Six-phase AC drive system with reduced common-mode voltage M.B.R. Corr&a',2, C.B. Jacobina', C.R. da Silva'", A.1I.N. Lima', E. R. C. da Silva'

'Laborat6rio de EletrBnica Industrial e Acionamento de hlhquinas Departamento de Engenharia EIBtrica, Universidade Federal da Paraiba

Caixa Postal 10105, 58109-970 Campina Grande, PB, Brazil Fax: ++55(83)310-1015 Email mbeltrao@dee.ufcg.edu.br

'Centro Federal de E d u c q b Tecnol6gica de Alagoas - CEFET - AL - Brazil 3Departamento de Eletricidade, Centro Federal de E d u c q b Tecnol6gica - CEFET - CE - Brazil

ABSTRACT tioned papers the authors adopt a six-phase machine con- stituted of two three-phase windings individually wye con- nected hut only j14] discusses the symmetrical six-phase machine is discussed. Despite the positive results in t e r m

This paper investigates a six-phase machine drive WnfiSration that Permits to reduce the common- mode voltage. The drive system consists of a Six- of common-mode voltage elimination, that work did not Phase inverter feeding a sk-phase induction tor, which is adequate to

investigate PWM strategies to command the system. a high power 'YS- This paper investigates several approaches to generate

t em with current rating restrictions. Several pulse- pWM control considerine reduction of either mean or in- width modulation techniques for generating gate signals t o command the power switches of the six- phase inverter are presented. The proposed strate- gies permits to reduce either the mean value or the instantaneous value of the common-mode voltage. Simulation and experimental results demonstrates the feasibility of the proposed solution.

I. INTRODUCTION

Adopting a six-phase machine it is possible to design a high power induction motor drive system with half of the rated current of an equivalent threephase scheme. A six- phase machine improves the reliability of the system since loss of one of the phase does not prevent the motor from the starting or running. Besides that, a six-phase system permits to minimize or even eliminate the effects of the common-mode voltage generated by PWM operation [l]. The common-mode voltages create coupling currents flow- ing through the motor parasitic capacitances toward the rotor iron. These currents find their way via the motor bearings back to the grounded stator case and form the so-called bearing currents 12-41, which cause bearing dete- rioration [5]. The wmmon-mode currents also cause a w n - ducted electromagnetic interference, activate the ground fanlt detection circuits and heat the conduit where it flows.

Alternatives to eliminate or at least to minimize the magnitude of the common-mode voltage is an important and very relevant topic. Recently, several papers addressed different strategies for reducing the common-mode voltage in PWM drive systems. Most of them focused the uti- lization of filters [S-S] or the active cancellation schemes based on new inverter topologies [SLlZ] . The dual-bridge inverter approach for feeding a doublewinding three-phase induction motor has been proposed to eliminate the mo- tor common-mode voltage [13,14]. In these two last men-

- stantaneous common-mode voltage in a six-phase induc- tion motor drive system.

11. SIX-PHASE MACHINE

The six-phase machine model in this research can be represented by six concentrated windings 60 degrees away from each other ~ see Fig. 1. This kind of machine can be derived from a conventional three-phase one by rewinding or just reconnecting the original groups of windings.

Adopting a b e d coordinate reference frame, the math- ematical model that describes the dynamic behavior of a six-phase induction machine is given by

(1)

(2)

Xsdq = lsisdq + 1st.irdq (3) Xrdq = lrrisdq + lrirdq (4)

r e = 3PLar(iaq&d - i s d i r q ) (5)

(6)

(7)

(8)

