Methods of Estimating AC Losses in Superconducting MgB$_{2}$ Armature Windings With Spatial and Time Harmonics

Recent developments in low ac loss MgB<inline-formula><tex-math notation="LaTeX">$_{2}$</tex-math></inline-formula> conductors are of significant importance given renewed interest in fully superconducting (SC) machines. Evaluating ac losses in fully SC machines is a critical step in developing feasible designs. In fully SC machines, SC armature windings experience non-uniform rotating magnetic fields, with spatial and temporal harmonics, which has an undisputed impact on ac losses. Existing ac loss models in the literature, which have been validated for stationary sinusoidal external fields, were extended to constant amplitude rotating fields. There is not enough research on validating the ac loss models for rotating non-uniform magnetic fields with harmonics. This paper proposes simplified methods to estimate the ac losses in conductors with non-uniform rotational applied magnetic fields experienced by the armature in a machine’s environment. Extended analytical models are proposed to estimate the ac loss in single and multi-filament MgB<inline-formula><tex-math notation="LaTeX">$_{2}$</tex-math></inline-formula> conductors. The models are then compared against finite element analysis (FEA) results with Power Law loss estimation to evaluate model fidelity.


I. INTRODUCTION
F ULLY superconducting (SC) machines have the potential to achieve the high-power densities [1] required for direct drive wind turbine [2]- [6] and commercial electric aircraft propulsion [7]- [10]. In a fully SC generator, the armature coils will carry time-varying current while being exposed to a rotating magnetic field with some phase delay [11], [12]. These conditions will generate thermal losses within the SC armature coils, a phenomenon known as ac losses. Ac losses in the armature windings are the paramount challenge which Manuscript  must be overcome to make SC machines practical. These heat losses must be removed from the armature windings at cryogenic temperatures to keep the superconductors below their critical temperature at an applied field while carrying transport current. The increase in cryogenic loading adversely impacts the power density and efficiency of the machine. Therefore, ac loss is the most important design parameter considered in the design of a fully SC machine. Fully SC machines designers use analytical models to estimate ac losses in their machines and optimize the machine parameters for minimal losses [5], [13]. The loss estimation is performed using the actual field pattern in the armature winding of a superconducting machine obtained from a detailed transient simulation. The resultant field is evaluated in multiple locations of the armature cross-section as shown in Fig. 1. Here, the armature is separated into three radial sections and losses are evaluated in each section. The single point marked represents the field experienced by a single conductor located at that location. The losses in each conductor are estimated using the field observed at that location and then integrated over the volume of the armature winding to evaluate the total loss value. The armature windings in fully SC machines experience a non-uniform rotating magnetic fields with spatial and temporal harmonics which impact the ac losses. Various analytical ac loss models for the case of a stationary and sinusoidal applied magnetic field have been proposed and experimentally validated in the literature [14]- [16]. These models were extended to evaluate the case of a constant amplitude rotating applied magnetic field [12], [14], [17], [19]- [20]. Ac losses are impacted by the harmonics present in the applied field as well as in the transport current. Several works have focused on capturing the impact of harmonics in the transport current on ac losses [21]- [24].  Capturing the additional loss due to a sinusoidal transverse magnetic field with ac ripples has been attempted by various researchers [25]- [27] but remains a challenge for a non-uniform rotating magnetic field. A comparison between the constant rotating field and non-uniform rotating field experienced by a single conductor in an actively shielded fully SC machine is shown in Figs. 2 and 3. This shows how the spatial harmonics influence the field experienced by the armature conductors. Available analytical methods do not capture these effects on ac losses and therefore necessitate extended models to include losses associated with non-uniform rotating fields. Analytical models derived for this special scenario are modified from stationary field models with various assumptions [28]. There are several ongoing efforts to measure losses in HTS tapes experiencing rotating magnetic fields, but loss prediction models have yet to be validated for MgB 2 conductors [12], [29], [30].
This paper proposes simplified methods to estimate ac losses in superconductors with simulated current and field wave forms experienced by an armature conductor in a machine environment. Proposed analytical models are extended from available models in the literature to capture non-uniform rotational magnetic field with superposition of harmonics. Spatial and temporal harmonics obtained from a transient fully SC machine analysis are fed into the analytical model to estimate the losses. These results are then compared against finite element analysis (FEA) results to evaluate model fidelity. The analytical models are then refined to capture different parameters influencing ac losses in machine application. A 2D model of a single and multifilamentary conductor is used to capture hysteresis, coupling and eddy current losses. The Power Law method is used to model the SC filaments while metallic matrix and sheath materials are modeled with isotropic resistivities. The presence of magnetic sheath materials in the conductor also affect the ac losses and are analyzed by various researchers [31]- [35]. This paper assumes a non-magnetic sheath material in the conductor. The temperature dependence of the parameters is neglected with the assumption that the machine will be maintained at 20 K.

