France



1. Introduction&lt;br/&gt;Nowadays, Finite Element Model (FEM) is considered as the most accurate approach to design and study electrical machines as it is able to consider many physical phenomena. However, its computation time can constitute a real drawback when several machines have to be tested thus making analytical approaches more suitable. Analytical model that involves winding functions and airgap permeance remains one of the more accurate and widely used approach. It is based on different assumptions including that of a regular stator and rotor slot pitches. As such, analytical expressions leads to an accurate representation of the air gap permeance [1]–[3]. However, recent studies show that spatial harmonic content of the stator Magneto Motive Force (MMF) can be reduced using non-regular slot pitches [4], [5]. Therefore, to be able to quantify the interest of such proposition or even optimize the location of the slots with limited time calculation, an analytical approach which makes it possible to take into account any location of the slots is needed. To achieve that, a generic analytical expression to model the air gap permeance of electrical machines is proposed. It enables to accurately calculate the air gap permeance in all cases i.e. with regular and non-regular stator and/or rotor slot pitches. Results obtained using the proposed approach are compared to the ones given by FEM in the case of an example of slotless rotor electrical machine with regular and non-regular stator slot pitches.&lt;br/&gt;&lt;br/&gt;2. Analytical model permeance&lt;br/&gt;Quantifying the performance of an electrical machine is based on an accurate calculation of the magnetic flux density in the air gap whose permeance is the key quantity. To obtain the latter through a general analytical expression, the air gap can be split into three areas as showed in the Figure 1:&lt;br/&gt;- Mechanical air gap, &lt;i&gt;δ&lt;sub&gt;e&lt;/sub&gt;&lt;/i&gt;, of constant thickness all around the circumference of the machine&lt;br/&gt;- Additional air gap due to stator slots, &lt;i&gt;δ&lt;sub&gt;s&lt;/sub&gt;&lt;/i&gt;, which depends on &lt;i&gt;θ&lt;sub&gt;s&lt;/sub&gt;&lt;/i&gt;&lt;br/&gt;- Additional air gap due to rotor slots, &lt;i&gt;δ&lt;sub&gt;r&lt;/sub&gt;&lt;/i&gt;, which depend on &lt;i&gt;θ&lt;sub&gt;s&lt;/sub&gt;&lt;/i&gt; and &lt;i&gt;θ&lt;/i&gt;&lt;br/&gt;&lt;i&gt;θ&lt;/i&gt;&lt;i&gt;&lt;sub&gt;s&lt;/sub&gt;&lt;/i&gt; represents a given position in the airgap with respects to a fixed stator axis and &lt;i&gt;θ&lt;/i&gt; is the relative position of the rotor with respects to the stator. At steady state with a constant speed &lt;i&gt;Ω&lt;/i&gt;, &lt;i&gt;θ&lt;/i&gt;&lt;i&gt; = &lt;/i&gt;&lt;i&gt;Ω&lt;/i&gt;&lt;i&gt;.t&lt;/i&gt;. The total air gap thickness &lt;i&gt;g(θ&lt;sub&gt;s&lt;/sub&gt;, θ)&lt;/i&gt; is then the sum of these three air gaps allowing to obtain the permeance through the calculation of &lt;i&gt;g&lt;sup&gt;-1&lt;/sup&gt;(θ&lt;sub&gt;s&lt;/sub&gt;, θ)&lt;/i&gt;.&lt;br/&gt;Air gap permeance calculation is usually made considering one regular slot pitch but if the latter is non-regular, it is more relevant to consider the entire air gap circumference. In the proposed approach, each additional air gap due to a stator slot &lt;i&gt;i&lt;/i&gt; or rotor slot &lt;i&gt;j&lt;/i&gt; is represented by an elementary function &lt;i&gt;δ&lt;/i&gt;&lt;i&gt;&lt;sub&gt;s,i&lt;/sub&gt;&lt;/i&gt;&lt;i&gt;&lt;sup&gt;elem&lt;/sup&gt;&lt;/i&gt;&lt;i&gt;(&lt;/i&gt;&lt;i&gt;θ&lt;/i&gt;&lt;i&gt;&lt;sub&gt;s&lt;/sub&gt;&lt;/i&gt;&lt;i&gt;,α&lt;sub&gt;i&lt;/sub&gt;)&lt;/i&gt; or &lt;i&gt;δ&lt;/i&gt;&lt;i&gt;&lt;sub&gt;rj&lt;/sub&gt;&lt;/i&gt;&lt;i&gt;&lt;sup&gt;elem&lt;/sup&gt;&lt;/i&gt;&lt;i&gt;(&lt;/i&gt;&lt;i&gt;θ&lt;/i&gt;&lt;i&gt;&lt;sub&gt;s&lt;/sub&gt;&lt;/i&gt;&lt;i&gt;,&lt;/i&gt;&lt;i&gt;θ&lt;/i&gt;&lt;i&gt;,&lt;/i&gt;&lt;i&gt;α&lt;/i&gt;&lt;i&gt;&lt;sub&gt;j&lt;/sub&gt;&lt;/i&gt;&lt;i&gt;) &lt;/i&gt;where &lt;i&gt;α&lt;sub&gt;i&lt;/sub&gt;&lt;/i&gt; and &lt;i&gt;α&lt;sub&gt;j&lt;/sub&gt;&lt;/i&gt; are the spatial shift phases related to the positions of the stator slot i and the rotor slot j with respect to reference stator and rotor axes respectively.The superposition of the elementary functions of all the slots enables to obtain the complete functions of &lt;i&gt;δ&lt;/i&gt;&lt;i&gt;&lt;sub&gt;s&lt;/sub&gt;&lt;/i&gt;&lt;i&gt;(&lt;/i&gt;&lt;i&gt;θ&lt;/i&gt;&lt;i&gt;&lt;sub&gt;s&lt;/sub&gt;&lt;/i&gt;&lt;i&gt;)&lt;/i&gt; and &lt;i&gt;δ&lt;/i&gt;&lt;i&gt;&lt;sub&gt;r&lt;/sub&gt;&lt;/i&gt;&lt;i&gt;(&lt;/i&gt;&lt;i&gt;θ&lt;/i&gt;&lt;i&gt;&lt;sub&gt;s&lt;/sub&gt;&lt;/i&gt;&lt;i&gt;,&lt;/i&gt;&lt;i&gt;θ&lt;/i&gt;&lt;i&gt;) &lt;/i&gt;and hence the global air gap as &lt;i&gt;g(θ&lt;sub&gt;s&lt;/sub&gt;, θ) = δ&lt;sub&gt;e&lt;/sub&gt;&lt;/i&gt; + &lt;i&gt;δ&lt;/i&gt;&lt;i&gt;&lt;sub&gt;s&lt;/sub&gt;&lt;/i&gt;&lt;i&gt;(&lt;/i&gt;&lt;i&gt;θ&lt;/i&gt;&lt;i&gt;&lt;sub&gt;s&lt;/sub&gt;&lt;/i&gt;&lt;i&gt;)&lt;/i&gt;+ &lt;i&gt;δ&lt;/i&gt;&lt;i&gt;&lt;sub&gt;r&lt;/sub&gt;&lt;/i&gt;&lt;i&gt;(&lt;/i&gt;&lt;i&gt;θ&lt;/i&gt;&lt;i&gt;&lt;sub&gt;s&lt;/sub&gt;&lt;/i&gt;&lt;i&gt;,&lt;/i&gt;&lt;i&gt;θ&lt;/i&gt;&lt;i&gt;)&lt;/i&gt;.&lt;br/&gt;&lt;br/&gt;3. Application&lt;br/&gt;In order to highlight the accuracy of the proposed model, a simple structure, i.e. slotless rotor, has been chosen. The proposed approach has then been used in the case of regular (Figure 2.a) and non-regular (Figure 2.b) slot pitch cases as proposed in the literature [4], [5]. The comparison between the magnetic flux density in the air gap obtained by both approaches under the assumption of linear behavior of the magnetic material are given in the Figure 2.c and 2.d. Results are very similar and show the accuracy of the proposed analytical expression of the air gap permeance.&lt;br/&gt;&lt;br/&gt;In the extended version, the analytical calculation of the air gap permeance will be presented in detail. Then, the proposed approach will be used in the case of non-regular rotor slot pitch and in the case of both non-regular stator and rotor slot pitches. Results will be given and discussed.

