Torque Performance of Axial Flux PM Fractional Open Slot Machine with Unequal Teeth

Torque Performance of Axial Flux PM Fractional Open Slot Machine with Unequal Teeth

E. Rakgati

University of Botswana

H. Kierstead

Botswana Power Corporation

E. Matlotse

University of Botswana

Abstract

Cogging torque ripple and torque ripple in electrical machines are generally considered as undesirable effects and results in rough operation, vibration and noise. This paper looks into minimization of these parasitic effects in axial flux permanent magnet machines with fractional open slots by employing a finite element coupled optimization procedure; particularly interest is paid to a single-layer machine with unequal teeth. Evaluation between the single-layer machine and its double-layer counterpart is highlighted, and the results show attractive performance of the single-layer machine with unequal teeth over its double-layer counterpart.

1. INTRODUC TION

   F

   F

   RACTIONAL slot permanent magnet (PM) machines are currently receiving increased attention for wind and electric vehicle applications. This is mainly attributed to their potential advantages in improved manufacturability, cost reduction and high power density levels over conventional radial flux machines. Amongst others, axial flux topologies with single-layer (SL) fractional open-slot windings are of particular interest as they are ideal options for pre-formed non-overlap modular coils. In certain pole/slot combinations, open slots show reduced ripple [1-4]. Certain PM machines with regular slot SL fractional windings show higher torque capacity than their double-layer (DL) counterpart [5-7]. The torque ripple of the former compares favourably to the latter when driven with trapezoidal wave currents, but with sinusoidal currents the effect is opposite [5]. Therefore this paper looks to investigate the possibility to improve the torque quality of SL machines under sinusoidal current

   excitation.

   To further enhance the torque performance of the SL PM

   machines, novel topologies of SL fractional slot machines with unequal teeth have been introduced in [8]. By the addition of unequal teeth, the slots become irregularly distributed and the winding factor in SL machines becomes adjustable, allowing for enhancement of the winding factor, an aspect not possible with DL structures. The typical method is to increase the tooth width around which the coil is wound and decrease width of the remaining teeth as shown by adapting Fig. 1a to 6. By this adaptation, the coil can link higher magnetic flux and better magnetic exploitation is achieved [9]. Nearly all the work done in this regard [8-12] is applied for trapezoidal wave currents, whereby the winding factor is fully maximised leading higher torque capacity and quality when compared to DL machines. In [11] and [12] a similar occurrence of increased capacity but with inferior quality as in [5] is reported when these topologies are driven with sinusoidal currents. Additionally the works [8-12] deal with radial flux structures with semi-enclosed slots, and the effects of the magnet pitch are not treated for except in [10]. In [2], an open-slot axial flux machine is presented with unequal teeth, but the machine presented is driven by trapezoidal currents and is modelled as a radial flux structure. Of all the works found, none specifically deal with particular optimization of machine parameters for the objective analysis of torque quality.

   This work aims to objectively improve torque quality in open-slot axial flux PM machines driven by sinusoidal currents, by full FE-coupled optimization of machine parameters affecting torque. The work involves comparative analysis of a SL and DL machine, in which both are optimized objectively for torque quality.

2. MACHINE TOPOLOGIES

   Fig. 1 shows the sectional layout of two fractional open

   slot (30-pole/36-slot) axial flux permanent magnet machines, in which (a) is of single-layer topology while (b) is of double- layer topology. The 30-pole/36-slot combinations are popular due to their high fundamental winding factors, high lowest common multiples (LCM), and high greatest common divisors (GCD). The machines were previously optimized separately for maximum torque density under sinusoidal excitation. Data of the machines is presented in Table 1.

   (a)

   (b)

   The cross sections of axial flux machines are not the same across their stack length and present no 2D symmetry as radial flux machines do, thus the full 3D modelling. The downside to full 3D modelling is heavy computational time required. To simplify this, axial flux machines can be modelled as 2D linear structures as in Fig. 1, normally based upon their average radius dr as in Fig. 2. This approach is suitable for solving most issues but when dealing with instantaneous torque profiling, more precision is required. To overcome this and avoid full 3D modelling, an alternative approach, adopted in this paper is multi-slice or quasi 3D modelling. The method involves modelling several linear 2D models, based on several diameter lengths along the machine stack, and taking the average of these.

