Conventional Magnets for Accelerators Lecture 2

Transcription

1 Conventional Magnets for Accelerators Lecture 2 Ben Shepherd Magneti cs and Radi ati on Sources Group ASTeC Daresbury Laboratory ben.shepherd@stfc.ac.uk Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

2 Contents lecture 2 The introduction of currents Coi l economic optimisation-capital/running costs Summary of the use of permanent magnets (PMs) Remnant fields and coercivity Behaviour and application of PMs The magnetic circuit Steel requirements: permeability and coercivity Backleg and coil geometry: 'C', 'H' and 'window frame' designs Classical solution to end and side geometries the Rogowsky rolloff Magnet design using FEA software FEA techniques and codes Opera 2D, Opera 3D Judgement of magnet suitability in design Magnet ends computation and design Some examples of magnet engi neeri ng Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

3 Current Affairs FIELDS DUE TO COILS (CONTINUED) Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

4 Coi l geometry Standard design is rectangular copper (or aluminium) conductor, with cooling water tube. Insulation is glass cloth and epoxy resi n. Amp-turns (NI) are determined, but total copper area (A copper ) and number of turns (N) are two degrees of freedom and need to be deci ded. Heat generated in the coil is a function of th e RMS current density: jj rrrrrr = NNII rrrrrr AA cccccccccccc Optimum j rms determined from economic criteria. Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

5 I rms depends on current waveform With an arbitrary waveform of period τ, the power generated in the coil is: WW = RR ττ ττ II 2 2 tt dddd = RRII rrrrrr 0 In a DC magnet the II rrrrrr = II DDDD For a pure AC sine wave II rrrrrr = 1 II 2 pppppppp For a discontinuous waveform the integration is over the whole of I a single period. peak A typical waveform for a booster synchrotron is a biased sin wave: If II rrrrrr = II 2 DDDD + 1 II 2 AAAA 2 II DDDD = II AAAA = 1 2 II pppppppp II rrrrrr = 3 2 II DDDD = II pppppppp Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn I DC I AC

6 Current density (j rms ) - optimisation Advantages of low j rms : 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 Lifetime cost Chosen value of j rms is an optimisation of magnet capital against power costs. 0.0 capital total running Current density j Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

7 Number of turns, N 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, neat terminals Thin interconnections- low cost and flexible More insulation in coil;larger coil volume; increased assembly costs High voltage power supply safety problem s Small N (high current) Large, bulky term inals Thick, expensive connections High percentage of copper in coil; more efficient use of available space High current power supply greater losses Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

8 Examples of typical turns/current From the Diamond 3 GeV synchrotron source: Dipole: N (per magnet): 40 I max 1500 A Voltage (circuit): 500 V Quadrupole: N (per pole) 54 I max 200 A Voltage (per m agnet): 25 V Sextupole: N (per pole) 48 I max 100 A Voltage (per m agnet): 25 V Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

9 An attractive proposition PERMANENT MAGNETS Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

10 Resi dual fi elds Remnant field B R : val ue of B at H = 0 Coercive force H C : negative value of field at B = 0 Residual field: the flux density in a gap at I = 0 II = 0: HH. dddd = 0 So: HH ssssccccss λλ + (HH gggggg )gg = 0 BB gggggg = μμ 0 ( HH ssssccccss ) λλ gg BB gggggg μμ 0HH cc λλ gg Where: λ is path length in steel g is gap height Because of presence of gap, residual field is determined by coercive force H C (A/m) and not remnant flux density B R (Tesla). B R -H C Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

11 Use of permanent magnet (PM) materials From previ ous sli de: HH. dddd = 0 BB gggggg = μμ 0 ( HH PPPP ) λλ gg ; PM materia l so the PM material B operates in the 2 nd quadrant of the hysteresis loop. H PM is polarised - magnetised only in the easy direction! Optimum B, H to give maximum energy density = BH; Note the factor of ½ missing! H c B r H Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

12 Energy density (BH) in PM available since 1900: Vacuumschmelze data BH in second quadrant for m ateri als avai lable today. Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

13 PM used in lattice magnets? ASTeC has been working with CERN on the design of 42,000 quadrupoles for the drive beam i n CLIC: The power consumption for the EM version will be ~8 MW Total power load limit to air within the tunnel is only 150 W/m A PM quadrupole would potentially have many advantages: Vastly reduced electrical power Ecologically green Very low operating costs No cooling water needed Very low power to air Problem: how to vary the strength of the quadrupole by a factor of 12? Solution: mechanical change to the PM geometry Problem: pole position is fixed and must be stable to around 20 µm Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

14 Pole and yoke fixed, PM moves Solution At 8% strength (3.5 T/m) At 100% strength (43 T/m) Lab prototype meets specification Design for mounting in CLIC drive beam Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

