Recent Developments of Electrical Drives - Part 8 pdf

An approach based on magnets segmentation is introduced for minimizing the cogging torque of surface-mounted permanent magnet motors.. Even if this solution allows to minimize the ripple

56 Kolehmainen

Table 2 Comparison of nominal load results

current Only copper losses in stator winding are taken account in efficiencyη calculations Other losses are relatively small

In the table first calculated U-shapeA results are calculated with the design with same total magnet length and thickness per pole than with V-shape design The second calculated results U-shapeB are calculated with design with longer and thinner magnets per pole Magnets thickness, width, and area are also shown in the table with unit of mm Dimensions

of V-shape design are real dimensions of one magnet and for U-shape designs dimension are values which corresponds values of V-shape design

The maximum output torque with the first U-shapeA design is smaller than with the V-shape design and it has also smaller load angle difference This is due to smaller reluctance torque and effect of iron bridges between the magnets Torque and reluctance torque curves are shown in Fig 10

Figure 10 Torque and reluctance torque of motors with V- and U-shape designs as a function of load

angle

I-5 Finite Element Analysis of PMSM with Buried Magnets 57

Figure 11 Power factor as a function of torque.

Reluctance torque is larger with V-shape than with U-shapeA design, because the mag-netic structure of rotor By comparing torques of U-shapeA and U-shapeB designs can also see the effect of decreasing of magnet thickness Reluctance and maximum torque is smaller with thicker magnets

Power factors of V-shape and U-shape designs are shown as a function of torque in Fig 11 Power factor of the motor with the U-shapeA design is larger up to the nominal point and with the higher torque it is smaller Nominal torque of the motors is 875 Nm and usually the motors are used with partial loads with different speeds Hence, the motor with the U-shape magnets is usually in the torque range with better power factor Also the maximum torque decreases because of the smaller reluctance With the longer and thinner magnets in the V-shapeB design there is smaller maximum torque and higher power factor with nominal load as can be expected

There is significant difference of torques between V- and U-shapeA designs This is shown

in the Fig 12 With the U-shapeA design, the oscillation of torque is with the frequency of magnets going over stator phase With the V-shape design oscillation frequency is two times

of frequency with U-shapes, because two magnets go over one stator phase with V-shape

Figure 12 Torque oscillations of V-shape and U-shapeA designs.

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design while one magnet going over one stator phase with U-shapeA design In addition the amplitude is smaller with V-shapes

Conclusion

It is shown that the PM motor with the U-shape magnets in every second pole works as well as the conventional PM motor with the V-shape magnets in every pole Asymmetrical structure of pole pairs in this design cause no asymmetry to the magnetic field of air gap This design gives a possibility to get higher flux densities with the same amount of magnets The number of magnet pieces is also reduced

Torque oscillation with U-shapeA design is too high compared to V-shape design This could be avoided with using different stator slots or iron structure near the magnets and air gap

In conclusion, this new solution gives more possibilities to produce buried permanent motors with better power factor and efficiency

References

[1] J Salo, T Heikkil¨a, J Pyrh¨onen, T Haring, “New Low-Speed High-Torque Permanent Magnet Synchronous Machine With Buried Magnets”, International Conference Electrical Machines (ICEM 00), Vol 3/3, Espoo, Finland, August 28–30, 2000, pp 1246–1250

[2] F Libert, J Soulard, J Engstr¨om, “Design of a 4-pole Line Start Permanent Magnet Synchronous Motor”, International Conference Electrical Machines (ICEM 02), Brugge, Belgium, August 25–28, 1998, p 173

