ACCELERATOR MAGNETS Bending Field (Dipole) and Magnetic

ACCELERATOR MAGNETS Bending Field (Dipole) and Magnetic

Transverse Motion and the Normalized Gradient
ACCELERATOR MAGNETS
B 0
r
ϕ
Stephan Russenschuck
CERN, AT-MEL-EM
1211 Geneva 23, Switzerland
O rb it
ρ
y
v
P a rtic le
R
z
s
x
Displaced particle at ρ = R + x with field Bz = B0 +
1
dx
1 d
p0
+ ( 2 − k)x = 0
p0 ds
ds
R
For x R, e/p0 = −1/(B0R) and k = − B1R
0
Bending Field (Dipole) and Magnetic Rigidity
B
∂ Bz ∂ x x=0 x
∂ Bz ∂ x x=0
Quadrupole
vz × B
3.3356p[GeV/c] = R[m] B0[T]
Remark 1: Consider the “filling factor” in accelerators between 0.6 and 0.7.
B
x
(vz × B)
y
(vz × B)
LHC in the LEP Tunnel (Virtual Reality)
Higher Order Multipole Fields: Sextupole
x
(vz × B)
B
y
(vz × B)
Remark 2: Magnet systems are the most costly item in an accelerator
The 26 GeV Proton Synchrotron (PS) in its Tunnel
Multipole Field Errors and Beam Parameters
b1 , a1
b2
a2
b3
a3
b4
a4
b5
a5
b7
b6 , a6 , a7 , >
Closed orbit distortions
β -beating
Vertical dispersion, linear coupling
Off-momentum β beating
Chromatic coupling inducing detuning
Dynamic aperture and detuning
Dynamic aperture at injection
Dynamic aperture and detuning
Off-momentum dynamic aperture
Dynamic aperture at injection
Dynamic aperture at injection
Magnet Metamorphosis
High Energy Ring Accelerator (HERA) at DESY
(H-Magnet)
|Btot| (T)
2.65
2.50
2.35
2.21
2.06
1.91
1.77
1.62
1.47
1.32
1.18
1.03
0.88
0.74
0.59
0.44
0.29
0.15
0.10
-
2.8
2.65
2.50
2.35
2.21
2.06
1.91
1.77
1.62
1.47
1.32
1.18
1.03
0.88
0.74
0.59
0.44
0.29
0.15
N ⋅ I = 24000 A
B1 = 0.3 T
Bs = 0.065 T
Fill.fac. 0.98
Magnet Metamorphosis
Magnet Metamorphosis
(Cryogenic or Super-Ferric Magnet)
(C- Core, LEP Dipole)
|Btot| (T)
|Btot| (T)
2.65
2.50
2.36
2.21
2.06
1.92
1.77
1.62
1.47
1.33
1.18
1.03
0.88
0.73
0.59
0.44
0.29
0.15
0.00
-
2.8
2.65
2.50
2.36
2.21
2.06
1.92
1.77
1.62
1.47
1.33
1.18
1.03
0.88
0.73
0.59
0.44
0.29
0.15
N ⋅ I = 4480 A
2.65
2.50
2.36
2.21
2.06
1.91
1.77
1.62
1.47
1.33
1.18
1.03
0.88
0.74
0.58
0.44
0.29
0.14
0.37
B1 = 0.13 T
Bs = 0.042 T
Fill.fac. 0.27
-
2.8
2.65
2.50
2.36
2.21
2.06
1.91
1.77
1.62
1.47
1.33
1.18
1.03
0.88
0.74
0.58
0.44
0.29
0.14
N ⋅ I = 96000 A
B1 = 1.18 T
Bs = 0.26 T
Magnet Metamorphosis
Magnet Metamorphosis
(Window Frame Dipole)
(LHC Coil-Test Facility
|Btot| (T)
|Btot| (T)
2.652 -
2.8
2.505 -
2.652
2.357 -
2.505
2.210 -
2.357
2.063 -
2.210
1.915 -
2.063
1.768 -
1.915
1.621 -
1.768
1.473 -
1.621
1.326 -
1.473
1.178 -
1.326
1.031 -
1.178
0.884 -
1.031
0.736 -
0.884
0.589 -
0.736
0.442 -
0.589
0.294 -
0.442
0.147 -
0.294
0.
