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### A.C. Magnet Systems

Neil Marks,

CI, ASTeC, U. of Liverpool,

The Cockcroft Institute,

Daresbury,

Warrington WA4 4AD,

U.K.

Tel: (44) (0)1925 603191

Fax: (44) (0)1925 603192

Philosophy

- Present practical details of how a.c. lattice magnets differ from d.c. magnets.
- Present details of the typical qualities of steel used in lattice magnets.
- 3. Give a qualitative overview of injection and extraction techniques as used in circular machines.
- 4. Present the standard designs for kicker and septum magnets and their associated power supplies.

Contents

- a) Variations in design and construction for a.c. magnets;
- Effects of eddy currents;
- ‘Low frequency’ a.c. magnets
- Coil transposition; eddy loss; hysteresis loss;
- Properties and choice of steel;
- Inductance in an a.c. magnet;
- b) Methods of injecting and extracting beam;
- Single turn injection/extraction;
- Multi-turn injection/extraction;
- Magnet requirements;
- c) ‘Fast’ magnets;
- Kicker magnets-lumped and distributed power supplies;
- Septum magnets-active and passive septa;
- Some modern examples.

Differences to d.c. magnets

- A.c magnets differ in two main respects to d.c. magnets:
- In addition to d.c ohmic loss in the coils, there will be ‘ac’ losses (eddy and hysteresis); design goals are to correctly calculate and minimise a.c. losses.
- Eddy currents will generate perturbing fields that will affect the beam.
- 3. Excitation voltage now includes an inductive (reactive) component; this may be small, major or dominant (depending on frequency); this must be accurately assessed.

A.C. Magnet Design

- Additional Maxwell equation for magneto-dynamics:
- curl E = -dB/dt.
- Applying Stoke’s theorem around any closed path s enclosing area A:
- curl E.dA = E.ds = V loop
- where Vloop is voltage around path s;
- - (dB /dt).dA = - dF/dt;
- Where F is total flux cutting A;
- So: Vloop = -dF/dt
- Thus, eddy currents are induced in any conducting material in the alternating field. This results in increased loss and modification to the field strength and quality.

Eddy Currents in a Conductor I

dx

- Rectangular cross section
- resistivity ,
- breadth 2 a ,
- thickness ,
- length l ,
- cut normally by field B sin t.
- Consider a strip at +x, width x , returning at –x (l >>x).
- Peak volts in circuit = 2 xl B
- Resistance of circuit = 2 l/( x )
- Peak current in circuit = x B x /
- Integrate this to give total Amp-turns in block.
- Peak instantaneous power in strip = 2 x2lw2 B2 x /
- Integrate w.r.t. x between 0 and a to obtain peak instantaneous power in block = (2/3) a3lw2 B2 /
- Cross section area A = 2 a
- Average power is ½ of above.
- Power loss/unit length = w2 B2 A a2/(6 r ) W/m;
- a 10x10 mm2 Cu conductor in a 1T 50Hz sin. field, loss = 1.7 kW/m

B sin wt

t

l

-a -x 0 x a

Cross section A

Eddy Currents in a Conductor II

- Circular cross section:
- resistivity ,
- radius a ,
- length l ,
- cut normally by field B sin t.
- Consider a strip at +x, width x , returning at –x (l >>x).
- Peak volts in circuit = 2 xl B
- Resistance of circuit = 2 l/{ 2 (a2-x2)1/2 x }
- Peak current in circuit = 2x B (a2-x2) 1/2 x /
- Integrate this to give total Amp-turns in block.
- Peak instantaneous power in strip = 4 x2lw2 B2 (a2-x2) 1/2 x /
- Integrate w.r.t. x between 0 and a to obtain peak instantaneous power in block = (p/4) a4lw2 B2 /
- Cross section area A = pa
- Average power is ½ of above.
- Power loss/unit length = w2 B2 A a2/(8 r ) W/m;

dx

a

x

R

q

B sin w t

t

Eddy Currents in a cylindrical vacuum vesseltotal flux cutting circuit at angle q:

Wall conductivity r

Ie = - 2 wt R2 B (cos wt) / r

Geometry of cylindrical vacuum vessel,

It can be seen that the eddy currents vary as the square of the cylindrical radius R and directly with the wall thickness t.

g

R

0

x

Magnet geometry around vessel radius R.

