Even-phase multi-motor vector controlled drive withsingle inverter supply and series connection of statorwindings

E. Levi, M. Jones and S.N. Vukosavic

Abstract: Vector control principles enable independent flux and torque control of an AC machineby means of only two stator d-q axis current components. This means that in AC machines with aphase number greater than three there exist additional degrees of freedom, which are nowadaysused to enhance the overall torque production of a multi-phase machine through injection of higherstator current harmonics. However, these additional degrees of freedom can be used to controlother machines independently within a multi-motor drive system. To do so, it is necessary toconnect in series stator windings of all the multi-phase machines, with an appropriate phasetransposition. A vector control algorithm is then applied to each machine separately, total inverterphase current references are created by summation of individual machine phase current references,and supply to the stator windings of the multi-machine set is provided from a single currentcontrolled voltage source inverter (VSI). The concept is introduced in the paper using the generaltheory of electrical machines, and all the possible situations that may arise for an even number ofphases are examined. Although an induction motor drive is considered throughout, the concept isequally applicable to all AC machines with sinusoidal stator and rotor flux distribution. Its mainadvantage is the potential for saving in the number of inverter legs, compared to an equivalent 3-phase motor drive system. The saving depends on the number of phases and, for an even phasenumber, comes into existence when the number of phases is X8. Various even phase numbers areconsidered in detail and appropriate connection diagrams are given. Verification of the proposedmulti-phase multi-motor drive is provided by simulation of a 10-phase 4-motor system.

1 Introduction

Industrial electrical drive applications often require anumber of variable speed drives. Examples include textileand paper manufacturing, robotics, traction, electricvehicles, etc. In the vast majority of cases these multi-motordrive systems require independent control of individualmotors. The standard solution in such situations is to use aset of 3-phase motors, which are supplied from their ownPWM VSI and all the inverters are connected to thecommon DC link. Either vector control or direct torquecontrol can be used for independent control of the motorswithin a group to obtain a high performance [1]. If a multi-motor system consists of k 3-phase machines, this approachrequires 3k inverter legs. Numerous solutions have thereforebeen proposed for reducing the total number of inverter legsrequired in a multi-machine system [2–4]. The two solutionsanalysed in detail in [4] require only 2k and 2k+1 inverterlegs, respectively. However, both configurations lead to asubstantial increase in the total harmonic distortion and toa reduced voltage capability. The authors conclude that theconfiguration with 2k+1 legs offers a better performancethan the configuration with 2k legs. Both are, however,inferior with respect to the standard solution with 3k legs.

In certain applications, such as traction, it is possible touse a multi-motor system that consists of k 3-phase motorsconnected in parallel and supplied from a single PWM VSI

[5–7]. Vector control is used in this case and it is aprerequisite that all the motors develop an identicalelectromagnetic torque [7]. Owing to parallel connectionand supply from a single inverter, the whole multi-machinesystem always has to operate with the same supply voltageand frequency, so that the means for independent control ofindividual motors within the multi-machine system do notexist. Very much the same applies to another parallelconnection of k 3-phase motors, supplied this time from asingle PWM current source inverter [8]. Since V/f control isused in [8], the motors are always subjected to the samesupply voltage and frequency conditions.

Supply of variable speed electric drives from invertersenables substitution of the standard 3-phase configurationwith an appropriate multi-phase (n-phase) configuration.Ever since the inception of the first multi-phase (5-phase)inverter-fed induction motor drive in 1969 [9], the interest inmulti-phase drive systems has been steadily increasing, asevidenced by recent survey papers [10, 11]. Majoradvantages of using a multi-phase machine instead of a 3-phase machine are detailed in [12] and can be summarisedas follows: higher torque density, greater efficiency, reducedtorque pulsation, greater fault tolerance, reduction in therequired rating per inverter leg and therefore simpler, morereliable power conditioning equipment. Additionally, noisecharacteristics of the drive improve as well [13]. Highertorque density in a multi-phase machine is possible since,apart from the fundamental spatial field harmonic, spaceharmonic fields can be used to contribute to the total torqueproduction [12, 14–17]. This advantage stems from the factthat vector control of the machine’s flux and torque,produced by the interaction of the fundamental fieldcomponent and the fundamental stator current component,requires only two stator currents (d-q current components).

The authors are with the School of Engineering, Liverpool John MooresUniversity, Liverpool L3 3AF, UK

r IEE, 2003

IEE Proceedings online no. 20030424

doi:10.1049/ip-epa:20030424

Paper first received 23rd October 2002 and in revised form 19th February 2003.Originally published online: 23rd May 2003

580 IEE Proc.-Electr. Power Appl., Vol. 150, No. 5, September 2003

In a multi-phase machine, with at least five phases, there aretherefore additional degrees of freedom, which can beutilised to enhance the torque production through injectionof higher order stator current harmonics. The concept isequally applicable to any AC machine type and has beensuccessfully demonstrated for induction [15, 17] andsynchronous reluctance [14, 16] machines. In a 5-phasemachine the third harmonic current injection can be used[14, 15, 17], while in a nine-phase machine it is possible touse injection of the third, the fifth and the seventh currentharmonic [16]. This is so since injection of any specificcurrent harmonic again requires two current components,similar to the torque/flux production due to fundamentalharmonic. In general, the torque density increases as thenumber of phases increases, up to 15 phases, but no furtherincrease is achieved beyond this [12].

As already noted, vector control of a multi-phasemachine requires only two currents if only the fundamentalof the field is utilised. Although the remaining degrees offreedom can be used to enhance the overall motor torqueproduction, they can also be used in an entirely differentmanner, as the basis for a multi-motor multi-phase drivesystem development. This paper attempts to develop thisidea in a systematic manner. An n–phase AC machine isconsidered, such that the number of phases is an evennumber. It is shown that, by connecting the multi-phasestator windings of the k machines in series, with anappropriate phase transposition, it becomes possible torealise a completely independent vector control of themachines even though only one inverter is used. There areno restrictions on the type of AC machine used within thedrive system, on machines’ power ratings, loading andoperating speed. The initial idea behind this concept wasindicated for the first time in [18], where the notion of an n-dimensional space for an n-phase machine [19] was appliedin the analysis and only a 2-motor, 5-phase system wasstudied. The concept has never been examined for an evennumber of phases and is developed here using the generaltheory of electric machines [20]. The necessary phasetransposition in the series winding connection is establishedby analysing the properties of the decoupling transforma-tion matrix. A so-called connectivity matrix is introducedand connection diagrams are given for some characteristicphase numbers. Classification of all the even phase numbersis further provided, with regard to the maximum number ofconnectable machines and the number of phases ofindividual machines in the group. The concept is verifiedby simulation of a 10-phase 4-motor drive system in thetorque mode of operation. Its main advantages andshortcomings are assessed in the concluding section of thepaper. A corresponding study for an arbitrary odd numberof phases has also been made and the results are to bereported in the near future.

