ACCELERATOR MAGNETS

Stephan RussenschuckCERN, AT-MEL-EM

1211 Geneva 23, Switzerland

Bending Field (Dipole) and Magnetic Rigidity

B vz × B

3.3356p[GeV/c] = R[m] B0[T]

Remark 1: Consider the “filling factor” in accelerators between 0.6 and 0.7.

Transverse Motion and the Normalized Gradient

ϕ ρ

Displaced particle at ρ = R + x with field Bz = B0 + ∂Bz∂x

∣∣∣x=0

x

1p0

dds

(p0

dxds

)+ (

1R2 − k)x = 0

For x R, e/p0 = −1/(B0R) and k = − 1B0R

∂Bz∂x

∣∣∣x=0

Quadrupole

B (vz × B)x (vz × B)y

Higher Order Multipole Fields: Sextupole

B (vz × B)x (vz × B)y

Multipole Field Errors and Beam Parameters

b1, a1 Closed orbit distortions

b2 β-beating

a2 Vertical dispersion, linear coupling

b3 Off-momentum β beating

a3 Chromatic coupling inducing detuning

b4 Dynamic aperture and detuning

a4 Dynamic aperture at injection

b5 Dynamic aperture and detuning

a5 Off-momentum dynamic aperture

b7 Dynamic aperture at injection

b6, a6,a7, > Dynamic aperture at injection

LHC in the LEP Tunnel (Virtual Reality)

Remark 2: Magnet systems are the most costly item in an accelerator

The 26 GeV Proton Synchrotron (PS) in its Tunnel

High Energy Ring Accelerator (HERA) at DESY

Magnet Metamorphosis

(C- Core, LEP Dipole)

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)

N ⋅ I = 4480A B1 = 0.13T Bs = 0.042T Fill.fac. 0.27

Magnet Metamorphosis

(H-Magnet)

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)

N ⋅ I = 24000A B1 = 0.3T Bs = 0.065T Fill.fac. 0.98

Magnet Metamorphosis(Cryogenic or Super-Ferric Magnet)

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)

N ⋅ I = 96000A B1 = 1.18T Bs = 0.26T

Magnet Metamorphosis

(Window Frame Dipole)

0. 0.147-

0.147 0.294-

0.294 0.442-

0.442 0.589-

0.589 0.736-

0.736 0.884-

0.884 1.031-

1.031 1.178-

1.178 1.326-

1.326 1.473-

1.473 1.621-

1.621 1.768-

1.768 1.915-

1.915 2.063-

2.063 2.210-

2.210 2.357-

2.357 2.505-

2.505 2.652-

2.652 2.8-

|Btot| (T)

N ⋅ I = 360000A B1 = 2.08T Bs = 1.04T

Magnet Metamorphosis

(Tevatron Dipole)

0 21 42 63 84 105 126 147 168 189 210

0. 0.147-

0.147 0.294-

0.294 0.442-

0.442 0.589-

0.589 0.736-

0.736 0.884-

0.884 1.031-

1.031 1.178-

1.178 1.326-

1.326 1.473-

1.473 1.621-

1.621 1.768-

1.768 1.915-

1.915 2.063-

2.063 2.210-

2.210 2.357-

2.357 2.505-

2.505 2.652-

2.652 2.8-

|Btot| (T)

N ⋅ I = 471000A B1 = 4.16T Bs = 3.39T

Magnet Metamorphosis

(LHC Coil-Test Facility

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)

N ⋅ I = 960000A B1 = 8.33T Bs = 7.77T

Magnet Metamorphosis

(LHC Main Dipole)

0. 0.147-

0.147 0.294-

0.294 0.442-

0.442 0.589-

0.589 0.736-

0.736 0.884-

0.884 1.031-

1.031 1.178-

1.178 1.326-

1.326 1.473-

1.473 1.621-

1.621 1.768-

1.768 1.915-

1.915 2.063-

2.063 2.210-

2.210 2.357-

2.357 2.505-

2.505 2.652-

2.652 2.8-

|Btot| (T)

