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Kazuhiro Kajikawa, Yuto Mametsuka, Masahiro Furukakoi, Taketsune Nakamura, [Akihiro Kikuchi](https://orcid.org/0000-0002-5044-7156), Tomohito Miura

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[Numerical and Theoretical Analyses of Losses in Armature Windings of Motors for Liquid Hydrogen Pumps](https://mdr.nims.go.jp/datasets/5c483e65-215f-4303-98b9-913f8d209d57)

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RESEARCHJournal of Superconductivity and Novel Magnetism (2025) 38:229https://doi.org/10.1007/s10948-025-07069-5with equivalent absorption, thereby reducing net emissions to zero [1]. It has been expected to use hydrogen as one of key technologies to realize the carbon neutrality up to now. It is well known that the hydrogen does not exist alone in the earth but is included in water, organic compounds, etc., 1  IntroductionMany governments around the world have officially pledged to achieve carbon neutrality by the middle of the 21st century, aiming to offset greenhouse gas emissions Kazuhiro Kajikawa, Yuto Mametsuka, Masahiro Furukakoi, Taketsune Nakamura, Akihiro Kikuchi, Tomohito Miura contributed equally to this work. 
 Kazuhiro Kajikawakajikawa@rs.socu.ac.jpYuto Mametsukaf124620@ed.socu.ac.jpMasahiro Furukakoifurukakoi@rs.socu.ac.jpTaketsune Nakamuranakamura.taketsune.2a@kyoto-u.ac.jpAkihiro KikuchiKIKUCHI.Akihiro@nims.go.jpTomohito Miurat-miura@torishima.co.jp1  Department of Electrical Engineering, Sanyo-Onoda City University, 1-1-1 Daigaku-Dori, 756-0884 Sanyo-Onoda, Yamaguchi, Japan2  Department of Electrical Engineering, Kyoto University, 1-30 Goryo-Ohara, Nishikyo-ku, 615-8245 Kyoto, Kyoto, Japan3  National Institute for Materials Science, 1-2-1 Sengen,  305-0047 Tsukuba, Ibaraki, Japan4  Torishima Pump Mfg. Co., Ltd., 1-1-8 Miyada-cho,  569-8660 Takatsuki, Osaka, JapanAbstractIn order to develop submerged pumps to transfer liquid hydrogen, high temperature superconducting (HTS) motors to drive them as one of key components are a promising candidate with low electromagnetic loss and large specific power density per unit mass. However, since armature windings wound using HTS wires have a couple of problems such as three-phase unbalanced currents and frequency limitations, HTS wires are used only for rotor windings of the motors and metallic cables have to be applied to their armature windings. Therefore, numerical analyses to evaluate loss properties in multi-strand metal twisted cables for armature windings of HTS motors cooled at liquid hydrogen temperature are car-ried out by means of a two-dimensional finite element method. On the basis of a clarified physical mechanism of losses generated in the windings arranged within a slot of armature iron core, the obtained numerical results are also reproduced with theoretical expressions of the Joule loss for an alternating transport current and the eddy-current loss for an externally applied AC magnetic field. The influences of losses on the frequency and number of strands in metal twisted cables are investigated quantitatively.Keywords  Armature winding · Eddy current loss · Joule loss · Finite element method · TheoryReceived: 2 September 2025 / Accepted: 15 October 2025 / Published online: 30 October 2025© The Author(s) 2025Numerical and Theoretical Analyses of Losses in Armature Windings of Motors for Liquid Hydrogen PumpsKazuhiro Kajikawa1 · Yuto Mametsuka1 · Masahiro Furukakoi1 · Taketsune Nakamura2 · Akihiro Kikuchi3 · Tomohito Miura41 3https://doi.org/10.1007/s10948-025-07069-5http://crossmark.crossref.org/dialog/?doi=10.1007/s10948-025-07069-5&domain=pdf&date_stamp=2025-10-30Journal of Superconductivity and Novel Magnetism (2025) 38:229so that a series of supply chains for the hydrogen to be pro-duced, stored, transported and utilized has to be established safely and stably. Although the hydrogen itself has a poor volume density, the liquefaction is one of realistic solutions for the effective use of hydrogen, especially during its stor-age and transportation. The application of high temperature superconductors (HTSs) with the liquid hydrogen such as a power cable  [2], generator  [3], superconducting magnetic energy storage (SMES)  [4], active magnetic refrigeration (AMR) [5], magnets [6–8], level sensors [9–11], and so on, has been fabricated and tested so far. The submerged pumps for liquid hydrogen driven by HTS motors, which will be necessary as one of infrastructures in future hydrogen society, have also been evaluated experimentally  [12–14], where MgB2 wires have been applied to