Magnets for FFAGs - Cockcroft Web...Magnets for FFAGs; Neil Marks FFAG School 2011 Present a short overview of electro - magnetic technology as used in particle accelerators, considering

Magnets for FFAGs; Neil Marks FFAG School 2011

Magnets for FFAGs

Neil Marks,STFC- ASTeC / U. of Liverpool,

Daresbury Laboratory,Warrington WA4 4AD,

U.K.Tel: (44) (0)1925 603191Fax: (44) (0)1925 603192

[email protected]


‘Omnis Gallia in tres partes divisa est’

GAIVS IVLIVS CÆSAR (100BC - 15 March 44BC)De Bello Gallico, Book 1, Ch 1.

Likewise, this presentation!


Present a short overview of electro- magnetic technology as used in particle accelerators, considering only warm magnets(ie superconducting magnets excluded) in three main sections:

i) DC magnets used in general accelerator lattices;

ii) pulsed magnets used in injection and extraction systems;

iii) specialised FFAG issues (including practical examples in determining pole face geometry);

(this section will also be delivered in the main workshop so will only be presented here if time permits).

Objectives


Section 1: ‘General’ Accelerator Magnet Theory and Practice.• Maxwell's 2 magneto-static equations;

Fields in free space:• Solutions in 2D with scalar potential (no currents); cylindrical

harmonics; field lines and potential for dipole, quadrupole, sextupole;Introduction of steel:

• Ideal pole shapes for dipole, quad and sextupole, and combined function magnets, significance and use of contours of constant scalar potential;Introduction of currents:

• Ampere-turns in dipole, quad and sextupole; coil economic optimisation-capital/running costs;Practical Issues:

• FEA techniques - Modern codes- OPERA 2D; TOSCA; judgement of magnet suitability in design.

Contents – Section 1.


Section 2: Injection and Extraction system pulsed magnets;• Methods of injection and extraction;• Septum magnets;• Kicker magnets and power supplies.

Section 3: Geometry of specialised FFAG magnets.• Example of lattice magnet requirements for a 27 cell FFAG (Pumplet*);• Development of a suitable pole for two lattice magnets;• A recent development at DL to save space and operating cost of an of-

centre multipole FFAG magnet.

* Acknowledging design by Grahame Rees, ASTeC, RAL.

Contents – Sections 2 & 3.


SECTION 1General Accelerator Magnet Theory

and Practice


Maxwell’s equations: ∇.B = 0 ;∇ ∧ H = j ; j = 0;

So then we can put: B = - ∇φ

So that: ∇2φ = 0 (Laplace's equation).

Taking the two dimensional case (ie constant in the z direction) and solving for cylindrical coordinates (r,θ):

φ = (E+F θ)(G+H ln r) + Σn=1∞ (Jn r n cos nθ +Kn r n sin nθ

+Ln r -n cos n θ + Mn r -n sin n θ )

No currents, no steel:


The scalar potential simplifies to:

φ = Σn (Jn r n cos nθ +Kn r n sin nθ),with n integral and Jn,Kn a function of geometry.

Giving components of flux density:

Br = - Σn (n Jn r n-1 cos nθ +nKn r n-1 sin nθ)Bθ = - Σn (-n Jn r n-1 sin nθ +nKn r n-1 cos nθ)

In practical situations:


This is an infinite series of cylindrical harmonics; they define the allowed distributions of B in 2 dimensions in the absence of currents within the domain of (r,θ).

Distributions not given by above are not physically realisable.

Coefficients Jn, Kn are determined by geometry (remote iron boundaries and current sources).

Note that this formulation can be expressed in terms of complex fields and potentials.

