Space Phasor Theory and Control of Multiphase Machines

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Electr Eng (2014) 96:79–94DOI 10.1007/s00202-013-0278-6

ORIGINAL PAPER

Space phasor theory and control of multiphase machinesthrough their decoupling into equivalent three-phase machines

Luis Serrano-Iribarnegaray

Received: 12 December 2011 / Accepted: 2 August 2013 / Published online: 23 August 2013© Springer-Verlag Berlin Heidelberg 2013

Abstract This paper first introduces the space phasor the-

ory from a general mathematical viewpoint and, above all,from a physical one as well. The theory is then applied tothe particular case of constant airgap multiphase machines(MMs) fed by arbitrary voltage waveforms. It is mathemat-ically proven, and thereafter checked by simulations too,that these machines can be split into a set of equivalentthree-phase machines mechanically coupled but electricallyindependent. This electrical decoupling leads immediatelyto develop the very fast torque control schemes of the MMs.All of them boil down to a mere extension of those alreadyused in the homologous three-phase machines.

Keywords Multiphase machines · Torque control ·Space phasor theory · GAFTOC principle

1 Introduction

Theelectrical machine user is mainlyinterested in itsexternalquantities (voltages, currents, speed, torque). On the otherhand, understanding the machine behavior in depth needsthe analysis of its main internal quantities (current sheet,induction wave, etc.). In this respect, it should be said thatthe classical theory shows clear differences in its analysismethod, depending on which of these two quantity types isunder survey.

As for the external quantities, a number of precise tech-niques (geometric constructions, equivalent circuits, etc.)were early developed, although most of them were only validfor particular applications. However, as for the internal quan-

L. Serrano-Iribarnegaray (B)Universitat Politécnica de Valencia, Valencia, Spaine-mail: [email protected]

tities, what predominated were rather a graphical description

andanintuitivereasoning.Inaddition,therewasnoanalyticaltool which showed in a precise and easy way how the evolu-tion of the machine internal quantities results in changes of its external ones. Thus, the correlation between both quantitytypes became blurred, and with the increasing mathemati-cal formalism it was gradually set aside in many books onelectrical machines which focus, almost exclusively, on theirexternal quantities.

After Kapp’s proposal in [1], the theory of electricalmachines in steady statemakes use of time phasors1 to repre-sent sinusoidal waves in time. Representing the induction orthe m.m.f. spatial waves in steady state through phasors was

sometimes used, although mainly as a mere tool for graphicalexplanations. In spite of that, as early as the 1910s Dreyfusproposed to characterize through phasors the m.m.f. spatialwaves in transient state and to take them as essential toolsfor dynamic studies [3]. Yet, his method, restricted to m.m.f.space waves, could hardly compete with Park’s one [4,5].

Park introduced his complex quantities to simplify thetransient equations of three-phase machines. Thereafter,Kron indicated that Park’s equations could be interpretedas “a transformation of the coordinates axes from the mov-ing terminals to the stationary polar and interpolar space forthe purpose of eliminating the troublesome cos θ term”. ([5],

p. 355). He also indicated in passing that there was a sec-ond interpretation for the Park’s current complex quantity: itcould be considered as a “current linear density wave” ([5],p. 354].

The first interpretation is of operational type and general,for it can be applied to all of the variables in Park’s equations.Yet, this is not the case with the second one. There is indeed

1 Time phasors, after Steinmetz’s work [2], have been mathematicallycharacterized in A.C. circuits theory by means of complex quantities.

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80 Electr Eng (2014) 96:79–94

no difficulty in defining and visualizing a current sheet spacewave and to correlate it with Park’s current complex quan-tity. However, in his later highly abstract and mathematicalpublications, Kron does not seem to have been interested indefining or searching for similar correlations between other(if at all existing) machine space waves and Park’s voltageand flux linkage complex quantities. In any case, when he

generalized Park’s work, he decidedly insisted on the firstinterpretation. Thesefacts greatly influenced the GeneralizedMachine Theory (GMTh) development, in which the conceptof coordinates transformation (and, more in general, the con-cept of matrix transformations) played from the beginning anoutstanding role.

In the 1950s, the authors in [6] emphasized that when ana-lyzing the transients in A.C. machines, it was important toprovide a picture and a physical insight into the phenom-ena. To this end, the interpretation of Park’s current complexquantity as a symbolical representation of a transient spacewaveismuchmoresuitablethanthatgivenintheGMTh.This

interpretation is the one they promoted and widely spread,and leads immediately to analyze the transient state in termsof sinusoidal space waves. And since in that context the mainvariables are sinusoidal waves (although their amplitude andspeed are not constant), Dreyfuss’s old idea of represent-ing these space waves by means of “space vectors” emergesagain. It is worth mentioning that resorting to space vectors(thus, for three-phase machines without space harmonics) aninteresting proposal was made in [7] several decades laterto relate a machine transient condition to the propagationprocess in space of distributed magnetic fields.

The Space Vector Theory (SVTh) in [6] aroused some-

how as an alternative and reaction to the GMTh’s exacer-bated mathematical formalism, which was most pronouncedin some books, where the machine is viewed completely“from outside”, as a “black box” to which different matrixtransformations are applied. By contrast, one of the mainideas in [6] was to look at the machine “from the inside”, tostart its dynamic analysis on the base of its space waves, andto proceed in this direction as far as possible.

However, the SVTh achieved his goal only in a modestscope. Indeed, their authors, who always assume the hypoth-esis of no space harrmonics, analyzed first in all detail [6,pp. 61–67] from a physical perspective the dynamic m.m.f.

space wave produced by a three-phase winding. They char-acterized it in an original manner by a graphical tool whichthey called “current space vector”, i. The expression of i hasadirect correlation with the phase currents and coincides withthe current complex quantity that Park had introduced yearsagofollowing a differentpath. Later on, as they could notfindspace quantities related to the phase flux linkage and voltagetime quantities, theyintroduced the formulae for the so-calledvoltage (u) and flux linkage () space vectors by a meremathematical analogy to the i space vector [6, p. 75]. Obvi-

ously, due to this merely mathematical definition, u and were restricted to three-phase machines. However, although,as just said, there was no physical meaning for them in termsof space waves, the authors in [6] profusely used these u and space vectors (together with the i vector) in a graphicalmanner to explain the machine equations. This new graphi-cal viewpoint of approaching and illustrating the transients,

in contrast to the abstract matrix transformation perspective,provided a much better insight into the phenomena, even inthe sophisticated cases when machines are fed through elec-tronic converters. That is why space vectors became in addi-tion very useful for electronic control studies in three-phasemachines, and as early as the 1970s they were widely usedin Central Europe in numerous books on this subject [8–11].The important influence of the new space vector viewpointon the development of modern control methods for three-phase machines was also pointed out in [12]. It should beadded that the approach in [6] was mainly spread in Germanlanguage and remained practically unnoticed in the English

literature until the eighties of the past century. (See, e.g., theprefaces to books [13,14].)

Combining the approach in [6], restricted to three-phasemachines without space harmonics, with the work in [15],Stepina showed that it was also possible to characterizethrough space phasors (this correct term was introduced byhim) the m.m.f. wave produced by a multiphase windingincluding its space harmonics. This was an important con-tribution. Relying on it, he tried to develop a general the-ory encompassing all kinds of machines (arbitrary airgapstructure and arbitrary number of space harmonics). Unfor-tunately, his general formulae [16,17] are incorrect (the start-

ing point of his deductions is erroneous for machines with nosmall airgap, since in this case, contrary to his assumption,the airgap reluctance is different for differentm.m.f. harmon-ics, as already proven in [18], pp. 916 and 938 or in [19], p.102]. Moreover, the fundamental u and space phasors arefully unknown in his work [16,17]. Yet, partial aspects of this work are very interesting, especially for design studiesof single phase inductions motors (small airgap machines inwhich, moreover, phasors u and are not needed). His veryvaluable contribution in this field [20] should be underlined.

The approach actually underlying Park’s work, the so-called magnetic coupling circuit approach (MCCA), was

very much enhanced by [4,5] and became the approach over-whelmingly used all over the world in machine transientanalysis. Put it simply: the MCCA regards the machine as anetwork made of resistances and inductances, many of whichvary with the rotor position. Park applied in [4,5] the MCCAto three-phase machines (m = 3)takingonlyintoaccountthefundamental space wave. The method was extended first tomachines with m > 3 in the GMTh and later on to machineswith arbitrary number of space harmonics and phases (e.g.,[21]).

