Pole Combinations for

Study of the Number of Slots/Pole Combinations for Low Speed Permanent Magnet Synchronous Generators M.V. Cistelecan*, Mircea Popescu**, Mihail Popescu *

Research Institute for Electrical Machines, 050881 Bucharest, 45 Bd.T.Vladimirescu, ROMANIA ** SPEED Laboratory, University of Glasgow, UK

Abstract -The paper is evaluating the main performances of the low speed, multi-pole permanent magnet synchronous generator provided with fractional, tooth concentrated windings. The constructive aspects versus running characteristics as cogging torque and full load/no-load voltage are presented for surface mounted permanent magnets generators. Appreciations are made about the terminology in the specific field in order to prevent the wander of the defined notions from the classical acceptance. The armature reaction mmf is analyzed as electrical fundamental and fractional space harmonics and experimental facts are reported for the small power synchronous generators up to 10 kW/200 rpm. Indexing terms: fractional tooth concentrated winding, magneto-motive force (mmf), permanent magnet, low speed synchronous generator, wind energy conversion

I. INTRODUCTION Permanent magnet (PM) synchronous generators are used as preferred solution for applications having low speed primary mover. Such applications are wind turbines and micro-hydro-turbines. In order to decrease the dimensions of the electrical energy converter, the low speed primary mover requires a mechanical gear box to increase the speed. Low speed directly driven permanent magnet synchronous generators are also characterised by a minimum periodically maintenance and additionally losses of the gear box [1, 2]. This type of generator is inevitably large from the power density point of view, but has some advantages compared to the high speed machines coupled to a speed multiplier [3]. The wind turbine speed can be controlled via the angle of the blades. Two main features of this type of electrical machine need further investigations and experimentally confirmations. Firstly, due to the interaction between the magnets and the variable permeance of the stator, the corresponding torque ripple component, i.e., the cogging torque, may be very important. Having in mind the wind turbine applications, if the rated power occurs at a rated speed of the wind that is on average 10 m/sec, the wind turbine has to be started (at noload) at a wind speed of 2.5 – 3 m/sec. Taking into account

This work was partially supported by Romanian National Authority for Scientific Research (ANCS) on contract basis.

that the delivered power varies with the wind speed at the power of 3, in an optimised design the cogging torque should be less than 1.5 – 2.5% from the rated torque. Secondly, due to the high number of poles fractional slots/pole configurations have to be used. This will also minimise the cogging torque. The number of slot per pole and phase is in the range of 0.25 – 0.5. In this case the phase coils are so-called tooth-concentrated double layer coils, being much simpler than the windings for integer slots/pole configurations. Unfortunately, these windings have relatively low fundamental winding factor and with an important harmonics content that will cause extra-heating, additional losses and vibration. Previous works [4] showed that the electromagnetic properties of the PM machines are highly dependent on the number of slots per pole and phase as well the shape of the magnets, the stator slots and the slot opening. A large amount of permanent magnet material has to be used if the maximum air gap flux density is higher than 0.8 – 0.85 T. The pull-out torque versus the magnet weight is highest when the the magnets pole arc width is in the range 0.65 – 0.8 p.u., as referred to the pole pitch. The cogging torque can be minimised by using several methods, such as: fractional slots/pole winding, suitable magnet pole arc width and small slot opening width. However, because the problem of selecting fractional slots/pole geometry and the corresponding windings remains partly unsolved [5], it is important to study the fractional slot machines carefully, using actual prototype machines. In this paper, some practical experience will be reported based on three phase generators of 5 , 7.5, 10 kW at 200-240 rpm for wind energy conversion. From the winding point of view the multi-pole magnetic structure imply a fractional three phase winding; the investigated number of slots per pole is 36/24 (q=0.5), 36/26 (q=0.462) and 36/30 (q=0.4). Some experimental facts will be reported in the paper regarding the spatial space harmonics of the mmf armature reaction, the wave-shape of the line to line voltage, the influence of the number of slot/pole/phase (q) and the length of the air gap on the cogging torque and the full load to no-load voltage. Some clarifications are made about the specific terminology that is necessary for these special electric machines.

