LHC Project Note XXX
2010-06-30
Per.Hagen@cern.ch
Magnetic model of MQWA and MQWB
P. Hagen for the FiDeL team
CERN, Technology Department
Keywords: Normal-conducting Magnets, Magnetic Field Model, Harmonics, LHC.
1. Introduction
Function in the machine: The resistive, water-cooled MQW magnets are used together with
the MQTLH magnets (Q6) to provide the optics used in IR3 and IR7, where collimators are
installed to clean the beam. In IR3 the off-momentum particles are removed by using high
dispersion and low beta functions. In IR7 the beam halo from drifting particles are “shaved
off” by using high beta and low dispersion functions. Undesired particle showers are created
in the collimation process. Normal-conducting magnets are therefore used instead of
superconducting magnets to avoid the issue of particle showers from the collimators inducing
quench. The MQW quadrupoles have twin 46 mm apertures. The unusual choice of common
iron yoke is due to space constraints in the tunnel. The trade-off is that the two apertures must
be powered in series. That is, the modulus of the main field must be the same in both
apertures to achieve reasonable field quality. The nominal operational current for MQWA is
defined as 710 A and 600 A for MQWB. The nominal gradients are 35 T/m and ~30 T/m,
respectively [1-4].
Fig. 1: MQW cross-section, drawing (right) and picture (left).
Numbers and variants: The difference between MQWA and MQWB is only the way the
current flows through the two apertures (polarity). In the symmetric “A” mode they work like
the lattice quadrupoles, i.e., “F/D” or “D/F” (if beam 1 is horizontally focused, then beam 2 is
horizontally defocused, and vice versa). In the anti-symmetric “B” mode the apertures are
This is an internal CERN publication and does not necessarily reflect the views of the LHC project management.
“F/F” or “D/D” which means both beams sees either focus or defocus in a given plane. Six
MQW quadrupole magnets constitute optical element Q4 or Q5, located on both sides of each
IR. Among the six magnets, 5 operate in A mode, and one magnet operates in B mode. Both
MQWA and MQWB are used to provide the nominal optics, see Figs. 2 and 3.
Fig. 2: Optical functions in IR3 for beam 1, x-plane (V6.503).
Fig. 3: Optical functions in IR7 for beam 1, x-plane (V6.503).
An optical requirement is that both beams see the same sequence of focusing and defocusing
fields in the insertion region. The “A” magnets are part of a regular FODO structure which
has an antisymmetric constraint on their strengths (n = 3 or 4):
𝑘A (QnL ) = −𝑘A (QnR )
The “B” magnets introduce extra symmetric focusing in around the IP:
𝑘B (QnL ) = 𝑘B (QnR )
The number of MQW in LHC is 2 IR × 2 sides of IR × 2 optical quadrupole functions × 6
magnets = 48 magnets. 48 + 4 spare magnets were produced. One has 48 x 5/6 = 40 MQWA
and 48 x 1/6 = 8 MQWB.
Naming convention: MQW magnets are identified by consecutive production numbers 1 to
55. The magnets numbers 2, 3 and 4 do not exist. The MTF naming scheme is
-2-
HCMQWm_001-AM0000nn where m is the operational mode (A or B), nn is the magnet
number. AM is manufacturer code for Alstom Canada.
Expected operational cycles, range of current and operational temperature: The injection
current for MQWA is in the range of 35-38 A, corresponding to a gradient range of 1.9-2.0
T/m (see Table I). The MQWA inside Q4 on the left side of IR are powered in series with the
Q4 right side. The same is true for Q5. The optical strength scales linearly with the particle
momentum during ramp (as for all magnets). The gradient does not change once collision
energy has been reached. That is, the optical functions remain constant during the LHC
squeeze. Table I shows the scenario for a few selected energy levels. The values are based
upon LHC optics V6.503 and on the revised FiDeL parameters for 2010.
The LHC design optics assigns little strength to the MQWB, but each one has an individual
power supply although the actual design uses symmetry around IP. The strength has not
changed since LHC optics V6.500. The injection current is as low as 2 A for MQWB inside
Q5.L7 and Q5.R7 which gives a gradient of 0.05 T/m. The operational range for the other
MQWB is in the range 12-30 A with a gradient range of 0.5-1.5 T/m. Table II shows the
scenario for a few selected energy levels. The values are based upon LHC optics V6.503 and
on the revised FiDeL parameters for 2010.
Table I: Operational currents and gradients for MQWA circuits based on FiDeL 2010 parameters.
CIRCUIT
RQ5.LR3
RQ4.LR3
RQ5.LR7
RQ4.LR7
E = 450 GeV
E = 3.5 TeV
E = 5 TeV
E = 6 TeV
E = 7 TeV
I (A)
G (T/m) I (A)
G (T/m) I (A)
G (T/m) I (A)
G (T/m) I (A)
G (T/m)
36.73
1.96
291.98
15.22
417.31
21.75
502.10
26.10
593.19
30.45
34.84
1.86
277.91
14.49
397.13
20.70
477.31
24.84
561.05
28.98
37.67
2.00
299.06
15.59
427.49
22.27
514.74
26.73
610.33
31.18
37.01
1.97
294.11
15.34
420.37
21.91
505.89
26.29
598.26
30.68
Table II: Operational currents and gradients for MQWB circuits based on FiDeL 2010 parameters.
CIRCUIT
RQT5.L3
RQT4.L3
RQT4.R3
RQT5.R3
RQT5.L7
RQT4.L7
RQT4.R7
RQT5.R7
E = 450 GeV
E = 3.5 TeV
E = 5 TeV
E = 6 TeV
E = 7 TeV
I (A)
G (T/m) I (A)
G (T/m) I (A)
G (T/m) I (A)
G (T/m) I (A)
G (T/m)
-30.24
-1.46
-221.28
-11.35 -315.29
-16.21 -377.96
-19.46 -440.64
-22.70
22.04
1.03
157.35
8.04
223.94
11.49
268.34
13.78
312.75
16.08
-22.04
-1.03
-157.35
-8.04 -223.94
-11.49 -268.34
-13.78 -312.75
-16.08
30.24
1.46
221.28
11.35
315.29
16.21
377.96
19.46
440.64
22.70
2.11
0.05
9.50
0.38
12.63
0.54
14.73
0.65
16.82
0.76
11.74
0.50
76.82
3.87
108.88
5.53
130.26
6.64
151.63
7.74
-11.74
-0.50
-76.82
-3.87 -108.88
-5.53 -130.26
-6.64 -151.63
-7.74
-2.11
-0.05
-9.50
-0.38
-12.63
-0.54
-14.73
-0.65
-16.82
-0.76
Summary of manufacturing parameters: the MQW have been designed and produced in
collaboration with TRIUMF, Canada [5]. They were built by Alstom, Canada. The series
production took place in the period 2002-2003. All magnets were initially measured at CERN
using rotating coils. Extended measurements were done in 2010 to characterise the transfer
function. The main parameters are summarized in Table III. The design values are shown in ()
parenthesis if different.
-3-
Table II: Main parameters of MQW based upon measurements, design values within () if different.
