Accelerator Magnets
Neil Marks,
STFC- ASTeC / U. of Liverpool,
Daresbury Laboratory,
Warrington WA4 4AD,
U.K.
Tel: (44) (0)1925 603191
Fax: (44) (0)1925 603192
n.marks@stfc.ac.uk
Room Temperature Magnets
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CAS Prague 2014
Contents
Theory
•
Maxwell's 2 magneto-static equations;
With no currents or steel present:
•
Solutions in two dimensions with scalar potential (no currents);
•
Cylindrical harmonic in two dimensions (trigonometric formulation);
•
Field lines and potential for dipole, quadrupole, sextupole;
Introduction of steel:
•
Ideal pole shapes for dipole, quad and sextupole;
•
•
Field harmonics-symmetry constraints and significance;
Significance and use of contours of constant vector potential;
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Contents (cont.)
Three dimensional issues:
•
Termination of magnet ends and pole sides;
•
The ‘Rogowski roll-off’
Introduction of currents:
•
Ampere-turns in dipole, quad and sextupole;
•
Coil design;
•
Coil economic optimisation-capital/running costs;
Practical Issues:
•
Backleg and coil geometry- 'C', 'H' and 'window frame' designs;
•
FEA techniques - Modern codes- OPERA 2D and 3D;
•
Judgement of magnet suitability in design.
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Magnets - introd u ction
Dip oles to bend the beam :
Qu ad ru poles to focu s it:
Sextu poles to correct
chrom aticity:
We shall establish a formal
approach to describing
these magnets.
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N o cu rrents, no steel:
Maxw ell’s equ ations:
.B = 0 ;
H=j;
Then we can put:
B = - 
So that:
2 = 0
(Laplace's equation).
Taking the two dimensional case (ie constant in the z
direction) and solving for cylindrical coordinates (r,):
 = (E+F )(G+H ln r) + n=1 (Jn r n cos n +Kn r n sin n
+Ln r -n cos n  + M n r -n sin n  )
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In p ractical situ ations:
The scalar potential sim plifies to:
 = n (Jn r n cos n +Kn r n sin n),
w ith n integral and Jn ,Kn a fu nction of geom etry.
Giving components of flux density:
Br = - n (n Jn r n-1 cos n +nKn r n-1 sin n)
B = - n (-n Jn r n-1 sin n +nKn r n-1 cos n)
Then to convert to Cartesian coord inates:
x = r cos ;
y = r sin ;
and
Bx = - / x;
By = - / y
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Significance
This is an infinite series of cylind rical harm onics;
they d efine the allow ed d istribu tions of B in 2
d im ensions in the absence of cu rrents w ithin the
d om ain of (r,).
Distribu tions not given by above are not physically
realisable.
Coefficients Jn , Kn are d eterm ined by geom etry (rem ote
iron bou nd aries and cu rrent sou rces).
N ote that this form u lation can be expressed in term s of
com plex field s and potentials.
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Dipole field n=1:
Cylindrical:
 =J1 r cos  +K1 r sin .
Br = J1 cos  + K1 sin ;
B = -J1 sin  + K1 os ;
Cartesian:
 =J1 x +K1 y
Bx = -J1
By = -K1
So, J1 = 0 gives vertical dipole field:
 = const.
B
K1 =0 gives
horizontal
dipole field.
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Qu ad ru pole field n=2:
Cylindrical:
 = J2 r 2 cos 2 +K2 r 2 sin 2;
Br = 2 J2 r cos 2 +2K2 r sin 2;
B = -2J2 r sin 2 +2K2 r cos 2;
J2 = 0 gives 'normal' or
‘upright’ quadrupole
field.
K2 = 0 gives 'skew'
quad fields (above
rotated by /4).
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Cartesian:
 = J2 (x2 - y 2)+2K2 xy
Bx = -2 (J2 x +K2 y)
By = -2 (-J2 y +K2 x)
Line of constant
scalar potential
Lines of flux
density
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Sextu pole field n=3:
Cylindrical;
 = J3 r 3 cos 3 +K3 r 3 sin 3;
Br = 3 J3r 2 cos 3 +3K3r 2 sin 3;
B= -3J3 r 2 sin 3+3K3 r 2 cos 3;
-C
+C
+C
Cartesian:
 = J3 (x3-3y 2x)+K3(3yx2-y 3)
Bx = -3J3 (x2-y 2)+2K3yx
By = -3-2 J3 xy +K3(x2-y 2)
J3 = 0 giving ‘upright‘
sextupole field.
Line of constant
scalar potential
-C
-C
+C
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Lines of flux
density
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Variation of By on x axis (u pright field s).
By
Dipole;
constant field :
x
Qu ad ru pole;
linear field
By
x
Sextu pole:
qu ad ratic variation:
100
By = G Q x;
G Q is quadrupole gradient
(T/m).
By
x
By = G S x2;
G S is sextupole gradient
(T/m 2).
0
0
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20
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Alternative notation:
(u sed in m ost lattice program s)
k n xn
B (x) = B  
n!
n 0

