 Open Access
 Total Downloads : 438
 Authors : E. Rakgati, H. Kierstead, E. Matlotse
 Paper ID : IJERTV2IS100643
 Volume & Issue : Volume 02, Issue 10 (October 2013)
 Published (First Online): 01112013
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
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 singlelayer machine with unequal teeth. Evaluation between the singlelayer machine and its doublelayer counterpart is highlighted, and the results show attractive performance of the singlelayer machine with unequal teeth over its doublelayer counterpart.

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 singlelayer (SL) fractional openslot windings are of particular interest as they are ideal options for preformed nonoverlap modular coils. In certain pole/slot combinations, open slots show reduced ripple [14]. Certain PM machines with regular slot SL fractional windings show higher torque capacity than their doublelayer (DL) counterpart [57]. 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 [812] 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 [812] deal with radial flux structures with semienclosed slots, and the effects of the magnet pitch are not treated for except in [10]. In [2], an openslot 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 openslot axial flux PM machines driven by sinusoidal currents, by full FEcoupled 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.

MACHINE TOPOLOGIES
Fig. 1 shows the sectional layout of two fractional open
slot (30pole/36slot) axial flux permanent magnet machines, in which (a) is of singlelayer topology while (b) is of double layer topology. The 30pole/36slot 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 multislice 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 airgap 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) Singlelayer
with equal teeth, and (b) Doublelayer.
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 airgap 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 pp Cogging
Per Unit pp Ripple

TORQUE ANALYSIS

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

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 coenergy 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/coggingpp , T/ripplepp 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 singlelayer 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 doublelayer machine as is found in [5], [1112].
employed, for the DL machine a linear search was done and for the SL machines the Powells optimization algorithm used.


RESULTS

Double Layer Machine: Magnet pitch and slot width
As doublelayer 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

(b)
Fig. 6. Flow charts of design technique for a) Double layer machine b) Singlelayer 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.


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 pp Cogging
Per Unit pp 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 pp Cogging
Per Unit pp 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.


CONCLUSIONS
By full FEcoupled 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.
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