(9)

d dt d d t

Vsdq = rsiadq -&dp

Vrdq = Trirdq + -&dq - j W v & d q

d dt

- d . d t d dt d . d t

vsry = r& + L-iszy

vrry - r&, + L,--1,,,

vso0' = r s i so~ + L3-iSd

r&,, + lor- - l~oo~ V l O O ' =

where Vsdq = Vsd + j V q , idq = i rd + ji,,, and X s d q = Aad + j A S q are the voltage, current and flux dq vectors of the stator, respectively; v,.~ = u..+jvay, is,, = i9*+jiSv, and A,,, = A,, + jA,, are the voltage, current and flux xy no-torque vectors of the stator, respectively; vsoo, = v,,+jv.,~, is,,, = i so+j i80 , , and Aao0, = A,,+jA,o, are the voltage, current and flux 00' homopolar no-torque vectors

07803-7811-21031517.W 82003 EE!? 1852

Fig. 1. Six-phase AC drive system

of the stator, respectively, (the equivalent rotor variables are obtained by replacing the subscript s by 7 ) ; Te is the electromagnetic torque; w, is the angular frequency of the rotor; r, and r, are the stator and rotor resistances; l , , l,,, I, and I,, are the self and leakage inductance of the stator and rotor, respectively; I,, is the mutual inductance and P is the number of pair of poles of the machine.

The dqxyoo' stator variables of the above model can be determined from the 123456 variables by using the trans- forming equation given by

ws123456 = Awsdqzyoo, (10)

with W s i z 3 4 5 ~ = [WJI W.2 Ws3 Ws4 Ws5 U d T , Wsdqryoo'

[Wad was wST zugy wg. w..rjT; A is a 6x6 matrix that performs a two-phase real component transformation 1151. Vectors We123456 and Wsdq.yoo' can be or voltage, or cur- rent, or flux vectors.

111. VOLTAGE IN THE SIX-PHASE DRIVE

Consider that the conduction state of the power switches of the inverter of Fig. l(b), is associated to the binary mi- ables qi (i = l, 2,3,4,5,6), where qi = l indicates a closed switch and qi = 0 an opened one. The output voltages can be calculated by

E v , ~ = v ~ o - u , I o = ( ~ ~ , - ~ ) ~ - v " ~ o fo r i=1 ,3 ,5 (11)

E V . ~ = V ~ O - V ~ Z O = ( ~ ~ ~ - ~ ) - - U , ~ O f o r i = 2 , 4 , 6 (12) 2

where E is the DC-bus voltage; vnl0 represents the voltage between the neutral point of the wye connected coils assc- ciated with phase 1 , 3 and 5 (nl) and the DC bus midpoint 0; v,zO represents the voltage between t,he neutral point of the wye connected coils associated with phase 2, 4 and 6 (n2) and the DC bus midpoint 0.

Assuming that is a symmetric machine one, the common-mode voltage (vcm) is given by

(13)

(14)

Un10 + vn20 U,, = , or

2'10 + 2'20 + 2'30 + 2'40 + VS0 + VGO

6 Ucm =

This expression is valid either for calculation of the mean or the instantaneous values of U,. To obtain vem = 0 instantaneously implies that three among of the six upper (lower) switches must he turned-on while all the rest must be turned-off, at same time.

In the vector modulation technique, the problem of s e lecting the appropriate switching sequence can be better understood if the phase voltage quantities (vs123~56) are transformed into dqzyoo' coordinates (Vdqryoo') at the sta- tionary reference frame. As discussed before, this transfor- mation is given by

V s 1 2 ~ 5 6 = AVadqZyod (15)

with Vs123456 = [Vel V S Z U83 Us4 21.5 U d T , Vadqzyoo' = [Ved vsd U,, us0 U,,,,]*. Considering that the double winding machine is wye connected without neutral con- nection, the homopolar components ug0 and vso, are null because the machine is symmetric and is, = ieo, = 0, the voltage components u8d, v , ~ , U,, and uOy can be obtained from (15) by using (11) and (12). Note that variables z and y do not generate electromagnetic torque but they generate harmonics in the phase variable.