2 Modelling of an n-phase AC machine

2.1 General remarksAn n-phase machine, such that the spatial displacementbetween any two consecutive stator phases equals a¼ 2p/n, is under consideration. Although the type of ACmachine is irrelevant, it is assumed that the machine is aninduction motor. Both stator and rotor windings aretreated as n–phase, for the sake of generality. It isassumed that the spatial distribution of the stator androtor flux is sinusoidal, since the intention is to controltorque production due to fundamental harmonic only.All the inductances within the stator and the rotor n-phase winding are therefore constants and the mutualinductances between the stator and rotor phases containonly the fundamental harmonic. Rotor winding is takenas referred to the stator winding. All the other standardassumptions of the general theory of electrical machinesapply. The phase number can be odd or even. In thispaper only even numbers of phases are considered.

It is important to note that only the true n-phasemachines are discussed here. This means that the mth phaseand the (n/2+m)th phase are positioned 180 degrees apartand are supplied with currents in phase opposition. Eachsuch an n-phase winding can be reconfigured into so-calledsemi-n-phase winding [10], with an effective number ofphases equal to n/2 (the customary 3-phase machine is inessence a semi-6-phase machine, [10]). While this reconfi-guration halves the number of phases and would thereforehalve the number of inverter legs, it would simultaneouslyhalve the number of connectable machines as well.Furthermore, if the original even phase number n is suchthat n/2 is a prime number, semi-n-phase winding becomesa winding with an odd number of phases (these phasenumbers are, as already noted, beyond the scope of thispaper). If the original phase number n is a power of two,reconfiguration into a semi-n-phase winding requires aneutral conductor for operation with the balanced system ofcurrents [10] (a semi-4-phase machine is customarily knownas a 2-phase machine; since the currents are 90 degreesapart, the 2-phase system requires three wires). It is for thesereasons that the analysis in the paper is restricted to true n-phase systems (n is even), with spatial displacement betweenany two consecutive windings equal to a¼ 2p/n.

2.2 Decoupling transformationThe machine model in phase variable form is transformedusing a decoupling (Clark’s) transformation matrix [20]. De-coupling transformation substitutes the original set of nphase currents with a new set of n transformed currents. Ade-coupling transformation matrix for an arbitrary evenphase number can be given in power invariant form with[20]

C ¼ffiffiffi2

n

r

abx1

y1

x2

y2

� � �xn�4

2

yn�42

0þ0�

1 cos a cos 2a cos 3a :::: cos 3a cos 2a cos a0 sin a sin 2a sin 3a ::::: � sin 3a � sin 2a � sin a1 cos 2a cos 4a cos 6a ::::: cos 6a cos 4a cos 2a0 sin 2a sin 4a sin 6a ::::: � sin 6a � sin 4a � sin 2a1 cos 3a cos 6a cos 9a ::::: cos 9a cos 6a cos 3a0 sin 3a sin 6a sin 9a ::::: � sin 9a � sin 6a � sin 3a::::: ::::: ::::: ::::: ::::: ::::: ::::: :::::

1 cosn� 2

2

� �a cos 2

n� 2

2

� �a cos 3

n� 2

2

� �a ::::: cos 3

n� 2

2

� �a cos 2

n� 2

2

� �a cos

n� 2

2

� �a

0 sinn� 2

2

� �a sin 2

n� 2

2

� �a sin 3

n� 2

2

� �a ::::: � sin 3

n� 2

2

� �a � sin 2

n� 2

2

� �a � sin

n� 2

2

� �a

1=ffiffiffi2

p1=

ffiffiffi2

p1=

ffiffiffi2

p1=

ffiffiffi2

p::::: 1=

ffiffiffi2

p1=

ffiffiffi2

p1=

ffiffiffi2

p

1=ffiffiffi2

p�1=

ffiffiffi2

p1=

ffiffiffi2

p�1=

ffiffiffi2

p::::: �1=

ffiffiffi2

p1=

ffiffiffi2

p�1=

ffiffiffi2

p

2666666666666666666664

3777777777777777777775

ð1Þ

IEE Proc.-Electr. Power Appl., Vol. 150, No. 5, September 2003 581

The first two rows in (1) define stator current componentsthat will lead to fundamental flux and torque production(a�b components; stator to rotor coupling appears only inthe equations for a-b components), while the last two rowsdefine the two zero sequence components. In between, thereare (n–4)/2 pairs of rows which define (n–4)/2 pairs of statorcurrent components, termed further on x-y components. Aswill be shown shortly, x-y pairs of current components willplay an important role in realising independent control ofthe machines connected in series.

Equations for pairs of x-y components are completelydecoupled from all the other components, and stator torotor coupling does not appear either [20]. These compo-nents do not contribute to torque production whensinusoidal distribution of the flux around the airgap isassumed. Zero sequence components will not exist in anystar-connected multi-phase system without neutral conduc-tor. This means that at most (n–2)/2 machines (for an evenphase number) can be connected in series. To do so it isnecessary to ensure that the flux/torque producing currentsof one machine do not produce flux and torque in all theother machines of the group. A simple series connection ofstator windings is obviously not going to achieve this goal.An appropriate phase transposition is therefore requiredwhen connecting the windings in series, as discussed shortly.

