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This paper attempts to present a quasi-3D finite element method technique for modeling and analyzing tubular linear permanent magnet machines (TLPM) with translator eccentricity. Normal tubular linear machines can be modeled effectively using 2D axisymmetric models, but with the introduction of the eccentricity factor into the geometry the axis symmetry is no longer valid, and hence 3D modeling techniques become the only way for studying the machine. However, sometimes 3D modeling can become time-consuming and requires a huge amount of resources especially for non-linear problems, so the interest in modeling techniques that can represent the problem at hand with less computation time and relatively accurate results is high. A quasi-3D technique was specifically developed to be able to represent the eccentricity in TLPMs. The finite element method (FEM) was used for modeling. Eccentricity analysis has been carried on two different TLPM structures using the proposed quasi-3D, and the results from the quasi-3D model were compared to those obtained using the 3D FEM model. Both the open circuit and the on-load operating conditions were studied with the magnetic saturation taken into account. The behavior and the accuracy of the proposed model varied between the two studied structures and for the different operating conditions, and with very low computation time and cost the results obtained from the quasi-3D model were satisfactory in general except for certain cases.

International audience; The aim of this paper is to assess the efficacy of non-conformal meshing in mesh-based generated reluctance network modeling. A representative semi-numerical example is introduced to demonstrate the possibilities and accuracy offered by a non-conformal meshing. In a conformal mesh, two reluctance block elements share the same branch. This makes the computation faster and more accurate but is not convenient for motion processing or mesh relaxation and it does not solve the air-gap modeling problem (rotor/stator connectivity). To solve this, different approaches are used. The interpolation approach developed in this paper aims to render motion processing independent of discretization. This approach is also applied to couple different meshes (spatial discretization) and makes it possible to connect a lumped parameter model to a meshed one. The model is divided into independent zones that are connected via the interpolation coupling. Several reluctance network (RN) non-conformal meshes are presented and global quantities are compared to a fine-meshed finite-element model. Magnetic saturation is considered in the iron parts. Comparisons are provided under open-circuit and on-load configurations for flux linkage and force. The main advantage is to overcome the limitations of generic RN for which the movement modeling requires remeshing at each position. The drawbacks of the method are the increased number of variables (additional interface nodes) and a certain loss of accuracy due to the interpolation function order.

Permanent magnet flux switching machines are gaining more and more interest due to their relatively good characteristics. Indeed, the presence of all magnetic field sources in the stator (armature windings, permanent magnets and/or field windings), which implies a completely passive rotor, makes it suitable for a large variety of applications [1] [2]. Different flux switching machines topologies have been studied in scientific literature. In this contribution, a relatively new modeling approach [3] based on the coupling of mesh based generated reluctance networks (MBGRN) and analytical models (AM), based on the formal solution of Maxwell’s equations, is used for the accurate prediction of cogging force of a linear tubular flux switching machine. This type of machines could be favorably used in oceanic renewable energy conversion. Figure 1(a) illustrates how the two approaches are combined for the case of a tubular linear flux switching structure. In this example the analytical solution is used for modeling the mechanical air-gap, and inner and outer airs, and the RN method is used to model the moving and static armatures. In order to have a more generic approach, the stator and moving armatures are modeled using mesh based generated reluctance network (MBGRN) technique [3]. This technique, as classical RN method, can be used with a minimum number of reluctances for regions where flux tubes are not highly affected by topology changes. As for finite elements analyses (FEA), the studied domain in MBGRN should be finely meshed in some regions (air-gap for example) and coarsely meshed in other regions. However, in contrast to FEA, the mesh relaxation in MBGRN can be done more easily conducting to reduced system matrix dimensions, and consequently reduced computation time. Indeed, while in FEA two adjacent elements should share an edge, it is no more necessary for RN method, as illustrated in Fig. 1(b). This new modeling approach has been used to analyze the performance of different electromagnetics devices (2D and 3D) [3]–[7]. In this contribution, this technique is used for the computation of cogging force of a linear tubular flux switching permanent magnet structure, something which has never been reported yet, to the best of our knowledge, in scientific literature. The goal is to further extend the investigation of this technique for the computation of relatively sensitive quantities, such as the cogging force, in more complex structures. Another goal is to highlight advantages of HAM as compared to other modeling techniques, and more particularly FEA, in order to obtain reduced order equations system [8] [9].