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Interesting investigations on converted fed multiphasemachines (MMs) have been known for several decades (e.g.,[22]). But nowadays MMs in conjunction with power elec-tronics are becoming more usual. This trend, very proba-bly, will increase in the future (see, e.g., the surveys [23,24]with more than 140 and 220 references, respectively). Yet,MMs cannot be analyzed by means of the SVTh in [6] for,

as already indicated, it is restricted to three-phase machineswithout space harmonics. Since, besides these two limita-tions, the u and space vectors, even in this simple caseof three-phase machines, are introduced in the same formalway as in the MCCA (i.e., lacking physical interpretation),it could be argued that the advantages and usefulness of try-ing to extend the new path opened by Kovacs and Racz [6]were clearly questionable. And that, in summary, as to theMMs, it was much better to simply let aside digressions onspace waves andto further develop thetheory on thepowerfulbase of the MCCA with the help of matrix transformations.In light of these facts, it can hardly surprise that the great

and expansive force of the SVTh in Central and East Europeduring the second half of the past century has gradually van-ished. Actually it matches quite better the facts to state that,even before the MCCA was applied to MMs, it already had“absorbed and digested” the SVTh to a great extent withinits mathematical formalism. In view of the SVTh limitationsand the undeniable power and success of the MCCA, need-less to say that in today’s publications on MMs, the old (andonly partially successful) efforts in favor of a space waveapproach have been fully abandoned worldwide. They areregarded as a waste of time and considered to be doomed tofailure in MMs analysis and control.

This paper holds just the opposite thesis. It states that fortransients analysis and especially for developing and under-standing the control schemes of MMs, it is highly advanta-geous to make use of the space waveapproach. Yet to find outand show the very high potential of this approach, a startingpoint and procedure quite different from those used in [6]areneeded. This results in the space phasor theory (SPhTh) aspresented in this paper.

The structure of this paper is as follows: the fundamentalidea underlying the SPhTh is presented in chapter 2. Chapter3 introduces the concepts of a dynamic polyphase system of sequence “g” and its associated dynamic time phasor. Chap-

ter 4 defines the concept of dynamic space phasor, introducesthe u and phasors and analyzes them in all detail. It showsthat each one of the different u and phasors existing inthe MM is related to one specific and independent group of MM space harmonics. This fact (valid for the i phasors too) isextensively discussed, since it constitutes the hard core of thelater MM decoupling into independent machines. Chapter 5analyzes in a similar manner, but now much more briefly,the different MM i space phasors. The conclusions and for-mulae in chapters 4 and 5, of general validity, are applied to

the particular case of constant airgap MMs fed by arbitraryvoltages (chapter 6). It is then proven that the MM can bedecomposed into a set of mechanically coupled machines,equal to the original one, but each of them fed exclusivelyby just one voltage space phasor, so that the problem of con-trolling the MM is tantamount to control a set of equivalentthree-phase machines mechanically coupled, but electrically

independent. This theoretical conclusion, also reinforced bysimulations, is then used (chapter 7) to deduce the very fasttorque control schemes of the MMs, which become a mereextension of those applied to three-phase machines. In thisregard, it is also shown that the GAFTOC principle is thetruly (although often unknown) driving idea behind all of theschemes. This fact, combined with the space wave approach,enables explaining in a rigorous, and at the same time veryphysical and didactic, manner how the whole MM controlsystem actually works.

2 Fundamental idea underlying the space phasor theory

Let it be very long (negligible end effects) salient polemachine with axial conductors at the stator surface. As

−→B = rot(

−→A ) (1)

the total flux linkages of an arbitrary stator single turn, “ab”(Fig. 1) of any stator phase is given by

ab =

S

−→B ·

−→dS =

S

rot(−→A ) ·

−→dS =

−→

A · −→dl

= (Aza − Azb) · l = l

y=a,b

±Az,y (2)

that is, ab is the sum of the vector potential values, Az, atthe positions of the two turn conductors, multiplied by themachine length, l. (Notice that A and B are constant in theaxial direction, z, since, as usual, the magnetic field distri-bution in the machine is considered to be bidimensional.)Therefore, the flux linkages of the whole phase are simplythe algebraic sum of the A values at its conductor positions.The sign ± in (2) corresponds to the direction attached toeach conductor (aa or aa in Fig. 1) when moving all along

the phase. The conductors are assumed to be of negligiblecross section. Notice that Eq. (2) always holds, no matter therotor shape or whether the magnetic circuit be saturated ornot. Thus, if the magnetic vector potential space wave at thestator surface is known, the phase flux linkages are obtained immediately in the most general case.

Analogously, if the space distribution (the space wave) of the stator electric potential difference in axial direction,ϕ z , isknown (potential difference between the two ends of an axialconductor), the voltage of any stator phase k is obtained by

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82 Electr Eng (2014) 96:79–94

ab

a

'ab

'b zb

A

za A

l

Fig. 1 Flux linking a stator coil of a salient pole machine

simply adding up the electric potential difference values atall of its conductor positions:

uk(t ) =

y=a,b,c,d...

± ϕz,y

(3)

Letus now reproduceone of the Kron’s basic ideas (see intro-duction to [25]): “The terminology and presentation of manyengineers actually assumes that electricity may be trans-ported across a network as if it were a package of merchan-

dise…”.Notoneofthewritersonthetheoryofsuchnetworksever stops for a minute to ask the key question: In what trulybasic respect does an electric network differ from the largevarietyofnon-electricnetworks?Evenalaymanfeelsinstinc-tively that transporting electric current across a networkrequires a different mechanism from transporting a pack-age of butter across the same network “An electric networkdiffers from all other types of non-electric network in that anelectric network is always surrounded by a dynamic electro-magnetic field of its own creation…” (underlined by Kron).

In keeping with this fundamental fact (which, surprisingly,many engineers tend to ignore completely), the SPhTh in

this paper considers that a rotating electrical machine canbe regarded as an electromechanical device that, when con-nected to an electric source, produces electromagnetic (field)waves with a restricted propagation capacity, namely, theyare forced to turn inside the airgap. These space waves, espe-cially the scalar electric potential difference and the vectormagnetic potential waves (and the linear current sheet wave)can be characterized very easily by space phasors (the u, and i phasors, respectively). These space phasors:

– enable to formulate the MM transient equations in a verysimple manner, giving at the same time a very powerful

and physical insight into the machine transient behavior.– make possible the fast decoupling of the MM into a set of

equivalent three-phase machines (chapter 6). This turnsout to be extremely useful for developing the MM torquecontrol schemes (chapter 7).

Thus,theSPhThinthispaper fully rejects space phasors tobe mere abstract mathematical entities (an idea often foundin papers on three-phase machines, let alone in MMs). Itsstarting point is just the opposite: space phasors are always

to be introduced representing well-defined machine internal

quantities (space waves). And what else these space quanti-ties could be, if not Maxwell’s field theory quantities? Fur-thermore, they must be the most significant and powerfulones. These requirements are fully met by A and ϕ whichare, by far, the most important quantities in electromagnetics[26,27].

According to the statement above, it can hardly surprisethat with the help of (just the space phasor assigned to A),all of the modern control methods for d.c. and three- phasemachines (e.g., field oriented or direct torque control) can bededuced in a simple and systematic way, proving that all of them are particular variants of a very simple and much moregeneral principle [28].

How to get an intuitive picture of A in simple cases isshown in a didactic manner, e.g., in [29]. Nevertheless, sincethe average electrical engineer is less familiar with flux link-ages than with voltage, this last quantity will be used in thenext chapter to introduce some previous concepts which are

essential in the SPhTh.

3 Dynamic polyphase systems and dynamic time

phasors of sequence “g”

Since all the essential features of the SPhTh can be illustratedwith systems having an odd number of phases, henceforththis number, m, will be assumed odd unless otherwise stated.Likewise, phase 1 axis will be assumed to be always placedon the real axis. Term γ stands for 2π/m.

Let it be a three-phase symmetrical winding with arbitraryvoltages, u1(t ), u2(t ), u3(t ), but without homopolar compo-nents. It must be possible to obtain these voltages by meansof the projection of a “rotating vector”2 (the voltage phasoruA) onto the phase axes, that is (e stands for “real part of”):

u1(t ) = e−→

u A

; u2(t ) = e

−→u Ae

− jγ

u3(t ) = e

−→u Ae

− j2γ

(4)

Indeed, since the sum of the phase voltages is zero, there areonly two independent equations in (4), which just determinemagnitude and angle of uA. After a very simple calculation,

it follows−→u A = u A(t )e

jε(t ) =2

3

u1(t ) + u2(t )e

jγ + u3(t )e j 2γ

(5)

Let it be now a five-phase winding with arbitrary voltages,but again without homopolar components. Since there are

2 The term “rotating vector” should be interpreted in this paper froma graphical viewpoint, as often done in the literature in similar cases.From a mathematical viewpoint, these “rotating vectors” (as well astime and space phasors) are complex time-varying quantities.