**References:**

[1] D. Zarko, D. Ban, et T. A. Lipo, « Analytical calculation of magnetic field distribution in the slotted air gap of a surface permanent-magnet motor using complex relative air-gap permeance », &lt;i&gt;IEEE Transactions on Magnetics&lt;/i&gt;, vol. 42, n&lt;sup&gt;o&lt;/sup&gt; 7, p. 1828-1837, juill. 2006, doi: 10.1109/TMAG.2006.874594.&lt;br/&gt;[2] Z. Q. Zhu, D. Howe, E. Bolte, et B. Ackermann, « Instantaneous magnetic field distribution in brushless permanent magnet DC motors. I. Open-circuit field », &lt;i&gt;IEEE Transactions on Magnetics&lt;/i&gt;, vol. 29, n&lt;sup&gt;o&lt;/sup&gt; 1, p. 124-135, janv. 1993, doi: 10.1109/20.195557.&lt;br/&gt;[3] B. Gaussens, E. Hoang, O. de la Barriere, J. Saint-Michel, M. Lecrivain, et M. Gabsi, « Analytical Approach for Air-Gap Modeling of Field-Excited Flux-Switching Machine: No-Load Operation », &lt;i&gt;IEEE Transactions on Magnetics&lt;/i&gt;, vol. 48, n&lt;sup&gt;o&lt;/sup&gt; 9, p. 2505-2517, sept. 2012, doi: 10.1109/TMAG.2012.2196706.&lt;br/&gt;[4] J. Marault, F. Gillon, M. Hecquet, et A. Tounzi, « Optimization of the MMF spatial harmonic content to design electrical machine winding », &lt;i&gt;International Journal of Applied Electromagnetics and Mechanics&lt;/i&gt;, vol. Preprint, n&lt;sup&gt;o&lt;/sup&gt; Preprint, p. 1-16, janv. 2020, doi: 10.3233/JAE-209508.&lt;br/&gt;[5] N. Bekka, M. E. H. Zaïm, N. Bernard, et D. Trichet, « A Novel Methodology for Optimal Design of Fractional Slot With Concentrated Windings », &lt;i&gt;IEEE Transactions on Energy Conversion&lt;/i&gt;, vol. 31, n&lt;sup&gt;o&lt;/sup&gt; 3, p. 1153-1160, sept. 2016, doi: 10.1109/TEC.2016.2552546.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/3648c4c383f1913c4405fe942990185c.jpg)
Fig 1. Air gap permeance function

![](https://assets.underline.io/uploads/markdown_image/1/image/a457905edd7422a022008b7f3cdcb912.jpg)
Fig 2. 18 slots, 10 poles simplified structure, (a) cross section with regular slot pitch, (b) ) cross section with non-regular slot pitch, (c) air gap flux density waveform for regular slot pitch, (d) air gap flux density waveform for non-regular slot pitch

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 A general analytical expression to model the air gap permeance of electrical machines

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1. Introduction Nowadays, Finite Element Model (FEM) is considered as the most accurate approach to design and study electrical machines as it is able to consider many physical phenomena. However, its computation time can constitute a real drawback when several machines have to be tested thus making analytical approaches more suitable. Analytical model that involves winding functions and airgap permeance remains one of the more accurate and widely used approach. It is based on different assumptions including that of a regular stator and rotor slot pitches. As such, analytical expressions leads to an accurate representation of the air gap permeance [1]–[3]. However, recent studies show that spatial harmonic content of the stator Magneto Motive Force (MMF) can be reduced using non-regular slot pitches [4], [5]. Therefore, to be able to quantify the interest of such proposition or even optimize the location of the slots with limited time calculation, an analytical approach which makes it possible to take into account any location of the slots is needed. To achieve that, a generic analytical expression to model the air gap permeance of electrical machines is proposed. It enables to accurately calculate the air gap permeance in all cases i.e. with regular and non-regular stator and/or rotor slot pitches. Results obtained using the proposed approach are compared to the ones given by FEM in the case of an example of slotless rotor electrical machine with regular and non-regular stator slot pitches. 2. Analytical model permeance Quantifying the performance of an electrical machine is based on an accurate calculation of the magnetic flux density in the air gap whose permeance is the key quantity. To obtain the latter through a general analytical expression, the air gap can be split into three areas as showed in the Figure 1: - Mechanical air gap, δe, of constant thickness all around the circumference of the machine - Additional air gap due to stator slots, δs, which depends on θs - Additional air gap due to rotor slots, δr, which depend on θs and θ θs represents a given position in the airgap with respects to a fixed stator axis and θ is the relative position of the rotor with respects to the stator. At steady state with a constant speed Ω, θ = Ω.t. The total air gap thickness g(θs, θ) is then the sum of these three air gaps allowing to obtain the permeance through the calculation of g-1(θs, θ). Air gap permeance calculation is usually made considering one regular slot pitch but if the latter is non-regular, it is more relevant to consider the entire air gap circumference. In the proposed approach, each additional air gap due to a stator slot i or rotor slot j is represented by an elementary function δs,ielem(θs,αi) or δrjelem(θs,θ,αj) where αi and αj are the spatial shift phases related to the positions of the stator slot i and the rotor slot j with respect to reference stator and rotor axes respectively.The superposition of the elementary functions of all the slots enables to obtain the complete functions of δs(θs) and δr(θs,θ) and hence the global air gap as g(θs, θ) = δe + δs(θs)+ δr(θs,θ). 3. Application In order to highlight the accuracy of the proposed model, a simple structure, i.e. slotless rotor, has been chosen. The proposed approach has then been used in the case of regular (Figure 2.a) and non-regular (Figure 2.b) slot pitch cases as proposed in the literature [4], [5]. The comparison between the magnetic flux density in the air gap obtained by both approaches under the assumption of linear behavior of the magnetic material are given in the Figure 2.c and 2.d. Results are very similar and show the accuracy of the proposed analytical expression of the air gap permeance. In the extended version, the analytical calculation of the air gap permeance will be presented in detail. Then, the proposed approach will be used in the case of non-regular rotor slot pitch and in the case of both non-regular stator and rotor slot pitches. Results will be given and discussed.

