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29/07/2014 Accelerator Magnets Neil Marks, STFC- ASTeC / U. of Liverpool, Daresbury Laboratory, Warrington WA4 4AD, U.K. Tel: (44) (0)1925 603191 Fax: (44) (0)1925 603192 [email protected] Room Temperature Magnets CAS Prague 2014 N eil Marks Contents Theory • Maxwell's 2 magneto-static equations; With no currents or steel present: • Solutions in two dimensions with scalar potential (no currents); • • Cylindrical harmonic in two dimensions (trigonometric formulation); Field lines and potential for dipole, quadrupole, sextupole; Introduction of steel: • Ideal pole shapes for dipole, quad and sextupole; • • Field harmonics-symmetry constraints and significance; Significance and use of contours of constant vector potential; Room Temperature Magnets N eil Marks CAS Prague 2014 1 29/07/2014 Contents (cont.) Three dimensional issues: • • Termination of magnet ends and pole sides; The ‘Rogowski roll-off’ Introduction of currents: • Ampere-turns in dipole, quad and sextupole; • • Coil design; Coil economic optimisation-capital/running costs; Practical Issues: • Backleg and coil geometry- 'C', 'H' and 'window frame' designs; • FEA techniques - Modern codes- OPERA 2D and 3D; • Judgement of magnet suitability in design. Room Temperature Magnets N eil Marks CAS Prague 2014 Magnets - introd uction Dipoles to bend the beam : Quad rupoles to focus it: Sextupoles to correct chrom aticity: We shall establish a formal approach to describing these magnets. Room Temperature Magnets N eil Marks CAS Prague 2014 2 29/07/2014 N o cu rrents, no steel: .B = 0 ; Maxw ell’s equations: H=j; Then we can put: B = - So that: 2 = 0 (Laplace's equation). Taking the two dimensional case (ie constant in the z direction) and solving for cylindrical coordinates (r,): = (E+F )(G+H ln r) + n=1 (Jn r n cos n +Kn r n sin n +Ln r -n cos n + M n r -n sin n ) Room Temperature Magnets N eil Marks CAS Prague 2014 In practical situations: The scalar potential sim plifies to: = n (Jn r n cos n +Kn r n sin n), w ith n integral and Jn ,Kn a function of geom etry. Giving components of flux density: Br = - n (n Jn r n-1 cos n +nKn r n-1 sin n) B = - n (-n Jn r n-1 sin n +nKn r n-1 cos n) Then to convert to Cartesian coord inates: x = r cos ; y = r sin ; and Bx = - / x; By = - / y Room Temperature Magnets N eil Marks CAS Prague 2014 3 29/07/2014 Significance This is an infinite series of cylind rical harm onics; they d efine the allow ed d istributions of B in 2 d im ensions in the absence of currents w ithin the d om ain of (r,). Distributions not given by above are not physically realisable. Coefficients Jn , Kn are d eterm ined by geom etry (rem ote iron bound aries and current sources). N ote that this form ulation can be expressed in term s of com plex field s and potentials. Room Temperature Magnets CAS Prague 2014 N eil Marks Dipole field n=1: Cylindrical: =J1 r cos +K1 r sin . Br = J1 cos + K1 sin ; B = -J1 sin + K1 os ; Cartesian: =J1 x +K1 y Bx = -J1 By = -K1 So, J1 = 0 gives vertical dipole field: = c o n s t. B K1 =0 gives horizontal dipole field. Room Temperature Magnets N eil Marks CAS Prague 2014 4 29/07/2014 Quad rupole field n=2: Cylindrical: = J2 r 2 cos 2 +K2 r 2 sin 2; Br = 2 J2 r cos 2 +2K2 r sin 2; B = -2J2 r sin 2 +2K2 r cos 2; J2 = 0 gives 'normal' or ‘upright’ quadrupole field. Line of constant scalar potential K2 = 0 gives 'skew' quad fields (above rotated