Perturbation field generated by eddy currents- Note:
- that if the vacuum vessel is between the poles of a a ferro-magnetic yoke, the eddy currents will couple to that yoke; the yoke geometry therefore determines the perturbing fields;
- this analysis assumes that the perturbing field is small compared to the imposed field.

Using: Be= m0 Ie/g;

Amplitude ratio between perturbing and imposed fields at X = 0 is:

Be(0)/B = - 2 m0 wt R2 / r g;

Phase of perturbing field w.r.t. imposed field is:

qe = arctan (- 2 m0 wt R2 / r g )

Distributions of perturbing fields

variation with horizontal position X

- Cylindrical vessel (radius R):
- Be(X)
- Rectangular vessel (semi axies a, b):
- Be(X)
- Elliptical vessel (semi axies a, b):
- Be(X)

Stainless steel vessels – amplitude.

- Example: Ratio of amplitude of perturbing eddy current dipole field to amplitude of imposed field as a function of frequency for three values of s.s. vessel wall thickness (R = g/2):

Calculation invalid in this region.

Stainless steel vessels – phase.

- Phase change (lag) of dipole field applied to beam as a function of frequency for three values of vessel wall thickness (R = g/2):

Calculation invalid in this region.

‘Low frequency’ a.c. magnets

- We shall deal separately with ‘low frequency’ and ‘fast’ magnets:
- ‘low frequency’
- – d.c. to c 100 Hz:
- ‘fast’ magnets
- – pulsed magnets with rise times from 10s ms to < < 1 ms.
- (But these are very slow compared to r.f. systems!)

0

~10

c d

a

b

b a

c d

a b

d c

b a

Coils for up to c 100 Hz.- Coil designed to avoid excessive eddy currents. Solutions:
- a) Small cross section copper per turn; this give large number of turns - high alternating voltage unless multiple conductors are connected in parallel; they must then be ‘transposed’:
- b) ‘Stranded’ conductor (standard solution in electrical engineering) with strands separately insulated and transposed (but problems locating the cooling tube!):
- Flux density at the coil is predicted by f.e.a. codes, so eddy loss in coils can be estimated during magnet design.

Two examples:

- Note that eddy loss varies as w2 ; B2, (width)2 and cross-section area.
- NINA :E = 5.6 GeV;
- w = 53 Hz;
- Bpeak = 0.9 T.
- ISIS: E = 800 MeV;
- w = 50 Hz;
- Bpeak ≈ 0.2 T.

}c 10mm x 10 mm solid conductor with cooling hole.

Steel Yoke Eddy Losses.

- At 10 Hz lamination thickness of 0.5mm to 1 mm can be used.
- At 50Hz, lamination thickness of 0.35mm to 0.65mm are standard.
- Laminations also allow steel to be ‘shuffled’ during magnet assembly, so each magnet contains a fraction of the total steel production; - used also for d.c. magnets.

To limit eddy losses, steel core are laminated, with a thin layer (~2 µm) of insulating material coated to one side of each lamination.

Steel hysteresis loss

Steel also has hysteresis loss caused by the finite area inside the B/H loop:

Loss is proportional to B.dH

integrated over the area

within the loop.

Steel loss data

- Manufacturers give figures for total loss (in W/kg) in their steels catalogues:
- for a sin waveform at a fixed peak field (Euro standard is at 1.5 T);
- and at fixed frequency (50 Hz in Europe, 60 Hz in USA);
- at different lamination thicknesses (0.35, 0.5, 0.65 & 1.0 mm typically)
- they do not give separate values for eddy and hysteresis loss.
- Accelerator magnets will have:
- different waveforms (unidirectional!);
- different d.c. bias values;
- different frequencies (0.2 Hz up to 50 Hz).
- How does the designer calculate steel loss?

Comparison between eddy and hysteresis loss in steel:

- Variation with:Eddy loss Hysteresis loss
- A.c. frequency: Square law Linear;
- A.c. amplitude: Square law Non-linear-depends on level;
- D.c. bias: No effect Increases non-linearly;
- Total volume of steel: Linear Linear;
- Lamination thickness: Square law No effect.

Choice of steel

- \'Electrical steel\' is either \'grain oriented\' or \'non-oriented‘:
- Grain oriented:
- strongly anisotropic,
- very high quality magnetic properties and very low a.c losses in the rolling direction;
- normal to rolling direction is much worse than non-oriented steel;
- stamping and machining causes loss of quality and the stamped laminations must be annealed before final assembly.

