Unit 8 Electromagnetic design Episode I
description
Transcript of Unit 8 Electromagnetic design Episode I
USPAS January 2012, Superconducting accelerator magnets
Unit 8Electromagnetic design
Episode I
Helene Felice and Soren PrestemonLawrence Berkeley National Laboratory (LBNL)
Paolo Ferracin and Ezio TodescoEuropean Organization for Nuclear Research (CERN)
USPAS January 2012, Superconducting accelerator magnets Unit 8: Electromagnetic design episode I – 8.2
QUESTIONS
Which coil shapes give a perfect field ?
How to build a sector coil with a real cable, and with good field quality for the beam ? Where to put wedges ? 41° 49’ 55” N – 88 ° 15’ 07” W
USPAS January 2012, Superconducting accelerator magnets Unit 8: Electromagnetic design episode I – 8.3
CONTENTS
1. How to generate a perfect fieldDipoles: cos, intersecting ellipses, pseudo-solenoidQuadrupoles: cos2, intersecting ellipses
2. How to build a good field with a sector coilDipolesQuadrupoles
USPAS January 2012, Superconducting accelerator magnets Unit 8: Electromagnetic design episode I – 8.4
1. HOW TO GENERATE A PERFECT FIELD
Perfect dipoles - 1Wall-dipole: a uniform current density in two walls of infinite height produces a pure dipolar field
+ mechanical structure and winding look easy- the coil is infinite- truncation gives reasonable field quality only for rather large
height the aperture radius (very large coil, not effective)
-60
0
60
-40 0 40
+
+-
-
-60
0
60
-40 0 40
+
+-
-
A wall-dipole, cross-section A practical winding with flat cablesA wall-dipole, artist view
USPAS January 2012, Superconducting accelerator magnets Unit 8: Electromagnetic design episode I – 8.5
1. HOW TO GENERATE A PERFECT FIELD
Perfect dipoles - 2Intersecting ellipses: a uniform current density in the area of two intersecting ellipses produces a pure dipolar field
- the aperture is not circular- the shape of the coil is not easy to wind with a flat cable
(ends?)- need of internal mechanical support that reduces available
aperture
-60
0
60
-40 0 40
+
+-
-
-60
0
60
-40 0 40
+
+-
-
Intersecting ellipses A practical (?) winding with flat cables
USPAS January 2012, Superconducting accelerator magnets Unit 8: Electromagnetic design episode I – 8.6
1. HOW TO GENERATE A PERFECT FIELD
Perfect dipoles - 2Proof that intersecting circles give perfect
fieldwithin a cylinder carrying uniform current j0, the field is perpendicular to the radial direction and proportional to the distance to the centre r:
Combining the effect of the two cylinders
Similar proof for intersecting ellipses
-60
0
60
-40 0 40
+
+-
-
From M. N. Wilson, pg. 28
200 rj
B
sj
rrrj
B y 2coscos
200
221100
0sinsin2 2211
00
rrrj
Bx
USPAS January 2012, Superconducting accelerator magnets Unit 8: Electromagnetic design episode I – 8.7
1. HOW TO GENERATE A PERFECT FIELD
Perfect dipoles - 3Cos: a current density proportional to cos in an annulus - it can be approximated by sectors with uniform current density
+ self supporting structure (roman arch)+ the aperture is circular, the coil is compact+ winding is manageable
-60
0
60
-40 0 40
+
+-
-j = j0 cos
An ideal cos A practical winding with one layer and wedges[from M. N. Wilson, pg.
33]
A practical winding with three layers and no
wedges[from M. N. Wilson, pg.
