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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