LHC Project Note XXX
2009-11-30
Per.Hagen@cern.ch
Magnetic model of the normal-conducting magnets MBW, MBXW(H)
and MCBW(H/V)
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 MBXW magnets (optical function D1) are used in
combination with the MBRC magnet (D2) to reduce the beam separation from 194 mm in the
arcs to 0 mm around the high luminosity experiments ATLAS (IP1) and CMS (IP5). One
MBXW magnet is used in the orbit bump around the LHC-b muon spectrometer (IP8). The
MBW magnets (optical functions D3 and D4) are used in the cleaning regions, IR3 and IR7,
to increase the beam separation from 194 mm in the arcs to 224 mm. The MCBW H/V
magnets are used in the cleaning regions for orbit correction (horizontal / vertical). Each
corrector acts on one beam. The “V” type is just a tilted “H” corrector [1]. Normalconducting magnets are needed in these places to avoid the issue of particle showers from the
IP or collimators inducing quench. See Appendix A for optics calculations. The magnetic
length is 3.4 m for MBW and MBXW, 1.7 m for MCBW, and the nominal field is 1.3-1.4 T
(see Table I), for a nominal current of about 700 A.
Fig. 1: MBXW cross-section.
This is an internal CERN publication and does not necessarily reflect the views of the LHC project management.
Fig. 2: MBW cross-section.
Table I: Main parameters of MBW, MBXW and MCBW based upon measurements, design values within () if
different
Magnet type
Magnetic length
Beam separation
Aperture (gap height)
Operating temperature
Nominal field
Nominal current
Resistance
Inductance
Power dissipation at Inom
(m)
(mm)
(mm)
(C)
(T)
(A)
(mohm)
(mH)
(kW)
MBW
3.4
194-224
52
< 65
1.44 (1.42)
720
52 (55)
178 (180)
27 (29)
MBXW
MCBW
3.43 (3.4) 1.73 (1.7)
0-27
224
63
52
< 65
< 65
1.38 (1.28) 1.0 (1.1)
750 (690) 500 (550)
58 (60)
48 (60)
144 (145)
42 (50)
33 (29)
10 (14)
Numbers and variants: The MBXW and MBW have similar “H” design and are made from
the same materials by BINP (Novosibirsk, Russia). The MBXW has a common bore for the 2
beams, whilst the MBW has two bores. The field direction is vertical (“up” or “down”) for
both beams. The MCBW has a single bore and the field is vertical for H and horizontal for V.
Several magnets are connected in series to achieve the needed integrated bending strength.
They operate as one optical element: 6 MBXW in series operate as D1, and 2 or 3 MBW
magnets in series operate as either D3 or D4. The MCBW act alone with a dedicated power
supply.
The number of MBXW in LHC is “2 IP × 2 sides of IP × 6 magnets + 1 for IP8 = 25
magnets”. 25 + 4 spare magnets were produced. The number of MBW for IR3 is “2 sides × 2
optical functions × 3 magnets = 12 magnets” and for IR7 is “2 sides × 2 optical functions × 2
magnets = 8 magnets”. This gives 20 MBW used in LHC. 20 + 4 spares were produced. There
are 20 MCBW correctors. 16 are used in the cleaning sections: “2 IP × 2 sides of IP × 2 beams
× (1 H + 1 V)”. MCBWH.5L8.B1 has replaced a faulty superconducting orbit corrector. So
there are 3 spares.
-2-
Naming convention: The MBW and MBXW magnets are identified by consecutive production
numbers 1 to n. The MTF naming schemes are respectively HCMBW__001-BI0000nn,
HCMBXW_001-BI0000nn and HCMCBW_001-BI0000nn. BI is the manufacturer code for
BINP.
Expected operational cycles, range of current: The operational current range (proton energy
range 450 GeV to 7 TeV) for MBW is around 41-643 A, corresponding to the field range
0.08-1.3 T. The operational current range for MBXW is 43-685 A, corresponding to the field
range 0.08-1.3 T. The optical strength scales linearly with the particle momentum during ramp
(true for all magnets). The field does not change once collision energy has been reached. That
is, the optical functions remain constant during the LHC squeeze. These numbers are based
upon LHC optics V6.503 released in 2008. The MCBW correctors do not have any predefined
optical strength.
The operational currents and fields are given in Tables II-V for the circuits. The FiDeL model
parameters were revised slightly in 2009 so we show original values from 2008 and the ones
from 2009, as well as the difference between them (units) in Tables VI-VII.
Table II: Operational currents and fields for MBW circuits based upon FiDeL 2008.
CIRCUIT
RD34.LR3
RD34.LR7
E = 450 GeV
I (A)
B (T)
40.97
0.08
40.97
0.08
E = 3.5 TeV
I (A)
B (T)
318.68
0.65
318.68
0.65
E = 5 TeV
I (A)
B (T)
455.34
0.93
455.34
0.93
E = 6 TeV
I (A)
B (T)
547.13
1.11
547.13
1.11
E = 7 TeV
I (A)
B (T)
642.65
1.30
642.65
1.30
Table III: Operational currents and fields for MBXW circuits based upon FiDeL 2008.
