@01063762-M. v. ZAGIRNYAK and s. a. NASAR,-Fields and Permeances of Flat Rectangular and Cylindrical DC Electromagnetic Structures

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Fields and Permeances of Flat Rectangular andCylindrical DC Electromagnetic StructuresM. v. ZAGIRNYAK A N D s. A. NASAR, FELLOW, IEEE

Abstmct-Analytical solutions to the field problem of flat rectangularand axially symm etric cylindrical dc electrom agnetic systems are pre-sented. It is shown that the flat configuration is a special case of thegeneral so luti on . The results are presented in normalized forms as per-meance functions. Calculated results are compared with those obtainedexperimentally.

I. INTRODUCTIOND EXCITED electromagnetic systems are extensivelyused in dc linear motors, acyclic machines, electromag-netic pulleys, and electromagnetic couplings [l -[4] . In de-signing such systems, a precise determination of th e magneticfield distribution within the winding zone is necessary. Mostanalytical solutions available in the literature [2 ], [ 5 ] - re car-ried out in rectangular coordinates for cylindrical systems.Otherwise, analog or graphical solutions are obtained in cylin-drical coordinates [3], [ 6 ] ,171. It has been fou nd that theseapproaches lead to errors which may be unacceptable in cer-tain cases [ 3 ] , 8 ] . Furthermore, some of themethods, al-though yield acceptable results, require. innumerable physicalstructures t o o btain the effects of parameter variation over alimited range [8].This paper presen ts an explicit analytical so lution to hefield problem of rectangular as well as axially symmetric cylin-drical electromagnetic systems. The solution to the fla t co n-figuration is considered as a special case of th e general solu-tion. T he end results are presented in normalized form s as per-meance func tion s, which ma y be readily used indesigningaxially sym metric cylindrical and flat electromagnets. A corre-

I lation between calculated and measured permeances isgiven tovalidate the assum ption underlying the analy tical results.11. ANALYSIS

A . The Physical Model and SimpliDing AssumptionsThe model of the flat and the cylindrical electromagneticsystems to be analyzed is shown in Fig. I(a), where the z-axisis the axis of sym metry for the cylindrical system. The coordi-

nates for th e rectangular model are shown in parentheses inFig. 1 a). The B-field across the opening of the window is as-sumed to be uniform as given in Fig. l(b). Other assumptionsare that the permeability of the iron tends t o infinity; the ex-citing current is dc and is uniformly d istributed over the crossManuscript received Ju ne 5, 19 84 ; revised O ctober 10, 1984.M. V. Zagirnyak s with the Department of E lectrical Machines andApparatuses, Machine Design Institut e, Voroshilovgrad 348034U.S.S.R.S. A. Nasar is with the Department of Electrical Engineering, Univer-sity of Kentucky, Lexington,KY 40506-0046.

rm( b )Fig. 1. Investigated model of electromag netic system. (a) Cross sectio nthrough theelectromagneticsystem. (b) Plot of function B,.(z) at

F =ro +a.section of the exciting coil of the electromag net; and , the cur-rent (and hence the current density J a n d the magnetic vectorpotential) is in a direction perpendicular to the cross sectionshown in Fig. l(a). Hence, for the rectangular flat mod el, thevector potential is only z-directed. For th e cylindrical modelthe vector potential is only $-directed. In bo th cases the vec-tor potential is denoted by the scalar A .B. Field Equations

The vector potential A in the window is governed by thePoisson equationV 2 A=-poJ (1)

where po is the permeability of free space. In two dimensions(1) takes the following forms.1) Flat Rectangular Model:

2) Cylindrical Model:

0018-9464/85/0300-1193$01.000 1985 IEEE


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1 1 9 4 IEEE TRANSACTIONS ON MAGNETICS, VOL. M A G - 2 1 , NO. 2 , M A R C H 1 9 8The corresponding boundary conditions are as follows. evaluating the constants al ,A , , C, , an d D , from the boun1 ) Rectangular Model: ary conditi0n.s and substituting in (8) yields, after som

l o = y < a I y= o

2) Cylindrical Model:=0 , r o < r < r o + a

In (4), B o f is the tangential comp onent of the m agnetic fluxdensity at the interface between the airgap and the pole piece,and can be determined from A mpere's law, which yields

By using the method of separation of variables [ l ], [ 9 ] ,[lo] , he solution to ( 2 ) may be written as

.cosh (2nny lb ) cos (2nnx lb ) . (7)The constant C is unknown, but it does not have to be deter-mined for the problem a t hand [ 9 ] l o ] .