   The axial flux machines in this paper are modelled as 1/6th sections with negative boundary conditions and an air-gap element as shown in Fig. 1. The torque performances of these machines are calculated by both the Maxwell stress tensor and virtual work methods given by

   Fig. 1. Base machine 1 and 2 models; (a) Single-layer

   with equal teeth, and (b) Double-layer.

   T pravg L

   2

   2

   0

   2 B B d

   2 B B d

   1 r

   and

   By initially optimizing each machine separately, leading to two base machines, each topology is in its optimum before

   T dW r

   VW ds avg

   W r ,

   s avg

   the analysis. This provides a fair base upon which to begin from, as compared to using one base machine which holds either a SL or DL winding; as the dimensions of a single base machine could be more suited to one winding type over the other.

   TABLE I

   DESIGN DATA OF TWO BASE MACHINES

   where p is the pole pairs, ravg the average airgap radius, L the machine axial length, Br and B the flux density components from the macro air-gap element, W is the magnetic co- energy, and s some small displacement.

   	Single Layer	Double Layer
   Stator outer diameter	330.0 mm	330 mm
   Total axial length	55 mm	55 mm
   Diameter Ratio	0.619	0.652
   Magnet arc to pitch ratio	0.915	0.9
   Slot to teeth width ratio	0.653	0.563
   Teeth width ratio	1	1

   3

   3

   Power Density 4366.39

   kW/m

   6343.55

   kW/m3

   Average Torque 361 Nm 334 Nm Per Unit p-p Cogging

   Per Unit p-p Ripple

3. TORQUE ANALYSIS

   1. Finite Element Modeling

      Fig. 2. Axial flux machine.

      The instantaneous torque of machine 1, calculated by the two methods is shown in Fig. 2, and the results agree well with only about 0.3 % in difference.

      361		362
      360		361
      359		360
      358 357		359 358

      361		362
      360		361
      359		360
      358 357		359 358

   2. Optimization for Torque Quality

   Torque [Nm]

   Torque [Nm]

   In the two machines, the parameters principally affecting the torque ripple are found to be, as shown in Fig. 3 the (i) Magnet arc to pole pitch ratio rf, (ii) Slot to teeth width ratio kd (inner+outer teeth), and (iii) Inner to outer tooth width ratio cp, (applicable only to single layer machines).

   Coil Span Tooth Pitch

   Slot Width

   Inner Tooth

   Inner Tooth

   Outer Tooth

   Outer Tooth

   (Fixed)

   356

   0 50 100 150

   Electrical Position [Deg]

   (b)

   357

   A A B B C C

   Fig. 3. Maxwell stress tensor and co-energy method (co- energy approximately 1 Nm less)

   Pole Pitch

   (Fixed)

   PM Pole

   1

   T [Nm]

   T [Nm]

   0.5

   0

   -0.5

   -1

   Fig. 5. Representation section of single layer pm machine

   with unequal teeth.

   DL Optimization Procedure

   SL

   The optimization procedure involves first definition of the

   objective function defining the optimizations is given by,

   n

   n

   F ypar wii ,

   0 30 60 90 120 150 180

   Electrical Pos [deg]

   (a)

   370

   i1

   where ypar is the value to be maximised, in this work three main objectives are persued which are T/coggingp-p , T/ripplep-p and T. Penalty factors i and their respective weighting factors wi are also included, so as the objective function does not to violate the limits of secondary functions.

   360

   T [Nm]

   T [Nm]

   350

   340

   330

   320

   SL

   DL

   The optimization algorithm then varies the selected machine parameters hunting for a maximum, while all other machine parameters are kept constant. Due to the machines having different torque capabilities at each case, for fair basis of comparison, torque results are compared on a per unit system based on the average machine torque in each case. The

   subsequent flow charts in Fig. 6 illustrate the methods

   0 30 60 90 120 150 180

   Electrical Pos [deg]

   (b)

   Fig. 4. Instantaneous torque waveforms of the two base machines, a) cogging torques, b) torque ripple.

   From the initial instantaneous torque waveforms of the machines, Fig. 4, the single-layer machine has per unit cogging torque. The single layer machine also possesses higher torque capability (5.74%), but with higher per unit ripple content than the double-layer machine as is found in [5], [11-12].

   employed, for the DL machine a linear search was done and for the SL machines the Powells optimization algorithm used.

4. RESULTS

   1. Double Layer Machine: Magnet pitch and slot width

      As double-layer topologies cannot use unequal teeth, the optimization parameters are limited to only two. The linear search method of Fig. 6a is applied to obtain the surface plots of Fig. 7. The points of minimum cogging, minimum torque ripple and maximum torque are presented in Table 2.