by /4). Room Temperature Magnets Cartesian: = J2 (x2 - y 2)+2K2 xy Bx = -2 (J2 x +K2 y) By = -2 (-J2 y +K2 x) Lines of flux density N eil Marks CAS Prague 2014 Sextupole field n=3: Cylindrical; = J3 r 3 cos 3 +K3 r 3 sin 3; Br = 3 J3r 2 cos 3 +3K3r 2 sin 3; B= -3J3 r 2 sin 3+3K3 r 2 cos 3; -C +C +C -C -C +C Room Temperature Magnets N eil Marks Cartesian: = J3 (x3-3y 2x)+K3(3yx2-y 3) Bx = -3J3 (x2-y 2)+2K3yx By = -3-2 J3 xy +K3(x2-y 2) J3 = 0 giving ‘upright‘ sextupole field. Line of constant scalar potential Lines of flux density CAS Prague 2014 5 29/07/2014 Variation of By on x axis (upright field s). By Dip ole; constant field : x Qu ad ru p ole; linear field By x Sextu p ole: qu ad ratic variation: 100 By = G Q x; G Q is quadrupole gradient (T/m). By x By = G S x2; G S is sextupole gradient (T/m 2). 0 -20 0 20 Room Temperature Magnets CAS Prague 2014 N eil Marks Alternative notation: (used in most lattice programs) B (x) = B k n n 0 x n n! magnet strengths are specified by the value of kn; (normalised to the beam rigidity); order n of k is different to the 'standard' notation: dipole is quad is n = 0; n = 1; etc. k0 (dipole) k1 (quadrupole) m-1; m-2; k has units: Room Temperature Magnets N eil Marks etc. CAS Prague 2014 6 29/07/2014 Introd u cing iron yokes and poles. What is the id eal pole shape? • Flux is norm al to a ferrom agnetic surface w ith infinite : curl H = 0 therefore H .d s = 0; in steel H = 0; 1 therefore parallel H air = 0 therefore B is norm al to surface. • Flux is norm al to lines of scalar potential, (B = - ); • So the lines of scalar potential are the id eal pole shapes! (but these are infinitely long!) Room Temperature Magnets N eil Marks CAS Prague 2014 Equations of id eal poles Equations for Id eal (infinite) poles; (Jn = 0) for ‘upright’ (ie not skew ) field s: D ipole: y= g/ 2; (g is inter-pole gap). Quadrupole: xy= R2/ 2; Sextupole: 3x2y - y 3 = R3; R R is the ‘inscribed radius’ of a multipole magnet. Room Temperature Magnets N eil Marks CAS Prague 2014 7 29/07/2014 ‘Pole-tip’ Field The radial field at pole centre of a m ultipole m agnet: BPT = G N R(n-1) ; Quad rupole: BPT = G Q R; sextupole: BPT = G S R2; etc; Has it any significance? - a quad rupole R = 50 m m ; G Q = 20 T/ m ; BPT = 1.0 T; i) Beam line – round beam r = 40 mm; pole extends to x = 65 mm; B y (65,0) = 1.3 T; OK ; ii) Synchrotron source – beam ± 50 mm horiz.; ± 10 mm vertical; pole extends to x = 80 mm; By (80,0)= 1.6 T; perhaps OK ??? ; iii) FFAG – beam ± 65 mm horiz.; ± 8 mm vertical; Pole extends to x = 105 mm; By (105, 0) = 2.1 T . Room Temperature Magnets CAS Prague 2014 N eil Marks Com bined function m agnets 'Com bined Function Magnets' - often d ipole and quad rupole field com bined (but see later slid e): A quad rupole m agnet w ith physical centre shifted from m agnetic centre. B n = 0 B , x Characterised by 'field ind ex' n, +ve or -ve d epend ing cuervatu re of w h eof r egrad th e r a d iu s o f c ur visa rad tu r iu e so of f th beam a nthe d B is th e c e n tr a l d ip o l is ient; on d irection 0 beam ; d o not confuse w ith harm onic n! Bo is central d ip ole field I f p h y s ic a l a n d m a g n e tic c e n tr e s a r e s e p a r a te d b y X Room Temperature Magnets CAS Prague 2014 N eil Marks T hen B th e r e f o r e X 0 0 0 B X ; 0 x = / n; 2 8 29/07/2014 Typ ical com bined d ip ole/ qu ad ru p ole SRS Booster c.f. dipole ‘F’ type -ve n ‘D ’ type +ve n. Room