DESIGN OF ACCELERATOR MAGNETS

S. RussenschuckCERN, 1211 Geneva 23, Switzerland

AbstractAnalytical and numerical field computation methods for the design of conven-tional and superconducting accelerator magnets are presented. The field in theaperture of these magnets is governed by the Laplace equation. The conse-quences for the field 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 fields for charged particles

A charged particle moving with velocity v through an electro-magnetic field 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 it’s energy by

∆E =∫ r2

r1

Fdr = e

∫ r2

r1

(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 field cannot contribute to a change in theparticle’s energy. However, if forces perpendicular to the particle trajectory are needed, magnetic fieldscan serve for the guiding and the focusing of particle beams. At relativistic speed, electric and magneticfields have the same effect on the particle trajectory if E = c B. A magnetic field of 1 T is then equivalentto an electric field of strength E = 3 · 108 V/m. A magnetic field of one Tesla strength can easily beachieved with conventional magnets (superconducting magnets on an industrial scale can reach up to 10T), whereas electric field strength in the Giga Volt / meter range are technically not to be realized. This isthe reason why for high energy particle accelerators only magnetic fields are used for guiding the beam.

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

∆E =13ε0

e2E4

(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:

∆Ep∆Ee

=(

mec2

mpc2

)4=(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 field in the dipole (bending) magnets. For a particle travelling on a circularclosed orbit in an uniform bending field, the resulting Lorentz force e v B has to equal the centrifugal

118

force mv2

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

p109eV

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 field 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 forthe LHC. All field calculations were performed using the CERN field computation program ROXIE. Thefield representations in the iron yokes are to scale, the size of the field vectors changes with the differentfield levels. Fig. 1 (left) shows the (slightly simplified) C-Core dipole for LEP. The advantage of C-Coremagnets is an easy access to the beam pipe, but they have a higher fringe field and are less rigid than theH-Type magnets as shown in fig. 1 (right). Additional pole shims can be applied in order to improve thefield quality in the aperture. The maximum field in the LEP dipoles is about 0.13 T. In order to reducethe effect of remanent iron magnetization, the yoke is laminated with a filling factor of only 0.27. It canbe seen that the field 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: Magneticfield strength in the iron yoke andfield vector presentation of accelerator magnets. Left: C-Core dipole

(N · I = 2× 5250 A , B1 = 0.13 T) with a filling factor of the yoke laminations of 0.27. Right: H-magnet (N · I = 12000

A, B1 = 0.3 T, Filling factor of yoke laminations 0.98)

If the excitational current is increased above a density of about 10A/mm2 one has to switch tosuperconducting coils. Neglecting the quantum-mechanical nature of the superconducting material, it issufficient 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 thefieldshape 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 thefield quality in the aperture is

119

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: Magneticfield strength in the iron yoke andfield vector presentation of accelerator magnets. Left: H-magnet with

increased excitation current (N · I = 48000 A , B1 = 1.17 T). Right: Window frame geometry. (N · I = 180000 A ,

B1 = 2.28 T). Notice the saturation of the poles in the H-magnet.

decreased due to the increasing fringefield. This can be avoided by constructing so-called window-framemagnets as shown infig. 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 difficult. The advantage is abetterfield quality and that pole shims can be avoided.

It can be seen, that at higherfield levels thefield quality in the aperture is increasingly affectedby the coil layout. Superconducting window-frame magnets are receiving considerable attention latelyas highfield (14-16 T) dipoles. As the coil winding is easier for window frame magnets than for theso-calledcosΘ magnets (shown infig. 3, left), the application of the mechanically less stable materialswith higher critical current density (e.g.Nb3Sn) becomes feasible. The LHC superconducting magnetsare of thecosnΘ type. The advantage of thecosΘ (dipole) magnets is that thefield outside the coildrops with1/r2 and therefore the saturation effects in the iron yoke are reduced. Fig. 3 (right)finallyshows the CTF (coil-test-facility) used for testing the manufacturing process of the LHC magnets. Noticethe large difference between thefield in the aperture (8.3 T) and thefield in the iron yoke (max. 2.8 T),which has merely the effect of shielding the fringefield.

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: Magneticfield strength in the iron yoke andfield vector presentation of accelerator magnets. Left: So-called cosΘ

magnet (N · I = 330000 A , B1 = 3.0 T). Right: LHC single aperture coil test facility (N · I = 480000 A , B1 = 8.33 T).

Notice that even with increasedfield in the aperture thefield strength in the yoke is reduced in the cosΘ magnet design.

120

3 Field quality in accelerator magnets

The quality of the magneticfield is essential to keep the particles on stable trajectories for about 108 turns.The magneticfield errors in the aperture of accelerator magnets can be expressed as the coefficients of theFourier-series expansion of the radialfield component at a given reference radius (in the 2-dimensionalcase). In the 3-dimensional case, the transversefield components are given at a longitudinal positionz0 or integrated over the entire length of the magnet. For beam tracking it is sufficient to consider thetransversefield components, since the effect of the z-component of thefield (present only in the magnetends) on the particle motion can be neglected. Assuming that the radial component of the magneticfluxdensityBr at a given reference radiusr = r0 inside the aperture of a magnet is measured or calculatedas a function of the angular positionϕ, we get for the Fourier-series expansion of thefield

Br(r0, ϕ) =∞∑

n=1

(Bn(r0) sin nϕ+An(r0) cosnϕ), (7)

with

An(r0) =1π

∫ π

−πBr(r0, ϕ) cos nϕdϕ, (n = 1, 2, 3, ...) (8)