2.3 Rotational transformationSince stator to rotor coupling appears only in the equationsfor a-b components regardless of the machine type, andtorque production due to the fundamental field componentis therefore entirely governed by a-b components, rotationaltransformation is applied to a-b equations only [20].Assuming transformation into an arbitrary commonreference frame (ys¼

Roadt), the transformation matrices

for stator and rotor variables are:

Ds ¼

cos ys sin ys� sin ys cos ys

1

:::::

1

26666664

37777775

Dr ¼

cos b sin b

� sinb cos b

1

:::::

1

26666664

37777775

ð2Þ

where b¼ ys–y and y is the instantaneous rotor angularposition.

Upon application of the decoupling transformation (1)and rotational transformation (2), an n–phase inductionmachine is described with the following voltage equilibriumand flux linkage equations (p�d/dt):

vds ¼ Rsids � oacqs þ pcds vdr ¼ 0 ¼ Rridr � ðoa � oÞcqr þ pcdrvqs ¼ Rsiqs þ oacds þ pcqs vqr ¼ 0 ¼ Rriqr þ ðoa � oÞcdr þ pcqrvx1s ¼ Rsix1s þ pcx1s vx1r ¼ 0 ¼ Rrix1r þ pcx1rvy1s ¼ Rsiy1s þ pcy1s vy1r ¼ 0 ¼ Rriy1r þ pcy1rvx2s ¼ Rsix2s þ pcx2s vx2r ¼ 0 ¼ Rrix2r þ pcx2rvy2s ¼ Rsiy2s þ pcy2s vy2r ¼ 0 ¼ Rriy2r þ pcy2r

voþs ¼ Rsioþs þ pcoþs voþr ¼ 0 ¼ Rrioþr þ pcoþrvo�s ¼ Rsio�s þ pco�s vo�r ¼ 0 ¼ Rrio�r þ pco�r

ð3Þ

cds ¼ ðLls þ LmÞids þ Lmidr cdr ¼ ðLlr þ LmÞidr þ Lmidscqs ¼ ðLls þ LmÞiqs þ Lmiqr cqr ¼ ðLlr þ LmÞiqr þ Lmiqscx1s ¼ Llsix1s cx1r ¼ Llrix1rcy1s ¼ Llsiy1s cy1r ¼ Llsiy1r

cx2s ¼ Llsix2s cx2r ¼ Llrix2rcy2s ¼ Llsiy2s cy2r ¼ Llriy2r

coþs ¼ Llsioþs coþr ¼ Llrioþrco�s ¼ Llsio�s co�r ¼ Llrio�r

ð4Þwhere Lm¼ (n/2)M and M is the maximum value of thestator to rotor mutual inductances in the phase variablemodel. Torque equation is given with

Te ¼ PLm½idriqs � idsiqr ð5Þ

3 Series connection of multi-phase statorwindings

Since only one pair of stator a-b (d-q) current components isrequired for the flux and torque control in one machine,there is a possibility of using the remaining degrees offreedom ([(n–2)/2–1] pairs of stator x-y current components)for control of other machines that are to be connected inseries with the first machine. However, if the control of themachines with series connected stator windings is to bedecoupled one from the other, it is necessary that the flux/torque producing currents of one machine do not produceflux and torque in all the other machines in the group. Inother words, the connection of stator windings of k¼(n�2)/2 multi-phase machines must be such that what onemachine sees as the d-q axis stator current components theother machines see as x-y current components and viceversa. It then becomes possible to completely independentlycontrol speed (position, torque) of (n�2)/2 machines whilesupplying the drive system from a single current-controlledvoltage source inverter. Achievement of the stated goalrequires an appropriate phase transposition when connect-ing the stator windings in series. Phase transposition meansshift in connection of the phases 1, 2, y, n of the firstmachine to the phases 1, 2, y, n of the second machine,etc., where 1, 2, 3yn is the flux/torque producing phasesequence of the given machine according to the spatialdistribution of the phases within the stator winding. Therequired phase transposition follows directly from thedecoupling transformation matrix (1).

According to (1), phase 1 of all the machines will beconnected directly in series (the first column in (1)). Thephase transposition for phase 1 is therefore 0 degrees andthe phase step is zero. However, phase 2 of the first machinewill be connected to phase 3 of the second machine, whichwill be further connected to phase 4 of the third machine,and so on. The phase transposition moving from onemachine to the other is the spatial angle a, and the phasestep is 1. This follows from the second column of thetransformation matrix. In a similar manner, phase 3 of thefirst machine (the third element in the first row of (1) withspatial displacement of 2a) is connected to the phase 5 of thesecond machine, which further gets connected to phase 7 ofthe third machine, and so on. The phase transposition is 2aand the phase step is 2. This follows from the third columnof (1). Further, phase 4 of the first machine needs to theconnected to phase 7 of the second machine, which getsconnected to phase 10 of the third machine, and so on. Herethe phase step is 3 and the phases are transposed by 3a. This

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

582 IEE Proc.-Electr. Power Appl., Vol. 150, No. 5, September 2003

corresponds to the fourth column in (1). For phase 5 of thefirst machine, the phase transposition equals 4a and thephase step is 4, for phase 6 the phase step is 5 and phasetransposition equals 5a, and so on. If a number obtained inthis way is greater than the number of phases n, thenresetting is performed by deducting j� n, j¼ 1, 2, 3y, fromthe number so that the resulting number belongs to the set[1, n]. It should be noted that for an n-phase supply system(n is even) not all of the k-connectable machines will be n-phase. This means that for some machines in the group theresulting set of the utilised phase numbers in the connectiondiagram, obtained in the described way, will be a subset ofthe set [1, n]. The statement is clarified in what follows, byexamining certain phase numbers in more detail.

This explanation enables construction of a connectiontable, which is further called a connectivity matrix. Theconnectivity matrix and corresponding connection diagramare given next for some selected even phase numbers. Flux/torque producing phase sequence for any particularmachine is denoted in the connectivity matrices withnumbers 1, 2, 3yn, while the notation a,b,c,d,y is usedin corresponding connection diagrams.

The minimum even number of phases that will enableseries connection is n¼ 6. The corresponding connectivitymatrix, obtained on the basis of the given procedure, isgiven in Table 1. As can be seen from the last row of thismatrix, only phases 1,3 and 5 of the second machine areutilised. As spatial displacement between these phases is1201, it follows that the second machine is a 3-phasemachine rather than a 6-phase machine. The correspondingconnection diagram is given in Fig. 1. Note that the flux/torque producing currents of the 6-phase machine mutuallycancel at the connection points with the 3-phase machine.This means that the 3-phase machine will not suffer fromany adverse effects due to the series connection with the 6-phase machine. This, of course, does not hold true for the 6-phase machine.