Temperature Magnets N eil Marks CAS Prague 2014 N IN A Com bined function m agnets Room Temperature Magnets N eil Marks CAS Prague 2014 9 29/07/2014 Pole for a com bined d ipole and quad . Physical and magnetic centres displacmen t from Horizontal are separated true quad Then by centre B X is 0 0 x' B X 0 x 2 x' y R therefore x' x X As Pole equation 0 R y is 2 2 where g is the half gap at the physical rewritten as Room Temperature Magnets centre n 1 n x y g 1 - or / 2 n x 1 1 of the magnet y x g 1 B 0 N eil Marks B x 1 CAS Prague 2014 Other com bined function m agnets: • d ipole, quad rupole and sextupole; • d ipole & sextupole (for chrom aticity control); • d ipole, skew quad , sextupole, octupole; Generated by • pole shapes given by sum of correct scalar potentials - am p litu d es bu ilt into p ole geom etry – not variable! OR: • m ultiple coils m ounted on the yoke - am plitud es ind epend ently varied by coil currents. Room Temperature Magnets N eil Marks CAS Prague 2014 10 29/07/2014 The SRS multipole magnet. Could d evelop: • vertical d ipole • horizontal d ipole; • upright quad ; • skew quad ; • sextupole; • octupole; • others. Room Temperature Magnets CAS Prague 2014 N eil Marks The practical pole in 2D Practically, poles are finite, introducing errors; these appear as higher harm onics w hich d egrad e the field d istribution. H ow ever, the iron geometries have certain sym m etries that restrict the nature of these errors. Dipole: Room Temperature Magnets Quad rupole: N eil Marks CAS Prague 2014 11 29/07/2014 Possible sym m etries. Lines of sym m etry: Dipole: Quad Pole orientation d etermines w hether pole is upright or skew . y = 0; x = 0; y = 0 Ad d itional sym m etry im posed by pole ed ges. x = 0; y=x The ad d itional constraints im posed by the sym m etrical pole ed ges lim its the values of n that have non zero coefficients Room Temperature Magnets CAS Prague 2014 N eil Marks Dipole sym m etries Type Symmetry Constraint Pole orientation () = -(-) all Jn = 0; Pole ed ges () = ( -) Kn non-zero only for: n = 1, 3, 5, etc; + + + - So, for a fully sym m etric d ipole, only 6, 10, 14 etc pole errors can be present. Room Temperature Magnets N eil Marks CAS Prague 2014 12 29/07/2014 Quad rupole sym m etries Type Symmetry Constraint Pole orientation () = -( -) All Jn = 0; () = -( -) Kn = 0 all od d n; () = (/ 2 -) Kn non-zero only for: n = 2, 6, 10, etc; Pole ed ges So, a fully sym m etric quad rupole, only 12, 20, 28 etc pole errors can be present. Room Temperature Magnets N eil Marks CAS Prague 2014 Sextupole sym m etries. Type Symmetry Constraint Pole orientation () = -( -) () = -(2/ 3 - ) () = -(4/ 3 - ) All Jn = 0; Kn = 0 for all n not m ultiples of 3; Pole ed ges () = (/ 3 - ) Kn non-zero only for: n = 3, 9, 15, etc. So, a fully sym m etric sextupole, only 18, 30, 42 etc pole errors can be present. Room Temperature Magnets N eil Marks CAS Prague 2014 13 29/07/2014 Su m m ary For p erfectly sym m etric m agnets, the ‘allow ed ’ error field s are fu lly d efined by the sym m etry of scalar p otential and the trigonom etry: D ipole: + Quadrupole: - + + + - - D ipole: only dipole, sextupole, 10 pole etc can be present Sextupole: only sextupole, 18, 30, 42 pole, etc. can be present Room Temperature Magnets N eil Marks Quadrupole: only quadrupole, 12 pole, 20 pole etc can be present CAS Prague 2014 Vector p otential in 2 D We have: and Expand ing: B = cu rl A (A is