Significance


To obtain these equations in Cartesian coordinates, expand the equations for φ and differentiate to obtain flux densities;

cos 2θ = cos2θ – sin2θ; cos 3θ = cos3θ – 3cosθ sin2θ; sin2θ = 2 sinθ cosθ; sin3θ = 3sinθ cos2θ – sin3θ;

cos 4θ = cos4θ + sin4θ – 6 cos2θ sin2θ;sin 4θ = 4 sinθ cos3θ – 4 sin3θ cosθ;

etc (messy!);x = r cos θ; y = r sin θ;

and Bx = - ∂φ/∂x; By = - ∂φ/∂y

Cartesian coordinates:


Cylindrical: Cartesian:φ =J1 r cos θ +K1 r sin θ. φ =J1 x +K1 yBr = J1 cos θ + K1 sin θ; Bx = -J1

Bθ = -J1 sin θ + K1 os θ; By = -K1

So, J1 = 0 gives vertical dipole field:

K1 =0 gives horizontal dipole field.

Bφ = const.

Dipole field n=1:


Cylindrical: Cartesian:φ = J2 r 2 cos 2θ +K2 r 2 sin 2θ; φ = J2 (x2 - y2)+2K2 xyBr = 2 J2 r cos 2θ +2K2 r sin 2θ; Bx = -2 (J2 x +K2 y)Bθ = -2J2 r sin 2θ +2K2 r cos 2θ; By = -2 (-J2 y +K2 x)

J2 = 0 gives 'normal' or ‘upright’ quadrupole field.

K2 = 0 gives 'skew' quad fields (above rotated by π/4).

Lines of flux density

Line of constant

scalar potential

Quadrupole field n=2:


Cylindrical; Cartesian:φ = J3 r3 cos 3θ +K3 r3 sin 3θ; φ = J3 (x3-3y2x)+K3(3yx2-y3)Br = 3 J3r2 cos 3θ +3K3r2 sin 3θ; Bx = -3J3 (x2-y2)+2K3yxBθ= -3J3 r2 sin 3θ+3K3 r2 cos 3θ; By = -3-2 J3 xy +K3(x2-y2)

Line of constant scalar potential

Lines of flux density

+C

-C

+C

-C

+C

-C J3 = 0 giving 'normal' or ‘right’ sextupole field.+C

-C

+C

-C

+C

-C

Sextupole field n=3:


Dipole; constant field:

Quad; linear variation:

Sextupole: quadratic variation:

x

By

00

By

x

x

By

Summary: variation of By on x axis.


B (x) = B ρ

k n xn

n!n=0

∞∑

magnet strengths are specified by the value of kn; (normalised to the beam rigidity);

order n of k is different to the 'standard' notation:

dipole is n = 0;quad is n = 1; etc.

k has units:k0 (dipole) m-1;k1 (quadrupole) m-2; etc.

Alternative notation:


What is the ideal pole shape?•Flux is normal to a ferromagnetic surface with infinite µ:

•Flux is normal to lines of scalar potential, (B = - ∇φ);

•So the lines of scalar potential are the ideal pole shapes!

(but these are infinitely long!)

curl H = 0

therefore ∫ H.ds = 0;

in steel H = 0;

therefore parallel H air = 0

therefore B is normal to surface.

µ = ∞

µ = 1

Introducing iron yokes and poles.


Equations for Ideal (infinite) poles;(Jn = 0) for normal (ie not skew) fields:Dipole:

y= ± g/2;(g is inter-pole gap).Quadrupole:

xy= ±R2/2;Sextupole:

3x2y - y3 = ±R3;

R

Equations of ideal poles


'Combined Function Magnets' - often dipole and quadrupole field combined (but see next-but-two slide):

A quadrupole magnet withphysical centre shifted frommagnetic centre.

Characterised by 'field index' n,+ve or -ve dependingon direction of gradient;do not confuse with harmonic n!

B

n = - ρΒ 0

∂B∂x

,

ρ is radius of curvature of the beam;

Bo is central dipole field

Combined function magnets


Combined dipole/quadrupoleCombined function (large dipole & small quadrupole) :• beam is at physical centre• flux density at beam = B0;• gradient at beam = ∂ B/∂x;• magnetic centre is at B = 0.• separation magnetic to physical centre = X0

magnetic centre,x’ = 0

physical centrex = 0

X0x

x’


Pole for a combined dipole and quad.

1

xB

0Bx - 1 g y asrewritten

magnet theof centre physical at the gap half theis g where

1n x - 1 g y or

1n x - 1 n 2

2R y isequation Pole

0X x ' x As2 / 2R y x' therefore

0X x B 0B Then

x'is centre quad truefromt displacmen Horizontal 0Xby separated are centres magnetic and Physical

−

∂∂±=

−±=

−±=

+=±=

∂∂=

ρ

ρρ


i) dipole, skew quad, sextupole, octupole ( in the SRS) ii) dipole & sextupole (for chromaticity control);iii) dipole, quadrupole sextupole and octupole (and more!);

Other combinations:

# i) generated by multiple coils mounted on a yoke; amplitudes independently varied by coil currents.

# ii) and iii) generated by pole shapes given by sum of correct scalar potentials, hence amplitudes built into pole geometry (but not variable); important for FFAG magnet design – see section 3.

Combined function magnets.


Vector potential in 2D

By definition: B = curl A (A is vector potential);and div A = 0Expanding: B = curl A = (∂Az/ ∂y - ∂Ay/ ∂z) i + (∂Ax/ ∂z - ∂Az/ ∂x) j + (∂Ay/ ∂x - ∂Ax/ ∂y) k;

where i, j, k, are unit vectors in x, y, z.In 2 dimensions Bz = 0; ∂ / ∂z = 0;So Ax = Ay = 0;and B = (∂Az/ ∂y ) i - (∂Az/ ∂x) jIn a 2D problem, A is in the z direction, normal to thePlane of the problem.Note: div B = ∂2Az/ ∂ x ∂y - ∂2Az/ ∂x ∂y = 0;


Total flux between 2 points.

In a two dimensional problem the magnetic flux between two points is proportional to the difference between the vector potentials at those points.

B

Φ

A1 A2

Φ∝ (A2 - A1);

for proof see next slide.


Proof

Consider a rectangular closed path, length λ in z direction at (x1,y1) and (x2,y2); apply Stokes’ theorem:

x

yz

(x1, y1) (x2, y2)

λ

BA

ds

dSΦ = ∫ ∫ B.dS = ∫ ∫ ( curl A).dS = ∫ A.ds

But A is exclusively in the z direction, and is constant in this direction.So:∫ A.ds = λ A(x1,y1) - A(x2,y2);

Φ = λ A(x1,y1) - A(x2,y2);


Practically, poles are finite, introducing errors; these appear as higher harmonics which degrade the field distribution.However, the iron geometries have certain symmetries that restrict the nature of these errors.

Dipole: Quadrupole:

The practical pole in 2D


Lines of symmetry:Dipole: Quad

Pole orientation y = 0; x = 0; y = 0determines whether poleis normal or skew.

Additional symmetry x = 0; y = ± ximposed by pole edges.

The additional constraints imposed by the symmetrical pole edges limits the values of n that have non zero coefficients

Possible symmetries.


+φ

-φ

Type Symmetry ConstraintPole orientation φ(θ) = -φ(-θ) all Jn = 0;

Pole edges φ(θ) = φ(π -θ) Kn non-zero only for:n = 1, 3, 5, etc;

So, for a fully symmetric dipole, only 6, 10, 14 etc pole errors can be present.

+φ +φ

Dipole symmetries


Type Symmetry Constraint

Pole orientation φ(θ) = -φ( -θ) All Jn = 0;

φ(θ) = -φ(π -θ) Kn = 0 all odd n;

Pole edges φ(θ) = φ(π/2 -θ) Kn non-zero only for:n = 2, 6, 10, etc;

So, a fully symmetric quadrupole, only 12, 20, 28 etc pole errors can be present.

Quadrupole symmetries


Type Symmetry Constraint

Pole orientation φ(θ) = -φ( -θ) All Jn = 0;φ(θ) = -φ(2π/3 - θ) Kn = 0 for all n φ(θ) = -φ(4π/3 - θ) not multiples of 3;

Pole edges φ(θ) = φ(π/3 - θ) Kn non-zero only for: n =

3, 9, 15, etc. So, a fully symmetric sextupole, only 18, 30, 42 etc pole errors can be present.

Sextupole symmetries.


