Validating the Physics-Driven Lumped-Element Model of the LHC Main Dipole Magnet

Measuring the complex impedance of a superconducting magnet as a function of frequency provides valuable insight into its electrodynamics. In particular, the characteristic features of some non-conform behaviour, such as an insulation fault, may be easier to assess when performing impedance measurements rather than observing time-domain signals. A physics-driven equivalent circuit model of a superconducting magnet has been recently developed, whose parameters are derived using solely measured geometric and material properties. This contribution describes its validation against impedance measurements of a spare LHC superconducting main dipole, performed at the CERN magnet test facility. The proposed model includes lumped-elements capturing individual physical phenomena, such as superconducting filament magnetization, inter-filament and inter-strand coupling currents, eddy currents in the strand copper matrix and various magnet components, and stray capacitances. It is possible to predict the impact of different physical effects in different frequency ranges and compare simulations to experimental results. It is shown that the validated model can accurately reproduce the magnet's impedance in a frequency range up to 5 kHz in the different conditions considered.

Validating the Physics-Driven Lumped-Element Model of the LHC Main Dipole Magnet M. Janitschke , M. Bednarek , E. Ravaioli , A.P. Verweij , G. Willering , M. Wozniak , and U. van Rienen , Member, IEEE Abstract-Measuring the complex impedance of a superconducting magnet as a function of frequency provides valuable insight into its electrodynamics.In particular, the characteristic features of some non-conform behaviour, such as an insulation fault, may be easier to assess when performing impedance measurements rather than observing time-domain signals.A physics-driven equivalent circuit model of a superconducting magnet has been recently developed, whose parameters are derived using solely measured geometric and material properties.This contribution describes its validation against impedance measurements of a spare LHC superconducting main dipole, performed at the CERN magnet test facility.The proposed model includes lumped-elements capturing individual physical phenomena, such as superconducting filament magnetization, inter-filament and inter-strand coupling currents, eddy currents in the strand copper matrix and various magnet components, and stray capacitances.It is possible to predict the impact of different physical effects in different frequency ranges and compare simulations to experimental results.It is shown that the validated model can accurately reproduce the magnet's impedance in a frequency range up to 5 kHz in the different conditions considered.Index Terms-Accelerator magnet, AC-losses, frequencydomain, impedance measurements, superconducting coil.

I. INTRODUCTION
M EASUREMENTS of the complex impedance as a func- tion of the frequency are often used across the electrical engineering domain to better understand and describe complex systems.They can provide valuable insights into the systems' behaviour that are difficult or impossible to obtain with timedomain measurements.
Transfer Function Measurements (TFM) of superconducting magnets could be used to detect the characteristic features of some non-conform behaviour of the coil-winding pack.This was used in the past for short-circuit detection in non-planar coils in fusion stellarators [1], [2] and in CICC model coils [3] and to monitor and control the resin impregnation process in Nb  coils [4].In order to correctly identify non-conform behaviour, it is crucial to properly understand and interpret the measured signals in a TFM.Moreover, reconstructing the magnet behaviour in an appropriate simulation model can achieve a capability to explore non-nominal behaviour and failure scenarios, which might be difficult to realize in a laboratory set-up.
Frequency-domain models utilizing equivalent lumpedelement circuit models were previously derived [5], [6], [7], [8], [9], [10], [11].However, these might lack physical interpretability, are not easily able to include further non-linear effects, and cannot be easily adapted to simulate non-conform behaviour such as for example inter-turn shorts in the coil winding pack.
A physics-driven equivalent circuit model of a superconducting magnet was developed, aiming to reproduce the behaviour of a superconducting magnet in the time-and frequency-domain.As such, the model includes different physical phenomena occurring in the magnet, such as superconducting filament magnetization, inter-filament and inter-strand coupling currents, eddy currents in various magnet components, and stray capacitances.The equivalent parameters used to model these phenomena are either based on analytically-derived equations or Finite-Element models (FEM).
An extensive impedance measurement campaign was recently conducted on a spare LHC main superconducting dipole at the CERN magnet test facility.This contribution describes the measurements and, utilizing the measured transfer functions, validates the derived model for the LHC superconducting main dipole in the frequency domain.[12], [13], [14], [15] TABLE II MAIN PARAMETER OF THE MAGNETS COMPONENTS [12], [13]

II. THE LHC MAIN DIPOLE
The LHC main superconducting dipole [12], [13], [14] is composed of two apertures connected in series, each with a differential inductance of L Ap =49 mH at nominal conditions.The cross-section of one aperture is shown in Fig. 1.The magnet has a magnetic length of l mag =14.3 m and generates a dipole field of B Nom =8.0 T at nominal current I Nom =11.85 kA and operating temperature T Nom =1.9 K.The coils are wound using two different Nb-Ti Rutherford cables for the outer and inner layers, whose parameters are summarized in Table I.
Each coil block is separated from each other with metallic coil wedges (W).Moreover, the coils are covered for manufacturing reasons with 1 mm thick coil-protection sheets (CPS) before collaring.The coils are separated from the beam region by a 1.5 mm thick cold-bore (CB).Since these components are made of low-resistivity materials and are not laminated, considerable eddy currents are developed therein during a fast transient.Table II summarises their volumes, materials and assumed resistivities ρ [Ωm], and the components are highlighted in Fig. 1.Unlike magnets installed in the LHC machine, a beam-screen was not present during the measurements.

