Difference in Unbalanced Magnetic Force of Fractional-Slot ... · non-overlapping fractional-slot machines, some machines with pole number close to slot number, e.g. 8-pole/9-slot,

$Page 1: Difference in Unbalanced Magnetic Force of Fractional-Slot ... · non-overlapping fractional-slot machines, some machines with pole number close to slot number, e.g. 8-pole/9-slot,$
154 CES TRANSACTIONS ON ELECTRICAL MACHINES AND SYSTEMS, VOL. 1, NO. 2, JUNE 2017

Abstract—This paper presents a comparative investigation into unbalanced magnetic force (UMF) of asymmetric permanent magnet machines without rotor eccentricities, particularly focusing on the difference between internal- and external-rotor topologies. The asymmetric field distribution results in radial and tangential asymmetric force waves. Although the radial and tangential stresses are in different direction, the UMF components they produce are nearly aligned. The UMF from asymmetric radial force wave can be additive or subtractive to that from asymmetric tangential force wave. Investigation shows that for the same pole slot number combination, if the UMFs due to radial and tangential force waves are additive in internal rotor machine, they are subtractive in the external rotor counterpart, and vice versa. Investigation reveals a general rule determining whether additive or cancelling: for a UMF produced by any two field harmonics, they are additive if the higher order is produced by the outer part outside the airgap, but cancelling if the higher order is produced by the inner part. Therefore, for a machine with pole number 2p=3k+1, they are additive if it is an external-rotor machine, but otherwise subtractive. On the other hand, for a machine with pole number 2p=3k-1, they are subtractive if it is an external-rotor machine, but otherwise additive. For the UMF due to armature reaction only, they are subtractive for external-rotor machines, but otherwise additive. The investigation is carried out by an analytical model and validated by finite element analysis.

Index Terms—Analytical model, electric machines, fractional-slot machines, permanent magnet machines, unbalanced magnetic force.

I. INTRODUCTION

ERMANENT MAGNET (PM) machines are widely used due to high torque density and efficiency. The employment of

fractional-slot windings in such machines make them even more attractive for lower torque ripple. The non-overlapping fractional-slot windings can further increase their torque

This work was supported in part by the National Natural Science Foundation of China under Grants 51677169 and 51637009. L. J. Wu, Youtong Fang, and Xiaoyan Huang are with College of Electrical

Engineering, Zhejiang University, China. (e-mail: [email protected]). Z. Q. Zhu is with the Department of Electronic and Electrical Engineering,

University of Sheffield, Mappin Street, Sheffield, S1 3JD, UK, (e-mail: [email protected]). Y. T. Fang is a professor with the College of Electrical Engineering,

Zhejiang University, China. (e-mail: [email protected]). X. Y. Huang is a with the College of Electrical Engineering, Zhejiang

University, China, (e-mail: [email protected]).

density and efficiency due to shorter end windings. Among non-overlapping fractional-slot machines, some machines with pole number close to slot number, e.g. 8-pole/9-slot, have very high winding factor. However, these machines are asymmetric and therefore may have unbalanced magnetic force (UMF), even when there are no stator and rotor eccentricities. The UMF is detrimental to the bearing life, noise and vibration, and therefore, thorough understanding of UMF is very important in the application of these machines. The UMF results from the asymmetric field distribution [1],

which can be due to asymmetric winding and/or stator slot layout. The symmetry mentioned here is not diametrical symmetry, but rotational symmetry instead, which means the cross-section of a machine looks the same after rotation of certain number of degrees (less than one full turn). For example, a 6-pole/9-slot machine is diametrically asymmetric, but rotationally symmetrical. Hence, it has no UMF. According to this rule, the PM machines with pole and slot numbers differed by one do not have rotational symmetry and hence UMF exists. The PM machines having pole and slot numbers differed by two are rotational symmetrical if all teeth wound windings are employed. They becomes rotational asymmetric if they are wound with alternative teeth windings [2]. These machines are attractive for their high torque density, high efficiency, high flux weakening capability, and low torque ripple, but suffer from the UMF. Many papers analyzed the UMF of PM machines without

