Magnetic arrays

Electromagnet arrays which can provide selected field patterns in either two or three dimensions, and in particular, which can provide single-sided field patterns in two or three dimensions. These features are achieved by providing arrays which have current densities that vary in the windings both parallel to the array and in the direction of array thickness.

FIELD OF THE INVENTION 
This invention relates to magnetic arrays and more particularly to magnetic 
array structures which are extensions of Halbach magnet arrays. 
BACKGROUND OF THE INVENTION 
In a typical magnetic array, such as those used for linear motors, 
alternate magnet segments in the array are magnetically oriented in 
directions which are 180.degree. rotated from each other about an axis 
perpendicular to the direction in which the array extends. Such arrays 
produce magnetic fields which are more or less symmetrically distributed 
on both sides of the array. However, there are applications where a 
modified, controlled magnetic field or force pattern may be desired. In 
particular, since in a typical application only the field on one side of 
the array is utilized, half of the field is typically wasted. Further, in 
some applications there may be a need to shield the field on the unused 
side of the array. Magnetic arrays which provide a selectively controlled 
field pattern, and more particularly, which are adapted to provide a 
single-sided field pattern, are therefore desirable. 
One type of magnetic array which has been available for some years and 
which provides a magnetic field which is limited to one side of the array 
is discussed in several articles by K. Halbach ("Design of Permanent 
Multipole Magnets With Oriented Rare Earth Cobalt Material", Nuclear 
Instruments and Methods, Vol. 169, No. 1, pp. 1-10, 1980; and "Application 
of Permanent Magnets In Accelerators and Electron Storage Rings", Journal 
of Applied Physics, Vol. 57, No. 8 pp. 3605-3608, 1985). These permanent 
magnet arrays differ from standard arrays in that each adjacent magnet 
segment is oriented around an axis perpendicular to the direction in which 
the array extends by an angle which differs from that of the adjacent 
segment by a selected angle, for example 45.degree. or 90.degree., which 
results in magnetic axes which are both array normal and array parallel. 
As taught by Halbach, the angle of rotation and the direction of rotation 
are the same for each adjacent pair of magnet segments. An added advantage 
of these arrays is that they provide a field which is a factor of 
.sqroot.2 stronger than the field for more conventional magnet arrays with 
the same volume. These magnet arrays will sometimes be referred to 
hereinafter as Halbach magnets or Halbach arrays. 
However, while Halbach arrays have some interesting properties, they have 
received little attention and only limited utilization outside of particle 
physics applications. One potential reason for this is that, while many 
applications for magnet arrays, such as motors and postioners, require 
electromagnetic array, Halbach arrays have heretofore only been 
implemented in permanent magnet form. A need therefore exists for an 
electromagnetic analog of Halbach arrays which would permit single sided 
fields to be electromagnetically generated. 
Further, it would be desirable if electromagnetic Halbach array could be 
structured to make the field more uniformly sinusoidal on the strong side 
of the array and to minimize stray fields on the weak side of the array. 
Such fields would preferably also be easily switched from one side of the 
array to the other. More generally, it would be desirable if an 
electromagnetic array could be structured to provide selected magnetic 
field patterns as appropriate for a particular application. 
Further, Halbach arrays have heretofore been single dimensional arrays. In 
order to enhance the utility of such arrays, it would be desirable if the 
principles of such arrays could be adapted to provide three-dimensional 
field patterns, and more generally, if selected three-dimensional field 
patterns could be generated by properly structuring magnetic arrays. 
SUMMARY OF THE INVENTION 
In accordance with the above, this invention provides electromagnet arrays 
which can provide selected field patterns in either two or three 
dimensions, and in particular, which can provide single-sided field 
patterns in two or three dimensions. These features are achieved by 
providing arrays which have current densities that vary in the windings 
both parallel to the array and in the direction of array thickness. 
For preferred embodiments, the electromagnet array has a plurality of 
adjacent electromagnet segments which extend in at least one dimension, 
with each electromagnet segment having a magnetic orientation about an 
axis perpendicular to the dimension which is rotated by a selected angle 
from the orientation of each adjacent electromagnet segment. The angle of 
rotation and the direction of rotation are preferably the same for each 
adjacent pair of electromagnets and are such that the array has magnetic 
axes which are both normal and parallel to the dimension in which the 
array extends. Electric currents are selectively applied to the 
electromagnetic segments. Currents may be applied to the segments in a 
predetermined sequence to achieve commutation or for other purposes. 
Windings in at least two different phases may be provided for each segment 
with appropriate currents being applied to the windings of each phase. 
