Lecture 5Jack Tanabe

Old Dominion UniversityHampton, VAJanuary 2011

[1] Halbach, K., FIRST ORDER PERTURBATION EFFECTS IN IRON-DOMINATED TWO-DIMENSIONAL SYMMETRICAL MULTIPOLES, “Nuclear Instruments and Methods”, Volume 74 (1969) No. 1, pp. 147-164. [2] Halbach, K., and R. Yourd, TABLES AND GRAPHS OF FIRST ORDER PERTURBATION EFFECTS IN IRON-DOMINATED TWO-DIMENSIONAL SYMMETRICAL MULTIPOLES, LBNL Internal Report, UCRL-18916, UC-34 Physics, TID 4500 (54thEd.), May 1969.

Perturbations

• The subject of Perturbations is covered in chapter 4 of the text and is one of the more important subjects covered in this course. This is because the performance of an accelerator lattice is dominated by the quality and reproducibility of the magnets fabricated/installed in the lattice.

• Perturbations are characterized by the multipole content of a magnet. A good magnet is characterized by the harmonic content of its integrated field. – A perfect dipole is characterized by F1= C1 z

– A perfect quadrupole is characterized by F2= C2 z2

– A perfect sextupole is characterized by F3= C3

• Tables of coefficients and expressions are presented which make it possible to compute the magnitude of error multipoles resulting from fabrication, assembly and pole excitation errors associated with magnet manufacture. – Pole excitation errors can result from shorted coil turns,

errors in winding the coils or other sources. • The other sources of pole excitation errors are poles

which differ in length from other poles.• Pole excitation errors can also be introduced

intentionally in order to produce trim fields, those fields which do not normally exist in a particular yoke geometry.

• Physics requirements normally specify the maximum amplitude of the various multipole errors. As a magnet designer, these multipole errors must be translated into fabrication/assembly tolerances. Examples of tolerance calculations are given in Section 4.3.5 of the text.

Effect of Mechanical Fabrication Errors on Error Multipole Content

• In the previous lecture, we showed that the field distribution in a magnet can be characterized by a function of the complex variable, z. In particular;

Nn

nn

NN zCzCF

Nn

nn zCsErrorField

ComponentFieldlFundamentazC NN

Random Multipole Errors Introduced by Pole Excitation

and Pole Placement Errors • Random multipole errors are introduced if the poles are

improperly excited or assembly errors which displace poles are introduced. If one can identify these errors, one can predict the multipole content of the magnet. The means for calculating these errors are summarized in two papers published by Klaus Halbach. The first paper describes the derivation of the relationships, the second computes and tabulates the coefficients used to calculate the multipole errors from the perturbations derived in the first paper.

• A portion of a table from UCRL-18916 by Halbach and Yourd is reproduced below. This table is for quadrupoles, N=2.

n

N

Cn j i

1 1.99 • 10-1 -4.25 • 10-1 7.46 • 10-2 1.76 • 10-12 2.50 • 10-1 -5.16 • 10-1 2.14 • 10-1 5.00 • 10-13 1.57 • 10-1 -2.88 • 10-1 2.88 • 10-1 6.60 • 10-14 0 6.76 • 10-2 2.31 • 10-1 5.00 • 10-15 -2.05 • 10-2 1.08 • 10-1 1.08 • 10-1 1.91 • 10-16 0 -4.45 • 10-2 2.87 • 10-2 07 1.61 • 10-2 -1.04 • 10-2 1.04 • 10-2 -3.06 • 10-28 0 1.28 • 10-2 1.56 • 10-2 09 -1.90 • 10-3 1.25 • 10-2 1.25 • 10-2 7.53 • 10-310 0 6.37 • 10-3 5.81 • 10-3 0

n

N

Cn rd i

n

N

Cn ad

n

N

Cn r

n

• This table of coefficients is used to estimate the error multipole due to excitation, radial, azimuthal and rotation errors in the location of a pole on the horizontal axis. The imaginary term in the denominator for the excitation (j) and radial (rd) terms indicate that errors in these quantities introduce skew terms. The tabulated coefficients are computed for poles centered on the positive horizontal axis. For the normal multipole magnet (one whose axis is rotated away from the horizontal axis), the expression for calculating the error multipole normalized to the desired fundamental, evaluated at the pole radius, is

given by;

inn

N

n eCN

n

H

H *

*where is the angle of the pole(s) which

have been perturbed.

