AbstractCalculations 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.

 

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Calculations of harmonics of quadrupole magnets

Budker Institute of Nuclear PhysicsRybitskaya Tatyana

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:

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Software used for calculationsMERMAID Mermaid applies first order Finite

Elements Method on the mesh consisted of triangle prisms for field computation. Created at Budker Institute of Nuclear physics.

OPERAA finite element environment for the complete analysis and design of electromagnetic applications in 3 dimensions. Product of company COBHAM

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Magnetic field expansion

1

1

0

)cossin(),,(0

0n

nn

nz

z

nBnArr

dzzrB

r

z

φ

1

1

0

)sincos(),,(0

0n

nn

nz

z

r nBnArr

dzzrB

4

2

10B

Aa n

n

4

2

10BB

b nn

On radius r0 = 2.5 cm

Skew harmonics

Normal harmonics

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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

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

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Shimming of magnet

K.HalbachFirst order perturbation effect in iron-dominated two dimensional symmetric multipoles // NIM vol.74, 1969.

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Relative changing of harmonic volume. Bumps along profile has the same area.

β

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

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Calculations of chamfers of long magnets

Without chamfer

Current 125 A

MERMAID10

Sextupole harmonics of current

First measurements of sextupole harmonics of first long magnet

9807-0001 measurements

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a3 current dependence was quite big

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.

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Steel properties measured at DESY

Calculations of a3 current dependence for long magnets

MERMAID

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If the worst case variation of a3 can be 2.5 units.

Δ=1.1

Δ=1.5

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

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Harmonics correction by bronze bars

+ ∆l

- ∆l/2- ∆l/2 - ∆l

+ ∆l

+ ∆l

- ∆l

Calculations ∆l = 20 µm

OPERA15

Widths of two insertions can correct a3 and b4

One bar width is increased

Two bars widths are increased simultaneously

Harmonics correction by nose pieces

+ ∆l

+ ∆l

+ ∆l

+ ∆l

Calculations ∆l = 150 µm

OPERA16

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.

Correction of allowed harmonics on ready magnet

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Nose pieces with modified profile

Calculations of magnetic field of middle length magnets

(9810 and 9813)

Chamfer 12.8 mm Chamfer 9.6 mm

MERMAID18

Measurements of magnetic field of magnet 9813-0001

Chamfer 13mm Chamfer 10mm

modified profile

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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.

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THE END

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

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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?

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