      Set Initial Start

      Values & Range: kd, rf, cp

      Set Initial Start

      Values & Range: kd, rf, cp

      Update

      kd, rf, cp

      Update

      kd, rf, cp

      360

      Set kd Range

      Set rf Range

      Set kd Range

      Set rf Range

      Call FE Rotate machine Record cog/rip

      Call FE Rotate machine Record cog/rip

      Update kd and rf

      Call FE Rotate machine Record cog/rip

      Update kd and rf

      340

      Torque [Nm]

      Torque [Nm]

      320

      300

      280

      1

      0.9

      0.8

      Optimization Algorithm

      Optimization Algorithm

      0.7

      0.5

      0.6

      0.8

      0.7

      End of kd / rf

      range? No

      Objective

      Function Max? No

      PM Pole Arc Ratio

      0.4

      (c)

      Slot Width Ratio

      END

      END

      END

      END

      Yes Yes

      1. (b)

         Fig. 6. Flow charts of design technique for a) Double- layer machine b) Single-layer machine

         Per Unit Cogging

         Per Unit Cogging

         0.05

         Fig. 7. Surface plot for double layer machine a) peak to peak cogging torque b) peak to peak torque ripple c) average torque.

   2. Single Layer: Magnet pitch, slot width, and unequal teeth

   In a SL 30 pole, 36 slot machine with equal teeth the winding factor is limited to 0.966, but by introducing unequal teeth, winding factors up to unity can be obtained. For the torque quality investigation of this machine, the method of Fig. 4b was used. Fig. 6 shows the machine model with unequal teeth, and Table 2 the optimization result.

   0

   1

   0.9

   0.8

   0.6

   0.7

   0.8

   TABLE II OPTIMIZATION RESULTS

   PM Pole Arc Ratio

   0.7

   (a)

   0.4

   0.5

   		Slot Width Ratio	PM Pole Arc Ratio	Teeth Ratio	Winding Factor	Torque	Per Unit p-p Cogging	Per Unit p-p Ripple
   PM 1	DL Min Cogging	0.45	0.76	0.5	0.945	318.5	0.1	1.33
   	DL Min Ripple	0.65	0.92	0.5	0.945	331.7	0.12	0.33
   	DL Max Torque	0.55	0.95	0.5	0.945	345	2.9	3.36
   PM 2	SL Min Cogging	0.623	0.91	0.5	0.966	364	0.31	1.3
   	SL Min Ripple	0.65	0.91	0.5	0.966	359.5	0.64	0.77
   	SL Max Torque	0.562	0.915	0.5	0.966	368.7	1.4	2.6
   PM 3	SLu Min Cogging	0.623	0.91	0.5	0.966	364	0.31	1.3
   	SLu Min Ripple	0.653	0.91	0.508	0.9677	359.7	1.05	0.44
   	SLu Max Torque	0.555	0.93	0.583	0.9864	377	5.11	3.84

   		Slot Width Ratio	PM Pole Arc Ratio	Teeth Ratio	Winding Factor	Torque	Per Unit p-p Cogging	Per Unit p-p Ripple
   PM 1	DL Min Cogging	0.45	0.76	0.5	0.945	318.5	0.1	1.33
   	DL Min Ripple	0.65	0.92	0.5	0.945	331.7	0.12	0.33
   	DL Max Torque	0.55	0.95	0.5	0.945	345	2.9	3.36
   PM 2	SL Min Cogging	0.623	0.91	0.5	0.966	364	0.31	1.3
   	SL Min Ripple	0.65	0.91	0.5	0.966	359.5	0.64	0.77
   	SL Max Torque	0.562	0.915	0.5	0.966	368.7	1.4	2.6
   PM 3	SLu Min Cogging	0.623	0.91	0.5	0.966	364	0.31	1.3
   	SLu Min Ripple	0.653	0.91	0.508	0.9677	359.7	1.05	0.44
   	SLu Max Torque	0.555	0.93	0.583	0.9864	377	5.11	3.84

   Slot Width Ratio

   Per Unit Ripple

   Per Unit Ripple

   0.05

   0

   1

   0.9

   0.6

   0.7

   0.8

   0.8

   PM Pole Arc Ratio

   0.7

   (b)

   0.4

   0.5 Slot Width Ratio

   As can be seen from the results, by adjusting the teeth

   ratio, a higher winding factor is obtainable for SL machines. From the results, minimum cogging torque is obtained by the DL machines, but suffers with a low average torque for this case. The SL machines have the same level of cogging as in both cases the optimization algorithm finds the same point, and it can be noted that unequal teeth provide no advantages for cogging torque reduction in this case.