The d,q components can be combined into a single com- plex voltage vector given by V = v d + j V , , where v d

and V, are the dq components of vector generated by the converter. The combination of the switch states origi- nates 64 different vectors in the plane dq as shown in Fig. 2(a). Sixty-four vectors are also generated in plane zy, in which the zy components are combined in the vector V = V, +jVy [Fig. 2(b)J. In order to create an easy repre- sentation for a vector V in the plane dq (zy), the identiiien 'a' and 'b' me introduced such that 'a' is associated with switches ( q l , 93. g5) and 'b' with ( q z , q4, qe) , assumingval- ues from 0 to 7 as described in table l(a) and l(b). Thus Vob can represent any vector in the dq as well in the xy plane.

Only 20 of 64 vectors in Fig. 2(a) or 2(b) have a zero common-mode voltage and two among them are null vector (1+0 and 1/07). For better an identification of these vectors in the dq and xy planes a box was drawn around them. The

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(b) Fig. 2. Voltage vectors and sectors io dq plane for the six-phase inverter.

TABLE 1. Definition of the vector identifiers a and b.

(a) b I O 1 2 ' 3 4 5 6 7 q 2 O l l 0 0 0 1 1

0 0 0 0 1 1 1 1 (b)

64 vectom permit to identify 12 sectors but this number is reduced to 6 if only null common-mode voltage vectom are

considered. The decision of working with 12 or 6 sectors is taken accordingly with the choice of the PWM patterns as will be described in next section.

Iv. VECTOR MODULATION FOR THE SIX-PHASE DRIVE

The objective of the modulation is to determine the switch command sequence to obtain the reference V* = uZd + ju:, required by the machine control. The voltage U, and u t must be made null in a PWM period in or- der to avoid phase-voltage distortion. Determination of the time interval associated to each vector has some pecu- liar procedure but, in general, follows the basic concept of the vector modulation [XI. In this work two basic distinc- tiom are made accordingly with the characteristic of the dg plane division in either, 30' sectors or 60" sect,ors.

30' Sectors

The 30' sectors in the dq plane can be identified by Sao with S3, = 1, ..., 12. Despite the amount of available vec- tors at each point of the dq plane, a given reference voltage vector (v') must be synthesized as a weighted sum of vec- tors. In another words, assuming any reference vector in a 3 0 O sector of dq plane, it is necessary to solve the following equation

In this equation, mr, m and n are expressed as a function of S ~ O in such a way that

m = S30, if SsO is odd or m = S30 + 1, if S30 is even (17) n = 530, if 5'30 is even or la = S3, + 1, if S ~ O is odd (18)

with s30 + 1 = 1 if S ~ O = 12; vo corresponds to any Vob null vector, v,, = Um,d + jw,,,,, (small vectors), v, = u,d + jv,, (large vectors) and v, = vnd + jv,, (mean vectors) are any of the non-null Vaa vector in the boundary of sector S30 with its location indicated in the dq plane by the index m/, m and n, respectively; finally, T is the PWM period. Note that m! is numerically equal t o m , i.e., if m = 1 + m, = 1. As an example, using Fig. 2(a) and assuming that 530 = 1 (m/ = m = 1 and n = 2) it could be possible to choose vo = V77, v,,,, = V ~ B , v, = Vl6 and v, = V.6. Another essential equation to solve the system is the time restriction, which is given by

T = t o + 1,. + tm f t n (19)

Since (16) is a vectorial equation it can be split into two equations if one takes into account the real and imaginary part separately. Together with (19) the split equation con- s t i t u t e a system of three equation with four variables to be solved. As vm, = iv, and vo = 0, then (16) can be rewritten as

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Now the calculation of 9 + t, and t, can be easily done by

(22) umdvzq - limqufd

%d%q - UmqUnd t, =

To obtain t,,, and t, there are many possibilities. In this study it was convenient to make an association of these variables with the modulation index ( M ) in such a way that

(23) -- tm - M + + t ,

with M = m, Note that (23) reaulates the . . - " use of the small and large vectors accordingly with the modulation index. Thus, tm, and t, can be calculated by

and finally, it is possible calculate to from (19).