magnet strengths are specified by the value of kn;
(normalised to the beam rigidity);
order n of k is different to the 'standard' notation:
dipole is
quad is
n = 0;
n = 1; etc.
k0 (dipole)
k1 (quadrupole)
m-1;
m-2;
k has units:
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etc.
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Introd u cing iron yokes and poles.
What is the id eal pole shape?
• Flu x is norm al to a ferrom agnetic su rface w ith infinite :

1
cu rl H = 0
therefore  H .d s = 0;
in steel H = 0;
therefore parallel H air = 0
therefore B is norm al to surface.
• Flu x is norm al to lines of scalar potential, (B = - );
• So the lines of scalar potential are the id eal pole shapes!
(bu t these are infinitely long!)
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Equ ations of id eal poles
Equ ations for Id eal (infinite) poles;
(Jn = 0) for ‘upright’ (ie not skew ) field s:
D ipole:
y=  g/ 2;
(g is inter-pole gap).
Quadrupole:
xy= R2/ 2;
Sextupole:
3x2y - y 3 = R3;
R
R is the ‘inscribed radius’ of a multipole magnet.
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‘Pole-tip ’ Field
The radial field at pole centre of a m u ltipole m agnet:
BPT = G N R(n-1) ;
Qu ad ru pole: BPT = G Q R; sextu pole: BPT = G S R2; etc;
Has it any significance?
- a qu ad ru pole R = 50 m m ; G Q = 20 T/ m ; BPT = 1.0 T;
i)
Beam line – round beam r = 40 mm;
pole extends to x = 65 mm; By (65,0) = 1.3 T; OK ;
ii)
Synchrotron source – beam ± 50 mm horiz.; ± 10 mm vertical;
pole extends to x = 80 mm; By (80,0)= 1.6 T; perhaps OK ??? ;
iii)
FFAG – beam ± 65 mm horiz.; ± 8 mm vertical;
Pole extends to x = 105 mm; By (105, 0) = 2.1 T .
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Com bined fu nction m agnets
'Com bined Fu nction Magnets' - often d ipole and
qu ad ru pole field com bined (bu t see later slid e):
A qu ad ru pole m agnet w ith
physical centre shifted from
m agnetic centre.
B




B

 ,
n = -


  x 
 0
Characterised by 'field ind ex' n,
+ve or -ve d epend ing
 is rad iu sofofthe
cu rvatu
re of
theB is
where
the radius of curvature
beam
and
 isient;
on d irection
of grad
0
beam ;
d o not confu se w ith harm onic n!
Bo is central d ipole field
If physical and magnetic centres are separated by X
0
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Typical com bined
d ipole/ qu ad rupole
SRS Booster c.f. dipole
‘F’ type -ve n
‘D ’ type +ve n.
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N IN A Com bined fu nction m agnets
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Pole for a com bined d ip ole and qu ad .
P hysica la nd ma gne ticce ntre sa re se pa ra te dby
X
0
Horizonta l displa cme nt from true qua d ce ntreis
x'
  B
X
B  
0   x  0
x' y   R 2 / 2
The n
the re fore
As
x'x  X
0
1
R 2 n  n x 
P ole e qua tionis
y 
12  
 