N ⋅ I = 2x944000A B1 = 8.32T Bs = 7.44T

LHC Main Dipol Cross-Section

1. Heat exchanger, 2. Bus bar, 3. Superconducting coil, 4. Vacuum Pipe und Beam-screen, 5. Cryostat,6. Thermal shield (55- 75 K), 7. Helium-Tank, 8. Super-Insulation, 9. Collars, 10. Yoke

Overview

Maxwell’s equations in integral formOne-dimensional field calculation for conventional magnets

Maxwell’s equations in differential formThe solution of Laplace’s equation

Harmonic fieldsComplex Potentials

Ideal pole shapes of conventional magnetsFeed down

Solution of Poisson’s equationThe field of line currents

Generation of pure multipole fieldsSensitivity to manufacturing errors

Numerical field computationSaturation of the iron yoke

Superconducting filament magnetization

Maxwell’s Equations in Integral Form

∮H ⋅ ds =

∫A(J +

∂ D∂ t

) ⋅ dA∮E ⋅ ds = −

∂∂ t

∫A

B ⋅ dA∫A

B ⋅ dA = 0∫A

D ⋅ dA =∫

VρdV

Constitutive Equations

B = µ H = µ0(H + M)D = εE = ε0(E + P)J = κ E + Jimp.

1D Field Calculation for a Conventional Dipole

∮H ⋅ ds =

∫A

J ⋅ dA

Hiron siron + Hgap sgap =1

µ0µrBiron siron +

1µ0

Bgap sgap = N I

µr 1 Bgap =µ0N Isgap

Warning 1: Check that the magnetic circuit contains no flux concentrationwhich increases the magnetic flux density above 1 T, as in this case

fringe fields can no longer be neglected.

1D Field Calculation for a Conventional Quadrupole

∮H ⋅ ds =

∫1

H1 ⋅ ds +∫

2H2 ⋅ ds +

∫3

H3 ⋅ ds = N I

Bx = gy By = gx ⇒ H =gµ0

√x2 + y2 =

gµ0

r

∫ r0

0Hdr =

gµ0

∫ r0

0rdr =

gµ0

r202

= N I ⇒ g =2µ0NI

r20

LEP Dipole and Quadrupole

-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)

Field Quality in the Dipole Aperture

Field Quality in the Dipole Aperture

|1 − ByBnom

y|

Definition of Field Quality in Accelerator Magnets

Fourier-series expansion of the radial component of the magnetic flux densityon a reference radius inside the aperture

Br(r0, ϕ) =∞∑

n=1

(Bn(r0) sin nϕ + An(r0) cos nϕ)

= BN(r0)∞∑

n=1

(bn(r0) sin nϕ + an(r0) cos nϕ)

An(r0) ≈1P

2P−1∑k=0

Br (r0, ϕk) cos nϕk , Bn(r0) ≈1P

2P−1∑k=0

Br (r0, ϕk) sin nϕk .

Remark 3: This definition is perfectly in line with magnetic fieldmeasurements using “harmonic coils” where the periodic flux variation

in tangential rotating coils is analyzed with a FFT.

Rotating Coil Measurement Setup at BNL (Brookhaven)

Maxwell’s Equations in Differential Form

curlH = J +∂ D∂ t

curlE = −∂B∂ t

div B = 0div D = ρ

Constitutive Equations

B = µ H = µ0(H + M)D = εE = ε0(E + P)J = κ E + Jimp.

Lemmata of Poincar e

The curl of an arbitrary vector field is source free div curlg = 0An arbitrary gradient field is curl free curl gradφ = 0

A source free field B, (div B = 0)can be expressed through a vector potential A in the form

B = curlA.

A curl free field H, (curlH = 0)can be expressed through a scalar potential φ in the form H = −gradΦm.

