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Abstract Calculations of Harmonics of Quadrupole Magnets Rybitskaya Tatyana, Budker Institute of Nuclear Physics, Russia Design of pole profile and chamfers of NSLSII quadrupole magnets manufactured at BINP are presented. Calculations of dependence skew-sextupole harmonic of current are shown. Way for correction the dependence is described. Calculations of harmonics correction by bronze bars, nose pieces and modified nose pieces are presented. 1

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Calculations of harmonics of quadrupole magnets Budker Institute of Nuclear Physics Rybitskaya Tatyana

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Plan of presentation Allowed harmonics Dependence of skew-sextupole harmonic of current Harmonics correction by bronze bars Harmonics correction by nose pieces Nose pieces with modified profile Calculations of: 3

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Software used for calculations MERMAID Mermaid applies first order Finite Elements Method on the mesh consisted of triangle prisms for field computation. Created at Budker Institute of Nuclear physics. OPERA A finite element environment for the complete analysis and design of electromagnetic applications in 3 dimensions. Product of company COBHAM 4

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Magnetic field expansion r z φ On radius r 0 = 2.5 cm Skew harmonics Normal harmonics 5

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Profile features of magnets calculated Flat parts of shims (necessary for stacking and gluing magnets with high precision). Long (9804/9807) and middle (9810/9813) length magnets have nose pieces. Chamfers are flat 45°. 45° L L magnetic axis chamfer Pole/ nose piece Flat part 6 chamfer

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Allowed harmonics in magnet center were minimized Yoke edges in 3D provide big b6 2D or magnetic field in magnet centre plane MERMAID Calculations of allowed harmonics Chamfers decrease b6 but at the same time increase b10 3D without chamfer 120 A 3D with chamfer 120 A 7

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Shimming of magnet K.Halbach First order perturbation effect in iron-dominated two dimensional symmetric multipoles // NIM vol.74, 1969. 8 Relative changing of harmonic volume. Bumps along profile has the same area. β

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α L H magnetic axis chamfer nose piece Calculations of chamfers of long magnets α - angle between magnetic axis and chamfer plane Nose pieces size: 15 mm along magnetic axis 34.45 mm perpendicular magnetic axis The sizes limit size of chamfer 9

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Calculations of chamfers of long magnets Without chamfer Current 125 A MERMAID 10

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Sextupole harmonics of current First measurements of sextupole harmonics of first long magnet 9807-0001 measurements 11 a3 current dependence was quite big

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Possible reasons of a3 current dependence Different permeability of halves. Two samples were cut from steel roll – one from beginning, another from middle of roll. Different packing factor of halves. Measurements shown maximum difference 0.25% Different halves lengths. Measurements shown maximum difference 0.2 mm. 12 Steel properties measured at DESY

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Calculations of a3 current dependence for long magnets MERMAID 13 If the worst case variation of a3 can be 2.5 units. Δ=1.1 Δ=1.5

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Corrections of harmonic a3 dependence of current The most effective way for correction is special shuffling. Additional way for correction : OPERA Face of nose piece Shift need additional correction Δ=1.94 14

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Harmonics correction by bronze bars + ∆l - ∆l/2 - ∆l + ∆l - ∆l Calculations ∆l = 20 µm OPERA 15 Widths of two insertions can correct a3 and b4 One bar width is increased Two bars widths are increased simultaneously

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Harmonics correction by nose pieces + ∆l Calculations ∆l = 150 µ m OPERA 16 Each nose pieces can be shifted independently. Advantage: In principal many harmonics can be corrected. Disadvantage: Nose pieces should be installed carefully not to reduce field quality.

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Correction of allowed harmonics on ready magnet 17 Nose pieces with modified profile

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Calculations of magnetic field of middle length magnets (9810 and 9813) Chamfer 12.8 mmChamfer 9.6 mm MERMAID 18

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Measurements of magnetic field of magnet 9813-0001 Chamfer 13mm Chamfer 10mm modified profile 19

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Summary 2D profile was chosen with allowed harmonics less then 0.7 units, as result chamfering increase b10 up to 2.5 units for short magnets. Material properties can increase a3 dependence of current up to 2.5 units. Bronze bars provide corrections of a3 and b4. Nose pieces provide correction of b3 and additional correction of a3 and b4. Using nose pieces with modified profile make integral fields and fields in the magnet center “good” at the same time. 20

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THE END 21

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Comparison measurements and calculations of long magnets without nose pieces and chamfers 22

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Question Why calculations and measurements has deviations? Different magnetic steel properties in program and in reality? Changes of permeability at the age of plates cause of stamping? Something else? 23

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