Bn(r0) =1π

∫ π

−πBr(r0, ϕ) sin nϕdϕ. (n = 1, 2, 3, ...) (9)

If the field components are related to the mainfield componentBN we get forN = 1 dipole,N = 2quadrupole, etc.:

Br(r0, ϕ) = BN (r0)∞∑

n=1

(bn(r0) sinnϕ+ an(r0) cosnϕ). (10)

TheBn are called thenormal and theAn theskew components of thefield given in Tesla,bn the normalrelative, andan the skew relativefield components. They are dimensionless and are usually given in unitsof 10−4 at a 17 mm reference radius. In practice theBr components are calculated in discrete points

ϕk =kπ

P− π (11)

k =0,1,2,..,2P -1 in the interval[−π, π) and a discrete Fourier transform is carried out:

An(r0) ≈ 1P

2P−1∑k=0

Br(r0, ϕk) cosnϕk, (12)

Bn(r0) ≈ 1P

2P−1∑k=0

Br(r0, ϕk) sin nϕk. (13)

The interpolation-error depends on the number of evaluation points and the amount of higher ordermultipole errors in thefield. For the multipoles up to the ordern = 13, 79 evaluation points (P = 40) aresufficient.

121

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 =∫

VρdV. (17)

The vectorfields E, H, D, B are called electric and magneticfield, and electric and magnetic induction(flux density), respectively. Eq. (14) is Ampere’s law as modified 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 = µ H = µ0( H + M), (18)

D = ε E = ε0(E + P ), (19)

J = σ E + Jimp., (20)

with the permeability of free spaceµ0 = 4π · 10−7 H/m and the permittivity of free spaceε0 = 8.8542 · 10−12 F/m.

5 Maxwell’s equations in vector notation

Thefield equations can be written in differential form as follows:

curl H = J +∂ D

∂t, (21)

curlE = −∂ B

∂t, (22)

div B = 0, (23)

div D = ρ. (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:∫

VdivgdV =

∫A

g · d A, (25)

which is called Gauss’ (Ostrogradskii’s) divergence theorem and∫Acurlg · d A =

∮g · ds, (26)

which is Stokes’ surface integral theorem.

122

6 Maxwell’s equations for magnetostatic problems

For magnetostatic problems the time derivative can be set to zero,∂∂t = 0, and Maxwell’s equations

reduce to

curl H = J, (27)

div B = 0, (28)

B = µ( H) H = µ0( H + M). (29)

7 Interface conditions

Bt2

Bn2

B2

Bt1

Bn1 B1α1

α2

n

Ht2

Ht1Ih

µ2

µ1

µ2

µ1

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

If we apply Ampere’s law in the integral form∮H · ds =

∫A

J · d A , (30)

to the loop displayed infig. 4 (left), and leth → 0, then the enclosed current is zero, as in an infinitesimalsmall rectangle there cannot be a currentflow. Therefore

Ht1 = Ht2, (31)

i.e.,

n× ( H1 − H2) = 0. (32)

Because of∮

B · d A = 0 we get at the interface

Bn1 = Bn2, (33)

i.e.,

n · ( B1 − B2) = 0. (34)

Now

tanα1tanα2

=Bt1Bn1

Bt2Bn2

=µ1Ht1

µ2Ht2=

µ1µ2

. (35)

Forµ2 µ1 it follows thattanα1 tanα2. Therefore for all anglesπ/2 > α2 > 0we gettanα1 ≈ 0,see alsofig. 4 (right). Thefield 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.

123

8 One-dimensional field computation for conventional magnets

Consider the magnetic (dipole) circuit shown infig. 5 (left). With Ampere’s law in the integral form∮H · ds = ∫A J · d A , we can write

Hiron siron +Hgap sgap = N I , (36)

1µ0µr

Biron siron +1µ0

Bgap sgap = N I , (37)

With µr 1 we get the easy relation

Bgap =µ0N 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 gapfield and the required excitational current can be derived.

For the quadrupole we can split up the integration path as shown infig. 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

H1 · ds+∫3

H3 · ds = N I . (39)

As we will see later, in a quadrupole thefield is defined by it’s gradientg with Bx = gy andBy = gx.Therefore the modulus of thefield along the integration path 1 is

H =g

µ0

√x2 + y2 =

g

µ0r. (40)

Along the x-axis thefield integral is zero becauseH ⊥ s. Therefore∫ r0

0Hdr =

g

µ0

∫ r0

0rdr =

g

µ0

r202= N I, (41)

g =2µ0NI

r20. (42)

Notice that for a givenNI the field decreases linearly with the gap length of the dipole, whereas thegradient in a quadrupole magnet is inverse proportional to the square of the aperture radiusr0.

124

8.1 Permanent magnet excitation

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

Hiron siron +Hgap sgap +Hmag smag = 0. (43)

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

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

In the absence of fringefields we get with the pole surfaceAgap and the magnet surfaceAmag:

BmagAmag = BgapAgap = µ0HgapAgap, (44)

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

Hgapsgap = −Hmagsmag, (45)

1µ0

BmagAmagAgap

sgap = −Hmagsmag, (46)

Bmag

µ0Hmag=−smagsgap

AgapAmag

= P, (47)

where P is called the permeance coefficient which (forAgap = Amag ) becomes zero forsgap smag(open circuit) and becomes−∞ for smag sgap (short circuit). The case ofAmag > Agap is usu-ally referred to as the“flux concentration” mode. The permeance coefficient defines 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 withsgap = 2smag and an open circuitwith a smarium cobalt magnet (remanentfield 0.9T).

