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M e a s u r e m e n t T e c h n i qu e s , V o L 3 7 , N o . 6 , 1 9 9 4

D E T E R M I N A T I O N O F T H E D I P O L E M A G N E T I C M O M E N T O FA P R O D U C T

B . M . S m i r n o v U D C 6 2 1 . 3 1 7 . 4 2 1

Desc r ibed is the de terminat ion o f the dipole magnet ic mom ent of a produ ct by measur ing the magnet icinduct ion vec tors be tween tw o se lec ted points in space lay ing on a s ingle l ine that passes through the produc t ,e .g . , through i ts geometr ical center. The sys tematic error of the de termined param eter , due to the e f fec t o fhigher-order mag net ic moments, i s es t imated and recommendations are given on the dis tance be tween m agnet ictransducers used to measure the ma gnet ic dipole moment.Determ in a t i o n o f th e d ip o le m ag n e t i c mo m en t o f a mag n e t i c f i e l d so u rce is u se fu l in q u a l i t y co n t ro l o f f e r ro mag n e t i c

p ro d u c t s , i n me asu rem en t s o f ro c k samp les , i n t e s t in g mag n e t i c co mp at ib i l it y o f f e r ro mag n e t i c an d e l ec t r i ca l o b jec t s , e t c . , [ 1,2 ] .

In [2 , 3 ] t h e mag n e t i c f i e l d o f a samp le o r p ro d u c t a t th e t r an sd u cer l o ca t i o n is ap p ro x im ated b y t h e m ag n e t i c f i e l d o fa d ip o le . T o f i n d t h e mag n e t i za t i o n o f ro ck samp les t h ey a r e g iv en a sp ec ia l sh ap e su ch as , f o r ex am p le , o f an e lo n g a t ede l li p so id [3 ] . T h e samp le i s p l aced a t a d i s tan ce o f a t l eas t t h r ee t imes i ts m ax imu m l i n ea r s i ze f ro m th e mag n e t i c t r an sd u cer ,and the magnet ic indu ct ion i s meas ured w i th a systemat ic er ro r o f less than 5 % assuming that the d ipo le sourc e i s located att h e mag n e t i c cen t e r o f t h e samp le [3 ] . Ho we v er , t h e p ro d u c t may h av e a sh ap e d i f f e r en t f ro m an e l l i pso id . A s a r esu l t , a t ad i s tan ce b e twe en t h e p ro d u c t an d m ag n e t i c t r an sd u cer eq u a l t o t h r ee times t h e max im u m l i n ea r s i ze o f t h e p ro d u c t , t h e m ag n e t i cf i e ld ma y h av e a m u l t ip o l e ch arac t e r so t h a t th e e r ro r o f t h e d ip ol e mag n e t i c mo m en t can b e as h ig h as sev era l t en s o f o n ep ercen t . O n t h e o th e r h an d , t h e i n s t ru men ta l e r ro r i n c reases wi th t h e d i s tan ce t o t h e t r an sd u cer e sp ec i a l l y i f p ro d u c tmag n e t i za t i o n i s weak .

He re we p ro p o se a meth o d t h a t , i n co mp ar i so n wi th o th e r solu ti o n s [2 , 3 ] , r ed u ces b o th t h e i n s t ru men ta l e r ro r d u e t oth e smal l d is t an ce b e tween t h e p ro d u c t an d t r an sd u cer , an d t h e sy s t emat i c e r ro r d u e t o weak en in g o f t h e mag n e t i c f i e l d e f f ec td u e t o mag n e t i c mo men t s o f h ig h er o rd er s .

As in [4 ] , the p roposed method i s based on assuming that , s ince the locat ion of the magnet ic f ie ld source i s no t known,th e mu l t ip o l e i s l o ca t ed a t t h e g eo m et r i ca l cen t e r o f t h e p ro d u c t. Th e co m p o n en t s o f th e m ag n e t i c i n d u c t i o n v ec to r a r e m easu redon a s t raigh t line passing the geome tr ical cen ter , the num ber spat ial po in ts on the l ine depending on the o rde r o f magnet icmo m en t s u sed t o ch arac t e r i ze t h e f i e l d so u rce . U n l ike i n [4 ], t h e d esc r i b ed meth o d o f f i n d ing t h e d ip o le m ag n e t i c mo me n tr eq u i r es p re l imin ary measu reme n t o f th e ex t e rn a l mag n e t i c f i e l d co mp o n en t s wi th o u t t h e p ro d u c t . Th e ex t e rn a l m ag n e t i c f i e l din ev ery mag n e t i c t r an sd u cer s i s co mp en sa t ed [5] . N ex t , t h e p ro d u c t i s i n ser t ed an d t h e mag n e t i c i n d u c t i o n v ec to r co m p o n en t sa re m easu red . Th e p ro j ec t i o n o f t h e mag n e t i c i n d u c t io n v ec to r a t t he i - t h p o in t i n sp ace can b e w r i t ten as [4]

