Magnets for High Intensity Proton Synchrotrons
J.-F. Ostiguy, V. Kashikhin and A. Makarov
Fermi National Accelerator Laboratory, Batavia, IL, USA1
Abstract. Recently, there has been considerable interest at Fermilab for the Proton Driver, a future high intensity
proton machine. Various scenarios are under consideration, including a superconducting linac. Each scenario
present some special challenges. We describe here the magnets proposed in a recent study, the Proton Driver
Study II, which assumes a conventional warm synchrotron, roughly of the size of the existing FNAL booster, but
capable of delivering 380 kW at 8 GeV.
INTRODUCTION
One of the principal considerations in designing a high
intensity proton synchrotron is to limit losses to prevent activation. Typically this translates into a requirement on the maximum fractional particle losses: on the
order of 10~4 to as low as 10~6. In order to provide sufficient dynamic stability, it is necessary to limit both space
charge induced tune shift and tune spread. This is accomplished in two ways: by keeping the machine circumference small and by spreading out the proton distribution
transversely. The former strategy implies rapid cycling,
the latter implies large aperture; both have an impact on
magnet design.
Aperture is a principal cost driver since magnet overall
size, fabrication, power consumption and power supply
hardware costs are directly proportional to stored magnetic energy. Magnet physical aperture, as opposed to
available beam aperture, is determined not only by space
charge considerations but also by the type of vacuum
pipe employed. Because of the very substantial eddy current induced losses associated with rapid cycling, a conducting beam pipe generally cannot be used. One possible solution is a ceramic beam tube with thin conducting
strips disposed on its inner surfaces to minimize beam
impedance. This is costly in terms of physical aperture
because for mechanical reasons, ceramic walls have to
be considerably thicker than metallic walls. The Proton
Driver magnets sidestep the difficulty by putting the entire magnet inside an evacuated enclosure an approach
employed for the Fermilab Booster magnets. One drawback is that due a larger desorption area, achieving satisfactory vacuum in order to prevent gas scattering induced
losses requires special care.
The presence of a large energy dependent space charge
tune shift and tune spread dictates the need for tight
tune control during the entire acceleration cycle. For this
1
Work supported by the U. S. Department of Energy under contract
number DE-AC02-76CH03000.
reason, dipole/quadrupole tracking errors are a special
concern. A tracking error is equivalent to momentum
offset error and results in a tune shift of magnitude
JA(G/B)1
^uncorrected I (G/B) \
where j is the ratio the gradient to main dipole field.
Note that the tune variation is proportional to the uncorrected chromaticity because, in the context of a focusing
error, there is no closed orbit error and the chromaticity correction sextupoles have no effect. The magnitude
of the tolerable tune shift is debatable. At ISIS (RAL,
U.K.), the ability to control the tune to within a part in
100 proved necessary, mostly to stay clear of specific
resonances at extraction. While it is conceivable that this
criterion can be relaxed, it provides a sense of what needs
to be achieved.
Some rapid cyclic machines (e.g. the Fermilab
Booster) use combined function magnets. This economizes space and naturally provides good tracking as
long as the operating field is kept below ~ 1 T. The
Fermilab Proton Driver study II assumes an aggressive
1.5T bending field and separate function magnets. The
field strength is determined by two requirements. First,
the circumference ratio between the Proton Driver and
the Main Injector should be a simple rational fraction
to allow synchronous beam transfers. Second, the total
circumference should be as small as possible in order
to minimize the space charge tune shift. Dipoles and
quadrupoles are on a common bus; the residual tracking
error (on the order of a percent) is handled by an independently powered active quadrupole correction system
capable of compensating for the tune shift dependence
on energy during acceleration.
