Download Intto to Design & Fab of Iron Dominated Magnets

INTRODUCTION to the DESIGN and
FABRICATION of IRONDOMINATED ACCELERATOR
MAGNETS
Cherrill Spencer, Magnet Engineer
SLAC National Accelerator Laboratory
Menlo Park, California, USA
Lecture # 1 of 2
Mexican Particle Accelerator School, October 2011
Overview of my Two Lectures, part 1
Lecture 1
• Purpose of my lectures on electromagnets
– And steps of producing accelerator magnets
• How Maxwell’s Equations help us design
magnets for particle accelerators
• The steps of designing accelerator magnets
• Computer modelling to make a detailed
design
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MePAS, Cherrill Spencer, Magnet Lecture #1
Guanajuato. 1st October 2011
Overview of my Two Lectures, part 2
Lecture 2
• Choice of materials and fabrication techniques
– Fabricating steel yoke
– Fabricating coils
• Assembling the whole magnet, connecting it to
power and cooling sources
• Testing & magnetically measuring a magnet
• Installing magnets in a beam line : alignment
• Resources where you can find out much more
about making accelerator magnets
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MePAS, Cherrill Spencer, Magnet Lecture #1
Guanajuato. 1st October 2011
Purpose of my lectures; what I cannot
do in the time available
Why am I smiling?
Give you an overview of topic, not enough time
to go into much detail. Can answer your
questions outside of the lecture time.
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Tell you about
designing and
fabricating
accelerator magnets.
What a magnet
engineer does in
order to produce
magnets.
MePAS, Cherrill Spencer, Magnet Lecture #1
Guanajuato. 1st October 2011
Steps in producing accelerator magnets
•
•
•
Accelerator physicist creates a beam lattice,
knows purpose of each magnet. Develops set of
requirements for each magnet: integrated field
strength, aperture size, approximate effective
length, how it operates. Gives requirements to a
magnet engineer (ME)
ME begins discussions with power supply (PS)
specialists and facilities personal re sizes of PS,
cooling water, size of tunnel
ME performs the design of each style of magnet,
on paper, with computer modeling [more details later]
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MePAS, Cherrill Spencer, Magnet Lecture #1
Guanajuato. 1st October 2011
More steps in producing accelerator
magnets
•
•
•
•
•
•
ME works with a mechanical designer to produce
details drawings of all the parts of the magnet
ME continues discussions with PS specialist and
facilities personal. Interfaces between magnet and rest
of world must be agreed upon: power, water, supports.
ME finds a magnet fabrication vendor; negotiates
on choice of materials, cost, schedule
ME monitors fabrication, monitors quality control
data, answers questions from vendor
ME supervises magnetic measurements -> field quality
ME discusses magnetic data with AP->can be installed
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MePAS, Cherrill Spencer, Magnet Lecture #1
Guanajuato. 1st October 2011
Review some electromagnetism facts
• Electromagnetism is concerned with the long range
interactions between electric charges
– Electric interactions occur between static OR moving
charges
– Magnetic interactions occur only between moving charges
• By definition a FIELD is the FORCE created by a
system of charges on one unit charge, q=1, having a
velocity vector v=1, parallel to the velocity vector of a
system of charges. Two fields derived from 2 kinds of force.
– electric force coming from the electric field, E
– magnetic force coming from the magnetic field, B, according to
F = q(E + v x B)
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MePAS, Cherrill Spencer, Magnet Lecture #1
Guanajuato. 1st October 2011
Electric currents create magnetic fields
Compass needle is affected by the bar magnet. It is
forced into position: there is a magnetic field produced
by the bar magnet. Can draw where lines of force are
depending on orientation of the compass needle.
Definition: lines of force, also called flux lines, or lines of
magnetic induction, go from North to South.
These lines have no beginning and no end; North poles
and South poles always exist in pairs.
MAXWELL’S 2nd EQUATION:
∙B = 0 is math representation of above facts
Electric current [positive ions moving] creates
a magnetic field around the wire in a plane
perpendicular to the wire: lines of force are B.
MAXWELL’S 4th EQUATION
x B = µ0 J is math representation of this
in free space with non-magnetic materials.
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MePAS, Cherrill Spencer, Magnet Lecture #1
Guanajuato. 1st October 2011
Units. Creating shapely magnetic fields.
B is also called the Magnetic Flux Density and its units are Weber/meter2
One Weber/meter2 = one TESLA. We will use Tesla.
J is a Current Density: amps/meter2
B in Tesla
Amps
J in
m2
 is the permeabili ty of air  1
 0 is the permeabili ty of vacuum  4  10
-7
Tm
Amp
I can manipulate the current in space into strange shapes and the B flux
will have its own shapes. What kind of shape do I want in magnets in a
particle accelerator?
As I wrote Maxwell’s equations they are partial differential equations, do
not yield the B field directly, we have to do some integration with some
boundary conditions to be able to calculate values for B.
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MePAS, Cherrill Spencer, Magnet Lecture #1
Guanajuato. 1st October 2011
What shapes of magnetic fields do accelerator
physicists want to guide & focus their beams?
Explanation here of LH figure
Is a magnetic field pattern for a
quadrupole that will focus the beam.
Length of arrows indicate strength of
field, at that point along an axis.
Are not flux lines
Can achieve this field shape with just
currents but its easier and the currents
needed for the same B are smaller if we use
some ferromagnetic material to help shape
the B
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MePAS, Cherrill Spencer, Magnet Lecture #1
Guanajuato. 1st October 2011
Representation of electromagnetic fields
by potential functions
• When dealing with fields, is easier to represent
them by POTENTIALS. Can do this with either a
vector potential, A or a scalar potential Φ
• These potentials are caused by sources such as
currents. Pay attention to where the field is relative to the source.
• e.g. Electric field intensity derived from Φ through
E= -Φ
• Similarly, in a region where J=0, B can be derived
from the scalar potential [in cartesian coords]:
B = -Φ. But also  ∙B = 0, so 2Φ=0 (Laplace Eq)
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MePAS, Cherrill Spencer, Magnet Lecture #1
Guanajuato. 1st October 2011
Equipotential surfaces and ferromagnetism
• Metallic surfaces are equipotential surfaces for electric
fields, what is the equivalent for magnetic fields? Consider
ferromagnetic material such as iron or steel (=iron plus very
small percentages of elements such as Carbon,
Manganese, Nickel, Sulphur)
• Electrons in Fe give an Fe atom a magnetic moment, i.e. it
has spin. In a collection of Fe atoms it is energetically
favourable for the spins of adjacent atoms to be parallel ->
say the domain is magnetized, it has a N and a S pole
• But the many domains in a piece of iron are randomly
oriented, so overall it has no magnetization. M=0
• If APPLY an external magnetic field then it will make the
domains change orientation, piece of iron becomes magnetized.
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MePAS, Cherrill Spencer, Magnet Lecture #1
Guanajuato. 1st October 2011
Magnetizing a piece of steel
• The applied field, usually created by an external
excitation current is called H: the magnetic field
intensity. Has same units as M: ampere-turns/m
• As H is increased so does the M, as domains get
larger and rotate to align with H’s direction.
Eventually all the domains are parallel to the
applied field and SATURATION has been reached.
• The overall flux density B is related to H and M:
B = µ0(M + H)
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MePAS, Cherrill Spencer, Magnet Lecture #1
Guanajuato. 1st October 2011
Hysteresis loop : variation of M, B with H
Saturation of iron occurs
By definition the permeability is
ratio between B and H
µ = B/H
To have a dimensionless
parameter put µ = µr µ0
µr is the relative permeability
and µ0 is the vacuum perm. It
takes all the units!