II. TOOTH CONCENTRATED WINDINGS A. Theory The tooth concentrated windings have the coil pitch equal to one. Therefore, the manufacturing of such three phase windings demands a value of q equal or less than ½, otherwise the shortening factor would be too low. To verify this assertion, we can analyze one simple 3-phase winding with Z=6 slots and two poles (q=1, “concentrated winding” in the classical acceptance of the word). If the coil pitch is y=3, the diametral winding will have the fundamental winding factor kw1=1. If y=2, kw1=0.866 will result. If the winding is “tooth concentrated”, y=1 and the winding factor becomes kw1 = 0.5 leading to a double the number of turns/phase for the same motor performances. In the limit case of q=½ (number of slots equal to 1.5 times the number of poles), there is no “distribution” of the coils, the fundamental winding factor will be kw1 =0.866 as it is the pitch factor. The most important aspect to be emphasized is the general equation of the pitch factor which can be simply expressed for a tooth concentrated windings as: q=

Z ; 6p

y = 1;

yτ =

Z ; ⇒ 2p

k y = sin 90 o

300 y (1) = sin yτ q

It is clear that the only value leading to the maximum value (1) of the pitch factor is q=1/3, not a favorable value due to the unbalanced resulting winding. Also, to limit the minimum value of ky to 0.866 q must be in the range ¼ …½. However there are values of q in a well defined range leading to the pitch factor bigger than a given value: TABLE I MAX. AND MIN. VALUES OF SHORTENING FACTOR KY Range of k y Minimum q Maximum q ky.> 0.866 0.25 0.50 ky > 0.90 0.2589 0.4676 ky > 0.95 0.2773 0.4178

Fig. 1. Pitch factor ky versus number of slot/pole/phase

The allowed range for q in order to obtain for example ky > 0.95 is quite large (Fig. 1). If a machine with 40 poles is to be built, the number of slots should be in the range of 33-48. This means there are 6 possibilities, taking into account the

values multiple of 3. It is obvious that 39 or 42 slots will lead to the maximum value of the winding factor. Related to these windings some misconceptions that have been created. The advantages of using of ‘tooth concentrated” windings are often exaggerated and the disadvantages are neglected. They are presented as windings that require new methods of analysis, new methods of calculating the winding factors of the fundamental wave and of the space harmonics. In fact the tooth concentrated windings were imposed as only possibility for the low speed permanent magnet multi-pole machines where the pole pitch expressed in slots number is close to one. As the methods of analysis and synthesis are concerned, these windings do not need any new methods and no new methods can be claimed. Calculating the space harmonics factors is done exactly using the old rules but in the new conditions of the pole pitch which is close to the tooth pitch. The classical star of the slots (star of the coils) is perfectly valid and in fact is used in different “new methods” of developing the “new concentrated windings” for both electrical fundamental and space harmonics [6, 7]. The fact that the general rules of developing classical windings are also applied for the “new” windings too has been presented in [8]. One should note the following features: -the pole number is an even number; -the offset between two consecutive phase windings has to be the same for the whole stator winding; -number of poles can not be equal to the number of slots; -number of the slots must be a multiple of number of phase. The following counter-example is given in order to clarify the so-called “end winding length” advantage of tooth concentrated windings. Let us compare two situations regarding the same machine: -72 slots, 24 poles, q=1, coil pitch y= 3 slots (concentrated, but not “tooth concentrated” winding, fundamental winding factor kw1=1) -36 slots, 24 poles, q=0.5, coil pitch y=1 slot (tooth concentrated winding, fundamental winding factor kw1=0.866, equal to pitch factor) To compare the two machines at the same armature lengths and PM structure, we have to take into account not only the end winding length, but the necessity to put 15% more turns on the stator winding to compensate the lower winding factor. The result of the comparison is depending on the length of the armature and the percentage of the length of the end winding to the total length of the turn. The advantage of the “tooth concentrated” winding from this point of view is not obvious and should be analyzed from case to case. The requirement of a minimized cogging torque imposes a fractional q and this is what will make the difference. However the “tooth concentrated” windings produces much more greater space harmonics of armature reaction compared to the common used distributed windings. Due to these harmonics the tooth concentrated windings can not be used without serious design problems to the low speed induction motors where the rotor cage strongly interact (as a mirror) with the stator waves. It is not the case of PM

