Lecture 2: Magnetization, cables and ac losses
Magnetization
• superconductors in changing fields
• magnetization of wires & filaments
fine filaments in LHC wire
• coupling between filaments
Cables
• why cables?
• coupling in cables
• effect on field error in magnets
AC losses
Rutherford cable
• general expression
• losses within filaments
• losses from coupling
Martin Wilson Lecture 3 slide1
JUAS Febuary 2013
Superconductors in changing magnetic fields
I
φ
Ḃ
E
E
Faraday’s Law of EM Induction
 Edl   B dA  
• changing field
→ changing flux linked by loop
→ electric field E in superconductor
→ current Ir flows around the loop
• change stops
→ electric field goes to zero
→ superconductor current falls back to Ic (not zero)
→ current circulates for ever persistent current
Martin Wilson Lecture 3 slide2
Ic
Ir
changing magnetic fields
on superconductors
→ electric field
→ resistance
→ power dissipation
JUAS Febuary 2013
Persistent screening currents
• screening currents are in
addition to the transport current,
which comes from the power supply
• like eddy currents but, because no
resistance, they don't decay
simplified slab model
infinite in y and z
• dB/dt induces an electric field E which drives
the screening current up to current density Jr
• dB/dt stops and current falls back to Jc
• so in the steady state we have persistent J = +Jc
or J = -Jc or J = 0 nothing else
• known as the critical state model or Bean
London model
J
• in the 1 dim infinite slab geometry, Maxwell's
equation says
B y
J
B
x
  o J z   o J c
x
B
Martin Wilson Lecture 3 slide3
• so a uniform Jc means a constant field
gradient inside the superconductor
JUAS Febuary 2013
The flux penetration process
plot field profile across the slab
B
fully penetrated
field increasing from zero
Bean London critical state model
• current density everywhere is Jc or zero
• change comes in from the outer surface
Martin Wilson Lecture 3 slide4
field decreasing through zero
JUAS Febuary 2013
Magnetization of the Superconductor
for cylindrical filaments the inner current boundary
is roughly elliptical (controversial)
When viewed from outside the sample,
the persistent currents produce a
magnetic moment.
Problem for accelerators because it
spoils the precise field shape
We can define a magnetization
(magnetic moment per unit volume)
M 
V
J
J
J
B
I .A
V
NB units of H
for a fully penetrated slab
when fully penetrated, the magnetization is
Ms 
B
2a
a
1
J a
M s   J c x dx  c
a0
2
Martin Wilson Lecture 3 slide5
4
2
Jc a 
Jc d f
3π
3π
where a, df = filament radius, diameter
Note: M is here defined per unit volume of NbTi filament
to reduce M need small d - fine filaments
JUAS Febuary 2013
Magnetization of NbTi
M
Bext
Hysteresis
like iron, but
diamagnetic
M
H
iron
Martin Wilson Lecture 3 slide6
Magnetization is important because
it produces field errors and ac losses
JUAS Febuary 2013
Coupling between filaments
recap
Ms 
2
Jc d f
3π
• reduce M by making fine filaments
• for ease of handling, filaments are
embedded in a copper matrix
• coupling currents flow along the
filaments and across the matrix
• reduce them by twisting the wire
• they behave like eddy currents and
produce an additional magnetization
dB 1  pw 
Me 
dt ρt  2π 
Me 
• but in changing fields, the filaments are
magnetically coupled
• screening currents go up the left filaments
and return down the right
Martin Wilson Lecture 3 slide7
2
μ
τ o
2ρt
2 dB
τ where
μo dt
 pw 
 2π 
 
2
per unit volume of wire
rt = resistivity across matrix, pw = wire twist pitch
JUAS Febuary 2013
Transverse resistivity across the matrix
Poor contact to filaments
J
r t  r Cu
1  λsw
1  λsw
Thick copper jacket
include the
copper jacket
as a resistance
in parallel
Martin Wilson Lecture 3 slide8
Good contact to filaments
J
where lsw is the fraction of superconductor
in the wire cross section (after J Carr)
J
r t  r Cu
1  λsw
1  λsw
Some complications
Copper core
resistance in
series for part
of current
path
JUAS Febuary 2013
Computation of current flow across matrix
B
B
calculated using
the COMSOL
code by
P.Fabbricatore et
al JAP, 106,
083905 (2009)
Martin Wilson Lecture 3 slide9
JUAS Febuary 2013
Two components of magnetization
Magnetization
1) persistent current within the filaments
M s  λsu
Mf depends on B
2
J c(B) d f
3π
where lsu = fraction of
superconductor in the unit cell
External field
Me
Ms
2) eddy current coupling
between the filaments
dB 1  pw 
M e  lwu
dt ρt  2π 
Me depends on dB/dt
or
Magnetization is averaged over the unit cell
Martin Wilson Lecture 3 slide10
M e  lwu
2
2 dB
μ p 
τ where τ  o  w 
μo dt
2ρt  2π 
2
where lwu = fraction of wire in the section
JUAS Febuary 2013
Measurement of magnetization
In field, the superconductor behaves just
like a magnetic material. We can plot the
magnetization curve using a magnetometer.
It shows hysteresis - just like iron only in
this case the magnetization is both
diamagnetic and paramagnetic.
M
d1 d2

