Magnetisation and demagnetisation of trapped field stacks in a superconducting machine for electric aircraft

This research presents a comprehensive and innovative approach to investigating the magnetisation and cross-field demagnetisation behaviour of high-temperature superconducting (HTS) coated conductors (CCs) in practical superconducting machines. This study introduces several novel contributions, including the operation of the machine in propulsion energy conversion mode, the exploration of harmonics interaction in a real electric machine environment involving CCs, and the extraction of these harmonics as cross-field components. A 2D electromagnetic-thermal coupled numerical model employing the finite element method has been developed and validated against experimental data to simulate a partially superconducting machine. Upon magnetisation, the HTS stacks effectively operate as trapped field magnets, generating rotor fields for motor operation. With a peak magnetic flux density of 462 mT of the trapped field stacks (TFSs) in the air gap, the average values of the fundamental and fifth harmonics of the tangential magnetic flux density experienced by the TFSs were observed to be 25 mT and 1.75 mT, respectively. The research has thoroughly examined the impact of cross-field demagnetisation parameters including amplitude and frequency on the demagnetisation of TFSs. Furthermore, the study has also investigated the magnetisation losses occurring in various layers of HTS tapes, encompassing the HTS layer, magnetic substrate layer, and silver stabiliser at different amplitudes and frequencies. Two tape structures, namely a semi-homogenised model and a multi-layered model, have been analysed in terms of magnetisation loss. Additionally, insights into the shielding effect and skin effect at high frequencies were obtained, offering valuable information on the performance of HTS TFSs exposed to high frequency scenarios especially in high-speed machines for electric aircraft. The research outcomes are anticipated to provide valuable knowledge for the design and optimisation of HTS rotors employing TFSs in superconducting machines, contributing to the advancement of superconducting machine technology.


Introduction
Electric aircraft are gaining popularity due to growing concerns over energy scarcity and environmental issues.To address the demand for efficient propulsion systems, superconducting motors equipped with high-temperature superconducting (HTS) materials, have emerged as an attractive option.These motors offer compact size, minimal losses, and high-power density, making them well-suited for electric aircraft propulsion [1][2][3].To realise the potential of superconducting motors in electric aircraft, several consortium projects have been launched, including the Advanced Superconducting Motor Experimental Demonstrator project (ASuMED) [4], aiming to construct the first fully superconducting motor prototype capable of achieving the power densities and efficiencies required for future large civil aircraft with hybrid-electric distributed propulsion; and the Advanced Superconducting and Cryogenic Experimental powertraiN Demonstrator (ASCEND) [5], aiming at building a cryogenic electric propulsion system for aircraft.Both of the two projects have raised high-power density goals: 20 kW • kg −1 motor power density for ASuMED [4], and 30 kW • kg −1 for the powertrain system composed of electric motors and power electronics in the case of ASCEND [5].However, no infield tests have been successfully conducted to demonstrate these targets since the series of fundamental challenges remain unsolved, including e.g., significant AC losses in armature windings built with HTS-coated conductors (CCs) [6][7][8][9][10], excessive cooling requirements [11,12], and demagnetisation of HTS CCs [13][14][15][16].Among those unsolved difficulties, the demagnetisation of ReBCO CCs in practical synchronous superconducting machines will be studied in this paper.
In synchronous superconducting machines, the traditional rotor's wound field coils or permanent magnets (PMs) can be substituted by superconducting coils [17][18][19][20], superconducting bulks [21-24], or stacked CCs [25][26][27].While wound field coils pose challenges due to the need for continual current sources typically provided by current leads [28] and the utilisation of brushes and slip rings [28,29], stacked HTS CCs offer advantages over superconducting bulk materials for trapped field magnets (TFMs) due to their higher mechanical strength and shape flexibility [9,30].In contrast to bulk HTS materials, stacked HTS tapes exhibit a relatively lower vulnerability to demagnetisation, attributed to their inherent characteristics such as high aspect ratio and ultra-thin thickness [25,31].Nonetheless, despite these advantages, the demagnetisation phenomenon remains a primary challenge that needs to be addressed for the practical application of stacked HTS tapes in electric machines.Therefore, mitigating demagnetisation effects is crucial to ensure the successful integration and optimal utilisation of stacked HTS tapes in electric machine systems.
To magnetise the stacks composed of yttrium barium copper oxide (YBCO) CCs for use as TFMs in superconducting machines, two methods are commonly employed: field cooling [32-35] and pulsed field magnetisation (PFM) [21,24,26,27,[36][37][38][39].Among these, the in-situ magnetisation process, PFM is preferred for electrical machine applications due to its portability, affordability, and compactness [40].However, during the magnetisation, the heat produced by PFM will hinder the realisation of the full trapped field and flux potential in CCs [40].While zero field cooling and flux pumping have also been reported as general magnetisation methods for CCs [41-46], they have not been employed extensively in superconducting machines.
Cross-field demagnetisation of CCs has been widely reported in superconducting machines [14,25,[47][48][49][50][51][52].This phenomenon is influenced by higher-order harmonic waves present in the air gap due to the stator's alternating arrangement of slots and teeth.These spatial harmonic components are not in sync with the rotor field and can lead to demagnetisation of PMs [53].While previous studies have explored crossfield demagnetisation of CCs, they often focused on explaining the physics involved without considering the practical scenarios in real electric machines [14].For cross-field demagnetisation, the interaction of abundant harmonics in an actual electric machine environment with CCs has not been fully explored.
Many studies [25,47,51,52,[54][55][56] investigating the demagnetisation of superconducting bulks or stacked tapebased CCs have employed sinusoidal [25,47,51] or triangleshape [52,[54][55][56] magnetic fields to simulate the harmonics present in the machine air gap.While these approaches provide indirect references to potential demagnetisation effects, they oversimplify the complexity of harmonics in real machines.In actual machines, harmonics can consist of multiple waves with varying magnitudes and directions, making the analysis more intricate.Therefore, a reasonable method involves applying the magnetic field extracted from the machine's airgap to the CCs, which are magnetised within the same machine but isolated from it during demagnetisation process.The extracted magnetic field can be decomposed into vertical and horizontal directions relative to the surfacemounted magnets.It should be noted that this paper specifically focuses on the demagnetisation analysis of rotor surfacemounted magnets in synchronous machines.Other types of magnet arrangements are not considered or discussed in this study.
The studied superconducting machine prototype is a synchronous machine derived from a surface-mounted PM motor with fractional slot concentrated winding (FSCW).In the superconducting machine used for this study, surface-mounted PMs have been replaced by TFMs composed of stacked CCs, representing a key characteristic of this superconducting motor design.For the trapped field stack (TFS), it is crucial to consider the effect of cross-field demagnetisation caused by transverse magnetic fields, particularly in FSCW machines where additional sub-and super-space harmonic components exist in the air gap due to winding configurations [57].As a result, PMs mounted on the rotor surface are more susceptible to air gap harmonics in FSCW machines, making the demagnetisation effect of magnetised TFMs a significant consideration in this study.The cylindrical geometry and large diameter of the machine introduce one of the key differences in this study compared to previously published works, namely the utilisation of curved and wide CCs, leading to a different distribution of magnetic fields and current density.This can significantly impact the magnetisation and demagnetisation processes [47].
The HTS machine is constructed based on a conventional machine design, featuring iron yokes in the stator and rotor.While air-cored machines can achieve higher magnetic loading and utilise the full potential of CCs [58], they experience more air gap harmonics caused by the stator compared to conventional iron-cored machines, leading to more pronounced demagnetisation of CCs.However, conventional iron-cored machines face limitations in their magnetic loading capacity due to iron saturation levels.
Compared with prior investigations focusing on superconducting motors employing TFMs, this research offers a comprehensive multi-stage approach covering the entire process from magnetisation to demagnetisation of ReBCO CCs within a practical electrical machine.Unlike previous studies that mostly concentrated on either the magnetisation or demagnetisation of ReBCO CCs, this work addresses both aspects alongside motor operation.While two pioneering research works [26,27], have explored CC demagnetisation in a real machine, they mainly operated the machine as a generator driven by a direct current motor, leaving a gap in understanding the demagnetisation during practical motor operation.This paper builds upon the measured data of the same HTS machine reported in [26].Its novel contributions encompass operating the machine in propulsion energy conversion mode, considering the interaction of abundant harmonics in a practical electric machine environment with CCs, and extracting these harmonics as cross field components.This approach reflects real-world scenarios of practical superconducting machines.Furthermore, this work presents detailed insights into the demagnetisation process, encompassing variations in current density fluctuations within HTS layers in response to varying cross field amplitudes and frequencies, as well as the analysis of magnetisation losses in distinct layers of the ReBCO CC through numerical investigations.These aspects, which were previously constrained by the physical limitations of actual superconducting machines in real world measurements, are comprehensively addressed in this study.

Methodology
The complete workflow of this research is illustrated in figure 1.
First, a 2D finite element method (FEM) model of the electric machine was constructed based on the geometry parameters provided in [26], as shown in figure 1(a).The simulation parameters related to individual tape characteristics and stack configuration were adjustable, while machine geometry and superconducting material properties remain constant.The final goal was to establish a well-calibrated model that could accurately trap the field, resulting in the same back electromotive force (emf) as measured in the experiment in [26].This numerical model effectively represented the stack assembly within the superconducting machine.
For the magnetisation and validation stage, PFM was employed to facilitate magnetisation.The machine model, referred to as Machine 1, was set up with a single stack aligned to the iron tooth at the 12 o'clock position, around which the magnetisation coil was wound.To investigate the flux penetration and heat generation during magnetisation, a thermal module was coupled to the electromagnetic model.
Following magnetisation, the rotor, containing eight superconducting stacks, rotated synchronously with the stator's magnetic field generated by three-phase alternating currents.The challenge lay in maintaining the current density induced by the magnetisation field within the stacks.While Brambilla et al [59] proposed a FEM model coupling an A-formulation containing normal conductors and an H-formulation domain incorporating CCs for modelling rotating machines with CCs, this approach was not applicable to our work.The reason is that in [59], field windings were utilised in the rotor, imposing constant external current densities directly onto the CCs.However, this method did not yield satisfactory results in our complex machine model due to the intricate physics and geometry involved.In our attempt to establish a rotating mesh for the rotor, including the magnetised CCs where supercurrents flow, the simulation solver encountered convergence issues.Even when the rotor remained stationary and the stator rotated in the opposite direction, the same problem persisted.Moreover, the coupling boundary between the Adomain and the H-domain proved problematic for magnetic flux travelling from the magnetising coils to the superconducting stack during magnetisation.Therefore, the H-formulation has been employed for the entire model during magnetisation and demagnetisation.
To examine the machine as a motor, a novel modelling approach was adopted in this research.The four winding coils wound on the machine teeth on the horizontal and vertical lines were used for PFM and eight identical HTS stacks were evenly positioned on the rotor surface, forming eight poles with alternate polarities, as shown in figure 1(c).The trapped magnetic field in the radial direction was transferred to a ring PM in a second machine model with identical geometry-Machine 2. In Machine 2, the trapped field underwent rotation using an equation that defines the rotation speed and the initial phase relative to the local cylindrical coordinate  Ring shape PM on the rotor surface A Motor/Generator system, as depicted in figure 1(d).In this stage, the primary objective was to extract the major harmonics potentially causing demagnetisation of the CCs.These extracted harmonics were subsequently applied to the magnetised stack as a crossfield demagnetisation source.Table 1 summarises the two submodels of the studied machine in detail.
In the demagnetisation stage, PFM was conducted once more as the initial step, as illustrated in figure 1(e).After completing the magnetisation, the magnetic flux density and current density of the CC stack are transferred to a second model in figure 1(f) as initial values for demagnetisation.In this model, only the stack is retained, and all other regions are designated as air.Subsequently, the extracted harmonics of tangential flux densities acquired from motor operation were applied to the azimuthal direction of the airgap boundary within the machine.
Figure 2 demonstrates a flow chart presenting the complete process described above.

Modelling method for model validation
This section describes the modelling method used for model validation, which consists of two steps.In the first step, a magnetisation model was built using the H-formulation.The second step is featured by a rotation model using a self-defined equation to measure the induced back-emf on a machine coil.

Machine model with one superconducting stack
As the first part of the workflow, magnetisation of a single superconducting stack in the machine was simulated in the model Machine 1.The geometries of the motor and tape stacks have been taken from [26].For the modelling of PFM, a FEM based electromagnetic-thermal coupled model was constructed in the commercial software COMSOL Multiphysics 6.0.For the electromagnetic modelling, the H-formulation [60,61] was utilised for the whole machine following the governing equations where µ is the magnetic permeability, ρ is the resistivity of the materials, E 0 presents the characteristic electric field with , and the field-dependent exponent n(B) was taken from [33].The anisotropic critical current density employed in equation ( 2) is magnetic field and temperature dependent [30,62,63].
Given the non-negligible heat generation during PFM, a complementary thermal model was established to incorporate heat transfer dynamics during the PFM and coupled to the electromagnetic model, forming a bi-directional connection.The thermal model contributes real-time temperature data to the electromagnetic model.The electromagnetic model delivers the current density J and electrical field E of the stack.Figure 3 demonstrates the coupled electromagnetic and heat transfer models for the machine model.
The governing equation for the heat transfer module: where ρ m is the mass density, C p denotes the specific heat capacity, k is the thermal conductivity, and Q is the heat generation power density.The heat capacity, thermal conductivity of the CCs were derived from [64].Q is the volumetric heat source for the thermal model, defined by The heat transfer module was specifically applied to the HTS stack domain, as the PFM occurred in a brief timeframe.Consequently, only the heat generated within the stack was considered.Additionally, a convective heat transfer with a heat transfer coefficient of 100 W • ( m 2 • K ) −1 was employed by imposing the convective heat flux boundary on the HTS stack boundaries [48].The ambient temperature was fixed at the liquid nitrogen boiling temperature, namely 77 K.The stator and rotor yokes were composed of silicon iron M270-35A, with the B-H curve from the iron region applied in the magnetisation model.The remaining components of the machine were modelled as air.
In previous studies [27, 36], the same stacked CC was treated as a homogenous bulk and its equivalent engineering critical current density was scaled down based on the volume fraction of the superconducting material.However, in high frequency environments, the electromagnetic behaviour of HTS stacks will become more complicated, necessitating the consideration of the multi-layer structure of HTS CCs [65,66].The AMSC tapes employed, as per [30], are Rolling-Assisted Biaxially Textured Substrate (RABiTS) YBCO CCs with Ni-5at.%Wmagnetic substrates.However, given that silver stabilisers have a negligible effect on the magnetisation results [67], a semi-homogenised model was built.For a single tape, both the superconducting layer and the magnetic substrate layer were taken into account, as depicted in figure 4(a).For clarity, the curvature and the real aspect ratio are not presented.A similar structure can be found in [68][69][70][71].However, the thickness of the superconducting layer was artificially expanded to 7 µm, making the total thickness of both superconducting layer and substrate layer equal to the thickness of a single tape.This artificial expansion technique of superconducting layer were also used in [25, 72,73], where superconducting layer's thickness was expanded to 100 µm.This approach aids in achieving better convergence by limiting the smallest element size and faster simulation speeds by reducing the number of mesh elements without sacrificing accuracy [70].The effectiveness of the thickness expansion lies in maintaining a sufficiently large aspect ratio, preserving the behaviour of the CCs as if they were infinitely thin strips, with the current distribution front progressing along the abplane of the superconducting layer [70].With this modification, the critical current density of the HTS layer was scaled down to 6.16 It was assumed that the layers are positioned infinitesimally close to each other, effectively eliminating any air gaps between the layers.This approach serves to reduce the total number of mesh elements and prevent convergence issue arising from tiny elements within air gaps between layers.The fitting function outlined in [74] was utilised to accurately represent the field-dependent behaviour of the relative magnetic permeability of the Ni-W ferromagnetic substrate of the RABiTS YBCO CC.This function is based on the experimental findings reported in [75], written as )) ) . ( Figure 4(b) demonstrates the mesh configuration of the curved nine-layer stack.Structured (mapped) meshes, which have a high aspect ratio and hence are capable of reducing the number of mesh elements in and between CCs [76], have been implemented in superconducting layers and substrate layers.

PFM
The iteratively magnetising pulsed field method with reduced amplitude (IMRA) [26] has been employed as PFM approach in this study.The PFM process involves the utilisation of a capacitor bank to generate the required pulsed current.The voltage of the capacitor bank is adjustable, consequently enabling control over the pulsed current magnitude.In alignment with [26], a sequence of ten pulses has been applied, wherein the voltage decreases from 50 V to 5 V in 5 V steps.The waveform of the current pulse employed in this work is extracted from [26].To closely mirror the experimental conditions, a simulation duration of 300 s, with a relaxation interval of 30 s following each pulse, has been executed.The simulation design facilitates the implementation of the ten pulses, each generating a reduced magnetisation field.
The magnetisation circuit is characterised by a capacitor (C), an inductor (L), and a resistor (R).Considering that the rising and falling edges of the input current pulses are in the order of ms, each pulse has low frequency components (no higher than 100 Hz), and thus the impedance of the entire circuit can all be seen as constant for the varying input voltage (in other words, the frequency components of the studied current pulses with different amplitudes are similar), which enables the current magnitude to be linear to the voltage level of the capacitor bank.For the other voltage levels, the waveform and pulse width are same as those of the 40 V and only the magnitudes are scaled up or down depending on the voltage magnitude.
Due to the large number of degrees of freedom (DOF) in the FEM model (about 40 000 for the one eighth model) and the significant small step size, the simulation time for the IMRA lasts for approximately 45 h using a Dell computer featured by Intel(R) Xeon(R) CPU E5-1620 v4 @ 3.50 GHz and 32GB memory.

Rotation after magnetisation
In terms of numerical modelling, it is very time-consuming to simulate the HTS rotor using moving mesh in COMSOL.To reduce the simulation time, another innovative point in this paper is that the time-spatial magnetic fields generated by the rotating HTS TFSs have been described by derived analytical formulae, which were applied through boundary conditions.Figure 5 demonstrates the rotation machine model in COMSOL 6.0. Figure 5 In the equation, B r,TF (x) is the trapped field along the arc length, which would be shown in the magnetisation result section.B r0 is a constant to control the magnitude of the copied magnetic field, f rot is the rotation frequency, φ 0 represents the starting position of the copied magnetic field in the machine, and φ sys and r sys stand for the angular and radial distances defined in the cylindrical system, respectively.

Model validation
This section is divided into two subsections.Section 4.1 will present the magnetisation results of the IMRA.Section 4.2 will involve the validation of the numerical model.

Magnetisation results
Figure 6 illustrates the distribution of current density (figure 6(a)) and temperature (figure 6(b)) within the stack at four different time points: the peak and end of the first pulse, the end of the first flux relaxation, which is also the very beginning of the second pulse, and the end of the whole magnetisation.The critical current density within the expanded 7 µm superconducting layer, denoted as J c0 , is 6.16 Notably, the highest temperature is observed at both edges of the stack, reaching 79 K during the peak of the first pulse, a value akin to the modelled result of 79.1 K in [27].Following each relaxation phase, the stack's average temperature reverts to 77 K.
From figure 6 it can be seen that the penetration of the magnetisation flux starts from the both edges to the centre of the stack, which agrees well with the finding of Brandt [77] and the conclusions drawn in [47].In addition, the positioning of the magnetisation coil above the stack leads to the penetration occurring from the top layer to the bottom layer of the stack, resembling the magnetisation process associated with a single vortex coil [78], enhanced by a soft iron yoke [79,80].However, it is important to note that homogenous penetration of the HTS stack from the top to the bottom is hindered by the shielding effect of the upper layers on the lower ones [39].This shielding effect contributes to the absence of a current density distribution, forming a trapezoidal shape of area in the middle of the stack.Notably, at pulse peaks, the temperatures at both edges significantly surpass the temperature at the centre of the stack, resulting in a lower critical current density at the edges compared to the stack's centre.Consequently, the current density at both edges is slightly lower than that the centre of the stack due to the limitation of the J c (T).
Figure 7 demonstrates the radial field profiles on an arc situated 1 mm above the stack surface within the airgap at the end of each pulse.The positive field direction points from the centre to the outer edge of the motor and vice versa.The trapped field takes on an M-shaped pattern, corresponding to the under-magnetised scenarios simulated in [36].In cases where the stack is fully magnetised, the trapped field's shape resembles a cone in the middle of the CCs as presented in [40,81].Figure 7 clearly presents that, from the initial pulse to the final pulse, the magnetic field successively decreases along the stack's edge while increasing at the stack's centre.Since the magnetic field emerges due to the circulating current within the stack, the field variation also indicates that the current distribution is driven from the edge to the centre of the stack.The graphical depiction showcases that the fundamental form of the trapped magnetic field establishes itself with the first pulse, which stands as the most crucial pulse within the overall magnetisation process.Subsequent pulses further enhance both the overall magnetic field and the current density.

Rotation after magnetisation
The comparison between the experiment and simulation results of the induced back-emf is presented in figure 8.The experimental data show asymmetrical positive and negative peaks, indicating that the trapped field in the experiment does not have a symmetry pattern.This asymmetry could be attributed to the cumulative error that occurs during the ten pulses of magnetisation due to the generated axial torques, which necessitate repositioning of the rotor after each relaxation period [27].
Additionally, it is important to highlight that the modelling approach employed in this study, which involves extending the superconducting layer by a factor of seven, can account for the discrepancy between the simulation and experimental outcomes, as illustrated in figure 8.This is particularly relevant when considering the electro-thermal properties of HTS stacks and accounting for the complex dependence of J c (B, T).While the relationship might not follow a simple linear correlation geometrically, the fact that the simulated back EMF magnitude aligns well with experimental results lends support to the usefulness of this simplification strategy.By adopting this approach, the electro-mechanical performance of the superconducting machine can be modelled with reduced computational complexity while maintaining a reasonable level of accuracy.The commendable agreement between the simulation and experimental back-emf data indicates successful validation of the FEM machine model featuring HTS TFSs.Consequently, this model can now be harnessed for further investigations and analyses pertaining to motor performance.

Operation in motor mode
This section is divided into two subsections.Section 5.1 covers the magnetisation of eight identical stacks using the validated model, while in section 5.2, the trapped field obtained from the magnetisation process was transferred to the ring magnet of the Machine 2 model, which was then operated as a motor with varying current and motor speed.By analysing extracted harmonics, the study assessed their impact on the stack behaviour and overall machine performance.

Magnetisation of eight stacks of CCs
This study is focused on the demagnetisation of CCs in a practical superconducting machine environment, rather than maximising the magnetic loading or trapped field in the airgap.Since the machine in this study is designed as a 12-slot/8-pole synchronous motor with surface PMs, eight TFMs are evenly placed on the rotor with alternate poles.
In preparation for motor operation, a crucial preliminary task involves magnetising the eight superconducting stacks depicted in figure 1(c).A practical magnetisation method utilising the symmetrical structure of the stator geometry to magnetise all rotor stacks is to utilise the four coils directly positioned above the four stacks on both the vertical and horizontal axes.The magnetisation employs the same pulse current profile as in the validation stage and one single pulse with a peak value of 2000 A was applied.Figure 9 depicts the magnetic flux density distribution of the eight poles after magnetisation.
Notably, the magnetisation current applied to the eight stacks yields a significantly higher magnetisation field magnitude than the saturation level of typical iron cores.The iron teeth possess a substantially lower relative permeability when saturated, compared to the linear zone of their B-H curve.However, as the relative permeability of the iron approaches and stabilises at one, the magnetic field within the iron core continues to increase in conjunction with the pulse current.
When a superconductor is fully magnetised, its trapped field exhibits approximately triangular waveform.The field's magnitude transitions from positive to negative when observed from the centre towards both edges of the superconductor, as shown in figure 10(a).Figure 10(b) depicted the magnetic field distribution at the edges of a stack using red arrows.Notably, To enhance the motor operation performance in the current machine model, The following improvements have been made based on the validated model.: 1. Number of CC layers: adjusted to eliminate any noticeable distortion and reduce the introduction of harmonics caused by CCs themselves.2. Width of CCs: extended so that there is no clearance is between any two adjacent stacks, resulting in no magnetic flux leakage between the stacks and the increase of the critical current in proportion to the tape width.
Figure 11 depicts the modifications result in the formation of eight poles with alternate polarities without any flux leakage between adjacent poles.The circular angle is adopted for the x-axis, originating from the 12 o'clock position and progressing clockwise.The four poles characterised by positive flux density correlate with the stacks located along the horizontal and vertical centre axes of the motor, corresponding to angles of 0 • , 90 • , 180 • , 270 • .Irrespective of the tape layer number, these poles have been fully magnetised.In contrast, the four stacks positioned along the motor's diagonal axes, specifically at angles of 45 • , 135 • , 225 • , and 315 • , and characterised by negative flux density, cannot achieve full penetration when the tape layer number exceeds four.The magnetisation of different stack layer number for a stack on the diagonal axis has been illustrated in the inset in figure 11.The partial penetration is attributed to the thin iron closure sheet under stator slots.This closure sheet becomes oversaturated when exposed to the high magnetisation field during magnetisation, which hindered the full penetration of the magnetic field into the stacks.Finally, it has been determined that a four-layer stack of AMSC tapes with a width of 45 mm, as shown in figure 12.According to the magnetisation results illustrated in figure 11, the field trapped capacity for a modified four-layer stack is 0.462 T without significant distortion.This configuration ensures optimal trapping of the magnetic field by the superconducting stacks.

Motor operation with trapped flux pattern
The 12-slot, 8-pole FSCW machine utilised in this study features non-overlapped coils and a double-layer winding arrangement.The winding connections for the three phases, as per [82], are shown in figure 13(a).To simulate the motor operation, the magnetic flux density pattern corresponding to a stack with a layer number of four, as depicted in figure 11, has been assigned as the remanent flux density to the ring magnet in Machine 2. The remanent flux density pattern rotates in the  same manner as described by equation ( 6).The three parameters B r0 , f rot , and φ 0 in equation ( 6) are adjusted accordingly for specific motor operation conditions.Figure 13(b) shows the magnetic flux distribution of the eight poles throughout the machine during the motor run.
The harmonics analysis of the motor has been performed under various operating conditions by running the motor at various stator currents and rotor speeds, as listed in table 2, while keeping the rotor remanent flux density constant.
Figure 14 showcases the motor torque characteristics at 1000 rpm. Figure 14(a) illustrates the torque waveforms in the time domain, while figure 14(b) depicts the most significant harmonic amplitudes extracted from fast Fourier transform (FFT) analysis-namely, the average value T Ave and the sixth harmonics.The analysis indicates that the torque waveforms at different rotor speeds exhibit similar shapes and magnitudes, as confirmed by the FFT results in figure 14(b).This consistency is attributed to that motor torque is predominantly determined by the stator current.Furthermore, figure 14(b) highlights that the sixth harmonic's magnitude remains consistent across varying rotor speed for a given stator current.
When analysing the harmonics in the air gap, the focus is on the radial and tangential magnetic fields experienced by a single stack on the rotor surface.These fields can be decomposed into their radial and tangential components with respect to the stack, as depicted in figure 15.The radial magnetic field in the airgap is denoted as B r and the tangential field (i.e., cross field) as B phi .The size of the stack in the figure is modified for better visualisation.
Figures 16 and 17 present the waveforms and harmonic analyses of the average radial and tangential magnetic flux densities experienced by a rotor stack.Similar to torque, the while this percentage rises to about 7% for the tangential flux density.
The presence of fifth harmonics is linked to the configuration of the PMs for a synchronous motor or TFMs for a superconducting motor.Notably, optimising the shape of the TFM can lead to the elimination of harmonics [83].However, the primary objective of this paper was not centered on the harmonics' elimination.Instead, our focus was directed towards the behaviour investigation of the existing superconducting motor.
Table 3 provides the ranges of the radial and tangential harmonics experienced by a single stack at the speed of 1000 rpm.The harmonics vary with the stator current but they remain constant to the rotor speed.Therefore, only one speed point

Cross-field demagnetisation
The demagnetisation analysis aims to investigate the effects of harmonics on the demagnetisation process of CCs.Understanding the demagnetisation process is crucial for designing robust and reliable superconducting machines that can effectively harness the benefits of CCs while ensuring their long-term stability.During the demagnetisation stage, it is essential to isolate the magnetised stack from the motor frame and only subject it to the influence of applied cross fields.To achieve this, all elements except the stack should be set as air in the simulation model.To maintain continuity and accuracy in the simulation, the mesh elements from the magnetisation stage should be copied one-to-one to the demagnetisation stage, allowing for the transfer of current densities and magnetic fields from the last step of the magnetisation as initial values for the demagnetisation phase, ensuring a smooth transition and reliable simulation results.Figure 18 depicts the method used for applying the cross field during the demagnetisation process.To reduces the computation load, only one quarter of the entire machine geometry was considered, resulting in fewer elements in the demagnetisation model.To ensure that the applied field was parallel to ab-plane of the stack, the cross field was applied to the boundaries of the airgap using the local base vector systemboundary system.Specifically, the tangential component of the upper and lower boundaries parallel to stack, as well as the normal component of the left and right boundaries, were assigned with the cross-field harmonics.The configuration can be seen in figure 18(a).Before demagnetisation, the magnetic field direction of the magnetised stack remained unchanged, as shown in figure 18(b).During demagnetisation, the applied cross field interacted with the existing magnetic field, resulting in a synthetic magnetic field with oscillating direction, as depicted in figure 18(c).The applied sine waves for tangential flux density harmonics contain the fundamental and the fifth harmonics and can be calculated by From table 3 we can conclude that the amplitude of the fundamental wave of the tangential harmonics is 14 times greater than the amplitude of the fifth harmonics.In the following simulation, this value relationship was always kept between the fundamental and the fifth harmonics.
The parallel penetration field of a single CC based on the slab model [84,85], can be calculated through which is equal to 27 mT for µ 0 being the vacuum permeability with µ 0 = 4π × 10 −7 H • m −1 , J c0 = 6.16 × 10 9 A • m −2 , and d = 7 µm.As per [15], the trapped field tends to stabilise and reach an asymptotic value after a substantial number of cycles, when the applied cross field remains below the parallel penetration field of a single tape.In our case, the extracted cross fields provided in table 3 are evidently lower than the calculated parallel penetration field.Therefore, it is reasonable to anticipate a motor operation characterised by long-time stability under the conditions outlined in section 5.2.Two tape structures were considered and utilised in the demagnetisation simulations: the semi-homogenised model and the multi-layered model [65].The semi-homogenised tape consists of a superconducting layer and a substrate layer, while the multi-layered model retains the original structure of the tape, composed of a superconducting layer, two silver stabilisers, and a substrate layer.The semi-homogenised model was used in the previous sections as the silver stabiliser does not have a significant influence on magnetisation and motor operation.However, the presence of the silver layer could have a significant impact on demagnetisation, particularly at high frequencies [47], which are of interest for high speed machines, such as generators for aerospace [86].Therefore, the multi-layered model is examined in this section to investigate its demagnetisation characteristics.For cross field demagnetisation, the critical current J c (B, T) in equation ( 3) contains only perpendicular field dependency as per [30].Therefore, the field orientation of the critical current density should be adjusted to parallel field-dependency as the external field would be applied to the ab-plane of the HTS stack.Due to the absence of specific data for the parallel field orientation, the adjustment was made based on the database in [87,88] as well as the general formula for the magneto-angular dependence of J c0 presented in [89].
To investigate the effect of amplitude and frequency of cross field on the stack demagnetisation, a series of simulations were conducted, as summarised in table 4. In contrast to other studies that commonly employed a fixed number of demagnetisation cycles [16,52], this research adopts a different approach, where a series of simulations with varying frequency and amplitude of the cross field were conducted over a certain time duration.This departure from the conventional approach was motivated by the fact that in previous studies, the applied cross field typically falls within the low-frequency range, where the time required to complete the same number of cycles is comparable.However, as the frequency increases and reaches the kHz range, the period length of a cycle decreases accordingly.Consequently, when a fixed number of cycles is applied, the simulation time becomes significantly longer for lower frequencies compared to higher frequency.This discrepancy in simulation time may lead to misleading conclusions suggesting that demagnetisation is more pronounced at lower frequencies than at higher frequencies.In addition, in engineering, it appears more sensible to quantify the power dissipation per unit time generated in HTS tapes compared to the losses accumulated during a fixed number of AC losses.Considering the relatively slower simulation speed associated with higher frequencies, a fixed duration of 15 ms was set for each simulation, corresponding to a complete cycle at a frequency of 66.7 Hz, but equating to 300 cycles at a frequency of 20 kHz.