In the case of an 8-phase system it is possible to connectthree machines in series. The connectivity matrix is the onegiven in Table 2. The first and the third machines are 8-phase; however, the second machine is 4-phase since onlyphases 1, 3, 5, 7 are utilised (and spatial displacement istherefore 901). It is important to note that, when connectingthe machines in series to the source, all the machines withthe highest phase number must come first. This means thatthe actual sequence of connection of the three machines tothe source has to be M1, M3, M2, as shown in theconnection diagram in Fig. 2. This is so since flux/torqueproducing currents of the machine with a higher phasenumber cancel when entering the machine with the lowerphase number (for example, in the 6-phase case of Fig. 1,phase currents a1 and d1 of the 6-phase machine are inphase opposition, so that their sum at the point of entry intophase a2 of the second machine is zero).

A 10-phase case is illustrated in Table 3 and Fig. 3. It isnow possible to connect four machines in series. Two ofthem are 10-phase (M1 and M3), while the remaining two(M2 and M4) are 5-phase. Such a situation will exist alwayswhen the even phase number n is such that n/2 is a primenumber, as discussed in the following section. Once more,the two machines with the higher phase number have to beconnected at first in series with the source. Five-phasemachines are then added at the end of the chain, as shownin Fig. 3.

In the three cases illustrated so far it was possible toconnect the maximum possible number k¼ (n–2)/2 ofmachines. Indeed, for any even phase number, one expects,on the basis of the considerations given so far, that the

Table 1: Connectivity matrix for 6-phase drive system

A B C D E F

M1 1 2 3 4 5 6

M2 1 3 5 1 3 5

Table 2: Connectivity matrix for 8-phase drive system

A B C D E F G H

M1 1 2 3 4 5 6 7 8

M2 1 3 5 7 1 3 5 7

M3 1 4 7 2 5 8 3 6

Fig. 1 Connection diagram for 6-phase 2-motor system

Fig. 2 Connection diagram for 8-phase drive system

Table 3: Connectivity matrix for 10-phase drive system

A B C D E F G H I J

M1 1 2 3 4 5 6 7 8 9 10

M2 1 3 5 7 9 1 3 5 7 9

M3 1 4 7 10 3 6 9 2 5 8

M4 1 5 9 3 7 1 5 9 3 7

IEE Proc.-Electr. Power Appl., Vol. 150, No. 5, September 2003 583

number of connectable machines will be k¼ (n–2)/2. This is,however, not always the case. Consider, for example, a 12-phase system. The connectivity matrix is shown in Table 4.Machines M1 and M5 are 12-phase, machine M2 is 6-phase,machine M3 is 4-phase, and machine M4 is 3-phase. Hence,it is not possible to connect all five machines in series sincethe ratio 4/3 is not an integer (an attempt to connect a 4-phase machine to a 3-phase machine leads to short-circuiting of all the terminals). At most four machines canbe connected in series: two 12-phase, followed by the 6-phase and 3-phase. The ordering is M1, M5, M2, M4.

Table 5 summarises the situation which arises for all theeven phase numbers up to 18. Bold notation applies to thephase numbers such that n/2 is a prime number. As can beseen from the table, only k/2 machines are n-phase, whilethe remaining k/2 machines are n/2–phase. In the 18-phasecase the number of connectable machines is smaller than

eight (at most seven), since the ratio 9/6 is not an integer. Asimilar situation arises in the 20-phase case (Table 6), whereboth 4-phase and 5-phase machines appear. There are four20-phase machines (M1, M3, M7, M9), two 10-phasemachines (M2, M6), two 5-phase machines (M4, M8) andone 4-phase machine (M5). At most eight machines can beconnected using the sequence M1, M3, M7, M9, M2, M6,M4, M8 (four 20-phase, two 10-phase and two 5-phase), buteight machines can be connected in series with the oddnumber of phases n¼ 17, which requires three fewerinverter legs.

4 Number of connectable machines

There are three different situations that may arise,depending on the properties of the phase number n. At

Fig. 3 Series connection of two 10-phase and two 5-phase machines to 10-phase source

Table 4: Connectivity matrix for 12-phase system

A B C D E F G H I J K L

M1 1 2 3 4 5 6 7 8 9 10 11 12

M2 1 3 5 7 9 11 1 3 5 7 9 11

M3 1 4 7 10 1 4 7 10 1 4 7 10

M4 1 5 9 1 5 9 1 5 9 1 5 9

M5 1 6 11 4 9 2 7 12 5 10 3 8

Table 5: Phase numbers of individual machines for supply phase numbers up to 18

Number of supply phases 6 8 10 12 14 16 18

Number of phases and ordering of connectablemachines (before re-ordering)

6 8 10 12 14 16 18

3 4 5 6 7 8 9

8 10 4 14 16 6

5 3 7 4 9

12 14 16 18

7 8 3

16 18

9

584 IEE Proc.-Electr. Power Appl., Vol. 150, No. 5, September 2003

least one case for each of them has been illustrated in thepreceding section:

(a) Let the number n/2 be a prime number. The numberof machines that can be connected in series is

k ¼ ðn� 2Þ=2 ð6Þ

The number of phases of k individual machines will be asfollows: k/2 machines will be n-phase and k/2 machines willbe n/2-phase. The ordering of machines has to follow therule that higher numbers of phases comes first. Hence, thefirst k/2 machines are n-phase, while the subsequent k/2machines are n/2-phase. This means that one-half of themachines have an even number of phases, while the resthave an odd number of phases. The phase numbersbelonging to this category are n¼ 6, 10, 14, 22, 26, 34, 38,46, 58, 62, 74, etc. The 6-phase and the 10-phase case wereelaborated in the preceding section.

