Recent Developments of Electrical Drives - Part 9 pps

Varying the magnet span influences not only the average torque but also the cogging torque and the ripple torque.. Cogging torque waveforms forαm= 152◦.For a segmented magnet machine N=

370

390

410

430

450

470

120 125 130 135 140 145 150 155 160 165 170 175 180

Magnet Span[Electrical degree]

1 CSMM 2 RSMM 3 RSMM 4 RSMM 5 RSMM 6 RSMM

Figure 6 Average torque vs magnet span for each structure.

allows to reduce the amplitude of the cogging torque The slot-opening angle is less than half a slot pitchτs

Assuming that the phase currents are function of the electrical rotor position θ

(self-controlled PM motor), the electromagnetic torque Tem, which is the sum of the cogging

torque, Tcand the current-magnets interaction torque Te −i, can be expressed by:

Tem(θ) = Tc(θ) + Te −i(θ)

= Tc(θ) + 1

q



j =1

where:

is rotor angular speed.

ijand ejare the current and back EMF of the jth phase

q is the phase number (q= 3)

The phase currents are assumed to be sinusoidal while the back EMF contains harmonics These harmonics are at the origin of the pulsating component of Te −i, called ripple torque

Tr So the total pulsating torque Tcris the mean value of Te −iis the average torque Tav.

Fig 6 shows the average torque vs magnet pole span for the PM motors with N elementary magnet blocks (N = 1, , 6) per pole It is obvious that the nonsegmented PM motor (CSMM) in which a curved magnet per pole is used, has the highest average torque since the magnet volume is more important and the air-gap is constant The PM motor with two magnets blocks per pole (two RSMM) has the lowest average torque because the average air-gap is more important which affects the air-gap flux density The four, five, and six RSMM configurations present almost the same average torque for a given value of magnet pole span Within sight of Fig 6, one finds a classical result, which shows that on one hand, beyondαm≈ 165◦the profit in average torque is weak compared to the cost generated by

the increase of the magnet volume On the other hand, forαm< 145◦the average torque is

relatively weak

Varying the magnet span influences not only the average torque but also the cogging torque and the ripple torque As these two pulsating torque components depend differently

on the magnet span, a compromise should be made in order to minimize the total pulsating

0 0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

Magnet Pole Span [Electrical degree]

(a)

0 0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

Magnet Pole Span [Electrical degree]

(b)

Figure 7 Cogging torque factor vs magnet pole span.

torque Tcr In order to carry out a comparative study of various configurations, we use the next criteria:

CTF= Tcpp

Tav

; PTF=Tcrpp

Tav

(13)

where CTF and PTF are respectively the Cogging Torque Factor and the Total Pulsating Torque Factor Tcpp and Tcrpp are respectively the peak-to-peak cogging torque and the peak-to-peak total pulsating torque

In order to illustrate clearly the effect on the magnets subdivision on the cogging torque, the results are presented on Fig 7(a,b) The CTF obtained for the CSMM configuration (N= 1) presents several minima (Fig 11a) achieved for the following values of the magnet pole span: γ = αm

1 = (n/2 + 0.17)τs with n= 10 to 14 In [5] the authors have found

αm= (n + 0.17)τs These two results are not in contradiction, because the studied stator has an odd slot number per pole pair which doubles the number of minima of the cogging torque

Figure 8 Cogging torque waveforms forαm= 152◦.

For a segmented magnet machine (N= 1) with rectangular magnet blocks (RSMM), the number of magnet edges per pole pair increases, there is no more universal rule giving the optimal values ofγ = αm

N minimizing the cogging torque As an example, for the case (N= 5) presented on Fig 7(a), the few minima are obtained for either γ = αm

5 = (n/2 −

0.17)τsorγ = αm

5 = (n/2 + 0.14k)τswith k an odd number

The results obtained for the other cases (N= 2, 3, 4, and 6) are gathered in Fig 7(b) because they present similar shapes Two common minima are clearly distinguished for the magnet pole spanαm≈ 135◦andαm≈ 152◦.

From the investigations presented above, it is clear that one cannot extract a general rule that reduces the cogging torque rate However, one can affirm that there is some con-figurations offering the possibility to reduce considerably the cogging torque as shown in Fig 8 Indeed, forαm= 152◦, the weakest cogging torque is achieved with six RSMM per

pole However the most important criterion is to maximize the average torque and reduce

as possible the total pulsating torque So we present in Fig 9 the total pulsating torque factor (PTF) evolution vs the magnets span for different configurations (N= 1, , 6).

The six curves have the same shape but the amplitude of the PTF varies slightly according

to segmentation number N Referring to Fig 9, the best choice (PTF≈ 1.2%) should be

N= 3 (αm≈ 165◦) or N= 4 (αm≈ 160◦) According to Fig 6 these two configurations

lead nearly to the same average torque but N= 4 corresponds to lower magnet volume Even if the most important criterion is to maximize the average torque and to reduce as possible the total pulsating torque, a special care must be taken to the reduction of lower torque harmonics (6 and 12) As these harmonics are due the low harmonics (5, 7, 11, and 13) of back EMF, we study in the following the simultaneous effects of the magnet span and the segmentation in N blocks on their amplitudes

Fig 10 presents the magnitude evolution of the main back EMF harmonics (7th, 11th, and 13th) Note that the fifth harmonic of the back EMF is null thanks to the adopted fractional winding

For the seventh harmonic, all the curves are almost identical (except for three RSMM structure), which shows that the segmentation has not a real influence on the seventh

har-monics For this latter, the minimum is obtained for a magnet pole span of 155◦ The magnet

segmentation seems to have a significant effect on the 11th and 13th harmonics as shown in Fig 10(b,c) However, all the curves shown on Fig 10(b,c) present two minima which do not coincide with the seventh harmonic one (Fig 10a) Then to reduce the amplitude of the sixth torque harmonic, one has to choice a magnet pole span such as the seventh harmonic

of the back EMF is weak compared to 11th and 13th

For the studied machine with an odd number of slots per pole pair, the choice of an adapted winding allows to suppress the fifth EMF harmonic The choice of an appropriate span (αm≈ 155◦) allows to make a good compromise between the increase of the average torque

and the reduction of the sixth torque harmonic This can be achieved with a segmentation number N equal to 4, 5, or 6 magnet blocks

Among these values (N= 4, 5, 6), for the studied machine, the choice of N = 4 leads

not only to the weakest value of the total pulsating torque (Fig 9) but also to the weakest value of the cogging torque (Fig 11)

Taking into account the obtained results in the case of the studied machine, we showed that the winding type, the stator slots number and the magnet pole span remain the main parameters acting on the principal performances (Average Torque, sixth torque harmonic)

of the machine For the appropriate choice of these main parameters, a well-adapted choice

of the segmentation number of blocks allows to reduce the cogging torque and the total pulsating torque as well

Conclusion

PM motors are finding expanded use in high power directly driven applications where torque smoothness is essential Cogging torque in PM motors is among the undesired effects contributing to the motor’s output ripple, vibration, and noise It can be substantially reduced by the combination of several well-known techniques

For manufacturing and cost reasons, in large permanent magnet motor, each rotor pole

is often realized with several elementary magnet blocks with the same polarity (magnets segmentation) In this paper we have shown that the choice of the magnet blocks number over a pole must be considered as an optimization parameter acting on local phenomena such as the cogging torque and higher torque harmonics

0

0,5

1

1,5

2

2,5

3

3,5

4

120 125 130 135 140 145 150 155 160 165 170 175 180

Magnet Pole Span[Electrical Degree]

1 CSMM

2 RSMM

3 RSMM

4 RSMM

5 RSMM

6 RSMM

Figure 9 Pulsating torque factor vs magnet pole span.

0,2

0,4

0,6

0,8

1

1,2

1,4

Magnet Pole Span[Electrical Degree]

1 CSMM

2 RSMM

3 RSMM

4 RSMM

5 RSMM

6 RSMM

(a)

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

120 125 130 135 140 145 150 155 160 165 170 175 180

Magnet Pole Span[Electrical Degree]

1 CSMM

2 RSMM

3 RSMM

4 RSMM

5 RSMM

6 RSMM

(b)

0

0,05

0,1

0,15

0,2

0,25

0,3

0,35

120 125 130 135 140 145 150 155 160 165 170 175 180

Magnet Pole Span[Electrical Degree]

1 CSM M

2 RSM M

3 RSM M

4 RSM M

5 RSM M

6 RSM M

(c)

Figure 10 7th, 11th, and 13th harmonics back EMF vs magnet span.

Figure 11 Cogging torque waveforms for a magnet span corresponding to 155◦(electrical degree).

The technique that consists on the choice of an appropriate number of magnet blocks over a magnet pole cannot be done without considering the main parameters, which impose the principal machine performances such as the average torque

In addition to the reduction of the cogging torque and high torque harmonics Another important effect of magnet subdivision is to reduce eddy currents inside the magnets This may be achieved by the choice of a segmentation number around 6

References

[1] A Arkkio, N Bianchi, S Bolognani, T Jokinen, F Luise, M Rosu, “Design of Synchronous

PM Motor for Submersed Marine Propulsion Systems”, International Conference on Electrical Machines (ICEM 2002), Paper No 523, Brugge, Belgium, August 25–28, 2002

[2] T.M Jahns, W.L Soong, Pulsating torque minimization techniques for permanent magnet AC motors drives—a review, IEEE Trans Ind Electron., Vol 43, No 2, pp 321–330, 1996 [3] J.-P Martin, F Meibody-Tabar, B Davat, “Multiple-phase Permanent Magnet Synchronous Machine Supplied By VSIs Working Under Fault Conditions”, IEEE Industry Applications Conference, 2000, 35th IAS Annual Meeting, Roma, Italy, October 2000

[4] L Parsa, L Hao, H.A Toliyat, “Optimization of Average and Cogging Torque in 3-Phase IPM Motor Drives”, IEEE Industry Applications Conference, 2002, 37th IAS Annual Meeting, Vol

1, October 13–18, 2002, pp 417–424

[5] T Li, G.R Slemon, Reduction of cogging torque in permanent magnet motors, IEEE Trans Magn., Vol 24, No 6, pp 2901–2903, 1988

[6] T Ishikawa, G.R Slemon, A method of reducing ripple torque in permanent magnet motors without skewing, IEEE Trans Magn., Vol 29, No 2, pp 2028–2031, 1993

[7] K.-C Lim, J.-K Woo, G.-H Kang, J.-P Hong, G.-T Kim, Detent force minimization techniques

in permanent magnet linear synchronous motors, IEEE Trans Magn., Vol 38, No 2, pp 1157–1160, 2002

[8] S.-M Hwang, J.-B Eom, Y.-H, Jung, D.-W Lee, B.-S Kang, Various design techniques to reduce cogging torque by controlling energy variation in permanent magnet motors, IEEE Trans Magn., Vol 37, No 4, pp 2806–2809, 2001

[9] D.C Hanselman, Effect of skew, pole count and slot count on brushless motor radial force, cogging torque and back EMF, IEE Proc Electron Power Appl., Vol 144, No 5, pp 325–330, 1997

[10] D Howe, Z.Q Zhu, The influence of finite element discretization on the prediction of cogging torque in permanent magnet excited motors, IEEE Trans Magn., Vol 28, No 2, pp 1080–1083, 1992

[11] F Henrotte, G Deli´ege, K Hameyer, “The Eggshell Method for the Computation of Elec-tromagnetic Forces on Rigid Bodies in 2d and 3d”, Proceedings of the 10th Biennial IEEE Conference on Electromagnetic Field Computation, CEFC’2002, June 2002, p 30

[12] D Meeker, Finite Element Method Magnetics Software, www.http://femm.foster-miller.com

