Development and assessment of simplified analytical method for current distribution among REBa2Cu3O y parallel conductors in armature windings for fully superconducting rotating machines

Fully superconducting rotating machines employing REBa2Cu3O y (REBCO, RE = rare earth elements or Y) superconducting armature and field coils, are particularly interesting for aircraft applications, owing to their high output power density (kW kg−1). To achieve high current capability in superconducting coils, we have proposed a cabling design for transposed multi-strand parallel conductors. In the parallel conductor design, the REBCO strands are insulated from each other, except for both terminal ends, and transposed during the winding process to achieve uniform current distribution by cancellation of interlinkage magnetic flux between the strands. In this study, a simplified analytical method considering inductances was developed based on Laplace’s equation in cylindrical coordinates to roughly calculate the current distributions of multi-strands under armature coil conditions. The validity of the analytical method was investigated through current distribution measurements of the sample coils wound with two-strand parallel conductors. Consequently, the analytical method was validated with approximately 10% deviation under the experimental coil conditions. To establish a more accurate analysis method, certain improvements are needed.

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Introduction
Superconducting REBa 2 Cu 3 O y (REBCO, RE = rare earth elements or Y) coated conductors are a possible candidate for various power applications such as electric motors, generators, power transmissions, and applications in magnets for fusion and accelerators. Some of these applications strongly require the enhancement of the current capability of REBCO conductors. For example, large-size transformers [1,2] and mega watt (MW)-class rotating machines, particularly for propulsion systems in aircraft applications [3,4], require a current capacity of several kA, and future fusion magnets require a current capacity in the 100 kA class [5,6]. Several different multi-strand cabling designs, including a unique coil design, have been demonstrated [2,[7][8][9][10][11][12] using commercially REBCO tapes. Some of them have complex wire configurations that decrease the engineering current density.
Our research group has proposed a cabling design for the transposed parallel conductors [13,14] and the laser-scribing of the wires into a multi-filamentary structure [15] to realize large current capacity and low AC loss. In addition, the cooling method of immersion in subcooled liquid nitrogen can result in high thermal stability because of the high specific heat and cooling capacity. The high current capability, low loss, and high stability achieved via this strategy have been proven in 3φ-66/6.9 kV-2 MVA superconducting transformers [16].
Our next challenge is to apply this strategy to a fully superconducting rotating machine system, and this study focuses on the technique of transposed parallel conductors. In the parallel conductor design, the REBCO strands are insulated from each other, except for both terminal ends, and transposed during the winding process to achieve uniform current distribution by cancellation of the interlinkage magnetic flux between the strands. In our previous studies, the transposition points and patterns were theoretically investigated by evaluating the self and mutual inductances of the strands in cylindrical coil conditions [17,18]. We eventually determined the systematic transposition concept and achieved uniform current distribution under various coil conditions: various numbers of turns, strands, and layers. For instance, in the case of the four-strand solenoidal coils with interlayer-transpositions, at least 16 layers must be wound to realize a uniform current distribution; that is, the strands should be transposed 15 times between the layers. This systematic transposition concept is significantly helpful in the design of power transformers, pulse coils, and other applications using cylindrical coils. Furthermore, the systematic transposition concept can be used for designing rotating machines; however, coil configuration and electromagnetic conditions are different from those of cylindrical coils. Superconducting rotating machines usually employ long coils, such as race-track-shaped coils, in the axial direction. In addition, the armature coils in the stator are exposed to the rotating magnetic fields by a field coil and other phase armature coils.
Our goal is to build the systematic transposition concept for the armature-winding conditions, which can aid in cabling design including the transposition patterns for fully superconducting rotating machines. To achieve this goal, as a first step, a simplified analytical method to calculate the current distribution among the multi-strand parallel conductors is required. The simplified analytical method facilitates a quick description of the current distribution properties for many transposition patterns. Furthermore, we propose a relatively low computational cost method to investigate the current distribution, instead of a numerical calculation based on the finite element method [19]. This is thus a convenient method for those working in the field. Therefore, the objective of this study is to develop a simplified analytical method, to roughly calculate the current distributions of the strands in the armature-winding conditions. Furthermore, the developed analytical method was validated via simple condition experiments.
Analytical expressions for magnetic fields were derived based on Laplace's equation in cylindrical coordinates, and consequently, the self and mutual inductances of the strands were calculated. The individual strand current contributions among the constituent strands were assumed to be determined only by the self and mutual inductances. For REBCO superconducting tapes, the non-linear relationship between the electric field and current density can be described by the E-J power law [20,21]. Furthermore, the E-J power law characteristics of REBCO tape are dependent on the value and angle of the local magnetic field [22]. When the load factor of the transport current to the critical current is approximately 100%, the voltage owing to the E-J power law in the superconducting tape should be considered to accurately calculate the current distributions. Under conventional operating conditions such as transport current having a sufficient margin against critical current and high frequency, the non-linear resistive component in the electric circuit is negligible. The sample coils composed of two-strand parallel conductors were set under singlephase arrangement for a two-pole rotating machine, and their branch currents were measured by Rogowski coils by applying AC transport current. The two simplest possible parallel conductors with no transposition in the coil were used to remove complicated fabrication factors to the best extent possible. For example, the location of transpositions in the actual windings results in spatial deviations from the analytical model.