µr =1 for air
µr for iron: is not constant varies
:1 to ~2000
USE THIS PROPERTY OF IRON
IN MAKING MAGNETS
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MePAS, Cherrill Spencer, Magnet Lecture #1
Guanajuato. 1st October 2011
Consider the magnetic field in the
aperture of an accelerator magnet
2Φ=0
Solve Laplace’s equation by separating variables and
imposing boundary conditions that Φ be periodic in the
angular coord and be finite at r =0. We know we want
our focusing magnet to have 4 fold symmetry.- HOW TO
MAKE THIS?
Series expansion of the scalar
potential. Then take negative value
of gradient of potential -> field
For a given r the m-th
term has m maxima
and m minima as a
function of azimuthal
angle . These angular
positions may be
regarded as the
locations of magnetic
poles.
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MePAS, Cherrill Spencer, Magnet Lecture #1
Guanajuato. 1st October 2011
Continue analysis of 2D fields we
could produce in a magnet
Use De Moivre theorem to get into
cartesian coords
Study real and imaginary parts
and if make phi a constant
then that defines an ideal pole
shape for a magnet with 2m
poles.
m=2 gives a quadrupole
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MePAS, Cherrill Spencer, Magnet Lecture #1
Guanajuato. 1st October 2011
Potentials and field components for
2 D multipoles up to m=5
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MePAS, Cherrill Spencer, Magnet Lecture #1
Guanajuato. 1st October 2011
Consider equations in table
V= phi
Pole shape is
a
HYPERBOLA
with its tip at
some radius r
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MePAS, Cherrill Spencer, Magnet Lecture #1
Guanajuato. 1st October 2011
So a whole quadrupole looks like this
The field varies linearly with the
the distance from the magnet
center. It focuses the beam along
one plane while defocusing the
beam along the orthogonal
plane. An F or focusing
quadrupole focuses the particle
beam along the horizontal plane
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MePAS, Cherrill Spencer, Magnet Lecture #1
Guanajuato. 1st October 2011
We have dealt with iron pole shapes, now work
out how much current we need
B = µµ0 J
Maxwell’s 4th equation in differential form
x H = J
Use H, will be more convenient
Apply Stoke’s Theorem and differential form can be
written in integral form.
Stokes Theorem - The line integral of a potential function
around a closed boundary is equal to the area integral of
the source distribution within that closed boundary.
x
Suppose the “source” is a simple current
Circumference
of circle = 2π r
I   H  dl

B

2 r
0
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MePAS, Cherrill Spencer, Magnet Lecture #1
Guanajuato. 1st October 2011
Use integral of H along flux path to
calculate current needed to create B
B’, the gradient is constant
Along Path 1,
H r
and
Consider path 2 Biron ≈B’h, Liron >10h and
µiron>1000, so integral along path 2 is ~0.01path1
and can be ignored for estimating current
Br   B' r
H r  
B' r

0
Therefore
h
 H  dl  
Path1
0
B' rdr

0
B' h 2

2 0
 H  dl   H  dl  0
Path2
Finally
Path3
B' h 2
 H  dl  NI  2 0
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MePAS, Cherrill Spencer, Magnet Lecture #1
Guanajuato. 1st October 2011
Use 2D computer modeling program to define
pole and coil shapes, where to place them
Define boundaries of
steel core and coil in
plane at right angles to
beam direction.
Beam
passing
here
Would be too difficult and
tedious to create a pole-tip
shape and place some coils
near it and calculate by hand
the field distribution in the
aperture where beam will pass
In 1960s family of computer
codes called POISSON
developed to solve
Poisson’s equation and
calculate the fields in a
combination of steel and coil
shapes input by the user.