rotor eccentricities [1-13]. It was found that the 8-pole/9-slot machine has much larger UMF than the 10-pole/9-slot machine in [2, 11, 13]. The additive and cancelling effects between UMFs due to radial and tangential force waves were firstly revealed in [2, 13]. It is an important determining factor in the difference of UMF between machines with different pole/slot number combinations. In [2], the conclusion was expanded generally that for the same slot number Ns, the machine having a pole number 2p=Ns-1 has larger UMF than the machine with 2p=Ns+1. But the conclusion is derived from the internal-rotor machines. It was shown in [14] that for the external-rotor topology, the 8-pole/9-slot machine has much smaller UMF than the 10-pole/9-slot machine. This paper will present a comparative investigation into the

UMF of radial flux asymmetric PM machines with different pole/slot combinations but no rotor eccentrics, particularly focusing on the difference between the internal and external

Difference in Unbalanced Magnetic Force of Fractional-Slot PM Machines between Internal

and External Rotor Topologies

L. J. Wu, Z. Q. Zhu, Fellow, Royal Academy of Engineering, Y. T. Fang, and X. Y. Huang

P

WU et al. : DIFFERENCE IN UNBALANCED MAGNETIC FORCE OF FRACTIONAL-SLOT PM MACHINES BETWEEN INTERNAL 155 AND EXTERNAL ROTOR TOPOLOGIES

rotor topologies. This paper will show that for the same pole slot number combination, if the UMFs due to radial and tangential force waves are additive in internal rotor machine, they are subtractive in the external rotor counterpart, and vice versa. Further investigation will reveal by the first time a simple rule governing the additive and cancelling effects in the UMF, which is generally applicable to both internal/external-rotor topologies. The only determining factor is whether the higher order field harmonic among two main contributing field harmonics is produced by the machine’s inner or outer part separated by the airgap. This simple and general rule reveals the additive and cancelling effects in any UMF component resulting from any two field harmonics. Therefore, it can indicate the additive and cancelling effects in the UMF of machines with all kinds of pole/slot number combinations and internal/external-rotor topologies. It not only indicates these effects in the resultant UMF, but also reveals them in the UMF due to armature reaction only.

II. MACHINES

The permanent magnet brushless AC (BLAC) machines with 4 different slot/pole number combinations, 8-pole/9-slot (8p9s), 10-pole/9-slot (10p9s), 4-pole/6-slot (4p6s), and 8-pole/6-slot (8p6s), are to be analyzed. For each slot/pole number combination, there are two options, internal- and external-rotor topologies. All these machines use the same magnet material of which remanence is 1.2T and recoil permeability is 1.05. The magnet thickness is 3mm and the pole-arc to pole-pitch ratio is 1. Other dimensions are shown in TABLE I and TABLE II for the external- and internal-rotor topologies, respectively. All machines have a nominal speed 400rpm, a rated torque 5Nm and a rated current 10Apeak. Fig. 1 shows their equal potential distributions, which are asymmetric.

(a) (b)

(c) (d)

(e) (f)

(g) (h)

Fig. 1. FE predicted equal potential distribution. (a) Internal-rotor, 8p9s. (b) Internal-rotor, 10p9s. (c) External-rotor, 8p9s. (d) External-rotor, 10p9s. (e) Internal-rotor, 4p6s. (f) Internal-rotor, 8p6s. (g) External-rotor, 4p6s. (h) External-rotor, 8p6s.

TABLE I MAJOR PARAMETERS OF EXTERNAL-ROTOR MACHINES

Machine 8p9s 10p9s 4p6s 8p6s Rotor outer diameter (mm) 100 100 100 100 Rotor inner diameter (mm) 82 84 76 82 Rotor yoke height (mm) 6 5 9 6 Stator outer diameter (mm) 80 82 74 80 Stator inner diameter (mm) 20 20 20 20 Stator yoke height (mm) 10 10 6.5 5.25 Slot opening (mm) 2 2 2 2 Tooth body width (mm) 10 9 13 10.5 Number of turns/phase 81 78 97 87