III. MODELLING APPROACH
A lumped-element network model has been developed, which aims to reproduce a superconducting magnet's electrical behavior.The model includes the magnet's inductances and stray capacitances as well as a coil resistance in case it is in the normal state.Each lossy, coupling effect is represented by a loop with an inductance L ec [H] and resistance R ec [Ω], coupled with a mutual inductance M ec [H] to the magnets' inductance [16].Moreover, all coupled inductors are also coupled with each other.As an example, a simplified electrical scheme of one dipole, showing the two apertures, each coupled to two coupling loops is shown in Fig. 2.
To derive the equivalent parameter of each coupling loop, the induced currents I ec [A] and power loss P ec [W] in the respective lossy volume, as well as the time constant τ ec [s] of the loss with j the imaginary unit and ω rad s the angular frequency.The time constants of each loop are summarized in Table III.The following paragraphs will describe the implementation of the main coupling effects.

A. Persistent Currents and Magnetization (PC)
Once the generated field penetrates into the superconducting filaments, instantaneous currents are induced in these, which oppose the penetration field and can persist until the superconductor transitions back into the normal conducting state [17], [18].
The effect on the magnet's differential inductance can be described in first approximation by the stored power in those currents [19], [20]: where we used the relation B = μ 0 (M + H) [T], with the vacuum permeability μ 0 H m .Power loss due to magnetization heat for the small ΔB in the TFM is expected to be negligible.
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The slope dM/dH, which is proportional to dI Ec /dI Ex [21], is approximately the same when the filaments are in virgin or magnetized state.Hence, the effect on the differential inductance is expected to be independent of the magnetization state.As these currents are instantaneous, it is also expected that the differential inductance of the magnet once transitioned into superconducting state, will reduce significantly [21], [22].
Due to the nature of this phenomenon, the equivalent resistance element is replaced by a current source.The equivalent inductance is derived by comparing the stored and lost power, and the equivalent network powers of the model [21].

B. Conductor Losses (CL)
The time-varying magnetic field generated by the sinusoidal excitation generates induced currents between the superconducting filaments.These currents are called inter-filament coupling currents and generate a field that opposes the applied field change.As these currents close through the conducting copper matrix of the strand, they generate a transitory loss (IFCL) [16], [23], [24], [25], [26].Similarly, currents are also induced between the cable strands.These currents are called inter-strand coupling currents, closing through the strands' cross-contact resistances R C [Ω] and hence causing losses (ISCL) [16], [23].
The superconducting filaments are usually enclosed in a copper sheath.The time-varying magnetic field applied to the strands also causes eddy currents to flow in this outer sheath of copper, generating additional losses (CML) [26].
In order to reduce model complexity, the derived parameters for all strands/turns are first binned based on the local magnetic field and then lumped together.The equivalent parameters for each bin are then the bin-averaged R ec and L ec , with a mutual coupling of M ec = n t n=1 M 2 ec,n with M ec,n being all n t mutual inductances in a bin, coupled to the inductance of the magnet's aperture.In this work, we used two bins per aperture, one for the outer and one for the inner layer.
Based on the calculated time constants of the three phenomena, IFCL and ISCL are expected to reduce the differential inductance in the lower frequency range, while the eddy currents in the outer copper sheath impact at higher frequencies, above 1 kHz.

C. Eddy Currents in the Magnets Metal Components (MC)
As the copper coil wedges, cold-bore, and the coil-protection sheets are made of conducting materials, the time-varying excitation field also induces eddy currents in these elements.The position and geometry of these components are complex, and there is no analytical derivation of induced current and loss.In order to include these effects, the respective components were modeled in a 2-dimensional FEM in COMSOL Multiphysics © .The time constants as well as frequency-dependent induced currents and losses were exported to calculate the equivalent frequency-dependent coupling parameters.The mutual coupling between the metal components is also simulated but assumed to be frequency-independent.Since the time constants of the eddy currents in these elements are smaller than a few ms, it is expected that they impact the magnet differential inductance for frequencies above 1 kHz.