Demagnetisation with semi-homogenised model
Figure 19 depicts the demagnetisation results for the semihomogenised model, where the frequency and amplitude were varied as per table 4. The evolution of the trapped field under different frequency and amplitude is illustrated from figures 19(a)-(e).Figure 19(f) showcases the corresponding demagnetisation rates, defined as the percentage reduction of the trapped field after demagnetisation (B t ) compared to the field after magnetisation (B 0 ).
where B 0 is 0.462 T for the four-tape stack.
The results indicate that the decay rate of the trapped field increases with both the amplitude and frequency of the applied field, which leads to the fact that the highest decay rate can reach 80% at the frequency of 20 kHz and the amplitude of 150 mT.Furthermore, the growth of decay rate from low to high amplitude is more pronounced at higher frequencies.Notably, the trapped field reduction is not dominated by flux creep decay in the simulated 15 ms, since the stack has gone through a period of 30 s flux relaxation after magnetisation.
Compared to studies conducted on a superconducting bulk sample in [54], where the trapped field decayed to only 10% and almost 2% after one single cycle for the ratio of the cross field amplitude B cf to the parallel penetration field B P being equal to 0.98 and 1.53, respectively, our study demonstrates a significantly low decay rate.Specifically, the decay rate remains below 1% for the amplitude of 25 mT (B cf /B P = 0.93), and less than 10% for the amplitude of 50 mT (B cf /B P = 1.9) across all frequency ranges.This confirms that, unlike HTS bulks, stacked HTS tapes exhibit a relatively higher resistance to demagnetisation.
The magnetisation power loss (W/m) caused by the transverse field accounts for the majority of the AC loss during the demagnetisation process and it can be calculated by [47] Figure 20 depicts the magnetisation loss characteristics in the HTS layer (Q HTS ) and substrate layer (Q HTS ), as well as the total magnetisation loss (Q Tot ) of the four-tape stack.In figures 20(a) and (b), both Q HTS and Q Sub exhibit a positive correlation with the frequency and amplitude of the applied cross field.This relationship is also evident in Q Tot with respect to the frequency and amplitude, as shown in figure 20(c).However, an important observation is that the proportion of Q HTS in Q Tot decreases with frequency, but increases with amplitude.In simpler terms, at lower amplitudes (e.g., 25 mT), the magnetisation loss is predominantly governed by Q HTS at lower frequencies (⩽1 kHz), while Q Sub dominates at higher frequencies (⩾10 kHz) due to the skin effect.This is consistent with the findings from [47].Conversely, at higher amplitudes (⩾100 mT), Q HTS emerges as the dominant contributor across all frequency ranges.Furthermore, it is noteworthy that high frequencies have a more pronounced impact on Q Sub compared to Q HTS .
Owing to the high aspect ratio of the HTS tapes (>6000), visualising the variation in current density distribution using conventional 2D graphs is challenging, which is a common manner in studies employing bulks or narrower CCs.Consequently, the analysis focus shifts to analysing the current density fluctuations in HTS layers.A circulating current is induced in the HTS stack after the magnetisation.In the context of a 2D model, both positive and negative currents flow simultaneously within the HTS layers.During the demagnetisation process, an external cross field aligned parallel to the ab-plane of the HTS stack was applied, subjecting all four HTS layers to the same alternating external field.Consequently, currents are induced in the HTS layers to oppose the change in the external field.
To examine the influence of the cross field on the four HTS layers, the positive current density in the HTS layers normalised to the critical current density J c0 , denoted J norm , is employed and calculated as (11) where J z represents the current density component in z direction and J c0 = 6.16 Figure 21 showcases the J norm fluctuation of the four HTS tapes at two different frequencies, 66.7 Hz (figures 21(a)-(d)) and 20 kHz (figures 21(e)-(h)), with various cross field amplitudes in a complete cross field cycle.The four HTS layers are labelled as HTS-1 to HTS-4 in figure 12. Notably, a clear distinction between the results at different frequencies is evident, with higher frequencies exhibiting more significant current density fluctuations.Additionally, the fluctuation of J norm increases in all four HTS layers as the amplitude of the applied cross field increases.
However, the top and bottom HTS layers (HTS-1 and HTS-4) exhibit larger variations in J norm values compared to the middle two layers (HTS-2 and HTS-3) throughout the complete working cycle of the applied cross field.At lower amplitudes (25 mT and 50 mT), the fluctuations in the normalised current density values in the inner layers are negligible compared to those in the outer layers.The current density fluctuations in the inner layers increase with the amplitude, yet they consistently remain smaller in magnitude compared to the fluctuations in the outer layers.This phenomenon is attributed to a shielding effect of the outermost layers on the inner layers, aligning with [66].
The shielding effect plays a crucial role in mitigating the impact of the cross field on the inner layers, thereby influencing the distribution of magnetisation loss within the stack.Consequently, the outer layers undergo a higher level of magnetisation loss compared to the inner layers.As the amplitude of the external cross field increases, the fluctuation of current densities within the HTS layers of the outer tapes increases with the external field.However, the normalised current density in the inner two tapes exhibit a noticeable waveform shift compared to the outer tapes.This shift can be attributed to the electromagnetic interaction between the inner and the outermost HTS layers.The combined effect of the skin effect, which drives the current distribution in the silver layer towards the stack's ends [66], and the shielding effect can have an important influence on the cross-field demagnetisation, particularly in high frequency regions.