Magnetic field distribution
This section presents an outline of the derivation of the formulae describing the magnetic field distributions assuming a two-pole rotating machine. A somewhat similar derivation for a superconducting generator was conducted by Kirty [23]. Figure 1(a) shows the two-dimensional basic model for magnetic field calculation with back yoke. The first step in the analysis is to determine the magnetic field generated by a current distribution, as shown in figure 1(a). This current density (J) is assumed to be uniform between the limits of the included angle (θ a ) for simplicity. The current distribution across the superconducting tape width drastically changes depending on the transport current and applied magnetic fields. For instance, a non-uniform current distribution is observed across commercial REBCO tapes at low current; however, a radical crossover to a uniform distribution is observed at the critical current [24,25]. Nevertheless, the current sharing property between strands of parallel conductors in a coil is not significantly affected by the current distribution across the single tape width. This is because it exerts a minimal impact on the magnetic field distributions in the whole winding from a macroscopic perspective. It should be kept in mind that the above must be considered in coils, particularly those with a small number of turns. The current distribution (K(θ)) represented as a Fourier series of sinusoidally distributed currents, is given by where n is natural number, θ is the angle, and θ p is the coil pitch angle. The coefficient of the nth term is given by In a current-free region, we may consider the magnetic field as the gradient of a scalar potential ( − → H = −∇ϕ ) with ϕ subject to Laplace's equation in cylindrical coordinates as follows: We are assuming a system that is constant in z, and the solutions to equation (3) considering equation (1) have a general form for ϕ inside and outside the current sheet as follows: for r < r a , for r a < r < r o , The solution is completed by fitting equations (4) and (5) to the boundary conditions, which are: The magnetic permeability is assumed as infinity in the back yoke, that is, ∂ϕ o ∂θ = 0 when r is the inner radius of the back yoke (r o ). (iii) The magnetic flux is continuous at the shell (r = r a ), that is, Applying these boundary conditions to equations (4) and (5) and using equation (1), we can obtain the expressions for ϕ inside and outside the current sheet as follows: (7) Using equations (6) and (7), we can obtain the magnetic field in the radial direction as follows: for r < r a , If the current sheet has a thickness in the radial direction, as shown in figure 1(b), by integrating equations (8) and/or (9) for r a over the range from the inner radius of the winding (r ai ) to the outer radius of the winding (r ao ), the magnetic field in the radial direction at any point can be calculated. In addition, J for this condition is given by where I a is the current and N a is the number of turns in the element. Here, we can substitute equation (10) into equation (2). Further, the obtained K n is placed into equations (8) and (9). Using the obtained B ri and B ro , the expressions that describe the magnetic field in the radial direction (B r ) for the three regions are as follows: (i) for r less than the inner radius of the winding (r < r ai ) (ii) for r outside the winding inside the back yoke (r ao < r < r o ) Here, θ is the angle, r is the radius, and ψ is the angular displacement. The current density (J) is assumed to be uniform between the limits of the included angle (θa, θ a1 , θ a2 , θ in , and θout). θp, θ p1 , θ p2 , θ p, in , and θp, out are coil pitch angles for each winding. θuvw is the different-phase axis displacement from the axis of another phase. ro is the inner radius of the back yoke. ra, r ai , r ai, in , and r ai, out are the inner radii for each winding. rao, r ao, in , and rao, out are the outer radii for each winding.