From eq. 44 and 45 we derive

BmagAmagsmag = µ0HgapAgap−Hgapsgap

Hmag. (48)

Therefore

Hgap =

√(Amagsmag)(−BmagHmag)

µ0(Agapsgap)=

√V OLmag(−BmagHmag)

µ0V OLgap. (49)

For a given magnet volume, the maximum air gapfield can be obtained by dimensioning the magneticcircuit in such a way thatBmagHmag is maximum. It is usually said that this implies operating thepermanent magnet withmaximum energy density. At this point we have to be careful, however:

125

mag

µ0Hmag

= 0.5 T

= −0.4 T

Bmag

µ0Hmag

B = 0.2 T

= 0 T = −0.7 T0.000 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

µ0Hmag

= 0.9 T

Fig. 7: Magnetic circuit with zero air gap (left), a circuit with sgap = 2smag (middle) and an open circuit (right) with a smarium

cobalt magnet (remanent field 0.9T). Notice the values for µ0Hmag and Bmag on the demagnetization curve. For the case with

sgap = 2smag the numerical calculation yields µ0Hgap = 0.197 T, µ0Hmag = −0.4 T.

• The relation between the magnetization and the field is not linear (and not even unique). It dependsnot only on the external field 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 fields cannot be neglected.

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

curl B = µ0( Jfree + Jmag) = µ0 Jfree + µ0curl M. (50)

Hence

curlB − µ0 M

µ0= Jfree. (51)

For the magnetized media we therefore have

H =B − µ0 M

µ0(52)

or

B = µ0( H + M). (53)

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

B = µ0 H + µ0χmH = µ0(1 + χm) H = µ0µr

H = µ H, (54)

where µr is the relative permeability and χm 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

126

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

div H = divB − µ0 M

µ0= − div M = ρmag. (55)

In eq. 55 a fictitious magnetic charge density ρmag = − div M was introduced with the field starting andending on these fictitious magnetic charges, c.f. fig. 8.

MBH

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

Coming back to the magnetic circuit as shown in fig. 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 fig. 9.

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

H =NI

2πr. (56)

B

H−µ0H

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

127

The induced voltage in the pick-up coil is

Ui =dφ

dt=

dB

dtA. (57)

UI

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 (∫

Uidt = 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 fig. 10, the power needed for exciting the field is

dW

dt= IN

dφ

dt= INA

dB

dt=

IN

2πrA2πr

dB

dt= HV

dB

dt. (58)

Therefore1V

dW

dt= H

dB

dt, (59)

W

V=∫ B2

B1

HdB. (60)

For a complete cycle we get

W

V=∮

HdB =∮

Hµ0(dH + dM) = µ0

∮HdM. (61)

As a consequence, the working point with maximal BmagHmag 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 fields

We will now show that in the aperture of a magnet (two-dimensional, current free region) both themagnetic scalar-potential as well as the vector-potential can be used to solve Maxwell’s equations:

H = −gradΦ = −∂Φ∂x

ex − ∂Φ∂y

ey, (62)

B = curl(ezAz) =∂Az

∂yex − ∂Az

∂xey, (63)

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

128

9.1 Magnetic scalar potential

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

H = Hs + Hm (64)

the curl free part Hm arises from the induced magnetism in ferromagnetic materials and the source freepart Hs is the field generated by the prescribed sources (can be calculated directly by means of BiotSavart’s law). With curlHm = 0 it follows that

H = −gradΦm + Hs (65)

and we get:

div B = 0 (66)

divµ(−gradΦm + Hs) = 0 (67)

divµgradΦm = div µ Hs (68)

While a solution of eq. 68 is possible, the two parts of the magnetic field Hm and Hs 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, curlH = 0 and the field canbe represented by a total scalar potential

H = −gradΦ (69)

and therefore we get

−µ0 div gradΦ = 0 , (70)

∇2Φ = 0 . (71)

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

∇ = (∂

∂x,

∂

∂y,

∂

∂z) (72)

and the Laplace operator

∆ = ∇2 = ∂2

∂x2+

∂2

∂y2+

∂2

∂z2. (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 = µ0( H + M), (74)

H =1µ0

curl A− M, (75)

1µ0

curl curl A = J + curl M (76)

1µ0

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

129

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

A0 = A+ gradψ. (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 thedivergence 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 = −µ0( J + curl M) (80)

In the two-dimensional case with no dependence on z, ∂∂z = 0 and J = Jz , A has only a z-component

and the Coulomb gauge is automatically fulfilled. Then we get the scalar Poisson differential equation

∇2Az = −µ0Jz . (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

∂2Az

∂x2+

∂2Az

∂y2= 0. (82)

and in cylindrical coordinates

r2∂2Az

∂r2+ r

∂Az

∂r+

∂2Az

∂ϕ2= 0. (83)

9.3 Field harmonics

A solution of the homogeneous differential equation (83) reads

Az(r, ϕ) =∞∑

n=1

(C1nrn + C2nr−n)(D1n sinnϕ+D2n cosnϕ). (84)

Considering that the field is finite at r = 0, the C2n have to be zero for the vector-potential inside theaperture of the magnet while for the solution in the area outside the coil all C1n vanish. Rearrangingeq. (84) yields the vector-potential in the aperture:

Az(r, ϕ) =∞∑

n=1

rn(Cn sinnϕ−Dn cosnϕ), (85)

and the field components can be expressed as

Br(r, ϕ) =1r

∂Az

∂ϕ=

∞∑n=1

nrn−1(Cn cosnϕ+Dn sinnϕ), (86)