Remark 4: The field in the aperture of accelerator magnetsis curl and source free

Warning 2: The Lemmata of Poincare hold only for simply connected domains withconnected boundaries.

Maxwell’s “House”

Warning 3: Only for sub-domains with continuous material parameters.

Maxwell’s “Facade”

curl1µ

curlA = J1µ0

curl curlA = 0 ∇ 2A − grad div A = 0 ∇ 2Az = 0

div µgradΦm = 0 µ0 div gradΦm = 0 ∇ 2Φm = 0

r2∂2Az

∂r2 + r∂Az

∂r+

∂2Az

∂ϕ2 = 0

General Solution of the Laplace Equation

Az(r , ϕ ) =∞∑

n=1(Enrn + Fnr−n)(Gn sin nϕ + Hn cos nϕ)

Br (r , ϕ ) =1r

∂Az

∂ϕ=

∞∑n=1

nrn−1(Cn sin nϕ + Dn cos nϕ)

nrn−1Cn = Bn nrn−1Dn = An

Bϕ(r , ϕ ) = −∂Az

∂r= −

∞∑n=1

nrn−1(Dn sin nϕ − Cn cos nϕ)

Normal (Skew) Dipole (n=1)

Br = C1 cos ϕ + D1 sin ϕ Bϕ = −C1 sin ϕ + D1 cos ϕ

Bx = Br cos ϕ − Bϕ sin ϕ By = Br sin ϕ + Bϕ cos ϕ

Bx = C1 By = D1

Normal (Skew) Quadrupole (n=2)

Br = 2C2r cos 2ϕ + 2D2r sin 2ϕBϕ = −2C2r sin 2ϕ + 2D2r cos 2ϕBx = 2C2x + 2D2yBy = −2C2y + 2D2x

Remark 5: Notice that we have not yet addressed the problemof how to create such field distributions.

Normal (Skew) Sextupole (n=3)

Br = 3C3r2 cos 3ϕ + 3D3r2 sin 3ϕBϕ = −3C3r2 sin 3ϕ + 3D3r2 cos 3ϕBx = 3C3(x2 − y2) + 6D3xyBy = −6C3xy + 3D3(x2 − y2)

Remark 6: The treatment of each harmonic separately is a mathematical abstraction.In practical situations many harmonics will be present (to be minimized).

Analytical Field Computation for Superconducting Magnets

Special solution of the inhomogeneous (Poisson) differential equation

∇ 2A = J

Fundamental solution of the Laplace Operator ∇ 2 (Green’s Function)

(2D) G =1

2πln |r −r ′|

(3D) G =1

4π1

|r −r ′|

Vector potential of a line current (2D)

Az = −µ0I2π

ln(|r −r ′|

a)

The Field of Line Currents (2D)

R = |r −r ′|

r ′ = (ri , Θ)

r = (r0, ϕ)

ϕΘ

Iz

Az

y

z x

r0 < ri

Az = −µ0I2π

ln(|r −r ′|

a)

Br == −µ0I2π

∞∑n=1

(r n−10r ni

)(sin nϕ cos nΘ − cos nϕ sin nΘ)

Bn(r0) = −µ0I2π

r n−10r ni

cos nΘ An(r0) =µ0I2π

r n−10r ni

sin nΘ

The Imaging Method

Coil in non-saturated yoke Imaged coil

0 20 40 60 80 100 120 140 160

0 20 40 60 80 100 120 140 160

Field Harmonics in SC Magnet

Bn(r0) = −ns∑i=1

µ0Ii2π

rn−10rni

(1 +

µr − 1µr + 1

(ri

RYoke)2n

)cos nΘi

Scaling laws

Bn(r1) = (r1/r0)n−1Bn(r0)

Influence of the iron yoke (non-saturated)ri = 43.5mm, RYoke = 89mm: B1 - 19% , B5 - 0.07%

Remark 7: Analytical field optimization can be used for higher order harmonics, finiteelement calculations (for the estimation of saturation effects in the iron yoke) only

needed for the lower order harmonics.