(b) Consider next the number of phases n such that n/2 isnot a prime number but satisfies the condition

n ¼ 2m; m ¼ 3; 4; 5 . . . ð7ÞThe number of machines that can be connected remains tobe given with (6). Once again, not all k machines are of thephase number equal to n. In the example illustratedpreviously, n¼ 8, k¼ 3. However, only two machines are8-phase, while the third one is 4-phase. Hence, for thegeneral case of mZ3, the phase numbers of the machinesthat can be connected in series will be

n; n=2; n=22; . . . ; n=2m�2 ð8ÞThis case arises when the phase number of the multi-drivesystem takes values of n¼ 8, 16, 32, 64, etc.

(c) The third possible case arises for all the other even n.The number of machines that can be connected is

koðn� 2Þ=2 ð9ÞSeries connection of machines whose phase number ratio isnot an integer is not possible. This case was illustrated for

n¼ 12 and 20 in the preceding section. Again, among thesek machines only a certain number is with n phases. Theother machines have phase numbers equal to n/2, n/3, n/4yas appropriate. There are at least three different phasenumbers among the multi-machine set. Phase numbers thatbelong to this group are 12, 18, 20, 24, 28, etc.

Cases (b) and (c) can be regarded as sub-cases of a moregeneral case for which n/2 is not a prime number. The totalnumber of connectable machines is, however, not the same,((6) and (9), respectively). A summary of all possiblesituations for an even number of phases is provided inTable 7. Only phase numbers that enable series connectionof the maximum number of connectable machines (i.e. cases(a) and (b)) are of practical value.

5 Vector control of the multi-motor system

A standard method of achieving indirect rotor flux orientedcontrol of a current-fed AC machine is considered here. It isassumed that there is a rotor position sensor attached toeach machine of the group. The basic form of the vectorcontroller is the same as for a 3-phase machine of the sametype and the only difference is in the co-ordinatetransformation, where n phase current references aregenerated by means of the co-ordinate transformationdescribed with (1) and (2), instead of three. The indirectvector controller for operation in the base speed (constantflux) region is illustrated in Fig. 4 for an n-phase inductionmachine. The form of the vector controller is the same for apermanent magnet synchronous and synchronous reluc-tance motor, provided that the constant K1 is set to zeroand that an appropriate value is assigned to the stator d-axiscurrent reference.

Current control is performed in the stationary referenceframe, using inverter phase currents. It is important tonotice that the presented form of the vector-controlledmulti-phase multi-motor drive system is valid when thecurrent control is exercised in the stationary referenceframe. This is so since minimisation of the inverter phase

Table 6: Connectivity matrix for 20-phase drive system

A B C D E F G H I J K L M N O P Q R S T

M1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

M2 1 3 5 7 9 11 13 15 17 19 1 3 5 7 9 11 13 15 17 19

M3 1 4 7 10 13 16 19 2 5 8 11 14 17 20 3 6 9 12 15 18

M4 1 5 9 13 17 1 5 9 13 17 1 5 9 13 17 1 5 9 13 17

M5 1 6 11 16 1 6 11 16 1 6 11 16 1 6 11 16 1 6 11 16

M6 1 7 13 19 5 11 17 3 9 15 1 7 13 19 5 11 17 3 9 15

M7 1 8 15 2 9 16 3 10 17 4 11 18 5 12 19 6 13 20 7 14

M8 1 9 17 5 13 1 9 17 5 13 1 9 17 5 13 1 9 17 5 13

M9 1 10 19 8 17 6 15 4 13 2 11 20 9 18 7 16 5 14 3 12

Table 7: Number of connectable machines and their phase order for an even system phase number

n¼an even number, Z6

Number of connectable machines Number of phases of machines

n/2¼prime number k ¼ n�22

k/2 are n-phase and k/2 are n/2-phase

n/2aprime number n¼ 2m, m¼ 3,4,5y k ¼ n�22

n; n2 ;

n22 ; � � � ; n

2m�2

all other even numbers ko n�22

n, n/2, n/3, n/4y as appropriate

IEE Proc.-Electr. Power Appl., Vol. 150, No. 5, September 2003 585

current errors through inverter switching automaticallygenerates appropriate voltages required for compensationof the additional voltage drops in the machines, caused bythe flow of x-y current components. The concept can beextended to current control in the rotating reference frame.This is, however, beyond the scope of this paper.

Either ramp-comparison or hysteresis current control canbe used. Generation of individual machine phase currentreferences is done first, using Fig. 4 (superscript Mj standsfor the machine under consideration, M1 to Mk):

i�ðMjÞ1 ¼ffiffiffi2

n

ri�ðMjÞds cosfðMjÞ

r � i�ðMjÞqs sinfðMjÞr

h i

i�ðMjÞ2 ¼ffiffiffi2

n

ri�ðMjÞds cosðfðMjÞ

r � aÞi�ðMjÞqs sinðfðMjÞr � aÞ

h i

i�ðMjÞn ¼ffiffiffi2

n

ri�ðMjÞds cosðfðMjÞ

r � ðn� 1ÞaÞh

�i�ðMjÞqs sinðfðMjÞr � ðn� 1ÞaÞ

ið10Þ

Inverter reference currents are further built, respecting theappropriate connection diagram for the given number ofinverter phases. Inverter reference current creation has totake into account the existence of machines with differentphase numbers within the group. Taking as an example the6-phase inverter with a 6-phase and a 3-phase machine,illustrated in Fig. 1 and Table 1, the inverter currentreferences are determined with:

i�A ¼ i�a1 þ 0:5i�a2 i�B ¼ i�b1 þ 0:5i�b2i�C ¼ i�c1 þ 0:5i�c2 i�D ¼ i�d1 þ 0:5i�a2i�E ¼ i�e1 þ 0:5i�b2 i�F ¼ i�f1 þ 0:5i�c2

