Magnetic frequency response of High-Luminosity Large Hadron Collider beam screens

Magnetic fields used to control particle beams in accelerators are usually controlled by regulating the electrical current of the power converters. In order to minimize lifetime degradation and ultimately luminosity loss in circular colliders, current-noise is a highly critical figure of merit of power converters, in particular for magnets located in areas with high beta-function, like the High-Luminosity Large Hadron Collider (HL-LHC) insertions.However, what is directly acting upon the beam is themagnetic field and not the current of the power converter, which undergoes several frequency-dependent transformations until the desired magnetic field, seen by the beam, is obtained. Beam screens are very rarely considered when assessing or specifying the noise figure of merit, but their magnetic frequency response is such that they realize relatively effective low pass filtering of the magnetic field produced by the system magnet-power converter. This work aims at filling this gap by quantifying the expected impact of different beam screen layouts for the most relevant HL-LHC insertion magnets. A well-defined postprocessing technique is used to derive the frequency response of the different multipoles from multiphysics finite element method (FEM) simulation results. In addition, a wellapproximated analytical formula for the low-frequency range of multi-layered beam screens is presented.


I. INTRODUCTION
In the framework of the High-Luminosity Large Hadron Collider (HL-LHC) project, numerous components of the accelerator will be upgraded during the third LHC long shutdown [1,2].The main focus lies on the interaction region (IR) of LHC Point 1, where the ATLAS experiment is located, and Point 5 occupied by the CMS detector.The layout, for either side of Point 1 or Point 5, is depicted in Fig. 1.New beam screens will be installed inside the so called inner triplet (IT) quadrupole magnets, namely Q1, Q2a, Q2b, and Q3 (see Fig. 2), together with the separation-recombination dipoles D1 and D2 [3].
All these magnets are installed at locations with high beta function and the beam is therefore particularly sensitive to any changes in the magnetic field of these magnets.The impact of the field fluctuations due to power supply noise, or the so called ripple, on the beam lifetime has been studied in the past in particular at the Super Proton Synchrotron (SPS) at CERN [4][5][6][7], the Hadron Electron Ring Facility (HERA) at DESY [8,9] and the TEVATRON at FERMILAB [10].In the case of the SPS a tune ripple of 10 −4 turned out to be acceptable while experiences at HERA showed that a tune ripple of 10 −5 for low frequencies and 10 −4 for high frequencies could lead to a significant decrease in lifetime.Typically, a few distinct frequencies feature much higher amplitudes and therefore, the focus of these studies was also to highlight the impact of a few distinct frequency lines.It was derived theoretically [8] as well as proven experimentally [4,5] that several frequencies in the noise spectrum are significantly more harmful than a single one.This knowledge was then applied to the LHC [11] and more recently to the HL-LHC [12][13][14] resulting in strict tolerances for the noise generated by the power converters.The HL-LHC studies revealed that a tune modulation due to power converter ripple in the quadrupoles in the range of 10 −5 to 10 −6 for specific frequencies lead to a visible decrease of the dynamic aperture and thus an expected degradation of the beam lifetime, see Fig. 3.Those studies approximated the magnet circuit with a simple RL-series circuit, but did not include the impact of the beam screen shielding that could change the figures on power converter noise tolerances.The minimization of the tune modulation to such small values, especially for the HL-LHC IT and separation dipoles where β-functions reach about 20 km (up to 40 km in pushed configurations), represents a technological challenge.
The detailed analysis of the frequency dependent shielding effect of the beam screen presented in this paper, was motivated by the need to specify the current and voltage stability of the power converters for the HL-LHC IR magnet [12][13][14][15].In general, two regimes can be distinguished for power converter ripples: (i) current control where f ≤ f 0 and the current of the power converter is directly controlled, (ii) voltage control where f > f 0 and the voltage of the power converter is directly controlled, assuming f 0 as a parameter of the power converter regulation ranging from few hundreds mHz to few Hz for HL-LHC.As it will be shown shortly, the transfer function of the beam screen for the main field component will not introduce any attenuation up to roughly 10 Hz and therefore the shielding effect of the beam screen only contributes in the voltage control regime.In this case, the following model for the transfer function from the voltage of the power converter to the magnetic field seen by the beam can be assumed: T B m to B b ðfÞ and to the best of the authors' knowledge a rigorous characterization of T B m to B b ðfÞ has never been presented in literature prior to this publication.The paper is structured as follows: Sec.II is dedicated to the description of the different HL-LHC beam screens, the FEM modeling and the simulation details.Sec.III presents the post-processing technique used to derive the frequency behavior of the relevant multipole components.In Sec.IV a new analytically approximated formulation is derived.The novelty of this low-frequency formula compared to earlier publications [16][17][18][19] is that it also applies to multi-layer conductive shells like those constituting the beam screens foreseen for HL-LHC.Furthermore, T B m to B b ðfÞ is properly defined in multipole terms as it is generally done in magnetic measurements for the DC characterization of particle accelerator magnets.Finally, a similar study for the LHC main dipoles and quadrupoles is presented in Sec.V for comparative purposes.