0.147
-
2.65
2.50
2.36
2.21
2.06
1.91
1.77
1.62
1.47
1.32
1.18
1.03
0.88
0.74
0.59
0.44
0.29
0.15
0.00
N ⋅ I = 360000 A
B1 = 2.08 T
-
2.8
2.65
2.50
2.36
2.21
2.06
1.91
1.77
1.62
1.47
1.32
1.18
1.03
0.88
0.74
0.59
0.44
0.29
0.15
N ⋅ I = 960000 A
Bs = 1.04 T
B1 = 8.33 T
Bs = 7.77 T
Magnet Metamorphosis
Magnet Metamorphosis
(Tevatron Dipole)
(LHC Main Dipole)
|Btot| (T)
|Btot| (T)
2.652 -
2.8
2.652 -
2.8
2.505 -
2.652
2.505 -
2.652
2.357 -
2.505
2.357 -
2.505
2.210 -
2.357
2.210 -
2.357
2.063 -
2.210
2.063 -
2.210
1.915 -
2.063
1.915 -
2.063
1.768 -
1.915
1.768 -
1.915
1.621 -
1.768
1.621 -
1.768
1.473 -
1.621
1.473 -
1.621
1.326 -
1.473
1.326 -
1.473
1.178 -
1.326
1.178 -
1.326
1.031 -
1.178
1.031 -
1.178
0.884 -
1.031
0.884 -
1.031
0.736 -
0.884
0.736 -
0.884
0.589 -
0.736
0.589 -
0.736
0.442 -
0.589
0.442 -
0.589
0.294 -
0.442
0.294 -
0.442
0.147 -
0.294
0.147 -
0.294
0.
0.147
0.
0.147
-
N ⋅ I = 471000 A
0
21
B1 = 4.16 T
42
63
84
105
126
147
168
189
Bs = 3.39 T
210
-
N ⋅ I = 2x944000 A
B1 = 8.32 T
Bs = 7.44 T
LHC Main Dipol Cross-Section
Maxwell’s Equations in Integral Form
∂D
) ⋅ dA
(J +
A ∂t
⋅ d s = − ∂
⋅ dA
E
B
∂
t
A
⋅ dA
= 0
B
A
⋅ dA
=
ρdV
D
⋅ d s =
H
A
V
Constitutive Equations
+ M)
= µ0(H
= µH
B
D = ε E = ε0(E + P)
+ Jimp.
J = κ E
1. Heat exchanger, 2. Bus bar, 3. Superconducting coil, 4. Vacuum Pipe und Beam-screen, 5. Cryostat,
6. Thermal shield (55- 75 K), 7. Helium-Tank, 8. Super-Insulation, 9. Collars, 10. Yoke
Overview
1D Field Calculation for a Conventional Dipole
Maxwell’s equations in integral form
One-dimensional field calculation for conventional magnets
Maxwell’s equations in differential form
The solution of Laplace’s equation
Harmonic fields
Complex Potentials
Ideal pole shapes of conventional magnets
Feed down
⋅ d s =
H
J ⋅ d A
A
Solution of Poisson’s equation
The field of line currents
Generation of pure multipole fields
Sensitivity to manufacturing errors
Numerical field computation
Saturation of the iron yoke
Superconducting filament magnetization
1
1
Biron siron + Bgap sgap = N I
µ0 µr
µ0
µ NI
µr 1
Bgap = 0
sgap
Hiron siron + Hgap sgap =
Warning 1: Check that the magnetic circuit contains no flux concentration
which increases the magnetic flux density above 1 T, as in this case
fringe fields can no longer be neglected.