15 Dynamic model Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn Colour code: red = high field; blue = low field

16 Yoking aside THE M AGNETIC CIRCUIT Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

17 Permeability of low silicon steel saturati on regi on: µ drops off µ B (T) Parallel to rolling direction Normal to rolling direction. Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

18 Flux at the pole and in the circuit Flux in the yoke includes the gap flux and stray flux, which extends (approx) one gap width on either side of the gap. b Approximate value for total flux in the backleg of magnet length λ: FF = BB gggggg bb + 2gg λλ g Width of backleg is chosen to limit B yoke and hence maintain high μ. Note: FEA codes give values of vector potential (A z ); hence values of total flux can be obtained. g Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

19 Magnet geometry Dipoles can be C core, H core, or Window frame 'C' Core: Advantages: Easy access Classic design Di sadvantages: Pole shims needed Asymmetric (small) Less rigid Shim Th e shim is a small, additional piece of ferromagnetic 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. Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

20 A typical C cored Dipole Cross-section of the Diam ond storage ring dipole Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

21 H-core and window frame magnets H core Advantages: Symmetric More rigid Di sadvantages: Still needs shims Access problems 'Window Frame' Advantages: High quality field No pole shim Symmetric & rigid Di sadvantages: Major access problems Insulation thickness Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

22 Window frame dipole 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 J up outer face of conductor H is uniform, vertical and H with same value in the middle of the gap perfect dipole field Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

23 The practical window frame dipole Insulation added to coil: B increases close to coil insulation surface B decreases close to coil insulation surface best compromise Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

24 PM magnets - position of materi al Using permanent magnet materials, the PM goes in series (at any convenient position) in the magnet circuit; shims still needed: Shim detail Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

25 Design of a PM magnetic circuit The magnetic circuit has to be designed to match the m ateri al s. characteri sti cs; the PM must operate close to its optimum energy density: eg: VACODYM 745 at (circa) H = -550 k A/ m ; B = 0.7 T (in the material) Adjust width to give B 0.7 T in material The energy provided by the PM will match the magnetic energy in the gap (beware the factor of ½) Adjust height to provide the required Amps across the gap Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

26 Diamond storage ring quadrupole The yoke support pieces in the horizontal plane need to provide space for beamlines and are not ferromagnetic. Error harmonics include n = 4 (octupole), a finite permeability error. An open-sided quadrupole Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

27 Typical pole designs 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. Di pole A 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. A When high fields are present, chamfer angles must be small, and tapering of poles may be necessary. Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

28 Assessing pole design A first assessment can be made by just examining B y (x) within the required good field region. Note that the expansion of B y (x) at y = 0 is a Taylor series: Also note: δδbb yy (xx) δδxx So quadrupole gradient BB yy xx = bb nn xx nn 1 = bb 2 + 2bb 3 xx + gg bb 2 = δδbb yy(xx) δδxx in a quadrupole But sextupole gradient gg ss bb 3 = 2 δδ2 BB yy (xx) in a sextupole δδxx 2 So coefficients are not equal to differentials for n = 3 etc. Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn nn=1 = bb 1 + bb 2 xx + bb 3 xx 2 + dipole quadrupole sextupole

29 Assessing an adequate design A simple judgment of field quality is given by plotting: Dipole: Quadrupole: Sextupole: BB yy xx BB yy (0) 1 ddbb yy (xx) dddd dd 2 BB yy (xx) ddxx 2 ΔBB(xx) BB 0 Δgg(xx) gg(0) Δgg 2 (xx) gg 2 (0) Typical acceptable variation inside good field region: ΔBB(xx) BB % Δgg(xx) gg(0) Δgg 2 (xx) gg 2 (0) 0.1% 1.0% Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

30 How do we terminate a pole end? For a pole with B 1.2 T saturation and non-linear behaviour will result if a square end is used: A smooth roll-off is needed at pole edges (transverse); and at the magnet ends (in the 3 rd dimension). But what shape? Solution provided by Walter Rogowski Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

31 How derived? Rogowski calculated electric potential lines around a flat capaci tor plate: Y Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn X

32 Blown-up version Y The central heavy line is for ϕ = 0.5. X Rogowski showed that this was the fastest changing line along which the field intensity was monotonically decreasing. Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

33 Then applied to magnet ends Conclusion: Recall that a high µ steel surface is a line of constant scalar potential. Hence, a magnet pole end using the ϕ = 0.5 potential line provides the maximum rate of increase in gap with a monotonic decrease in flux density at the surface, i.e. no saturation Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

34 The equation The 'Rogowski' roll-off: Equation: yy = gg 2 + gg ππππ αααααα exp 1 gg gg 2 is dipole half gap y = 0 is centre line of gap α is a parameter controlling gradient at x = 0 (~ 1) Diamond dipole end Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