The winding pattern may be triangular or may be rectangular and the 
direction in which current flows may be selectively reversed to change the 
side of the array on which the magnetic field appears. The windings of the 
electromagnetic segments may also be formed of a material which is 
superconducting below a selected temperature and a suitable means may be 
provided for maintaining the windings for at least the segments to which 
current is being applied below the selected temperature. Superconducting 
winding embodiments are particularly suitable for use in maglev 
applications where electromagnets would be provided in the track to 
interact with the array to provide lift. For preferred embodiments, the 
selected angle by which the magnet orientation is rotated between adjacent 
segments is not more than 90.degree.. 
Embodiments are also provided wherein permanent magnet or electromagnet 
segments extend in two dimensions, the array being formed of a matrix of 
rows and columns of magnet segments, with each magnet having a 
three-dimensional magnetic orientation which is the sum of the magnetic 
orientations for the segments for its corresponding row and column. In 
particular, a plurality of magnet segments may be arranged in a 
two-dimensional matrix array of rows and columns, with each segment having 
at least one adjacent row segment and at least one adjacent column 
segment. Each of the segments has a three-dimensional magnetic orientation 
which is the vector sum of a nominal magnetic orientation for the 
segment's row and a nominal magnetic orientation for the segment's column. 
Adjacent segments in each column have a nominal magnetic orientation about 
a row axis which is rotated in a selected direction by an angle which is 
preferably not more than 90.degree. in a selected direction from the 
nominal column magnetic rotation of each adjacent segment of the column, 
and adjacent segments in each row have nominal magnetic orientation about 
eight column axes which is rotated in a selected direction by an angle of 
not more than 90.degree. in a selected direction from the nominal row 
magnetic orientation of each adjacent segment in the row.

DETAILED DESCRIPTION 
FIG. 1 shows a prior art permanent magnet array IO developed by Klaus 
Halbach. The array differs from standard magnetic arrays in that the 
magnetic orientation between adjacent magnetic segments 12A-12G differ by 
90.degree., rather than by 180.degree.. For the embodiment shown in FIG. 
1, the magnetic orientations between successive segments 12 rotate in a 
clockwise direction when moving from left to right. As previously 
indicated, standard magnetic arrays with adjacent segments differing in 
magnetic orientation by 180.degree. have substantially symmetric magnetic 
fields on the top of the array (i.e., the side facing upward in the x 
direction) and the bottom of the array. Conversely, the array shown in 
FIG. 1, which will sometimes be referred to hereinafter as a Halbach 
magnet or Halbach array, has a strong side and a weak side, there being 
little if any magnetic field on the weak side. For the magnets rotating in 
the clockwise direction as shown in FIG. 1, the strong side of the array 
is on the bottom side 20 and the weak side of the array is the top side 
22. If the rotation between successive segments 12 were in the 
counterclockwise direction, the strong side would be on the top and the 
weak side on the bottom. Halbach arrays also have the advantage of 
providing a 42 times stronger field than that of more conventional 
ironless magnet arrays of the same volume. 
While of significant academic interest, Halbach arrays have heretofore 
found only limited commercial application, for example, in undulators and 
wigglers of particle accelerators. One reason for this is that the use of 
permanent magnets for these arrays has limited their versatility and 
adaptability for applications where magnetic arrays are typically used. In 
particular, electromagnet arrays are frequently required in various 
positioner and motor applications, including maglev applications. An 
electromagnet capable of providing the single-sided field of Halbach 
arrays is therefore desirable. 
FIG. 2A is a front view of the magnet array shown in FIG. 1 and FIG. 2B is 
a similar view of an electromagnet analog 14 of this array. However, to 
implement the winding pattern shown in FIG. 2B, with the X's indicating 
windings going into the paper and the dots indicating windings coming out 
of the paper requires true surface currents, and to achieve true surface 
currents requires infinite current density in the wires, which is 
obviously not possible. Other winding patterns for the electromagnets are 
therefore required. 
FIG. 2C shows an array 16 having a triangular winding pattern which is 
particularly advantageous for this application. In particular, this 
pattern provides varying currents through the thickness of the magnet 
which permits the field to be controlled. It is also a relatively simple 
pattern to wind and results in shorter lead lengths which minimize power 
dissipation for a given field strength. The triangular winding pattern 
array shown in FIG. 2C produces a single-sided magnetic field pattern. 
