Analysis of Optimal Stator Concentric Winding Patterns Design

Wen Ouyang*, Ahmed. EL-Antably**, Surong Huang***, T.A.Lipo*

*Department of Electrical and Computer EngineeringUniversity of Wisconsin-Madison

1415 Engineering DriveMadison, WI 53706

" Allison TransmissionDivision of General Motors

P.O.Box 502650Indianapolis, Indiana 46250

* * *Department of AutomationShanghai University

Shanghai, 200072, P.R.China

Abstract- Most small induction motors in current use employconcentric windings which have equal numbers of turns per coil.An optimal choice of turns can achieve better distribution ofMMF in the gap while keeping the turns in each slot essentiallythe same. In this paper, the air gap Magnetic Motive Force (MMF)harmonics of electrical machine created by the stator concentricwindings with sinusoidal current injection are investigated. Theair gap MMF is modeled ideally by a linear winding function,which simplifies the MMF expression significantly by neglectingthe slot effects and saturation for uniform air-gap machines. Byevaluating the harmonic property of the synthesized three phaseair gap MMF waveform, optimal winding patterns can bedeveloped for conventional choices of phase belts.

Keywords - Concentric Windings, MMF, Harmonics,Induction Motor Windings

I. INTRODUCTION

Concentric windings are widely used in electrical machinesusing random wound coils. Random wound coils are moreamenable for factory automation compared with the copperbars used for higher power rating machines. They can benested in a manner such that the entire phase group windingscan be inserted into slots in one operation, while theoverlapping nature of the more traditional lap winding makesthem difficult to insert.

In the concentric winding coil fabrication, the coils for onephase belt are usually wound in a very simple manner, i.e., withequal turns for each coil. However, the automation processnowadays makes it quite easy for the coils to be wound moreflexibly without much cost increase. In this paper, the optimalconcentric winding patterns for various phase belts and slotnumber/pole/phase are investigated based on the windingfunction principle. Although the slot shape and statorsaturation effect also contribute significantly to the air gap fluxdistribution, the general expression based on the windingampere-turns for the air gap MMF distribution is theconventional means for distinguishing the harmonics due towinding layout from those due to saturation and by slotpermeance variations[1].

II. WINDING FUNCTION

In order to analyze the air gap field distribution by the statorexciting current for the uniform air-gap machine, a linear modelbased on ampere-turns is illustrated in Figure 1 [2]. Theassumptions are made here for the problem simplification:

1) g = so for the iron sections

2) g = to for the uniform air gap

3) Saturation and slot effects are neglected

4) Air gap flux is distributed uniformly between slots

5) Balanced sinusoidal current injection and symmetricwinding layout for the periodic flux distribution in the air gap.

Ni

(A=Ho) g(p PO))

LBJ

BRL=>x

Fig.1: Air gap flux model

As shown in Fig. 1, the distribution of air gap magneticmotive force (MMF) determines the air gap flux shape directlyin this model. The basic relation equations between MMF andflux density are provided in (1):

MMF= N i= RL BL.XP * d+ RR BR. (L -xP).d (1)

where:

N i: product ofwinding turns and current.

RL and RR: reluctances for the air gap sections at left andright side of the slot.

BL and BR: uniformly distributed flux for the air gap sectionsat left and right side of the slot.

d: depth of the model (directed normally into the paper).

With the reference set to the slot position, noting the flux isbalanced at left and right side of the slot for continuity (Gauss'Law), the flux densities for BL and BR can be obtained:

94

BNi L X XPg L

BR= 0Ni xP

q B(xN 3 Ex )[cos(O -(3n 2)(2) B(x,O0)=l n2rsn xjg VOIVj=l nl n / g T

(3) -1)cos(O +

By applying superposition principle, the air gap fluxdistribution for an arbitrary winding position can be easilyobtained as illustrated in Figure 2.

| Nlil N2i2

B

x. ~~~~IB1(x) x

Bt B1 (x)+B2(x)

x

Fig.2: Air gap flux by superposition

Thus, the air gap flux for the concentric winding can beconveniently defined. An example is provided in Figure 3.

Fig.3: Air gap flux for a typical concentric winding

The Fourier series for the B1(x) can be expressed as (4)utilizing the even function property.

B1(x) = ao + Eancos x (4)n=l1

where:

ao =-fB1 (x) dx = 0 (5)

an 2 B1 (x) cosTx dx = 2 Nil (t) sin-xn=- I~~~~~~~~~~~~~~~~~~~2 (6)

Note that (2) and (3) are used in the calculation of (6).