I-7 OVERLAPPING MESH MODEL

FOR THE ANALYSIS OF ELECTROSTATIC MICROACTUATORS

WITH ECCENTRIC ROTOR

Piotr Rembowski and Adam Pelikant

Institute of Mechatronics and Information Systems, Technical University of Lodz, Poland,

ul Stefanowskiego 18/22, 90-924 Lodz, Poland

prembowski@poczta.onet.pl, apelikan@mail.p.lodz.pl

Abstract The numerical model for three-dimensional field analysis of electrostatic micromotors

with stator and rotor symmetry axes located in the different points has been presented The results of the numerical tests confirm the thesis about the correctness of the model Short CPU time is obtained even with quite big number of mesh elements

Introduction

The paper presents numerical model for three-dimensional field analysis of electrostatic micromotors with stator and rotor symmetry axes located in different points Due to a very small size of micromachines it is impossible to place the rotor in such a position that would provide ideal symmetrical air gap between electrodes There is no algorithm which fully covers this kind of asymmetry Solving this problem through commercial applications leads

to mesh generating for each single analyzed position, which means increased time of the analysis The application of the mesh overlapping lets one avoid repeated mesh generating for the whole model and decrease the time of computation Reduction of this time can be obtained by using separated submeshes for both stator and rotor generated only once, and only recalculating the part describing the air gap

Possible applications

There are many technical solutions with purposely designed nonsymmetrical air gap One of the most important applications is the possibility of using ferroelectrics in the construction

of the wobble motor (Fig 1) This solution results in a considerable increase of torque In this kind of motors the air gap asymmetry is so important that it must not be omitted in computation

With the exception of a few approximated analytical modeling cases, there are no known algorithms for an efficient solution that deals with big asymmetries of the air gap in the electrostatic microactuators, especially with ferroelectrics

2006 Springer.

Figure 1 SEM photo of a 100μm-diameter, 2.5 μm-gap, wobble micromotor with a free bearing.

A new solution was developed to compute models with nonsymmetrical air gap This was possible due to extending mesh overlapping model

What is more, it is possible to apply the algorithm for models with leant rotor rotation axis The sloping can result from technical inaccuracy as well as constructor’s intention So far the problem of mesh overlapping in all three dimensions has not been so far considered

in literature

Model

A numerical, finite element method based algorithm has been constructed to solve problems mentioned above The integral form of the second Maxwell equation with application of the Gauss law (1) is a base to formulate mathematical equation describing the analyzed object







S

The equation (1) applied in finite element method with the approximation on each mesh element with weigh functionsλ i leads to formula (2), where Vi means the values of the potentials of the nodes



j



S j

ε j grad





 e

λ i V i



Assuming cylindrical coordinate system, the second degree polynomial as approximating function in a single element was in form (3):

a r ( ϑr)z + a r ( ϑr) + a (ϑr)z + a r z + a r + a (ϑr) + a z + a = 0 (3)

In the consequence of the above one gets a system of linear equations (4) with symmetric, well conditioned, positive definite matrix with 27 nonzero elements in each row

Mesh overlapping

In the overlapping mesh model the stator and the rotor are represented by two separate meshes An air gap is included in both of them There is a common region consisting of at least one common layer along all the height of the model Values of potentials of peripheral nodes are determined by boundary conditions and linear approximation based on values

in neighboring nodes of the other mesh There are two cases of solving the problem: first when the centers of the stator and the rotor are shifted by a distance which is smaller than one third of the air gap width (Fig 2) and the second when the shift is larger (Fig 3)

In both cases the rotor and stator meshes cover the air gap on the area whose width is equal to the smallest distance between the rotor and the stator electrodes In connection with the above, in the former case both meshes cover the whole area of the air gap and one single layer of elements (the last one) can be used for mesh overlapping

In the latter there is a need to extend one of the meshes (by adding additional layers of nodes) to cover the whole area of the air gap In this case it is necessary to use more layers

of elements in mesh overlapping computation

In the symmetrical model both meshes have a common surface, in the air gap area, along all the height of the model In this case nodes of the stator and the rotor meshes for overlapping bounds have only different angleθ (Fig 4) In this figure nods belonging to the rotor mesh have numbers starting from the letter i , and nods belonging to the stator mesh have numbers starting from the letter j The letters k and n signify the numbers of nodes in

rotor and stator meshes for the constant radius

Figure 2 Generated meshes for the shit less than one third of the air gap width.