Proceeding in a similar fashion, the general solution to (3)may be written as1 mA ( r , z )=- 5L p J r 2 + a l r az t [ A , co s ( k n z )

n = l+Bn sin (knz) l [Cn 1 ( knr )+DnK1 (knr)l (8)

where a l ,a 2 , k,, A , , B , ,C, ,an d D , are undetermined con-stants, and I , (k,r) an d K 1 k , r) are m odified Bessel function sof t he first order and the first and second kind, respectively.The conditions listed in (5) are used to evaluate the undeter-mined constants. First, because A ( r , z ) s an even function,B,

manipulation,1 a2 ab23 2 n26A ( r , z )=-- poJr2 t- oJror +- - oJ

(1 1In (1 l) , a2 is unknown, but it does not have to be evaluate(for the same reason as C f in (7)) 191, [lo] . The form of amay be obtained using (7), if (7) is considered as a limitincase of (1 l) , for identical conditions, asro -+-. Le t

r = r o +y. (1 2Then using expressions for the limits of Bessel functions [12gives:

Substituting (13) in (1 1) shows that the expression under thsummation n (11) is identical to that in (7). Hence, takinthe limit asro tend s to infinity givesa2 =(Caro - By : ) p o ~ (14

where Ca is aconstant ndependent of the magnet systeparameters ro and R.Equations (7) and (1 1) provide all the information needefor th e analysis and calculations of the magnetic field distribution in th e flat and cylindrical dc magnet systems, as discussein the next section.in (8) is zero. The characteristic value k , is usually obtainedfrom the boundary conditions on B(r) [111 as the roots of atransce nden tal equa tio n involving Bessel function s. However For a unit stack length Of the f la t mapetYstemy the fluxethis procedure is rather involved and th e resulting expressions @c , @a>@s, an d $81 Y shown in Fig- l(a>>may be expressed iare cumbersome [1 1 1 . In this paper the boundary conditions terms Of the vector potential ason B(z) are used to determine k,. Accordingly, the first con-dition (5) yields (1 5

111. EQUIVALENT PERMEANCEALCULATIONS

k , =2 n n / b .Expressing B, of Fig. l( b) in a Fourier series as


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Z A G I R N Y A K A N D N A S A R : D C E L E C T R O M A G N ET I C S T R U CT U R E S 1195

In the cylindrical str ucture , the flux in the annular cylinderof outer radius a t the point labeled 2 in Fig. l(a) and innerradius at the point 1 is given by [ 9 ], l l ]@1,2 = 2 n [ r , A ' ( r 1 , z , ) - r 2 A ( r 2 , z , ) l . ( 2 0 )

Hence the various fluxes of Fig. l(a), may be w ritten as

=2 n [ r A ( r ,0) - R A ( R , O)]. ( 2 4 )The equivalent permeance Ai , for the flux @i,, in this casebecomes

where R i is the radius corresponding to he lux @ic, andbecomesR c = R = r o a ,

fo r 6,R a =R S 1=R ' =$ ( R t o) =9 ( 2 r 0 t a),

fo r @a an d @slan d

R , =o , fo r 9,.' Now, (15)-(18) may be substituted in (19)a nd (21)-(24) in(25) to ob tain the equivalent permeances. Hence the follow-ing results are obtained.I ) Flat Rectangular Model:

2 bA , = - 5n26 n = l ( 2 n -A, =- - M s (y)(-1)j" 1n26 n = l n 2

* sin (T) oth(T)nna

a b " 12 b n26 n = l n 2A,, =- - - ine)anh (y) ( 2 9 )

2) CylindricalModel:

As a special case it may be verified that th e results for the fla trectangular model m ay be obtained from he correspondingexpressions for the cylindrical model by using the limit asro -+-.Based o n the preceding analysis equivalent permeances werecalculated for the two types of electromagnetic structures.Figs. 2-5 show these permeances as functions of the variousparameters of th e electromagnetic systems. In the numericalcalculations, it was foun d adequate to take n = 1000.IV . EXPERIMENTALE RI FI CAT I O N

The validity of the expressions for the leakage flux calcula-tions obtained n the preceding sections was verified experi-


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1196 IEEE TRANSACTIONS O N MAGNETICS, VOL. MAG-21, NO. 2, MARCH 19

0.5

0.45

0.4

0.J 5

0.3

0.18

0.16

0.043

0.0420 5 I I 5 2 2.5 3.5Fig. 2. Equivalent permeance for flux from internal pole iece surface.

I I I I IFig. 3. Equivalent permeance for flux from the end (lateral) side sur-face of winding window.mentallyfor th e cylindrical mod el ofFig. l(a).The experi-mentalset-up isshown in Fig. 6 . The measurements of themagnetic quantities-the fluxes and potential differences-werecarried out by using search coils, ballistic galvonome ter, andmagnetic potentio mete r (Rogovskys belt). From th e measure-me nts of the search coils, the leakage fluxes are obtained as

0.85

0.7

0.550.4

0.25

0.15

0.050.0350 2

0.01

0.005

0 I 1.5 2 2.5 3 3.5Pig. 4 . Equivalent permeance for flux on core surface.

2 .oI 6

I.?

I o0.8

0.6

0 5

0.4

0.3

0.25

0.2 b /6~ I 15 2.5 3 3.51:ig. 5. Equivalent permeance for flux through surface of the coil vecal middle cross section.

= - $2; GC=$2 - $3 ; an d A =$1 - $4 (seeFig. l(a)The equivalent permeances, obtained experimentally, are cculated from(3


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1197

: Parameter Permeance A sermeance Xermeance X amodel ion i n Error,b/C; a/6 R / 6 1R/6 1 r/ P Cornput.( Exper., %omput. I xper .I %omput Exper. I % Error, prror ,

where Vi s the magnetic potential difference correspondingoa radius & .and lux @i.Theexperimental andcom puted results are comp ared inTable I, which also shows the results for an actual workingmodel of an electromagn etic pulley. It is seen that the agree-ment between these results is excellent, except when t he per-meance is extremely small. Some errors m ay be attributed tothe m ethod of easurements.V. CONCLUSION

Normalized permeance functions for dc electromagnetic sys-tems of flat and axially symm etric cylindrical configurationsar eobtained.These solutions areobtainedby solving theboundary-valueproblem in a simplified mann er. To aid thedesigner, the results are presented in terms of the parametersof the m agnetic systems. Calculated results are compared withexperimental results to verify the validity of the simplifiedanalysis. It is shown that he flat configuration is a specialcase of the general solution for theylindrical structure.

REFERENCES[11 J. I.Dolinsky, Calculation of pulling electromagnets. Herald ofKharkovsky polytechnical institute, Problems of Contact Ap-paratus of Automatics, vol. 28 , issue 3, pp. 56-62,1968 (inRussian).[2) R. D. Smolkin and K. M. Nakonechny, Mathematical modellingand some parameter calculation of electromagnetic separators:

Axially symmetric magnetic system with assistance of integretorEGDA-9/60, Proc. I . Gipromachugleobogashenie,vol. 3 , Mos-kow, Nedra, pp. 345-357,1971 (in Russian).[31 R. D. Smolkin, 0. P. Saiko, and R. G. Ustinova, Calculation ofplane meridian magnetic system of pulley electromagnetic sepa-rators, Electrotechnika (Electrical Eng.),vol. 6 , pp. 37-41,1980 (in Russian).[4] S. A.Nasar and I. Boldea, Linear MotionElectric Machines.New York: Wiley Interscience, 1976.[ 5 ] S. A. Swann and J. W. Salmon, Effective resistance and reac-tance of a rectangular conductor placed in a semi-closed slot,Proc. IEE ,vol. 110, no. 9, pp. 1652-1662, Sep. 1963.[ 6 ] N. P. Rychintsev, E. M. Tmochenko, and A. G. Frolov, Theory,Calculation and Design of Impact Electromagnetic Machine.Moskow, Nauka: Science, 1970 (inRussian).[71 B. K. Bul et al., Basis ofElectrical Apparatus T heory. Moskow:Vischay shkola, 1970 (in Russian).[81 V. D. Kartashan and M. V. Zagirnyak, Investigation and calcula-tion of permeances of electromagnetic pulley open axially s y m -metric magnetic systems,zvestya vuzov (transactions of highesteducation establishments on electromechanics), vol. 7, pp. 765-770 ,1977 .[91 V. A. Govorkov, Electric and Magn etic Fields. Moskow: Energy,196 8 (in Russian).[ lo] K. J. Binns and P. J. Lawrenson,Analysis and Computation ofElectric and M agnetic Field Problems. Oxford: Pergamon, 1973.[111 B. L.Alievsky and A. G. Sherstiuk, Leakage field in cylindricalslot of exciting axially symmetric system, (inRussian), Trans.USSR Academy of Sci. on POWeF and Transport, vol. 1 , pp. 119-129 ,1980 .[121 M . Abramowitz and A. Stegun Editors,Handbook ofMathemati-ca l Function with Formulns, Graphs, and Mathematical Tables.New York: Dover, 1972.

@01063762-M. v. ZAGIRNYAK and s. a. NASAR,-Fields and Permeances of Flat Rectangular and Cylindrical DC Electromagnetic Structures

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