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IEEE TRANSACTIONS ON MAGNETICS, VOL. MAG-22, NO. , SEPTEMBER 1986 1069

Tomy Sebastian and Gordon R. Sl-n M.A. rcahmanUniversity of Toronto -rial University of NewfowdlandToronto, Canada M5S 1pL4 St. John's, Newfoundland A1B 3x5A b s t r a c t wc's u p p l y an d s h a f t posi t ion s e n s i n g can be cons t ruc ted f Ll ku s i n g r a d i a l l y - directel h i g h e n e r g y magnets OR i t sr o to r . T h i s paper p r e s e n t s e q u i v a l e n t c i r c u i t m o d e l s ' L g d 3fu r such m o t o r s , The model parameters ar e related t ot h e m o t o r s truc ture and dimensions. Computed values of 1parameters are compared w i t h those measured frm two d-axis q-axise x p e r i m e n t a l m o t o r s w i t h different rotor construction. Fig. 3 Flux linkage m e l s

A perm anen t m agne t m o t o r w i t h p o w e r semiconductor

1 LTIntroductionD e v e l o p m e n t s i n power electronics and in h i g he n e r g y permanent magnet (PM)materiais s u c h as NeodymiumBoron Iron have enhanced t h e a p p l i c a t i o n o f PM s y nc h ro -n o u s motors i n v a r i a b l e speed drives, position controls y s t e m s and robotics. A motor w i t h a power electronics u p p l y and s h a f t positidn s e n s i n g does n o t n e d asquirrel cage w i n d i n g since it is always i nsynchronism. A l s o , t h e r e s i d u a l f l u x density o f t h enewer PM materials is h i g h enough t o use r a d i a l l y -oriented magnets f i x e d to t h e s u r f a c e o f t h e iron coreo f the ro tor . Th e stator design is s i m i l a r t d that o fan i n d u c t i o n m o t o r ' .M a g n e t i c Circuit and Flux Linkage M o d e l sCross-sections of tw o 4 - p o l e PM m o t o r s are shownin Fig. I . U s i n g the approach h i c h is c o n v e n t i o n a l in

Electric Circuit ModelsWith a rotor angul ar velocity v , t h e f l u x l i n k a g e

A d i n t h e d-axis generates a v o l t a g e Ad i n theq-axis, where v ' = v p / 2 a d p is t h e number o f poles .A l s o the f lux l i n k a g e A q generates a voltage - A qv' int h e d - a x i s . The electric equivalent circuits for+ h e t w o axes ar e then shown in Fig. 4 where Ld=L1+L,drLq= L 1 t L Oq an d i,'=i,l,~/l~. fl,s t h e s t a t o r

Fig. 4 E l e c t r i c &valent circuitsw i n d i n g resistance.and are v a l i d as long as t h e a i r ga p i n t h e q - a x i sregion i s l o n g enough t o a v o i d s a t u r a t i o n i n t h e s t a t o rteeth +

Th e c i r c u i t s o f Fig. 4 have constant parameters

Th e m o t o r torque is g iven b y

Consider a 3-phas e winding w i t h N t u r n s per po l eFig. 1 Whan i c a l layouts cf 4x0 4 pole R x n m e n t s i n u s o i d a l l y d i s t r i b u t e d for each p h a s e , For p h a s e

magnet synchronous notors currents i,, f b an d i,, the phase mmfs can be resolvede l e c t r i c machine a n a l y s i s , t h e stator w i n d i n g s may be i n t o tw o s i n u s o i d a l l y d i s t r i b u t e d components Fd ereplaced b y s i n u s o i d a l l y - d i s t r i b u t e d e q u i v a l e n t centred on t h e d - a x i s o f t h e rotor pole and F qe on t h ew i n d i n g s a l o ng t h e d and q a x e s o f symmetry of t h e q - a x i s , as shown in Fig 5 ( a ) . The e f f e c t i v e l e n g t hr o t o r , The equivalent m a g n e t i c circuit models fo r t h e of the ai r gap 4 0 is shown a s f u n c t i o n o f the angle et w o a x e s a r e as shown i n F i q . 2. Because o f th e l i n e a r measured f r m t he d - a x i s , i n Fig. T ( b ) . Th e m a g n e t s8-H characteristic o f the newer h i g h energy PM are assum4 to be partly imbedded in the ro tor i ron .I n determining t h e f l u x d u e to stator mm f , t h e magnets

th e r a d i a l l e n g t h o f t h e magnet d i v i d e d b y t h e re la t iveis of the order o f 1.05. The a i r gap f l u x density is

v.rd may be assumed to be replaced by an a i r space equal toFd , f L 5 F h l F q magnet p e r m i a b i l i t y Prwhere v r f a r h i g h energy materials

