Calculation of Magnetic Field of Two Coaxially Located Circular Permanent Magnets Using a Method of Physical Analogies

Transcription

1 World Applied Sienes Journal 3 (): , 3 ISSN IDOSI Puliations, 3 DOI: 589/idosiwasj3335 Calulation of Magneti Field of Two Coaxially Loated Cirular Permanent Magnets Using a Method of Physial Analogies Vyaheslav Evgenevih Vavilov Ufa State Aviation Tehnial University, Ufa, Russia Sumitted: Jun 6, 3; Aepted: Jul 4, 3; Pulished: Jul 8, 3 Astrat: A modern updating stage of an eletromehanial energy onverter, inluding the reation of power eletromehanial systems with high-oerivity permanent magnets (PES with HPM), whih omprise magneti earings on permanent magnets (MBPM) and hyrid magneti earings (HMB), puts forward an urgent sientifi and engineering prolem - simplifiation of design proedure of PES with HPM at the simultaneous tendeny of its auray improvement To solve this prolem, a mathematial tool to alulate power harateristis of MBPM was developed in the work As distint from the familiar mathematial expressions to determine power harateristis of MBPM, the mathematial tool in the artile was developed using a method of physial analogies y means of replaement of a real system of permanent magnets with a winding system with opposite diretion urrents This physial transformation is permissile taking into onsideration the allowane aout the permanene of magneti ondutivity of permanent magnets Thanks to this physial transformation the mathematial tool eomes signifiantly simpler and, onsequently, more aeptale for analytial alulations It was presented a omparison of the developed mathematial tool with the experimental investigations of the other authors, whilst the disrepany etween the experimental data and the alulated ones did not exeed %, proving the adequay of the results, otained in the artile Key words: Method of physial analogies Magneti earings on permanent magnets Permanent magnets INTRODUCTION Fundamental Priniples: A alulation model is ased on the general variant of onstrutive exeution of MBPM: Permanent magnets are effetively used in all with ylinder permanent magnets, loated oaxially one industries: from household applianes to aviation another, figure The usage of permanent magnets takes a speial plae in Due to omplexity of alulations, the work deals with magneti levitation units of the rotor: magneti earings ordinary for suh kind of tasks allowanes: the value of on permanent magnets (MBPM) or hyrid magneti nonmagneti gap is muh less than the radius of its urve; earings [-4] magneti ondutivity of nonmagneti gap equals to The works [5, 6] deal with the development of alulation proedures of MBPM using the integral methods y means of viewing of the magneti field pattern of permanent magnets in three-dimensional spae Undoutedly, all these proedures have highly preise results; however, the omplexity of the analytial analysis of MBPM parameters limits their pratial implementation That is why the author suggests another proedure for MBPM alulation y means of the method of physial analogies, when the atual system of MBPM is replaed with the wiring with urrent Fig : A alulation model of MBPM Corresponding Author: Vavilov, Ufa State Aviation Tehnial University, Karl Marx Str,, 45, Ufa, Russia 345

2 World Appl Si J, 3 (): , 3? Adl = l (6) where is a magneti flow in a working gap; l length Integrating the magneti flow along the length µ I? = lln p + d d (7) Fig : A alulation wiring for repulsive fore the vauum permeaility µ, magneti ondutivity of permanent magnets is onstant; external magnet is rigidly fixed As it was mentioned aove, the real magneti system of MBPM is replaed y a wiring with urrent, Figure Main Part: When onsidering a magneti field etween the wiring, figure, the funtion of magneti flow has the following view [7]: The expression for magneti field energy:?i W = where W is the magneti field energy; I- urrent intensity; magneti flow Magneti flow energy in the gap etween wires is determined, taking into onsideration the expressions (7), as follows: µ I W = lln p + d d (8) (9) VM = ( I ln r+ I ln r ) + C p () The expression for repulsive fore: where I, I are the urrents of the first and seond wires; r, r is the distane up to intensity measurement point; C is a onstant of integration Let the vetor potential A = at the origin of the z oordinates, then the onstant of integration C= [8], () is taken as: V M A = = I ln p r r As per the figure : I + d ln, p d () r = d (3) r = d + (4) where is a nonmagneti gap; d is the wire thikness; Vetor potential is determined as follows: as: (5) F = W () After differentiating the expression of magneti energy in the gap in magnitude of movement, the repulsive fore for the wiring is determined: I µ F = l p d + w = D mid d, Dext + D D in mid = w I µ F = l, p d + () A transfer from the wiring to the winding system, equivalent to MBPM, is exeuted y multiplying y the numer of wires, inluded to the system in radial diretion: where is the average diameter, m () (3) The expression (3) is for the repulsive fores of the winding with plan-parallel air-gap magneti field For the atual MBPM model it is important to determine the repulsive fores for the ylindrial gap 346