synchronous machines where the high energy magnets, such as sintered NdFeB, have high value of electrical resistivity (a typical value is in the range 1.1 – 1.5 Ωmm2/m, ten times higher than pure iron which has the resistivity six times higher than copper). The iron hub as support of the PMs is far enough to the inner surface of the stator, however some additional eddy current losses in the iron hub are reported and analyzed in case of low speed direct driven big power wind generators [9]. B. Terminology Some specific aspects are to be taken into consideration. The opinion of the authors is that the classical notions should be modified or re-interpreted only in well justified cases. In the following only two problems are emphasized. a) One should note the necessity to make difference between the terms “tooth concentrated” and classical “concentrated” windings, having different meanings. “Concentrated winding” is that winding having one diametrical coil per each polar pitch, all the coils having the same emf as phase, the addition being pure algebraically. In classical acceptance “concentrated winding” is in opposition with “distributed winding” in which the induced EMF has different phases in different coils, the addition being made vectorially, leading to the so called “distribution factor”. In this acceptance, it results that “tooth concentrated” winding may be concentrated (example double layers, Z=12, 2p=8, m=3, q=0.5) or distributed (example double layers, Z=12, 2p=10, m=3, q=0.4). b) The well known “single layer” or “double layers” winding became in the new literature [8] windings with “all teeth wound”, respective “alternate teeth wound”. The new terminology is more complicated and there is no reason to accept it, even the double layer in the specific case may be done by sharing the slot on the width (not only on the height as in the classical windings).

the air gap and to compensate the field due to armature reaction in the load. The design allows for high value of the air gap flux density up to 0.9 – 1 T. B. Analysis of the armature reaction air gap mmf Due to the high number of poles, the winding is sometimes manufactured with tooth coils. Depending on the ratio between the number of slots, Z and the number of poles, 2p the winding is shortened or lengthened. Closer the numbers Z and 2p lead to closer shortening factor to one (1). The double layer winding has some advantages regarding the minimisation of some space mmf space-harmonics [6]. Except the electrical fundamental mmf that is rotating synchronously with the PM system, all other space harmonics mmfs will create torque ripples and additional eddy current losses [7]. To optimize the electrical system one can analyze and select carefully the combination between the slots and poles numbers [9].

III. EXPERIMENTAL AND COMPUTED RESULTS A. Magnetic design Some attempt were made in order to reduce the cogging torque in a given stator lamination configuration and surface mounted permanent magnets in the rotor. Three possibilities of laying the magnets were experimented as it is presented in figure 2 (uniform distributed, pairs of magnets having the distance between them bigger and smaller respectively, only one pair of magnets having bigger distance). The decreasing of the cogging torque is not significant, however in the last two attempts there is a decreasing in the no load emf, the result of the layout of the magnets is equivalent with a smaller fundamental winding factor. The progressively increasing of the air gap for a given machine was experimented too (results in figure 3). The magnets taken into consideration are NdFeB type with a remanence flux-density of 1.15 T and coercitivity field-strength of 8360 A/cm as typical values. The magnets are glued on a ferromagnetic hub being alternatively magnetized. Each magnet has to cover the magnetization of

Fig. 2. Three configurations of the magnetic system

The phase mmf can be analysed starting from the input current and the actual distribution of the coil branches in the slots. For that purpose an angular reference, a system should be defined. If the geometrical space angle of the slot “k” is θk and it contains Nck conductors of phase A, w is the total number of turns per phase, c is the number of parallel path, kwν the ν-order phase winding factor and ϕνA the space angle

of the ν-order space harmonic, the general equations of the winding distribution are [10]:

cogging torque [kgfm]

1.5



= τ dC + τ dV

0.5

0.0 0

1

2

th e a i r g a p [m m ]