Magnet type
Magnetic length
Beam separation (mm)
Aperture
Operating temperature
Nominal gradient
Nominal current
Resistance (both apertures)
Inductance (both apertures)
Power dissipation at Inom
MQWA
MQWB
(m) 3.12 (3.108) 3.13 (3.108)
224
224
(mm)
46
46
(C)
< 65
< 65
(T/m)
34.9 (35)
30.7 (30)
(A)
710
600
(mohm)
37
37
(mH) 64 (28+28) 64 (28+28)
(kW)
19
13 (14)
2. Layout
Slots and positions: the 40 MQWA and 8 MQWB are located in IR3 and IR7 according to
Table III and Figs. 4 and 5, which refer to the installation in 2008. In addition, there are 4
spares, i.e., magnets 5, 10, 38 and 51. Magnets 2, 3 and 4 do not exist.
The field measurements have been done with longitudinal axis CS (connection side) to NCS
(non connection side). Many of the MQW magnets on the left side of “IP” have been
vertically -rotated in the tunnel w. r. t. this orientation. The reason for the rotation is to better
shield the power cables from radiation coming from the collimators. The transformation of
harmonics when doing a vertical rotation is described in Appendix of [6]. For the normal
MQW quadrupoles the equations become:
𝑏 ′ n = (−1)n−2 𝑏n
𝑎′ n = (−1)n−1 𝑎n
These equations are valid for both apertures as they have a common reference system. Care
must be taken when using the data in MAD as there are several definitions of the coordinate
system for beam 2 [7].
Circuits: The two apertures in a given magnet are connected in series. All 5 A magnets in one
Q4 or Q5 quadrupole are connected in series. In addition, the Q4 on left side of IR is
connected in series with Q4 on right side. The same holds for Q5. This minimises the number
of power supplies for the A magnets, taking into account that the gradient is asymmetric (odd
function) mirrored around the IR. In total one has 4 circuits for the A type. The “B” magnets
have individual power supplies, for a total of 8 circuits.
-4-
IR3 MOMENTUM CLEANING
Q5.L3
A
A
Q4.L3
A v B
A
A




A
A
A
B
A
A





A
A


Q4.R3
A
A
B
A
Q5.R3
A
A
A
A
B
A

Fig. 4: Block diagram of IR3 showing magnet grouping and vertical rotations.
IR7 BETATRON CLEANING
Q5.L7
Q4.L7
A
A
A v B
A
A
A
A
A
B
A
A











A
A

Q4.R7
A
A
B
A
Q5.R7
A
A
A
A
B
A
Fig. 5: Block diagram of IR7 showing magnet grouping and vertical rotations.
-5-
Table III: MQW slot allocation, position (s) in beam 1 direction and vertical rotation.
SLOT
MQWA.E5L3
MQWA.D5L3
MQWA.C5L3
MQWB.5L3
MQWA.B5L3
MQWA.A5L3
MQWA.E4L3
MQWA.D4L3
MQWA.C4L3
MQWB.4L3
MQWA.B4L3
MQWA.A4L3
MQWA.A4R3
MQWA.B4R3
MQWB.4R3
MQWA.C4R3
MQWA.D4R3
MQWA.E4R3
MQWA.A5R3
MQWA.B5R3
MQWB.5R3
MQWA.C5R3
MQWA.D5R3
MQWA.E5R3
MQWA.E5L7
MQWA.D5L7
MQWA.C5L7
MQWB.5L7
MQWA.B5L7
MQWA.A5L7
MQWA.E4L7
MQWA.D4L7
MQWA.C4L7
MQWB.4L7
MQWA.B4L7
MQWA.A4L7
MQWA.A4R7
MQWA.B4R7
MQWB.4R7
MQWA.C4R7
MQWA.D4R7
MQWA.E4R7
MQWA.A5R7
MQWA.B5R7
MQWB.5R7
MQWA.C5R7
MQWA.D5R7
MQWA.E5R7
S
6514.1728
6517.9728
6526.0128
6529.8128
6533.6128
6537.4128
6616.8458
6624.8858
6628.6858
6632.4858
6636.2858
6640.0858
6689.3558
6693.1558
6696.9558
6700.7558
6704.5558
6712.5958
6792.0288
6795.8288
6799.6288
6803.4288
6811.4688
6815.2688
19851.7964
19855.5964
19859.3964
19863.1964
19866.9964
19870.7964
19910.0864
19913.8864
19922.5864
19926.3864
19930.1864
19933.9864
20054.3384
20058.1384
20061.9384
20065.7384
20074.4384
20078.2384
20117.5284
20121.3284
20125.1284
20128.9284
20132.7284
20136.5284
MAGNET
HCMQWA_001-AM000055
HCMQWA_001-AM000012
HCMQWA_001-AM000049
HCMQWB_001-AM000036
HCMQWA_001-AM000022
HCMQWA_001-AM000013
HCMQWA_001-AM000033
HCMQWA_001-AM000039
HCMQWA_001-AM000008
HCMQWB_001-AM000050
HCMQWA_001-AM000042
HCMQWA_001-AM000023
HCMQWA_001-AM000021
HCMQWA_001-AM000032
HCMQWB_001-AM000009
HCMQWA_001-AM000024
HCMQWA_001-AM000011
HCMQWA_001-AM000043
HCMQWA_001-AM000048
HCMQWA_001-AM000034
HCMQWB_001-AM000035
HCMQWA_001-AM000053
HCMQWA_001-AM000014
HCMQWA_001-AM000017
HCMQWA_001-AM000001
HCMQWA_001-AM000028
HCMQWA_001-AM000030
HCMQWB_001-AM000046
HCMQWA_001-AM000047
HCMQWA_001-AM000045
HCMQWA_001-AM000027
HCMQWA_001-AM000054
HCMQWA_001-AM000007
HCMQWB_001-AM000006
HCMQWA_001-AM000040
HCMQWA_001-AM000016
HCMQWA_001-AM000019
HCMQWA_001-AM000029
HCMQWB_001-AM000052
HCMQWA_001-AM000041
HCMQWA_001-AM000037
HCMQWA_001-AM000015
HCMQWA_001-AM000020
HCMQWA_001-AM000018
HCMQWB_001-AM000025
HCMQWA_001-AM000031
HCMQWA_001-AM000044
HCMQWA_001-AM000026
-6-
V ROTATED
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
3. Series measurements
All the MQW magnets were measured in CERN after delivery. Each magnet was measured
both in A and B mode. The measurements are described in detail in [8-10].
Device: Measurements are done with a 750-mm-long mole having 5 rotating coils, 41 mm
diameter, inserted into a guiding tube of stainless steel. Measurements were done at 5
positions to cover the integral. Repeated local measurements were done at position 3 to find
the B versus I curves. The pre-cycle done to stabilize the magnet is shown in Fig. 6. It is the
same for both “A” and “B” mode. Fig 7 shows the demagnetisation cycle which was
performed in-between changing “A/B” modes..
900
800
700
Current (A)
600
500
400
300
200
100
0
1
6
11
16
21
26
Time (s)
Fig. 6: MQW pre-cycle where Imin=20A, Imax=810A, dI/dt=100A/s, 10s wait time between ramps, I = 0 at end
of pre-cycle when measurement cycle starts.
600
400
Current (A)
200
0
0
20
40
60
80
100
120
140
160
180
-200
-400
-600
-800
Time (s)
Fig. 7: MQW demagnetisation-cycle where Imin / Imax = -710, 500, -300, 120, -40, 12, -30A, dI/dt=100A/s, 10s
wait time between ramps, I = 0 at end of demagnetisation cycle.