1D Field Calculation for a Conventional Quadrupole
Field Quality in the Dipole Aperture
2
1
⋅ d s =
H
Bx = gy
1
⋅ d s +
H
1
By = gx
2
3
⋅ d s +
H
2
3
g
H=
µ0
⇒
g r0
g r02
=NI
Hdr =
rdr =
µ0 0
µ0 2
0
r
0
⋅ d s = N I
H
3
x2 + y2 =
⇒
LEP Dipole and Quadrupole
g=
g
r
µ0
2µ0NI
r02
Field Quality in the Dipole Aperture
B
y
|
|1 − B nom
y
A (Tm)
|Btot| (T)
0.039 -
0.041
1.969 -
2.078
0.037 -
0.039
1.859 -
1.969
0.034 -
0.037
1.750 -
1.859
0.032 -
0.034
1.641 -
1.750
0.030 -
0.032
1.531 -
1.641
0.028 -
0.030
1.422 -
1.531
0.025 -
0.028
1.313 -
1.422
0.023 -
0.025
1.203 -
1.313
0.021 -
0.023
1.094 -
1.203
0.018 -
0.021
0.984 -
1.094
0.016 -
0.018
0.875 -
0.984
0.014 -
0.016
0.766 -
0.875
0.012 -
0.014
0.656 -
0.766
0.009 -
0.012
0.547 -
0.656
0.007 -
0.009
0.438 -
0.547
0.005 -
0.007
0.328 -
0.438
0.003 -
0.005
0.219 -
0.328
0.864 -
0.003
0.110 -
0.219
-0.13 -
0.864
0.664 -
0.110
Definition of Field Quality in Accelerator Magnets
Maxwell’s Equations in Differential Form
Fourier-series expansion of the radial component of the magnetic flux density
on a reference radius inside the aperture
Br (r0 , ϕ ) =
∞
(Bn (r0 ) sin nϕ + An (r0) cos nϕ )
n=1
= BN (r0 )
∞
(bn (r0 ) sin nϕ + an (r0) cos nϕ )
n=1
An (r0) ≈
2P−1
1 Br (r0, ϕk ) cos nϕk ,
P
k =0
Bn (r0) ≈
2P−1
1 Br (r0, ϕk ) sin nϕk .
P
k =0
Remark 3: This definition is perfectly in line with magnetic field
measurements using “harmonic coils” where the periodic flux variation
in tangential rotating coils is analyzed with a FFT.
= J + ∂ D
curlH
∂t
= −∂ B
curlE
∂t
div B = 0
= ρ
div D
Constitutive Equations
+ M)
= µ (H
= µH
B
0
+ P)
= ε (E
= εE
D
0
+J
J = κ E
imp.
Rotating Coil Measurement Setup at BNL (Brookhaven)
Lemmata of Poincaré
The curl of an arbitrary vector field is source free div curlg = 0
An arbitrary gradient field is curl free curl gradφ = 0
(div B
= 0)
A source free field B,
in the form
can be expressed through a vector potential A
B = curlA.
(curlH
= 0)
A curl free field H,
= −gradΦm .
can be expressed through a scalar potential φ in the form H
Remark 4: The field in the aperture of accelerator magnets
is curl and source free
Warning 2: The Lemmata of Poincaré hold only for simply connected domains with
connected boundaries.
Maxwell’s “House”
d t
General Solution of the Laplace Equation
0
q
f
k
E
D
e
∞
c u r l
m
B
c u r l
n=1
J
A
- g r a d
∞
(En r n + Fn r −n )(Gn sin nϕ + Hn cos nϕ )
Az (r , ϕ ) =
d iv
1 ∂ Az n−1
=
nr (C n sin nϕ + Dn cos nϕ )
Br (r , ϕ ) =
r ∂ϕ
H
n=1
nr n−1Cn = Bn
- g r a d
d iv
∞
∂ Az
=−
Bϕ (r , ϕ ) = −
nr n−1(Dn sin nϕ − Cn cos nϕ )
∂r
f m
0
nr n−1Dn = An
n=1
-d t
Warning 3: Only for sub-domains with continuous material parameters.