35 M AGNET DESI GN Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

36 Computer codes A number of computer codes are available: e.g. the Vector Fields (Cobham) codes Opera 2D and 3D. These have: finite elements with variable triangular mesh multiple iterations to simulate steel nonlinearity extensi ve pre- and post-processors cross-platform compatibility Technique is iterative: operafea.com calculate flux generated by a defi ned geom etry adjust the geometry until required distribution is achi eved Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

37 Design Procedures Opera 2D The model is set up in 2D using a GUI to define regions : steel regi ons coils (including current density) a background region which defines the physical extent of the model the symmetry constraints on the boundaries the B-H curves for the steel and other materials m esh i s generated and data saved Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

38 Model of ALICE quadrupole steel yoke coil symmetry lines background region Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

39 With mesh added coarse mesh fine mesh Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

40 Central gradi ent: 1.79 T/ m Flux lines (equi potenti als) Field in model Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

41 Calculation Sol ver - either: linear,using a predefined constant permeability for a si ngle calculati on Useful for coil-only models nonlinear, which is iterative with steel permeability set according to B in steel calculated on previous iteration Essential for iron-dominated magnets Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

42 Post-Processor: Data Display Opera 2D uses pre-processor model for many options for displaying field amplitude and quality: field lines graphs contours gradi ents harm oni cs (from a Fouri er analysi s around a predefined circle) Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

43 2D quadrupole gradient quality on x axis Field quality: ±2.5x10-4 within ±40mm Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

44 Opera 3D model of CLARA quadrupole 3 symmetry planes, 1/ 8 th of magnet modelled Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

45 Fields in 3D model Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

46 Harmonics indicate magnet quali ty The amplitude and phase of the harmonic components in a magnet provide an assessment: when accelerator physicists are calculating beam behaviour in a lattice when designs are judged for suitability when the manufactured magnet is measured to judge acceptability of a manufactured magnet Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

47 The third dimension magnet ends Fringe flux will be present at the magnet ends so beam deflection continues beyond magnet end: z B 0 B y The magnet s strength is given by BB yy zz dddd along the m a gn et, the integration including the fringe field at each end. 1 Th e magnetic length is defined as BB BB yy zz dddd over the same 0 integration path, where B 0 is the field at the azimuthal centre. Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

48 Magnet End Fields and Geometry Necessary to terminate the magnet in a controlled way: to define the length (strength) to prevent saturation in a sharp corner (see diagram) to maintain length constant with x, y to prevent flux entering normal to lamination (AC) The end of the magnet is therefore chamfered (a Rogowski roll-off if high field), increasing the gap (or inscribed radius) and lowering the field as the end is approached. Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

49 Pole profi le adjustment As the gap is increased, the size (area) of the shim is increased, to give some control of the field quality at the lower field. This is far from perfect! Transverse adjustment at end of dipole Transverse adjustment at end of quadrupole Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

50 The NINA magnet ends Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

51 Calculation of end effects with 2D codes FEA model in longitudinal plane, with correct end geometry (including coil), but 'idealised' return yoke: + - This will establish the end distribution; a numerical integration will give the 'B' length. Provided ddbb yy is not too large, single 'slices' in the transverse plane can be used dddd to calculate the radial distribution as the gap increases. Again, numerical integration will give BB. dddd as a fun ction of x. This technique is less satisfactory with a quadrupole, but end effects are less critical with a quad. Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

52 End geometries - dipole Simpler geometries can be used in some cases. The Diamond dipoles have a Rogowski roll-off at the ends (as well as Rogowski roll-offs at each side of the pole). See photographs to follow. This gives small negative sextupole field in the ends which will be compensated by adjustments of the strengths in adjacent sextupole magnets this is possible because each sextupole will have its own individual power supply. Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

53 MAGNET EXAMPLES Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

54 Diamond Dipole Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

55 Diamond dipole ends Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

56 Diamond Dipole end Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

57 Simplified end geometries - quadrupole Diam ond quadrupoles have an angular cut at the end; depth and angle were adjusted using 3D codes to give optimum integrated gradi ent. Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

58 Diamond W quad end Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

59 Opera 3D results - different depths 45 end chamfers on Δgg gg 0 integrated through magnet and end fringe field (0.4 m long WM quad) End chamfering - Diamond W quad Fractional deviation mm Cut 7 mm Cut 6 mm Cut 4 mm Cut No Cut Thanks to Chris Bailey (DLS) who performed this working using OPERA 3D. Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn X

60 Sextupole ends It is not usually necessary to chamfer sextupole ends (in a DC magnet). Diam ond sextupole end: Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

61 Sexy pi cs of sextupoles Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

62 Further Reading CERN Accelerator School on Magnets Bruges, Belgium; June Uni ted States Parti cle Accelerator School Magnet and RF Cavity Design, January stin-magnets.shtml J.D. Jack son, Classi cal Electrodynam i cs J.T. Tanabe, Iron Dom i nated Electrom agnets Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn