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Pulsed Superconducting Magnets (for
accelerators)
Plan of the Lectures
Martin N Wilson (Rutherford Lab  Oxford Instruments  consultant)
1. Introduction to Superconductors
• where to find more information
• properties of superconductors, critical field, critical
temperature & critical current density
• screening currents and the critical state model
2. Magnetization, AC Losses & Filamentary Wires
4. AC Losses in Magnets - and Training
•
•
•
•
summation of ac loss and refrigeration load
temperature rise and temperature margin
measurement of ac loss
training - energy releases in the magnet
conductor design for stability
•
•
•
•
irreversible magnetization and hysteresis loops
field errors and flux jumping
5. Quenching and some other Accelerators
ac losses in terms of magnetization
• the quench process
fine filaments & composite wires, coupling & twisting • decay times and temperature rise
3. Cables and Materials Manufacture
• propagation of the resistive zone
• why accelerators need cables
• quench protection schemes
• coupling currents in cables: anisotropy
• case study: LHC protection
• interstrand resistance
• pictures of superconducting accelerators
• manufacture of wire and cable
Martin Wilson Lecture 1 slide 1
'Pulsed Superconducting Magnets' CERN Academic Training
May 2006
Pulsed Superconducting Magnets- why bother?
Superconductivity Abolishes Ohm’s Law !
no Ohm's law
• no power consumption (except refrigeration)
 lower power bills
• ampere turns are cheap, so don’t need iron (except shielding)
 higher magnetic fields
 higher energies and /or smaller rings
 reduced capital cost
• high current density
 compact windings
 high gradients
 higher luminosity
but
• superconductors suffer losses when the field changes
 rise in temperature (and there's not much margin)
 increase in refrigeration load
Martin Wilson Lecture 1 slide 2
'Pulsed Superconducting Magnets' CERN Academic Training
May 2006
Some useful references
Superconducting Magnets
• Superconducting Accelerator Magnets: KH Mess, P
Schmuser, S Wolf., pub World Scientific, (1996) ISBN
981-02-2790-6
• High Field Superconducting Magnets: FM Asner, pub
Oxford University Press (1999) ISBN 0 19 851764 5
• Case Studies in Superconducting Magnets: Y Iwasa,
pub Plenum Press, New York (1994), ISBN 0-30644881-5.
• Superconducting Magnets: MN Wilson, pub Oxford
University Press (1983) ISBN 0-019-854805-2
• Proc Applied Superconductivity Conference: pub as
IEEE Trans Applied Superconductivity, Mar 93 to 99,
and as IEEE Trans Magnetics Mar 75 to 91
• Handbook of Applied Superconductivity ed B Seeber,
pub UK Institute Physics 1998
Cryogenics
• Helium Cryogenics Van Sciver SW, pub Plenum 86
ISBN 0-0306-42335-9
• Cryogenic Engineering, Hands BA, pub Academic
Press 86 ISBN 0-012-322991-X
• Cryogenics: published monthly by Butterworths
• Cryogenie: Ses Applications en Supraconductivite, pub
IIR 177 Boulevard Malesherbes F5017 Paris France
Martin Wilson Lecture 1 slide 3
Materials Mechanical
• Materials at Low Temperature: Ed RP Reed & AF Clark,
pub Am. Soc. Metals 1983. ISBN 0-87170-146-4
• Handbook on Materials for Superconducting Machinery
pub Batelle Columbus Laboratories 1977.
• Nonmetallic materials and composites at low
temperatures: Ed AF Clark, RP Reed, G Hartwig pub
Plenum
• Nonmetallic materials and composites at low
temperatures 2, Ed G Hartwig, D Evans, pub Plenum
1982
• Austenitic Steels at low temperatures Editors R.P.Reed
and T.Horiuchi, pub Plenum1983
Superconducting Materials
• Superconductor Science and Technology, published
monthly by Institute of Physics (UK).
• Superconductivity of metals and Cuprates, JR Waldram,
Institute of Physics Publishing (1996) ISBN 0 85274
337 8
• High Temperature Superconductors: Processing and
Science, A Bourdillon and NX Tan Bourdillon,
Academic Press, ISBN 0 12 117680 0
'Pulsed Superconducting Magnets' CERN Academic Training
May 2006
Materials data web sites
• Cryogenic properties (1-300 K) of many solids,
including thermal conductivity, specific heat, and
thermal expansion, have been empirically fitted and
the equation parameters are available free on the
web at www.cryogenics.nist.gov.
• Plots and automated data-look-up using the NIST
equations are available on the web for a fee from
www.cpia.jhu.edu.
• Other fee web sites that use their own fitting
equations for a number of cryogenic material
properties include: www.cryodata.com (cryogenic
properties of about 100 materials),
and www.jahm.com (temperature dependent
properties of about 1000 materials, many at
cryogenic temperatures).
• Commercially supplied room-temperature data are
available free online for about 10 to 20 properties of
about 24,000 materials at www.matweb.com.
Cryodata Software Products
GASPAK
properties of pure fluids from the triple
point to high temperatures.
HEPAK
properties of helium including superfluid
above 0.8 K, up to 1500 K.
STEAMPAK
properties of water from the triple point to
2000 K and 200 MPa.
METALPAK, CPPACK, EXPAK
reference properties of metals and other
solids, 1 - 300 K.
CRYOCOMP
properties and thermal design calculations
for solid materials, 1 - 300 K.
SUPERMAGNET
four unique engineering design codes for
superconducting magnet systems.
KRYOM
numerical modelling calculations on
radiation-shielded cryogenic enclosures.
thanks to Jack Ekin of NIST for this information
Martin Wilson Lecture 1 slide 4
'Pulsed Superconducting Magnets' CERN Academic Training
May 2006
The critical surface of niobium titanium
• Niobium titanium NbTi is the standard
‘work horse’ of the superconducting magnet
business
Jc
• it is a ductile alloy
qc
Bc2
Current density (kA.mm-2)
• picture shows the critical surface, which is
the boundary between superconductivity and
normal resistivity in 3 dimensional space
• superconductivity prevails everywhere
below the surface, resistance everywhere
above it
• we define an upper critical field Bc2 (at zero
temperature and current) and critical
temperature qc (at zero field and current)
which are characteristic of the alloy
composition
• critical current density Jc(B,q) depends on
processing
Martin Wilson Lecture 1 slide 5
'Pulsed Superconducting Magnets' CERN Academic Training
May 2006
Critical current density A.mm-2
The critical line at 4.2K
• because magnets usually work in
boiling liquid helium, the critical
surface is often represented by a curve
of current versus field at 4.2K
104
Nb3S
n
103
• but it is brittle intermetallic compound
with poor mechanical properties
NbTi
• note that both the field and current
density of both superconductors are
way above the capability of
conventional electromagnets
102
10
• niobium tin Nb3Sn has a much higher
performance in terms of critical
current field and temperature than
NbTi
Conventional
iron yoke
electromagnets
Magnetic field (Tesla)
Martin Wilson Lecture 1 slide 6
'Pulsed Superconducting Magnets' CERN Academic Training
May 2006
Filamentary composite wires
• for reasons that will be described later,
superconducting materials are always
used in combination with a good
normal conductor such as copper
• to ensure intimate mixing between the
two, the superconductor is made in the
form of fine filaments embedded in a
matrix of copper
• typical dimensions are:
• wire diameter = 0.3 - 1.0mm
• filament diameter = 10 - 60mm
• for electromagnetic reasons, the
composite wires are twisted so that the
filaments look like a rope (see Lecture
3 on filamentary conductors and
cables)
Martin Wilson Lecture 1 slide 7
'Pulsed Superconducting Magnets' CERN Academic Training
May 2006
Critical properties: temperature and field 1
Critical Temperature qc
3.5k Bq c  2D(0)
where kB is Boltzmann's constant
and D(0) is the energy gap (binding
energy of Cooper pairs) of at q = 0
Critical Field Bc:
Type 1 superconductors show the
Meissner effect. Field is expelled when
sample becomes superconducting
It costs energy to keep the field out. Critical
field happens when the condensation energy
of the superconducting state is just equal to
the energy penalty of keeping the field out.
Bc 2
 Gn  Gs
2m o
where G is the Gibbs Free Energy of the
normal/superconducting state. BCS theory
says
Gn (0)  Gs (0)  1 / 2 N F (D(0)) 2
where NF is the density of states at the
Fermi surface of metal in normal state calculate it from:
  2 / 3 2 N F k B 2
where  is Sommerfeld coefficient of
electronic specific heat
C  q  Aq 3
Martin Wilson Lecture 1 slide 8
'Pulsed Superconducting Magnets' CERN Academic Training
May 2006
Critical properties: temperature and field 2
combining the previous equations:
1
3m 0  2
1
1
3.5 2
Bc (0)  
 q c  7.65x10 4  2q c

 2  2
'thermodynamic critical field' Bc
so like the critical temperature, Bc is
defined by the 'chemistry'
typically for NbTi  ~ 103 J m-3 K-1
so if q = 10 K Bc = 0.24T
Conclusion:
Type 1 superconductors are useless
for magnets!
Note however: Meissner effect is not total, the magnetic field actually penetrates a small
distance l the London Penetration Depth.
Another characteristic distance is the coherence length z
- the minimum distance over which the electronic state can change from superconducting to
normal
Martin Wilson Lecture 1 slide 9
'Pulsed Superconducting Magnets' CERN Academic Training
May 2006
Critical properties: type 2 superconductors
Theory of Ginsburg, Landau, Abrikosov and
Gorkov GLAG
kl/x
defines the ratio
If k > 1/2 the magnetic field can penetrate
in the form of discrete fluxoids - Type 2
a single fluxoid encloses flux
I
o 
  h / 2e
h
 2 1015Webers
2e
where h = Planck's constant,
e = electronic charge
upper
critical field
Bc 2  2k Bc
thus the upper critical field
for NbTi:
in the
‘dirty limit'
Bc 2  3.1 10 3  nq c
 ~ 900 J m -3 K-2
Martin Wilson Lecture 1 slide 10
k  2.4 10 
6
1
2
n ~ 65 x10 -8 W m
qc = 9. K
n
where n is the normal
state resistivity
- best superconductors
are best resistors!
hence
Bc2 = 18.5 T
'Pulsed Superconducting Magnets' CERN Academic Training
May 2006
Critical properties: current density
Fluxoids consist of resistive cores with supercurrents circulating round them.
precipitates of a Ti in Nb Ti
spacing between the fluxoids
1
o  2
2
d 

 3 B
 22nm at 5T
a uniform distribution of fluxoids gives no net
current, so Jc= 0,
but a gradient produces a net current density
CurlB  mo J
• gradients must be produced by inhomogeneities
in the material, eg dislocations or precipitates
• process is known as flux pinning
• flux pinning is an irreversible lossy process
Martin Wilson Lecture 1 slide 11
fluxoid lattice at 5T on the same scale
'Pulsed Superconducting Magnets' CERN Academic Training
May 2006
Critical properties: a summary
•
Critical temperature: choose the
right material to have a large energy
gap or 'depairing energy'
•
Critical field: choose a Type 2
superconductor with a high critical
temperature and a high normal state
resistivity
•
Critical current density: mess up
the microstructure by cold working
and precipitation heat treatments
- this is the only one where we have
any control
Similar effects in high
temperature superconducting
materials : fluxoid lattice in
BSCCO
Martin Wilson Lecture 1 slide 12
'Pulsed Superconducting Magnets' CERN Academic Training
May 2006
Upper critical fields of metallic superconductors
Note: of all the metallic
superconductors, only NbTi is ductile.
All the rest are brittle intermetallic
compounds
2000
Jc (A/mm2)
MgB2 at 5K
MgB2 at 5K
NbTi at 5K
1000
0
0
B (T)
4
MgB2
Martin Wilson Lecture 1 slide 13
'Pulsed Superconducting Magnets' CERN Academic Training
May 2006
8
High temperature superconductors
80
• many superconductors with critical
temperature above 90K - BSCCO and YBCO
70
• operate in liquid nitrogen?
Upper critical field Bc2 (T)
60
50
YBa2Cu3O7
40
'YBCO'
B2212
30
NbTi
20
10
0
0
20
40
60
80
100
Critical temperature Tc (K)
Martin Wilson Lecture 1 slide 14
'Pulsed Superconducting Magnets' CERN Academic Training
May 2006
High temperature superconductors
YBCO structure
Conduction layers consist of two CuO2
layers separated by yttrium atoms.
The charge layer consists of copper,
barium and oxygen atoms
Note: this structure makes the properties
highly anisotropic
Martin Wilson Lecture 1 slide 15
'Pulsed Superconducting Magnets' CERN Academic Training
May 2006
Irreversibility line - a big disappointment
Unlike the metallic superconductors, HTS do not
have a sharply defined critical current.
At higher temperatures and fields, there is an
'flux flow' region, where the material is resistive
- although still superconducting
The boundary between flux pinning and flux
flow is called the irreversibility line


q
metallic
q
Oxide HTS
M agnetic Field
Normal Resistivity
Flux Flow
Flux Pinning
Temperature
Martin Wilson Lecture 1 slide 16
'Pulsed Superconducting Magnets' CERN Academic Training
May 2006
Engineering current density
In designing a magnet, what really
matters is the overall 'engineering'
current density Jeng
J eng 
current
 J supercon  lmetal  lwinding
unit cell area
lmetal 
fill factor in the wire
insulation
NbTi
Cu
1
1  mat )
where mat = matrix : superconductor ratio
typically:
for NbTi mat = 1.5 to 3.0 ie lmetal = 0.4 to 0.25
for Nb3Sn mat ~ 3.0
ie
lmetal ~ 0.25
for B2212 mat = 3.0 to 4.0 ie lmetal = 0.25 to 0.2
lwinding takes account of space occupied by
insulation, cooling channels, mechanical
reinforcement etc and is typically 0.7 to 0.8
So typically Jeng is only 15% to 30% of Jsupercon
Martin Wilson Lecture 1 slide 17
'Pulsed Superconducting Magnets' CERN Academic Training
May 2006
Importance of (engineering) current density: (1)
solenoids
• the field produced by an infinitely long
solenoid is
B  mo J e t
• in solenoids of finite length the central
field is
B  mo f J e t
where f is a factor less than 1,
typically ~ 0.8
• so the thickness (volume, cost) of a
solenoid to produce a given field is
inversely proportional to the
engineering current density Je
Martin Wilson Lecture 1 slide 18
'Pulsed Superconducting Magnets' CERN Academic Training
May 2006
Importance of (engineering) current density: (2)
dipoles
I
II
Je = 37.5
Amm-2
field produced
by a perfect
dipole is
B
B  mo J e
t
2
LHC dipole
Je = 375 Amm-2
120mm
660mm
9.5x105 Amp turns
9.5x106 Amp turns
=1.9x106 A.m per m
=1.9x107 A.m per m
Martin Wilson Lecture 1 slide 19
'Pulsed Superconducting Magnets' CERN Academic Training
May 2006
Some engineering current densities
600
Nb3Sn at 4.2K
NbTi at 1.9K
500
Je A/mm
2
NbTi at 4.2K
B2212 at 4.2K
400
B2212 at 35K
300
200
100
0
0
Martin Wilson Lecture 1 slide 20
5
10
B Tesla
15
20
25
'Pulsed Superconducting Magnets' CERN Academic Training
May 2006
Measurement of critical current
•
•
this sample
holder is placed
in the bore of a
superconductin
g solenoid,
usually in
liquid helium
boiling at 4.2K
at each field
level the
current is
slowly
increased and
voltage across
the test section
is measured
Martin Wilson Lecture 1 slide 21
'Pulsed Superconducting Magnets' CERN Academic Training
May 2006
Resistive transition 1
When measured sensitively, the boundary between
superconducting and resistive states is not sharp, but
slightly blurred.