Numerical methods and mostly finite element analysis (FEA) are popular in electromagnetic modeling because they are standardized methods and because of the availability of a large number of commercial software. However, the computation time may be prohibitive, especially for 3D models. Analytical models are fast but restricted in terms of magnetic saturation evaluation and geometry complexity. Magnetic Equivalent Circuits (MEC) are an excellent compromise as they are fast and take into account magnetic saturation. They are not as generic as FEA and if parameters vary, MEC models have to be readjusted [1–6]. Thus, model development durations are longer for MEC than for FEA. For more genericity, authors in [7–9] have demonstrated the equivalence between the equations of reluctance network (RN) and FEA method and the possibility to form RN matrix system by means of FEA. This paper presents the load analysis of a Permanent Magnet Linear Machine (PMLM) as a case study for a direct flux tube MEC method using an automatic Mesh Based Generated Reluctance Network (MGRN). The aim is to show that it is possible to quickly build a semi-numerical model of studied structures with an automatically generated MEC with accurate results on global and local quantities. Comparisons of iron losses evaluated by the use of the MGRN model show that results are in a very good agreement with those obtained byAnsys-Maxwell FEA [10] used as a reference. This approach can be used in a tool that allows the automated processing of an arbitrary geometry providing an accurate model in a shorter amount of time than needed for building a dedicated model. MESH BASED GENERATED RELUCTANCE NETWORK Geometry is discretized in a number of bidirectional elementary reluctance blocks (Fig. 1). The model is divided into zones according to geometry and material proprieties. Discretization can be made finer or coarser according to the desired precision. Central nodes of element blocks are connected via their branches. The advantage of using bidirectional block elements is that no previous knowledge of flux paths is necessary. Each mesh of the network of the final MEC will describe a possible path for flux to go through. The second advantage is to have access to normal and tangential components of local quantities such as flux density and magnetic field. Thus, forces can be evaluated using the Maxwell stress tensor and iron losses by means of Bertotti model [11]. Scalar magnetic potential is used to formulate and solve the nodal based matrix equation: $[\mathrm{U}] =[\mathrm{P}] ^{-1} [\Phi_{s}][\mathrm{P}]$ Permeance matrix [U] Scalar magnetic potential in each node $[\Phi_{s}]$ Sum of flux sources for each node n Total number of nodes The approach for solving the equation system as well as PMLM proprieties are given in [12]. A non-linear B-H curve is used for the ferromagnetic material and saturation is addressed with an iterative method as the value of permeability is adjusted in every ferromagnetic block element till convergence of the algorithm. The framework for space discretization, boundary conditions and PM sources assignment are explained in [13]. The PMLM is supplied with a 3-phase sinewave current and distribution of sources of magneto-motive force (MMF) due to coil currents is determined at each step. Fig 1 illustrates the geometry of the PMLM and the MMF distribution.Sources distribution MMF sources due to PM need to be placed on the branches of the magnetization direction (y axis) in the elements through all layers of the PM zones. Similarly, MMF sources due to coil currents are distributed on all elements of the teeth and slots. Total MMF according to position (along the x axis) is the sum of the MMF created by each coil. The ratio (height of element block to height of coil zone) is used to weight the MMF sources on each branch in each block of the winding zone (Fig. 1): mmf elem =(1/ 2)×E h Z h )× e(x node )e Total mmf according to position $\mathrm{x}_{node}$ Position of the central node of block mmf $_{elem}$: MMF on the branch of the block E h Element height Z h Zone height Saturation The permeability in the reluctances of the blocks in the ferromagnetic parts is initialized with the linear part of the B-H curve. Convergence criteria is magnetic energy in each block as a product of flux density and magnetic field at each step is compared to the one of the previous iteration till the difference no longer exceeds a pre-defined value. At each displacement step, the magnetic permeability value of the block elements are those of the previous magnetic state in the whole model i.e. the previous displacement step. \textit{Motion} A relative moving zone is defined including mover, PM and lower airgap region. The airgap is discretized in a way that the desired movement step is equal to the size of an element block in the airgap. This avoids non-conformal meshing problems and makes it easier to handle changes in matrix topology. \textit{Iron losses} The total iron losses In the MGRN model are calculated in post-processing and are the sum of hysteresis losses, eddy current and excess losses as shown in the equation: $\mathrm{P}= \mathrm{k}_{h} \quad f \mathrm{B}_{m}^{2} \quad + (\pi \sigma \mathrm{d}^{2}f^{2}/ 6 ) \int($ dB$/ \mathrm{d}\unicode{0x03B8} ) ^{2} \mathrm{d}\unicode{0x03B8} + (\pi \sigma \mathrm{d}^{2}f^{2}/ 6 ) \int\vert$ dB$/ \mathrm{d}\unicode{0x03B8} \vert ^{1.5} \mathrm{d}\unicode{0x03B8}$ Flux density values in each node of each reluctance block element are evaluated in the MGRN model using the same B(H) curve and material proprieties as implemented in the Ansys-Maxwell FEA model. Fig 2 shows comparisons of total iron losses (in all ferromagnetic parts i.e. stator and mover) vs frequency for the MGRN and the FEA models used as a reference at $\mathrm{I}_{max}=5\mathrm{A}$. It is shown that the results are in very good agreement.

