Magnetic Model of the CERN PS Accelerator

1 24 February, 2012

Magnetic Model of the CERN PS Accelerator

ECOLE DOCTORALEDOCTORAL SCHOOL

PROGRAMME DOCTORAL EN PHYSIQUEDOCTORAL PROGRAM IN PHYSICS

Mariusz Juchno

Miniworkshop on PS Main Magnet field issues

2

Objectives To develop a model of the magnetic field

inside the PS magnets, capable of accurately recreating the magnetic field along the beam trajectory.

Implement and validate the magnetic model inside existing optical model of the PS accelerator.

24 February, 2012Miniworkshop on PS Main Magnet field issues

3

Methodology

24 February, 2012

Investigation of the field development inside the PS magnet Broad numerical analysis in 2D and 3D Magnetic measurements

Derivation of quasi-static formulas of the field components.

Implementation of the magnetic model in existing optical model the PS accelerator. Simulation of the optical parameters with MAD-X

model. Beam-based measurements (tune and chromaticity). Verification and calibration of the magnetic model. Optical model enhancements.


4

Proton Synchrotron main magnetic unit

24 February, 2012

Combined-function magnet with hyperbolic pole shape Dipole field – guiding Quadrupole field – focusing Higher component are also

present due to saturation Focusing and defocusing half

(alternating-gradient focusing) 5 C-shaped block in each half Wedge shaped air gaps

between blocks Complex geometry of coils

system In total 100+1 main units of

four different types.


5

Coils of the PS magnet

24 February, 2012

Main coil Dipole and quadrupole field

mostly Figure-of-eight loop

Adjusts quadrupole field but also contributes to dipole field

Pole-face windings (PFW) Separately for focusing and

defocusing half Each winding has narrow and

wide circuit Corrects higher components of

the field

PFW Powering upgrade Five currents (If8, IpfwFN, IpfwFW, IpfwDN,

IpfwDW) insted of three (If8, IpfwF, IpfwD) Control of the four beam parameters

Qh, Qv, ξh, ξv

One current remains free for controlling an additional physical parameter

Possibility of exploring new working points

Debalancing PFW narrow and wide circuits leads to strong nonlinearities !!!


6

Investigating contributions of separate circuits

24 February, 2012

2D quasi-static numerical analysis (OPERA) of the magnetic field inside the PS magnet.

Range of operations: Injection pinj = 2.12 GeV/c Extraction pextr = 26 GeV/c

Current range: Main coil Imc = 400-5500 A (ΔImc = 250 &

500 A) Figure-of-eight loop If8 = ±1200 A (ΔIf8 =

600 A) Pole-face windings Ipfw = ±200 A (Δ Ipfw =

100 A)Miniworkshop on PS Main Magnet field issues

7

Contribution of auxiliary circuits

24 February, 2012

-0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05-14

-12

-10

-8

-6

-4

-2

0

2

4x 10

-5 Dipole contribution of auxiliary coils in defocusing half-unit

r [m]

B

[T

/A]

figure-eight

wide pole-facenarrow pole-face

n+w pole-face

-0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0x 10

-4 Dipole contribution of auxiliary coils in focusing half-unit

r [m]

B

[T

/A]

figure-eight


n+w pole-face

-0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05-3

-2.5

-2

-1.5

-1

-0.5

0

0.5x 10

-3 Quadrupole contribution of auxiliary coils in defocusing half-unit

r [m]

G

[T

m-1

/A]

figure-eight


n+w pole-face

-0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05-0.5

0

0.5

1

1.5

2

2.5

3x 10

-3 Quadrupole contribution of auxiliary coils in focusing half-unit

r [m]

G

[T

m-1

/A]

figure-eight


n+w pole-face

Focusing Defocusing

DipoleContribution

ΔB [T/A]

QuadrupoleContribution

ΔG [Tm-1/A]


8

Contribution of auxiliary circuits

24 February, 2012

-0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05Sextupole contribution of auxiliary coils in defocusing half-unit

r [m]

S

[T

m-2

/A]

figure-eight

wide pole-face

narrow pole-face

n+w pole-face

figure-eight [mTm-2/A]

-0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05Sextupole contribution of auxiliary coils in focusing half-unit

r [m]

S

[T

m-2

/A]

figure-eight


n+w pole-face


-0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05-1.5

-1

-0.5

0

0.5

1

1.5Octupole contribution of auxiliary coils in defocusing half-unit

r [m]

O

[T

m-3

/A]

figure-eight


n+w pole-face


-0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05-1.5

-1

-0.5

0

0.5

1

1.5Octupole contribution of auxiliary coils in focusing half-unit

r [m]

O

[T

m-3

/A]

figure-eight


n+w pole-face


Focusing Defocusing

SextupoleContribution

ΔS [Tm-2/A]

