A three-dimensional field solution for radially polarized ... - IEEE Xplore

IEEE TRANSACTIONS ON MAGNETICS, VOL. 31, NO. 1, JANUARY 1995

844

A Three-Dimensional Field Solution for Radially Polarized Cvlinders J

E. P. Furlani, S . Reznik and A. Kroll

Abstract-A three-dimensional field solution is presented for radially polarized permanent-magnet cylinders. The derived field formulae are evaluated in terms of finite sums of elementary functions. They are readily programmed and ideal for performing rapid parametric studies of the field distribution outside of cylinders made from rare earth materials such as NdFeB. The theory is demonstrated with some sample calculations that are verified by use of three-dimensional finite element analysis.

I. INTRODUCTION ADIALLY polarized magnetic cylinders are used for myriad applications including magnetic encoders, couplings, and the controlled movement of small magnetic particles in the electrophotographic process (Fig. I ) [1]-[3]. For many of these applications, the magnet can be considered to be in free-space since its surrounding area is absent of any other materials that would otherwise perturb or contribute to the field structure. Therefore, the field problem reduces to that of the magnet itself which is essentially three-dimensional, and while various numerical techniques such as finite element analysis are applicable, they tend to be awkward for the kind of parametric analysis that is often desired [4]. In this article, a three-dimensional field solution is presented. The analysis is based on the assumption that the cylinder can be modeled as an ideal magnet with the following second quadrant constitutive equation

R

B

=

pO(H

+ M).

(1)

This equation is valid for numerous materials including NdFeB . The solution method entails the use of the vector potential and involves the closed-form integration of the free-space Green’s function over one of two spatial variables. The resulting field formulae are expressed in terms of definite integrals of elementary functions over the remaining spatial variable. These remaining integrations are evaluated numerically and the field values are ultimately computed by evaluating discrete sums of elementary functions. Thus, the field formulae are readily programmed and ideally suited for rapid parametric studies

Manuscript received July 9 , 1993; revised July 6, 1994. The authors are with Eastman Kodak Company, Rochester, NY 146535305. IEEE Log Number 94056 18.

Fig. 1 . Radially polarized cylindrical magnet.

of the field strength at any point outside of the cylinder. The authors have used these formulae extensively to study edge effects and to optimize geometric dimensions and/or the magnetic polarization relative to obtaining a prespecified field distribution over a given spatial region. The theory is demonstrated and tested using both a two-dimensional analytical algorithm and finite element analysis (FEA). Lastly, it is worth noting that similar three-dimensional field solutions have been derived for axiallypolarized multipole disks and bipolar cylinders [5], (61. In addition, a two-dimensional analytical field solution has been derived for cylindrical geometries with arbitrary cross-sectional polarization [7]. 11. THEORY There are numerous techniques for computing the field due to permanent magnets. The approach taken is to model the magnet as a distribution of equivalent currents. The analysis starts with the magnetostatic field equations for current free regions,

V x H = O ,

(2)

and

V - B = O , (3 ) where H is the magnetic field strength and B is the magnetic flux density. In magnetic materials, the two fields H and B are related to the physical magnetization M ,

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FURLANl et a l . : RADIALLY POLARIZED CYLINDERS

845

It is well-known that the two first-order field equations reduce to the second-order equation

B=VxA.

I iI 1 iI I i;

(6)

If the magnetization is confined to a given volume V and falls abruptly to zero outside of this volume, then the solution to (5) can be written in the following integral form: PO A(7) = -

47r

S

JM(?’)

VI?-

d3X,

7’1

V x M

(volume current density),

(8) bottom surface =

and jM= M x n

(surface current density),

1 1 1 1 1 1

I l I

1 l 1

I I I1

1 1 11

I

1 1

Fig. 2. Sector geometry and orientation.

where S defines the surface of the magnet, and J M andjM are equivalent volume and surface current densities given by JM =

I I I I I I

(9)

respectively. For the problem at hand, it is assumed that the magnetization is strictly along the radial (r) direction

I

r,(l)

Ir’ Ir,(2)

8,(l)

I8 I8,(2)

z’

(14)

= Z,(l),

r,(l)

Ir’ Ir,(2)

(15)

z,(l) Iz 5 z,(2),

M ( 7 ) = +Mi., where the k term takes into account the altemating polarity of adjacent poles. It follows from (8) that the volume current density is zero

and r,(l)

Ir’ Ir,(2)

z,(l)

Iz Iz,(2).

(16) JM =

V x M = 0,

(1 1)

and, therefore, (7) reduces to The unit normals for these surfaces are as follows:

i The B field is computed from (12) using (6). However, instead of determining B directly for the entire magnet, it is simpler to first evaluate (12) for a single sector (Fig. 2), and then compute B for the sector, and, lastly, compute the total field as a superposition of the contributions from all the sectors. To this end, consider the sector shown in Fig. 2. As a first step, it is necessary to determine the functional form of jM for the various surfaces. From (9) and (lo), it follows that j M = 0 on the back ( r = R , ) and front ( r = R2) of the sector as the magnetization and surface normal are either parallel or antiparallel for these surfaces. There are four remaining surfaces to consider:

top surface =

I

r,(l) 5 r’ O,(l) 5 8

z’

= z,(2),

(13)

-2

(bottom surface)

-8

(left side)

8

(right side),

and, therefore, assuming a radial magnetization

the corresponding surface current densities are given by

f - ~ 8 (top surface)

Ir,(2) I8,(2)

(top surface)

.iM

=

I

M8 -Mz^

\ Mf

(bottom surface) (left side) (right side).

(19)

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IE13E TRANSACTIONS ON MAGNETICS. VOL. 31. NO. I , JANUARY 1095

Taking into account the results of (13)-(19), (12) can be rewritten

and

3) A,( 3)

)

r,' dtl ' d t - 1 . .-dj )

In this expression, and for the remainder of this paper, the subscript s denotes the contributions due to a single sector. Note that the unit vector 8 in the integrand of (20) is itself a function of 0 ,

e = -sin

(e)a + cos (e)y,

s": 1""'

x

dr' dz' = 0.d j

(21)

therefore, (20) can be written in terms of dinates,

c

CLoM (-1)j 4a j-I

=-

r,(l)

1

17

-

-r ' 1

I

d r ' d z ' . (28) @'=Ov(,)

Equations (26), (27), and (28) give the vector potential due to a single sector; the field for the sector B,( ?) can be obtained using (6). In addition, since the total field is a superposition of the fields due to all the sectors, it can be expressed in the following symbolic form: N p ~ e

B ( 7)

=

.\

c (-l)(J+93s(7), =

I

(29)

where, once again, the ( - l)(s I ) term takes into account the alternating polarity of adjacent poles. Expressions for the three field components Br, Be, and B, are derived below. For these derivations it is useful to introduce the Green's function notation +

POM A s ( 7 )= C (-1)'

47r

'=I

?,(I)

' 1

which in cylindrical coordinates ( r , 8 ,

17 - 7'1 e ' = ( j , ( j )

dz'dr'

G ( r , 8,

Z;

r',

e',

z ) reduces to

z')

(3 1) The field components are derived in the following three sections. (22) This can be rewritten in terms of cylindrical coordinates by computing the following projections: L - I ~7, ~ )= ( A , ( 3)

*

i,

(23)

A @ . , ( ? )= A , ( ? )

*

8,

(24)

A. The Radial Component (B,)

The radial field component due to a single sector follows from (6):

Br,s(3)

=

l a a - /IZ,,( 3) - - Ae.,( 3). r a0 az

-

(32)

Substituting (27) and (28) into (32) yields Applications of these projections to (22) results in the following explicit forms: Ar.,s(7 )

PoM 47r j

= -

c (-I)"+

I)

= I

&(I)

17

-

7'1

r' de' dr', z'=Zy(j)

(27)

The integrations in Z' in the first term and r' in the second term can be expressed in closed form using the (44) and (45) of the Appendix. Also, the contributions from all the sectors need to be taken into account in accordance with

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FURLANI et al.: RADIALLY POLARIZED CYLINDERS

(29). This results in the following field expression:

~ ~ (= 7__ PoM ) 47T

C C C

Np’ie

[ !I:; *

(-l)(s+l+j+k)

s=lj=lk=l

r’ sin (6 - 19,(j))

Ii(r, 8,

2;

Similar to the radial case, the integral in r’ in the first term and z’ in the second term can be written in closed-form by use of (44) and (45) of the Appendix. The total azimuthal field component is obtained as a superposition of contributions from all the sectors, and is given by, PoMN@e

C .C C

BO(?) =

r’, e s ( j ) ,z,(k)) dr’

(-l)(s+’+k)

s=l j = 1 k = l

842)

j

+ e

041)

(z - z , ( j ) > cos (e - e r )

z2(r, e,

Z;