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four independent voltages, it seems at first sight that the twoindependent voltage phasors

−→u A = u A(t )e

jε A(t ); −→

u B = u B(t )e jε B(t ) (6)

would suffice to determine the voltage of any phase (sum of the two phasor projections onto the phase). Yet, this way allthe voltages would actually be derived from the projections

of only one effective phasor (sum of uA and uB). Therefore,and for the purpose of specifying the phase voltages, it isnecessary to impose the additional condition that uA anduB also differ from one another as to the way in which theirprojections over the winding phases take place. This could bedone in many ways. The most direct one is to image that thephase positions are exchanged in cyclic manner dependingon the phasor considered (e.g., for uA the phases are placedin their actual sequence, 1, 2, 3, 4, 5, whereas for uB they areassumed3 to be in the sequence 1, 3, 5, 2, 4). Mathematicallywe get:

u1(t ) = e−→

u Ae− j

0

+ e−→

u B e− j

0

u2(t ) = e

−→u Ae

− j·1γ

+ e

−→u B e

− j·1·2γ

u3(t ) = e

−→u Ae

− j ·2γ

+ e

−→u B e

− j·2·2γ

u4(t ) = e

−→u Ae

− j ·3γ

+ e

−→u B e

− j3·2γ

u5(t ) = e

−→u Ae

− j 4γ

+ e

−→u B e

− j4·2γ

(7)

Solving (7) with the additional condition of no homopolarvoltage components, a very simple calculation yields:

−→u A = u A(t )e jε A(t ) =2

5

5 x =1

u x (t )e j ( x −1)γ

−→u B = u B(t )e

jε B(t ) =2

5

5 x =1

u x (t )e j ( x −1)2γ (8)

If there is a homopolar component, u0(t ), in the phase volt-ages, this value must be simply added to the right of theexpressions in (7). Of course, u0(t ) can also be obtained bytheprojections of a third phasor. The condition as to thephasesequence in this case reads obviously that all the phase axescoincide. The expression for this homopolar voltage phasoris:

−→u0 = u 0(t ) =

1

m

m x =1

u x (t ) (9)

3 Two other cyclicchanges seem possible: instead of phase 3, we couldplace phase 4 or 5 behind phase 1. The first change leads to a phasoruB conjugate of the uB in (8) The second one simply corresponds to theinversephasesequence1,5,4,3,2,1.Sothereisactuallyonlyoneeffec-tive symmetrical and cyclic phases exchange. Analogous conclusionsare obtained irrespective of the phases number.

u0,uA and uB are called voltage phasors of sequence 0, 1and 2, respectively. It is important to keep in mind that, obvi-ously, only phasors of the same sequence can be vectoriallycombined and added up. In other words, through the phasorsof sequence 0, 1 and 2, the original five- voltage system hasbeen decomposed into threeindependent systems (originatedby three independent phasors of different sequence).

Analogously, to characterize the arbitrary voltages of aseven-phase winding, three independent voltage phasors,uA, uB and uC plus one homopolar phasor, u0 (given by 9,with m = 7) are needed. The phase sequence for the phasorprojections are (1, 2, 3, 4, 5, 6, 7), (1, 3, 5, 7, 2, 4, 6) and(1,4,7,3,6,2,5)for uA,uB and uC, respectively. Extending(7) to the projections of the three phasors uA,uB and uC,andsolving the system, it quickly follows by analogy to (8):

−→u A =

2

7

7 x =1

u x (t )e j ( x −1)γ −→u B =

2

7

7 x =1

u x (t )e j ( x −1)2γ

−→uC = 27

7 x =1

u x (t )e j ( x −1)3γ (10)

This process can be easily extended to symmetrical wind-ings with an arbitrary m . In that case (m − 1)/2 phasors of sequences “g” = 1, 2, . . . (m − 1)/2 plus a homopolar volt-age phasor given by (9) are needed. The expression for thegeneral phasor of sequence “g” is:

−→u

g =

2

m

m x =1

u x (t )e j ( x −1)gγ (11)

Next, the concept of a dynamic m-phase system (DmPhS) of sequence “g” will be introduced. By definition, the m quan-tities (m currents, m voltages, etc.) of a m-phase symmetricalwinding are said to constitute a DmPhS of sequence “g”, if they meet the following equations:

x 1(t ) = x (t ) cos

ε(t ) − g · 0 ·

2π

m

x 2(t ) = x (t ) cos

ε(t ) − g · 1 ·

2π

m

. . .

x m(t ) = x (t ) cosε(t ) − g · (m − 1) ·2π

m (12)

x (t ) and ε(t ) can be arbitrary time functions. For g = mq+ 1, mq + 2, mq + 3, etc., where q is any positive naturalnumber, the values in (12) are the same than for g = 1, 2, 3,etc., respectively. When g = mq or zero, the system is calledthe homopolar DmPhS.

A DmPhS of sequence “g” only has two independent vari-ables, x (t ) and ε(t ). Thus, it can be fully characterized bya new mathematical tool that will be called in this paper“dynamic time phasor of a DmPhS of sequence g”

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−→ X

g

= x (t )e jε(t )

g

(13)

Variables x (t ) and ε(t ) correspond with the instantaneousamplitude and position of the dynamic phasor in the com-plex plane. Notice that the quantity of any phase with itsaxis placed at the actual position “ y” is obtained by simplyprojecting the phasor onto the axis that would belong to this

phase after having performed with sequence “g” the cyclicexchange of phases referred above. Mathematically:

x y(t ) = e−→

X e− jg ( y−1)2πm

(14)

The sum of currents, voltages, etc., of several DmPhSs, S1,S2, etc., of thesame sequence, “g”, produces a DmPhS whichalso has the sequence “g”. The dynamic phasor of the resul-tant DmPhS equals the vectorial sum of the dynamic phasorsassociated to S1, S2, etc. (This is by total analogy to the clas-sical polyphase sinusoidal systems, which are a very simplefamily of DmPhSs, the phasors of which turn at constantspeed with constant amplitude.) Notice that although func-tions x (t ) and ε(t ) in (12) may be arbitrary and very differentfor the different systems S1, S2, etc., the statement above onthe vectorial sum of dynamic phasors always holds.

Sequences “g” and (m–g) are called complementarysequences. Any DmPhS of sequence “g” defined by its pha-sor (13) can be converted into a DmPhS of sequence (m–g)according to the formula (* stands for conjugate complex):−→

X

g

=

−→ X ∗

m−g

(15)

Indeed, according to (12), the quantity in the general phase“y” for the DmPhS, X , in (13), of sequence “g”, is

x y(t ) X g

= x (t ) cos

ε(t ) − g( y − 1) ·

2π

m

(16)

and the quantity in the same phase “y” for the DmPhS, X ∗,of sequence (m–g)

x y(t ) X ∗m−g

= x (t ) cos

−ε(t ) − (m − g) · ( y − 1) ·

2π

m

= x (t ) cos

−ε(t )+g( y −1) ·

2π

m

= x y(t )

X g

(17)

These formulae show that DmPhSs with equal or comple-

mentary sequences can be added up, and the result can becharacterized by just one dynamic time phasor.

Notice on the other hand that the “voltage phasors of sequence g” previously introduced (see text after 9) are sim-ply dynamic time phasors of sequence “g”.

In summary, in this section, it has been shown how todecompose a m-phase system of arbitrary time quantitiesinto its independent DmPhSs (e.g. Eq. 7 for m = 5) and howto calculate their corresponding dynamic time phasors (e.g.,Eqs. 8 and 9 for m = 5). The dynamic time phasors can be

regarded as a powerful extension of the Kapp’s steady-state

time phasors. Their amplitude and speed may vary follow-ing an arbitrary law. They incorporate the phase sequenceconcept and are very useful for dealing (in analytical andgraphical manner) with time quantities of arbitrary evolutionin multiphase windings. The suitable tools for dealing withmachine space quantities of arbitrary evolution are intro-

duced in the next section.

4 Space phasor concept. Phasors u and ψ and their

associated space waves. Correlating phase quantities

with airgap waves

Studies on electrical machines require operating with cer-tain quantities which are spatially distributed (current sheet,induction, etc.). Dynamic space phasors are very suitableto this task. By definition, a dynamic space phasor is an ori-ented segment in the complex plane that represents thespatial

sinusoidal distribution of an internal machine quantity. Thephasor always points to the positive maximum of the wave(in the case of bipolar waves) and its modulus is equal to thewave’s amplitude. Both the wave amplitude and speed mayvary in an arbitrary manner

Usually the internal quantity is not sinusoidal. Then a har-monic space phasor is assigned to each space harmonic of its Fourier expansion. To this end, a domain transformationis defined in such a manner that any angle, α, in the machinedomain becomes an angle ν α (ν = absolute harmonic order)in its corresponding phasorialdomain. Noticethat in this wayevery multipolar wave is characterized by only one space

phasor (the same coordinate in the phasorial domain cor-responds to all its positive crests in the machine; Fig. 2).Since all the harmonic space waves become bipolar waves intheir corresponding phasorial domains, this transformation(which boils down to transform the mechanical angles intoelectrical ones) turns out to be very useful, not only for themathematical treatment, but also for the physical and graph-ical interpretation of later equations.