II. SIMULATION SETUP A. Beam screen versions
The new beam screen is an octagonally-shaped pipe made of high nitrogen-high manganese stainless steel (P506) [20] whose main function is to shield by means of tungsten-based inserts the superconducting magnets from debris coming from the collisions, screening the cold mass from beaminduced heating, and ensuring the vacuum levels required for the beam lifetime.The beam screen is placed within the cold bore made of stainless steel 316 LN of the new superconducting magnets [2,21].Cross sections of the different configurations are depicted in Fig. 4 and their geometrical dimensions are reported in Table I.
The internal side of the assembly is colaminated with 75 μm of high purity copper (RRR 100) to lower the beam impedance.On top of the longitudinal flat surfaces of the beam screen four tungsten-based inserts and four cooling tubes are placed in an alternate way.The inserts are laid on the octagonal pipe to avoid detrimental residual stress during the cool-down [22], while the tubes are laserwelded.The temperature of the beam screen is expected to be between 60 and 80 K while the cold bore is maintained at 1.9 K.The heat load is therefore intercepted at a higher temperature by the inserts and then transferred to the cooling tubes through small copper links.The evacuation of heat at these temperatures rather than at 1.9 K lowers dramatically the energy consumption required by the cryogenic system.In addition, the desired vacuum performance of the screen fitted with pumping slots is ensured [23].As discussed in [16,17], the filtering effect of the beam screen depends on the electrical conductivity σ, where a larger electrical conductivity leads to a stronger filtering effect (lower cutoff frequencies).
As σ is inversely proportional to temperature, the worst case scenario within the scope of this work is set to be at 80 K for the Q1, Q2 (in the following Q2 will be used both for Q2a and Q2b), Q3 and D1 beam screens.Similarly, a 20 K threshold is assumed for the D2 beam screen that will be kept at the LHC temperature (i.e., 4.5 K up to 20 K).

B. Physics of the model
The magnetic frequency response of the beam screen is computed through a two-dimensional numerical model implemented in the commercial FEM platform COMSOL Multiphysics [24].The model hinges on the well known Maxwell's equations, which are considered in their differential form.For clarity, the relevant quantities, symbols, and units are summarized in Table II.
Such a model was originally developed to study the mechanical behavior of the beam screen during a magnet quench [25].In this model, the magnet coils generating the harmonic field have not been accounted for in the FEM discretization as this would have increased considerably the complexity and the computational load of the simulation.The magnetic field input has been considered more conveniently through the reduced magnet vector potential (RMVP) formulation [26].This formulation is based on the magnetic vector potential by which B can be expressed as: In turn, the magnetic vector potential is the sum of the reduced potential, A red , and the known background field, A ext : The strategy is to solve only for A red .
Therefore, for a time-harmonic study, the governing equation in the conducting region becomes: where the external current density J e is 0 for the application treated in this paper.