0 2 0 ~ a ~ ) 1 ( 1 )~ o ~ M x m + M y m . .~ . M z m - - ;B x i - - ~ r . ~ O xOv ~ r mr t ~ I~~t~ ~ Mx mBu i-- ~ OxOyn Z~ |

02 0 ~ ) 1 (2 )

B zi = T~P~ m=lZ M~m O x d z q-Mvrn, dtjOz -~- Mzm ~ ,

w h e r e / z 0 = 4 7 r- 1 0 - 7 H/m ; M x m , Mym, an d M zm are p ro j ec t io n s o f t h e m- th d ip o le m ag n e t i c mo m en t o n t h e ax es o f t h e sy s t emo f co o rd in a t es Oxyz wi th an o r i g in a t p o in t O co in c id in g , fo r ex am p le , w i th t h e g eo m et r i ca l cen t e r o f t h e p ro d u c t ,

T ran s l a t ed f ro m Izm er i t e l ' n ay a Tek h n ik a , No . 6 , p p . 3 9 -4 1 , Jun e , 1 9 94 .

0 5 4 3 -1 9 7 2 /9 4 /3 7 0 6 -0 6 6 7 5 1 2 .5 0 9 P l en u m P u b l i sh in g Co rp o ra t i o n 667

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_~ .(2/%i r ~ ~~ /,7 / o

YAFig . 1 . Spat ial d is t r ibu t ion of ma gnet ict r an sd u cer s a t measu r in g p o in t 1 an d 2 ,an d o f t h r ee p o in t s A , O , C wh e re d ip o l eso u rces wi th mag n e t i c mo m en t s M 4 , M o ,.rM c a re located .

1 1 ~ 1 ( 0 e 0 y ~ 1r,--~- ~ -7- + ~ < 7 7 ,~~ ~ . 4 Zo,, o-7-/ -7- ;

r = ~/x2 + y 2 + z 2 i s t he mo d u lu s o f t he r ad iu s v ec to r eq u a l t o th e d i s tan ce f ro m th e mu l t i p o le r e f e r en ce p o in t t o t h e ch o senp o in t o f o b se rv a t i o n ; x , y , z a r e p ro j ec t i o n s o f th e r ad iu s v ec to r r o n t h e sy s t em o f co o rd in a t es Oxyz; Xom, Yore , an d Zorn arep ro j ec ti o n s o f t h e seg me n t b e tween t h e p o in t o f l o ca t io n o f th e m- th d ip ol e an d t h e mu l t i po l e r e f e r e n ce p o in t, an d n = 1 , 23 . . . . an d i = 1 , 2 , 3 . . . . a r e o b se rv a t i o n p o in ts a t wh ich th e mag n e t i c i n d u c t io n v ec to r co mp o n en t s a r e m easu red .

By m easu r in g t h e m ag n e t i c i n d u c t io n v ec to r co m p o n en t s a t i p o in ts o f sp ace l ay in g o n o n e s t r a ig h t l i n e o n t h e sameside f rom the product and so lv ing a system of 3 i equat ions (1 )- (3) , i t i s possib le to f ind the magnet ic moment can wi th asy s temat i c e r ro r d ep en d in g o n t h e e f f ec t o f mag n e t i c mo m en t s o f t h e ( i+ 1 ) -t h o rd er an d h ig h er Th i s sy s t em o f eq u a t io n sp ro v id es an ex p l i c i t so lu t io n fo r t h e d ip o le m ag n e t i c mo m en t . I n p rac t i ca l wo rk i t is f r eq u en t l y su f f i c ien t t o ap p ro x im ate t h e

mag n e t i c f i e ld so u rce b y t h e d ip o l e an d q u ad ru p o l e mag n e t i c m o men t s .Le t u s as su me th a t t h e mag n e t i c f i e ld so u rce i s d esc r i b ed b y mag n e t i c mo men t s o f t h e f i r s t an d seco n d o rd er . Th e

p ro j ec t io n s o f th e d ip o l e mag n e t i c mo m en t v ec to r can t h en b e o b t a in ed f ro m (1 ) -(3 ) b y measu r in g t h e co m p o n en t s o f ma g n e t i cinduct ion at two po in ts in space fo r two posi t ions o f the p roduct .