dt
dt
integrate
Δφ1  Δφ2  ΔM
B
Note the minor loops, where
field and therefore screening
currents are reversing
Two balanced search coils connected in series opposition, are placed within the bore of a superconducting solenoid.
With a superconducting sample in one coil, the integrator measures DM when the solenoid field is swept up and down
Martin Wilson Lecture 3 slide11
JUAS Febuary 2013
Magnetization measurements
RRP Nb3Sn
wire with
50m
filaments
flux jumping at low
field caused by large
filaments and high Jc
NbTi wire for RHIC
with 6m filaments
5105
Magnetization M (A/m) .
Magnetization M A/m .
16000
0
-5105
-3
Martin Wilson Lecture 3 slide12
total
magnetization
0
reversible
magnetization
-16000
0
field B Tesla
5
-4
0
Field B (T)
JUAS Febuary 2013
4
Fine filaments for low magnetization
Accelerator
magnets need
the finest
filaments
- to minimize
field errors and
ac losses
single stack
double stack
Martin Wilson Lecture 3 slide13
Typical diameters are in
the range 5 - 10m.
Even smaller diameters
would give lower
magnetization, but at the
cost of lower Jc and more
difficult production.
JUAS Febuary 2013
Cables - why do we need them?
• for accurate tracking we connect
synchrotron magnets in series
• for a given field and volume, stored energy of a
magnet is fixed, regardless of current or inductance
• for rise time t and operating current I
, charging voltage is
V
RHIC
1 2 B2
E  LI 
vol
2
2μo
L I 2E

t
It
E = 40kJ/m, t = 75s, 30 strand cable
cable I = 5kA, charge voltage per km = 213V
wire I = 167A, charge voltage per km = 6400V
FAIR at GSI E = 74kJ/m, t = 4s, 30 strand cable
cable I = 6.8kA, charge voltage per km = 5.4kV
wire I = 227A, charge voltage per km = 163kV
• so we need high currents!
the RHIC tunnel
• a single 5m filament of NbTi in 6T carries 50mA
• a composite wire of fine filaments typically has 5,000 to 10,000 filaments, so it carries 250A to 500A
• for 5 to 10kA, we need 20 to 40 wires in parallel - a cable
Martin Wilson Lecture 3 slide14
JUAS Febuary 2013
Cable transposition
• many wires in parallel - want them all to carry same current
zero resistance - so current divides according to inductance
• in a simple twisted cable, wires in the centre have a higher
self inductance than those at the outside
• current fed in from the power supply therefore takes the low
inductance path and stays on the outside
• outer wires reach Jc while inner are still empty
• so the wires must be fully transposed, ie
every wire must change places with every
other wire along the length
inner wires  outside outer wire  inside
• three types of fully transposed cable
have been tried in accelerators
- rope
- braid
- Rutherford
Martin Wilson Lecture 3 slide15
JUAS Febuary 2013
Rutherford
cable
• Rutherford cable succeeded where others failed
because it could be compacted to a high density (88 94%) without damaging the wires, and rolled to a
good dimensional accuracy (~ 10m).
• Note the 'keystone angle', which enables the cables to
be stacked closely round a circular aperture
Martin Wilson Lecture 3 slide16
JUAS Febuary 2013
Manufacture of Rutherford cable
'Turk's head' roller die
puller
finished
cable
superconducting
wires
Martin Wilson Lecture 3 slide17
JUAS Febuary 2013
Coupling in Rutherford cables
B
• Field transverse
coupling via crossover resistance Rc
Ra
Rc
• Field parallel coupling via
adjacent resistance Ra
usually negligible
• Field transverse
coupling via adjacent
resistance Ra
B
crossover resistance Rc
adjacent resistance Ra
B
Martin Wilson Lecture 3 slide18
JUAS Febuary 2013
Magnetization from coupling in cables
B`
• Field transverse
coupling via
crossover
resistance Rc
1 Bt c
1 Bt 2 c 2
M tc 
p N(N  1) 
p 2
120 Rc b
60 ρc
b
2b
2c
where M = magnetization per unit volume of cable, p= twist pitch, N = number of strands
Rc Ra = resistance per crossover rc ra = effective resistivity between wire centres
• Field transverse
coupling via adjacent resistance Ra
1 Bt c 1 Bt p 2
M ta 
p 
6 Ra b 48 r a Cos 2θ
where q = slope angle of wires Cosq ~ 1
• Field parallel
coupling via adjacent resistance Ra
• Field transverse
ratio crossover/adjacent
M pa
1 B p b 1 B p p 2 b 2

p 
8 Ra c 64 ρa cos 2θ c 2
(usually
negligible)
M tc Ra N(N  1)
R

 45 a
M ta Rc
20
Rc
So without increasing loss too much can make Ra 50 times less than Rc - anisotropy
Martin Wilson Lecture 3 slide19
JUAS Febuary 2013
Cable coupling adds more magnetization
filament magnetization Mf depends on B
Magnetization
M s  λsu
External field
Mtc
2
J c(B) d f
3π
coupling between filaments Me
depends on dB/dt
dB 1  pw 
M e  lwu
dt ρt  2π 
Me
Ms
2
coupling
between wires
in cable depends on dB/dt
1 Bt c
M tc  lcu
p N(N  1)
120 Rc b
1 Bt c
M ta  lcu
p
6 Ra b
where lcu = fraction of cable in the section
Martin Wilson Lecture 3 slide20
JUAS Febuary 2013
Controlling Ra and Rc
1000
bare copper
untreated Staybrite
• surface coatings on the wires are used
to adjust the contact resistance
• the values obtained are very sensitive
to pressure and heat treatments used in
coil manufacture (to cure the adhesive
between turns)
• data from David Richter CERN
Cored Cables
• using a resistive core allows
us to increase Rc while
keeping Ra the same
Resistance per crossover Rc 
nickel
oxidized Stabrite
100
10
1
0.1
0
50
100
150
200
Heat treatment temperature C
250
• thus we reduce losses but
still maintain good current
transfer between wires
• not affected by heat
treatment
Martin Wilson Lecture 3 slide21
JUAS Febuary 2013
Magnetization and field errors – an extreme case
Magnetization is important in accelerators because it produces field error. The effect is worst
- DB/B is greatest
at injection because
- magnetization, ie DB is greatest at low field
skew quadrupole error
300
6 mT/sec
13 mT/seec
19 mT/sec
200
skew
quadrupole
error in
Nb3Sn dipole
which has
exceptionally
large
coupling
magnetization
(University of
Twente)
100
0
-100
-200
-300
0
Martin Wilson Lecture 3 slide22
1
2
Field B (T)
3
4
5
JUAS Febuary 2013
AC loss power
E
Faraday's law of induction
d
 Edl  dt  BdA
J
A
Jc
loss power / unit length in slice of width dx
p( x )  E J c dx 
dB
J c dx
dt
P  B M
B
total loss in slab per unit volume
a
dx
a
a
1
1 dB
a
P   p( x )dx 
J c  x dx B J c  B M
a0
a dt 0
2
x
for round wires (not proved here)
J
B
4 
2 
P  B M 
B Jc a 
B Jc d f
3π
3π
also for coupling magnetization
1
Pe  B M e  B 2
ρt
Martin Wilson Lecture 3 slide23
2
B 2
 pw 
 2π    2
o
JUAS Febuary 2013
Hysteresis loss
T2
B2
T2
loss over a field ramp
B1
B
2
dB
Q  M
dt   MdB
dt
T1
B1
loss per ramp

independent of B
T1
• in general, when the field changes by dB the magnetic field energy changes by
(see textbooks on electromagnetism)
• so work done by the field on the material
dE  HdB
W   o HdM
• around a closed loop, this integral must be the energy dissipated in the material
Q   o HdM   o MdH
hysteresis loss
per cycle
(not per ramp)
Martin Wilson Lecture 3 slide24
M
H
just
like
iron
JUAS Febuary 2013
Integrated loss over a ramp
1) Screening currents within filaments
B2
Jc and M vary
with field
Q   M(B)dB
B1
round
wire
J
J o Bo
Kim Anderson
J c ( B) 
( B  Bo )
approximation
good at low field, less so at high field
 B2  B0 
2 2 J o Bo
2
Q
d
dB

d
J
B
ln

f
f o o 
3π B1 (B  Bo)
3π
B

B
0 
 1
B
2
M s(B) 
J c (B)d f
3π
loss per ramp, independent of B
2) Coupling currents between filaments
1
M e  B
ρt
 pw 
 2π 
2
B
(B2  B1 )2 1  pw 

