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Electromagnetical efficiency of the six-phase winding/Seisafazes apvijos elektromagnetinis efektyvumas.

Introduction

Currently the six-phase voltage system is used in practice only in very rare cases. The opinion prevails that the three-phase voltage system optimally satisfies all the requirements of consumers of electrical power [1-3]. Therefore the impact of the six-phase and generally multiphase voltage systems on the operation of respective power consumers has not been studied widely.

At the present time the multi-phase voltage system (m> >3) is mostly used only in the current rectification circuits since the pulsation of the rectified current decreases with the increase of the number of phases.

The three-phase voltage system can be transformed to the multi-phase voltage system by using inductivity or gate-based power transducers. For example, in order to obtain the six-phase voltage system it is required to have the three-phase transformed of the required power rating containing one primary ant two secondary windings of identical parameters [2]. The primary windings of such transformer are star-connected and two secondary three-phase windings are star-connected with a common neutral node (the endpoints of the first secondary winding are short-circuited with the start points of the second secondary winding). Thus the primary windings A, B, C together with the secondary windings [a.sub.1], [b.sub.1], [c.sub.1] form a zero vector group of the transformer, and the same primary winding together with the secondary windings [a.sub.2], [b.sub.2], [c.sub.2] form a sixth vector group. So the windings of the secondary circuit create the symmetrical six-phase voltage system which can be used to supply the six-phase synchronous and asynchronous alternating current motors. Such motors are not manufactured yet in any country since it is not clear if they would have any advantages (or only disadvantages) compared to the three-phase alternating current motors. Only a few research topics regarding the analysis of the six-phase alternating current motors and their respective components were found [4,5].

The aim of this work is to investigate the six-phase winding of particular parameters by determining its electromagnetic efficiency and to compare it with the three-phase winding of analogous parameters.

The object of the research

The two-layer preformed six-phase winding was used in the research; winding parameters were: number of poles 2p = 2, number of sections in the group q = 2, number of slots in the stator magnetic circuit Z = 2p m q = 2 x 6 x 2 = = 24, pole pitch [tau] = Z/(2p) = 24/2 = 12, winding span y = = 5 [[tau]/6] = 5 x [12/6] = 10, magnetic circuit slot pitch in electrical degrees [alpha] = 360[degrees]p/Z = 360[degrees] x [1/24] = 15[degrees].

The distribution of the sides of active coils of this winding into the slots of magnetic circuit is given in Table 1.

The distribution of separate phase coils of analyzed six-phase winding into the slots of magnetic circuit based on parameters indicated in Table 1 is given in Table 2.

According to data of Tables 1 and 2 the development of electrical circuit of two-layer preformed six-phase winding is obtained (Fig. 1, a).

Research results

Assume that the relative quantities of the electric current amplitude values of considered six-phase winding are

[I.sup.*.sub.m1] = [I.sup.*.sub.m2] = ... = [I.sup.*.sub.m6] = 1. (1)

Then the instantaneous current values of the phase windings in the moment of time t = 0 would be:

[i.sup.*.sub.1] = [I.sup.*.sub.m1] sin [omega]t = 0; (2)

[i.sup.*.sub.2] = [I.sup.*.sub.m2] sin([omega]t - [2[pi]/6]) = [I.sup.*.sub.m2] sin(-60[degrees]) = -0,866; (3)

[i.sup.*.sub.3] = [I.sup.*.sub.m3] sin([omega]t - [4[pi]/6]) = [I.sup.*.sub.m3] sin(-120[degrees]) = -0,866; (4)

[i.sup.*.sub.4] = [I.sup.*.sub.m4] sin([omega]t - [6[pi]/6]) = [I.sup.*.sub.m4] sin(-180[degrees]) = 0; (5)

[i.sup.*.sub.5] = [I.sup.*.sub.m5] sin([omega]t - [8[pi]/6]) = [I.sup.*.sub.m5] sin(-240[degrees]) = 0,866; (6)

[i.sup.*.sub.6] = [I.sup.*.sub.m6] sin([omega]t - [10[pi]/6]) = [I.sup.*.sub.m6] sin(-300[degrees]) = 0,866. (7)

[FIGURE 1 OMITTED]

The relative size of effective conductors of one slot of magnetic circuit of the considered two-layer preformed six-phase winding with q = 2

[N.sup.*.sub.g] = [N.sup.*.sub.1]/q = 1/2 = 0,5; (8)

here [N.sup.*.sub.1] = 1--the relative size of effective conductors of magnetic circuit slot of the concentrated multi-phase winding.