Bϕ(r, ϕ) = −∂Az

∂r= −

∞∑n=1

nrn−1(Cn sinnϕ−Dn cosnϕ). (87)

130

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

Bx = Br cosϕ−Bϕ sinϕ, (88)

By = Br sinϕ+Bϕ cosϕ. (89)

For the dipole field (n=1) we get

Br = C1 cosϕ+D1 sinϕ, (90)

Bϕ = −C1 sinϕ+D1 cosϕ, (91)

Bx = C1, (92)

By = D1. (93)

This is a simple, constant field distribution according to the values of C1 and D1. Notice that we have notyet addressed the conditions necessary to obtain such a field distribution. For the pure quadrupole (n=2)we get from eq. 86 and 87:

Br = 2 r C2 cos 2ϕ+ 2 rD2 sin 2ϕ, (94)

Bϕ = −2 r C2 sin 2ϕ+ 2 rD2 cos 2ϕ, (95)

Bx = 2(C2x+D2y), (96)

By = 2(−C2y +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 fields 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:

Br = 3 r C3 cos 3ϕ+ 3 rD3 sin 3ϕ, (98)

Bϕ = −3 r C3 sin 3ϕ+ 3 rD3 cos 3ϕ, (99)

Bx = 3C3(x2 − y2) + 6D3xy, (100)

By = −6C3xy + 3D3(x2 − y2). (101)

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

By = D1 + 2D2x+ 3D3x2 + 4D4x3 + ... (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 coefficients Cn 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 coefficients are not known at this stage. They are defined through the (given) bound-ary conditions on some reference radius or can be calculated from the Fourier series expansion of the(numerically) calculated field (ref. eq. 7) in the aperture using the relations

An = nrn−10 Cn and Bn = nrn−1

0 Dn. (103)

131

10 Ideal pole shapes of conventional magnets

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

dϕ = − E · ds = −(Exdx+ Eydy + Ezdz) = (104)(∂ϕ

∂xdx +

∂ϕ

∂ydy +

∂ϕ

∂zdz

)= (gradϕ) · ds.

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

dϕ = gradϕ · ds = 0 (105)

i.e., the gradient is perpendicular to the equipotential. With the field lines (lines of constant vectorpotential) leaving highly permeable materials perpendicular to the surface (ref. chapter 7), the lines oftotal magnetic scalar potential define 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 field:

Φ = C1x+D1y. (106)

So C1 = 0 , D1 = 0 gives a vertical (normal) dipole field, C1 = 0 , D1 = 0 yields a horizontal (skew)dipole field. The equipotential surfaces are parallel to the x-axis or y-axis depending on the values of Cand D and results in a simple flux density distribution used for bending magnets in accelerators. For thequadrupole:

Φ = C2(x2 − y2) + 2D2xy (107)

with C2 = 0 giving a normal quadrupole field and with D2 = 0 giving a skew quadrupole field (which isthe above rotated by π/4). The quadrupole field is generated by lines of equipotential having hyperbolicform. For the C2 = 0 case, the asymptotes are the two major axes.

In practice, however, the magnets have a finite pole width (due to the need of a magnetic fluxreturn yoke and space for the coil). To ensure a good field quality with these finite approximations ofthe ideal shape, small shims are added at the outer ends of each pole. The shim geometry has to beoptimized (using numerical field computation tools) while considering, that with an increasing height ofthe shim saturation occurs at high excitation and leads to the field distribution being strongly dependenton the magnet excitation level. On the other hand, with a very thin and long shim the nature of the fieldgenerated 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 ofconstant vector-potential (for the dipole) and magnetic field modulus in the iron yoke for the quadrupole.The field quality in the dipole was improved by adding shims on the pole surface. In case of thequadrupole, however, the pole shape is defined 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 field of superconducting magnets

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

132

x

y

y = const.

Shim

x

y

xy = const.

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

-0.13 0.864-

0.864 0.003-

0.003 0.005-

0.005 0.007-

0.007 0.009-

0.009 0.012-

0.012 0.014-

0.014 0.016-

0.016 0.018-

0.018 0.021-

0.021 0.023-

0.023 0.025-

0.025 0.028-

0.028 0.030-

0.030 0.032-

0.032 0.034-

0.034 0.037-

0.037 0.039-

0.039 0.041-

A (Tm)

0.664 0.110-

0.110 0.219-

0.219 0.328-

0.328 0.438-

0.438 0.547-

0.547 0.656-

0.656 0.766-

0.766 0.875-

0.875 0.984-

0.984 1.094-

1.094 1.203-

1.203 1.313-

1.313 1.422-

1.422 1.531-

1.531 1.641-

1.641 1.750-

1.750 1.859-

1.859 1.969-

1.969 2.078-

|Btot| (T)

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

magnetic field modulus (right).

distribution. It is reasonable to focus on the fields generated by line-currents, since the field of any currentdistribution over an arbitrary cross-section can be approximated by summing the fields of a number ofline-currents distributed within the cross-section. For a set of ns of these line-currents at the position(ri,Θi) carrying a current Ii, the multipole coefficients are given by [15]

Bn(r0) = −ns∑i=1

µ0Ii

2πrn−10

rni

(1 +

µr − 1µr + 1

(ri

Ryoke)2n)cosnΘi, (108)

An(r0) =ns∑i=1

µ0Ii

2πrn−10

rni

(1 +

µr − 1µr + 1

(ri

Ryoke)2n)sinnΘi, (109)

where Ryoke is the inner radius of the iron yoke with the relative permeability µr. The field of any currentdistribution over an arbitrary cross-section can be approximated by summing the fields of a number ofline-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.