II. METHODS TO EVALUATE AC LOSS WITH HARMONICS
Various analytical models have been proposed in the literature to evaluate ac losses in multi-filamentary MgB 2 superconductors [36]- [39]. These approaches use conductor geometric properties and the applied field and current characteristics to estimate losses per unit volume. Carr proposed an extension to models for stationary fields to estimate losses in constant amplitude rotating fields [14]. These models are used in [40]- [42] to estimate losses. Wilson [15] proposed an analytical model to estimate losses in multi-filamentary conductors. This approach uses a loss function Γ(β) to capture the dependence of hysteresis loss on the applied field strength. These models were used to estimate losses in several applications [43], [44]. Hysteresis and coupling losses are the dominant ac loss factors in armature windings. To accommodate the influence of non-uniform rotating fields, two methods are proposed: the superposition of harmonic losses and the integration of instantaneous losses. The superposition of harmonics method evaluates the total losses by adding the individual losses contributed by each harmonic, similar to how iron losses are often calculated [45]. Using a Fourier transform, any applied transverse field to a conductor can be written as the summation of two spatially orthogonal alternating sinusoidal fields with a 90 o phase difference: where, x and y represent the two orthogonal axes considered, f is the fundamental frequency, ω f if the fundamental angular velocity, i is the order of harmonic, B xi,m is the i th harmonic field amplitude chosen in the x direction, B yi,m is the i th harmonic field amplitude chosen in the y direction, and t is the time. To find the hysteresis losses generated by each harmonic, losses are multiplied by a factor of R. This is computed in a similar manner to the uniform rotating-field models in [14] by comparing the resultant harmonic vector waveform to the fundamental pure sinusoidal waveform. Evaluation of the resultant R factor is as follows: where˙ B i is the time derivative of the i th harmonic applied field and B m,i corresponds to the peak magnitude of the resultant rotating field. Once the R factor is found, hysteresis losses can be evaluated as follows: where N is the total number of harmonics, J c,i (B m,i ) is the critical current density corresponding to the i th harmonic peak field, and both d f and λ are conductor properties tabulated in Table I. Eddy current and coupling losses are also calculated for individual harmonics and added together. Since there is a phase difference between the tangential and normal components, major and minor axis peaks of the resultant waveform are used to estimate the losses as shown in Fig. 4. The total loss can be evaluated as: where k = 4 and n = 2 are constants determined for a circular conductors, D 0 and L p are conductor properties tabulated in Table I. The instantaneous loss method evaluates the energy loss at each instant and integrates over a period to calculate the average power loss. For a circular SC wire with diameter D 0 and external applied transverse field B, the cyclic hysteresis loss P h can be written as: where, |Ḃ| is the absolute value of the time derivative of the applied field, J c is the critical current density at peak, B, and T is the period. For a rotating, non-uniform magnetic field, an additional angle derivative of the rotating field, B dθ dt , was added to capture the rotating field effects.
where θ indicates the rotational direction. Instantaneous losses are evaluated from FEA results based on the rotational field experienced by a conductor in the armature. Field data can be given in the normal and tangential referenced coordinate system or in any random orthogonal coordinate system as shown in Fig. 5. Two consecutive resultant vector fields V 1 and V 2 are denoted as follows: then, the angle between the two consecutive field vectors is evaluated as: 11) and the instantaneous loss is evaluated as: By integrating over one cycle, the cyclic energy loss is evaluated as: Multiplying that energy loss by the average frequency gives an average hysteretic ac loss for the SC wire: where T is the period. Similarly, eddy current loss can be evaluated as: and coupling current loss can be evaluated as:

