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Mauricio Lopes – FNAL
Stored Energy, Magnetic Forces and Dynamic Effects
US Particle Accelerator School – Grand Rapids, MI – June 2012
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Outlook
US Particle Accelerator School – Grand Rapids, MI – June 2012 2
• Magnetic Forces
• Stored Energy and Inductance
• Fringe Fields
• End Chamfering
• Eddy Currents
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Magnetic Forces
US Particle Accelerator School – Grand Rapids, MI – June 2012 3
• We use the figure to illustrate a simple example.
w a
x
B0
j = uniform current density
F
F
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Magnetic Force on a Conductor
US Particle Accelerator School – Grand Rapids, MI – June 2012 4
We use the expression, dvBjF Integrated over the volume of the conductor.
hLdxdvw
xBB
w
B
wh
NIj=
hBNI
0
0
0
0
0
L is the length into the paper.
0
2
02
0
2
2
0
00
2
2
0
0
0
0
22
hLBw
w
hLBxdx
w
hLB
hLdxw
xB
w
BB dvjF
w
0
2
0
2
B
hL
F
Area
Forcepressurerepulsive
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Magnetic Force on a Pole
US Particle Accelerator School – Grand Rapids, MI – June 2012 5
We use the expression, dvHF
h
Bρ 0where the magnetic charge density is given by,
The force is in the same direction as the H vector and is attractive.
dv=aLdyh
yBH
0
0
0
2
0
2
0
2
2
0
00
2
2
0
0
00
2=
2==
aLBh
h
aLBydy
h
aLBaLdy
h
yB
h
BdvHF
h
0
2
0
2
B
aL
F
Area
Forcepressureattractive
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Pressure
US Particle Accelerator School – Grand Rapids, MI – June 2012 6
The magnitudes of the repulsive pressure for the current and the attractive pressure at the pole are identical. In general, the pressures parallel to the field lines are attractive and the forces normal to the field lines are repulsive. The pressure is proportional to the flux density squared.
0
2
0
2
Bpressure
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Example
US Particle Accelerator School – Grand Rapids, MI – June 2012 7
Let us perform a calculation for a flux density of 5 kG = 0.5T.
27
2
0
2
0 472,991042
5.0
2 m
NewtonsBpressure
atmosphere 143.14472,99 220.5T @
in
lb
m
Newtonpressure
f
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Magnet Stored Energy
US Particle Accelerator School – Grand Rapids, MI – June 2012 8
• The magnet stored energy is given by; HBdvU2
1
the volume integral of the product of H and B.
Consider a window frame dipole field (illustrated earlier) with uniform field in the space between the coil. If we ignore the field in the coil and in the iron,
volumemagneticB
U 2 0
2
0
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Magnet Inductance and Ramping Voltage
US Particle Accelerator School – Grand Rapids, MI – June 2012 9
Inductance is given by, 2
2
I
UL
tI
URI
t
I
I
URI
dt
dILRIV
222
In fast ramped magnets, the resistive term is small.
volumeB
tItI
UV
0
2
0
2
22
volume=ahL
N
hBI
0
0
t
NaLBahL
B
tN
hBV
0
0
2
0
0
0
1
Substituting,
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Units and Design Options
US Particle Accelerator School – Grand Rapids, MI – June 2012 10
The units are,
VoltVoltm
m
WebersTm
t
NaLBV
sec
sec
secsec
2
2
2
0
t
NaLBV
0
Given the field = B0, pole width = a, Magnet Length = L and ramp time, t, the only design option available for changing the voltage is the number of turns, N.
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Other Magnet Geometries
US Particle Accelerator School – Grand Rapids, MI – June 2012 11
Normally, the stored energy in other magnets (ie. H dipoles, quadrupoles and sextupoles) is not as easily computed. However, for more complex geometries, two dimensional magnetostatic codes will compute the stored energy per unit length of magnet.
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Effective Length
US Particle Accelerator School – Grand Rapids, MI – June 2012 12
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
-300 -250 -200 -150 -100 -50 0 50 100 150 200 250 300
Bn
(au
)
Z (mm)
Ideal
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
-300 -250 -200 -150 -100 -50 0 50 100 150 200 250 300
Bn
(au
)
Z (mm)
Ideal Real
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
-300 -250 -200 -150 -100 -50 0 50 100 150 200 250 300
Bn
(au
)
Z (mm)
Ideal
Real
Effective length
𝐿𝑒𝑓𝑓 = 𝐵𝑛(𝑧)+∞
−∞𝑑𝑧
𝐵𝑛(𝑧 = 0)
z
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Fringe Fields and Effective Lengths
• Often, canonical rules of thumb are adopted in order to estimate the effective length of magnets.