vector potential); d iv A = 0 B = cu rl A = (A z / y - A y / z) i + (A x/ z - A z / x) j + (A y / x - A x/ y) k; w here i, j, k, are u nit vectors in x, y, z. In 2 dimensions Bz = 0; / z = 0; So A x = A y = 0; and B = (A z / y ) i - (A z / x) j A is in the z direction, normal to the 2 D problem. N ote: d iv B = 2A z / (x y) - 2A z / (x y) = 0; Room Temperature Magnets N eil Marks CAS Prague 2014 14 29/07/2014 Total flux betw een tw o points DA In a tw o dimensional problem the m agnetic flux betw een tw o points is proportional to the d ifference betw een the vector potentials at those points. B F (A2 - A1) A1 A2 F Proof on next slid e. Room Temperature Magnets CAS Prague 2014 N eil Marks Proof: Consid er a rectangular closed path, length l in z d irection at (x 1,y 1) and (x2,y 2); apply Stokes’ theorem : A F = B.dS = ( cu rl A).dS = A.ds Bu t A is exclu sively in the z d irection, and is constant in this d irection. dS l So: A.d s = l { A(x1,y 1) - A(x2,y 2)}; (x1, y 1) B ds (x2, y 2) y z x F = l { A(x1,y 1) - A(x2,y 2)}; Room Temperature Magnets N eil Marks CAS Prague 2014 15 29/07/2014 Contours of constant A Therefore: i) Contou rs of constant vector p otential in 2D give a grap hical rep resentation of lines of flu x. ii) These are u sed in 2D FEA analysis to obtain a grap hical im age of flu x d istribu tion. iii) The total flu x cu tting the coil allow s the calcu lation of the ind u ctive voltage p er tu rn in an ac m agnet: V = - d F/d t; iv) For a sine w ave oscillation frequ ency w: Vpeak = w (F/2) Room Temperature Magnets N eil Marks CAS Prague 2014 In 3D – pole end s (also pole sid es). Fringe flu x w ill be p resent at p ole end s so beam d eflection continu es beyond m agnet end : z 1. 2 B0 By 0 0 The magnet’s strength is given by By (z) dz along the magnet , the integration including the fringe field at each end; The ‘magnetic length’ is defined as (1/B0)( By (z) dz ) over the same integration path, where B0 is the field at the azimuthal centre. Room Temperature Magnets N eil Marks CAS Prague 2014 16 29/07/2014 End Field s and Geom etry. At high(ish) fields it is necessary to terminate the pole (transverse OR longitudinal) in a controlled way:; •to prevent saturation in a sharp corner (see diagram); •to maintain length constant with x, y; •to define the length (strength) or preserve quality; •to prevent flux entering normal to lamination (ac). Longitudinally, the end of the magnet is therefore 'chamfered' to give increasing gap (or inscribed radius) and lower fields as the end is approached. Room Temperature Magnets N eil Marks CAS Prague 2014 Classical end or sid e solution The 'Rogowski' roll-off: Equation: y = g/2 +(g/a) [exp (ax/g)-1]; g/2 is dipole half gap; y = 0 is centre line of gap; a = 1.5 a ( ̴ 1); parameter to control a = 1.25 the roll off; a = 1.0 With a = 1, this profile provides the maximum rate of increase in gap with a monotonic decrease in flux density at the surface; For a high By magnet this avoids any additional induced non-linearity Room Temperature Magnets N eil Marks CAS Prague 2014 17 29/07/2014 Introd uction of currents Now for j 0 H=j; To expand, use Stoke’s Theorum: for any vector V and a closed curve s : V.ds = curl V.dS Apply this to: curl H = j ; dS ds V then in a magnetic circuit: H.ds = N I; N I (Ampere-turns) is total current cutting S Room Temperature Magnets N eil Marks CAS Prague 2014 Excitation cu rrent in a d ip ole B is approx