Inductance
The next step in this analysis is to obtain the flux linked by each winding using equations (11)- (13). We consider the differential winding element shown in figure 1(b), at radius (r) and angular displacement (ψ ).
Using equation (13), the flux linkage due to the selfmagnetic field (λ a ) is given by where l a is the effective length. Therefore, self-inductance is given by for n ̸ = 2 components: for n = 2 component: where X is (r ai / r ao ). The windings #1 (red winding area) and #2 (green winding area) are assumed to have the same r ai and r ao and different coil pitch, as shown in figure 1(b). Using equation (13), the flux linked by magnetic field generated by winding #2 for winding #1 (λ a12 ) is given by (18) Therefore, the mutual inductance between windings #1 and #2 is given by for n ̸ = 2 components: for n = 2 component: The inside winding is assumed to have r ai, in (<r ai, out ), r ao, in (<r ao, out ), coil pitch (θ p, in ), and armature-winding included angle (θ in ), as shown in the blue winding area in figure 1(b). The outside winding is assumed to have r ai, out , r ao, out , θ p, out , and θ out , as shown in the green winding area in figure 1(b). Using equation (11), the flux linked by magnetic field generated by outside winding for inside winding (λ aio ) is given by Therefore, the mutual inductance between the inside and outside windings is given by for n ̸ = 2 components: for n = 2 component: where Y is (r ai, in / r ao, in ) and Z is (r ai, out / rao, out). In the case of mutual inductance between different-phase windings (yellow winding area), each equation should be multiplied by cos (nθuvw). If a magnetic shield is assumed as a fully reflecting shield such as a superconducting shield instead of a back yoke, the term ro is negative rather than positive for equations (16), (17), (20), (21), (24) and (25). In addition, in the case of no magnetic shield, ro is substituted for infinity.

Coil specification and winding
The REBCO superconducting tapes were provided by Fujikura Ltd. The total thickness with the insulation film, width, and critical current (Ic) are ∼200 µm, 4.0 mm, and over 200 A, at 77.3 K in self-field, respectively. The specifications of sample coils are listed in table 1. Figure 2 shows the sample coil with 48 turns. The two insulated REBCO strands were co-wound into a racetrack-type coil with double-pancake winding. The two parallel strands were stacked in the thickness direction and had no-transposition in the coil. The total number of turns of the sample coils were 16, 32, and 48. We did not check the Ic of the sample coils; however, the superconducting current with no-voltage was confirmed up to 30 A in liquid nitrogen. Figure 3 shows the sample coil arrangement. This coil arrangement assumed the single-phase (U and U'-phase) arrangement for the armature coils of a two-pole rotating machine without back yoke. The U and U'-phase coils facing each other were placed at a distance of 30 cm. The REBCO strands of each coil were soldered individually. Two connected combinations were prepared: no-transposition and transposition. The notransposition connection was the combination that connected the outer strands to each other and the inner strands to each other. The transposition connection was the combination that connected the outer strand of one side and the inner strand of the other. The V (V') and W (W') phase coils will be placed at a position rotated 120 • and 240 • from the position of U (U') phase coils, respectively (area enclosed by the square with dotted lines). In addition, a field coil with a shaft will be placed at the center part (area enclosed by the dotted circle). Figure 4(a) shows the two-dimensional model of the sample coil arrangement of figure 3. The red area indicates the winding area of the straight part for the sample coils with 48 turns. The experimental setup had no-back yoke for simplicity; therefore, ro was substituted for infinity in equations (16), (17), (20), (21), (24) and (25) during the calculation. The inset enclosed by the blue line ( figure 4(b)) shows the enlarged view of the one-side winding area of the U-phase coil. The black dashed line indicates the actual winding area for the sample coil; here, the analytical winding area is slightly different from the actual winding area. The inset enclosed by the green line (figure 4(c)) shows the superconducting layer areas of the two parallel conductors in the inner most winding for figure 4(b). It should be     noted that the aspect ratio is made different from the actual one for better viewing. The actual superconducting layer thickness is ∼2 µm. In the analysis, for the polar coordinate system, the superconducting layer area was described by radial coordinates in the range of 152-156 mm and an included angle of ∼1.299 × 10 −5 rad. One turn formed by a single strand was represented as one elemental inductance and the self and mutual inductances of all turns were then calculated using equations (15), (19) and (23) considering the winding structure. In addition, the inductances of the same strand in each turn and each coil were added, considering the transposition [26], to obtain the total inductance of each strand: total self-inductance of the one strand (L t1 ) and another strand (L t2 ), and total mutual inductance between each strand (M t12 and M t21 ). Figure 5 shows the equivalent circuit for the analysis. Finally, the current ratios (I r1 and I r2 ) of the strands are expressed as 26) where I total is the total current and V is the applied voltage.