ð11Þ

Similarly, for the 10-phase system, illustrated in Fig. 3and Table 3, the inverter phase current references aregoverned with the following expressions:

i�A ¼ i�a1 þ i�a2 þ 0:5i�a3 þ 0:5i�a4 i�F ¼ i�f1 þ i�f2 þ 0:5i�a3 þ 0:5i�a4i�B ¼ i�b1 þ i�d2 þ 0:5i�b3 þ 0:5i�c4 i�G ¼ i�g1 þ i�i2 þ 0:5i�b3 þ 0:5i�c4i�C ¼ i�c1 þ i�g2 þ 0:5i�c3 þ 0:5i�e4 i�H ¼ i�h1 þ i�b2 þ 0:5i�c3 þ 0:5i�e4i�D ¼ i�d1 þ i�j2 þ 0:5i�d3 þ 0:5i�b4 i�I ¼ i�i1 þ i�e2 þ 0:5i�d3 þ 0:5i�b4i�E ¼ i�e1 þ i�c2 þ 0:5i�e3 þ 0:5i�d4 i�J ¼ i�j1 þ i�h2 þ 0:5i�e3 þ 0:5i�d4

ð12Þ

Inverter phase current references will be built in this mannerfor any phase number n. It is only necessary to form anappropriate connectivity matrix and the correspondingconnection diagram in order to arrive at an equation of theform given in (11) and (12). Since current control isperformed in the stationary reference frame using totalinverter phase currents, inverter output phase voltages willbe of the form required to minimise the phase currenterrors. Inverter output phase voltages are again governedwith the appropriate connection diagram for the givennumber of phases. For the six-phase case already con-sidered, the inverter output phase voltages follow directlyfrom Fig. 1:

vA ¼ va1 þ va2 vB ¼ vb1 þ vb2 vC ¼ vc1 þ vc2vD ¼ vd1 þ va2 vE ¼ ve1 þ vb2 vF ¼ vf1 þ vc2

ð13Þ

Similarly, for the 10-phase inverter the voltages aredetermined with the connection diagram of Fig. 3:

vA ¼ va1 þ va2 þ va3 þ va4 vF ¼ vf1 þ vf2 þ va3 þ va4vB ¼ vb1 þ vd2 þ vb3 þ vc4 vG ¼ vg1 þ vi2 þ vb3 þ vc4vC ¼ vc1 þ vg2 þ vc3 þ ve4 vH ¼ vh1 þ vb2 þ vc3 þ ve4vD ¼ vd1 þ vj2 þ vd3 þ vb4 vI ¼ vi1 þ ve2 þ vd3 þ vb4vE ¼ ve1 þ vc2 þ ve3 þ vd4 vJ ¼ vj1 þ vh2 þ ve3 þ vd4

ð14Þ

6 Simulation verification of the multi-phasemulti-motor system

The concept of the multi-phase multi-motor drive system,developed in this paper, is verified by performing simulationof a 10-phase 4-motor drive, consisting of four inductionmotors. Relevant per-phase equivalent circuit parametersand other data of the machines are given in the Appendix(section 10.1) – all the four machines are assumed to havethe same per-phase equivalent circuit parameters andratings. The system consists of two 10-phase and two 5-phase machines, and inverter current reference generation isdescribed with (12). The current controlled PWM inverter istreated as ideal in simulation, so that the inverter phasecurrent references are equated to the inverter output phasecurrents. Stator currents of all the machines are thereforeknown and stator phase voltages are obtained byreconstruction. Inverter output phase voltages are thencalculated using (14). All the four induction machines arerepresented for simulation purposes with the appropriatephase variable models. The machine models obtainableusing the general theory of electrical machines are thereforenot utilised and such an approach leads to an ultimateproof of the concept. The torque mode of operation isexamined.

Excitation of all the four machines is initiated simulta-neously, by ramping the rotor flux reference from zero totwice the rated value in the time interval from zero to 0.01 s.The rotor flux reference is brought back to the rated value(1.797 Wb for the 10-phase and 1.27 Wb for the 5-phasemachines) in the time interval from 0.05 to 0.06 s in thelinear manner and is further kept unchanged. A forcedexcitation, leading to a faster build-up of the rotor flux inthe machines, is obtained in this way. On completion of theexcitation transient, different torque commands, of differingduration, are applied to the four machines in different timeinstants. Application and removal of the torque commandis in all the cases ramp-like, with a ramp duration of 0.01 s.A rated torque command (16.667 Nm) is applied to the

Fig. 4 Indirect vector controller for an n-phase induction motorK1 ¼ 1=ðT �

r i�dsÞ

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

586 IEE Proc.-Electr. Power Appl., Vol. 150, No. 5, September 2003

10-phase machine (IM1) at 0.5 s and is removed at 0.7 s. Atorque command equal to 2/3 of the rated torque (i.e. 11.11Nm) is applied to the second 10-phase machine (IM2) at0.45 s and removal commences at 0.65 s. Rated torquecommand (8.33 Nm) is applied to the 5-phase machine(IM3) at 0.4 s and the removal is initiated at 0.65 s.Torque command for the second 5-phase machine (IM4)is applied at 0.3 and the removal starts at 0.55 s (3

4of the

rated torque, i.e. 6.25 Nm). Load torque is zero for allmachines.

Fig. 5 shows the rotor flux reference and correspondingresponse of the rotor flux in the four machines. Theexcitation process of any of the four machines is notdisturbed in any way by the presence of the other machinesin the group. Rotor flux in any of the machines attains thereference value, confirming the absence of any x-y rotor fluxcomponents. On completion of the excitation, transientrotor flux remains at a constant value equal to the reference,regardless of what happens with any of the four machinesfurther on. Torque reference and torque response (whichare indistinguishable from one another due to assumed idealcurrent feeding) are shown in Fig. 6. The torque response ofany of the four machines is not affected at all by thepresence of the other machines. Furthermore, torquecontrol of any of the four machines is completely decoupledfrom the flux control. Thus, not only is full decoupled fluxand torque control of any particular machine achieved, butalso the completely independent control of all the machinesresults, due to the introduced phase transposition, aspredicted by the theoretical considerations. Correspondingspeed responses are shown in Fig. 7; they are perfectlysmooth and the fastest possible due to the achievedcomplete decoupling of the torque control of the fourmachines.

Stator phase current references for the four machines areillustrated by means of Fig. 8, where the traces for phase aare shown. They are of familiar waveform, met in the caseof any vector controlled 3-phase machine, and aresinusoidal in final steady states. However, total invertercurrent references, shown in Fig. 9 for the first four phasesof the inverter, are highly distorted since they aredetermined with the summation given in (12), whichaccounts for the phase transposition.