**References:**

[1] D. Zarko, D. Ban, et T. A. Lipo, « Analytical calculation of magnetic field distribution in the slotted air gap of a surface permanent-magnet motor using complex relative air-gap permeance », IEEE Transactions on Magnetics, vol. 42, no 7, p. 1828-1837, juill. 2006, doi: 10.1109/TMAG.2006.874594. [2] Z. Q. Zhu, D. Howe, E. Bolte, et B. Ackermann, « Instantaneous magnetic field distribution in brushless permanent magnet DC motors. I. Open-circuit field », IEEE Transactions on Magnetics, vol. 29, no 1, p. 124-135, janv. 1993, doi: 10.1109/20.195557. [3] B. Gaussens, E. Hoang, O. de la Barriere, J. Saint-Michel, M. Lecrivain, et M. Gabsi, « Analytical Approach for Air-Gap Modeling of Field-Excited Flux-Switching Machine: No-Load Operation », IEEE Transactions on Magnetics, vol. 48, no 9, p. 2505-2517, sept. 2012, doi: 10.1109/TMAG.2012.2196706. [4] J. Marault, F. Gillon, M. Hecquet, et A. Tounzi, « Optimization of the MMF spatial harmonic content to design electrical machine winding », International Journal of Applied Electromagnetics and Mechanics, vol. Preprint, no Preprint, p. 1-16, janv. 2020, doi: 10.3233/JAE-209508. [5] N. Bekka, M. E. H. Zaïm, N. Bernard, et D. Trichet, « A Novel Methodology for Optimal Design of Fractional Slot With Concentrated Windings », IEEE Transactions on Energy Conversion, vol. 31, no 3, p. 1153-1160, sept. 2016, doi: 10.1109/TEC.2016.2552546.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/3648c4c383f1913c4405fe942990185c.jpg)
Fig 1. Air gap permeance function

![](https://assets.underline.io/uploads/markdown_image/1/image/a457905edd7422a022008b7f3cdcb912.jpg)
Fig 2. 18 slots, 10 poles simplified structure, (a) cross section with regular slot pitch, (b) ) cross section with non-regular slot pitch, (c) air gap flux density waveform for regular slot pitch, (d) air gap flux density waveform for non-regular slot pitch

poster

Research into geometrical frustration over the past two decades has revealed fundamentally new exotic behavior [1]. The magnetic interactions of the structure, coupled with the specific geometry causes frustration in the system, referred to as artificial spin ices. Several geometries exist, but for the purpose of this research, the kagome arrangement was used. In general, the geometry of these frustrated systems follows the ice rule, that being two interacting moments point in while two point out. The kagome however follows the modified ice rule of two moments pointing in while one points out or vice versa. These systems appear as 2D analogues to the 3D structure and it is the lattice configuration along with the specific frustration created that causes these exotic behaviors. Presented here are the magnetoresistances obtained for kagome magnetically frustrated systems in a single vertex and lattice format. Both geometries are compared in a magnetically connected format and a magnetically disconnected format in order to observe the exchange and dipolar energies. The structures were simulated with parameters for permalloy, using BORIS [2], and dimensions of 1µm long, 100nm wide and 10nm thick. The aim was to evaluate the magneto resistance, which was done assuming the current density aligns with the long axis and a basic resistor network. By comparing the simulated data for the connected and disconnected single vertex with the measurements taken previously for a lattice, a greater understanding of the effect of the disconnected versus connected structures can be achieved. Measurements of the kagome lattice showed that the disconnected lattice had a change in magnetoresistance that was 2~3 times greater than the connected lattice, experimentally. The simulations of the single vertex’s magnetoresistance differ by approximately an order of magnitude to those of the experiments. While the simulations are not for the full lattice, they shed some light on the experimental results obtained and the single vertex interactions that occur within the lattice. The simulation of the hybrid vertex appears to mirror that of the experimental data obtained for the hybrid lattice while the simulation of the connected vertex appears to differ somewhat from the experimental data obtained for the connected lattice. This highlights the implications of the experimental work in that the connected kagome has more complex interactions throughout the lattice while the magnetically disconnected hybrid is rather a simplified system. The subtle difference in the geometries of the connected and disconnected lattices aids in the explanation of the effects and interplay of both the spin structures and the current density. This work demonstrates the potential for using simulations to expose magnetic interactions in a lattice type structure and other interesting geometries, along with the enhanced magnetoresistance signals that were obtained.

**References:**

References: [1] Nisoli, C., Moessner, R., & Schiffer, P. (2013). Colloquium: Artificial spin ice: Designing and imaging magnetic frustration. Reviews of Modern Physics, 85(4), 1473. [2] Lepadatu, Serban. (2019). Boris Computational Spintronics User Manual.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/bc02a8200bfe4ba73c8d18a80397bf3d.jpg)
Figure 1. SEM images of an electrically connected but magnetically disconnected (hybrid) lattice (left). Also shown is the standard honeycomb lattice for comparison.

![](https://assets.underline.io/uploads/markdown_image/1/image/65475fb3f43153712262934a633adf6d.jpg)
Figure 2: Simulated single vertex and experimental lattice for the magnetically disconnected, electronically connected (hybrid) kagome plotted against the applied magnetic field. Data shown is the positive part of a full sweep of the magnetic field i.e. from a positive applied field through to the negative of the same applied field.

Micromagnetic simulations of artificial spin ice lattices and vertices

The micromagnetic configuration of a ferromagnetic sample results from the balance between different energy terms, in particular the magnetic anisotropy, the exchange energy and the magnetostatic energy. A very large variety of domain structures is observed depending on the intrinsic properties of the ferromagnetic material (anisotropy, exchange, spontaneous magnetization) and on its shape. Of particular interest are domains presenting a cylindrical symmetry that appear in thin films or patterned cylindrical micro/nanostructures. As examples, magnetic bubble domains have been extensively studied in micron-thick rare-earth orthoferrites films where they were considered for magnetic recording applications. A theory linking the static stability of these structures to both the material properties and sample geometry was developed, highlighting the requirement that the uniaxial anisotropy must overcome the stray field energy in order to stabilize the bubble. Stable cylindrical target domains have also been observed in a variety of patterned magnetic thin films, characterized by multiple nested concentric magnetic rings. Such micromagnetic configurations have been reported in systems with out-of-plane magnetization such as high magnetic anisotropy Co and FePt films [1] or perpendicular anisotropy multilayers [2,3], as well as in low anisotropy NiFe and Ni films [4,5] in which the magnetization is mostly in-plane with a weak out-of-plane component. Several numerical simulation studies have been undertaken to investigate these magnetic configurations [1-3,5-7]. Most of these simulations assumed the presence of a strong perpendicular anisotropy. Numerical simulations with low anisotropy systems have received less attention [4,5], and still require to consider a rather large perpendicular uniaxial anisotropy - attributed to built-in magnetostriction or resulting from a columnar grain structure [8] - to explain the formation of bubble or target domains. In this paper, we present experimental and numerical results obtained on patterned micron-size permalloy disks showing magnetic target domains. We conducted a thorough investigation of the evolution of the magnetic structure as a function of the Py thickness. Arrays of microdisks were patterned by optical lithography, e-beam evaporation and lift-off step. Disks diameter is constant at 1.3 µm and the thicknesses was varied from 90 nm to 400 nm. The MFM imaging of the remanent magnetization for disks in the as deposited state are shown in Figure 1. Thinner disks exhibit the expected vortex structure (Fig. 1a, b). For thicker disks, imaging reveals a structure with spiraling or concentric ring domains with alternating, up and down, perpendicular component of the magnetization (Fig. 1c - 1h). Micromagnetic numerical simulation have been carried out where we consider that the disks exhibit a columnar magnetic grain structure with variable inter-grain coupling and, in contrast with previous calculations, no contribution from magnetocrystalline or intrinsic uniaxial anisotropies. In our case, the inter-grain coupling intensity σ allows for the modulation of the demagnetizing field energy, giving rise to various magnetic configurations that will be described in details (Fig. 2). It is found that, for the thinnest disks, despite the columnar structure, the most stable magnetic structure is the vortex structure. Above a critical thickness the lowest energy structure evolves into a target domain structure, with a number of concentric domains that decreases as the thickness increases. It is also observed that, in addition to the target structure, less symmetrical spiral or disordered metastable structures can be observed. These observations closely match the experimental results for disks with similar aspect ratio, indicating that, together with σ, these are the two governing parameters in this case. We will also present details of the different energy contributions and describe the internal structure of the magnetization. This model based on a columnar texture of the film constitutes an alternative to explain the formation of target domain structures without relying on the existence of a perpendicular anisotropy difficult to justify in materials such as permalloy.

**References:**

[1] C. Moutafis, S. Komineas and C. A. F. Vaz, Phys. Rev. B 76, 104426 (2007). [2] F. Garcia, H. Westfahl and J. Schoenmaker, Appl. Phys. Lett. 97, 022501 (2010). [3] T. Liu, V. Puliafito and F. Montaigne, J. Phys. D: Appl. Phys. 49, 245002 (2016). [4] P. Eames and E. Dan Dahlberg, J. Appl. Phys. 91, 7986 (2002). [5] G. D. Skidmore, A. Kunz and C. E. Campbell, Phys. Rev. B 70, 012410 (2004). [6] J. K. Ha, R. Hertel and J. Kirschner, Europhys. Lett. 64, 810 (2003). [7] S. Komineas, C. A. F. Vaz and J. A. C. Bland, Phys. Rev. B 71, 060405 (2005). [8] S. Finizio, S. Wintz and D. Bracher, Phys. Rev. B 98, 104415 (2018).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/45753e3a6b6a4c36a98fc4ede0ad4c75.jpg)
Fig. 1: MFM images of permalloy disks with 1.3 µm diameter, in the as-deposited state, with different thicknesses. (a, b) Vortex structure; (c, e) Four concentric ring domains structure; (d, f) Spiraling domains structure; (g, h) Three concentric ring domains structure.

![](https://assets.underline.io/uploads/markdown_image/1/image/f9c8a31341f86902d48f4d2469dc767b.jpg)
Fig. 2: (a) Calculated magnetic structure in the median plane of the disk (plane at half thickness), as a function of the thickness and σ coupling parameter. (b) Examples of metastable configurations with disordered, spiral or incomplete ring domains, obtained with σ = 0.1 and 180 nm thickness (top); σ = 0.2 and 150 nm (middle); and σ = 0.1 and 150 nm (bottom). The red/blue colors refer to the ±Mz magnetization component.

Concentric target domains in magnetic nanodisks

Recent advances on the stabilization and manipulation of chiral magnetic configurations hosted by systems that consist in alternating atomic layers of ferromagnetic and non-magnetic materials hold promise of innovative spintronic devices. The low dimensionality of the systems promotes spin-orbit driven interfacial effects[1] like the antisymmetric Dzyaloshinskii-Moriya interaction (DMI) and interfacial magnetic anisotropy (iMAE), whose relative strengths may be tuned to achieve stable nanometer sized magnetic objects with fixed chirality called magnetic skyrmions[2], which are proposed as information carriers in future spin-orbitronics technologies. While in most of the cases this is obtained by engineering complex multilayers stacks in which interlayer dipolar fields become relevant[3], we consider here a simple epitaxial trilayer in which a ferromagnet (namely Cobalt, Co), with variable thickness, is embedded between a heavy metal and graphene[4]. The latter enhances the perpendicular magnetic anisotropy of the system, promotes a Rashba-type DMI, and can sustain very long spin diffusion length. The work consists in the development of a layer-resolved micromagnetic model capable to account for the interfacial nature of the interactions[5], which leads to macroscopic parameters that depend on the thickness of the ferromagnetic layer. We demonstrate that our model correctly reproduces the experimental magnetization configurations and the spin reorientation transition (SRT) of a gr/Co(tCo)/Pt sample[4]. Next, we study the stabilization of magnetic skyrmions in patterned magnetic nanodots with a diameter of 256 nm. In particular, we analyze the thickness-dependent stability of the skyrmionic state in contrast with a single domain state. Our results demonstrate that, for samples with Co thicknesses larger than 3.6 nm, intrinsic mixed Bloch-Néel (predominantly Bloch-type) skyrmions are stabilized in 256 nm wide dots. For thicknesses larger than ~5 nm, the results show that the geometrical confinement imposed by the nanodot promotes the stabilization of chiral spin textures similar to skyrmions but with complex and fluctuating domain walls.