Choice of steel (cont).

- Non-oriented steel:
- some anisotropy (~5%);
- manufactured in many different grades, with different magnetic and loss figures;
- losses controlled by the percentage of silicon included in the mix;
- high silicon gives low losses (low coercivity), higher permeability at low flux density but poorer magnetic performance at high field;
- low (but not zero) silicon gives good performance at high B;
- silicon mechanically ‘stabilises’ the steel, prevents aging.

Solid steel

- Low carbon/high purity steels:
- usually used for solid d.c. magnets;
- good magnetic properties at high fields
- but hysteresis loss not as low as high silicon steel;
- accelerator magnets are seldom made from solid steel; (laminations preferred to allow shuffling and reduce eddy currents)

Comparisons

- Property: DK-70: CK-27: 27 M 3: XC06 :
- Type Non- Non- Grain- Non-
- oriented oriented oriented oriented
- Silicon content Low High - Very low
- Lam thickness 0.65 mm 0.35 mm 0.27 mm Solid
- a.c. loss (50 Hz):
- at 1.5 T peak 6.9 W/kg 2.25 W/kg 0.79 W/kg Not suitable
- Permeability:
- at B=1.5 T 1,680 990 > 10,000 >1,000
- at B=1.8 T 184 122 3,100 >160

The ‘problem’ with grain oriented steel

- In spite of the
- obvious advantage,
- grain oriented is
- seldom used in
- accelerator magnets
- because of the mechanical
- problem of keeping B
- in the direction of the grain.

Difficult (impossible?) to make each limb out of separate strips of steel.

n turns,

current I

Magnet Inductance- Definition:
- Inductance: L = n F /I
- Dipole Inductance.
- For an iron cored dipole:
- F = B A = µ0 n I A/(g +l/µ);
- Where: A is total area of flux (including gap fringe flux);
- l is path length in steel;
- g is total gap height
- So: Lm = µ0 n2 A/(g +l/µ);
- Note that the f.e.a. codes give values of vector potential to provide total flux/unit length.

Inductances in series and parallel.

Two coils, inductance L, with no mutual coupling:

Inductance in series = 2 L:

Inductance in parallel = L/2:

ie, just like resistors.

But

- Two coils, inductance L, on the same core (fully mutually coupled):
- Inductance of coils in series = 4 L

n is doubled, n2 is quadrupled.

Inductance of coils in parallel = L

same number of turns, cross section of conductor is doubled.

The Injection/Extraction problem.

- Single turn injection/extraction:
- a magnetic element inflects beam into the ring and turn-off before the beam completes the first turn (extraction is the reverse).
- Multi-turn injection/extraction:
- the system must inflect the beam into
- the ring with an existing beam circulating
- without producing excessive disturbance
- or 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

magnetic element

injected beam

Single turn – simple solution

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

B

t

injection – fast fall

extraction – fast rise

Problems:

i) rise or fall 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 ms.

iv) magnets are inductive fast rise (fall) means (very) high voltage.

x

Multi-turn injection solutions- Beam can be injected by phase-space manipulation:
- a) 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:

septum

0 field deflect. field

turn 2

turn 3

turn 1 – first injection

turn 4 – last injection

stored beam

injected beam

next injection after 1 damping time

Multi-turn injection solutions- b) Inject into outer region of phase space - damping coalesces beam into the central region before re-injecting (high energy leptons only):

c) inject negative ions through a bending magnet and then ‘strip’ to produce a p after injection (H- to p only).

beam movement

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

septum

- 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.

Magnet requirements

- Magnets required for injection and extraction systems.
- i) Kicker magnets:
- pulsed waveform;
- rapid rise or fall times (usually << 1 ms);
- 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

Fast Magnet & Power Supplies

- Because 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:
- Distributed circuit: magnet and power supply made up of delay line circuits.
- 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 Magnets

- Kicker 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.

Typical geometry:

Kickers - Distributed System

- Standard (CERN) delay line magnet and power supply:

Power Supply Thyratron Magnet Resistor

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

Distributed System -mode of operation

- the first delay line is charged to by
- the 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.