OctupoleContribution

ΔO [Tm-3/A]


9

Formulas of the field model

24 February, 2012

Field multipoles in the Taylor coefficients [T/mn-1]

Main and auxiliary field multipoles

Total multipole component

0

1

1

xx

ny

n

n dxxBd

xB

mcmcmcnfmcmcmcn INTINB ,,

auxnauxnfauxauxauxn FTINB ,,, gT f 0

Linear field transfer function

aux auxauxauxnmcmcauxn INfINF ,,

Equivalent magnetomotiveforce

aux auxnmcntotn BBB ,,,


10

Formulas of the field model

24 February, 2012

Circuit efficiency function [1/mn-1]

Main circuit efficiency

Circuit saturation

1,'1 nnmcn ggn

0

1

1

xx

nmc

n

n dxxdx

i nom

isnin

innn NI

NINIsNI tanh1

210

0

0

Fecorecore

gapgap

gapcore

gapmc

AlR

AgR

RR

R

Efficiency functions example (dipole component)


11

Effective magnetic length corrections

24 February, 2012

3D numerical analysis

-3 -2 -1 0 1 2 3-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4Dipole field, current range 500 A (blue) - 6000 A (red)

B [

T]

z [m]


12


24 February, 2012


-3 -2 -1 0 1 2 3-0.2

0

0.2

0.4

0.6

0.8

1

1.2


B [

T]

z [m]

-3 -2 -1 0 1 2 3-6

-4

-2

0

2

4

6Quadrupolar field, current range 500 A (blue) - 6000 A (red)

G [

T/m

]

z [m]


13


24 February, 2012


-3 -2 -1 0 1 2 3-0.2

0

0.2

0.4

0.6

0.8

1

1.2


B [

T]

z [m]

-3 -2 -1 0 1 2 3-6

-4

-2

0

2

4

6Quadrupolar field, current range 500 A (blue) - 6000 A (red)

G [

T/m

]

z [m]

-3 -2 -1 0 1 2 3-15

-10

-5

0

5

10

15Sextupolar field, current range 500 A (blue) - 6000 A (red)

S [

T/m

2 ]

z [m]

Correction components Magnet ends Junction Block gaps

Contain the pole-face angle effect

High variation of the sextupolar distribution

Octupolar component ?


14


24 February, 2012

Bare machine corrections

0 0.2 0.4 0.6 0.8 1 1.2 1.40.03

0.035

0.04

0.045

0.05

0.055

0.06

0.065

0.07

0.075Bending length correction

B [T]

l B

[m

]

old measurement F

old measurement D3D model F

3D model D


15


24 February, 2012


0 0.2 0.4 0.6 0.8 1 1.2 1.40.03

0.035

0.04

0.045

0.05

0.055

0.06

0.065

0.07


B [T]

l B

[m

]

old measurement F


3D model D

0 0.2 0.4 0.6 0.8 1 1.2 1.4-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04Gradient length correction

B [T]

l G

[m

]

old measurement F


3D model D


16


24 February, 2012


0 0.2 0.4 0.6 0.8 1 1.2 1.40.03

0.035

0.04

0.045

0.05

0.055

0.06

0.065

0.07


B [T]

l B

[m

]

old measurement F


3D model D

0 0.2 0.4 0.6 0.8 1 1.2 1.4-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04Gradient length correction

B [T]

l G

[m

]

old measurement F


3D model D

0 0.2 0.4 0.6 0.8 1 1.2 1.4-12

-10

-8

-6

-4

-2

0

2

B [T]

S [

T/m

]

Integral of the sextupolar reference field along the magnet length

Sfref

Sdref

Sfcor

Sdcor

Validation with beam-based measurements

Quadrupolar correction discrepancy (JUNCTION)

Strong sextupolar correction

Radial position dependency!

Auxiliary coils! Octupolar component?


17

Magnet representation in the optical model

24 February, 2012

Official optics Static elements length SBEND

Bare machine 14 GeV/c quadrupolar component

No pole-face angle MULTIPOLE

Beam-based fit JUNCTION=DRIFT

Model optics Dynamic elements length

– effective length correction

SBEND Up to K2 from the model Integrated pole-face angle

effect MULTIPOLES

K3 (and higher?) No JUNCTION element Beam-based matched

effective lengths corrections?