Le t t h e o r i g in o f co o rd in a t es O co in c id e wi th th e g eo me t r i ca l cen t e r o f t h e p ro d u c t . Th ree -co mp o n en t mag n e t i ct ransducers are located along the O x ax is at po in ts 1 (x G 0 ; 0 ) and 2 (x2 ; 0 ; 0 ) so that the mag net ic ax es o f these t ra nsduce rsa re co l l i n ea r wi th ax es o f t h e sy s t em o f co o rd in a t es O x y z . To ap p ro x ima te t h e mag n e t i c f i e l d so u rce o f t h e p ro d u c t b y t h emag n e t i c mo m en t s o f th e f i r s t an d seco n d o rd e r i n (1 ) - (3 ) fo r 1/rrn, we l imi t o u r se lv es t o n = 1 , 2. T h e d ip o l e mag n e t i c v ec to rco mp o n en t s a r e t h en g iv en b y

4 ~. 1 ~ 4 (4)M~ := 2 b to b ( r l B z l - - r 2 B ~ ) ;4 = 1 4 4M~,~- ~.t---~- 7 ( q B y l - - r z B y Q ; (5 )4 3 1M~= ~to b (r~e~ --r~B~ z), (6)

w h e r e b = r 2 - r l , r 1 = I x l I , a n d r 2 = I x 2 } 9Le t u s as su me, as i n [4 ] , t h a t th e p ro d u c t h as t h e fo rm o f a 2 -m lo n g s t r in g A C (F ig . 1 ) . The s t r ing l ies along the Oy

ax is an d i t s g eo met r i ca l cen t e r co in c id es wi th p o in t O wh ich i s t he o r i g in o f t h e C ar t es i an sy s t em o f co o rd in a t es O x y z . At p o in t sA an d O wi th t h e co o rd in a t es A(0 , - 1 m, 0 ) an d O(0 , 0 , 0 ) a r e two d ip o l e f i e ld so u rces wi th mag n e t i c mo m en t s M A{0, 5A ' m 2 , 0 } an d .~0 {0 , 5 A 'm 2 , 0} . Th ree -c o mp o n en t mag n e t i c t r an sd u cer s a r e p l aced a t p o in t s 1 (1 .5 m , 0 , 0 ) an d 2 (2 m , 0 , 0 )on the O x ax i s . S o lv in g t he d i r ec t p ro b l em an d ap p ly in g t h e we l l -k n o wn eq u a t i o n fo r t h e mag n e t i c i n d u c ti o n v ec to r o f a d ip o l ef ield source [3 ] :

( 7 )

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A.m2/mT

/ .

IIIr IO ' l r ~ c 0,37 0,5 " ) , 0

, b / r tF ig . 2 . D e p e n d e n c e o f t h e fu n c tio n 0 = g m / aB o n b / r I fo r d i s t a n c er 1 = 1 m b e tw e e n t r a n s d u c e r 1 a n d th e m u l t ip o le r e fe re n c e p o in t .

w e g e t B x l = 1 1 8 . 1 6 n T , B y 1 = - 1 5 4 . 7 n T , Bz l = O , Bx 2 = 5 3 . 6 6 n T , B y 2 = - 8 0 . 3 8 8 n T , a n d Bz2 = O.With a s y s t e m a t i c e r ro r d u e to th e e f f e c t o f m a g n e t i c m o m e n t s o f th e th i rd o rd e r a n d h ig h e r , t h e v e c to r o f t h e d ip o le

m a g n e t i c m o m e n t o f t h e p r o d u c t i s f o u n d f r o m ( 4 )- ( 6) a s~ rp { 2 , 6 A . m 2 ; 1 0 . 0 6 A . m 2 , 0 }.

T h e n o m in a l v a lu e i s I ~ rp [ = 1 0 . 3 9 A - m 2 . T h e a c tu al v a lu e o f t h e m o d u lu s o f t h e d ip o le m a g n e t i c m o m e n t v e c to rM = 1 0 A . m 2 . A c c o r d i n g l y , t h e r e la t i ve e r r o r c a lc u l at e d f r o m 6 M = ( M p - M ) / M i s 3 . 9 % . T h e r e l a t iv e e r ro r o f th e d ip o lem a g n e t i c m o m e n t v e c t o r f o u n d b y m e a s u r i n g o n l y " B 1 a n d r I o r B 2 an d r 2 w o u ld b e 4 4 o r 3 2 % re s p e c t iv e ly .