Maxwell’s “Facade”
1
= J
curl curlA
µ
1
=0
curl curlA
µ0
− grad div A
=0
∇2 A
J
A
c u r l
m
H
d iv
- g r a d
f m
0
div µ gradΦm = 0
r2
µ0 div gradΦm = 0
∂ 2 Az
∂ Az ∂ 2 Az
+
+
r
=0
∂r2
∂r
∂ϕ 2
Br = C1 cos ϕ + D1 sin ϕ
Bϕ = −C1 sin ϕ + D1 cos ϕ
Bx = Br cos ϕ − Bϕ sin ϕ
By = Br sin ϕ + Bϕ cos ϕ
Bx = C 1
c u r l
B
Normal (Skew) Dipole (n=1)
∇ 2 Az = 0
∇2Φm = 0
By = D 1
Normal (Skew) Quadrupole (n=2)
Br
Bϕ
Bx
By
=
=
=
=
2C2r cos 2ϕ + 2D2r sin 2ϕ
−2C2r sin 2ϕ + 2D2r cos 2ϕ
2C2x + 2D2y
−2C2y + 2D2x
Analytical Field Computation for Superconducting Magnets
Special solution of the inhomogeneous (Poisson) differential equation
= J
∇2A
Fundamental solution of the Laplace Operator ∇2 (Green’s Function)
1
ln |r − r |
2π
1
1
G =
4π |r − r |
(2D)
G =
(3D)
Vector potential of a line current (2D)
µ I |r − r |
Az = − 0 ln(
)
2π
a
Remark 5: Notice that we have not yet addressed the problem
of how to create such field distributions.
Normal (Skew) Sextupole (n=3)
Br
Bϕ
Bx
By
=
=
=
=
The Field of Line Currents (2D)
3C3r 2 cos 3ϕ + 3D3r 2 sin 3ϕ
−3C3r 2 sin 3ϕ + 3D3r 2 cos 3ϕ
3C3(x 2 − y 2) + 6D3xy
−6C3xy + 3D3(x 2 − y 2)
y
r0 < ri
Az
R = |r − r |
r = (r0, ϕ )
ϕ
Iz
r = (ri , Θ)
Θ
z
x
Az = −
µ0I |r − r |
)
ln(
2π
a
∞
Br == −
Remark 6: The treatment of each harmonic separately is a mathematical abstraction.
In practical situations many harmonics will be present (to be minimized).
µ0I r0n−1
( n )(sin nϕ cos nΘ − cos nϕ sin nΘ)
2π
ri
n=1
Bn (r0 ) = −
µ0I r0n−1
cos nΘ
2π rin
An (r0 ) =
µ0I r0n−1
sin nΘ
2π rin
Ideal (cos n θ ) Current Distribution
The Imaging Method
Coil in non-saturated yoke
Imaged coil
Bn (r0 ) = −
ns i=1
ro
ri
2π
0
Dipole
0
20
40
60
80
100
120
140
160
0
20
40
60
80
100
120
140
µ0J0 cos mΘ r0n−1
µr − 1 ri 2n
(
)
1
+
cos nΘi r dΘdr
2π
rin
µr + 1 RYoke
Quadrupole
160
Field Harmonics in SC Magnet
Ideal Current Distribution Approximated with Coil-Blocks
ns
µ0Ii r0n−1
µr − 1 ri 2n
Bn (r0) = −
(
1+
cos nΘi
)
2π rin
µr + 1 RYoke
i=1
Scaling laws
Bn (r1) = (r1/r0)n−1Bn (r0)
Influence of the iron yoke (non-saturated)
ri = 43.5 mm, RYoke = 89 mm: B1 - 19% , B5 - 0.07%
Remark 7: Analytical field optimization can be used for higher order harmonics, finite
element calculations (for the estimation of saturation effects in the iron yoke) only
needed for the lower order harmonics.
Dipole
Quadrupole
Coil Winding
B3 and B5 Contribution of Strand Current
µ0Ii r0n−1
µr − 1 ri 2n
(
Bn (r0) = −
)
1+
cos nΘi
2π rin
µr + 1 RYoke
B3 CONTR. (T)
B5 CONTR. (T)
(*E-3)
(*E-3)
0.892 -
0.998
0.317 -
0.354
0.786 -
0.892
0.279 -
0.317
0.680 -
0.786
0.242 -
0.279
0.574 -
0.680
0.204 -
0.242
0.468 -
0.574
0.166 -
0.204
0.362 -
0.468
0.129 -
0.166
0.257 -
0.362
0.091 -
0.129
0.151 -
0.257
0.054 -
0.091
0.045 -
0.151
0.016 -
0.054
-0.60 -
0.045
-0.20 -
0.016
-0.16 -
-0.60
-0.58 -
-0.20
-0.27 -
-0.16
-0.95 -
-0.58
-0.37 -
-0.27
-0.13 -
-0.95
-0.48 -
-0.37
-0.17 -
-0.13
-0.59 -
-0.48
-0.20 -
-0.17
-0.69 -
-0.59
-0.24 -
-0.20
-0.80 -
-0.69
-0.28 -
-0.24
-0.90 -
-0.80
-0.32 -
-0.28
-1.01 -
-0.90
-0.35 -
-0.32
Sensitivity to Coil Block Displacements
µ0Ii r0n−1
µr − 1 ri 2n
1+
cos nΘi
(
Bn (r0) = −
)
2π rin
µr + 1 RYoke
ri 2n
∂ Bn (r0)
µ0Ii nr0n−1
1+(
sin nΘi
=−
)
∂ Θi
2π rin
RYoke
ri 2n
∂ Bn (r0) µ0Ii nr0n−1
1−(
cos nΘi
=
)
∂ ri
2π rin+1
RYoke
Increase of the azimuthal coil size by 0.1 mm produces (in units of 10−4):
b1 = −14.