International audience; Research on continuum manipulators is increasingly developing in the context of bionic robotics because of their many advantages over conventional rigid manipulators. Due to their soft structure, they have inherent flexibility which makes it a huge challenge to control them with high performances. Before elaborating a control strategy of such robots, it is essential to reconstruct first the behavior of the robot through the development of an approximate behavioral model. This can be kinematic or dynamic depending on the conditions of operation of the robot itself. Kinematically, two types of modeling methods exist to describe the robot behavior, quantitative methods describe a model-based method; and qualitative methods, describe a learning-based method. In kinematic modeling of continuum manipulator, the assumption of constant curvature is often considered to simplify the model formulation. In this work, a quantitative modeling method is proposed, based on the Pythagorean Hodograph (PH) curves. The aim is to obtain a 3D reconstruction of the shape of the continuum manipulator with variable curvature, allowing the calculation of its inverse kinematic model. It is noticed that the performances of the PH-based kinematic modeling of continuum manipulators are considerable regarding position accuracy, shape reconstruction and time-cost of the model calculation, than other kinematic modeling methods, for two cases: free load manipulation and variable load manipulation. This modeling method is applied to Compact Bionic Handling Assistant (CBHA) manipulator for validation. The results are compared with other inverse kinematic models developed in case of CBHA manipulator.

The aim of this paper is, first, to extend the application of the hybrid analytical modeling (HAM) approach for the analysis of tubular linear machines (TLPM), something that has never been reported yet in the scientific literature. The HAM consists of a strong coupling between the reluctance network and the Fourier series solution of magnetic scalar potential. In order to assess the merits and capabilities of the HAM, it is applied to the study of the cogging force in linear machines and compared with finite element analyses (FEA). The cogging force is a global quantity, which is very sensitive to magnetic field computation accuracy and constitutes, therefore, a good measure of the accuracy of the HAM as compared with FEA. The HAM should require less computation time as compared with FEA, and is then more suitable for predesign stage. Three different structures are analyzed: a flat linear machine, a TLPM with internal mobile armature, and, finally, a TLPM with external mobile armature. The analyses are conducted, considering the effect of magnetic saturation.

In highly concurrential markets the time to market is reduced, which implies a need of design tools having relatively high accuracy to computation time ratio. It is to address this problematic that the presented modelling approach has been developed. A general framework for the analysis and design of electromagnetic devices is presented. This framework is based on a new hybrid analytical model (HAM) [1]-[4]. The HAM can be applied to different electric machines types (synchronous and asynchronous) and structures (rotating with radial field or axial field, linear flat or tubular structures, or multi-degree-of-freedom structures). It is based on a direct coupling of analytical solution of Maxwell's equations (AM) with reluctance networks (RN). This new technique allows to combine advantages of both methods and should help reduce computation time as compared to finite elements method [3]. This modeling approach is relatively simple and allows the accurate prediction of performances of electromagnetic devices.

In low- to medium-power-rated brushless machines, noise of electromagnetic origin is generally prevailing over mechanical and aerodynamic sources. The appropriate combination of pole and slot of a permanent magnet synchronous motor is one of the keys to its performances among which stands noise reduction. This paper investigates configurations regarding this aspect for distributed overlapping and single-layer concentrated nonoverlapping winding configurations. Chosen combinations are 8 poles 48 slots (8p48s), 8p72s, 46p48s and 50p48s. The air-gap flux density is obtained by 2-D finite element analysis (FEA), allowing to compute the electromagnetic pressure. It then serves as input of mechanical and acoustical 3-D FEA models of the stator with carter.

Fast design procedures of electrical machines require the use of adapted modeling approaches. This is particularly true for axial flux permanent magnet (AFPM) machines, which have an intrinsic 3-D electromagnetic structure. Indeed, accurate modeling of AFPM machines requires the use of 3-D finite-element analysis (3-D FEA), which is highly time consuming, especially at first design stages. This paper presents a new quasi-3-D analytical model (AM) that allows modeling accurately AFPM machines while saving a huge amount of time as compared with 3-D FEA. This new approach combines a multislice AM, which takes into account a part of the 3-D effects, and considers the end-effects by a simple and effective radial dependence modeling of the magnetic field in AFPM machines.

This paper describes an analytical technique for prediction of open circuit eddy-current loss in armature windings and permanent magnets (PM) of series double excitation synchronous machines. First, a 2-D exact analytical solution of open circuit magnetic field distribution in idealized geometry of series double excitation synchronous machine is established. It involves solving Maxwell's equations in stator slots, air gap, PM region, and rotor slots. Then, magnetic vector potential solutions in the stator slots and PM regions are respectively used for prediction of resistance limited eddy currents in armature windings and permanent magnets. This analytical model is then used to estimate eddy-current loss in armature windings and permanent magnets. The validity of the developed model, which is also applicable to conventional designs of permanent-magnet machine, is verified by time-stepped transient finite-element analysis (FEA). The developed model is then used to quantify the effectiveness of segmenting the magnets and armature windings in reducing the eddy-current loss.

This paper presents an exact 2-D analytical model for predicting the armature reaction magnetic field in idealized structures of permanent-magnet tubular linear machines with surface-inset mounted magnets. The armature reaction magnetic field distributions is analytically derived and compared to finite-element analyses. The analytical solution allows the prediction of the machine inductance and reluctance force in closed forms, and facilitates the evaluation of any possible partial irreversible demagnetization of the magnets. It can also be used to estimate resistance limited eddy-current losses in armature windings and permanent magnets. The developed analytical model can be advantageously used for the analysis and design of a class of linear tubular machines.

We present an exact 2-D analytical model for predicting the magnetic field in idealized structures of surface mounted permanent-magnet tubular linear machines that account for the stator slotting effect. The open-circuit and armature reaction magnetic field distributions are analytically derived and compared to finite-element analyses. The on-load magnetic field calculation is based on the superposition of the component fields due to permanent magnets and armature field reaction. Our model can be advantageously used for the analysis and design of a class of linear tubular machines.

We describe an exact 2-D analytical model for predicting the open circuit magnetic field in idealized structures of wound field and series hybrid excitation permanent-magnet synchronous machines. This technique can be used for either internal or external rotor radial-field machine topologies. It involves analytical solution of Maxwell's equations in stator and rotor slots and annular air-gap/magnet regions. We compare results from this analytical model to corresponding finite-element analyses. We then use it to estimate the cogging torque, induced back-electromotive force, and to study the effects of rotor skewing.

This paper presents an analytical model for prediction of eddy current loss in armature windings of permanent magnet brushless AC machines. The developed model can either be used in the case of internal or external rotor radial-field machines topologies. First, a 2-D exact analytical solution of magnetic field distribution in an actual geometry of slotted surface mounted PM radial flux synchronous machines is established. It involves solution of Maxwell's equations in slots, airgap and PM's region. Then, magnetic vector potential solution in the slots is used for prediction of resistance limited eddy current in armature windings. Finally, results from this analytical model are compared to corresponding finite element analyses.

This paper presents an analytical solution for prediction of the armature reaction magnetic field in slotted surface mounted permanent magnet radial flux synchronous machines. This technique is used in the case of internal and external rotor radial-field machines. The magnetic field expressions are developed in both slots regions and magnetic airgap region leading to an exact calculation of the effects of slotting on the airgap magnetic quantities. Results from this analytical model are compared to corresponding finite element analyses. This analytical model is then used to estimate the self and mutual inductances.

In this paper, the suitability of a class of electric machines for vehicle traction applications is discussed. These machines, which are known as hybrid excitation synchronous machines, combine permanent-magnet (PM) excitation with wound field excitation. The goal behind the principle of hybrid excitation is to combine the advantages of PM excited machines and wound field synchronous machines. It is shown that these machines have good flux weakening capability compared with PM machines, and that they constitute an energy-efficient solution for vehicle propulsion.

Accurate modeling of axial flux permanent magnet (AFPM) machines requires the use of three dimensional finite-element method (3D FEA). However, 3D FEA is high time consuming especially at first design stages. This paper discusses the validity domain of a multislice analytical model (quasi-3D model) of AFPM machines with trapezoidal magnets. This analytical model (AM) considers the machine as composed of several annular machines, each one being modeled using a two dimensional model. A detailed parametric study investigates the influence of certain design features on the validity domain of proposed approach. The goal is to have a fast pre-design tool with a fairly good accuracy as compared to 3D FEA.