Accelerator Magnet

8/9/2019 Accelerator Magnet

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DESIGN OF ACCELERATOR MAGNETS

S. Russenschuck CERN, 1211 Geneva 23, Switzerland

Abstract

Analytical and numerical eld computation methods for the design of conven-tional and superconducting accelerator magnets are presented. The eld in theaperture of these magnets is governed by the Laplace equation. The conse-quences for the eld quality estimation and the ideal pole shapes (for conven-tional magnets) and ideal current distributions (for superconducting magnets)are described. Examples of conventional (LEP) and superconducting (LHC)dipoles and quadrupoles are given.

1 Guiding elds for charged particles

A charged particle moving with velocity v through an electro-magnetic eld is subjected to the Lorentzforce

F = e( v B + E ). (1)

While the particle moves from the location r1 to r2 with v = drdt , it changes its energy by

E = r2

r 1

F dr = e r2

r 1( v B + E )dr . (2)

The particle trajectory dr is always parallel to the velocity vector v. Therefore the vector v B isperpendicular to dr, i.e., (v B ) dr = 0 . The magnetic eld cannot contribute to a change in theparticles energy. However, if forces perpendicular to the particle trajectory are needed, magnetic eldscan serve for the guiding and the focusing of particle beams. At relativistic speed, electric and magneticelds have the same effect on the particle trajectory if E = c B . A magnetic eld of 1 T is then equivalentto an electric eld of strength E = 3 108 V/m. A magnetic eld of one Tesla strength can easily beachieved with conventional magnets (superconducting magnets on an industrial scale can reach up to 10T), whereas electric eld strength in the Giga Volt / meter range are technically not to be realized. This isthe reason why for high energy particle accelerators only magnetic elds are used for guiding the beam.

A charged particle forced to move along a circular trajectory looses energy by emission of photonsaccording to

E = 13 0

e2E 4

(m0c2)4R (3)

with every turn completed [24], where R is the curvature of the trajectory and E is the energy of thebeam. A comparison between electron and proton beams of the same energy yields:

E p E e

=mec2

mpc24

= 0.511MeV938.19MeV

4

= 8 .8 10 14 . (4)

For the heavier protons, synchrotron radiation is therefore not a limiting factor, however, the maximumenergy is limited by the eld in the dipole (bending) magnets. For a particle travelling on a circularclosed orbit in an uniform bending eld, the resulting Lorentz force e v B has to equal the centrifugal


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force mv2

R . With the bending radius R given in meters and the magnetic ux density given in Tesla, theparticle momentum p = mv (given in GeV/c) is therefore determined by:

p 109eV

0.2998 109 ms= R B e (5)

and therefore

p = R B 0.2998. (6)

The factor 0.2998 comes from the change of units kgms 1 GeV/ c. The maximum energy in cir-cular lepton machines is limited by the synchrotron radiation and in linear colliders by the maximumachievable electric eld in the accelerator structure. In circular proton machines the maximum energy isbasically limited by the strength of the bending magnets.

2 Conventional and superconducting magnets

Figs. 1 - 3 show a methamorphosis between the conventional dipoles for the LEP (Large electronpositron collider) and the single aperture dipole model used for testing the dipole coil manufacture for

the LHC. All eld calculations were performed using the CERN eld computation program ROXIE. Theeld representations in the iron yokes are to scale, the size of the eld vectors changes with the differenteld levels. Fig. 1 (left) shows the (slightly simplied) C-Core dipole for LEP. The advantage of C-Coremagnets is an easy access to the beam pipe, but they have a higher fringe eld and are less rigid than theH-Type magnets as shown in g. 1 (right). Additional pole shims can be applied in order to improve theeld quality in the aperture. The maximum eld in the LEP dipoles is about 0.13 T. In order to reducethe effect of remanent iron magnetization, the yoke is laminated with a lling factor of only 0.27. It canbe seen that the eld is dominated by the shape of the iron yoke.

0.00 0.15-0.15 0.29-0.29 0.44-0.44 0.59-0.59 0.73-0.73 0.88-0.88 1.03-1.03 1.18-1.18 1.33-1.33 1.47-1.47 1.62-1.62 1.77-1.77 1.92-1.92 2.06-2.06 2.21-2.21 2.36-2.36 2.50-2.50 2.65-2.65 2.8-

|Btot| (T)

0.10 0.15-0.15 0.29-0.29 0.44-0.44 0.59-0.59 0.74-0.74 0.88-0.88 1.03-1.03 1.18-1.18 1.32-1.32 1.47-1.47 1.62-1.62 1.77-1.77 1.91-1.91 2.06-2.06 2.21-2.21 2.35-2.35 2.50-2.50 2.65-2.65 2.8-

|Btot| (T)

Fig. 1: Magnetic eld strength in the iron yoke and eld vector presentation of accelerator magnets. Left: C-Core dipole

(N I = 2 5250 A , B1 = 0 .13 T) with a lling factor of the yoke laminations of 0.27. Right: H-magnet ( N I = 12000A, B1 = 0 .3 T, Filling factor of yoke laminations 0.98)

If the excitational current is increased above a density of about 10 A/ mm2 one has to switch tosuperconducting coils. Neglecting the quantum-mechanical nature of the superconducting material, it issuf cient to notice that the maximum achievable current density in the superconducting coil is by thefactor of 1000 higher than in copper coils. Magnets where the coils are superconducting, but the eldshape is dominated by the iron pole are called super-ferric. Fig. 2 (left) shows the H-Type (super-ferric)magnet with increased excitation. The poles are starting to saturate and the eld quality in the aperture is


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0.37 0.14-0.14 0.29-0.29 0.44-0.44 0.58-0.58 0.74-0.74 0.88-0.88 1.03-1.03 1.18-1.18 1.33-1.33 1.47-1.47 1.62-

1.62 1.77-1.77 1.91-1.91 2.06-2.06 2.21-2.21 2.36-2.36 2.50-2.50 2.65-2.65 2.8-

|Btot| (T)

0.02 0.16-0.16 0.31-0.31 0.45-0.45 0.60-0.60 0.75-0.75 0.89-0.89 1.04-1.04 1.19-1.19 1.33-1.33 1.48-1.48 1.63-

1.63 1.77-1.77 1.92-1.92 2.07-2.07 2.21-2.21 2.36-2.36 2.51-2.51 2.65-2.65 2.8-

|Btot| (T)

Fig. 2: Magnetic eld strength in the iron yoke and eld vector presentation of accelerator magnets. Left: H-magnet with

increased excitation current ( N I = 48000 A , B 1 = 1 .17 T). Right: Window frame geometry. ( N I = 180000 A ,B 1 = 2 .28 T). Notice the saturation of the poles in the H-magnet.

decreased due to the increasing fringe eld. This can be avoided by constructing so-called window-framemagnets as shown in g. 2 (right).

The disadvantage of window-frame magnets is that the synchrotron radiation is partly absorbed inthe (superconducting) coils and that access to the beam pipe is even more dif cult. The advantage is abetter eld quality and that pole shims can be avoided.

It can be seen, that at higher eld levels the eld quality in the aperture is increasingly affectedby the coil layout. Superconducting window-frame magnets are receiving considerable attention latelyas high eld (14-16 T) dipoles. As the coil winding is easier for window frame magnets than for theso-called cos magnets (shown in g. 3, left), the application of the mechanically less stable materialswith higher critical current density (e.g. Nb3Sn) becomes feasible. The LHC superconducting magnetsare of the cosn type. The advantage of the cos (dipole) magnets is that the eld outside the coildrops with 1/r 2 and therefore the saturation effects in the iron yoke are reduced. Fig. 3 (right) nallyshows the CTF (coil-test-facility) used for testing the manufacturing process of the LHC magnets. Noticethe large difference between the eld in the aperture (8.3 T) and the eld in the iron yoke (max. 2.8 T),which has merely the effect of shielding the fringe eld.

0.20 0.34-0.34 0.47-0.47 0.61-0.61 0.75-0.75 0.88-0.88 1.02-1.02 1.16-1.16 1.29-1.29 1.43-1.43 1.57-1.57 1.70-1.70 1.84-1.84 1.98-1.98 2.11-2.11 2.25-2.25 2.39-2.39 2.52-2.52 2.66-2.66 2.8-

|Btot| (T)

0.00 0.15-0.15 0.29-0.29 0.44-0.44 0.59-0.59 0.74-0.74 0.88-0.88 1.03-1.03 1.18-1.18 1.32-1.32 1.47-1.47 1.62-1.62 1.77-1.77 1.91-1.91 2.06-2.06 2.21-2.21 2.36-2.36 2.50-2.50 2.65-2.65 2.8-

|Btot| (T)

Fig. 3: Magnetic eld strength in the iron yoke and eld vector presentation of accelerator magnets. Left: So-called cos

magnet ( N I = 330000 A , B1 = 3 .0 T). Right: LHC single aperture coil test facility ( N I = 480000 A , B 1 = 8 .33 T).Notice that even with increased eld in the aperture the eld strength in the yoke is reduced in the cos magnet design.