Ideal (cos n θ) Current Distribution

Bn(r0) = −ns∑i=1

∫ ro

ri

∫ 2π

0

µ0J0 cos mΘ2π

r n−10r ni

(1 +

µr − 1µr + 1

(ri

RYoke)2n

)cos nΘi rdΘdr

Dipole Quadrupole

Ideal Current Distribution Approximated with Coil-Blocks

Dipole Quadrupole

B3 and B5 Contribution of Strand Current

Bn(r0) = −µ0Ii2π

r n−10r ni

(1 +

µr − 1µr + 1

(ri

RYoke)2n

)cos nΘi

-1.01 -0.90-

-0.90 -0.80-

-0.80 -0.69-

-0.69 -0.59-

-0.59 -0.48-

-0.48 -0.37-

-0.37 -0.27-

-0.27 -0.16-

-0.16 -0.60-

-0.60 0.045-

0.045 0.151-

0.151 0.257-

0.257 0.362-

0.362 0.468-

0.468 0.574-

0.574 0.680-

0.680 0.786-

0.786 0.892-

0.892 0.998-

(*E-3)

B3 CONTR. (T)

-0.35 -0.32-

-0.32 -0.28-

-0.28 -0.24-

-0.24 -0.20-

-0.20 -0.17-

-0.17 -0.13-

-0.13 -0.95-

-0.95 -0.58-

-0.58 -0.20-

-0.20 0.016-

0.016 0.054-

0.054 0.091-

0.091 0.129-

0.129 0.166-

0.166 0.204-

0.204 0.242-

0.242 0.279-

0.279 0.317-

0.317 0.354-

(*E-3)

B5 CONTR. (T)

Sensitivity to Coil Block Displacements

Bn(r0) = −µ0Ii2π

r n−10r ni

(1 +

µr − 1µr + 1

(ri

RYoke)2n

)cos nΘi

∂Bn(r0)∂Θi

= −µ0Ii2π

nrn−10r ni

(1+(

ri

RYoke)2n

)sin nΘi

∂Bn(r0)∂ri

=µ0Ii2π

nrn−10

r n+1i

(1−(

ri

RYoke)2n

)cos nΘi

Increase of the azimuthal coil size by 0.1 mm produces (in units of 10−4):

b1 = −14. b3 = 1.2 b5 = 0.03

Specified tolerances on coils: ± 0.025mm

Coil Winding

Curing Press and Curing Mold

Complex Potentials

H = −gradΦm = −∂Φm

∂xex −

∂Φm

∂yey

B = curlAz =∂Az

∂yex −

∂Az

∂xey

Cauchy-Riemann DEQ

∂Az

∂y= −µ0

∂Φm

∂x∂Az

∂x= µ0

∂Φm

∂y

W (z) = Az(x , y) + iµ0Φm(x , y) z = x + iy

−dWdz

= −∂Az

∂x− iµ0

∂Φm

∂x= By(x , y) + iBx(x , y)

Cartesian Field Components

W (z) = Az(x , y) + iµ0Φm(x , y) z = x + iy

By + iBx =∞∑

n=1

(Bn + iAn)(z0

r0)n−1 = −

µ0I2π

∞∑n=1

zn−10zn |z0| < |z|

Ideal (Iron) Pole Shapes

Feed down

∞∑n=1

cn

(zr0

)n−1

=∞∑

n=1

c′n

(z ′

r0

)n−1

= By + i Bx

z = z ′ + ∆z z = x + iy

c′n =

∞∑k=n

ck(k − 1)!

(k − n)! (n − 1)!

(∆zr0

)k−n

c′2 = c2 + 2c3

(∆zr0

)+ 3c4

(∆zr0

)2

+ …

• Measurement of magnetic axis in dipole by powering the coil as a quadrupole.

• The feed down effect can be used to center the measurements coilby minimization of the b10 which can only occur as feed down from b11.