Fig.3 Influence of the air gap on the cogging torque

ϕνA =

1

ν

  ∑ N ck cos νθ k   k∈{K A } arctg

2

   +  ∑ N ck sin νθ k     k ∈{K A }

∑N

ck

sin νθ k

∑N

ck

cos νθ k

k ∈{K A } k ∈{K A }

2

  ;  

(2)

The set of the armature’s slots containing conductors of phase A was noted by KA. The number of conductors Nck is including the sign (±) depending on the current sense in the respective slot. Taking into account that generally the currents of the three phase machines are time delayed by 2π/3 the resulting air gap mmf can be written as a sum of CW and CCW rotating waves: Frez (θ , t) =

3wI 2

π

∞

∑[ ν =1

k wdν

ν

sin(νθ − ω t) +

k wiν

ν

sin(νθ + ω t)]

(3)

where the winding factors kwν are: kwdν =



p  τ d =   k  wp 

1.0

2 cwk w ν A =

general case in the paper this coefficient is defined as having two components, the first is time-constant (τdC) and can be calculated exactly as in the literature, the second is timedependent (τdV) and its magnitude can characterize the degree of unbalance of the machine:

2 2 C1ν + C22ν ; 3

k wiν =

2

 k wdν  2  k wdi  2  k k  +  + 2 wdν 2 wiν cos 2ω t  =  ∑ ν ν≠p   ν   ν  cos 2ωt

(5)

The presence of the time-dependent coefficient τdV in the equations is associated only to the unbalanced machines, for example when q has a denominator multiple by the number of phases. In the paper, as the studied windings are balanced, the coefficients τdC is used as main criteria of the design. C. Cogging torque experimental results Two winding configurations with Z/2p equal to 36/26 and 36/30 have been manufactured in double layers. In figure 4 it is presented the 36 slots, 26 poles winding arrangement with all the phase coils series connected (there is for this configuration also the two parallel paths possibility). The armature reaction mmf (the mmf produced only by the load current in the phase windings) is computed and represented in figure 5 taking into account the fundamental wave (ν=13) and only the biggest two sub-harmonics (ν=5 and ν=7). It is interesting to note that the main harmonic is modulated by the sub-harmonics and the magnitude of the resulting rotating wave has no constant magnitude under different poles. The armature reaction space harmonics will be damped by magnetic parts of the rotor by additional losses due to the eddy currents [9]. Some experimental facts as cogging torque is concerned, are reported in Table 2 for the investigated machines.

2 D12ν + D22ν 3

1 2π 2π C1ν = [kwAν cosϕνA + k wBν cos(ϕνB + ) + k wCν cos(ϕνC − )]; 2 3 3 1 2π 2π C2ν = − [k wAν sinϕνA + kwBν sin(ϕνB + ) + k wCν sin(ϕνC − )] 2 3 3 1 2π 2π D1ν = [k wAν cosϕνA + kwBν cos(ϕνB − ) + k wCν cos(ϕνC + )]; 2 3 3 1 2π 2π D2ν = − [kwAν sinϕνA + kwBν sin(ϕνB − ) + k wCν sin(ϕνC + )] 2 3 3

(4)

Fig.4. Winding diagram for 36 slots/26 poles, double layer, series connection. The six terminals are considered allowing star or delta connection

The amplitudes of the magnetic waves calculated by (2) are calibrated by the factors 2/3 in order to be comparable by the phase MMF winding factors. The essential assumption for the previous equation is that the system of the phase currents keep on being symmetric despite the spatially unbalanced distribution of the phase conductors. A remarkable property of the three phase windings is the differential reactance coefficient (τd) that characterizes the space harmonic content of the air gap mmf. A general method of calculation the τd for pure waves is given in [11] starting from the magnetic energy stored in the air gap. In the most

Fig.5. Armature reaction air gap mmf (resulting wave as the sum between 5-th, 7-th and 13-th order) for 36 slots/26 poles winding

