a scaling law for planar and helical superconducting

A Scaling Law for Planar and Helical Superconducting Undulators

SRI 2010 Workshop 3:Superconducting undulators and other new ID sourcesSeptember 20 – 21, 2010 at the APS Conference Center

Suk Hong Kim

Magnetic Devices Group

Advanced Photon Source

S.H. Kim ANL/APS Magnetic Devices Group SRI-2010 Workshop 3: SCUs and other ID Sources, 9/20-21/2010 at the APS 29/23/2010

032 sin( / 4)( )

sinh( / 2)n

NI nB

n nkg

032 cosh( ) cosh(3 )

sinh( / 2) 3sinh(3 / 2)2y

NI ky kyB

kg kg

2( )k

2

0 exp ,g g

B a b c

Introduction

Iron dominated magnet [1]

0

0

0

0

0

0

2( )

3( )

pole

Quad

pole

Sext

NIB r

r

NIB r

r

Electromagnetic undulator [2]

Permanent magnet undulator [3]

(0.07 0.7)g

SmCo5: Br = 0.9 T, a = 3.33, b = 5.47, c = 1.8

NdFeB: Br = 1.1 T, a = 3.44, b = 5.08, c = 1.54

/ 2

0

sin( / )2 (1 )

( / )

kh kg

r

mB B e e

m

B0 = const jl

Planar superconducting undulator

[1] G.E. Fisher, AIP Conf. Proc.No. 153, AIP, New York (1987) p.1122.[2] R.P. Walker, Nucl. Instrum. Methods, A237,366 (1985); CERN 98-04 (1998) p. 129.[3] K. Halbach, Nucl. Instrum. Methods, 187 (1981) 109; J. Phys. C1 44, 211 (1983).

Outline

Introduction

Analytical expression of planar (electromagnetic) undulator

j scaling and Ampere’s law

Data on pole-gap dependence and coil dimensions to support the j scaling

Characteristics, j scaling, and analytical expression of bifilar helical undulator

An example of the j scaling for a superconducting undulator(SCU): Why it is much more difficult with “short-period” SCUs

Summary


Planar undulator


2( )k

1,3,

4( ) sin( )cos( )

2x

n

j aj z nk nkz

n

Analytical expressions [1]

per unit height of the coils

Schematic 2D cross section in (x = 0)

When SCU dimensions are scaled, with l as the reference, B0 remains unchanged for a constant jl. The on-axis peak field depends approximately on exp(-kg/2).The derived equation Bn predicted the third harmonic of the on-axis field correctly. Surprisingly, the jl scale holds for undulators with nonlinear magnetic materials.

0

0

(0, ) sin( )

(0, ) cos( )

y

z

B z B kz

B z B kz

[1] S.H. Kim, Nucl. Instrum. Methods, A 546 , 604 (2005).

1,3,

1,3,

/ 2

2

( , ) cosh( )sin( )

( , ) sinh( )cos( )

2sin( ) 1

( ) 2

y n

n

z n

n

nkg nkbon

B y z B nky nkz

B y z B nky nkz

j aB nk e e

n

/ 2

0 2

2sin( ) 1

2

kg kbo j aB k e e

Does Ampere’s law explain the jl scaling ?

S.H. Kim ANL/APS Magnetic Devices Group SRI-2010 Workshop 3: SCUs and other ID Sources, 9/20-21/2010 at the APS 59/23/2010 5

For a constant jl, if the flux density distributions remain unchanged in the above two geometries (which include the nonlinear magnetic poles and flux-return yokes), then the jl scaling may be understandable with Ampere’s law.Numerical analysis showed that the flux density, as well as the permeability,

distributions of the two were identical [1].

1

2

NI

1

2NI

[1] Opera, Vector Fields Software, Cobham Technical Services, Aurora, IL 60505, USA


For g/ > 0.4, B0 is proportional to exp(-kg/2) within 1%. The above equation, which has a correction term, holds for g/ > 0.06 within 1%.

0 0

1( , ) ( ,0.5)exp[ ( 0.5 {1 exp[ 2.4 ( 0.] 5) )]}n

g gB

gjj B j

( 0.0016)

Peak field B0 dependence on g/

for nonlinear cases

Eq. (1)


When the pole gap is reduced, the coil maximum field increases, which, in turn, reduces the SC critical current.Therefore, the correction term

for g/ > 0.4 in Eq. (1) may be neglected.