Summary of ‘allowed harmonics’ in fully symmetric magnets with no dimensional errors:

Fundamental geometry

‘Allowed’ harmonics

Dipole, n = 1 n = 3, 5, 7, ......( 6 pole, 10 pole, etc.)

Quadrupole, n = 2 n = 6, 10, 14, ....(12 pole, 20 pole, etc.)

Sextupole, n = 3 n = 9, 15, 21, ...(18 pole, 30 pole, etc.)

Octupole, n = 4 n = 12, 20, 28, ....(24 pole, 40 pole, etc.)

Summary: allowed harmonics.


Now for j ≠ 0 ∇ ∧ H = j ;

To expand, use Stoke’s Theorum:for any vector V and a closed curve s :

∫V.ds =∫∫ curl V.dS

Apply this to: curl H = j ;

dS

dsV

then in a magnetic circuit:

∫ H.ds = N I;

N I (Ampere-turns) is total current cutting S

Introduction of currents


µ >>

g

λ

1

NI/2

NI/2

B is approx constant round the loop made up of λ and g, (but see below);

But in iron, µ>>1,and Hiron = Hair /µ ;So

Bair = µ0 NI / (g + λ/µ);

g, and λ/µ are the 'reluctance' of the gap and iron.

Approximation ignoring iron reluctance (λ/µ << g ):

NI = B g /µ0

Excitation current in a dipole


For quadrupoles and sextupoles, the required excitation can be calculated by considering fields and gap at large x. For example: Quadrupole:

y

x B

Pole equation: xy = R2 /2On x axes BY = gx;where g is gradient (T/m).

At large x (to give vertical lines of B):

N I = (gx) ( R2 /2x)/µ0ie

N I = g R2 /2 µ0 (per pole).

The same method for a Sextupole,

( coefficient gS,), gives:

N I = gS R3/3 µ0 (per pole)

Excitation current in quad & sextupole


In air (remote currents! ), B = µ0 H B = - ∇φ

Integrating over a limited path(not circular) in air: N I = (φ1 – φ2)/µoφ1, φ2 are the scalar potentials at two points in air.Define φ = 0 at magnet centre;then potential at the pole is:

µo NI

Apply the general equations for magneticfield harmonic order n for non-skewmagnets (all Jn = 0) giving:

N I = (1/n) (1/µ0) Br/R (n-1) R nWhere:

NI is excitation per pole;R is the inscribed radius (or half gap in a dipole);term in brackets is magnet strength in T/m (n-1).

y

φ = 0

φ = µ0 NI

General solution-magnets of order n


Standard design is rectangular copper (or aluminium) conductor, with cooling 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.

Current density:j = NI/Acopper Optimum j determined from economic criteria.

Coil geometry


Advantages of low j:• lower power loss – power bill is decreased;• lower power loss – power converter size is decreased;• less heat dissipated into magnet tunnel.

Advantages of high j:• smaller coils;• lower capital cost;• smaller magnets.

Chosen value of j is anoptimisation of magnet capital against power costs.

0.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0

Current density j

Life

time

cost

running

capital

total

Current density - optimisation


The 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) Small, neat terminals. Large, bulky terminals

Thin interconnections-hence low Thick, expensive connections.cost and flexible.

More insulation layers in coil, High percentage of copper inhence larger coil volume and coil volume. More efficient useincreased assembly costs. of space available

High voltage power supply High current power supply.-safety problems. -greater losses.

Number of turns per coil-N


From the Diamond 3 GeV synchrotron source:Dipole:

N (per magnet): 40;I max 1500 A;Volts (circuit): 500 V.

Quadrupole:N (per pole) 54;I max 200 A;Volts (per magnet): 25 V.

Sextupole:N (per pole) 48;I max 100 A;Volts (per magnet) 25 V.

Examples-turns & current


Dipoles can be ‘C core’ ‘H core’ or ‘Window frame’''C' Core:Advantages:

Easy access;Classic design;

Disadvantages:Pole shims needed;

Asymmetric (small);Less rigid; Shim

The ‘shim’ is a small, additional piece of ferro-magnetic material added on each side of the two poles – it compensates for the finite cut-off of the pole, and is optimised to reduce the 6, 10, 14...... pole error harmonics.

