Computation of the Three-Dimensional Magnetic Field From Solid ...

IEEE TRANSACTIONS ON MAGNETICS, VOL. 43, NO. 10, OCTOBER 2007

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Computation of the Three-Dimensional Magnetic Field From Solid Permanent-Magnet Bipolar Cylinders By Employing Toroidal Harmonics Jerry P. Selvaggi1 , Sheppard Salon1 , O.-Mun Kwon2 , and M. V. K. Chari1 Electrical, Computer, and Systems Engineering Department, Rensselaer Polytechnic Institute, Troy, NY 12180-3590 USA Raser Technologies, Orem, UT 84057 USA We present an analytical method, employing toroidal harmonics, for computing the three-dimensional (3-D) magnetic field from a circular cylindrical bipolar permanent magnet. Bipolar magnets are those which are polarized perpendicular to the axis of the cylinder. We take a completely analytical approach in order to facilitate parametric studies of the external 3-D magnetic field produced by bipolar magnets. The results of our analysis are verified by comparing them to previously published results. The application of toroidal harmonics are ultimately shown to be well-suited for both parametric studies as well as numerical computation. Index Terms—Bipolar magnet, observation cylinder,

-function, toroidal function.

I. INTRODUCTION

H

ARD ferromagnets are those magnets whose magnetization is, for all practical purposes, independent of the applied field strength, provided the externally applied field is not too great. These permanent magnets are used throughout industry in a wide range of applications [1]–[3]. Many closedform solutions exist for the magnetic field from a rectangular parallelepiped permanent magnet and these solutions are used for both computation as well as for parametric studies. However, when dealing with circular cylindrical permanent magnets, closed form solutions are not easily found. In fact, most employ a combination of numerical and analytical techniques [4]–[6]. The advantage of a completely analytical solution over a purely numerical one is that it allows one to perform numerous parametric studies rather quickly. This has definite advantages in design, optimization, etc. Circular cylindrical permanent magnets are a common design and bipolar magnets [4] represent but one type of circular cylindrical permanent magnet that can be analyzed by the method developed in this paper. The computation of the external magnetic field from a permanent magnet will be found by employing toroidal functions [7], [8]. These functions have gained some attention in both the physics and engineering communities [9]–[20]. Although their usefulness for solving various problems which exhibit circular cylindrical symmetry has been known for many years [21]–[25], not much can be found in the literature. One of the reasons, we believe, is because many of the problems which can be solved using toroidal functions can also be formulated with the more widely known elliptic integral formulation. However, these functions offer some analytical and numerical advantages over other methods [12]. II. BASIC THEORY AND FORMULATION Fig. 1 represents a simplified model of a circular cylindrical bipolar permanent magnet.

Fig. 1. Solid bipolar cylinder.

The magnetization vector, , is assumed, for this problem, to be directed in the -direction as shown in Fig. 1. This assumption does not limit the technique. In fact, the toroidal functions can be used to solve for the magnetic field from circular cylindrical permanent magnets which have an array of different magnetization vectors. However, it is most useful for uniform magnetization vectors or for those which vary azimuthally. The magnetic field from a permanent magnet can be solved by the introduction of a magnetic scalar potential function and the concept of magnetic charge density [26]. One can write the magnetic scalar potential function, valid for some external point from a circular cylindrical permanent magnet as shown in Fig. 1, as (1)

Digital Object Identifier 10.1109/TMAG.2007.902995 Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org.

where the normal unit vector on the cylindrical shell is , and the normal vectors on the top and the bottom cylindrical caps are , respectively. The charge densities can be com-