33]
wedgeCable block
Artist view of a cos magnet
[from Schmuser]
USPAS January 2012, Superconducting accelerator magnets Unit 8: Electromagnetic design episode I – 8.8
1. HOW TO GENERATE A PERFECT FIELD
Perfect dipoles - 3Cos theta: proof – we have a distribution
The vector potential reads
and substituting one has
using the orthogonality of Fourier series -60
0
60
-40 0 40
+
+-
-j = j0 cos
)cos()( 0 mjj
)cos(2
),(0
00
mmj
Am
z
1
2
00
00 )](cos[)cos(12
),(n
n
z dnmn
jA
1 0
0 )](cos[12
),(n
n
z nn
jA
USPAS January 2012, Superconducting accelerator magnets Unit 8: Electromagnetic design episode I – 8.9
1. HOW TO GENERATE A PERFECT FIELD
Perfect quadrupolesCos2: a current density proportional to cos2 in an annulus - approximated by sectors with uniform current density and wedges(Two) intersecting ellipses
Perfect sextupoles: cos3 or three intersecting ellipsesPerfect 2n-poles: cosn or n intersecting ellipses
-60
0
60
-40 0 40
++
- - j = j 0 cos 2
- -
++
-60
0
60
-40 0 40
++
- -
- -
++
Quadrupole as an ideal cos2 Quadrupole as two intersecting ellipses
USPAS January 2012, Superconducting accelerator magnets Unit 8: Electromagnetic design episode I – 8.10
CONTENTS
1. How to generate a perfect fieldDipoles: cos, intersecting ellipses, pseudo-solenoidQuadrupoles: cos2, intersecting ellipses
2. How to build a good field with a sector coilDipolesQuadrupoles
USPAS January 2012, Superconducting accelerator magnets Unit 8: Electromagnetic design episode I – 8.11
2. HOW TO BUILD A GOOD FIELD: SECTOR COILS FOR DIPOLES
We recall the equations relative to a current line we expand in a Taylor series
the multipolar expansion is defined as
the main component is
the non-normalized multipoles are
.)(2
)(0
0
zzI
zB
nref
refn z
RRI
C
0
0
2
0
0
0
01
cos2
1Re2 z
Iz
IB
-40
0
40
-40 0 40
z 0=x 0+iy 0
x
y
z=x+iy
B=B y+iB x
1
11
00
0
1
1
00
0
22)(
n
n
ref
nref
n
n
Riyx
zR
zI
zz
zIzB
1
1
1
1
)(n
n
refnn
n
n
refn R
ziABRzCzB
USPAS January 2012, Superconducting accelerator magnets Unit 8: Electromagnetic design episode I – 8.12
2. HOW TO BUILD A GOOD FIELD: SECTOR COILS FOR DIPOLES
We compute the central field given by a sector dipole with uniform current density j
Taking into account of current signs
This simple computation is full of consequencesB1 current density (obvious)B1 coil width w (less obvious)B1 is independent of the aperture r (much less obvious)
For a cos,
ddjI +
+
-
-
a
w
r
wj
ddj
Bwr
r 2cos
24 0
2/
0
20
1
a
a
a
sin2cos2
2 001 wjddjB
wr
r
0
0
0
01
cos2
1Re2 z
Iz
IB
USPAS January 2012, Superconducting accelerator magnets Unit 8: Electromagnetic design episode I – 8.13
2. HOW TO BUILD A GOOD FIELD: SECTOR COILS FOR DIPOLES
A dipolar symmetry is characterized byUp-down symmetry (with same current sign)Left-right symmetry (with opposite sign)
Why this configuration?Opposite sign in left-right is necessary to avoid that the field created by the left part is canceled by the right oneIn this way all multipoles except B2n+1 are canceled
these multipoles are called “allowed multipoles”Remember the power law decay of multipoles with order (Unit 5) And that field quality specifications concern only first 10-15 multipoles
The field quality optimization of a coil lay-out concerns only a few quantities ! Usually b3 , b5 , b7 , and possibly b9 , b11
+
+
-
-
r
w
...)(4
5
2
31
refref RzB
RzBBzB
...101)( 4
4
52
2
34
1refref Rzb
RzbBzB
USPAS January 2012, Superconducting accelerator magnets Unit 8: Electromagnetic design episode I – 8.14
2. HOW TO BUILD A GOOD FIELD: SECTOR COILS FOR DIPOLES
Multipoles of a sector coil
for n=2 one has
and for n>2
Main features of these equationsMultipoles n are proportional to sin ( n angle of the sector)
They can be made equal to zero !Proportional to the inverse of sector distance to power n
High order multipoles are not affected by coil parts far from the centre
rwRj
B ref 1log)2sin(02 a
a
a
a
a
wr
r
nnref
wr
rn
nref
n ddinRj
ddinRj
C 11
01
0 )exp()exp(
22
nrwr
nnRj
Bnnn
refn
2)()sin(2 221
0 a
USPAS January 2012, Superconducting accelerator magnets Unit 8: Electromagnetic design episode I – 8.15
2. HOW TO BUILD A GOOD FIELD: SECTOR COILS FOR DIPOLES
First allowed multipole B3 (sextupole)
for a=/3 (i.e. a 60° sector coil) one has B3=0
Second allowed multipole B5 (decapole)
for a=/5 (i.e. a 36° sector coil) or for a=2/5 (i.e. a 72° sector coil)
one has B5=0
With one sector one cannot set to zero both multipoles … let us try with more sectors !
wrrjR
B ref 113
)3sin(20
3a
33
40
511
5)5sin(
wrr
jRB ref a
+
+
-
-
a
w
r
USPAS January 2012, Superconducting accelerator magnets Unit 8: Electromagnetic design episode I – 8.16
2. HOW TO BUILD A GOOD FIELD: SECTOR COILS FOR DIPOLES
Coil with two sectors
Note: we have to work with non-normalized multipoles, which can be added together
Equations to set to zero B3 and B5
There is a one-parameter family of solutions, for instance (48°,60°,72°) or (36°,44°,64°) are solutions
wrrjR
B ref 113
3sin3sin3sin 1232
03
aaa
33
1234
05 )(
115
5sin5sin5sinwrr
jRB ref aaa
0.0
5.0
10.0
15.0
20.0
25.0
30.0
35.0
40.0
45.0
50.0
a1
a2
a3
0)5sin()5sin()5sin(0)3sin()3sin()3sin(
123
123
aaaaaa
USPAS January 2012, Superconducting accelerator magnets Unit 8: Electromagnetic design episode I – 8.17
2. HOW TO BUILD A GOOD FIELD: SECTOR COILS FOR DIPOLES
Let us try to better “understand” why (48°,60°,72°) has B3=B5=0
Multipole n sin nClose to 0, all multipoles are positive72° sets B5=0The wedge is symmetric around 54° preserves B5=0 It sets B3=0 since the slice [48 °,60 °] has the same contribution as [60°,72°]
hoping that now is a bit more clear …
otherwise just blindly trust algebra !
0.0
5.0
10.0
15.0
20.0
25.0
30.0
35.0
40.0
45.0
50.0
B3>0
B3<0
-1.0
-0.5
0.0
0.5
1.0
0 10 20 30 40 50 60 70 80 90
B3B5
-1.0
-0.5
0.0
0.5
1.0
0 10 20 30 40 50 60 70 80 90
B3
B5
Series9
USPAS January 2012, Superconducting accelerator magnets Unit 8: Electromagnetic design episode I – 8.18
2. HOW TO BUILD A GOOD FIELD: SECTOR COILS FOR DIPOLES
One can compute numerically the solutions of There is a one-parameter familyInteger solutions
[0°-24°, 36°-60°] – it has the minimal sector width [0°-36°, 44°-64°] – it has the minimal wedge width[0°-48°, 60°-72°][0°-52°, 72°-88°] – very large sector width (not useful)
one solution ~[0°-43.2°, 52.2°-67.3°] sets also B7=0
0.0
5.0
10.0
15.0
20.0
25.0
30.0
35.0
40.0
45.0
50.0
a1
a2
a3
0)5sin()5sin()5sin(0)3sin()3sin()3sin(
123
123
aaaaaa
0
15
30
45
60
75
90
60 65 70 75 80 85 90a3 (degrees)
Ang
le (d
egre
es)
-3
-2
-1
0
1
2
3
B7 (
units
)
a1 a2 Wedge thickness B7a1 a2
USPAS January 2012, Superconducting accelerator magnets Unit 8: Electromagnetic design episode I – 8.19
2. HOW TO BUILD A GOOD FIELD: SECTOR COILS FOR DIPOLES
We have seen that with one wedge one can set to zero three multipoles (B3, B5 and B7)
What about two wedges ?
