conventional magnets for accelerators lecture 2 · ben shepherd, astec cockcroft institute:...

Ben Shepherd, ASTeC Cockcroft Inst i tute: Convent ional Magnets, Autumn 2016 1

Convent ional Magnets for AcceleratorsLecture 2Ben Shepherd

Magnetics and Radiation Sources GroupASTeC

Daresbury Laboratory

[email protected]


Contents – lecture 2

• The introduct ion of currents• Coi l economic optimisation-capi tal / running costs

• Summary of the use of permanent magnets (PMs)• Remnant fields and coercivi ty• Behaviour and appl ication of PMs

• The magnetic ci rcui t• Steel requi rements: permeabi l i ty and coercivi ty• Backleg and coi l geometry: 'C', 'H' and 'window frame' designs• Classical solut ion to end and side geometries – the Rogowsky

rol loff• Magnet design using FEA software

• FEA techniques and codes – Opera 2D, Opera 3D• Judgement of magnet sui tabi l i ty in design• Magnet ends – computation and design

• Some examples of magnet engineering


FIELDS DUE TO COILS (CONTINUED)

Current Affai rs


Coi l geometry

Standard design is rectangular copper (or aluminium) conductor, wi th cool ing water tube. Insulation is glass cloth and epoxy resin.

Amp-turns (NI) are determined, but total copper area (Acopper) and number of turns (N) are two degrees of freedom and need to be decided.

Heat gen era ted in th e coil is a fun ction of th e RMS curren t den s ity:

𝑗𝑗𝑟𝑟𝑟𝑟𝑟𝑟 =𝑁𝑁𝐼𝐼𝑟𝑟𝑟𝑟𝑟𝑟𝐴𝐴𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑟𝑟

Optim um jrms determ in ed from economic criteria .


𝐼𝐼𝑟𝑟𝑟𝑟𝑟𝑟 = 32𝐼𝐼𝐷𝐷𝐷𝐷 = 1

232𝐼𝐼𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝

IDC

IAC

0

With an arbi trary waveform of period τ , th e power gen era ted in th e coil is :

In a DC m agn et th e 𝐼𝐼𝑟𝑟𝑟𝑟𝑟𝑟 = 𝐼𝐼𝐷𝐷𝐷𝐷For a pure AC s in e wave 𝐼𝐼𝑟𝑟𝑟𝑟𝑟𝑟 = 1

2𝐼𝐼𝑐𝑐𝑐𝑐𝑝𝑝𝑝𝑝

For a discon tin uous waveform th e in tegra tion is over th e wh ole of a s in gle period.A typical waveform for a booster synchrotronis a biased sin wave:

𝐼𝐼𝑟𝑟𝑟𝑟𝑟𝑟 = 𝐼𝐼𝐷𝐷𝐷𝐷2 + 12𝐼𝐼𝐴𝐴𝐷𝐷2

If 𝐼𝐼𝐷𝐷𝐷𝐷 = 𝐼𝐼𝐴𝐴𝐷𝐷 = 12𝐼𝐼𝑐𝑐𝑐𝑐𝑝𝑝𝑝𝑝

Ipeak

I rms depends on current waveform

𝑊𝑊 =𝑅𝑅𝜏𝜏�0

𝜏𝜏𝐼𝐼2 𝑡𝑡 𝑑𝑑𝑡𝑡 = 𝑅𝑅𝐼𝐼𝑟𝑟𝑟𝑟𝑟𝑟2


Current densi ty (jrms) - opt imisat ion

0.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0

Current density j

Life

time

cost

running

capitaltotal

Advantages of low jrms:• lower power loss – power bi l l is decreased• lower power loss – power converter size is decreased• less heat dissipated into magnet tunnel.

Advantages of high j:• smal ler coi ls• lower capi tal cost• smal ler magnets

Chosen value of jrms is anoptimisation of magnet capi tal against power costs.