DIPOLES
The Proton Driver dipole is a conventional H-magnet
design. Field homogeneity is preserved at high excitation
by profiled pole edges and by the presence of circular
CP642, High Intensity and High Brightness Hadron Beams: 20th ICFA Advanced Beam Dynamics Workshop on
High Intensity and High Brightness Hadron Beams, edited by W. Chou, Y. Mori, D. Neuffer, and J.-F. Ostiguy
© 2002 American Institute of Physics 0-7354-0097-0/02/$ 19.00
83
TABLE 1. Proton Driver II Main Dipole Magnet Parameters
PeakDipole Field
Good Field Aperture
Physical Aperture
Field Homogeneity
Magnet Length
Cycle Frequency
Peak Current
Conductor Dimensions
Conductor cooling hole diameter
No of turns/pole (3 conductors & top/bottom coils in parallel)
Lamination Thickness
Lamination Material
DC Resistance
Stored Energy
Coil Losses
Core Losses
Core mass
Peak Terminal Voltage
Water Pressure Drop
Water How
Water Temperature Rise
1.5
101.6 x 152.4
101.6x273.1
±0.0005
5.72
15
5170
20.2 x 15
10
12
0.35
Si-Fe M17
18
4.7
0.063
115
16.3
37,000
4.85
10
1.7
17
Eddy currents induced in conductors are potentially
more problematic. If the current distribution is nonuniform, resistive losses can be substantial; furthermore,
the field homogeneity can be affected. The latter problem
is largely eliminated by avoiding to place conductors into
or near to the magnet mid-plane.
For a rectangular conductor immersed in a uniform
time-varying magnetic field BQsm(cQt), it can be shown
that the power losses are given by
Tesla
2
Hz
A
mH
MJ
kW
kW
kg
kV
bar
= co2B2)Aa2/l6p
(3)
where A is the conductor area, a is the conductor width
and p is the resistivity. Clearly, eddy currents can be reduced either by reducing conductor cross-section or the
magnetic field in which the coils are immersed. Reducing
individual conductor area increases the number of turns
N and therefore the magnet terminal voltage, since the
inductance scales like N2. To keep the voltage at a reasonable level, it becomes necessary to connect multiple
turns in parallel. The Proton Driver II magnet assumes
groups of three conductors connected in parallel in each
coil and the top and bottom coils also connected in parallel. The electrical and mechanical connections in the end
regions are shown in detail in Figure 1. Note that the conductors are not transposed. Although transposition would
reduce losses slightly, it would also render connections in
the end region very complex and cumbersome.
In the Proton Driver Study I, eddy current power loss
problems were completely side-stepped by the use of a
special water cooled stranded conductor. Aside from the
technology required to produce reliable electrical and
mechanical joints, the main drawback of the stranded
conductor is its high cost ( about an order of magnitude
more expensive than conventional copper conductor).
For study II, the possibility of using conventional solid
water cooled conductors has been revisited. Figure 3
presents the result of an eddy current computation.
Eddy current losses reach a substantial level in the
conductors closest to the edge of the pole, in the fringe
field region. To take advantage of the fact that the magnetic field is predominantly vertical in that region, a
rectangular (as opposed to square) conductor is employed. Relatively high losses affect approximately 20%
of the total coil cross section; however, localized heating should be prevented by good thermal contact between conductors. Conventional water-cooled solid copper coil is a well-understood technology. Compared with
stranded conductors, the trade-off is reduced operational
efficiency vs reduced up-front fabrication costs.
holes in the center of the poles. The magnet cross-section
is shown in Figure 1; a list of relevant parameters is
presented in Table 1. The dipoles are excited so as to
produce a magnetic field strength of the form
B(t) = B0-Bl cos(cot + 0) + 0.12551 sm(2cot + 20)
(2)
where BQ is the injection field, Bl is the magnitude of the
fundamental component, co/2n = / = 15 Hz and 0 is a
constant phase factor. The second harmonic component
is introduced to reduce the maximum value of dB/dt,
which determines the peak RF accelerating voltage and
the overall RF system costs.
In an iron dominated dipole, good field homogeneity
can be achieved over the entire extent of the physical vertical aperture g, that is, all the way to the pole surfaces.
However, field homogeneity over the horizontal extent
of the aperture requires a certain amount of pole overhang w. For a given field homogeneity, the required horizontal extent of the physical aperture is minimized by
shimming the pole pieces edges. The optimality of a design can be accessed by comparing the achieved physical horizontal extent to a theoretical estimate developed
by Klaus Halbach [2]. Figure 2 presents calculated field
homogeneities achieved by the Proton Driver II dipole
magnetFor this magnet, the ratio g/w ~ 0.6 and we note
that the achieved homogeneity is in good agreement with
the prediction from Halbach's formula.
Eddy Currents
In a rapid cycling magnet, the presence of eddy currents is a source of technical difficulties. Eddy currents
are induced both in the magnetic core and in the conductors. In the core, they are largely suppressed by a laminated core construction which impedes their flow in the
longitudinal direction. As long as the lamination thickness is smaller than the skin depth, their is little impact
on field quality and principal effect is to increased losses.
Quadrupoles
The Proton Driver quadrupoles are four-fold symmetric magnets. Both horizontal and vertical focusing
84
-[247 rm,]
7
-[27'L]
I"
"I
[102°mm]
*1 [ — — 1 - 5 0 0 X
FIGURE 1. Proton Driver dipole cross-section and coil detail. Note the circular holes in the center of the poles. Note also the
FIGURE
1.1. Proton
dipole
cross-section
holes in
in the
the center
center of
of the
thepoles.
poles.Note
Notealso
alsothe
the
FIGURE
Proton
Driver
dipole
cross-section
and coil
coil detail.
detail. Note
Note the
the circular holes
parallel
connections
at Driver
one
of the
coils
extremities.and
parallel
parallelconnections
connectionsatatone
oneof
ofthe
the coils
coils extremities.
extremities.
file reaches approximately 1.5 T, and saturation begins
file reaches approximately
approximately 1.5
1.5 T, and
and saturation
saturation begins
begins
to affect the linearity of the relation between quadrupole
to affect
the
linearity
of
the
relation
between
quadrupole
affect
linearity of the relation between quadrupole
strength and excitation. Note that saturation in backleg
strength and excitation.
excitation. Note
Note that
that saturation
saturation in
in backleg
backleg
region also affect
nonlinearity; however,
however, this
this effect
effect can
can
affect nonlinearity;
nonlinearity;
region also affect
however, this
effect can
be
minimized
by
adjusting
the
backleg
width.
Note
that
be minimized by adjusting
adjusting the
the backleg
backleg width.
width. Note
Notethat
that
for
symmetricquadrupole
quadrupolesaturation
saturationdoes
doesnot
not
for aa four-fold
four-fold symmetric
symmetric
quadrupole
saturation
does
not
four-fold
adversely
impact
field
quality
since
all
harmonics
except
adversely impact field
field quality
quality since
since all
all harmonics
harmonicsexcept
except
those
4n (8n-pole)
(8n-pole) are
are suppressed:
suppressed:the
thefirst
firstalalthose of
4n
(8n-pole)
are
suppressed:
the
first
alof order
order 4n
lowed
harmonic
is
the
12-pole.
At
8
GeV,
the
deviation
lowed harmonic is the
the 12-pole.
12-pole. At
At 88 GeV,
GeV,the
thedeviation
deviation
is
order
of 2.0%.
2.0%.
the order
is on
on the
order of
of
2.0%.