Let it be a sinusoidal wave with hp pole pairs distributedover the stator cylindrical surface of a salient pole machine(for instance, an induction wave, a wave of electrical poten-

5ε= 5ε =5ε = ...1 2 3ε2 ε1

Fig. 2 Space wave of five pole pairs in the machine domain (left ) andits corresponding space phasor in the phasorial domain (right )

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Electr Eng (2014) 96:79–94 85

·hp X e jhpα −

hp X hpα

hpα −hp proj X over hpα

Fig. 3 Geometric interpretation of Eq. (20)

tial difference in axial direction, etc.). If x hp (t ) is the instan-taneous amplitude of the space wave and ε(t ) defines anyof its instantaneous positive crests in the machine, the waveexpression becomes:

x hp (α, t ) = x hp (t ) cos {hp [ε(t ) − α)]} (18)

On the other hand, this wave, as just stated, is fully character-ized in its phasorial domain by its space phasor, whose most

general expression is:−→ X hp = x hp (t )e

j hp ε(t ) (19)

From (19) and (18) it follows immediately

x hp (α, t ) = e

−→ X hp e

− j phα

(20)

that is, the quantity (e.g., induction, electric potential dif-ference in axial direction, etc.) at any point (or at any axialstraight line) of the stator surface specified by the coordi-nate α in the machine domain is given by the projection of the space phasor onto the straight line defined by h pα in the

phasorial domain (Fig. 3).Therefore, the voltage of a stator phase k due exclusivelyto one electric potential wave with 2hp poles is equal to thesum of the projections of the wave space phasor, u hp , ontothe Z k phase conductorstranslatedinto the phasorial domain.This sum becomes (see Appendix A):

u(t )phase k

wave hp = Z k e

−→uhp ξ hp ,k

∗

(21)

where complex winding factor, ξ hp,k , in (21) is a complexconstantwhosemodulusequalstheoneoftheclassicwindingfactor for the harmonic of relative order h. (More details inAppendix A).

In summary, (21) states that the k-phase instantaneousvoltage due exclusively to the wave with hp pole pairs is sim-

ply the projection of the wave phasor onto the corresponding

phasorial axis of phase k (multiplied by a constant).Let it be a 2p poles salient pole machine with a m-phase

symmetrical winding on the stator. Let us choose as absciseaxis the symmetry axis of first phase, and define ξ h,wndg asthe winding factor of relative order h of this first phase. Dueto winding symmetry, the phase in the generic position “k ”reproduces the configuration of phase 1 but with an angular

displacement in the machine 2π · (k − 1)/( p · m). Therefore,its complex winding factor in the phasorial domain becomes:

ξ hp

phasek =

ξ hp

phase 1 e

j h p2π(k −1)

p·m =ξ h,wnd ge j h (k −1) 2πm

(22)

By simply applying (21)and (22), it follows that the instanta-

neous phase voltages of a m-phase stator winding due exclu-sively to the axial electric potential difference space wavewith hp pole pairs characterized by its phasor (19) become:

u(t )phas1

wave hp = Z ξ h,wnd uhp (t ) cos[h pε (t )]

[u(t )fase 2]wave hp = Z ξ h,wnd uhp (t ) cos

h pε (t )−h · 1 ·

2π

m

. . .

[u(t )fase m]wave hp = Z ξ h,wnd uhp (t ) cos

h pε (t )−h(m −1)

2π

m

(23)

For instance, let it be a seven-phase (m = 7) winding,and assume the general case of space waves with arbitrary

changes in their amplitude and speed. According to (23),each of the waves with p, 8p, 15p, etc., pole pairs (that is,each of the harmonic waves of relative order h = 7q + 1 withq = 0, 1, 2, 3, etc.) produces a voltage DmPhS of sequence1. Likewise, each of the waves of order h = 7q + 2; h =7q +3, etc. produces a voltage DmPhS of sequence 2, 3, etc.,respectively.

Therefore, for a seven-phase stator winding, all of the elec-

tric potential difference waves in axial direction at the stator

surface can be classified into seven families. The instanta-neousamplitudesandpositionsofthedifferentwavesbelong-ing to the same family are, in the general case, quite different

fromoneanother,andsoareaswelltheircorrespondingspacephasors. However, all of them originate DmPhSs of the samesequence which can be added up to give a resultant DmPhSof this same sequence. Thus, the combined action of all of the waves of a family on the phase voltages can be character-

ized by just one equivalent or effective voltage space phasor.

Or alternatively, the resultant voltage DmPhS produced by

a whole family of waves can be imagined to be produced by

only one effective or equivalent space wave, the phasor of

which is just the effective phasor of the waves family.Moreover, the two voltage DmPhSs produced by the fam-

ilies with h = 7q +1 and h = 7q +6 are DmPhSs of comple-

mentary sequence. Therefore, they can be added up so that just one space phasor suffices to characterize the combinedaction of both families. The same statement applies to thefamilies with h = 7q +2 and h = 7q + 5 as well as to thefamilies with h = 7q + 3 and h = 7q + 4. Thus, the com-bined effect of all of the electrical potential difference waveson the voltages of the seven phases is fully characterized by

just four effective voltage space phasors.This reasoning can be extended to symmetrical wind-

ings with arbitrary phase number. For instance, to obtain the

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86 Electr Eng (2014) 96:79–94

voltages of a five-phase winding due to all the machine spacewaves, three effective voltage space phasors (one of whichbeing the effective homopolar space phasor) suffice in themost general case.

As just seen, the set of effective voltage space phasorsdetermines the instantaneous voltage of any winding phase

in a very simple way: by projecting the phasors onto the

phase axis and summing up the projections. The inverseproblem (determining the effective voltage space phasorsout of the instantaneous phase voltages) is quite simple, too.Notice that each one of these space phasors produces oneof the independent DmPhSs into which the voltage systemof the polyphase winding can be decomposed (simply com-pare Eqs. 23 and 12). Therefore, from a merely mathematicalviewpoint, obtaining the effective space phasors out of the

phase voltages boils down to get the dynamic time phasors

which define the polyphase voltage system. This problemhas already been solved in Sect. 3 (see especially its finalremarks). Anyway, it is useful to make an additional com-

ment on the physical meaning of the solution obtained thisway by means of an example.

Let it be again a salient pole machine with, for instance, afive- phase stator winding without homopolar components.In this case, two effective stator voltage space phasors, uAand uB must be computed. Their expressions are given in(8). Phasors uA and uB symbolize (at a given scale) twoeffective stator electrical potential difference space waves,with different pole pair numbers, p and 3 p, respectively.(More precisely, these two effective phasors characterize thetotal action on the phase voltages of two different sets of actual stator electric potential difference space waves. The

first phasor refers to the families with waves of relative orderh = 5q +1 and h = 5q + 4. The second one to the familieswith h = 5q+3 and h = 5q+2.) Amplitude and speed of both effective phasors vary in due manner to give the phasevoltages.

In summary, for an m-phase stator symmetrical winding,the numerous and actual stator electric potential differencespace waves which appearat thestator cylindricalsurface canbe classified, in the most general case, into m families. For modd,(m−1)/2of these families have a complementaryfamily.Yet, the combined action on the stator phase voltage of allthe waves of two complementary families can be accounted

for by just one effective stator voltage space phasor. Thus,(m − 1)/2 effective independent space phasors (plus onehomopolar phasor) suffice to characterize the action of all of

the stator electric potential waves. These phasors fully definethe instantaneous phase voltages and, conversely, accordingto (11), they are determined out of the stator phase voltagesas follows:

−−−−−−−−−→ufamiles g,(m−g) =

2

m

m x =1

u x (t )e j ( x −1)g 2πm (24)

The statement above as well as the corresponding equation(24) hold, no matter the rotor shape or whether the magneticcircuit be saturated or not.

Analogous statements apply to the magnetic vector poten-tial waves over the cylindricalstator surface. Again, the com-bined action on the stator phase flux linkages of all of themagnetic potential space waves of two complementary fam-

iliescanbeaccountedforbyjustoneeffectivestatormagneticvector potential space phasor, families. Its expression can beobtained out of the flux linkages of the stator phases, no mat-ter the rotor shape or whether the machine be saturated ornot. Operating as done with the electric potential waves, itfollows, by analogy with (24),

−−−−−−−−−→families,g,m−g =

2

m

m x =1

x (t )e j ( x −1)g 2πm (25)

It is convenient to insist on the physical content of (23). Itstates that, given a m-phase winding, any electric (or vector

magnetic) potential space wave with hp pole pairs always produces in the winding a voltage (or flux linkage) DmPhS of

a sequence “g”, no matter the changes in the wave amplitude

and speed. Sequence “g” only depends on the wave polesnumber. Waves with different poles number but belonging tothe same family produce DmPhSs of the same sequence.

It is worth mentioning that even in the simple case of three-phase machines, and even in good books (e.g. [30], p.286 or [31], p. 21]), it has been explicitly stated that pha-sors u and lack a physical interpretation. Yet assigning aphysical meaning to the u phasor was already done in [32],where the u phasordefinition was carried outfrom a field per-

spective (axial electric potential difference) and a circuit per-spective (average value of conductors voltages). The seconddefinition was later used to also introduce the phasor and,unfortunately, it was the one almost exclusively used by thisauthor in his later publications on the SPhTh (e.g., [33,34]).The procedure and definitions in those publications are initself legitimate and correct, but they are useful for transientsanalysis only if the space harmonics are neglected, other-wise, the expressions for u and become too complex, asexplicitly acknowledged in [33,34]. Therefore, those spacephasor definitions and their corresponding developments inpreviousauthor’spublications which do not coincide with the

onesgivenhere or in[35] are to be replaced in due manner. Inother words, the “field perspective” in [32] is the truly suit-able one and the one to be chosen for introducing the spacephasors definitions and formulae, as done and explained indetail in this paper.