One can set to zero five multipoles (B3, B5, B7 , B9 and B11) ~[0°-33.3°, 37.1°- 53.1°, 63.4°- 71.8°]
0)3sin()3sin()3sin()3sin()3sin( 12345 aaaaa
0)5sin()5sin()5sin()5sin()5sin( 12345 aaaaa
0)7sin()7sin()7sin()7sin()7sin( 12345 aaaaa
0)9sin()9sin()9sin()9sin()9sin( 12345 aaaaa
0)11sin()11sin()11sin()11sin()11sin( 12345 aaaaa
One wedge, b3=b5=b7=0 [0-43.2,52.2-67.3]
Two wedges, b3=b5=b7=b9=b11=0 [0-33.3,37.1-53.1,63.4- 71.8]
USPAS January 2012, Superconducting accelerator magnets Unit 8: Electromagnetic design episode I – 8.20
2. HOW TO BUILD A GOOD FIELD: SECTOR COILS FOR DIPOLES
Let us see two coil lay-outs of real magnetsThe Tevatron has two blocks on two layers – with two (thin !!) layers one can set to zero B3 and B5 The RHIC dipole has four blocks
0
20
40
60
0 20 40 60x (mm)
y (m
m)
RHIC main dipole - 1995Tevatron main dipole - 1980
0
20
40
60
0 20 40 60x (mm)
y (m
m)
USPAS January 2012, Superconducting accelerator magnets Unit 8: Electromagnetic design episode I – 8.21
2. HOW TO BUILD A GOOD FIELD: SECTOR COILS FOR DIPOLES
Limits due to the cable geometryFinite thickness one cannot produce sectors of any widthCables cannot be key-stoned beyond a certain angle, some wedges can be used to better follow the arch
One does not always aim at having zero multipoles
There are other contributions (iron, persistent currents …)
RHIC main dipoleOur case with two wedges
0
20
40
60
0 20 40 60x (mm)
y (m
m)
USPAS January 2012, Superconducting accelerator magnets Unit 8: Electromagnetic design episode I – 8.22
2. HOW TO BUILD A GOOD FIELD: SECTOR COILS FOR DIPOLES
Case of two layers, no wedges
There solutions only up to w/r<0.5There are two branches that join at w/r0.5, in a1 70° and a2 25°Tevatron dipole fits with these solutions
wrwrwrrB
211)3sin(11)3sin( 213 aa
3323315 )2(1
)(1)5sin(
)(11)5sin(
wrwrwrrB aa
0
50
0 50
a1
a2
r
w
0
20
40
60
80
0 20 40 60 80x (mm)
y (m
m)
72o
37o
Tevatron dipole: w/r0.20
0
15
30
45
60
75
90
0.0 0.1 0.2 0.3 0.4 0.5 0.6w/r (adim)
Ang
le (d
egre
es)
a1 - first branch a2 - first branch
a1 - second branch a2 - second branch
a1 a2
a2a1
Tevatron main dipole
USPAS January 2012, Superconducting accelerator magnets Unit 8: Electromagnetic design episode I – 8.23
CONTENTS
1. How to generate a perfect fieldDipoles: cos, intersecting ellipses, pseudo-solenoidQuadrupoles: cos2, intersecting ellipses
2. How to build a good field with a sector coilDipolesQuadrupoles
USPAS January 2012, Superconducting accelerator magnets Unit 8: Electromagnetic design episode I – 8.24
2. HOW TO BUILD A GOOD FIELD: SECTOR COILS FOR QUADRUPOLES
We recall the equations relative to a current line we expand in a Taylor series
the multipolar expansion is defined as
the main quadrupolar component is
the non-normalized multipoles are
.)(2
)(0
0
zzI
zB
1
11
00
0
1
1
00
0
22)(
n
nn
n
n
Rz
zR
zI
zz
zIzB
1
1
1
1
)()(n
n
nnn
n
n RziAB
RzCzB
nref
refn z
RRI
C
0
0
2
-40
0
40
-40 0 40
z 0=x 0+iy 0
x
y
z=x+iy
B=B x+iB y
20
020
02
2cos2
1Re2 z
RIz
RIB refref
USPAS January 2012, Superconducting accelerator magnets Unit 8: Electromagnetic design episode I – 8.25
2. HOW TO BUILD A GOOD FIELD: SECTOR COILS FOR QUADRUPOLES