Number of turns, NThe value of number of turns (N) is chosen to match power supply and interconnection impedances.Factors determining choice of N:

Large N (low current) Small N (high current)Sm all, n ea t term in als Large, bulky term in als Th in in tercon n ection s - low cos t an d flexible

Th ick , expen s ive con n ection s

More in sula tion in coil; la rger coil volum e; in creased as sem bly cos ts

High percen tage of copper in coil; m ore efficien t use of available space

High voltage power supply –safety problem s

High curren t power supply –grea ter los ses


Examples of typical turns/current

From the Diamond 3 GeV synchrotron source:

Dipole: N (per magnet): 40Imax 1500 AVoltage (ci rcui t): 500 V

Quadrupole: N (per pole) 54Imax 200 AVoltage (per magnet): 25 V

Sextupole: N (per pole) 48Imax 100 AVoltage (per magnet): 25 V


PERMANENT MAGNETSAn attractive proposi t ion


‘Residual ’ f ieldsRemnant f ield BR: value of B at H = 0Coercive force HC: negative value of field at B = 0Residual f ield: the flux densi ty in a gap at I = 0

-HC

BR

𝐼𝐼 = 0: ∫𝐻𝐻.𝑑𝑑𝑑𝑑 = 0So: 𝐻𝐻𝑟𝑟𝑠𝑠𝑐𝑐𝑐𝑐𝑠𝑠 𝜆𝜆 + (𝐻𝐻𝑔𝑔𝑝𝑝𝑐𝑐)𝑔𝑔 = 0

𝐵𝐵𝑔𝑔𝑝𝑝𝑐𝑐 = 𝜇𝜇0(−𝐻𝐻𝑟𝑟𝑠𝑠𝑐𝑐𝑐𝑐𝑠𝑠)𝜆𝜆𝑔𝑔

𝐵𝐵𝑔𝑔𝑝𝑝𝑐𝑐 ≈𝜇𝜇0𝐻𝐻𝑐𝑐𝜆𝜆𝑔𝑔

Wh ere: λ is pa th len gth in s teelg is gap h eigh t

Because of presence of gap, residual f ield is determined by coercive force HC (A/m) and not remnant f lux densi ty BR (Tesla).


Use of permanent magnet (PM) materials

From previous sl ide:𝐻𝐻.𝑑𝑑𝑑𝑑 = 0

𝐵𝐵𝑔𝑔𝑝𝑝𝑐𝑐 = 𝜇𝜇0(−𝐻𝐻𝑃𝑃𝑃𝑃) 𝜆𝜆𝑔𝑔;

so th e PM m ateria loperates in the 2nd quadrantof the hysteresis loop.

H

Br

Hc

B HPM

material

PM is polarised -magnet ised only in the

‘easy’ direct ion!

Optimum B, Hto give

maximum energy densi ty

= BH;Note the factor of ½ missing!


Vacuumschmelze dataEnergy densi ty (BH) in PM avai lable since 1900:

BH in second quadrant for materials avai lable today.


PM used in lat t ice magnets?ASTeC has been work ing wi th CERN on the design of 42,000quadrupoles for the drive beam in CLIC:• The power consumption for the EM version wi l l be ~8 MW• Total power load l imit to air wi thin the tunnel is only 150 W/m• A PM quadrupole would potent ial ly have many advantages:

– Vast ly reduced electr ical power– Ecological ly ‘green’– Very low operat ing costs– No cool ing water needed– Very low power to air

Problem: how to vary the strength of the quadrupole by a factor of 12? Solut ion: mechanical change to the PM geometryProblem: pole posi t ion is f ixed and must be stable to around 20 µm


Solut ionPole and yoke f ixed,

PM moves

At 8% strength (3.5 T/m) At 100% strength (43 T/m)

Lab prototype meets speci f icat ion Design for mounting in CLIC drive beam


Dynamic model

Colour code: red = high field; blue = low field


THE MAGNETIC CIRCUITYoking aside


Permeabi l i ty of low si l icon steel

10

100

1000

10000

0.0 0.5 1.0 1.5 2.0 2.5B (T)

µ

Parallel to rolling direction Normal to rolling direction.

‘saturation’ region:µ drops off


Flux at the pole and in the ci rcui t

Flux in the yoke includes the gap flux and stray flux, which extends (approx) one gap width on ei ther side of the gap.

g

g

b

Approxim ate va lue for tota l flux in th e backleg of m agn et len gth λ:

𝐹𝐹 = 𝐵𝐵𝑔𝑔𝑝𝑝𝑐𝑐 𝑏𝑏 + 2𝑔𝑔 𝜆𝜆

Width of backleg is ch osen to lim it Byoke an d h en ce m ain ta in h igh μ.

Note: FEA codes give values of vector poten tia l (Az); h en ce values of tota l flux can be obta in ed.