III. AC LOSS VERIFICATION
Several numerical analysis methods utilizing FEA tools have been proposed to evaluate the ac losses in SC cables under sinusoidal external field [46], [47]. Numerical models for MgB 2 conductors in high power transmission cables are presented in [48]. A 12 filament superconductor with 1.1 mm wire outer diameter is modeled and self-field losses are evaluated. In this paper the Power Law is used to compute the spatial receptivity of the superconductor and subsequent losses within the finite element method. A fourth-order polynomial fit is used to extrapolate J c vs. B data for an example conductor provided by HyperTech Research, and Kim-Anderson constants are used in calculating the electric field: where B is the applied magnetic field, J is the current density, E is the electric field, and constants a 0 − a 4 , n 0 , B 1 , J c and E c are unique to the specific conductor chosen and are evaluated from conductor experimental data. A 2D FEA simulation was used to evaluate the losses on a single SC conductor. Losses are evaluated by integrating (25) in the SC filament area and copper area. The MgB 2 conductor is placed in the center of a rotating magnetic field and is surrounded by a copper matrix. Similarly, a multi-filament cable is also placed in the center of the rotating magnetic field. Specifications of the MgB 2 conductors used in this study are provided in Table I. Air-gap field harmonic data obtained from a transient analysis of the proposed 2.5-MW CHEETA fully SC motor is used to verify the proposed model [13]. Air-gap field harmonic data of this machine is tabulated in Table II. For the initial analysis, single filament and multi-filament conductors are modeled and analyzed in FEA. Fig. 6 shows the current density variation across the conductors at partial and full penetration. The behaviour of a SC element under rotating magnetic fields is quite different from that due to a stationary field. When the field is increased from an initial value of zero, the induced magnetic fields in the SC element are aligned in a direction such to oppose the applied field. However, after one rotation there is a lag between the induced magnetic field and the external applied field, similar to a boundary-edge effect, at the interface between the copper and superconductor. This trend can also be seen in the multi-filamentary conductor as a collective pattern. A 2D flux density distribution across the multi-filamentary conductor is shown in Fig. 6(b) and (d). From the figure, this phenomenon happens at both the bundle level and at the individual filament level based on the local flux density. It can also be seen that the coupling currents dominate in the multi-filament conductors and thus require a detailed 3-D analysis for loss estimation. To simplify the model for comparison, the following scenarios are considered to evaluate the losses on a single filament conductor: r Scenario 1: Assume a fundamental frequency with a constant rotating field (ideal sinusoidal waveform). Shown in Fig. 3(a). r Scenario 2: Assume a fundamental frequency with a nonuniform rotating field caused by a phase angle difference. Similar to Fig. 3(b). r Scenario 3: Assume a fundamental frequency with a nonuniform rotating field caused by an amplitude difference. Similar to Fig. 3(c). r Scenario 4: Assume a fundamental frequency with a nonuniform rotating field caused by a phase angle and amplitude difference. Similar to Fig. 3(b). r Scenario 5: Assume a non-uniform field with 1 st and 3 rd order harmonics with amplitude and phase variation. Similar to 3(d). These scenarios are also simulated for a multi-filament conductor at 10 Hz. Fig. 7 provides the evaluated instantaneous loss components using FEA modeling. This clearly shows that the constant rotating field has constant losses and all other scenarios create cyclic losses. The instantaneous loss variation shows a close relationship to the applied non-uniform rotating field thus providing support for the ratio constant, R.   depicts the evaluated average loss components using analytical models compared to those from FEA. The results show that for single-filament cases, hysteresis and eddy current losses closely align for both of the analytical approaches until harmonics are introduced. When the harmonics are introduced, instantaneous loss evaluation closely matches the FEA simulation. The same analytical models are extended to evaluate the losses in a multi-filament conductor. Fig. 9 shows the average cyclic losses estimated using both of the proposed methods. We can see a similar trend to that of the single conductor case, where the instantaneous loss estimation method shows moderate increments compared to the superposition of harmonics method. Due to the significant coupling currents, a 3D FEA model is necessary to evaluate FEA ac losses in multi-filament conductor and will be explored in future work.

IV. CONCLUSION AND FUTURE WORKS
AC losses under non-uniform rotating fields are evaluated in single filament and multi-filament MgB 2 armature conductors. The introduction of harmonics in the applied field necessitates adjustments to analytical models to capture the additional impact on losses. Simplified analytical models are proposed to include non-uniform rotating field with harmonics. FEA models are used to validate the extended analytical methods and estimate losses under an applied non-uniform rotational magnetic field. In the future, 2D winding packs and 3D FEA analysis results for the MgB 2 conductor will be analyzed to further validate the proposed computations.