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A.C. Magnets Neil Marks, CCLRC, Daresbury Laboratory, Warrington WA4 4AD. [email protected] Neil Marks; DLS/CCLRC Cockcroft Institute 2005/6, © N.Marks, 2006 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. Present an overview of the design and operation of power supply systems, both d.c. (for storage rings) and cycling (for cycling accelerators). 4. Give a qualitative overview of injection and extraction techniques as used in circular machines. 5. Present the standard designs for kicker and septum magnets and their associated power supplies. Neil Marks; DLS/CCLRC Cockcroft Institute 2005/6, © N.Marks, 2006 Contents Core Syllabus 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; ‘Extension’ Power supply systems – d.c. and a.c.; Injection and extraction schemes; ‘Fast’ magnets; Kicker magnets-lumped and distributed power supplies; Septum magnets-active and passive septa; High frequency ferromagnetic material; Neil Marks; DLS/CCLRC Cockcroft Institute 2005/6, © N.Marks, 2006 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 a.c. losses; minimise a.c. losses. 2. Excitation voltage now includes an inductive (reactive) component; this may be small, major or dominant (depending on frequency); this must be accurately assessed. Neil Marks; DLS/CCLRC Cockcroft Institute 2005/6, © N.Marks, 2006 Equivalent circuit of a.c. magnet Rac Rdc Lm Im Cleakage Neil Marks; DLS/CCLRC Cockcroft Institute 2005/6, © N.Marks, 2006 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. Neil Marks; DLS/CCLRC Cockcroft Institute 2005/6, © N.Marks, 2006 Eddy Currents in a Conductor I x Rectangular cross section resistivity , B sin t 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 / -a -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. l 0 x a Cross section A Power loss/unit length = 2 B2 A a2/(6 ) W/m; a 10 mm x 10 mm Cu conductor in a 1 T sin field, loss = 1.7 kW/m Neil Marks; DLS/CCLRC Cockcroft Institute 2005/6, © N.Marks, 2006 Eddy Currents in a Conductor II x 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. a x Power loss/unit length = 2 B2 A a2/(8 ) W/m; Neil Marks; DLS/CCLRC Cockcroft Institute 2005/6, © N.Marks, 2006 Eddy Currents in a cylindrical vacuum vessel total flux cutting circuit at angle q: B sin t Wall conductivity 2 ( R cos q )( B sin t ); /t 2 (R cos q )(B cos t); voltage round unit length of loop (top only) : V q 2 (R cos q )(B cos t); R resistance of unit length of loop : 2 / ( R q) q eddy currents in top and bottom loops : Ie 2 R 2 (cos q) B (cos t) q / ; total eddy current in cylinder is integral of above w.r.t. q between 0, p/2 : Ie = - 2 R2 B (cos t) / 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. Neil Marks; DLS/CCLRC Cockcroft Institute 2005/6, © N.Marks, 2006 Perturbation field generated by eddy currents m= Field perturbations Be(X) generated by the eddy currents will depend on the vessel’s magnetic environment. If the imposed field is generated by coils on a ferro-magnetic yoke, eddy currents will couple to that yoke. To establish the amplitude and distribution of the field perturbation, a yoke geometry has to be considered. g R 0 x Magnet geometry around vessel radius R. 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 ) Neil Marks; DLS/CCLRC Cockcroft Institute 2005/6, © N.Marks, 2006 Distributions of perturbing fields Cylindrical vessel: Be(X) 2R Bcos t R 2 - X2 . g 2 R 2 Bcos t X2 X4 X6 5 X8 1 .......... .. 2 4 6 8 g 8R 16 R 128 R 2R Rectangular vessel (semi axies a, b): Be(X) 2 m 0 (B cos t) (a 2 X 2 ) ab g 2 Elliptical vessel (semi axies a, b): Be(X) m0 B cos t 2 g Neil Marks; DLS/CCLRC 2 1/2 2 2 1/2 2 tan -1 (a b ) ( a - X ) 2 2 2 2 1/ 2 (a 2 b 2 ) b X (a / b 1) a a2 b Cockcroft Institute 2005/6, © N.Marks, 2006 ‘Low frequency’ a.c. magnets 1.5 We shall deal separately with ‘low frequency’ and ‘fast’ magnets: ‘low frequency’ – d.c. to c 100 Hz: 0 0 50 100 150 ‘fast’ magnets frequency (Hz) – pulsed magnets with rise times from 10s ms to < 1 ms. 200 time (ms) (But these are very slow0 compared to r.f.