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IEEE TRANSACTIONS ON MAGNETICS, VOL. 43, NO. 10, OCTOBER 2007

puted from (2) and (3) where the equivalent magnetic volume charge density and the equivalent magnetic surface charge density are given by (2) and (3), respectively. For a uniform magnetization vector, , the volume charge density, , is identically zero. , is also identically zero on the The surface charge density, circular cylindrical caps. This is true because the dot-product between the normal vector on either cap and the -directed magnetization vector is zero. However, on the circular cylin, the magnetic surface charge density is drical surface at given by

is called a Legendre function of the The function second kind and of half-integral degree [28]–[30] or a toroidal function of zeroth order [7]. They are also referred to as -functions [25]. Equation (7) represents a Fourier series expansion of the inverse distance function in circular cylindrical coordinates whose weighting coefficients are -functions. The -function can be written as a summation given by [16], [25], [35] (9) for where . In (8), only the term survives the all integration. This leads to the following magnetic scalar potential function: (10) where

(4) The distance between the source point, located on the surface of the circular cylinder, and the field point, expressed in cylindrical coordinates, is given by (5) where

represents the field or observation point and represents the source point. Employing (4) and (5) allows one to write (1) as

(6) Equation (6) can be, and usually is, reduced to an elliptic integral. However, we propose an alternative approach [16], [27], [28]. This approach relies on the fact that the inverse distance between the source point and the observation point in cylindrical coordinates can be expanded in terms of a toroidal harmonic expansion [23], [25], [29]–[33]. This is given, in its most general form, by (7) and is the where , and 2 for all . Neumann factor [34] which is 1 for . One For the specific problem considered in this paper, may notice that (6), with the help of (7), can immediately be formulated in terms of toroidal harmonics without any algebraic manipulations that are usually associated with an elliptic integral formulation. Employing (7) allows (6) to be written as

(11) Employing (11), (10) becomes

(12) The interchange of summation and integration is valid since the . A detailed series in (11) is uniformly convergent for exposition of this and many other mathematical properties is given in [25]. The integral in (12) can be expressed in terms of hypergeometric functions [35]. This leads to the final form for the magnetic scalar potential:

(13)

(8)

Equation (13) represents the external magnetic scalar potential from a finite circular cylindrical dipolar permanent magnet.

SELVAGGI et al.: COMPUTATION OF THE 3-D MAGNETIC FIELD FROM SOLID PERMANENT-MAGNET BIPOLAR CYLINDERS

One can do a simple check to see that the magnetic scalar potential represented by (13) reduces to zero for all points which lie outside the magnet but on the -axis. Likewise, the three components of the magnetic flux density at points external to the bipolar cylinder, but which lie on the -axis, are given by

(14) (15)

(16) is satisfied by the compoAlso, one can show that the nents of (14) through (16) that make up the magnetic flux density vector for all points on the -axis and outside the dipolar cylinder. The hypergeometric functions in (13) can be evaluated for in (13), one obtains any value of . For example, when

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The total magnetic scalar potential is (21) where is large enough to approximate (13). The hypergeometric functions given in (13) can be converted to appropriate summation formulas. If this is done, then one could use the result to do more effective numerical computations. However, the main focus of this paper is to find a formulation that can be used for parametric studies and this is more readily accomplished by pursuing an analytical approach. Also, specialized mathematical tools such as Mathematica [36], Maple [37], and others are well suited for handling the more complicated algebraic manipulations that are always required when analytical solutions are sought. This is particularly true if one is dealing with the more advanced mathematics that one usually encounters in engineering and physics. In fact, Mathematica and Maple were used for both checking and developing all the results in this paper. Once the magnetic scalar potential is known, the external magnetic flux density can be computed from

(22)

(17) Likewise, the magnetic scalar potential for terms up to and interm is given by cluding the

Taking the gradient of (13) in cylindrical coordinates results in somewhat lengthy expressions for each component of magnetic flux density. However, these expressions are easily computed and plotted using Mathematica or Maple. The external magnetic term in (21), for example, are field components for the given by

(18) where sum. Also

includes the

and

contribution to the

(19)

(23)

and (20) In fact, one can obtain closed-form expressions, in terms of elementary functions, for the external magnetic scalar potential function, , for any value of in (13). However, as increase, the algebra becomes intractable. This is no longer a stumbling block with the advent of powerful symbolic mathematical tools.