133

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 fields in supercon-ducting magnets is given. The iron yoke is represented by image currents (the second term in the paren-theses). At low field level, when the saturation of the iron yoke is low, this is a sufficient method foroptimizing the coil cross-section. Under that assumption some important conclusions can be drawn:

• For a coil without iron yoke the field errors scale with 1/rn 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 field. 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 I0 at (r0,Θ0) and at (r0,−Θ0))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 field (coil field plus iron magnetization) is fora non-saturated yoke (µr 1) approximately (1+(Ryoke

r )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 influence of the yoke is only0.07%.

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

12 The generation of pure multipole fields

Consider a current shell ri < r < re with a current density varying with the azimuthal angle Θ, J(Θ) =J0 cosmΘ, then we get for the Bn components

Bn(r0) =∫ re

ri

∫ 2π

0−µ0J0r

n−10

2πrn

(1 +

µr − 1µr + 1

(r

Ryoke)2n)cosmΘcosnΘ rdΘdr . (110)

134

With∫ 2π0 cosmΘcosnΘdΘ = πδm,n(m,n = 0) it follows that the current shell produces a pure 2m-

polar field and in the case of the dipole (m = n = 1) one gets

B1(r0) = −µ0J02

((re − ri) +

µr − 1µr + 1

1R2yoke

13(r3e − r3i )

). (111)

Obviously, since∫ 2π0 cosmΘsinnΘdΘ = 0, all An components vanish. A shell with cosΘ and cos 2Θ

dependent current density is displayed in figure 15. Because of |B| =∑n

√B2

n +A2n the modulus of

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

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

|B| = µ0J02

(re − ri). (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 insufficient angle to allow for perfect sector geometries. Thereforethe shells are subdivided into coil-blocks, separated by copper wedges. The field 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 fig. 15.

135

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 field computation

For the calculation of the saturation-induced field errors of the lower-order (b2 − b5), which vary as afunction of the excitational field, numerical techniques such as the finite-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 hardlyinfluenced by saturation effects. Fig. 16 (left) shows the transfer function B/I as a function of theexcitation from injection to nominal field level including the saturation effects and the persistent currentmultipoles. Fig. 17 shows the relative permeability of the iron yoke for both injection and nominal fieldlevel.

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

b2

b3

b410

b510

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 field

level. 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 finite difference method for

136

1.002 208.2-

208.2 415.5-

415.5 622.7-

622.7 830.0-

830.0 1037.-

1037. 1244.-

1244. 1451.-

1451. 1659.-

1659. 1866.-

1866. 2073.-

2073. 2280.-

2280. 2488.-

2488. 2695.-

2695. 2902.-

2902. 3109.-

3109. 3317.-

3317. 3524.-

3524. 3731.-

3731. 3938.-

MUEr

1.002 208.2-

208.2 415.5-

415.5 622.7-

622.7 830.0-

830.0 1037.-

1037. 1244.-

1244. 1451.-

1451. 1659.-

1659. 1866.-

1866. 2073.-

2073. 2280.-

2280. 2488.-

2488. 2695.-

2695. 2902.-

2902. 3109.-

3109. 3317.-

3317. 3524.-

3524. 3731.-

3731. 3938.-

MUEr

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

at nominal field level. Change in permeability results in the saturation dependent multipole content as shown in fig. 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 finite 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 finite 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 fields had been developed by Newman, Turner and Trowbridge thatwas based on the magnetization integral equation and was applied to magnet design. Nevertheless, forthe superconducting magnet design, it was necessary to find more appropriate formulations which do notrequire the modeling of the coils in the finite-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 fine mesh.

The method of coupled boundary-elements/finite-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 fields. This implies that the air regions need not to be meshed at all.

The principle steps in numerical field 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 finite elements.

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

137

• In case of the finite 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/finite-element method, the two domains are coupled throughthe 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 sufficient to concentrate on someaspects of the formulations. The function approximation with finite or boundary elements and the solu-tion techniques will only be explained very briefly, as this report cannot replace a lecture on finite-elementtechniques. Nevertheless, as the magnetostatic problem is one of the most simple cases, the basic con-cepts of numerical field 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, fig. 18, consisting of two different domains: Ωi the iron regionwith permeability µ and Ωa the air region with the permeability µ0. The regions are connected to eachother at the interface Γai. Furthermore, each volume is bounded by a surface Γ (sometimes denoted ∂Ω)itself consisting of two different parts ΓH and ΓB with their outward normal vector n. The elementarymodel problem as shown in fig. 18 is a mixed boundary value problem. The non-conductive air regionΩa may also contain a certain number of conductor sources J which do not intersect the iron region Ωi.