We compute the central field given by a sector quadrupole with uniform current density j
Taking into account of current signs
The gradient is a function of w/r For large w, the gradient increases with log w
Field gradient [T/m]Allowed multipoles
ddjI
-50
0
50
-50 0 50
+
+
+
+
--
- -
r w
...)(9
10
5
62
refrefref RzB
RzB
RzBzB ....101)( 8
8
104
4
64
refref Rzb
RzbGzzB
refRBG 2
rwRj
ddRj
B refwr
r
ref 1ln2sin42cos
28 0
02
02 a
a
20
020
02
2cos2
1Re2 z
RIz
RIB refref
USPAS January 2012, Superconducting accelerator magnets Unit 8: Electromagnetic design episode I – 8.26
2. HOW TO BUILD A GOOD FIELD: SECTOR COILS FOR QUADRUPOLES
First allowed multipole B6 (dodecapole)
for a=/6 (i.e. a 30° sector coil) one has B6=0
Second allowed multipole B10
for a=/10 (i.e. a 18° sector coil) or for a=/5 (i.e. a 36° sector coil)
one has B5=0The conditions look similar to the dipole case …
44
50
611
6)6sin(
wrrjR
B ref a
88
80
1011
10)10sin(
wrrjR
B ref a
USPAS January 2012, Superconducting accelerator magnets Unit 8: Electromagnetic design episode I – 8.27
2. HOW TO BUILD A GOOD FIELD: SECTOR COILS FOR QUADRUPOLES
For a sector coil with one layer, the same results of the dipole case hold with the following transformation
Angels have to be divided by two Multipole orders have to be multiplied by two
ExamplesOne wedge coil: ~[0°-48°, 60°-72°] sets to zero b3 and b5 in dipolesOne wedge coil: ~[0°-24°, 30°-36°] sets to zero b6 and b10 in quadrupoles
The LHC main quadrupole isbased on this lay-out: one wedgebetween 24° and 30°, plus one on the outer layer for keystoning thecoil
0
20
40
0 20 40 60 80x (mm)
y (m
m)
LHC main quadrupole
USPAS January 2012, Superconducting accelerator magnets Unit 8: Electromagnetic design episode I – 8.28
2. HOW TO BUILD A GOOD FIELD: SECTOR COILS FOR QUADRUPOLES
ExamplesOne wedge coil: ~[0°-12°, 18°-30°] sets to zero b6 and b10 in quadrupoles
The Tevatron main quadrupole is based on this lay-out: a 30° sector coil with one wedge between 12° and 18° (inner layer) to zero b6 and b10 – the outer layer can be at 30° without wedges since it does not affect much b10
One wedge coil: ~[0°-18°, 22°-32°] sets to zero b6 and b10 The RHIC main quadrupole is based on this lay-out – its is the solution with the smallest angular width of the wedge
Tevatron main quadrupole
0
20
40
0 20 40 60 80x (mm)
y (m
m)
RHIC main quadrupole
0
20
40
0 20 40 60 80x (mm)
y (m
m)
USPAS January 2012, Superconducting accelerator magnets Unit 8: Electromagnetic design episode I – 8.29
3. CONCLUSIONS
We have shown how to generate a pure multipolar field
We showed cases for the dipole and quadrupole fields
We analysed the constraints for having a perfect field quality in a sector coil
Equations for zeroing multipoles can be givenLay-outs canceling field harmonics can be foundSeveral built dipoles and quadrupoles follow these guidelines
Coming soonPutting together what we saw about design and superconductor limits computation of the short sample field
USPAS January 2012, Superconducting accelerator magnets Unit 8: Electromagnetic design episode I – 8.30
REFERENCES
Field quality constraintsM. N. Wilson, Ch. 1P. Schmuser, Ch. 4A. Asner, Ch. 9Classes given by A. Devred at USPAS