UNITS
Length
:m
Flux densityUNITS
:T
-1
Field
strength : A
Length
: mm
-1
Potential
Flux density : Wb
:T m
-1 -1
Conductivity
Field strength: S
: Amm
-2
Source
density: A
m m-1
Potential
: Wb
-1
Conductivity : W
:Sm
Power
-2
Source density
:Am
Force
:N
Power
Energy
: J: W
Force
:N
Mass
: kg
Energy
:J
Mass
: kg
1.0E-04
1.0E-04
0.0
0.0
-1.0E-04
-1.0E-04
-2.0E-04
-2.0E-04
-2.0E-04
-3.0E-04
-3.0E-04
-3.0E-04
-4.0E-04
-4.0E-04
-5.0E-04
-5.0E-04
-5.0E-04
-6.0E-04
-6.0E-04
-6.0E-04
X coord
Y coord
X coord
Y coord
0.0 0.005
0.015
0.025
0.035
0.045
0.055
0.00.00.0
0.00.005
0.0
0.0
0.005 0.00.015
0.015 0.0
0.045 0.0
0.055
0.025
0.035
0.045
0.055
0.0 0.0
0.0
0.0
0.0
0.0
0.0
Homogeneity of BY w.r.t. value -1.51657145 at (0.0,0.0)
Homogeneity
w.r.t.
value-0.16300944
-1.51657145atat(0.0,0.0)
(0.0,0.0)
Homogeneity
ofof
BYBY
w.r.t.
value
0.065
0.0
0.065
0.065
0.0
0.075
0.0
0.075
0.0
Homogeneity of BY w.r.t. value -0.16300944 at (0.0,0.0)
PROBLEM DATA
dipfinal.tr
LinearPROBLEM
elements DATA
dipfinal.tr
XY
symmetry
Linearpotential
elements
Vector
XY symmetry
Magnetic
fields
Vector potential
Transient
solution
Magnetic
fields s
Time
= 0.0333333
Transient
solution
53214
elements
Time =nodes
0.0333333 s
26734
53214
elements
227
regions
26734 nodes
227 regions
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CONCLUSIONS
CONCLUSIONS
CONCLUSIONS
FIGURE
FIGURE2.2.2. Proton
ProtonDriver
Driverdipole
dipolefield
fi eldhomogeneity
homogeneity at
atminiminiFIGURE
Proton
Driver
dipole
field
homogeneity
at
minimum
mumand
andmaximum
maximumexcitations.
excitations.
mum
and
maximum
excitations.
Y [m]
Y [m]
224
188 190 193 196 199 202 205 224
208 211 214 217 220 223
188
189 190
192 193195 196198 199201 202204 205207208210211213214216217219220222223
189
191 192
194 195
197 198
200 201
203 204
206 207
209 210
212 213
215 216
218 219
221 222
191 194 197 200 203 206 209 212 215 218 221
154 157 160 163 166 169 172 175 178 181 184 187
153 154
156 157159 160162 163165 166168 169171172174175177178180181183184186187
152 153
155 156
158 159
161 162
164 165
167 168
170 171
173 174
176 177
179 180
182 183
185 186
152 155 158 161 164 167 170 173 176 179 182 185
118 121 124 127 130 133 136 139 142 145 148 151
117 118
120 121123 124126 127129 130132 133135136138139141142144145147148150151
116 117
119 120
122 123
125 126
128 129
131 132
134 135
137 138
140 141
143 144
146 147
149 150
116 119 122 125 128 131 134 137 140 143 146 149
0.17
0.17
0.16
0.16
0.15
0.15
0.14
0.14
0.13
0.13
0.12
0.12
0.11
0.11
81
84
87
90
93
96
99
102 105 108 111 114 2
99
80 81
82
83 84
85
86 87
88
89 90
91
92 93
94
95 96
97
98 100
101 102
103
104 105
106
107 108
109
110 111
112
113 114
115 2
0.1
80 82
83 85
86 88
89 91
92 94
95 97
98 100
101 103
104 106
107 109
110 112
113 115
0.1
46
49
52
55
58
61
64
67
70
73
76
79
0.09
0.09
225
45 4648 49 51 52 54 55 57 58 60 61 63 64 66 67 69 70 72 73 75 76 78 79
44 45
47 48
50 5153 5456 5759 6062 6365 6668 6971 7274 7577 78
0.08 225
44
47
50
53
56
59
62
65
68
71
74
77
0.08
8
10
11 13
14 16
17 19
20 22
23 25
26 28
29 31
32 34
35 37
38 40
41 43
0.07
8 9 10
1112 13
1415 16
1718 19
2021 22
2324 25
2627 28
2930 31
3233 34
3536 37
3839 40
4142 43
0.07
9
12
15
18
21
24
27
30
33
36
39
42
0.06
0.06
0.05
0.05
0.19
0.21
0.23
0.25
0.27
0.29
0.31
0.33
0.35
0.
0.19
0.21
0.23
0.25
0.27
0.29
0.31
0.33
0.35
X [m] 0.
Component: J
X [m]
Component: J
-2.3225E+07
2669852.0
2.85644E+07
-2.3225E+07
2669852.0
2.85644E+07
2
1
2
1
High
operation
is aaa serious
seriousconcern.
concern.The
Thedipole
dipole
voltage operation
High voltage
operation is
is
serious
concern.
The
dipole
magnets
have
a
maximum
terminal
voltage
on
the
order
magnets have a maximum
maximum terminal
terminal voltage
voltageon
onthe
theorder
order
of
5
kV;
in
the
proposed
resonant
configuration,
the
maxof 5 kV; in the proposed
proposed resonant
resonant configuration,
configuration,the
themaxmaximum
voltage
to
ground
reaches
approximately
3.3
kV.
imum voltage to ground
ground reaches
reaches approximately
approximately 3.3
3.3kV.
kV.
Another
source
of
concern
in
the
fact
that
the
entire
magAnother source of concern
concern in
in the
the fact
fact that
thatthe
theentire
entiremagmagnet
placed
inside an
an evacuated
evacuatedenclosure.
enclosure.Special
Special
net is
is be
be placed
placed inside
inside
an
evacuated
enclosure.
Special
precautions
be needed
needed during
duringthe
theassembly
assemblyprocess
process
precautions will
will be
be
needed
during
the
assembly
process
to
avoid
excessive
degassing.
excessive
degassing.
to avoid excessive degassing.
interesting
An
avenue for
forfuture
futureR&D
R&Dwould
wouldbe
bean
anininAn interesting
interesting avenue
avenue
for
future
R&D
would
be
an
inof
super-ferric
magnet
technology.
In
recent
vestigation
super-ferric
magnet
technology.
In
recent
vestigation of super-ferric magnet technology. In recent
years,
generation
of Nb-Ti
Nb-Tisuperconducting
superconductingcable
cable
years, aa new
new generation
generation of
of
Nb-Ti
superconducting
cable
has
developed
for applications
applicationsin
inpower
powergeneration
generation
has been
been developed
developed for
for
applications
in
power
generation
and transmission. The
insulatedfilaments
filamentswith
with
The cable
cable has
has insulated
insulated
filaments
with
diameter as small as
embeddedin
inaaaCu-Ni
Cu-Nialloy
alloy
a diameter
µ m embedded
embedded
in
Cu-Ni
alloy
as 0.1
0.1 jum
matrix. Superconducting
Superconducting coils
wouldresult
resultin
insubstantial
substantial
matrix.
coils would
would
result
in
substantial
savings
in
overall
magnet
size
and
power
consumption;
savings in overall magnet size and
and power
power consumption;
consumption;
these savings
savings may
these
may be
enoughto
tooffset
offsetthe
theadadbe substantial
substantial enough
enough
to
offset
the
additional
costs
and
complexity
engendered
by
the
ditional costs and complexity engendered
engendered by
by the
thecryocryocryogenic system.
system.
genic
UNITS
Length
:m
UNITS
Flux
density : T
Length
: m -1
Field
Fluxstrength
density : A
: Tm -1
Potential
m-1
Field strength: Wb
:Am
Conductivity
:S
m-1 m-1
Potential
: Wb
-2
Source
density: A
Conductivity
: Smm-1
Power
:W
Source density
: A m-2
Force
:N
Power
:W
Energy
: J: N
Force
Mass
: kg
Energy
:J
Mass
: kg
PROBLEM DATA
dipfinal.tr
PROBLEM DATA
Linear
elements
dipfinal.tr
XY
symmetry
Linear
elements
Vector
potential
XY symmetry
Magnetic
fields
Vector potential
Transient
Magneticsolution
fields
Time
= 5.0E-04
s
Transient
solution
Time =elements
5.0E-04 s
53214
53214nodes
elements
26734
26734
nodes
227
regions
227 regions
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FIGURE
Eddycurrent
currentdistribution
distributionin
inthe
theconductors.
conductors.Note
Note
FIGURE
3.3.3. Eddy
FIGURE
Eddy
current
distribution
in
the
conductors.
Note
thatthe
thedistribution
distributionisisismost
mostuneven
unevenin
inconductors
conductors located
located near
near
that
that
the
distribution
most
uneven
in
conductors
located
near
thepole
poleedge.
edge.
the
the
pole
edge.
quadrupolesare
areidentical
identicaland
andthe
theaperture
aperture radius
radius is
is set
set
quadrupoles
quadrupoles
are
identical
and
the
aperture
radius
is
set
to
accommodate
a
rectangular
good
field
region
of
the
totoaccommodate
accommodate aa rectangular
rectangular good
good field
field region
region of
of the
the
samesize
sizeas
asthe
thedipoles’.
dipoles'.AA
Acommon
common current
current bus
bus proprosame
same
size
as
the
dipoles’.
common
current
bus
providesdynamical
dynamicaltracking
trackingbetween
between the
the quadrupole
quadrupole gragravides
vides
dynamical
tracking
between
the
quadrupole
gradient
and
dipole
field.
The
number
of
turns
and
length
dient
dientand
anddipole
dipole field.
field. The
The number
number of
of turns
turns and
and length
length
thequadrupole
quadrupoleare
areselected
selected so
so as
as to
to match
match optical
optical
ofofofthe
the
quadrupole
are
selected
so
as
to
match
optical
andphysical
physicalrequirements
requirementsatatatinjection.
injection. At
At higher
higher enerenerand
and
physical
requirements
injection.
At
higher
energies,
the
gradient/dipole
strength
ratio
must
not
deviate
gies,
gies,the
thegradient/dipole
gradient/dipole strength
strength ratio
ratio must
must not
not deviate
deviate
bymore
morethan
thanaaapercent
percentor
ortwo;
two; this
this effectively
effectively limits
limits
by
by
more
than
percent
or
two;
this
effectively
limits
the
maximum
achievable
gradient.
For
the
required
aperthe
themaximum
maximumachievable
achievablegradient.
gradient.For
For the
the required
required aperaperture, when the quadrupole pole tip field reaches 0.84 T,
ture,when
whenthe
thequadrupole
quadrupole pole
pole tip
tip field
field reaches
reaches 0.84
0.84 T,
ture,
the field at the edges of the truncated hyperbolic prothe field
field atat the
the edges
edges of
of the
the truncated
truncated hyperbolic
hyperbolic proprothe
REFERENCES
REFERENCES
i.
1.
2.
2.
85
The Proton
Design Study,
The
Proton Driver
Study,Fermilab
FermilabTM-2136,
TM-2136,
Driver Design Study,
Fermilab
TM-2136,
December 2000.
December 2000.
A.W. Chao and M. Tigner Eds., Handbook of Accelerator
A.W. Chao and M. Tigner Eds.,
Eds., Handbook
Handbook of
of Accelerator
Accelerator
Physics and Engineering, World Scientifi c, 1999.
Physics and Engineering, World
World Scientific,
Scientific, 1999.
1999.