squirrel-cage wind-ings in the rotor.In order to realize the submerged pumps for liquid hydro-gen, it is important to reduce losses in armature windings of the drive motor cooled at cryogenic temperature, which have three-phase alternating currents to generate a rotating magnetic field inside and then rotate the rotor. Generally, two types of conductors are considered for motor armature windings: HTS wires and low-resistivity metallic cables. Fully superconducting motors, where HTS materials are applied to both rotors and stators, have been fabricated and tested up to now  [14–18]. When HTS wires are used for the armature windings energized with a widely used voltage drive inverter, however, it has been found that the risk of burnout of them might arise due to unbalanced three-phase currents and therefore the HTS wires could not be applied to them [19]. It has also been reported that, at frequencies of several hundred hertz, the weights and losses in the armature windings of motors cooled to liquid hydrogen temperature using fine multi-strand metallic cables with high residual resistivity ratios (RRRs) are lower than those estimated with HTS wires  [20]. It is well known that the electrical resistivity of metals decreases with temperature, and that the Joule loss for an alternating transport current also decreases with resistivity. On the other hand, the eddy-current loss for an externally applied AC magnetic field increases with decreasing the resistivity and can be suppressed by using thin wires, which also leads to avoid the skin effect at cryo-genic temperature. As a result, it has recently begun evalu-ating losses in multi-strand metal cables under cryogenic conditions. [20–25].In this study, the numerical analyses to evaluate loss prop-erties in multi-strand metal twisted cables arranged within iron core slots are carried out by means of a two-dimen-sional finite element method. The obtained numerical results are also reproduced with another numerical model and derived theoretical expressions. The influences of losses on frequency and number of strands in metal twisted cables at liquid hydrogen temperature are investigated quantitatively.2  Theoretical Expression of Joule Loss in Round WireLet us derive a theoretical expression of Joule loss in a round wire made of a normal metal with a radius rs and an infinite length in the z-direction. Figure 1 shows the local current density Jz  and magnetic field Hθ inside the round wire, to which the transport current It = Ie ejωt is applied, where Ie is the root mean square (RMS) value of transport current and ω (= 2πf) is the angular frequency with the frequency f. In this case, the current density Jz  and magnetic field Hθ satisfy the modified Bessel differential equations,∂2Jz∂u2 + 1u∂Jz∂u− Jz = 0, � (1)∂2Hθ∂u2 + 1u∂Hθ∂u−(1 + 1u2)Hθ = 0, � (2)Fig. 1  Local current density Jz  and magnetic field Hθ  inside round wire made of normal metal with radius rs and infinite length, to which transport current It is applied 1 3229  Page 2 of 12Journal of Superconductivity and Novel Magnetism (2025) 38:229obtained easily from Maxwell’s equations, where the nor-malized variable u is given by u = αr with the parameter α represented byα =√jωµ0ρs= 1 + jλs,� (3)where j is the imaginary unit and λs is the skin depth given by λs =√2ρs/(ωµ0) with the resistivity ρs.The exact solutions for (1) and (2) can be expressed as [26]Jz = c1 I0(u) ejωt, � (4)Hθ = c2 I1(u) ejωt, � (5)by using the constants of integration, c1 and c2, having a relationship c1 = αc2, and the modified Bessel function of the first kind, In(u), which is simply represented byIn(u) = 1πˆ π0eu cos q cos nq dq,� (6)for an integer n, where q is the integral variable. The boundary condition is given by Hθ|r=rs= It/(2πrs), so that the time-domain formulations of the transport current It, current density Jz , and magnetic field Hθ are ultimately obtained by transforming the corresponding phasor repre-sentations asIt(t) =√2 Im[Ie ejωt]=√2 Ie sin ωt, � (7)Jz(r, t) = Jersλs√G21(r) + G22(r)G23(rs) + G24(rs)× sin[ωt + π4+ φ(r) − ψ(rs)],� (8)Hθ(r, t) = Hm√G23(r) + G24(r)G23(rs) + G24(rs)× sin[ωt + ψ(r) − ψ(rs)] ,� (9)where Je = Ie/(πr2s) and Hm =√2 Ie/(2πrs). The func-tions, G1, G2, G3 and G4, are expressed asG1(r) = 1πˆ π0erλs cos q cos(rλscos q)dq, � (10)G2(r) = 1πˆ π0erλs cos q sin(rλscos q)dq, � (11)G3(r) = 1πˆ π0erλs cos q cos(rλscos q)cos q dq, � (12)G4(r) = 1πˆ π0erλs cos q sin(rλscos q)cos q dq. � (13)The functions, φ and ψ, are also given byφ(r) = arctanG2(r)G1(r), � (14)ψ(r) = arctanG4(r)G3(r). � (15)By using (8), the local power dissipation p per unit volume can becomep(r) = ω2πˆ 2πω0ρsJ2z (r, t) dt= ρsJ2e2r2sλ2sG21(r) + G22(r)G23(rs) + G24(rs).