Lecture 7 Jack Tanabe Old Dominion University Hampton, VA January 2011 Magnetic Measurements
Dec 23, 2015
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
dxBAx
AB yy
LAdxBLVdt y Therefore, substituting;
where A, the vector potential is a function of the rotation angle, .
sec2
2
VoltWebersm
m
Webers
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.
radiusdt
dBLradius
dt
dBLV 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
nn
n
n
nin
n
innin
nn
ninzC
ezC
ezeCzCzF
n
n
sin cos
Rewriting;
n
n
n nzC
zFA
cos
Re
The Vector Potential is, therefore;
n
n
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
zinCzCdz
diziFB
111
1* '
n
n
n
n
n
nnynxn
ninnzCn
ninnzCiniBBB
1cos 1sin
1sin 1cos1
1*
n
nn
nny
nx
n
nzCnLL
B
B
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.
n
n
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
gra
tor
Outp
ut
(v-s
ec.
)
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
n
n
nn
n
n
nnzCL
nzCLVdt
sin sin cos cos
cos
n
n
nn
n
n
nn
zCLb
zCLa
sin
cos
Equating common terms,
22
22
nn
n
neff
NN
N
Neff
bazCL
bazCL
Na
b
a
b
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
n
nN
N
N
nn
n
n
nzCNzC
nzCA
cos cos
cos
Nnn
n
nN
N
N 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
r
r
outer
inner
M
Mand
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
21
Multipole Index
Sen
sitiv
ities
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
gra
tor
Outp
ut
(mv-s
ec.
)
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
nn
neffeffy
x
n
nzCnLL
B
B
1cos
1sin1
the amplitude of the fundamental field is,
1
122 N
neffNyNxeffN 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;
N
N
outereff
NN
N
effN
NeffeffNSrML
barNLrCNLLB
1
221
11
1
Tmm
Webers
meter
Volt
SrM
baNLB
Nouter
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 srM
banLB
1
22
@ 1
22
22
@ 1 NNn
nnN
reffN
effn
baNs
banS
LB
LB
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 rr
r
B
B
1
1
Therefore,
and
Nn
NNn
nnN
reffN
effn
r
r
baNs
banS
LB
LB
1
0
22
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
Sensi
tivit
ies
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 N
n
N
nNnn
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-04 810 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 Spectrum
15Q-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.
Multipole Errors as Vectors
Q15-001 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.
Sk
ew
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
15
Q-0
01
15
Q-0
02
15
Q-0
03
15
Q-0
04
15
Q-0
05
15
Q-0
06
15
Q-0
07
15
Q-0
08
15
Q-0
09
15
Q-0
10
15
Q-0
10
15
Q-0
12
15
q-0
13
15
Q-0
14
15
Q-0
15
15
Q-0
16
15
Q-0
17
15
Q-0
18
15
Q-0
19
15
Q-0
20
15
Q-0
21
15
q-0
22
15
Q-0
23
15
Q-0
24
15
Q-0
25
15
Q-0
26
15
Q-0
27
15
Q-0
28
15
Q-0
29
15
q-0
30
Magnets
|B6
/B2
| @
32
mm
.
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
15
Q-0
01
15
Q-0
02
15
Q-0
03
15
Q-0
04
15
Q-0
05
15
Q-0
06
15
Q-0
07
15
Q-0
08
15
Q-0
09
15
Q-0
10
15
Q-0
10
15
Q-0
12
15
q-0
13
15
Q-0
14
15
Q-0
15
15
Q-0
16
15
Q-0
17
15
Q-0
18
15
Q-0
19
15
Q-0
20
15
Q-0
21
15
q-0
22
15
Q-0
23
15
Q-0
24
15
Q-0
25
15
Q-0
26
15
Q-0
27
15
Q-0
28
15
Q-0
29
15
q-0
30
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
15
Q-0
01
15
Q-0
02
15
Q-0
03
15
Q-0
04
15
Q-0
05
15
Q-0
06
15
Q-0
07
15
Q-0
08
15
Q-0
09
15
Q-0
10
15
Q-0
10
15
Q-0
12
15
q-0
13
15
Q-0
14
15
Q-0
15
15
Q-0
16
15
Q-0
17
15
Q-0
18
15
Q-0
19
15
Q-0
20
15
Q-0
21
15
q-0
22
15
Q-0
23
15
Q-0
24
15
Q-0
25
15
Q-0
26
15
Q-0
27
15
Q-0
28
15
Q-0
29
15
q-0
30
Magnets
B3
/B2
@ 3
2 m
m.
(Re
al
an
d S
ke
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 r
zBB
n
n
rnny
n
n
rnnx
nr
zBB
nr
zBB
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 r
zBB
r
n
n
r
nny
n
n
r
nnx
nr
z
B
B
B
B
nr
z
B
B
B
B
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
nr
z
B
B
B
B
B
B
nr
z
B
B
B
B
B
B
2
221
2B
B
and 0
22
0 r
yx
r
z
x
y1tan 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
0.5
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
• 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
0.5
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
55
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.