Concerning the Physics of Halbach Arrays

actorrattleΠολεοδομικά Έργα

15 Νοε 2013 (πριν από 3 χρόνια και 9 μήνες)

97 εμφανίσεις


1

Concerning the Physics of Halbach Arrays


James

R. Creel

Department of Physics, University of Texas at Arlington,

Arlington, TX 76019

(Published Spring of 2006)


Concerning the physics of magnetic levitation (maglev), particle
accelerators, and laser appli
cations, Halbach arrays have played an
incidental role. Discovered by Mallinson in 1973 as a “magnetic curiosity”
but years later independently discovered by Klaus Halbach, these
magnetic arrays had the ability to created a one
-
sided flux of
magnetization.

Originally, these arrays were recognized to make
significant improvements in magnetic tape technology, but later were seen
by Halbach as a novel way to enhance the effects of particle accelerators
through guiding technology. Throughout this discussion, Ha
lbach arrays
will be explored for their uniqueness and applications current and future.



I. INTRODUCTION


If you have ever used any kind of
magnet with iron fillings, you know that
some interesting “things” happen when the
filings come into contac
t with the field of
the magnet. These “things” that happen
show the invisible field of the magnet by
aligning the iron filings parallel to the field
lines

(Fig 1)
.






FIG
. 1. Bar magnet with Iron Filings


It is also well known that magnets
have North

and South Poles, and that like
poles repeal and opposite poles attract. It
was from this, mistakenly, that a magnet
was said to produce lines of force. This
two pole system of a magnet is interesting
in that unlike free electrical charges, the
poles of a

magnet cannot be broken up. It
is not possible to find a magnetic north (or
south) pole just wandering in space.
Hitherto, if you were to divide any magnet
in half, you would just have to smaller
magnets with the same poles as your
original magnet.

Pierr
e Curie, a French physicist,
discovered that in order for a permanent
magnet (e.g. bar magnet) to lose its
magnetic field, it must be heated to a
certain temperature (the Curie point). This
reason for this is because of the intrinsic
properties of the magn
et, its magnetic
moments. These moments in a magnet are
initially aligned in random directions and a
net uniform direction after the introduction
of a magnetic field.

Modeling a magnet field lines as arrows,
these arrows would start on a North Pole
of the

magnet and end on the South. As
you can see, a net of arrows in the same
direction shows either magnetic
characteristics of the magnet substance (a
piece of metal, etc.) or the presence of a
magnetic field. Magnetic moments can be
aligned by taking a Ferr
omagnetic material
past the Curie point, applying an external

2

magnetic field, and then bringing it back
below the Curie point

(Fig 2)
.








FIG
. 2. Magnetic Moments


Modeling a magnet field lines as arrows,
these arrows would start o
n a North Pole
of the magnet and end on the South. As
you can see, a net of arrows in the same
direction shows either magnetic
characteristics of the magnet substance (a
piece of metal, etc.) or the presence of a
magnetic field. Magnetic moments can be
ali
gned by taking a Ferromagnetic material
past the Curie point, applying an external
magnetic field, and then bringing it back
below the Curie point

[1
],
[2
]
.


II.
BACKGROUND


Halbach arrays, similar to some
discoveries in the past, came into being
under two

different scientists. In the early
70’s a scientist and engineer named John
Mallinson noted a ‘magnetic curiosity’
with a particular permanent magnet
combination of dipole magnets. He also
realized that this ‘magnetic curiosity’
could be very useful with
magnetic tape
technology. Mallinson showed that
because the x and y components of the
magnetic flux are out of phase with each
other, a one
-
sided flux is nearly produced.
Though, the name preceding the array is
clearly Halbach. Klaus Halbach, a
physicist,
discovered the same effect later
that decade and published his results.
Halbach saw these arrays (Fig. 3) as an
application to particle accelerator design
which gave significant contributions to the
field

[3]
.



FIG
. 3. Halbach Array


The uniqueness of these arrays
does not only lie with the fact of the nearly
one
-
sided f
lux (only one sided flux for
“perfect world” examples). Another
interesting and unique quality of these
arrays is that they array is stronger than its
individual components, because field lines
can be thought of as being somewhat
superimposed. Therefore, i
f the array was
made up with the strongest permanent
magnets in existence, the array would
produce an even stronger magnetic field!


III. TECHNICAL ANALYSIS


In order to find the field associated
with a Halbach array, we need to think of
a planar structure

[
4
],

(Fig 4.)
, of thickness
d
, lying in the
x
,
z
plane.




FIG
.4. Planar Structure


We can take the magnetization of the array
to be two sinusoids in quadrature: Eq. (1
-
3)



m
x
= m
0
sin(kx),




(1)


X

Z

-
Y


3

m
y

= m
0
cos(kx),



(2)


m
z
=0.





(3)


Next, we proceed with solving the
boundary value problems for the scalar
potentials above and below the sheet.

Potentials
wit
hin the sheet obey Poisson’s
formula
:

2
inside
m
0
kcos
kx

a
s well as
Laplace’s equation above and below

the
sheet
2
above
0

and
2
below
0
.
Since we know the particular solutions to
Poisson’s Equation is


m
0
k
cos
kx
we can
get the general solutions:



above
Aexp
ky
Bexp
ky
cos
kx
, (4)


inside
Cexp
ky
Dexp
ky
mo
k
cos
kx

(5)


below
Eexp
ky
Fexp
ky
cos
kx
.