[3] Flux2D Software, www.cedrat.com

I-6 DESIGN TECHNIQUE FOR

REDUCING THE COGGING

TORQUE IN LARGE SURFACE-MOUNTED

MAGNET MOTORS

R Lateb1, N Takorabet1, F Meibody-Tabar1, J Enon2

and A Sarribouette2

1INPL–GREEN, 2 avenue de la forˆet de Haye, 54516 Vandoeuvre-l`es-Nancy, France

ramdane.lateb@ensem.inpl-nancy.fr, Nouredinne.Takorabet@ensem.inpl.nancy.fr,

Farid.Meibody-Tabar@ensem.inpl-nancy.fr

2Converteam, 4 rue de la Rompure, 54250 Champigneulles, France

jacques.enon@converteam.com, alain.saribouette@converteam.com

Abstract An approach based on magnets segmentation is introduced for minimizing the cogging

torque of surface-mounted permanent magnet motors The authors show that the magnet segmen-tation has also an effect on the ripple torque especially on its high order harmonics However, this technique has a small effect on the main performances of the motor such as the average torque So, the segmentation number is chosen according to the choice of the magnet span and the stator winding An approach based on Fourier analysis is used to justify the numerical results obtained by finite elements method

Introduction

The use of high power surface-mounted permanent magnet (PM) motors in different in-dustrial applications (windmill generator, marine propulsion, traction ) is very attractive thanks to their high torque density [1] The design of PM motors must take into account the requirements of such applications One of the most important constraints is the mechanical shaft vibrations, especially at low speeds, that can be avoided by reducing the amplitude of torque harmonics This can be achieved by using a wide range of techniques proposed by sev-eral authors [2] Some of these techniques are based on modifying the current waveforms to cancel torque pulsations for any PM motor with known electromotive force (EMF) waveform [3] If the EMF is not sinusoidal, high dynamic current waveform is required which is diffi-cult to apply by high power inverters The other techniques are based on structural solutions The first structural solution consists in an adapted choice of the stator winding for a given number of stator slots [4] Even if this solution allows to minimize the ripple torque rate due the interaction between stator currents and rotor magnets, it has no effect on the cogging torque since the stator slots design is chosen As well known in PM motors,

S Wiak, M Dems, K Kom˛eza (eds.), Recent Developments of Electrical Drives, 59–72.

2006 Springer.

60 Lateb et al.

cogging torque arises from the interaction between magnets and slotted iron stator Another solution consists to optimize the magnets pole angular width (span) Even if this geometrical parameter allows to reduce the cogging torque, it is often used to maximize the average torque and to minimize the ripple torque rate for an imposed current waveform It has shown that a classical requirement to eliminate the cogging torque first harmonic is to choice a magnet pole span almost a multiple of slot pitch [5–7] It’s obvious that motors with closed slots or slotless stator have no cogging torque These solutions, which lead to mechanical difficulties, can be approached by introducing magnetic wedges in slots that reduces significantly the cogging torque It has been shown that choosing equal tooth and slot width may diminish the fundamental component of the cogging torque [5] In fact, for

a given slot pitch, there is several values of tooth width which minimize the cogging torque

A teeth-pairing design with different tooth widths as well as the teeth notching design

is other techniques that allow to reduce the cogging torque [8] One of the most popular techniques to reduce cogging torque is to skew the stator lamination stack or rotor magnets [9] Ideally, the cogging torque is vanished with a skewing angle of integer multiple of the cogging torque period Stator skewing is less interesting because of a more complex construction To make easier the rotor manufacturing, the rotor skewing may be done by placing the PM axially skewed by several discrete steps [9] All the techniques cannot be related in this paper but one can find a large bibliography in [2]

In high power PM motors, the magnet pole span is too large for being realized in only one block For technical and cost considerations, each rotor pole is often realized with several elementary magnet blocks with the same polarity (magnet segmentation) In this paper, the authors present the effect of the magnets segmentation on the machine perfor-mances One can expect that the segmentation of the magnets does not modify signifi-cantly the main performances since the magnet span conserves the same value However, the magnets segmentation modifies locally the air-gap magnetic field distribution which leads not only to a significant modification of the cogging torque but also the back EMF harmonics

In the second section, using analytical approaches, we present different techniques re-ducing the cogging torque For the sake of the analyze of the cogging torque minimization, the analytical method is useful but not enough precise to evaluate its exact shape and its amplitude In the third section we present the used numerical finite elements method In the last part of this paper, for different values of the magnet span, we analyze the influence

of magnets segmentation on the performances (Average torque, Harmonics of back EMF, cogging torque, Pulsating torque) of a surface-mounted PM motor

The obtained results and the carried out analysis highlight that for a given value

of a magnet pole span, it exists an optimal number of magnet blocks per pole for which the best compromise between average torque and pulsating torque rate may be achieved

Used techniques for cogging torque minimization

The cogging torque being caused by the interaction between the rotor magnets and the stator teeth The main parameters which affect significantly its shape and amplitude are: the

I-6 Design Technique for Reducing the Cogging Torque 61

(a)

(b)

stator

rotor

τs

τp=6τs

stator

rotor

τs

=6.5τs

Figure 1 Two cases of symmetric distribution: (a) two poles, 12 slots; (b) two poles, 13 slots.

number of stator teeth, the magnet pole span, and the magnet segmentation In the following

we detail how to choose these parameters to minimize the cogging torque

Number of slot

In a PM motor with Nspstator slots per pole pair, the contribution Tcmof one magnet to the cogging torque is of the form:

Tcm(θ) =∞

h =1

Where Th, is the Fourier coefficient of the hth harmonic.θ is the electrical angular position.

To illustrate the interest of an odd number of stator slots per pole pair, we consider a pole pair of a PM motor as shown in Fig 1 The cases of Nsp= 12 (Fig 1a) and Nsp= 13 (Fig 1b) are considered

In Fig 1(a), we can observe that each magnet has the same relative position with re-spect to the stator teeth The cogging torque per pole pair is twice the contribution of one magnet

Tcp(θ) = 2∞

h =1

In the second case, the magnets have not the same relative position with respect to the stator teeth Since one magnet has a half slot pitch electrical shift angle θ0= π

N , the

62 Lateb et al.

cogging torque per pole pair becomes:

Tcp(θ) =∞

h =1

Th

 sin(hNspθ) + sin(hNsp(θ − θ0))

(3)

Hence:

Tcp(θ) =∞

h =1

Th

 sin(hNspθ)(1 + cos(hNspθ0)

− cos(hNspθ) sin(hNspθ0)

(4)

By replacing the expression ofθ0in (4), one can show that the fundamental of the cogging torque (h= 1) is eliminated as well as all odd harmonics Indeed, in this case only the even harmonics (2× Nsp) of the Fourier decomposition subsist

In the general case, one can demonstrate that for a symmetric distribution of Npmagnets and Ns slots in the motor, the fundamental frequency of the cogging torque is the least common multiple (LCM) of Npand Ns

High power low speed PM motors offer the possibility to use a large number of slots per pole that allows to increase the frequency of the cogging torque first harmonic Moreover,

as illustrated above, using an odd number of slots per pole pair doubles the frequency of the cogging torque

Trailing edge

magnet

Produced torque at the trailing edge with the tooth 1.

Leading edge

magnet

magnet

Figure 2 Simple model of cogging torque mechanism.

I-6 Design Technique for Reducing the Cogging Torque 63

Magnet span The motor cogging torque is mainly caused by the interactions of magnets edges (trailing and leading edges) and stator teeth Thus, the study of the cogging torque can be reduced

to the analysis of these interactions

By ignoring the effect of rotor curvature, magnet leakage flux, and fringing flux and by bringing back to a rectangular field problem, one can represent the mechanism of cogging torque production by the simple model illustrated in Fig 2, where the dashed lines represent the area in which the main part of magnetic energy is stored

For the three magnet positions shown in Fig 2, the variation of the magnetic energy under the tooth 1 during the passage of the trailing edge is illustrated, while the energy under the other teeth (2, 3, 4) doesn’t vary The produced torque due to the passage of the trailing edge is proportional to this energy variation It is a periodic function that can be expressed as:

Ttrailing(θ) = T0+∞

h =1

T

Let’s defineαmas the electrical shift angle between the leading and trailing edges The torque produced by the leading edge is the opposite of the one produced by the trailing edge shifted byαm

Tleading(θ) = −T0−∞

h =1

T

hcos(hNsp(θ − αm)) (7) Then the expression of the cogging torque, produced by this magnet, becomes:

Tcm(θ) =∞

h =1

T h

 cos(hNspθ) − cos(hNsp(θ − αm))

(8)

Under the assumptions mentioned above, from (8) it can be easily shown that the cogging torque can be eliminated by choosing:

αm= 2πk

Nsp

with k an integer andτsthe electrical slot pitch angle

This condition, indicating that the angleαmmust be a multiple of a slot pitch, is obtained under the mentioned assumptions, which don’t take into account the effect of rotor curvature, magnet leakage flux and fringing flux Considering these phenomena, a zero cogging torque cannot be achieved However, according to different authors, using finite element analysis, the cogging torque may be minimized for a magnet span ofαm= (n + 0.14)τsby ignoring the effect of rotor curvature [5] or (n+ 0.17)τsby considering the effect of rotor curvature [6] or (n+ 0.25)τsfor linear motors [7]

Magnets segmentation

As previously said, 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 the following, a curved shape magnet is used for each

64 Lateb et al.

magnet

αs

air-gap

γ

rotor

Figure 3 Representation of an elementary RSMM.

nonsegmented rotor pole and rectangular cross-section magnet blocks are used for realizing each pole of a segmented permanent magnet machine For the sake of simplicity, we will use the next abbreviations:

rRSMM: Rectangular Surface-Mounted Magnets with parallel magnetization

rCSMM: Curved Surface-Mounted Magnet per pole

For a CSMM motor, the mechanical air-gap is constant (CSMM is delimited by the red dashed lines in Fig 3), while in the RSMM motor the air-gap varies

Fig 3 shows the geometrical representation of an elementary RSMM The relation between the whole magnet pole spanαmand the elementary block magnets spanγ is:



αm= Nγ

where N is the number of elementary magnet blocks forming each pole,αs the opening angle of the elementary magnet facing the air-gap, andβ half of the opening angle of the

slit between two elementary magnet blocks The span of each elementary magnet block is expressed in term of the slot pitch in the same way adopted above:



γ = (n ± ε)τs

where n is an integer

Using the results given in [6], the optimal span of each elementary magnet block should be such asε ≈ 0.17 However this result has been obtained for CSMM motor For rectangular

magnets (RSMM) motors the optimum value ofε will be probably modified.

Numerical analysis

Finite elements method is used for the computation of the machine characteristics In order

to increase the precision of the results and especially the computation of the cogging torque,

a finer mesh is applied all around the air-gap [10], at each rotor position the meshing is renewed Thanks to geometrical and electrical symmetries, only one pole pair of the machine

is considered, it allows a minimum time consuming In addition, for each design we made the calculations for 60 different rotor positions over a slot pitch The number of nodes is more than 40,000 nodes The computations take into account the saturation of iron core

I-6 Design Technique for Reducing the Cogging Torque 65

Figure 4 PM motor with six rectangular magnet blocks.

The cogging torque is computed through the Maxwell weighted stress tensor method [11] and the used software for the computation is FEMM [12]

Fig 5 shows a cross-section view of a PM motor in three configurations where each magnet pole is divided in two, three, or four blocks In the following, the performances

of the segmented PM motor having N elementary magnet blocks (N= 1, , 6) per pole

are computed (Fig 6) These performances will be presented vs the total magnet span

αm= Nγ.

Results

The computations are performed for a 6 MW, 16-poles, 170 rpm, surface-mounted PM motor supplied by sinusoidal waveform currents An adapted stator winding with a fractional slot number per pole and per phase (15 slots per pole pair) allows to reduce considerably the space harmonics of stator magnetomotive force (MMF), especially the fifth one which is totally cancelled Another advantage of using an odd slot number over a pole pair is to increase the pulsation of the cogging torque So, for the studied topology the cogging torque period is 2π/30 electrical degrees Magnetic wedges used in the slot openings (isthmus),

Figure 5 Cross-section view of one pair pole of the PM motors with: (a) two magnet blocks per pole;

(b) three magnet blocks per pole; (c) four magnet blocks per pole