CIRCUIT
RD1.LR1
RD1.LR5
E = 450 GeV
I (A)
B (T)
43.17
0.08
43.17
0.08
E = 3.5 TeV
I (A)
B (T)
335.76
0.65
335.76
0.65
E = 5 TeV
I (A)
B (T)
479.80
0.92
479.80
0.92
E = 6 TeV
I (A)
B (T)
577.47
1.11
577.47
1.11
E = 7 TeV
I (A)
B (T)
685.21
1.29
685.21
1.29
Table IV: Operational currents and fields for MBW circuits based upon FiDeL 2009.
CIRCUIT
RD34.LR3
RD34.LR7
E = 450 GeV
I (A)
B (T)
40.93
0.08
40.93
0.08
E = 3.5 TeV
I (A)
B (T)
318.32
0.65
318.35
0.65
E = 5 TeV
I (A)
B (T)
454.85
0.93
454.90
0.93
E = 6 TeV
I (A)
B (T)
546.63
1.11
546.68
1.11
E = 7 TeV
I (A)
B (T)
642.39
1.30
642.45
1.30
Table V: Operational currents and fields for MBXW circuits based upon FiDeL 2009.
CIRCUIT
RD1.LR1
RD1.LR5
E = 450 GeV
I (A)
B (T)
43.12
0.08
43.10
0.08
E = 3.5 TeV
I (A)
B (T)
335.34
0.65
335.22
0.65
E = 5 TeV
I (A)
B (T)
479.24
0.92
479.06
0.92
E = 6 TeV
I (A)
B (T)
576.97
1.11
576.74
1.11
E = 7 TeV
I (A)
B (T)
684.99
1.29
684.66
1.29
Table VI: Difference between operational currents and fields for MBW: FiDeL 2009 minus 2008 (units).
CIRCUIT
RD34.LR3
RD34.LR7
E = 450 GeV
I (units) B (units)
-11
0
-10
0
E = 3.5 TeV
I (units) B (units)
-11
0
-10
0
E = 5 TeV
I (units) B (units)
-11
0
-10
0
E = 6 TeV
I (units) B (units)
-9
0
-8
0
E = 7 TeV
I (units) B (units)
-4
0
-3
0
Table VII: Difference between operational currents and fields MBXW: FiDeL 2009 minus 2008 (units).
CIRCUIT
RD1.LR1
RD1.LR5
E = 450 GeV
I (units) B (units)
-13
0
-16
0
E = 3.5 TeV
I (units) B (units)
-12
0
-16
0
E = 5 TeV
I (units) B (units)
-12
0
-15
0
-3-
E = 6 TeV
I (units) B (units)
-9
0
-13
0
E = 7 TeV
I (units) B (units)
-3
0
-8
0
Summary of manufacturing parameters: the main parameters of the MBW, MBXW and
MCBW are shown in Table I.
2. Layout
Slots and positions: the 20 MBW are located in IR3 and IR7 according to Table VIII. The 24
MBXW are located in IR1 and IR5 according to Table IX. These tables are based upon the
installation in 2008.
The field measurements have been done from the CS (connection side) to the NCS (non
connection side), as usual. Half of the MBW magnets 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 the Appendix of Ref. [2]. For the normal MBW dipoles the
equations becomes:
𝑏 ′ n = (−1)n−1 𝑏n
𝑎′ n = (−1)n−2 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 [3].
Circuits: The two apertures in the MBW magnet are connected in series. All MBW magnets in
the optical functions D3 and D4 are connected in series, both sides of the “IP”. This
minimises the number of power converters by taking into account that the integrated field
strength is the same (full symmetry). Similarly all MBXW on both sides of the IP are in
series. This gives a total of 2 circuits for D1 (MBXW) and 2 for D3 + D4. See Tables VIII and
IX.
Table VIII: MBW slot allocation, optical function, nominal beam separation, position (s) in beam 1 direction and
vertical rotation.
