In this contribution the power capability of shifted inductances axes permanent magnet machines (SIAPMM) is studied. Both low speed (maximum torque) and high speed (flux weakening) operations are analyzed. The goal is to start from a common ground already defined in some contributions dedicated to power capability of synchronous machines [1-3], and extend it to SIAPMM. The concept of SIAPMM have been introduced recently [4-8]. These machines can be regarded as the combination of non-salient PM rotor and a variable reluctance (VR) rotor under the same stator [Fig. 1(a)]. Two Park’s referential frames can be then defined [Fig. 1(b)], one for the PM rotor part, and a second for the VR rotor part, shifted by an angle β [Fig. 1(b)]. The maximum torque can be increased by adequately choosing angle β. Instead of computing the "armature current / EMF" phase shift ψ maximizing the torque, as classically done, for SIAPMM the starting point consists of imposing the optimal angle value ψ = 0, and determining the optimal β angle [Figs. 1(c) and 1(d)]. It can be shown that β = π/4, if ρ &lt; 1, and β = -π/4, if ρ &gt; 1, where ρ is the saliency ratio (= Lq/Ld). Fig. 2(a) shows the percentage of "maximum torque" increase in the (Ldn, ρ) plane, of the SIAPM machines as compared to classical PM machines having the same ρ, and PM excitation flux linkage ΦPM. Ldn is the normalized value of d axis inductance, defined as (Ld×Imax/ΦPM), where Imax is the maximum armature current. The percentage increase is computed as (TSIAPM-TPM)×100/TPM. Fig. 2(b) compares power capabilities of a SIAPMM, and a classical PM machine, with (Ldn = 2.5, ρ = 1.5). The SIAPMM have a better power capability as compared to classical PM machine. In the full version, the power capability of SIAPMM is studied in the entire (Ldn, ρ) plane. The optimal flux weakening strategies to be adopted considering the values of Ldn and ρ will be defined.

![](https://assets.underline.io/uploads/markdown_image/1/image/4a5fd7b19e2e9a52505d43526cfaedf7.jpg)

Fig. 1. SIAPM machines principal. (a) Structure of a SIAPM machine. (b) Park’s referential frame definition. (c) Classical PM machine (β = 0). (d) SIAPM machine (β = π/4).

![](https://assets.underline.io/uploads/markdown_image/1/image/a04db9d7e5b4c28ffd4f848dce128a9a.jpg)

Fig. 2. Power capabilities of SIAPM machines. (a) Torque increase (low speed operation). (b) Operation over a large speed range (Ldn = 2.5, ρ = 1.5).

Power capability of shifted inductances axes permanent magnet machines

Magnetic gears (MGs) have emerged in the last 20 years with promising torque densities [1]-[5]. Some of their potential advantages are reduced maintenance and improved reliability. Axial field MGs offer multiple possible advantages over radial field configurations [6]. However, modeling axial field structures requires a 3D geometrical model to properly represent its performance. The purpose of this work is to present a 3D permeance network modeling technique for axial field flux focusing magnetic gears. Permeance network (PN) is a very well-known modeling technique. It’s based on the concept of flux tubes where the geometry is divided into blocks with a central node, and then the network equations are generated using Gauss’s law, which states that the sum of all fluxes into a node must be zero [7][8]. The unknowns to be solved then are magnetic scalar potential values at every node. The nodes are connected through a lumped component which can be permeance or a source. Figure 1 shows the MG structure to be modeled with the elementary permeance block. It consists of three parts, a high-speed rotor (HSR), a low-speed rotor (LSR), and a stationary part. The versatility of the model will be demonstrated by changing multiple variables and calculating different quantities. Figure 2 compares the magnetic induction components calculated using the 3D PN model and a mean radius 2D finite element method model (FEM). The calculated components are at the LSR air gap side. The permanent magnets and the ferromagnetic materials were considered in their linear case with µrf = 10000, µrpm = 1, and Br = 1.25T. The differences between the 2D model and the proposed 3D model are visible, where we can see a difference in the amplitudes and shapes which demonstrates the importance of the radial component in the axial field structures. The full work will demonstrate more the potential of the proposed model by performing linear and non-linear calculations and showing how this model can be very suitable for early design stages.