Commutation may be achieved by successively energizing the electromagnetic 
segments 17A-17G of array 16 in some predetermined pattern. The pattern 
for this array, which would be similar to that shown in FIG. 6, but not as 
well formed and with the weak and strong sides reversed, has a strong side 
20 and a weak side 22 which is dictated by both the triangular winding 
pattern, which gives a particular depth profile to the magnetic field, and 
by the direction of rotation of magnetic orientation for successive 
segments 17 of the array. This results in the array having both array 
normal and array parallel magnetic axes which are spacially distributed so 
as to produce the substantially single-sided field. However, with a 
different winding pattern on the array which produces a different depth 
profile for the magnetic field of each segment, a different selected 
magnetic field profile may be obtained. In particular, parameters can be 
selected which can be solved using Fourier analysis to produce a winding 
pattern to achieve desired field patterns, as will be discussed in greater 
detail later. The field pattern selected will depend on application. 
FIG. 3 shows an array 24 having another winding pattern which produces 
results substantially similar to those for the array 16 shown in FIG. 2C. 
This array has rectangular segments 26, each of which is made up of a pair 
of adjacent windings with adjacent segments being rotated in the same 
manner as for array 16. As for the earlier array, bottom side 20 is the 
strong side and upper side 22 is the weak side. While two windings have 
been shown per segment for the array 24, and this configuration is 
preferable for the array, particularly where commutation is desired, in 
applications where commutation is not required, each segment 26 could be 
formed of a single winding oriented in the manner shown. In FIG. 3, the 
length L indicates the periodicity of the array (i.e., the length of the 
array before the winding pattern repeats). The winding pattern of FIG. 3 
also provides the current density varying in both the X and Z direction 
required to achieve a controlled magnetic field pattern in general, and a 
substantially single-sided magnetic field in particular. 
For either the array 16 of FIG. 2C or the array 24 of FIG. 3, a Fourier 
analysis shows that if the spacial period L is four times as long as the 
thickness T of the array, then the fundamental component of the field on 
weak side 22 will be substantially equal to zero. To the extent this 
condition exists, the fundamental field on the weak side is approximately 
equal to zero. However, non-zero higher order components of the field 
still exist on both sides of the array. The magnetic field resulting from 
these higher order components is generally negligible for the disclosed 
embodiments. However, even if the L=4T relationship is not absolutely met 
for the array, the increase in the weak side magnetic field will not be 
substantial and for most cases a substantially single-sided field will 
still be provided. 
Halbach recognized that a more uniformly sinusoidal field on the strong 
side could be achieved by having the magnetic orientation continuously 
rotate in the selected direction along the array, rather than by having 
discrete 90.degree. changes in the field. However, with either permanent 
magnets or electromagnets, it is increasingly expensive to assemble the 
arrays as the angle between adjacent segments decreases. This places a 
practical limit on the quality of waveform which can be achieved, and in 
particular on the degree of weak side suppression which can be obtained. 
FIG. 4 illustrates a technique which may be utilized to achieve 
significantly enhanced resolution for electromagnetic embodiments 
resulting both in a more perfectly sinusoidal magnetic field on the strong 
side and enhanced suppression of harmonics, such suppression further 
smoothing the wave shape on the strong side and reducing stray field on 
the weak side. In particular, FIG. 4 shows an array of the type shown in 
FIG. 2C which has been modified so as to be a two-phase array. The 
windings for phase 1 of a single period of the array are shown on line a 
of FIG. 4, while the windings for a phase 2, which are 90.degree. behind 
those of phase 1 are shown on line b. Line c shows the actual windings for 
the two-phase array which is a combination of the phase 1 and phase 2 
windings on lines a and b, respectively. The sine waves for the fields 
resulting from the two phases for the array on line c at FIG. 4 combine to 
provide a magnetic field which more closely approximates that for 
continuous rotation in field direction, with a smoother sinusoidal field 
on the strong side and enhanced suppression of harmonic caused fields on 
the weak side. 
A further reduction in harmonics, resulting in even closer approximations 
to the ideal of continuous field direction rotation can be achieved with 
more complicated winding patterns providing three-phase operation, 
four-phase operation, . . . n phase operation. While for most 
applications, the incremental benefits from going beyond two phases 
probably do not warrant the added cost of the array, and the added 
complexity of control, for n phase operation, for applications where any 
magnetic field on the weak side is not acceptable and/or where a very 
sinusoidal magnetic field on the strong side is required, the option is 
available to increase the number of phases for the array until the desired 
field pattern is achieved. 
In operation, all n phases would typically be applied simultaneously to 
each active segment of a multi-phase array, with the relative current 
applied to the coil for each phase being controlled to achieve the desired 
forces. FIG. 5 illustrates a simple circuit which may be utilized to 
control the currents applied for each phase of each segment or stage 30 as 
a function of the desired magnetic fields in the X and Z directions for 
such stage. Referring to FIG. 5, a circuit 32 is provided which contains 
the commutation laws to be applied for determining currents as a function 
of desired fields. An example of how such laws may be applied with 
two-phase commutation in a somewhat different environment are taught in 
"Magnetic Arrays For Synchronous Machines", D. L. Trumper, M. E. Williams, 
and T. H. Nguyen, Proceedings of the IEEE IAS 28th Annual Meeting, pp. 
9-18, October, 1993. Device 32 may be a general purpose microprocessor or 
other processor programmed to perform the current determinations in 
accordance with the commutation laws, may be special purpose circuitry 
designed to perform this function, may be a computer which is programmed 
to perform this function in conjunction with other functions as part of a 
system, or may be a hybrid hardware/software device. The inputs to device 
32 are desired forces which would normally also be computer determined in 
accordance with a predetermined algorithm and the outputs are resultant 
currents for each phase for a particular magnet segment 30. The output 
currents appearing on lines 36-1 through 36-N are applied through 
corresponding power operational amplifier or other suitable devices 38-1 
to 38-N to the coil 40-1 to 40-N for the phase. Typically, a single 
commutation law circuit or device 32 would be utilized to determine the 
currents to be applied for the various phases for all of the 
segments/stages of an array 28. 
FIG. 6 illustrates the magnetic field pattern for the two-phase array 28 of 
FIG. 4. In this drawing, it is seen that, for this array, the strong side 
and weak side are reversed from that for the arrays of FIG. 2C and FIG. 3, 
that with some limited exception the magnetic flux lines 42 on the strong 
side of the array have a fairly smooth sinusoidal shape and that the stray 
fields 44 on the weak side of the array are minimal. As discussed earlier, 
further improvement in the wave shape on the strong side and further 
suppression of field on the weak side may be achievable by increasing the 
number of phases for each segment 30 of the array. 
FIG. 7 illustrates an alternative two-phase array 46. Line a illustrates 
various rectangular elements of this array with lines f and g indicating 
the elements on line a which are used for each of the two phases. Thus, 
phase 1 uses elements A--A , C--C , F--F , G--G , and J--J . Similarly, 
phase 2 utilizes elements B--B , D--D , E--E , H--H , and I--I . Lines b, 
c, d and e show illustrative current directions, with x being into the 
paper and the dots being out of the paper for the various winding at 
successive time intervals in order to achieve commutation with movement 
from left to right. 
While as previously discussed, permanent magnet Halbach arrays are 
advantageous in that they provide a fundamental field which is stronger by 
a factor of .sqroot.2 than conventional permanent magnet arrays, this 
improved performance is not true for electromagnetic arrays which normally 
produce a single-sided field having a magnitude which is roughly 53% of 
that for a conventional pattern with the same consumption of electric 
power. Therefore, such single-sided windings are not likely to be used in 
power sensitive applications. 
However, notwithstanding the above, such magnet arrays do have a number of 
advantages which may allow for their use in appropriate applications. 
Among the advantages are: 
(1) no fundamental of the magnetic field on the weak side 22; 
(2) near sinusoidal magnetic field pattern on the strong side 20; 
(3) shorter turns for the electromagnetic windings which reduces power 
dissipation and also makes the motor simpler to manufacture; 
(4) simpler driving circuit since only two phases is required; 
(5) less magnetic field shielding material for certain applications such as 
maglev and electric vehicles (this application will be described later); 
(6) easy manufacture by modules (each of the segments 30 is an identical 
module and the segments differ only in the direction in which they are 
oriented for assembly). 
Therefore, for applications where the above advantages are more important 
than power consumption, the electromagnets described above may be 
utilized. However, in applications where superconducting coils may be 
utilized, the increased power is no longer a factor and the advantages of 
the arrays discussed above, and in particular array 28, would make the use 
of such arrays very attractive. In particular, in maglev applications one 
problem is protecting passengers from magnetic fields which appear on the 
unused side of the magnets and from other stray magnetic fields. Such 
fields may pose a health risk to passengers, especially those using field 
sensitive devices such as pacemakers. Thus, extensive shielding must be 
provided between the magnets and the passenger compartments which 
significantly increases both the cost and the weight of the vehicle. This 
invention, with its one-sided fields, should permit a significant 
reduction in the shielding requirements for such applications and should 
be practical for such applications since superconducting coils are 
frequently already used. 