Thus, the air gap flux generated by one winding coil can beexpressed as (7) which combine the space position andassumed sinusoidal current injection angle 0.[3]

c 2pu0Ni(O) . nT nfTB(x,) (0 x)cos x

n=l nlg T T(7)

The actual air gap flux can then be synthesized from all ofthestator winding coils by superposition. With different turnnumbers Nj in each slot and q coils for each pole, the fullexpression in closed form for the three phase air-gap flux can

be derived as (8).

Note that the xj denotes the different coil spans, and I is thecurrent amplitude.

By selecting the turns number of each coil of the concentricwinding, the harmonic content in the resulting air gap MMF(flux) waveform can be minimized, which leads to thepossibility of an optimal winding pattern.

III. OPTIMAL WINDING PATTERNS

To optimize the concentric winding patterns, the minimumTHD is set as the objective function combined with the balanceof coil turns in each slot (the total turns difference for each slotis limited to 1 or 2). Four typical cases with standard phasebelts of 60, 90, 120, and 180 degree are briefly discussed in thispaper.

600phase belt

Table 1 describes the concentric winding layout under onepole span for 60 phase belt with q =3. The minus sign denotesthe reversed current direction for the windings in a given slot.'0' means no corresponding phase windings are in that slot.

Table.1: Winding layout for 60 degree phase belt, q 3.Slot# 1 2 3 4 5 6 7 8 9

Phase C 0 0 0 0 0 0 c c cPhase B b b b 0 0 0 0 0 0Phase A 0 0 0 -a -a -a 0 0 0

Optimal winding patterns for the 60 degree phase belt and q3 with different total coil turns per pole per phase are listed in

Table 2. The winding fundamental coefficient Kwl andwinding THD (considering up to the 99th harmonic) are alsoprovided. The optimal pattern for this case with only one layerof winding converges to an equal number of turns Ntotal foreach slot.

Table.2: Optimal winding patterns for 60 degree phase belt, q = 3.Ntotal NI N2 N3 Kwl THD

9 3 3 3 0.9598 0.113410 3 4 3 0.9638 0.118311 4 3 4 0.9561 0.115112 4 4 4 0.9598 0.113413 4 5 4 0.9629 0.1166

14 5 4 5 0.9569 0.114215 5 5 5 0.9598 0.113416 5 6 5 0.9623 0.115617 6 5 0.9574 0.113818 6 6 6 0.9598 0.1134

19 6 7 6 0.9619 0.115120 7 6 7 0.9578 0.1136

21 7 7 7 0.9598 0.113422 7 8 7 0.9616 0.114723 8 7 8 0.9581 0.113424 8 8 8 0.9598 0.113425 8 9 8 0.9614 0.114526 9 8 9 0.9583 0.113427 9 9 9 0.9598 0.113428 10 8 10 0.9569 0.114229 10 9 10 0.9584 0.113330 10 10 10 0.9598 0.1134

95

(8)

P.-_

Figure.4 illustrates the harmonic spectrum for the windingconfiguration of 60 phase belt, q = 3, with one turn coil per slotand with unit current injection. The harmonic with negativeamplitude, such as 19th, 35th, rotates in the opposite directioncompared with the fundamental. Since the layouts basicallyfollow the patterns of equal turns in each slot, the fundamentalwinding coefficient and wind THTD are maintained at valuesclose to each other. Basically, the symmetric three phasewinding and balance current injections can make thesynthesized air-gap MMF sinusoidal naturally. With equalturns coil in each slot, the irregularity created by the unequalstep height (unequal turns) is thus reduced to the minimum. Theincidental harmonics with this winding configuration, such as

5th, 7th 11th ..., also reach their minimum values with an equalturns pattern.

a 10 29 30 40 i X X 1 D

Figure 4: Typical harmonic spectrum for 60 phase belt, q 3, a1=2.75

For the higher order harmonics, it is interesting to note thatthe harmonics mainly concentrates around the multiples of theslot number covered by one pole pair. In this case, one pole pairspans 18 slots, resulting the attenuated harmonics of 17th, 19th,35th137th, .... These characteristic slot harmonics will appear inthe normal way at the order as described in equation (9) forthree phase machines [1].

hk 6kq +±1 (9)

900phase belt

The typical concentric winding layout under one pole spanfor 90 phase belt and q =2 is described in Table.3. Note that thewindings from different phases begin to share one slot.

Table.3: Winding configuration for 90 degree phase belt, q = 2.Slot# 1 2 3 4 5 6PhaseC 0 0 0 c T CPhaseB b b 0 0 0 -bPhase A 0 -a -a -a 0 0

The optimized winding patterns for 900 phase belt with q 2are provided in Table 4. The pattern numbers in the slots aredistributed in a symmetric manner due to the constraints of abalanced three phase winding configuration and balancedcurrent injection, which is also coincident with the total turnsnumber difference limit requirement.