constant round the loop made up of l and g, (but see below); But in iron, and So >>1, Hiron = Hair / ; l N I/ 2 g N I/ 2 1 Bair = 0 NI / (g + l/); g, and l/ are the 'reluctance' of the gap and iron. Approximation ignoring iron reluctance (l/ << g ): NI = B g /0 Room Temperature Magnets N eil Marks CAS Prague 2014 18 29/07/2014 Excitation current in quad & sextupole For quadrupoles and sextupoles, the required excitation can be calculated by considering fields and gap at large x. For example: Quadrupole: y Pole equation: xy = R2 /2 On x axes BY = GQx; where G Q is gradient (T/m). B x The sam e m ethod for a Sextupole, ( coefficient G S,), gives: At large x (to give vertical lines of B): N I = (GQx) ( R2 /2x)/0 N I = GQ R2 /2 0 (per pole). Room Temperature Magnets N I = G S R3/ 30 (per pole) CAS Prague 2014 N eil Marks General solu tion -m agnets ord er n B = 0 H B = - In air (remote currents! ), y Integrating over a lim ited path (not circular) in air: N I = (1 – 2)/ o 1, 2 are the scalar potentials at tw o points in air. Define = 0 at m agnet centre; then potential at the pole is: o N I = 0 NI Apply the general equations for m agnetic =0 field harm onic ord er n for non -skew m agnets (all Jn = 0) giving: N I = (1/ n) (1/ 0) Br / R (n-1) R n Where: N I is excitation per pole; R is the inscribed rad ius (or half gap in a d ipole); term in brackets is m agnet grad ient G N in T/ m (n-1). Room Temperature Magnets N eil Marks x B CAS Prague 2014 19 29/07/2014 Coil geom etry Stand ard d esign is rectangular copper (or alum inium ) cond uctor, w ith cooling w ater tube. Insulation is glass cloth and epoxy resin. Am p -turns (N I) are d eterm ined , but total copper area (A copper ) and num ber of turns (N ) are tw o d egrees of freed om and need to be d ecid ed . Room Temperature Magnets Current d ensity: j = N I/ A copper Optim um j d eterm ined from economic criteria. CAS Prague 2014 N eil Marks Current d ensity - optim isation Ad vantages of low j: • low er pow er loss – pow er bill is d ecreased ; • low er pow er loss – pow er converter size is d ecreased ; • less heat d issipated into m agnet tunnel. 15.0 • smaller coils; • low er capital cost; • smaller magnets. Lifetime cost Ad vantages of high j: Chosen value of j is an optim isation of magnet capital against pow er costs. Room Temperature Magnets N eil Marks total capital running 0.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 Current density j CAS Prague 2014 20 29/07/2014 N um ber of turns per coil-N The value of num ber of turns (N ) is chosen to m atch pow er supply and interconnection imped ances. Factors d eterm ining choice of N : Large N (low current) Small N (high current) Sm all, neat term inals. Large, bu lky term inals Thin interconnections-hence low cost and flexible. Thick, exp ensive connections. More insulation layers in coil, hence larger coil volum e and increased assem bly costs. H igh percentage of copper in coil volum e. More efficient use of space available H igh voltage pow er supply -safety problem s. H igh current pow er supply. -greater losses. Room Temperature Magnets CAS Prague 2014 N eil Marks Exam ples-turns & cu rrent From the Diam ond 3 GeV synchrotron source: Dipole: N (per m agnet): I m ax Volts (circuit): 40; 1500 500 A; V. Quad rupole: N (per pole) 54; I m ax 200 Volts (per m agnet): 25 A; V. N (per pole) 48; I m ax 100 Volts (per m agnet) 25 A; V. Sextupole: Room Temperature