Magnet geometryDipoles can be ‘C core’, ‘H core’, or ‘Window frame’'C' Core:Advantages:• Easy access• Classic designDisadvantages:• Pole shims needed• Asymmetric (small)• Less rigid Shim

Th e ‘sh im ’ is a sm all, addition al p iece of ferrom agn etic m ateria l added on each s ide of th e two poles – it com pen sates for th e fin ite cut-off of th e pole, an d is optim ised to reduce th e 6-, 10-, 14-… pole error h arm on ics .


A typical ‘C’ cored Dipole

Cross-section of the Diamond storage ring dipole


H-core and ‘window frame’ magnets

‘H core’Advantages:• Symmetric• More rigidDisadvantages:• Sti l l needs shims• Access problems

'Window Frame'Advantages:• High qual i ty field• No pole shim• Symmetric & rigidDisadvantages:• Major access problems• Insulation thickness


Window frame dipoleProviding the conductor is continuous to the steel ‘window frame’ surfaces (impossible because coi l must be electrical ly insulated), and the steel has infini te μ, th is m agn et gen era tes perfect dipole field .

Providin g curren t den s ity J is un iform in con ductor:• H is un iform an d vertica l

up outer face of con ductor• H is un iform , vertica l an d

with s am e value in th e m iddle of th e gap

perfect dipole field

J

H


The pract ical window frame dipole

Insulation added to coi l :

B increases close to coi l insulation surface

B decreases close to coi l insulation surface

best compromise


PM magnets - posi t ion of material

Using permanent magnet materials, the PM goes in series (at any convenient posi t ion) in the magnet ci rcui t; shims sti l l needed:

Shim detail


Design of a PM magnet ic ci rcui t

The magnetic ci rcui t has to be designed to match the material ’s. characterist ics;the PM must operate close to i ts optimum energy densi ty:eg: VACODYM 745 at (ci rca) H = -550 kA/m;

B = 0.7 T (in the material)

Adjust height to

provide the required

Amps across the

gap

Adjust width to give B ̴ 0.7 T in material

The energy provided by the PM wi l l match the magnet ic energy in the gap (beware the factor of ½)


Diamond storage ring quadrupole

The yoke support pieces in the horizontal plane need to provide space for beamlines and are not ferromagnetic.

Error harmonics include n = 4 (octupole), a fini te permeabi l i ty error.

An open-sided quadrupole


Typical pole designsTo compensate for the non-infini te pole, shims are added at the pole edges. The area and shape of the shims determine the ampl i tude of error harmonics which wi l l be present.

A

A

Dipole Quadrupole

Th e des ign er optim ises th e pole by ‘predictin g’ th e field resultin g from a given pole geom etry an d th en adjus tin g it to give th e required quality.

Wh en h igh fields a re presen t,ch am fer an gles m us t be sm all, an d taperin g of poles m ay be n ecessary.


Assessing pole designA fi rst assessment can be made by just examining By(x) wi thin the requi red ‘good field’ region.Note that the expansion of By(x) at y = 0 is a Taylor series:

Also note:𝛿𝛿𝐵𝐵𝑦𝑦(𝑥𝑥)𝛿𝛿𝑥𝑥

= 𝑏𝑏2 + 2𝑏𝑏3𝑥𝑥 + ⋯

So quadrupole gradien t 𝑔𝑔 ≡ 𝑏𝑏2 = 𝛿𝛿𝐵𝐵𝑦𝑦(𝑥𝑥)𝛿𝛿𝑥𝑥

in a quadrupole

But s extupole gradien t 𝑔𝑔𝑟𝑟 ≡ 𝑏𝑏3 = 2 𝛿𝛿2𝐵𝐵𝑦𝑦(𝑥𝑥)𝛿𝛿𝑥𝑥2

in a s extupole

So coefficien ts are n ot equal to differen tia ls for n = 3 etc.

