectures on Clifford(geometric) algebras and applications’ (Birkhauser, 2003)

[4] MENTI A., ZACHARIAS T., MILIAS-ARGITIS J.: ‘Geometric algebra:a powerful tool for representing power undernonsinusoidal conditions’, IEEE Trans. Circuits Syst., 2007,54, (3), pp. 601–609

[5] PENFIELD P., SPENCE R., DUINKER S.: ‘A generalized form ofTellegen’s theorem’, IEEE Trans. Circuit Theory, 1970,CT-17, (3), pp. 302–305

[6] VAN VALKENBURG M.E.: ‘Analisis de Redes’ (EditorialLimusa, Mexico, 1977)

[7] EBLING J., SCHEUERMANN G.: ‘Clifford Fourier transform onvector fields’, IEEE Trans. Vis. Comput. Graphics, 2005, 11,(4), pp. 469–479

[8] VITKOV M.G.: ‘Zakoni elektricheskich cepey dlyavzveshennich tokov’, Elektrichestvo, (5), pp. 73–75

10 Appendix: GCGA10.1 Generalised complex geometricproduct

Let us consider C to be the complex vector space and Cln theClifford algebra in n-dimensional real space. We define theset

CCln ¼X

1�i1,...,ik�n


( )

where the coefficients �Ci1i2,...,ik[C and the basis

si1i2,...,ik[ :Cln. It is trivial that CCln is a vector space over R.

For two multivectors ~C ¼ �Ci1i2,...,iksi1i2,...,ik

and~C0¼ �C

0

j1 j2, ... , jssj1 j2, ... , js

, where complex numbers associatedto each multivector are

�Ci1i2,...,ik¼ �Ci1i2,...,ik

�� e jai1 i2,...,ik

and

�C0j1 j2, ... , js¼ �C0j1 j2, ... , js

�� e j a0j1 j2, ... , js�b0j1 j2, ... , js

389


390

&

www.ietdl.org

we define a new geometric product termed ‘generalisedcomplex geometric product’, g

g: <a,a0Wg

(29)

The letter ‘g’ represents the usual geometric product and<a,a0

is an application in the complex planes associated to anymultivector product; it is given by

<a,a0�C, �C

0� �

¼e�2j a0j1 j2,...,js

�ai1 i2,...,ik

ifPkh¼1

ih .Ps

m¼1

jm

1 otherwise

8<: (30)

This new product for multivectors ~C and ~C0

is given by

�Csi1i2,...,ikg �C0� sj1j2,...,js

¼ �C �C0si1i2,...,ik

sj1j2,...,js

and the basis transposition holds that

�C0 �Csj1 j2, ... , js

si1 i2, ... , ik

� �¼ (�1)ks

<a,a0�C , �C

0� �

�C

� �C0si1 i2, ... , ik

sj1 j2, ... , js(31)

If ai1i2,...,i ¼ a0j1j2,...,js8si1i2,...,ik

, sj1j2,...,js, then

<a,a ¼ IdC (32)

and the ‘generalised complex geometric product’, (29), is theclassic geometric product.

In this context, we define the ‘reverse’ element as

�C � si1i2,...,ik

� �y¼ (�1)k(k�1)=2 �Csikik�1,...,i1

(33)

where (†) is the ‘reverse’ operation.

The Institution of Engineering and Technology 2008

The ‘conjugated’ operation (�) is given by

�C � si1i2,...,ik

� ��¼ �C

�� si1i2,...,ik

(34)

10.2 Properties

The structure CCln, g

is a complex GA because thefollowing properties are fulfilled: associative, distributivewith respect to the sum and contraction.

10.3 Norm definition

The norm, value or magnitude, of a multivector ~C is a uniquescalar, k ~Ck, calculated by

k ~Ck2¼ k ~C( ~C

y)�l0 (35)

10.4 Squared value of apparent power

j ~Sj2 ¼Xp¼q

A20 þ

Xp=q

D2pq ¼

Xp[N <Lq[N <Mp=q

j ~U pg ~I�

q j2

¼Xp,q

j( ~U pg ~I�

p þ~U q g ~I

�

q )j2 þXp=q

j( ~U pg ~I�

q �~U q g ~I

�

p)j2

¼Xp,q

[U 2pI 2

p þ U 2q I 2

q þ 2U qU pI pI q cos(aq � ap)]

þ[U 2pI 2

q þ U 2q I 2

p � 2U qU pI pI q cos (aq � ap)]

( )

¼X

p[N <L

U 2p

Xq[N <M

I 2q

(36)


Geometric algebra: a multivectorial proof of Tellegen's theorem in multiterminal networks

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