Then the relative quantity of turns of single coil of two-layer winding

[N.sup.*.sub.c] = [N.sup.*.sub.g]/2 = 0,5/2 = 0,25. (9)

The expression of conditional change of magnetomotive force in the slot of two-layer winding is the following

[F.sup.*.sub.g] = [F.sup.*] + [F.sup.*]' = [i.sup.*.sub.i][N.sup.*.sub.c] + [i.sup.*.sub.i]' [N.sup.*.sub.c]; (10)

here [F.sup.*], [F.sup.*]'--changes of magnetomotive forces induced in the slot of active sides of upper and lower coils; [i.sup.*.sub.i], [i.sup.*.sub.i]'--relative quantities of instantaneous current values flowing through the upper and lower sides of coils located in the slots of respective phase windings.

From formulas (2/7), (9) and (10) and also no the base of Fig. 1, a, the conditional changes of magnetomotive force in the slots of magnetic circuit are determined (Table 3).

According to the results in Table 3 the spatial distribution of instantaneous rotating magnetomotive force in the set moment of time (t = 0) is determined (Fig 1, b). The obtained results allow to state that the symmetric six-phase current system in the distributed symmetric six-phase winding will induce the stair-shaped curves of magnetomotive force which move in space and periodically vary over time. The step functions of such rotating magnetomotive force will have only odd harmonics since they will be symmetrical in respect of coordinate axes in any moment of time. After application of the principle of superposition such amplitudes of odd harmonics of instantaneous rotating magnetomotive force can be calculated according to the following equation [6]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

here [F.sub.is]--conditional height of the i-th rectangle of the stair-shaped curve of instantaneous rotating magnetomotive force; [[beta].sub.i]--width of the i-th rectangle, expressed in electrical degrees of the fundamental space harmonic; k--the number of rectangles which constitute the stair-shaped curve of instantaneous rotating magnetomotive force; v--series number of odd space harmonic.

On the base of Table 4 and Fig. 1, b the parameters of positive half-period of instantaneous rotating magnetomotive force of the considered winding are the following: k = 5; [F.sub.1s] = 0,433; [F.sub.2s] = 0,433; [F.sub.3s] = 0,433; [F.sub.4s] = 0,2165; [F.sub.5s] = 0,2165; [[beta].sub.1] = 165[degrees]; [[beta].sub.2] = 135[degrees]; [[beta].sub.3] = = 105[degrees]; [[beta].sub.4] = 75[degrees]; [[beta].sub.5] = 45[degrees].

Based on these parameters of rotating magnetomotive force function the conditional magnitudes [F.sub.sv] and relative magnitudes [f.sub.v] of the space harmonics of magnetomotive force created by six-phase winding were calculated using expression (11) (Table 4).

In essence the magnetomotive forces of the higher-order space harmonics have a negative impact on the operation of the alternating current electrical machines, consequently they and their absolute relative magnitudes can be considered as negative. Negative relative magnitudes of magnetomotive force of the higher-order harmonics can be combined into a single equivalent relative magnitude which has to be compensated by magnetomotive force of the fundamental space harmonic. On the basis of this reasoning the six-phase windings, similarly as three-phase windings, from the electromagnetical point of view can be evaluated using the electromagnetical efficiency factor which is expressed as [7]

[k.sub.ef] = 1 - [square root of [[infinity].summation over (v=1)] [f.sub.v] - 1]; (12)

here [f.sub.v]--the relative magnitude of the v-th harmonic of rotating magnetomotive force.

On the basis of results of [f.sub.v] presented in Table 5 the electromagnetical efficiency factor of two-layer preformed six-phase winding calculated according to expression (12) is [k.sub.ef] = 0,930. This determined factor will be compared to the electromagnetical efficiency factor of the analogous three-phase winding.

For the purpose of comparison of analyzed winding the two-layer preformed three-phase winding of the following parameters was used: number of poles 2p = 2, number of sections in the group q = 2, number of slots of stator magnetic circuit Z = 2p m q = 2 x 3 x 2 = 12, pole pitch [tau] = Z/(2p) = 12/2 = 6, winding span y = 5 [[tau]/6] = 5 x [6/6] = 5, magnetic circuit slot pitch in electrical degrees [alpha] = 360[degrees]p/Z = 360[degrees] x [1/12] = 30[degrees]. The development of the electrical circuit of this winding and the spatial distribution of its rotating magnetomotive force in the moment of time t = 0 is shown in Fig. 2 [8].