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4 Field equations

Maxwell s equations for the stationary case read in SI (MKS) units:

H ds = A ( J + D

t ) d A, (14)

E ds =

t A B d A, (15)

A B d A = 0 , (16) A D d A = V dV. (17)

The vector elds E, H, D, B are called electric and magnetic eld, and electric and magnetic induction(ux density), respectively. Eq. (14) is Amp` ere s law as modi ed by Maxwell to include the displace-ment current distribution and eq. (15) is Faraday s law of electromagnetic induction. Eq. (17) is Gauss fundamental theorem of electrostatics. The constitutive equations are:

B =

H = 0(

H +

M ), (18) D = E = 0( E + P ), (19)

J = E + J imp ., (20)

with the permeability of free space 0 = 4 10 7 H/m and the permittivity of free space0 = 8 .8542 10 12 F/m.

5 Maxwell s equations in vector notation

The eld equations can be written in differential form as follows:

curl H = J + D

t , (21)

curl E = Bt

, (22)

div B = 0 , (23)

div D = . (24)

Eq. (21) - (24) are Maxwell s equations in SI units and vector notation which is mainly due to O. Heavi-side in the 1880s, who also eliminated the vector-potential and the scalar potential in Maxwell s originalset of equations. The link between Eq. (14) - (17) and (21) - (24) is given through the integral theorems:

V div gdV = A g d A, (25)which is called Gauss (Ostrogradskii s) divergence theorem and A curl g d A = g ds, (26)

which is Stokes surface integral theorem.


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6 Maxwell s equations for magnetostatic problems

For magnetostatic problems the time derivative can be set to zero, t = 0 , and Maxwell s equationsreduce to

curl H = J , (27)

div B = 0 , (28)

B = ( H ) H = 0( H + M ). (29)

7 Interface conditions

B t2Bn2

B2

B t1

Bn1 B1 1

2

n

H t2

H t1I h

2

1

2

1

Fig. 4: On the interface conditions for permeable media.

If we apply Amp` ere s law in the integral form

H ds = A J d A , (30)to the loop displayed in g. 4 (left), and let h 0, then the enclosed current is zero, as in an in nitesimalsmall rectangle there cannot be a current ow. Therefore

H t1 = H t2 , (31)

i.e.,

n ( H 1 H 2) = 0 . (32)

Because of B d A = 0 we get at the interfaceBn1 = Bn2 , (33)i.e.,

n (

B1 B2) = 0 . (34)

Now

tan 1tan 2

=B t1B n1B t2B n2

= 1H t12H t2

= 12

. (35)

For 2 1 it follows that tan 1 tan 2 . Therefore for all angles / 2 > 2 > 0 we get tan 1 0,see also g. 4 (right). The eld exits vertically from a highly permeable medium into a medium withlow permeability. We will come back to this point when we discuss ideal pole shapes of conventionalmagnets.


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8 One-dimensional eld computation for conventional magnets

Consider the magnetic (dipole) circuit shown in g. 5 (left). With Amp` ere s law in the integral form

H ds = A J d A , we can writeH iron s iron + H gap sgap = N I , (36)10r B

iron s iron + 10 Bgap sgap = N I , (37)

With r 1 we get the easy relation

Bgap = 0N I

sgap. (38)

Fig. 5: Magnetic circuit of a conventional dipole magnet (left) and a quadrupole magnet (right). Neglecting the magnetic

resistance of the iron yoke, an easy relation between the air gap eld and the required excitational current can be derived.

For the quadrupole we can split up the integration path as shown in g. 5 (right). From the originto the pole (part 1), along an arbitrary path through the iron yoke (part 2), and back along the x-axis (part3). Neglecting again the magnetic resistance of the yoke we get

H ds = 1 H 1 ds + 3 H 3 ds = N I . (39)As we will see later, in a quadrupole the eld is de ned by it s gradient g with Bx = gy and By = gx.Therefore the modulus of the eld along the integration path 1 is

H = g0 x2 + y2 =

g0

r. (40)

Along the x-axis the eld integral is zero because H s. Therefore

r 0

0Hdr =

g0

r 0

0rdr =

g0

r 202

= N I, (41)

g = 20NI

r 20. (42)

Notice that for a given NI the eld decreases linearly with the gap length of the dipole, whereas thegradient in a quadrupole magnet is inverse proportional to the square of the aperture radius r0 .


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8.1 Permanent magnet excitation

For a magnetic circuit with permanent magnet excitation as shown in g. 6 we can repeat the exercisewith

H iron s iron + H gap sgap + H mag smag = 0 . (43)

Fig. 6: Dipole with permanent magnet excitation. Neglecting the magnetic resistance of the iron yoke, an easy relation between

the air gap eld and the required size of the permanent magnet can be derived.

In the absence of fringe elds we get with the pole surface Agap and the magnet surface Amag :

Bmag Amag = Bgap Agap = 0H gap Agap , (44)

For r 1 we can again neglect the magnetic resistance of the yoke and from eq. 43 it follows that

H gap sgap = H mag smag , (45)1

0Bmag

AmagAgap

sgap = H mag smag , (46)

Bmag0H mag

= smag

sgapAgapAmag

= P, (47)

where P is called the permeance coef cient which (for Agap = Amag ) becomes zero for sgap smag(open circuit) and becomes for smag sgap (short circuit). The case of Amag > A gap is usu-ally referred to as the ux concentration mode. The permeance coef cient de nes the point on thedemagnetisation curve, i.e., the branch of the permanent magnet hysteresis curve in the second quadrant.

Fig. 7 shows a magnetic circuit with zero air gap, a circuit with sgap = 2smag and an open circuitwith a smarium cobalt magnet (remanent eld 0.9T).

From eq. 44 and 45 we derive

Bmag Amag smag = 0H gap Agap H gap sgapH mag. (48)

Therefore

H gap = (Amag smag )( Bmag H mag )0(Agap sgap ) = V OLmag ( Bmag H mag )0V OLgap . (49)For a given magnet volume, the maximum air gap eld can be obtained by dimensioning the magneticcircuit in such a way that Bmag H mag is maximum. It is usually said that this implies operating thepermanent magnet with maximum energy density . At this point we have to be careful, however:


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mag

0H mag

= 0.5 T= 0.4 T

Bmag 0H mag

B = 0.2 T= 0 T = 0.7 T

0.000 0.102-

0.102 0.204-

0.204 0.307-

0.307 0.409-

0.409 0.511-

0.511 0.614-

0.614 0.716-

0.716 0.818-0.818 0.921-

0.921 1.023-

1.023 1.126-

1.126 1.228-

1.228 1.330-

1.330 1.433-

1.433 1.535-

1.535 1.637-

1.637 1.740-

1.740 1.842-

1.842 1.945-

|Btot| (T)

Bmag 0H mag

= 0.9 T

Fig. 7: Magnetic circuit with zero air gap (left), a circuit with sgap = 2 smag (middle) and an open circuit (right) with a smariumcobalt magnet (remanent eld 0.9T). Notice the values for 0 H mag and Bmag on the demagnetization curve. For the case withsgap = 2 smag the numerical calculation yields 0 H gap = 0 .197 T, 0 H mag = 0 .4 T.

The relation between the magnetization and the eld is not linear (and not even unique). It dependsnot only on the external eld but also on the history of how it was applied.

Because of the history dependence of the magnetization, the stored magnetic energy is not! W =0.5BHV which holds only for linear material. We will have to study in detail the hysteresis effectsof hard ferromagnetic material.

For larger gap sizes the fringe elds cannot be neglected.

In the presence of magnetized domains the magnetic eld can be calculated as in vacuum, if all currents(including the magnetization currents) are explicitly considered

curl B = 0( J free + J mag ) = 0 J free + 0curl M. (50)

Hence

curl B 0 M

0= J free . (51)

For the magnetized media we therefore have

H = B 0 M

0(52)

or B = 0( H + M ). (53)

For linear material we get the relation between B and H as:

B = 0 H + 0 m H = 0(1 + m ) H = 0r H = H, (54)

where r is the relative permeability and m is called magnetic susceptibility. On the other hand, eq.53 is valid for all non-linear media, e.g., permanent magnets, where the magnetization persists without


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exterior eld. The magnetic induction B is always source free, i.e., div B = 0 , but the magnetic eld H is not:

div H = div B 0 M

0= div M = mag . (55)

In eq. 55 a ctitious magnetic charge density mag = div M was introduced with the eld starting and

ending on these ctitious magnetic charges, c.f. g. 8.

M B H

Fig. 8: Field, induction and magnetization in permanent magnets.

Coming back to the magnetic circuit as shown in g. 6 the conclusions are formally correct, aslong as the magnetization M is constant. This is indeed the case for so-called rare earth material likesamarium cobalt SmCo5 or neodymium iron boron Nd2Fe14B but not for iron alloys with aluminum andnickel (Alnico) or Ferrite (e.g., Fe2O3), see g. 9.

Now we shall consider a torus of ferromagnetic material ( g. 10) with N excitational windingsthat excite the eld

H = NI 2r

. (56)

B

H 0H

Fig. 9: Demagnetization curves for different permanent magnet materials.


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The induced voltage in the pick-up coil is

U i = ddt

= dB

dt A. (57)

U I r

Coercivity

Remanence

Virgin curve

Saturation

H

B

Fig. 10: Schematic setup for the measurement of hysteresis effects in ferromagnetic material (left). Hysteresis curve for a hard

ferromagnetic material (right).

Time integration ( U i dt = BA) yields the corresponding values of I and U and consequently thehysteresis curve for H and B . The surface spanned by the hysteresis curve are the magnetization losses.For the torus in g. 10, the power needed for exciting the eld is

dW dt

= IN ddt

= INAdBdt

= IN 2r

A2rdBdt

= HV dBdt

. (58)

Therefore1V

dW dt

= H dBdt

, (59)

W

V =

B2

B 1HdB. (60)

For a complete cycle we get

W V

= HdB = H0(dH + dM ) = 0 HdM. (61)As a consequence, the working point with maximal Bmag H mag guarantees that the air gap induc-

tion is maximum for a given aperture and magnet volume. However, this is not! the state with maximumenergy density in the permanent magnet material.

9 Harmonic elds

We will now show that in the aperture of a magnet (two-dimensional, current free region) both the

magnetic scalar-potential as well as the vector-potential can be used to solve Maxwell s equations:

H = grad = x

ex y

ey , (62)

B = curl( ez Az ) = Az

y ex

Azx

ey , (63)

and that both formulations yield the Laplace equation. These elds are called harmonic and the eldquality can be expressed by the fundamental solutions of the Laplace equation. Lines of constant vector-potential give the direction of the magnetic eld, whereas lines of constant scalar potential de ne theideal pole shapes of conventional magnets.


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9.1 Magnetic scalar potential

Every vector eld can be split into a source free and a curl free part. In case of the magnetic eld with

H = H s + H m (64)

the curl free part H m arises from the induced magnetism in ferromagnetic materials and the source freepart H s is the eld generated by the prescribed sources (can be calculated directly by means of Biot

Savart s law). With curl H m = 0 it follows that H = grad m + H s (65)

and we get:

div B = 0 (66)

div ( grad m + H s) = 0 (67)

div grad m = div H s (68)

While a solution of eq. 68 is possible, the two parts of the magnetic eld H m and H s tend to be of similarmagnitude (but opposite direction) in non-saturated magnetic materials, so that cancellation errors occurin the computation. For regions where the current density is zero, however, curl H = 0 and the eld canbe represented by a total scalar potential

H = grad (69)

and therefore we get

0 div grad = 0 , (70)

2 = 0 . (71)

which is the Laplace equation for the scalar potential. The vector-operator Nabla (Hamilton operator) isdened as

= ( x

, y

, z

) (72)

and the Laplace operator

= 2 = 2

x 2 +

2

y2 +

2

z 2. (73)

The Laplace operator itself is essentially scalar. When it acts on a scalar function the result is a scalar,when it acts on a vector function, the result is a vector.

9.2 Vector-potential

Because of div B = 0 a vector potential A can be introduced: B = curl A. We then get

curl A = 0( H + M ), (74)

H = 10

curl A M, (75)

10

curl curl A = J + curl M (76)

10

( 2 A + grad div A) = J + curl M, (77)


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Since the curl (rotation) of a gradient eld is zero, the vector-potential is not unique. The gradientof any (differentiable) scalar eld can be added without changing the curl of A:

A0 = A + grad . (78)

Eq. (78) is called a gauge-transformation between A0 and A. B is gauge-invariant as the transformationfrom A to A0 does not change B . The freedom given by the gauge-transformation can be used to set the

divergence of A to zero

div A = 0 , (79)

which (together with additional boundary conditions) makes the vector-potential unique. Eq. (79) iscalled the Coulomb gauge, as it leads to a Poisson type equation for the magnetic vector-potential. There-fore, from eq. 77 we get after incorporating the Coulomb gauge:

2 A = 0( J + curl M ) (80)

In the two-dimensional case with no dependence on z, z = 0 and J = J z , A has only a z-componentand the Coulomb gauge is automatically ful lled. Then we get the scalar Poisson differential equation

2Az = 0J z . (81)

For current-free regions (e.g. in the aperture of a magnet) eq. (81) reduces to the Laplace equation,which reads in Cartesian coordinates

2Azx 2

+ 2Az

y 2 = 0 . (82)

and in cylindrical coordinates

r 2 2Azr 2

+ rA zr

+ 2Az

2 = 0 . (83)

9.3 Field harmonics

A solution of the homogeneous differential equation (83) reads

Az (r, ) =

n =1(C 1n r n + C 2n r

n )(D 1n sin n + D 2n cos n). (84)

Considering that the eld is nite at r = 0, the C 2n have to be zero for the vector-potential inside theaperture of the magnet while for the solution in the area outside the coil all C 1n vanish. Rearrangingeq. (84) yields the vector-potential in the aperture:

Az (r, ) =

n =1r n (C n sin n D n cos n), (85)

and the eld components can be expressed as

B r (r, ) = 1r

A z

=

n =1nr n 1(C n cos n + Dn sin n), (86)

B(r, ) = A zr

=

n =1nr n 1(C n sin n D n cosn). (87)


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Each value of the integer n in the solution of the Laplace equation corresponds to a different ux distri-bution generated by different magnet geometries. The three lowest values, n=1,2, and 3 correspond to adipole, quadrupole and sextupole ux density distribution. The solution in Cartesian coordinates can beobtained from the simple transformations

Bx = Br cos B sin , (88)

By = Br sin + B cos. (89)

For the dipole eld (n=1) we get

B r = C 1 cos + D1 sin, (90)

B = C 1 sin + D1 cos, (91)

Bx = C 1, (92)

By = D1. (93)

This is a simple, constant eld distribution according to the values of C 1 and D1. Notice that we have notyet addressed the conditions necessary to obtain such a eld distribution. For the pure quadrupole (n=2)we get from eq. 86 and 87:

B r = 2 r C 2 cos2 + 2 r D2 sin 2, (94)

B = 2 r C 2 sin 2 + 2 r D2 cos2, (95)

Bx = 2( C 2x + D2y), (96)

By = 2( C 2y + D2x). (97)

The amplitudes of the horizontal and vertical components vary linearly with the displacements from theorigin, i.e., the gradient is constant. With a zero induction in the origin, the distribution provides linearfocusing of the particles. It is interesting to notice that the components of the magnetic elds are coupled,i.e., the distribution in both planes cannot be made independent of each other. Consequently a quadrupolefocusing in one plane will defocus in the other.

Repeating the exercise for the case of the pure sextupole (n=3) yields:

B r = 3 r C 3 cos3 + 3 r D3 sin 3, (98)

B = 3 r C 3 sin 3 + 3 r D3 cos3, (99)

Bx = 3C 3(x2 y2) + 6 D3xy, (100)

By = 6C 3xy + 3 D3(x2 y2). (101)

Along the x-axis (y=0) we then get the expression for the y-component of the eld:

By = D1 + 2 D2x + 3 D3x2 + 4 D4x3 + ... (102)

If only the two lowest order elements are used for steering the beam, forces on the particles are eitherconstant or vary linear with the distance from the origin. This is called a linear beam optic. It has tobe noted that the treatment of each harmonic separately is a mathematical abstraction. In practical situ-ations many harmonics will be present and many of the coef cients C n and Dn will be non-vanishing.A successful magnet design will, however, minimize the unwanted terms to small values. It has to bestressed that the coef cients are not known at this stage. They are de ned through the (given) bound-ary conditions on some reference radius or can be calculated from the Fourier series expansion of the(numerically) calculated eld (ref. eq. 7) in the aperture using the relations

An = nr n 1

0 C n and Bn = nrn 10 Dn . (103)


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10 Ideal pole shapes of conventional magnets

From the theory of electrostatics we remember that the potential difference between two points (close inspace) is

d = E ds = (E x dx + E y dy + E z dz) = (104)

xdx +

ydy +

zdz = (grad ) ds.

