STUDY OF THE UNIFORM MAGNETIC FIED DOMAINS (3D) IN THE CASE OF THE HEMHOTZ COIS FORIN ENACHE, GHEORGHE GAVRIĂ, EMI CAZACU, Key wods: Unifom mgnetic field, Helmholt coils. Helmholt coils e used to estblish unifom mgnetic field with the known vlue necessy fo diffeent pplictions fom the technicl field, such s: the clibting of the mgnetic field senso, electomgnetic immunity (EMI) testing esech egding the mgnetic field effects ginst humn body. Genelly the pcticl elition of the Helmholt coils is bsed on the clssic study of the mgnetic field, nmely the mgnetic field computtion only on the coils xis. In this ppe e shown on the bsis of the mgnetic field nlyticl computtion in ny point of the spce (3D), tht thee e othe domins in which the mgnetic field is unifom.. INTRODUCTION oth in the field of the technicl pplictions nd in physics nd in the field of the medicl pptus elition it is necessy to be ceted unifom mgnetic field. One of the vints fequent utilied is tht which uses the Helmholt coils. The system Fig., is mde fom two thin cicul coils, with the equl dius, plced co-xil to the distnce opposite one nothe, cossed by the sme totl cuent I, in the sme diection. In the mjoity of the specilty ppes [, 3, ] the study of the unifom mgnetic field domins is done only on the O xis. The question is if thee e othe domins in which the mgnetic field is unifom. Such study cn be done only with the nlysis methods (nlyticl o numeicl) which pemit 3D visulition of the clcultion esults. The pesent ppe hs just such object, utiliing, fo exmple the cylindicl coodintes (, ϕ, ). Mility Technicl Acdemy, 8-83 Geoge Cosbuc lvd, 7575 uchest, Romni, floinenche@mt.o, gvil_g@mt.o Politehnic Univesity of uchest, 33 Spliul Independenţei, 0600 uchest, Romni, ccu@elth.pub.o Rev. Roum. Sci. Techn. Électotechn. et Éneg., 53,, p. 89 97, ucest, 008
90 Floin Enche, Gheoghe Gvilă, Emil Ccu Fig. The Helmholt coils.. CASSIC STUDY (D) OF THE HEMHOTZ COIS In point plced on the common xis of the coils, to the distnce fom the meditoy plne, the mgnetic field stength (xil) is: ( ) ( ) I H. () 3 3 It is obseved without difficulty tht in the middle of the distnce 0 the field hs n exteme point fo ny distnce between coils: d H d 0 fo 0. () An optimum distnce cn be chosen, which ssues s possible lge domin with unifom mgnetic field. With tht end in view it is set the condition s well s the second ode diffeentil d H/d to be nnulled in the middle of the distnce: dh 3I d ( ( ) ) R 5 5, (3)
3 Unifom mgnetic field domins (3D) fo the Helmholt coils 9 d H 3I d 5 5 ( ( ) ) ( ( ) ) 5I ( ). 7 7 ( ( ) ) ( ) Fo 0 the nnulment condition is obtined: ( ) 0 5, () tht is. Noting H 0 I/(), the field obtined in the cente of single coil, the following vlues esults: H( 0).3 H0, H( ).5H0 0.9958H( 0 ), (5) H.356H 0.975H 0. Theefoe, in n impotnt egion the field is constnt in pctice, the egion sie depending on the unifomity degee which is equied. 0 3. STUDY OF THE (3D) DOMAINS WITH UNIFORM MAGNETIC FIED Fo the nlyticl detemintion of the mgnetic field genetes by the Helmholt coils in the whole spce, it is clculted fistly the vecto mgnetic potentil poduced by flt cicul coil utiliing the diect method by mens of iot-svt-plce fomul nd then the supeposition theoem is pplied. The expessions of the vecto mgnetic potentil components in cylindicl system of coodintes (, ϕ, ), with the oigin in the cente of the Helmholt coil nd the O xis long the Helmholt coil xis, e: A ϕ 0, µ NI [( ( ) ) K( ) π ( ) ( ) µ NI ( ) ( ) ) E( )] π ( ) ( ) ( ) K E [ ) ] A A 0,, (6)
9 Floin Enche, Gheoghe Gvilă, Emil Ccu in which:., (7) The mgnetic flux density is clculted with the known eltion A ot. It follows: [ ] [ [ ] [ ] [ ], E K π E K π µ 0,, K E π K E π µ µ µ ϕ NI NI NI NI (8) In the eltions (6) nd (8) the tems K(), espectively E(), e the complete elliptic integl of the fist kind, espectively of the second kind. Fo the detemintion of the unifomity domins of the mgnetic field it is clculted fistly the mgnetic flux density modulus with the eltion: ϕ. (9) Developing in Tylo seies (until the tems of the fouth ode inclusive) the mgnetic flux density modulus in the cente of the Helmholt coil (, ) (0, 0) it follows:
5 Unifom mgnetic field domins (3D) fo the Helmholt coils 93 µ NI ( ) 90 5 68 8 8 3 3 5 8 ( ) ( ) ( ) ( ) ( ) 6( ) 80 ( ) 35 0 90. 3 837 68 (0) The mgnetic field is unifom if the tems of the second nd the fouth ode fom Tylo seies development e nnulled. Fo exmple the nnulment condition fo the tem in is. In Tble e pesented the nnulment conditions fo the tems fom the eltion (0). Tble Annulment conditions Tem Condition 5 360 ± 0 906 0 65 ± 5 6569 0 6 ± 7 Fo the effectution of the nlyticl clculi which hve led to the eltions (6), (8) nd (0) the MAPE pogm fo the symbolic mnipultion of the expession hve been utilied. The min 3D unifomity egions of the mgnetic field function of the diffeent nnulment conditions of the tems fom Tylo seies development of the mgnetic flux density modulus e given bck in the Fig.. to Fig. 7. The vlues of the pmetes e: mpee-tuns of coils NI 00 A, the coils dius 0 cm.
9 Floin Enche, Gheoghe Gvilă, Emil Ccu 6 Fig. 3D vition of the mgnetic flux density in n xil section of Helmholt coil fo. Fig. 3 3D vition of the mgnetic flux density in n xil section of Helmholt coil fo 5. Fig. Mgnetic flux density vition long O xis of Helmholt coil in cse (fo diffeent vlues of the coodinte ). Fig. 5 Mgnetic flux density vition long O xis of Helmholt coil in cse (fo 3 diffeent vlues of the coodinte ).
7 Unifom mgnetic field domins (3D) fo the Helmholt coils 95 Fig. 6 Mgnetic flux density vition long O xis of Helmholt coil in cse 5 (fo diffeent vlues of the coodinte ). Fig. 7 Mgnetic flux density vition long O xis of Helmholt coil in cse 5 (fo 3 diffeent vlues of the coodinte ). Fo the compison nd fo the esults vlidtion the mgnetic field is clculted lso numeic, utiliing the nlysis pogm of the electomgnetic field with finite elements FEMM.0 (Fig. 8 to Fig. 3). Fig. 8 Mgnetic field spectum in n xil section of Helmholt coil in cse. Fig. 9 Mgnetic field spectum in n xil section of Helmholt coil in cse 5.
96 Floin Enche, Gheoghe Gvilă, Emil Ccu 8 Fig. 0 Mgnetic flux density vition long O xis of Helmholt coil in cse (fo 3 diffeent vlues of the coodinte ). Fig. Mgnetic flux density vition long O xis of Helmholt coil in cse (fo 3 diffeent vlues of the coodinte ). Fig. Mgnetic flux density vition long O xis of Helmholt coil in cse 5 (fo 3 diffeent vlues of the coodinte ). Fig. 3 Mgnetic flux density vition long O xis of Helmholt coil in cse 5 (fo 3 diffeent vlues of the coodinte ).. CONCUSIONS The ppe pesents new ppoch egding the clculus of the unifomity egions of the mgnetic field geneted by Helmholt coils. Fom the effectuted nlysis it is obseved tht, function de imposed conditions fo the nnulment of one o nothe fom the tems of Tylo seies development of the mgnetic flux density modulus, new egions of unifom
9 Unifom mgnetic field domins (3D) fo the Helmholt coils 97 mgnetic field ppe, in compison with the egion mde evident by the clssic study. The loclition of these new egions with the unifom mgnetic field cn be useful in mny pplictions fom the physics nd technique, fo exmple the electomgnetic lens designing. It hs to be specified tht this study cn be done only nlyticl, the numeicl methods being used only fo the vlidtion of the obtined esults. The complex nlyticl clculi fom this study hve been effectuted using the fcilities offeed by the symbolic mnipulto MAPE. In ddition this pogm offes the possibility fo the effectution of the ccute numeicl clculi nd contins seies of function with which cn be elied suggestive gphicl epesenttions D, D nd 3D (vition cuves, field spect, equipotentil sufces, etc.) The study will be continued by the utilition of clssic nonline optimition pogm o with genetic lgoithms wht pemits the detemintion of the geometicl conditions necessy fo the obtining of the lgest egions with the unifom mgnetic field. Received on 6 Novembe 007 REFERENCES. C.I. Mocnu, Teoi câmpului electomgnetic, Edit. Didctică şi Pedgogică, uchest, 99.. Gh. Gvilă, Electosttic, Edit. Didctică şi Pedgogică, uchest, 998. 3. A. Mou, Complemente de teoi câmpului electomgnetic, Edit. Mtix Rom, uchest, 003.. Gh. Gvilă, Elemente de electocinetică şi electodinmică, Edit. Mtix Rom, uchest, 007. 5. Iin Muntenu, D. Ion, F. Enche, Symbolic Computtion in Electomgnetics: A Clssoom Expeiment, Poceedings of CEFC 00, Milwukee, Wisconsin, U.S.A., June 7, 000. 6. Gh. Gvilă, F. Enche, Computtion of the Electosttic Fields in the Cse of the Tue Electic Chges Distibutions, 5th Intentionl Confeence on Electomechnicl nd Powe Systems (SIEMEN 005), Octobe 6 8, 005, Chişinău, Poceedings, Vol., pp. 00 007. 7. D. Ion, F. Enche, Symbolic computtion in electomgnetics, Poceedings of 7th Intentionl Wokshop Symbolic Methods nd Applictions to Cicuit Design (SMACD 00), Sini, Romni, Octobe 0, 00, pp. 5 50. 8. K.M. Hel, M.. Hnsen, K. M. Richd, Mple V ening Guide, Spinge Velg, 996.