Table.4: Optimal winding patterns for 90 degree phase belt, q= 2.Ntotal N1 N2 N3 Kwl THD

5 1 3 1 0.9464 0.16336 2 2 2 0.9107 0.16337 2 3 2 0.9234 0.14808 2 4 2 0.9330 0.14809 2 5 2 09405 0.154610 3 4 3 0.9196 0.150811 3 5 3 0.9269 0.146812 3 6 3 0.9330 0.1480

13 3 7 3 0.9382 0.152014 4 6 4 0.9234 0.148015 4 7 4 0.9286 0.146716 4 8 4 0.9330 0.148017 4 8 5 0.9291 0.150418 5 8 5 0.9256 0.147119 5 9 5 0.9295 0.146820 5 10 5 0.9330 0.148021 5 10 6 0.9298 0.149322 6 10 6 0.9269 0.146823 6 1 1 6 0.9301 0.146924 6 12 6 0.9330 0.148025 6 12 7 0.9303 0.148726 7 12 7 0.9279 0.146727 7 13 7 0.9305 0.147028 7 14 7 0.9330 0.148029 7 14 8 0.9307 0.148330 8 14 8 0.9286 0.1467

For comparison, the spectrum of the winding patterns withequal turns in slots (10-10-10) and optimized pattern (8-14-8)are provided in Figure 5. One of the most significantcontributions of the optimized winding pattern is theelimination of the harmonics of 5th and 7th, which benefits themachine efficiency, torque ripple, noise and vibrations, etc,although the total THID deduction looks small. Meanwhile, thewinding coefficient is also increased from 0.9107 (same as1-1-1) to 0.9286, which means a machine design with moreeffective material usage.

u~~~~ ~ ~ ~ ~ ~ ~ ~ ~~~~~~~~~~~~

26~~~~~~~~

T D ' "'"1'.2609Figure 5.a: Harmonic spectrum for q=2, winding pattern: 10-10-10.

THD=16.33%, al=26.09

X0 a0 0 4h so go i B

Figure 5.b: Harmonic spectrum for q=2, winding pattern: 8-14-8.THD=14.67%, a1=26.6.

Figure.6: Fundamental amplitude comparison of equal turns pattern andoptimized patterns for 90° phase belt, q = 4.

96

l . . '| ~~~~~:[ EqLual Turns Pattern |

l ' . | l Cl~~~~~~~ptimirized Pat:tern||

L-------- -~ ~~~ ~~~rn-- ---r---- ---- - -- -- -- -----

5~ ~~~ ~~~ ~~l05 20 25 3

:1:

Moreover, the fundamental of the air gap MMF is enhancedby the optimal winding pattern as well. Figure.6 illustrates thefundamental enhancement by the optimized patterns for the 900phase belt with q 4. The benefit of fundamental enhancementof the optimal pattern is obviously exhibited with an increasedwinding turns number, which directly contributes to the torquee

g 1@ 2@ ~~ ~ ~~~30 J qDproduction. Figure 7.b: Harmonic spectrum for q 2, winding pattern: 4-11-11-4.

THD=14.67%, a1=25.691200phase belt Similarly, the 5th and 7th harmonics are eliminated by the

The three phase winding configuration under one pole pitch optimal winding patterns. One can note also, that thefor 120 degree phase belt with q = 2 is provided in Table 5. fundamental component is strengthened.

Table.5: Winding configuration for 120 degree phase belt, q = 2.Slot # 1 2 3 4 5 6 180°phase belt

Phase B b |b b b c For completeness of the discussion, the optimal windingPhase A 0 -a -a -a -a 0 patterns for the 180 degree phase belt are also investigated. The

three phase winding configuration under one pole pitch span isThe optimized winding patterns for this winding illustrated in Table 7.

configuration are listed in Table 6. The symmetric property is Table.7: Winding configuration for 180 degree phase belt, q 2.still shared by those patterns. However, with an increased slot Slot # 1 2 3 4 5 6number under one pole pitch span, the fundamental coefficient Phase C -c -c c c c C

of the winding has deteriorated due to the short pitch effect. It Phase B b b b b -b -bis also interesting to note that the THDs converge to almost the r Phase A -a -a -a a -a -asame value, which is actually the combined effect of the The optimized patterns and the typical winding harmonicsintrinsic harmonics as depicted by the plots after optimization. spectrum are provided in Table 8 and Figure 8.