� (16)Therefore, the total power dissipation PJ  per unit length can be obtained asPJ =ˆ rs02πrp(r) dr = P0G23(rs) + G24(rs)׈ rsλs0rλs[G21(r) + G22(r)]d(rλs),� (17)where P0 is the power dissipation per unit length in the case where the current uniformly flows inside the round wire, which is given by P0 = ρsJ2e Ss with the cross-sectional area Ss(= πr2s) of the round wire. The validity of (17) is confirmed in Section 4.3  Numerical Calculations of LossesFour kinds of metallic cables considered in this study are listed in Table 1. The number of strands, Ns, in twisted cables is 7, 19 or 37 as well as 1 for a single solid round wire. The diameter of the single wire is 0.812 mm, which corresponds to 20 American wire gauge (AWG), whereas the diameter 2rs of strands in 7-, 19- and 37-strand cables are 0.255 mm, 0.143 mm and 0.101 mm, respectively, which correspond to 30, 35 and 38 AWG. Although the diameters of strands, 2rs, are different from each other, the outer diameters 2rh in the multi-strand cables are identical to that for the single wire by adjusting the distances d between adjacent layers 1 3Page 3 of 12  229Journal of Superconductivity and Novel Magnetism (2025) 38:229used [27], and no magnetic-field dependence of resistivity is considered here  [28]. The characteristic frequencies fs, at which the skin depth becomes equal to the radius rs of strand, are 257 Hz for the single wire, and 2.60, 8.27 and 16.6 kHz for the 7-, 19- and 37-strand cables, respectively. The RMS value of transport current applied to the cable, Ie, is fixed at 20 A.Figure 2 shows the partial cross-sectional view of arma-ture windings arranged within slots of a stator icon core separated from a rotor iron core by a mechanical gap.The region surrounded by a dotted line is focused on for numerical analyses in this study. Figure 3 shows three kinds of numerical analysis models used here. The electro-magnetic fields in these models are numerically calculated based on the H-formulation [29–32] by using the COMSOL Multiphysics® software  [33]. Figure  3(c) is called “a sta-tor model”, where a realistic situation of armature windings arranged within one of slots is considered for numerical analysis. w and h are the width and depth of iron core slot, respectively. s and b are also the width and depth of slot opening, respectively. p is the pitch between the slots, and δ is the gap length between the cores. N cables are arranged vertically and horizontally within the iron core slot, follow-ing several combinations of periodic patterns. There are six layers within the slot (m = 6), and the number of turns in each layer, denoted as N1 to N6, is 3 or 4. In the case of Fig. 3(c), the total number of turns becomes N = 20. The same transport current It is applied to each turn. The multi-strand cable is used for turn under consideration, and the single wires are used except for it. Every possible pattern of in the twisted cables. In this case, the volume fractions of strands, ϕ(= Nsr2s/r2h), vary from 100% in the single wire to 57.2% in the 37-stand cable, and decrease monotonically with increasing the number of strands. The resistivity of copper, 0.167 nΩ·m, at 20 K and 0 T with RRR of 100 is Table 1  Specifications of metallic cablesNumber of strands, Ns1 7 19 37Diameter of strand, 2rs0.812 mm 0.255 mm 0.143 mm 0.101 mm(20 AWG) (30 AWG) (35 AWG) (38 AWG)Distance between adjacent layers in cable, d- 0.0235 mm 0.0243 mm 0.0175 mmOuter diameter of cable, 2rh0.812 mm 0.812 mm 0.812 mm 0.812 mm(20 AWG) (20 AWG) (20 AWG) (20 AWG)Volume fraction of strands, ϕ100% 69.0% 58.9% 57.2%Resistivity of strand, ρs [27]0.167 nΩ·m 0.167 nΩ·m 0.167 nΩ·m 0.167 nΩ·mCharac-teristic frequency, fs257 Hz 2.60 kHz 8.27 kHz 16.6 kHzRMS value of applied current, Ie20 A 20 A 20 A 20 AFig. 2  Partial cross-sectional view of armature windings arranged within slots of stator icon core separated from rotor iron core by mechanical gap. The region sur-rounded by a dotted line is focused on for numerical analyses in this study 1 3229  Page 4 of 12Journal of Superconductivity and Novel Magnetism (2025) 38:229to be infinite, and the tangential components Ht of magnetic fields are equal to zero. On the other hand, the boundaries indicated by red lines has non-zero tangential components. In the case of Fig. 3(c), the tangential magnetic fields Hg  in the gap between the cores are given by Hg = NIt/(2δ) from Ampère’s law. As is customary in the H-formulation, configuration is taken into account to obtain a final result. The multi-strand cable is twisted, so that the currents in each layer of the cable become identical to each other. The boundaries indicated by blue lines represent the surfaces of iron cores, where the magnetic fields become perpendicular to them if the magnetic permeability of iron core is assumed Fig. 3  Numerical analysis models called (a) lone model, (b) simple model and (c) stator model. The blue lines represent the surfaces of iron cores, where the magnetic fields become perpendicular to them if the magnetic permeability of iron core is assumed to be infinite. The Dirichlet boundary conditions are also applied on the red lines 1 3Page 5 of 12  229Journal of Superconductivity and Novel Magnetism (2025) 38:229the vacuum region is virtually treated as a conductor with extremely high resistivity ρv, which is set to 1 × 10−5 Ω·m here [30]. The numerical parameters for the stator model are listed in Table 2.Figure 3(b) is called “a simple model”, which is used to understand the physical mechanism of losses in the cables within the iron core slot. A unit cell with the width w0 and height h0, which contains only one cable in its center, is considered first, and some of unit cells are vertically stacked to form an iron core slot. The number of unit cells, which is equal to the number N of turns within the iron core slot, is set to 1, 3 or 5. The turns are counted from the deepest point within the iron core slot as 1st to N-th turn. The cable configuration and boundary conditions similar to those in Fig.  3(c) are applied to the simple model. Figure  3(a) is called “a lone model”, where a single cable is isolated in the vacuum, and the transport current It and an external transverse magnetic field Be are simultaneously applied to it. The outer diameter of vacuum region, 2rv , is set to be ten times larger than the cable diameter 2rh. The boundary con-dition for the lone model is given by Ht = It/(2πrv). By using the lone model, the loss in each turn within the iron core slot tries to be reproduced later. The numerical param-eters for the lone and simple models are listed in Table 3.Figure  4 shows the frequency dependence of the com-puted losses per unit length for simple models with 1, 3 and 5 turns within the iron core slot. It is found that the numeri-cal results of losses are constant in a small range of fre-quency, and the Joule losses are dominant. The Joule losses Table 2  Numerical parameters for stator modelNumber of turns arranged within slot, N 20Number of layers within slot, m 6Number of turns in each layer, Ni (i = 1, · · ·, m) 3 or 4Slot width, w 4.8 mmSlot depth, h 7.2 mmWidth of slot opening, s 1.2 mmDepth of slot opening, b 0.4 mmPitch between slots, p 9.6 mmGap length between cores, δ 0.4 mmTable 3  Numerical parameters for lone and simple modelsDiameter of vacuum region in 8.12 mm                         lone model, 2rvResistivity of vacuum region, ρv 1 × 10−5 Ω·mNumber of turns arranged within slot, N 1, 3, 5Slot width of unit cell, w0 1.2 mmSlot depth of unit cell, h0 1.2 mmWidth of slot opening, s 0.4 mmDepth of slot opening, b 0.4 mmPitch between slots, p 2.4 mmGap length between cores, δ 0.4 mmFig. 4  Frequency dependence of numerical results of losses per unit length for simple models with (a) single turn, (b) 3 turns and (c) 5 turns within iron core slot 1 3229  Page 6 of 12Journal of Superconductivity and Novel Magnetism (2025) 38:2291st turns of 3- and 5-turn models are same, but the loss in 1-turn model is larger than them. To give another example, the losses in 2nd turns for 3- and 5-turn models are also same, but the losses in their 3rd turns are different from each other. These loss properties can be understood on the basis of the physical mechanism illustrated in Fig.  6 [31, 34]. Figure 6(a) shows the typical profiles of magnetic lines of force within the iron core slot, where five cables are placed inside as an example. Since the permeability of iron core is assumed to be infinite, the magnetic lines of force become perpendicular to its surfaces. Figure 6(b) also shows the pro-files of magnetic lines of force if the iron core were entirely increase with the number of strands due to the reduction of their volume fractions. On the other hand, the numerical results of losses increase with frequency in a larger range of frequency, and the eddy-current losses become dominant. The eddy-current losses decrease with increasing the num-ber of strands due to the reduction of their strand diameters.Figure 5 compares the calculated losses per unit length for each turn in simple models using single round wires and cables composed of 7, 19 and 37 strands. It can be seen that the numerical results of losses in turns, counted from deep-est point within the iron core slot, are identical except for turns nearest to the slot opening. For example, the losses in Fig. 5  Comparison between numerical results of losses per unit length for each turn in simple models with (a) single solid round wires, (b) 7-strand cables, (c) 19-strand cables and (d) 37-strand cables. 