Magnet geometry


Cross section of the Diamond storage ring dipole.

Typical ‘C’ cored Dipole


‘H core’:Advantages:

Symmetric;More rigid;

Disadvantages:Still needs shims;Access problems.

''Window Frame'Advantages:

High quality field;No pole shim;Symmetric & rigid;

Disadvantages:Major access problems;Insulation thickness

H core and window-frame magnets


Providing the conductor is continuous to the steel ‘window frame’ surfaces (impossible because coil must be electrically insulated), and the steel has infinite µ, this magnet generates perfect dipole field.

Providing current density J is uniform in conductor:• H is uniform and vertical up outer face of conductor;• H is uniform, vertical and with same value in the middle of the gap;• → perfect dipole field.

J

H

Window frame dipole


Insulation added to coil:

B increases close to coil insulation surface

B decrease close to coil insulation surface

best compromise

Practical window frame dipole.


‘Diamond’ storage ring quadrupole.

The yoke support pieces in the horizontal plane need to provide space for beam-lines and are not ferro-magnetic.

Error harmonics include n = 4 (octupole) a finite permeability error.

Open-sided Quadrupole


To compensate for the non-infinite pole, shims are added at the pole edges. The area and shape of the shims determine the amplitude of error harmonics which will be present.

A

A

Dipole: Quadrupole:

The designer optimises the pole by ‘predicting’ the field resulting from a given pole geometry and then adjusting it to give the required quality.

When high fields are present,chamfer angles must be small, and tapering of poles may be necessary

Typical pole designs


A first assessment can be made by just examining By(x) within the required ‘good field’ region.Note that the expansion of By(x) y = 0 is a Taylor series:

By(x) = ∑n =1

∞bn x (n-1)

= b1 + b2x + b3x2 + ………dipole quad sextupole

Also note:∂ By(x) /∂ x = b2 + 2 b3x + ……..

So quad gradient g ≡ b2 = ∂ By(x) /∂ x in a quadBut sext. gradient gs ≡ b3 = 2 ∂2 By(x) /∂ x2 in a sext.So coefficients are not equal to differentials for n = 3 etc.

Assessing pole design


A simple judgement of field quality is given by plotting:

•Dipole: By (x) - By (0)/BY (0) (∆B(x)/B(0))•Quad: dBy (x)/dx (∆g(x)/g(0))•6poles: d2By(x)/dx2 (∆g2(x)/g2(0))

‘Typical’ acceptable variation inside ‘good field’ region:

∆B(x)/B(0) ≤ 0.01%∆g(x)/g(0) ≤ 0.1%∆g2(x)/g2(0) ≤ 1.0%

Is it ‘fit for purpose’?


Computer codes are now used; eg the Vector Fields codes -‘OPERA 2D’ and ‘OPERA 3D’.These have:

• finite elements with variable triangular mesh;• multiple iterations to simulate steel non-linearity;• extensive pre and post processors;• compatibility with many platforms and P.C. o.s.

Technique is iterative:• calculate flux generated by a defined geometry;• adjust the geometry until required distribution is achieved.

Design computer codes.


Pre-processor:The model is set-up in 2D using a GUI (graphics user’s interface) to define ‘regions’:

• steel regions;• coils (including current density);• a ‘background’ region which defines the physical

extent of the model;• the symmetry constraints on the boundaries;• the permeability for the steel (or use the pre-

programmed curve);• mesh is generated and data saved.

Design Procedures – OPERA 2D


Model of Diamond storage ring dipole


With mesh added


Pole profile, showing shim and Rogowski side roll-off for Diamond 1.4 T dipole.:

Close-up of pole region.


Diamond s.r. dipole: ∆B/B = By(x)- B(0,0)/B(0,0); typically ± 1:104 within the ‘good field region’ of -12mm ≤ x ≤ +12 mm..

2 D Dipole field homogeneity on x axis


Transverse (x,y) plane in Diamond s.r. dipole;

contours are ±0.01%

required good field region:

2 D Dipole field homogeneity in gap


OPERA 3D model of Diamond dipole.