Coil max. field Bm dependence on g/

Optimized coil dimensions?Bigger coil dimensions make

higher coil maximum fields, which reduce the SC critical current densities.Therefore, the spread of the

peak fields reduced, and the dependence of the achievable peak fields on coil dimensions is relatively small.

g/l= 0.5 NbTi SC

Helical undulator


Helical Solenoid [1]

Helical undulator [2]

Helical Undulators with coil dimensions (a, b) [3]

0 1 20 0 0 0 1 0

( )( ) ( )

I IB kr K kr K kr

(Current I in a filament wire) (k = 2/

0 1 2( )z

I IB

For I1 and I2 in opposite directions in each helix:

0ˆˆ( ) sin( ) cos( )kz B r kz kz B

0 1 1

1,3,5..

ˆ[ ( ) ( ) sin ( )n

n n

n

B r I nkr I nkr n kz

B

ˆ ˆ( 2 / ) ( )cos ( ) 2 ( )cos ( )]n nkr I nkr n kz z I nkr n kz

0

0

00 0 1

2sin( ) { ( ) ( )} .

2

r b

r

j a drB k krK kr K kr

2

0

( / 2)( ) ,

!( )!

m

m

xI x

m

( 1)m

[1] W.R. Smythe, Static and Dynamic Electricity (McGraw-Hill, New York, 1939), p. 272.[2] B.M. Kincaid, J. Appl. Phys. 48, 2684 (1977); J.P. Blewett and R. Chasman, abid. 48, 2692 (1977).[3] S.H. Kim, Nucl. Instrum. Methods, A 584, 266 (2008).

0

0 0 0 1 0

0

0 0 0 0 1 0

{ ( ) ( )}

2{ ( ) ( )}

tr

IB kr K kr K kr

IB kr K kr K kr

0

0 0 1 0 0

2sin ( ) ( )

2

n

n n

j nkaB kr K nkr K nkr

On-axis field has no higher harmonic fields, while the off-axis field components do.For a constant j, B0 remains unchanged when length dimensions are scaled. How about with steel poles?

8


period = 12 mm, coil r0 = 3.15 mm coil dimension b = 3.84 mm j(coil) = 1kA/mm2

(BOP – BEq)/BEq < 0.1%

Analytical equation agrees with model coil calculations within 0.1%.Higher harmonic coefficients of the on-axis field are less than 10-6 .

[ ] exp( 0.95 )K kr

Helical undulator:Model calculation to verify the analytical results

0

00 0 10.8 sin( ) ( ) ( )

2

r b

r

ka drB j krK kr K kr

0

2/

B T

j kA mm

mm

Helical undulator: Scaling law for steel poles


= 12, j0

= 1.0 kA/mm2, B0

= 0.877286 T, steel poles

= 24, j0

= 0.5 kA/mm2, B0

= 0.876782 T, steel poles, DB0

= 5x10-4

= 12, j0

= 1.0 kA/mm2, B0

= 0.577 T, air poles

Off-axis field components follow the j scaling within 0.1%.

On-axis field and coil maximum field for the two periods follow the j scaling.

When the off-axis field components are normalized to the on-axis fields for the steel poles

and air poles, the field components follow the j scaling for r < 3 mm.

How to use the jl scaling law and why it is so difficult with “short-period” planar and helical SCUs


Bm = 3.6 T, jc = 1.55 (@ 16

B0 = 1.25, K = 1.868 (@ 16

j 2j)(/2: same Bm, B0 data

K 0.5K (1.8680.934)

g 0.5g (8 mm4 mm)

jc(SC)-curve not 2 jc(SC)-curve

jc = 1.55 1.8, B0 = 1.25 0.9 T

K = 0.934 0.672

B0(g1) = (B0)exp{-(g1-4)/8}

B0(6 mm) = 0.41 T

New K = 0.305

All at NbTi jc at 4.2 K.

an example: Scale from l=16 to l= 8 mm

Don’t be discouraged. If you increase the period from 16 to only 20 mm, there will be big rewards!

o

o

l = 8 mm, 1.0 2.0 3.0 4.0 5.0 6.0 7.0

l = 20 mm, 0.4 0.8 1.2 1.6 2.0 2.4 2.8

Summary


The derived analytical expressions of B0 for planar and helical undulators show that, when length dimensions are scaled according to a period ratio, the field remain unchanged for a constant j00 jnn.

The j scaling law extends to the distributions of the flux density and permeability for the whole region of SCUs with nonlinear poles and yokes.

With one data set for B0 and Bm, B0 for different periods may be calculated.

– B0 dependence on coil dimension is insignificant.

For g/ > 0.15, the peak field varies as

As an example for using the j scaling law, it was shown why achieving a “required” B0 for a “shorter period” is an issue.

1 10 0 exp

g g g gB B

a scaling law for planar and helical superconducting

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