1


n
x

or
y   g  1  

whe reg is the half gap a t the physica lce ntreof the ma gne t
1


x   B 


re writte na s
y  g 1  B   x 
0


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Other com bined fu nction m agnets:
• d ipole, qu ad ru pole and sextu pole;
• d ipole & sextu pole (for chrom aticity control);
• d ipole, skew qu ad , sextu pole, octu pole;
Generated by
• pole shapes given by su m of correct scalar potentials
- am plitu d es bu ilt into pole geom etry – not variable!
OR:
• m u ltiple coils m ou nted on the yoke
- am plitu d es ind epend ently varied by coil cu rrents.
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The SRS m u ltipole m agnet.
Cou ld d evelop:
• vertical d ipole
• horizontal d ipole;
• u pright qu ad ;
• skew qu ad ;
• sextu pole;
• octu pole;
• others.
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The practical pole in 2D
Practically, poles are finite, introducing errors;
these appear as higher harm onics w hich d egrad e the
field d istribu tion.
H ow ever, the iron geom etries have certain
sym m etries that restrict the natu re of these errors.
Dipole:
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Qu ad ru pole:
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Possible sym m etries.
Lines of sym m etry:
Dipole:
Qu ad
Pole orientation
d eterm ines w hether pole
is u pright or skew .
y = 0;
x = 0; y = 0
Ad d itional sym m etry
im posed by pole ed ges.
x = 0;
y=x
The ad d itional constraints im posed by the sym m etrical
pole ed ges lim its the valu es of n that have non zero
coefficients
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Dipole sym m etries
Type
Symmetry
Constraint
Pole orientation
() = -(-)
all Jn = 0;
Pole ed ges
() = ( -)
Kn non-zero
only for:
n = 1, 3, 5, etc;
+
+
+
-
So, for a fu lly sym m etric d ipole, only 6, 10, 14 etc pole
errors can be present.
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Qu ad ru pole sym m etries
Type
Symmetry
Constraint
Pole orientation
() = -( -)
All Jn = 0;
() = -( -)
Kn = 0 all od d n;
() = (/ 2 -)
Kn non-zero
only for:
n = 2, 6, 10, etc;
Pole ed ges
So, a fu lly sym m etric qu ad ru pole, only 12, 20, 28 etc
pole errors can be present.
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Sextu pole sym m etries.
Type
Symmetry
Constraint
Pole orientation
() = -( -)
() = -(2/ 3 - )
() = -(4/ 3 - )
All Jn = 0;
Kn = 0 for all n
not m u ltiples of 3;
Pole ed ges
() = (/ 3 - )
Kn non-zero only
for: n = 3, 9, 15, etc.
So, a fu lly sym m etric sextu pole, only 18, 30, 42 etc pole
errors can be present.
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Su m m ary
For perfectly sym m etric magnets, the ‘allow ed ’ error field s are fu lly
d efined by the sym m etry of scalar potential  and the trigonom etry:
D ipole:
+
Quadrupole:
-
+
+
+
-
-
D ipole: only dipole, sextupole,
10 pole etc can be present
Sextupole: only sextupole, 18, 30, 42
pole, etc. can be present
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Quadrupole: only quadrupole, 12
pole, 20 pole etc can be present
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Vector potential in 2 D
We have:
and
Expand ing:
B = cu rl A
(A is vector potential);
d iv A = 0
B = cu rl A =
(A z / y - A y / z) i + (A x/ z - A z / x) j + (A y / x - A x/ y) k;
w here
i, j, k, are u nit vectors in x, y, z.
In 2 dimensions
Bz = 0;
 / z = 0;
So
A x = A y = 0;
and
B = (A z / y ) i - (A z / x) j
A is in the z direction, normal to the 2 D problem.
N ote:
d iv B = 2A z / (x y) - 2A z / (x y) = 0;
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Total flu x betw een tw o p oints  DA
In a tw o dimensional problem the m agnetic flu x betw een
tw o points is proportional to the d ifference betw een the
vector potentials at those points.
B
F  (A2 - A1)
A1
A2
F
Proof on next slid e.
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Proof:
Consid er a rectangu lar closed path, length l in z
d irection at (x1,y 1) and (x2,y 2); apply Stokes’ theorem :
A
F =   B.dS =   ( cu rl A).dS =  A.ds
Bu t A is exclu sively in the z
d irection, and is constant in this
d irection.
B
dS
l
So:
 A.d s = l { A(x1,y 1) - A(x2,y 2)};
(x1, y 1)
ds
(x2, y 2)
y
z
x
F = l { A(x1,y 1) - A(x2,y 2)};
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Contou rs of constant A
Therefore:
i) Contou rs of constant vector potential in 2D give a graphical
representation of lines of flu x.
ii) These are u sed in 2D FEA analysis to obtain a graphical image of
flu x d istribu tion.
iii) The total flu x cu tting the coil allow s the calcu lation of the ind u ctive
voltage per tu rn in an ac m agnet:
V = - d F/d t;
iv) For a sine w ave oscillation frequ ency w:
Vp eak = w (F/2)
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In 3D – pole end s (also pole sid es).