TABLE II EXPERIMENTAL RESULTS COGGING TORQUE ANALYSIS Max. Cogging Starting Rated Z/(2p) q Magnet Rated torque wind power width torque, Cogging torque, % speed and speed to pole kgfm kgfm m/s kW/rpm pitch ratio 5/240 72/24 1 0.75 20.28 1.67 8.23 4.35 5/240 36/24 0.5 0.75 20.28 1.08 5.32 3.76 5/240 36/26 0.462 0.813 20.28 0.35 1.72 2.58 7.5/200 36/26 0.462 0.813 36.50 0.65 1.78 2.61 10/200 36/26 0.462 0.813 48.67 1.03 2.12 2.77 Comparing the last three lines in the table containing machines with exactly the same cross section but the ratio of the length 1:2:3, it is clear that the cogging is almost proportional to the length. The main measure to keep low the cogging torque is neither the magnet width to pole pitch ratio, nor the skewing the stack of lamination but is the proper choice of the stator slot per pole and phase, defining the magnetic structure periodicity. IV. CONCLUSIONS The number of slots/pole combination theoretical and experimental analysis is made with a focus on the low speed PM synchronous generators. The fractional slots/pole windings exhibit an important space mmf harmonics content and thus torque ripple and additional core losses are present. The low speed multipole machines will have inevitably fractional winding and both mmf upper and sub-harmonics will appear from spatial point of view. These space harmonics, especially those rotating in opposite side related to the PM system will induce eddy currents in the iron components of the rotor leading to additional load losses. To reduce these space harmonics it is compulsory to use double instead of single layer winding. Above the whole philosophy of the permanent magnet width or the ratio of permanent magnet width to pole pitch, the cogging torque is essentially determined by the proper

combination of number of slots and number of pole pairs. It is necessary for these two numbers to have no common divisors. ACKNOWLEDGMENT The authors would like to thank to the Romanian R&D National Programme “AGRAL” and to the Research Agency “AMCSIT-Politechnica” for the permission and support for presenting this paper. REFERENCES [1] Grauers, A.: “Directly driven wind turbine generators”, ICEM’96, Vigo (Spain), Sept. 1996, vol.2, pp.417-422 [2] Lampola, P.: “Electromagnetic design of an unconventional directly driven permanent magnet wind generator”, In Proceedings of the International Conference on Electrical Machines (ICEM'98), Istanbul, Turkey, 2-4 September, 1998, Vol. 3, pp. 1705-1710 [3] Fathiyah, R., Mellot, R., Panagoda, M.: “Windmill design optimization through component costing:, http://www.mth.msu.edu/Graduate/msim/MSIMProjectReports/MCP1.M ay.2001.report.doc. [4] Lampola, P.: “Directly driven low speed PM generators for wind power applications”, PhD Thesis, Acta Politechnica Scandinavica, Electrical engineering series, No.101, Espoo, 2000. [5] Salminen Pia, Niemala, M., Pyrhonen, J., Mantere, J.: “Performance analysis of fractional slot wound PM-motors for low speed applications”, IEEE Conference IAS-2004, pp. 1032-1037. [6] Mehdi, T. Abolhassani: A new concentrated windings surface mounted permanent magnet synchronous machine for wind energy application”, IEMDC-2005, p.931-936 [7] Magnussen, F., Sadarangani, C.: “Winding factors and Joule losses of permanent magnet machines with concentrated windings”, IEMDC2003, p.333-339 [8] Skaar, S.E., Krovel, O., Nilssen, R.: “Distribution, coil span and winding factors for PM machines with concentrated windings”, ICEM-2006, Chania (Greece), Sept.2006, paper 346 [9] Nakano, M., Kometani, H., Kawamura, M.: “Permanent magnet dynamo electric machine and permanent magnet synchronous generator for wind power generation”, US Patent 6,894,413/May 17, 2005 [10] Cistelecan, M., Cosan, B., Popescu, M.: “Tooth concentrated fractional windings for low speed three phase a.c. machines”, ICEM’06, Chania, Greece, Sept. 2006, paper 362 [11] Heller, B., Hamata, V.: “Harmonic field effects in induction machines”, Academia Publishing House, Prague, 1977.