Formulae (24)and(25) coincide with the so-called instan-taneous symmetrical components (ISC) of voltages and fluxlinkages. Although these ISC have been profusely used allover the world for more thanhalfa century, it has beenimpos-sible so far to attach a clear physical interpretation to them.

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Electr Eng (2014) 96:79–94 87

They keep on being introduced directly as a mere (and fortu-nate) transformation of variables that simplify the solution of the system equations. Yet, according to previous paragraphs,they have a deep and double physical meaning: effective volt-age and flux linkage dynamic space phasors associated withthe different (and independent) space waves families (firstinterpretation: Machine viewpoint). Dynamic (and decou-

pled) time phasors which fully describe (by simple projec-tion) the time evolution, in the most general case, of phasevoltagesand flux linkages of a m-phase symmetrical winding(second interpretation: Circuits viewpoint. See Eq. 11).

5 Current space phasors and their associated space

waves

Any phase A placed in the cylindrical stator ( D = innerdiameter) of an electrical machine produces a current sheetspace wave that can be split into space harmonics. If i A(t )

is the phase current, the space phasor, a, that characterizesthe harmonic of relative order h of the current sheet waveproduced by the phase A is [35]

−−−→ah· p, A =

2 Z Aπ D

i A(t )ξ h· p, A (26)

The current sheet space phasor of a polyphase winding is thesum of the space phasors of all of its phases. Thus, the spaceharmonic of relative order h of the current sheet wave pro-duced by a m-phase symmetric winding fed by arbitrary cur-rents can be represented by its space phasor, whose expres-sion, deduced immediately from (26) and (22), is:

−−−−→ahp ,wnd =

phases

−→ahp =

2 Z

π Dξ hp,wnd

mk =1

ik (t )e j h(k −1) 2πm

(27)

Instead of the current sheet space phasor it is often advan-tageous to use the so-called current space phasor, definedas:

−−−−→ihp,wnd g =

ξ hp ,wnd gξ hp ,wnd g2

m

mk =1

ik (t )e j h(k −1) 2πm (28)

Notice that both phasors only differ by a constant and there-

fore they represent (at different scales) the same space quan-tity. For h = 1 (fundamental wave) and m = 3, the currentspace phasor in (28) results in the Park’s current vector.

As readily expected, like the u phasor (electric potential)and the phasor (magnetic vector potential), also the i pha-sor has a direct relationship with one of the Maxwell’s fieldtheory quantities. The current sheet (i phasor) at any pointof the stator surface is equal to the tangential componentof the magnetic field intensity, Ht, at this point, assumingthe classical hypothesis of µFe to be infinite (e.g., [30], p.

342) Furthermore, the vectorial product of the space phasorsof voltage (related with the axial electric field) and current(related with the tangential magnetic field) is directly relatedwith the well- known Poynting vector, S = E × H. Start-ing from S the different electrical machines and their mainformulae can also be deduced, as done in a systematic andbeautiful manner in [36]. This allows a deep physical insight

into the machine behavior from a perspective quite differentfrom the usual one.

In view of the relationships between DmPhSs and wavesfamilies for phasors u and , it is only logical to expect thata current DmPhS applied to a symmetrical m-phase windingwill produce current sheet waves belonging to specific fami-lies. And this is just what happens: if the sequence of the cur-rent DmPhS is “g”, all the waves are of order h = qm ± g(see Appendix B). For instance, a current DmPhS of g = 2applied to a seven-phase winding only produces space wavesof relative order h = 2, 5, 9, 12, etc.

Finally, notice that, like Eqs. (24), (25), (28) always holds

too, no matter the rotor shape or whether the magnetic circuitbe saturated or not.

6 Decoupling a constant airgap multiphase machine

into independent machines. Theoretical proof and

validation through simulations

This section refers to the following constant airgap MMs:inductionmachines(IM),doubly fed asynchronous machines(DFAM) and permanent magnet synchronous machines(PMSM) with a stator winding constituted by m phases sym-

metrically distributed.The hard core of the problem of establishing the machine

equations actually lies on developing the relationshipsbetween current sheet andflux linkage space phasors (inotherwords, between currents and fluxes).

Contrary to Eqs. (24), (25) and (27), of general validity,the relationship between current sheet and magnetic vectorpotential space waves (or flux linkage waves,) changes forevery machine type and, for some machines, it turns out to bevery complex, even if the phase leakage flux is considered (aswill be done here too) proportionalto thephasecurrent. How-ever, the solution is simple for those machines in which the

two classic hypotheses of ideal magnetic circuit and constantairgapof negligible width canbe assumed. In such a case, anysinusoidal current sheet wave results in just one flux linkage

wave (or in just one airgap induction wave) with the same

pole number (there are no saturation or reluctance inductionwaves). The relationship between these current sheet andinduction waves is very simple: their amplitudes are propor-tional and the induction wave always lies 90◦ apart from itsexciting current sheet wave in the phasorial domain. Thesespace waves relationships are very simple to write down by

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88 Electr Eng (2014) 96:79–94

means of space phasors, no matter the machine harmonicsand phases number, as has been shown in detail in [35].

Yet, it is not at all necessary to develop the machine equa-tions to prove in a precise and rigorous way that the MM canbe decoupled into a set of independent machines. This will

be done in the next paragraphs.Let it be an ideal IM with an arbitrary phase number, m ,

in stator and rotor, and let us apply to the stator an arbi-trary current DmPhS, Si, of sequence g. Assume first thatthe rotor circuit is open. Si produces a current sheet wavesfamily with hest = qm ± g (see Appendix B), which in turn,as above commented, produces a homologous family (thesame h values) of stator and of airgap vector potential spacewaves. This last waves family generates a stator commonflux linkages DmPhS of sequence g. Therefore, the electro-motive forces (EMFs) induced in the stator phases constituteat any moment a DmPhS of sequence g too. And the m-phasewinding voltages system, S V (sumoftheEMFsandthephaseimpedance voltage drops systems) also constitutes a DmPhS

of sequence g. In summary, a current DmPhS of sequenceg can only produce flux linkages and voltages DmPhSs of

the same sequence. (Of course, if instead Si the input is thevoltage system S V, the output will be then the current systemSi).

Assume the rotor winding be closed now (It should beunderlined that in the more simple case of a PMSM, this sec-ond step of the analysis is unnecessary). At the first moment,the airgap flux linkage space waves family (due to the statorcurrents) induces in the rotor an EMF DmPhS of sequence g,which in turn, as there are no external rotor voltages,4 pro-duces a rotor current DmPhS of sequence g too. This happens

no matter the changes in the amplitude and speed of withrespect to the rotor (see comments after Eq. 25). The rotorcurrent DmPhS produces (see again Appendix B) a family of rotor current sheet waves with hrot = qm ± g. (Notice thatthe general rotor family term h rot coincides with the one of the stator family, h str.) Each pair of stator and rotor currentsheet waves of the same order (or their phasors expressed ina common reference frame) combines to give at any momenta machine resultant wave of the same order, so that we get amachine resultant waves family with hmach = qm ± g. Thisresultant current sheet waves family originates a homolo-gous airgap waves family which induces an EMF DmPhS

of sequence g in stator and another in the rotor. This mod-ifies the stator and rotor currents in the next moment, but,obviously, they keep on being two DmPhSs of sequence g.Thus, they produce two homologous families of current sheetwaves with h = qm ± g, which combine together, and theprocess repeats again. In summary, in constant airgap sym-

4 Of course, the reasoning in the text also applies to a doubly fedasyncronous machine provided that the external rotor voltages systembe an arbitary DmPhS, S V,rot of the same sequence, g.

metrical MMs with equal stator and rotor phase number,

two arbitrary stator and rotor voltage DmPhSs of the same

sequence, g, produce at any moment stator and rotor current

DmPhSs of sequence g

Ontheotherhand,itiswellknownthatonlytheinteractionof stator and rotor space waves with the same pole numbercan produce torque. This means that only stator and rotor

voltage DmPhSs of the same sequence can produce torque(hest = hrot only in families associated to DmPhSs of thesame sequence).

Therefore, each one of the two arbitrary stator and rotorvoltage systems of the MM analyzed above has to beexpanded into (m + 1)/2-independent DmPhSs (m odd), oneof which being the homopolar system (g = 0). The MM canbe regarded as a set of (m + 1)/2 mechanically coupled but

electrically independent machines with the same windings.Stator and rotor of machines 0, 1, 2, etc., are supplied by the(stator and rotor) voltage DmPhSs of sequence g = 0, 1, 2,,etc., respectively. The MM torque (or current) equals the sum

of the torques (or currents) produced by all of the indepen-dent machines. Notice that this statement applies to thoseIMs or DFAMs in which mstr = m rot, as well as to the muchmore simple case of the PMSM.

