An Approach to the Multivectorial Apparent Power in Terms of a Generalized Poynting Multivector

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 . REFERENCES 1. Maxwell, J. C., “A dinamical heo y o he elec omagne ic ield,” Phil. T ans. o he Royal Socie y, Vol. 155, 459–512, London, 1865. 2. S einme z, C. 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