P og ess In Elec omagne ics Resea ch B, Vol. 15, 401–422, 2009
AN APPROACH TO THE MULTIVECTORIAL APPAR-
ENT POWER IN TERMS OF A GENERALIZED POYNT-
ING MULTIVECTOR
M. Cas illa and J. C. B a o
Elec ical Enginee ing Depa men
Uni e si y o Se illa
Escuela Uni e si a ia Poli ´ecnica, Vi gen de A ica 7
Se illa 41011, Spain
M. O d´o˜nez
Applied Ma hema ics Depa men
Uni e si y o Se illa
Escuela Uni e si a ia Poli ´ecnica, Vi gen de A ica 7
Se illa 41011, Spain
J. C. Mon a˜no
Spanish Resea ch Council (CSIC)
Reina Me cedes 10, Se illa 41012, Spain
Abs ac —The pu pose o his pape is o explain an exac
de i a ion o appa en powe in n-sinusoidal ope a ion ounded on
elec omagne ic heo y, un il now unexplained by simple ma hema ical
models. The aim is o explo e a new ool o a igo ous ma hema ical
and physical analysis o he powe equa ion om he Poyn ing Vec o
(PV) concep . A powe ul ma hema ical s uc u e is necessa y and
Geome ic Algeb a o e s such a cha ac e is ic. In his sense, PV has
been e o mula ed om a new Mul i ec o ial Euclidean Vec o Space
s uc u e (CGn-R3) o ob ain a Gene alized Poyn ing Mul i ec o (˜
S).
Consequen ly, om ˜
S, a sui able mul i ec o ial o m ( ˜
Pand ˜
D) o he
Poyn ing Vec o co esponds o each componen o appa en powe .
In pa icula , his amewo k is essen ial o he cla i ica ion o he
connec ion be ween a Complemen a y Poyn ing Mul i ec o (˜
D) and
he powe con ibu ion due o c oss- equency p oduc s. A simple
applica ion example is p esen ed as an illus a ion o he p oposed
powe mul i ec o analysis.
Co esponding au ho : M. Cas illa ([email p o ec ed]).
402 Cas illa e al.
1. LIST OF SYMBOLS (NOMENCLATURE)
n-sinusoidal = non-sinusoidal o mul i-sinusoidal.
R= eal numbe s
E3= Euclidean ec o space
C= complex ec o space
Vn= linea space o e eal numbe s
Gn= Cli o d algeb a in n-dimensional eal space
CGn= complex Cli o d Algeb a
Φ = ope a o
Γ = ime-domain equency-domain ans o m
CG
n-R3= ime gene alized Euclidean space
CGn-R3= equency gene alized Euclidean space
~
1X,~
1Y,~
1Z= Euclidean canonical basis
~
1X,Y,Z = gene ic uni a y ec o o E3
σ1,...,k = Cli o d algeb a canonical basis
IdC= iden i y ope a ion
˜
z( ) = ins an aneous geome ic ec o (˜
z∈ CG
n-R3)
˜
e( ) = ins an aneous elec ic ield geome ic ec o
˜
h( ) = ins an aneous magne ic ield geome ic ec o
˜
d( ) = ins an aneous displacemen ield geome ic ec o
˜
b( ) = ins an aneous magne ic induc ion ield geome ic ec o
˜zX,Y,Z = componen s o ˜
z( )
˜
zp=p- h ha monic componen o ˜
z( )
˜
ZX,Y,Z= spa ial componen s o ˜
Z
˜
Z= spa ial geome ic phaso (˜
Z∈ CGn-R3)
˜
Zp= spa ial p- h ha monic componen o ˜
Z
˜
Z= geome ic phaso ( ˜
Z∈ CGn)
˜
Zp=p- h ha monic componen o ˜
Z
˜
Zpq = bi ec o componen o ˜
Z
˜
E= elec ic ield geome ic phaso
˜
H= magne ic ield geome ic phaso
˜
D= displacemen ield geome ic phaso
˜
B= magne ic induc ion ield geome ic phaso
˜
S= gene alized Poyn ing mul i ec o (GPM)
˜
P= Poyn ing mul i ec o (PM)
˜
D= complemen a y Poyn ing mul i ec o (CPM)
Up=p- h ha monic ol age ms alue
Ip=p- h ha monic cu en ms alue
⊗= classic geome ic p oduc
¯= gene alized geome ic p oduc in CGn
P og ess In Elec omagne ics Resea ch B, Vol. 15, 2009 403
◦= gene alized geome ic p oduc in CG
n-R3
·= inne p oduc
∧= ou e p oduc
⊕= di ec sum
+ = classic sum o scala s and also di ec sum o mul i ec o s
j = imagina y uni
∗= conjuga ed ope a ion
†= e e se ope a ion
hi0= scala pa
hi2= bi ec o pa
˜
S= appa en powe mul i ec o
k˜
Sk= no m, alue o magni ude o mul i ec o ˜
S
˜
Ω·= complex scala
˜
Ω∧= complex bi ec o
ωp, ωq= ha monic equencies
αp= phase angle o p- h ol age geome ic phaso
αq= phase angle o q- h cu en geome ic phaso
ϕq= phase angle be ween q- h ol age and q- h cu en geome ic
phaso s
˜
δ= ela i e quali y index mul i ec o (RQI )
PF = powe ac o
2. INTRODUCTION
2.1. Mo i a ion
One o he undamen al issues in powe sys em analysis is ela ed wi h
he elec omagne ic heo y in o de o explain he ene gy ans e in
an elec ic ci cui . Hence, his pape es ablishes an elec omagne ic
ounda ion o he powe equa ion ep esen a ion. Fo his goal, a new
Gene alized Poyn ing Mul i ec o is p oposed.
2.2. Li e a u e Re iew
The elec ical ci cui s in n-sinusoidal ope a ion can be analyzed by
means o ma hema ical ools ha a e much simple o handle han
elec omagne ic heo y based on Maxwell equa ions [1]. Howe e , i
is also ue ha hese equa ions ully explain in e ac ions be ween
elec ic and magne ic ields and he e o e explain e e y elec omagne ic
phenomenon, including he ene gy ans e in an elec ic sys em. In
n-sinusoidal ope a ion, he dis o ed elec omagne ic ields can be
ep esen ed as sums o se ies o ha monics. Each ha monic componen
404 Cas illa e al.
o he ield is go e ned by Maxwell equa ions and sa is ies he Poyn ing
Theo em.
I is ele an o classi y he con ibu ion o hese equa ions o he
elec ic powe heo y in o ollowing lines o hough :
a) Ci cui heo y analysis: Fi s , is he mos commonly used
app oach. I analyzes cu en s, ol ages and ci cui elemen p ope ies.
In his sense, ci cui heo y, uled by simple equa ions based on Ohm’s
law, can be ega ded as a e y pa icula case o elec omagne ic
heo y, and powe heo y was de eloped mainly om ci cui analysis.
Elec ical componen s o powe sys ems a e conside ed as elemen s o
ci cui s and hei elec omagne ic beha iou is desc ibed by means
o ol ages and cu en s o elemen e minals. Ci cui heo y can
explain only he powe lows be ween componen s, and i is unable
o e eal hei spa ial dis ibu ion. Ne e heless, no phenomena such
as hys e esis losses o skin e ec s can be explained by ci cui heo y.
These a e phenomena o he elec omagne ic ield cha ac e is ics.
Fo his i s app oach, Complex Algeb a [2] p o ides an ini ial
p ocedu e o sol e he p oblem, despi e i s limi a ion o he pu ely
sinusoidal case. The n-sinusoidal ope a ion imposes he subs i u ion
o he Complex Algeb a app oach wi h a new ep esen a ion model
and he e o mula ion o he ene gy balance. Conside able esea ch
e o s ha e been di ec ed owa ds he ep esen a ion o appa en
powe in a ious ways [3–9]. Speci ically, in [9], he au ho s use
Geome ic Algeb a o de ine a mul i ec o powe based on he
decomposi ion o he ins an aneous cu en in o he ac i e and eac i e
componen s. I should be no ed ha hei app oach does no
dis inguish be ween eac i e and dis o ion powe om a ma hema ical
iewpoin . Fu he mo e, none o he a o emen ioned pape s leads o a
ep esen a ion ha could be conside ed uni e sally sa is ac o y.
b) Elec omagne ic heo y analysis: This second alid me hod
analyzes he ene gy low using he Poyn ing Theo em (PT), and
he e o e he Poyn ing Vec o (PV) should be conside ed, since i
ep esen s he b idge be ween elec omagne ic heo y and ci cui
heo y [10]. These ools a e undamen al concep s o elec omagne ic
heo y wi h espec o ene gy low. The goal is o in es iga e he na u e
o he non-ac i e powe and some p og ess has undeniably been made.
Nume ous aluable con ibu ions ha e appea ed in he li e a u e [11–
17], each shedding mo e ligh on some aspec s o he p oblem. F om
among hem, [12, 13] mas e ully explain he physical mechanism o
ene gy p opaga ion in elec ic powe sys ems, [15] econside s he
bases o elec omagne ism in o de o ind a physical in e p e a ion
o he powe equa ion, and [16] uses he PV o illus a e he na u e o
powe low in elec ic ci cui s using elec omagne ic ields. Howe e ,
P og ess In Elec omagne ics Resea ch B, Vol. 15, 2009 405
c i ics o PV calcula ions [17] a gue ha elec omagne ic heo y is
useless o p ac ical applica ions o elec ic powe heo y. Agains his
e e ence, i is ou iew ha he powe equa ion can be based and
in e p e ed h ough a new o mula ion o he Poyn ing Vec o and
ha o he aspec s conce ning he elec omagne ic ield in n-sinusoidal
ope a ion and hei di ec ela ion wi h powe heo y ha e ye o be
ho oughly in es iga ed. Thus, he pu pose o his pape is o ad ance
ene gy low analysis by using a new ma hema ical s uc u e o he
ep esen a ion o he powe equa ion in single-phase ci cui s unde n-
sinusoidal ope a ion. In his way, a comple e solu ion o he powe
equa ion analysis p oblem o linea /non-linea ci cui s based on a
Gene alized Poyn ing Mul i ec o (˜
S), is p esen ed.
To his end, p ima ily ou wo k in oduces a CG
n-R3ma hema ical
s uc u e based on Cli o d Algeb a o he de ini ion o he dis o ed
elec ic and magne ic ield in ensi ies (˜
e,˜
h), which a e ime geome ic
ields associa ed o an Euclidean di ec ion. F om hese de ini ions i
is possible o ob ain he quan i ies called spa ial geome ic phaso s
(˜
E,˜
H) in he CGn-R3s uc u e in equency domain. Consequen ly,
ou wo k is aimed a showing how an elec ic and magne ic ield can
be associa ed wi h he elemen s o Cli o d Algeb as [21, 22] o a new
o mula ion and in e p e a ion o powe heo y in his amewo k.
This second app oach is mo e gene al and undamen al ha he
i s app oach based on ci cui heo y, and i has he addi ional
ad an age o p o iding a physical insigh in o he spa ial dis ibu ion
o he powe low.
Finally, his pape add esses he need o unde s and he
mul idimensional cha ac e o elec ic powe heo y and i s ela ion
o he elec omagne ic heo y.
2.3. Con ibu ions
The pape is conce ned wi h a ep esen a ion o he powe equa ion
unde non-sinusoidal condi ions om elec omagne ic heo y. The
appa en powe concep is be e unde s ood i a Cli o d ec o space
is used o he ep esen a ion o he dis o ed elec ic and magne ic
ield in ensi ies. This gene a es a la ge linea space called Gene alized
Euclidean Space CG
n-R3, which will be u ilized in his pape o a new
ep esen a ion o he powe equa ion. This objec i e canno be eached
on he Complex Algeb a amewo k.

406 Cas illa e al.
3. MATHEMATICAL FOUNDATIONS: GEOMETRIC
EUCLIDEAN SPACES
3.1. Time Domain: Gene alized Euclidean Space CG
n-R3
In o de o in oduce he ins an aneous quan i ies o elec ic and
magne ic ields, in his sec ion we de ine a new s uc u e o he
ime domain ha we ha e named Gene alized Euclidean Space,CG
n-
R3, whose coe icien s belong o he Complex Geome ic Algeb a CGn
cons uc ed in [18]. Le n~
1X,~
1Y,~
1Zobe he “canonic” basis o he
Euclidean space E3. A gene ic elemen o CG
n-R3is gi en by
˜
z( ) = ˜zX~
1X+ ˜zY~
1Y+ ˜zZ~
1Z(1)
whe e each componen in (1) is in he o m ˜z( ) = kej[α( )+θ]σa∈ CG
n,
k≥0 and σais a basis elemen o CGns uc u e [18].
Thus, he CG
n-R3s uc u e is a CG
n ec o space whose inne
p oduc is de ined by
˜
z( )·˜
w( ) = h˜zX,˜w∗
Xi0+h˜zY,˜w∗
Yi0+h˜zZ,˜w∗
Zi0(2)
whe e, ˜
z( ) = ˜zX~
1X+ ˜zY~
1Y+ ˜zZ~
1Z,˜
w( ) = ˜wX~
1X+ ˜wY~
1Y+ ˜wZ~
1Z.
Mo eo e , om (D1) he no m o ˜
z( ) is gi en by
k˜
z( )k=X
i=X,Y,Z h˜zi,˜z∗
ii0(3)
Now we de ine he ou e p oduc in his s uc u e as
˜
z( )∧(−˜
w( )) = 

~
1X~
1Y~
1Z
˜zX˜zY˜zZ
−˜wX−˜wY−˜wZ

= (h−˜zY,˜wZi2+h˜zZ,˜wYi2)~
1X
+ (h˜zX,˜wZi2+h−˜zZ,˜wXi2)~
1Y
+ (h−˜zX,˜wYi2+h˜zY,˜wXi2)~
1Z(4)
Based on (3) and (4), he Geome ic Algeb a CG
n-R3is de ined by he
ollowing geome ic p oduc
˜
z( )◦˜
w( ) = ˜
z( )·˜
w( ) + ˜
z( )∧˜
w( ) (5)
3.2. F equency Domain: Gene alized Euclidean Space
CGn-R3
Le Φ : CG
n→ CGn, Φ ¡k ej[α( )+θ]σa¢=k ejθσa, be he ope a o
ha enables he ans o ma ion be ween ime-domain and equency-
domain. We de ine CGn-R3as
Φ(CG
n)~
1X+ Φ(CG
n)~
1Y+ Φ(CG
n)~
1Z(6)
P og ess In Elec omagne ics Resea ch B, Vol. 15, 2009 407
whe e a gene ic elemen o his space is Φ(˜za) = ˜
Za. No e ha CGn-
R3can also be seen as
CGn-R3=n˜
ZX~
1X+˜
ZY~
1Y+˜
ZZ~
1Z:˜
Z∈ CGno
whe e ˜
Z=X
p
¯
Zpσp,¯
Zp∈ C and σp∈ Gn
Ob iously, CGn-R3is a CGn(complex-geome ic) ec o space and he
mul iplica ion ule o wo ec o s ˜
Z,˜
W∈ CGn-R3is gi en by
˜
Z◦˜
W=˜
Z·˜
W+˜
Z∧³−˜
W´(7)
whe e (7) is he es ic ion om CG
n-R3→ CGn-R3.
The nes ing o he geome ic Euclidean spaces deno ed by CG
n-R3,
CGn-R3, and CGna e g aphically illus a ed in Fig. 1.
Figu e 1. Nes ed geome ic Euclidean ec o spaces.
The undamen al concep s o Gene alized Complex Geome ic
Algeb a CGna e gi en in [18] and u he esea ch abou Geome ic
Algeb a can be ound in [21, 22].
4. DISTORTED PERIODIC ELECTRIC AND
MAGNETIC FIELDS: BASIC CONCEPTS
A pe iodic elec omagne ic ield is dis o ed i , simul aneously wi h he
undamen al ha monic o he ield, he highes ha monic componen s
a e p esen . In his way, i dis o ed ec o ield unc ions sa is y
Di ichle ’s condi ions, hen hey can be de eloped in o Fou ie se ies,
namely:
e( )=X
p
ep( ),d( )=X
p
dp( ),h( )=X
q
hq( ),b( )=X
q
bq( ) (8)
408 Cas illa e al.
whe e ep,dp,hq,bq, a e ha monics o he ield ec o s. Each ha monic
componen in he equency domain o a pe iodic elec omagne ic ield
sa is ies Maxwell’s equa ions
∇×Hp=Jp+jωpDp
∇×Ep=−jωpBp
∇·Dp=ρp
∇·Bp= 0
(9)
whe e Ep,Hp,Dp,Bpa e complex phaso s o he p- h ha monic o he
ields.
One o he mos impo an consequences o he i s wo
Maxwell equa ions is Poyn ing’s heo em, which desc ibes he low
o elec omagne ic ene gy in space and o a olume enclosed by
a su ace s. This can be s a ed as
ZZ −(e×h)nds =ZZZ e·jd +ZZZ µh·∂b
∂ +e·∂d
∂ ¶d (10)
whe e nis he uni ec o o hogonal o he in ini esimal su ace ds,e
and ha e he ins an aneous in ensi y o he elec ic and magne ic
ields, dand ba e he ins an aneous lux densi ies o hese ields
espec i ely, and jis he ins an aneous cu en densi y. The heo em
simply means ha he inc ease in s o ed ene gy in he ields plus he
ohmic losses wi hin a olume, equal he in low o a ec o e×hac oss
he su ace bounding ha olume. The ec o e×his known as he
Poyn ing Vec o (PV), and gi es he powe densi y a a poin on he
su ace in e ms o he elec ic and magne ic ields a ha poin . I s
physical meaning is also known [23].
The equi alen complex Poyn ing heo em o a sys em in linea
media is gi en by
−ZZ(Ep×Hp)nds =ZZZ EpJ∗
pd +jωpZZZ[BpH∗
p−EpD∗
p]d (11)
The ene ge ic in e p e a ion o (11) is as ollows
¯
Sp=Pp+jQp(12)
whe e
•¯
Spis a complex appa en powe o he p- h ha monic ecei ed by
he sys em enclosed in he su ace “s”.
•Ppis he ac i e powe o he p- h ha monic ecei ed by he sys em.
•Qpis he eac i e powe o he p- h ha monic ecei ed by he
sys em.
P og ess In Elec omagne ics Resea ch B, Vol. 15, 2009 409
One can eadily obse e ha
Pp=ZZZ EpJ∗
pd (13)
Qp= 2ωpZZZ ·BpH∗
p
2−EpD∗
p
2¸d (14)
whe e Pp ep esen s ha monic losses in Joules and Qpis associa ed o
he a e age alues o he p- h ha monic magne ic and elec ic ene gies
accumula ed in he olume [15]. Ano he ep esen a ion o (13)
and (14) is gi en
Pp= Re ·−I(Ep×H∗
p)¸nds (15)
Qp= Im ·−I(Ep×H∗
p)¸nds (16)
5. POWER FLOWS IN DISTORTED
ELECTROMAGNETIC FIELDS: GENERALIZED
POYNTING MULTIVECTOR ( ˜
S)
The ollowing no a ion is adop ed o de ine he elec ic and magne ic
ields in he CG
n-R3 amewo k:
˜
ep=|˜
ep|ej(ωp +θp)σp~
1X,Y,Z,˜
hq=|˜
hq|ej(ωq +γq)σq~
1X,Y,Z (17)
whe e ˜
ep( ) and ˜
hq( ) a e called ins an aneous elec ic and magne ic
complex-geome ic ields espec i ely.
Obse e ha he classic ins an aneous ields can be de i ed om
he eal (o imagina y) pa o he p ojec ions gi en by he scala
p oduc (2) as ollows:
ep( ) = Im(˜
ep·σp) = Im{|˜
ep|ej(ωp +θp)~
1X,Y,Z}(18)
hq( ) = Im(˜
hq·σq) = Im{|˜
hq|ej(ωq +γq)~
1X,Y,Z}(19)
In o de o ob ain he geome ic phaso s, i is necessa y o apply he
Φ ope a o on hese quan i ies (see Sec ion 3.2).
Φ(˜
ep) = |˜
ep|ejθpσp~
1X,Y,Z =˜
Ep(20)
Φ(˜
hq) = |˜
hq|ejγqσq~
1X,Y,Z =˜
Hq(21)
whe e ˜
Epand ˜
Hqa e called ha monic “spa ial geome ic phaso s”
o he elec ic and magne ic ha monic ields espec i ely and e i y
416 Cas illa e al.
By igno ing eddy cu en s, line impedance, inging e ec s, hen (32)
and (33) can be exp essed as
˜
E=1
lE
(200ej0σ1+ 100ej0σ2)~
1X(43)
˜
H∗=1
lH
(10ej30σ1+ 5e−j45σ2+ 10e−j60σ3)~
1Y(44)
Howe e , om (35) and (36), i ollows ha
ZZ
sX
p
~
1Z·˜
Pds = [(1732 + 353.5) + j(1000 −353.5)]σ0~
1Z
= (2085.5 + j646.5)σ0~
1Z(45)
The e o e, om (38),
ZZ
SX
p6=q
˜
1Z·˜
Dds=(−158.9−j1207.1)σ12 +(1000−j1732)σ13 +(500−j866)σ23 (46)
This example s a es ha Re{˜
Ω·
1}= 1732σ0, Re{˜
Ω·
2}= 353.5σ0,
Im{˜
Ω·
1}=j1000σ0, Im{˜
Ω·
2}=−j353.5σ0and ha he linea complex
bi ec o componen becomes ˜
Ω∧
12 = (−158.9−j1207.1)σ12, as well as
he nonlinea complex bi ec o componen s ˜
Ω∧
13 = (1000 −j1732)σ13,
˜
Ω∧
23 = (500 −j866)σ23 wi h hei co esponding di ec ions and senses.
On he o he hand, he ms alues o ol age and cu en a e gi en
by k˜
Uk2= 2002+ 1002= 5 ·104and k˜
Ik2= 102+ 52+ 102= 225
espec i ely. The alues o P=kRe{˜
Ω·}k2,kIm{˜
Ω·}k2,k˜
Ω∧k2a e
ound o add up o
k˜
Sk2=P2+kIm{˜
Ω·}k2+k˜
Ω∧k2= 11.25 ·106(47)
The e o e, appa en ol -ampe es k˜
Ska he e minals a e ound om
he ela ion k˜
Sk2=k˜
Uk2k˜
Ik2= 11.25·106. Finally, om (41) and (42)
we ob ain he ela i e quali y index and powe ac o espec i ely
˜
δ= 1 + j646.5σ0
2085.5σ0
+(−158.9−j1207.1) σ12+(1000−j1732) σ13+(500−j866) σ23
2085.5σ0
°
°
°˜
δ°
°
°= 1.608
PF =1
°
°
°˜
δ°
°
°
= 0.62 (48)

P og ess In Elec omagne ics Resea ch B, Vol. 15, 2009 417
The me hodology in he abo e example di e s g ea ly o ha o ci cui
heo y. Fu he mo e, unlike he ci cui heo y app oach, i can be
applied o sol e and unde s and he ope a ion o elec ic sys ems
designed o wo k in he equency domain.
8. CONCLUSION
The sugges ion ha he powe equa ion should be ounded on
elec omagne ic heo y is analyzed in his pape . This goal emains
unexplained by simple ma hema ical models used in classical heo y.
To his end, we p opose a Gene alized Poyn ing Mul i ec o (˜
S)
based on Cli o d Algeb as, which is decomposed in o a Poyn ing
Mul i ec o (˜
P) and a Complemen a y Poyn ing Mul i ec o (˜
D).