SLOT
FUNCTION BEAM SEP S
MAGNET
V ROTATED
MBW.F6L3
D4.L3
194.0 6473.2548 HCMBW__001-BI000004
MBW.E6L3
195.4 6477.4898 HCMBW__001-BI000012
MBW.D6L3
198.8 6481.7248 HCMBW__001-BI000016
MBW.C6L3
D3.L3
219.2 6499.7478 HCMBW__001-BI000005 Yes
MBW.B6L3
222.6 6503.9828 HCMBW__001-BI000017 Yes
MBW.A6L3
224.0 6508.2178 HCMBW__001-BI000019 Yes
MBW.A6R3
D3.R3
224.0 6821.0718 HCMBW__001-BI000020
MBW.B6R3
222.6 6825.3068 HCMBW__001-BI000014
MBW.C6R3
219.2 6829.5418 HCMBW__001-BI000018
MBW.D6R3
D4.R3
198.4 6847.5648 HCMBW__001-BI000021 Yes
MBW.E6R3
195.4 6851.7998 HCMBW__001-BI000013 Yes
MBW.F6R3
194.0 6856.0348 HCMBW__001-BI000001 Yes
MBW.D6L7
D4.L7
194.1 19779.7494 HCMBW__001-BI000008
MBW.C6L7
195.7 19783.9844 HCMBW__001-BI000010
MBW.B6L7
D3.L7
221.4 19819.1389 HCMBW__001-BI000007 Yes
MBW.A6L7
224.0 19824.0739 HCMBW__001-BI000011 Yes
MBW.A6R7
D3.R7
224.0 20164.2509 HCMBW__001-BI000002
MBW.B6R7
222.1 20169.1859 HCMBW__001-BI000006
MBW.C6R7
D4.R7
195.7 20204.3404 HCMBW__001-BI000003 Yes
MBW.D6R7
194.1 20208.5754 HCMBW__001-BI000009 Yes
-4-
Table IX: MBXW slot allocation, optical function, nominal beam separation, position (s) in beam 1 direction and
vertical rotation
SLOT
MBXW.A4R1
MBXW.B4R1
MBXW.C4R1
MBXW.D4R1
MBXW.E4R1
MBXW.F4R1
MBXW.F4L5
MBXW.E4L5
MBXW.D4L5
MBXW.C4L5
MBXW.B4L5
MBXW.A4L5
MBXW.A4R5
MBXW.B4R5
MBXW.C4R5
MBXW.D4R5
MBXW.E4R5
MBXW.F4R5
MBXW.F4L1
MBXW.E4L1
MBXW.D4L1
MBXW.C4L1
MBXW.B4L1
MBXW.A4L1
FUNCTION BEAM SEP
D1.R1
0.0
0.8
3.6
8.4
15.3
24.1
D1.L5
24.1
14.5
8.4
3.6
0.8
0.0
D1.R5
0.0
0.8
3.6
8.4
15.3
24.1
D1.L1
24.1
15.3
8.4
3.6
0.8
0.0
S
61.3220
65.5880
69.8540
74.1200
78.3860
82.6520
13246.7896
13251.0556
13255.3216
13259.5876
13263.8536
13268.1196
13390.7636
13395.0296
13399.2956
13403.5616
13407.8276
13412.0936
26576.2312
26580.4972
26584.7632
26589.0292
26593.2952
26597.5612
MAGNET
V ROTATED
HCMBXW_001-BI000021
HCMBXW_001-BI000020
HCMBXW_001-BI000024
HCMBXW_001-BI000023
HCMBXW_001-BI000025
HCMBXW_001-BI000028
HCMBXW_001-BI000013
HCMBXW_001-BI000004
HCMBXW_001-BI000012
HCMBXW_001-BI000010
HCMBXW_001-BI000011
HCMBXW_001-BI000005
HCMBXW_001-BI000006
HCMBXW_001-BI000002
HCMBXW_001-BI000007
HCMBXW_001-BI000016
HCMBXW_001-BI000003
HCMBXW_001-BI000008
HCMBXW_001-BI000022
HCMBXW_001-BI000014
HCMBXW_001-BI000017
HCMBXW_001-BI000019
HCMBXW_001-BI000015
HCMBXW_001-BI000018
3. Measurements
The requirements of measurements during production are described in detail in [4-6].
Transfer function: Each magnet is measured with a Hall probe array (19 probes positioned
horizontally). The field integral is evaluated by measuring many points in the longitudinal
direction, covering 4 m to include head effects and with a measurement point for each 2 mm.
The MBW magnet is cycled 4 times from 0 to 810 A, ramp rate 5 A/s and flat-top of 30 s to
stabilise the field (reproducibility). The MBW integral field measurements are evaluated for
the currents 100, 500, 720 and 810 A. The MBXW magnet is cycled 4 times from 0 to 835 A,
ramp rate 5 A/s and flat top of 25 s. The MBXW integrated field measurements are evaluated
for the currents 100, 500, 730 and 830 A. The MCBW magnet is cycled 4 times from 0 to 650
A, ramp rate 25 A/s and flat top of 25s. The field integral are evaluated for the currents 100,
500, 550 and 600 A,
The Hall probe measurements are calibrated with absolute values to a NMR measurement of
the MBW-, MBXW-2 and MCBW-1 magnets. As we can see in Table VIII and IX the beam
separation is dependent on the longitudinal position. For this reason, the field is measured
horizontally to cover +/- 45 mm of aperture. However, for the FiDeL parameters we use only
the measurements at the central position as we otherwise would have to take the actual slot
into account for each magnet. The goals of the Hall measurements were to achieve an absolute
error below 5 units.
-5-
Field harmonics: 3 magnets of each family were measured at CERN with rotating coils: MBW
1, 9 and 15 and MBXW 1, 9 and 20. These measurements give us an estimate of the
multipoles of the series production. The field errors of MCBW are only a fraction of units and
can therefore be ignored given the fact that it is a corrector.
4. MBW transfer function
4.1 INITIAL DECISIONS
Table X shows the integrated transfer function for all the Hall measurements. There is no
significant difference between aperture 1 and 2 so we can safely take the average value of the
two for the purpose of computing the FiDeL parameters. These data served as the only source
for fitting the REFPARM in 2008. This time we also want to make comparison with all the
rotating coil measurements. Please note that 640 A is the maximum foreseen current: at this
level, saturation is around 1%.
Table X: Statistics summary of MBW Hall probe measurements, integrated field, average and sigma expressed in
units of geometric value at 100 A.