For this three-phase winding the relative magnitudes of the number of effective conductors of single magnetic circuit slot and number of coil turns are the same as in case of the analyzed six-phase winding. The instantaneous values of the phase winding currents expressed in relative magnitudes in the moment of time t = 0 will be the following:

[i.sup.*.sub.U] = [I.sup.*.sub.mU] sin [omega]t = 0; (13)

[i.sup.*.sub.V] = [i.sup.*.sub.mV] sin([omega]t - [2[pi]/3]) = [i.sup.*.sub.mV] sin(-120[degrees]) = -0,866; (14)

[i.sup.*.sub.W] = [i.sup.*.sub.mW] sin([omega]t - [4[pi]/3]) = [i.sup.*.sub.mW] sin(-240[degrees]) = 0,866. (15)

[FIGURE 2 OMITTED]

The changes of magnetomotive force in the slots of magnetic circuit calculated according to expression (10) are graphically illustrated in Fig. 2, b. The harmonic analysis of instantaneous rotating magnetomotive force function of three-phase winding was performed on the basis of expression (11). The parameters of negative half-period of this magnetomotive force space function are such: k = 3; [F.sub.1s] = -0,2165; [F.sub.2s] = -0,433; [F.sub.3s] = -0,2165; [[beta].sub.1] = 180[degrees]; [[beta].sub.2] = 120[degrees]; [[beta].sub.3] = 60[degrees] [8].

Conditional magnitudes [F.sub.sv] and relative magnitudes [f.sub.v] of the space harmonics of rotating magnetomotive force created by the three-phase winding are given in Table 5.

On the basis of calculation results of [f.sub.v] presented in Table 5 the electromagnetical efficiency factor of two-layer preformed three-phase winding calculated according to expression (12) is [k.sub.ef] = 0,856.

After comparing the electromagnetical efficiency factors of two windings it is determined that this factor for the six-phase winding is 8,64% higher that in case of the three-phase winding. Furthermore, the conditional magnitude of rotating magnetomotive force of the fundamental harmonic of the six-phase winding is 1,983 times higher compared to the three-phase winding.

Conclusions

1. The theory of the three-phase windings is suitable for the creation of the six-phase windings of alternating current electrical machines.

2. Since in the symmetrical six-phase voltage system the angles between adjacent phase vectors is 60[degrees], thus the beginnings of the phases of the six-phase winding are allocated in space every 60 electrical degrees.

3. The rotating magnetomotive force created by the six-phase winding becomes closer to the sine distribution due to increased number of steps in its spatial distribution, compared to the rotating magnetomotive force of the three-phase winding of analogous parameters.

4. The electromagnetical efficiency factor of the six-phase winding ([k.sub.ef] = 0,930) is 8,64% higher than of the three-phase winding of analogous parameters ([k.sub.ef] = 0,856).

5. The amplitude value of the fundamental harmonic of rotating magnetomotive force of the six-phase winding ([F.sub.m1] = 1,767) is almost two times higher that the amplitude value of rotating magnetomotive force of the three-phase winding ([F.sub.m1] = 0,891).

References

[1.] Krause P. C., Wasynczuk O., Sudhoff S. D. Analysis of Electric Machinery.--The Institute of Electrical and Electronics Engineers, New York: McGraw-Hill, 1995.--564

[2.] [TEXT NOT REPRODUCIBLE IN ASCII], 2002-606 c.

[3.] [TEXT NOT REPRODUCIBLE IN ASCII], 1989-399 c.

[4.] Glukhov D. M., Muravleva O. O. Multiphase Induction Motors for a Variable Speed Drive // The 9th International Scientific and Practical Conference "Modern Technique and Technology".--Tomsk, TPU Press, 2003.--P. 128-130.

[5.] Bugenis S. J., Vanagas J., Gecys S. Optimal Phase Number of Induction Motor with the Integrated Frequency Converter. // Electronics and Electrical Engineering.--Kaunas: Technologija, 2008.--No. 8(88).--P. 67-70.

[6.] Buksnaitis J. Trifaziij zadinimo apvijij kuriamij magnetiniij laukij analizo // Elektronika ir elektrotechnika.--Kaunas: Technologija, 2003.--Nr. 1(43).--P. 43-46.

[7.] Buksnaitis J. New Apprroach for Evaluation of Electromagnetic Properties of Three-Phase Windings // Elektronika ir elektrotechnika.--Kaunas: Technologija, 2007.--Nr. 3(75).--P. 31-36.