Equipotentials are surfaces where is constant. For a path d s along the equipotential it therefore results

d = grad ds = 0 (105)

i.e., the gradient is perpendicular to the equipotential. With the eld lines (lines of constant vectorpotential) leaving highly permeable materials perpendicular to the surface (ref. chapter 7), the lines of total magnetic scalar potential de ne the pole shapes of conventional magnets. As in 2D (with absenceof magnetization and free currents) the z-component of the vector potential and the magnetic scalarpotential both satisfy the Laplace equation, we already have the solution for a bipolar eld:

= C 1x + D

1y. (106)

So C 1 = 0 , D1 = 0 gives a vertical (normal) dipole eld, C 1 = 0 , D1 = 0 yields a horizontal (skew)dipole eld. The equipotential surfaces are parallel to the x-axis or y-axis depending on the values of C and D and results in a simple ux density distribution used for bending magnets in accelerators. For thequadrupole:

= C 2(x2 y2) + 2 D2xy (107)

with C 2 = 0 giving a normal quadrupole eld and with D2 = 0 giving a skew quadrupole eld (which isthe above rotated by / 4). The quadrupole eld is generated by lines of equipotential having hyperbolicform. For the C 2 = 0 case, the asymptotes are the two major axes.

In practice, however, the magnets have a nite pole width (due to the need of a magnetic uxreturn yoke and space for the coil). To ensure a good eld quality with these nite approximations of the ideal shape, small shims are added at the outer ends of each pole. The shim geometry has to beoptimized (using numerical eld computation tools) while considering, that with an increasing height of the shim saturation occurs at high excitation and leads to the eld distribution being strongly dependenton the magnet excitation level. On the other hand, with a very thin and long shim the nature of the eldgenerated will change and different harmonics are being generated. Fig. 11 shows the pole shape of aconventional dipole and quadrupole magnet, with magnetic shims.

Fig. 12 shows the cross-section of the LEP dipole and quadrupole magnets with iso-surfaces of constant vector-potential (for the dipole) and magnetic eld modulus in the iron yoke for the quadrupole.The eld quality in the dipole was improved by adding shims on the pole surface. In case of the

quadrupole, however, the pole shape is de ned as a combination of a hyperbola, a straight section andan arc. The points at which the segments are connected was found in an optimization process not onlyconsidering the multipole components in the cross-section, but also to provide for a part compensationof the end-effects.

11 Coil eld of superconducting magnets

For coil dominated superconducting magnets with elds well above one Tesla the current distribution inthe coils dominate the eld quality and not the shape of the iron yoke, as it is the case in conventionalmagnets. The problem therefore remains how to calculate the eld harmonics from a given current


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x

y

y = const.Shim

x

y

xy = const.

Fig. 11: Pole shape of a conventional dipole and quadrupole magnet.

-0. 13 0.864-

0 .86 4 0. 00 3-

0 .00 3 0. 00 5-

0 .00 5 0. 00 7-

0 .00 7 0. 00 9-

0 .00 9 0. 01 2-

0 .01 2 0. 01 4-

0 .01 4 0. 01 6-

0 .01 6 0. 01 8-

0 .01 8 0. 02 1-

0 .02 1 0. 02 3-

0 .02 3 0. 02 5-

0 .02 5 0. 02 8-

0 .02 8 0. 03 0-

0 .03 0 0. 03 2-

0 .03 2 0. 03 4-

0 .03 4 0. 03 7-

0 .03 7 0. 03 9-

0 .03 9 0. 04 1-

A (Tm)

0. 66 4 0 .1 10-

0. 11 0 0 .2 19-

0. 21 9 0 .3 28-

0. 32 8 0 .4 38-

0. 43 8 0 .5 47-

0. 54 7 0 .6 56-

0. 65 6 0 .7 66-

0. 76 6 0 .8 75-

0. 87 5 0 .9 84-

0. 98 4 1 .0 94-

1. 09 4 1 .2 03-

1. 20 3 1 .3 13-

1. 31 3 1 .4 22-

1. 42 2 1 .5 31-

1. 53 1 1 .6 41-

1. 64 1 1 .7 50-

1. 75 0 1 .8 59-

1. 85 9 1 .9 69-

1. 96 9 2 .0 78-

|Btot| (T)

Fig. 12: Cross-section of the LEP dipole and quadrupole magnets with iso-surfaces of constant vector-potential (left) and

magnetic eld modulus (right).

distribution. It is reasonable to focus on the elds generated by line-currents, since the eld of any currentdistribution over an arbitrary cross-section can be approximated by summing the elds of a number of line-currents distributed within the cross-section. For a set of ns of these line-currents at the position(r i , i ) carrying a current I i , the multipole coef cients are given by [15]

Bn (r 0) = n s

i=1

0I i2

r n 10r ni

1 + r 1r + 1

( r iRyoke

)2n cos ni , (108)

An (r 0) =n s

i=10I i2

rn 10r ni

1 + r 1r + 1

( r iRyoke

)2n sin ni , (109)

where Ryoke is the inner radius of the iron yoke with the relative permeability r . The eld of any currentdistribution over an arbitrary cross-section can be approximated by summing the elds of a number of line-currents distributed within the cross-section. As superconducting cables are composed of singlestrands with a diameter of about 1 mm, a good computational accuracy can be obtained by representingeach cable by two layers of equally spaced line-currents at the strand position. Thus the grading of thecurrent density in the cable due to the different compaction on its narrow and wide side is automaticallyconsidered.


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0 20 40 60 80 100 120 140 160

0 20 40 60 80 100 120 140 160

Fig. 13: Field vectors for a superconducting coil in a (non-saturated) iron yoke of cylindrical inner shape (left) and the repre-

sentation of the iron yoke with image current (right).

With equations (108) and (109), a semi-analytical method for calculating the elds in supercon-ducting magnets is given. The iron yoke is represented by image currents (the second term in the paren-theses). At low eld level, when the saturation of the iron yoke is low, this is a suf cient method foroptimizing the coil cross-section. Under that assumption some important conclusions can be drawn:

For a coil without iron yoke the eld errors scale with 1/r n where n is the order of the multipoleand r is the mid radius of the coil. It is clear, however, that an increase of coil aperture causesa linear drop in dipole eld. Other limitations of the coil size are the beam distance, the elec-tromagnetic forces, yoke size, and the stored energy which results in an increase of the hot-spottemperature during a quench.

For certain symmetry conditions in the magnet, some of the multipole components vanish, i.e.for an up-down symmetry in a dipole magnet (positive current I 0 at (r 0, 0) and at (r0, 0))no An terms occur. If there is an additional left-right symmetry, only the odd B1, B3 , B5 , B7, ..components remain.

The relative contribution of the iron yoke to the total eld (coil eld plus iron magnetization) is fora non-saturated yoke ( r 1) approximately (1+(

R yoker )

2n ) 1 . For the main dipoles with a meancoil radius of r = 43.5 mm and a yoke radius of Ryoke = 89 mm we get for the B1 component a19% contribution from the yoke, whereas for the B5 component the in uence of the yoke is only0.07%.

It is therefore appropriate to optimize for higher harmonics rst using analytical eld calculation, andinclude the effect of iron saturation on the lower-order multipoles only at a later stage.

12 The generation of pure multipole elds

Consider a current shell ri < r < r e with a current density varying with the azimuthal angle , J () =J 0 cosm, then we get for the Bn components

Bn (r 0) = r e

r i 2

0

0J 0r n 1

02r n

1 + r 1r + 1

( r

Ryoke)2n cos mcos n r dd r . (110)


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With 2

0 cosmcos nd = m,n (m, n = 0) it follows that the current shell produces a pure 2m -polar eld and in the case of the dipole ( m = n = 1 ) one gets

B1(r 0) = 0J 0

2(r e r i) +

r 1r + 1

1R2yoke

13

(r 3e r3i ) . (111)

Obviously, since

2

0 cosmsin nd = 0 , all An components vanish. A shell with cos and cos 2

dependent current density is displayed in gure 15. Because of |B | = n B 2n + A2n the modulus of

Fig. 14: Shells with cos (left) and cos 2 (right) dependent current density.

the eld inside the aperture of the shell dipole without iron yoke is given as

|B | = 0J 0

2 (r e r i). (112)

12.1 Coil-block arrangements

Usually the coils do not consist of perfect cylindrical shells because the conductors itself are eitherrectangular or keystoned with an insuf cient angle to allow for perfect sector geometries. Thereforethe shells are subdivided into coil-blocks, separated by copper wedges. The eld generated by this coillayout has to be calculated with the line-current approximation of the superconducting cable. Real coil-geometries with one and two layers of coil-blocks are shown in g. 15.