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 flux is zero, the application of a total A-formulation in thedomain Ω = Ωa ∪Ωi automatically satisfies eq. (28). Ampere’s law (27) then takes the form

curl1µcurl A = J in Ω. (113)

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

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

B · n = curl A · n = 0 on ΓB . (115)

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

138

Ωi

Ωa

Ωa

µrµ0

µ0

µ0

ΓH H × n = 0 (Ht = 0)

ΓH

ΓB

B · n = 0

(Bn = 0)

Γai

Γai

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

total vector-potential is displayed. The non-conductive air region Ωa contains a certain number of conductor sources J which

do not intersect the iron region Ωi.

conditions have to be satisfied (continuity of Bn and Ht):

Bi · ni + Ba · na = 0 on Γai, (116)

Hi × ni + Ha × na = 0 on Γai, (117)

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

1µ(curl Ai)× ni +

1µ0

(curl Aa)× na = 0 on Γai. (118)

The normal vector ni on the boundary between iron and air is pointing out of the iron domain Ωi andna is pointing out of the air domain Ωa. The solution of the 3D (vector) boundary value problem is notunique, however. Introducing a penalty term subtracted from eq. (113), yields

curl1µcurl A− grad

1µdiv A = J inΩ. (119)

139

With the additional boundary conditions

A · n = 0 on ΓH , (120)

A× n = 0 on ΓB , (121)1µdiv A = 0 on ΓB , (122)

1µ0

div Aa =1µdiv Ai on Γai, (123)

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

1µdiv A = 0 in Ω. (124)

The complete formulation for the vector-potential reads

curl1µcurl A− grad

1µdiv A = J inΩ, (125)

A · n = 0 on ΓH , (126)1µdiv A = 0 on ΓB , (127)

A× n = 0 on ΓB , (128)1µ(curl A)× n = 0 on ΓH , (129)

1µ0

div Aa − 1µdiv Ai = 0 on Γai, (130)

1µ(curl Ai) × ni +

1µ0

(curl Aa) × na = 0 on Γai, (131)

A continous on Γai. (132)

14.1 The weighted residual

The domain Ω = Ωa ∪ Ωi is discretized into finite-elements in order to solve this problem numerically.For the approximate solution of A defined on the nodes of the finite element mesh, the differentialequation (125) is only approximately fulfilled:

curl1µcurl A− grad

1µdiv 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 Ω, i.e., ∫

Ωwa · R dΩ = 0, a = 1, 2, 3 (134)

with the vector weighting functions

w1 =

w1

00

, w2 =

0w2

0

, w3 =

00w3

, (135)

140

where w1, w2, w3 are arbitrary (but known) weighting functions. The vector weighting functions wa

obey the homogeneous boundary conditions

wa · n = 0 on ΓH , (136)

wa × n = 0 onΓB . (137)

Forcing the weighted residual to zero yields∫Ω

wa ·(curl

1µcurl A− grad

1µdiv A

)dΩ =

∫Ω

wa · J dΩ, a = 1, 2, 3. (138)

14.2 The weak form

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

(curl

1µcurl A

)· wa dΩ =

∫Ω

1µcurl A · curlwadΩ−

∮Γ

1µ

(curl A× n

)· wadΓ, (139)

∫Ω

(−grad 1

µdiv A

)· wa dΩ =

∫Ω

1µdiv A divwadΩ−

∮Γ

1µdiv A(n · wa)dΓ, (140)

the weighted residual of (125) can be transformed to∫Ω

1µcurl A · curlwadΩ−

∫ΓH

1µ

(curl A× n

)· wadΓH +

∫Ω

1µdiv A divwadΩ−

∫ΓB

1µdiv A(n · wa)dΓB −

∫Γai

(1µdiv Ai(ni · wa) +

1µ0

div Aa(na · wa))dΓai −

∫Γai

(1µ(curl Ai × ni) +

1µ0

(curl Aa × na))· wadΓai =

∫Ω

wa · J dΩ, (141)

with a = 1,2,3. Due to the boundary conditions (127),(129)-(129), all the boundary integrals in eq. (141)vanish and therefore∫

Ω

1µcurlwa · curl AdΩ+

∫Ω

1µdivwadiv A dΩ =

∫Ω

wa · J dΩ (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 first derivative of the shape functionsis a constant and the second derivative vanishes. On the element boundary we find a jump in the firstderivative and a Dirac δ-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 C0 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 satisfied in the integral average senseover the domain Ω ,i.e., in the weak sense.

In two dimensions, with ∂∂z = 0, the Coulomb gauge is automatically fulfilled and eq. (142)

further reduces to ∫Ω

1µcurlw3 · curl Az dΩ =

∫Ω

w3 · Jz dΩ. (143)

141

With the relation curlGz = gradGz × ez it follows:∫Ω

1µgradw3 · gradAz dΩ =

∫Ω

w3 · Jz dΩ. (144)

The essential boundary condition A×n = 0on ΓB takes the easy form Az = 0on ΓB and the boundarycondition on ΓH , A · n = 0 is automatically fulfilled 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. fig. 19.

Fig. 19: Finite element mesh of the LHC main 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. Simplifications

of the coil geometry yield inaccurate field quality estimates.

15 Coupled BEM-FEM method

The disadvantage of the finite-element method is that only a finite domain can be discretized, and there-fore the field calculation in the magnet coil-ends with their large fringe-fields requires a large number ofelements in the air region. The relatively new boundary-element method is defined on an infinite domainand can therefore solve open boundary problems without approximation with far-field boundaries. Thedisadvantage is that non-homogeneous materials are difficult to consider. The BEM-FEM method cou-ples the finite-element method inside magnetic bodies Ωi = ΩFEM with the boundary-element methodin the domain outside the magnetic material Ωa = ΩBEM, by means of the normal derivative of thevector-potential on the interface Γai between iron and air. The application of the BEM-FEM method tomagnet design has the following intrinsic advantages:

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

142

• The BEM-FEM coupling method allows for the direct computation of the reduced vector potentialAr instead of the total vector potential A. Consequently, errors do not influence the dominatingcontribution As due to the superconducting coil.

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

• The surrounding air region need not be meshed at all. This simplifies the preprocessing and avoidsartificial boundary conditions at some far-field-boundaries. Moreover, the geometry of the perme-able parts can be modified 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 field problems.