DefocusingHalf-unit(SBEND)

FocusingHalf-unit(SBEND)

Drift space

(DRIFT)

Drift space

(DRIFT)

Defocusing higher order components

(MULTIPOLE)


(MULTIPOLE)

DefocusingHalf-unit(SBEND)

FocusingHalf-unit(SBEND)

Drift space(DRIFT)

Drift space

(DRIFT)

Junction(SBEND)

Defocusing higher order components(MULTIPOLE)


(MULTIPOLE)


18

Flowchart: corrections for the basic case

24 February, 2012

Iaux

ΔIaux=0Btr

Magnetic model+Optimisation

Btr(Imc,Iaux)=(1.091 BF+0.909

BD)/2

RequiredImc

Field control loop

Magnetic model

K0, K1, K2, K3

for T, U, R and S

magnets

Measured Q and ξ

for ΔIaux=0 and Δp/p=0

MAD-X+MATCHING

Effective length

correction for K1 and

K2


19

Flowchart: chromaticity analysis for ΔIaux

24 February, 2012

Iaux

ΔIaux

Btr

Magnetic model+Optimisation

Btr(Imc,Iaux)=(1.091 BF+0.909

BD)/2

RequiredImc

Field control loop

Magnetic modelK0, K1, K2, K3

for ΔIaux

Δp/p, correctionsfor ΔIaux=0

MAD-X(PTC)

Q and ξas function of

Δp/p, Imc and

Iaux+ΔIaux


20

Nonlinear chromaticity (14 GeV)

24 February, 2012

-0.01 -0.008 -0.006 -0.004 -0.002 0 0.002 0.004 0.006 0.008 0.016.1

6.15

6.2

6.25

6.3

6.35

6.4

6.45

p/p

Qh,Q

v

Figure-of-eight loop variation (If8

= 50.0A)

-0.01 -0.008 -0.006 -0.004 -0.002 0 0.002 0.004 0.006 0.008 0.016.1

6.15

6.2

6.25

6.3

6.35

6.4

p/p

Qh,Q

v

Focusing narrow PFW variation (Ifn

= 5.0A)

-0.01 -0.008 -0.006 -0.004 -0.002 0 0.002 0.004 0.006 0.008 0.016.15

6.2

6.25

6.3

6.35

6.4

p/p

Qh,Q

v

Focusing wide PFW variation (Ifw

= 5.0A)

-0.01 -0.008 -0.006 -0.004 -0.002 0 0.002 0.004 0.006 0.008 0.016.15

6.2

6.25

6.3

6.35

6.4

p/p

Qh,Q

v

Defocusing narrow PFW variation (Idn

= 5.0A)

-0.01 -0.008 -0.006 -0.004 -0.002 0 0.002 0.004 0.006 0.008 0.016.15

6.2

6.25

6.3

6.35

6.4

p/p

Qh,Q

v

Defocusing wide PFW variation (Idw

= 5.0A)

Figure-of-eight variation Wide focusing PFW variation Wide defocusing PFW variation

Narrow focusing PFW variation Narrow defocusing PFW variation

ΔIaux=5A ΔIaux=5A

ΔIaux=5A ΔIaux=5AΔIaux=50A

If8=543.3A If=43.5A Id=-52.56AMeasurement data: matrix measurement campaingMiniworkshop on PS Main Magnet field issues

21

14 GeV Transfer Matrices

24 February, 2012

Δ Ifn Δ Ifw Δ Idn Δ Idw Δ If8

Δ Qh 0.00457 0.00486

-0.00250

-0.00321

-0.00125

Δ Qv

-0.00235

-0.00312 0.00447 0.00481 0.00128

Δ ξh 0.14514-

0.04095 0.08578-

0.01965 0.00076

Δ ξv

-0.09837 0.02351

-0.11875 0.03079

0.00023


Δ Qh 0.00462 0.00473

-0.00252

-0.00313

-0.00184

Δ Qv

-0.00247

-0.00317 0.00458 0.00477 0.00191

Δ ξh 0.12792-

0.02221 0.07440-

0.01440 0.00000

Δ ξv

-0.08729 0.01300

-0.10619 0.02190

0.00000


Δ Qh 0.00288 0.00525

-0.00360

-0.00292

-0.00126

Δ Qv

-0.00119

-0.00333 0.00602 0.00436 0.00127

Δ ξh 0.12698-

0.02163 0.09310-

0.02615 0.00072

Δ ξv

-0.08603 0.01115

-0.12886 0.04016

0.00025


Δ Qh 0.00283 0.00455

-0.00314

-0.00268

-0.00121

Δ Qv

-0.00128

-0.00322 0.00512 0.00410 0.00121

Δ ξh 0.11215-

0.01152 0.07699-

0.01671 0.00079

Δ ξv

-0.07358 0.00768

-0.10599 0.02229

-0.0003

3

Predicted in 1974 Reproduced with the model

Measured matrix Reproduced with the modelfor dp/p= -0.002

In the model MRP=0 for dp/p=0 BUT in reality MRP≠0 for dp/p=0


22


24 February, 2012

-0.01 -0.005 0 0.005 0.016

6.05

6.1

6.15

6.2

6.25

6.3

6.35

p/p

Qh,Q

v


= 5.0A)