(24)

(25)

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IEEE TRANSACTIONS ON MAGNETICS, VOL. 43, NO. 10, OCTOBER 2007

Fig. 2. Magnetic scalar potential 8

.

Fig. 3. Radial induction field B

.

Equations (17), (23), (24), and (25) yield accurate results for the far-field solution. In fact, the far field solution is measured at only a few radii lengths from the permanent magnet. The term, for this case, may be all that is needed for design purposes. Of course, if one is interested in making computations close to the source, then more terms in the series solution of (21) will be needed. Two basic questions need some consideration. First, how many terms does one need to compute? Second, what inforterm? Remember, mation can be gleaned from only the term can be computed very rapidly. This is very the useful when a parametric study is undertaken. Although it yields an accurate far-field solution, what information, if any, can be gained by using it to compute the near-field solution? III. ILLUSTRATIVE EXAMPLES Example 1: Consider, for example, a solid circular cylindrical dipolar permanent magnet with a radius of 1/2 inch, a length of A/m, and modeled as shown in 1 inch, a magnetization of Fig. 1. All dimensions were converted to the mks system of units before computations were carried out. In order to compute the components of the magnetic flux density in cylindrical coordinates at any arbitrary point in space external to the permanent magnet, (21) and (22) are employed. Consider enclosing the permanent magnet in a hypothetical observation cylinder. This cylinder is simply a grid of observation points where the magnetic scalar potential and the three components of the magnetic flux density are computed. In this example, a total of 4453 points were used. Each cap has 1460 observation points and the cylindrical shell has 1533 observation points. The maximum radius of the observation cylinder is 0.55 in and it has a length of 1.1 in. The permanent magnet is centered at the origin as shown in Fig. 1 and the observation cylinder completely encloses the magnet. As a first step, the magnetic scalar potential and all of the term in (21). In field components were computed for the other words, (17), (23), (24), and (25) were plotted, using Tecplot [38], on the three-dimensional (3-D) grid described above. term, the magnetic scalar poFigs. 2–5 represent, for the , the radial component, , the axial component, , tential,

Fig. 4. Axial induction field B

Fig. 5. Azimuthal induction field B

.

.

, of the magnetic flux density, and the azimuthal component, respectively. term is, one needs to In order to see how accurate the make the same computation for a larger value of in (21). Of course, one will not know, without a more detailed mathematical error analysis, what value of will yield a solution which is considered to be accurate enough for ones purposes. However,

SELVAGGI et al.: COMPUTATION OF THE 3-D MAGNETIC FIELD FROM SOLID PERMANENT-MAGNET BIPOLAR CYLINDERS

Fig. 6. Magnetic scalar potential 8

Fig. 7. Radial induction field B

.

.

because the toroidal functions are rapidly convergent [39]–[42], one is hopeful that not too many terms will be required. In fact, Snow [25] gives a detailed exposition of the mathematical convergence properties of the -function and Fettis [42] discovered a technique for rapidly computing toroidal functions by using their asymptotic expression. This makes their use for numerical computation highly attractive. in (21). The Consider, for example, the choice of scalar potential and the field components, for this value of , are shown in Figs. 6–9. term does not yield the One can easily see that the most accurate solutions when the observation cylinder is close to the magnetic source. However, it does yield a field pattern . In other words, which looks very similar to that of the magnitudes of the scalar potential and the field components may change but the field pattern is more-or-less the same. Ten terms in the series were computed in order to generate Figs. 6–9. Higher values of will, of course, increase the accuracy. However, one may not need to compute a large number of terms. In other words, only a few terms were actually needed to give a fairly accurate calculation for this particular example. One can easily check to see whether, for example, the magnetic scalar is accurate. Fig. 10 represents the magpotential for in (21). netic scalar potential for

Fig. 8. Axial induction field B

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.