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

Ωi = ΩFEM

Ωa = ΩBEM

Ωa = ΩBEMM

µ0ΓH ( H × n = 0)

ΓB

( B · n = 0)

Γai = ΓBEMFEM

ΓBEMFEM

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

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

sources J which do not intersect the iron region Ωi. The finite-element method inside the magnetic body Ωi = ΩFEM is coupled

with the boundary-element method in the domain outside the magnetic material Ωa = ΩBEM, by means of the normal derivative

of the vector-potential on the interface Γai = ΓBEMFEM between iron and air.

143

15.1 The FEM part

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

1µ0

curl curl A = J + curl M inΩi, (145)

1µ0

(−∇2 A + grad div A) = J + curl M inΩi. (146)

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

− 1µ0∇2 A = J + curl M inΩi, (147)

A · n = 0 on ΓH , (148)1µ0

div A = 0 on ΓB , (149)

A× n = 0 on ΓB , (150)1µ(curl A)× n = 0 on ΓH , (151)

1µ0

div Aa − 1µ0

div Ai = 0 on Γai, (152)

1µ0

(curl Ai − µ0 M)× ni +1µ0

(curl Aa)× na = 0 on Γai. (153)

Eq. (153) is the continuity condition of Hti = Hta on the interface between iron and air. Forcing theweighted residual to zero yields

−∫Ωi

1µ0∇2 A · wadΩi =

∫Ωi

( J + curl M) · wadΩi a = 1, 2, 3. (154)

w1 =

w1

00

, w2 =

0w2

0

, w3 =

00w3

. (155)

The weighting functions wa obey again the homogeneous boundary conditions

wa · n = 0 on ΓH , (156)

wa × n = 0 onΓB . (157)

With Green’s first theorem∫Ωi

∇2 A · wadΩi = −∫Ωi

grad( A · ea) · gradwa dΩi +∮Γ

∂ A

∂ni· wa dΓ (158)

and the identity ∫Ωi

curl M · wadΩi =∫Ωi

M · curlwadΩi −∮Γ( M × ni) · wa dΓ (159)

144

we get for the weak form

1µ0

∫Ωi

grad( A · ea) · gradwa dΩi − 1µ0

∫ΓB

(∂ A

∂ni− (µ0 M × ni)

)· wa dΓB

− 1µ0

∫ΓH

(∂ A

∂ni− (µ0 M × ni)

)· wa dΓH − 1

µ0

∫Γai

(∂ A

∂ni− (µ0 M × ni)

)· wa dΓai =

∫Ωi

M · curlwa dΩi +∫Ωi

wa · J dΩi (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

1µ0

∫Ωi

grad( A · ea) · gradwa dΩi − 1µ0

∮Γai

(∂ A

∂ni− (µ0 M × ni)

)· wa dΓai =

∫Ωi

M · curlwa dΩi (161)

with a = 1,2,3. The continuity condition of Ht, eq. (153), i.e.,

1µ0

(curl Ai − µ0 M)× ni +1µ0

(curl Aa)× na = 0 on Γai (162)

on the boundary between iron and air is equivalent to

∂ Ai∂ni

− (µ0 M × ni) +∂ Aa∂na

= 0, (163)

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

QΓai= −∂ ABEMΓai

∂na(164)

is given. If the domain Ωi is discretized into finite-elements Ωj (C0-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

([KΩiΩi

] [KΩiΓai

]0[

KΓaiΩi

] [KΓaiΓai

] [T])

AΩi

AΓai

QΓai

=

(00

)(165)

with all nodal values of AΩi, AΓai

and QΓaigrouped in arrays

AΩi

=(A(1)Ωi

, A(2)Ωi

, . . .),

AΓai

=(A(1)Γai

, A(2)Γai

, . . .),

QΓai

=(Q(1)Γai

, Q(2)Γai

. . . .).

(166)

The subscripts Γai and Ωi 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 the

nonlinear material. All the matrices in (165) are sparse.

145

15.2 The BEM part

By definition, the BEM domain Ωa contains no iron, and therefore M = 0 and µ = µ0. Eq. (145) thenreduces to

∇2 A = −µ0 J (167)

As Cartesian coordinates are used, eq . (167) decomposes into three scalar Poisson equations to besolved. For an approximate solution of these equations and the weighted residual forced to zero yields:∫

Ωa

∇2AwdΩa = −∫Ωa

µ0JwdΩa. (168)

Employing Green’s second theorem∫Ω(φ∇2ψ − ψ∇2φ)dΩ =

∮Γ(φ

∂ψ

∂n− ψ

∂φ

∂n)dΓ (169)

yields ∫Ωa

A∇2wdΩa = −∫Ωa

µ0Jw dΩa +∫Γai

A∂w

∂nadΓai −

∫Γai

∂A

∂nawdΓai . (170)

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

w = u∗ =1

4πR. (171)

With

∂w

∂na= q∗ = − 1

4πR2(172)

and

∇2w = −δ(R) (173)

we get the Fredholm integral equation of the second kind:

Θ4π

A+∫Γai

QΓaiu∗ dΓai +

∫Γai

AΓaiq∗ dΓai =

∫Ωa

µ0Ju∗ dΩa. (174)

As it is common practice in literature on boundary element techniques, e.g. [7], the notation u∗ forweighting 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 ∈ Ωa (e.g. on the reference radiusfor the field harmonics) have to be computed from (174) as soon as the components of the vector potentialAΓai

and their normal derivatives QΓaion the boundary Γai are known. Θ is the solid angle enclosed by

the domain Ωa in the vicinity of r0.

