Ananalysisof3-dimensionalmagneticfielddistributionsinasmall-sizedsynchronousmotorwithapermanentmagnetrotor Electricity&Magnetismfields

Physics

Electricity & Magnetism fields

Okayama University Year 1983

An analysis of 3-dimensional magnetic field distributions in a small-sized synchronous motor with a permanent

magnet rotor

T. Misaki H. Tsuboi

Okayama University Okayama University

This paper is posted at eScholarship@OUDIR : Okayama University Digital Information Repository.

http://escholarship.lib.okayama-u.ac.jp/electricity and magnetism/145

AN ANALYSIS O F 3-DIMENSIONAL MAGNETIC F I E F DISTRIBUTIONS I N A SMALL-SIZED SYNCHRONOUS MOTOR WITH A PERMANENT MAGNET ROTOR

T. Misaki and H. Tsuboi

A b s t r a c t

-

T h i s p a p e r d e s c r i b e s a n a n a l y s i s o f 3- d i m e n s i o n a l m a g n e t i c f i e l d d i s t r i b u t i o n s i n a small- sized synchronous motor with a permanent magnet rotor

and a s i m u l a t i o n o f the r o t o r b e h a v i o r . Here, t h e concept of a s u r f a c e m a g n e t i c c h a r g e i s i n t r o d u c e d , a n d t h e n t h e m a g n e t i c f i e l d d i s t r i b u t i o n s are computed by u s i n g t h e i n t e g r a l e q u a t i o n method. Next, the rotor

displacement i s computed b y u s i n g Newmark's f3-

parameter method. By t h e u s e o f t h e s e t e c h n i q u e s ,

s i m u l a t i o n o f t h e r o t o r b e h a v i o r i s performed. The r e s u l t s of t h e s i m u l a t i o n a r e examined i n c o n t r a s t t o t h o s e of t h e e x p e r i m e n t s .

INTRODUCTION

AS i s g e n e r a l l y known, a small-sized synchronous motor with a permanent magnet rotor i s w i d e l y u s e d a s an i n d u s t r i a l timer. I n t h e d e s i g n o f t h i s m o t o r , it i s v e r y i m p o r t a n t t o d e t e r m i n e t h e m a g n e t i c f i e l d

d i s t r i b u t i o n . I n t h i s p a p e r , t h e a u t h o r s wish to r e p o r t c o m p u t a t i o n s o f 3 - d i m e n s i o n a l m a g n e t i c f i e l d

d i s t r i b u t i o n i n t h e m o t o r a n d d e s c r i b e s i m u l a t i o n o f t h e r o t o r b e h a v i o r .

H e r e , t h e c o n c e p t o f a s u r f a c e m a g n e t i c c h a r g e i s i n t r o d u c e d , a n d th e n the m a g n e t i c f i e l d d i s t r i b u t i o n i s computed b y u s i n q t h e i n t e g r a l e q u a t i o n method. I n t h i s method, each curved surface on which t h e c h a r g e is d i s t r i b u t e d i s d i v i d e d i n t o a number of curved s u r f a c e e l e m e n t s . The s u r f a c e c h a r g e d i s t r i b u t i o n i s o b t a i n e d u s i n g b y e q u a t i o n s t h e f o r n u m e r i c a l

c o m p u t a t i o n ; t h e n , t h e m a g n e t i c f i e l d d i s t r i b u t i o n and t h e r o t o r t o r q u e a r e d e t e r m i n e d . N e x t , t h e r o t o r

displacement i s computed by means of Newmark's 6-

parameter method. By t h e u s e of t h e s e t e c h n i q u e s , w e were able t o s i m u l a t e t h e r o t o r b e h a v i o r . L a s t l y , t h e

s i m u l a t i o n r e s u l t s are v a l i d a t e d t h r o u g h t h e experiments.

DESCRIPTION O F THE MATHEMATICAL MODEL

A n a l y s i s o f t h e M a g n e t i c F i e l d

F o r t h e p u r p o s e o f computing 3-dimensional

m a g n e t i c f i e l d d i s t r i b u t i o n s i n a s m a l l - s i z e d s y n c h r o n o u s m o t o r , t h e c o n c e p t o f s u r f a c e m a g n e t i c

charge i s i n t r o d u c e d . Here, t h e p o l a r i z a t i o n o f t h e m a g n e t i c m a t e r i a l i n t h e ' m a g n e t i c f i e l d i s r e p l a c e d b y t h e m a g n e t i c c h a r g e d i s t r i b u t i o n o n t h e m a t e r i a l - b o d y s u r f a c e , and t h e n t h e i n t e g r a l e q u a t i o n s f o r t h e

magnetic charge density+are formed [ll- [ 4 ]

.

The f i e l u v e c t o r , H , d u e to th e m a g n e t i c c h a r g e d e n s i t y , a , i s d e s c r i b e d by t h e e q u a t i o n :

T

where

2

i s t h e p o s i t i o n v e c t o r from t h e s o u r c e p o i n t t o t h e f i e l d p o i n t , ' ^S i s t h e a r e a of t h e m a t e r i a l - body s u r f a c e and Po i s t h e p e r m e a b i l i t y o f f r e e s p a c e . The f i e l d v e c t o r , &

,

due t o t h e permanent magnet is d e s c r i b e d b y t h e e W a t i o n :

The a u t h o r s are p r e s e n t l y w i t h t h e D e p a r t m e n t of E l e c t r i c a l E n g i n e e r i n g , Okayama U n i v e r s i t y , Tsushima, Okayama 700, Japan.

where i s t h e m a g n e t i c c h a r g e d e n s i t y on t h e

permanent magnet surface,sM

.

The f i e l d v e c t o r , $ I

,

due t o t h e c u r r e n t - c a r r y i n g c o n d u c t o r s is d e s c r i b e d b y t h e e q u a t i o n :

where

i

i s t h e v e c t o r of t h e c u r r e n t d e n s i t y and V i s t h e volume of t h e c u r r e n t - c a r r y i n g c o n d u c t o r s .

T h e r e f o r e , t h e f i e l d v e c t o r ,

Zp,

a t any f i e l d p o i n t , P, i s computed by t h e f o l l o w i n g e q u a t i o n .

3

The p r i n c i p l e of f l u x c o n t i n u i t y i s a p p l i e d a t a n y p o i n t , C , on t h e b o u n d a r i e s o f two magnetic m a t e r i a l s . T h a t is:

where pl and U2 are the p e r m e a b i l i t y t h e o f

r e s p e c t i v e $des o f 2 h e b o u n d a r i e s of t ! s magnetic m a t e r i a l s , Hl and ~2 are t h e f i e l d vect+ors of t h e r e s p e c t i v e s i d e s o f t h e b o u n d a r i e s , a n d nc is t h e normal u n i t v e c t o r a t any point,C. From (51, t h e s i m u l t a n e o u s e q u a t i o n s f o r m a g n e t i c c h a r g e d e n s i t y are obtained by:

r c o l { O k l = i f 0 1 ' ( 6 )

where [C,] i s t h e c o e f f i c i e n t m a t r i x , {Okj i s t h e v e c t o r o f t h e m a g n e t i c c h a r g e d e n s i t y , a n d

{fo)

i s t h e v e c t o r d e t e r m i n e d b y OM and

i.

According t o t h i s method, each boundary on which

t h e m a g n e t i c c h a r g e i s d i s t r i b u t e d i s d i v i d e d in t o many c u r v e d s u r f a c e e l e m e n t s [l].

d e t e r m i n e d b y t h e f o l l o w i n g e q u a t i o n [51.

The c u r r e n t i n t e n s i t y , I , i n t h e s t a t o r c o i l is

- dQ, f R I = V

d t

where ^@ i s t h e q u a n t i t y of t h e i n t e r l i n k e d flux, R i s t h e v a l u e of the r e s i s t a n c e o f the s t a t o r c o i l and v is t h e a p p l i e d v o l t a q e . Also, @ and

i

r e s p e c t i v e l y , a r e g i v e n b y

( 9 ) where A i s t h e i n t e r l i n k a g e a r e a and S, i s t h e a r e a o f t h e c r o s s s e c t i o n of t h e c o i l .

The d i f f e r e n c e e x p r e s s i o n of ( 7 ) is obtained by t h e f o l l o w i n g e q u a t i o n .

where A t i s t h e s t e p w i d t h in te r m s o f time. We can 0018-9464/83/1100-2585$0l.CN

0

1983 IEEE

2586

deduce from (81, (9) and (lo), that the additional equation due to the current density,

i ,

is

Thus, the final simultaneous equations for the magnetic charge density and the current density are

formed from ( 6 ) and ( 7 ) . That is:

where

By solving these equations, the magnetic charge density and the current density are obtained.

-.

Simulation of Rotor Behavior

Next, if T and

e

represent the torque and angular displacement of the rotor, respectively, the rotor

torque equation can be written as

where J is the moment of inertia and TF is the Coulomb friction torque [ 6 ]

.

On the other hand, the magnitude of T is given by

T = L M d o M +~ +~cis - t

(14) where d is the distance between $he calculating point and the rotation axis; and t is the tangential

unit vector. The magnitude of TIP is determined experimentally. When applying Newmark’s &parameter method ( B=1/6 1 to (13), the angular displacement, 3t+at

,

and the angular velocity, (dB/dt)t+At

,

are described by the equations:

J

Therefore, we can predict by computation the rotor behaviour by using (131, (14) and ( 1 5 ) .

SIMULATION PROCEDURE Figure 1 shows the procedure for simulating

rotor behavior. In this case, the inverse matrix of the coefficient matrix is computed in order to

simplify the computation procedure and therefore

reduce the computation time. Thus, the simulation of the rotor behavior is carried out by utilizing three programs: the field computation program, the rotor

torque Computation program and the rotor displacement computation program.

I I I

I

^{t +}

I

⁰

t

Read the geometric and electric data of the motor

I

I

Form matrix [C] and compute IC]-’

I

Read initial conditions

I

c

Compute {f} due to the magnetization of permanent magnet rotor

I I

Compute = ~ C I -

E d

b

1

I

Compute new value of T

1

I

~~

Compute new values of 3 , d8/dt and d28/dt2

1

I

^{t f t + A t}

I 1

I

Print and store results

I

Fig. 1. Flow Chart Illustration of Simulation Procedure

COMPUTATION RESULTS

The small-sized synchronous motor of Fig. 2 was chosen as a numerical example. The permanent magnet rotor is magnetized as shown in Fig. 3 . The

In this case, we used curved surface triangular elements and curved surface rectangular elements

[I].

The latter elements were used-for cylindric surfaces only. The charge distribution on these elements can be

approximated by a linear formula. The number of nodes, on which unknown values of the magnetic charge density are defined, is 437. Table 1 shows the parameters of the motor. Figure 5 shows the distributions of flux density. Both computational and experimental results of the rotor behavior are shown in Fig. 6 . in this case, the stepwidth, At

,

^is 0.000521 sec. The respective results proved to be coincident.

(a) stator

STATOR

COI'L

/ ^ROT'OR \

\ ^{STATOR A} / ^{STATOR B}

Fig. 2. Cross Section of the Small-sized Synchronous Motor

Fig. 3. Permanent Magnet Rotor

Table 1. Parameters of the Motor

J

OM 0.2

N-m 2.65

x

^10-4

TF

kg-m2 2.31 X 10-8

R 4.34

x

¹⁰³

^Q

T

V

60. HZ f

50. V

(b) rotor

Fig. 4. Arrangement of Surface Elements

(b) v=50 [VI

Fig. 5 . Distribution of Flux Density

2588

20

10

H

-10 -20

experimental results

- - - -

computational results

1/60 [secl

angular displacement of the rotor

1/60 [secl

(c) current intensity of the stator coil

CONCLUSION In this paper, both the computation of 3- dimensional magnetic field distribution and the simulation of rotor behavior were described. The

simulation results provide adequate explanation of the experimental fact within the limits of this

investigation. Due to the fact that we employed the computer simulation method instead of trial

manufacture, both the labor and material costs of design this motor were remarkably reduced.

We feel that this method is practicable for the designing of small-sized synchronous motors.

REFERENCES

T. Misaki, H. Tsuboi, K. Itaka and T. Hara:

"Computation of Three-Dimensional Electric Field Problems by a Surface Charge Method and Its

Application to Optimum Insulator Design" IEEE Transaction on Power Apparatus and Systems, Vol.

PAS-101, No. 3, March 1982, pp. 627-634 M. H. Lean and A. Wexler: "Accurate Field

Computation with the Boundary Element Method"

IEEE Transaction on Magnetics, Vol. MAG-18, No.

2, 1982, pp. 331-335

2 . K. Chow, Y. T. Lee and A. Owen: "An Integral- Equation/Singularity-Method Approach for 3-D

Electromagnetic Field Determination in the End Region of a Turbine-Generator'' IEEE Transaction on Magnetics, V o l . MAG-18, No. 2, 1982, pp.

J. H. McWhirter, J. J. Oravec and R. W. Haack:

"Computation of Magnetostatic Field in Three- Dimensions Based on Fredholm Integral Equations"

340-345

IEEE Transaction on Magnetics, V O l . MAG-18, No.

2, 1982, pp. 373-378

T. Nakata and N. ^' Takahashi: "Direct Finite Element Analysis of Flux and Current

Distributions under Specified Conditions" 1 s Transaction on Maqnetics, Vol. MAG-18, No. 2 , Benjamin G. Kuo, Automatic Control Systems, 2nd Edition, New Jersey: Prentice-Hall, 1967

1982, pp. 325-330

Fig. 6. Computation and Experimental Results of the Rotor Behavior

4