-0.01 -0.005 0 0.005 0.016

6.05

6.1

6.15

6.2

6.25

6.3

6.35

p/p

Qh,Q

v


= 2.0A)

-0.01 -0.005 0 0.005 0.016

6.05

6.1

6.15

6.2

6.25

6.3

p/p

Qh,Q

v


= 2.0A)

-0.01 -0.005 0 0.005 0.016

6.05

6.1

6.15

6.2

6.25

6.3

6.35

p/p

Qh,Q

v


= 2.0A)

-0.01 -0.005 0 0.005 0.016

6.05

6.1

6.15

6.2

6.25

6.3

6.35

p/p

Qh,Q

v


= 2.0A)



ΔIaux=2A

ΔIaux=2A


If8=-0.018 A Ifn=-0.015A Ifw=-14.545A Idn=-4.669A Idw=-8.235A Measurement data: A.

HuschauerMiniworkshop on PS Main Magnet field issues

23

Nonlinear chromaticity (3.5 GeV)

24 February, 2012

-0.01 -0.008 -0.006 -0.004 -0.002 0 0.002 0.004 0.006 0.008 0.016.15

6.2

6.25

6.3

6.35

6.4

p/p

Qh,Q

v


= 7.0A)

-0.01 -0.008 -0.006 -0.004 -0.002 0 0.002 0.004 0.006 0.008 0.016.1

6.15

6.2

6.25

6.3

6.35

6.4

p/p

Qh,Q

v


= 1.5A)

-0.01 -0.008 -0.006 -0.004 -0.002 0 0.002 0.004 0.006 0.008 0.016.15

6.2

6.25

6.3

6.35

6.4

p/p

Qh,Q

v


= 2.0A)

-0.01 -0.008 -0.006 -0.004 -0.002 0 0.002 0.004 0.006 0.008 0.016.15

6.2

6.25

6.3

6.35

6.4

6.45

6.5

p/p

Qh,Q

v


= 2.0A)

-0.01 -0.008 -0.006 -0.004 -0.002 0 0.002 0.004 0.006 0.008 0.016.15

6.2

6.25

6.3

6.35

6.4

p/p

Qh,Q

v


= 2.0A)



ΔIaux=1.5A ΔIaux=2A


Iaux=0A Measurement data: matrix measurement

campaingMiniworkshop on PS Main Magnet field issues

24


24 February, 2012

-0.01 -0.008 -0.006 -0.004 -0.002 0 0.002 0.004 0.006 0.008 0.016.14

6.16

6.18

6.2

6.22

6.24

6.26

6.28

6.3

6.32

p/p

Qh,Q

v


= 140.0A)

-0.01 -0.008 -0.006 -0.004 -0.002 0 0.002 0.004 0.006 0.008 0.01

6.1

6.15

6.2

6.25

6.3

6.35

6.4

p/p

Qh,Q

v


= 20.0A)

-0.01 -0.008 -0.006 -0.004 -0.002 0 0.002 0.004 0.006 0.008 0.016.14

6.16

6.18

6.2

6.22

6.24

6.26

6.28

6.3

6.32

p/p

Qh,Q

v


= 20.0A)

-0.01 -0.008 -0.006 -0.004 -0.002 0 0.002 0.004 0.006 0.008 0.01

6.1

6.15

6.2

6.25

6.3

p/p

Qh,Q

v


= 15.0A)

-0.01 -0.008 -0.006 -0.004 -0.002 0 0.002 0.004 0.006 0.008 0.016.16

6.18

6.2

6.22

6.24

6.26

6.28

6.3

6.32

6.34

p/p

Qh,Q

v


= 20.0A)



ΔIaux=20A

ΔIaux=15A

ΔIaux=20A

ΔIaux=20A

ΔIaux=140A

If8=1370.8A If=205.1A Id=80.4AMeasurement data: matrix measurement campaingMiniworkshop on PS Main Magnet field issues

25

What next?

24 February, 2012

Further validation with the beam-based measurements

Real-time magnetic measurements with a prototype coil

Effective length corrections Understanding discrepancies Investigating radial position dependency Implementing auxiliary coils dependency

Detailed nonlinear chromaticity analysis Consolidation with the up to date (official)

optics model


26

Possible error sources

24 February, 2012

Random errors Manufacturing tolerances

Numerical estimation by introducing random displacements within manufacturing tolerances (Monte-Carlo)

Coils position Pole shape Blocks alignement

Systematic errors Magnetic field related displacement

Poles atractiontion (Th. Zickler, Deformation Measurements on the PS Main Magnets)

Lorentz forces (coils, eddy currents) Main coil terminals


Magnetic Model of the CERN PS Accelerator

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