+ C c g i v e n i n Fig. S ( c ) an d (d ) where, for the d - a x i s ,Only t he fundamenta l space component o f f luxd e n s i t y produces a net f l u x l i n k age i n the s i n u s o i d a l l ymaterials , t h e magnet c a n be modelled a s a m a g n e t o - d i s t r i b u t e d stator w i n d i n g s . For t h e direct a x i s , t h i s

motive force (mmf)source F i n series w i t h a l inear fundamental space component h as a maximum v a l u e R l g dreluctance h d . Rgd and Rgg a r e t h e l i n e a r reluctances wheredue to t h e a i r g a p s i n th e i r ec t and q u a d r a t u r e axes n/Lan d Rm q is t h e reluctance due to t h e m a g n e t i n t h e Blgd=4_ f Bgde sin 0 d e (2 1q - a x i s . The pa ths for t h e s t a t o r l e a k a g e f l u x e s i n t h e a 0d an d g a x e s are represented symbolically by R l . It is convenient to v i s u a l i z e t h e d and q - a x i sThe parameters o f t h e magnet ic circuits a r e not e q u i v a l e n t w i n d i n g s as hav ing / 3 / 2 Ns t u r n s pereas i ly determinable because t h e m m f s are d i s t r i b u t e d pa le -phas e each i n converting fran m f - f l u x densityan d t h e m agne t s cover o n l y a part o f t h e pole p i t c h . variables to f l u x Linkage-current variables . TheThese magnetic circuits are however u s e f u l i n deter- direct a x i s f l u x linkage due to t h e fundamentalm i n i n g t h e structure o f the f l u x l i n k a g e m o d e l s shown component Blgd is t h e n given byin Fig. 3 , These are d u a l in f o r m to t h emagnetic circuits. Each mm f source becomes a current A d = / ? IT Ns 81gd ( 3 )source and each reluctance becomes an i n v e r s e l y -proportional inductance carrying t h e same s u b s c r i p t where 1 is t h e ax ia l l e n g t h of t h e ro t o r and r is i t s

n o t a t i o n , The p a r a l l e l combination of L g d an d 1, is a i r gap r a d i u s . The i nduc tance Lo d is t h e n g i b e n by

d-ads q- axis f & d F vFig. 2 Ylgnetic circuit m d e h

deno ted a s Lode [ I ] 'Od = od/id where id i s t h e peak v a l u e of Fd 80018-94~/86/Q900-1~9$01.0001986 EEE

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1070s insoidal current source a t fundamental f requency

T h e f a c t o r ~ 2 / 3 anpensates f b r t h e turns r a t i o betweeone phase an d t h e e q u i v a l e n t d w i n d i n g . The d a x i st h r o u g h the magnet is taken a s the reference resultingi n the phase a n g l e of zero,

-

The s t a t o r cu r ren t i n phase a leads t h e e f f e c t i v emagnet c u r r e n t by a n angle f3 an d is related to t h ea x i s c u r r e n t s o f F i g . 3 a n d 4 b yI = (id LO+^^ kou J3 ( 6 )An a l t e r n a t i v e formo f the e q u i v a l e n t c i r c u i t is

f o r these models, the torque isgiven i n F i g . 6(b) where E, = L o I, and L=LL+LoT=3I Lo I, Im s i n B (7 12

E x p e r i m e n t a l R e s u l t s for Non-Inset Ma g n e tsA motor was fabricated u s i n g a s t a n d a r d 5-hp

i n d u c t i o n motor s t a t o r and neodymium m agne t s s l i g h t l ynotched i n t o t h e i r o n core as shown for Machine #/I i nFig. I . The e q u i v a l e n t c i r c u i t parameters of thismotor were measured and ar e compared w i t h computed

'" teBmD v a l u e s i n Table I . It is noted t h a t t he q u a d r a t u r e

.. - -.. . . . .PARAMETER COMPUTED I EXPERIMENTAL

Ld mhLq rnhi,' A

4.755.04

88.5

4.024.0

88 .4Fig. 5 - Mrtf and Flux IXnsity Space Waveformd i v i d e d b y f l 2 S' S i m i l a r expressions may b e usedt o determine t h e q u a d r a t u r e axis i n d u c t a n c e LOq. Thestator l e a k a g e inductance L 1 may b e c a l c u l a t e d a s f o ran i n d u c t i o n m o t o r . [2]Determination o f Eauivalent Source Current of Maanet

