Lecture 7Jack Tanabe

Old Dominion UniversityHampton, VAJanuary 2011

Magnetic Measurements

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.

Voltage in a Coil

dxt

BL

dAt

BVoltageV

y

y

dxBAxAB yy

LAdxBLVdt y Therefore, substituting;

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

sec2

2

VoltWebersm

mWebers

mTeslamdxBLUnits y

dxBL

dxdtt

BL

VoltageIntegratedVdt

y

y

Measurement System Schematic

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

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

mechanical system.

radiusdtdBLradius

dtdBLV effeff

radiusdBLVdt eff

radiusBLradiusdBLVdt effeff

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

Theory LAdxBLVdt y

where L is the coil length and A is the vector potential, a function of the rotation angle .

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

ezeCzCzFn

n

sin cos

Rewriting;

nn

n nzC

zFA

cos

Re

The Vector Potential is, therefore;

nn

n nzCLLAVdt cos

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.

Field Integral

nn nin

nnini

n

nn

nnn

ezCinezeCin

zinCzCdzdiziFB

111

1* '

n

nn

nn

nnynxn

ninnzCn

ninnzCiniBBB

1cos 1sin

1sin 1cos1

1*

n

nnn

ny

nx

nn

zCnLLBB

1cos 1sin1

Equating the real

and imaginary parts of the expression;

Let us take just one term of the infinite series.

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

nn

n nzCLLAVdt cos

-2.0

-1.5

-1.0

-0.5

0.0

0.5

0 45 90 135 180 225 270 315 360

Angle (deg.)

Inte

grat

or O

utpu

t (v-

sec.

)

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

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. nbnaVdt nn sin cos

nnn

nn

nn

nn

nnzCL

nzCLVdt

sin sin cos cos

cos

nn

nn

nn

nn

zCLb

zCLa

sin

cos

Equating common terms,

22

22

nnn

neff

NNN

Neff

bazCL

bazCL

Nab

ab

nn

N

NN

1

1

tan

tan

n

n

n

nn a

b

cos sin tan

n

nn a

b1tan or, finally,

Separately, for the fundamental and error terms;

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

Nnn

nnN

NN nzCLNzCL

LAVdt

cos cos

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 null the fundamental field, that is, to measure the error fields in the absence of the large fundamental signal.

• Consider the illustrated coil. r1

r2

r3

r4

M inner turnsM inner turns M outer turnsM outer turns

n

nnn

noutereffouternrrCMLVdt cos31

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

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

n

nnn

innernn

outerndcompensatenrrMrrMCLVdt cos4231

Define the following parameters:

1

31 r

r

2

42 r

r

1

2

rr

outer

inner

MM

and

n

nnnnn

noutereffdcompensatenrCMLVdt cos 1 1 211

nnnns 1 1 21 We define the coil sensitivities;

n

nnn

nouterdcompensatensrCLMVdt cos1 then,

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

Why one under the fundamental?

22

2212 1 1 sConsider the

quadrupole, N=2

12

111 1 1

NNNNs

1 1 211 s

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

2= 0.625,= ,2. ,5.0 21

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 21Multipole Index

Sens

itivitie

s

Qsens

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.)

Inte

grat

or O

utpu

t (m

v-se

c.)

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.

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

nnS 1 1

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

222

122 2cos SrCMLVdt outereff

11121 cos SrCMLVdt outereff

For the Quadrupole;

• Recalling the expression for the magnetic field components,

n

nnneffeff

y

x

nn

zCnLLBB

1cos 1sin1

the amplitude of the fundamental field is,

11

22 NneffNyNxeffN rCNLBBLB

21 2cos NN

NoutereffNSrCMLVdt

NN

NoutereffNNN

SrCMLbaVdt 122

NN

outereff

NNN SrML

baC

1

22 Solving,

• 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

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.

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.

• 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.

• The desired good field radius is 32 mm., the maximum 10 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

022

22

@ 0

Dipole Measurements

• The quadrupole coil configuration can also be used to measure a dipole magnet. Since the coil has no quadrupole sensitivity in the bucked configuration, 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.

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

s

s

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

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

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

Sens

itivi

ties

Ssens

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, N, 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

063 , modPhase Rel. corrected n

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

• 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 and cos 2 terms from the uncompensated measurements.

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

• 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.

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| @

32

mm

.

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.

Skew

Bn/

B2

@ 3

2 m

m.

y3

y4

y5

y6

y7

y8

y9

y10

y11

y12

y13

y14

y15

y16

y17

y18

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

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

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

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.

1

00

n

rnn rzBB

n

n

rnny

n

n

rnnx

nrzBB

nrzBB

1sin

1cos

1

0

1

0

0

0

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

n

Therefore,

022

0 rzBB

r

n

n

r

nny

n

n

r

nnx

nrz

BB

BB

nrz

BB

BB

1sin

1cos

2

022

2

022

0

0

2

18

3

2

02

18

3 22

1

18

3

2

02

18

3 22

1sin

1cos

0

0

nn

n

r

n

n

nxy

nn

n

r

n

n

nxx

nrz

BB

BB

BB

nrz

BB

BB

BB

2

221

2BB

and 0

22

0 ryx

rz

xy1tan Where

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

0.50.5

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

• The iso-error plot is replotted for only the allowed multipoles (n=6, 10, 14 and 18) and the first three unallowed multipoles (n=3,4 and 5). It can be seen that it is virtually identical with the previous plot, indicating that the unallowed 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

0.10.5

0.5

0.50.5

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

• When the iso-error curve is replotted with the unallowed 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 unallowed multipole 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

Lecture 8

• Lecture 8, describes techniques and principles for core fabrication. These descriptions are extremely important since the performance and quality of the magnetic field are dominated by the iron core of the manufactured magnets. – The requirements are for the full population of magnets required

for the synchrotron, not only for the individual magnets. – This important subject is covered in chapter 9 of the text.

• Lecture 8 also describes magnet assembly and electrical bussing.

• Finally, fiducialization, installation and alignment are briefly described. These subjects are covered in chapter 12 of the text.