-7-
The precision in the transfer function measurements were estimated to 20 units at 40 A and 10
units for nominal current, the precision on b3 to 1 unit, and 10% accuracy for the other
harmonics. The measurement sequence changed during production due to issues with
reproducibility. At the end of the production all measurements were normalised to the cycles
in Figs. 6-7.
Available and missing measurements: Two types of measurements: local and integral. Type
A: the integral measurements were done for 40, 200, 710 and 810 A. Magnet 10 (a spare) was
not measured until 2009. The local measurements of position 3 have more currents: 30, 40,
100, 150, 200, 300, 400, 500, 600, 650, 710, 750 and 810 A. All A type magnets have local
measurements. Type B: Integral measurements were done for: 40, 200 and 600 A. Magnet 40
was not measured and operates in “A” mode. The local measurements in position 3 were done
with the currents 30, 40, 200 and 600 A, and repeated several times. All magnets except
magnet 40 and magnet 5 were measured. Unfortunately measurements for 30 A is missing for
magnet 36 and 51. They operate in “B” mode in LHC. All data has been normalised w. r. t.
feed-down setting to zero ninth order harmonics.
Rejected or faulty measurements: Local measurement of magnet 5 in B mode has been
rejected during data processing.
Use of the measurements in FiDeL: The geometric is computed through the integral
measurements. The local measurements are used to obtain the saturation and residual
magnetization parameters. The local and integral measurements are normalized to the
geometric and cross-checked to have an estimate of the precision of the measuring system.
4. MQWA transfer function based upon series measurements
4.1 INITIAL DECISIONS
Transfer function, ap1 + ap2 (T/A)
Fig 8 shows the transfer function for all local measurements. There is a systematic difference
between the apertures. Since the two apertures are connected in series, we decided to process
them together, i.e., to use the average transfer function of local measurements for fitting rather
than individual magnets.
0.00092
0.00090
0.00088
0.00086
0.00084
0.00082
0.00080
0.00078
0.00076
0
200
400
Current (A)
600
800
Fig. 8: TF for local measurements (both apertures plus dotted line for average)
-8-
ΔTransfer function, ap2-ap1 (units)
0.0
-5.0
-10.0
-15.0
-20.0
-25.0
-30.0
0
200
400
Current (A)
600
800
Fig. 9: Difference between TF in the aperture 1, and TF in aperture 2, in units. TF ap1 = 10 000 units
4.2 GEOMETRIC
Fig. 10 shows the integral measurements for all magnets. We use the integral measurement at
200 A to define the geometric component, see Table IV. We use a different geometric for
each circuit. Since the difference between each circuit is within 2 units we could have used the
same value for all.
0.00285
Integrated Transfer function
(T.m/A) ap1 + ap2
0.00280
0.00275
0.00270
0.00265
0.00260
0.00255
0.00250
0.00245
0.00240
0.00235
0
200
400
600
Current (A)
Fig. 10: TF for integral measurements (both apertures)
-9-
800
Table IV: TF geometric, average and spread for the MQWA circuits
CIRCUIT
RQ5.LR3
RQ4.LR3
RQ4.LR7
RQ5.LR7
TF (T.m/A) TF (units)
0.00275454
-1.1
0.00275492
0.3
0.00275534
1.8
0.00275452
-1.1
Average
Stdev
0.00275483
1.4
The agreement between the local and integral measurements is shown in Fig. 11 and in Tables
V and VI. Data are normalized, in units, using the geometric values at 200 A as reference.
Local measurements have a lower transfer function at high currents of about 50-70 units w.r.t.
integral measurements. Measurements at 40 A agree within 3 units, and match at 200 A by
definition.
Integrated Transfer function (units)
400
200
0
-200
-400
-600
-800
-1000
-1200
-1400
0
200
400
600
800
Current (A)
Fig. 11: Average of local TF (circles) and integral TF (triangles), expressed in units of geometric
Table V: TF average and spread in local measurements of MQWA
I (A)
TF ave (units) TF sigma (units)
30
175
32
40
93
27
100
-13
15
150
-6
14
200
0
13
300
1
14
400
-14
14
500
-44
17
600
-142
19
650
-292
20
710
-577
20
750
-806
20
810
-1184
20
- 10 -
Table VI: TF average and spread in integral measurements of MQWA
I (A)
TF ave (units) TF sigma (units)
40
96
24
200
0
10
710
-631
18
810
-1249
16
5. Extra measurements in 2010
The measurements of beta-beating in LHC convinced us that the MQWA transfer function
was wrong by 1.5 to 2.0% at injection. The spare magnets 10 and 51 were measured during
spring 2010. The measurement campaign is described in [11].
Device: Measurements were done using SSW (single stretch wire) since we only needed to
correct the integral transfer function. The pre-cycle with 4 different Imin to characterise the
transfer function (“a” to “d”) is shown in Fig. 12 and Table VII. The pre-cycle type “b” is the
one actually used in LHC since the power converters are not stable at lower currents.
800.0
700.0
Current (A)
600.0
500.0
400.0
300.0
200.0
100.0
0.0
0.0
500.0
1000.0
1500.0
2000.0
2500.0
3000.0
Time (s)
Fig. 12: MQWA pre-cycle where Imin=20.1A, Imax=710A, dI/dt=2A/s, 10s wait time between ramps.
Table VII: MQWA pre-cycle parameters (4 different Imin)
Pre-cycle type
No of periods
ramp rate (A/s)
I_MIN (A)
t_MIN (s)
I_FLAT_TOP (A)
t_FLAT_TOP (s)
a
b
4
2
20
10
710
10
- 11 -
c
4
2
25
10
710
10
d
4
2
30
10
710
10
4
2
35
10
710
10
6. MQWA transfer function based upon extra 2010 measurements
6.1 INITIAL DECISIONS
Table VIII shows that the new measurements of MQWA 51 are consistent with the series
measurements. The average discrepancy is about 10 units. The measurement reproducibility is
in the range 2 to 15 units with largest spread at injection and nominal currents. The MQWA
10 was not measured during series so it cannot be used for comparison.
Table VIII: Comparison of MQWA 51 measurements. (SSW – RC) / RC [units]
I
a
b
200
710
0
-6
Aperture 1
c
0
-6
d
0
-6
a
0
-6
b
-21
-15
Aperture 2
c
d
-21
-15
-14
-14
-18
-15
Fig. 13 and 14 show the shape of the transfer function with the new pre-cycles. We can see
that it has a significant Imin dependency for ramp-up. The ramp-down TF curves follow very
much the same path (no Imin dependency). Table IX summaries it numerically by taking the
average value of the 4 apertures. The residual magnetisation is much stronger than in the
series measurement. We find that the residual magnetisation is in reality 2% stronger at 30A
and 1% at 40A compared to series measurements.
800
a
TF (units)
600
b
400
c
200
d
0
-200
-400
-600
-800
0
200
400
600
Current (A)
Fig. 13: MQWA 10 TF (units) after removing geometric at 200A, pre-cycles: a,b,c,d.
- 12 -
800
700
a
TF (units)
600
b
500
c
400
d
300
200
100
0
-100
0
50
100
150
200
Current (A)
Fig. 14: Zoom of Fig 13. for the current range 0-200A.
Table IX: Average MQWA TF (units) after removing geometric at 200A, pre-cycles: a,b,c,d.
The column “b sigma” is spread (units) for pre-cycle b.
