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A.C. Magnet Systems

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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

- 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.

- 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.

- 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.

Rac

Rdc

Lm

Im

Cleakage

- 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.

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

- 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

dq

R

q

B sin w t

t

total 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.

m =

g

R

0

x

Magnet geometry around vessel radius R.

- 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 )

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)

- 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.

- 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.

time (ms)

- 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

d c

c d

a

b

b a

c d

a b

d c

b a

- 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.

- 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.

- 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 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.

- 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?

- 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.

- '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.

- 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.

- 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)

- Property: DK-70:CK-27: 27 M 3: XC06 :
- Type Non- Non- Grain- Non-
- orientedoriented oriented oriented
- Silicon content LowHigh - Very low
- Lam thickness 0.65 mm0.35 mm 0.27 mm Solid
- a.c. loss (50 Hz):
- at 1.5 T peak 6.9 W/kg2.25 W/kg 0.79 W/kg Not suitable
- Permeability:
- at B=1.5 T 1,680990 > 10,000 >1,000
- at B=1.8 T 184122 3,100 >160

- 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.

F

n turns,

current I

- 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.

Two coils, inductance L, with no mutual coupling:

Inductance in series = 2 L:

Inductance in parallel = L/2:

ie, just like resistors.

- 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.

- 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

- 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’

x

- 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

dynamic aperture

stored beam

injected beam

next injection after 1 damping time

- 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).

extraction channel

beam movement

- ‘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.

- 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

- 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.

- 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:

- 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:

- 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 resistorZ,
- to prevent reflection.

EEV

HV = 80kV

Peak current 15 kA

repetition 2 kHz

Life time ~3 year

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

ie it is a pseudo-distributed line

- 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.

- 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!

- 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!

I = (V/Z) (1 – exp (-Z t /L)

- 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.

- 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..

C

The extra capacitor C improves the pulse substantially.

- Example calculated for the following parameters:

mag inductance L = 1 mH;

rise timet = 0.2 ms;

resistorR = 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.

- 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

- 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 %

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

- 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.

- Classical:Eddy current:
- Excitationd.c or low frequency pulse;pulse at > 10 kHz;
- Coilsingle turn includingsingle or multi-turn on
- front septum;backleg, room for
- large cross section;
- Coolingcomplex-water spiralsheat generated in in thermal contact with shield is conducted to septum;base plate;
- Yokeconventional steelhigh frequency material (ferrite or thin steel lams).

- 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 to10 mm;
- Excitation25 µs,
- half sinewave;
- Skin depth in copper at 20 kHz0.45 mm

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

- 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.’

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