Diamond dipole poles


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

Simplified end geometries - quadrupole


It is not usually necessary to chamfer sextupole ends (in a d.c. magnet). Diamond sextupole end:

Sextupole ends


SECTION 2Injection and extraction magnets.


The Injection/Extraction problem.Single turn injection/extraction:a magnetic element inflects beam into the ring and turns-off before the beam completes the first turn (extraction is the reverse).

Multi-turn injection/extraction:the system must inflect the beam intothe ring with an existing beam circulatingwithout producing excessive disturbanceor loss to the circulating beam.

Accumulation in a storage ring:A special case of multi-turn injection - continues over many turns (with the aim of minimal disturbance to the stored beam).

straight section

injected beam

magnetic element


Single turn – simple solution

A ‘kicker magnet’ with fast turn-off (injection) or turn-on (extraction) can be used for single turn injection.

injection – fast fall extraction – fast rise

Problems:i) rise or fall time will always be non-zero → loss of beam;ii) single turn inject does not allow the accumulation of high current;iii) in small accelerators revolution times can be << 1 µs.iv) magnets are inductive → fast rise (fall) means (very) high voltage.

B

t


Multi-turn injection 1 - general

Beam can be injected by phase-space manipulation:Inject into an unoccupied outer region of phase space with non-integer tune which ensures many turns before the injected beam re-occupies the same region (electrons and protons):

eg – Horizontal phase space at Q = ¼ integer:x

x’

septum

turn 1 – first injection turn 2 turn 3turn 4 – last injection

0 field deflect. field

Then the beam has to be moved back from the septum magnet!


Multi-turn injection – 2

Lepton storage rings: Inject into outer region of phase space – damping (slow?) coalesces beam into the central region before re-injecting.

dynamic aperture

injected beam next injection after 1 damping timestored beam

Protons:Inject negative ions through a bending magnet and then ‘strip’ to produce a positive ion after injection (eg H- to p).


Multi-turn extraction solution‘Shave’ particles from edge of beam into an extraction channel whilst the beam is moved across the aperture:

beam movement

extraction channel

Points:•some beam loss on the septum cannot be prevented;•efficiency can be improved by ‘blowing up’ on 1/3rd or 1/4th integer resonance.

septum


Magnet requirements

Magnets required for injection and extraction systems.i) Kicker magnets:•pulsed waveform;•rapid rise or fall times (usually << 1 µs);•flat-top for uniform beam deflection.

ii) Septum magnets:•pulsed or d.c. waveform;•spatial separation into two regions;•one region of high field (for injection deflection);•one region of very low (ideally 0) field for existing beam;•septum to be as thin as possible to limit beam loss.

Septum magnet schematic


Kicker Magnet & Power SuppliesBecause of the demanding performance required from these systems, the magnet and power supply must be strongly integrated and designed as a single unit.

Two alternative approaches to powering these magnets:

1) Distributed circuit: magnet and power supply made up of delay line circuits.

2) Lumped circuits: magnet is designed as a pure inductance; power supply can be use delay line or a capacitor to feed the high pulse current.


High Frequency Kicker MagnetsKicker Magnets:•used for rapid deflection of beam for injection or extraction;•usually located inside the vacuum chamber;•rise/fall times << 1µs.•yoke assembled from high frequency ferrite;•single turn coil;•pulse current ∼ 104A;•pulse voltages of many kV.

beam

Conductors

Ferri te Core

Typical geometry:


Kickers - Distributed SystemStandard (CERN) delay line magnet and power supply:

dc

L, C L, C

Z 0

Power Supply Thyratron Magnet Resistor

The power supply, interconnecting cables and terminating resistor are matched to the surge impedance of the delay line magnet:


Distributed System -mode of operation•the first delay line is charged to bythe d.c. supply to a voltage : V;

•the thyratron triggers, a voltages wave: V/2 propagates into magnet;

•this gives a current wave of V/( 2 Z )propagating into the magnet;

•the circuit is terminated by pure resistor Z,to prevent reflection.