Aperture I (A)
1
1
1
1
2
2
2
2
100
500
720
810
100
500
720
810
TF (Tm / A) TF ave (units)
TF sigma (units)
0.00692104
0
4
0.00690243
-27
4
0.00672390
-285
14
0.00648078
-636
21
0.00692110
0
4
0.00690242
-27
3
0.00672368
-285
14
0.00648070
-636
20
100
0
MBW 1 TF (units)
-100
-200
-300
-400
-500
-600
-700
0
200
400
600
800
Current (A)
Fig. 3: Comparison of all the measurements of the MBW 1 magnet after removing geometric at 100A and
converting to units. Original NMR measurement (red circles), 2 separate rotating coil measurements magnet at
CERN (green box and blue triangles), Hall integral (black diamonds).
-6-
Fig. 3 shows all measurements we have of MBW 1. The rotating coil measurements are all
local ones from the central part. The measurements agree within 20 units over the current
range. Fig. 4 shows the plots of all local rotating coil measurements together. We decide to
use the average (dotted line) for making one fit for all the MBW magnets. Table XI gives the
average and spread expressed in units of geometric at 100 A. The local rotating coil
measurements have about 50 units less saturation w. r. t. the Hall measurements at 720 A (244
units instead of 294 units), which is already beyond the maximum operational current of 650
A (see Tables X and XI).
50
MBW 1,9,15 TF (units)
0
-50
-100
-150
-200
0
200
400
600
800
Current (A)
Fig. 4: All local measurements of MBW 1, 9 and 15 plotted together for comparison, circles = RC measurement,
average = dotted line.
Table XI: Statistics summary of all local MBW rotating coil measurements, average and sigma expressed in units
of geometric value at 100 A.
I (A)
30
40
100
150
200
300
400
500
600
650
700
720
750
810
TF ave (units) TF sigma (units)
12
5
-3
5
0
0
4
3
6
5
6
4
1
1
-11
3
-38
4
-82
6
-184
9
-244
14
-347
17
-590
20
-7-
4.2 GEOMETRIC
We use the integral Hall measurement at 100 A to define the geometric component. We give
each magnet its individual value for the purpose of being as accurate as possible for
simulations in MAD-X. Since the magnets are connected in series on only 2 circuits we
expect that the geometric will be practically the same for both circuits: this is the case, within
1 unit, see Table XII.
Table XII: TF geometric, average and spread for the MBW circuits
CIRCUIT
RD34.LR3
RD34.LR7
TF (Tm / A) TF (units)
0.00692186
0.00692121
Average
Stdev
0.5
-0.5
0.00692154
0.7
4.3 SATURATION COMPONENT
There are no dynamic, time-dependent components for normal-conducting magnets. The fits
are done according to the equations in [7]. We use only the saturation component to fit the TF
curve in Fig. 4.
𝑇𝐹 = 𝛾 𝑔𝑒𝑜 + 𝑇𝐹 𝑠𝑎𝑡
𝑇𝐹 𝑠𝑎𝑡 = −
𝜎𝑚
𝐼 − 𝐼0
[1 + 𝑒𝑟𝑓 (𝑆
)]
2
𝐼𝑛𝑜𝑚
In principle we should also have included the residual magnetisation component, which can
be seen in Table XI as 12 units at 30 A. However, the power supply cannot reproduce the precycle done for the measurements as it can only go slightly lower than the injection current.
Therefore we do not know how strong this effect is.
The fit is based on the average of the Hall measurements merged with average of local
measurements in Table X and XI. We give more weight to the Hall measurements at high
current, assuming the local RC measurements underestimate head effects. The measurements
with current < 300A are not considered. We notice that in Table XI the values for currents
between 150 and 400 A are slightly positive. This could be because we have defined the
geometric at too low current where the residual magnetisation is still present. But we need to
use the value at 100 A in order to calibrate the curve to the absolute Hall integral value at 100
A. The alternative would have been to define the geometric at 500 A, but we assume to have
some saturation at this current.
-8-
100
0
TF (units)
-100
-200
-300
-400
-500
-600
-700
0
200
400
600
800
Current (A)
Fig. 5: Fit of the MBW TF component saturation (units), rotating coil measurements (triangles), FiDeL curve
(dotted line), Hall probe measurements (diamonds).
Fig 5. illustrates that we have some discrepancy between the RC and Hall integral
measurements at high current. The fit goes almost exactly through the Hall measurements
points at 100 and 810 A. There is a difference of 30 units between fit and Hall measurement
at 720 A.
The FiDeL fit parameters are shown in Table XIII and XIV. The residual fit error is shown in
Fig. 6. This is just based on the difference between the FiDeL and the measured values
without considering uncertainty of the measurements and estimates. The error of the fit is
below 20 units in the operational current range.
Table XIII: FiDeL TF 2008 fit parameters for MBW saturation
Parameter
s
I0
s
Inomref
Units
(T m /A)
(A)
(adim)
(A)
Value
0.00167790
900.9500
3.5982
720
Table XIV: FiDeL TF 2009 fit parameters for MBW saturation
Parameter
s
I0
s
Inomref
Units
(T m /A)
(A)
(adim)
(A)
-9-
Value
0.00161642
896.2477
3.5510
720
30
Residual error (units)
20
10
0
-10
-20
-30
-40
30
230
430
630
830
Current (A)
Fig. 6: Error of the fit of the MBW TF, 2009 parameters (units).
5. MBW field errors
The local rotating coil measurements provide the field harmonics. We have to assume that the
integral harmonics are equal to the local values, since the information about the heads are not
available. We have only measurements for three magnets: MBW 1, 9 and 15. Table XV
summaries the situation. We decide to model b3, b5 and b7 since the average is somewhat
significant and larger than the spread. These harmonics are also expected (“allowed”) from
design point of view. It is assumed the field errors in the cleaning insertions are of low
importance since the beta-function is moderate. Anyhow we choose to model the field errors
for the sake of completeness for MAD-X simulations. We decide to only use the geometric
value for the harmonics. In fact we use the average value of 40 and 600 A since the harmonics
seems rather constant in this current region. See last row in Table XV.
Table XV: Average and standard deviation for the MBW harmonics.
Ap
I (A)
b2
a2
b3
a3
b4
a4
b5
a5
b6
a6 b7
a7
b8
a8
b9
a9 b10 a10 b11 a11
ave
stdev
1
1
40.0
0.3
1.8
0.1
0.1
1.5
0.4
0.0
0.1
0.0
0.4
0.0 -0.4
0.0 0.3
0.0 0.0 0.0 -0.3 0.0 0.0 0.0 -0.1 0.0 0.0 0.0 0.0 0.0
0.0 0.4 0.1 0.2 0.0 0.2 0.0 0.0 0.0 0.0 0.0 0.0 0.0
ave
stdev
1
1
600.0
0.7
1.6
0.1
0.3
1.3
0.4
0.0 -0.1
0.1 0.5
0.0 -0.3
0.0 0.3
0.0 0.0 0.0 -0.3 0.0 0.0 0.0 -0.1 0.0 0.0 0.0 0.0 0.0
0.1 0.4 0.1 0.2 0.1 0.2 0.0 0.0 0.0 0.0 0.0 0.0 0.0
ave
stdev
1
1
810.0
1.5
1.3
0.2 -0.5
0.2 0.8
0.0 -0.2
0.2 1.0
0.1 -0.6
0.1 0.3
0.0 0.1 0.0 -0.3 0.0 0.0 0.0
0.1 0.4 0.1 0.2 0.1 0.2 0.0
ave
stdev
2
2
40.0
-0.5
1.8
0.0
0.1
1.4
0.5
0.0
0.1
0.0
0.5
0.0 -0.4 -0.1 0.1 0.0 -0.3 -0.1 0.1 0.0 -0.1 0.0 0.0 0.0 0.0 0.0
0.1 0.2 0.1 0.5 0.1 0.2 0.1 0.2 0.1 0.0 0.0 0.0 0.0 0.0 0.0
ave
stdev
2
2
600.0
-0.7
1.6
0.0
0.1
1.3
0.5
0.0
0.2
0.1
0.5
0.1 -0.3
0.1 0.2
0.0 0.0 0.1 -0.3 -0.1 0.1 0.0
0.2 0.4 0.2 0.2 0.1 0.2 0.1
0.0 0.0 0.0 0.0 0.0 0.0
0.1 0.0 0.1 0.0 0.1 0.0
ave
stdev
geo
2
2
810.0
-1.4
1.1
0.2 -0.4 -0.1
0.2 0.8 0.3
1.4
0.3
0.8
0.1 -0.5 -0.1 0.0 0.0 -0.2 -0.1 0.1 0.0
0.2 0.2 0.1 0.3 0.2 0.2 0.1 0.2 0.1
-0.4
-0.3
0.0 0.0 0.1 0.0 0.1 0.0
0.2 0.0 0.2 0.0 0.1 0.0
- 10 -
0.0 0.0 -0.1 0.0 0.1 0.0
0.1 0.0 0.1 0.0 0.1 0.0
6. MBXW transfer function
6.1 INITIAL DECISIONS
Table XVI shows the integrated transfer function for all the Hall measurements. It depends on
the calibration of one single NMR measurement of MBXW 1. The spread increases with the
current but is low in most of the operational range. Please note that at the maximum
operational current of 680 A the saturation is about 2%.
Table XVI: Statistics summary of MBXW Hall probe measurements, integrated field, average and sigma
expressed in units of geometric value at 100 A
I (A)
TF (Tm / A) TF ave (units)
TF sigma (units)
100
0.00655273
0
7
500
0.00653464
-28
4
750
0.00627223
-428
14
830
0.00606208
-749
16
100
0
MBXW 1 TF (units)
-100
-200
-300
-400
-500
-600
-700
-800
0
200
400
Current (A)
600
800
Fig. 7: Comparison of the measurements of the MBXW 1 magnet after removing geometric at 100A and
converting to units. Original NMR measurement (red circles), 2 separate rotating coil measurements magnet at
CERN (green box and blue triangles), Hall integral (black diamonds).
Fig. 7 shows all measurements we have of MBW 1. The rotating coil measurements are all
local ones from the central part. One can see that the Hall measurement has been calibrated to
the NMR. The measurements agree within 30 units over the operational current range. The
NMR measurement shows stronger saturation. The two rotating coil measurements were done
after the magnet arrived at CERN. They fully overlap in Fig. 7. Fig. 8 shows the plots of all
local rotating coils measurements together. Table XVII gives the average and spread
expressed in units of geometric at 100 A. The NMR measurement was used for the FiDeL
model parameters in 2008. This time we add the local RC measurements for the shape of the
TF curve.