(6)


Matching the b
oundary conditions for the
potential on either side of the sheet, the
fields and potentials must go to zero as y
becomes infinite
above
below
0
. The
tangential fields must match on the sheet
upper and lower surfaces:


above
inside
when
y
0

(7)


below
inside
when
y
d


(8)


Also, the normal flux density must be
continuous on the upper and lower
surfaces by:


below
y
inside
y
m
0
cos
kx
when
y
d

(9)

above
y
inside
y
m
0
cos
kx
when
y
0.


(10)

Rewriting our original solutions to match
these boundary conditions gives:



above
0

(11)


inside
m
0
k
exp
ky
1
cos
kx

(12)


below
m
0
k
1
exp
kd
exp
ky
cos
kx



(13)


Below, in Fig 5, we have the graphical
representation of the equations given
above. The distance from top to bottom of
the array is d

[5
]
.



FIG. 5. Halbach Array with Field Lines


IV. APPLICATIONS


The Halbach array
has found its way into
many physics disciplines. Discussed
before, Halbach arrays found their
beginning in particle physics. Scientists
were able to use these arrays as particle
guiders in particle accelerators.
One area
where these arrays are found in is
accelerators in the beam intersection
region of a
PEP
-
II
β
-
factory

[6
]
.
Another
application of science uses Halbach arrays
as a braking system for roller coasters.
Most roller coasters that go high speed
need very efficient and strong braking
systems which

can lead to strong
electromagnets exerting a force against
each other to stop the coaster. Utilizing
Halbach arrays, the need for
electromagnets is cut in half, likewise the
cost of running them.

They can also be
employed in levitation of the cars [
7
].

Th
ese arrays have also found their
way into optics. Yet again, because of the
strong magnetic fields produced by these
arrays, levitating stages for samples can be
built which reduce outside vibration when
taking samples

as well as the capability of
getting
to micro order sample positioning

4

resolution

[8
]
.

Levitation abilities alone
now give the arrays a place in the area of
maglev train technology and other general
levitating objects. Because of the
sinusoidal varying field, a design can be
made in which a 1
00
-
passenger train can
levitate just by forward motion!
Analysis
has also been made on how these arrays
would fair as the main components to a
generator. S.M. Jang

[9
] analyzes

the
benefits and drawbacks of using
a Halbach
array.
A recent Scientific Americ
an article
has given Halbach arrays an even greater
role in magnetic levitation by the
possibility
of helping a shuttle into space
[10]
,
[11
].

Perhaps more ways of using
Halbach arrays will be explored in the
future. This may be accomplished by
looking at
magnetism not only in its
classical component but in its quantum
component as well. One person who might
be called a pioneer in this respect would
be J. H. Van Fleck. [
12
]


6. ACKNOWLEDGEMENTS


I would like to thank my classmates and
my professor of Semina
r 4117. I think that
this class helped me to understand the
proper procedures and criteria that go
along with a technical paper and
presentation. I know that this will be quite
a valuable asset in my life.



REFERENCES


[1]
Schafer, Rudolph, and Alex Huber
t.
Magnetic
Domains
. Springer, 2001. 484
-
534.

[2]
Watanabe, Kazuo, and Misuhiro Motokawa.

Materials Science in Static High Magnetic
Fields
. Springer, 2002. 125
-
200.

[3]
K. Halbach. “
Physical and Optical Properties
of Rare Earth Cobalt Magnets
,” Nucl. Inst.
and Methods 187. pp.109
-
117(1981).

[4]
Shute, H.A.; Mallinson, J.C.; Wilton, D.T.;
Mapps, D.J., "
One
-
sided fluxes in planar,
cylindrical, and spherical magnetized
structures
,"
Magnetics, IEEE Transactions on

, vol.36, no.2pp.440
-
451, Mar 2000

[5]
Campbell
, A

M. "Forces Between Arrays
o
f
Magnets and Superconductors,”


IOP

(2002). 14 Oct. 2005
<http://www.iop.org/EJ/abstract/0953
-
2048/15/5/323/>.

[6]

Results from A Prototype Permanent

Magnet Dipole quadrupole Hybrid for the
Pep
-
II
β
-
Factory
,” Conceptual Design

Report,
SLAC
-
PUB
-
7562

x.

UCRL
-
ID
-
1 14055, UC
-
I
IRPA
-
Y3
-
01, June 1997
.

[7]“
Maglev System. Transrapid, High Tech for

Flying on the Ground
,” Transrapid
International,
<
http://www.transrapi
d.de
>
.


[8]
Kawato, Y.; Won
-
jong Kim;



A novel multi
-
DOF precision positioning
methodology using two
-
axis Hall
-
effect
sensors
,”

American Control Conference, 2005.
Proceedings of the 2005

8
-
10 June 2005
Page(s):3042
-

3047 vol. 5

[9]
Seok
-
Myeong
L
ang
,
Sung
-
Ho
Lee "
Design and
Analysis of Three Types for Permanent
Magnet Linear Synchronous Machine
,”


IEEE Transactions
on Magnetics,
Vo1.38,
No.5, September
2002

[10]
Pope, David. “
Halbach Arrays Enter the
Maglev Race
,” December 1998 V4, N4
<
http://www.aip.org/tip/INPHFA/vol
-
4/iss
-
4/p12.pdf
>.


5

[11]
I.
-
K. Kim, R. Kratz, and D. Doll, “
General
Atomics
U
rban
M
aglev
T
echnology
Development
,” in 17th International

Conference on Magnetically Levi
tated
Systems and Linear Drives (MAGLEV

2002),
(Lausanne,

Switzerland), Sep.

3
-
5, 2002.

[12]Van Fleck, J.H. “
Quantum Mechanics,

T
he
Key to Understanding Magnetism
,” Nobel
Lecture, 8 December 1977. Harvard
University, Cambridge, Massachusetts, USA.