Magnet Physical assembly

- Magnet:
- Usually capacitance is introduced along the length of the magnet, which is split into many segments:

ie it is a pseudo-distributed line

Power supplies for distributed systems.

- Can be:
- a true ‘line’ (ie a long length of high voltage coaxial cable);
- or a multi-segment lumped line.
- These are referred to as ‘pulse forming networks’ (p.f.n.s) and are used extensively in ‘modulators’ for:
- linacs;
- radar installations.

Parameters

- The value of impedance Z (and therefore the added distributed capacitance) is determined by the required rise time of current:
- total magnet inductance = L;
- capacitance added = C;
- surge impedance Z0 = (L/C);
- transit time (t) in magnet = (LC);
- so Z0 = L/t;
- for a current pulse (I), V = 2 Z I ; = 2 I L / t .
- The voltage (V/2) is the same as that required for a linear rise across a pure inductance of the same value – the distributed capacitance has not slowed the pulse down!

Suitability of distributed system:

- Strengths:
- the most widely used system for high I and V applications;
- highly suitable if power supply is remote from the magnet;
- this system is capable of very high quality pulses;
- other circuits can approach this in performance but not improve on it;
- the volts do not reverse across the thyratron at the end of the pulse.
- Problems:
- the pulse voltage is only 1/2 of the line voltage;
- the volts are on the magnet throughout the pulse;
- the magnet is a complex piece of electrical & mechanical engineering;
- the terminating resistor must have a very low inductance - problem!

Distributed power supply– lumped magnet

I = (V/Z) (1 – exp (-Z t /L)

Example of such a distributed kicker system

- SNS facility (Brookhaven)– extraction kickers:
- 14 kicker pulse power
- supplies & magnets;
- operated at a 60 Hz
- repetition rate;
- kicks beam in 250 nS;
- 750nS pulse flat top.

Kickers – Lumped Systems.

- The magnet is (mainly) inductive - no added distributed capacitance;
- the magnetmust be very close to the supply (minimises inductance).

I = (V/R) (1 – exp (- R t /L)

i.e. the same waveform as distributed power supply, lumped magnet systems..

Resulting Waveform

- Example calculated for the following parameters:

mag inductance L = 1 mH;

rise time t = 0.2 ms;

resistor R = 10 W;

trim capacitor C = 4,000 pF.

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

Performance: at t = 0.1 ms, current amplitude = 0.777 of peak;

at t = 0.2 ms, current amplitude = 1.01 of peak.

The maximum ‘overswing’ is 2.5%.

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

Septum Magnets – ‘classic’ design.

- Often (not always) located inside the vacuum and used to deflect part of the beam for injection or extraction:

- 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

Multiple septa

- These engineering problems can be partially overcome by using multiple septa magnets (the septa can get thicker as the beams diverge).
- eg – KEK (3 GeV beam):
- Operation: DC
- Beam: H+
- Energy: 3.0 GeV
- Field strength: 0.41067 T (SEPEX-1)
- 0.75023 T (SEPEX-2)
- 0.87418 T (SEPEX-3)
- 1.00530 T (SEPEX-4)
- Effective length: 0.9 m
- Field flatness: +/- 0.1 %

‘Opposite bend’ septa magnets

- KEK also use ‘opposite bend’ septum magnets at 50 GeV:

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.

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 on
- front 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 high frequency material (ferrite or thin steel lams).

Example

- Skin depth in material: resistivity r;
- permeability m;
- at frequency w
- is given by: d = (2 r/wµµ0 )
- 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

Design of the EMMA septum magnet

Inner steel yoke is assembled from 0.1mm thick silicon steel laminations, insulated with 0.2 mm coatings on each side.

‘Out of Vacuum’ designs.

- Benefits in locating the magnet outside the vacuum.
- But a (metallic) vessel has to be inserted inside the magnet -the use of an eddy current design (probably) impossible.
- eg the upgrade to the APS septum (2002):
- ‘The designs of the six septum magnets required for the APS facility have evolved since operation began in 1996. Improvements .. have provided
- better injection/extraction performance and extended the machine reliability...’
- ‘Currently a new synchrotron extraction direct-drive septum with the core out of vacuum is being built to replace the existing, in-vacuum eddy-current-shielded magnet.’

‘New’ APS septum magnet.

Synchrotron extraction septum conductor assembly partially installed in the laminated core.

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