1

rsw,e r , z , ( j ) ) der

. (34)

The functions I , and Z2 are as specified in the Appendix. The remaining integrals in r’ and 8 ’ can be evaluated numerically using standard methods, and the resulting field formula is ultimately expressed and as a finite sum of elementary functions. The various parameters that appear in (34) are defined in accordance with the sector geometry of Fig. 2 as follows: K

OS(l) = -(2s - 3), Npok

Again, the functions I , and Z2 are as specified in the Appendix, and the remaining integral in r’ and 8 ’ can be evaluated numerically resulting in a finite sum of elementary functions. The various parameters that appear in (38) are defined in (35).

a

8,(2) = -(2s - l), &le

s = 1,

2,

*

, N,,,,

RI (m)

(inner radius),

r,(2) = R2(m)

(outer radius),

r,(l)

=

z,(2) = top of sector (m),

z, (1)

= bottom of sector (m).

(35) Note that (34) gives the radial field component at any point outside of the magnet. B. The Azimuthal Component (Bo) The azimuthal component of the field also follows from (6):

Again, substituting (26) and (28) into (36) yields L

c (-1)j 47~

PO BO.,(?) = -

j=l

C. The Axial Component (B,) The axial component also follows from (6):

Substituting (26) and (27) into (39), and collecting like terms yields PO M B,,,(?) = 4 7 ~j

(-1); = ~

[ i;;

s~.m ~ r rcos eS(i)

I?

(e

-

-

?’I3

er)

1

de’ dr’

zg=z,(j)

The r’ integration can be expressed in closed form by use of (45) and (46) of the Appendix. Once again, the total field component is superposition of all the sector components,

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1EE.E TRANSACTIONS ON MAGNETICS, VOL. 31. NO. I , JANUARY 1995

Equation (41) gives the axial field component for the entire magnet. Note that care must be taken when evaluating (42) since its functional dependence is dependent on the relation between the source and field points.

M = 4.3 x lo5 A/m, RI

=

2 cm (inner radius),

R2 = 4 cm (outer radius), 111. RESULTS The field expressions (34), (38), and (41) were tested using both an analytical two-dimensional algorithm and FEA. The reader should note that the predicted data is displayed as solid or dotted lines in the accompanying figures, whereas the test data is identified by the symbol. For the two-dimensional test, an infinitely long cylindrical shell with 10 alternating poles was modeled. The inner and outer radii were taken to be R I = 2 cm and R2 = 4 cm, and the magnetization was set to M = 4.3 X lo5 A/m. The two-dimensional analytical model derived in [8] was used to compute the reference field data. For the three-dimensional model, the height of the cylinder was taken to be 80 cm, in accordance with the two-dimensional approximation. The field components B, and Bo were computed at a radius r = 5 cm for a series of angular values 0 = 0, 3, 6 , . . * , 36" (i.e., from the center of one pole to the center of the neighboring pole). The corresponding field data from the two models is compared in Fig. 3. For the three-dimensional analysis Simson's integration was used and the number of mesh points in the r r and 6 integrations ( N , and No) were both set to 30. It took roughly 15 s of run time on a 486/25 MHz PC to compute each set of data (24 field values). Next, the derived field expressions were tested using a three-dimensional cylindrical geometry with an axisymmetric polarization (Fig. 4). The MAXWELL software from ANSOFT Corp. was used for the FEA. The parameters used in the analysis are as follows:

+

h = 4 cm (height).

(43)

The value of M corresponds to a bonded NdFeB material. It is important to note that for this analysis the cylinder is taken to be symmetrically positioned heightwise with respect to the x - y plane (i.e., z = 0 corresponds to the middle of cylinder). The field values were computed along several different lines. Specifically, B, and Bz were computed at the points z = 0, 5 , 10, * , 60 mm, with the radial variable set to r = 50 mm. In all, 24 field values were computed and these are compared with the corresponding FEA data in Fig. 5 . The axial component B, was also computed along the z axis where the radial component is zero (Fig. 6). Both field components were also computed along the line z = 4 0 mm at the points r = 0, 5 , 10, , 60 mm. This data is shown in Fig. 7. For this analysis the Simpson integration mesh parameters N , and No were set to 20 and 360, respectively, and it took roughly 20 s of run time on a 486125 MHz PC to compute each set of data (24 field values).

-

IV. CONCLUSION A three-dimensional field solution has been derived for radially polarized cylinders. The derivation is based on the assumption of ideal magnetization and is applicable to rare earth materials such as NdFeB. The resulting field expressions are easily implemented in any programming environment. Moreover, they require only seconds of run time on a modern workstation. Consequently, they enable

~

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FURLAN! er a l . : RADIALLY POLARIZED CYLINDERS

A“--

ldag)

Fig. 3 . B, and B, vs. 8 , r

=

50 mm.

Fig. 6. B; vs. z along the L axis (+ = FEA).

Fig. 4. Axisymmetric polarization.

100-

0.

M =Y

,

-100-

-200-

a z:

Fig. 7 . B, and B; vs. r, L

=

40 mm.

Er (+ = FW

-700a-1m. + *

goo,0

1

0 I

8, (+ = FW

2

0 I

3

,0

~

,0

5 ,0

(

1

where

0

I , (r, e,

Z;

r’, e (Z -

rapid parametric studies of field strength, and are therefore useful for the analysis and optimization of radially polarized cylindrical geometries.

APPEND Ix This section contains closed-form expressions for three integrals that appear at various stages of the analysis. The first integral is as follows:

Z,(W Z A W G ( ~ ,e, z ; rr, e ’ , + rr2 - 2rrr cos (e -

zs(w

e’) ( r 2 + r t 2 - 2rrr COS (e - e ’ ) z r2

0)

-1

2 ( z - ZS(kN2 (r = r’ COS (e -

e’)

= 1,

z

+ z,(k)). (44)

According to (44),the functional form of I , depends on the relationship between the coordinates of the field point (unprimed) and the source point (primed). In particular, I , is evaluated in terms of the upper ratio as long as r 2 rr2 - 2rr‘ cos (0 - e ’ ) # 0. However, if this condition is violated, then the lower ratio is used. Note that this analysis does not include the case when the field and source point are one and the same.

+

IEEE TRANSACTIONS ON MAGNETICS, VOL. 31, NO. I. JANUARY 1995

850

The second integral is

where

The definition of function 1, also depends on the relation between the source and field points. Specifically, if r 2 sin2 (0 - e ’ ) ( z - 2’)’ # 0 then the first term in (45) is used. When this condition is violated, the remaining three cases apply as indicated above. Note that the third term is well defined since r,(k) # 0 for this case (i.e., the field and source points cannot coincide). The third and last integral is

+

+ In [12G-’(r,8 , z ; r,v(k),e ’ , z‘) + 2r,T(k)( r 2sin2 (e - e r ) + (Z - z ~ +) 0~)

( p = 1 when r < r,v(l),p

( r # r,,(k), COS

In (r,,(k))

(e

-

=

e’)

(r = 0, z

=

2 when r > r,v(2))

= -1,

z’).

z = z’)

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FURLANI er al.: RADIALLY POLARIZED CYLINDERS

The conditions for the dependence of Z3 are the same as those for Z2,as stated above. The above integrals are special cases of general forms that can be found in [9].

REFERENCES [ I ] L. B. Schein, Electrophotography and Development Physics, vol. 14, Springer Series in Electrophysics, New York: Springer-Verlag, 1988. [2] L. R. Moskowitz, Permanent Magnet Design and Application Handbook, Florida: Krieger, 1986. [3] R. J. Parker, Advances in Permanent Magnetism, New York: Wiley, 1990. [4] S. R. H. Hook, Computer-Aided Analysis and Design of Electromagnetic Devices, New York: Elsevier, 1989. [5] E. P. Furlani, “A three-dimensional field solution for axially polarized multipole Disks,’’ J. Magn. Magn. Mat., 135, pp. 205-214, 1994.

E. P. Furlani, S. Reznik, and W. Janson, “A three-dimensional field solution for bipolar cylinders,” IEEE Trans. Magn., vol. 30, no. 5 , p. 2, Sept. 1994. I. D. Mayergoyz, E . P. Furlani, and S. Reznik, “The computation of magnetic fields of permanent magnet cylinders used in the electrophotographic process,” J . Appl. Phys., vol. 7 3 , pp. 5440-5442, 1993. J. K. Lee and T. S. Lewis, “Epoxy-bonded NdFeB magnetic rollers: Experimental and theoretical analysis,” J . Appl. Phys., vol. 70, pp. 6615-6617, 1991. 1. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals Series and Products, New York: Academic Press, 1990.

E. P. Furlani biography not available at time of publication S. Reznik biography not available at time of publication. A. Kroll biography not available at time of publication.