![](https://assets.underline.io/uploads/markdown_image/1/image/7fc302ac3e86324c301eb2a6485110d5.jpg)

Fig.1 Flux focusing axial field magnetic gear with p1=6, n2=24, and p3=18 pole-pairs.

![](https://assets.underline.io/uploads/markdown_image/1/image/742a5eb474df0b9d992e46d479febd24.jpg)

Fig.2 Axial and radial magnetic induction components calculated using the 3D PN and a 2D FEM model at the LSR airgap side

3D Permeance Network Modeling for an Axial Field Flux Focusing Magnetic Gear

Tubular linear permanent magnet machines (TLPM) have gained increasing popularity over the past years due to their versatile nature. Whether used as actuators for linear motion applications or as generators for energy production applications, these machines can provide accurate and precise motion with relatively compact structures and simpler control systems [1]-[3]. Like any other mechanical application, TLPMs might suffer from different structural deformations during either operation or assembly. The moving part of the machine called the translator is the most vulnerable to such mechanical deformations. One commonly occurring deformation happens at the level of the translator shaft and is usually called a translator eccentricity where the translator shaft is no longer aligned perfectly with the central axis of the machines's cylindrical structure. Eccentricity can take different forms and has been studied previously by some authors [4]-[7]. In [4] [5], eccentricity analysis was done using analytical and 3D FEM modeling and the calculated results were compared and validated by experimental results. In [6], eccentricity effects were studied for a surface mounted TLPM structure using the 3D finite element method (3D FEM). The study was continued in [7] by analyzing the eccentricity effects on another TLPM structure, also using 3D FEM, and comparing the eccentricity effects on the two structures. The authors of [6] [7] noted the extensive amount of time and effort required to compute some of the 3D FEM models and generate the results. Hence, an alternative modeling technique for eccentricity analysis that can provide relatively good results with less computation time is needed. The aim of this paper is to present a quasi-3D modeling technique that can represent the 3D problem using the superposition of multiple 2D models. It will be shown that the proposed model takes much less computation time and effort and produce relatively accurate results in comparison to the 3D model. Translator eccentricity unbalances the radial electromagnetic forces acting on the translator by forming a non-uniform air gap between it and the primary. Static eccentricity, which is a common eccentricity form, is the state of having the translator axis displaced by a constant distance ε from the primary axis while both axes remain parallel to each other. This work only focuses on the static eccentricity form leaving other forms for future works. The proposed quasi-3D model is achieved by dividing the 3D structure in the tangential direction (θ direction) into a certain number of sections (see Fig. 1). Since the air gap is non-uniform around the translator in the tangential direction, there is a different air gap value e (Fig. 1) for each θ along the tangential direction where θ∈[0, 2π]. Hence, each section in the tangential direction is modeled as a standalone 2D linear machine having an air gap equal to the air gap value at the central angle of the corresponding section. Then, for each 2D linear machine model, the magnetic induction in the air gap is calculated where it will have two components x and y corresponding to the axial (z-direction) and normal (r-direction) cylindrical components of the 3D model respectively. The tangential component can be associated with a z component which is impossible to determine in a 2D model that has only x and y directions, and hence it will be neglected. After gathering the results from all the 2D modeled sections, using the Maxwell stress tensor the electromagnetic forces acting on the translator in the axial and radial directions are calculated. Both the 2D and 3D models used are finite element method based models, and for this work, they were constructed using the FLUX commercial software. The quasi-3D model will be used to study two TLPM structures having static eccentricity. The first one is a surface-mounted PM structure, the second one is an inserted PM flux focusing structure, and the two have similar dimensions. Both open circuit and on load operations will be studied, and the two cases of linear and non-linear ferromagnetic materials will be considered as well. Studying the different cases will provide an insight into the accuracy and efficiency of the proposed quasi-3D model. Fig. 2 shows a comparison between the quantities (cogging force and attraction force) calculated using the quasi-3D and the 3D FEM models for the surface mounted TLPM having an eccentricity value ε equal to 0.75mm (75% of the air gap value). For this comparison, the ferromagnetic materials were defined in their linear case with a constant μr = 10000. Also, the relative permeability of the PM was defined as μr = 1 and its magnetic remanence as Br = 1T. Noting that the attraction force computed is acting along the axis aligned with the angles having the minimum and maximum air gap values. The results shown in Fig. 2 are relatively accurate and acceptable especially that the solution of 65 time steps of the quasi-3D model with 32 tangential divisions took only around 15 minutes with much less construction effort, while it took around 3 days to solve 33 time steps of the 3D FEM model with much more construction and computation efforts. Computation time differences will be even more noticeable and remarkable when solving non-linear models that will be presented and compared in the full paper release.

**References:**

[1] N. Bianchi, S. Bolognani, D. D. Corte and F. Tonel, "Tubular linear permanent magnet motors: an overall comparison," in IEEE Transactions on Industry Applications, vol. 39, no. 2, pp. 466-475 (2003) [2] J. Wang, W. Wang and K. Atallah, "A Linear Permanent-Magnet Motor for Active Vehicle Suspension," in IEEE Transactions on Vehicular Technology, vol. 60, no. 1, pp. 55-63 (2011) [3] L. Szabo, C. Oprea, I. Viorel and K. A. Biro, "Novel Permanent Magnet Tubular Linear Generator for Wave Energy Converters," IEEE International Electric Machines & Drives Conference, pp. 983-987 (2007) [4] F. Marignetti, P. Cancelliere, and M. Scarano, “Analytical formulation of the no-load magnetic field in IPM tubular machines with translator eccentricity,” in IECON 2006 - 32nd Annual Conference on IEEE Industrial Electronics, pp. 829–835 (2006) [5] L. Cappelli, Y. Coia, F. Marignetti and Z. Q. Zhu, "Analysis of Eccentricity in Permanent-Magnet Tubular Machines," in IEEE Transactions on Industrial Electronics, vol. 61, no. 5, pp. 2208-2216 (2014) [6] D. Sarr Lo, H. Lawali Ali, Y. Amara, G. Barakat, and F. Chabour, “3D Finite Element Analysis of Eccentricity in a Tubular Linear Permanent Magnet Machine,” in 2018 XIII International Conference on Electrical Machines (ICEM), Alexandroupoli, p. 1996–2001 (2018) [7] H. Diab, Y. Amara and G. Barakat, "Comparison of Two Tubular Linear Permanent Magnet Machines with Translator Eccentricity," 2020 International Conference on Electrical Machines (ICEM), Gothenburg, pp. 612-618 (2020)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/2fedcffd641e7ac4c796e19d941f5391.jpg)
Fig.1 A 3D TLPM structure divided in the tangential direction with a 2D axial cross-section view with dimensions

![](https://assets.underline.io/uploads/markdown_image/1/image/094e73a6fefce9ab02b34bf22374e05b.jpg)
Fig.2 Comparison of the cogging and attraction forces generated by the quasi-3D and 3D FEM models

 Translator Eccentricity Analysis in Tubular Linear Machines using Quasi-3D Finite Element Method Modeling

**Session chair:** Mohammad Sedigh Toulabi, University of Windsor



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Power capability of shifted inductances axes permanent magnet machines

3D Permeance Network Modeling for an Axial Field Flux Focusing Magnetic Gear

Translator Eccentricity Analysis in Tubular Linear Machines using Quasi-3D Finite Element Method Modeling

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