3 World Appl Si J, 3 (): , 3 Gap variations are taken as follows: / / = x osq y sin q (4) Solution of integrals (7), (8) results in the following expressions: / x x = where is the relative movement along the axis x; / y y = where is the relative movement along the axis y Then (3) taking into onsideration (4) is determined: = fr os( q) q, / d + ( x os q) I µ f r = w l p where (9) I µ w F = l p / / d + ( x osq y sin q) x y, F = F + F (5) Fore harateristi in MBPM working gap is determined y two onstituents, then: x-diretion fore is as follows: (6) os( q) = fr q, + os q () where = d + ; = x The onditions of MBPM working apaity an e presented as the following inequalities: > x, () I µ w = l os( ) p q q / d + ( x os q) y -diretion fore is determined as follows: I µ w Fy = l sin( ) p q q / d + ( y sin q) (7) (8) x < > () The expressions (),() show, that MBPM is ale to work when there is no mehanial ontat etween the internal and external rings Taking into aount the integration limits and (),() the expression () is the following [9]: q ( ) tg q ( artg + F ) x = fr +, (3) ( + ) ( ) = fr + ( artg artg ) (4) Equivalent onditions of wiring and permanent magnets: Iw J =, l (5) where J is the magnetization intensity of the permanent magnet Taking into onsideration () the equivalent onditions are determined as follows: 347

4 World Appl Si J, 3 (): , 3 p J = dl ID mid (6) Then the expression (4): 4 J µ 3 ( ) ( ) ( artg + artg = l + ) p (7) y-diretion fore is determined in the same way: sin( q) Fy = fr q, + sin q) (8) where = y Taking into aount the integration limits, the expression (8) is the following q tg + q F =f ( ( artg )), + y r p ( + ) ( ) Fy = fr ( artg artg ) (9) (3) Then, likewise the expression (7), y-diretion fore, taking into onsideration (3), is determined in the following way: 4 J µ 3 p ( + ) ( ) Fy = l ( artg artg ) p (3) Total fore in the MBPM gap, as per (6) is the following: ( + ) + ( artg ( ) artg ) + 4 J µ 3 F = l, p p ( + ) + ( artg ( ) artg ) (3) 348

5 World Appl Si J, 3 (): , 3 In ase of approximation of the real harateristi of permanent magnet y the linear dependeny with the relation of residual indution to oeritive fore B r H = and due to the allowane for ferromagneti areas µ =, r demagnetizing harateristi of the permanent magnet is the following: B = B + µ H (33) r The magneti field in the permanent magnet onsists of external, applied field and internal field (magnetization): B = µ ( H + J) (34) As all vetors are direted along the axis of permanent magnet: Fig 3: Dependene of x-diretion fore on the shift along the axis x B = µ (H + J) (35) Then taking into onsideration the expression (35), the magnetization of permanent magnets: J = B r µ (36) The expression (3), taking into onsideration (36), is determined as follows: p ( ) + ( artg ( ) artg ) + 3 4Br l F = µ p p ( + ) (37) + ( artg ( ) artg ) Based on the expressions (4), (7) for the analysis and theoretial evaluation of the developed mathematial tool, x, y-diretion fores were alulated for the MBPM with the geometrial parameters of rings: l=,699 m, d=33 m, =,775m, B r =,385 T Figure 3, as an example, shows the dependeny of x-diretion fore on the variation of x shift Figure 4, as an example, shows the dependeny of y-diretion fore on the variation of y shift Fig 4: Dependene of y-diretion fore on the shift along the axis y As a result of the analysis of the dependeny of x-diretion fore on the shift along the axis x in the working gap of MBPM, Figure 3, it was determined that with the twofold shift inrease the fore on the part of the gap redues y 56% If the shift along the axis y inreases twofold, y-diretion fore on the part of the gap redution inreases y 56% and on the part of gap inrease it redues y 56% In ase of stati alane (y=) y - diretion fore = Thus, the shift results in the disalane of eletromagneti fores of PM, providing a neessity of strit ontrol over the shift of magneti rings of MBPM Experimental investigations of MBPM with radially magnetized magneti rings with parameters l l=,699 m, l =,635 m, d=,33 m, B r=,385 T are represented in [] The analysis of experimental results, otained in [] and analytial alulations as per (3), Figure 4 showed 349

6 World Appl Si J, 3 (): , 3 the disrepany y,9% A value of this disrepany is Swann, MK, J Watkins and KR Bornstein, 996 the reason not only for inauray, ontriuted y Present limits of operation of produt luriated and allowanes, ut also for the fat, that the rings of unequal magneti earings in pumps Proeedings of the 4-th length were used in the experimental investigations international Pump Users Symposium, pp: -7 (l-l =64 m) Thus, the otained mathematial tool is 3 Johan K Fremerey, Permanentmagnetishe simpler for analytial alulations, than the one, presented Lager: Jülih Forshungshungszentrum Jülih in the works [5,6] and at the same time, it has the adequate 4 Shweitzer, G, 9 Appliations and Researh auray, as onfirmed y the experimental data of other Topis for Ative Magneti Bearings [Internet authors [] Resoure], Pro IUTAM-Symp on Emerging Trends in Rotor Dynamis, Marh 3-6, Indian Institute of ACKNOWLEDGEMENT Tehnology, Delhi, India Springer-Verlag - Aess Mode: wwwmgsh/ The author expresses gratitude to his heads of referene_puliationshtml researh, DEngS, Professor Ismagilov FR and 5 Ravaund, R, G Lemarquand and V Lemarquand, DEngS, Professor Khairullin IKh 9 Fore and Stiffness of Passive Magneti CONCLUSIONS Bearings Using Permanent Magnets Part : Axial Magnetization IEEE Transations on Magnetis, 45: 7 The work deals with the development of mathematial 6 Ravaund, R, G Lemarquand and V Lemarquand, tool, providing with the alulations of magneti 8 Analytial Calulation of the Magneti Field earings on permanent magnets and differing from Created y Permanent-Magnet Rings IEEE the familiar one y the simpliity of alulations Transtions on Magnetis, 44(8) The developed mathematial tool an e the asis for 7 Neyman, LR, 98 Theoretial Bases of Eletrial reation of engineering alulation proedure of Engineering: Textook for Institutes in Two Volumes, magneti earings on permanent magnets Neyman, LR and KS Demyrhan Leningrade, A omparative analysis of the results ased on the rd Energoizdat, Vol, 3 Revised Edition, pp: 46 presented expressions with experimental 8 Govorkov, VA, 96 Eletri and Magneti Fields, investigations of other authors showed the nd Revised Edition, Mosow: Gosenergoizdat, disrepany y,9%, what onfirms possile use of pp: 463 this tool in pratie 9 Bronshtein, IN, 986 Referene Book on Mathematis for Engineers and Students of REFERENCES Institutes, Bronshtein, IN and KA Semendyaev 3th Revised Edition, Mosow: Nauka, pp: 544 Johnson, D, P Pillay and M Malengret, High Jansen, R and E Di Russo, 996 Passive Magneti speed PM motor with hyrid magneti earing for Bearing with Ferrofluid Stailization NASA, NASA kineti energy storage Conferene Reord - IAS TM-754 Annual Meeting (IEEE Industry Appliations Soiety), :