TABLE II

MAJOR PARAMETERS OF INTERNAL-ROTOR MACHINES

Machine 8p9s 10p9s 4p6s 8p6s Stator outer diameter (mm) 100 100 100 100 Stator inner diameter (mm) 53 53 45 53 Rotor outer diameter (mm) 51 51 43 51 Stator yoke height (mm) 4.4 4.4 7.1 4.5 Slot opening (mm) 2 2 2 2 Tooth body width (mm) 8.7 8.7 11.1 7.1 Number of turns/phase 126 126 166 137

III. CALCULATION METHOD

According to the Maxwell stress tensor, the radial force wave can be given by:

( )2 20/2/rB Bασ µ= − (1)

where Br and Bα are the radial and tangential components of resultant airgap flux density, respectively. Although they can be calculated by the analytical model [15] or FE analysis, they are calculated by the analytical model in this paper. The tangential force wave can be calculated by:


0/rBBατ µ= (2)

for the internal-rotor topology and

0/rBBατ µ= − (3)

for the external-rotor topology. The x and y components of UMF F of a machine with an

active length la can be calculated by integrating the Maxwell stress tensor along a circular path in the airgap of radius r:

x x xF F Fσ τ= + (4)

y y yF F Fσ τ= + (5)

where Fτx and Fτy are the x and y components of UMF due to tangential force wave Fr:

2

0sinx aF rl d

π

τ τ αα= − ∫ (6)

2

0cosy aF rl d

π

τ τ αα= ∫ (7)

and Fσx and Fσy are the x and y components of UMF due to radial force wave Fσ:

2

0cosx aF rl d

π

σ σ αα= ∫ (8)

2

0siny aF rl d

π

σ σ αα= ∫ (9)

for the internal-rotor topology and

2

0cosx aF rl d

π

σ σ αα= − ∫ (10)

2

0siny aF rl d

π

σ σ αα= − ∫ (11)

for the external-rotor topology. If the resultant flux density Br and Bα in (7) and (10)-(11) are

replaced by the armature reaction flux density Bar and Baα, the foregoing equations for force waves and UMF will give values due to armature reaction only and correspondingly, the symbols Fτx, Fτy, Fσx, Fσy, Fr and Fσ for UMF components will be replaced by Faτx, Faτy, Faσx, Faσy, Far and Faσ, respectively.

IV. 9-SLOT MACHINES

The UMFs of four 9-slot machines are compared in Fig. 2. The UMF repeats every half revolution so that its waveform is shown for only 180 elec. deg. rotor position. For the internal-rotor topology, the 8p9s machine has much larger UMF than the 10p9s machine. However, for the external-rotor, the 10p9s machine has much larger UMF than the 8p9s machine.

Fig. 2. Analytically predicted UMFs of 9-slot machines.

This interesting phenomenon is revealed by the analytical model. The comparison between the 8p9s and 10p9s internal rotor machines was done in [2, 13]. It shows that the UMFs due to radial and tangential force waves are additive in the 8p9s internal-rotor machine, but cancelling in the 10p9s internal-rotor machine. This results in a very large UMF in the 8p9s machine but much smaller UMF in the 10p9s machine. However, as seen in Fig. 3, regarding the external rotor machines, the UMFs from radial and tangential force waves are canceling in the 8p9s machine, but additive in the 10p9s machine. Hence, the 8p9s external-rotor machine has much smaller UMF than the 10p9s counterpart. The additive and cancelling effects in the UMF are summarized in TABLE III.