FIG. 8 illustrates a maglev application where the train or other vehicle 50 
having a passenger compartment therein has a magnetic array 28 mounted 
thereunder with two-phase electromagnetic segments 30. While eight 
segments are shown for array 28 in FIG. 8, the number of segments in the 
array would vary with application. While some shielding 52 might still be 
required between the electromagnet array 28 and the passenger compartment 
of vehicle 50, the single-sided field 18 for such array would 
significantly reduce such shielding requirements. Further, since the coils 
for array 28 could be superconducting coils, the power consumption penalty 
which normally exists for such arrays would not be a problem. As with 
existing maglev devices, the electromagnetic array 28 would interact with 
electromagnetic coils 54 on track 56. The coils would be mounted on a 
structural support 58 which is the other portion of the track. 
In the discussion to this point, the field pattern provided by all of the 
electromagnet has been a generally single-sided field with a strong side 
having a generally sinusoidal shape. However, the capability of designing 
electromagnetic arrays having current density patterns which vary through 
the thickness T of the array and, in particular which can be designed by 
providing a winding pattern for the array which matches the sum of a 
series of complex exponentials, for example via a Fourier transform in 
space, permits a variety of specialized field patterns to be achieved. 
Thus, one could start with a desired field pattern, express such field 
pattern as a sum of complex exponentials in space, and then generate a 
winding pattern utilizing techniques known in the art which achieves or 
matches such a series of complex exponentials. FIG. 9 shows an 
illustrative winding pattern for such a complex field pattern which 
pattern has a periodicity L. 
In the discussion so far, Halbach arrays and other arrays of either a 
permanent magnet or electromagnetic variety have been linear arrays. FIG. 
10 illustrates an array 60 using Halbach magnet principles which generates 
fields in three dimensions. The segments for the matrix 60 of magnet 
segments shown in FIG. 10 are oriented as follows: 
N=oriented in the X direction out of the paper 
N+arrow=oriented in the direction of the arrow and in the X direction at a 
45.degree. angle out of the paper 
arrow=oriented in the direction of the arrow 
S+arrow=oriented in the direction of the arrow and in the X direction at a 
45.degree. angle into the paper 
S=oriented in the X direction into the paper 
blank=no magnetic field (all magnetic fields cancel) 
The orientation for the magnets shown in FIG. 10 are determined as the sum 
of Halbach arrays preceding from the bottom of the figure to the top of 
the figure in the Y direction and Halbach arrays preceding from the left 
of the figure to the right of the figure in the Z direction. Thus, the 
segment 62 is the sum of a magnetic field facing upward in the Y direction 
for a Halbach array made up of the left most column of magnetic segments 
and an orientation to the right for the Halbach array formed by the bottom 
row of segments. The segment 64 is the sum of the field pointing upward 
for the first magnet of the Halbach array which constitutes the second 
column and a field which is rotated 90.degree. for the bottom row to point 
out of the paper in the X direction. This results in the segment having 
the indicated orientation. Segment 66 has been rotated for both its row 
and column to point upward in the X direction and thus has an upward or 
north orientation. By following the above process, the orientations shown 
for the remaining magnet segments in the array can be determined. 
While because of improved power efficiency, it is assumed that the array 60 
shown in FIG. 10 is formed of permanent magnet segments, in suitable 
applications the array could also be formed of electromagnet segments. 
Since the array provides magnetic fields which vary in three dimensions 
with position, forming generally sinusoidal mountains and valleys in the X 
direction, array 60 may for example find application in positioning 
devices with six degrees of freedom, array 60 interacting in such 
applications with suitably activated electromagnetic coils. For example, 
the array 60 could be mounted to a platen to be positioned with the 
electromagnets being on a stator. Further, while FIG. 10 assumes Halbach 
arrays in the two dimensions, a particular desired three-dimensional field 
pattern may require different winding patterns to be added. These 
different winding patterns to be added would generally be determined as 
discussed in conjunction with FIG. 9. 
A number of different embodiments have therefore been disclosed for 
utilizing Halbach magnetic array concepts to develop various useful 
electromagnetic and three dimensional permanent magnet arrays and to 
develop other arrays having current densities which vary both parallel to 
and normal to the array. Further, while both the angular variations and 
the direction of rotation between successive segments for Halbach type 
arrays have been the same for a given array, and this is normally 
preferred, these are not limitations on the invention and may not be the 
case for some special applications. Thus, while the invention has been 
particularly shown and described above with respect to various preferred 
embodiments and variations on such embodiments have been discussed, it is 
apparent that the foregoing and other changes in form and detail may be 
made therein by those skilled in the art without departing from the spirit 
and scope of the invention. 
Various mathematical proofs for some embodiments of the invention are 
provided in the attached appendix which is not to be printed.