Demagnetisation with multi-layered model
During demagnetisation, it is essential to consider the effect of silver stabilisers, particularly at high frequencies.The magnetisation loss, specifically the eddy current loss in the silver layer, can make a non-negligible contribution to the total loss [30].To examine the influence of the multi-layered model on demagnetisation, it is crucial to adjust the simulation model to a four-tape stack with the original structure in [30].To investigate the impact of high frequency on demagnetisation within the multi-layered model, the same set of simulations with varying frequencies and amplitudes was conducted.Figure 22 presents the magnetisation losses in different layers of the superconducting stack, the total magnetisation loss, and the percentage of HTS layer loss in the total loss.A comparison with figure 20 reveals that in the multi-layered structure simulation, Q HTS exhibits a slight increase at lower frequencies (⩽1 kHz) and a slight decrease at higher frequencies (⩾10 kHz).Notably, the silver stabiliser demonstrates a linear relationship between magnetisation loss and the amplitude and frequency of the cross field.The magnetisation loss in silver stabilisers (Q Ag ) is significantly higher than Q Sub , attributed to the much higher (almost 100 times) electrical conductivity of silver compared to Ni-5at.%W at 77 K. Furthermore, Q Tot of the multi-layered structure is higher than that in the semi-homogenised model, particularly at higher frequencies (⩾10 kHz).For example, at 20 kHz and 150 mT, Q Tot of the multi-layered model is 11.8% higher than that in the semi-homogenised model.Additionally, the proportion of Q HTS in Q Tot is lower in the multi-layered model compared to the semi-homogenised model, particularly at higher frequencies, where the rapid increase of with frequency was not considered.Notably, at high frequencies (>1 kHz), the magnetisation loss in the non-superconducting parts of the stack, specifically the silver layer, dominates the total loss due to skin effect.

Conclusion
This paper presents a systematic exploration of the magnetisation and cross-field demagnetisation of HTS TFSs in a practical partially superconducting machine, by proposing a novel analysis approach based on the FEM modelling.The study introduces novelty and contributions characterised through the exploration of harmonics interaction during practical motor operation and the utilisation of multilayer modelling for the demagnetisation analyses.A unique modelling approach to simplify motor operation without moving mesh is also presented.Lastly, this paper reveals the impact of high-frequency harmonics on demagnetisation.
This research work commenced with the validation of a superconducting machine model through experimental results, ensuring the accuracy and reliability of the model.The validated model was then utilised to examine the typical electro-mechanical performance of the studied HTS machine employing TFSs as alternatives to traditional magnets in PM synchronous machines.A novel method was employed to simulate the motor operation by using analytical formulae to reduce the number of DOFs so that the simulation efficiency can be significantly enhanced.Due to the geometry design of the machine, eight stacks, each containing four tapes, were employed for the motor operation, to maximise the trapped field ability of HTS CCs.
The demagnetisation analyses were based on the extracted harmonics experienced by the TFSs during the motor operation.For the harmonics extracted from the machine airgap, the fundamental and fifth harmonics of the tangential magnetic field are dominant according to the FFT results and were utilised for the demagnetisation study.The average values of the fundamental and fifth harmonics were 25 mT and 1.75 mT, respectively, for the 462 mT peak value of the magnetic flux density in the air gap.
In the demagnetisation analyses, two aspects were examined: field decay and magnetisation loss.A series of simulations with varying frequency and amplitude of the cross field were conducted over a time duration of 15 ms.Below the parallel penetration field of the employed AMSC tape, being 27 mT, the trapped field exhibits minimal changes across all frequencies.When the applied cross field exceeds the penetration field, the demagnetisation decay rate of the trapped field increases in correlation with the frequency and amplitude of the applied cross field.Notably, the highest decay rate reached almost 80% with an amplitude of 150 mT at 20 kHz.Regarding the analyses for magnetisation loss, two tape structures were considered: a semi-homogenised model and a multi-layered model.The semi-homogenised model, disregarding the presence of silver stabilisers, was employed for the model validation and the motor operation due to its negligible impact on the magnetisation and motor operation.However, in the demagnetisation process, the multi-layered structure with silver stabilisers provides a more accurate representation compared to the semi-homogenised model because the skin effect and shielding effect due to the electromagnetic interaction between normal conducting and superconducting layers of the TFSs have to be considered at high frequencies.
The magnetisation loss within various layers of the HTS stack has been thoroughly analysed across a wide range of frequencies from 66.7 Hz to 20 kHz and amplitudes from 25 mT to 150 mT as far as the cross field is concerned.The results indicate that magnetisation loss in all layers as well as the total magnetisation loss increases with the frequency and amplitude of the applied cross field.Within the frequency range below 1 kHz, the magnetisation loss in the HTS layer predominantly governs the total loss.However, in higher frequency ranges, (above 1 kHz), the non-superconducting layers, particularly the silver layer, take precedence in determining the total loss due to the skin effect.Furthermore, it is noteworthy that the outermost HTS layers exhibit a shielding effect on the inner tapes, resulting in higher magnetisation loss in the outer layers compared to the inner HTS layers.This shielding effect becomes more pronounced at higher frequencies.
The study significantly contributes to the in-depth understanding of the magnetisation and demagnetisation characteristics of HTS stacks in practical superconducting machine applications.The novel approach for motor operation developed in this research provides a time-saving model for accurately modelling superconducting motors with TFSs.By considering the impact of frequency and amplitude of cross field, the analysis offers valuable knowledge of the performance and stability of superconducting stacks, specifically in the context of electric aircraft.Furthermore, the study explores the interaction of different layers within a single HTS stack at varying frequencies, especially at high frequencies, thereby providing crucial insights into the performance of superconductors for high-speed machine applications.Overall, the paper serves as a valuable analysis and design tool for superconducting electric machines, particularly those applied to electric aircraft and other transport systems, which can inspire more relevant research work in the wider energy conversion community.