Fringe flu x w ill be present at pole end s
so beam d eflection continu es beyond
m agnet end :
z
1. 2
B0
By
0
0
The magnet’s strength is given by
 By (z) dz
along the
magnet , the integration including the fringe field at each end;
The ‘magnetic length’ is defined as
(1/B0)( By (z) dz )
over the same integration path, where B0 is the field at the azimuthal
centre.
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End Field s and Geom etry.
At high(ish) fields it is necessary to terminate the pole
(transverse OR longitudinal) in a controlled way:;
•to prevent saturation in a sharp corner (see diagram);
•to maintain length constant with x, y;
•to define the length (strength) or preserve quality;
•to prevent flux entering normal
to lamination (ac).
Longitudinally, the end of the
magnet is therefore
'chamfered' to give increasing gap
(or inscribed radius) and lower
fields as the end is approached.
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Classical end or sid e solu tion
The 'Rogowski' roll-off: Equation:
y = g/2 +(g/a) [exp (ax/g)-1];
g/2 is dipole half gap;
y = 0 is centre line of gap;
a = 1.5
a ( ̴ 1); parameter to control
a = 1.25
the roll off;
a = 1.0
With a = 1, this profile
provides the maximum rate
of increase in gap with a
monotonic decrease in flux
density at the surface;
For a high By magnet this avoids any
additional induced non-linearity
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Introd u ction of cu rrents
Now for j  0
H=j;
dS
To expand, use Stoke’s Theorum:
for any vector V and a closed
curve s :
V.ds = curl V.dS
Apply this to:
curl H = j ;
V
ds
then in a magnetic circuit:
 H.ds = N I;
N I (Ampere-turns) is total current cutting S
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Excitation cu rrent in a d ipole
B is approx constant round the
loop made up of l and g, (but see
below);
But in iron,
and
So
>>1,
Hiron = Hair / ;
l
NI/2
g
  1
NI/2
Bair = 0 NI / (g + l/);
g, and l/ are the 'reluctance' of the gap and iron.
Approximation ignoring iron reluctance (l/ << g ):
NI = B g /0
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Excitation cu rrent in qu ad & sextu p ole
For quadrupoles and sextupoles, the required
excitation can be calculated by considering fields and
gap at large x. For example:
Quadrupole:
y
Pole equation:
xy = R2 /2
On x axes
BY = GQx;
where G Q is gradient (T/m).
x
At large x (to give vertical
lines of B):
N I = (GQx) ( R2 /2x)/0
N I = GQ R2 /2 0 (per pole).
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B
The sam e m ethod for a
Sextupole,
( coefficient G S,), gives:
N I = G S R3/ 30 (per pole)
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General solu tion-m agnets ord er n
B = 0 H
B = - 
In air (remote currents! ),
y
Integrating over a lim ited path
(not circu lar) in air:
N I = (1 – 2)/ o
1, 2 are the scalar potentials at tw o points in air.
Define  = 0 at m agnet centre;
then potential at the pole is:
o N I
 = 0 NI
Apply the general equations for m agnetic
=0
field harm onic ord er n for non -skew
m agnets (all Jn = 0) giving:
N I = (1/ n) (1/ 0) Br / R (n-1) R n
Where:
N I is excitation per pole;
R is the inscribed rad ius (or half gap in a d ipole);
term in brackets  is m agnet grad ient G N in T/ m (n-1).
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x
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Coil geom etry
Stand ard d esign is
rectangu lar copper (or
alu m iniu m ) cond u ctor, w ith
cooling w ater tu be. Insu lation
is glass cloth and epoxy resin.
Am p -tu rns (N I) are
d eterm ined , bu t total copper
area (A copper ) and nu m ber of
tu rns (N ) are tw o d egrees of
freed om and need to be
d ecid ed .
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Cu rrent d ensity:
j = N I/ A cop p er
Optim u m j
d eterm ined from
economic criteria.
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Cu rrent d ensity - optim isation
Ad vantages of low j:
• low er pow er loss – pow er bill is d ecreased ;
• low er pow er loss – pow er converter size is d ecreased ;
• less heat d issipated into m agnet tu nnel.
15.0
• smaller coils;
• low er capital cost;
• smaller magnets.
Chosen valu e of j is an
optim isation of m agnet
capital against pow er costs.
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Lifetime cost
Ad vantages of high j:
total
capital
running
0.0
0.0
1.0
2.0
3.0
4.0
5.0
Current density j
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6.0
N u m ber of tu rns p er coil-N
The valu e of nu m ber of tu rns (N ) is chosen to m atch
pow er su pply and interconnection im ped ances.
Factors d eterm ining choice of N :
Large N (low current)
Small N (high current)
Sm all, neat term inals.
Large, bu lky term inals
Thin interconnections-hence low