Example Calculation• Suppose we construct a 35 mm radius quadrupole whose first pole is

radially offset by 1 mm. What is the effect on the n=3 (sextupole) multipole error.

35

1

r

position azimuthal =

0

rd

288.0

i

rdC

N

n n where n=3.

434

3

23

43

*

*

1023.81023.8

35

1288.0

iii

iin

rdnin

nN

n

eee

eieiNi

CneC

N

n

H

H

Meaning of the Result

• The calculation means that the n=3 (sextupole) error multipole normalized to the fundamental field at the pole radius (35 mm) is approximately 0.8% due to the radial displacement of the first pole at /4 by 1 mm.

• Carrying the calculation further to determine the phases;

4sin

4cos1023.8

1023.8

3

43

35@2

33

i

eiH

iHH i

mmy

yx

• Further simplification;

2

2

2

21023.8 3

35@2

33

35@2

33

i

H

HiH

iH

iHH

mmy

yx

mmy

yx

iH

iHH

mmy

xy

11082.5 3

35@2

33

This means that the real component of the sextupole error is out of phase with the fundamental field and that a positive skew component of the sextupole field also exists.

Evaluation at the Required Good Field Radius

• We recall that the field for an n multipole varies as zn-1. Therefore if the good field radius is r0=30 mm, the n=3 normalized multipole error can be evaluated at this radius.

35

30

3530

3530

35@2

33

2

35@2

33

30@2

33

mmy

xy

mmy

xy

mmy

xy

H

iHH

H

iHH

H

iHH

Other Errors

• The coefficient table can be used for other errors.

excitation

excitation =

j

0r

position radial =

rd

radiansrotation = pole

0r

position azimuthal =

ad

The Error Multipole Spectrum• In general, the table of coefficients includes entries for all the

multipole indices. Therefore, although the sample calculation was performed only for the sextupole error field component, all the multipole errors due to the radial misalignment of the first pole exist.

• These errors include the error in the fundamental (n=2) as well as the

n=1 dipole field.

Real and Skew Components of Error Multipole due to 1 mm Radial Displacement of Pole #3 for a Quadrupole

-1.5E-02

-1.0E-02

-5.0E-03

0.0E+00

5.0E-03

1.0E-02

1 2 3 4 5 6 7 8 9 10

Multipole Index

Re

& S

k B

n/B

2 @

Po

le R

adiu

s

Re35mm

Im35mm

Error Amplitudes as a Function of Radius

Error Multipole Spectrum due to 1 mm Radial Displacement of Pole #1 for a Quadrupole

1.E-04

1.E-03

1.E-02

1.E-01

1 2 3 4 5 6 7 8 9 10

Multipole Index

|Bn

/B2|

@ R

=35

and

32

mm

.

Error35

Error30

Lesson• The lesson from this sample calculation is not the detailed

calculation of the multipole error, but the estimate of the mechanical assembly tolerances which must be met in order to achieve a required field quality.

• In general, the coefficient is <0.5. Therefore in order to achieve a field error at the pole radius of 5 parts in 10000 (a typical multipole error tolerance), the following tolerance illustrated in the calculation must be maintained.

*

*

PoleRadiustCoefficien

H

H

N

n

inchmm

mm

tCoefficien

PoleRadius

H

H

requiredN

n

0014.0035.0

5.0

35105 4

*

*

A very small error.

Another LessonThe Magnet Center

• We note that all the multipole errors are introduced by mechanical assembly errors. In particular, we look in detail at the dipole error term introduced by assembly errors.

• For the pure quadrupole field, the expression for the complex function is;

22 zCzF

If the magnet center is shifted by an amount z, the expression becomes;

22 zzCzF

zCizCizzCiziFH 222'* 222

The first term in this expression is the quadrupole field (a linear function of z). The second term is a constant and, therefore, is the dipole field.