Magnets N eil Marks CAS Prague 2014 21 29/07/2014 Magnet geom etry Dipoles can be ‘C core’ ‘H core’ or ‘Wind ow fram e’ ''C' Core: Ad vantages: Easy access; Classic d esign; Disad vantages: Pole shim s need ed ; Asym m etric (sm all); Less rigid ; S h im d e t a il The ‘shim ’ is a sm all, ad d itional p iece of ferro -m agnetic m aterial ad d ed on each sid e of the tw o p oles – it com p ensates for the finite cu t-off of the p ole, and is op tim ised to red u ce the 6, 10, 14...... p ole error harm onics. Room Temperature Magnets CAS Prague 2014 N eil Marks Flux in the gap. Flux in the yoke includ es the gap flux and stray flux, w hich extend s (approx) one gap w id th on either sid e of the gap. b Thus, to calculate total flux in the back-leg of magnet length l: g F =Bgap (b + 2g) l. Wid th of backleg is chosen to lim it Byoke and hence m aintain high . Room Temperature Magnets N eil Marks g CAS Prague 2014 22 29/07/2014 Steel- B/ H cu rves -for typical silicon steel laminations N ote: • the relative perm eability is the grad ient of these curves; • the low er grad ient close to the origin - low er perm eability; • the perm eability is m axim um at betw een 0.4 and 0.6 T. Room Temperature Magnets N eil Marks CAS Prague 2014 Typical ‘C’ cored Dipole Cross section of the Diam ond storage ring d ipole. Room Temperature Magnets N eil Marks CAS Prague 2014 23 29/07/2014 H core and w ind ow -fram e m agnets ''Wind ow Fram e' Ad vantages: H igh qu ality field ; N o p ole shim ; Sym m etric & rigid ; Disad vantages: Major access p roblem s; Insu lation thickness ‘H core’: Ad vantages: Sym m etric; More rigid ; Disad vantages: Still need s shim s; Access p roblem s. Room Temperature Magnets N eil Marks CAS Prague 2014 Wind ow fram e d ip ole Provid ing the cond uctor is continuous to the steel ‘w ind ow fram e’ surfaces (im possible because coil m ust be electrically insulated ), and the steel has infinite , this m agnet generates perfect d ipole field . Provid ing current d ensity J is uniform in cond uctor: • H is uniform and vertical up outer face of cond uctor; J • H is uniform , vertical and H w ith sam e value in the m id d le of the gap; • perfect d ipole field . Room Temperature Magnets N eil Marks CAS Prague 2014 24 29/07/2014 Practical w ind ow fram e d ip ole. Insulation ad d ed to coil: B increases close to coil insulation surface Room Temperature Magnets B d ecrease close to coil insulation surface N eil Marks best com prom ise CAS Prague 2014 Open-sid ed Quad rupole ‘Diam ond ’ storage ring quad rupole. The yoke support pieces in the horizontal plane need to provid e space for beam -lines and are not ferrom agnetic. Error harm onics includ e n = 4 (octupole) a finite perm eability error. Room Temperature Magnets N eil Marks CAS Prague 2014 25 29/07/2014 Typical pole d esigns To com pensate for the non -infinite pole, shim s are ad d ed at the pole ed ges. The area and shape of the shim s d eterm ine the amplitud e of error harmonics w hich w ill be present. Dipole: Quad rupole: A The d esigner optim ises the pole by ‘pred icting’ the field resulting from a given pole geom etry and then ad justing it to give the required quality. Room Temperature Magnets A When high field s are p resent, cham fer angles m u st be sm all, and tap ering of p oles m ay be necessary CAS Prague 2014 N eil Marks Pole-end correction. 