Magnetic Measurement and Magnet Tutorial, Part 3 - SLAC

SLAC–WP–035 PSN WE08 May 2003

Magnetic Measurement and Magnet Tutorial, Part 3 ("Lecture 10")*

J. Tanabe Stanford Linear Accelerator Center, Stanford University, Stanford, California 94309

Invited talk presented at 13th International Magnetic Measurement Workshop

Stanford, California

May 19-22, 2003

* Work supported in part by Department of Energy contract DE–AC03–76SF00515.

Lecture 10Jack Tanabe

Santa Barbara, CA

Magnetic Measurements

13th International Magnetic Measurement Workshop, May 19-22, 2003, Stanford, California

1WE08 - Tanabe

Introduction

• Magnetic measurements, like magnet design, is a broad subject. It is the intention of this lecture to cover only a small part of the field, regarding the characterization of the line integral field quality of multipole magnets (dipoles, quadrupoles and sextupoles) using compensated rotating coils. Other areas which are not covered are magnet mapping, AC measurements and sweeping wire measurements.


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Voltage in a Coil


3WE08 - Tanabe

∫

∫

∂∂

=

∂∂

==

dxt

BL

dAt

BVoltageV

y

y

∫=⇒∂∂= dxBA

xA

B yy

( )θLAdxBLVdt y == ∫∫Therefore, substituting;

where A, the vector potential is a function of the rotation angle, θ.

( )sec2

2 ×==×=

××=∫VoltWebersm

mWebers

mTeslamdxBLUnits y

∫∫∫

∫

=∂

∂=

=

dxBL

dxdtt

BL

VoltageIntegratedVdt

y

y

⇒


4WE08 - Tanabe

Measurement System Schematic


5WE08 - Tanabe

Digital Integrator• The Digital Integrator consists of two

elements.– Voltage to Frequency Converter.– Up-Down (Pulse) Counter.

-15

-10

-5

0

5

10

15

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

High Voltage

Low Voltage


6WE08 - Tanabe

Using an Integrator on a Rotating Coil• Using an integrator simplifies the requirements on the

mechanical system.

radiusdtdB

LradiusdtdB

LV effeffθ

θ∂∂==

radiusdBLVdt eff ×=

( ) radiusBLradiusdBLVdt effeff ×=×= ∫∫ θ

The use of an integrator measures the angular distribution of the integrated field independent of the angular rotation rate of the coil.


7WE08 - Tanabe

Theory

( )θLAdxBLVdt y == ∫∫where L is the coil length and A is the vector potential, a function of the rotation angle θ.

The magnetic field can be expressed as a function of a complex variable which can be expressed, in general as ;

( ) ∑=+= nn zCiVAzF

( )( )

( ) ( )[ ]∑∑

∑∑

+++=

=

==+

nnn

n

ninn

innin

nn

ninzC

ezC

ezeCzCzF

n

n

sin cos

ψθψθ

ψθ

θψ

Rewriting;

( )[ ]( )∑ +=

=

nn

n nzC

zFA

cos

Re

ψθThe Vector Potentialis, therefore;


8WE08 - Tanabe

( ) ( )∑∫ +== nn

n nzCLLAVdt cos ψθθ

Therefore, when we are measuring the integrated Voltage, we are actually measuring the real part of the function of a complex variable.

We are measuring the rotational distribution of the integrated Vector Potential, AL. We really want to measure the distribution of the Field Integral.


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Field Integral

( )( ) ( )( )nn nin

nnini

n

nn

nn

ezCinezeCin

zinCzCdzd

iziFB

111

1'*

ψθθψ +−−−

−

==

===

( )( ) ( )( )[ ]( )( ) ( )( )[ ]n

nn

nn

nyx

ninnzCn

ninnzCiniBBB

1cos 1sin

1sin 1cos*1

1

ψθψθ

ψθψθ

+−++−−=

+−++−=−=−

−

( )( )( )( )

+−+−

−=

−

n

nnn

y

x

nn

zCnLLBB

1cos 1sin1

ψθψθEquating the real

and imaginary parts of the expression;

Let us take just one term of the infinite series.


10WE08 - Tanabe

• In order to fully characterize the line integral of the magnetic field distribution, we need to obtain only |Cn| and ψn from the measurement data.

( ) ( )nn

n nzCLLAVdt cos ψθθ +==∫

-2.0

-1.5

-1.0

-0.5

0.0

0.5

0 45 90 135 180 225 270 315 360

Angle (deg.)

The graph illustrates the output from a quadrupole measurement. The integrator is zeroedbefore the start of measurement and the graph displays the result of a linear drift due to DC voltage generated in the coil.


11WE08 - Tanabe

Fourier Analysis• In principal, it is possible to mathematically characterize

the measured data by performing a Fourier analysis of the data. – The Fourier Analysis is performed after the linear portion of the

curve is subtracted from the data.

∑∫ += θθ nbnaVdt nn sin cos

( )

( )∑

∑∫−=

+=

nnn

nn

nn

nn

nnzCL

nzCLVdt

sin sin cos cos

cos

ψθψθ

ψθ

nn

nn

nn

nn

zCLb

zCLa

sin

cos

ψ

ψ

−=

=Equating common terms,


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22

22

nnn

neff

NNN

Neff

bazCL

bazCL

+=

+=

−=

−=

−

−

Nab

ab

nn

N

NN

1

1

tan

tan

ψ

ψ

n

n

n

nn a

b−== cos sin

tanψψψ

−= −

n

nn a

b1tan ψor, finally,

Separately, for the fundamental and error terms;


13WE08 - Tanabe

Fundamental and Error Fields• In general, the Fourier analysis of measurement data will

include as many terms as desired. The number of terms is only limited by the number of measurement points. – Earlier, we introduced the concept of the fundamental and error

fields. The Vector potential can be expressed in these terms.

( )

( ) ( )∑

∑

≠

+++=

+=

Nnn

nnN

NN

nn

nn

nzCNzC

nzCA

cos cos

cos

ψθψθ

ψθ

[ ]( ) ( )∑

∫

≠

+++=

=

Nnn

nnN

NN nzCLNzCL

LAVdt

cos cos ψθψθ

θ


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Compensated (Bucked) Coil• The multipole errors are usually very small

compared to the amplitude of the fundamental field. Typically they are < 10-3 of the fundamental field at the measurement radius. – The accuracy of the measurement of the multipole

errors is often limited by the resolution of the voltmeter or the voltage integrator.

• Therefore, a coil system has been devised to nullthe fundamental field, that is, to measure the error fields in the absence of the large fundamental signal.


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• Consider the illustrated coil. r1

r2

r3

r4

M inner turnsM inner turns M outer turnsM outer turns

( ) ( ) ( )∑∫ +−=n

nnn

noutereffouternrrCMLVdt cos31 ψθ

Two sets of nested coils with Mouter and Minner number of turns to increase the output voltage for the outer and inner coils, respectively, are illustrated.

( ) ( ) ( )∑∫ +−=n

nnn

ninnereffinnernrrCMLVdt cos42 ψθ


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Compensated Connection• The two coils are connected in series opposition.

( ) ( ) ( )[ ] ( )∑∫ +−−−=n

nnn

innernn

outerndcompensatenrrMrrMCLVdt cos4231 ψθ

Define the following parameters:

1

31 r

r≡β2

42 r

r≡β1

2

rr≡ρ

outer

inner

MM≡µand

( ) ( )( ) ( )( )[ ] ( )∑∫ +−−−−−=n

nnnnn

noutereffdcompensatenrCMLVdt cos 1 1 211 ψθβρµβ

( )( ) ( )( )nnnns 1 1 21 βρµβ −−−−−≡We define the coil sensitivities;

( ) ( )∑∫ +=n

nnn

nouterdcompensatensrCLMVdt cos1 ψθthen,


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Compensation (Bucking)

• The sensitivities for the fundamental (n=N) and the multipole one under the fundamental (n=N-1) are considered.

( )( ) ( )( )NNNNs 1 1 21 βρµβ −−−−−=

Why one under the fundamental?

( )( ) ( )( )22

2212 1 1 βρµβ −−−−−=sConsider the

quadrupole, N=2

( )( ) ( )( )12

111 1 1 −−

− −−−−−= NNNNs βρµβ

( )( ) ( )( ) 1 1 211 βρµβ −−−−−=s


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• The classical geometry which satisfies the conditions fornulling the N=2 and N=1 field components in the compensated mode have the following geometry.

2= 0.625,= ,2. ,5.0 21 µρββ ==

Homework, show that s1 and s2 are zero for these values, compute the balance of the sensitivities and compare with the graph.

Quadrupole Coil Sensitivities

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1 3 5 7 9

11

13

15

17

19

21

Multipole Index

Qsens


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Compensated Measurements• Quadrupole measurements using the coil in the

compensated configuration are typically as illustrated in the figure.

-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

0 45 90 135 180 225 270 315 360

Angle (deg.)


20WE08 - Tanabe

Bucking Ratio• In the illustrated example of the compensated

measurements, two properties can be readily seen.– The drift is present. Usually, it is a larger portion of the

signal than in the uncompensated measurements. This is because the DC voltage, usually due to thermocouple effects, is a larger fraction of the small compensated coil measurements.

– The signal is dominated by a quadrupole term. This is because of coil fabrication errors so that the quadrupole sensitivity is only approximately zero. The quality of the compensation is measured as a bucking ratio.

( )( )

buckedNN

unbuckedNN

ba

baBucking

22

22

Ratio +

+=

Achieving a Bucking Ratio > 100 indicates a well fabricated coil.


21WE08 - Tanabe

Uncompensated Measurements• The magnet is also measured with the rotating coil

wired in the uncompensated condition to measure the fundamental field integral and the multipole one below the fundamental.

( ) ( )( ) ( )∑∫ +−−=n

nnn

noutereffteduncompensanrCMLVdt cos 1 11 ψθβ

( )( )nnS 1 1 β−−=

Where the sensitivities in the uncompensated condition are designated by capital S.

( ) ( )222

122 2cos ψθ +=∫ SrCMLVdt outereff

( ) ( )11121 cos ψθ +=∫ SrCMLVdt outereff

For the Quadrupole;


22WE08 - Tanabe

• Recalling the expression for the magnetic field components,

( )( )( )( )

+−+−

−=

−

n

nnneffeff

y

x

nn

zCnLLBB

1cos 1sin1

ψθψθ

the amplitude of the fundamental field is,

11

22 −=+= NneffNyNxeffN rCNLBBLB

( ) ( )21 2cos ψθ +=∫ NN

NoutereffNSrCMLVdt

NN

NoutereffNNN

SrCMLbaVdt 122 =+=∫

NN

outereff

NNN SrML

baC

1

22 +=Solving,


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• Substituting into the expression for the fundamental amplitude;

NN

outereff

NNN

effNNeffeffN

SrML

barNLrCNLLB

1

22111

1

+==

−−

Tmm

Webersmeter

VoltSrMbaN

LBNouter

NN

reffN ==−=+

= .sec

1

22

@ 1


24WE08 - Tanabe

Normalized Field Errors• The separate multipole field errors, normalized to the

fundamental field amplitude can be computed from the measurement data.

nn

noutereffnnn

srCMLbaVdt 122 =+=∫

nouter

nn

reffn srMban

LB1

22

@ 1

+=

22

22

@ 1 NNn

nnN

reffN

effn

baNs

banSLBLB

+

+=

an and bn are from the compensated measurements and aN and bN are from the uncompensated measurements.


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Reference Radius

• The expression for the normalized error multipole is evaluated at the outside radius of the inner coil, r1. This radius is limited by measurement coil fabrication constraints and, in general, is substantially smaller than the pole radius and generally smaller than the desired radius of the good field region, which might be > 80% of the pole radius. Therefore, the expression for the normalized error multipole is re-evaluated at a reference radius, r0.


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• The figure illustrates a 35 mm. pole radius quadrupole with a compensated rotating coil installed in the gap. The coil housing is < 35 mm. so that it will fit between the four poles. A half cylinder sleeve is placed around the housing to center the coil. As a result of these mechanical constraints, the maximum coil radius is < 27 mm.


27WE08 - Tanabe

• The desired good field radius is 32 mm., the maximum 10 σ beam radius. Therefore, in order to compute the field quality at this radius, the normalized field errors are recomputed at the required r0.

1 −∝ nn rB 1 −∝ N

N rBand

NnN

n

N

n rrr

BB −

−

−

=∝ 1

1

Therefore,

andNn

NNn

nnN

reffN

effn

rr

baNs

banSLBLB

−

×+

+=

1

0

22

22

@ 0


28WE08 - Tanabe

Dipole Measurements

• The quadrupole coil configuration can also be used to measure a dipole magnet. Since the coil has no quadrupole sensitivity in the buckedconfiguration, a quadrupole error must be evaluated using the unbucked configuration. Since a quadrupole multipole is not an allowed multipole for a symmetric dipole magnet, this does not usually present a serious problem. However, if the dipole design constraints requires that the symmetry conditions be violated (ie. a “C” shaped dipole), the evaluation of the small quadrupole error present in this geometry may be marginal.


29WE08 - Tanabe

Sextupole Measurements• For sextupole measurements, it is desirable to

make s3 and s2=0 for the compensated coil.

( ) ( )( ){ }( )

( ) ( )( ){ }( ) 0 1 1

1 1

0 1 1

1 1

22

221

22

2212

32

331

32

3313

=−−−=−−−−−=

=+−+=

−−−−−=

βρµββρµβ

βµρβ

βρµβ

s

s

This set of equations is difficult to solve algebraically. Therefore, the equations are solved transcendentally.


30WE08 - Tanabe

• One of many solutions to these equations are, 2= 0.77987,= ,83234.0 ,79139.0 21 µρββ ==

The compensated sensitivities for these parameters are illustrated.

Sextupole Coil Sensitivities

-1.2

-0.8

-0.4

0.0

0.4

0.8

1.2

1 3 5 7 9

11 13 15 17 19 21

Multipole Index

Ssens


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Relative Phase• The calculation of the phase angles is based on an

arbitrary mechanical angular shaft encoder zero datum, adjusted by aligning the measurement coil. Therefore, a phase of the fundamental field, ψN, is always present. This angular offset can introduce large errors since small angular offsets between this datum and the zero phase of the fundamental field can result in large errors in the relative phase of the multipole error with respect to the quadrupole zero datum. Therefore, one normally computes a relative phase with respect to a zero phase for the fundamental field.

index field lfundamenta=index multipoleerror =

where= measured measured corrected Nn

Nn

Nnn ψλλ −

( )o063 , modPhase Rel. corrected nλ=


32WE08 - Tanabe

A one page summary of the multipoles for 15Q-001 measured at approximately 81 Amps is reproduced in the table. These measurements were made at IHEP in the PRC.

Magnet ID: 15Q-001 Polarity: DFile Name: a150181t2

Norm.I(A): 81

n PHI[n] Angle PHI[n]/PHI[2] Coil Coef.[n] B[n]/B[2] Rel Phase(*10E-08 V.S) (dgr.)

u1 311879.602 297.248 4.1040E-02 1.0400E-01 4.2581E-031 1831.573 296.431

u2 7599033.299 181.496 1.0000E+00 1.0000E+00 1.0000E+002 52902.217 143.7273 3341.497 3.276 4.3973E-04 7.6002E-01 3.3420E-04 914 375.048 167.623 4.9355E-05 2.4505E+00 1.2094E-04 1655 252.797 260.508 3.3267E-05 1.6031E+00 5.3329E-05 1676 195.521 335.883 2.5730E-05 3.3999E+00 8.7479E-05 1517 202.765 89.396 2.6683E-05 3.2245E+00 8.6041E-05 1748 29.374 268.986 3.8655E-06 5.5016E+00 2.1266E-05 2639 127.338 104.702 1.6757E-05 6.0269E+00 1.0099E-04 8

10 1004.305 5.026 1.3216E-04 9.0748E+00 1.1993E-03 17811 32.07 268.79 4.2203E-06 1.0631E+01 4.4866E-05 35112 2.155 74.915 2.8359E-07 1.4855E+01 4.2128E-06 6613 10.023 258.199 1.3190E-06 1.7978E+01 2.3713E-05 15814 49.045 2.676 6.4541E-06 2.4003E+01 1.5492E-04 17215 4.321 62.461 5.6862E-07 2.9496E+01 1.6772E-05 14116 2.354 168.007 3.0978E-07 3.8279E+01 1.1858E-05 15617 8.098 115.978 1.0657E-06 4.7340E+01 5.0449E-05 1318 80.486 6.305 1.0592E-05 6.0330E+01 6.3899E-04 173

Sample Quadrupole

Measurements


33WE08 - Tanabe

• Two measurements are made at each current, one with the coil connected in the uncompensated mode and one in the compensated mode. The integrated voltage for each magnet is Fourier analyzed and the amplitudes of each coefficient are listed. The u1 and u2 amplitudes (PHI[n] in 10E-8 V-sec.) are the amplitudes of the coefficients for the cos θ and cos 2 θ terms from the uncompensated measurements.

• The balance of the amplitudes are the coefficients of the cos n θ terms from the compensated coil measurements.


34WE08 - Tanabe

• The next four columns include measured and computed values. – Angle The absolute phase angle of the nth Fourier

term with respect to the shaft encoder zero datum. The same datum is used for both the uncompensated and compensated measurements.

– PHI[n]/PHI[2] The ratio of the compensated nth Fourier coefficient to the uncompensated 2nd Fourier coefficient.

– Coil Coef.[n] The coil sensitivities computed from the design radii of the various measurement coil wire bundles.

– B[n]/B[2] The computed (using the coil sensitivities) absolute value of the ratio of the multipole amplitude to the quadrupole field amplitude, evaluated at 32 mm.


35WE08 - Tanabe

Multipole Spectrum15Q-001 Multipoles @ 81 Amps

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Multipole Index

|Bn

/B2|

@ 3

2 m

m.


36WE08 - Tanabe

Multipole Errors as VectorsQ15-001 Multipole Vectors

-0.0005

0

0.0005

0.001

0.0015

-0.0015 -0.001 -0.0005 0 0.0005

Re Bn/B2 @ 32 mm.

Ske

w B

n/B

2 @

32

mm

.

y3

y4

y5

y6

y7

y8

y9

y10

y11

y12

y13

y14

y15

y16

y17

y18


37WE08 - Tanabe

Distribution of n=6 Multipole ErrorsQ-15 n=6 Multipoles

1.E-05

1.E-04

1.E-03

1.E-02

15Q

-001

15Q

-002

15Q

-003

15Q

-004

15Q

-005

15Q

-006

15Q

-007

15Q

-008

15Q

-009

15Q

-010

15Q

-010

15Q

-012

15q-

013

15Q

-014

15Q

-015

15Q

-016

15Q

-017

15Q

-018

15Q

-019

15Q

-020

15Q

-021

15q-

022

15Q

-023

15Q

-024

15Q

-025

15Q

-026

15Q

-027

15Q

-028

15Q

-029

15q-

030

Magnets

|B6/

B2|

@ 3

2 m

m.

at 81A

at 89A


38WE08 - Tanabe

Distribution of n=10 Multipole ErrorsQ-15 n=10 Multipols

1.E-05

1.E-04

1.E-03

1.E-02

15Q

-001

15Q

-002

15Q

-003

15Q

-004

15Q

-005

15Q

-006

15Q

-007

15Q

-008

15Q

-009

15Q

-010

15Q

-010

15Q

-012

15q-

013

15Q

-014

15Q

-015

15Q

-016

15Q

-017

15Q

-018

15Q

-019

15Q

-020

15Q

-021

15q-

022

15Q

-023

15Q

-024

15Q

-025

15Q

-026

15Q

-027

15Q

-028

15Q

-029

15q-

030

Magnets

|B10

/B2|

@ 3

2 m

m.

at 81A

at 89A


39WE08 - Tanabe

Distribution of n=3 First Random Multipole Errors

Q-15 n=3 Multipoles

-5.E-04

-4.E-04

-3.E-04

-2.E-04

-1.E-04

0.E+00

1.E-04

2.E-04

3.E-04

4.E-04

15Q

-001

15Q

-002

15Q

-003

15Q

-004

15Q

-005

15Q

-006

15Q

-007

15Q

-008

15Q

-009

15Q

-010

15Q

-010

15Q

-012

15q-

013

15Q

-014

15Q

-015

15Q

-016

15Q

-017

15Q

-018

15Q

-019

15Q

-020

15Q

-021

15q-

022

15Q

-023

15Q

-024

15Q

-025

15Q

-026

15Q

-027

15Q

-028

15Q

-029

15q-

030

Magnets

B3/

B2

@ 3

2 m

m. (

Rea

l and

Ske

w)

Re3

Im3


40WE08 - Tanabe

Iso-Errors

• The normalized multipole errors and their phases provide information regarding the Fourier components of the error fields. Often, however, one wants to obtain a map of the field error distribution within the required beam aperture. This analog picture of the field distribution can be obtained by constructing an iso-error map of the field error distribution. This map can be reconstructed from the normalized error Fourier coefficients and phases.


41WE08 - Tanabe

( )1

00

−

∆=∆

n

rnn rz

BB ⇒

( )[ ]

( )[ ]n

n

rnny

n

n

rnnx

nrz

BB

nrz

BB

1sin

1cos

1

0

1

0

0

0

ψθ

ψθ

+−∆=∆

+−∆=∆

−

−

where is the phase angle of the multipole error with respect to the zero phase for the fundamental (quadrupole) field.

n ψ

Therefore, 0

220 r

zBB

r=

( )[ ]

( )[ ]n

n

r

nny

n

n

r

nnx

nrz

BB

B

B

nrz

BB

BB

1sin

1cos

2

022

2

022

0

0

ψθ

ψθ

+−∆

=∆

+−∆

=∆

−

−


42WE08 - Tanabe

( )[ ]

( )[ ] 2

18

3

2

02

18

3 22

1

18

3

2

02

18

3 22

1sin

1cos

0

0

Σ

Σ

=+−∆

=∆

=∆

=+−∆

=∆

=∆

∑∑

∑∑

=

−

=

=

−

=

nn

n

r

n

n

nxy

nn

n

r

n

n

nxx

nrz

BB

BB

B

B

nrz

BB

BB

BB

ψθ

ψθ

∑∑ +=∆ 2

221

2BB

and 0

22

0 ryx

rz +

=xy1tan −=θWhere


43WE08 - Tanabe

The computationsand contour mapare programmed using MatLab.

15Q01 at 81 Amps.

-40 -30 -20 -10 0 10 20 30 40

-30

-20

-10

0

10

20

30

x (mm)

y

15Q001 DB/B2 (X104) at 81 Amps

0.1

0.5

0.5

0.5 0.5

0.5

0.5

1

1

11

11

1

1

1

1

2

22

2

2

2

2

2

2

2

55

5

5

5

5

5

5

5

10

10

10

10

10

10

10

10


44WE08 - Tanabe

• The iso-error plot isreplotted for only the allowed multipoles (n=6, 10, 14 and 18) and the first threeunallowed multipoles (n=3,4 and 5). It can be seen that it is virtually identical with the previous plot, indicating that theunallowed multipole errors > 6 are not important.

-40 -30 -20 -10 0 10 20 30 40

-30

-20

-10

0

10

20

30

x (mm)

y

15Q001 DB/B2 (X104) at 81 Amps

0.1

0.5

0.5

0.5

0.5

0.5

0.5

11

1 1

11

1

1

1

1

1

2

2

2

2

2

2

2

2

2

2

5

5

5

5

5

5 5

5

5

5

10

10

10

10

10

10

10

10

15Q001 at 81 Amps


45WE08 - Tanabe

• When the iso-error curve is replotted with theunallowed multipole errors reduced to zero and the allowed multipole phases adjusted to eliminate the skew terms, the ∆B/B <1x10-4 region is dramatically increased. This illustrates the importance of the first three unallowedmultipole errors which are primarily the result of magnet fabrication and assembly errors.

15Q01 at 81 Amps. Unallowed multipole errors = 0. No skew phases for allowed multipoles.

-40 -30 -20 -10 0 10 20 30 40

-30

-20

-10

0

10

20

30

x (mm)

y

15Q001 DB/B2 (X104) at 81 Amps

0.1

0.1

0.1

0.10.1

0.1

0.10.1

0.50.5 0.5

0.50.50.5

1

11

1

1

1

1

2

22

2

222

5

5

5 5

5

555

5

10

10

10

10

10

10

10

10


46WE08 - Tanabe

Date post:	09-Feb-2022
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Magnetic Measurement and Magnet Tutorial, Part 3 - SLAC

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