• Rewriting the expression as the sum of two fields;

yixCizCiHHH 22*1

*2

* 22

yixCiiHHH yx 211*1 2

zC

yixCi

H

iHH yx

2

2

*2

11

2

2

000

@

*2

1

@

*2

1

@

*2

11

000

r

y

r

xi

r

yixi

H

Hi

H

H

H

iHH

r

y

r

x

r

yx

0

0

2

10

2

10

r

x

r

y

H

Hry

H

Hrx

Evaluating the quadrupole field at the pole radius, r0;

Equating the real and imaginary parts of the expression, the magnetic center shift can be evaluated from the dipole field.

• These relationships may be more easily visualized with a figure

H2 r0

H2 r0

H1y

H1x

r0

r0

x

y

Effect of a Pole Excitation Error on the Magnetic Center

• One of the many issues faced by the NLC project is the stability of the magnetic center of adjustable hybrid permanent magnet quadrupoles.

• A sample calculation is made to compute the required pole excitation precision.

excitation

excitation =

j

1-10.991= Ni

jnCn For n = 1 (dipole error)

inn

N

n eCN

n

H

H *

*

where

• Suppose we have a 1% error on the excitation of a single pole in the first quadrant.

4sin

4cos002.0 01.0199.0 4

*2

*1

iieieCN

n

H

H iin

n

0014.00014.0707.0707.0002.02

11

2

11

iii

H

iHH

iH

iHH

y

xy

y

yx

Equating the real and imaginary terms,

0014.02

1 y

y

H

H

0014.02

1 y

x

H

H0

2

10

02

10

0.0014

0.0014 -

0

0

rH

Hry

rH

Hrx

r

x

r

y

The Four Piece Magnet Yoke• The ideal assembly satisfies the

rotational symmetry requirements so that the only error multipoles are allowed multipoles, n=6, 10, 14 ... However, each segment can be assembled with errors with three kinematic motions, x, y and (rotation). Thus, combining the possible errors of the three segments with respect to the datum segment, the core assembly can be assembled with errors with 3x3x3=27 degrees of freedom.

x1

y1

1

x2

y2

2

x3

y3

3

datum

The Two Piece Magnet Yoke

• This assembly has the advantage that the two core halves can be assembled kinematically with only three degrees of freedom for assembly errors. Thus, assembly errors are more easily measured and controlled.

x1

y1

1

Coefficients for a Two-Piece Quadrupole

1 0 0 2.49 • 10-12 3.02 • 10-1 -7.30 • 10-1 03 0 0 -9.33 • 10-14 -9.56 • 10-2 -3.27 • 10-1 05 0 0 -2.70 • 10-16 -4.06 • 10-2 -6.29 • 10-2 07 0 0 -4.33 • 10-28 1.81 • 10-2 2.20 • 10-2 09 0 0 1.06 • 10-210 8.22 • 10-3 9.01 • 10-3 011 0 0 5.12 • 10-312 -3.77 • 10-3 -3.94 • 10-3 013 0 0 -1.31 • 10-314 -1.74 • 10-3 -1.78 • 10-3 015 0 0 -9.41 • 10-416 8.14 • 10-4 8.23 • 10-4 0

xi

C

N

n n

y

C

N

n n

nC

N

nn

Two Piece Quadrupole Error Computations• The computations of the multipole error fields due to assembly errors of the

two piece quadrupole are similar but simpler than the computations for the four piece quadrupole. In the expressions below, h is the pole radius.

• The error terms are evaluated at the pole radius.

nN

n CN

n

H

H

*

*

h

xx

h

yy

=Rotational Perturbation

h

xitCoefficien

H

H

xN

n *

*

h

ytCoefficien

H

H

yN

n *

*

*

*

tCoefficien

H

H

RotationN

n

Why i?

• Referring to the table; – The shear motion of the top half of the magnet

with respect to the bottom introduces skew even multipole errors.

– The vertical motion introduces real even multipole errors.