Program makes
mesh of small
triangles
ME inputs material properties and
flux boundary conditions into
“Automesh”
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MePAS, Cherrill Spencer, Magnet Lecture #1
Guanajuato. 1st October 2011
Figure showing what POISSON program
produces, plus printout of field values
TABLE FOR FIELD COEFFICIENTS FOR ATF2 QD0 with 2.5cm rad
w/0.25"sideshim SQ end 7 Dec07
NORMALIZATION RADIUS = 2.50000
(BX - I BY) = I * SUM N*(AN + I BN)/R * (Z/R)**(N-1)
N N(AN)/R
N(BN)/R
ABS(N(CN)/R) RATIO 2Npole/4 pole
2 -3.0151E+03 0.0000E+00
3.0151E+03 1.0000E+00
6 -1.7329E+01 0.0000E+00
1.7329E+01 5.7473E-03
10 -1.7196E+02 0.0000E+00 1.7196E+02 5.7033E-02
14 -1.4998E+01 0.0000E+00 1.4998E+01 4.9743E-03
18 4.1147E+00 0.0000E+00
4.1147E+00 1.3647E-03
MePAS, Cherrill
Spencer, Magnet Lecture #1
22 -6.2010E+00 0.0000E+00 6.2010E+00
2.0566E-03
st
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Guanajuato. 1 October 2011
Typical set of magnet requirements from AP
For A DIPOLE MAGNET
• Nominal bending angle
Theta = 97.2434 mrad
• Nominal integral dipole field at 9 GeV B0L = 29.193 kG-m
• Effective length (approx value)
L = 1.800 m
• Height of aperture
25.4 mm
• Operating field range
from 15.5 to 16.6 kG
• Integrated strength operating range 27.90 to 29.88 kG-m
• Minimum full pole width for beam
210 mm
• Dipole field variation relative to average dipole field in four bends in eor e+ chicane
– DeltaB0/Baver = +/-0.1%
• Multipole field tolerances at R = 10 cm (for “coupled” beams)
–
–
–
–
B1/B0 = +/-0.21%
B2/B0 = +/-0.27%
B3/B0 = +/-0.30%
B4/B0 = +/-0.33%
(quadrupole)
(sextupole)
(octupole)
(decapole)
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MePAS, Cherrill Spencer, Magnet Lecture #1
Guanajuato. 1st October 2011
Shape of top right hand part of new sector 10 dipole: my design in
response to the requirements given to me by an AP
Total width of core: 30” [76.2cm]
Total width of
poletip at gap
is 26.0 cm
Low carbon solid
steel core
Low carbon steel core
Main coil: 20 turns
of 0.44”sq hollow
copper conductor
with 0.186”  hole
Trim coil: 88 turns
of AWG#10 solid
copper wire: <5%
of main coil
Need model only
one quarter of
the whole
magnet
1.27cm (0.5”)
half-gap
2.54cm (1”) wide, 30º chamfer improves the multipoles in the gap to well
below the specs
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MePAS, Cherrill Spencer, Magnet Lecture #1
Guanajuato. 1st October 2011
POISSON 2D MODEL of new style dipole showing
magnetic flux lines, with trim working
In POISSON,
because of
symmetry of top and
bottom halves and
left and right halves,
need to model only
one quarter of the
whole magnet
Note the flux lines are perpendicular to the mid-plane of the gap where beam will
pass [into the page] and enter the steel core at right angles to edge of steel
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MePAS, Cherrill Spencer, Magnet Lecture #1
Guanajuato. 1st October 2011
Magnet Lecture #1 Homework Question
0.30m
0.17m
Develop an equation
that connects the
path integral of B
through this section
of a dipole and the
total current in the
section of the coil
enclosed by the
path.
0.025m
If the half gap is
0.025m and B is to
be one Tesla, what
is the value of NI
needed
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MePAS, Cherrill Spencer, Magnet Lecture #1
Guanajuato. 1st October 2011
Photo of a dipole with “dog-ear” coils
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MePAS, Cherrill Spencer, Magnet Lecture #1
Guanajuato. 1st October 2011