Approaches to Permanent Magnet Circuit Design

The advent of high-coercivity /high remanence magnet materials has greatly extended the range of applicability of permanent magnets so that many formerly impracticable devices are now viable. This paper discusses some of the more fruitful approaches to the use of these materials in magnetic circuit design and how application of comparatively few and simple techniques can result in a large number of useful devices. The approaches discussed are estimation of permeances, magnetic cladding, equivalent pole densities and current sheets, analytic application of Maxwell’s equations, the magnetic moment rotation theorem and magnetic “mirrors”. Some novel devices resulting from these applications are cited.


Introduction
Permanent magnet design can be rather difficult because solutions for fields of arbitrary configuration are rarely analytic.Fortunately, many of the most useful geometries possess sufficient symmetry to make a solution or an acceptable approximation to a solution analytic.For less tractable configurations, there are rougher approaches to obtaining initial approximations that form starting points for computer-aided iterative improvements.These approaches were simplified by the advent of the rigid ( j @ f c > B R ) , high energy product materials because they provide constant magnetomotive force regardless of the circuits in which they are placed.Therefore, it is now possible to provide a surprisingly extensive variety of field distributions by use of only a few relatively simple computational tools and architectural concepts.This paper discusses some of the more fruitful of these and gives examples of some of the resulting structures.

Estimation of Permeances
An old technique, whose usefulness was limited before the availability of rigid permanent magnets, is an analogy with electrical circuit theory in which the analogues of the electromotive force V, the current I and the conductance G are the magnetomotive force F, the flux Qr and the permeance P, respectively [1,2].We then have a magnetic Ohm's law, i.e.Qr is equal to the product of P and F in analogy to the usual electric expression where I is equal to the product of G and V.
The magnetic form is more difficult to use since flux paths are not clearly defined as are the wires that form the current paths in electrical circuits.However, the former can be estimated by a division of the space in and about the structure into paths bounded by cylindrical, spherical and a rigid (squared looped) magnet is given by the product of the remanence and magnet length, we have the formal equivalent of an electric circuit whose conductances and driving EMFs are known and hence the Qr's can be calculated by Ohm's law and all the other algorithms of electrical circuit theory.
This procedure can be surprisingly accurate if total fluxes are determined because errors accrued in the permeance estimation tend to cancel each other.For detailed field calculations, it is less satisfactory but still constitutes a good basis for rough "back of envelope" calculations that at least provide a starting configuration which can be refined by a computer-aided iterative process.The method can also provide estimates of field magnitudes and leakage fluxes in feasibility studies.
Figure 2 shows the cross section of a cylindrically symmetric speaker-type magnet together with its computer generated flux plot.The estimation of permeances (EOP)

Cladding
The EOP method suggests that when magnetic structures provide field to enclosed spaces, one should be able to eliminate flux leakage by reduction of the outer surface of the structure to an equipotential.This can be accomplished by a "cladding" of the flux-supply magnets with other magnets of orientation and dimensions tailored to confine the flux [2-51.It is analogous to the placement of "bucking" batteries in branches of electrical circuits from which current is to be excluded.An example is shown in Fig. 3 in the form of an ordinary horse shoe magnet.It is desirable to confine all of the generated flux to the gap between the pole pieces.For confinement to occur in the electric circuit analogue, no current may flow through the branches parallel to G,.Such magnetomotive force to those of the supply magnets can be placed on the sides of the supply magnets to prevent flux flow from there by reducing the entire outer surface of the structure to an equipotential.
A particularly useful structure suggested by the cladding procedure is that of the permanent magnet solenoid [3-51.Such structures provide uniform, longitudinal flux densities of several tenths of a tesla over long distances without bulky electric solenoids, energy expenditure or current sources.Permanent magnet solenoids are pictured in Figs. 4 and 5. Flux is supplied by tubular magnet A to the working space via iron pole pieces B at the ends of the cylindrical working chamber W. Inwardly and/or outwardly magnetized dadding magnets C confine the flux to the interior of the structure by eliminating all potential differences between the working space and the exterior, thereby reducing the entire outer surface to an equipotential from which no flux lines can emanate.This is important when close packing with field sensitive instruments is necessary as in aerospace vehicles.The many applications of cladding include electron-beam devices