**References:**

[1] A. Soumyanarayanan, N. Reyren, A. Fert and C. Panagopoulos, Nature, Vol. 539, p. 509-517 (2016) [2] A. Fert, N. Reyren and V. Cros, Nature Reviews Materials, Vol. 2, p. 170301 (2017) [3] W. Legrand, J.-Y. Chauleau, D. Maccariello et al., Science Advances, Vol. 4(7), p. eaat0415 (2018) [4] F. Ajejas, A. Gudín, R. Guerrero et al., Nano Letters, Vol. 18(9), p. 5364-5372 (2018) [5] P. Olleros-Rodríguez, R. Guerrero, J. Camarero et al., Applied Materials and Interfaces, Vol. 12(22), p. 25419-25427 (2020)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/b0a17000d2dcf3624f3186e212665bbe.jpg)
Fig.1 Sketch of modelled magnetic nanodots at selected cobalt thicknesses. The stabilized spin textures are represented by the arrows at the top of the structures. The color of the arrows represents the out-of-plane component of the magnetization. The color of the surface represents the X-component of the magnetization.

Micromagnetic Study of Stable Skyrmions in Dot-Patterned Graphene-Based Magnetic Trilayers

Magnetic skyrmions in ferromagnetic materials are mainly stabilized by the interplay of several magnetic energy contributions, i.e. exchange, dipole, anisotropy, and Zeeman. In particular, the asymmetric Dzyaloshinskii-Moriya exchange interaction (DMI) introduces chiral canting between neighboring spins and favors skyrmion stability [1]. Skyrmions are particle-like chiral spin textures found in magnetic films and patterned nanostructures with out-of-plane anisotropy and are considered to be potential candidates as information carriers in next-generation data storage devices [2]. The challenge of the experimental study of skyrmions is to find an efficient technique for forming stable skyrmions. In particular, the outstanding issues are the reproducible generation, stabilization, and confinement of skyrmions at room temperature [2,3,4]. It is well known that various magnetic textures can be stabilized by geometrical confinement using artificial nanostructures, however, there are several key challenges like reproducible skyrmion formation that still need to be addressed. We present two methods of skyrmions formation in nanodots and antidots arrays. For these first structures, we have shown the formation of skyrmion using the tip of a magnetic force microscope [3], and during the remagnetization process for antidots arrays [4]. Here, we present the results of the investigation of a skyrmion formation in Pt/Co/Au nanodots. We demonstrate that the high magnetic momentum probe induces individual skyrmions during the scan. A specific path of the MFM tip causes a systematic change in the state of the magnetic domains during scanning. The process of annihilation of remaining domains is ruled by magnetostatic interactions where the magnetic domains are simultaneously repelled from the edges of the nanodots and from the tip field. Micromagnetic simulations have shown that specific conditions restricting the movement of magnetic domains must be met in order to form the magnetic skyrmion [3]. In this work, we present a method to form nanometer-sized magnetic skyrmions in an array of magnetic topological defects in the form of an antidot lattice. In particular, the introduction of the holes in a thin magnetic film modifies the spatial distribution of the effective field, which has a profound influence on the formation mechanism of the skyrmions. The position of skyrmion formation was recognized at the saddle point of the lattice as a result of the spatial distribution of the effective field. With micromagnetic simulations, we elucidate the steps of the skyrmion formation process and showed that this process depends on the antidot lattice parameters. This behavior is confirmed with scanning transmission x-ray microscopy measurements [4]. The work was supported by National Science Centre of Poland, Projects Nos. UMO-2017/27/N/ST3/00419 and UMO-2018/30/Q/ST3/00416.

**References:**

[1] Fert et al. "Magnetic skyrmions: advances in physics and potential applications", Nat. Rev. Mater. 2, 17031 (2017). [2] C Back et al. "The 2020 skyrmionics roadmap", J. Phys. D: Appl. Phys. 53 363001 (2020). [3] Iu. V Vetrova et al. "Skyrmion formation in nanodisks using magnetic force microscopy tip", arXiv preprint arXiv:2002.11685. [4] S. Saha et al. "Formation of Néel-type skyrmions in an antidot lattice with perpendicular magnetic anisotropy", Phys. Rev. B 100, 144435.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/f61ffb0cc45868d21b91dd067ad9f3f4.jpg)
Fig. 1: Simulated images of the evolution of the magnetic configuration in nanodots for random initial state and different positions of the tip field. x (nm) Simulations were performed for 400 nm disks with 4 repetition of the stack with DMI=1.0 mJ/m 2 . The red circle represents the z-component of the magnetic field induced by the high-momentum tip. The amplitude of tip field was 200 mT at center of the spot, and spot diameter was equal to 25 nm. The colors represent the orientation of the magnetization.

Néel type skyrmions formation in nanodots and antidot arrays.

Skyrmions in magnetic thin films are solitonic excitations that could open new possibilities for spintronic devices. The recent successes obtained in ferromagnetic films have shown some limitations, with dipolar-coupling induced large skyrmion sizes and topology induced gyrotropic deflection. Antiferromagnetic films and their artificial equivalents that are ferrimagnets and synthetic antiferromagnets could overcome these limitations. In this talk, we explore the skyrmion dynamics in such materials. Separating them into two antiferromagnetically coupled sublattices, we show that the dynamics can be described by two coupled Thiele equations. This framework is used to address the consequences of the small skyrmion size and the angular momentum compensation. It provides a simple approach to describe this problematic. Small skyrmions, that are possible due to the vanishing total magnetization, are shown to be significantly slower (velocity proportional to radius) than larger ones (size independent velocity). This result is a direct consequence of the skyrmion topology, which induces an excess of dissipation per size unit at small sizes. Angular momentum compensation is shown to cancel the gyrotropic deflection, without the need to modify the topology and maximizes the velocity. In a track, where the edge repulsion cancels the gyrotropic deflection, the angular momentum compensation has no effect and skyrmions travel faster that non confined skyrmions. From an applied perspective, the data rate is given by the velocity to radius ratio and correspond to the inverse of the time needed for a skyrmion to jump from one site to another. This shows that small skyrmions in ferrimagnetic films would still be preferable to large ones, providing fast and dense devices.

**References:**



**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/255c7d94b2bc6cbde230f79959b9396a.jpg)
(a) Variation of the skyrmion velocity in a CoGd thin film and (b) variation of the skyrmion velocity to skyrmion radius ratio (proportionnal to an application data rate), when the skyrmion is confined in a track, as a function of the alloy compostion, close to the angular momentum (LS) composition (79.4%), and for different DMI values. The points are the results of simulations, and the lines are the result of the model. Note that the skyrmion size increases with DMI at a given composition, and that the size is minimual at the magnetic moment compensation (77.5%).

Smaller and faster skyrmions in compensated ferrimagnets and synthetic antiferromagnets

In this last decade, experimental results and their understanding have led to magnetic skyrmions becoming promising candidates for the storage and processing of digital information [1]. However, some key barriers such as reducing their size, enhancing their stability at room temperature, or their low-power manipulation under electrical current have not yet been overcome, meaning that they are rarely integrated in actual devices [2]. To surmount these limitations, it has been proposed to study these magnetic textures in antiferromagnets or synthetic antiferromagnets (SAFs) instead of the conventional ferromagnetic layers that are usually considered [3,4]. In this more complex system, multiple ferromagnetic layers are coupled antiferromagnetically through a non-magnetic spacer layer via the RKKY interaction (Fig a) [4,5]. By switching to these systems, two crucial points need to be addressed. First, a drastic reduction of stray fields is expected in SAFs due to the anti-alignment of magnetization, which should reduce the size of the skyrmions [6]. Second, it is not possible to manipulate a conventional skyrmion in the driving force direction in a ferromagnet due to its topology that intrinsically induces a deflection known as the skyrmion Hall effect [7]. In SAF systems, since the magnetisation of the skyrmions in each magnetic layer are opposite, it is expected that the two deflections cancel each other out and the two skyrmions move in the current direction [3]. Even if skyrmions have recently been observed in SAFs [4,8,9], a clear and simple micromagnetic description of both the static and dynamic properties of such skyrmions is so far lacking to further understand this phenomenon. In order to improve the description of these properties and highlight the expected benefits of using SAFs rather than conventional ferromagnets, we simulate numerically the behavior of skyrmions in bilayer SAFs for a large range of parameters. We also adapt the ferromagnetic analytical formalism [10] to SAF systems. This leads to results that are in good agreement with the numerical simulations whilst also highlighting the underlying mechanisms. First, we investigate the stability region of SAF skyrmions and how their radii r evolve with micromagnetic parameters. We show that they are only slightly smaller than for ferromagnetic systems (Fig b), but that they exist for a larger range of parameters. Second, we study the dynamics of these skyrmions under spin currents. We find, as expected, that SAF skyrmions move at a faster velocity v than in a conventional ferromagnet (Fig c) for a given current density J, and move in the applied current direction (Fig d) with zero skyrmion Hall angle θSky. By using the analytical model based on the Thiele equation, we are able to recover these dynamics (Fig c and d) and identify the required condition for the vanishing of the topological deflection. Finally, we unbalanced the SAF by asymmetrizing the skyrmions in each layer in different ways in order to verify the condition for vanishing deflection. The results obtained highlight the qualitative benefits of SAF skyrmions and propose some methods for more quantitative optimizations.

**References:**

[1] The 2020 skyrmionics roadmap: C. Back et al., J. Phys. D: Appl. Phys. 53 (2020) [2] Magnetic skyrmions: advances in physics and potential applications: A. Fert, et al. Nat. Rev. Mat. 2, 17031 (2017) [3] Static and dynamical properties of antiferromagnetic skyrmions in the presence of applied current and temperature: J. Barker, et al. Phys. Rev. Lett. 116, 147203 (2016) [4] Room-temperature stabilization of antiferromagnetic skyrmions in synthetic antiferromagnets. W. Legrand et al., Nat. Mat. 19, 34–42 (2020) [5] Oscillatory magnetic exchange coupling through thin copper layers: S. S. P. Parkin, et. al . Phys. Rev. Lett. 66, 2152–2155 (1991) [6] Theory of isolated magnetic skyrmions: from fundamentals to room temperature applications: F.Büttner, et al,. Sci. Rep. 8 4464 (2018) [7] Nucleation, stability and current-induced motion of isolated magnetic skyrmions in nanostructures: J. Sampaio, Nat. Nano. 8, 839-844 (2013) [8] Imaging non-collinear antiferromagnetic textures via single spin relaxometry: A. Finco et al., arXiv:2006.13130v1 (2020) [9] Exploring different facets of magnetic skyrmion nucleation and dynamics in ultra-thin films: R. Juge, PhD Thesis, Université Grenoble-Alpes (2020) [10] The skyrmion-bubble transition in a ferromagnetic thin film: A. Bernand-Mantel et.al. SciPost Phys. 4, 027 (2018)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/060a538fb9d9988c2db214e60cb7057e.jpg)
(a) Sketch of the skyrmion structure and motion under current in ferromagnetic and SAF systems. Radii (b), velocity (c) and deflection (d) for a ferromagnetic skyrmion (in blue) and a SAF skyrmion (in red) with the same micromagnetic parameters versus the current density J. Points correspond to the micromagnetic numerical simulations and lines to the analytical model.

 Statics and Dynamics of Skyrmions in Synthetic Antiferromagnets : Benefits and Micromagnetic Understanding

Magnetic skyrmions are chiral, topologically-protected magnetic spin textures known for their rich physics and potential for future data storage devices. Recently, magnetic multilayers with interfacial Dzyaloshinskii-Moriya interactions (DMI) have attracted great attention because they can support stable Néel-type magnetic skyrmions at room temperature. Here, we report on micromagnetic simulations of the effect of initial magnetic field treatment on skyrmion formation in [Pt/Co/Ir]n magnetic multilayers. The multilayer system was modeled with MuMax3 using parameters determined from experimental measurements on [Pt/Co/Ir]n=3,5,8 multilayer thin films. The simulations show that the application and subsequent removal of a nearly in-plane magnetic field can transition the originally labyrinth domains into bubble skyrmions, which agrees well magneto-optical Kerr effect imaging results. Systematic simulations of the magnetic configuration of the [Pt/Co/Ir]n as a function of angle and magnitude of the applied magnetic field reveal that the formation of bubble skyrmions occurs for field magnitudes that are close to but below the in-plane saturating field and that an additional out-of-plane component will help the skyrmion formation. Skyrmion formation is observed over a wider magnetic field range as the repetition number is increased, which corresponds to increasing importance of the demagnetization field coupling. Work at Bryn Mawr College was supported by the National Science Foundation CAREER DMR-1053854 and DMR-1708790. Work at Colorado State University was supported by the National Science Foundation DMR-1709525 and work at Argonne National Laboratory was supported by the U.S. Department of Energy, Office of Science, Materials Science and Engineering Division.