I f t w o d i p o le s o u r c e s w i t h m a g n e t ic m o m e n t s M ~ { 0 , 5 A - m 2, 0} and M c{0 , 5 A.m 2 , 0} a re loca ted a t po in ts A(0 ,- 1 m , 0 ) a n d C (0 , 1 m , 0 ) s y m m e t r i c a l ly w i th r e s p e c t to th e p o in t O , s o lv in g th e d i r e c t p ro b le m a n d (7 ) , w e g e t t h e m a g n e t i ci n du c ti o n v e c to r s B I { 0 , - 1 3 . 1 3 n T , 0 } an d B 2 { 0 , - 3 5 . 7 8 n T , 0 } . W i t h a n e r r o r d ue t o t h e e f f e c t o f m a g n e t i c m o m e n t s o fth e th i rd o rd e r a n d h ig h e r , t h e m a g n e t i c m o m e n t d ip o le v e c to r i s g iv e n b y (4 ) - (6 ) a s Md { 0 , 1 0 . 1 2 A . m 2 , 0 } a n d I ~ rd [ =1 0 . 1 2 A . m 2 . T h e r e l a t iv e e r ro r o f t he d ip o le m a g n e t i c m o m e n t m o d u lu s i s ~iM = 1 . 2 % . T h e m o d u l u s c a lc u l a te d b y m e a s u r i n go n ly B I a n d r 1 o r B 2 a n d r 2 w o u ld b e 9 5 . 6 o r 7 1 . 4 % re s p e c t iv e ly .

T o re d u c e th e s y s t e m a t i c e r ro r o f d e t e rm in a t io n o f t h e d ip o le m a g n e t i c m o m e n t o f a p ro d u c t , t h e p o in t s a t w h ic h th em a g n e t i c i n d u c t io n v e c to r c o m p o n e n t s a re m e a s u re d s h o u ld b e s l e c t e d a t a g re a t e r d i s t a n c e f ro m th e p ro d u c t s u r fa c e o r m o rem e a s u re m e n t p o in t s s h o u ld b e u se d .

E q u a t io n s (4 ) a n d (5 ) g iv e th e d e p e n d e n c e o f th e s t a n d a rd d e v ia t io n o f th e r e s u l t s o f d ip o le m a g n e t i c m o m e n tm e a s u re m e n t s o n th e s t a n d a rd d e v ia t io n o f t he r e s u l ts o f m e a s u re m e n t o f t h e m a g n e t i c i n d u c t io n v e c to r c o m p o n e n t s a t sp a t i alpo in ts 1 and 2 (see F ig . 1 ) tha t can be wri t ten as

a M = 0 a B , ( 8 )

w h e re a M i s t h e s t a n d a rd d e v ia t io n o f t h e r e s u l ts o f d ip o le m a g n e t i c m o m e n t m e a s u re m e n t ,

0 = 2 t to ~3 4 g r r - ~ -- Jb ) s ' l /~r l = I x l I , c~B i s t h e s t a n d a rd d e v ia t io n o f th e r e s u l ts o f m e a s u re m e n t o f th e m a g n e t i c i n d u c t io n v e c to r c o m p o n e n t s a t s p a c ep o in t s 1 a n d 2 ( se e F ig . 1 ). In th e e x a m p le s d i s c u s s e d a b o v e , 6 8 = 0 . 5 a ~ A . m Z /n T . E x p re s s io n (8 ) s h o w s th a t a M d e p e n d s o nb / r 1 . T h u s a M = 0 . 8 a B A . m 2 /n T fo r r 1 = 1 . 5 m , b = 0 . 1 1 5 a n d fo r r 1 = 1 . 5 m , b = 1 . 5 m .

A n a ly s i s o f (8 ) fo r e x t r e m u m s h o w s th a t 0 h a s a p e a k a t b = 0 . 3 7 r 1 . F ig u re 2 s h o w s 0 a s a fu n c t io n o f b / r I fo r r 1 =l m .

T h u s , t h e p r o p o s e d m e t h o d m a k e s i t p o s s i b l e t o f i n d t h e d i p o l e m a g n e t i c m o m e n t o f a p r o d u c t b y m e a s u r i n g t h ec o m p o n e n t s o f th e m a g n e t i c i n d u c t io n v e c to r a t tw o s p a t i a l p o in t s w h o s e d i s t a n c e f ro m th e p ro d u c t s u r fa c e i s c o m m e n s u ra b le

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with i ts maximum linear size. The systematic error o f the measu red parameter dep ends on the effect of magne tic mom ents othe (i + 1)-th order and higher.

R E F E R E N C E S

1 .

2.3.4o5.

S. A. Skoro dum ov and Yu. P. Oboishev, Interfe rence-F ree Magnetic M easure men t Instruments [in Russian]l~nergoizdat, Leningrad (1981).B. M. Sm irnov, Abstracts o f the Conference-Seminar on Magnetic P urity and Electrom agnetic Com patibil i ty in th"Mars-94" Projec t [ in Russian] , NP O V NIIM D. I . Mendeleeva , Leningrad (1991).A. A. Lo gache v and V. G . Zakharo v, Magnetic Prospecting [in Russian], Ned ra, Lening rad (1979).B. M. Smirnov, Izmer. Tekh., No. 5, 55 (1992).Yu. V. A fanas 'ev, N. V. Studentsov, and A. P. Shchelkin, M agnetometric Trans ducers and Instruments [in Russian]t~nergiya, Leningrad (1972), p. 172.

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