A.C. Magnet Systems - The Cockcroft Institute · AC Mags; Lecture to Cockcroft Institute, ... 1. Present practical ... R ac . Neil Marks; ASTeC, U. of Liverpool, CI.

Neil Marks; ASTeC, U. of Liverpool, CI. AC Mags; Lecture to Cockcroft Institute, November/December 2009

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

1. Present practical details of how a.c. lattice magnets differ

from d.c. magnets.

2. 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:

1. 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.

2. 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.


Equivalent circuit of a.c. magnet

Lm Rdc

Cleakage

Im

Rac


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.

A.C. Magnet Design


Eddy Currents in a Conductor I

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 x l 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 x2 l 2 B2 x /

Integrate w.r.t. x between 0 and a to obtain peak instantaneous power in block = (2/3) a3 l 2 B2 /

Cross section area A = 2 a

Average power is ½ of above.

Power loss/unit length = 2 B2 A a2/(6 ) W/m;

a 10x10 mm2 Cu conductor in a 1T 50Hz sin. field, loss = 1.7 kW/m

x

l

-a -x 0 x a

B sin t

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 x l 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 x2 l 2 B2 (a2-x2) 1/2 x /

Integrate w.r.t. x between 0 and a to obtain peak instantaneous power in block

= (p/4) a4 l 2 B2 /

Cross section area A = pa

Average power is ½ of above.

Power loss/unit length = 2 B2 A a2/(8 ) W/m;

x

a

x


Eddy Currents in a cylindrical vacuum

vessel

R

q

q

B sin t

Geometry of cylindrical vacuum vessel,

:/2 0,between w.r.t.above

of integral iscylinder in current eddy total

;/ t) (cos B ) (cos R 2 I

:loops bottom and in top currentseddy

) R ( / 2

:loop oflength unit of resistance

t); cos )(B cos (R 2 V

:only) (top loop oflength unit round voltage

t); cos )(B cos (R 2 t /

);t sin B )( cos R ( 2

2e

pq

qq

q

q

q

q

total flux cutting circuit at angle q:

Ie = - 2 R2 B (cos t) /

It can be seen that the eddy currents vary as the square of the cylindrical radius R and

directly with the wall thickness t.

Wall

conductivity


Perturbation field generated by eddy

currents

Magnet geometry around vessel

radius R.

g

m =

R

x 0

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 R2 / g;

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

qe = arctan (- 2 m0 R2 / g )


Distributions of perturbing fields

Cylindrical vessel (radius R):

Be(X)

Rectangular vessel (semi axies a, b):

Be(X)

Elliptical vessel (semi axies a, b):

Be(X)

m

2/12222

1/2221/2221-

22

2 0

a 1) - b/ (a X b

) X - a ( )b a(tan

)b (a

b a

g

t cos B 2

m ab

2

)Xa(

g

t) cos (B 2

220

. X - R g

t Bcos oR2 22

m

m ............

R 128

X 5

R 16

X

R 8

X

R 2

X 1

g

t Bcos R o2

8

8

6

6

4

4

2

22

variation with horizontal position 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):

0.0001

0.001

0.01

0.1

1

1 10 100 1000

Frequency (Hz)

Pertu

rb

ati

on

/im

po

sed

fie

ld .

thickness= 0.25 mm thickness = 0.5 mm thickness = 1 mm

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):

0.01

0.1

1

10

100

1 10 100 1000

Frequency (Hz)

Ph

ase

ch

an

ge i

n f

ield

ap

pli

ed

to

bea

m ;

(d

eg

rees)

thickness = 0.25 mm thickness = 0.5 mm thickness = 1 mm

Calculation

invalid in this

region.


0

1.5

0 50 100 150 200

frequency (Hz)

‘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!) time (ms)

0 ~10


c d

b a b a

d c

d c

b a

c d

a b

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 2 ; B2, (width)2 and cross-

section area.

NINA : E = 5.6 GeV;

= 53 Hz;

Bpeak = 0.9 T.

ISIS: E = 800 MeV;

= 50 Hz;

Bpeak ≈ 0.2 T.

Transposed, stranded conductor.

Cooling tube.

}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?

0

3

0 10


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.

B

Rolli ng

direction .

Difficult (impossible?) to make

each limb out of separate strips

of steel.


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.

F

n turns,

current I


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

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 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.

B

t


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:

x

x’

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

septum

0 field deflect. field


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):

dynamic aperture

injected beam next injection after 1 damping time stored beam

c) inject negative ions through a bending magnet and then ‘strip’ to produce a p after

injection (H- to p only).


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 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.

beam

Conductors

Ferrite Core

Typical geometry:


Kickers - Distributed System

Standard (CERN) delay line magnet and power supply:

dc

L, C L, C

Z 0

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.


EEV Thyratron CX1925

EEV

HV = 80kV

Peak current 15 kA

repetition 2 kHz

Life time ~3 year


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

Ldc

R = Z

Z0

0

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.

kicker magnet inductance 0.76 -0.8 uH

magnet current 2 - 2.5 kA

blumlein PFN Voltage 35 kV

pulse current rise time 200nS

current pulse width 750 nS

pulse repetition 60 Hz


Extraction systems layout


Kicker p.f.n simulation model


Simulated current waveform


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)

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


Improvement on above

Ldc

R

C

The extra capacitor C improves the pulse substantially.


Resulting Waveform

Example calculated for the following parameters:

Pulse Waveform

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 ms

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.


An 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


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.

- +

Single or multi turncoil

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 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 ;

permeability m;

at frequency

is given by: d = (2 /µµ0 )

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 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.

A.C. Magnet Systems - The Cockcroft Institute · AC Mags; Lecture to Cockcroft Institute, ... 1. Present practical ... R ac . Neil Marks; ASTeC, U. of Liverpool, CI.

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