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 Room Temperature Magnets N eil Marks Transverse adjustment at end of quadrupole CAS Prague 2014 26 29/07/2014 Assessing p ole d esign A first assessm ent can be m ad e by just exam ining By (x) w ithin the required ‘good field ’ region. N ote that the exp ansion of By (x) y = 0 is a Taylor series: By (x) = n =1 {b n x (n-1)} = b 1 + b 2x + b 3x2 + … … … d ip ole qu ad sextu p ole Also note: By (x) / x = b 2 + 2 b 3x + … … .. So quad grad ient G b 2 = By (x) / x in a quad But sext. grad ient G s b 3 = ½ 2 By (x) / x2 in a sext. So coefficients are not equal to d ifferentials for n = 3 etc. Room Temperature Magnets CAS Prague 2014 N eil Marks Is it ‘fit for purpose’? A simple jud gement of field quality is given by plotting: • D ipole: • Quad: • 6poles: {By (x) - By (0)}/ BY (0) d By (x)/ d x d 2By (x)/ d x2 (DB(x)/ B(0)) (Dg(x)/ g(0)) (Dg 2(x)/ g 2(0)) ‘Typical’ acceptable variation insid e ‘good field ’ region: DB(x)/ B(0) Dg(x)/ g(0) Dg 2(x)/ g 2(0) Room Temperature Magnets N eil Marks 0.01% 0.1% 1.0% CAS Prague 2014 27 29/07/2014 Design com puter cod es. Com puter cod es are now used ; eg the Vector Field s cod es -‘OPERA 2D’ and 3D. These have: • • • • finite elements w ith variable triangular mesh; m ultiple iterations to sim ulate steel non -linearity; extensive pre and post processors; com patibility w ith m any platform s and P.C. o.s. Technique is iterative: • calculate flux generated by a d efined geom etry; • ad just the geom etry until required d istribution is achieved . Room Temperature Magnets N eil Marks CAS Prague 2014 Design Proced u res – OPERA 2D Pre-p rocessor: The m od el is set-up in 2D using a GUI (graphics user’s interface) to d efine ‘regions’: • steel regions; • coils (includ ing current d ensity); • a ‘background ’ region w hich d efines the physical extent of the m od el; • the sym m etry constraints on the bound aries; • the perm eability for the steel (or use the preprogram m ed curve); • m esh is generated and d ata saved . Room Temperature Magnets N eil Marks CAS Prague 2014 28 29/07/2014 Mod el of Diam ond storage ring d ipole Room Temperature Magnets N eil Marks CAS Prague 2014 With m esh ad d ed Room Temperature Magnets N eil Marks CAS Prague 2014 29 29/07/2014 Close-u p of p ole region. Pole profile, show ing shim and Rogow ski sid e roll-off for Diam ond 1.4 T d ipole.: Room Temperature Magnets N eil Marks CAS Prague 2014 Diam ond qu ad ru p ole: a sim p lified m od el N ote – one eighth of Qu ad ru p ole cou ld be u sed w ith op p osite sym m etries d efined on horizontal and y = x axis. Bu t I have often exp erienced d iscontinu ities at the origin w ith su ch1/ 8 th geom etry. Room Temperature Magnets N eil Marks CAS Prague 2014 30 29/07/2014 Calculation of end effects using 2D cod es 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 dBY/dz is not too large, single 'slices' in the transverse plane can be used to calculated the radial distribution as the gap increases. Again, numerical integration will give B.dl as a function of x. This technique is less satisfactory with a quadrupole, but end effects are less critical with a quad. Room Temperature Magnets N eil Marks CAS Prague 2014 Calcu lation. Data Processor: either: • linear - w hich uses a pred efined constant perm eability for a single calculation, or • non-linear - w hich is iterative w ith steel perm eability set accord ing to B in steel calculated on previous iteration. Room Temperature