Transcript Superconducting Accelerator Magnets

Accelerator Magnets
Luca Bottura
CERN Division LHC, CH-1211 Geneva 23, Switzerland
[email protected]
What you will learn today

SC accelerator magnet design
 Complex field representation in 2-D
 Multipoles and symmetries
 Elements of magnetic design

SC accelerator magnet construction
 Coil winding and assembly, structures
 LHC dipole

Field errors in SC accelerator magnets
 Linear and non linear contributions
 SC cable magnetization effects
 Interaction with current distribution
Accelerators

What for ?





a microscope for nuclear physics
X-ray source (lithography, spectrography, …)
cancer therapy
isotopes transmutation
Operation modes
 fixed target
 collider
Evolution

Livingston plot:
particle energy in
laboratory frame vs.
commissioning year
 steady increase
 main jumps happen
through technology
development
Why high energy ?

Shorter wavelength
 Increase resolution

Higher mass
 New particles

Explore early universe time, corresponding to
high energy states
Linear accelerators

Sequence of
 accelerating stations (cavities), and
 focussing elements (quadrupoles)
 E and C proportional to length
accelerated
beam
Circular accelerators

Sequence of
 accelerating stations (cavities),
 bending and focussing elements (magnets)
Energy limits

Bending radius:
E GeV 


rm 
0.3qBT 
 Example : a 1 TeV (E=1000 GeV) proton (q=1) is bent by a
5 T field on a radius r = 667 m

Synchrotron radiation:
E 4 GeV  1
dEkeV   88.5
r m m 4
 Example : a proton (m = 1840) with 1 TeV (E=1000 GeV)
bent on r = 667 m, looses dE = 0.012 keV per turn
Cost considerations
C  C1 r  C2 rB n  C3dE
Total cost:
 C1 – civil engineering, proportional to length
 C2 – magnetic system, proportional to length and field
strength
 C3 – installed power, proportional to the energy loss per turn
civil engineering
magnetic system
installed power
total
30
25
20
15
10
civil engineering
magnetic system
installed power
total
30
25
cost (a.u.)
cost (a.u.)

20
15
10
5
5
0
0
10
100
1000
radius (m)
10000
1
10
field (T)
100
CERN accelerator complex
Accelerator operation
coast
coast
It
12000
dipole current (A)
10000
8000
injection
beam
dump
I  et
energy
ramp
preinjection
6000
4000
preparation
and access
2000
0
-4000
injection
phase
-2000
0
2000
time from start of injection (s)
4000
I  t2
Bending
Uniform field (dipole)
(a)
ideal
y
real
z
x
Focussing
Gradient field (quadrupole)
de-focussing
focussing
(b)
y
z
x
FODO cell

Sequence of:
 focussing (F) – bending (O) – defocussing (D) – bending (O)
magnets
 additional correctors (see LHC example)
MB_
MSCB
MQT
MCDO
lattice dipole
lattice sextupole+orbit corrector
trim quadrupole
spool-piece decapole-octupole
MQ
MO
MQS
MCS
lattice quadrupole
lattice octupole
skew trim quadrupole
spool-piece sextupole
Magnetic field

2-D field (slender magnet), with components
only in x and y and no component along z
 Ignore z and define the complex plane s = x + i y
Complex field function:
 B is analytic in s

B  By  iBx
 Cauchy-Riemann conditions:
 ReB   ImB 

0
x
y
 ReB   ImB 

0
y
x
Field expansion

B is analytic and can be expanded in Taylor
series (the series converges) inside a currentfree disk
 Magnetic field expansion:
 s
B y  iBx   C n 
R
n 1
 ref
 Multipole coefficients:
Cn  Bn  iAn





n 1
Multipole magnets
Normal quadrupole
Normal dipole
y
y
B1
B2
x
x
Skew dipole
Skew quadrupole
y
A1
y
x
A2
x
Normalised coefficients

Cn : absolute, complex multipoles, in T @ Rref
Cn  Bn  iAn
 cn
: relative multipoles, in units @ Rref
 High-order multipoles are generally small, 100 ppm and less
of the main field
Cn
An 
4  Bn
  bn  ian
c n  10
 10 
i
Bm
Bm 
 Bm
4
Current line

Field and harmonics of a current line I located
at R = x + iy
 Field:
 Multipoles:
0 I 1
B
2 s  R
0 I
Cn  
2Rref
y
 Rref

 R



n
B
I
R
s
x
Magnetic moment

Field and harmonics of a moment m = my + mx
located at R = x + iy
m
 Field:
 0m *
B
2
2 s  R 
y
R
s
 Multipoles:
m*
C n  n 0
2
2Rref
 Rref

 R



n 1
x
B
Effect of an iron yoke - I

Current line
R’
y
 Image current:
 1
I 
I
 1
2
Riron
R 
R*
R
I
x
I’
Effect of an iron yoke - m

Magnetic moment
R’
y
 Image moment:
  1 R yoke
m  
m*
2
 1 R
2
2
Riron
R 
R*
m
R
x
m’
Magnetic design - 1
y

Field of a cos(pq) distribution
-J
+J
J  J 0 cos pq 
 Field:
B
 0 J  Rref 
 Multipoles: B p  
x
p 1


2  R 
 0 J 0  Rref 


2  R 
 s

R
 ref
p 1




p 1
Magnetic design - 2

Field of intersecting circles (and ellipses)
y
-J
 uniform field:
B  By  
 0 Jd
2
+J
x
d
Magnetic design - 3

Intersecting ellipses to generate a quadrupole
 0 Ja
 0 Jb
 0 J a  b 

 Bx  a  b y  a  b y  a  b y

 Jb
 Ja
 J a  b 
 By   0 x  0 x  0
x
ab
ab
ab

y
+J
P
-J
 uniform gradient:
g
 0 J a  b 
ab
x
b
a
Magnetic design - 4

Approximation for the ideal dipole current
distribution…
Rutherford cable
y
P
+J
-J
x
d
Magnetic design - 5

… and for the ideal quadrupole current
distribution…
y
+J
Rutherford cable
P
-J
x
b
a
Magnetic design - 6

Uniform current shells
dipole
quadrupole
y
y
-J
-J
+J
+J
x
x
+J
-J
Tevatron dipole
pole
midplane
2 current
shells (layers)
HERA dipole
wedge
2 layers
LHC dipole
LHC quadrupole
Winding in blocks
B
B
Allowed harmonics

Technical current distribution can be considered
as a series approximation:
y
y
y
-J
+J
+J
x
-J
-J
+J
=
x
+J
+-J
x
-J
+J
B
=
B1
+
+…
B3
+…
Symmetries

Dipole symmetry:
 Rotate by  and change sign to the current – the dipole is
the same

Quadrupole symmetry:
 Rotate by /2 and change sign to the current – the
quadrupole is the same

Symmetry for a magnet of order m:
 Rotate by /m and change sign to the current – the magnet
is the same
Allowed multipoles

A magnet of order m can only contain the
following multipoles (n, k, m integer)
n = (2 k + 1 ) m
 Dipole
• m=1, n={1,3,5,7,…}: dipole, sextupole, decapole …
 Quadrupole
• m=2, n={2,6,10,…}: quadrupole, dodecapole, 20-pole …
 Sextupole
• m=3, n={3,9,15,…}: sextupole, 18-pole …
Dipole magnet principle
Dipole magnet designs
4 T, 90 mm
6.8 T, 50 mm
3.4 T, 80 mm
4.7 T, 75 mm
LHC dipole
LHC dipole design
8.3 T, 56 mm
Superconducting coil
B
B
Rutherford cable
superconducting
cable
SC strand
SC filament
Collars
175 tons/m
85 tons/m
F
Iron yoke
heat exchanger
flux lines
bus-bar
gap between
coil and yoke
saturation
control
Coil ends
B
Cryostated magnet
Ideal transfer function

For linear materials (=const), no movements
(R=const), no eddy currents (dB/dt=0)
BI

Define a transfer function:
B
T   const
I
T1 
B
B1
B
 const ; T2  2  const ; T3  3  const …
I
I
I
Transfer function
Transfer function (T/kA)
0.715
MBP2N1
geometric (linear)
contribution
T = 0.713 T/kA
0.7125
saturation
dT = -6 mT/kA (1 %)
0.71
persistent currents
dT = -0.6 mT/kA (0.1 %)
0.7075
0.705
0
5000
Current (A)
10000
Saturation of the field
saturated region
(B > 2 T)
effective iron boundary
moves away from the
coil: less field
Normal sextupole
30
b3 (units @ 17 mm)
MBP2N1
20
aperture 1
aperture 2
10
0
partial compensation
of persistent currents
at injection
-10
-20
-30
0
5000
Current (A)
10000
Persistent currents
Field change DB
 Eddy currents Jc with
t=  persistent
 Diamagnetic moment at
each filament:
MDCJc*Dfil
 Jc(B,T)  MDC(DB,B,T)