[8.] Buksnaitis J. Kintamosios sroves trifaziij elektros masimj apvijij elektromagnetinis efektyvumas: monografija.--Lietuvos zemes ukio universitetas, Kaunas: Technologija, 2007.--196 p.

Received 2011 10 19

Accepted after revision 2011 12 15

J. Buksnaitis

Department of Agroenergetics, University of Aleksandras Stulginskis, Akademija, LT--53361 Kaunas distr., Lithuania, phone: +370 7 397529, e-mail: juozas.buksnaitis@asu.lt

doi: 10.5755/j01.eee.119.3.1352
Table 1. Distribution of the sides coils of two-layer preformed
six-phase winding into the slots of magnetic circuit

Phase change     U1       W2       X1       Z2       V1       U2

Number of coils  2        2        2        2        2        2
in a group

Slot No.  Z     1; 2     3; 4     5; 6     7; 8    9; 10    11; 12
          Z'   11; 12   13; 14   15; 16   17; 18   19; 20   21; 22

Phase change     Y1       X2       W1       V2       Z1       Y2

Slot No.  Z    13; 14   15; 16   17; 18   19; 20   21; 22   23; 24
          Z'   23; 24    1; 2     3; 4     5; 6     7; 8    9; 10

Table 2. The distribution of separate phase coils of
Analyzed winding into the slots of magnetic circuit

     1 phase              2 phase             3 phase
       (U)                  (X)                 (V)

[right arrow]1-11    [right arrow]5-15   [right arrow]9-19
             2-12                 6-16               10-20

[left arrow]21-21    [left arrow]15-1    [left arrow]19-5
            12-22                16-2                20-6

     4 phase              5 phase
       (Y)                  (W)

[right arrow]13-23   [right arrow]17-3
             14-24                18-4

[left arrow]23-9     [left arrow]3-13
            24-10                4-14

     1 phase              6 phase
        (U)                  (Z)

[right arrow]1-11    [right arrow]21-7
      2-12                 22-8

[left arrow]21-21    [left arrow]7-17
      12-22                8-18

Table 3. Conditional changes of magnetomotive force in the slots
of magnetic circuit in the moment of time t = 0

Slot No.           1       2       3       4       5       6       7

Conditional      0,216   0,216   0,433   0,433   0,433   0,433   0,433
  changes of
  magnetomotive
  force

Slot No.            8       9      10     11   12    13      14

Conditional       0,433   0,216   0,216   0    0    0,216   0,216
  changes of
  magnetomotive
  force

Slot No.          15      16      17      18      19      20      21

Conditional      0,433   0,433   0,433   0,433   0,433   0,433   0,216
  changes of
  magnetomotive
  force

Slot No.           22     23   24

Conditional       0,216   0    0
  changes of
  magnetomotive
  force

Table 4. Harmonic analysis results of rotating magnetomotive
force space function of two-layer preformed six-phase winding
with q = 2 and the relative magnitudes of its space harmonics

No. of space     1       5       7       11       13      17
harmonics

[F.sub.sv]     1,767   -0,02   0,011   -0,021   -0,018   0,005

[f.sub.v] =      1     0,011   0,006   0,012    0,010    0,003
  [f.sub.sv]/
  [f.sub.s1]

No. of space     19      23      25      29      31      35      37
harmonics

[F.sub.sv]     0,005   0,077   0,071   0,003   0,003   0,007   0,006

[f.sub.v] =    0,003   0,044   0,040   0,002   0,002   0,004   0,003
  [f.sub.sv]/
  [f.sub.s1]

Table 5. Harmonic analysis results of instantaneous rotating
magnetomotive force space function of two-layer preformed
three-phase winding with q = 2 and the relative magnitudes of its
space harmonics

No. of space      1        5       7       11       13      17
harmonics

[F.sub.sv]     -0,891   0,013   -0,009   0,081   -0,069   0,004

[f.sub.v] =       1      0,015   0,010    0,091   0,077    0,004
  [F.sub.sv]/
   [F.sub.s1]

No. of space    19      23      25      29      31      35      37
harmonics

[F.sub.sv]    0,003   0,039   0,036   0,002   0,002   0,025   0,024

[f.sub.v] =   0,003   0,044   0,040   0,002   0,002   0,028   0,027
  [F.sub.sv]/
  [F.sub.s1]
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Title Annotation:ELECTRICAL ENGINEERING/ELEKTROS INZINERIJA
Author:Buksnaitis, J.
Publication:Elektronika ir Elektrotechnika
Article Type:Report
Geographic Code:4EXLT
Date:Mar 1, 2012
Words:2765
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