F om Equa ions (34)–(40), bo h quan i ies a e conside ed as he
keys one o he b idge be ween elec omagne ic heo y and ci cui
heo y. Thus, he eal pa o he low o Poyn ing Mul i ec o (˜
P)
coincide wi h ac i e powe , and imagina y pa o he complex scala
coincides wi h he powe con ibu ion due o like- equency p oduc s.
The Complemen a y Poyn ing Mul i ec o (˜
D) is associa ed o he
complex bi ec o o o he powe con ibu ion due o c oss- equency
p oduc s. This analysis demons a es ha he powe equa ion can
be ounded on he mul i ec o ial concep o he Gene alized Poyn ing
Mul i ec o (˜
S). Consequen ly, he appa en , ac i e, and non-ac i e
powe s can be exp essed and di e en ia ed in e ms o ˜
S. The
applica ion o he p oposed Gene alized Poyn ing Mul i ec o (˜
S) o
powe heo y should indica e impo an ad ances o any eal u u e
esea ch in his a ea.
ACKNOWLEDGMENT
We would like o hank he Minis y o Educa ion and Science o
suppo ing his wo k as pa o a esea ch h ough p ojec DPI-2006-
17467-CO2-01.
APPENDIX A. GENERALIZED COMPLEX
GEOMETRIC PRODUCT IN CGn
We de ine as C he complex- ec o space, and Gn, he Cli o d algeb a
on n-dimensional eal space Vn. We de ine he se
CG=
n

X
k=1,2...n
¯
Z1...kσ1...k


(A1)
418 Cas illa e al.
whe e he coe icien s ¯
Z1...k ∈ C and he basis σ1...k ∈ Gn. Ob iously
CGnis a ec o space o e R. Acco ding o (A1) de ini ion, in he
complex- ec o case, we ob ain he ec o subspace [CGn]1=
n
P
p=1
¯
Zpσp,
whe e ¯
Zp∈ C and σp∈ Gn. The gene ic elemen ¯
Zpσp, is a p-
h complex- ec o , and can be ep esen ed by he geome ic phaso
˜
Zp= (ap+jbp)σp. In he complex-bi ec o case, we ob ain he ec o
subspace [CGn]2=P
p6=q
¯
Zpqσpq. The gene ic elemen ¯
Zpqσpq, is a pq- h
complex-bi ec o , and can be ep esen ed by ˜
Zpq = (apq +jbpq)σpq.
In he mos gene al o m, complex-mul i ec o s, we ob ain he ec o
subspace [CGn]k=P¯
Z12...kσ12...k. The elemen ¯
Z12...kσ12...k, is he
12 . . . k- h complex-mul i ec o , and may be ep esen ed by ˜
Z12...k =
(a12...k +j b12...k)σ12...k. The e o e, CGn(A1), also can be ep esen ed
as
CGn=C
|{z}
complex
scala
⊕[CGn]1
|{z}
complex
ec o s
⊕[CGn]2
|{z}
complex
bi ec o s
⊕···⊕ [CGn]n
| {z }
complex
pseudoscala
The s uc u e {CGn,¯} is a complex geome ic algeb a since he
ollowing p ope ies a e ul illed: associa i e, dis ibu i e wi h espec
o he sum and con ac ion.
APPENDIX B. PARTICULAR CASE: GENERALIZED
COMPLEX GEOMETRIC PRODUCT FOR COMPLEX
VECTORS (GEOMETRIC PHASORS)
Le {σ1, . . . , σn}be a ec o basis o CGn. Fo wo ec o s ˜
Zp=
¯
Zpσp(p∈Ω) and ˜
Z0
q=¯
Z0
qσq(q∈Ψ) whe e Ω,Ψ⊆ {1,2, . . . , n}, and
whe e complex numbe s associa ed o each ec o a e
¯
Zp=Zpejαp
¯
Z0
q=Z0
qejβq=Z0
qej(αq−ϕq)(B1)
we de ine a new geome ic p oduc e med “gene alized complex
geome ic p oduc ”,¯:
¯:¡<αp,αq,⊗¢(B2)
The symbol “⊗” ep esen s he classic geome ic p oduc [21]
and <αp,αqis an applica ion in he complex planes associa ed o any
mul i ec o p oduc when αp6=αq, and is gi en by
<αp,αq¡¯
Z0
p,¯
Z0
q¢=½e−2j(αq−αp)i p > q, p, q ∈N
1 o he wise, p and/o q /∈N(B3)
P og ess In Elec omagne ics Resea ch B, Vol. 15, 2009 419
whe e N= Ω ∩Ψ.