TABLE III ADDITIVE AND CANCELLING EFFECTS OF UMFS DUE TO RADIAL AND

TANGENTIAL FORCE WAVES IN 9-SLOT MACHINES

Machine topology 8p9s 10p9s

Internal-rotor Additive Cancelling

External-rotor Cancelling Additive

By way of example, the radial and tangential force

distributions at the 0 degree rotor position shown in Fig. 3 are analyzed and shown in Fig. 4-Fig. 5 for two 9-slot external rotor machines. The force distribution is not symmetrical. The first spatial order force distribution contributes to the UMF. The radial and tangential force distributions of the first spatial order have a phase shift of about 90 Mech. Deg., i.e. their peak values locate about 90 Mech. Deg. away in the circumferential position. Together with 90 Mech. Deg. angle difference from the normal and tangential directions, the UMFs due to radial and tangential force waves are either in the same direction, additive, or in the opposite direction, cancelling. It was revealed in [13] that the 8p9s and 10p9s machines have the opposite phase shift between radial and tangential force waves so that their additive and cancelling effects are opposite. The question is why the internal and external rotor topologies have opposite additive and cancelling effects as shown in TABLE III. This can be seen by comparing Fig. 4(b)-(c) with Fig. 6 and comparing Fig. 5(b)-(c) with Fig. 7. The tangential force waves of both topologies are similar. However, the radial forces of both topologies are in the opposite directions. This explains why the UMFs due to radial and tangential force waves Fr and Fσ are additive in the 8p9s internal-rotor machine, Fig. 6, but cancelling in the 8p9s external-rotor machine, Fig. 4(b)-(c).

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Unbalanced magnetic foce (N)

Rotor position (Elec. Deg.)

Fσx FσyFτx FτyFx Fy

(a)


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Fσx Fσy

Fτx Fτy

Fx Fy

(b)

Fig. 3. Analytically predicted UMF components of 9-slot machines. (a) 8p9s, external-rotor. (b) 10p9s, external-rotor.

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First order stress (kPa)

Total stress (kPa)

Position (Mech. Deg.)

σ

τ

σ,1st orderτ, 1st order

(a)

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x component of stress (kPa)

y component of stress (kPa)

σ

τ

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Fx (N)

Fy (N)

Fσ

Fτ

(b) (c) Fig. 4. Analytically predicted force distribution in 8p9s external-rotor machine. (a) Stress waveform. (b) The 1st spatial order force distribution on rotor outer surface denoted by circle. (c) UMF components.

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First order stress (kPa)

Total stress (kPa)

Position (Mech. Deg.)

σ

τ

σ,1st orderτ, 1st order

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σ

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Fx (N)

Fy (N)

Fσ

Fτ

(b) (c)

Fig. 5. Analytically predicted force distribution in 10p9s external-rotor machine. (a) Stress waveform. (b) The 1st spatial order force distribution on rotor outer surface denoted by circle. (c) UMF components.

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σ

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Fx (N)

Fy (N)

Fσ

Fτ

(a) (b)

Fig. 6. Analytically predicted the 1st spatial order force distribution and UMF on rotor outer surface in 8p9s internal-rotor machine. (a) The 1st spatial order force distribution on rotor outer surface denoted by circle. (b) UMF components.

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y component of stress (kPa) σ

τ

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0

40

80

120

Fx (N)

Fy (N)

Fσ

Fτ

(a) (b)

Fig. 7. Analytically predicted the 1st spatial order force distribution and UMF on rotor outer surface in 10p9s internal-rotor machine. (a) The 1st spatial order force distribution on rotor outer surface denoted by circle. (b) UMF components.

The difference in the additive and cancelling effects between machines with different slot/pole number combinations and different rotor topologies must be due to the difference in the field harmonics. Fig. 8-Fig. 11 show the field spectra of four 9-slot machines. The UMF results from the interaction between two field harmonics of spatial orders differed by one [13]. It can be seen in the figures, the UMF is mostly produced by the interaction between the 4th spatial order field harmonic and the 5th field harmonic in all four machines. Hence, the difference in the additive and cancelling effects must be due to the difference in the 4th PM and 5th armature reaction field harmonics. The difference in the field harmonic magnitude can be ruled out, since it does not make substantial difference in the UMF, but only quantitative difference. The only substantial difference in the 4th and 5th field harmonics between these 4 machines is in the locations of the parts producing these field harmonics, either inner or outer. Therefore, the additive or canceling effect is closely related with the source part locations of the main contributing field harmonics.

(a)


(b)

Fig. 8. Airgap flux density spectra of internal-rotor 8-pole/9-slot machine. (a) PM field. (b) Armature reaction.

(a)

(b)


(a)

(b)

Fig. 10. Airgap flux density spectra of external-rotor 8-pole/9-slot machine. (a) PM field. (b) Armature reaction.

