Unit 9 Electromagnetic design Episode II

USPAS June 2007, Superconducting accelerator magnets

Unit 9 Electromagnetic design

Episode II

Soren Prestemon and Paolo FerracinLawrence Berkeley National Laboratory (LBNL)

Ezio TodescoEuropean Organization for Nuclear Research (CERN)

Unit 9: Electromagnetic design episode II – 9.2USPAS June 2007, Superconducting accelerator magnets

QUESTIONS

Given a material and an aperture, and a quantity of cable, what is the strongest dipole / quadrupole we can build ?

Can we have explicit equations ?Or do we have to run codes in each case ?


CONTENTS

1. Dipoles: short sample field versus material and lay-out

2. Quadrupoles: short sample gradient versus material and lay-out

3. A flowchart for magnet design


1. DIPOLES: FIELD VERSUS MATERIAL AND COIL THICKNESS

We recall the equations for the critical surfaceNb-Ti: linear approximation is good

with c~6.0108 [A/(T m2)] and B*c2~10 T at 4.2 K or 13 T at

1.9 KLHC production: c~6.7108 [A/(T m2)] and B*

c2~13.4 T at 1.9 K

),( *2, BBcj ccsc

0

2000

4000

6000

8000

0 5 10 15B (T)

j sc(A

/mm

2 )

Nb-Ti at 1.9 K

Nb-Ti at 4.2 K



The current density in the coil is lowerStrand made of superconductor and normal conducting (copper)

Cu-sc is the ratio between the copper and the superconductor, usually ranging from 1 to 2 in most cases

If the strands are assembled in rectangular cables, there are voids:

w-c is the fraction of cable occupied by strands (usually ~85%)

The cables are insulated: c-i is the fraction of insulated cable occupied by the bare cable (~85%)

The current density flowing in the insulated cable is reduced by a factor (filling ratio)

The filling ratio ranges from ¼ to 1/3The critical surface for j (engineering current density) is

scCuiccw

1

1

scjj )( *2 BBcj cc



Examples of filling ratio in dipoles (similar for quads)

Copper to superconductor ranging from 1.2 to 2.2Extreme case of D20: 0.43

Void fraction from 11% to 18% Insulation from 11% to 18%

Case of FNAL HFDA: 24% for insulation

scCuiccw

1

1scjj

Magnet Cu-Sc

w-c

c-i

MaterialTevatron MB 1.85 0.82 0.81 0.232 Nb-TiHERA MB 1.88 0.89 0.85 0.262 Nb-Ti

SSC MB inner 1.50 0.84 0.89 0.298 Nb-TiRHIC MB 2.25 0.87 0.84 0.226 Nb-Ti

LHC MB inner 1.65 0.87 0.87 0.286 Nb-TiFRESCA 1.60 0.87 0.88 0.293 Nb-Ti

MSUT inner 1.25 0.85 0.88 0.330 Nb3Sn

D20 inner 0.43 0.83 0.84 0.484 Nb3Sn

FNAL HFDA02-03 1.25 0.86 0.76 0.288 Nb3Sn

NED 1.25 0.83 0.84 0.309 Nb3Sn



We characterize the coil by two parameters

: how much field in the centre is given per unit of current density: ratio between peak field and central field

for a sector dipole or a cos , w for a cos dipole, =1

We can now compute what is the highest peak field that can be reached in the dipole in the case of a linear critical surface

jB jBBp

c

cBB c

ssp

1

*2

,

)( ,*2, sspccssp BBcjB

0

1000

2000

3000

0 5 10 15B (T)

j(A

/mm

2 ) j c= c(B *c2-B)

B p= j[B p,ss,j p,ss]



We can now compute the maximum current density that can be tolerated by the superconductor (short sample limit)

the short sample current is

and the short sample field is

jB jBBp

c

cBB c

ss

1

*2

c

cBB c

ssp

1

*2

,

c

cBj c

ss

1

*2

0

1000

2000

3000

0 5 10 15B (T)

j(A

/mm

2 ) j c= c(B *c2-B)

B p=j[B p,ss,j p,ss]



When we go for larger and larger coils we increase both (central field) and (peak field)

when c >>1 (large coil)

The interplay between these two factors constitutes the problem of the coil optimizationNote that the slope of the critical surface c defines the “large coil”Note that for large coils the filling factor becomes not relevant

c

cBB c

ss

1

*2

*2c

ss

BB



Examples

The quantity c is larger than 1 in the 6 analysed dipoles, and is around 5 for dipoles with large coil widths (SSC, LHC, Fresca)

This means that for these three dipoles we are rather close to the maximum field we can get with Nb-Ti

c

cBB c

ss

1

*2

*2c

ss

BB

Magnet c c

(adim) (A/T/m2) (adim) (T m2 /A) (adim)Tevatron MB 0.232 6.00E+08 1.13 1.23E-08 1.9HERA MB 0.262 6.00E+08 1.08 1.64E-08 2.8SSC MB 0.298 6.00E+08 1.05 2.14E-08 4.0

RHIC MB 0.226 6.00E+08 1.18 9.54E-09 1.5LHC MB 0.286 6.00E+08 1.03 2.38E-08 4.2FRESCA 0.293 6.00E+08 1.05 2.94E-08 5.5



Examples of (central field per unit of current density)

Good agreement (within 4%) between 5 dipoles (no grading) and [0°-48°,60°-72°] lay-outFive cases give 10-20% due to grading (see Unit 11)

0E+00

1E-08

2E-08

3E-08

4E-08

0 10 20 30 40 50equivalent width w (mm)

[T

m2 /A

]

TEV MB HERA MBSSC MB RHIC MB

LHC MB FrescaMSUT D20

HFDA NED

Grading

w0



Numerical evaluation of for different sector coilsTo compute the peak field one has to compute the field everywhere in the coil, and take the maximumOne can prove that the maximum is always on the border of the coil – useful to reduce the computation time

0

20

40

60

80

0 20 40 60 80x (mm)

y (m

m)

LHC main dipole – location of the peak field

0

20

40

60

80

0 20 40 60 80x (mm)

y (m

m)

RHIC main dipole – location of the peak field

0

20

40

60

80

0 20 40 60 80x (mm)

y (m

m)

Tevatron main dipole – location of the peak field



Numerical evaluation of for different sector coilsFor large widths, 1 This means that for very large widths we can reach B*

c2 !

1.0

1.1

1.2

1.3

1.4

1.5

1.6

0 20 40 60 80width (mm)

(

adim

)

1 layer 60

1 layer 48-60-72

1 layer 42.8-51.6-67

1 layer three blocks

*2c

ss

BB



Numerical evaluation of for different sector coilsFor interesting widths (10 to 30 mm) is 1.05 - 1.15The cos approx having =1 is not so bad for w>20 mmThe [0°-48°,60°-72°] is well fit by with a~0.045

B

B p

1.0

1.1

1.2

1.3

1.4

1.5

1.6

0 20 40 60 80width (mm)

(

adim

)

1 layer 601 layer 48-60-721 layer 42.8-51.6-671 layer three blocksFit

w

arrw 1~),(



Examples of ratio between peak field and central field)

Agreement with the semi-analytical fit found for the [0°-48°,60°-72°] lay-out is good (within 2% in the analysed cases)

w

arrw 1~),(

1.0

1.1

1.2

1.3

0.0 0.5 1.0 1.5 2.0equivalent width w/r

[

adim

]

TEV MB HERA MB

SSC MB RHIC MB

LHC MB Fresca

MSUT D20

HFDA NED



We now can write the short sample field for a sector coil as a function of

Material parameters c, B*c2

Cable parameters Aperture r and coil width wfor a [0°-48°,60°-72°] coil, a=0.045 0=6.6310-7 [Tm/A]

for Nb-Ti c~6.0108 [A/(T m2)] and B*c2~10 T at 4.2 K or 13 T at

1.9 K

Cos model:=210-7 [Tm/A]

arwc

wcB

w

arwc

wcBB cc

ss

0

0*2

0

0*2

111

w0

w

arrw 1~),(

wc

wcBB c

ss

1

*2

c

cBB c

ss

1

*2



Evaluation of short sample field in sector lay-outs and cos model for a given aperture (r=30 mm)

Tends asymptotically to B*c2, as B*

c2 w/(1+w), for wSimilar results for different position of wedges

0

2

4

6

8

10

0 20 40 60 80width (mm)

Bss

(T

) 1 layer 601 layer 48-60-721 layer 42.8-51.6-671 layer three blocksCos theta

Nb-Ti 4.2 K - B *c2 =10 T



Dependence on the apertureFor very large aperture magnets, one has less field for the same coil thicknessFor small apertures it tends to the cos model

0

2

4

6

8

10

0 20 40 60 80equivalent width (mm)

Bss

(T

) r=10 mmr=30 mmr=60 mmr=120 mmCos theta

Nb-Ti at 4.2 K B *c2 =10 T =0.35



Case of Nb3SnThe critical surface is not linear, it can be solved numericallyThe saturation for large widths is slower (due to the shape of the surface)

0

5

10

15


Bss

(T

)

r=30 mm r=60 mm

r=120 mm Cos theta

Nb-Ti at 4.2 K =0.35

Nb3Sn at 4.2 K =0.35


CONTENTS





2. QUADRUPOLES: GRADIENT VERSUS MATERIAL AND COIL THICKNESS

The same approach can be used for a quadrupoleWe define

the only difference is that now gives the gradient per unit of current density, and in Bp we multiply by r for having T and not T/mWe compute the quantities at the short sample limit for a material with a linear critical surface (as Nb-Ti)

Please note that is not any more proportional to w and independent of r !

r

w1log0

jG jrrGBp

r

cr

cBB c

cp

1

*2

,

cr

cBj cc

1

*2

cr

cBG c

ss

1

*2



Please note that is not any more proportional to w and independent of r !

Actual values fit well with the constant for [0°-24°,30°-36°] lay-out 0=6.6310-7 [Tm/A], as for the dipoles …

r

w1log0

0.E+00

1.E-07

2.E-07

3.E-07

4.E-07

5.E-07

6.E-07

0 0.25 0.5 0.75 1 1.25aspect ratio weq/r (adim)

[T

m/A

]

ISR MQ TEV MQHERA MQ SSC MQLEP I MQC LEP II MQCRHIC MQ RHIC MQY

LHC MQ LHC MQMLHC MQY LHC MQXALHC MQXB

current grading[0°-30°] lay-out


0

20

40

0 20 40 60 80x (mm)

y (m

m)


The ratio is defined as ratio between peak field and gradient times aperture (central field is zero …)

Numerically, one finds that for large coils Peak field is “going outside” for large widths

RHIC main quadrupole

LHC main quadrupole

0

20

40

0 20 40 60 80x (mm)

y (m

m)

1.1

1.2

1.3

1.4

1.5

0 1 2 3w/r (adim)

(

adim

)

[0,30]

[0-24,30-36]

[0-18,22-32]



The ratio is defined as ratio between peak field and gradient times aperture (central field is zero …)

The ratio depends on w/rA good fit is

a-1~0.04 and a1~0.11 for the [0°-24°,30°-36°] coil

A reasonable approximation is ~0=1.15 for ¼<w/r<1

r

wa

w

rarw 11 1),(

1.1

1.2

1.3

1.4

1.5

0 1 2 3w/r (adim)

(

adim

)

[0,30]

[0-24,30-36]

[0-18,22-32]



Comparison for the ratio between the fit for the [0°-24°,30°-36°] coil and actual values

1.0

1.1

1.2

1.3

1.4

1.5

0.0 0.5 1.0 1.5 2.0aspect ratio w eq/r (adim)

[a

dim

]

ISR MQ TEV MQ HERA MQSSC MQ LEP I MQC LEP II MQCRHIC MQ RHIC MQY LHC MQLHC MQM LHC MQY LHC MQXBLHC MQXA

current grading



We now can write the short sample gradient for a sector coil as a function of

Material parameters c, B*c2

(linear case as Nb-Ti)Cable parameters Aperture r and coil width w

Relevant feature: for very large coil widths w the short sample gradient tends to zero !

r

wa

w

rarw 11 1),(

cr

cBG c

ss

1

*2

r

w1ln0

r

wcr

r

wa

w

ra

r

wcB

cr

cBG

cc

ss

1ln11

1ln

1011

0*2*

2



Evaluation of short sample gradient in several sector lay-outs for a given aperture (r=30 mm)

No point in making coils larger than 30 mm !Max gradient is 300 T/m and not 13/0.03=433 T/m !! We lose 30% !!

0

100

200

300

400

0 10 20 30 40 50 60 70 80 90 sector width w (mm)

Gss

(T

/m)

[0,30]

[0-24,30-36]

[0-18,22-32]

G * =B *c2 /r

-30%



Dependence of of short sample gradient on the aperture

Large aperture quadrupoles go closer to G*=B*c2/r

Very small aperture quadrupoles do not exploit the sc !!Large aperture need smaller ratio w/r

For r=30-100 mm, no need of having w>r

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0w/r (adim)

Gss

/G* [

adim

]

r=10 mm r=30 mm

r=60 mm r=120 mm

Nb-Ti



Case of Nb3Sn – in this case one can solve the equations numerically

Gain is ~50% in gradient for the same aperture (at 35 mm)Gain is ~70% in aperture for the same gradient (at 200 T/m)

0

100

200

300

400

500

0 10 20 30 40 50 60 70 80 90 100Aperture radius r [mm]

Gra

dien

t [T

/m]

Nb3Sn at 1.9 K

Nb-Ti at 1.9 K

+47%

+52%

+58%

=0.35


CONTENTS





3. A FLOWCHART FOR MAGNET DESIGN - DIPOLES

Having an apertureThe technology gives the maximal field that can be reached

Nb-Ti: 7-8 T at 4.2 K, 11 T at 1.9 K (80% of B*c2)

Nb3Sn: 17-20 T ?

Having an aperture and a fieldOne can evaluate the thickness of the coil needed to get the field using the equations for a sector coilCost optimization – higher fields costs more and more $$$ (or euro)

0

2

4

6

8

10


Bc

(T)

r=30 mm

r=60 mm

r=120 mm

cos theta

Nb-Ti at 4.2 K

0

0*2

1

carw

wcBB c

ss


3. A FLOWCHART FOR MAGNET DESIGN - DIPOLES

Having aperture, field and cable thicknessSurface necessary to get that field can be estimated (i.e. number of turns) orWedges are put to optimize field quality

Sometimes the cable is given, but not the fieldHaving the cable one can make a quick estimation of the short sample field that can be obtained with 1/2/3 … layersThen the actual lay-out with wedges is made

At first order one aims at zero multipoles – pure fieldSuccessive optimizations

Effect of iron (Unit 11) and persistent currents (Unit 15) is included – fine tuning of cross-sectionBalance of optimization between low and high fieldOnce built, an additional correction is usually needed (Unit 20)

Models are not enough precise (in absolute) but work well in relative


4. CONCLUSIONS

We wrote equations for computing the short sample field or gradient

This gives the dependence on the aperture, coil thickness, filling ration, material

Episode IIICan we make better ?What about iron ?Other lay-outs


REFERENCES

Field quality constraintsM. N. Wilson, Ch. 1P. Schmuser, Ch. 4A. Asner, Ch. 9Classes given by A. Devred at USPAS

Electromagnetic designS. Caspi, P. Ferracin, “Limits of Nb3Sn accelerator magnets“, Particle Accelerator Conference (2005) 107-11.L. Rossi, E. Todesco, Electromagnetic design of superconducting quadrupoles , Phys. Rev. ST Accel. Beams 9 (2006) 102401.Classes given by R. Gupta at USPAS 2006, Unit 3,4,5,6S. Russenschuck, et al, CERN 99-01 (1999) 18-30 and 82-92.

Numerical methods to calculate magnetic fieldS. Russenschuck, “Electromagnetic design of superconducting accelerator magnets”, CERN 2004-008 (2004) 71-117.


ACKNOWLEDGEMENTS

B. Auchmann, L. Bottura, A. Den Ouden, A. Devred, P. Ferracin, V. Kashikin, A. McInturff, T. Nakamoto, N. Schwerg, S. Russenschuck, T. Taylor, C. Vollinger, S. Zlobin, for kindly providing magnet designs … and perhaps others I forgotS. Caspi, L. Rossi for discussing magnet design, grading, and other interesting subjects …

No frogs have been harmed during the preparation of these slides


APPENDIX AAN EXPLICIT EXPRESSION FOR NB3SN

Case of Nb3Sn – an explicit expressionAn analytical expression can be found using a hyperbolic fit

that agrees well between 11 and 17 Twith c~4.0109 [A/(T m2)] and b~21 T at 4.2 K, b~23 T at 1.9 K Using this fit one can find explicit expression for the short sample field

and the constant are the same as before (they depend on the lay-out, not on the material)

0

2000

4000

6000

8000

0 5 10 15 20 25B (T)

j sc(A

/mm

2 )

Nb-Ti at 1.9 K

Nb-Ti at 4.2 K

Nb3Sn at 1.9 K

Nb3Sn at 4.2 K

1

B

bcjc

11

4

2

c

bcBss

Unit 9 Electromagnetic design Episode II

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Transcript of Unit 9 Electromagnetic design Episode II