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Fig. 15: Coil-block arrangements made of Rutherford type cable with grading of current density due to the keystoning of the

cable. LHC main dipole model coil with two layers wound from different cable.

13 Numerical eld computation

For the calculation of the saturation-induced eld errors of the lower-order ( b2 b5), which vary as afunction of the excitational eld, numerical techniques such as the nite-element method (FEM) have tobe applied. Fig. 16 (right) shows the variation of the lower-order multipoles as a function of the excitationcurrent in the two-in-one main dipoles of the LHC. It can be seen that already the b4 component is hardlyinuenced by saturation effects. Fig. 16 (left) shows the transfer function B/I as a function of theexcitation from injection to nominal eld level including the saturation effects and the persistent currentmultipoles. Fig. 17 shows the relative permeability of the iron yoke for both injection and nominal eldlevel.

0 2000 4000 6000 8000 10000 12000

0.705

0.706

0.707

0.708

0.709

B0

I (A)

T/kA

0 2000 4000 6000 8000 10000 12000

-6

-4

-2

0

2

4

6

b 2

b 3

b 410

b 510

I (A)

x 10 -4

Fig. 16: Left: Transfer function B/I in the LHC main dipole as a function of the excitation from injection to nominal eldlevel. Right: Variation of the lower-order multipoles as a function of the excitation current.

Magnets for particle accelerators have always been a key application of numerical methods inelectromagnetism. Hornsby [11], in 1963, developed a code based on the nite difference method for


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1 .0 02 2 08 .2-

2 08 .2 4 15 .5-

4 15 .5 6 22 .7-

6 22 .7 8 30 .0-

8 30 .0 1 03 7.-

1 03 7. 1 24 4.-

1 24 4. 1 45 1.-

1 45 1. 1 65 9.-

1 65 9. 1 86 6.-

1 86 6. 2 07 3.-

2 07 3. 2 28 0.-

2 28 0. 2 48 8.-

2 48 8. 2 69 5.-

2 69 5. 2 90 2.-

2 90 2. 3 10 9.-

3 10 9. 3 31 7.-

3 31 7. 3 52 4.-

3 52 4. 3 73 1.-

3 73 1. 3 93 8.-

MUEr

1 .0 02 2 08 .2-

2 08 .2 4 15 .5-

4 15 .5 6 22 .7-

6 22 .7 8 30 .0-

8 30 .0 1 03 7.-

1 03 7. 1 24 4.-

1 24 4. 1 45 1.-

1 45 1. 1 65 9.-

1 65 9. 1 86 6.-

1 86 6. 2 07 3.-

2 07 3. 2 28 0.-

2 28 0. 2 48 8.-

2 48 8. 2 69 5.-

2 69 5. 2 90 2.-

2 90 2. 3 10 9.-

3 10 9. 3 31 7.-

3 31 7. 3 52 4.-

3 52 4. 3 73 1.-

3 73 1. 3 93 8.-

MUEr

Fig. 17: Left: Permeability in the iron yoke of the LHC main dipole at injection eld level. Right: Permeability in the iron yoke

at nominal eld level. Change in permeability results in the saturation dependent multipole content as shown in g. 16.

the solving of elliptic partial differential equations and applied it to the design of magnets. Winslow [21]created the computer code TRIM (Triangular Mesh) with a discretization scheme based on an irregulargrid of plane triangles by using a generalized nite difference scheme. He also introduced a variationalprinciple and showed that the two approaches lead to the same result. In this respect, the work can beviewed as one of the earliest examples of the nite element method applied to the design of magnets. ThePOISSON code which was developed by Halbach and Holsinger [10] was the successor of this code andwas still applied for the optimization of the superconducting magnets for the LHC during the early designstages. Halbach had also, in 1967, [9] introduced a method for optimizing coil arrangements and poleshapes of magnets based on the TRIM code, an approach he named MIRT. In the early 1970 s a generalpurpose program (GFUN) for static elds had been developed by Newman, Turner and Trowbridge thatwas based on the magnetization integral equation and was applied to magnet design. Nevertheless, for

the superconducting magnet design, it was necessary to nd more appropriate formulations which do notrequire the modeling of the coils in the nite-element mesh. The integral equation method of GFUNwould be appropriate, however, it leads to a very large (fully populated) matrix if the shape of the ironyoke requires a ne mesh.

The method of coupled boundary-elements/ nite-elements (BEM-FEM), developed by Fetzer,Haas, and Kurz at the University of Stuttgart, Germany, combines a FE description using incompletequadratic (20-node) elements and a gauged total vector-potential formulation for the interior of the mag-netic parts, and a boundary element formulation for the coupling of these parts to the exterior, whichincludes excitational coil elds. This implies that the air regions need not to be meshed at all.

The principle steps in numerical eld computation are:

Formulation of the physical laws by means of partial differential equations. Transformation of these equations into an integral equation with the weighted residual method.

Integration by parts in order to obtain the so-called weak integral form. Consideration of thenatural boundary conditions.

Discretization of the domain into nite elements.

Approximation of the solution as a linear-combination of so-called shape functions.


21/36

In case of the nite element (Galerkin) method, the shape functions in the elements and the weight-ing functions of the weak integral form are identical.

In case of the boundary-element method, another partial integration of the weak integral formresults in an integral equation. Using the fundamental solution of the Laplace operator as weightingfunctions yields an algebraic system of equations for the unknowns on the domain boundary. Incase of the coupled boundary-element/ nite-element method, the two domains are coupled through

the normal derivatives of the vector-potential on their common boundary.

Consideration of the essential boundary conditions in the resulting equation system.

Numerical solution of the algebraic equations. A direct solver with Newton iteration is used in the2D case; the domain decomposition method with a M (B ) iteration [8] is applied in the 3D case.

In order to understand the special properties of these methods and the reasoning which leads to theirapplication in the design and optimization of accelerator magnets, it is suf cient to concentrate on someaspects of the formulations. The function approximation with nite or boundary elements and the solu-tion techniques will only be explained very brie y, as this report cannot replace a lecture on nite-elementtechniques. Nevertheless, as the magnetostatic problem is one of the most simple cases, the basic con-

cepts of numerical eld computation can be explained by means of the most commonly used methodwith the total vector potential formulation, and triangular elements with linear shape functions.

14 Total vector-potential formulation

Consider the elementary model problem, g. 18, consisting of two different domains: i the iron regionwith permeability and a the air region with the permeability 0 . The regions are connected to eachother at the interface ai . Furthermore, each volume is bounded by a surface (sometimes denoted )itself consisting of two different parts H and B with their outward normal vector n. The elementarymodel problem as shown in g. 18 is a mixed boundary value problem. The non-conductive air regiona may also contain a certain number of conductor sources J which do not intersect the iron region i.

Subsequently, the Cartesian coordinate system is always used, as only in this case the vectorLaplace operator, decomposes into three scalar Laplace operators acting on the three components of thevector-potential.

As the divergence of the magnetic ux is zero, the application of a total A-formulation in thedomain = a i automatically satis es eq. (28). Amp` ere s law (27) then takes the form

curl 1

curl A = J in . (113)

Because of the iron saturation, the permeability depends on the magnetic eld and therefore B =( H ) H . The boundary conditions read:

H n = 1 (curl A) n = 0 on H , (114)

B n = curl A n = 0 on B . (115)

Eq. (114) is the homogeneous Neumann boundary condition on H where H is normal to the boundary.Surface current densities do not appear as long as nite conductivity and continous time dependency isassumed. Eq. (115) is the homogeneous Dirichlet boundary condition ( B is parallel to the boundary B ,no ctitious magnetic surface charge density). The far- eld boundary is also part of B . The condition(115) is equivalent to At , i.e., A n = 0 on B [ ? ]. At the interface ai between i and a interface


22/36

i

a

a

r 0

0

0

H H n = 0 (H t = 0)

H

B

B n = 0

(Bn = 0)

ai

ai

Fig. 18: Elementary model problem for the numerical eld calculation of superconducting magnets. In the iron domain the

total vector-potential is displayed. The non-conductive air region a contains a certain number of conductor sources J whichdo not intersect the iron region i .

conditions have to be satis ed (continuity of Bn and H t ):

B i ni +

Ba na = 0 on ai , (116) H i ni + H a na = 0 on ai , (117)

where ni and na are the outer normals associated with the respective subregions. The normal componentof the magnetic ux density Bn is continuous due to the chosen shape functions of the nodal nite-elements. For the tangential component of the magnetic eld intensity H t , the equation (117) written interms of the vector-potential is:

1

(curl Ai) ni + 10

(curl Aa ) na = 0 on ai . (118)

The normal vector ni on the boundary between iron and air is pointing out of the iron domain i and

na is pointing out of the air domain a . The solution of the 3D (vector) boundary value problem is notunique, however. Introducing a penalty term subtracted from eq. (113), yields

curl 1

curl A grad 1

div A = J in . (119)