IV. MEASUREMENT SET-UP
One spare main dipole was measured at the CERN magnet test facility.The measurements were performed with a gain/phase analyzer, which generates a sinusoidal excitation signal of amplitude V AC = 10 V with a maximum peak-to-peak current of I Ex = 1 A. This corresponds to a maximum field change in the centre of the aperture of about 0.7 mT.To reduce parasitic effects, the output signal of the generator is sent through an isolation amplifier, which isolates the measured signal from the ground and possible noise sources.
The schematic of the measurement set-up is shown in Fig. 3.
is then measured between the respective voltage taps of interest.To reproduce the conditions in the LHC, all measurements were repeated with one resistor R Par =100 Ω in parallel to both apertures [12].The complex impedance was measured in the 1 Hz to 100 kHz frequency range with 20 points per decade.Using the complex-valued measurements, the impedance is calculated as . In order to measure the effect of different magnetization states of the superconducting filaments, the impedance was also measured just after cool-down ("virgin" magnet) and after one magnetic cycle to nominal current and back.This cycle ensured that nearly all filaments were completely saturated, as the field was well above the penetration field of the dipole strands B p = μ 0 J C (T,B)d f π [T], with J c A m 2 the critical current density of the filament [23].The measurements were also done continuously during the natural warm-up.

V. MODEL VALIDATION
The measurements were used to validate the proposed features of the model in the frequency range below 5 kHz, which is most relevant for accelerator magnets, operating in the range of a few mHz.The measured impedance magnitude over this frequency range compared to the simulated one is shown in Fig. 4. In order to reflect the real conditions in the LHC, simulations and measurements with R Par are also shown.The agreement between simulation and measurement is good throughout the entire frequency range, with an average error below 3.5%.The average error with respect to a simulation not including non-linear effects would range above 50%.
The measured transfer functions for the case of a virgin and fully magnetized magnet are shown in Fig. 5.As expected by the model, both measurements are identical at all measured frequencies, with an error within the error range of the measurement set-up.The TFM also validates the persistent current and magnetization model feature over temperature.The measured inductance and capacitance, calculated from the low-and highfrequency reactances, respectively, are shown in Fig. 6.Due to thermal contractions and permittivity changes, the capacitance and inductance drop from about 350 nF to 330 nF and from about 50 mH to the nominal 49 mH, respectively.One can observe the sudden drop of inductance, due to superconducting effects in the filaments, once the magnet coil transitions into the superconducting state.
Finally, Fig. 7 shows the contributions of the effects to the final simulation results.The persistent currents reduce the differential inductance of the magnet at low frequencies and explain the  measured, reduced inductance with an error of less than 1%.The conductor losses impact the impedance of the magnet only at slightly higher frequencies, in the range of a few Hz up to about 1 kHz.This can be observed by the simulated impedance kink in these frequencies, including the conductor loss contribution.The conductor losses and the filament currents provide a very good agreement between measurement and simulation at frequencies up to about 1 kHz.However, the impedance above 1 kHz is strongly influenced by the eddy currents in the magnet's metal components due to their smaller time constants.

VI. CONCLUSION AND OUTLOOK
A physics-driven lumped-element electrical model, which includes multiple non-linear, interdependent coupling effects in a superconducting magnet, was described.A transfer function measurement campaign on a spare LHC main dipole at the CERN magnet test facility was described and provided data to validate the model in the frequency domain.It is shown that the proposed model is capable of accurately reproducing the LHC main dipole transfer function in a frequency range up to 5 kHz.Moreover, the model accurately captures the effects due to instantaneous superconducting filament magnetization states, which were observed in the temperature-dependent measurements.The validated model can be used to investigate further behaviour of the dipole, for example, to simulate different Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
electrical non-conformities and to evaluate their impact on the magnet performance.

Fig. 1 .
Fig. 1.Cross-section of one aperture of the LHC main dipole showing the coil turns, the copper wedges, cold-bore and coil-protection sheets.

Fig. 2 .
Fig.2.Equivalent lumped-element network model, showing the modeling approach with two equivalent loops, coupled to the two magnet apertures.

Fig. 3 .
Fig. 3. Schematic showing the measurement set-up.The impedances are measured across various voltage taps (here shown for aperture 1).

Fig. 4 .Fig. 5 .
Fig. 4. Measured impedance magnitude of the LHC main dipole versus the simulation of the equivalent network model, with and without R Par .

Fig. 6 .
Fig. 6.Measured inductance of the two apertures and cubic spline-fitted capacitance from different voltage taps to ground.

Fig. 7 .
Fig. 7. Measured complex impedance of the LHC main dipole versus the simulated impedance of the equivalent network model.

TABLE I MAIN
CONDUCTOR PARAMETERS OF THE LHC MAIN DIPOLE

TABLE III TIME
CONSTANTS OF THE CONSIDERED EFFECTS mechanism, are needed.For a harmonic, sinusoidal excitation with the magnet current I Ex [A], the equivalent parameters read: