1
Massachusetts Institute of Technology22.68J/2.64J
Superconducting Magnets
⇓
May 1, 2003
•Lecture #9 – AC LossesAC LossesJoint Losses
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• Hysteresis:– Intrinsic to superconductor and because Type II, when
exposed to Be, is in mixed state.– Theory agrees reasonably well with experiment.
• Coupling:– 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.• Eddy-current:
– 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.
AC Loss Components
eh BP ∝
2ec BP ∝
2ee BP ∝
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• 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.
Parameters Affecting AC Losses
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• 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
Implications for Superconductor Design
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Relevant Superconducting Wires are Complex Composites
Typical SSC Nb-47wt.%Tistrand (OST manufacture).
Typical reacted ITER Nb3Snstrand (IGC manufacture).
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Field Penetration in a Slab –Bean Model
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AC Loss ExpressionsHysteresisAssumption: Bean-London model applies, i.e., Jc ≠ f(B)
Hp – Field required to fully penetrate a slab:
dJH cp 2
=
Wh/V -Loss per cycle per unit volume:
⎥⎥⎦
⎤
⎢⎢⎣
⎡−⎟
⎟⎠
⎞⎜⎜⎝
⎛=
≥
⎟⎟⎠
⎞⎜⎜⎝
⎛=
≤
2332
32
2
3
2
p
mpo
h
pm
p
mpo
h
pm
HHH
VW
HH
HHH
VW
HH
µ
µ
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Simplified AC Loss ExpressionsNormalize to full-penetration field loss:
Then:
⎥⎥⎦
⎤
⎢⎢⎣
⎡−⎟
⎟⎠
⎞⎜⎜⎝
⎛=
≥
⎟⎟⎠
⎞⎜⎜⎝
⎛=
≤
23
,1
,1
3
p
m
o
h
p
m
p
m
o
h
p
m
HH
WW
HH
HH
WW
HH
2
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poo HVW µ=
And for:⎟⎟⎠
⎞⎜⎜⎝
⎛≈>>
p
m
o
h
p
m
HH
WW
HH 3 , 1
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Field Penetration in a Slab –Kim Model
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Schematic for Magnetization Measurement Using Pick-Up Coils
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Area Under MagnetizationLoop is Proportional to DissipatedEnergy/cycle ∝ ∆B Jc deff
M ∝ Jc
Technical Type II SuperconductorsDisplay a Magnetic Hysteresis
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Calorimetric Measurement Method
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Calorimetric Measurement Method
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Field Penetration in a Slab – Bean Model with Transport Current
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Effect of Transport Current
Transport current will increase the loss for the same magnetic field change if the ∆H > Hp(i).
( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛−=
c
tcp I
IdJIH 12
( ) ( )iHiH
dJH
IIi
pop
cpo
c
t
−=
=
=
1
2
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Effect of Transport Current
)0(32 2
poo HVW µ=
( )
( )( )
( )
( ) ( ) ( ) ( ) ( )23
3
10)(
031
,1
0
,1
iH
iHHHi
WiW
iHH
HH
WiW
iHH
p
p
p
m
o
h
p
m
p
m
o
h
p
m
+⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛−⎟
⎟⎠
⎞⎜⎜⎝
⎛+−=
≥
⎟⎟⎠
⎞⎜⎜⎝
⎛=
≤
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Effect of Transport Current
( ) ,1)0(
or 1)(
iHH
iHH
p
m
p
m −≤≤
( ) ( ) ( )( )
( )( )
( ) LossTransport 0
Loss Shielding 0
Where
Loss Total 0
3
3
=
⎟⎟⎠
⎞⎜⎜⎝
⎛=
⎟⎟⎠
⎞⎜⎜⎝
⎛=+=
o
t
p
m
o
s
p
m
o
t
o
s
o
h
WiW
HH
WiW
HH
WiW
WiW
WiW
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Effect of Transport Current
( ) ,1)0(
or 1)(
iHH
iHH
p
m
p
m −≥≥
( ) ( ) ( ) ( ) ( )( )( ) ( )
( ) ( ) ( )( )( ) ( )
( )( )
( )( ) LossTransport i 00
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Loss Shielding i1 00
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Where
Loss Total i1 00
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2
23
23
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛−⎟
⎟⎠
⎞⎜⎜⎝
⎛=
−⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛−⎟
⎟⎠
⎞⎜⎜⎝
⎛+−=
+⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛−⎟
⎟⎠
⎞⎜⎜⎝
⎛+−=+=
p
p
p
m
o
t
p
p
p
m
o
s
p
p
p
m
o
t
o
s
o
h
HiH
HH
WiW
HiH
HHi
WiW
HiH
HHi
WiW
WiW
WiW
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Bean Model with Transport CurrentShielding Loss in a
Slab
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Bean Model with Transport Current
Transport Loss in a Slab
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Bean Model with Transport Current
Total Loss in a Slab vsTransport Current
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Bean Model with AC Transport Current
and Field in Synchronization
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Bean Model with AC Transport CurrentTerminal Voltage and Transport Current versus
Time
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Bean Model with AC Transport CurrentTerminal Voltage as a Function of
Transport Current
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Bean Model Circular Filament in a Transverse Field
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Transverse Flux Penetration into a Circular Superconducting Filament
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Bean Model Circular Filament in a Transverse FieldNumerical Solution
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Bean Model Elliptical
Filament in a Transverse Field
Numerical Solution
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Bean Model Elliptical
Filament in a Transverse Field
Numerical Solution
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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)
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Coupling Losses
Twisting filaments also necessary to reduce coupling lossesPower dissipation ∝ (Twist Pitch)2
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Effective Matrix ResistivityAfter W.J. Carr
NbTi 11
SnNb 11
3
mf
feff
mf
feff
ρλλ
ρ
ρλλ
ρ
−+
=
+−
=
λf = volume fraction of superconducting filamentsρm = matrix resistivity [Ω-m]
Coupling Loss Power:τ
µo
BVolP 22
=
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AC Losses in Multistrand Cables
Round Multistage Cable
Braid
Flat (Rutherford) Cable
Reasons for Making Multistrand Cables:• Increase current capacity• Reduce AC and transient coupling losses• Mechanical rigidity
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AC Losses in Cables
Electromagnetic Analysis of AC Losses in Large Superconducting CablesGeneral Loss Components::
Hysteresis (magnetization) in superconducting filamentsCoupling (intrastrand)Eddy (stabilizer)Coupling (inerstrand in sub-cables and cable-cable in built-up conductors)
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AC LossesElectromagnetic Analysis of AC Losses in Large Superconducting Cables
Code output is calibrated against specific, well-controlled small scale laboratory experiments:
Useful for code calibration with limited boundary conditions
Does not capture all full-size magnet operating conditions
Linear ramp and sinusoidal AC field simulated
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AC LossesElectromagnetic Analysis of AC Losses in Large Superconducting Cables
Modeling a >1000 superconducting strand 5 stage cable
Mathematical calculation of strand position over a last-stage transposition twist pitch
Individual strand locations in a 1/6th petal
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AC Losses- General Solution
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CSMC Measured ResultsTypical distribution of coupling loss tau
for 18 layers x 2-in-hand =36 conductors
20 s dump, 36.8 kA, 6/26/2000
0
50
100
150
200
250
300In
sert A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B
CS 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15 16 16 17 17 18 18
coup
ling
time
cons
t tau
, ms
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.
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CSMC Measured ResultsReduction of AC coupling losses with cycles
Inner coil
0
50
100
150
200
250
0 20 40 60 80 100 120Np
τ (m
s) 1A3A6A
Layer
Cou
plin
g Ti
me
Con
stan
t
Outer coil
0
50
100
150
200
250
300
0 5 10 15 20 25 30 35 40
Np
τ (m
s) 13A15A18A
Layer
Pulse Number
Cou
plin
g Ti
me
Con
stan
t
Thesis: Cycling affects effective coupling time constant by changes in strand contact pressure distribution and interfacial resistance.
An important (and often unknown a priori) parameter is the effective transverse conductivity between and among the wires and cable stages.
A separate lab-scale experimental program is used to determine this parameter.
Thesis: Cycling affects effective coupling time constant by changes in strand contact pressure distribution and interfacial resistance.
An important (and often unknown a priori) parameter is the effective transverse conductivity between and among the wires and cable stages.
A separate lab-scale experimental program is used to determine this parameter.
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Joule Dissipation, Gsl
Splice (Joint) Losses
2tslsl IRG =
It = Transport current through the joint [A]Rsl = Joint Resistance [Ω]
sl
ct
ct
ctsl al
RARR ==
Rct = contact resistance [Ω-m2]a = conductor width (joint width) [m]lsl = splice length [m]
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• 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.
Splice (Joint) Losses
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Joint Orientation to Field2.2 Operation in varying field
coupling currents between strands and eddy currents in the copper soles are induced in the joint
• transverse field variation (dBv/dt)
⇒ high intercable losses ⇒ worst orientation⇒ low eddy current losses⇒ risk of current saturation / flux jump• normal field variation (dBu/dt)
⇒ interstrand losses ⇒ controlled losses⇒ high eddy current losses (high RRR)
• longitudinal field variation (dBw/dt)⇒ low interstrand losses ⇒ best orientation⇒ low eddy current losses
II
I
u (normal)
v (transverse)
w (longitudinal)Current inducedby dBw/dt
Current inducedby dBv/dt
Current inducedby dBu/dt
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US CSMC Joint Sample
Stainless steel clamp/ joint box
Nilo alloy wedges
Glidcop
Monel transition pieces
Stainless steel eyeglass piece
Incoloy conduit
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PTF Cryostat Sample Plumbing
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JA CSMC Butt Joint
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JA CSMC Butt Joint