b3 = 1.2
b5 = 0.03
Specified tolerances on coils: ± 0.025 mm
Curing Press and Curing Mold
Feed down
Complex Potentials
∞
n=1
= −gradΦm = − ∂ Φm ex − ∂ Φm ey
H
∂x
∂y
∂
A
∂
Az
z
= curlAz =
ex −
ey
B
∂y
∂x
n−1 n−1
∞
z
z
cn
=
cn
= By + i Bx
r0
r0
n=1
z = z + ∆z
z = x + iy
k −n
∞
∆z
(k − 1)!
ck
cn =
(k − n)! (n − 1)! r0
k =n
Cauchy-Riemann DEQ
∂ Az
∂ Φm
= −µ0
∂y
∂x
∂ Az
∂ Φm
= µ0
∂x
∂y
W (z) = Az (x, y) + i µ0Φm (x, y)
−
c2
∂ Az
∂ Φm
dW
=−
− i µ0
= By (x, y) + iBx (x, y)
dz
∂x
∂x
Cartesian Field Components
By + iBx =
∞
n=1
∞
n=1
Ideal (Iron) Pole Shapes
+ 3 c4
∆z
r0
2
+…
• The feed down effect can be used to center the measurements coil
by minimization of the b10 which can only occur as feed down from b11.
• The dipole magnetic axis has to be well aligned with respect to the closed orbit: ± 0.1
mm for both ∆x and ∆y systematic and 0.5 mm for both σx and σy random (r.m.s.).
• Alignment tolerances of MCS and MCDO correctors w.r.t. MB: 0.3 mm radially
Mechanical Axis Measurement for Spool Piece Alignment
z = x + iy
z0
µ0I z0n−1
(Bn + iAn )( )n−1 = −
r0
2π
zn
= c2 + 2 c3
∆z
r0
• Measurement of magnetic axis in dipole by powering the coil as a quadrupole.
z = x + iy
W (z) = Az (x, y) + i µ0Φm (x, y)
|z0 | < |z|
Excitation Cycle of the LHC Dipoles
Saturation Effect in LHC Dipoles
1 2 0 0 0
I[A ]
8
1 0 0 0 0
B [T ]
8 0 0 0
6
R a m p -u p
MUEr
6 0 0 0
4
4 0 0 0
2
2 0 0 0
0
In je c tio n
1 0 0 0
0
2 0 0 0
3 0 0 0
t[s ]
4 0 0 0
5 0 0 0
0
Field Variation with Excitation
MUEr
3731. -
3938.
3731. -
3938.
3524. -
3731.
3524. -
3731.
3317. -
3524.
3317. -
3524.
3109. -
3317.
3109. -
3317.
2902. -
3109.
2902. -
3109.
2695. -
2902.
2695. -
2902.
2488. -
2695.
2488. -
2695.
2280. -
2488.
2280. -
2488.
2073. -
2280.
2073. -
2280.
1866. -
2073.
1866. -
2073.
1659. -
1866.
1659. -
1866.
1451. -
1659.
1451. -
1659.
1244. -
1451.
1244. -
1451.
1037. -
1244.
1037. -
1244.
830.0 -
1037.
830.0 -
1037.