If we measure Jc with
voltage taps across the
sample, we see that the
voltage rises gradually.
To define Jc, we must
therefore define a
measurement
sensitivity in terms of
electric field or
effective resistivity.
Commonly used definitions are   10-14 Wm or E = 1 mV.m-1
Overall Current density 10 8 A.m-2
Martin Wilson Lecture 1 slide 22
Critical current defined at this level is about what you would
expect the conductor in a resin impregnated solenoid to
achieve. At higher resistivity, self heating starts to raise the
internal temperature and reduce the critical current
'Pulsed Superconducting Magnets' CERN Academic Training
May 2006
Resistive transition 2
It has been found empirically that the resistive
transition may be represented by a power law
J 
 ( J )  o  
Jo 
n
a) perfect filament
b) mid sausaging
c) bad sausaging
where n is called the resistive transition index.
The effect is partly intrinsic and partly geometrical.
'Sausaging of the filaments, forces current to cross the
copper matrix as critical current is approached.
resistive transition can be the main source of decay in
persistent magnets
'n' is often taken as a measure of quality - look for n > 50
HTS conductors so far have low n ~ 5 - 10
Martin Wilson Lecture 1 slide 23
'Pulsed Superconducting Magnets' CERN Academic Training
May 2006
Screening currents and the critical state
model
• when a superconductor is subjected to a
changing magnetic field, screening currents
are induced to flow
• screening currents are in addition to the
transport current, which comes from the
power supply
• they are like eddy currents but, because
there is no resistance, they don't decay
• usual model is a superconducting slab in a
changing magnetic field By
• assume it's infinitely long in the z and y directions
- simplifies to a 1 dim problem
• dB/dt induces an electric field E which causes
screening currents to flow at critical current
density Jc
• known as the critical state model or Bean model
• in the 1 dim infinite slab geometry, Maxwell's
equation says
J
B y
x
J
B
x
Martin Wilson Lecture 1 slide 24
 m o J z  m o J c
• so uniform Jc means a constant field gradient
inside the superconductor
• everywhere in the superconductor the current
density is either Jc or zero
'Pulsed Superconducting Magnets' CERN Academic Training
May 2006
The flux penetration process
plot field profile across the slab
B
fully penetrated
field increasing from zero
field decreasing through zero
Martin Wilson Lecture 1 slide 25
'Pulsed Superconducting Magnets' CERN Academic Training
May 2006
The flux penetration process
plot field profile across the slab
B
fully penetrated
field increasing from zero
field decreasing through zero
Martin Wilson Lecture 1 slide 26
'Pulsed Superconducting Magnets' CERN Academic Training
May 2006
Lecture 1: concluding remarks
• superconducting magnets burn no power (dc); they produce high fields and high gradients
- but in changing fields they suffer ac losses
• three types of superconductor
- type 1: unsuitable for high field
- type 2: good for high field - but must work hard to get current density
- HTS: good for high field & temperature - but current density still a problem in field
• all superconducting accelerators to date use type 2, usually NbTi (45 years after its discovery)
• performance of type 2 superconductors is described by the critical surface in B q J space,
- properties in B and q space are reversible properties determined by the chemistry
- properties in J space are irreversible and lossy, determined by flux pinning inhomogeneities
• engineering current density is what counts for magnet performance
• changing magnet fields induce persistent currents in superconductors
• persistent currents may be described by the Bean critical state model,
- we shall see they are a major cause of ac losses
Martin Wilson Lecture 1 slide 27
'Pulsed Superconducting Magnets' CERN Academic Training
May 2006