1 3Page 7 of 12  229Journal of Superconductivity and Novel Magnetism (2025) 38:229here. Figure 7 shows the comparison between the calculated results with the simple and lone models, plotted with closed and open symbols, respectively. The parameter β in the lone model is fixed at 0.5, 1.5, 2.5, 3.5 or 4.5. It can be seen that the numerical result of loss in each turn of the simple model below the characteristic frequency can be well reproduced by using the lone model except for turns nearest to the slot opening, to which the slightly enhanced magnetic fields are applied and whose loss becomes somewhat larger than that for the lone model.Next, in order to reproduce the numerical results of losses for the lone models plotted with the open symbols in Fig. 7 theoretically, it is considered that the loss generated in a single cable can simply be divided into two components, the Joule loss PJ  cased by only the transport current It and the eddy-current loss Pe caused by only the external magnetic field Be.Let us explain how to estimate the Joule loss PJ  theoreti-cally at first. The red symbols in Fig. 8 are obtained from the lone model with a single round wire carrying only the transport current It and exposed to no external magnetic field. The red curve in Fig. 8 is also drawn with the theo-retical expression (17), and have a good agreement with the corresponding numerical results with the lone model. Therefore, the theoretical expression (17) for the round wire is extended to the cases of multi-strand twisted cables as fol-lows. The multi-strand cable with N strands of 2rs in diame-ter is regarded as a homogenized single round wire with the identical outer diameter of 2rh, where the subscripts s and h denote the physical quantities for strand and homogenized removed and the infinite number of mirror images of wind-ings were arranged at even intervals of the slot width w0. It can be seen that the profiles of magnetic lines of force in Fig. 6(a) have a good agreement with those in Fig. 6(b). The magnetic field B0 generated by a single layer of horizontal array of mirror images is given byB0 = µ0Itw0,� (18)based on an analogy with those for an infinite solenoid. In this case, the magnetic fields generated by the windings gradually increase from zero at the deepest point within the slot to NB0 near the slot opening. In fact, however, it has to be mentioned that the reason why the losses for turns nearest to the slot opening in Fig. 5 become larger is that the mag-netic fields applied to them are enhanced somewhat larger than NB0 due to the shape effect around the slot opening.4  Numerical and Theoretical Reproductions of LossesIn order to reproduce the numerical results of losses obtained with the simple models shown in Fig.  5 numerically, the lone models with single cables carrying the transport cur-rent It and simultaneously exposed to an external magnetic field Be = βB0 are used. Only the cases of 5 turns within the iron core slots for the simple models are focused on Fig. 6  Typical profiles of magnetic lines of force (a) within iron core slot and (b) for infinite arrays of mirror images 1 3229  Page 8 of 12Journal of Superconductivity and Novel Magnetism (2025) 38:229wire with ρh = ρs/ϕ and Jh = ϕJs for the homogenized wire, respectively. In this case, the power dissipation P0 per unit length for low frequency can express aswire, respectively. The similar expression (17) for the homogenized wire can be used by replacing both the resis-tivity ρs and current density Js for the single solid round Fig. 7  Comparison among numerical results of losses per unit length for each turn in simple models, those for lone models and theoretical values with (a) single solid round wires, (b) 7-strand cables, (c) 19-strand cables and (d) 37-strand cables 1 3Page 9 of 12  229Journal of Superconductivity and Novel Magnetism (2025) 38:229Bi = µ0NiItw, i = 1, · · ·, m,� (22)and it is exposed to an external magnetic field ofB′i =i−1∑k=1Bk + Bi2, i = 1, · · ·, m.