Figure 1 .
Figure 1.Working process from the magnetisation to the demagnetisation of the proposed approach.

Figure 2 .
Figure 2. Working flow of this research work: from model validation and magnetisation to demagnetisation.

Figure 3 .
Figure 3. Setup of the electromagnetic and thermal modules of the machine for the pulsed field magnetisation.

Figure 4 .
Figure 4. Numerical model of the superconducting machine with the semi-homogenised model of HTS stack (a) schematic sketch of the nine-layer stack structure (not drawn to scale) and (b) mesh for one eighth of the machine.
(a) demonstrates the exact setup and figure 5(b) shows the magnetic flux line distribution, in which a good symmetry between the left and right half sections of the machine is evident.The magnetised stack starts rotating from the original position during magnetisation, aligned to the coil teeth at 12 o'clock, The rotational speed is set to 650 rpm [26].The most important step is to set up a rotation function for the magnetic field of the magnetised stack.The equation should be compatible to a random form of the field.The remanent flux density of the ring-shape PM on the rotor surface is defined as an equation of radius (r), azimuth (φ ), and time (t) set up in cylindrical coordinate system (sys):

Figure 5 .
Figure 5. Machine model setup for back emf measurement (a) rotation model setup (b) magnetic flux density distribution with the trapped magnetic field copied from magnetised stack at the beginning of the rotation.

Figure 6 .
Figure 6.Comparison at the first pulse peak, first pulse end, first relaxation end, and end of the magnetisation of the (a) current density and (b) temperature of the stack.

Figure 7 .
Figure 7.Comparison of the trapped field profiles 1 mm above stack surface at the end of every flux relaxation.

Figure 8 .
Figure 8.Comparison of experimental result and simulation result.

Figure 9 .
Figure 9. Magnetic flux density distribution of 8 stacks in the whole machine.

Figure 10 .
Figure 10.Radial magnetic flux density distribution on the edge of stack after magnetisation (a) a fully magnetised superconductor and magnetic field distribution (b) leaked magnetic flux on the edge of the saturated superconducting stack.

Figure 11 .
Figure 11.Magnetisation of eight poles in the motor with varying amount of CC layers in each stack.

Figure 12 .
Figure 12.Stack configuration of the semi-homogenised model for motor operation and demagnetisation study based on the magnetisation results in figure 11.For clarity, the curvature and the real aspect ratio is not presented.

Figure 13 .Table 2 .Figure 14 .
Figure 13.Operation of the machine as a motor (a) excitation of stator winding and setup of rotor magnet remanent flux density in motor operation (b) flux distribution of the eight poles in motor operation.

Figure 15 .
Figure 15.Radial and tangential magnetic flux density experienced by a stack.

Figure 16 .
Figure 16.Radial magnetic field experienced by the four-layer stack in the airgap (a) waveform of Br with stator current varying from 10 A to 60 A at a rotor speed of 1000 rpm (b) FFT of Br with stator current varying from 10 A to 60 A and rotor speed from 1000 rpm to 6000 rpm.

Figure 17 .
Figure 17.Tangential magnetic field experienced by the four-layer stack in airgap (a) waveform of B phi with stator current varying from 10 A to 60 A at a rotor speed of 1000 rpm (b) FFT of B phi with stator current varying from 10 A to 60 A and rotor speed from 1000 rpm to 6000 rpm.

Figure 18 .
Figure 18.Application of cross field in air gap (a) setup in the FEM model (b) direction of the trapped field before cross field demagnetisation (c) direction of the trapped field during cross field application.

Figure 19 .
Figure 19.Cross field demagnetisation for 0.015 s for different frequency and amplitude of the cross field (a)-(e) decay of the normalised trapped field with the frequency and the amplitude listed in table 4 (f) variation of the decay rate of demagnetisation with frequency and amplitude of cross field.

Figure 20 .
Figure 20.Magnetisation loss (a) in HTS layer, (b) in substrate layer, and (c) total magnetisation loss in the 4-tape stack with semi-homogenised model under the applied cross field with different frequencies and amplitudes.

Figure 21 .
Figure 21.Fluctuation of the normalised current density in each HTS layer of the four-tape stack with semi-homogenised model under the cross field at the frequency of 66.7 Hz and 20 kHz with different amplitude (a)-(d) current density fluctuation of the tapes HTS-1 to HTS-4 under the cross field with 0.025 T, 0.05 T, 0.1 T, 0.15 T at 66.7 Hz (e)-(h) current density fluctuation of the tapes HTS-1 to HTS-4 under the cross field with 0.025 T, 0.05 T, 0.1 T, 0.15 T at 20 kHz.

Figure 22 .
Figure 22.Magnetisation loss in (a) HTS layer, (b) substrate layer, (c) silver layer, and (d) total magnetisation loss in the 4-tape stack with multi-layered model under the applied cross field with different frequencies and amplitudes.

Table 1 .
Description of the two machine model variants.

Table 3 .
Extracted tangential flux density harmonics experienced by a single stack on rotor surface.

Table 4 .
Range of parameters variation for cross field demagnetisation.