-Jc
MDC +Jc
DB
Persistent currents
Measurement in MBP2O1 - Aperture 1
0
b3 (units @ 17 mm)
b3 (units @ 17 mm)
20
10
0
-10
injection
-20
1.2
-2
1
-4
0.8
-6
0.6
-8
0.4
-10
0.2
200 400 600 800
0
minimum current (A)
-30
0
500
1000
current (A)
1500
hysteresis crossing: no overshoot
possible, operation of correctors !
b5 (units @ 17 mm)
30
Persistent currents
Measurement in MTP1N2 - Aperture 2
30
1.8 K
b3 (units @ 17 mm)
20
4.2 K
10
0
-10
injection
-20
-30
0
500
1000
current (A)
a +0.1 K temperature
increase gives a +0.17
units change on b3 (1.7
units/K)
1500
Coupling currents
resistive contact
at cross-overs Rc
 dB/dt
Field ramp dB/dt
 Eddy currents Ieddy
IeddydB/dt

Coupling current effects
Normal quadrupole during ramps
2
50 A/s
B3 -B3geometric (Gauss @ 10 mm)
B2-B2geometric (Gauss @ 10 mm)
4
Normal sextupole during ramps
35 A/s
20 A/s
2
10 A/s
0
-2
-4
-6
MTP1N2
MTP1N2
50 A/s
35 A/s
1
20 A/s
0
10 A/s
-1
-2
-3
-8
0
2
4
6
field (T)
8
10
0
2
4
6
field (T)
allowed and non-allowed multipoles !
8
10
Decay and Snap-back
LHC operation cycle
decay
5
1500
5000
0
-2000
0
2000
4000
6000
4
1300
3
1100
2
900
snap-back
1
700
0
500
1500
time from beginning of injection (s)
0
500
1000
time from beginning of injection (s)
dipole current (A)
10000
b3 (units @ 17 mm)
dipole current (A)
15000
Decay and snap-back
Snap-back at
the start of the
acceleration
ramp
b3 (units @ 17 mm)
0
-2
-4
-6
decay during
injection
-8
MBP2N1
-10
700
750
current (A)
800
Decay
exponential fit
ti = 900 s
b3 (units @ 17 mm)
0
-2
-4
-6
decay during
simulated 10,000 s
MBP2N1
injection
-8
-10
0
5000
10000
time from start of injection (s)
15000
Snap-back
snap-back fit:
Db3 [1-(I-Iinj)/DI]3
Db3= 3.7units
DI = 27A  DB = 19 mT
decay
snap-back
Decay and SB physics

Current distribution is not uniform in the cables
 joints
 supercurrents
 I/Ic

non-uniform dB/dt
Current distribution changes in time , causing a
variable rotating field...
Decay and SB physics
Magnetization of a typical LHC strand
… the local field
change in turn
affects the
magnetization of
the SC filaments:
average M
decreases
(decay)
0.07
0.06
0.05
0.04
net decrease of
magnetization
M (T)
0.03
0.02
0.01
0
-0.01
DB
-0.02
DB
-0.03
0
0.2
0.4
0.6
field (T)
0.8
1
Decay and SB physics

The magnetization state is reestablished as soon as the
background field is increased
(snap-back)

The background field change necessary is of the
same order of the internal field change in the
cable
  100 A change in current imbalance
  10 mT average internal field change (vs. 5…20 mT
measured)
A demonstration
Demonstration experiment
at Twente University.
Courtesy of M. Haverkamp
NbTi strand
Copper strands
B
measured
0
0.01
-0.002
+DI
+2DI
0
-DI
M (T)
Strand Magnetization M (T)
0.02
+DI
-0.004
-0.006
-0.01
-0.008
0.48
-0.02
0
0.2
0.4
Background Field B (T)
0.6
computed
0.5
0.52
B (T)
0.54
A bit of reality…
Field quality reconstructed from measurements
performed in MBP2N1
 Plot of homogeneity |B(x,y)-B1|/B1 inside the
aperture of the magnet:





blue  OK (1  10-4)
green  so, so (5  10-4)
yellow  Houston, we have a problem (1  10-3)
red  bye, bye (5  10-3)
… the measurement
Transfer function (T/kA)
0.715
12000
8000
0.7125
0.71
0.7075
0.705
6000
0
5000
Current (A)
10000
4000
30
MBP2N1
2000
0
-4000
-2000
0
2000
time from start of injection (s)
4000
b3 (units @ 17 mm)
dipole current (A)
10000
MBP2N1
20
aperture 1
aperture 2
10
0
-10
-20
-30
0
5000
Current (A)
10000
Sony Playstation III (or Tracking
the LHC...)
Field homogeneity
reconstructed from
measurements
Rref = 17 mm
operating current
coil of MBP2N1