– Dipole fringe field length = 1 half gap at each end
– Quadrupole fringe field length = 1/2 pole radius at each end.
– Sextupole fringe field length = 1/3 pole radius at each end.
US Particle Accelerator School – Grand Rapids, MI – June 2012 13
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Three Dimensional Fringe Fields
The shape of the three dimensional fringe field contributes to the integrated multipole error of a magnet.
US Particle Accelerator School – Grand Rapids, MI – June 2012 14
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• Dipole Fringe Field
– Typically, the fringe field is longer at the center of the magnet and drops off near the edges.
– This distribution is approximately quadratic and the integrated multipole field looks like a sextupole field.
h
h = "canonical" fringe field length
edge of pole
Fringe field length is typically longer in the center for an "unchamfered" pole end.
z
B
z
x
US Particle Accelerator School – Grand Rapids, MI – June 2012 15
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Virtual Field Boundary
US Particle Accelerator School – Grand Rapids, MI – June 2012 16
beam
Magnet
virtual field boundary
tan1
Fx
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• The photograph shows a removable insert with a machined chamfer installed on the SPEAR3 prototype gradient magnet.
• The shape of the chamfer depth was determined empirically and was approximately parabolic. It was designed to reduce the integrated sextupole field.
• The chamfer shape was machined onto the end packs of subsequently manufactured production magnets.
US Particle Accelerator School – Grand Rapids, MI – June 2012 17
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Quadrupole 3-Dimensional Fringe Field
• The quadrupole “gap” is
largest at its center. The
gap decreases as the
distance from its center
(g1>g2>g3>g4>g5>g6).
Since the fringe field is
roughly proportional to the
magnet gap, it is longest
near the magnet pole center. g1g2
g3g4 g5 g6
x
y
US Particle Accelerator School – Grand Rapids, MI – June 2012 18
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Quadrupole Chamfer
• The quadrupole pole chamfer is a straight angled cut, which shortens the pole at its center. The angle is cut such that the pole is longer near its edge.
• Again, this cut was determined empirically by trial an error, minimizing the n=6 integrated multipole error.
US Particle Accelerator School – Grand Rapids, MI – June 2012 19
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-10
-8
-6
-4
-2
0
2
4
0 3 4 5 6
B6
B10
B14
end chamfer length (mm)
R = 25 mm
Bn/B
2. 1
04
Harmonics as function of the end chamfer length
US Particle Accelerator School – Grand Rapids, MI – June 2012 20
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Ideal end chamfer length as function of quadrupole length
US Particle Accelerator School – Grand Rapids, MI – June 2012 21
R=20mm
3.0
3.5
4.0
4.5
5.0
5.5
6.0
100 150 200 250 300 350 400 450 500 550 600
Quadrupole length (mm)
Ide
al e
nd
ch
am
fer le
ng
th
(mm
)
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Effective length as function of the end chamfer length
US Particle Accelerator School – Grand Rapids, MI – June 2012 22
Q530
563.0
563.5
564.0
564.5
565.0
565.5
566.0
566.5
567.0
0 1 2 3 4 5 6
Chamfer length (mm)
Eff
ecti
ve l
en
gth
(m
m)
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Multipoles as function of the coil excitation
US Particle Accelerator School – Grand Rapids, MI – June 2012 23
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Dynamic effects
US Particle Accelerator School – Grand Rapids, MI – June 2012 24
dt
BE
h
w
g t h
w
g tk B
x
y
jB .̂. tieBo
jBj ˆˆ
x
E
z
E zx
oz Bi
x
E
xBidxBiEz oo EJ
xBiEJ ozz
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Dynamic effects
US Particle Accelerator School – Grand Rapids, MI – June 2012 25
o
e
Feo
gBdl
Bdl
BI
dlHI
..
.