– The rotational motion introduces real odd multipole errors.

h=pole radius

xx

x

hh=pole radius

yy

y

h

h=pole radius

2

Experiments

• Computations using the coefficients for the two piece magnet have been compared to experiments where the upper half of a magnet was intentionally displaced with respect to its nominal position.

A

B

C

D

E

F

Water Connection

Search Coil

Measurement Points

Vertical Perturbation

A = .015"

B = .005"

C = .011"

Ave. = .0103"

D = .015"

E = .007"

F = .009"

Ave. = .0103"

Left Side Right Side

A = .018"

B = .012"

C = .021"

Ave. = .0170"

D = .018"

E = .013"

F = .019"

Ave. = .0167"

Left Side Right Side

5 mil shims

Normal Assembly

Vertical Perturbation

161514131211109876543-15

-10

-5

0

5

dy meas rldy calc rldy meas im

for Vertical Perturbation

Multipole Number

Norm

alized

Mu

ltip

ole

(p

art

s in

10000)

• Vertical and Rotational Motion

A = .021"

B = .012"

C = .018"

Ave. = .0170"

D = .015"

E = .006"

F = .013"

Ave. = .0113"

Left Side Right Side

5 mil shims

14.5 inch

161514131211109876543-10

-5

0

5

dyrot meas rldyrot calc rldyrot meas im

for Vertical and Rotational Perturbation

Multipole Number

Norm

alized

Mu

ltip

ole

(

part

s in

10000)

• Shear and Vertical Motion

A = .016"

B = .009"

C = .015"

Ave. = .0133"

D = .016"

E = .010"

F = .016"

Ave. = .0140"

Left Side Right Side

5 mil shims

161514131211109876543-10

-5

0

5

dydx meas rldydx calc rl

for Vertical and Shear Perturbation

Multipole Number

Norm

alized

Real C

om

pon

en

t of

the

M

ult

ipole

Err

or

(part

s in

10000)

• Skew term due to vertical and shear perturbations.

161514131211109876543-5

0

5

dydx meas imdydx calc im

for Vertical and Shear Perturbations

Multipole Number

Norm

alized

Im

ag

inary

Com

pon

en

t of

the M

ult

ipole

Err

or

(part

s in

10000)

Other Applications

• In crowded lattices, there is often insufficient room to place all the desired magnetic elements. An occasional solution to this problem, employed both at ALS and at APS, is to provide trim windings on a sextupole yoke in order to obtain horizontal and vertical steering fields and skew quadrupole fields without introducing a sextupole field. (The controls sextupole fields want to be independent of the horizontal and vertical steering and the skew quadrupole controls.) The design of such trim windings and the evaluation of the field quality which results when employing these techniques exploit Klaus Halbach’s perturbation coefficients.

• The table of coefficients for N=3 (sextupole) for pole excitation error, , is reproduced from Klaus Halbach’s perturbation paper.

1 9.79E-022 1.56E-013 1.67E-014 1.33E-015 7.09E-0267 -1.34E-028 -1.07E-02910 9.13E-0311 9.72E-031213 -1.01E-0314 -1.18E-031516 1.63E-0317 2.07E-031819 -1.12E-0420 -1.70E-042122 3.25E-0423 4.65E-0424

iC

N

nx n

n

n

inn

N

n eCN

n

H

H *

*

inn

inn

N

n exiei

C

N

ni

H

H

*

*

sextupole

pole

NI

NI=

tCoefficienxn

where

and

Pole Perturbed of Angle=

Vertical Steering Trim(Horizontal Flux Lines)

• Vertical steering trim (horizontal flux lines) is achieved in a sextupole yoke by exciting the four horizontal poles.

Pole 1 = 30°

Pole 2 = 90°

Pole 3 = 150°

Pole 4 = 210°

Pole 5 = 270°

Pole 6 = 330°

+NI

+NI

-NI

-NI

Field Direction

• The formulation of the expressions follows:

poles

inn

poles

inn

nxnynynx

N

n

jj iexexi

H

iHH

iH

iHH

H

H

33*

*

6

11cos

6

7cos

6

5cos

6cos

6

11sin

6

7sin

6

5sin

6sin

6

11sin

6

7sin

6

5sin

6sin

6

11cos

6

7cos

6

5cos

6cos

nnnni

nnnn

nnnni

nnnn

iiepoles

in j

• Equating real and imaginary terms for the horizontal steering trim;

poles

inn

ny jiexH

H

3

Re

poles

inn

nx jiexH

H

3

Im

• Tabulating the results;

n1 9.79E-02 0 3.4641 0.33912 1.56E-01 0 0 03 1.67E-01 0 0 04 1.33E-01 0 0 05 7.09E-02 0 -3.4641 -0.24566 0 0 0 07 -1.34E-02 0 -3.4641 0.04648 -1.07E-02 0 0 09 0 0 0 010 9.13E-03 0 0 011 9.72E-03 0 3.4641 0.033712 0 0 0 013 -1.01E-03 0 3.4641 -0.003514 -1.18E-03 0 0 015 0 0 0 016 1.63E-03 0 0 017 2.07E-03 0 -3.4641 -0.007218 0 0 0 019 -1.12E-04 0 -3.4641 0.000420 -1.70E-04 0 0 021 0 0 0 022 3.25E-04 0 0 023 4.65E-04 0 3.4641 0.001624 0 0 0 0

jinie Re jinie Im

iC

N

nx nn

3 B

Bxn

Required Vertical Steering Trim Excitation

• From the table:

3391.0 3

H

H nx

3391.0 3391.0

3

1

3

1

sextupole

xx

NI

NI

B

B

H

H

2

"

2

""

""'

2

3@

2

hBBB

xBxdxBB

xBdxBB

h

0

3

6

"

hB

NI sextupole and

• Substituting;

6

"3391.0

2

"

0

321

3

1

hB

NI

hB

B

B

B xx

0

1

3391.03

hBNINI x

eeringVerticalSt

n1 12 03 04 05 -0.72426 07 0.13698 09 010 011 0.099312 013 -0.010314 015 016 017 -0.021118 019 0.001120 021 022 023 0.004724 0

x

n

B

B

1

From the table, we note that the n=5 multipole error is >70% of the fundamental (horizontal dipole) field at the pole radius.

Horizontal Steering Trim(Vertical Flux Lines)

• Horizontal steering trim can be achieved by exciting all six of the sextupole poles.

+NI+NI

-NI-NI

Field Direction

+2 NI

-2 NI

The excitation of the vertical poles is twice the excitation of the horizontal poles.

• Again, we can formulate the field in terms of the excitations of the various poles.

poles

inn

nxny jiexH

iHH

3

6

11cos

2

3cos2

6

7cos

6

5cos

2cos2

6cos

6

11sin

2

3sin2

6

7sin

6

5sin

2sin2

6sin

nnn

nnn

i

nnn

nnn

iepoles

in j

• Tabulating the results,

n1 9.79E-02 6 0 0.5874 12 1.56E-01 0 0 0 03 1.67E-01 0 0 0 04 1.33E-01 0 0 0 05 7.09E-02 6 0 0.4254 0.72426 0 0 0 07 -1.34E-02 -6 0 0.0804 0.13698 -1.07E-02 0 0 0 09 0 0 0 010 9.13E-03 0 0 0 011 9.72E-03 -6 0 -0.0583 -0.099312 0 0 0 013 -1.01E-03 6 0 -0.0061 -0.010314 -1.18E-03 0 0 0 015 0 0 0 016 1.63E-03 0 0 0 017 2.07E-03 6 0 0.0124 0.021118 0 0 0 019 -1.12E-04 -6 0 0.0007 0.001120 -1.70E-04 0 0 0 021 0 0 0 022 3.25E-04 0 0 0 023 4.65E-04 -6 0 -0.0028 -0.004724 0 0 0 0

jinie Re jinie Im

iC

N

nx n

n

y

n

B

B

1y

yn

B

B

3

Required Horizontal Steering Trim Excitation

• From the table:

5874.0 3

H

H ny

5874.0 5874.0

3

1

3

1

sextupole

yy

NI

NI

B

B

H

H

2

" 2

3@

hBBB h

0

3

6

"

hB

NI sextupole and

• Substituting;

6

"5874.0

2

"

0

32

1

3

1

hB

NI

hB

B

B

B yy

0

1

5874.03

hBNINI y

Steeringhorizontal

Skew Quadrupole Trim(Horizontal Flux Lines)

• Skew quadrupole trim field can be achieved by exciting two of the sextupole poles.

