Computation of the External Magnetic Field, Near-Field or Far-Field

IEEE TRANSACTIONS ON MAGNETICS, VOL. 43, NO. 4, APRIL 2007

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Computation of the External Magnetic Field, Near-Field or Far-Field, From a Circular Cylindrical Magnetic Source Using Toroidal Functions Jerry P. Selvaggi1 , Fellow, IEEE, Sheppard Salon1 , O.-Mun Kwon1 , Fellow, IEEE, M. V. K. Chari1 , and Mark DeBortoli2 Electrical College of Engineering, Computer, and Systems Engineering Department, Rensselaer Polytechnic Institute, Troy, NY 12180-3590 USA Lockheed Martin, Inc., Schenectady, NY 12301-1072 USA

A method is developed for computing the magnetic field from a circular or noncircular cylindrical magnetic source. A Fourier series expansion is introduced which yields an alternative to the more familiar spherical harmonic solution, Elliptic integral solution, or Bessel function solution. This alternate formulation coupled with a method called charge simulation allows one to compute the external magnetic field from an arbitrary magnetic source in terms of a toroidal expansion which is valid on any finite hypothetical external observation cylinder. In other words, the magnetic scalar potential or the magnetic field intensity is computed on a exterior cylinder which encloses the magnetic source. Also, one can compute an equivalent multipole distribution of the real magnetic source valid for points close to the circular cylindrical boundary where the more familiar spherical multipole distribution is not valid. This method can also be used to accurately compute the far field where a finite-element formulation is known to be inaccurate. Index Terms—Charge simulation, permanent-magnet motor, Q-function, toroidal function.

I. INTRODUCTION

problem has been found [6]–[8] which allows for a wide range of applications which will be discussed.

A

METHOD using a Fourier series expansion, derived from a circular cylindrical Green’s function [1], will be introduced in order to characterize the magnetic field from a circular cylindrical or noncircular cylindrical magnetic source in terms of an equivalent multipole distribution. Charge simulation [2], [3] is used to replace an arbitrary magnetic source with fictitious magnetic charges on a circular cylinder, and this point charge distribution becomes the new magnetic source. Once a magnetic point charge distribution is found which accurately reproduces the external magnetic field produced by the original source, one can apply the magnetic form of Coulomb’s law [4] to compute the magnetic field from a charged cylinder at any point in space external to the source. In order to apply the magnetic form of Coulomb’s law in circular cylindrical coordinates, one needs to find an appropriate expansion valid for a circular cylindrical coordinate system. This expansion is known [1], [5] but has not been extensively utilized. One of the reasons for its lack of use has to do with the fact that the Green’s function in circular cylindrical coordinates requires the accurate evaluation of an infinite integral over a product of Bessel or modified Bessel functions. This, in theory, can be handled numerically, but an analytical solution would allow one to compute an equivalent multipole distribution with greater accuracy. An analytical solution to this

II. FORMULATION The reciprocal distance, in cylindrical coordinates, between , and the observation point, , the source point, is given by (1) where represents the position vector of the source point and is the position vector of the observation or field point as illustrated in Fig. 1. There are a number of ways in which the reciprocal distance can be represented, and one way utilizes the expansion of the free-space Green’s function in cylindrical coordinates. This expansion leads to the expression [1]

(2) where and are modified Bessel functions of the first and second kind, respectively. Equation (1) is valid if . Let (3) Writing (3) in terms of real quantities gives

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(4)

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(9) For magnetic field problems which exhibit cylindrical symmetry, one may also need to know the higher order derivatives of . More specifically, the expansion for for will be required. This expansion is given by [14]

Fig. 1. Inverse distance in cylindrical coordinates.

The coefficient is called the Neumann factor [9]. The Neumann factor can be expressed in terms of the Kronecker delta function [5]. This is represented by (5) if and for . where The infinite integral in (3) has been evaluated [10]–[12] and is given by (6)

(10) are the associated toroidal functions. Nowhere . Also, one can write [14] tice that (10) reduces to (7) for

Utilizing (6), one can rewrite (4) as [13]–[15] (7) , and is called a Legendre function of the second kind and of half-integral degree or a toroidal function of zeroth order. They are also referred to as Q-functions [14]. Equation (7) represents a Fourier series expansion of the inverse distance function in circular cylindrical coordinates whose weighting coefficients are Q-functions. It has been shown that the Q-function can be written as [8] where

(8) where

for all

. It is often necessary, in electromagnetic field problems, to know not only the inverse distance function but its gradient. The inverse distance is useful when formulating the electric scalar potential, the magnetic scalar potential, or the magnetic vector potential in terms of an integral. However, in magnetostatic problems, it is necessary to compute the magnetic field intensity or the magnetic flux density . This requires a knowledge of the gradient of the inverse distance function. In cylindrical coordinates, one must find an expression for the gradient of (7). This is given by

(11) for

. For example, if

, (11) reduces to

(12) Equations (10) and (11) are quite useful for solving electromagnetic radiation problems in circular cylindrical coordinates. The Green’s function expansion in cylindrical coordinates for charges can now be written as (13) where represents the position vector of the th source point and is the position vector of the observation or field point. From (13) and from the magnetic form of Coulomb’s law [4], one can compute the corresponding magnetic scalar potential at some field point, , as shown in Fig. 1. This is given by

SELVAGGI et al.: COMPUTATION OF THE EXTERNAL MAGNETIC FIELD

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Fig. 3. Total 8

Fig. 2. Permanent magnet pole arrangement.