Be stands for the magnetic field due to the eddy currents
w
1
2
3 g tk
31
21
321
2 III
II
IIII
JdxdyI i
2
0
12
1wtBixdxBitI
w
kook
2
32
1k
w
tw
oo twtgBixdxBigI
k
22
2 gtwgw
g
tBiB kkoo
e
Be (x=0) (theoretical) 0.0129 (T)
Be (x=0) (FEMM) 0.0127 (T)
Bo(T) 0.145
frequency (Hz) 100
tk (mm) 3
g (mm) 25
w (mm) 28
(MS/m) 1.334
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Real-case example
US Particle Accelerator School – Grand Rapids, MI – June 2012 26
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Real-case example
US Particle Accelerator School – Grand Rapids, MI – June 2012 27
thickness = 1 mm
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0.00
-30 -20 -10 0 10 20 30x (mm)
By
(T
)
400 Hz
350 Hz
300 Hz
250 Hz
200 Hz
150 Hz
100 Hz
90 Hz
80 Hz
70 Hz
60 Hz
50 Hz
40 Hz
30 Hz
20 Hz
10 Hz
thickness = 2 mm
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0.00
-30 -20 -10 0 10 20 30x (mm)
By (
T)
400 Hz
350 Hz
300 Hz
250 Hz
200 Hz
150 Hz
100 Hz
90 Hz
80 Hz
70 Hz
60 Hz
50 Hz
40 Hz
30 Hz
20 Hz
10 Hz
thickness = 3 mm
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0.00
-30 -20 -10 0 10 20 30x (mm)
By (
T)
400 Hz
350 Hz
300 Hz
250 Hz
200 Hz
150 Hz
100 Hz
90 Hz
80 Hz
70 Hz
60 Hz
50 Hz
40 Hz
30 Hz
20 Hz
10 Hz
thickness = 4 mm
-0.070000
-0.060000
-0.050000
-0.040000
-0.030000
-0.020000
-0.010000
0.000000
-30 -20 -10 0 10 20 30x (mm)
By
(T
)
400 Hz
350 Hz
300 Hz
250 Hz
200 Hz
150 Hz
100 Hz
90 Hz
80 Hz
70 Hz
60 Hz
50 Hz
40 Hz
30 Hz
20 Hz
10 Hz
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x = 20 mm
0.00
0.20
0.40
0.60
0.80
1.00
0 100 200 300 400
f (Hz)
By(f
)/B
y(0
Hz)
4 mm
3 mm
2 mm
1 mm
US Particle Accelerator School – Grand Rapids, MI – June 2012 28
x = 0 mm
0.00
0.20
0.40
0.60
0.80
1.00
0 100 200 300 400
f (Hz)
By(f
)/B
y(0
Hz)
4 mm
3 mm
2 mm
1 mm
x = -20 mm
0.00
0.20
0.40
0.60
0.80
1.00
0 100 200 300 400
f (Hz)
By(f
)/B
y(0
Hz)
4 mm
3 mm
2 mm
1 mm
Real-case example
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Phase delay
US Particle Accelerator School – Grand Rapids, MI – June 2012 29
or
2
radians
kt
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Attenuation due to lamination on the material
US Particle Accelerator School – Grand Rapids, MI – June 2012 30
)2/tan(2
kdkdo
ik 2
= 2200.o
= 9.93 MS/m
Example
Bo
Bmin Bmean
d
2
minBBmean
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Real-case example summary
US Particle Accelerator School – Grand Rapids, MI – June 2012 31
Vacuum chamber thickness
2 mm 3 mm
Core lamination thickness
0.60 0.54
0.5 mm 0.67 0.40 0.36
1.0 mm 0.46 0.27 0.25
@400 Hz
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Induced sextupolar field due to eddy currents
US Particle Accelerator School – Grand Rapids, MI – June 2012 32
xBEJ zz
h
w
g t h
w
g t B
x
y
2xBtI ko
BhI
2)( xt
B
h
txB k
o
t
B
Bh
tFk k
o
122
dxxw
gxF
2/11
0
2
2
2 12
F=1 when g<<w, and F=2 for a circular chamber
)()()( tfBBBtB iei
The sextupole strength (or moment) is defined by
)cos(12
1)( ttf
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Example – Alba booster dipole
US Particle Accelerator School – Grand Rapids, MI – June 2012 33
11.4592 (m)
o 410-7 (H/m)
1.344 (MS/m)
Freq. 3 (Hz)
Be 0.029 (T)
Bi 0.873 (T)
20 mm
30 mm
17 mm
50 mm
VC1
VC2
F
VC1 1.5739
VC2 1.2141
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Summary
US Particle Accelerator School – Grand Rapids, MI – June 2012 34
This chapter showed a collection of “loose-ends” calculations:
• Magnetic Forces • Stored Energy and Inductance • Fringe Fields • End Chamfering • Eddy Currents
Those effects have an impact in the magnet design but also need to be taken into consideration into the magnet fabrication, power supply design, vacuum chamber design and beam optics.
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US Particle Accelerator School – Grand Rapids, MI – June 2012 35
Magnetic measurements