Phase voltages of individual machines are illustrated forphase a in Fig. 10. As can be seen from this figure, phasevoltages of all the four machines start changing at theinstant of the application of the torque command to IM4

(0.3 s). Since IM4 is a 5-phase machine, its accelerationaffects all the other machines in the group. The flow of x-ystator current components through the windings of all fourmachines is responsible for this. Waveforms of phasevoltages show a certain amount of distortion, caused by theseries connection. It should be noted that the two 5-phasemachines are affecting only one another, while the two 10-phase machines are affected by all the other machines in thegroup.

Inverter output phase voltages, obtained by means of(14), are displayed in Fig. 11 for the first four phases.They are heavily distorted since they represent anappropriate sum of four individual machine phasevoltages, of different fundamental frequencies and alreadyhaving some amount of distortion due to the x-ycomponents of the machines’ phase voltages. However,generation of such distorted inverter voltage waveformspresents no difficulty in practice, provided that fastinverter current control with a sufficiently high inverterswitching frequency is used. The practical questions of howmuch the DC link voltage should be, as a function of the

machines’ rated voltages, and how much the voltage reservefor good current control needs to be, are left for futureinvestigation.

Fig. 5 Rotor flux reference and rotor flux response of fourmachines in 10-phase drive system

Fig. 6 Torque references and torque responses of four machines

Fig. 7 Speed responses of 4-motor drive system

IEE Proc.-Electr. Power Appl., Vol. 150, No. 5, September 2003 587

Fig. 8 Phase a current references of four machinesFig. 9 Inverter phase current references for first four phases

Fig. 10 Stator phase a voltages of four machines Fig. 11 Inverter output phase voltages for first four phases

588 IEE Proc.-Electr. Power Appl., Vol. 150, No. 5, September 2003

7 Conclusion

The paper develops a concept for a multi-motor drivesystem, which enables independent control of a certainnumber of multi-phase machines with even numbers ofphases, although the whole system is supplied from a singlecurrent-controlled voltage source inverter. The stator multi-phase windings have to be connected in series with anappropriate phase transposition, in order to achieve theindependent control of the machines in the system. Theconcept is developed in a systematic manner, using generaltheory of electrical machines, and is valid regardless of thetype of AC machine, so that different machine types can beused within the same multi-motor drive system. Thenecessary phase transposition in the stator winding connec-tion is established by analysing the properties of thedecoupling transformation matrix, and a so-called con-nectivity matrix is formed for selected phase numbers.Corresponding connection diagrams are further developedon the basis of the connectivity matrix. The concept is ingeneral applicable to any phase number greater than orequal to five. Considerations in this paper are restricted toan even phase number. An even number of phases and theprevious odd number enable, at best, connection of thesame number of machines in series, and therefore the oddphase number saves a larger number of inverter legscompared to an equivalent 3-phase drive system. However,as shown in the paper, an even phase number always leadsto a multi-motor system with at least two different phasenumbers. All the machines in the group with a smallernumber of phases are not affected by series connection tothe machines with a larger phase number. The negativeconsequences of the series connection, discussed in the nextparagraph, are therefore less pronounced for an even phasenumber than for an odd phase number.

An obvious and major drawback of the concept is anincrease in the stator winding losses due to the flow of theflux/torque producing currents of all the machines throughstator windings of some of the machines (note that rotorwinding losses are not affected). Similarly, stator iron losseswill increase as well (although to a much lesser extent), dueto the increased phase voltage of the individual machinescaused by the flow of x-y current components. This willdecrease the efficiency of the affected machines in the multi-motor system and will yield an overall reduction in the totalefficiency of the drive system, when compared to anequivalent 3-phase counterpart. Reduction in the efficiencyis expected to be smaller for a system with an even phasenumber, when compared to the equivalent multi-motorsystem with an odd phase number. In the 10-phase caseconsidered in detail, the two 5-phase machines only affectone another, and do not suffer from any adverse effectscaused by connection to the two 10-phase machines. Ofcourse, the two 10-phase machines are affected by thepresence of all the other machines.

It has to be noted that some of the advantages of themulti-phase machines, which exist in the case of a singlemulti-phase motor drive, have been lost in the proposedmulti-motor multi-phase system. For example, torquedensity cannot be increased by injection of higher statorcurrent harmonics, since all the available degrees of freedomare used to control other machines in the group. Similarly,fault tolerance is completely lost for the same reason.

All the possible even phase numbers are examined(disregarding the physical feasibility of large phase numbers)in order to establish which of them offer a potential forconnection of the largest number of machines. It is shownthat the maximum number of connectable machines results

when the phase number n is either such that n/2 is a primenumber or is a power of two.

The concept is verified by simulation of a 4-motor 10-phase drive system. Torque mode of operation is examinedand it is shown that completely decoupled and independentvector control of the four machines is possible with theproposed series connection. The major advantage of such amulti-drive system is the saving in the required number ofinverter legs (when compared to an equivalent multi-motor3-phase drive), thus leading to an increase in reliability. The10-phase system requires 10 inverter legs, while thecorresponding 4-motor 3-phase system would ask for 12inverter legs. The other extremely useful feature of theconcept is the ease of implementation within a single DSP.It is necessary to execute the required number of vectorcontrol algorithms in parallel, calculate the individual motorphase current references, and then give at the output of theDSP inverter current references, obtained by summationand respecting the connection diagram. Current control isfurther executed in the stationary reference frame, usingeither hysteresis or ramp comparison inverter phase currentcontrol.

8 Acknowledgment

The authors gratefully acknowledge support provided forthe work on this project by the EPSRC, under the standardresearch grant number GR/R64452/01, and by SemikronLtd. Mr. M. Jones acknowledges financial support providedfor his PhD studies by the IEE, through the IEE RobinsonResearch Scholarship.