Q   M(B)dB 
(T2  T2 ) ρt  2π 
B1
B2
T2
2
B2
linear ramp
B1
T1
3) Coupling currents between wires in cable
1 Bt 2 c 2
M tc 
pc 2
60 ρc
b
Martin Wilson Lecture 3 slide25
2
2
2
(B

B
)
1
p
c
2
1
c
Q   M tc(B )dB 
2
(T

T
)
r
60
b
2
2
c
B1
loss per ramp ~ 1/DT
B2
all per unit volume
- volume of what?
JUAS Febuary 2013
The effect of transport current
• in magnets there is a transport current, coming from the
power supply, in addition to magnetization currents.
• because the transport current 'uses up' some of the
available Jc the magnetization is reduced.
• but the loss is increased because the power supply does
work and this adds to the work done by external field
total loss is increased by factor (1+i2)
E
where i = Imax / Ic
 B  Bo 
2
2
d f J o Bo ln  2
(1  i )
3π
 B1  Bo 
B
usually not such a
big factor because
• design for a
margin in Jc
plot field profile across the slab
Jc
• most of magnet is
in a field much
lower than the
peak
B
Martin Wilson Lecture 3 slide26
JUAS Febuary 2013
AC losses from coupling
1) Persistent currents within filaments
Ps 
2
lsu B J c d f
3π
between filaments
Magnetization
2) Coupling between filaments within the wire
1
Pe  lwu B 2
ρt
 pw 
 2π 
2
within filaments
External field
3) Coupling between wires in the cable
between wires
1 Bt2 c
Ptc  lcu
pc N(N  1)
120 Rc b
1 B t2 c
Pta  lcu
pc
6 Ra
b
Martin Wilson Lecture 3 slide27
2
1 B p b
Ppa  lcu
pc
8 Ra
c
JUAS Febuary 2013
Summary of losses - per unit volume of winding
1) Persistent currents
in filaments
power
W.m-3
2
Ps  lsu M f B  lsu
J c ( B ) d f B
3π
loss per per ramp J.m-3
Qs  lsu
 B  Bo 
2
d f J o Bo ln  2

3π
B

B
o 
 1
where lsu , lwu , lcu = fractions of superconductor, wire and cable in the winding cross section
2) Coupling currents between
filaments in the wire
3) Coupling currents
between wires in
the cable
don't forget the
filling factors
Martin Wilson Lecture 3 slide28
power
W.m-3
transverse field crossover
resistance power W.m-3
 2  p 2
B
Pe lwu M e B  λwu
 
ρt  2π 
1 Bt2 c
Ptc  lcu
p N(N  1)
120 Rc b
transverse field adjacent
resistance power W.m-3
1 Bt2 c
Pta  lcu
p
6 Ra b
parallel field adjacent
resistance power W.m-3
2
1 B p b
Ppa  lcu
p
8 Ra c
JUAS Febuary 2013
Concluding remarks
• changing magnetic fields drive superconductor into resistive state  losses – leave persistent currents
• screening currents produce magnetization (magnetic moment per unit volume)
 lots of problems - field errors and ac losses
• in a synchrotron, the field errors from magnetization are worst at injection
• we reduce magnetization by making fine filaments - for practical use embed them in a matrix
• in changing fields, filaments are coupled through the matrix  increased magnetization
- reduce it by twisting and by increasing the transverse resistivity of the matrix
• accelerator magnets must run at high current because they are all connected in series
- combine wires in a cable, it must be fully transposed to ensure equal currents in each wire
• wires in cable must have some resistive contact to allow current sharing
- in changing fields the wires are coupled via the contact resistance
- different coupling when the field is parallel and perpendicular to face of cable
- coupling produces more magnetization  more field errors
• irreversible magnetization  ac losses in changing fields
- coupling between filaments in the wire adds to the loss
- coupling between wire in the cable adds more
Martin Wilson Lecture 3 slide29
never forget that magnetization
and ac loss are defined per unit
volume - filling factors
JUAS Febuary 2013