Table.6: Optimal winding patterns for 120 degree phase belt, q = 2.Ntotal NI N2 N3 N4 Kwl THD

4 1 1 1 1 0.8365 0.16335 1 2 1 1 0.8624 0.17436 1 2 2 1 0.8797 0.14807 1 2 3 1 0.8920 0.15828 1 3 3 1 0.9012 0.14689 1 3 4 1 0.9084 0.154010 1 4 4 1 0.9142 0.148011 1 4 5 1 0.9189 0.1531

12 2 4 4 2 0.8797 0.148013 2 4 5 2 0.8863 0.150614 2 5 5 2 0.8920 0.146815 2 5 6 2 0.8969 0.149316 2 6 6 2 0.9012 0.146817 2 6 7 2 0.9050 0.149018 2 7 7 2 0.9084 0.147319 3 6 7 3 0.8842 0.149020 3 7 7 3 0.8883 0.147021 3 8 7 3 0.8920 0.148122 3 8 8 3 0.8953 0.146723 3 8 9 3 0.8984 0.147824 3 9 9 3 0.9012 0.146825 3 9 10 3 0.9038 0.147926 3 10 10 3 0.9062 0.1471

27 4 9 10 4 0.8892 0.147828 4 10 10 4 0.8920 0.146829 4 10 11 4 0.8945 0.147430 4 11 11 4 0.8969 0.1467

Table.8: Optimal winding patterns for 180 degree phase belt, q = 2.Ntotal NI N2 N3 N4 N5 N6 Kwl THD

4 0 1 1 1 1 0 0.8365 0.15855 0 1 1 2 0 1 0.7727 0.15856 0 1 2 2 1 0 0.8797 0.14307 0 2 1 2 1 1 0.7540 0.14308 0 1 3 3 1 0 0.9012 0.14189 0 2 2 3 1 1 0.8011 0.141810 1 2 2 2 2 1 0.7210 0.14181 1 1 3 1 2 2 2 0.6554 0.141812 2 3 1 1 3 2 0.6008 0.141913 2 4 0 1 3 3 0.5546 0.141914 0 2 5 5 2 0 0.8920 0.141815 0 3 4 5 2 1 0.8325 0.141816 1 3 4 4 3 1 0.7805 0.141817 1 4 3 4 3 2 0.7346 0.141818 1 3 5 5 3 1 0.8011 0.141819 2 3 5 4 4 1 0.7589 0.141820 2 4 4 4 4 2 0.7210 0.141821 2 5 3 4 4 3 0.6866 0.141822 0 3 8 8 3 0 0.8953 0.141723 0 4 7 8 3 1 0.8564 0.141724 1 4 7 7 4 1 0.8207 0.141725 1 5 6 7 4 2 0.7879 0.141726 2 5 6 6 5 2 0.7576 0.1417

27 2 6 5 6 5 3 0.7295 0.141728 3 6 5 5 6 3 0.7035 0.141729 3 7 4 5 6 4 0.6792 0.141730 4 7 4 4 7 4 0.6566 0.1418

j,L

Figure 7.a: Harmonic spectrum for q 2, winding pattern: 7-8-8-7

THD 15.88o, a l24.21

Figure 8.a: Harmonic spectrum for q 2, winding pattern: 5-5-5-5-5-5.

THD 16.33o,al=18.448

97

jI .3 -.........M.....,

_

Figure 8.b: Harmonic spectrum for q 2, winding pattern: 4-7-4-4-7-4.THD= 14.67%, a1=18.81

Basically, the optimal patterns for the 180 degree phase beltshare similar properties as discussed before, while the windingfundamental coefficient further deteriorates. However theoverall result can be integrated into the optimization objectivesto obtain a higher fundamental coefficient for a givenapplication.

IV. CONCLUSION

This paper investigates optimal winding patterns for theconcentric three phase windings based on a traditional uniformair gap MMF harmonic analysis. The optimized patternscontribute to the reduction of TH4Ds, especially the eliminationof 5th and 7t MMF harmonics, which benefits the machineefficiency, noise and vibration. Meanwhile, the enhanced airgap MMF fundamental component favors torque production,and the higher winding coefficient makes the design of themachine materials more effective. The prevalence of automaticwinding fabrication processes should make it possible toproduce the concentric windings with different coil turns atnegligible additional cost.

REFERENCE

[1] T.A.Lipo, "Introduction to AC Machine Design", University ofWisconsin, 2004.

[2] Sang-Baeck Yoon, "Optimization of Winding Turns for ConcentricWindings with 180 and 120 degree phase belts", Report, University ofWisconsin, May, 1999.

[3] Hamid A. Toliyat, T.A. Lipo, J.C. White, "Analysis of a ConcentratedWinding induction machine for Adjustable Speed Drive Applications",IEEE Transaction on Energy Conversion, Vol.6, No.4, Dec, 1991.

98