Equivalent Distributions of Pole DensitiedCurrent Sheets
A distribution of rigidly magnetized bodies produces the same magnetic field everywhere as would a distribution of magnetic poles in which the polar volume density is given by [61 p = -V .$ (1) and the surface pole density by where ri is the unit vector normal to the surface where Q is being calculated.Also equivalent would be an electric current distribution with current densities for volumes and surfaces given by Pole densities can then be placed in Coulomb's law and integrated over all space to find the field at any point of interest.Similarly, current densities can be placed into the Law of Biot and Savart and integrated.This method is of limited usefulness because the integrals obtained from it are rarely analytic.However, for certain high symmetry situations, it can be very convenient and fruitful.
Examples are determination of the on-axis fields of radially and axially magnetized cylindrical annular rings as illustrated in Fig. 6.Such rings can be stacked axially so that the resulting sequence alternates between radially and axially oriented rings, as in Fig. 7A.This stack is an exceptionally efficient arrangement for the production of alternating axial fields for electron beam focussing in travelling wave tubes (TWT's).At the field levels generally needed in TWTs, such stacks can be made as much as one or two orders of magnitude lighter and smaller than a conventional stack that employs iron pole pieces sandwiched between opposed axially oriented rings, as in Fig. 7B[7].

Boundary Conditions on Components with Ungorm Intemal H , B a n d M
A method developed by Abele [8-121 affords the design of shells that supply and confine uniform transverse fields to cylindrical spaces of arbitrary cross sectional shape.A detailed description of this method can be found in references [8-121 and only a brief qualitative description is given here.First, the specified boundary of the cylindrical working space is approximated by line segments if it has curved portions and if not, used as is to form the inner boundary of a magnetic shell whose outer border, also consisting of line segments, is to be determined.The correct assumption is then made that the resulting shell can be composed of irregular blocks over each of which H, B and M are uniform.Then by application of Maxwell's equations at all boundaries, invocation of the requirements that the field exterior to the structure be zero everywhere, and that the field in the interior working space be uniform and of specified magnitude and direction, a family of solutions of varying efficiency can be obtained.One of these can then be chosen on the basis of manufacturing convenience, mass efficiency or the architectural peculiarities of the application.Figure 8 illustrates a square cylinder designed for Magnetic Resonance Imaging (MRI).

Rotation Theorem
A simple computational tool is the rotation theorem [13] which states that if in any two-dimensional dipole distribution, all the dipoles are rotated through an angle 8, the field everywhere will retain the same magnitude but will be reoriented through an angle -8.This is useful in determination of the best array about a cylindrical working space to provide a transverse field there.If the sides of the encompassing magnetic tubes are to be of similar form, as are the segments of a circular shell, the theorem can be used to find the magnetic orientation of each segment for coherent addition of their fields.The segments traversed by the diameter in the desired field direction should be magnetized in that direction.For each other segment to contribute the same magnitude in the same direction, the rotation theorem implies that its magnetization should be at an angle of 29 to the desired field where 8 is the angular coordinate of the segment.
Figure 9 shows the resulting structure which has a very high field to mass efficiency and is therefore often called a "magic" ring.In fact, it can attain fields of arbitrarily high strengths if one is willing to expend the necessary material.The field to mass dependence is logarithmic viz.
where ro and ri are the outer and inner radii, respectively.Therefore, the usual practical upper limit to the working flux density is about 2.0 Tesla with presently available materials.Higher remanences B would increase this limit proportionately.The "magic" ring also confines flux to its interior and is therefore amenable to close packing and placement in areas where stray fields would be a nuisance.
The square cross section of the structure in Fig. 8 which also produces a uniform field in its interior cavity can be derived with the aid of the rotation theorem.The configuration of the upper segment (Fig. 10) of the square is obtained from the application of Maxwell's equations at the inner and outer boundaries and from the requirements of a square shape, uniform internal field and lack of extemal fields.The lower segment is then obtained readily from the anti-symmetry of the strucwe which makes the directions of magnetization in the side forming the bottom of the configuration the inverse mirror image of the top side.The magnetization orientations in the other wo sides are found by the dipolar rotation theorem.If the side shown in Fig. 11A had its magnetization vectors arranged relative to its sides as in the top side, the resulting field would point to the left as in shown.To have the field reenforce that of the top side, it would have to be rotated 90' to make it point straight down as in Fig. 11B.The theorem tells us that for this to be accomplished, the magnetization vectors must all be rotated -90' to form the configuration shown in Fig. 8 and of which Fig. 11B is a part.A permanent magnet material of remanence 1.0 Teslas produces a flux density of 0.293 Teslas and there is no limit to the intensity that can be attained by successive circumscription of similar structures so that the total flux density is 0.293 n Teslas where n is the number of layers used.
Such structures are particularly useful as MRI magnets because of their cavity shape which is convenient for the accommodation of human bodies and because they can be compensated for small defects incurred in manufacture and assembly.This is accomplished by the placement of small dipoles at the inner or outer periphery of the structure, usually in the inner comers.The magnet is constructed from sectional slices that are compensated individually.The field that an ideal slice should generate is calculated and then compared with the measured actual field.The nature of the deviation field then determines the magnitudes and directions of the compensating dipoles needed [15,16].
With the aid of the rotation theorem, supply magnets of other polygonal cross sections can be attained as well.

