Vector Potential and Stored Energy of a Quadrupole Magnet Array
advertisement
Proceedings of the 1999 Particle Accelerator Conference, New York, 1999
VECTOR POTENTIAL AND STORED ENERGY OF
A QUADRUPOLE MAGNET ARRAY
S. Caspi, LBNL, Berkeley, CA
Abstract
The quad current density J (A/m) (a form that satisfies the
conservation condition
), is :
!
$
The vector potential, magnetic field and stored energy of a
quadrupole magnet array are derived. Each magnet within
the array is a current sheet with a current density proportional to the azimuthal angle 2 and the longitudinal periodicity
. Individual quadrupoles within the array
are oriented in a way that maximizes the field gradient.
The array does not have to be of equal spacing and can be
of a finite size, however when the array is equally spaced
and is of infinite size the solution can be simplified. We
note that whereas, in a single quadrupole magnet with a
current density proportional to cos2 the gradient is pure,
such purity is not preserved in a quadrupole array.
#
6
34
E
F
7
)
A
9
B
9
:
<
>
?
<
>
?
#
'
)
*
,
-
.
%
0
/
5
G
1
.
)
A
9
C
D
C
D
B
<
<
7
M
0
1
H
I
0
L
*
O
P
Q
%
1
K
7
S
T
U
W
Y
Z
T
U
W
\
]
^
`
Z
b
[
c
G2,m is gradient at z=0 and L denotes the half period.
Quadrupoles are combined into an array with a center to
center spacing of 2S and alternating current direction that
maximizes the gradient (Fig. 1 and 2).
1 INTRODUCTION
g
It has been proposed that commercial electricity can be
generated economically from ion beam-driven fusion of
deuterium and tritium in tiny target pellets [1]. A leading driver candidate is a high energy, high current heavy
ion accelerator. To achieve high currents it is generally
desirable to accelerate multiple beams in parallel through
a low impedance accelerating structure; a long pulse induction linac can be designed to do this. Efficient transport of beam current in the multibeam accelerator would
be accomplished with multiple channel superconducting
quadrupole magnets operating in a DC mode with warm
bore[2].
d
f
coil i,j
e
2s
first octant
beam
TYPE II
The vector potential and the magnetic field have been
derived for an array of quadrupole magnets with a thin
cos(2 ) current sheet at a radius r=R. [3][4]. The field
strength within each coil varies purely as a Fourier sinusoidal series of the longitudinal coordinate z in proportion
to m z, where
, L denotes the half-period,
and m is an integer associated with the longitudal harmonic. The analysis is based on the expansion of the
vector potential in the region external to the windings of
a single quad, and the use of the “Addition Theorem” to
revise the expansion to one around any arbitrary point in
space.
self coil i=j=0
Figure 1: Cross section showing
current density arrangement
Email: s_caspi@lbl.gov
This work was supported by the Director, Office of Energy Research,
Office of High Energy and Nuclear Physics, High Energy Physics
Division, U. S. Department of Energy, under Contract No. DEAC03–76SF00098.
0-7803-5573-3/99/$10.00@1999 IEEE.
Figure 2: View of a 3x3 quadrupole array. The
windings (of constant current) correspond to three terms
m=1,2,3 which provide axial free space between arrays.
Based on such a current distribution the resulting vectorpotential
and magnetic field
within the bore R of
2793
i
j
h
h
Proceedings of the 1999 Particle Accelerator Conference, New York, 1999
k
k
l
Á
each quad, are:
The format used here for and was specifically chosen
to avoid a singularity that may rise when L is large (e.g.
when the 3d problem reduces to 2d).
x
t
t
x
u
v
w
v
w
y
z
{
z
|
}
~
¬
­
r
k
~
l
m
n
2 ANALYSIS
oq
x
x
§
u
ª
«
q
w
u
¨
©
q
q
u
¡
¢£¤
¥
¦
x
®
u
ª
w
¨
Consider
a
quadrupole
with its
center
at
as shown in Fig.
3.
The expansion of the vector potential in the region
outside the current sheet ( >R) and around that center is,
³
x
§
u
q
u
ª
q
³
§
q
´
§
·
x
t
t
x
u
v
w
v
w
y
z
{
z
|
}
~
¬
­
r
k
¯
n
¹
Å
¹
Å
¡
Æ
~
l
´
·
¥
qs
oq
x
x
§
u
ª
«
q
w
u
©
¨
±
q
§
q
»
u
¹
Ã
¹
u
¢£
¡
¤
¥
x
»
k
u
u
®
ª
u
È
¡
w
¨
²
l
Ç
n
Â
¸
x
§
u
q
u
ª
É
q
u
v
w
q
¥
¦
»
»
u
qs
u
w
u
w
u
Ê
x
t
¨
¨
Æ
Æ
t
µ
µ
±
r
k
§
l
±
n
u
¹
»
¹
Ã
x
u
v
w
v
w
y
{
z
z
|
}
~
u
u
»
¡
k
l
Ë
n
È
²
Â
~
o
x
x
¸
u
§
u
ª
w
u
¡
¨
É
u
v
w
s
¥
u
¦
»
u
w
u
»
w
«
u
Ê
¨
¨
Æ
Æ
µ
µ
k
²
l
±
n
±
§
u
¹
»
u
»
u
u
¹
¡
Ã
³
È
³
¨
Æ
µ
¬
u
v
w
Â
§
x
´
§
´
u
·
w
¸
³
¤
¹
«
The relations between the components of the vector
around
and the above components
are,
³
µ
®
¢
¯
l
m
l
l
±
§
x
´
§
´
ª
Å
u
n
²
³
·
²
x
¸
¹
§
Å
l
l
Å
Ë
Å
¡
l
Ç
u
±
µ
Å
x
´
v
w
v
w
¥
Å
¦
u
»
u
»
º
¨
Ë
l
m
n
l
Ç
l
¸
¾
¿
³
³
¼
À
«
È
È
¬
l
¯
n
l
l
Ë
Ç
§
x
´
§
´
u
·
w
¸
¹
³
«
È
È
³
¤
µ
®
l
±
n
l
±
¢
x
§
u
n
²
³
´
§
´
·
²
x
¸
¹
coil
§
u
µ
x
´
i,j
w
v
w
v
w
¥
u
u
»
»
º
¨
©
β
¸
¾
¿
¼
À
Where In and Kn are the “modified” Bessel functions
of the first and second kind of order n, and the prime
denotes differentiation with respect to the argument. The
summation i,j is carried over the quads in the first octant
of the array.
The magnetic field components are,
x
²
²
±
u
¹
Ã
¹
¸
¾
¼
m
n
Á
u
»
¿
À
x
u
v
w
Â
v
w
ψ
Ñ
Ñ
Î
r θ
φ
Ð
Si,j
Ñ
θ0
Í
Ó
Ô
Õ
Ö
×
Ò
x
x
§
u
ª
u
¡
w
u
©
¨
x
¥
²
±
²
§
u
¹
¹
Ã
¸
¼
u
»
¿
À
x
¯
n
Á
u
v
w
Â
v
w
x
¾
R
Figure 3: The geometry associated
with the Addition Theorem[5].
If we wish to consider an ininite size array where the eight
fold symmetry exists for each and every quad located at
,
, we shall add the contributions of quads with
their centers at
and consider farther summations to be within the first
octant only (quads on the symmetry line will require a
weight factor of 1/2).
Once the vector potential has been derived, the field components within the bore R can then be calculated from,
Ò
§
sφ)
Si,jr co
2
+ r -2
ρ =√(Si,j
2
Ï
Ó
×
Ô
Ø
Ù
Õ
Ö
Ú
Û
Ý
Ô
Þ
Ù
ß
à
Ù
Õ
Ö
á
Û
Ó
Ò
Ó
×
Ô
Ø
×
â
Ù
Õ
Ö
Ú
u
ª
w
¨
x
§
u
¡
ã
u
ã
ê
ä
u
å
ç
è
ª
x
¥
²
±
²
§
u
¹
¹
¸
Ã
¾
¼
Finally the arithmatic is check assuring the divergence of
the vector potential and field are zero
n
Á
±
u
»
¿
À
x
u
v
w
Â
v
w
x
x
§
u
u
ª
w
u
¡
¨
ã
ã
ê
ç
å
ë
ç
¥
Û
2794
ä
å
ë
Proceedings of the 1999 Particle Accelerator Conference, New York, 1999
3 STORED ENERGY
and the current density components are,
The stored energy can be calculated by integrating the
product of current density and vector potential
:
ï
í
ð
ò
ó
ô
õ
î
ì
ï
í
B
ò
Z
W
C
ì
ï
ö
ö
÷
í
ø
ø
N
U
û
L
M
(
O
D
F
B
û
ö
õ
í
í
C
$
>
>
?
@
>
?
@
ST
T
ú
û
T
T
ì
T
T[
M
D
F
D
F
B
ø
û
C
\
>
õ
ï
ú
í
ù
û
ü
ü
ý
ü
þ
ÿ
õ
ú
ï
î
í
ì
û
ì
ü
ì
þ
þ
>
î
ü
T
T
T
T
TV
T
ì
(the current density is per unit length and the unit of energy
is ).
Applying the orthogonality relations, the stored energy in
a single quad is,
í
(
)
(
+
í
ø
$
&
$
&
!
"
"
õ
ù
ý
,
(
÷
ó
-
.
$
&
ú
$
(
0
$
&
(
1
&
!
"
where the second term in the square bracket corresponds
to the contributions that arises from all neigboring coils
in the array.
4 SIMULATION OF CURRENT
DENSITY AND FLOW LINES
Figure 4: View of flow lines over a half period
quad (M=1,2,3). These special cases reveal
the reduction in crowding between magnets at
the expense of an increased non-linear field.
To generate flow lines we make use of a technique first
demonstrated by J. Laslett and W. Fawley of this laboratory. The character of the flow lines (Figure 4) for
a quadrupole magnet n=2 with a current density
õ
í
ô
ì
3
ø
í
B
ö
û
÷
D
F
û
D
B
ø
G
F
C
C
&
5
7
8
&
:
ó
<
>
?
@
>
?
@
>
>
ô
H
ö
5 CONCLUSION
õ
will follow by integrating the differential equation,
, so that
I
H
ø
I
J
K
The 3D expressions for the vector potential, field and
energy have been derived. We note that neighboring coils
within the array give rise to harmonic terms (m) which do
not exist in a single quad with cos(2 ) current density. We
also point out that the coefficients associated with
in the vector potential drop out in the expressions for the
field and energy.
3
ø
í
ó
õ
û
û
>
?
>
@
?
@
3
ø
í
D
F
]
&
>
ó
`
^
a
b
_
where
denotes the value of at z=0.
In a special case, we may choose special values for J02,m
such that,
û
(
P
O
[1]
[2]
$
ú
ú
õ
í
í
N
L
M
O
÷
6 REFERENCES
(
P
Q
O
(
P
Q
û
$
$
ú
where M corresponds to the number of m terms and J02
is a constant.
With that, the flow lines reduce to the simple expression,
[3]
[4]
õ
ú
û
û
ø
>
?
>
@
?
@
D
M
F
7
>
<
[5]
2795
Entire issue, “Fusion technology”, Vol 13, No. 2, 1998.
K. Chow A. Faltens S. Gourlay R. Hinkins R. Gupta E. Lee
A. McInturff R. Scanlan C. Taylor S. Caspi, R. Bangerter and
D. Wolgast, “A superconducting quadrupole magnet array for a
heavy ion fusion driver”, to be published in IEEE Trans. on Applied
Superconductivity, October 1998.
S. Caspi, “The 3d vector potential, magnetic field and stored energy
in a thin cos(2theta) coil array”, Lawrence Berkeley Laboratory,
LBNL-40564, SC-MAG-598, July 1997.
S. Caspi, “Multipoles in cos(2theta) coil arrays - type ii”, Lawrence
Berkeley Laboratory, SC-MAG-596, June 1997.
G.N. Eatson, “Theory of bessel functions”, Cambridge University
Press, page 361.