For the discretization of the boundary Γai into individual boundary elements Γai,j , again C0-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

146

boundary. The components of AΓaiand QΓai

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

Θ4π

A = As −

QΓai

· g− AΓai

· 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 rj . This procedure iscalled point-wise collocation and yields a linear system of equations,[

G]

QΓai

+[H]

AΓai

=

As. (176)

In (176),

As

contains the values of the source vector potential at the nodal points rj , 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 field problem can be obtained by complementing the FEM de-scription (165) with the BEM description (176) which results in

[KΩiΩi

] [KΩiΓai

]0[

KΓaiΩi

] [KΓaiΓai

] [T]

0[H] [

G]

AΩi

AΓai

QΓai

=

00As . (177)

Equation (176) gives exactly the missing relationship between the Dirichlet data

AΓai

and the Neu-

mann data

QΓai

on the boundary Γai. It can be shown [12] that this procedure yields the correct

physical interface conditions, the continuity of n · B and n× H across Γai.

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 magneticfield, 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 offilaments 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 sufficient 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 field quality produced by the coil. The coil must beshaped to make the best use of the superconducting cable (limited by the peak field to which it is exposed)while producing a dipole field with a highest possible field 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 field by about 10% but also reducesthe stored energy and shields the fringe field. The dipole magnet, its connections, and the bus-bars are

147

enclosed in the stainless steel shrinking cylinder closed at its ends and form the dipole cold-mass, acontainment filled with static pressurized superfluid 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 fig. 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) finite element mesh of collar and yoke for theLHC main dipole, the magnetic field strength, the magnetic vector potential and the relative permeabilityin the iron yoke at nominal field level.

Table 1 shows the field 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 influences the the higherorder multipoles b7 - b11. These values are therefore a measure for the accuracy of the method.

148

0.779 0.450-

0.450 0.899-

0.899 1.348-

1.348 1.797-

1.797 2.247-

2.247 2.696-

2.696 3.145-

3.145 3.595-

3.595 4.044-

4.044 4.493-

4.493 4.942-

4.942 5.392-

5.392 5.841-

5.841 6.290-

6.290 6.740-

6.740 7.189-

7.189 7.638-

7.638 8.087-

8.087 8.537-

|Btot| (T)

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

field strength in collar and yoke

-0.41 -0.37-

-0.37 -0.34-

-0.34 -0.31-

-0.31 -0.28-

-0.28 -0.24-

-0.24 -0.21-

-0.21 -0.18-

-0.18 -0.14-

-0.14 -0.11-

-0.11 -0.83-

-0.83 -0.50-

-0.50 -0.17-

-0.17 0.015-

0.015 0.048-

0.048 0.081-

0.081 0.113-

0.113 0.146-

0.146 0.179-

0.179 0.212-

A (Tm)

1.002 208.2-

208.2 415.5-

415.5 622.7-

622.7 830.0-

830.0 1037.-

1037. 1244.-

1244. 1451.-

1451. 1659.-

1659. 1866.-

1866. 2073.-

2073. 2280.-

2280. 2488.-

2488. 2695.-

2695. 2902.-

2902. 3109.-

3109. 3317.-

3317. 3524.-

3524. 3731.-

3731. 3938.-

MUEr

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

16.2 Magnet extremities

The coil must so be shaped as to make the best use of the superconducting cable (limited by the peakfield to which it is exposed), while producing a dipole field with a highest possible field 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. fig. 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 fields are displayed in fig. 25 where the 3D coil representation, and thefull 3D model are shown.

In order to reduce the peak field 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-fields. 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 field

149

Single coil Both coils

(Analytical in common

calculation) collar and yoke

inj. nom

b2 - 1.04 2.97

b3 4.145 5.95 6.03

b4 - -0.13 -0.30

b5 -0.881 -0.59 -0.58

b6 - 0.01 -

b7 0.615 0.57 0.68

b8 - - -

b9 0.101 0.10 0.10

b10 - - -

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 field 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 µr = 2000. The two-in-one configuration creates additional b2, b4, .. field errors which

also vary as a function of the excitation from injection to nominal field 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 field with its non-saturated iron yoke. It is therefore still impossible toapply mathematical optimization techniques to the 3D field calculation with iron yoke. However, as theadditional effect from the fringe field on the field quality is low, it is sufficient 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 field and the reduced fieldfrom the iron magnetization. Fig. 26 shows the field 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.

150

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 flux density at nominal current along a line at x = 97mm, y = 43.6 mm (above the beam-axis of aperture 2

on a radius between the inner and the outer layer coil) from z = -200 mm inside the magnet yoke to z = 200 mm outside the

yoke. The iron yoke ends at z = -80 mm, the onset of the coil-end is at z = 0. Left: coil field, Middle: reduced field from iron

magnetization, Right: total field. Notice the different scales and the relatively small contribution from the yoke.

151

REFERENCES

[1] Armstrong, A.G.A.M. , Fan, M.W., Simkin, J., Trowbridge, C.W.: Automated optimization ofmagnet 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 fields, John Wiley & Sons, 1992

[5] C.J. Collie: Magnetic fields 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 field 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 field 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 fields in superconducting magnets, Electrical Engineering, 1999

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

[15] Mess, K.H., Schmuser, P., Wolff S.: Superconducting Accelerator Magnets, World Scientific, 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 field 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 Scientific Statistical Computing, 1986

152

[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

153