C. Boundary conditions
To account for the magnetic-field interactions, it is necessary to model a medium surrounding the beam screen.Therefore, a cylindrical domain has been created around the assembly.Symmetry conditions are used to lower the computational load of the magnetic simulations.A quarter of the beam screen is sufficient to fully characterize the behavior of the assembly.Figures 5(a) and 5(b) show the boundary conditions for a quarter of the beam screen inserted in a dipole and quadrupole magnet, respectively.
The magnetic insulation boundary condition can be expressed as: and the perfect magnetic conductor boundary condition as: The former implies that the magnetic field is zero in the normal direction to the boundary.Therefore, the field can only be tangential.The latter has the opposite effect, i.e., the magnetic field can only be perpendicular to the selected boundary while the tangential component has to be zero.Considering the magnetic field distribution of the dipolar field, see Fig. 5(a), B can be truncated along the y-axis by the magnetic insulation condition and along the x-axis by the perfect magnetic conductor condition.Instead, the quadrupole field can be truncated at π 4 and 3 4 π angle by the perfect magnetic conductor conditions, see Fig. 5(b).
In the text B is referred as magnetic field or induction.
To use the reduced-field formulation, the total field A has to be equal to the background field A b along the outer boundaries of the air domain, see Figs. 5(a) and 5(b).This condition translates into:

D. Domain discretization
In electromagnetic problems the mesh discretization depends mainly on the skin depth of the physical domains.The skin depth for a conductive material is: It is recommended that at least two linear elements per skin depth are used to capture the variation of the fields [27].If the skin depth is much shorter than the geometrical domain it can be replaced by an impedance boundary condition.In the study presented in this paper, the inner copper layer at 20 K has the highest electrical conductivity, namely 6 × 10 9 S m −1 , and therefore the shortest skin depth amongst all the beam screen materials.At 1000 Hz, the highest frequency analyzed in this study, the skin depth of copper at 20 K is 206 μm which is larger than the thickness of the copper layer itself (75 μm).However, considering that the computational time of the simulations is within a few minutes, two quadratic type elements are used to mesh the copper layer, as depicted in Fig. 6.

E. Electrical resistivity of the copper layer
The results presented in this paper take into account the magnetoresistivity of copper, whose variation of electrical resistivity due to the magnetic field is not negligible in comparison with the other materials of the beam screen.For the quadrupole magnets, the magnetoresistivity is considered by implementing in the material properties of COMSOL a data sheet containing the electrical conductivity as a function of the magnetic field.The LHC beam screen operates in a magnetic field range between 3.69 to the LHC main quadrupole (MQ) and of 132.6 T m −1 for the quadrupole HL-LHC magnets.The value of electrical resistivity of the dipole magnets has been set constant over the whole beam screen region.The magnetic field considered for the HL-LHC dipole magnets is 5.6 T, while for the LHC is 8 T. In general, the effect of the magnetoresistivity is proportionally more significant at 20 K than at 80 K as the intrinsic component of the phonons resistivity is very small at lower temperatures [28].In fact, for a magnetic field of 5.6 T the electrical resistivity is 56% higher at 20 K, while it is only 6% higher at 80 K.For a magnetic field of 8 T the electrical resistivity is 65% higher at 20 K, while it is 9% higher at 80 K.The assumption adopted for the electrical resistivity of copper is as follows.If B < 0.1 T the electrical resistivity is based on the Matthiessen's formulation [29]: where ρ 0 is the residual resistivity which is determined by the chemical and physical imperfections of the material, ρ i is the intrinsic resistivity affected by the electrons interacting with the phonons, and ρ i0 is a term to account for some observed deviation from experimental data.Considering ρð273KÞ ¼ 15.53 nΩ m, these quantities read to: The constants are: P 1 ¼ 1.171 × 10 −17 , P 2 ¼ 4.49, P 3 ¼ 3.841 × 10 10 , P 4 ¼ 1.14, P 5 ¼ 50, P 6 ¼ 6.428, P 7 ¼ 0.4531.If B > 0.1 T, the electrical resistivity in the direction perpendicular to the magnetic field can be described by the empirical Kohler's rule [29,30]: where ρ T ðBÞ is the electrical resistivity as a function of the magnetic field at a specific temperature, ρ T ð0Þ is the electrical resistivity with no magnetic field at a specific temperature, SðTÞ is the ratio between the electrical resistivity at 273 K and a specific temperature with no magnetic field, i.e., ρ 273K ð0Þ=ρ T ð0Þ.