� (23)P0 = ρsJ2s NsSs = ρhJ2hSh,� (19)with the cross-sectional area Sh(= πr2h) of the homoge-nized wire. Figure 8 already shows the comparison between the calculated results of losses in the lone models and the theoretical curves proposed here. It is found that the pro-posed theoretical procedure for estimating the Joule loss in the multi-strand cables carrying the transport current has a good agreement with the corresponding numerical results. The small discrepancy between them at very high frequency might be due to the concentration of current around the cable surface based on the skin effect.Let us move on to theoretical estimation of the eddy-cur-rent loss Pe. The theoretical expression of eddy-current loss We per unit volume per cycle for an infinite slab with the thickness of 2D is given by [35]We = πB2m2µ0∆sinh(2∆) − sin(2∆)cosh(2∆) + cos(2∆)≃ 2πB2m3µ0∆2, ∆ ≪ 1,� (20)where Bm is the amplitude of external magnetic field and ∆ = D/λs. If the cross-sectional area 4D2 of a square with the sides 2D is equal to that for the strand, Ss, with the diameter of 2rs, the theoretical expression of the eddy-current loss Pe per unit length of the multi-strand cable can be obtained asPe = NsSsfWe = Ns6ρs(πSsfBm)2.� (21)It has been well known that the eddy-current loss is propor-tional to the second power of both the frequency f and field amplitude Bm, and inversely proportional to the resistivity ρs.Figure 7 already shows the comparison between the cal-culated results of losses in the lone models with the single cables in vacuum and the theoretical curves drawn with solid lines. It can be seen that the numerical results of losses obtained from the lone models can be well reproduced by simple sums of theoretical values for the Joule and eddy-current losses below characteristic frequencies.Figure 9 shows the comparison between the calculated results of losses with the stator models and the theoretical curves. The symbols represent the numerical results of total losses for all of the turns. The solid lines obtained from the proposed theoretical procedures almost agree with the numerical results. In this case, it is considered that each layer of horizontal array of cables generates a magnetic field on its lower side in Fig. 3(c) asFig. 9  Comparison between numerical results of total losses with stator models and theoretical values. The former are plotted with symbols, whereas the latter are drawn with solid lines Fig. 8  Comparison between numerical results of Joule losses with lone models and theoretical values. The former are plotted with symbols, whereas the latter are drawn with solid lines 1 3229  Page 10 of 12Journal of Superconductivity and Novel Magnetism (2025) 38:229References1.  Chen, J.M.: Carbon neutrality: Toward a sustainable future. The Innovation (2021). ​h​t​t​p​​s​:​/​​/​d​o​i​​.​o​​r​g​/​​1​0​.​1​​0​1​6​​/​j​.​​x​i​n​n​.​2​0​2​1​.​1​0​0​1​2​72.  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It is also found that the losses in the multi-strand cables at several hundred hertz are much smaller than those for the single round wires.5  ConclusionThe armature windings of HTS motors driven by voltage from a commonly used inverter should be constructed using metallic cables to manage both unbalanced three-phase cur-rents and frequency limitations. The losses in multi-strand metal twisted cables within one of iron core slots for arma-ture were calculated numerically using the finite element method. The obtained results were well reproduced using both another numerical analysis model in vacuum and theo-retical procedures to estimate the losses for transport cur-rent and external magnetic field. Additionally, the losses in the numerical analysis model simulating the actual armature were estimated using the finite element method, and almost agreed with theoretical predictions.Acknowledgements  This article is based on results obtained from a project, JPNP23004, subsidized by the New Energy and Industrial Technology Development Organization (NEDO).Author Contributions  All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by K.K., Y.M. and M.F. The first draft of the manuscript was written by K.K. and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript. Funding was originally planned, requested and secured by T.M., T.N., K.K. and A.K.Funding  Open Access funding provided by Sanyo-Onoda City Uni-versity.Data Availability  No datasets were generated or analysed during the current study.DeclarationsCompeting interests  The authors declare no competing interests.Open Access   This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adap-tation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. 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