Kickers – Lumped Systems.•The magnet is (mainly) inductive - no added distributed capacitance;•the magnet must be very close to the supply (minimises

inductance).

Ldc

R

I = (V/R) (1 – exp (- R t /L)


Improvement on above

Ldc

R

C

The extra capacitor C improves the pulse substantially.


Resulting WaveformExample calculated for the following parameters:

0

0.2

0.4

0.6

0.8

1

1.2

0.00E+00 2.00E-07 4.00E-07 6.00E-07

Time µs

mag inductance L = 1 µH;rise time t = 0.2 µs;resistor R = 10 Ω; trim capacitor C = 4,000 pF.

The impedance in the lumped circuit is twice that in the distributed! The voltage to produce a given peak current is the same in both cases.

Performance: at t = 0.1 µs, current amplitude = 0.777 of peak;at t = 0.2 µs, current amplitude = 1.01 of peak.The maximum ‘overswing’ is 2.5%.

This system is much simpler and cheaper than the distributed system.


EMMA kicker magnet – ferrite cored lumped system.


EMMA Injection Kicker Magnet Waveform


Septum Magnets – ‘classic’ design.Often (not always) located inside the vacuum and used to deflect part of the beam for injection or extraction:

Yoke.

Single turn coil

Beam

The thin 'septum' coil on the front face gives:•high field within the gap,•low field externally;

Problems:•The thickness of the septum must be minimised to limit beam loss;•the front septum has very high current density and major heating problems


Septum Magnet – eddy current design.•uses a pulsed current through a backleg coil (usually a poor design feature) to generate the field;•the front eddy current shield must be, at the septum, a number of skin depths thick; elsewhere at least ten skin depths;•high eddy currents are induced in the front screen; but this is at earth potential and bonded to the base plate – heat is conducted out to the base plate;•field outside the septum are usually ~ 1% of field in the gap.

- +

Single or multi turn

Eddy currentshield


Comparison of the two types.Classical: Eddy current:

Excitation d.c or low frequency pulse; pulse at > 10 kHz;

Coil single turn including single or multi-turn onfront septum; backleg, room for

large cross section;

Cooling complex-water spirals heat generated in in thermal contact with shield is conducted to septum; base plate;

Yoke conventional steel laminations high frequency material (ferrite or very thin steel lams).


ExampleSkin depth in material: resistivity ρ;

permeability µ;at frequency ω

is given by: d = √(2 ρ/ωµµ0 )

Example: EMMA injection and extraction eddy current septa:

Screen thickness (at beam height): 1 mm;" " (elsewhere) – up to 10 mm;

Excitation 25 µs, half sinewave;

Skin depth in copper at 20 kHz 0.45 mm


Location of EMMA septum magnets


Design of the EMMA septum magnetInner steel yoke is assembled from 0.1mm thick silicon steel laminations, insulated with 0.2 µm coatings on each side.


SECTION 3FFAG Pole Design.


Are FFAG magnets ‘complex’?

Yes – often complex – but no more difficult than ‘conventional’ synchrotron magnets!

Consideration in pole design:i) what is the lattice specification – either variation on axis of By vs x, or , better still, the harmonic components of the By?ii) then (for the magnet designer) – what basic type of magnet is it (dipole, quadrupole, sextupole, etc).iii) finally, what procedure should be used to establish the pole profile?

See next slide.


Detailed ProcedureTo determine the ‘perfect’ pole:

i) establish coefficients of Taylor series (harmonic amplitudes) up to 3rd or 4th order (or higher) to ‘fit’ the specified By(x) curve;ii) develop the equation for scalar potential φ (x,y) by summing the scalar potentials for all harmonics (equations for φ (x,y) shown in slide 10,11 and 12);iii) generate a table of (x,y) values corresponding to a fixed value of φ (an iso-potential line) – this is one of the (infinite number) of‘perfect’ pole shapes (not taking account of pole edge or end effects).

Procedure illustrated-the determination of poles for ‘complex’ magnets in a FFAG – ‘Pumplet'


FFAG ‘Pumplet’ (*)

(*) as specified by Grahame Rees.