622.7 -
830.0
622.7 -
830.0
415.5 -
622.7
415.5 -
622.7
208.2 -
415.5
208.2 -
415.5
1.002 -
208.2
1.002 -
208.2
Objectives for the ROXIE Development
Automatic generation of coil geometry
Field computation specially suited for magnet design (BEM-FEM)
No meshing of the coil
No artificial boundary conditions
Confine numerical errors to the iron magnetization
Higher-order quadrilateral finite-elements
Parametric mesh generator
6
B0
x 10
0.709
-4
4
b3
2
10b4
0.708
0
0.707
-2
b2
Mathematical optimization
-4
0.706
10b5
-6
CAD/CAM interfaces
0.705
0
2000
4000
6000
8000
10000
12000
I (A)
0
2000
4000
6000
8000
10000
12000
I (A)
Preparation and Measurement of Collared Coils at BNN, Germany
Field Computation with the BEM-FEM Coupling Method
Ω1
BIRON
R1
AΓ1
B
FEM-domains
C
C
C
C
C
C
IRON C
R2
C
y
Bi
BS,i
BS,i
C
Ω2
AΓ2
BS,i
Ω3
IS
C
C
BEM-domain
Calculation of Fringe Fields in a LHC Dipole Model
Field Quality Calculations for Collared Coils
y
A (Tm)
0.211 -
0.236
0.186 -
0.211
0.161 -
0.186
0.136 -
0.161
0.111 -
0.136
0.087 -
0.111
0.062 -
0.087
0.037 -
0.062
0.012 -
0.037
-0.12 -
0.012
-0.37 -
-0.12
-0.62 -
-0.37
-0.87 -
-0.62
-0.11 -
-0.87
-0.13 -
-0.11
-0.16 -
-0.13
-0.18 -
-0.16
-0.21 -
-0.18
-0.23 -
-0.21
z
x
b2 = −0.239
b3 = 2.742
b4 = −0.012
b5 = −0.733
Source field
Reduced field
Total field
Field and Magnetization in the LHC Dipole Coil
|B| (T)
M (A/m)
(*E3)
8
2
6
1.5
8
|B|
|B|
T
6
T
T
1
4
4
|B|
0.5
2
2
1.390 -
1.466
-6.99 -
-5.93
1.314 -
1.390
-8.05 -
-6.99
1.238 -
1.314
-9.12 -
-8.05
1.162 -
1.238
-10.1 -
-9.12
1.086 -
1.162
-11.2 -
-10.1
1.010 -
1.086
-12.3 -
-11.2
0.934 -
1.010
-13.3 -
-12.3
0.859 -
0.934
-14.4 -
-13.3
0.783 -
0.859
-15.5 -
-14.4
0.707 -
0.783
-16.5 -
-15.5
0.631 -
0.707
-17.6 -
-16.5
0.555 -
0.631
-18.7 -
-17.6
0.479 -
0.555
-19.7 -
-18.7
0.403 -
0.479
-20.8 -
-19.7
0.328 -
0.403
-21.9 -
-20.8
0.252 -
0.328
-22.9 -
-21.9
0.176 -
0.252
-24.0 -
-22.9
0.100 -
0.176
-25.0 -
-24.0
0.024 -
0.100
-26.1 -
-25.0
Bz
Bx
0
Bz
Bz
Bx
Bx
0
0
-0.5
-2
-2
By
-1
-4
-4
-1.5
By
-6
By
-6
-2
-200
-160
-120
-80
-40
0
40
80
120
160
200
-200
-160
-120
mm
-80
-40
0
40
80
120
160
200
-200
-160
-120
-80
mm
-40
0
40
80
120
160
200
mm
Hysteresis in Superconductor Magnetization
Generated B3 Field Errors
Bn (r0 ) = −
0.06
Bn (r0 ) =
0.04
0.02
B3 Contr. of Istrand (T)
µ0Ii r0n−1
cos nΘ
2π rin
µ0 r0n−1
n(mr sin nθ + mθ cos nθ )
2π rin+1
B3 Contr. of Mstrand (T)
µ0 Μ [Τ]
(*E-3)
0.00
-0.02
strand measurement
-0.04
initial state curve
strand magnetization
filament magnetization
-0.06
-1.6
-1.2
-0.8
-0.4
0.0
0.4
0.8
1.2
1.6
Β [Τ]
(*E-3)
0.182 -
0.204
0.058 -
0.066
0.161 -
0.182
0.050 -
0.058
0.139 -
0.161
0.043 -
0.050
0.117 -
0.139
0.035 -
0.043
0.095 -
0.117
0.027 -
0.035
0.074 -
0.095
0.019 -
0.027
0.052 -
0.074
