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CERN-ATS-2010-166

01/07/2010

EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH

CERN - ACCELERATORS AND TECHNOLOGY SECTOR

NUMERICAL FIELD CALCULATION IN SUPPORT OF

THE HARDWARE COMMISSIONING OF THE LHC

Nikolai Schwerg, Bernhard Auchmann, Stephan Russenschuck

CERN, Technology Department, Geneva, Switzerland

The hardware commissioning of the Large Hadron Collider (LHC) required the testing and qualification of the

cryogenic and vacuum system, as well as the electrical systems for the powering of more than 10000 superconducting

magnets. Non-conformities had to be resolved within a tight schedule. In this paper we focus on the role that

electromagnetic field computation has played during hardware commissioning in terms of analysis of magnet quench,

electromagnetic force calculations in busbars and splices, as well as field-quality prediction for the optimization of

powering cycles.

Presented at the Compumag 2009 Conference

22-26 November 2009, Florianópolis, Brasil

Geneva, Switzerland

CERN/ATS 2010-166

February 2010

Abstract

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12. DEVICES AND APPLICATIONS1

Numerical Field Calculation in Support of

the Hardware Commissioning of the LHC

Nikolai Schwerg∗, Bernhard Auchmann∗, Stephan Russenschuck∗

∗CERN/TE, Geneva, Switzerland, Email: bernhard.auchmann@cern.ch

Abstract—The hardware commissioning of the Large Hadron

Collider (LHC) required the testing and qualification of the

cryogenic and vacuum system, as well as the electrical systems for

the powering of more than 10000 superconducting magnets. Non-

conformities had to be resolved within a tight schedule. In this

paper we focus on the role that electromagnetic field computation

has played during hardware commissioning in terms of analysis

of magnet quench, electromagnetic force calculations in busbars

and splices, as well as field-quality prediction for the optimization

of powering cycles.

Index Terms—Superconducting magnets, quench calculation,

Lorentz forces

I. INTRODUCTION

W

tal theories by studying collisions of counter-rotating proton

beams with a center-of-mass energy of 14 tera electron volts

(TeV). Physicists hope to prove the Higgs mechanism for

generating elementary particle masses of quarks, leptons, and

the W and Z bosons. The LHC reuses civil engineering

infrastructure from the Large Electron Positron collider (LEP)

at CERN that straddles the Swiss French border near Geneva.

The existing tunnel is 3.8 m wide and a has circumference

of about 27 kilometers. With a given radius of the acceler-

ator tunnel, the maximum achievable particle momentum is

proportional to the operational field in the bending magnets.

Superconducting dipole magnets cooled to 1.9 K with a

nominal field of 8.33 T, will allow energies of up to 7 TeV

per proton beam.

During the design and construction of the LHC, an under-

taking of more than 20 years, various challenges had to be met

in all domains of physics and engineering. The requirements

on field uniformity in the apertures of the superconducting

magnets also posed a challenge to numerical field computation

and optimization techniques. Field computation needs to reach

an accuracy of six digits inside the magnet aperture. Yet the

modeling should be flexible and fast in order to allow for

optimization routines to determine optimum design parame-

ters. The coupling method of boundary-elements and finite

elements (BEM-FEM) corresponds to both requirements, as

the magnet apertures are contained in the BEM regions. The

coil fields can be computed to machine precision from Biot-

Savart law. Only the nonlinear iron yoke needs to be modeled

by a finite-element mesh.

After an ambitious hardware commissioning phase a first

beam was circulated on September 10, 2008. In this paper we

review the role that numerical field computation plays during

ITH the Large Hadron Collider (LHC), the particle

physics community aims at testing various fundamen-

the hardware commissioning phase, i.e., the period when all

circuits of magnets in the LHC tunnel were tested and their

behavior validated.

II. 3-D FORCE- AND PEAK-FIELD-CALCULATIONS

Electromagnetic forces on interconnection busbars were

identified during commissioning to be an important issue for

the long-term reliability and the electro-mechanical integrity

of the machine. While field-quality calculations for long accel-

erator magnets can be carried out to highest precision in 2-D

calculations, the interconnect regions between magnet coils

and neighbouring magnets require a 3-D approach. Applying

the BEM-FEM technique, the finite-element modeling could

be restricted to the nonlinear ferromagnetic yoke. The involved

layout of cosine-theta-type coil-ends and the busbar routing

was described in a Biot-Savart type model built from straight

line-current segments.

The Biot-Savart model was generated from basic building

blocks such as easy- and hardway bend and twists. The

twist-pitch of the Rutherford-type cable is neglected. An

automatic connection routine computes the transformations

that are required for a continuous interconnection of individual

components. The complete BEM-FEM model consists of about

600,000 line-current segments in the BEM domain, and only

about 60,000 finite elements. The calculation of forces on the

interconnection busbars takes approximately two hours on a

2.8 GHz Xeon processor machine.

Despite the relatively low number of finite elements in

the model, the accuracy of peak-field calculations on the

superconducting cable exceeds by far the precision that could

be achieved in a pure finite-element model. The reason is

that the local field distribution in the cables is determined

by the current flow in individual strands, which cannot be

adequately represented in a FEM approach. We note that the

forces pulling in longitudinal direction (along the magnet axis)

on the so-called half-moon interconnects (the 180-degree arcs

above both aperture), see Fig. 1 (left) that connect individual

coils in a magnet assembly are on average 240 N.

III. QUENCH SIMULATION

A. Active and Passive Quench Protection

Above a certain limit on the temperature, current density

and magnetic flux density, superconductors show a transition

between the superconducting and normal conducting state.

The range of parameters that marks a transition is called

the critical surface. A quench is a transition that causes an

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12. DEVICES AND APPLICATIONS2

Fig. 1.

the main-bending dipoles of the LHC.

Electromagnetic model of the interconnections at the end regions of

amount of resistive losses, which cannot be absorbed by

the cooling system. Quench detection and magnet protection

against overheating and voltage peaks during a quench are

important issues in the design of superconducting magnets [1].

An incipient quench is detected by the resistive voltage rise

across the normal zone, which must be distinguished from

the induced voltage during the ramping of the magnet. This is

accomplished by a bridge detection system. Magnet protection

schemes can be classified in two groups. Passive protection

schemes may include a diode or a resistor connected in parallel

to the magnet, but principally they rely on a strong stabilization

of the conductor, such that the magnet can withstand the

current decay without overheating. A stabilized conductor is a

conductor with a copper-to-superconductor ratio large enough

so that, in case the superconductor quenches, the copper can

take over the current for long enough to ramp down the magnet

safely [1].

We speak of active protection when measures are taken

to speed up the normal-zone propagation and the current

decay in the magnet. The current decay is determined by the

propagation speed of the normal zone and by the external

electrical circuit connected to the magnet. Due to the high

inductance of superconducting magnet circuits, the current

cannot be switched off instantaneously, and therefore the

power supply is short circuited with a free-wheeling diode. The

current-decay rate is given by the inductance and resistance of

the remaining circuit.

Active protection relies on quench heaters and/or on an

energy extraction system. We will be concerned with pro-

tection by heaters. The heaters cause a resistive transition in

the covered coil windings, ensuring that the stored energy

is dissipated over a larger fraction of the coil volume. The

rising resistance decreases the discharge time constant and

thus reduces the hot-spot temperature. A quench-heater circuit

consists of a stainless-steel heater strip co-laminated with

polyimide insulation, a thyristor, and an aluminum electrolytic

capacitor bank. Upon trigger, the capacitor bank is discharged

across the resistance of the heater strip.

As the normal zone propagates along the superconducting

cable, the coil-winding scheme of the magnet with its internal

and external connections must be taken into account. The

electrical connection of the twin-aperture LHC main dipole is

shown in Fig. 2. Current entering the A terminal of the magnet

will flow first through the lower, outer-layer coil of the inner

aperture. The inner aperture is the one at the right-hand side

of an observer looking downstream of the circulating Beam 1.

The current will then enter the lower inner, the upper inner,

and the upper outer coils, before it will be directed to the outer

aperture. Figure 2 also shows the quench-heater circuits and

the outer-layer cables covered by the heaters. Only the high-

field heaters (circuits 211, 221, 111, 121) are usually powered,

the low-field heaters are mounted for redundancy1.

Typical values for the LHC main dipole protection system

are as follows: the threshold voltage is 0.1 V; the quench-

heaters are triggered after a delay of 10 ms needed for signal

validation [2].; the time constant for the dissipated heater

power is 37 ms; measurements indicate that a heater-provoked

quench at 1.5 kA occurs at around 80 ms after the trigger. At

nominal current, the delay reduces to 35 ms [3].

B. Modeling Quench

The field distribution in the coil is calculated by means of

the coupling method between boundary and finite elements

(BEM-FEM). For accelerator magnets, the numerical field

computation can be often restricted to two dimensions. The

rapid current decay during a quench creates losses due to the

interfilament coupling currents (IFCC) and the interstrand cou-

pling currents (ISCC). The coupling-current time constants are

influenced by the copper resistivity, the contact resistances, and

geometric cable parameters, all of which are input parameters

to the simulation.

In the superconducting state, the working point lies below

the critical surface. At every time step, the cable’s temperature

margin to the critical surface is evaluated as a function of

the peak field and current density. The copper resistivity is

calculated as a function of the temperature and of the average

magnetic flux density in the cable cross section. The average

magnetic flux density is taken because of the transposition of

the strands within the cable and the 2-D approximation of the

field problem. The dissipated power in each cable is the sum

of resistive losses due to the current in the copper matrix of

the normal zone, losses due to coupling currents (IFCCs and

ISCCs), the heat transfered from the quench heaters, and beam

losses. The coupling-current losses can be neglected once the

cable has quenched. Quench heaters are characterized by the

maximum heater power, a delay of heat transfer and the time

constant of the exponential decay. These parameters can be

used to validate the model with the measured thermal coupling

between the heaters and the coil [4].

The simulation of thermal processes at cryogenic tempera-

tures is an intricate problem. Material properties at cryogenic

temperatures and under pressure are often difficult to know

to adequate precision. The highly nonlinear behavior of these

parameters lead to an ill-conditioned numerical problem in

quench simulations. Moreover the problem generally is ill-

posed, as there are more model parameters than validation

criteria [5] such as measurements of the current decay and

signals at the voltage taps or quench antenna.

The electromagnetic and thermal models exhibit different

time constants. Moreover, field calculation is computationally

1The naming convention for the quench heater circuits is as follows: XYZ

where X is the aperture (1 or 2), Y is the position at the coil (1 for left and

2 for right), and Z stands for the low or high field region (1 for high, 2 for

low)

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12. DEVICES AND APPLICATIONS3

A

B

Anode right

Outer (1)

Inner (2)

LO

LI

LI

LO

UO

UI

UO

UI

YT111

!!!#

YT112

!!"#

YT121

!"!#

YT211

"!!#

YT122

YT212

YT221

""!#

YT222

"""#

!""##"!"#

$#%#

&#'#

(#

)#

*#

"#

!#

+!#

+"#

+,#

Fig. 2.

LHC main dipole magnet, rotated by 90 degrees, together with the position

of quench heaters. The arrow indicates the turn where the quench is triggered

in our simulations.

Winding scheme and internal connections in the double aperture

more expensive than the solution of the coupled electrical and

thermal network equations. Hence a weak coupling between

the electromagnetic and thermal models is the most efficient

method to solve the multi-physics problem. The integrated

numerical model allows to study the impact of different effects

such as quench-back, normal zone propagation, quench heater

performance, local field distribution, and iron saturation [4].

C. Quench Simulation for Hardware Commissioning

In the commissioning phase of the LHC, a quench heater

connection broke after the cool-down of a dipole magnet to

the operation temperature [6]. As a replacement of a magnet

in the string of superconducting magnets is very costly and

time consuming, the low-field quench heaters, that were built

in for redundancy, were to replace the broken specimen. By

means of quench simulation, an optimum powering scheme

for the remaining quench heater circuits was to be found. The

hot-spot temperature, the maximum voltage to ground, and the

maximum electric field in the coil windings during the quench

were identified as decisive criteria.

The study consists of eight different heater-powering

schemes. Using the heater numbering in Figure 2, the cases

are summarized in Tab. I. Case 1 represents the hypothetical

case in which no quench heaters fire. Case 3 shows the nominal

protection scheme. Four power supplies are available to fire the

quench heaters. One supply may be used to fire two heaters

in parallel. This case is indicated above by brackets. In all

simulations, the quench is assumed to originate in the inner

most turn of coil 2; see the arrow in Fig. 2. The broken heater

is assumed to be number 221.

TABLE I

QUENCH HEATER POWERING SCHEMES; COMPARE FIG. 2.

Case

1

2

3

4

5

6

7

8

Heater configuration

No heaters

(112+122), 121, 211, (212+222)

111, 121, 211, 221 (LHC operation conditions)

111, 121, 211, 222

111, 121, 212, 222

112, 121, 211, 222

112, 122, 212, 222

111, 121, 211, (212+222)

-1400

-1200

-1000

-800

-600

-400

-200

0

200

400

600

V

0 20 40 60 80 100 120 140 160

Coil winding number

Potential to ground

Quench heater position

A

B

X Y Z

1

2

3

4

5

2

3

Fig. 3. Electric potential of the coil windings for different powering schemes

(Case 2, 3, and 5) of the quench heaters.

The model parameters for the quench heaters have been

determined in [7]. In case of two heaters connected in parallel,

the time constant of the heat pulse halves. Also, the energy

that is deposited in the conductors which are covered by the

heaters is reduced by a factor two.

The simulation results are summarized in Table II, which

shows the peak temperature, the quench load, the maximum

voltage to ground, and the maximum electrical field between

adjacent conductors. For Case 1 the simulation was stopped

when a temperature of 600 K was reached, at which point the

magnet is considered to be damaged.

The quench load, i.e., the squared current integrated over

time, is a function of quench current, quench-heater delay,

and the time constant. Low-field heaters have a longer delay

time (about 20 ms longer) as a consequence of the larger

temperature-margin. Moreover, due to magneto-resistive ef-

fects, the resistivity of the conductors which are fired by low-

field heaters is lower than that of conductors covered by high-

field heaters. In conclusion, the use of low field heaters in

replacement for high-field heaters increases the quench load,

which is reflected in a rising peak temperature. The peak

temperature is further increased if hot-spot movement occurs,

e.g., if the temperature in conductors covered by quench

heaters increases faster than the temperature at the quench

origin.

The voltage to ground as a function of winding number is

displayed for the Cases 2, 3, and 5 at the time of maximum

TABLE II

PEAK TEMPERATURE, QUENCH LOAD, MAXIMUM VOLTAGE TO GROUND,

AND ELECTRIC FIELD IN THE LHC MAIN DIPOLE AS A FUNCTION OF THE

QUENCH-HEATER POWERING SCHEME.

Case

Tmax

K

600

280

244

267

299

299

440

260

Quench load

106A2s

64

33

32

33

35

35

42

34

Umax

V

576

296

441

768

1248

273

415

668

Emax

MVm−1

1.19

0.44

0.90

1.13

1.75

0.62

0.38

1.06

1

2

3

4

5

6

7

8

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12. DEVICES AND APPLICATIONS4

amplitude in Fig. 3. For the simulation, we assume that the A

terminal is connected to ground while terminal B is connected

to the cold by-pass diode, yielding a static terminal voltage of

6 V for as long as the diode is conductive. A negative slope

in the graphs indicates predominantly inductive voltage in the

respective windings, whereas a positive slope is due to resistive

voltages. Figure 2 shows the winding scheme and the positions

of heaters, voltage taps, and connection terminals.

For the nominal heater setup, Case 3, we easily identify four

positive slopes due to high-field heaters; see Figure 3. The

voltage distribution along the winding is symmetrical w.r.t.

to the winding midpoint. While the peak voltage is of the

order of several hundred volts, the voltage difference between

the two apertures is negligible. In Case 5, one aperture is

quenched with high-field heaters, and the other one with low-

field heaters. The strong imbalance in resistive voltage creates

a voltage to ground of more than 1.2 kV. It can be said that

the outer aperture discharges the stored energy in the inner-

aperture coils, resulting in a greater temperature rise. In Case 2

all low-field heaters and two high-field heaters are employed.

This setting ensures a good balance of inductive and resistive

voltage over the entire winding, resulting in the lowest voltages

in the study.

Eventually the scheme of Case 8 was applied to replace the

broken heater. In this solution all remaining high-field heaters

are fired, as well as two low field heaters. The peak voltages

are higher than in other schemes. However, the firing of all

remaining high-field heaters results in a reduced quench load

and an optimized peak temperature.

IV. FIELD-QUALITY SIMULATION

The operation of the LHC requires that all field errors in the

superconducting magnets are compensated for by dedicated

corrector magnets, so that the integrated field error as seen by

the particle beam remains below a tolerated limit. Beam-based

measurements can serve as a feedback on the field quality and

as an input for automated controls of the corrector magnets.

Yet, for a large scale machine like the LHC, operators cannot

rely solely on feedback systems. For this reason, the Field

Description of the LHC (FiDeL) program collects measure-

ment data of all components in the the LHC and extracts

a fast online model, that yields the field quality at a given

time, magnet operating current, magnet ramp rate, magnet

temperature, and magnet powering history [8]. The modeling

is especially critical at very low currents, where the persistent

current magnetization of superconducting strands has a large

influence on the field quality. The FiDeL model is based

on measurements that were performed for quality assurance

during the production of the LHC magnets. Simulation is

performed if required for three reasons: 1) validation of the

FiDeL mathematical model; 2) counter-check measurement

data; 3) supply data where measurements are not available.

The FiDeL model is based on the identification and physical

decomposition of the effects that contribute to the total field in

the magnet aperture. Each effect is modeled by an appropriate

mathematical model. The physical behavior of the models

can be tested over the entire parameter space by means of

simulation. At the exception of decay- and snap-back effects,

the ROXIE software comprises all relevant effects. A database

of electromagnetic models of all LHC superconducting mag-

nets is used to compare model- and measurement results.

Suspicious measurement data can be tested and validated by

comparison. In the low-field region, where the persistent-

current effects on field quality are dominant, the ROXIE model

[9] can yield data for powering cycles that have not previously

been measured.

V. CONCLUSION

Numerical field computation is indispensable in the design

and manufacturing phase of accelerator magnets. Computa-

tional challenges, however, arrise also from the commissioning

phase. Understanding actual magnet behavior, contributing to

risk analysis, helping to improve the overall understanding

of the machine, and working on fall-back solutions in case

of equipment failure, are some of the tasks encountered

during the LHC commissioning. To keep the response time

to computation requests short, up-to-date numerical models of

all equipment must be maintained for all magnets installed in

the accelerator tunnel.

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