The above conclusions have been confirmed by numeroussimulations carried out with the space phasor model devel-oped in detail in [35]. The model applies to machines withany stator and rotor phase number, fed by arbitrary voltagewaveforms and taking into account the space harmonics.The author would like to emphasize that very much atten-tion was paid to a reliable model validation by comparingin [35] torque simulations with direct torque measurements,

that is, with precise measurements of mechanical (not elec-trical) quantities performed in a squirrel cage motor. In thissense, the great difficulties of measuring with accuracy pul-sating electromagnetic torques of several hundreds of hertzwere brought into light,the possible measurement techniqueswere critically reviewed and the solution chosen was exten-sively discussed. As far asthe author knows, it is the first timein theliterature that transient torques of hundreds of hertz dueto winding space harmonics have been directly measured andcompared with model simulations.

Figures 4, 5, 6, and 7 show, by way of example, the sim-ulation results of a direct on line no load starting-up of an

induction motor, “a”, with four poles and seven phases in sta-tor and rotor. The simulations were performed applying themodel in [35] first to motor “a” fed with the stator voltagesystem S1 defined as

uax (t ) = 300 cos [ωt − ( x − 1)2π/7]

+200cos[3ωt − 3( x − 1)2π/7]

+100cos[5ωt − 5( x − 1)2π/7]

= u bx (t ) + ucx (t ) + udx (t ) (29)

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Electr Eng (2014) 96:79–94 89

0 0.05 0.1 0.15 0.2 0.25-400

-200

0

200

400

600

T o r q u e ( N m )

Time (s)

Fig. 4 Machine “a” no load starting-up torque. Main machinedata Rstr = 0.41; Lstr = 100mH; Lσ str = 2.5 mH; Rrot =0.28 m; L rot = 14 µH; Lσ rot = 1.12µH; J = 0.03Kgm

2; hmax =

25. Full pitch stator and rotor coils placed in 56 and 28 slots

0 0.05 0.1 0.15 0.2 0.25-400

-200

0

200

400

Time (s)

T o r q u e ( N

m )

Fig. 5 Torques of machines “b” (blue), “c” (green) and “d” (red ).Remaining data as in Fig. 4 (color figure online)

where u ax (t ) stands for voltage at phase “x” ( x = 1–7) of

machine “a”. Thereafter, the model was applied to threemachines (b–d) mechanically coupled, all of them equalto machine “a”, and fed with the three-voltage DmPhSsubx (t ), ucx (t ), and ud x (t ) obtained from S1 in (29).

The actual data in Figs. 4 and 5 confirm that Ta is exactlyequal to Tb + Tc + Td at any moment. Analogous statementholds for the currents in Figs. 6 and 7. These results wereobserved in all simulations, no matter voltage input, loadvalue, space harmonics number or machine parameters, aspredicted by the theory.

It should be added that the authors in [37] have dealt withthe particular case of PMSM following a completely differ-

ent approach. Resorting to several analysis tools (endomor-phism, orthonormal base of eigenvectors, etc.), they arrive atdecoupling laws for PMSM fully equivalent to the results inthis paper.

If m str and m rot are not equal in the IM or in the DFAM,then the exact decoupling of the MMin the most general caseis not possible, as also confirmed by simulations (physicalexplanation: the field harmonics belonging to one only statorwaves family have their rotor counterparts distributed amongseveral rotor families; and conversely). Yet, even in these

0 0.05 0.1 0.15 0.2 0.25-150

-100

-50

0

50

100

150

S t a t o

r p h a s e 1 c u r r e n t ( A )

Time (s)

Fig. 6 Machine “a” stator phase 1 current during no load starting-up.Remaining data as in Fig. 4

0 0.05 0.1 0.15 0.2 0.25-60

-40

-20

0

20

40

60

80

Time (s)

S t a

t o r p h a s e 1

c u

r r e n t s ( A )

Fig. 7 Stator phase 1 currents in machines “b” (blue), “c” (green) and“d” (red ). Remaining data as in Fig. 4 (color figure online)

machines, if they are fed through a converter, the decouplingabove referred to keep on being valid with small or negligibleerror, as shown in the next section.

7 Fast torque control of multiphase machines

The general strategy (GAFTOC principle) to get a very fastmachine torque control is based on keeping the pulsationalEMFs of all of its phases null during transients and achievingthe torque changes by exclusively enhancing the rotationalEMFs [28]. Three-phase stator or rotor windings withouthomopolar currents produce only one set of flux linkagesspace waves of general order h = 3q ± 1 (that is, only onestr and one rot. phasors). Therefore, keeping the pulsa-

tional EMFs null in all stator or rotor phases is achieved ina three-phase machine by simply keeping constant the mag-nitude of its str or rot phasor (See Fig. 2 in [28]). Fieldorientation control (FOC) and direct torque control are sim-ply two particular methods of implementing the GAFTOCprinciple.

It has also been shown in the previous section thata constant airgap MM with m stator and rotor phasescan be split up into a machine with homopolar volt-ages (always to be avoided for well-known reasons) plus

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90 Electr Eng (2014) 96:79–94

rot

phasors

calculation

Ψ

1 1,a b

*

sAi

*

sBi

*

sE i

+

−

*

1sd i1 1,d q

1ϕ

Ψ

*

.rot

const Ψ =

*T

3 3,a b

3 3,d q

1 1

3 3

,

,

a b

a b

, ,

, ,

A B

C D E

*

1sqi

*

3sd i

*

3sqi

3ϕ

Ψ

rot references

distribution

Ψ −

Torque

references

distribution

*

1 .rot const Ψ =

*

3 .rot const Ψ =

*

1T

*

3T

. . I M

Fig. 8 Rotor FOC of a five-phase induction machine. Asterisks stand for reference value

(m −1)/2-independent machines. Each of these independent machines hasonly onestr andonerot , space phasor which

fully define the flux linkages of any phase, just as happensin three-phase machines without homopolar components.Thus, the MM (more concretely, PMSM, IM and DFAM)has been decoupled into a set of equivalent and electricallyindependent three-phase machines which are mechanicallycoupled.

Moreover, in converter-fed machines, it is not necessaryto take into account all the members of each waves family.

Indeed, it hasbeen theoretically justified as well as confirmedby numerous simulations in [35] that the field harmonicseffect on the transient torque of three-phase induction motorsis always negligible at low slips. Obviously this conclusioncan be extended to the different and independent waves fam-ilies of converter-fed MMs (small slips, even in transients).This means that in these MMs only the space harmonicswhich are head members of their groups (that is, the har-monics with the lowest pole number) need to be consideredas to the torque, since they alone are able to provide a non-negligible torque contribution [35].

Certainly, if mstr = mrot in the IM or the DFAM, the

exact decoupling of the MM is not possible. However, as tothe torque control, this fact is scarcely relevant, since only thehead members of the families should be considered. In fact,although often ignored, this simplified assumption under-lies all the control schemes in the literature on converter-fedthree-phase induction machines, where only the fundamentalstator and rotor waves are considered as to the torque pro-duction. (Notice that the three-phase stator winding produces

just one family of, e.g., waves. However, the multiphasesquirrel cage produces several families. Yet, only the head

members of the stator family and of the rotor main family aretaken into account in the torque equation.) In other words, if only the head members of each wave family are to be con-sidered (converter-fed MMs) then the effective decouplingof the MM still holds, no matter m str and m rot be equal ornot.

Thus, establishing the scheme for a very fast torque con-trol of the MMs of type IM, DFAM and PMSM boils down toapplying the GAFTOC principle to each one of the indepen-dent machines into which the MM can be split up. In other

words, any of the schemes in [28] for three-phase machinescan be extended directly to their homologous MMs. By wayof example, Fig. 8 (a mere extension of Fig. 3 in [28]) showsthe rotor FOC of a five-phase induction motor.

The scheme in Fig. 8 is associated with two fictitious inde-pendent machines, each of them with only one rot pha-sor. Stator current and voltages of both machines are known(ISC of the actual five-phase machine), as well as their elec-tric parameters and rotor positions (equal to the ones of theactual machine). On the other hand, obtaining in a three-phase machine its rot phasor (actually only its angle ϕwith respect to the stator is needed) from rotor position, sta-

tor currents and machine parameters is a classical problemsolved a long time ago. In this sense, the block in Fig. 8 “rotphasors calculation”representsthe setof operations to gettherot phasor of each one of the two fictitious machines (itscounterpart in Fig. 3 of [28] must calculate only one rotphasor, of course). The block “Torque references distribu-tion” simply specifies the constant percentages of the totaltorque which must be provided by the fundamental and thethird space harmonics of the actual machine (these harmon-ics are the head members of the two sets of space waves).

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Electr Eng (2014) 96:79–94 91

An analogous statement holds for the block “Flux linkagesreferences distribution”.