cost and flexible.
Thick, expensive connections.
More insulation layers in coil,
hence larger coil volum e and
increased assem bly costs.
H igh percentage of copper in
coil volum e. More efficient use
of space available
H igh voltage pow er supply
-safety problem s.
H igh current pow er supply.
-greater losses.
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Exam ples-tu rns & cu rrent
From the Diam ond 3 GeV synchrotron sou rce:
Dipole:
N (p er m agnet):
I m ax
Volts (circu it):
40;
1500
500
A;
V.
Qu ad ru pole:
N (p er pole)
54;
I m ax
200
Volts (p er m agnet): 25
A;
V.
N (p er pole)
48;
I m ax
100
Volts (p er m agnet) 25
A;
V.
Sextu pole:
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Magnet geom etry
Dipoles can be ‘C core’ ‘H core’ or ‘Wind ow fram e’
''C' Core:
Ad vantages:
Easy access;
Classic d esign;
Disad vantages:
Pole shim s need ed ;
Asym m etric (sm all);
Less rigid ;
Shim
detail
The ‘shim ’ is a sm all, ad d itional piece of ferro-magnetic material
ad d ed on each sid e of the tw o poles – it com pensates for the finite
cu t-off of the pole, and is optim ised to red u ce the 6, 10, 14...... pole
error harm onics.
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Flu x in the gap.
Flu x in the yoke inclu d es the gap flu x and stray flu x,
w hich extend s (approx) one gap w id th on either sid e of
the gap.
b
Thu s, to calcu late total flu x
in the back-leg of m agnet
length l:
g
F =Bgap (b + 2g) l.
Wid th of backleg is chosen
to lim it Byoke and hence
m aintain high .
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g
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Steel- B/ H cu rves
-for typical silicon steel laminations
N ote:
• the relative perm eability is the
grad ient of these cu rves;
• the low er grad ient close to the
origin - low er perm eability;
• the perm eability is m axim u m at
betw een 0.4 and 0.6 T.
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Typical ‘C’ cored Dipole
Cross section of
the Diam ond
storage ring
d ipole.
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H core and w ind ow -fram e m agnets
''Wind ow Fram e'
Ad vantages:
H igh qu ality field ;
N o pole shim ;
Sym m etric & rigid ;
Disad vantages:
Major access problem s;
Insu lation thickness
‘H core’:
Ad vantages:
Sym m etric;
More rigid ;
Disad vantages:
Still need s shims;
Access problem s.
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Wind ow fram e d ipole
Provid ing the cond u ctor is continu ou s to the steel ‘w ind ow
fram e’ su rfaces (im possible becau se coil m u st be electrically
insu lated ), and the steel has infinite , this m agnet generates
perfect d ipole field .
Provid ing cu rrent d ensity J is
u niform in cond u ctor:
• H is u niform and vertical u p
J
ou ter face of cond u ctor;
• H is u niform , vertical and
H
w ith sam e valu e in the m id d le
of the gap;
•  perfect d ipole field .
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Practical w ind ow fram e d ipole.
Insu lation ad d ed to coil:
B increases
close to coil
insu lation
su rface
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B d ecrease
close to coil
insu lation
su rface
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best
com prom ise
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Op en-sid ed Qu ad ru p ole
‘Diam ond ’ storage ring
qu ad ru pole.
The yoke su pport
pieces in the horizontal
plane need to provid e
space for beam -lines
and are not ferrom agnetic.
Error harm onics
inclu d e n = 4 (octu pole)
a finite perm eability
error.
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Typical pole d esigns
To com pensate for the non -infinite pole, shim s are ad d ed
at the pole ed ges. The area and shape of the shim s d eterm ine
the am plitu d e of error harm onics w hich w ill be present.
Dipole:
Qu ad ru pole:
A
The d esigner optim ises the
pole by ‘pred icting’ the field
resu lting from a given pole
geom etry and then ad ju sting
it to give the requ ired
qu ality.
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A
When high field s are present,
chamfer angles mu st be small,
and tapering of poles m ay be
necessary
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Pole-end correction.
As the gap is increased, the size (area) of the shim is
increased, to give some control of the field quality at the
lower field. This is far from perfect!
Transverse adjustment at end of
dipole
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Transverse adjustment at end of
quadrupole
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Assessing pole d esign
A first assessm ent can be m ad e by ju st exam ining By (x)
w ithin the requ ired ‘good field ’ region.
N ote that the expansion of By (x) y = 0 is a Taylor series:
By (x) =


