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
Presented at the Compumag 2009 Conference
22-26 November 2009, Florianópolis, Brasil
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: firstname.lastname@example.org
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,
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
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
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
12. DEVICES AND APPLICATIONS2
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 .
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
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 .; 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 .
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 .
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  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
12. DEVICES AND APPLICATIONS3
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 .
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 . 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.
QUENCH HEATER POWERING SCHEMES; COMPARE FIG. 2.
(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)
0 20 40 60 80 100 120 140 160
Coil winding number
Potential to ground
Quench heater position
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 . 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
The voltage to ground as a function of winding number is
displayed for the Cases 2, 3, and 5 at the time of maximum
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.
12. DEVICES AND APPLICATIONS4 Download full-text
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 . 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
 can yield data for powering cycles that have not previously
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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