~10systems!) Neil Marks; DLS/CCLRC Cockcroft Institute 2005/6, © N.Marks, 2006 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’: a b c d b a d c d c b a c d a b 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. Neil Marks; DLS/CCLRC Cockcroft Institute 2005/6, © N.Marks, 2006 Two examples: Note that eddy loss varies as 2 ; B2 and (width)2 . NINA : ISIS: E = 5.6 GeV; = 53 Hz; Bpeak = 0.9 T. E = 800 MeV; = 50 Hz; Bpeak ≈ 0.2 T. Neil Marks; DLS/CCLRC Cooling tube. Transposed, stranded conductor. } c 10mm x 10 mm solid conductor with cooling hole. Cockcroft Institute 2005/6, © N.Marks, 2006 Steel Yoke Eddy Losses. To limit eddy losses, steel core are laminated, with a thin layer (~2 µm) of insulating material coated to one side of each lamination. 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. Neil Marks; DLS/CCLRC Cockcroft Institute 2005/6, © N.Marks, 2006 Steel hysteresis loss Steel also has hysteresis loss caused by the finite area inside the B/H loop: for a.c. excitation frequency f, power loss Phys = f { B B.dH + A - A W/m3. C B.dH B DB.dH - C CB.dH } B D D A Neil Marks; DLS/CCLRC Cockcroft Institute 2005/6, © N.Marks, 2006 Steel loss data Manufacturers give figures for total loss (in W/kg) in their steels catalogues: •for a sinusoidal waveform at a fixed peak field (European 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. 3 Accelerator magnets will have: •different waveforms (unidirectional!); •different d.c. bias values; •different frequencies (0.2 Hz up to 50 Hz). 0 0 How does the designer calculate steel loss? Neil Marks; DLS/CCLRC Cockcroft Institute 2005/6, © N.Marks, 2006 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. Neil Marks; DLS/CCLRC Cockcroft Institute 2005/6, © N.Marks, 2006 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. Neil Marks; DLS/CCLRC Cockcroft Institute 2005/6, © N.Marks, 2006 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. Neil Marks; DLS/CCLRC Cockcroft Institute 2005/6, © N.Marks, 2006 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) Neil Marks; DLS/CCLRC Cockcroft Institute 2005/6, © N.Marks, 2006 DK 70 (low silicon non-oriented steel) Losses at 50 Hz: W/kg vs peak induction (T) for 0.65 and 0.5 mm laminations. Neil Marks; DLS/CCLRC Magnetisation curves: B(T) vs H (A/m) Cockcroft Institute 2005/6, © N.Marks, 2006 CK 27 (high silicon non-oriented steel) Losses at 50 Hz: W/kg vs peak induction (T) for 0.5 and 0.35 mm laminations. Neil Marks; DLS/CCLRC Magnetisation curves: B(T) vs H (A/m) Cockcroft Institute 2005/6, © N.Marks, 2006 M4, M5, M6 grain oriented steel. Losses at 50 Hz: W/kg vs peak induction (T) for 0.27, 0.30 and 0.35 mm laminations. Neil Marks; DLS/CCLRC Magnetisation curves: B(T) vs H (A/m) Cockcroft Institute 2005/6, © N.Marks, 2006 Solid low carbon steel. Neil Marks; DLS/CCLRC Cockcroft Institute 2005/6, © N.Marks, 2006 Comparisons Property: DK-70: CK-27: 27 M 3: XC06 : Type Nonoriented Low 0.65 mm Nonoriented High 0.35 mm Grainoriented 0.27 mm Nonoriented Very low Solid Silicon content Lam thickness a.c. loss (50 Hz): at 1.5 T peak Permeability: at B=1.5 T at B=1.8 T 6.9 W/kg 2.25 W/kg 1680 184 990 122 Neil Marks; DLS/CCLRC 0.79 W/kg Not suitable > 10,000 3,100 >1,000 >160 Cockcroft Institute 2005/6, © N.Marks, 2006 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. Rolling direction. B Difficult (impossible?) to make each limb out of separate strips of steel. Neil Marks; DLS/CCLRC Cockcroft Institute 2005/6, © N.Marks, 2006 Magnet Inductance F Definition: n turns, L = n F /I Inductance: Dipole Inductance. current I 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. Neil Marks; DLS/CCLRC Cockcroft Institute 2005/6, © N.Marks, 2006 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. Neil Marks; DLS/CCLRC Cockcroft Institute 2005/6, © N.Marks, 2006 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. Neil Marks; DLS/CCLRC Cockcroft Institute 2005/6, © N.Marks, 2006