T a b l e I Computed and e x p e r i m e n t a l results fo r m o t o rwith non-inset m a g n e t sa x i s inductance is o n l y s l i g h t l y h i g h e r t h a n t h e direca x i s inductance (4-6%).Thus t h e steady s t a t e c i rcu i tmo d e l s o f Fig. 6 which assume e q u a l i t y o f t h ei n d u c t a n c e s are applicable.The current source e q u i v a l e n t i, o f t h e magnet i sd e t e r m i n e d by compar ing t h e fundamental space c m p o n e n tof f l u x d e n s i t y B lm t h a t t h e magnet produces i n the a i rg ap w i t h t h e value o f , l g d pr o d u c d by d i r e c t a x i s

s t a t o r current i d ' F o r a magne t m a t e r i a l w i t h coerciveforce H, an d a l i n ea r characteristic t h r o u g h o u t t h esecond 8 -H q u a d r a n t , t h e magnet mmf is F,=Hc, where1, is t h e r a d i a l length of th e m a g n e t . The a i r gapf l u x density due t o t h e magnet is t h e n given byF i g . 4(e), If 3 1 ~ s t h e f u n d amen t a l space c ompo n e n tof t h i s f l u x d e n s i t y , the current source i, i s e v a l -u a t e d as'= p Fm / g e over t h e - w i d t h of t h e magnet a s s h o w n i n

Exper iments were performed on the m o t o r w i t h as i n u s o i d a l current controlled s u p p l y . F i g . 7 s h o w s0 rcr) - CQ l _ C H t R TED --*

S Y M 8 O L S - E X P E R I M E N T R L 9Q.00 D E L .

i,=i(j 3I m 'Iyd (4 )S f r n d i f i e d Eauiva en t Circuit f o r Fundamental Freauencv1- 1 I .,A model g i v i n g t h e relat ionships o f t h efundamental f requency v o l t a g e s an d currents may bed e r i v e d f o r t h e special case where t h e magnets a r e o n l ys l i g h t l y inset i n the i r o n , For t h i s situation, thea ir g ap inductances t,d and L Oq are then each equal toto. Th e fundamental f requency components o f p h a s ev o l t a g e an d current of an inverter-driven motor can b eregarded a s three-phase b a l a n c e d s e t s for s t e a d y s t a t ean d fo r slowly v a r y i n g transient c ond i t ions , T h e motorc a n t h e n be represented by t h e per phase e q u i v a l e n tc ircu i t o f F ig . 6 ( a ) , The constant magnet current i i nt h e f l u x linkage model of F i g . 3 is r e p l a c d by a Fig, 7 Torcpe vs Stator cwrcnt of Motor w i t .non-Inset magnets mr vLx3.ous angles B

t h e measured torque v s stator current t o g e t h e r w i t hv a l u e s c o mp u te d u s i n g t h e equivalent circuits, Th ecomputed v a l u e s o f l i n e - l i n e voltage as a f u n c t i o n off o r several v a l u e s o f s ta to r current are plotted a l o n gwith exper imenta l results i n Fig. 8. Th e closecorrespondence o f ca lcu la ted an d exper imental v a l u e sconfirms the v a l i d i t y of the circuit models.Vs1

VBMotor w i t h Fullv-Inset MaqnetsAn a l te rna te Form of p e rm a n e n t maqnet motor a sshown f o r Machine #I2 i n F i g . 1 h a s i t s m a g n e t s f u l l y

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00RNGLE B I D E C R E E S IFig. 8, a l y t i c a L and mpximntal Line-Line Voltagevs mgleB fo r Various stator Currents at 700 rev/mhi nse t i n t o t h e rotor core. The ai r ga p i n thequadrature axis r e g i o n i s sm a l l , p r o v i d i n g o n l ymechanical clearance a s i s characteristic o f i n d u c t i o nm o t o r s . T y p i c a l l y , in an induct ion m o t o r , amagnctirlng current greater than a b o u t 0 3 f r a t e dvalue ma y be assumed t o produce essential l y c m p l e t es a t urat i on i n the s t a t o r teeth. As n o t e d p r e v i o u s l y ,the angle 8 is nomally i n excess o f 90" f o r motoring,Fig,, 9 ( a ) shows a t y p i c a l distribution o f t h e s t a t o rm m f Fse i n F e h t i o n t o th e magnet f lux d e n s i t y

A 1Od*ersU

Fig* 9 & F l u d a s i t y distribution fo r mtor W i t hdistribution shown i n Fig.9tb) for a magnetangular w i d t h o f 160-26 any value o f 8 b e t n e e n 6 an dt80-6 w i l L r e s u l t in p o s i t i v e mnf across t h e ironsection between 180- 6 an d 180+6 Fo r rated statorcurrent, t h e mnf will b e adequate t o saturate thes t a t o r teeth above t h e iron s e c t i o n for the range '6+3O c W 1%- 6 assuming s a t u r a t i o n occurs a t .S o fr a te d current . Assume t h e average a i r gap flux d e n s i t yi n this region is 8, as shown i n f i g c 9 ( c ) . W h i l e thef l u x d e n si ty i n t h i s reg i on is then i n s e n s i t i v e t o t h evalue o f F,a, the s t a t o r mnf can produce a f iux densitydistribution in t h e magnet region as shown b y t h ed o t t e d line. It is assumed that the sum o f t h i s f l u xdensity and tha t fa r t h e magnet does not s a t u r a t e t h es t a t o r t e e th . For the l imited o p e r a t i n g range w i t h30+S