60° Sectors

Another method that is exploited in this work is based in a dq plane split in 60° sectors. These sectors will be identified by& with Ss0 = 1, ..., 6and thereferencevector can be synthesized by

in which m = Seo with 2m + 1 = 1, if m = 6. The new dq plane is shown in Fig. 3 using only vectors that has common-mode voltage equal to zero.

Taking into account the time restriction given by

T = to + t2m-1, + tzm-1 + t z m t l ) + t z m t l (27)

and spliting (26) into two equations, one obtains a system of three equations and five variables to be solved. Using the relations between vectors given by vzm-lr = ;vzm--1 and vzm+ll = $vzm+l, it is possible to write (26) as

. . The vectorial equation can be solved by separating the d and q variables to obtain the following solutions

(29)

(30)

tzm-1' vZm+lqu:d - UZm+ldU:p 2 + tzm--1 =

7j- + t2*+1 =

~Zm--ldV2m+lq - UZm-lq~Zm+ld

tzmt1, uZm-ldu:q - vZm-lqu~d

%-ldVZm+lq - U2m-lqfJZm+ld

7

Fig. 3. dq plane split in six sector

(34) UZm-ldU& - vZm-1qV:d

VZm-ldVZm+lp - ~ Z m - l q ~ Z m t l d t zmt1 = M

The time interval of use of a null vector is given by (27) .

A . Mean common-mode voltage elimination Using a 30° sector mapping of dq plane it is impossible

to apply an instantaneous common-mode voltage strategy because all v,, vectors have non-null common-mode volt- age.

Case f: Using sk instantaneous non-zero common-mode vectors

Following the restriction imposed by = U& = uno = 0, it is necessary to establish a set of vectors that must be associated with the time interval to , t,,, 1 , and t, in each sector m of the dq plane. Note that to fulfill the previous restrictions it is necessary to evaluate not only the vectors in the dq plane but also those in the zy plane and the mean value of uno, as well. 'Thble (2) presents the vectors that must be applied during a time interval given by to/2, tm,/2, tJ2 and t, for each one of the 30 degrees sectors. The use of more than one vector at a same locus in the dq plane is a necessary condition to nullify u., vy and v,o. To better understand this rule, note that V,, is 180' apart from VI, in xy plane but that they are redundant vectors

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in dq plane. The time distribution of vectors given by (23) may be

different. As an example, when vnl = 11-2 = 0 it is possible to show that the relation between t,, and t, is

t,! = t, (35)

The main difference between these two solutions is that with (23) the maximum phase voltage is E/& while with (35) it is changed to E / 2 .

Case 2: Using only two instantaneous non-zero common-mode vectors

The instantaneous common-mode voltage reduction of case 1 can be achieved by replacing 4 vectors that present the highest U,, values. In terms of the dq plane, this pat- tern is identical to that of Case 1 and the calculation pro- cedure for the time intervals t o , t,,, t, and t, is the same BS in that case. The preselection sequence of vectors for Case 2 is presented in Table (3). To maintain the mean switching frequency equal to that in Case 1 the instan- taneous switching frequency must be reduced 25% with respect to that established in Case 1.

TABLE 3. Sequence of vecton far Case 2.

530 1 tal2 t,,/z t"12 tm tJ2 tm,/2 t o p

1 %7 %5 %6 VI6 VI1 v6, v70

2 %Oi vS6 1/26 1/21 VI1 VI2 v70

4 "07 1/23 v2Z v3Z v31 Val v70 3 Vnr Via Vzi Vzi 1/31 Vm

5 Vo7 Val v42 1132 v33 v23 v70 6 vo7 vsa h a h 3 v33 v34 Vi0 7 Vo7 Vm Vu V43 V53 &a V7o 8 vo7 vdS v4& vs4 vS3 vs3 viO 10 "0, vi4 VM V65 VSS vm v7o

12 "0, v6i Vsa Vi6 VIS Vas V7o

g v07 v63 v64 v54 v5S v4S v70

11 V07 v56 v66 v65 VIS VI4 v70

B. Instantaneous common-mode voltage elimination using the null vectors

In this case only 20 vectors are available and the number of Sectors is reduced to 6 . For all cases presented in this section the calculation of the time interval during which

a vector must be applied is achieved by using (31); (32), (33), (34) and (27).