Structures with Three Dimensional Magnetization Conjiguratwns
The cross section of any of the structures described in the last section can be rotated around its magnetic axis to form solids of revolution of cylindrical symmetry.An important example is the "magic" sphere (Fig. 12).Such structures yield flux densities one third larger than the "magic" rings from which they are derived, i.e.Therefore, the "magic" spheres are even more field efficient than the "magic" rings but they do not confine their flux completely.They have an extemal field of a dipole centered in the sphere with a strength of where V is the volume of the spherical shell and M is its magnetization.

Compensating Shells
A useful technique for the elimination of smy fields from structures with dipole moments is the inscripture of the structure in a uniformly magnetized spherical shell with a dipole moment equal and opposite to that of the structure.Since a uniformly magnetized shell has no interior field, it will not alter the internal field produced by the inscribed structure and yet will cancel its extemal field.Figure 12B shows a "magic" sphere so compensated,

Magnetic Mirrors
Any magnetic structure with a plane of anti-mirror symmetry can be cut in two along that plane and the halves can be placed on slabs of a perfect passive ferromagnet (PPn defined by a zero internal field [17].Figure 13A shows a magic hemisphere so placed on an iron slab that approximates a PPF.The anti-mirror image of the hemisphere in the iron takes the place of the missing half and produces the same cavity field as the p e n t sphere.advantage of fie "magic" igloo is that only half as many expensive, hud-tomanufacture pieces are needed as for the full sphere.Another The highest is that Of pemendur OT 2.3 T* a very high field for so small a structure. A perfect diamagnetic slab @ * , , = 0) could be Placed dong a Plane of ItlkKE sy"etry i d k a cut along that m e to form a "magic" igloo with its basal plane parallel to the fields (Fig. 13B).Unfortunate1y, the only such diamagnets are superconductors, but these are limited by their comparatively small lower critical fields, HC 1 .Diamagnetic and ferromagnetic slabs can be used together to form an eighth sphere of the same field as the parent as in

Manuscript received 15 Fig. 1
Fig. 1 Commonly used approximations to flux paths and their calculated peITIRMCeS.

9 Fig. 2
Fig. 2 Computer plot of flux lines produced by a cylindrically symmetric speaker magnet of E-shaped cross section in proximity with an iron plate.Pi's are the permeance paths and the other capital letters are critical flux junctures.crude approximations of the EOP method with regard to both the qualitative topology of the flux paths Pi and the quantitative locations of the flux branchings, junctures and centers as indicated by the capital letters.

Fig. 3
Fig. 3 Cladding of a horseshoe magnet and the electrical analog.

Fig. 6
Fig. 6 Calculation of on-axis field of a uniformly magnetized radially oriented magnet ring .

Fig. 7
Fig. 7 (A)Compact travelling wave tube (TWT) stack composed of altemately arranged array of axially and radially magnetized rings.(B) Conventional TWT stack with sandwiched iron pole pieces.

Fig. 8
Fig. 8 Square dipolar field source that generates a transverse field.Small arrows show orientations of constituent magnets.Large arrow is field direction in cavity.
Fig. 10 Upper section of a dipolar field source.

U.S. Government Work Not Protected by U.S. Copyright method, when applied to this structure yields a total flux of 21.8 pWb. The more precise finite element analysis by mmputer resulted in 21.2 pWb, a difference of less than 3%. h currents can
be prevented, by placement in those branches, of "bucking" batteries, VC of potential equal and opposite to that of the original supply battery.Analogously, radially oriented cladding magnets, FC of equal and opposite