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With the additional boundary conditions

A n = 0 on H , (120) A n = 0 on B , (121)

1

div A = 0 on B , (122)

10div Aa = 1 div Ai on ai , (123)

it can be proved that the the boundary value problem has a unique solution satisfying the Coulomb gauge

1

div A = 0 in . (124)

The complete formulation for the vector-potential reads

curl 1

curl A grad 1

div A = J in , (125)

A n = 0 on H , (126)1

div A = 0 on B , (127)

A n = 0 on B , (128)1

(curl A) n = 0 on H , (129)

10

div Aa 1

div Ai = 0 on ai , (130)

1

(curl Ai) ni + 10

(curl Aa ) na = 0 on ai , (131)

A continous on ai . (132)

14.1 The weighted residual

The domain = a i is discretized into nite-elements in order to solve this problem numerically.For the approximate solution of A dened on the nodes of the nite element mesh, the differentialequation (125) is only approximately ful lled:

curl 1

curl A grad 1

div A J = R (133)

with a residual (error) vector R . A linear equation system for the unknown nodal values of the vectorpotential A(k ) can be obtained by minimizing the weighted residuals R in an average sense over thedomain , i.e.,

wa R d = 0 , a = 1, 2, 3 (134)with the vector weighting functions

w1 =w100

, w2 =0

w20

, w3 =00

w3

, (135)


24/36

where w1, w2 , w3 are arbitrary (but known) weighting functions. The vector weighting functions waobey the homogeneous boundary conditions

wa n = 0 on H , (136)

wa n = 0 on B . (137)

Forcing the weighted residual to zero yields

wa curl 1 curl A grad 1div A d = wa J d, a = 1 , 2, 3. (138)14.2 The weak form

With the boundary conditions (136) and (137) for wa and the identities

curl 1 curl A wa d = 1 curl A curl wa d 1 curl A n wa d, (139)

grad

1

div A wa d =

1

div A div wa d

1

div A( n wa )d , (140)

the weighted residual of (125) can be transformed to

1 curl A curl wa d H 1 curl A n wa dH + 1 div A div wa d B 1 div A( n wa )d B ai 1 div Ai( ni wa ) + 10 div Aa ( na wa ) dai

ai 1 (curl Ai ni) + 10 (curl Aa na) wa dai = wa J d, (141)with a = 1,2,3. Due to the boundary conditions (127),(129)-(129), all the boundary integrals in eq. (141)

vanish and therefore

1 curl wa curl A d + 1 div wa div A d = wa J d (142)with a = 1,2,3. Eq. (142) is called the weak form of the vector-potential formulation because the secondderivatives have been removed and the continuity requirements on A have been relaxed at the expenseof an increase in the continuity conditions of the weighting functions. Only this makes possible the useof elements with linear shape functions. Inside this elements the rst derivative of the shape functionsis a constant and the second derivative vanishes. On the element boundary we nd a jump in the rstderivative and a Dirac -function for the second. Thus there would be a problem in eq. (142) if thesecond derivative was present. This level of continuity is termed as C 0 continous. The boundary valueproblem (125)-(132) is identical with the weak formulation (142) and the additional boundary conditions(126),(128) and (132) which have to be considered when the matrix of the linear equation system isassembled. Therefore these boundary conditions are also called essential , in contrast to the boundaryconditions (127),(129)-(132) which are incorporated in the weak formulation and are called naturalboundary conditions. The natural boundary conditions are only satis ed in the integral average senseover the domain ,i.e., in the weak sense.

In two dimensions, with z = 0 , the Coulomb gauge is automatically ful lled and eq. (142)further reduces to

1curl w3 curl Az d = w3 J z d. (143)


25/36

With the relation curl Gz = grad Gz ez it follows:

1 grad w3 grad Az d = w3 J z d. (144)The essential boundary condition A n = 0 on B takes the easy form Az = 0 on B and the boundarycondition on H , A n = 0 is automatically ful lled as n ez .

The current density J appears on the right hand side of the differential equations (144) or (142).In consequence, when using the FE-method for the solution of this problem the relatively complicatedshape of the coils must be modeled in the FE-mesh, c.f. g. 19.

Fig. 19: Finite element mesh of the LHCmain dipole coil. The mesh required for the accurate modeling of the coil is very dense,

resulting in large number of unknowns in particular if the surrounding iron yoke geometry has to be considered. Simpli cations

of the coil geometry yield inaccurate eld quality estimates.

15 Coupled BEM-FEM method

The disadvantage of the nite-element method is that only a nite domain can be discretized, and there-fore the eld calculation in the magnet coil-ends with their large fringe- elds requires a large number of elements in the air region. The relatively new boundary-element method is de ned on an in nite domainand can therefore solve open boundary problems without approximation with far- eld boundaries. The

disadvantage is that non-homogeneous materials are dif cult to consider. The BEM-FEM method cou-ples the nite-element method inside magnetic bodies i = FEM with the boundary-element methodin the domain outside the magnetic material a = BEM , by means of the normal derivative of thevector-potential on the interface ai between iron and air. The application of the BEM-FEM method tomagnet design has the following intrinsic advantages:

The coil eld can be taken into account in terms of its source vector potential As , which can beobtained easily from the lamentary currents I s by means of Biot-Savart type integrals without themeshing of the coil.


26/36

The BEM-FEM coupling method allows for the direct computation of the reduced vector potential Ar instead of the total vector potential A. Consequently, errors do not in uence the dominating

contribution As due to the superconducting coil.

Because the eld in the aperture is calculated through the integration over all the BEM elements,local eld errors in the iron yoke cancel out and the calculated multipole content is suf cientlyaccurate even for very sparse meshes.

The surrounding air region need not be meshed at all. This simpli es the preprocessing and avoidsarti cial boundary conditions at some far- eld-boundaries. Moreover, the geometry of the perme-able parts can be modi ed without regard to the mesh in the surrounding air region, which stronglysupports the feature based, parametric geometry modeling that is required for mathematical opti-mization.

The method can be applied to both 2D and 3D eld problems.

The elementary model problem for a single aperture model dipole (featuring both Dirichlet and Neumannbounds on the iron yoke) is shown in g. 20.

i = FEM

a = BEM

a = BEM M

0H ( H n = 0)

B

( B n = 0)

ai = BEMFEM

BEMFEM

Fig. 20: Elementary model problem for the numerical eld calculation of a superconducting (single aperture) model magnet. In

the iron domain the total vector potential is displayed. The non-conductive air region a contains a certain number of conductor

sources J which do not intersect the iron region i . The nite-element method inside the magnetic body i = FEM is coupledwith the boundary-element method in the domain outside the magnetic material a = BEM , by means of the normal derivative

of the vector-potential on the interface ai = BEMFEM between iron and air.


27/36

15.1 The FEM part

Inside the magnetic domain i a gauged vector-potential formulation is applied. Starting from eq. (76)a different (but equivalent) formulation is obtained:

10

curl curl A = J + curl M in i , (145)

10 (

2 A + grad div A) = J + curl M in i . (146)

Using the Coulomb gauge div A = 0 , the complete formulation of the problem reads

10

2 A = J + curl M in i , (147)

A n = 0 on H , (148)1

0div A = 0 on B , (149)

A n = 0 on B , (150)

1 (curl A) n = 0 on H , (151)

10

div Aa 10

div Ai = 0 on ai , (152)

10

(curl Ai 0 M ) ni + 10

(curl Aa) na = 0 on ai . (153)

Eq. (153) is the continuity condition of H t i = H t a on the interface between iron and air. Forcing theweighted residual to zero yields

i

10

2 A wa di =

i

( J + curl M ) wa di a = 1 , 2, 3. (154)

w1 =w100

, w2 =0

w20

, w3 =00

w3

. (155)

The weighting functions wa obey again the homogeneous boundary conditions

wa n = 0 on H , (156)

wa n = 0 on B . (157)

With Green s rst theorem

i 2 A wa di = i grad( A ea ) grad wa di + A

n i wa d (158)

and the identity

i curl M wa di = i M curl wa di ( M ni ) wa d (159)


28/36

we get for the weak form

10 i grad( A ea ) grad wa di 10 B

An i

(0 M ni) wa dB

10 H

An i

(0 M ni) wa dH 10 ai

An i

(0 M ni) wa dai =

i M curl wa di + i wa J di (160)with a = 1,2,3. With the boundary conditions (148)-(151), and taking into account that the current densityin the iron domain is zero, equation (160) further reduces to

10 i grad( A ea ) grad wa di 10 ai

An i

(0 M ni) wa dai =

i M curl wa di (161)with a = 1,2,3. The continuity condition of H t , eq. (153), i.e.,

10

(curl Ai 0 M ) ni + 10

(curl Aa) na = 0 on ai (162)

on the boundary between iron and air is equivalent to

Ain i

(0 M ni) + Aan a

= 0 , (163)

where ni is the normal vector on ai pointing out of the FEM domain i, and na is the normal vector onai pointing out of the BEM domain a . The boundary integral term on the boundary between iron andair ai in (161) serves as the coupling term between the BEM and the FEM domain. Let us now assumethat the normal derivative on ai

Qai = ABEMai

n a(164)

is given. If the domain i is discretized into nite-elements j (C 0-continuous, isoparametric 20-nodedhexahedron elements are used) and the Galerkin method is applied to the weak formulation, then a non-linear system of equations is obtained

K i i K i ai 0K ai i

K ai ai

T

Ai Aai Qai

=00

(165)

with all nodal values of Ai , Aai and Qai grouped in arrays

Ai = A(1)i , A

(2)i , . . . , Aai = A

(1)ai , A

(2)ai , . . . , Qai = Q

(1)ai , Q

(2)ai . . . . .