𝐵𝐵𝑦𝑦 𝑥𝑥 = �𝑛𝑛=1

∞

𝑏𝑏𝑛𝑛𝑥𝑥𝑛𝑛−1

= 𝑏𝑏1 + 𝑏𝑏2𝑥𝑥 + 𝑏𝑏3𝑥𝑥2 + ⋯

dipole quadrupole sextupole


Assessing an adequate design

A simple judgment of field qual i ty is given by plott ing:• Dipole: 𝐵𝐵𝑦𝑦 𝑥𝑥

𝐵𝐵𝑦𝑦(0)− 1 Δ𝐵𝐵(𝑥𝑥)

𝐵𝐵0

• Quadrupole: 𝑑𝑑𝐵𝐵𝑦𝑦(𝑥𝑥)𝑑𝑑𝑥𝑥

Δ𝑔𝑔(𝑥𝑥)𝑔𝑔(0)

• Sextupole: 𝑑𝑑2𝐵𝐵𝑦𝑦(𝑥𝑥)𝑑𝑑𝑥𝑥2

Δ𝑔𝑔2(𝑥𝑥)𝑔𝑔2(0)

‘Typical’ acceptable varia tion in s ide ‘good field’ region :Δ𝐵𝐵(𝑥𝑥)𝐵𝐵0

≤ 0.01%Δ𝑔𝑔(𝑥𝑥)𝑔𝑔(0)

≤ 0.1%Δ𝑔𝑔2(𝑥𝑥)𝑔𝑔2(0)

≤ 1.0%


How do we terminate a pole end?

For a pole wi th B ≥ 1.2 T sa tura tion an d n on -lin ear beh aviour will result if a square en d is used:

A sm ooth ‘roll-off’ is n eeded a t pole edges (tran sverse); an d a t th e m agn et en ds (in th e 3rd dim en s ion ).But wh at sh ape?Solution provided by Walter Rogowski


How derived?Rogowski calculated electric potent ial l ines around a flat capaci tor plate:

543210-1-2-3-4-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

X

Y


Blown-up version

0.50.0-0.5-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

X

Y

The central heavy l ine is for ϕ = 0.5.

Rogowski sh owed th a t th is was th e fas tes t ch an gin g lin e a lon g wh ich th e field in ten s ity was monotonical ly decreasing.


Then appl ied to magnet ends

Conclusion: Recal l that a high µ steel surface is a l ine of constant scalar potential . Hence, a magnet pole end using the ϕ = 0.5 poten tia l lin e provides th e m axim um ra te of increase in gap with a m on oton ic decrease in flux den s ity a t th e surface, i.e . n o sa tura tion .

10-1-20

1

2


The equation

The 'Rogowski ' rol l -off:

Equat ion: 𝑦𝑦 = 𝑔𝑔2

+ 𝑔𝑔𝜋𝜋𝜋𝜋

exp 𝜋𝜋𝜋𝜋𝑥𝑥𝑔𝑔− 1

𝑔𝑔2

is dipole h alf gap

y = 0 is centre l ine of gapα is a param eter con trollin ggradien t a t x = 0 (~ 1)

Diam on d dipole en d


MAGNET DESIGN


Computer codes

A number of computer codes are avai lable:e.g. the Vector Fields (Cobham) codes – Opera 2D and 3D. These have:

• fini te elements wi th variable triangular mesh• mult iple i terations to simulate steel nonl ineari ty• extensive pre- and post-processors• cross-platform compatibi l i ty

Technique is i terative:• calculate flux generated by a defined geometry• adjust the geometry unti l requi red distribution is

achieved

operafea.com


Design Procedures – Opera 2D

The model is set up in 2D using a GUI to define ‘regions’:

• steel regions• coi ls (including current densi ty)• a ‘background’ region which defines the

physical extent of the model• the symmetry constraints on the boundaries• the B-H curves for the steel and other materials• mesh is generated and data saved


Model of ALICE quadrupole

symmetry l ines background region

steel yoke

coi l


With mesh addedcoarse mesh

fine mesh


Field in model

Central gradient:1.79 T/m

Flux l ines(equipotentials)


Calculat ion

Solver - ei ther:• l inear, using a predefined constant permeabi l i ty for

a single calculation• Useful for coi l -only models

• nonl inear, which is i terative wi th steel permeabi l i ty set according to B in steel calculated on previous i teration• Essential for i ron-dominated magnets


Data Display – Opera 2D

Post-Processor:uses pre-processor model for many options for displaying field ampli tude and qual i ty:

• field l ines• graphs• contours• gradients• harmonics (from a Fourier analysis around a pre-

defined ci rcle)


2D quadrupole gradient qual i ty on x axis

Field qual i ty: ±2.5x10-4 wi thin ±40mm


Opera 3D model of CLARA quadrupole

3 symmetry planes,1/8th of magnet model led


Fields in 3D model


Harmonics indicate magnet qual i ty

The ampli tude and phase of the harmonic components in a magnet provide an assessment:

• when accelerator physicists are calculating beam behaviour in a latt ice

• when designs are judged for sui tabi l i ty• when the manufactured magnet is

measured• to judge acceptabi l i ty of a manufactured

magnet


The third dimension – magnet ends

Fringe flux wi l l be present at the magnet ends so beam deflection continues beyond magnet end:

z

By

The magnet’s strength is given by ∫𝐵𝐵𝑦𝑦 𝑧𝑧 𝑑𝑑𝑧𝑧 alon g th e m agn et, th e in tegra tion in cludin g th e frin ge field a t each en d.