- 11 -
100
MBXW 1,9,20 TF (units)
0
-100
-200
-300
-400
-500
-600
-700
-800
0
200
400
600
800
Current (A)
Fig. 8: All local measurements of MBW 1, 9 and 20 plotted together for comparison, dots = rotating coil
measurement, average = dotted line.
Table XVII: Statistics summary of all local MBXW RC measurements, average and sigma expressed in units of
geometric value at 100 A
I (A)
30
40
100
150
200
300
400
500
600
650
700
750
800
830
TF ave (units) TF sigma (units)
6
4
-3
13
0
0
7
3
6
3
5
2
-1
2
-12
2
-43
8
-107
22
-229
33
-397
39
-593
40
-718
39
6.2 GEOMETRIC
We use the integral Hall measurement at 100 A to define the geometric component. We give
each magnet its individual value for the purpose of being as accurate as possible for
simulations in MAD-X. Since the magnets, except MBXW 1 having its own power supply,
are connected in series on only 2 circuits we expect that the geometric will be practically the
same for both circuits, see Table XVIII.
- 12 -
Table XVIII: TF geometric, average and spread for the MBXW circuits
CIRCUIT
TF (Tm / A) TF (units)
RD1.LR1
0.00655124
RD1.LR5
0.00655370
RBXWH.L8
0.00656430
-7.9
-4.1
12.0
Average
Stdev
10.6
0.00655641
6.3 SATURATION COMPONENT
There are no dynamic, time-dependent components for normal-conducting magnets. The fits
are done according to the equations in [7]. We use only the saturation component to fit the TF
curve in Fig. 8. In principle we should also have included the residual magnetisation
component, which can be seen in Table XVIII as 6 units at 30 A. However, the power supply
cannot reproduce the pre-cycle done for the measurements as it can only go slightly lower
than the injection current. Therefore we do not know how strong this effect is.
𝑇𝐹 = 𝛾 𝑔𝑒𝑜 + 𝑇𝐹 𝑠𝑎𝑡
𝑇𝐹 𝑠𝑎𝑡 = −
𝜎𝑚
𝐼 − 𝐼0
[1 + 𝑒𝑟𝑓 (𝑆
)]
2
𝐼𝑛𝑜𝑚
The fit is based on the average of the Hall measurements merged with average of local
measurements in Table XVI and XVII. We give more weight to the Hall measurements at
high current, assuming the local RC measurements underestimate head effects. The
measurements with current < 400A are not considered. We have also used a ROXIE model for
the MBXW magnets. It gives significantly less saturation effect than what has been measured.
Therefore we do not use it.
Fig 9. illustrates that the fit matches well the Hall measurements, except at 500 A where there
is a difference of 20 units. We assume in general the RC measurements underestimate head
effects.
The FiDeL fit parameters are shown in Table XIX and XX. The residual fit error is shown in
Fig. 10. This is just based upon the difference between the FiDeL and the measured values
without considering uncertainty of the measurements and estimates.
Table XIX: FiDeL TF 2008 fit parameters for saturation.
Parameter
s
I0
s
Inomref
Units
(T m /A)
(A)
(adim)
(A)
Value
0.00075038
789.8300
4.3711
690
Table XX: FiDeL TF 2009 fit parameters for saturation.
- 13 -
Parameter
s
I0
s
Inomref
Units
(T m /A)
(A)
(adim)
(A)
Value
0.00077631
791.7549
4.2487
690
100
0
-100
TF (units)
-200
-300
-400
-500
-600
-700
-800
-900
0
200
400
600
800
Current (A)
Fig. 9: Fit of the MBXW TF component saturation (units), measurements (triangles), FiDeL curve (dotted line),
Hall probe measurements (diamonds), ROXIE estimate (squares).
Residual error (units)
40
30
20
10
0
-10
-20
-30
0
200
400
600
Current (A)
Fig. 10: Residual error of the fit of the MBXW TF (units).
- 14 -
800
7. MBXW field errors
The local RC measurements contain harmonics. We have to assume that the integral
harmonics are equal to the local values. We have only measurements for three magnets: MBW
1, 9 and 20. Table XXI summaries the situation. The field quality is excellent! We decide not
to model any. The only harmonic that shows up is b2, but the value is only -0.2 units
Table XXI: Average and standard deviation for the MBXW harmonics.
I (A) b2
ave
stdev
a2
b3
40.0
-0.2
0.1
-0.2
0.2
ave 500.0
stdev
-0.2
0.1
0.0
0.1
ave 750.0
stdev
ave 830.0
stdev
0.1
0.3
a3
b4
a4
b5
a5
b6
a6 b7 a7
b8
a8
b9
a9
b10 a10 b11 a11
0.0
0.0
0.0
0.0
0.0
0.0
0.1
0.2
0.0
0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0
0.0 -0.1
0.3 0.1
0.0
0.0
0.0
0.0
0.1
0.1
0.0
0.1
0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0
-0.2
0.1
0.0 -0.4 -0.1
0.1 0.3 0.1
0.0
0.0
0.0
0.0
0.1
0.2
0.0
0.1
0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0
-0.3
0.1
0.0 -0.9 -0.1
0.1 0.3 0.1
0.0
0.0
0.0
0.0
0.1
0.2
0.0
0.1
0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0
- 15 -
8. MCBW transfer function
8.1 INITIAL DECISIONS
Table XXII shows the integrated transfer function for all the Hall measurements. It depends
upon the calibration of one single NMR measurement of MCBW 1. The spread is unusually
low in all the operational range. Saturation is about 1% at the nominal current of 500 A.