• The dipole magnetic axis has to be well aligned with respect to the closed orbit: ± 0.1mm for both ∆x and ∆y systematic and 0.5mm for both σx and σy random (r.m.s.).

• Alignment tolerances of MCS and MCDO correctors w.r.t. MB: 0.3mm radially

Mechanical Axis Measurement for Spool Piece Alignment

Excitation Cycle of the LHC Dipoles

! " # $

" % $

" $

! & ' ( &

) )

Field Variation with Excitation

0 2000 4000 6000 8000 10000 12000

0.705

0.706

0.707

0.708

0.709

B0

I (A)

0 2000 4000 6000 8000 10000 12000

-6

-4

-2

0

2

4

6

b2

b3

b410

b510

I (A)

x 10-4

Saturation Effect in LHC Dipoles

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

Objectives for the ROXIE Development

Automatic generation of coil geometry

Field computation specially suited for magnet design (BEM-FEM)No meshing of the coil

No artificial boundary conditionsConfine numerical errors to the iron magnetization

Higher-order quadrilateral finite-elementsParametric mesh generator

Mathematical optimization

CAD/CAM interfaces

Field Computation with the BEM-FEM Coupling Method

IS

BS,i

Ω1

AΓ1 AΓ2

BIRONR1

BS,i

BIRONR2

BS,i

Ω2

BEM-domain

FEM-domains

Ω3

Bi

Field Quality Calculations for Collared Coils

-0.23 -0.21-

-0.21 -0.18-

-0.18 -0.16-

-0.16 -0.13-

-0.13 -0.11-

-0.11 -0.87-

-0.87 -0.62-

-0.62 -0.37-

-0.37 -0.12-

-0.12 0.012-

0.012 0.037-

0.037 0.062-

0.062 0.087-

0.087 0.111-

0.111 0.136-

0.136 0.161-

0.161 0.186-

0.186 0.211-

0.211 0.236-

A (Tm)

b2 = −0.239 b3 = 2.742 b4 = −0.012 b5 = −0.733

Preparation and Measurement of Collared Coils at BNN, Germany

Calculation of Fringe Fields in a LHC Dipole Model

y

x

z

Source field Reduced field Total field

-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

Hysteresis in Superconductor Magnetization

-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

-1.6 -1.2 -0.8 -0.4 0.0 0.4 0.8 1.2 1.6Β [Τ]

µ 0 Μ

[Τ]

strand measurement

strand magnetization

filament magnetization

initial state curve

Field and Magnetization in the LHC Dipole Coil

0.024 0.100-

0.100 0.176-

0.176 0.252-

0.252 0.328-

0.328 0.403-

0.403 0.479-

0.479 0.555-

0.555 0.631-

0.631 0.707-

0.707 0.783-

0.783 0.859-

0.859 0.934-

0.934 1.010-

1.010 1.086-

1.086 1.162-

1.162 1.238-

1.238 1.314-

1.314 1.390-

1.390 1.466-

|B| (T)

-26.1 -25.0-

-25.0 -24.0-

-24.0 -22.9-

-22.9 -21.9-

-21.9 -20.8-

-20.8 -19.7-

-19.7 -18.7-

-18.7 -17.6-

-17.6 -16.5-

-16.5 -15.5-

-15.5 -14.4-

-14.4 -13.3-

-13.3 -12.3-

-12.3 -11.2-

-11.2 -10.1-

-10.1 -9.12-

-9.12 -8.05-

-8.05 -6.99-

-6.99 -5.93-

(*E3)

M (A/m)

Generated B3 Field Errors

Bn(r0) = −µ0Ii2π

r n−10r ni

cos nΘ

Bn(r0) =µ0

2πr n−10

r n+1i

n(mr ′ sin nθ + mθ cos nθ)

-0.20 -0.18-

-0.18 -0.16-

-0.16 -0.14-

-0.14 -0.12-

-0.12 -0.99-

-0.99 -0.77-

-0.77 -0.55-

-0.55 -0.34-

-0.34 -0.12-

-0.12 0.009-

0.009 0.030-

0.030 0.052-

0.052 0.074-

0.074 0.095-

0.095 0.117-

0.117 0.139-

0.139 0.161-

0.161 0.182-

0.182 0.204-

(*E-3)

B3 Contr. of Istrand (T)

-0.83 -0.75-

-0.75 -0.67-

-0.67 -0.59-

-0.59 -0.51-

-0.51 -0.43-

-0.43 -0.35-

-0.35 -0.28-

-0.28 -0.20-

-0.20 -0.12-

-0.12 -0.43-

-0.43 0.003-

0.003 0.011-

0.011 0.019-

0.019 0.027-

0.027 0.035-

0.035 0.043-

0.043 0.050-

0.050 0.058-

0.058 0.066-

(*E-3)

B3 Contr. of Mstrand (T)

Macro-Photography of Cable and Strand

Inner (outer) layer LHC dipole coil:Filament diameter 7 (6) µm, Strand diameter 1.065 (0.825) mm

Warning 4:

Scaling Laws for Multipoles are Wrong in 3 Dimensions

Bn(r1) = (r1

r0)n−1Bn(r0) bn(r1) = (

r1

r0)n−Nbn(r0)

0 20 40 60 80 100 120 140 160 180 200

-80

-60

-40

-20

0

20

40

60

80

b3

b3 (scaled, wrong)

Scaling Laws Hold for Integrated Multipoles

∇ 2Φm(x , y , z) =∂2Φm(x , y , z)

∂x2 +∂2Φm(x , y , z)

∂y2 +∂2Φm(x , y , z)

∂z2 = 0

Φm(x , y) =∫ z0

−z0

Φm(x , y , z)dz

∇ 2Φm(x , y) =∂2Φm(x , y)

∂x2 +∂2Φm(x , y)

∂y2 = 0

Sufficient condition: Integration path is extended far enough away from (into) themagnet so that the axial component of the field has dropped to zero.

∂2Φm

∂x2 +∂2Φm

∂y2 =∫ z0

−z0

(∂2Φm

∂x2 +∂2Φm

∂y2

)dz =

∫ z0

−z0

(−

∂2Φm

∂z2

)dz = −

∂Φm

∂z

∣∣∣∣z0

−z0

= Hz|−z0 − Hz|z0

References

[1] M. Aleksa, S. Russenschuck and C. Vollinger Magnetic Field Calculations Including the Impact of Persistent Currents inSuperconducting Filaments IEEE Trans. on Magn., vol. 38, no. 2, 2002.

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

[3] Beth, R. A.: An Integral Formula for Two-dimensional Fields, Journal of Applied Physics, Vol. 38, Nr. 12, 1967

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

[5] Bryant P.J.: Basic theory of magnetic measurements, CAS, CERN Accelerator School on Magnetic Measurement andAlignment, CERN 92-05, Geneva, 1992

[6] CAS, CERN Accelerator School on Superconductivity in Particle Accelerators, Haus Rissen, Hamburg, Germany, 1995,Proceedings, CERN-96-03

[7] Kurz, S., Russenschuck, S.: The application of the BEM-FEM coupling method for the accurate calculation of fields insuperconducting magnets, Electrical Engineering, 1999

[8] Mess, K.H., Schmuser, P., Wolff S.: Superconducting Accelerator Magnets, World Scientific, 1996

[9] M. Pekeler et al., Coupled Persistent-Current Effects in the Hera Dipoles and Beam Pipe Correction Coils, Desy Reportno. 92-06, Hamburg, 1992

[10] Russenschuck, S.: ROXIE: Routine for the Optimization of magnet X-sections, Inverse field calculation and coil Enddesign, Proceedings of the First International ROXIE users meeting and workshop, CERN 99-01, ISBN 92-9083-140-5.

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