It is useful to make some additional comments on how thesystem in Fig. 8 actually works. The machine space wavesare characterized by space phasors in cartesian or polar coor-dinates (e.g., i phasors represent current sheet waves).Themain waves, by far, are the waves, which are forced by the

control structure to keep their amplitudes constant (this way,only rotational EMFs are induced in the windings: GAFTOC

principle).This task is particularly simple if operating withthe phasors in reference frames tied to the waves. Natu-rally, the changes in the space waves result in changes in theirassociated phase time waves (phase quantities). The valuesof these phase quantities u(t ), (t ), etc. are obtained by sim-ply projecting (in their phasorial domains) the set of u, ,etc. space phasors onto the phase axis and summing up theprojections, as proven in detail in chapter 4. This is just whatis done by the operations in block (a1 b1 a3 b3) → (A B CD E) which are presented in the literature as abstract matrix

transformations. Actually, they simply represent the way inwhich the machine internal quantities are correlated with itsexternal ones. The drawback in the MCCA of lacking toolsto perform this correlation in a precise and simple way wasunderlined at the beginning of the introduction section. Thiscorrelation is not a problem in the SPhTh (sum of phasorsprojections) so that one can go from space waves to phasetime quantities and conversely at any time.

Finally, notice that, as already said, Fig. 8 (an extensionof the first control scheme in [28]) is just one example. Allthe other three-phase control schemes in [28] for PMSM, IM and DFAM can be extended in a similar manner to MMs.

8 Conclusions

Kapp’s time phasors refer to sinusoidal time waves and areonly valid for steady states. For transient states analysis,Dreyfus already proposed in [3] to use m.m.f. space waves.Yet, his method could hardly compete with Park’s one.

The later SVTh by Kovacs and Racz [6], profusely usedin Central Europe in the second half of the past century, canbe regarded as a second and more vigorous attempt to placethe machine space waves at the central point of the study.

Unfortunately, the SVTh in [6] is restricted to three-phasemachines without field harmonics and has no physical inter-pretation for the essential u and phasors.

This paper is fully pervaded by the following driving idea:for analyzing transients in electrical machines and develop-ing their control schemes, it is highly advantageous to con-nect again with the old “wave approach”; that is, to connect with the way of thinking and the mental stream of analy-

sis that, flowing through the SVTh, goes back to Dreyfuss’swork (and even to Kapp’s phasor concept); a mental stream

(developed specially in Central Europe) that, as to transientsstudies, has been almost completely blocked in the last 30years by the undeniable power and success of the MCCA.Yet, to overcome this block, a starting point and procedureclearly different from the ones in [6] are necessary, namelyone has to go resolutely and directly to the physical roots of the machine behavior. And, as in any electric system, these

roots, ultimately, always are Maxwell’s field theory quanti-ties (see the, in author’s opinion, excellent book [26]). Toput it concretely: by contrast to the MCCA (machine as anetwork), the SPhTh in this paper states that a machine canalso be regarded as an electromechanical device which pro-duces electromagnetic waves inside its airgap. These spacewaves, especially the fundamental u, and i waves, can becharacterized very easily by space phasors which are to betaken as essential tools of the theory. Moreover, they also canbe correlated easily with the phase time quantities.

With regard to this second point, since a space wave, nomatter the changes in its speed and amplitude, always pro-

duces a DmPhS of a fixed sequence, it follows that alsoDmPhSs shouldbecome key elements of the SPhTh. To oper-ate with them a new tool called “dynamic time phasor of aDmPhS of sequence g” has been introduced in this paper. It can be regarded as a powerful extension of Kapp’s phasor .Dynamic time and space phasors, just as Kapp’s phasors, canbe dealt with mathematically (complex time-varying quan-tities) and graphically (vectorial addition), and also enabledrawing dynamic time and space diagrams. This way theSPhTh extends to transients problems a method of analysisand graphical representation very familiar to any electricalengineer. In this sense, space phasors are much more intu-

itive than abstract matrix transformations. And, conceptu-ally, much more suitable. For instance, the statements oftenfound that time quantities such as currents, fluxlinkages, etc.,are subjected to transformations of the coordinates axes canhardly be accepted from a correct physical viewpoint (onlyspace quantities can be subjected to space transformations).It is dubious that such statements can lead to a clear insightinto the machine dynamics and control. Yet, in view of theincreasing use of converter-fed MMs, accurate physical andgraphical descriptions of these processes become a must.

Each one of the different u and dynamic space phasorsexisting in the MM is related to one specific and indepen-

dent group of space harmonics. Likewise, a current DmPhSof sequence g (characterized by its dynamic time phasor)produces only current sheet space waves of relative orderh = qm ± g. These facts have been extensively discussedin the paper, since they constitute the hard core of the MM decoupling.

Thereafter, making use of these two analysis tools (dyna-mic space phasor in the hp-phasorial domain; dynamic time

phasor of a a DmPhS of sequence “g”), it is theoreticallyproven that constant airgap MMs (either PMSM or IM and

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92 Electr Eng (2014) 96:79–94

DFAM with mstr = m rot) can always be decoupled (m odd)into one machine with homopolar components plus a set of (m −1)/2-independent and equivalent three-phase machineswithout homopolar components. (This MM decoupling hadbeen previously proven in the literature only for the particu-lar case of PMSM.) The theoretical conclusions above havebeen also confirmed by numerous simulations with themodel

in [35]. (Notice that whereas the u and formulae and phys-ical interpretation for the general case are given in this paper,the whole set of machine equations for the particular caseof constant airgap machines has been deduced in detail in[35].) For converter-fed MMs the decoupling holds too, withsmall or negligible error, even if m str = mrot. This enablesestablishing directly the schemes for a very fast torque con-trol of the MMs, since they boil down to a mere extension of those already known for their homologous three-phase coun-terparts. Moreover, it is shown that the GAFTOC principle isthe truly driving idea behind all of these schemes. This fact,combined with the space wave approach, allows explaining

in a rigorous, and at the same time very physical and didactic,manner how the whole MM control system actually works.

Finally, this paper gives for the first time the interpretationof the u and phasors from a space wave perspective. Italso gives the double and deep physical meaning of the uand instantaneous symmetrical components in polyphasemachines and windings.

9 Appendix

9.1 A. Deduction of Eq. (21)

LetitbeaphaseAwithZaA conductors placed at (the straightline definedby)αa,ZbA conductors placed atαb, etc. Accord-ing to Eq. (20), the sum of the projections of phasor uhp ontoall of the series-connected conductors of phase A translatedinto the hp-phasorial domain becomes:

u(t )phas A

wav hp = e

−→uhp

± Z a Ae

− j phαa

± Z b Ae− j phαb ± · · ·

= Z AReal

−→uhpξ hp , A

∗

where ξ hp, A = (1/ Z ) ·

y=a,b,c... ± Z y e+ j hpα y is a complex

constant (the complex winding factor of relative order h of phase A). More details are in [35].

9.2 B. Space waves produced by a current DmPhS

Let it be a current DmPhS of sequence g as defined in (12).The current sheet space waves produced by this system areobtained by replacing in (28) the values of (12). Making useof Euler’s formula (2cosα = e jα + e− jα) it follows, after

simple mathematical manipulations:

−→ih =

1

m

ξ hξ h i(t )e jε(t )

y=m y=1

e j ( y−1)(h−g)2πm

+e− jε(t ) y=m

y=1

e j ( y−1)(h+g)2πm

Inside the bracket in the above equation, there is a first sum of m unitary vectors which are symmetrically distributed withan angle (h − g)∗2π/m between two consecutive vectors,and another analogous sum, but with an angle which is equalto (h + g)∗2π/m. Each one of these sums is different fromzero only if the angle between twoconsecutive vectors is zeroor a 2π multiple. Therefore, a current DmPhS of sequence gcan only produce current sheet space waves of relative orderh = qm ± g

9.3 C. Comparison between SPhTh and MCCA

Magnetic coupling circuit approach and space phasor theoryare two very different approaches (electric network with vari-able inductances; electromagnetic device with airgap fieldwaves) which nevertheless describe the same process. Yet,when two approaches with the same simplifying hypothesesare used for analyzing a process, they produce two equationsets which must be either equal or formally equivalent. TheMCCA has been successfully used all over the world forthe last 50 years. Thus, it would be sheer madness to presentthe SPhTh in this paper as a new method that, relying on thesame hypotheses as the ones of the MCCA, brings new and

improved equations which no one could have been able toobtain previously or for which there are no equivalent onesin the MCCA. In other words, in the end, the equations of both theories must be mathematically equivalent. But thisdoes not mean that both theories are equally suitable. In theauthor’s opinion, the following aspects are to be consideredin favor of the SPhTh:

Use of analytical tools with a clear physical content and

meaning

Space phasors are much more intuitive than matrix trans-formations. Theycharacterize the dynamic space distributionof a well-defined internal or space quantity.

Relating space waves to machine phase quantities

Space phasors show in a precise and easy way how theevolution of the machine internal quantitiesresults in changesof its external ones. (Projections of the space phasors into thephase axis in their corresponding phasorial domains.)

Simplicity of the model and of the methodological proce-

dure

The MCCA regards the machine as a system of magneti-cally coupled coils. A chain of abstract mathematical trans-formations (almost always lacking any underlying physical

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Electr Eng (2014) 96:79–94 93

explanation) is applied to decouple the variables and to sim-plify the machine structure. In the SPTh no abstract phasereduction or transformation matrices are required, and thedynamic equations are formulated in an easy way [35].