Fig. 9. Azimuthal induction field B

.

Fig. 10. Magnetic scalar potential 8

.

One can see that Figs. 10 and 6 are virtually identical. One may not need to compute many terms in the series given in (21). This is an important feature when a parametric study is undertaken because one wants to be able to make rapid changes in various parameters without having to wait a long time for the computations to complete. This is one reason why the finite-element

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Fig. 11.

IEEE TRANSACTIONS ON MAGNETICS, VOL. 43, NO. 10, OCTOBER 2007

B

and B

.

method and others are not well suited for parametric studies except for relatively simple problems. Also, the authors chose a large number of observation points and each point has an associated computation time. One could easily have chosen fewer observation points without loss of accuracy and this would lessen the computation time. Example 2: This example can be found in both Furlani’s book [43, pp. 222 and 223] and in [4]. It is an excellent way of verifying the method used in this paper. Consider a solid circular cylindrical dipolar permanent magnet with a radius of 2.54 mm, a length of 50 mm, a magnetization of 43 10 A/m, and modeled as shown in Fig. 1. Our purpose is compute a 2-D plot of the radial and azimuthal component of the magnetic flux density plane at a radius mm, and for an angular on the . Employing (13), (21), and (22), variation of one obtains the following 2-D plots of the radial and azimuthal (Fig. 11) and (Fig. 12), respectively. B-field for One can easily see how the solution quickly converges. The solution is identical to that obtained by Furlani using less than 15 terms in (21). IV. DISCUSSION AND CONCLUSION An alternate method has been proposed for computing the 3-D magnetic field from a bipolar circular cylindrical permanent magnet. This method relies on the use of toroidal harmonics. Toroidal functions, well known to those in fusion research, are seldom used by others. Specifically, the toroidal functions of zeroth order are well suited to handle finite circular cylindrical geometries. Most papers which deal with cylindrical magnets rely on a combination of analytical and numerical techniques to arrive at a 3-D solution. However, the approach taken in this paper yields a solution through a completely analytical approach. Although the final result was in the form of an infinite series, the series converges quite rapidly even for observation points relatively close to the magnetic source.

Fig. 12.

B

and

B

.

An example is given which illustrates the 3-D field pattern exhibited by each component of the magnetic flux density , and . Because the infinite series solution for converges rapidly, only a few terms are needed to produce an accurate field pattern. Since the main focus of this paper is to produce a rapid solution, the formalism developed allows one to change various parameters, such as magnetization, physical geometry, observation cylinder, etc., quickly enough to modify a design and, if desired, perform an optimization. Of course, the use of powerful symbolic mathematical tools such as Maple and Mathematica played an integral role in the application of the method developed in this paper. Before the advent of such tools, this technique would have been more difficult to apply. In other words, parametric studies are most easily accomplished when an analytical formalism is developed and this was made possible by using sophisticated symbolic mathematical algorithms. A second example, taken from Furlani [4], [43], is used to verify our results. This example clearly illustrates how quickly the toroidal harmonic expansion converges to the exact solution. A number of other techniques can be employed in solving for the 3-D field from a bipolar cylinder. One, in particular, due to Furlani [4], provides another viable approach for producing a rapid parametric study of the magnetic field. However, the authors felt that toroidal function approach offered a new perspective. It is also well suited for problems which have an azimuthally varying magnetization. In other words, if the magnetization vector, , were constant or a function of the azimuthal variable , then the toroidal harmonic approach would have some definite advantages because the azimuthal integration is usually quite simple. However, the authors have applied this technique for magnetization vectors which are dependent upon all three cylindrical coordinates. In this instance, the problem can still be solved, but the integrations become more complicated.

SELVAGGI et al.: COMPUTATION OF THE 3-D MAGNETIC FIELD FROM SOLID PERMANENT-MAGNET BIPOLAR CYLINDERS

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