(a)

(b)


TABLE IV compares the source parts of the main contributing field harmonics, the 4th and 5th spatial orders. The pattern in TABLE IV matches with that in TABLE III. The 8p9s internal-rotor and 10p9s external-rotor machines have the 4th spatial order field harmonic from the inner part and the 5th spatial order field harmonic from the outer part, TABLE IV. Their UMFs due to radial and tangential force waves are additive, TABLE III. On the contrary, the other two machines have the 4th spatial field harmonic from the outer part and the 5th spatial field harmonic from the inner part. Their UMFs due to radial and tangential force waves are cancelling. Based on the foregoing analysis, the correlation of the

additive and canceling effects in the UMF with the source locations of two field harmonics of spatial orders n and n+1 is generalized as: (1) If the nth spatial order field harmonic is produced by the inner part and the (n+1)th spatial order field harmonic is produced by the outer part, their produced UMFs due to radial and tangential force waves are additive; (2) If the nth spatial order field harmonic is produced by the outer part and the (n+1)th spatial order field harmonic is produced by the inner part, their produced UMFs due to radial and tangential force waves are cancelling.

TABLE IV COMPARISON BETWEEN SOURCE PARTS OF MAIN CONTRIBUTING FIELD

HARMONICS TO UMF IN 9-SLOT MACHINES

Field harmonic

Machine topology

8p9s 10p9s

4th Internal-rotor Inner part Outer part

External-rotor Outer part Inner part

5th Internal-rotor Outer part Inner part External-rotor Inner part Outer part

Although the UMF due to armature reaction only is very small in these four machines, the analysis of this part UMF can reveal the additive and cancelling effects in the UMF when both neighbouring order field harmonics are from the same source


part. As seen in Fig. 10-Fig. 9, the armature reaction field itself has many harmonics of orders differed by one. These field harmonics also produce the UMF. This part of UMF is the UMF due to armature reaction only. It is analysed by the analytical model and shown in Fig. 12. As can be seen, in both 8p9s and 10p9s internal-rotor machines, the UMFs due to radial and tangential force waves are additive, but they are cancelling in both external-rotor machines. Therefore, the statement in [13] that the UMFs due to radial and tangential force waves are always additive is valid only for internal-rotor machines. It can be concluded here that: (1) If both the nth and (n+1)th spatial order field harmonics are produced by the inner part, the UMFs due to radial and tangential force waves are canceling; (2) If both the nth and (n+1)th spatial order field harmonics are produced by the outer part, the UMFs due to radial and tangential force waves are additive.

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Faσx Faσy

Faτx Faτy

Fax Fay

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Faσx Faσy

Faτx Faτy

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(d)

Fig. 12. Analytically predicted UMF components due to armature reaction only in 9-slot machines. (a) 8p9s, internal-rotor. (b) 10p9s, internal-rotor. (c) 8p9s, external-rotor. (d) 10p9s, external-rotor.

The full picture correlation of the additive and canceling effects in the UMF with the source locations of two field harmonics of spatial orders n and n+1 is summarized in TABLE V. It is interesting to see that whether the UMFs due to radial and tangential force waves are additive or canceling is determined by the source location of the higher order field harmonic. If the higher order is from the outer part, the UMFs due to radial and tangential force waves are additive. Otherwise, they are canceling. By applying this simple criterion, the additive or cancelling effect between UMFs due to radial and tangential force waves can be easily found out just by the pole/slot number combination and internal/external-rotor topologies. It is summarised in TABLE VI and as follows. (a) The additive or cancelling effect in total UMF depends on the pole number and internal/external-rotor topologies. If 2p=3k+1, i.e p=3k-1, two main contributing field harmonics to UMF are the pth PM field harmonic and the (p-1)th armature reaction field harmonic, because the triplen number (p+1) does not exist in the armature reaction field harmonic orders. Hence, the pth PM field harmonic has higher spatial order than the other main contributing harmonic. According to the rules in TABLE V, the UMFs due to radial and tangential force waves are canceling if it is an internal-rotor machine, but otherwise additive. On the other hand, if 2p=3k-1, i.e p=3k+1, two main contributing field harmonics to UMF are the pth PM field harmonic and the (p+1)th armature reaction field harmonic, because the armature reaction field does not have the triplen order (p-1)th harmonic. Therefore, the higher order harmonic among two main contributing harmonics is from the armature. According to the rules in TABLE V, the UMFs due to radial and tangential force waves are additive if it is an internal-rotor machine, but otherwise cancelling. (b) The additive or cancelling effect in the UMF from armature reaction only is determined only by the internal/external-rotor topologies. According to the rules in TABLE V, the UMFs due to radial and tangential force waves are additive for internal-rotor machines, but cancelling for external-rotor machines.

TABLE V CORRELATION OF ADDITIVE AND CANCELLING EFFECTS IN UMF WITH SOURCE

PARTS OF TWO FIELD HARMONICS OF ORDERS DIFFERED BY ONE

Additive or canceling effect in UMF Source of the (n+1)th order

field harmonic Inner part Outer part

Source of the nth order field harmonic

Inner part Cancelling Additive Outer part Cancelling Additive

TABLE VI ADDITIVE AND CANCELLING EFFECTS OF UMFS DUE TO RADIAL AND

TANGENTIAL FORCE WAVES IN MACHINES WITH UMF

Pole number 2p=3k+1, i.e p=3k-1

2p=3k-1, i.e p=3k+1

Example pole/slot number combination

4p3s, 4p6s, 10p9s, 16p18s, 22p21s, …

2p3s, 8p6s, 8p9s, 20p18s, 20p21s, …

Total UMF Internal-rotor Cancelling Additive External-rotor Additive Cancelling

UMF due to armature reaction only

Internal-rotor Additive Additive

External-rotor Cancelling Cancelling


The general conclusions shown in TABLE V-TABLE VI will be verified again in the next section by analyzing four more machines with pole and slot numbers differed by two and alternative teeth wound windings.

V. 6-SLOT MACHINES

Four 6-slot machines with either 4 or 8 poles are analyzed by the analytical model. These machines have UMF when the alternate teeth wound windings are employed. Fig. 13-Fig. 16 shows the airgap field spectra. The main contributing field harmonics to the UMF can be found from these figures. They are shown in TABLE VII and TABLE VIII for the total UMF and UMF due to armature reaction only, respectively. By utilizing the rules obtained in the previous section upon

the main contributing field harmonics shown in TABLE VII-TABLE VIII, the additive and canceling effects in the UMF of these four machines can be deduced. They are summarized in Table IX.

(a)

(b)


(a)

(b)


(a)

(b)


(a)

(b)



TABLE VII

MAIN CONTRIBUTING FIELD HARMONICS TO TOTAL UMF AND THEIR SOURCE PARTS IN 6-SLOT MACHINES

Machine topology 4p6s 8p6s Source part

Internal-rotor 2nd 4th Inner part

1st 5th Outer part

External-rotor 1st 5th Inner part 2nd 4th Outer part

TABLE VIII

MAIN CONTRIBUTING FIELD HARMONICS TO UMF DUE TO ARMATURE REACTION ONLY AND THEIR SOURCE PARTS IN 6-SLOT MACHINES

Machine topology 4p6s 8p6s Source part

Internal-rotor 2nd 2nd Outer part

1st 1st Outer part

External-rotor 2nd 2nd Inner part 1st 1st Inner part

TABLE IX

ADDITIVE AND CANCELLING EFFECTS OF UMFS DUE TO RADIAL AND TANGENTIAL FORCE WAVES IN 6-SLOT MACHINES

UMF component Machine topology 4p6s 8p6s

Total UMF Internal-rotor Cancelling Additive External-rotor Additive Cancelling

UMF due to armature reaction only

Internal-rotor Additive Additive External-rotor Cancelling Cancelling

The analytically predicted UMFs of these four machines are shown in Fig. 17. The results confirm the predicted additive and cancelling effect in the total UMF shown in TABLE IX. In the 4p6s internal-rotor and 8p6s external-rotor machines, the UMFs due to radial and tangential force waves are cancelling, while in the other two machines, they are additive. The analytically predicted UMFs due to armature reaction only are shown in Fig. 18. Again, the results match with the predicted additive and canceling effects in the UMF due to armature reaction only shown in Table IX. For the two internal-rotor machines, the UMFs due to radial and tangential force waves are additive. On the contrary, they are canceling for the two external-rotor machines.

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Fig. 17. Analytically predicted UMF components of 6-slot machines. (a) 4p6s, internal-rotor. (b) 8p6s, internal-rotor. (c) 4p6s, external-rotor. (d) 8p6s, external-rotor.

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Faσx Faσy

Faτx Faτy

Fax Fay

(b)

-60

-40

-20

0

20

40

60

0 30 60 90 120 150 180



Faσx Faσy

Faτx Faτy

Fax Fay

(c)


-60

-40

-20

0

20

40

60

0 30 60 90 120 150 180



Faσx Faσy

Faτx Faτy

Fax Fay

(d)

Fig. 18. Analytically predicted UMF due to armature reaction only of 6-slot machines. (a) 4p6s, internal-rotor. (b) 8p6s, internal-rotor. (c) 4p6s, external-rotor. (d) 8p6s, external-rotor.

VI. VALIDATION

The analytical prediction is validated by the FE result for external rotor machines only in this paper, since the analytical model was validated for internal rotor machines in [13]. In the FE models, the iron material is set to have a very large permeability 104 to simulate the infinitely permeable iron material in the analytical model. The very fine mesh size is employed to achieve negligible numerical error. The FE predicted UMF waveforms are compared with the

analytical predictions in Fig. 19. Excellent agreement has been achieved. It should be noted that in the FE prediction, the slotting effect is accounted for, while it is neglected in the analytical model.

(a)

(b)

(c)

(d)

Fig. 19. Comparison between FE predicted and analytically predicted UMF waveforms. (a) 8p9s, external-rotor. (b) 10p9s, external-rotor. (c) 4p6s, external-rotor. (d) 8p6s, external-rotor.

VII. CONCLUSION

This paper has investigated the UMF of asymmetric PM machines focusing on the difference between internal/external-rotor topologies. Investigation has been carried out by an analytical model and validated by the FE simulation. Any two field harmonics with spatial order differed by one can interact to produce a UMF. It has been revealed that weather the UMFs due to radial and tangential force waves are additive or cancelling depends on the source part of the higher order harmonic. They are additive if the higher order is from the outer part, but otherwise cancelling. For a machine with pole number 2p=3k+1, the UMFs due to radial and tangential force waves are canceling if it is an internal-rotor machine, but otherwise additive. On the other hand, for 2p=3k-1, they are additive if it is an internal-rotor machine, but otherwise cancelling. For the UMF from armature reaction only, the UMFs due to radial and tangential force waves are additive for internal-rotor machines, but cancelling for external-rotor machines.

REFERENCES

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[2] Z. Q. Zhu, M. L. Mohd-Jamil, and L. J. Wu, "Influence of slot and pole number combinations on unbalanced magnetic force in PM machines with diametrically asymmetric windings," IEEE Transactions on Industry Applications, vol. 49, pp. 19-30, 2013.

[3] C. Bi, Z. J. Liu, and T. S. Low, "Analysis of unbalanced-magnetic-pulls in hard disk drive spindle motors using a hybrid method," IEEE Transactions on Magnetics, vol. 32, pp. 4308-4310, 1996.

[4] C. Bi, Z. J. Liu, and T. S. Low, "Effects of unbalanced magnetic pull in spindle motors," IEEE Transactions on Magnetics, vol. 33, pp. 4080-4082, 1997.

[5] S. X. Chen, T. S. Low, H. Lin, and Z. J. Liu, "Design trends of spindle motors for high performance hard disk drives," IEEE Transactions on Magnetics, vol. 32, pp. 3848-3850, 1996.

[6] S. X. Chen, Q. D. Zhang, Z. J. Liu, and H. Lin, "Design of fluid bearing spindle motors with controlled unbalanced magnetic forces," IEEE Transactions on Magnetics, vol. 33, pp. 2638-2640, 1997.

[7] Z. J. Liu, C. Bi, Q. D. Zhang, M. A. Jabbar, and T. S. Low, "Electromagnetic design for hard disk drive spindle motors with fluid film lubricated bearings," IEEE Transactions on Magnetics, vol. 32, pp. 3893-3895, 1996.


[8] N. Schirle and D. K. Lieu, "History and trends in the development of

motorized spindles for hard disk drives," IEEE Transactions on Magnetics, vol. 32, pp. 1703-1708, 1996.

[9] Z. Q. Zhu, D. Ishak, D. Howe, and J. Chen, "Unbalanced magnetic forces in permanent-magnet brushless machines with diametrically asymmetric phase windings," IEEE Transactions on Industry Applications, vol. 43, pp. 1544-1553, 2007.

[10] D. G. Dorrell, M. Popescu, C. Cossar, and D. Ionel, "Unbalanced magnetic pull in fractional-slot brushless PM motors," in IEEE Industry Applications Society Annual Meeting, 2008, pp. 1-8.

[11] Z. Q. Zhu, D. M. M. Jamil, and D. Howe, "Comparative study of unbalanced magnetic force in fractional slot PM brushless machines," presented at the Proc. 3rd Int. Conf. Ecological Vehicles and Renewable Energies, 2008.

[12] D. G. Dorrell, M. Popescu, and D. M. Ionel, "Unbalanced magnetic pull due to asymmetry and low-level static rotor eccentricity in fractional-slot brushless permanent-magnet motors With surface-magnet and consequent-pole rotors," IEEE Transactions on Magnetics, vol. 46, pp. 2675-2685, 2010.

[13] L. J. Wu, Z. Q. Zhu, J. T. Chen, and Z. P. Xia, "An analytical model of unbalanced magnetic force in fractional-slot surface-mounted permanent magnet machines," IEEE Transactions on Magnetics, vol. 46, pp. 2686-2700, 2010.

[14] L. J. Wu and Z. Q. Zhu, "Comparative analysis of unbalanced magnetic force in fractional-slot permanent magnet machines having external rotor topologies," presented at the IET International Conference on Power Electronics, Machines and Drives (PEMD2012), 2012.

[15] Z. Q. Zhu and D. Howe, "Instantaneous magnetic field distribution in permanent magnet brushless dc motors. Part IV: Magnetic field on load," IEEE Transactions on Magnetics, vol. 29, pp. 152-158, 1993.

L. J. Wu received the B.Eng. and M.Sc. degrees from Hefei University of Technology, Hefei, China, in 2001 and 2004, respectively, and the Ph.D degree from the University of Sheffield, Sheffield, U.K., in 2011, all in electrical engineering. From 2004 to 2007, he was an Engineer with Delta Electronics (Shanghai) Co, Ltd. From 2012-2016, he joined Siemens as a

design engineer focusing on wind power generators. Currently, he is a professor at Zhejiang University, China, and director of Zhejiang University and Shanghai Electric Wind Power Research Center. His main interests include permanent magnet machines and applications.

Professor Z. Q. Zhu, born on 23 Sept. 1962, Fellow of Royal Academy of Engineering, Fellow IEEE, Fellow IET, PhD, Professor at The University of Sheffield, UK. Major research interests include design, control, and applications of brushless permanent magnet machines and drives for applications ranging from automotive to renewable energy.

Y. T. Fang received the B.S. degree and Ph.D. degree in electrical engineering from Hebei University of Technology, Hebei, China, in 1984 and 2001 respectively. Currently, he is a professor with the College of Electrical Engineering, Zhejiang University, China. His research interests include the application, control, and design

of electrical machines.

X. Y. Huang received the B.E. degree, from Zhejiang University, Hangzhou, China, in 2003, and received the Ph.D. degree in electrical machines and drives from the University of Nottingham, Nottingham, U.K., in 2008. From 2008 to 2009, she was a Research Fellow with the University of Nottingham. Currently, she is a professor with the College of Electrical

Engineering, Zhejiang University, China, where she is working on electrical machines and drives. Her research interests are PM machines and drives for aerospace and traction applications, and generator system for urban networks.

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