0.011 -
0.019
0.030 -
0.052
0.003 -
0.011
0.009 -
0.030
-0.43 -
0.003
-0.12 -
0.009
-0.12 -
-0.43
-0.34 -
-0.12
-0.20 -
-0.12
-0.55 -
-0.34
-0.28 -
-0.20
-0.77 -
-0.55
-0.35 -
-0.28
-0.99 -
-0.77
-0.43 -
-0.35
-0.12 -
-0.99
-0.51 -
-0.43
-0.14 -
-0.12
-0.59 -
-0.51
-0.16 -
-0.14
-0.67 -
-0.59
-0.18 -
-0.16
-0.75 -
-0.67
-0.20 -
-0.18
-0.83 -
-0.75
Macro-Photography of Cable and Strand
Scaling Laws Hold for Integrated Multipoles
∇2Φm(x, y, z) =
∂ 2Φm(x, y, z) ∂ 2Φm(x, y, z) ∂ 2Φm(x, y, z)
+
+
=0
∂x2
∂y2
∂ z2
z0
Φm(x, y) =
Φm (x, y, z)dz
−z0
∇2Φm(x, y) =
∂ 2Φm(x, y) ∂ 2Φm(x, y)
+
=0
∂x2
∂y2
Sufficient condition: Integration path is extended far enough away from (into) the
magnet so that the axial component of the field has dropped to zero.
Inner (outer) layer LHC dipole coil:
Filament diameter 7 (6) µ m, Strand diameter 1.065 (0.825) mm
∂ 2Φm ∂ 2Φm
+
=
∂x2
∂y2
z0
−z0
z0 2 ∂ 2Φm ∂ 2Φm
∂ Φm
∂ Φm z0
dz =
−
dz = −
+
= Hz |−z0 − Hz |z0
∂x2
∂y2
∂ z2
∂ z −z0
−z0
Warning 4:
Scaling Laws for Multipoles are Wrong in 3 Dimensions
r1
Bn (r1) = ( )n−1Bn (r0 )
r0
r1
bn (r1) = ( )n−N bn (r0)
r0
References
[1] M. Aleksa, S. Russenschuck and C. Völlinger Magnetic Field Calculations Including the Impact of Persistent Currents in
Superconducting Filaments IEEE Trans. on Magn., vol. 38, no. 2, 2002.
[2] Bossavit, A.: Computational Electromagnetism, Academic Press, 1998
[3] Beth, R. A.: An Integral Formula for Two-dimensional Fields, Journal of Applied Physics, Vol. 38, Nr. 12, 1967
[4] Binns, K.J., Lawrenson, P.J., Trowbridge, C.W.: The analytical and numerical solution of electric and magnetic fields,
John Wiley & Sons, 1992
80
[5] Bryant P.J.: Basic theory of magnetic measurements, CAS, CERN Accelerator School on Magnetic Measurement and
Alignment, CERN 92-05, Geneva, 1992
b3 (scaled, wrong)
60
b3
40
[6] CAS, CERN Accelerator School on Superconductivity in Particle Accelerators, Haus Rissen, Hamburg, Germany, 1995,
Proceedings, CERN-96-03
20
[7] Kurz, S., Russenschuck, S.: The application of the BEM-FEM coupling method for the accurate calculation of fields in
superconducting magnets, Electrical Engineering, 1999
0
-20
[8] Mess, K.H., Schmüser, P., Wolff S.: Superconducting Accelerator Magnets, World Scientific, 1996
-40
[9] M. Pekeler et al., Coupled Persistent-Current Effects in the Hera Dipoles and Beam Pipe Correction Coils, Desy Report
no. 92-06, Hamburg, 1992
-60
[10] Russenschuck, S.: ROXIE: Routine for the Optimization of magnet X-sections, Inverse field calculation and coil End
design, Proceedings of the First International ROXIE users meeting and workshop, CERN 99-01, ISBN 92-9083-140-5.
-80
0
20
40
60
80
100
120
140
160
180
200
[11] Wilson, M.N.: Superconducting Magnets, Oxford Science Publications, 1983