Massachusetts Institute of Technology
22.68J/2.64J
Superconducting Magnets
⇓
May 1, 2003
•Lecture #9 – AC Losses
¾AC Losses
¾Joint Losses
1
AC Loss Components
• Hysteresis: Ph ∝ B e
•
•
– Intrinsic to superconductor and because Type II, when
exposed to Be, is in mixed state.
– Theory agrees reasonably well with experiment.
Coupling: Pc ∝ B e2
– Joule dissipation of a dB/dt-induced filament-to-filament
(intra-strand) coupling current in the matrix metal.
– Theory well-understood but exact computation not possible
because some parameter values are not well known.
– Inter-strand coupling between composite strands in a cable
is particularly difficult to calculate.
2
Eddy-current: Pe ∝ B e
– Joule dissipation of dB/dt-induced current in the resistive
matrix metal not occupied by a cluster of filaments.
– Theory, e.g. problem 2.7, useful.
2
Parameters Affecting AC Losses
• Critical Properties: Jc (B,T)
• Magnetic field:
– Bias (DC); amplitude (AC); frequency, phase.
– Orientation (transverse, parallel, rotating)
• Current:
– Bias (DC); Amplitude (AC); frequency, phase.
• Composite Material Properties:
– Resistivity; geometry.
• Multistrand Cables and Braids:
– Configuration; twist pitches; contact resistances.
3
Implications for Superconductor Design
• Decrease Hysteresis Loss:
– Reduce superconductor dimension
• Many small superconducting filaments
• Decrease Coupling Loss:
– Twist Wire → Twist filaments
– Reduce twist pitch
• Reduce wire size
– Increase transverse resistivity
• Add internal barriers
• Make Multistrand Cables and Braids:
– For high current
– Low AC losses
4
Relevant Superconducting Wires are Complex
Composites
Typical SSC Nb-47wt.%Ti
strand (OST manufacture).
Typical reacted ITER Nb3Sn
strand (IGC manufacture).
5
Field Penetration in a Slab –
Bean Model
6
AC Loss Expressions
Hysteresis
Assumption: Bean-London model applies, i.e., Jc ≠ f(B)
Hp – Field required to fully penetrate a slab:
Hp =
Jc
d
2
Wh/V -Loss per cycle per unit volume:
Hm ≤ H p
⎞
⎛
Wh 2
2 Hm
⎟
⎜
= µo H p ⎜
⎟
V
3
⎝ Hp ⎠
3
Hm ≥ H p
⎡ ⎛ Hm ⎞ ⎤
Wh 2
2
⎟ − 2⎥
= µ o H p ⎢ 3⎜⎜
⎟
V
3
⎣⎢ ⎝ H p ⎠ ⎥⎦
7
Simplified AC Loss Expressions
Normalize to full-penetration field loss:
Wo 2
= µ o H 2p
3
V
Then:
Hm
≤ 1,
Hp
Wh ⎛⎜ H m ⎞⎟
=
Wo ⎜⎝ H p ⎟⎠
3
Hm
≥ 1,
Hp
Wh ⎡ ⎛⎜ H m ⎞⎟ ⎤
− 2⎥
= ⎢3
Wo ⎢⎣ ⎜⎝ H p ⎟⎠ ⎦⎥
And for:
Hm
>> 1 ,
Hp
⎛ Hm ⎞
Wh
⎟
≈ 3⎜⎜
⎟
Wo
H
p
⎝
⎠
8
Field Penetration in a Slab –
Kim Model
9
Schematic for Magnetization
Measurement Using Pick-Up Coils
10
Technical Type II Superconductors
Display a Magnetic Hysteresis
M ∝ Jc
Area Under Magnetization
Loop is Proportional to Dissipated
Energy/cycle ∝ ∆B Jc deff
11
Calorimetric Measurement Method
12
Calorimetric Measurement Method
13
Field Penetration in a Slab
– Bean Model with
Transport Current
14
Effect of Transport Current
H p (I ) =
Jcd ⎛
I ⎞
⎜⎜ 1 − t ⎟⎟
2 ⎝
Ic ⎠
It
i=
Ic
H po =
J cd
2
H p (i ) = H po (1 − i )
Transport current will increase the loss for the same magnetic field
change if the ∆H > Hp(i).
15
Effect of Transport Current
Wo 2
= µ o H 2p ( 0)
V
3
Hm
≤ 1,
H p (i )
Wh (i ) ⎛⎜ H m ⎞⎟
=⎜
⎟
Wo
⎝ H p (0) ⎠
3
Hm
≥ 1,
H p (i )
⎡⎛ H m ⎞ ⎛ H p ( i ) ⎞ ⎤
Wh (i )
3
⎟−⎜
⎟ ⎥ (1 + i 2 )
= (1 − i ) + 3 ⎢⎜⎜
⎟ ⎜
⎟
Wo
⎣⎢⎝ H p (0) ⎠ ⎝ H p (0) ⎠ ⎥⎦
16
Effect of Transport Current
Hm
Hm
≤ 1 or
≤ (1 − i ),
H p (i )
H p ( 0)
Wh (i ) W s (i ) Wt (i ) ⎛⎜ H m ⎞⎟
=
+
=⎜
⎟
Wo
Wo
Wo
⎝ H p (0) ⎠
3
Total Loss
Where
W s (i ) ⎛⎜ H m ⎞⎟
=⎜
⎟
Wo
⎝ H p (0) ⎠
Wt (i )
=0
Wo
3
Shielding Loss
Transport Loss
17
Effect of Transport Current
Hm
Hm
≥ 1 or
≥ (1 − i ),
H p (i )
H p ( 0)
⎡⎛ H m ⎞ ⎛ H p (i ) ⎞ ⎤
Wh (i ) W s (i ) Wt (i )
3
⎟ ⎥ (1 + i 2 )
⎟−⎜
=
+
= (1 − i ) + 3 ⎢⎜⎜
⎟
⎟ ⎜
Wo
Wo
Wo
⎢⎣⎝ H p (0) ⎠ ⎝ H p (0) ⎠ ⎥⎦
Total Loss
Where
⎡⎛ H m ⎞ ⎛ H p (i ) ⎞ ⎤
W s (i )
3
⎟ ⎥ (1 − i 2 )
⎟−⎜
= (1 − i ) + 3 ⎢⎜⎜
⎟
⎟ ⎜
Wo
⎢⎣⎝ H p (0) ⎠ ⎝ H p (0) ⎠ ⎥⎦
Shielding Loss
⎡⎛ H m ⎞ ⎛ H p (i ) ⎞ ⎤ 2
Wt (i )
⎟⎥ i
⎟−⎜
= 6 ⎢⎜⎜
⎟
⎜
⎟
Wo
⎢⎣⎝ H p (0) ⎠ ⎝ H p (0) ⎠ ⎥⎦
Transport Loss
18
Bean Model with
Transport Current
Shielding Loss in a
Slab
19
Bean Model with
Transport Current
Transport Loss in a Slab
20
Bean Model with
Transport Current
Total Loss in a Slab vs
Transport Current
21
Bean Model with AC
Transport Current
and Field in
Synchronization
22
Bean Model with AC
Transport Current
Terminal Voltage
and Transport
Current versus
Time
23
Bean Model with AC
Transport Current
Terminal Voltage
as a Function of
Transport Current
24
Bean Model
Circular Filament in
a Transverse Field
25
Transverse Flux Penetration
into a Circular Superconducting Filament
26
Bean Model
Circular Filament in
a Transverse Field
Numerical Solution
27
Bean Model
Elliptical
Filament in a
Transverse Field
Numerical
Solution
28
Bean Model
Elliptical
Filament in a
Transverse Field
Numerical
Solution
29
Coupling Losses
Twisting the superconducting filaments in
the composite wire is necessary to
electrodynamically decouple them
Hysteresis losses
∝ filament diameter
(~1 µm)
Hysteresis losses
∝ strand diameter
(~1 mm)
30
Coupling Losses
Twisting filaments also necessary to reduce
coupling losses
Power dissipation ∝ (Twist Pitch)2
31
Effective Matrix Resistivity
After W.J. Carr
ρ eff =
ρ eff =
1− λf
1+ λf
1+ λf
1− λf
ρm
Nb3Sn
ρm
NbTi
λf = volume fraction of superconducting filaments
ρm = matrix resistivity [Ω-m]
Coupling Loss Power:
2
2B
P
τ
=
µo
Vol
32
AC Losses in Multistrand Cables
Braid
Round Multistage Cable
Flat (Rutherford) Cable
Reasons for Making Multistrand Cables:
• Increase current capacity
• Reduce AC and transient coupling losses
• Mechanical rigidity
33
AC Losses in Cables
Electromagnetic Analysis of AC Losses in Large Superconducting Cables
General Loss Components::
¾
¾
¾
¾
Hysteresis (magnetization) in superconducting filaments
Coupling (intrastrand)
Eddy (stabilizer)
Coupling (inerstrand in sub-cables and cable-cable in built-up conductors)
34
AC Losses
Electromagnetic Analysis of AC Losses in Large Superconducting Cables
Code output is calibrated against
specific, well-controlled small scale
laboratory experiments:
¾ Useful for code calibration with
limited boundary conditions
¾ Does not capture all full-size magnet
operating conditions
Linear ramp and sinusoidal AC field simulated
35
AC Losses
Electromagnetic Analysis of AC Losses in Large Superconducting Cables
Modeling a >1000 superconducting strand 5 stage cable
Individual strand locations
in a 1/6th petal
Mathematical calculation of
strand position over a laststage transposition twist pitch
36
AC Losses- General Solution
37
CSMC Measured Results
Typical distribution of coupling loss tau
for 18 layers x 2-in-hand =36 conductors
20 s dump, 36.8 kA, 6/26/2000
300
coupling time const tau, ms
250
200
150
100
50
B
A
9
B
A
8
A
B
8
B
A
7
A
B
7
B
A
6
A
B
6
B
A
5
A
B
5
B
A
4
A
B
4
B
A
3
A
B
3
B
A
2
A
B
2
B
A
1
A
B
CS 1
B
A
Insert
0
9 10 10 11 11 12 12 13 13 14 14 15 15 16 16 17 17 18 18
Effective cable coupling time constants vary greatly depending on cable twist pitches, magnitude
of Lorentz Force (J x B), surface coating, magnetoresistance, and field magnitude.
38
CSMC Measured Results
Reduction of AC coupling losses with cycles
250
200
Layer
τ (ms)
Coupling Time Constant
Inner coil
1A
3A
6A
150
100
Thesis:
Thesis:Cycling
Cyclingaffects
affectseffective
effective
coupling
couplingtime
timeconstant
constantby
by
changes
in
strand
contact
changes in strand contact
pressure
pressuredistribution
distributionand
and
interfacial
resistance.
interfacial resistance.
50
0
0
20
40
60
80
100
120
Np
300
¾¾ AAseparate
separatelab-scale
lab-scaleexperimental
experimental
program
programisisused
usedtotodetermine
determinethis
this
parameter.
parameter.
250
τ (ms)
Coupling Time Constant
Outer coil
Layer
200
13A
15A
18A
150
¾¾ An
Animportant
important(and
(andoften
oftenunknown
unknown
aapriori)
parameter
is
the
effective
priori) parameter is the effective
transverse
transverseconductivity
conductivitybetween
between
and
among
the
wires
and
and among the wires andcable
cable
stages.
stages.
100
50
0
0
5
10
15
20
25
30
35
40
Np
Pulse Number
39
Splice (Joint) Losses
Joule Dissipation, Gsl
G sl = Rsl I t2
It = Transport current through the joint [A]
Rsl = Joint Resistance [Ω]
Rsl =
Rct Rct
=
Act al sl
Rct = contact resistance [Ω-m2]
a = conductor width (joint width) [m]
lsl = splice length [m]
40
Splice (Joint) Losses
• Surfaces can be in contact under pressure with no solder
– then contact resistance depends on the surface condition.
(roughness, surface oxides, etc.)
– often silver plate.
• Surfaces are often soldered together
– Some mechanical integrity- but often not too strong.
– Rct is usually > ρsolder δsolder because of contact resistance
at solder-copper interface.
– Best to measure
• Don’t overheat joint during soldering because excessive
temperature can reduce Jc of NbTi.
41
Joint Orientation to Field
2.2
Operation in varying field
coupling currents between strands and eddy currents in the copper soles are induced in the joint
I
Current induced
by dBw/dt
w (longitudinal)
v (transverse)
Current induced
by dBv/dt
u (normal)
Current induced
by dBu/dt
I
I
•
transverse field variation (dBv/dt)
⇒
⇒
⇒
high intercable losses
low eddy current losses
risk of current saturation / flux jump
⇒
⇒
interstrand losses
high eddy current losses (high RRR)
•
longitudinal field variation (dBw/dt)
⇒
⇒
low interstrand losses
low eddy current losses
•
⇒ worst orientation
normal field variation (dBu/dt)
⇒ controlled losses
⇒ best orientation
42
US CSMC Joint Sample
Stainless steel clamp/
joint box
Nilo alloy wedges
Glidcop
Monel transition pieces
Stainless steel
eyeglass piece
Incoloy conduit
43
PTF Cryostat Sample Plumbing
44
JA CSMC Butt Joint
45
JA CSMC Butt Joint
46