Magnets N eil Marks CAS Prague 2014 31 29/07/2014 Data Disp lay – OPERA 2D Post-processor: uses pre-processor mod el for many options for d isplaying field am plitud e and quality: • • • • • field lines; graphs; contours; grad ients; harm onics (from a Fourier analysis around a pred efined circle). Room Temperature Magnets N eil Marks CAS Prague 2014 2 D Dipole field hom ogeneity on x axis Diam ond s.r. d ip ole: DB/ B = {By(x)- B(0,0)}/ B(0,0); typ ically 1:104 w ithin the ‘good field region’ of -12m m x +12 m m .. Room Temperature Magnets N eil Marks CAS Prague 2014 32 29/07/2014 2 D Flux d ensity d istribution in a d ipole Room Temperature Magnets N eil Marks CAS Prague 2014 2 D Dipole field hom ogeneity in gap Transverse (x,y) p lane in Diam ond s.r. d ip ole; contou rs are 0.01% requ ired good field region: Room Temperature Magnets N eil Marks CAS Prague 2014 33 29/07/2014 H arm onics ind icate magnet quality The am plitud e and phase of the integrated harm onic com ponents in a m agnet provid e an assessm ent: • w hen accelerator physicists are calculating beam behaviour in a lattice; • w hen d esigns are jud ged for suitability; • w hen the m anufactured m agnet is m easured ; • to jud ge acceptability of a m anufactured m agnet. Measurem ent of a m agnet after m anufacture w ill be d iscussed in the section on m easurem ents. Room Temperature Magnets N eil Marks CAS Prague 2014 End geom etries - d ip ole Sim pler geom etries can be used in som e cases. The Diam ond d ipoles have a Rogaw ski roll-off at the end s (as w ell as Rogaw ski roll-offs at each sid e of the pole). See photographs to follow . This give sm all negative sextupole field in the end s w hich w ill be com pensated by ad justm ents of the strengths in ad jacent sextupole m agnets – this is possible because each sextupole w ill have its ow n ind ivid ual pow er supply. Room Temperature Magnets N eil Marks CAS Prague 2014 34 29/07/2014 OPERA 3D m od el of Diam ond d ipole. Room Temperature Magnets N eil Marks CAS Prague 2014 Diam ond d ipole poles Room Temperature Magnets N eil Marks CAS Prague 2014 35 29/07/2014 2 D Assessm ent of quad rupole grad ient quality 0 .1 D iamond WM quadrupole: D d H y /d x (% ) graph is percentage variation in dBy/dx vs x at different values of y. 0 .0 5 0 -0 .0 5 Gradient quality is to be 0.1 % or better to x = 36 mm. -0 .1 0 4 8 12 16 20 24 28 32 36 x (m m ) y= 0 Room Temperature Magnets y = 4 mm y = 8 mm y = 1 2 mm y = 1 6 mm CAS Prague 2014 N eil Marks End cham fering - Diam ond ‘W’ quad Tosca resu lts d ifferent d ep ths 45 end cham fers on Dg/ g 0 integrated throu gh m agnet and end fringe field (0.4 m long WM qu ad ). Fractional deviation 0.002 0.001 X 5 10 15 20 25 30 35 -0.001 -0.002 8 7 6 4 mm Cut mm Cut mm Cut mm Cut No Cut Thanks to Chris Bailey (D LS) w ho performed this w orking using OPERA 3D . Room Temperature Magnets N eil Marks CAS Prague 2014 36 29/07/2014 Sim plified end geom etries - quad rupole Diam ond quad rupoles have an angular cut at the end ; d epth and angle w ere ad justed using 3D cod es to give optim um integrated grad ient. Room Temperature Magnets N eil Marks CAS Prague 2014 Sextupole end s It is not usually necessary to chamfer sextupole end s (in a d .c. m agnet). Diam ond sextupole end : Room Temperature Magnets N eil Marks CAS Prague 2014 37 29/07/2014 ‘Artistic’ Diamond Sextupoles Room Temperature Magnets N eil Marks CAS Prague 2014 38