Table XXII: Statistics summary of MCBW Hall probe measurements, integrated field, average and sigma
expressed in units of geometric value at 100 A
I (A) TF (Tm / A) TF ave (units) TF sigma (units)
100 0.00349836
0
5
500 0.00346060
-108
6
550 0.00342069
-222
6
600 0.00335983
-396
6
50
0
MCBW 1 TF (units)
-50
-100
-150
-200
-250
-300
-350
-400
0
200
400
600
800
Current (A)
Fig. 11: Measurements of the MCBW 1 magnet after removing geometric at 100A and converting to units.
Local rotating coil measurement at CERN (green triangles), Hall integral (black diamonds).
8.2 GEOMETRIC
We use the integral Hall measurement at 100 A to define the geometric component. Since the
spread is only 6 units in Table XXII we use the same value for all 20 magnets.
8.3 SATURATION COMPONENT
There are no dynamic, time-dependent components for normal-conducting magnets. The fits
are done according to the equations in [7]. We use only the saturation component to fit the TF
curve in Fig. 11. In principle we should also have included the residual magnetisation
component, but the measurement in Fig. 11 does not show any effect so we are missing data
to evaluate this effect. The data for the fit is shown in Table XXIII. We use the average of the
Hall measurements. In addition we add some extra current points from the initial NMR
measurement. The resulting TF curve is shown in Fig. 12 and the residual error in Fig. 13.
- 16 -
Table XXIV and XXV reveals that the FiDeL parameters are almost the same for 2008 and
2009.
𝑇𝐹 = 𝛾 𝑔𝑒𝑜 + 𝑇𝐹 𝑠𝑎𝑡
𝑇𝐹 𝑠𝑎𝑡 = −
𝜎𝑚
𝐼 − 𝐼0
[1 + 𝑒𝑟𝑓 (𝑆
)]
2
𝐼𝑛𝑜𝑚
Table XXIII: Data used to fit the MCBW transfer function, expressed in units of geometric value at 100 A
I (A) TF ave (units)
100
0
196
-8
244
-10
292
-12
340
-15
388
-20
436
-31
484
-71
500
-108
550
-222
600
-396
50
0
-50
TF (units)
-100
-150
-200
-250
-300
-350
-400
-450
0
100
200
300
400
500
600
Current (A)
Fig. 12: Fit of the TF component saturation (units), triangles = measurements, dotted line = FiDeL curve,
diamonds = Hall probe measurements.
- 17 -
Table XXIV: FiDeL TF 2008 fit parameters for saturation
Parameter
s
I0
s
Inomref
Units
(T m /A)
(A)
(adim)
(A)
Value
0.00049874
669.7600
3.2866
550
Table XXV: FiDeL TF 2009 fit parameters for saturation
Parameter
s
I0
s
Inomref
Units
(T m /A)
(A)
(adim)
(A)
Value
0.00049872
669.7253
3.2877
550
Residual error (units)
15
10
5
0
-5
-10
-15
0
200
400
600
Current (A)
Fig. 13: Residual TF 2009 fit error (units).
9. Summary and critical issues


All magnets have been measured individually with Hall probes. Each one of the three
families (MBW, MBXW and MBCW) shows a pretty low spread (a few units). For the
main magnets the geometric has been modelled individually, but the resulting
differences in the TF of the circuits are small: within 1 unit for the MBXW and within
10 units for the MBW. For the correctors one TF has been adopted. The precision of
this measurement, calibrated with NMR probes should be of a few units.
All magnets have a relevant saturation (between 1 and 2% at nominal current). A
systematic difference in the saturation component is observed between the Hall and
- 18 -
the rotating coil measurements (which do not include the magnet heads). Close to the
nominal current we can expect a larger error of the model, of the order of 10 units.

Field quality, measured with rotating coils, is very good and practically not sensitive to
the current. It has been modelled only for a few harmonics in the MBW.
Acknowledgements
We wish to thank the personnel in BINP and CERN doing the measurements, R. Wolf for
making the original analysis and REFPARM file for the 2008 start-up, L. Bottura, M. Buzio,
S. Ramberger, G. de Rijk and E. Todesco for discussions, equations and feedback.
References
[1]
[2]
[3]
[4]
O. Bruning, et al., CERN Report 2004-003 (2004)
R. Wolf, “Corrector Memo”, internal department memo, 2006
R. Wolf, “Field Error Naming Conventions For LHC magnets”, rev 3, EDMS 90250, 2001
G. de Rijk “Technical Specification for the magnet measurements of the MBW magnets for the LHC beam
cleaning insertions”, EDMS 310226, 2001
[5] G. de Rijk “Technical Specification for the magnet measurements of the MBXW magnets for the LHC
beam cleaning insertions”, EDMS 473793, 2002
[6] S. Ramberger et al “Normal-Conducting Separation and Compensation Dipoles for the LHC Experimental
Insertions”, EDMS 703192, 2006
[7] FiDeL Model specifications, EDMS 908232, 2008
- 19 -
Appendix A
We include the following simple treatment to better understand the optical issues involved.
The strength assigned to an optical element like a magnet is normalised with the design radius
for a particle with nominal momentum, and nominal bending field in the arc. A dipole has the
simple strength equation:
kb =
Bb
Bmb_arc ρ0
There is a good reason for this normalisation. The B field scales linearly with particle
momentum to preserve geometry. Dividing by Barc and ρ0 we obtain momentum invariance
and a circle with unit radius. For small angles (sinΘ ~ Θ) we can multiply the dipole field with
the magnetic length to get the deflection angle. Therefore the bending strength of an ideal arc
dipole can be approximated with a simple equation for the bending deflection:
θmbarc = k mb_arc lmb_arc =
2π
nmb_arc
Thanks to the normalised bending strength, we can now express small bending angles in the
machine using simple geometry (without explicit reference to magnetic fields). In addition the
deflection can be thought of as taking place in the middle of the magnet (discontinuity rather
than a smooth arc function). The principle is shown in Fig. 14.
Dn+1
IR region where no Barc field so s-axis is a straight line segment
Dn
ln
ln+1
Δh = bs/2
Θd
s-axis
Δs
Fig. 14: Simplified diagram for particle motion through two magnets changing the beam separation.
The beam goes through the Dn magnet where it receives a small deflection angle in the
middle of the magnet (ln/2). The beam travels in a straight line to next dipole magnet where
the deflection angle is cancelled. The s-axis is a straight line for the IR-region and is in the
mid-plane between the 2 beams. If we imagine that the same process takes place in the other
beam travelling from right to left in Fig. 11 (not shown for simplicity), the change in beam
separation is 2Δh so the deflection angle is:
Θd = tan−1 (
The bending strength for Dn is:
- 20 -
Δbs
)
2Δs
𝑘𝑑 =
1
𝛩
𝑙𝑛 𝑑
Example:
On the right side of IP1 (ATLAS) we have D1 equal to 6 x MBXW in series. The s-coordinate
of s(D1_mid) = 71.987 m and for s(D2_mid) = 157.900 m so Δs = 85.913 m. Δh = 0.194 / 2 m
because before D1 the beam separation is close to 0 and after D2 the standard arc-separation
is required. This gives:
Θd ~
𝑘𝐷1 =
0.194
2∗85.913
1
6 𝑙𝑀𝐵𝑋𝑊
𝑘𝐷2 =
1
𝑙𝑀𝐵𝑅𝐶
rad ~ 1.129 mrad
𝛩𝑑 ~
1.129
6 ∗ 3.4 ∗ 1000
𝛩𝑑 ~
1.129
9.45 ∗ 1000
As a rule of thumb we can expect that 1 m of superconducting dipole magnet in LHC (NbTi)
corresponds roughly to 2.5 m of a conventional magnet (max operational field of 8.5 T vs 3.5
T).
Finally we present the equations for computing the Δh more precisely. These equations are
based upon the notation in Fig. 11. The bending magnet forces the particle into a circular
motion with a bending radius ρD over an interaction region equal to the magnetic length. The
equation for motion on a circular path requires:
𝑠 2 + (∆ℎ − 𝜌𝐷1 )2 = 𝜌𝐷1 2
where s=0 when it encounters the magnet and s = lD1 at exit. Over this length the bending
angle has increased linearly in s from 0 to Θd. We relate the magnetic length and bending
radius:
𝑙𝐷1 = 𝜌𝐷1 sin 𝛩𝐷
Since Θd is small we find
𝜌𝐷1 =
𝑙𝐷1
𝛩𝐷
The increase in h for the interval 0 ≤ s ≤ lD1 can be written as:
2
𝑙𝐷
𝑠𝛩
𝛥ℎ1 (𝑠) = 1 [1 − √1 − ( ) ]
𝛩𝐷
𝑙𝐷1
The increase in h after exit of D1 and until it enters D2 (drift space) is given by Θd:
𝛥ℎ𝑑 (𝑠) = (𝑠 − 𝑙𝐷1 ) tan 𝛩𝐷
D2 will undo the bending angle. We find the equation Δh2 by rotating the equation for Δh1
around both the h and the s-axis:
- 21 -
𝑙𝐷
𝛥ℎ2 (𝑠) = − 𝛩 2 [1 − √1 − (
𝐷
(𝑠−𝑙𝐷2 )𝛩
𝑙𝐷2
2
) ]+
𝑙𝐷2
𝛩𝐷
[1 − √ 1 − 𝛩𝐷 2 ]
Note that also in D2 we have used s=0 for the start of the magnetic length. The last term in the
equation is due to the fact that we introduce the boundary condition for convenience:
𝛥ℎ2 (0) = 0
This allows us to express the total Δh as the sum of the 3 individual components and the beam
separation as 2Δh. Fig 12. shows the graph of beam separation right of IP1 using the
equations.
0.25
0.2
0.15
0.1
0.05
0
60
80
100
120
140
160
Fig. 15: Beam separation (m) on the right side of IP1 (ATLAS) as function of s (m).
- 22 -