Case 3: Using only two large zero common-mode voltage vectors

This solution can be straight derived from the general solution hy making t2m--1# = tz,+l, = 0. This consists of using (29) instead of (32) to calculate t 2 , - 1 and using (30) instead of (34) to calculate t2,+l. The resultant sequence of vector during the time interval application is presented in Table (4). Compared to Case 1, the instantaneous and mean switching frequencies are equal. The PWM pattern adopted in [14] to achieve a null common-mode voltage is a particular solution of this case.

TABLE 4. Sequence of vectors for Case 3.

t2m+l v2l v21 v43 v43 v65

tO/2 VOl VOl v01 VOl v07

Case 4: Using all zem common-mode voltage vectors

TABLE 5. Sequence of vectors for Case 4 - first period T.

t 2 m 4 v21 v43 v43 v65 v65

t Z n ~ + l / ~ v36 v32 v32 VU v34 VI6

t2m+1s/2 Via V36 V ~ Z v14

tO/2 v07 v07 v07 v07 v07

TABLE 6. Sequence of vectors for Case 4 - second period T

& 1 1 2 . 3 4 i5 6

t~m-112 V,I V,1 V,3 1/43 6 5

This PWM pattern uses all available zero common-mode voltage vectors in a 60' sector. It c m be achieved by using the general equation to calculate tzm-l,, t2,+1,,

and to. The vectors that must be applied are pre- sented in Tables (5) and (6). Now vz and vy assume not null mean values within a PWM period T , but the mean value is zero for a 2T period. Thus the sequence shown in Table ( 5 ) must be applied in the first period T and the sequence in Table (6) in the next one. This procedure increase the amount of vectors with null U, and vy com- ponents in a period T, which is desirable. As in Case 2

b / 2 v70 v i 0 v70 v70 v70 v70

t 2 m - l f / 2 .&1 h3 v23 bs &5 &I

t2m+1/2 VI6 v32 v32 v54 6 4 vl6

t2m+1'/2 %2 VI2 v34 v34 v56 v56

b / 2 h7 h 7 v07 v07 v07 v07

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this PWM pattern needs reduced the switching frequency in 25% compared with Case 1 to maintain the same mean switching frequency during a period of the fundamental voltage.

Case 5: Using only small zero common-mode voltage vectors

The vectors locus for this pattern is the most inner hexagon of dq plane and is only adequate for small modula- tion index. The calculation of the time interval application for each vector can be effected by using (29) and (30) with tz,,-l = tzm+t = 0 and (27). The recommended sequence of application of each vector is presented in Table (7). In t e r m of the switching frequency this pattern does not re- quire any adjustment when compared to Case 1.

TABLE 7. Sequence of veCtorS for Csse 5

1 4 i 1 2 3 4 5 6 to/4 v70 v70 v70 v70 v70 v70

tZm-1,/2 %I &3 v23 E5 E5 &I

t2m+1!/2 v12 VI2 f i 4 b4 v56 v56

t0/2 1/07 v07 v07 h 7 v07 1/07 t2m+lf/2 fi6 6 6 v 5 2 1/52 VI4 VI4

t2m-lf/2 &5 v41 b1 h3 %3 v25

v70 v70 v70 KO KO v70

V. SCALAR MODULATION

It is also possible to define an equivalent scalar modula- tion technique to control the six-phase inverter and derive the relations between the vector time and pulse-width pat- terns. The calculation of the pulse width can he performed

. ,=(1+2co)T,fori=l,2,3,4,5,6 2 E (36)

with 0 5 rc 5 T , which is equivalent to Iv& 5 E/2. Note that vcm = 0 is a consequence of u,10 + U,,ZO = 0, since U,, = *. Therefore, since U,, is made null, v,,l~ = -vnZO can be used as a homopolar voltage that is added to the phase voltage in order to modify the waveform of the modulating signal (UEO) and generate phase voltage in such a way that Iv;il 5 E/&. This procedure is similar to what had been done for three-phase VSI system [17].

by

VI. HARMONIC ANALYSIS

Usually, the choice between different PWM patterns must take into account some performance criteria, besides nullify ucm. In this work this was made by calculating the THD (Total Harmonic Distortion) of the dq components by

in which al is the amplitude of the dq fundamental volt- age, ax is the amplitude of the kth harmonic and p is the

OJ I 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 I

l a ) M

Fig. 4. Total harmonic distortion (THD)

number of harmonics used for calculation. These results can be useful to select the most appropriated strategy for any particular system. They are shown in Figs. 4(a) and 4(b). Note that except for Case 5 the maximum allow- able modulation index is 1. This is because of the adopted solution given by (23). If (35) is used instead of (23) the maximum modulation index is 0,86 for Case 1 and Case 2 and 0.75 for Case 4. The better performance of Case 4 when compared to Case 3 is due to the distribution of the application time of vectom as a function of the modulation index. Finally, except for Case 5, the proposed methods generate a maximum phase voltage equal to Ef ~6.

VII. EXPERIMENTAL RESULTS

Two proposed strategies for eliminating the common- mode voltage in AC drive system were tested experimen- tally. The strategies experimentally evaluated were Case 1 and Case 3. Figures 5(a) and 5(b) show the experimen- tal results for the mean and instantaneous common-mode voltage elimination of v,,,,, respectively. In spite of gener- ating a non-null instantaneous common-mode voltage by using Case 1, the vectorial development can be applied to improve the performance of system in which the common- mode voltage is not critical as for a three-phase system.

The result associated with Case 3 shows that the common-mode voltage is null. Also the vectorial solution shows that it is possible to generate phase voltage with amplitude higher than E/2.

VIII. CONCLUSION

In this paper, several PWM patterns are derived, which are based in the vectorial approach. The main purpose is to nullify the common-mode voltage in order to avoid the mechanical deterioration of a machine due to bearing currents. However, the theory is extensive for others goals.

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.. .. . . .. .. .. .

. .. .. . .

0.0s 0.1 0.15 0 2 0.25 t l-)

(a)

,mk ... . ... ; . . . ..... ~. . . . ... ,.. i... . . . . 4

0 - 0 1 0 1s 0 2 025

f fms)

(b)

Fig. 5. Experimental mmmon-mode voltage of the six-phase machine (a) with mean zero mmmon-mode voltage and - ca5e 1 (b) instant- ~ W U S zero mmmon-mode voltage - e m 3.

The study reveals that in spite of the vectors redun- dancy, it is possible define a simply procedure to calcu- late the time of application of a previously selected set of vectors. Accordingly with the preselection of vectors as well their sequence application the switching frequency can be modified due to a new generated pulse width pattern. Therefore, the PWM period must be changed to maintain a constant mean switching frequency. Another fundamental required analysis concem the x and 3 components. De- pending on the given solution for the sequence of vectors the infiuence of the instantaneous generated voltage can add undesirable harmonics to the phase variable.

The 30' based solutions reduce the instantaneous

common-mode voltage if the not null common-mode volt- age vectors of Case 1 are replaced by other with null common-mode voltage.

For the 60" based approach, in this paper, three PWhl vectorial patterns are proposed to feed the machine by US-

ing only instantaneous null common-mode voltage vectors. The vectorial resolution shows how to generate phase volt- age higher than El2 and how to improve the performance in terms of THD with null common-mode voltage.

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