Excellent suitability for electronic control studies

First: Space phasors provide a very good physical insightinto the machine behavior. This way, they enable to “untan-

gle” (with little mathematics) the machine structure, and todecouple the MM (and to understand its underlying physicalbasis!) into a set of equivalent and independent three-phasemachines (machines with one only space phasor in statorand rotor).

Second: Electromechanical energy conversion is con-cerned with motional emfs. Thus, pulsational emfs shouldbe considered a waste of resources (increasing in machineand converter sizes) and time (slower dynamic answer) andare always to be avoided. This simple idea constitutes theGAFTOC principle [28], from which all the other controlmethods can be deduced. Yet, implementing the GAFTOC

becomes very simple in the SPhTh: it boils down to keep con-stant the magnitude of every space phasor and to changeonly its rotational speed.

Graphical representation of transients

A sinusoidal space wave can always be graphically rep-resented by a space phasor, no matter the changes in itsamplitude and speed. Dynamic diagrams provide en excel-lent insight into transient processes.

Torque formula very simple and intuitive

The torque formula is quite simple and has a direct andvery physical interpretation: tendency to the alignment of two magnets.

Removing the abrupt gap between steady and transient

state studies

Space phasors make use of equations and diagrams for-mally analogous to the ones in steady state. Thus, they enableextending to transient problems a mathematical and graphi-calanalysis technique very familiar to anyelectrical engineerfrom his steady-state studies.

Physical interpretation of the instantaneous symmetrical

components

See the end of chapter 4.

9.4 D. Torque expression with space phasors

For the analytical description of the machine with space har-monics, the MCCA [21] and the SPhTh assume the hypothe-sis of µFe = ∞. In linear systems, the torque can be obtainedfrom the stored magnetic energy through the general expres-sion (e.g., [38], chapter 2)

T =∂Wmag∂λ

i1,i2...=cte

(30)

In the case of IMs and DFAMs with three-phase windings onstatorand rotor (PMSM is a more simplecase), thetorque dueto the interaction of the main waves produced by stator androtor (fundamental machine torque) can also be expressed bymeans of space phasors as

T (t ) =9

4 p L1,mut.

−−−→i1 p,rote

j pλ(t ) × −−−→i1 p,str (31)

That is, the torque is proportional (in a formal language) tothe vectorial product of the two fundamental current spacephasors of stator and rotor. Thus, the torque can be under-stood as the tendency to alignment of two magnets or of theirassociated current sheet waves. (For deduction of (31), see,e.g., [14]). Only space waves with the same pole pair num-ber, hp, produce torque. Therefore, in the general case of aconstant airgap MM with space harmonics, applying exactlythe same procedure as in [14] to each pair (same hp) of sta-tor and rotor harmonic current space phasors, given by Eq.(28), and summing up all of the harmonic torques, if follows

(λ: rotor-stator angle. L h,mut = stator–rotor phases maximalmutual inductance for the induction wave or order h. Moredetails are in [35]):

T (t ) =mestmrot

4

hmaxh=1

hpLh,mut.

−−−→ihp,rote

j hpλ(t ) × −−−→ihp,est

(32)

Equation (32) coincides with (30) for constant airgap MM.

References

1. Kapp G (1887) Induction Coils graphically treated. TheElectrician18:502–504 (524–525, 568–571)

2. SteinmetzChP (1893)Complexquantitiesand theiruse in electricalengineering. In: International electrical congress, Chicago, pp 33–74

3. Dreyfus L (1916) Ausgleichvorgänge beim plötzlichen Kurz-schluss von Synchrongeneratoren. Archiv für Elektrotech 5(H.4):103–140

4. Park RH (1929) Two-reaction theory of synchronous machines.AIEE Trans 48:716–730

5. Park RH (1933) Two-reaction theory of synchronous machines.AIEE Trans 52:352–355

6. Kovacs KP, Racz I (1959) Transiente Vorgänge in Wech-selstrommaschinen. Akademiai Kiado, Budapest

7. Holz J (1996) On the spatial propagation of transient magneticfields in AC machines. IEEE Trans Ind Appl 32(4):927–9378. Pfaff G (1971) Regelung Elektrischer Antriebe I. Oldenbourg,

Wien9. Leonhard W (1974) Regelung in der Elektrischen Antriebstechnik.

Teubner, Stuttgart10. Bühler H (1977) Einführung in die Theorie geregelter

Drehstromantriebe, vol I, II. Birkhaüser, Stuttgart11. Kleinrath H (1980) Stromrichtergespeiste Drehfeldma schinen.

Springer, Wien12. Leonhard W (1991) 30 years space vectors, 20 years field orienta-

tion, 10 years digital signal processing with AC-drives. Eur PowerElectron Drives J 1(1):13–20

1 3


16/16

94 Electr Eng (2014) 96:79–94

13. KovacsKP (1984)Transient phenomena in electric machines. Else-vier, Amsterdam

14. Leonhard W (1985) Control of electrial drives. Springer, Berlin15. Lax F, Jordan H (1940) Über die Fourier-Entwicklung der Felder-

regerkurve von schrittverkürzen Drehstromwicklungen beliebigerPhasenzahl. Archiv für Elektrotechnik 34(H. 10):591–597

16. Stepina J (1968) Fundamental equations of the space vector analy-sis of electric machines. Acta Tech CSAV 12:184–189

17. Stepina J (1979) Non-transformational matrixanalysis of electricalmachinery. Electric Mach Electron Mech, 255–268

18. Doherty RE, Nickle CA (1926) Synchronous machines. I. Anextension of Blondel’s two-reaction theory. II Steady state power-angle chraracteristics. Trans AIEE, 912–947

19. Müller G (1990) Elektrische Maschinen. Betriebsverhalten Elek-trischer Maschinen. VEB Verlag Technik, Berlin

20. Stepina J (1982) Die Einphasenasynchronmotoren. Springer, Wien21. Fudeh HR, Ong CM (1983) Modeling and analysis of induction

machines containing space harmonicspartsI, II andIII. IEEE TransPower Appar Syst PAS-102(8):2608–2615 (2616–2620; 2621–2628)

22. Bienek K (1983) Untersuchung der Asynchronmaschine mit dreiund sechs Wicklungssträngen am stromeinprägenden Wechsel-richter. Phd Thesis, Darmstadt

23. Levi E, Bojoi R, Profumo F, Tolyat HA, Williamson S (2007) Mul-tiphase induction motor drives. A technology status review. IETElectric Power Appl 1(4)

24. Levi E (2008) Multiphase electric machines for variable speedapplications. IEEE Trans Ind Electron 55:1893–1909

25. Kron G (1964) Tensor analysis of networks. Macdonald, London26. Simonyi K (1977) Theoretische Elektrotechnik. VEB Deutscher

Verlag der Wissenschaften, Berlin27. Feynmann RP, Leighton RB, Sands M (1964) The Feynmann lec-

turesonphysics,vol2.Addison-WesleyPub.Comp,Massachussets(See chapter 15.5)

28. Serrano-Iribarnegaray L, Martínez-Román J (2007) A unifiedapproach to the very fast torque control methods for dc and ac.machines. IEEE Trans Ind Electron 54:2047–2056

29. Rogowski W (1914) Wie kann man sich vom Wirbel eines Vek-torfeldes und von Vektorpotentiale eine Anschauung verschaffen?Archiv für Elektrotechnik 2(H. 6):234–245

30. Ivanov-Smolensky A (1983) Machines Electriques. MIR, Moscú31. Seefried E, Müller G (1992) Frequenzgesteuerte Dreh-

stromasynchronantriebe. Verlag Technik, München32. Serrano-Iribarnegaray L (1991) Significado físico del fasor espa-

cial U.Ecuaciones en régimen dinámico de las máquinas eléctricasrotativas. 2nd Portuguese-Spanish Electrical Engineering. Meet-ing. Coimbra, Plenary Session. vol I, pp 51–72

33. Serrano-Iribarnegaray L (1993) The modern space phasor theory.Parts I. Eur Trans Electr Power Eng 3(2):171–180

34. Serrano-Iribarnegaray L (1993) The modern space phasor theory.Parts II. Eur Trans Electr Power Eng 3(3):213–219

35. Echeverría-Villar JA, Martínez-Román J, Serrano-Iribarnegaray L(2012) Transient harmonic torques in induction machines: mea-surement and impact on motor performance. Electr Eng 94(2):67–80

36. Palit B (1980) Einheitliche Untersuchung der elektrischen Maschi-nen mit Hilfe des Poynting-Vektors und des elektromagnetischenEnegieflusses im Luftspaltraum. Zeitschrift für angewandte Math-ematik und Physik 31:384–412

37. Semail E, Bouscayrol A, Hautier JP (2003) Vectorial formalismfor analysis and design of polyphase synchronous machines. EurPhys J Appl Phys 22:207–220

38. Seely S (1962) Electromechanical energy conversion. Mc. GrawHill, New York

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Space Phasor Theory and Control of Multiphase Machines