**References:**



**Images:**

 The effects of field history on magnetic skyrmion formation in [Pt/Co/Ir]n multilayers.

Heat Assisted Magnetic Recording (HAMR) technology has reached an areal density of over 2Tb/in2, but the Signal to Noise Ratio (SNR) as well as the overall recording quality clearly decreases with the recording bit length (BL). In HAMR, recording patterns typically have a crescent shape due to the moving heat spot. The curvature of this crescent shape is of great significance as it is a significant source of transition broadening of the readback signal[1][2]. Various methods have been applied to determine and modify this curvature for better recording performance[3][4][5]. Thus, investigating how this curvature varies with different bit lengths and studying the change of this curvature during the HAMR writing process may help improve HAMR performance at the highest linear densities. In this digest, a detailed study of the dependence of curvature in HAMR transitions on bit length is presented. With the help of micromagnetic simulations, HAMR transition patterns for three different bit lengths, 10.5nm, 15nm and 21nm, were obtained and analyzed. For each bit length, 128 bits were written on multiple media and overlapped. Curvatures were then calculated using these averaged patterns. AC-erased thermal Exchange Coupling Composite (ECC) media was used, where a superparamagnetic layer with high Curie temperature forms the writing layer and a doped FePt layer with relatively low Curie temperature serves as the storage layer[6]. The change in curvature during the cooling process was also tracked for pure FePt media. Curvatures of DC-erased thermal ECC media were tested as well for comparison. Introduction of intergranular exchange was also implemented to find potential optimizations of HAMR system designs. Micromagnetic simulations in this study are guided by the Landau-Lifshitz-Gilbert equation. Renormalization group theory is applied to enable inclusion of atomic level phenomena[7]. Media dimension is 384nm x 48nm x 9nm. For a certain bit length, the 128 written bits were overlapped and averaged, as shown in Fig. 1. The left edge of the -1 bit (blue-colored) was chosen for curvature calculation. This curve was fitted to a parabolic curve and the reciprocal value of its second order coefficient was taken as its curvature. In HAMR, the cooling process could potentially modify the written bits as magnetic properties of the media materials, such as saturation magnetization and magneto-crystalline anisotropy, change greatly with temperature. Curvatures before and after the cooling process for both AC-erased thermal ECC media and pure FePt media with BL10.5nm, 15nm and 21nm are presented in Fig. 2. For thermal ECC media, curvatures before the cooling process for all the three bit lengths were close to each other, as shown with the cyan solid line in Fig. 2. After the bits were cooled down to room temperature, a decrease in curvature was seen for both BL10.5nm and 15nm while no obvious curvature change was observed for BL21nm. The BL10.5nm curvature experienced a bigger drop than the BL15nm curvature. This means that curvatures are reduced more during the cooling process for smaller bit lengths. The reason lies in the distance between the bit edge and the heat spot. When a -1 bit is just written, the left edge of this bit is about twice the bit length away from the heat spot. For BL10.5nm case, this distance is 21nm, which is much smaller than that for BL15nm and BL21nm cases. This implies that for BL10.5nm case, this bit edge is under a much higher temperature and spins here have a greater chance to be switched due to their reduced anisotropy fields. When the magnetic field switches as the head moves on, some spins at the center of this left bit edge area would get flipped, which will flatten the curve and thus cause the curvature to decrease. This leads to the interesting conclusion that the enhanced erase-after-write found in the thermal ECC media is actually beneficial owing to the resulting reduction in curvature. FePt media shows a very similar erase-after-write effect, with the 10.5 nm bits showing before and after cooling curvatures very close to the thermal ECC media. Interestingly, the 21 nm bits, while showing reduced erase-after-write as expected, have overall larger values of the curvature in comparison to the thermal ECC media, possibly owing to the differing magnetostatics. Curvatures for DC-erased thermal ECC media are found to be larger than those for AC-erased media in all the three BL cases. This is because the +1 background of DC-erased media provides stronger dipole interaction than the averaged 0 background of AC-erased media. Spins have a higher switching probability, especially in the middle of the track. For AC-erased thermal ECC media, when intergranular exchange was introduced, a decrease in curvature was seen in all the three BL cases. This results from the weaker intergranular exchange effect received by two transition corner areas than by the central areas due to the averaged 0 background. The two corners are then removed and the curvature is reduced.

**References:**

[1] Liu, Z. and Gilbert, I., IEEE Transactions on Magnetics, 55(3). (2019) [2] Tseng, C. Y. and Hurben, M. J., IEEE Transactions on Magnetics, 47(10), 2974–2976. (2011) [3] Hashimoto, M. and Salo, M., INTERMAG 2006 - IEEE International Magnetics Conference, 43(7), 448. (2006) [4] Zhu, J. G. J. and Li, H., IEEE Transactions on Magnetics, 53(2). (2017) [5] Scholz, W. and Batra, S., IEEE Transactions on Magnetics, 41(10), 2839–2844. (2005) [6] Liu, Z., Jiao, Y. and Victora, R. H., Applied Physics Letters, 108(23). (2016) [7] Victora, R. H. and Huang, P. W., IEEE Transactions on Magnetics, 49(2), 751–757. (2013)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/cb39a9a43bf6f4499492fde625d24769.jpg)
Fig. 1 Averaged HAMR transition patterns with BL 10.5nm.

![](https://assets.underline.io/uploads/markdown_image/1/image/bda63439307c1ae3b1ec0e0301bca139.jpg)
Fig. 2 Curvature change during HAMR cooling process. Solid lines are for thermal ECC media while dashed lines are for FePt media.

 Dependence of Curvature in HAMR Transitions on Bit Length

I.Introduction Spoke-Type permanent magnet synchronous motors (STPMSMs) are widely applied in the industry, especially for the electric vehicle where the high efficiency and high power density are required [1]. As a basic parameter of motor design, the inductance affects the performance, such as flux-weakening capability and control strategy [2]. It is necessary to accurately calculate the inductance. Various researches on inductance with analytical method have been carried out. However, the effect of saturation is normally neglected in the past. This paper presents an analytical method to calculate the d- and q-axis inductances accounting for saturation. II.Proposed Method The proposed analytical method is based on the subdomain model. Firstly, the STPMSM is divided into several types of simple subdomains. The analytical expression of each subdomain is derived and the field solution is obtained by the boundary conditions. Then, the lumped magnetic circuit model (LMCM) is established to calculate the stator reluctance. According to the iterative method, the flux linkage in the case of saturation is calculated. At last, the d- and q-axis inductances under saturation can be obtained. The solution process is summarized in Fig. 1. III.Results and Analysis The finite element method (FEM) is employed to validate the accuracy of the proposed model. Fig.2 shows the d- and q-axis inductances of a 18-slot/16-pole STPMSM. As the current increases, the effect of saturation on the inductance is more pronounced. Compared with the analytical method without saturation, the result of the proposed method is closer to that of FEM. Hence the proposed method is more advantageous for the inductance calculation. IV.Conclusion This paper presents an analytical method for inductance calculation which combines the advantages of subdomain model and LMCM. The effect of saturation on the inductance is fully considered, which improves the accuracy of the inductance calculation. The validity of the analytical technique is verified by FEM. In addition, the method can also be used in the inductance calculation for other types of motors.

**References:**

[1] W. Zhao, J. w. Kwon, X. Wang, T. A. Lipo, and B. i. Kwon, "Optimal Design of a Spoke-type Permanent Magnet Motor with Phase-group Concentrated-coil Windings to Minimize Torque Pulsations," IEEE Trans. Magn., vol. 53, no. 6, Jun. 2017, Art. no. 8104604. [2] J. Y. Kim, D. W. Kang, T. C. Jeong, D. H. Jeong, J. Lee, and S. G. Lee, "Optimal Rotor Design of an 150 kW-Class IPMSM by the 3-D Voltage-Inductance-Map Analysis Method," IEEE Trans. Magn., vol. 53, no. 11, NOV. 2017, Art. no. 8206805.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/9e5feb6a11a5aab0b29176ccc159f03d.jpg)

![](https://assets.underline.io/uploads/markdown_image/1/image/75139eb242956e0a98ed217fc6897d4c.jpg)

 Analytical Computation of Inductance for Spoke-Type Permanent Magnet Synchronous Motor Accounting for Saturation

I. Introduction Form-wound winding is one of the common forms of electrical machine windings, which owns many advantages, such as simple structure and good insulation. It is widely used in medium and large capacity electrical machines. In order to estimate the AC loss of the form-wound winding, researchers have proposed many analytical models and finite element models [1-6]. Among these analytical methods, many researches focused on the uneven current distribution inside the wire bars. Meanwhile, the circulating currents between parallel wire bars of one coil is usually neglected due to the torsion at the end winding. In fact, the torsion at the end winding cannot completely eliminate the circulating current, especially in the case of multi-bar parallel windings and high frequency conditions. It is a challenging task to establish a complex external circuit in order to simulate connections of wire bars. In this paper, an analytical calculation method to estimate the AC cooper loss of the form-wound winding, taking the circulating current effect into consideration, is presented. Firstly, the current of each wire bar is calculated by building the electric circuit model. Consequently, the current density distribution inside each wire bar is obtained by introducing the previous 1-D analytical method. Finally, the accuracy of the model is validated by FEM simulations. II. Analytical Model of Circulating Current Assuming that the main flux difference between parallel wire bars of one coil, the uneven current distribution inside the wire bar, the MMF drop inside the core and the magnetic field x component are neglected. The voltage equation of the pth wire bar of the ith coil is written as Eq. 1. Considering all voltage equations of wire bars in the same coil, two adjacent equations can be subtracted from each other, thus eliminating the induced voltage caused by main flux, as shown in Eq. 2. With total current constraint shown in Eq. 3, currents can be obtained. The mutual leakage inductances of the wire bars are the key parameters. As shown in Fig. 1 (b), the leakage inductance matrix of bars in one slot is firstly derived through the analytical method. By introducing the directional incidence matrix C between coil bars and slot bars, the mutual leakage inductance matrix of coil bars can be finally obtained, as shown in Fig. 1(c). It’s worth to mention that the positive and negative sides of one coil need to be considered separately to simulate the torsion at the end winding. III. Analytical Model of Uneven Current Distribution inside Bars The currents of bars calculated above are considered as inputs of the analytical model for calculating the current density inside bars. Assuming that the length of the slot opening is infinite and the flux lines are parallel to the slot bottom, the computation of the uneven current distribution inside bars can be simplified to 1-D problem. Based on the Maxwell equations and continuous boundary conditions and current constraints, the partial differential equation of the magnetic field can be derived and solved by analytical method [7]. IV. FEM Validation A 12-pole 72-slot PM generator with six phases is taken for FEM validation. The simplified structure of one coil is shown in Fig. 2(a). Due to the torsion at the end winding, the orders of bars placements in positive side and negative side of one coil are opposite.The currents of bars in one coil is shown in Fig. 2(b). It shows a good agreement between FEM and the proposed method results. Furthermore, the calculation error increases with frequency and the maximum amplitude error of current vectors is about 6.58% (@1000Hz).The current density distribution is shown in Fig. 2(c). The proposed method shows a high accuracy compared with the FEM result. Fig. 2(d) shows the variation curves of the AC loss factor with frequency, which are obtained by the FEM method, the conventional analytical method and the proposed method, respectively. These results show that the conventional method ignoring circulating current effect can cause significant error, especially in high-frequency conditions. Obviously, the accuracy of the AC copper loss estimation has been greatly improved by the proposed method in this case. V. Conclusion An analytical method for estimating form-wound winding loss considering circulating current effect was presented, the accuracy is validated by FEM method. For the case in this paper, the new proposed method greatly improved the accuracy compared with the conventional method. The proposed method provided a fast and accurate copper loss calculation approach for electrical machine optimization design.

**References:**

[1]P. L. Dowell, Proceedings of the Institution of Electrical Engineers, Vol. 113, No.8, p. 1387-1394 (1966) [2]A. S. Thomas, Z. Q. Zhu, and G. W. Jewell, IEEE Transactions on Magnetics, Vol. 45, No. 10, pp. 4748-4751 (2009) [3]L. J. Wu, Z. Q. Zhu and D. Staton, IEEE Transactions on Magnetics, Vol. 48, No. 7, p. 2138-2151 (2012) [4]Y. Wang, J. Pries and K. Zhou, IEEE Transactions on Magnetics, Vol. 56, No. 4, p. 1-9 (2020) [5]P. Mellor, R. Wrobel and N. Simpson, The Energy Conversion Congress and Exposition, p. 5563-5570 (2014) [6]H. Hamalainen, J. Pyrhonen and J. Nerg, IEEE Transactions on Magnetics, Vol. 49, No. 6, p. 2967-2973 (2013)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/58d991873f052535077a8fb8d9ba17d6.jpg)
Fig.1 Brief derivation of circulating current and inductance matrix

![](https://assets.underline.io/uploads/markdown_image/1/image/e0ba56abe6c49f44d04351c473c1e8ba.jpg)
Fig.2 Test model and results comparison

A general analytical expression to model the air gap permeance of electrical machines

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