This new p oduc o ec o s ˜
Zpand ˜
Z0
qis gi en by
¯
Zpσp¯¯
Z0
qσq=¯
Zp¯
Z0
qσpq (B4)
and he basis ansposi ion s a es
¡¯
Z0
q¯
Zpσqp¢= (−1)<αp,αq¯
Zp¯
Z0
qσpq (B5)
No e ha he ansposi ion ope a ion is in olu i e.
I αp=αq∀p,q∈N, hen
<αp,αp=IdC(B6)
and “¯”, (B2), will hen become he classic geome ic p oduc “⊗”. I
should be no ed ha when Cis es ic ed o eal numbe s, he classic
Cli o d Algeb a is ob ained.
In pa icula , o wo complex ec o s
˜
Z=X
p
Zpejαpσpand ˜
Z0=X
q
Z0
qej(−αq+ϕq)σq,
whe e he angles αpand (−αq+ϕq) iden i y he phase o he p- h
and q- h ha monics espec i ely, he gene alized complex geome ic
p oduc in linea ope a ion (p, q ∈N), can be w i en
˜
Z¯˜
Z0=X
p
ZpZ0
pejϕp+X
p<q
ej(αp−αq)ZpZ0
qejϕqσpq
+X
q<p
ej(αq−αp)ZqZ0
pejϕpσqp =X
p
ZpZ0
pejϕp
+X
p<qnej(αp−αq)ZpZ0
qejϕq−<αp,αqej(αq−αp)ZqZ0
pejϕpoσpq (B7)
whe e
<αp,αqej(αq−αp)ZqZ0
pejφpσq p =ej(αp−αq)ZqZ0
pejφpσqp
APPENDIX C. REVERSE AND CONJUGATED
OPERATIONS
We de ine he bi ec o e e se elemen as
¡¯
Zq pσq p¢†= (−1) ¯
Zpqσpq (C1)
whe e (†) is he “ e e se” ope a ion.
The “conjuga ed” ope a ion (∗) is gi en by
¡¯
Zpσp¢∗=¯
Z∗
pσp(C2)
420 Cas illa e al.
APPENDIX D. NORM DEFINITION
The no m, alue o magni ude, o a mul i ec o ˜
Zis he unique scala
°
°
°˜
Z°
°
°,Zcalcula ed by
°
°
°˜
Z°
°
°
2=h˜
Z(˜
Z†)∗i0(D1)
whe e we apply (∗) in C, and (†) in Gn.
APPENDIX E. TIME-DOMAIN FREQUENCY-DOMAIN
TRANSFORM: Γ-TRANSFORM
Le k:R→ CG
n, k( ) = Xkej(ωk +θk)σkbe a con inuous signal. The
Γ- ans o m o kis gi en by
Γ{ k( )}(ω) = 1
TZ
T
k( )e−jωk d =Xkejθkσk(E1)
whe e j2=−1.
Le ˜
:R→ CG
nbe a eal- alued mul i ec o unc ion. The e o e
˜
( ) = P
A∈P({1,...,n})∪0
A( ) wi h A( ) = kAej(ωA +θA)σA, whe e
P({1, . . . , n}) is he se o all he subse s o {1, . . . , n}.
Acco ding o he linea i y o he Γ- ans o m:
Γn˜
( )o(ω) = X
A∈P({1,...,n})∪0
Γ{ A( )}(E2)
and
Γn˜
( )o(ω) = X
A∈P({1,...,n})∪0
kAejθAσA=˜
F(ω) (E3)
whe e ˜
F(ω) is a geome ic phaso .
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