n =1
{b n x (n-1)}
= b 1 + b 2x + b 3x2 + ………
d ipole
qu ad
sextu pole
Also note:
 By (x) /  x = b 2 + 2 b 3x + ……..
So qu ad grad ient G  b 2 =  By (x) /  x in a qu ad
Bu t sext. grad ient G s  b 3 = ½ 2 By (x) /  x2 in a sext.
So coefficients are not equ al to d ifferentials for n = 3 etc.
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Is it ‘fit for pu rpose’?
A sim ple ju d gem ent of field qu ality is given by plotting:
• D ipole:
• Quad:
• 6poles:
{By (x) - By (0)}/ BY (0)
d By (x)/ d x
d 2By (x)/ d x2
(DB(x)/ B(0))
(Dg(x)/ g(0))
(Dg 2(x)/ g 2(0))
‘Typical’ acceptable variation insid e ‘good field ’ region:
DB(x)/ B(0) 
Dg(x)/ g(0)

Dg 2(x)/ g 2(0) 
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0.01%
0.1%
1.0%
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Design com pu ter cod es.
Com pu ter cod es are now u sed ; eg the Vector
Field s cod es -‘OPERA 2D’ and 3D.
These have:
•
•
•
•
finite elem ents w ith variable triangu lar m esh;
m u ltiple iterations to sim u late steel non -linearity;
extensive pre and post processors;
com patibility w ith m any platform s and P.C. o.s.
Techniqu e is iterative:
• calcu late flu x generated by a d efined geom etry;
• ad ju st the geom etry u ntil requ ired d istribu tion is
achieved .
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Design Proced u res –OPERA 2D
Pre-processor:
The m od el is set-u p in 2D u sing a GUI (grap hics
u ser’s interface) to d efine ‘regions’:
• steel regions;
• coils (inclu d ing cu rrent d ensity);
• a ‘backgrou nd ’ region w hich d efines the physical
extent of the m od el;
• the sym m etry constraints on the bou nd aries;
• the perm eability for the steel (or u se the preprogram m ed cu rve);
• m esh is generated and d ata saved .
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Mod el of Diam ond storage ring d ip ole
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With m esh ad d ed
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Close-u p of pole region.
Pole profile, show ing shim and Rogow ski sid e roll-off
for Diam ond 1.4 T d ipole.:
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Diam ond qu ad ru pole: a
sim plified m od el
N ote – one eighth of
Qu ad ru pole cou ld
be u sed w ith
opposite sym m etries
d efined on
horizontal and y = x
axis.
Bu t I have often
experienced
d iscontinu ities at the
origin w ith su ch1/ 8
th geom etry.
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Calcu lation of end effects u sing 2D cod es
FEA model in longitudinal plane, with correct end
geometry (including coil), but 'idealised' return yoke:
+
This will establish the end distribution; a numerical integration will give the
'B' length.
Provided dBY/dz is not too large, single 'slices' in the transverse plane can be
used to calculated the radial distribution as the gap increases. Again,
numerical integration will give  B.dl as a function of x.
This technique is less satisfactory with a quadrupole, but end effects are less
critical with a quad.
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Calcu lation.
Data Processor:
either:
• linear - w hich u ses a pred efined constant perm eability
for a single calcu lation, or
• non-linear - w hich is iterative w ith steel perm eability
set accord ing to B in steel calcu lated on previou s
iteration.
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Data Display – OPERA 2D
Post-processor:
u ses p re-processor m od el for m any op tions for
d isp laying field am plitu d e and qu ality:
•
•
•
•
•
field lines;
graphs;
contou rs;
grad ients;
harm onics (from a Fou rier analysis arou nd a pred efined circle).
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2 D Dipole field hom ogeneity on x axis
Diam ond s.r. d ipole: DB/ B = {By(x)- B(0,0)}/ B(0,0);
typically  1:104 w ithin the ‘good field region’ of -12m m  x  +12 m m ..
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2 D Flu x d ensity d istribu tion in a d ip ole
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2 D Dipole field hom ogeneity in gap
Transverse
(x,y) plane in
Diam ond s.r.
d ipole;
contou rs are
0.01%
requ ired
good field
region:
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H arm onics ind icate m agnet qu ality
The am plitu d e and phase of the integrated
harm onic com ponents in a m agnet provid e an
assessm ent:
• w hen accelerator physicists are calcu lating beam
behaviou r in a lattice;
• w hen d esigns are ju d ged for su itability;
• w hen the m anu factu red m agnet is m easu red ;
• to ju d ge acceptability of a m anu factu red m agnet.
Measu rem ent of a m agnet after m anu factu re w ill be
d iscu ssed in the section on m easu rem ents.
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End geom etries - d ipole
Sim pler geom etries can be u sed in som e cases.
The Diam ond d ipoles have a Rogaw ski roll-off at the end s
(as w ell as Rogaw ski roll-offs at each sid e of the pole).
See photographs to follow .
This give sm all negative sextu pole field in the end s w hich
w ill be com pensated by ad ju stm ents of the strengths in
ad jacent sextu pole m agnets – this is possible becau se each
sextu pole w ill have its ow n ind ivid u al pow er su pply.
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OPERA 3D m od el of Diam ond d ip ole.
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Diam ond d ipole poles
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2 D Assessm ent of qu ad ru pole grad ient qu ality
0.1
D iamond
WM
quadrupole:
Gradient
quality is to
be  0.1 % or
better to x =
36 mm.
D dHy/dx (%)
graph is
percentage
variation in
dBy/dx vs x
at different
values of y.
0.05
0
-0.05
-0.1
0
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4
8
12
16
20
24
28
32
x (mm)
y= 0
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y = 4 mm
y = 8 mm
y = 12 mm
y = 16 mm
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36
End cham fering - Diam ond ‘W’ qu ad
Tosca
resu lts d ifferent
d epths 45
end
chamfers
on Dg/ g 0
integrated
throu gh
m agnet
and end
fringe field
(0.4 m long
WM
qu ad ).
Fractional
deviation
0.002
0.001
X
5
10
15
20
25
30
35
-0.001
-0.002
8
7
6
4
mm Cut
mm Cut
mm Cut
mm Cut
No Cut
Thanks to Chris Bailey (D LS) w ho performed this w orking using OPERA 3D .
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Sim plified end geom etries - qu ad ru pole
Diam ond
qu ad ru poles
have an
angu lar cu t at
the end ;
d epth and
angle w ere
ad ju sted
u sing 3D
cod es to give
optim u m
integrated
grad ient.
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Sextu pole end s
It is not
u su ally
necessary
to cham fer
sextu p ole
end s (in a
d .c.
m agnet).
Diam ond
sextu p ole
end :
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‘Artistic’ Diam ond Sextu poles
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