I
a
30
35
38
40
70
100
200
400
500
561
610
710
b
139
115
87
69
-1
-13
0
-16
-48
-94
-187
-626
c
366
267
220
186
36
5
0
-17
-48
-94
-189
-627
d
573
382
319
281
56
12
0
-18
-50
-98
-190
-629
b sigma
507
402
354
74
16
0
-17
-48
-94
-193
-627
38
26
28
28
7
4
0
8
13
18
20
24
Table X and Fig. 15 show the new synthesised transfer function. We have used the geometric
component at 200A from the series integral measurements as well for the saturation. We have
measurement points for the residual magnetisation and the beginning of the saturation.
Table X: Synthesised MQWA TF average (units) and spread (units) after removing geometric at 200A.
I
TF ave TF sigma source
30
366
38 SSW 2010
35
267
26 SSW 2010
38
220
28 SSW 2010
40
186
28 SSW 2010
70
36
7 SSW 2010
200
0
0 all by def
300
0
14 RC ORIG
400
-17
8 SSW 2010
500
-48
13 SSW 2010
561
-94
18 SSW 2010
710
-631
18 RC ORIG
810
-1249
17 RC ORIG
- 13 -
why
pre-cycle
pre-cycle
pre-cycle
pre-cycle
pre-cycle
normalisation
all magnets
extra point
extra point
extra point
all magnets
all magnets
600
400
200
TF (units)
0
-200
-400
-600
-800
-1000
-1200
-1400
0
200
400
600
800
Current (A)
Fig. 15: MQWA FiDeL 2010 TF. The encircled measurement points encircled are from the 2010 measurements
(average of 2 magnets). The reminding are from the series measurements (average of all magnets)
6.2 GEOMETRIC
The FiDeL geometric values at 200A from the series measurement are used. See 4.2.
6.3 SATURATION AND RESIDUAL MAGNETIZATION COMPONENTS
We use the two components saturation and residual magnetization to fit the TF curve in Fig
15. There are no DC magnetization components for normal-conducting magnets, neither
time-dependent components, and there is no beam screen in the bore. The fits are done
according to the equations in [12]:
𝑇𝐹 = 𝛾 𝑔𝑒𝑜 + 𝑇𝐹 𝑠𝑎𝑡 + 𝑇𝐹 𝑟𝑒𝑠𝑖𝑑𝑢𝑎𝑙
𝑇𝐹 𝑠𝑎𝑡 = −
𝑇𝐹
𝜎𝑚
𝐼 − 𝐼0
[1 + 𝑒𝑟𝑓 (𝑆
)]
2
𝐼𝑛𝑜𝑚
𝑟𝑒𝑠𝑖𝑑𝑢𝑎𝑙
𝐼𝑖𝑛𝑗 𝑝𝑚
= 𝜌𝑚 ( )
|𝐼|
Saturation: the fit is based on the average of the available measurements in Table X. We use
the integral measurements from the series measurements (“RC ORIG” in Table X) and a few
extra points from the 2010 SSW measurements. The fit is illustrated in Fig. 16. We have also
validated the measurements against simulations with a ROXIE [13] model of the MQWA.
The agreement is very good, with less than 50 units difference at 810 A. ROXIE computations
relate to 2D cross-section while the integral measurements include the heads (end effects).
- 14 -
600
400
200
TF (units)
0
-200
-400
-600
-800
-1000
-1200
-1400
0
200
400
600
800
Current (A)
Fig. 16: Fit of the TF saturation (units), line = fit, triangles = measurements, diamonds = ROXIE simulation.
Residual: the fit is based exclusively on 2010 SSW measurements in Table X. Only currents
equal or less than 200 A are considered. The fit can be seen in Fig. 15. ROXIE cannot be used
for validating the low current measurements. It cannot deal with residual magnetisation of
materials, so it predicts that TF will be constant for low currents.
Table XI shows the fit parameters and Fig. 17 illustrates the fit error as function of current.
Table XI: FiDeL TF 2010 fit parameters for saturation and residual magnetisation.
Component
Saturation
Parameter
sigma
I0
s
Inomref
Residual mag rho
r
Iinjref
Value
0.00053814
766.2540
4.0882
710
0.00004855
2.7151
40
20
Residual error (units)
15
10
5
0
-5
-10
-15
-20
-25
0
200
400
600
800
Current (A)
Fig. 17: Residual TF fit error, triangles = measurement-fit points, dotted line = artificial smoothing.
- 15 -
7. MQWA field errors
Given the unusual design of the twin apertures we check the field errors for all harmonics
rather than assuming any symmetry. We need only to consider the integral data as the local
measurements were done for the same currents. The available data has been corrected for
feed-down using C9, i.e., the b9 a9 are set to zero. We consider modelling only harmonics
where the mean value of the measurement series has a significant value compared to the
corresponding spread. That is, we look at the ratio between the two. We need to process each
aperture separately to avoid being fooled by odd functions. Table XII summarises the
statistics. Similarly to the transfer function, we choose to give the FiDeL geometric value at
200 A. We conclude that we need to model saturation and residual magnetization for b1 b3 b4
b5 b6 b7. Table XIII shows the data after removing the geometric value. The b1 b3 b5 b7 are odd
functions so we need to fit each aperture separately. We give only geometric values at 200 A
for the other harmonics a1 ... a11, b8 ... b11. We use the same model parameters as for TF;
saturation and residual magnetisation. We use a numeric method to find the local minima of
the squared fit error. Since the number of free parameters is larger than the number of points
to fit, we add an artificial, intermediary measurement point at 600 A. We use the data from the
ROXIE simulation for this purpose. Figs. 18-27 show the FiDeL model curves, measurement
points and the predictions from the ROXIE model. The error bars is ±1 sigma of the measured
magnets at the given current.
The ROXIE predictions match saturation with 20% error, systematically underestimating
saturation. In the case of b4 there is no match, see Fig. 24, since ROXIE predicts no saturation
and the measured one is about 4 units. In this case we used an extra point at 455 A for the fit,
half-way between other measured points, assuming linear approximation.
The fitting parameters are summarised in Table XIV. The error associated to the fit is 10 units
for b1, 1 unit for b3, and less than 0.05 units for the other multipoles.
Table XII: Average and standard deviation for the MQWA harmonics.