+NI

+NI

Again, we can formulate the field in terms of the excitations of the various poles.

poles

inn

nxny jiexH

iHH

3

2

3cos

2cos

2

3sin

2sin

nni

nnie

poles

in j

• Tabulating the results,

n1 9.79E-02 0 0 0 02 1.56E-01 0 -2 -0.3120 13 1.67E-01 0 0 0 04 1.33E-01 0 2 0.2660 -0.85265 7.09E-02 0 0 0 06 0 -2 0 07 -1.34E-02 0 0 0 08 -1.07E-02 0 2 -0.0214 0.06869 0 0 0 010 9.13E-03 0 -2 -0.0183 0.058511 9.72E-03 0 0 0 012 0 2 0 013 -1.01E-03 0 0 0 014 -1.18E-03 0 -2 0.0024 -0.007615 0 0 0 016 1.63E-03 0 2 0.0033 -0.010417 2.07E-03 0 0 0 018 0 -2 0 019 -1.12E-04 0 0 0 020 -1.70E-04 0 2 -0.0003 0.001121 0 0 0 022 3.25E-04 0 -2 -0.0007 0.002123 4.65E-04 0 0 0 024 0 2 0 0

jinie Re jinie Im

iC

N

nx n

n

y

xn

B

B

3 x

xn

B

B

2

Required Skew Quadrupole Trim Excitation

• From the table:

3120.0 3

H

H ny

3120.0 3120.0

3

1

3

1

sextupole

yy

NI

NI

B

B

H

H

2

" 2

3@

hBBB h

0

3

6

"

hB

NI sextupole and

• Substituting;

6

"3120.0

2

"

'

0

323

2

hB

NI

hB

hB

B

B x

0

2

3120.03

'

hBNINI poleSkewQuadru

Predicted n=4, 8 and 10 Multipole Errors

• Applying the perturbation theory, the multipole errors normalized to the skew quadrupole field, evaluated at the pole radius, h, can be computed.

853.0@2

4

hxB

B069.0

@2

8

hxB

B059.0

@2

10

hxB

B

Magnetic measurements were performed on the SPEAR3 production sextupole magnets with skew quadrupole windings. These measurements were evaluated at the required good field radius, 32 mm. The predicted normalized multipole errors at 32 mm can be computed.

2

@2

8

@2

n

hxrx

n

h

r

B

B

B

B

• These prediction are compared with measurements.

00386.045

32059.0

8

32@2

10

mmxB

B

431.045

32853.0

2

32@2

4

mmxB

B

00892.045

32069.0

6

32@2

8

mmxB

B

Sextupole – Skew Quadrupole Field Error Measurements

Sextupole Skew Quadrupole "Allowed" Multipoles

1.E-03

1.E-02

1.E-01

1.E+00P

redi

cted

21s-

01

21s-

02

21s-

03

21s-

04

21s-

05

21s-

06

21s-

07

21s-

08

21s-

09

21s-

10

21s-

11

Magnet

|Bn

/B2s

kew

| @ 3

2 m

m.

n=4

n=8

n=10

Closure• In many ways, this is one of the most important lectures. It

is important that the student understands the chapter on Perturbations since successfully translating the performance of the mathematical design to the magnets manufactured and installed in a synchrotron requires that mechanical manufacturing and assembly errors translates into field errors which can threaten the performance of the synchrotron.

• Understanding the impact of mechanical fabrication and assembly errors on the magnet performance and thus, the physics impacts of these errors, can provide the understanding so that mechanical tolerances can be properly assigned.

Lecture 6

• The material covered in lecture 6 is covered in chapter 5 of the text.

• Please read this chapter prior to the next lecture. Homework covered in this chapter will be assigned.