Current distribution measurement
The experimental setup of the current distribution measurement is shown in figure 6. Rogowski coils were mounted on every strand to measure the branch current. The detailed specification of the Rogowski coils and measurement system have been provided in previous studies [26,27]. The mutual inductances of the Rogowski coils were ∼1.2 µH. The total transport current (I total ) was measured using a non-inductive shunt resistance. The measurements were performed in liquid nitrogen by applying an AC current with an amplitude of approximately 10 A and a frequency in the range of 10-500 Hz. As  shown in the two clamping connections and the soldering connection in figure 6, the experimental circuit included contact resistances Rt (=Rs + 2Rc), as shown in figure 7. The equivalent circuit is shown in figure 7. Here, Rs is the resistance at the soldering connection and Rc denotes the resistance at the clamping connection. We observed the following: Rs = ∼20 nΩ and Rc = 7.0-47.8 µΩ. Even when Rt was at a maximum, it was 1/1000 smaller than the inductive reactance with a 16-turn coil for a frequency higher than 300 Hz. The experimental equivalent circuit can practically ignore Rt, at least for a frequency higher than 300 Hz, which corresponds to the equivalent circuit for the analysis. The two soldering connection combinations were prepared as shown in the inset of figure 7, indicated by the aqua dashed line. The no-transposition connection was the combination that connected the outer strands to each other and the inner strands to each other. The transposition connection was the combination that connected the outer strand of one side and the inner strand of the other. In addition, the current distribution for a single coil was measured in this study.

Results and discussion
To verify the analytical expressions, the predicted current distributions for the strands were compared with the experimental values after checking the measured current distribution data. Figure 8 shows the typical data for the time dependence of the transport current for a frequency of 400 Hz in (a) the single U-phase coil and (b) the U and U'-phase coils connected by the no-transposition connection. The former was a 16-turn coil, which had the smallest inductance case, and the latter was a 48-turn coil, which had the largest inductance case in this experiment. The inset of figure 8(a) shows the frequency dependence of the current distribution ratio for the strands in the single U and U'-phase coils. The current distributions of the inner (I in ) and outer strands (Iout) were independent of the frequency higher than 100 Hz; therefore, we can exclude the contact resistances from this discussion. In addition, the sample coils had consistent quality because the I in and Iout values for each coil were almost equal in this region. In figures 8(a) and (b), the total current of the strands measured by the Rogowski coils (I in + Iout and I in-in + Iout-out) corresponded to I total evaluated via the shunt resistor; therefore, it was confirmed that the Rogowski coils were sufficiently accurate. I in with 7.0 A peak and Iout with 3.4 A peak exhibited the same phase, as shown in figure 8(a). However, Iout-out with 4.6 A peak had the reverse phase to I in-in with 15.7 A peak , as shown in figure 8(b). These phenomena can reasonably be described by the inductances predicted by the analytical expressions. The self-inductances of the inner and outer strands for the 16-turn single coil were 49.29 and 49.38 µH, respectively. Both selfinductances exceeded the mutual inductance of 49.24 µH. The self-inductances for the I in-in and Iout-out strands for the 48-turn coils were 782.2 and 784.3 µH, respectively. Here, the mutual inductance of 782.6 µH exceeded the former; hence, Iout-out exhibited a reversal phase. Figure 9 shows the dependence of the current distribution ratios on the number of turns under a frequency of 400 Hz in (a) the single coil and (b) the U and U'-phase coils connected with the no-transposition or transposition connection. The predicted current distribution ratios are indicated by solid lines. It can be observed that all the predicted lines roughly correspond with the measured plots. The maximum deviations were approximately 11% for the single coil with 32 turns and no-transposition with 48 turns cases. The standard deviations of the measured current distribution ratios were in the range of 0.30%-2.0% (average of 0.82%), which can be attributed to the measurement system error. As shown in figure 9(a), the current distribution ratios between the U and U'-coils for same number of turns were slightly different, owing to the heterogeneity of winding, which was in the range of 0.20%-2.3% (average of 0.86%). Therefore, the experimental values yielded errors in the approximate range of 1%-2%, which may be considered a reason for deviations between the analysis and experimental results. Another reason for this deviation is the difference between the analytical model and the actual winding areas, as indicated in figure 4. The other possible reasons are the non-uniform current distribution across the superconducting tape width and the lack of consideration for the internal alternating structure of coated conductors. Commercial REBCO tape usually has a layered architecture of Cu/Ag/REBCO/buffer layer/hastelloy/Ag/Cu. To establish a more accurate analysis method, these complex issues should also be considered. I in-out and I out-in exhibited approximately uniform current distributions and corresponded with the predicted line. We confirmed that the transposition effect can be described by analytical expressions. According to these results, an analytical model based on these analytical expressions can roughly describe the current distribution of the strands.

