IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 25, NO. 1, JANUARYIFEBRUARY 1989 107

Torque Production in Permanent-Magnet Synchronous Motors

Abstract-The investigation involves detailed analysis of the torque production in a permanent-magnet synchronous motor drive with rectangular current excitation. The analysis includes the prediction of cogging as well as commutation torques. The effect of skewing is considered in the study.

1. INTRODUCTION ITH the recent increasing demand of small-size and high-performance dc permanent-magnet (PM) brushless

motors, it has become important to evaluate and reduce torque pulsations. These pulsations are responsible for irregularities of speed, which can be compensated only to a certain extent by means of speed control and are caused by both cogging and commutation torques. The cogging torque is produced by the interaction between the rotor poles and the stator structure and is independent of the load current. The commutation torque is produced by the interaction of the stator current and the rotor poles. Both the cogging and commutation torques are directly dependent of the flux density distribution and the shape of the stator core.

Up to now accurate calculations of torque have been done by using finite-element techniques. This method is very time consuming, and unless a three-dimensional package is used the effects of skewing could not be incorporated into the analysis.

The purpose of this research is to develop an analytical model capable of accurately predicting both the cogging and the commutation torques and to analyze and discuss the effect of different physical parameters of the machine design on the torque pulsations. The analysis is given for both the surface- mounted and the buried mounted magnet type rotor structures.

11. MATHEMATICAL MODEL

Both cogging and commutation torques depend on the shape of the flux density distribution in the air gap. The analysis presented here takes advantage of the explicit expressions for the air-gap flux density distribution obtained from a two- dimensional model presented in previous papers [ 11, [2 ] .

Paper IPCSD 88-20, approved by the Electric Machines Committee of the IEEE Industry Applications Society for presentation at the 1987 Industry Applications Society Annual Meeting, Atlanta, GA, October 19-23. Manu- script released for publication May 20, 1988.

J. De La Ree is with the Department of Electrical Engineering, College of Engineering, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061.

N. Boules with the the Electrical Engineering Department, General Motors Research Laboratories, Warren, MI 48090-9055.

IEEE Log Number 8823807.

T H U C - W M I NU DRl@OC I W C R T C I

U T I M 1

Fig. I . Basic rectangular current drive power circuit.

A . Cogging Torque Analysis The cogging torque was estimated by calculating the rate of

change of total stored energy in the air gap with respect to the rotor angular position, i.e.,

(1) dW D d W dB 2 dy

T = - = - - .

This relation, of course, assumes that the energy stored in the iron of the machine is negligible or varies little with the rotor position:

W = 1 dw (2) (3)

dv = gdldy. (4)

For a buried PM machine, the flux density distribution in the

0093-9994/89/01OO-0107$01 .OO O 1989 IEEE

108 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 25, NO. 1, JANUARYIFEBRUARY 1989

Fig. 2. Airgap layout

air gap was obtained from [2] (see Fig. 2). OD

~ ( y ) = & ( i ) sin (jay). ( 5 ) i = 1 , 3 ; . .

Since this expression is derived from a model that assumes a smooth stator surface, the approach described in [3] was used to introduce the effect of the stator slotting into the analysis. In this approach the flux density under the slot is assumed to be zero while the entire flux over the slot pitch enters the stator under the tooth; thus

L2 y2 g

L l YI 2Po w = j j - ( l+U,/b,)

where a, = slot opening; b, = tooth width

Yl=yo+a, /2+Ztan CP

Y2=yo+u,/2+Ztan ++b,

1 2

L1= -- LR

1 2

CP = angle of skewing

L2=- LR

and

(9.1)

+ i Bn(i )B, , ( j ) sin (iay) sin (jay) i = 1,3,5,

j = 1 + 2 , i + 4 ,

and flux distribution.

where

The integral in (10) yields

1 8i2u2 tan CP

+ (cos (2iaY3) -cos @ay4)

j = i + Z , i + 4 ; . ,

(-cos ( ( i - j )aY, )+cos ( ( i - j )uY4)

+ cos ((i - j ) u Y 5 ) - cos ( ( i - j ) a Y 6 ) ) 1

~ ( i + j ) ~ tan CP - (-cos ( ( i + j ) a Y , )

+ cos ( ( i + j ) a Y 4 ) + cos ( ( i + j ) a Y 5 )

(12.0)

(12.1)

>> -cos ( ( i+ j )aY6)) Y3= Y0+a, /2+b,+L2 tan CP

Y4= Yo+a, /2+b,+L1 tan CP (12.2)

Y5= Yo+a, /2+L2 tan CP (12.3)

Y,= Y o + a , / 2 + L I tan CP. (12.4)

DE LA REE AND BOULES: PERMANENT-MAGNET SYNCHRONOUS MOTORS

-b a -E 1

109

Y

In case of no skewing (i.e., CP = 0), (12) reduces to

1 4 iu

-_ (sin (2iaY3 -sin (2iaY5))

j=i+Z, i+4;"

. (sin ((i-j)uY3) - sin ((i-j)aY5))

>> 1 u ( i + j ) (sin (( i + j ) U Y3) - sin (( i + j ) a Y, )) -~ Using (l), one obtains

(-sin (2iaY3) + sin (2iuY4) + sin (2ia Y,) - sin (2iu Y6))

(sin ((i-j)aY3)-sin ((i-j)uYd)

-sin ((i-j)aYS)+sin ((i-j)aY6))

1 (sin ((i+j)aY3) -

a ( i + j ) tan

- sin ((i+j)uY4) - sin ((i+j)uY5)

+sin ((i+j)uY6)) >> and with no skewing (a = 0) the torque equation is

i/ O /

TL=DP I",: B(x)J (x , I) dl dx. (16) J(x) is a current sheet representing the stator ampere turns,

as described in Fig. 3, and can be expressed as

J( x ) = J , [U ( x - XI) - U (x - X2)l (17) X , = X o + I t a n cP+a/3 (18)

X 2 = X o + l t a n + + a (19)

pole arc pole pitch *

U =

For a buried-magnet machine the flux density distribution is given by ( 5 ) . Evaluating the integral of (16), we have

- sin ( iuX4) - sin ( iaX,) + sin ( i d 6 ) ) X3=Xo+L2 tan @ + + I 3 (23.1)

X4=Xo+Ll tan 9 + ~ / 3 (23.2)

X5=Xo+L2tan@+u (23.3)

&=Xo+L, tan @ + U . (23.4)

In case of no skewing (a = 0), (43) becomes -cos ((i -j)uY5) - cos ((i+j)aY3)

+cos ((i+j)uY5)) . (15) 1 T/=DPJ,LR ( 2

(COS ( i ~ X 3 ) - COS i = 1,3;'.

> A similar analysis was performed for a surface-mounted PM

The total instantaneous commutating torque of the motor machine. The results are presented in Appendix I.

110

1

m Lo 4

," .a C

C 0

0

.6 L U Lo

0

z CI

: 4 U O

x 3

IL -

2

n

' 1 /

/

IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 25, NO. 1, JANUARYIFEBRUARY 1989

1 I i , i

I i I

0 10 20 30 40 50 60

Mechanical n i g l e in Degrees

Fig. 4. Flex density distribution, buried PM machine.

3

2

t

= 1 C

U

m L c " 0 m

m m

C

B - 1

-2

-3

A Skewing - 0 S l o t Pitch,

I

0 2 4 6 B 10

Mechanical Angle In Degrees

Fig. 5. Cogging torque, buried PM machine.

Results of a similar analysis for the commutating torque of a surface-mounted magnet machine are presented in Appendix II.

III. RESULTS To evaluate the results of the analytical model, the equations

were programmed and different parameters of the machine were varied. Given machine design parameters, the program determines the flux density distribution and its harmonic constituents, as well as the developed cogging and commutat- ing torques.

Fig. 4 shows the flux density distribution at the stator surface of the motor as calculated by the analytical model. Verification of the accuracy of this model was presented elsewhere [ 11, [2].

Fig. 5 shows the cogging torque, as calculated by the

12

1

z c

D 3

0 c

0 m 0 J

f 12.

11.5

Mechanical Angle in Degrees

Fig. 6. Commutating torque, buried PM machine.

3

2.5

2 y1 L D U m L

0 Y I al z

c 1.5

1

5

0

', Peak Cogging Torque '\,\ /

\\

\\ \

-\ \\ / Commutating Torque P u l s a t i o n Vmax = 12 6451 \ \

\\ \

/ -\ \\ Commutating Tor&e P u l s a t i o n Vmax = 12.6451 \

\

0 2 4 6 B 1

Skewing/Slot Pitch

Fig. 7. Peak cogging torque and commutating torque pulsation.

analytical model, for a stator structure with no skewing, 1/2 slot skewing, and 1 slot skewing. As was expected, cogging torque goes to zero for an stator skewing of 1 slot pitch.

Fig. 6 shows the commutating torque, as calculated by the analytical model, for a stator with no skewing, 1/2 slot skewing, and 1 slot skewing.

Fig. 7 shows cogging torque peak value and commutating torque pulsation as functions of skewing. It is interesting to notice that, even when the cogging torque goes to zero with the skewing, the average torque decreases given a large commu- tating torque pulsation.

IV. CONCLUSION An analytical method was developed to determine cogging

as well as commutating torque in a permanent-magnet machine of both buried and surface-mounted magnet types. The method takes advantage of the knowledge of the flux density distribu-

DE LA REE AND BOULES: PERMANENT-MAGNET SYNCHRONOUS MOTORS 1 1 1

tion in the air gap as a function of design parameters. The equations can be programmed easily and are very useful for parametric studies and optimization.

(sin ((i-j)uY3)-sin ((i-j)uY5))

1 -___ (sin ((i+j)pO3) - sin ((i+.j)pe5)))

p(i+.i) NOMENCLATURE

U, Slot opening. B Flux density distribution. bl Tooth width. D Inside diameter of the stator. g Air-gap length. J Current sheet. LR Rotor axial length. P Pairs of poles. + Angle of skewing. U Pole pitch.

APPENDIX I

D 1 6 ( i ~ ) ~ tan 9

+ (- cos (2ipe3) + cos (2ipf3,)

) + cos (2ipQ -cos (2ipe6))

j = i + 2 , i + 4; .

(-COS ((i-j)p&)+cos ( ( i - j )pe4)

+cos ((i-.j)pe5)-cOs ((i-j)pea))

D + (-cos ((i+j)P03) +COS ((i+j)$),)+cos ( ( i+j)p&)

2pqi+.jp tan +

--os (G+.j)PY6)))) (A. 1 )

d3 =do + (us + 2L2 tan + + 2b,)/D (A. 1 . 1 ) e,= eo+ (us+ 2L1 tan + 2b,)/D (A. 1 . 2 )

(A. 1.3)

(A. 1.4)

e5 = eo + (U, + 2 L~ tan +)/D 0, = Bo + (U, + 2L tan +)/D.

In case of no skewing (9 = 0), (A.l ) reduces to

1 4 ip

-- (sin (2ipe3 - sin (2ip0,))

(A. 2)

and from energy the expression for torque used was

aw

ae T=-

yielding

(sin (2ip03) - sin (2ip04)

+ sin (2ip&) + sin @@e6))

(sin ((i-j)p03)-sin ((i- j)pO,)

-sin ( ( i - j )p&) + sin ( ( i - j )P&)) D

2p( i+ j ) tan 9 (sin ((i+.j)@3)

- sin ((i +j )pe , ) - sin ((i + j ) p e 5 )

-

+ sin ((i +j)p66)) . ('4.4) >> With no skewing (+ = 0), the torque equation is

APPENDIX I1

/ m n

(B. 1 . 1 )

112 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 25, NO. 1, JANUARYIFEBRUARY 1989

In case of no skewing (4 = 0), (B.l)

/ m 1

TI= DPJ,LR ( 2 B,(i) A (sin i = 1,3;.. 1P

APPENDIX III

reduces

(iltlP3) -

to

sin

[2] N. Boules, Prediction of no-load flex density distribution in perma- nent magnet machines, IEEE Trans. Ind. Appl., vol. IA-21, no. 3, pp. 633-643, MaylJune 1985. H. R. Bolton, Y. D. Liu, and N. M. Mallinson, Investigation into a class of brushless dc motor with quasi-square voltage and currents, IEE Proc., vol. 133, pt. B, no. 2, Mar. 1986.

(B. 1.2)

[3]

(B. 1.3)

(B. 1.4) Jaime De La Ree was born in Hermosillo Sonora, Mexico, on December 17, 1957. He received the B.S. degree with distinction in electrical engineer- ing from the Instituto Tecnologico y de Estudios Superiores de Monterrey in 1980 and the M.S. and Ph.D. degrees from the University of Pittsburgh, Pittsburgh, PA, in 1981 and 1984, respectively.

In 1984 he joined the faculty of Virginia Poly- technic Institute and State University, Blacksburg, VA, where he is currently an Assistant Professor. His research interest is in the area of rotating

machinery.

Societies. Physical Data of Prototype Machine Dr. De La Ree is a member of Tau Beta Pi and Eta Kappa Nu Honorary

Magnet height, h,, Magnet radial depth, W,,

Air-gap length, g, Number of poles, 2p Magnet residual induction, B,, Magnet recoil permeability, pm = 1.38 x H/m. Tooth width, bt, Number of slots, Qs, = 36. Braunschweig, West Germany.

= 6.35 D. = 25.4 mm. = 49.7 mm. = 0.66 mm.

Pole pitch, U ,

Nady Boules (M80-SM83) was born on August 20, 1948, in Alexandria, Egypt. He received the B.S.E.E. degree and the M.S.E.E. degree from Cairo University, Egypt, in 1968 and 1973, respec- tively. He received the Dr.-Ing. degree in 1978 from the Technical University of Braunschweig,

From 1968 to 1974 he worked as an Instructor and Assistant Lecturer at Cairo Institute of Technol- ogy in Helwan, Egypt. In 1974 he joined the Technical University of Braunschweig, Braunsch-

weig, West G e m y , as a fellowship holder of the Deutsch Akademsch Austauschdienst. From 1978 to 1980 he worked as a Research Assistant at the Institute of Electrical Machines. Technical Universitv of Braunschweig, West

= 6. = 1.03 T.

= 5.95 mm.

ACKNOWLEDGMENT

The authors wish to express their appreciation to Dr. T. NeM and Dr. L. Jacovides, General Motors Research Labors- tories, for their valuable discussions and comments.

t11

Germany. From 1980 to 1982 he was a Senior Magnetics Engine& with Simmonds Precision Corporation, Norwich, NY. Since 1982 he has been with the General Motors Research Laboratories in Warren, MI, where he is now Section Manager for electromechanics in the electrical and electronics engineering department.

REFERENCES N. Boules, Field analysis of PM machines with buried magnet rotor, presented at Int. Conf. Electric Machines, Munich, West Germany, Sept. 8-10, 1986.