By(T) and x (T) specifications


Magnet bd – By curve fitting

Fit:

Series: b0 + b1x + b2 x2 + b3 x3; Coefficients: b0 = 0.04693; b1 = 2.9562 E-4; b2 = -2.9366 E-6; b3 = -1.6920 E-7;RMS fitting error: 3.67 E-5; 8:104 of mean (need to be better for actual project).

0.040

0.042

0.044

0.046

0.048

0.050

0.052

-20.0 -10.0 0.0 10.0 20.0

By (T)

x (mm)

Forth order fit to By vs x

Defined data

Fitted data


bd - lines of iso scalar potentialPairs of (x,y) and (x,-y) to give φ = 0.300000 T mm; this gives the poles shapes.


bd poles and vac vessel.

What By distribution does this give? Model using OPERA 2D.

-10.0

-8.0

-6.0

-4.0

-2.0

0.0

2.0

4.0

6.0

8.0

10.0

-40.0 -20.0 0.0 20.0 40.0

Pole - y vs x in mm


OPERA 2D model


By vs x at y = 0.

How does this compare with the specified data?


bd -comparison of OPERA 2D with defined By

RMS error (fitting + determining potentials + OPERA FEA) : 3.75 E-5

0.040

0.044

0.048

0.052

-20.0 -15.0 -10.0 -5.0 0.0 5.0 10.0 15.0 20.0

Comparison of By defined and OPERA 2D prediction

Defined data

OPERA 2D prediction


Magnet BF curve fit

0.000

0.010

0.020

0.030

0.040

0.050

-20.0 -15.0 -10.0 -5.0 0.0 5.0 10.0 15.0 20.0

BF fourth order curve fit

Defined data

Fitted data

What sort of magnet is this? dipole/quadrupole/sextupole? See next slide.


Magnet BF curve fit – extended.Extrapolated down to x = – 55 mm.

It’s a sextupole (with dipole, quadrupole and octupole components)magnetic centre at ∼ - 40 mm !!!

-0.010

0.000

0.010

0.020

0.030

0.040

0.050

-60.0 -50.0 -40.0 -30.0 -20.0 -10.0 0.0 10.0 20.0 30.0


Magnet BF pole arrangement with vac vessel.

0.0

10.0

20.0

30.0

40.0

50.0

60.0

70.0

80.0

-80.0 -70.0 -60.0 -50.0 -40.0 -30.0 -20.0 -10.0 0.0 10.0 20.0 30.0 40.0

Poles have: φ = 0.35 T mm;

Note all that wasted space at -70 < x < -25? See next slide but one!


OPERA Model of magnet BF


Comparison between OPERA BY prediction and defined data; top pole at full potential

0

0.01

0.02

0.03

0.04

0.05

0.06

-25 -20 -15 -10 -5 0 5 10 15 20 25

BY

(T)

x (mm)

OPERA PredictionDefined data


Solution to wasted space problem.Side poles have: φ = 0.35 T mm;BUT:Central poles have: φ = 0.01 T mm;

Central poles:i) are much closer tomedian line; ii) now require only 1/35 of the coilexcitation current.

STFC have now applied for a patent for this arrangement.

-40.0

-30.0

-20.0

-10.0

0.0

10.0

20.0

30.0

40.0

-60.0 -50.0 -40.0 -30.0 -20.0 -10.0 0.0 10.0 20.0 30.0 40.0

φ = 0.35φ = - 0.01


OPERA model of BF;(low potential central poles).


Comparison of By:between OPERA & defined data; reduced φ on top pole.

0.00E+00

1.00E-02

2.00E-02

3.00E-02

4.00E-02

5.00E-02

6.00E-02

-25.00 -20.00 -15.00 -10.00 -5.00 0.00 5.00 10.00 15.00 20.00 25.00

BY

(T)

x (mm)

OPERA PredictionDefined data


Finis

Thank you for listening;

Any questions?

Magnets for FFAGs - Cockcroft Web...Magnets for FFAGs; Neil Marks FFAG School 2011 Present a short overview of electro - magnetic technology as used in particle accelerators, considering

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