F. Time-domain vs frequency-domain study
The frequency response of the model has been validated in the time domain for the Q1 beam screen at 10, 100, and 1000 Hz.The comparison has been performed by considering the norm of the magnetic field at x ¼ 0, y ¼ 49.5 mm.A sinusoidal time-dependent magnetic field with a magnitude of 1 T has been chosen as the excitation for the time-dependent study.The same magnitude has been used for the quasi-static frequency study.After the transients of the time-domain simulations are vanished, an excellent agreement was found for all frequencies analyzed.Therefore, the study has been efficiently conducted through a stationary problem in the frequency domain with complex-valued solutions.The frequencies, in the range 0.1 to 1000 Hz with a logarithmically spaced grid of 37 points, have been computed through the direct solver MUMPS [31].The memory allocation factor used for such solver is 1.2 with the preordering algorithm based on the nested dissection.The relative tolerance for the solver to converge was set to 10 −3 .Figure 7 shows the magnetic field map of the D1 beam screen at 278 Hz using the stationary solver.

G. Contribution of the heat absorber wings
As shown in Fig. 8 for the Q1 beam screen, the tungstenbased inserts (also called heat absorbers) have some geometrical extensions on both sides known as tungsten wings.As these are not uniformly present along the beam axis, they require a 3D model.However, an effective workaround still allowing the use of a 2D simulation was found in [32].It consists of defining an equivalent electrical conductivity related to the numbers of wings along the whole length of the heat absorber.For the case study presented in this paper, the geometrical filling ratio of the wings is 20%, therefore, the electrical conductivity was also set to 20% of the tungsten alloy used in the heat absorbers.
For the D1 and Q2 cases, sharing the same type of beam screen, the percentage difference in terms of magnetic field FIG. 7. Magnetic field map around the D1-type beam screen at 278 Hz.The color map, in T, is normalized to the input source.The red vectors represent the magnetic field lines.
with and without the tungsten wings is given in Fig. 9.The magnitude of the magnetic field has been compared as a function of the frequency at x ¼ 0, y ¼ 0 of the D1 magnet and at x ¼ 0, y ¼ 30 mm for the Q2 magnet.It turns out that for D1 the shielding effect increases monotonically with frequency.This is due to the extra absorbing material of the wings.For Q2, at the chosen observation point, the shielding effect increases slightly up to 200 Hz but then decreases from 200 to 1000 Hz.The difference between these two profiles is due to the intrinsic distribution of the dipole and quadrupole magnetic field.However, in the case of the IT, this difference is deemed negligible for the purpose of this study.The wings of the heat absorbers are therefore not accounted for in the frequency response of the IT beam screens.

H. The inner copper layer
The nominal thickness of the inner copper layer for both LHC and HL-LHC beam screens is set to 75 μm (−8 μm=þ18 μm).Following colamination, the strip is partially annealed under a protective atmosphere with a partial O 2 pressure lower than 10 ppm.The annealing process improves the adhesion of the copper layer and ensures an appropriate compromise between mechanical properties (stress relieving) and electrical properties (low temperature surface resistance of the copper) of the finished product.As an indication, partial annealing in a continuous furnace under pure H 2 at 920 °C during 5 minutes has given satisfactory results during the LHC manufacturing runs.However, during the manufacturing process a diffusion mechanism takes place from the stainless steel to the copper layer which affects its purity and, therefore, the electrical conductivity.It turns out that the bottom 25 μm of the copper becomes polluted.Some electrical resistance measurements have been performed on the LHC beam screen after manufacturing showing an effective thickness of the copper layer between 50 μm and 80 μm [30].The analysis presented in this paper is conservative as it has been performed considering 50 μm.

III. POSTPROCESSING
2D multipole expansions of magnetic fields and the analysis of relative field components is common practice in studying imperfections of accelerator magnets [33].In this analysis, a formalism as well as practical equations are developed in order to define and extract relative field components for time varying fields in the frequency domain based on simulation data.

A. Frequency-dependent multipole analysis
A 2D translation-invariant, quasi-static magnetic field ½B x ðx; y; tÞ; B y ðx; y; tÞ can be expressed in a source-free region using a 2D multipole expansion defined by: B y ðx; y; tÞ þ iB x ðx; y; tÞ ¼ where B n , A n are the multipole components featuring an explicit time dependence since ∇ • B ¼ 0 and ∇ × B ¼ μϵ∂ t E ≈ 0 at low frequency.R is a convenient reference radius (typically 2=3 of the magnet aperture).For brevity, ( 14) can be rewritten using complex quantities, yielding: where Bðz; tÞ ¼ B y ðx; y; tÞ þ iB x ðx; y; tÞ, C n ðtÞ ¼ B n ðtÞ þ iA n ðtÞ and z ¼ x þ iy.
If the magnetic field Bðz; tÞ is calculated in M points fz k g ¼ z 0 ; …; z M−1 uniformly placed on a circle of radius R, (15) simplifies to: Under the assumption that multipole components of order larger than M=2 can be neglected (M ¼ 64 has been used in the following analysis to avoid any risk of aliasing effects) and M is a multiple of 2, the multipole components of the order up to M=2 can be efficiently extracted from simulations by applying a fast Fourier transform (FFT) on the complex signal fB 0 ðtÞ; …; B M−1 ðtÞg because: If the field is periodic with a frequency f also the multipole coefficients are periodic.The multipole components for each frequency f can be extracted from the fields calculated by a frequency domain simulations.The fields are normally given in the terms of in-phase (I) and quadrature (Q) components which are defined by: B y ðx; y; tÞ ¼ ℜfðB I y ðx; y; fÞ þ jB Q y ðx; y; fÞÞe j2πft g; ð18Þ B x ðx; y; tÞ ¼ ℜfðB I x ðx; y; fÞ þ jB Q x ðx; y; fÞÞe j2πft g; ð19Þ in which the complex variable j is used as the imaginary unit of the complex plane related to the frequency domain to distinguish it from i, related to complex plane associated to the 2D fields, and ℜ denotes the real part of the complex fields.Each multipole component will have an amplitude and a phase, therefore each multipole could be denoted by an I and Q component (B I n ðfÞ, B Q n ðfÞ, A I n ðfÞ, A Q n ðfÞ) or amplitude and phase components in a complex number: Since the time dependence is separable from the spatial dependence, the multipole analysis can be carried out separately for I and Q components of the field, resulting in I and Q components of each multipole coefficients as defined below: or using complex variables: When analyzing the field imperfections of a magnet, it is furthermore convenient to express higher order multipoles relative to the main multipole leading to: where BN ð0Þ is the main DC field component (e.g., B1 ð0Þ for a dipole, B2 ð0Þ for a quadrupole), Bn ðfÞ; Ān ðfÞ are the absolute AC multipole field components, and bn ðfÞ and ān ðfÞ are the relative AC multipole field components.Main field components scale exactly to numerical precision as the reference radius for low frequency with small deviations of the order of 10 −3 at high frequency.Higher orders are less precise but still acceptable in the whole spectrum.The ratio between different reference radii calculated from the ratio of the multipole components is always within 10 −3 of the expected value.The shielding effect of the beam screen and the cold bore is clearly visible on the main field component with different cutoff frequencies (see Table IV).Higher order multipole amplitudes in general increase initially with frequency as the shielding is not homogeneous in the region due to the geometry of the conductors that carry the eddy currents.They then decrease at high frequencies as the cold bore (normally contributing less to the shielding) reduces the field that generates the eddy currents in the beam screen.

IV. APPROXIMATED ANALYTICAL FORMULATION
A simplified analytical derivation was first presented in [16] for the case of a cylindrical, infinitely thin (and infinitely long) conductive shell.That formulation, which also took into account the magnet's multipole order, aims at evaluating T B m toB b ðfÞ from a beam pipe (as well as obtaining an equivalent circuit model [16]) and it has been used to estimate the potential effects of the HL-LHC beam screen [15].The expression presented for the cutoff frequency was then generalized, for dipolar field, to an arbitrary cross section in [18], where also elliptical and rectangular shapes are considered and [17] assuming a thin (but not infinitely thin) conductive shell together with experimental results for its validation.A formula for the transfer function of thick round beam pipe was also presented in [19].In this section, this formulations will be further extended to a more complex shape of the beam screen and validated with the simulation results obtained taking the complex geometry and material layering into account.

A. Low-frequency transfer function derivation
An analytical derivation, different from those found in [16,17] is presented here for the case of dipole field in order to better highlight its validity range.An infinitely long cylindrical thin conductive shield with a B field orthogonal to its axis is considered.A closed-form analytical solution exists for the shielding efficiency (SE) of such a simplified structure, as reported in [34].However, for the scope of this low-frequency characterization and assuming that the shield thickness Δ is small compared to the average radius ρ0 of the structure (averaged between inner and outer radii) the following expression holds for jγΔj ≪ 1 [34]: For the structure under analysis clearly μ r ≈ 1, so the shielding efficiency can be further simplified to: As the attenuation introduced by the beam screen is simply the inverse of the shielding efficiency, the magnitude of the frequency response is then given by: TðfÞ will be used in the following as a shorthand for T B m to B b ðfÞ.(30) represents the amplitude of the frequency response of a single pole low-pass filter with a cutoff frequency: This equation is identical to that presented in [17] for a dipole.

B. Generalization of the approximated formula
According to [16], as also detailed in Appendix, the cutoff frequency for an nth order multipole field is Only the cold bore can be approximated as an infinitely long cylinder for which (32) immediately applies.The expression for the cutoff frequency in (31), however, can also be rewritten in terms of the annulus area occupied by the shielding material with conductivity σ and generalized as: where A ¼ 2π ρ0 Δ is the cross-section area of the shield.On the other hand, the eddy currents for a nth order multipole (for a purely cylindrical conductive shell) have a magnitude proportional to jcos nθj as shown in [16] (and in Appendix).The Aσ product can be calculated summing up according to the magnitude of the eddy currents as follows: It is straightforward to determine that the proportionality constant k must be equal to π 2 in order, for (34) to be equal to 2π ρ0 Δσ.
For a noncylindrical conductive shell the magnitude of the eddy currents is no longer strictly proportional to j cos nθj; furthermore even for a cylindrical shell whose conductivity is not uniform such proportionality alone is not sufficient to describe the magnitude of the eddy currents.This is the case, as an example, of the magnetoresistive effect in the case of magnet of order 2n when n > 1 and it will be briefly addressed in VA.The derivation of the approximated formula will be based on the assumption of uniform conductivity.In the following we also assume a closed geometry with an arbitrary number of layers, such that the unwanted higher order components of the magnetic field are, in the worst case, at least one order of magnitude smaller than the nth component (Figs. 10 and 11) in the range of frequency of interest.Within this approximation, the induced magnetic vector potential is approximately proportional, and opposite, to the source.Therefore the corresponding current density of the induced potential has, approximately, the same azimuthal dependency of the source which, in absolute value, is proportional to j cos nθj.In other words the idea is to find a cylindrical approximation for a geometrically complex multi-layer beam screen for which, however, all the quantities in the integral (34) depend on θ.The error introduced by these approximations needs to be verified for specific case with detailed simulations as done in the following sections.

C. HL-LHC beam screens
By considering the octagonal shape of the HL-LHC beam screens and the different materials that constitute it, a weighted average area-conductivity product of the cross section of the structure can therefore be evaluated as follows: (i) using an equivalent circular approximation with radius ρϕ for each side of the octagon and then correct for the area with a factor F ϕ ¼ sin ðϕ=2Þ=ϕ=2 (ratio between the actual length of the side and the length of the arc; this applies for both ϕ ¼ α and ϕ ¼ 2β in Fig. 4, only for β in Fig. 16); (ii) for each sector forming the ðΔσÞ ϕ product as P j ϕ Δ j ϕ σ j ϕ where j ϕ represents the different material layers within the angle ϕ; (iii) integrating ðΔσÞ ϕ over 2π, sector by sector, weighing with π 2 j cos nθj; (iv) adding the conductivity product of the cold bore to the estimated one c Aσ BS (the cold bore area-conductivity product representing a small fraction of the overall one).By symmetry of the structure the general formula for the equivalent area-conductivity product can be easily expressed as: The constants introduced are Since the area-conductivity product of the cold bore is much smaller than the one of the beam screen, its cutoff frequency alone is much larger; one can alternatively think that higher order multipole components of the field would experience the filtering due to the cold bore whereas the main component would experience mostly the attenuation due to the beam screen.

D. Numerical validation
It must be pointed out here that the approximated formula in (28) is valid for jγΔj ≪ 1 which is a safe assumption for the −3 dB point.In order to also guarantee accuracy for higher frequencies, a more general formula would be needed, but this outgoes the scope of this paper.It can be stated, however, that as (28) is in turn derived from first order Taylor series of terms in sinh ðγΔÞ and cosh ðγΔÞ, the filtering effect at higher frequencies is going to be stronger than what can be calculated by means of (30); this holds true for magnets of any order.
From (35) and the constants in (36) the cutoff frequency can be estimated both for dipole and quadrupole configurations as: This equivalent cutoff frequency can then be calculated for all configurations of the beam screen in terms of the dimensions reported in Table I and conductivities listed in Table III [29,35,36].The results are summarized in Table IV together with a single pole fit [i.e., with the model described by Eq. ( 30)] of the relative multipole frequency response obtained in simulations.All the values reported in the table have been derived by means of the nonlinear least-square fit Trust-Region algorithm (disabling any robust option) as available in Curve Fitting Tool of MATLAB and all frequencies have been weighted equally.For comparison, the simulation results together with the best fit and the analytical formulas are shown in Figs. 12 and 13.
In Fig. 13(a) it can be observed that the simulated frequency response, in blue, drops significantly faster after 100 Hz.This is probably due to the 16 mm thick tungsten-based heat absorbers on the ρ β sides which conflicts with the assumption jγΔj ≪ 1.

V. LHC MAIN MAGNETS
For comparison, the simulation results of the LHC main dipoles and quadrupoles are presented in this section.The obtained results are also deemed interesting in order to better understand in which frequency range the ripple of the currently installed LHC power converters can impact the stability of the beam.The LHC beam screen is illustrated in Fig. 16.For this geometry the formula in (35) simplifies to: as the beam screen is actually circular within the angle α.
The results for the cutoff frequency are summarized in Table V based on the AC multipole analysis depicted in Fig. 14 and the simulations shown in Fig. 15.For the case of LHC quadrupoles, the part of the beam screen within the angle α sees an approximately constant B field of 4.65 T, whereas the part within the angle β sees, in the counterclockwise sense, a decreasing field whose minimum is of 3.69 T when β ¼ π=2.Here it is simply assumed that the conductivity is constant and equal to the one within angle α (worst case as it represent the lowest conductivity); this means that the calculated cutoff frequency is overestimated.As it can be seen from Table V, the accuracy of f0 is very good, even for the quadrupoles where the conductivity cannot be assumed uniform.A. LHC quadrupole with position dependent magneto-resistance Expression (34) can be generalized by considering that all the quantities involved depend on θ.For the specific case of LHC quadrupoles (n ¼ 2) the simplified expression in (38) can still be considered valid within angle α, whereas within angle β it is reasonable to assume that only the conductivity σ varies with the θ as the magnitude B of magnetic induction field is constant within angle α.These considerations are summarized in the following equation: The interpolated, position dependent, conductivity can be easily expressed as: where α ≤ θ ≤ π 2 , B max ¼ BðαÞ, B min ¼ Bð π 2 Þ, σ max ¼ σð π 2 Þ and σ min ¼ σðαÞ.Indeed the magnitude of the magnetic induction field goes as 1= sin θ for α ≤ θ ≤ π=2.The interpolation is based on the assumption that at high field the decrease of conductivity is linear with the magnitude of the field [30].
The values used are reported in the last two rows of Table III.With this more accurate approximation the estimation of the cutoff frequency can be improved as shown in Fig. 17 where the estimation error drops from þ7.4% (145 Hz) down to þ3% (139 Hz) with respect to the fitted value of 135 Hz.

VI. CONCLUSION AND FUTURE WORK
The magnetic transfer function of a beam screen has been rigorously defined according to the magnet type.Specific 2D FEM simulations in the frequency domain have been performed for different HL-LHC beam screen layouts and validated against those evaluated in the time domain.All simulations were performed at the maximum tolerated temperature and including magnetoresistive effects, which represent the worst case scenarios in terms of a current ripple being transferred to the magnetic field and then affecting the beam.The postprocessing technique proposed in this paper allowed the evaluation and comparison of the frequency response of the HL-LHC beam screen configurations as well as those of LHC main dipoles and quadrupoles.An easy-to-use approximated formula for the single pole cutoff frequency has also been generalized for the case of a multilayer beam screen and noncylindrical geometry which well matches the simulation on the considered geometries.Applying the formula to noncylindrical geometry is an approximation, with generally unknown accuracy.Once validated for a class of geometries, the formula can be used to quickly assess the cutoff frequency at different temperatures or the impact of different materials without requiring a complete simulation and the subsequent postprocessing.Furthermore, the proposed description is deemed to fully characterize T B m to B b ðfÞ.As such, it is considered an important contribution for the specification of power converter performance for the HL-LHC.Future work will focus on the characterization of the full transfer function from power converter voltage to the magnetic field applied to the particles.

APPENDIX: IDEAL THIN CONDUCTIVE SHELL INSIDE A PURE MONOPOLE
Following closely the analysis conducted in [16], and using the same notation, an oscillating current sheet at radius r ¼ b with frequency f gives a vector potential for r < b: The dependence on t will be omitted in the following.
If at r ¼ a (a < b) there is an infinitely thin sheet of conductor, a current is induced which can be written as: J 0 z ða; θ; fÞ ¼ σ s E z ¼ −j2πfσ s ½A z ða; θ; fÞ þ A 0 z ða; θ; fÞ; ðA3Þ where σ s is the surface conductivity of the sheet and A z þ A 0 z represents the total vector potential A tot z .A 0 z can be expressed, for r ≤ a, considering the new source J 0 z ða; θ; fÞ as:

FIG. 1 .FIG. 3 .
FIG.2.The new Q1 HL-LHC beam screen inserted in the MQXF magnet.Q2, Q3 magnets have the same cross section but are equipped with a larger beam screen.The red area represents the coils and the beam screen is placed in the aperture surrounded by the cold bore.

FIG. 4 .
FIG. 4. Different types of beam screens for the IR quadrupoles.
FIG. 5. Symmetry conditions (in blue and in red) applied on the straight edges to reduce the modelling domain in case of a dipolar (a) and quadrupolar (b) magnetic field.The field lines are shown in gray.For both cases the external vector potential is applied on the round edge.

Figures 10 and 11
Figures 10 and 11(and also Fig. 14 for LHC) show the amplitude and phase of the AC relative field components resulting from the beam screen of each IR magnets.For the symmetry of field and geometry b1 ðfÞ; b3 ðfÞ; b5 ðfÞ have the largest amplitudes for dipoles and b2 ðfÞ; b6 ðfÞ; b10 ðfÞ have the largest amplitudes for quadrupoles.Other components have negligible amplitudes or appear as numerical noise in the processed data.The analysis has been validated by computing the multipole components for different Figures 10 and 11(and also Fig. 14 for LHC) show the amplitude and phase of the AC relative field components resulting from the beam screen of each IR magnets.For the symmetry of field and geometry b1 ðfÞ; b3 ðfÞ; b5 ðfÞ have the largest amplitudes for dipoles and b2 ðfÞ; b6 ðfÞ; b10 ðfÞ have the largest amplitudes for quadrupoles.Other components have negligible amplitudes or appear as numerical noise in the processed data.The analysis has been validated by computing the multipole components for different

FIG. 10 .
FIG.10.AC multipole analysis for the beam screens of the HL-LHC triplet quadrupoles at 80 K.

FIG. 14 .
FIG. 14. AC multipole analysis for the beam screens of the LHC Main Dipoles.

TABLE I .
Characteristic angles [deg] and dimensions [mm] of the beam screens analyzed in this paper, for LHC see Fig. 16.

TABLE II .
Physical quantities of the constitutive equations governing the beam screen behavior subject to a time-harmonic signal.

TABLE III .
Conductivities of materials used for the evaluation of f0 at their relative temperatures.

TABLE IV .
Estimation of f0 for the different cases studied and comparison with numerical fit of the simulated data (assuming single pole behavior).(*): the fit with robust options activated gives better estimations between 59.4 Hz and 59.6 Hz and hence an error not greater than 10.7%.For the sake of uniformity though the fitted cutoff frequencies reported here have been calculated with the same algorithm and settings.
FIG. 12. Frequency response of the shielding effect inside the HL-LHC separation-recombination dipoles.FIG. 13.Frequency response of the shielding effect inside the HL-LHC quadrupoles.

TABLE V .
Estimation of f0 for the LHC main dipoles and quadrupoles and comparison with numerical fit of the simulated data (assuming single pole behavior).