Ap
I (A)
b1
a1
b3
a3
b4
a4
b5
a5
b6
a6
b7
a7
b8
a8
b9
a9 b10 a10 b11 a11
ave
stdev
1
1
40.0 -174.8 9.1
0.0 72.6 67.6
-1.9
3.7
-0.2
4.8
0.2 -0.1
1.8 1.4
2.6
0.5
0.0 -1.4 0.0 -0.6
0.8 0.4 0.2 0.2
0.0
0.1
0.1
0.1
0.0
0.2
0.0 0.0 1.0 0.0 0.0 0.0
0.0 0.0 0.1 0.0 0.0 0.0
ave
stdev
1
1
200.0 -138.9 5.0
0.1 56.7 61.7
3.5
3.0
0.0
5.2
0.7 -0.1
1.8 1.4
2.9
0.4
0.0 -1.1 0.0 -0.8
0.9 0.4 0.2 0.1
0.0
0.0
0.1
0.1
0.0
0.2
0.0 0.0 1.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0
ave
stdev
1
1
710.1
0.1
4.5 4.0 39.7
68.0 59.6 5.0
0.0
4.4
4.2 -0.1
1.8 1.4
5.8
0.5
0.0 -1.1 0.0 -1.8
0.8 0.4 0.2 0.2
0.0
0.1
0.0
0.1
0.0
0.2
0.0 0.0 1.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0
ave
stdev
1
1
810.1
0.1
-24.5 3.9 36.4
73.2 60.4 5.5
0.0
4.1
4.7 -0.1
1.8 1.4
5.7
0.6
0.0 -1.3 0.0 -1.8
0.8 0.4 0.2 0.2
0.0
0.1
0.0
0.1
0.0
0.2
0.0 0.0 0.9 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0
ave
stdev
2
2
40.0 155.8 20.9
0.0 64.8 68.4
7.3
4.6
0.9
5.5
0.7
1.6
0.0
1.3
-2.2 -0.1 -1.5 0.0
0.5 0.9 0.4 0.2
0.5
0.2
0.0
0.1
0.1 -0.1
0.1 0.2
0.0 0.0 1.0 0.0 0.1 0.0
0.0 0.0 0.1 0.0 0.0 0.0
ave
stdev
2
2
200.0 129.4 6.9
0.1 55.6 64.1
-2.9
3.6
0.4
5.6
1.1
1.6
0.0
1.3
-2.8 -0.1 -1.1 0.0
0.5 0.9 0.4 0.2
0.8
0.1
0.0
0.1
0.0
0.1
0.0
0.2
0.0 0.0 1.0 0.0 0.1 0.0
0.0 0.0 0.0 0.0 0.0 0.0
ave
stdev
2
2
710.1
0.1
-6.1 2.0 -38.9
67.4 58.8 5.2
-0.1
4.7
4.7
1.9
0.0
1.3
-5.6
0.6
0.0 -1.1 0.0
0.8 0.4 0.2
1.8
0.2
0.0
0.1
0.0
0.1
0.0
0.2
0.0 0.0 1.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0
ave
stdev
2
2
810.1
0.1
22.7 3.6 -35.6
72.6 58.5 5.6
-0.3
4.2
5.1
2.0
0.0
1.3
-5.6
0.6
0.0 -1.3 0.0
0.8 0.4 0.2
1.8
0.2
0.0
0.1
0.0
0.1
0.0
0.2
0.0 0.0 0.9 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0
- 16 -
Table XIII: Average and standard deviation for the MQWA harmonics after removing geometric at 200A.
I (A)
Ap
b7
b6
b5
b4
b3
b1
b10
0.2
0.1
0.0
0.0
2.9
0.3
0.0 -1.1
0.1 0.1
-0.1
0.0
4.0
0.7
2.8
0.3
-0.2 -1.0
0.1 0.1
-0.1
0.0
26.4 10.2
19.4 3.9
-0.5
0.1
0.6
0.3
-0.4 -0.3
0.1 0.1
0.0
0.0
2
2
710.1 -135.5 -36.0
20.5 4.1
3.5
0.7
-2.9
0.3
0.0
0.1
1.1
0.1
-0.1
0.0
2
2
810.1 -106.6 -32.7
25.8 4.9
4.0
0.9
-2.8
0.3
-0.2
0.1
1.0
0.1
-0.1
0.0
-5.5
3.1
-0.5
0.1
-0.3
0.2
1
1
710.1 143.4 36.2
18.3 3.8
3.5
0.6
ave
stdev
1
1
810.1 114.4 32.8
23.8 4.5
ave
stdev
2
2
ave
stdev
ave
stdev
ave
stdev
1
1
ave
stdev
40.0
40.0
-41.0
18.7
-0.3
0.1
200
b1 (units)
150
100
50
0
-50
-100
0
200
400
600
800
Current (A)
Fig. 18: Fit of b1 ap1 (units) after removing geometric at 200A, line = fit, triangles = integral measurements,
diamonds = ROXIE simulation.
- 17 -
100
50
b1 (units)
0
-50
-100
-150
-200
0
200
400
600
800
Current (A)
Fig. 19: Fit of b1 ap2 (units) after removing geometric at 200A, line = fit, triangles = integral measurements,
diamonds = ROXIE simulation.
50
40
b3 (units)
30
20
10
0
-10
-20
0
200
400
600
800
Current (A)
Fig. 20: Fit of b3 ap1 (units) after removing geometric at 200A, line = fit, triangles = integral measurements,
diamonds = ROXIE simulation.
- 18 -
20
10
b3 (units)
0
-10
-20
-30
-40
-50
0
200
400
600
800
Current (A)
Fig. 21: Fit of b3 ap2 (units) after removing geometric at 200A, line = fit, triangles = integral measurements,
diamonds = ROXIE simulation.
5.0
4.0
b4 (units)
3.0
2.0
1.0
0.0
-1.0
-2.0
0
200
400
600
800
Current (A)
Fig. 22: Fit of b4 (units)) after removing geometric at 200A, line = fit, triangles = integral measurements,
diamonds = ROXIE simulation.
- 19 -
3.5
3.0
b5 (units)
2.5
2.0
1.5
1.0
0.5
0.0
-0.5
-1.0
0
200
400
600
800
Current (A)
Fig. 23: Fit of b5 ap1 (units) after removing geometric at 200A, line = fit, triangles = integral measurements,
diamonds = ROXIE simulation.
1.5
1.0
0.5
b5 (units)
0.0
-0.5
-1.0
-1.5
-2.0
-2.5
-3.0
-3.5
0
200
400
600
800
Current (A)
Fig. 24: Fit of b5 ap2 (units) after removing geometric at 200A, line = fit, triangles = integral measurements,
diamonds = ROXIE simulation.
- 20 -
0.2
0.1
b6 (units)
0.0
-0.1
-0.2
-0.3
-0.4
-0.5
-0.6
0
200
400
600
800
Current (A)
Fig. 25: Fit of b6 (units) after removing geometric at 200A, line = fit, triangles = integral measurements,
diamonds = ROXIE simulation.
0.4
0.2
b7 (units)
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
-1.2
-1.4
0
200
400
600
800
Current (A)
Fig. 26: Fit of b7 ap1 (units) after removing geometric at 200A, line = fit, triangles = integral measurements,
diamonds = ROXIE simulation.
- 21 -
1.4
1.2
1.0
b7 (units)
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
0
200
400
600
800
Current (A)
Fig. 27: Fit of b7 ap2 (units) after removing geometric at 200A, line = fit, triangles = integral measurements,
diamonds = ROXIE simulation.
Table XIV: Harmonic fit parameters for saturation and residual magnetisation.
Component
Saturation
Parameter
sigma
I0
s
Inomref
Residual mag rho
r
Iinjref
Ave fit error
b1 ap1
b1 ap2
b3 ap1
b3 ap2
b4
b5 ap1
b5 ap2
b6
b7 ap1
b7 ap2
-128.9058
121.0436
-34.4976
34.3321
-3.9984
-2.8514
2.8454
0.1762
1.0406
-1.0286
626.4983
625.9282 622.1215 622.0595
671.8697 625.6147 625.4454 727.6331 621.8350 622.4412
24.6573
24.5051
22.7959
22.7881
15.6639
20.2559
20.3618
30
21.6830 21.7802
710
710
710
710
710
710
710
710
710
710
-41.0205
26.4604
-5.4793
10.2438
-0.4690
-0.2542
0.5977
-0.3545
0.1982
-0.3065
6.9109
6.6607
5.7820
6.3439
0.7022
5.7820
6.3439
5.7819
4.0504
4.1166
40
40
40
40
40
40
40
40
40
40
9.16
9.12
1.07
1.05
0.08
0.01
0.01
0.01
0.02
0.02
8. MQWB transfer function based upon series measurements
8.1 INITIAL DECISIONS
Given the fact that only 8 MQWB are installed in the LHC and that each magnet has its own
power supply with the 2 apertures connected in series, and that they operate within 3 different
ranges of low current we decide to model the transfer function individually for each magnet.
We add the 4 spare magnets having integral measurements in “B” mode in case they one day
will be needed.
8.2 GEOMETRIC
As for the MQWA, we use the integral measurements at 200 A to define the geometric
component, see Table XV. The spread between the magnets is not negligible (±15 units): this
justifies the need of a different geometric value for each magnet. See overview in Figs. 28-29.
- 22 -
Table XV: TF geometric, average and spread for the MQWB circuits
CIRCUIT MAGNET
TF (T.m/A) TF (units)
RQT5.L3
36 0.00275087
12.6
RQT4.L3
50 0.00274807
2.4
RQT4.R3
9 0.00274843
3.7
RQT5.R3
35 0.00274799
2.1
RQT5.L7
46 0.00275033
10.6
RQT4.L7
6 0.00274312
-15.6
RQT4.R7
52 0.00274457
-10.3
RQT5.R7
25 0.00274589
-5.5
Average
Stdev
0.00274741
9.8
Integral data show a 1.5% larger TF at 40 A due to residual magnetisation, and 1.5% smaller
at 600 A due to saturation (Table XVI). Local measurements are given in Table XVII: the
spread at 600 A is much larger, thus suggesting that some data should be rejected.
Table XVI: TF for integral measurements, expressed in units of geometric at 200 A.
I (A)
TF ave (units) TF sigma (units)
40
157
31
200
0
13
600
-151
15
Table XVII: TF for local measurements, before rejecting MQWB 5, expressed in units of geometric at 200 A.
I (A)
30
40
200
600
TF ave (units) TF sigma (units)
256
45
150
34
0
17
-124
41
Table XVIII: TF for local measurements, after rejecting MQWB 5, expressed in units of geometric at 200 A.
I (A)
TF ave (units) TF sigma (units)
30
259
41
40
156
30
200
0
10
600
-129
14
Results after data rejection of local measurement of MQWB 5 are shown in Table XVIII. We
have a very good match between integrals and locals at 40 A, and we have a systematic
difference of about 20 units as in the MQWA case, with a similar spread.
- 23 -
Transfer function, ap1 + ap2 (T/A)
0.00092
0.00091
0.00091
0.00090
0.00090
0.00089
0.00089
0.00088
0.00088
0.00087
0.00087
0.00086
0
100
200
300
Current (A)
400
500
600
Fig. 28: TF for local measurements (both apertures).
Integrated Transfer function
(T.m/A) ap1 + ap2
0.00282
0.00280
0.00278
0.00276
0.00274
0.00272
0.00270
0.00268
0
100
200
300
400
500
600
Current (A)
Fig. 29: TF for integral measurements (both apertures).
9. Extra measurements in 2010
The 2010 measurements of beta-beating in LHC convinced us that some of the MQWB
transfer functions were wrong by more than 200% at injection! The spare magnets 10 and 51
were measured during spring 2010. The measurement campaign is described in [11].
Device: Measurements were done using SSW (single stretch wire) since we only needed to
correct the integral transfer function. The pre-cycle is identical to the actual LHC pre-cycle
which take advantage of the bipolar power supply in order make sure the magnet is at the right
hysteresis branch at injection. See Fig. 30 and Table XIX.
- 24 -
I (A)
600
400
200
0
0
500
1000
1500
2000
-200
-400
-600
t (s)
Fig. 30: MQWB pre-cycle where Imin=-600A, Imax=+600A, dI/dt=5A/s, 10s wait time between ramps.
Table XIX: MQWB pre-cycle parameters
Pre-cycle
No of periods
ramp rate (A/s)
I_MIN (A)
t_MIN (s)
I_MAX (A)
t_MAX (s)
4
5
-600
10
600
10
10. MQWB transfer function based upon 2010 measurements
10.1 INITIAL DECISIONS
Table XX shows the comparison between the new SSW and the old RC measurements for the
MQWB 51. MQWB 10 was not measured during series measurements. We have good
agreement for 600 A, but the value at 200 A is surprisingly about 1% different.
Table XX: Comparison of MQWB 51 measurements. (SSW – RC) / RC [units]
I
Ap 1
200
600
Ap 2
-87
-8
-93
-15
Fig. 31 and Table XXI show the shape of the transfer function, as usual based upon
normalising the curve to the geometric component at 200A as usual. We can see from Table
XXI that the sign of the transfer function is opposite from previous measurements due to the
difference in pre-cycle. This was anticipated. But what comes a surprise is that the transfer
function is not flat at 200 A. It reaches its flattest point at 437 A. So the whole transfer
function has changed. This would then explain the difference in Table XX at 200A. We must
re-express the transfer function with a geometric component at 437 A. See Table XXII for the
new values. Unfortunately we do not have any more measurements close to 437 A to find
more precisely the peak. We remove the measurement points at 1 and 2 A for making the
FiDeL fit since these points give a negative field integral. See Fig. 32 for the shape of the
transfer function. The only measurement point from the series we could have re-used is the
one at 600 A. But since the spread is small at 600 A, we do not believe this is justified.
- 25 -
25000
MQWB avg
20000
15000
b
TF (units)
10000
5000
0
-5000
-10000
-15000 0
10
20
30
40
-20000
-25000
Current (A)
Fig. 31: MQWB average TF (units) for current range 0-40A, after removing geometric at 200A.
Table XXI: Average MQWB TF (units) and spread (units) after removing geometric at 200A.
I
TF avg
TF sigma
1
-20915
-826
2
-10715
-6765
5
-4323
354
10
-2155
122
15
-1416
71
20
-992
48
27
-704
42
40
-437
28
100
-111
13
200
0
8
312
29
8
437
31
11
600
-70
17
437
138
9
Table XXII: Average MQWB TF (units) after removing geometric at 437A.
I
5
10
15
20
27
40
100
200
312
437
600
TF ave
-4340
-2179
-1442
-1019
-733
-467
-141
-31
-2
0
-101
- 26 -
0
MQWB 10 +51TF (units)
-2000
-4000
-6000
-8000
-10000
0
100
200
300
400
500
600
-12000
Current (A)
Fig. 32: MQWB magnets 10+51, average TF (units), after removing geometric at 437A.
10.2 GEOMETRIC
The average FiDeL geometric value at 437A from the 2010 SSW measurements is used. This
means we use the average value of 2 magnets to represent the 8 magnets installed. See Table
XXIII.
Table XXIII: FiDeL TF 2010 fit parameters for geometric, saturation and residual magnetisation.
Component
Geo
Parameter Value
gamma
0.00273293
Igeoref
437
Saturation
sigma
0.00011784
I0
642.8172
s
11.0361
Inomref
600
Residual mag rho
-0.00018512
r
1.0401
Iinjref
30
10.3 SATURATION AND RESIDUAL MAGNETIZATION COMPONENTS
We use the two components saturation and residual magnetization to fit the TF curve in Fig
32. There are no DC magnetization components for normal-conducting magnets, neither
time-dependent components, and there is no beam screen in the bore. The fits are done
according to the equations in [12]:
𝑇𝐹 = 𝛾 𝑔𝑒𝑜 + 𝑇𝐹 𝑠𝑎𝑡 + 𝑇𝐹 𝑟𝑒𝑠𝑖𝑑𝑢𝑎𝑙
𝑇𝐹 𝑠𝑎𝑡 = −
𝜎𝑚
𝐼 − 𝐼0
[1 + 𝑒𝑟𝑓 (𝑆
)]
2
𝐼𝑛𝑜𝑚
- 27 -
𝑇𝐹
𝐼𝑖𝑛𝑗 𝑝𝑚
= 𝜌𝑚 ( )
|𝐼|
𝑟𝑒𝑠𝑖𝑑𝑢𝑎𝑙
Saturation: We have only two measure points from Table XXII (437A and 600A) to make the
fit. The reason is that we did not anticipate that the entire TF should change as function of the
pre-cycle so that we had to adjust the geometric to 437A. Fortunately we have confidence that
the values at 600 A represent the MQWB family (Table XX). Fig. 33 shows the fit of the
saturation. We use the ROXIE simulation of the MQWB 2D model to check. We have a very
good agreement, within 10 units at 600A. ROXIE computations relate to 2D cross-section
while the integral measurements include the heads (end effects).
0
MQWB 10 +51TF (units)
-2000
-4000
-6000
-8000
-10000
0
100
200
300
400
500
600
-12000
Current (A)
Fig. 33: Fit of the TF saturation (units), line = fit, triangles = measurements, diamonds = ROXIE simulation.
Residual: We use the 2010 SSW measurements in Table XXII for the fit, where current is less
geometric at 437A. The fit can be seen in left part of Fig. 32. ROXIE cannot be used for
validating the low current measurements. It cannot deal with residual magnetisation of
materials, so it predicts that TF will be constant for low currents.
The fit error is much bigger than for MQWA. See Fig. 34. It is about 10% at lowest
operational current 2A and 2% at 15A. There is too much spread and uncertainty in this nonlinear region to make any better estimates. The fit error undershoots with about 1% at 100A
and then stays within 10 units as the current increases.
- 28 -
Residual error (units)
1000
800
600
400
200
0
-200
0
200
400
600
Current (A)
Fig. 34: Residual TF fit error, triangles = measurement-fit points, dotted line = artificial smoothing.
11. MQWB field errors
We process the MQWB field errors following the same considerations as for MQWA. As
usual we use the FiDeL saturation and residual magnetisation components. We only have to
consider the integral data as the local measurements were done for the same currents. We
consider modelling only harmonics where the mean value of the measurement series has a
significant value compared to the corresponding sigma. That is, we look at the ratio between
the two. We need to process each aperture separately to avoid being fooled by systematic odd
functions. Table XXIV shows the statistics of the integral measurements. We conclude that
we need to model saturation and residual magnetization for b1 b3 b4 b5 b6 b7. The b1 b3 b5 b7 are
odd functions so we need to fit each aperture separately. We give only geometric values at
200 A for the other harmonics a1 a3 ... a11 b8 ... b11.
We should have expressed the field errors at 437 A as we did for the MQWB transfer
function, but lack of measurements prevent us from doing so. Fortunately the MQWB are few
and mostly to be regarded as corrector magnets. Therefore relative field errors will have small
effects on LHC.
We need to add an extra measurement point at higher current to be able to numerically fit the
saturation in a more sensible way (it would otherwise be an “unphysical” step function). We
use the value from ROXIE at 500 A. Table XXV shows the values used for the fitting curves
after removing geometric at 200A. Figs. 35-43 show the measurements, ROXIE model and
FiDeL fits. The fitting parameters are summarised in Table XXVI.
- 29 -
Table XXIV: Average and standard deviation for the MQWB integral harmonics.
Ap I (A)
b1
b3
40.01 -111.37
0.03
99.42
a3
b4
a4
b5
a5
b6
a6
b7
a7
b8
a8
b9
a9
b10
a10
b11
a11
ave
stdev
1
1
3.19 -2.00
3.92 4.31
0.99 -0.17
2.04 1.59
2.88 0.34 -1.57 -0.03 -0.79
0.33 0.86 0.41 0.18 0.18
0.00
0.10
0.04 -0.01 0.00 0.00
0.11 0.20 0.00 0.00
1.02 -0.01 -0.04
0.05 0.05 0.04
-0.02
0.01
ave
stdev
1 200.04 -218.84 -25.64 -2.09
1
0.08
96.87 3.12 5.56
1.46 -0.19
1.98 1.56
0.75 0.37 -1.30 -0.02
0.34 0.97 0.37 0.20
0.01
0.05
0.00 0.01 0.00 0.00
0.10 0.20 0.00 0.00
1.05 0.00 -0.11
0.02 0.05 0.02
-0.02
0.01
ave
stdev
1 600.09 -160.33 -11.01 -1.34
1
0.13
98.67 3.70 5.57
3.57 -0.19
1.87 1.55
1.98 0.30 -1.13 -0.03 -0.31 -0.01 -0.02 0.02 0.00 0.00
0.40 1.00 0.47 0.19 0.19 0.05 0.10 0.19 0.00 0.00
1.00 0.00 -0.09
0.04 0.05 0.02
-0.01
0.01
ave
stdev
2
2
40.00
0.03
31.75 -4.81 0.58
112.86 4.21 4.77
0.73
1.85
0.19 -2.93 -0.13 -1.59
1.36 0.63 0.73 0.43
0.05 -0.08 0.00 0.00
0.09 0.19 0.00 0.00
0.98 -0.01
0.14 0.05
0.08
0.03
-0.01
0.01
ave
stdev
2 200.02
2
0.07
196.51 24.80 -0.72
97.43 4.67 4.75
1.06
1.87
0.20 -0.85 -0.04 -1.31 -0.01 -0.04
1.34 0.54 0.72 0.42 0.19 0.17
0.04
0.09
0.02 -0.05 0.00 0.01
0.08 0.18 0.00 0.01
1.05 -0.01
0.02 0.04
0.12
0.02
-0.01
0.01
ave
stdev
2 600.09
2
0.13
146.16 10.42 -0.67
100.69 5.39 4.09
3.19
1.88
0.16 -2.09 -0.02 -1.15 -0.03
1.32 0.59 0.63 0.58 0.20
0.03
0.09
0.00 -0.04 0.00 0.00
0.08 0.18 0.00 0.00
1.00 -0.01
0.04 0.04
0.09
0.02
-0.01
0.01
0.00
0.19
0.07
0.17
0.84 -0.01
0.23 0.09
0.32
0.19
Table XXV: Average and standard deviation for the MQWB harmonics after removing geometric at 200A.
Ap I (A)
b1
b3
b4
ave
stdev
1
1
40
107.47 28.83 -0.48
14.45 2.23 0.26
ave
stdev
1
1
600
58.51 14.63 2.11
13.57 1.12 0.46
ave
stdev
2
2
ave
stdev
2
2
b5
b6
1.23
0.12
-50.35 -14.39 2.12 -1.24
12.01 1.00 0.31 0.09
- 30 -
b10
2.13 -0.27 -0.86 -0.03
0.16 0.10 0.07 0.03
0.17 -0.38 -0.05
0.29 0.03 0.03
40 -164.76 -29.61 -0.33 -2.08 -0.28
91.97 3.16 0.46 0.43 0.11
600
b7
0.16
0.29
0.88 -0.07
0.15 0.14
0.36 -0.05
0.03 0.03
140
120
b1 (units)
100
80
60
40
20
0
-20
-40
0
100
200
300
400
500
600
Current (A)
b1 (units)
Fig. 35: Fit of b1 ap1 (units) after removing geometric at 200A, line = fit, triangles = integral measurements,
diamonds = ROXIE simulation.
40
20
0
-20
-40
-60
-80
-100
-120
-140
-160
-180
0
100
200
300
400
500
600
Current (A)
Fig. 36: Fit of b1 ap2 (units) after removing geometric at 200A, line = fit, triangles = integral measurements,
diamonds = ROXIE simulation.
- 31 -
35
30
b3 (units)
25
20
15
10
5
0
-5
0
100
200
300
400
500
600
Current (A)
Fig. 37: Fit of b3 ap1 (units) after removing geometric at 200A, line = fit, triangles = integral measurements,
diamonds = ROXIE simulation.
5
0
b3 (units)
-5
-10
-15
-20
-25
-30
-35
0
100
200
300
400
500
600
Current (A)
Fig. 38: Fit of b3 ap2 (units) after removing geometric at 200A, line = fit, triangles = integral measurements,
diamonds = ROXIE simulation.
- 32 -
3.0
2.5
b4 (units)
2.0
1.5
1.0
0.5
0.0
-0.5
-1.0
0
100
200
300
400
500
600
Current (A)
Fig. 39: Fit of b4 (units) after removing geometric at 200A, line = fit, triangles = integral measurements,
diamonds = ROXIE simulation.
2.5
b5 (units)
2.0
1.5
1.0
0.5
0.0
-0.5
0
100
200
300
400
500
600
Current (A)
Fig. 40: Fit of b5 ap1 (units) after removing geometric at 200A, line = fit, triangles = integral measurements,
diamonds = ROXIE simulation.
- 33 -
0.5
b5 (units)
0.0
-0.5
-1.0
-1.5
-2.0
-2.5
0
100
200
300
400
500
600
Current (A)
Fig. 41: Fit of b5 ap2 (units) after removing geometric at 200A, line = fit, triangles = integral measurements,
diamonds = ROXIE simulation.
0.2
b7 (units)
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
0
100
200
300
400
500
600
Current (A)
Fig. 42: Fit of b7 ap1 (units) after removing geometric at 200A, line = fit, triangles = integral measurements,
diamonds = ROXIE simulation.
- 34 -
1.0
b7 (units)
0.8
0.6
0.4
0.2
0.0
-0.2
0
100
200
300
400
500
600
Current (A)
Fig. 43: Fit of b7 ap2 (units) after removing geometric at 200A, line = fit, triangles = integral measurements,
diamonds = ROXIE simulation.
Table XXVI: MQWB harmonic fit parameters for saturation and residual magnetisation.
Component
Saturation
Parameter b1 ap1
b1 ap2
b3 ap1
b3 ap2
b4
b5 ap1
b5 ap2
b6
b7 ap1
b7 ap2
sigma
-58.509756 50.351598 -14.628818 14.385909 -2.116363 -1.228817 1.235635 -0.167727 0.375636 -0.364455
I0
538.33
538.26
529.17
529.17
527.03
523.22
523.22
526.96
523.16
523.21
s
33
33
36
36
34
34
34 381887770 447723571 447717906
Inomref
600
600
600
600
600
600
600
600
600
600
Residual mag rho
107.625114 -137.636805 28.868311 -29.619868 -0.402909 2.130273 -2.076697 -0.271532 -0.863979
0.884742
r
7.31
7.83
6.75
6.88
5.29
5.78
6.34
4.97
5.35
6.15
Iinjref
40
40
40
40
40
40
40
40
40
40
Min fit error
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
Ave fit error
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
Max fit error
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
12. Summary and critical points

The TF of the MQWA has a large saturation of about 1.5%, which is modelled. The
spread between magnets is very low. At injection we have a strong magnetization
component of about 3%. This effect is based upon extra 2010 measurements of 2
magnets and is due to the Imin=20A in the pre-cycle. Geometric components are
modelled for each magnet. We have 3 different types of measurements: local and
integral RC measurements, integral SSW measurements.

Normal multipoles have a systematic component of saturation and magnetization. We
modelled an average b1, b3, b4, b5, b7 for the whole family. The ROXIE model is
systematically underestimating the saturation by about 20%, but matches well the
shape of the measured curve. The only exception is b4: measurements show a
saturation giving 4 units at high field, against a model result of less than 0.1 units. For
all other multipoles, a geometric component has been computed.

The TF of the MQWB does not fit the series measurements due to a different pre-cycle
need for reproducibility in LHC operation. The saturation effect is about 1.5% at 600
A. The MQWB magnets operate in 3 different current ranges and the dominating
- 35 -
effect is always the residual magnetisation which is about 20% at 10 A. The MQWB
TF is based upon the 2010 SSW measurements of 2 spare magnets.
13. Acknowledgements
We wish to thank D. Cornuet and M. Silva for the original measurements and reports, X.
Panagiota for making the original analysis and FiDeL REFPARM file for the 2008 LHC startup, G. de Rijk for the MQW design models and advices, B. Auchmann for help with ROXIE,
E. Todesco for discussions and feedback, R. Tomas Garcia for the beta-beat measurements
and analysis in LHC 2009-2010, the extra measurement campaign in 2009-2010 by M. Buzio,
J. Garcia Perez and R. Chritin.
References
D. I. Kaltchev, et al., “Optics Solutions for the Collimation Insertions of LHC”, PAC 1999.
O. Bruning, “Collimation Optics”, Presentation in LHC Machine Advisory Committee, meeting 14, 2003.
O. Bruning, et al., CERN Report 2004-003 (2004).
R. Assmann, et al., “The New Layout of the LHC Cleaning Insertions”, LHC Project Report 774, 2004
G. de Rijk, et al., “Construction and Measurement of the Pre-Series Twin Aperture Resistive Quadrupole
Magnet for the LHC Beam Cleaning Insertions”, IEEE Trans. Appl. Superconduct., Vol 12, pp 55-58, 2002
[6] R. Wolf, “Corrector Memo”, internal department memo, 2006
[7] R. Wolf, “Field Error Naming Conventions For LHC magnets”, rev 3, EDMS 90250, 2001
[8] D. Cornuet, et al., “Field Quality Measurements of the LHC Warm Twin Aperture Quadrupole Magnets”,
LHC Project Note 380, 2006, EDMS 703686, 2006.
[9] M. Silva, “MQW magnets: results from the measurement campaign”, EDMS 609363, 2005
[10] D. Cornuet, “Status of data retrieval for normal-conducting magnets”, presentation in FiDeL WG 03 April
2007, web site: http://cern.ch/fidel
[11] P. Hagen, “2010 measurement campaign and updated FiDeL models”, presentation in FiDeL WG 11 May
2010, web site: http://cern.ch/fidel
[12] FiDeL Model specifications, EDMS 908232, 2008
[13] ROXIE web site: : http://cern.ch/roxie
[1]
[2]
[3]
[4]
[5]
- 36 -