(166)

The subscripts ai and i refer to nodes on the boundary and in the interior of the domain, respec-tively. The domain and boundary integrals in the weak formulation yield the stiffness matrices K andthe boundary matrix T . The stiffness matrices depend on the local permeability distribution in thenonlinear material. All the matrices in (165) are sparse.


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15.2 The BEM part

By de nition, the BEM domain a contains no iron, and therefore M = 0 and = 0. Eq. (145) thenreduces to

2 A = 0 J (167)

As Cartesian coordinates are used, eq . (167) decomposes into three scalar Poisson equations to be

solved. For an approximate solution of these equations and the weighted residual forced to zero yields:

a 2Awda = a 0Jw da . (168)Employing Green s second theorem

( 2 2)d = ( n n )d (169)yields

a A2

wda = a 0Jw da + ai A wn a dai ai

An a wd ai . (170)

In eq. (170) it is already considered that all the boundary integrals on the far eld boundary BEM vanish. Now the weighting function is chosen as the fundamental solution of the Laplace equation,which is in 3D

w = u = 14R

. (171)

With

w

n a= q =

1

4R2 (172)

and

2w = (R) (173)

we get the Fredholm integral equation of the second kind:

4

A + ai

Qai u dai +

ai

Aai q d ai =

a

0Ju da . (174)

As it is common practice in literature on boundary element techniques, e.g. [7], the notation u for

weighting functions is used in eq. (174). The right hand side of eq. (174) is a Biot-Savart-type integralfor the source vector potential As.The components of the vector potential A at arbitrary points r0 a (e.g. on the reference radius

for the eld harmonics) have to be computed from (174) as soon as the components of the vector potential Aai and their normal derivatives Qai on the boundary ai are known. is the solid angle enclosed by

the domain a in the vicinity of r0 .For the discretization of the boundary ai into individual boundary elements ai ,j , again C 0-

continuous, isoparametric 8-noded quadrilateral boundary elements (in 3D) are used. In 2D 3-noded lineelements are used. They have to be consistent with the elements from the FEM domain touching this


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boundary. The components of Aai and Qai are expanded with respect to the element shape functions,and (174) can be rewritten in terms of the nodal data of the discrete model,

4

A = As Qai g Aai h . (175)

In (175), g results from the boundary integral with the kernel u , and h results from the boundary integralwith the kernel q . The discrete analogue of the Fredholm integral equation can be obtained from (175)by successively putting the evaluation point r0 at the location of each nodal point r j . This procedure iscalled point-wise collocation and yields a linear system of equations,

G Qai + H Aai = As . (176)

In (176), As contains the values of the source vector potential at the nodal points r j , j = 1 , 2, . . . .The matrices G and H are unsymmetrical and fully populated.

15.3 The BEM-FEM Coupling

An overall numerical description of the eld problem can be obtained by complementing the FEM de-scription (165) with the BEM description (176) which results in

K i i K i ai 0K ai i K ai ai T 0 H G Ai Aai Qai

=00 As

. (177)

Equation (176) gives exactly the missing relationship between the Dirichlet data Aai and the Neu-mann data Qai on the boundary ai . It can be shown [12] that this procedure yields the correctphysical interface conditions, the continuity of n B and n H across ai .

16 Field calculation of the LHC main dipole using BEM-FEM coupling

16.1 Cross-section

The coils of the LHC dipole magnets are wound of Rutherford-type cable of trapezoidal (keystoned)shape. The coil consists of two layers with cables of the same height but of different width. Both layersare connected in series so that the current density in the outer layer, being exposed to a lower magneticeld, is about 40 % higher. The conductor for the inner layer consists of 28 strands of 1.065 mm diameter,the outer layer conductor has 36 strands of 0.825 mm diameter. The strands are made of thousands of laments of NbTi material embedded in a copper matrix which serves for stabilizing the conductor andto take over the current in case of a quench.

The keystoning of the cable is not suf cient to allow the cables to build up arc segments. Thereforecopper wedges are inserted between the blocks of conductors. The size and shape of these wedges yieldthe necessary degrees of freedom for optimizing the eld quality produced by the coil. The coil must beshaped to make the best use of the superconducting cable (limited by the peak eld to which it is exposed)while producing a dipole eld with a highest possible eld homogeneity. As the beam pipe has to bekept free, the coil is wound in a so-called constant perimeter shape, around saddle shaped end-spacers.

The size and elastic modulus of each coil layer is measured to determine pole and coil-head shim-ming for the collaring. The required shim thickness is calculated such that the compression under thecollaring press is about 120 MPa. After the collaring rods are inserted and external pressure is released,the residual coil pre-stress is about 50-60 MPa on both layers. The collars (made of stainless steel) aresurrounded by an iron yoke which not only enhances the magnetic eld by about 10% but also reducesthe stored energy and shields the fringe eld. The dipole magnet, its connections, and the bus-bars are


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enclosed in the stainless steel shrinking cylinder closed at its ends and form the dipole cold-mass, acontainment lled with static pressurized super uid helium at 1.9 K. The cold-mass, weighing about24 tons, is assembled inside its cryostat, which comprises a support system, cryogenic piping, radiationinsulation, and thermal shield, all contained within a vacuum vessel.

The version (04.2000) cross-section of the magnet cold-mass in its cryostat is shown in g. 21.

Fig. 21: Cross-section of the dipole magnet and cryostat. 1. Heat exchanger 2. Bus bar 3. Superconducting coil, 4. Cold-bore

and beam-screen, 5. Cryostat, 6. Thermal shield (55 to 75 K), 7. Shrinking cylinder, 8. Super-insulation, 9. Collars, 10. Yoke

Fig. 22 and 23 show the quadrilateral (higher order) nite element mesh of collar and yoke for theLHC main dipole, the magnetic eld strength, the magnetic vector potential and the relative permeabilityin the iron yoke at nominal eld level.

Table 1 shows the eld errors calculated for the 2D magnet cross-section as shown in Fig. 22 and23. As can be expected from the analytical estimation, the iron yoke hardly in uences the the higherorder multipoles b7 - b11 . These values are therefore a measure for the accuracy of the method.


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0 .7 79 0 .4 50-

0 .4 50 0 .8 99-

0 .8 99 1 .3 48-

1 .3 48 1 .7 97-

1 .7 97 2 .2 47-

2 .2 47 2 .6 96-

2 .6 96 3 .1 45-

3 .1 45 3 .5 95-

3 .5 95 4 .0 44-

4 .0 44 4 .4 93-

4 .4 93 4 .9 42-

4 .9 42 5 .3 92-

5 .3 92 5 .8 41-

5 .8 41 6 .2 90-

6 .2 90 6 .7 40-

6 .7 40 7 .1 89-

7 .1 89 7 .6 38-

7 .6 38 8 .0 87-

8 .0 87 8 .5 37-

|Btot| (T)

Fig. 22: Left: Quadrilateral (higher order) nite element mesh of collar and yoke for the LHC main dipole. Right: Magnetic

eld strength in collar and yoke

-0 .4 1 - 0. 37-

-0 .3 7 - 0. 34-

-0 .3 4 - 0. 31-

-0 .3 1 - 0. 28-

-0 .2 8 - 0. 24-

-0 .2 4 - 0. 21-

-0 .2 1 - 0. 18-

-0 .1 8 - 0. 14-

-0 .1 4 - 0. 11-

-0 .1 1 - 0. 83-

-0 .8 3 - 0. 50-

-0 .5 0 - 0. 17-

-0 .1 7 0 .0 15-

0 .0 15 0 .0 48-

0 .0 48 0 .0 81-

0 .0 81 0 .1 13-

0 .1 13 0 .1 46-

0 .1 46 0 .1 79-

0 .1 79 0 .2 12-

A (Tm)

1 .0 02 2 08 .2-

2 08 .2 4 15 .5-

4 15 .5 6 22 .7-

6 22 .7 8 30 .0-

8 30 .0 1 03 7.-

1 03 7. 1 24 4.-

1 24 4. 1 45 1.-

1 45 1. 1 65 9.-

1 65 9. 1 86 6.-

1 86 6. 2 07 3.-

2 07 3. 2 28 0.-

2 28 0. 2 48 8.-

2 48 8. 2 69 5.-

2 69 5. 2 90 2.-

2 90 2. 3 10 9.-

3 10 9. 3 31 7.-

3 31 7. 3 52 4.-

3 52 4. 3 73 1.-

3 73 1. 3 93 8.-

MUEr

Fig. 23: Left: Magnetic vector potential; Right: Relative permeability in the iron yoke at nominal eld level

16.2 Magnet extremities

The coil must so be shaped as to make the best use of the superconducting cable (limited by the peak eld to which it is exposed), while producing a dipole eld with a highest possible eld homogeneity.As the beam pipe has to be kept free, the coil is wound on a winding mandrel around saddle-shapedend-spacers such that the two narrow sides follow curves of equal arc length.