(14) are the hypothetical magnetic charges whose dimenwhere sions are amperes meters. The advantage of this method over an Elliptic integral or Bessel function approach is due to the fact that the Green’s function in circular cylindrical coordinates for electro- or magnetostatic problems yields a direct relationship between Coulomb’s law and charge simulation. This relationship also holds for time-varying electromagnetic field problems as well. The magnetic field intensity is given by

(; ; z).

Fig. 4.

8 (; ; z) component.

Fig. 5.

8 (; ; z) component.

(15)

III. ILLUSTRATIVE EXAMPLE Consider a real six-pole permanent-magnet motor with each of its six poles having a constant magnetization vector oriented along the magnet’s axis. Fig. 2 represents the pole arrangement of the six-pole motor. This problem is a real application of a technique that has already been developed for a spherical coordinate system [2] but adjusted to handle problems which exhibit circular cylindrical symmetry. Using charge simulation, the motor is replaced with an equivalent point charge distribution on a cylinder (Charge Cylinder), and this charge distribution is used to calculate the scalar potential using (14) on another hypothetical cylinder (Observation Cylinder) located outside and concentric to the charge cylinder [8]. Fig. 3 is a plot of the total magnetic scalar potential. Figs. 4–6 represent the various components up to and including the term which contribute to the total magnetic scalar potential of a real six-pole motor under full load conditions. Terms higher than contribute substantially less to the total magnetic scalar potential. No term exists since must be zero. A term would imply a monopole contribution which is not possible for magnetic field problems. If one is dealing with a problem involving the computation of electric fields, then an term may or may not exist. The units of magnetic scalar potential are amperes. Fig. 6 shows that the term would be the dominant term in the far field; however, the and terms will

contribute in the near field. The authors have developed a multipole reference table which can be used qualitatively in order to predict which terms in the toroidal expansion will yield a spherical monopole, a spherical dipole, a spherical quadrupole, etc. This is shown in Table I. One can see from Table I that the term includes the monopole , the quadrupole , the hexadecapole , etc. Also, the term includes the dipole , the octupole , the tricontadipole , etc. Likewise, the term includes

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Fig. 6.


8 (; ; z) component. TABLE I MULTIPOLE REFERENCE TABLE

Fig. 7. Dead zone surrounding a real cylindrical magnetic source.

unable to capture the contribution to the external magnetic field close to the circular cylindrical magnetic source. This region where a spherical harmonic expansion is inapplicable will be called the “dead zone.” However, the toroidal expansion easily captures this region and allows one to compute an equivalent multipole distribution close to the circular cylindrical boundary. REFERENCES the quadrupole , the hexadecapole , the hexacontatetrapole , etc. The multipole reference table simply shows where each of the spherical multipoles reside within a given toroidal expansion. The multipole reference table shows that Fig. 4 has a spherical dipole as its its dominant contribution. Likewise, Fig. 5 has a spherical quadrupole as its dominant contribution, and Fig. 6 has a spherical octupole as its its dominant contribution. As discussed previously, no monopole contribution is allowed. One may notice that the term includes a quadrupole term as its dominant contribution. The term also has a quadrupole term can exist for magnetic contribution but since no term can possibly contribute to field problems, only the the quadrupole. IV. CONCLUSION A general technique has been introduced for describing circular cylindrical magnetic systems. The Fourier series expansion or the toroidal expansion whose coefficients are the Q-functions can be used to represent a noncylindrical magnetic source through the use of a method called charge simulation which allows for a direct application of the magnetic form of Coulomb’s law. In particular, the method of charge simulation allows one to map a noncircular cylindrical magnetic source onto a circular cylindrical boundary and then employ the toroidal expansion in order to accurately compute the external magnetic field. An example of a real six-pole permanent magnet motor was introduced in order to show how the toroidal expansion can be used to characterize a magnetic source in terms of its equivalent multipole distribution. The power of a toroidal expansion is in its ability to predict the equivalent multipole distribution close to the circular cylindrical magnetic source. This is illustrated in Fig. 7. Fig. 7 shows that a spherical harmonic expansion is

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Manuscript received May 1, 2006 (e-mail: [email protected]).