9 References

1 Belhadj, J., Belkhodja, I., De Fornel, B., and Pietrzak-David, M.:‘DTC strategy for multi-machine multi-inverter industrial system’.Presented at European Power electronics and appli-cations Conf.,Paper no. PP01002 EPE, 2001, Graz, Austria

2 Lee, B.K., Fahimi, B., and Ehsani, M.: ‘Overview of reduced partsconverter topologies for ac motor drives’. Proc. IEEE Powerelectronics specialists Conf., PESC, Vancouver, Canada, 17–21 June2001, pp. 2019–2024

3 Ledezma, E., Mcgrath, B., Muoz, A., and Lipo, T.A.: ‘Dual ac-drivesystem with a reduced switch count’, IEEE Trans. Ind. Appl., 2001, 37,(5), pp. 1325–1333

4 Jacobina, C.B., Oliviera, T.M., Correa, M.B. de R., Lima, A.M.N.,and Da Silva, E.R.C.: ‘Component minimized drive systems for multi-machine applications’. Presented at IEEE Power electronics specialistsConf., PESC, Paper no. 10028, 23–27 June 2002, Cairns, Australia

5 Kuono, Y., Kawai, H., Yokomizo, S., and Matsuse, K.: ‘A speedsensorless vector control method of parallel connected dual inductionmotor fed by a single inverter’. Presented at IEEE Ind. Appl. Soc.Annual Meeting, (IAS), Paper no. 29–04, Chicago, IL, 30 September–4 October 2001.

6 Matsumoto, Y., Ozaki, S., and Kawamura, A.: ‘A novel vectorcontrol of single-inverter multiple-induction motor drives for Shin-kansen traction system’. Proceedings of IEEE Applied powerelectronics Conf., (APEC), Anaheim, CA, 4–8 March 2001, pp.608–614

7 Eguiluz, P., Pietrzak-David, M., and De Fornel, B.: ‘Comparison ofseveral control strategies for parallel connected dual inductionmotors’. Presented at Power electronics and motion control Conf.,(EPE-PEMC), Paper no. T11–010, 9–11 September 2002, Cavtat,Croatia

8 Ma, J.D., Wu, B., Zargari, N.R., and Rizzo, S.C.: ‘A space vectormodulated CSI-based ac drive for multimotor applications’, IEEETrans. Power Electron., 2001, 16, (4), pp. 535–544

9 Ward, E.E., and H.arer, H.: ‘Preliminary investigation of an invertor-fed 5-phase induction motor’, Proc. Inst. Electr. Eng., 1969, 116, (6),pp. 980–984

10 Singh, G.K.: ‘Multi-phase induction machine drive research – asurvey’, Electr. Power Syst. Res., 2002, 61, (2), pp. 139–147

11 Jones, M., and Levi, E.: ‘A literature survey of state-of-the-art inmultiphase AC drives’. Proc. Universities power engineering Conf.,(UPEC), Stafford, UK, 9–11 September 2002, pp. 505–510

12 Hodge, C., Williamson, S., and Smith, S.: ‘Direct drive propulsionmotors’. Presented at Int. Conf. on Electrical machines, (ICEM),Paper no. 087, 25–28 August 2002, Bruges, Belgium

13 Golubev, A.N., and Ignatenko, S.V.: ‘Influence of number of stator-winding phases on the noise characteristics of an asynchronousmotor’, Russ. Electr. Eng., 2000, 71, (6), pp. 41–46

IEE Proc.-Electr. Power Appl., Vol. 150, No. 5, September 2003 589

14 Toliyat, H.A., Rahimian, M.M., and Lipo, T.A.: ‘A five phasereluctance motor with high specific torque’, IEEE Trans. Ind. Appl.,1992, 28, (3), pp. 659–667

15 Xu, H., Toliyat, H.A., and Petersen, L.J.: ‘Rotor field oriented controlof a five-phase induction motor with the combined fundamental andthird harmonic injection’. Proc. IEEE Applied power electronicsConf., (APEC), Anaheim, CA, 4–8 March 2001, pp. 608–614

16 Coates, C.E., Platt, D., and Gosbell, V.J.: ‘Performance evaluation ofa nine-phase synchronous reluctance drive’. Presented at IEEE Ind.App. Society Annual Meeting, IAS, Paper no. 49–05, 30 September–4October 2001, Chicago, IL

17 Kestelyn, X., Semail, E., and Hautier, J.P.: ‘Vectorial multi-machinemodeling for a five-phase machine’. Presented at Int. Conf. onElectrical machines (ICEM), Paper no. 394, 25–28 August 2002,Bruges, Belgium

18 Gataric, S.: ‘A polyphase Cartesian vector approach to control ofpolyphase AC machines’. Presented at IEEE Ind. Appl. Soc. AnnualMeeting (IAS), Paper no. 38–02, Rome, Italy, 8–12 October 2000.

19 Lipo, T.A.: ‘A Cartesian vector approach to reference frame theory ofAC machines’. Proc. Int. Conf. on Electrical machines (ICEM),Lausanne, Switzerland, 18–21 September 1984, pp. 239–242

20 White, D.C., and Woodson, H.H.: ‘Electromechanical energyconversion’, (John Wiley & Sons, New York, NY, 1959)

10 Appendix

10.1Per-phase equivalent circuit parameters of the 50 Hz 5-phase and 10-phase induction motors:

Rs ¼ 10O Rr ¼ 6:3OLls ¼ Llr ¼ 0:04H Lm ¼ 0:42H

Inertia and number of pole pairs: J¼ 0.03 kgm2, P¼ 2.Rated per-phase torque: 1.667 Nm. Rated rotor flux(RMS): 0.568 Wb. Rated per-phase voltage and current(RMS): 220 V and 2.1 A.

10.2 List of principal symbols

v, i, c voltage, current and flux linkage, respec-tively

ys, y transformation angle for stator variablesand rotor instantaneous angular position,respectively

oa, o angular speed of the common referenceframe and angular electrical speed ofrotor, respectively

Rs, Rr stator and rotor per-phase resistanceLls, Llr, Lm stator and rotor per-phase leakage in-

ductance, magnetising inductanceP, Te number of pole pairs and electromagnetic

torque of the machine, respectivelyp Laplace operatorfr instantaneous position of the rotor flux

Indices:

s, r stator and rotor, respectivelyd, q d-q axis components of voltages, currents

and flux linkages in the common refer-ence frame

x, y x-y components of voltages, currents andflux linkages

o+, o� zero sequence components of voltages,currents and flux linkages

n rated value

590 IEE Proc.-Electr. Power Appl., Vol. 150, No. 5, September 2003