Summary
For superconducting armature windings in fully superconducting rotating machines, a simplified analytical method considering only inductances was developed to calculate the current distributions between the multi-strand parallel conductors composed of REBCO superconducting tapes. Analytical expressions for self and mutual inductances of the strands were derived based on Laplace's equation in cylindrical coordinates. In addition, two-dimensional analytical modeling based on the analytical expressions was conducted. Furthermore, the developed analytical method was roughly validated via current distribution measurements for the sample coils wound with two-strand parallel conductors. Consequently, the validity of the analytical method was confirmed with approximately 10% deviation under the experimental coil conditions in this study. The experimental values yielded errors in the approximate range of 1%-2%, which may be a reason for the deviations between the analysis and experimental results. Another reason for this deviation is the difference between the analytical model and the actual winding areas, as indicated in figure 4. In addition, the current distribution across the superconducting tape width and the internal alternating structure of coated conductors may also contribute to the deviation. Since the E-J power law characteristics of the superconducting tape have not been considered, the developed analytical method is applicable under limited operating conditions, such as transport current having a sufficient margin against the critical current.
For experimental simplicity, the sample coils were prepared as racetrack configurations in this study. In the future, we will develop novel armature coil configurations, wherein the parallel direction of the REBCO tape face corresponds to the radial direction in the rotating machines, similar to a saddle-shaped configuration. Such a coil offers an advantage from the perspective of AC loss. According to the numerical calculations [28], the AC loss in the saddle-shaped coil was lower than that of the racetrack coil because the perpendicular magnetic fields applied to the flat face of the winding tapes were smaller in the saddle-shaped windings than those in the racetrack-shaped windings. Furthermore, the analytical model is expected to correspond better to such a coil and provide more accurate predictions.

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).

Appendix. Analytical expressions for general multipolar machines
This appendix presents information on the analytical expressions for multipolar machines that provide helpful context for calculating current distributions of general rotating machines. The main paper focused on only two-pole rotating machines. When multipolar machines are assumed, equation (1) is replaced by Kn [cos {nP(θ − θp)} − cos(nPθ)] (27) where P is the number of pole pairs of the winding. For instance, if a two-pole rotating machine is considered, P should be unity. If a four-pole rotating machine is assumed, P should be two. When multipolar machines are assumed, equation (2) is replaced by ) . (28) Therefore, the subsequent equations should contain P. For instance, self-inductance is given by for nP ̸ = 2 components: for nP = 2 component: ORCID iD S Miura  https://orcid.org/0000-0003-2841-286X