Th e magnetic length is defin ed as 1𝐵𝐵0∫𝐵𝐵𝑦𝑦 𝑧𝑧 𝑑𝑑𝑧𝑧 over th e sam e

in tegra tion path , wh ere B0 is th e field a t th e azim uth al cen tre.

B0


Magnet End Fields and Geometry

The end of the magnet is thereforechamfered (a Rogowski rol l -off i f high field), increasing the gap (or inscribed radius) and lowering the field as the end is approached.

Necessary to term in ate th e m agn et in a con trolled way:• to defin e th e len gth (s tren gth )• to preven t s a tura tion in a sh arp corn er (s ee diagram )• to m ain ta in len gth con s tan t with x, y• to preven t flux en terin g n orm al to lam in ation (AC)


Pole prof i le adjustment

As the gap is increased, the size (area) of the shim is increased, to give some control of the field qual i ty at the lower field. This is far from perfect!

Tran sverse adjus tm en t a t en d of dipole

Tran sverse adjus tm en t a t en d of quadrupole


The NINA magnet ends


Calculat ion of end effects wi th 2D codesFEA model in longi tudinal plane, wi th correct end geometry (including coi l), but 'ideal ised' return yoke:

+

-

Th is will es tablish th e en d dis tribution ; a n um erica l in tegra tion will give th e 'B' len gth .

Provided 𝑑𝑑𝐵𝐵𝑦𝑦𝑑𝑑𝑑𝑑

is n ot too la rge, s in gle 's lices ' in th e tran sverse p lan e can be used to ca lcula te th e radia l dis tribution as th e gap in creases . Again , n um erica l in tegra tion will give ∫𝐵𝐵.𝑑𝑑𝑑𝑑 as a fun ction of x.

This technique is less satisfactory wi th a quadrupole, but end effects are less cri t ical wi th a quad.


End geometries - dipole

Simpler geometries can be used in some cases.The Diamond dipoles have a Rogowski rol l -off at the ends (as wel l as Rogowski rol l -offs at each side of the pole).

See photographs to fol low.

This gives small negative sextupole field in the ends which wi l l be compensated by adjustments of the strengths in adjacent sextupole magnets – this is possible because each sextupole wi l l have i ts own individual power supply.


MAGNET EXAMPLES


Diamond Dipole


Diamond dipole ends


Diamond Dipole end


Simpl i f ied end geometries -quadrupole

Diamond quadrupoles have an angular cut at the end; depth and angle were adjusted using 3D codes to give optimum integrated gradient.


Diamond W quad end


End chamfering - Diamond ‘W’ quad

Opera 3D results -di fferent depths 45° en d ch am fers on Δ𝑔𝑔

𝑔𝑔0in tegra ted th rough m agn et an d en d frin ge field (0.4 m lon g WM quad).

5 10 15 20 25 30 35X

-0.002

-0.001

0.001

0.002

Fractionaldeviation

No Cut4 mm Cut

8 mm Cut

6 mm Cut7 mm Cut

Thanks to Chris Bailey (DLS) who performed this working using OPERA 3D.


Sextupole ends

It is not usual ly necessary to chamfer sextupole ends (in a DC magnet).Diamond sextupole end:


Sexy pics of sextupoles


Further Reading

• CERN Accelerator School on MagnetsBruges, Belgium; June 2009https:/ /arxiv.org/html/1105.5069v1

• Uni ted States Particle Accelerator SchoolMagnet and RF Cavi ty Design, January 2016http:/ /uspas.fnal.gov/materials/16Austin/austin-magnets.shtml

• J.D. Jackson, Classical Electrodynamics• J.T. Tanabe, Iron Dominated Electromagnets

https://arxiv.org/html/1105.5069v1

http://uspas.fnal.gov/materials/16Austin/austin-magnets.shtml

conventional magnets for accelerators lecture 2 · ben shepherd, astec cockcroft institute:...

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