A.C. Magnet Systems

1. 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

2. 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.

3. 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.

4. 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.

5. Rac Rdc Lm Im Cleakage Equivalent circuit of a.c. magnet

6. 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.

7. 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

8. 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

9. dq R q B sin w t t Eddy Currents in a cylindrical vacuum vessel 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.

10. m =  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 )

11. 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)

12. 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.

13. 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.

14. time (ms) ‘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

15. d c 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.

16. 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.

17. 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.

18. 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.

19. 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?

20. 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.

21. 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.

22. 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.

23. 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)

24. 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

25. 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.

26. F 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.

27. 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.

28. 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.

29. 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

30. 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.

31. x’ 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

32. dynamic aperture 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).

33. extraction channel 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.

34. 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

35. 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.

36. 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:

37. 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:

38. 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.

39. EEV Thyratron CX1925 EEV HV = 80kV Peak current 15 kA repetition 2 kHz Life time ~3 year

40. 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

41. 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.

42. 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!

43. 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!

44. Distributed power supply– lumped magnet I = (V/Z) (1 – exp (-Z t /L)

45. 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.

46. Extraction systems layout

47. Kicker p.f.n simulation model

48. Simulated current waveform

49. 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..

50. C Improvement on above The extra capacitor C improves the pulse substantially.