   In terms of torque ripple, the DL machine presents the smoothest case, but again with a low average torque. The single layer machine with unequal teeth is not too far off and has a much higher average torque.

   400

   Torque [Nm]

   Torque [Nm]

   380

   360

   340

   320

   300

   SLu SL

   DL

   SLu SL

   DL

   0 30 60 90 120 150 180

   Electrical Pos [deg]

   (c)

   For case of maximum torque, the single layer machine with unequal teeth is best, but with very poor torque quality.

   Fig. 8. Linear model of single layer machine with unequal teeth

   SL/Slu

   DL

   SL/Slu

   DL

   0.8

   Torque [Nm]

   Torque [Nm]

   0.4

   0

   -0.4

   -0.8

   0 30 60 90 120 150 180

   Electrical Pos [deg]

   (a)

   	SLu
   SL DL

   	SLu
   SL DL

   370

   Torque [Nm]

   Torque [Nm]

   360

   350

   340

   330

   320

   310

   0 30 60 90 120 150 180

   Electrical Pos [deg]

   (b)

   Fig. 9. Instantaneous torque waveforms for the three machines a) minimum cogging torque, b) minimum torque ripple, c) maximum torque.

5. CONCLUSIONS

By full FE-coupled optimization for torque quality, it can be noted that ultimately in this topology, DL machines provide the best torque quality, but suffer from lower torque capacity. Interestingly found is the advantages of using unequal teeth for single layer machines over conventional SL machines. By employing unequal teeth, torque ripple can be minimized and torque capability increased. In terms of cogging torque though, unequal teeth present no advantages.

REFERENCES

1. P. Salminen, J. Pyrhönen, F. Libert and J. Soulard, Torque Ripple of Permanent Magnet Machines with Concentrated Windings, ISEF, Spain, Sept. 2005.

2. R. Neji, S. Tounsi and F. Sellami, Contribution to the definition of a permanent magnet motor with reduced production cost for the electric vehicle propulsion, Euro. Trans. Electr. Power, 16: 437-460, 2006.

3. A. Rix, M. Kamper, R. Wang, Torque Ripple and Flux Density Variation Due to Stator Slot Opening of Concentrated Coil Permanent Magnet Machine, SAUPEC Conference, Cape Town, 2007.

4. P. Salminen, M. Niemelä, J. Pyrhönen and J. Mantere, High- Torque Low-Ripple Fractional-Slot PM-Motors, Proceedings of IEEE IEMDC, pp.144 148, May 2005.

5. D. Ishak, Z. Q. Shu and D. Howe, Comparison of PM Brushless Motors, Having Either All Teeth or Alternate Teeth Wound, IEEE Trans-EC 21(1): 95-103, Mar. 2006.

6. J. Colton, D. Patterson, J. Hudgins, Design of a Low-Cost and Efficient Integrated Starter-Alternator, Proceedings of IET- PEMD, pp.357 361, April 2008.

7. N. Bianchi, S. Bolognani, M. Dai Pre and G. Grezzani, Design Considerations for Fractional-Slot Winding Configurations

   of Synchronous Machines, IEEE Trans-IA 42(4):997-1006, Jul./Aug. 2006.

8. J. Cros and P. Viarouge, Synthesis of High Performance PM Motors with Concentrated Windings, IEEE Trans-EC 17(2): 248- 253, June 2002.

9. N. Bianchi, M. Dai Pre, L. Alberti and E. Fornasiero, Theory and design of fractional slot PM machines, IEEE-IAS Tutorial Course notes, 2007.

10. J. Cros, P. Viarouge, C. Gelinas, Design of PM Brushless Motors Using Iron-Resin Composites for Automotive Applications, IEEE-IAS Conference, vol.1, pp. 5-11, 1998.

11. Th. Koch and A. Binder, Permanent Magnet Machines with Fractional Slot Windings for Electric Traction, Proceedings of. ICEM 2002, Belgium.

12. D. Ishak, Z. Q. Zhu and D. Howe, Permanent-Magnet Brushless Machines with Unequal Tooth Widths and Similar Slot Pole Numbers, IEEE Trans-IA 41(2): 584-590, Mar./Apr. 2005.