This is called a constant-perimeter end, cf. g. 24, and it allows a more appropriate shaping of thecoil in the straight section than would be possible with race-track coils. The geometric models used forthe calculation of the magnetic elds are displayed in g. 25 where the 3D coil representation, and thefull 3D model are shown.

In order to reduce the peak eld in the coil-end and thus increase the quench margin in the regionwith a weaker mechanical structure, the magnetic iron yoke ends approximately 100 mm from the onsetof the ends. The BEM-FEM coupling method is therefore used for the calculation of the end- elds. Thecomputing time for the 3D calculation is in the order of 5 hours on a DEC Alpha 5/333 workstation. Theiterative solution of the linear equation system converges better in the case of a high excitational eld


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Single coil Both coils(Analytical in commoncalculation) collar and yoke

inj. nom

b2 - 1.04 2.97b3 4.145 5.95 6.03b4 - -0.13 -0.30b5 -0.881 -0.59 -0.58b6 - 0.01 -b7 0.615 0.57 0.68b8 - - -b9 0.101 0.10 0.10b10 - - -b11 0.598 0.62 0.62

Table 1: Field errors in the LHC main dipole (pre-series magnets) in units of 10 4 at 17 mm reference radius. For the

analytical eld computation, the iron yoke is considered by means of the imaging method, assuming an inner radius of 98 mm

and a constant relative permeability of r = 2000. The two-in-one con guration creates additional b2 , b4 , .. eld errors whichalso vary as a function of the excitation from injection to nominal eld level.

Fig. 24: Race-track coil (left) and constant perimeter type coil (right) as used for the LHC dipoles. Only one coil-block is

displayed; connections are not shown. Constant perimeter ends allow a more appropriate shape of the coil in the cross-section,

while keeping the space for the beam pipe.

than in the case of the injection eld with its non-saturated iron yoke. It is therefore still impossible toapply mathematical optimization techniques to the 3D eld calculation with iron yoke. However, as theadditional effect from the fringe eld on the eld quality is low, it is suf cient to calculate the additionaleffect and then partially compensate with the coil design, if necessary. It has already been explainedthat the BEM-FEM coupling method allows the distinction between the coil eld and the reduced eldfrom the iron magnetization. Fig. 26 shows the eld components along a line in the end-region of thetwin-aperture dipole prototype magnet (MBP2), 43.6 mm above the beam-axis in aperture 2 (on a radiusbetween the inner and outer layer coil) from z = -200 mm inside the magnet yoke to z = 200 mm outsidethe yoke. The iron yoke ends at z = -80 mm, the onset of the coil-end is at z = 0.


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y1,2

x1,2

z1,2

S

y

x

z

Fig. 25: Geometric model of the dipole coil-end and full 3D model of coils and two-in-one iron yoke.

-200 -160 -120 -80 -40 0 40 80 120 160 200

-6

-4

-2

0

2

4

6

8

Bx

By

Bz

|B|

T

mm-200 -160 -120 -80 -40 0 40 80 120 160 200

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Bx

By

Bz

|B|

T

mm-200 -160 -120 -80 -40 0 40 80 120 160 200

-6

-4

-2

0

2

4

6

8

Bx

By

Bz

|B|

T

mm

Fig. 26: Magnetic ux density at nominal current along a line at x = 97mm, y = 43.6 mm (above the beam-axis of aperture 2on a radius between the inner and the outer layer coil) from z = -200 mm inside the magnet yoke to z = 200 mm outside theyoke. The iron yoke ends at z = -80 mm, the onset of the coil-end is at z = 0. Left: coil eld, Middle: reduced eld from ironmagnetization, Right: total eld. Notice the different scales and the relatively small contribution from the yoke.


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REFERENCES

[1] Armstrong, A.G.A.M. , Fan, M.W., Simkin, J., Trowbridge, C.W.: Automated optimization of magnet design using the boundary integral method, IEEE Transactions on Magnetics, 1982

[2] Bathe, K.-J: Finite Element Procedures, Prentice Hall, 1996

[3] Bossavit, A.: Computational Electromagnetism, Academic Press, 1998

[4] Binns, K.J., Lawrenson, P.J., Trowbridge, C.W.: The analytical and numerical solution of electricand magnetic elds, John Wiley & Sons, 1992

[5] C.J. Collie: Magnetic elds and potentials of linearly varying current or magnetization in a planebounded region, Proc. of the 1st Compumag Conference on the Computation of ElectromagneticFields, Oxford, U.K., 1976

[6] Biro, O., Preis, K., Paul. C.: The use of a reduced vector potential Ar Formulation for the calcula-tion of iron induced eld errors, Proceedings of the First international ROXIE user s meeting andworkshop, CERN, March 1998.

[7] Brebbia, C.A.: The boundary element method for engineers, Pentech Press, 1978.

[8] Fetzer, J., Kurz, S., Lehner, G.: Comparison between different formulations for the solution of 3Dnonlinear magnetostatic problems using BEM-FEM coupling, IEEE Transactions on Magnetics,Vol. 32, 1996

[9] Halbach, K.: A Program for Inversion of System Analysis and its Application to the Design of Mag-nets, Proceedings of the International Conference on Magnet Technology (MT2), The RutherfordLaboratory, 1967

[10] Halbach, K., Holsinger R.: Poisson user manual, Techn. Report, Lawrence Berkeley Laboratory,Berkeley, 1972

[11] Hornsby, J.S.: A computer program for the solution of elliptic partial differential equations, Tech-nical Report 63-7, CERN, 1967

[12] Kurz, S., Fetzer, J., Rucker, W.M.: Coupled BEM-FEM methods for 3D eld calculations withiron saturation, Proceedings of the First International ROXIE users meeting and workshop, CERN,March 16-18, 1998

[13] Kurz, S., Russenschuck, S.: The application of the BEM-FEM coupling method for the accuratecalculation of elds in superconducting magnets, Electrical Engineering, 1999

[14] S. Kurz and S. Russenschuck, The application of the BEM-FEM coupling method for the accuratecalculation of elds in superconducting magnets, Electrical Engineering, 1999.

[15] Mess, K.H., Schm user, P., Wolff S.: Superconducting Accelerator Magnets, World Scienti c, 1996[16] G.D. Nowottny, Netzerzeugung durch Gebietszerlegung und duale Graphenmethode, Shaker Ver-

lag, Aachen, 1999.

[17] Russenschuck, S.: ROXIE: Routine for the Optimization of magnet X-sections, Inverse eld cal-culation and coil End design, Proceedings of the First International ROXIE users meeting andworkshop, CERN 99-01, ISBN 92-9083-140-5.

[18] Y. Saad, M.H. Schultz: GMRES: A Generalized Minimal Residual Algorithm For Solving Non-symmetric Linear Systems, SIAM Journal of Scienti c Statistical Computing, 1986


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[19] The LHC study group, Large Hadron Collider, The accelerator project, CERN/AC/93-03

[20] The LHC study group, The Large Hadron Collider, Conceptual Design, CERN/AC/95-05

[21] Winslow A.A.: Numerical solution of the quasi-linear Poisson equation in a non-uniform triangularmesh, Jorunal of computational physics, 1, 1971

[22] Wilson, M.N.: Superconducting Magnets, Oxford Science Publications, 1983

[23] E.J.N. Wilson, Introduction to Particle Accelerators, 1996-1997

[24] Wille, K.: Physik der Teilchenbeschleuniger und Synchrotronstahlungsquellen, Teubner, 1992

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