Hybrid analytical modeling of saturated electromagnetic devic-es: Integration of Fourier modeling and magnetic equivalent circuits

Modeling of the electromagnetic field distribution is vitally important in the design and optimization process of electromagnetic devices. In recent years, several advanced methods are being researched, e.g. Fourier modeling, charge modeling, Schwarz-Christoffel conformal mapping, boundary element method, etc [1]. Among these, Fourier modeling is particularly interesting for structures with periodicity, however, the drawback is the unavailability of including geometric details (regions smaller than the full periodicity) made of high-permeable material, e.g. slot shapes [2]–[3]. In [3], a hybrid analytical modeling (HAM) technique that integrates Fourier modeling and meshed magnetic equivalent circuit (MEC) is introduced to model complex structures in Cartesian coordinate system. It provides promising results under the assumption of linear material properties, however, such assumption is not often sufficient during the design of electromagnetic devices. Therefore, this paper further investigates the HAM technique to include nonlinear materials for saturated electromagnetic devices. To present the diverse applicability of the proposed method, it is applied to two electromagnetic examples, one in Cartesian coordinate system with permanent magnet excitation (geometry shown in Fig. 1(a)), and the other one in polar coordinate system with current excitation (geometry shown in Fig. 1(b)). Only 2D solutions are discussed in this paper, however, it could be extended to 3D modeling [4]. In the examples, periodicities are respectively applied in x-or θ-directions, while regions are divided in y-or r-directions. Only the airgap and outer air regions are modeled using Fourier expressions since Fourier modeling divides the geometry in periodical regions with homogeneous permeability and does not allow to model local saturation. The meshed MEC, on the other hand, is used for regions with nonlinear permeability (such as stator and rotor), since each mesh element has its unique permeability and hence, is able to take saturation into account. Inside Fourier and MEC regions, the expression of magnetic field is derived by solving the magnetostatic Maxwell equations. Each mesh-based element in MEC regions contains the information of dimension, permeability and magnetic source as presented in Fig. 1(c). In elements that represent soft magnetic materials, the permeability and magnetic source are defined by a locally linearized B-H characteristic in terms of differential permeability and remanent magnetic flux density, and are updated by an iterative approach [1]. The elements representing permanent magnets and coils have magnetomotive (MMF) sources related to magnetization and current. Particularly, the distribution of current-related MMF sources has to fulfill the Ampere’s law [5]. Between Fourier and MEC regions, boundary conditions are given such that continuity of magnetic field is ensured, i.e., consistent normal flux density and tangential magnetic intensity in both spatial frequency and space domains [3]. Details of model formulations, iterative algorithm and field solutions will be discussed in the full paper. The results calculated by this modeling technique are validated by finite element analysis (FEA). Excellent agreement is achieved in both examples as can be observed in the flux density waveforms in Fig. 2(a) and Fig. 2(b), respectively. Additionally, the flux density waveforms when a linear B-H curve would be applied for the iron materials are shown as reference. The HAM technique has several advantages with respect to FEA. It has the potential to be faster than FEA since Fourier regions are not meshed, while for the airgap, a high mesh density is required in FEA to obtain an accurate flux density distribution. Additionally, since the expression of the magnetic field in the airgap is obtained analytically, force and torque calculations are straightforward and fast since the Maxwell stress tensor can be solved analytically. Furthermore, it offers the advantage of fast dimension and topology sweeping without redrawing geometries, therefore is able to simplify and accelerate the design and optimization procedures for saturated electromagnetic devices, resulting in a new computational and design framework. Future research will focus on implementation for different motor topologies, improvement of meshing and iterative algorithms to reduce computation time, and derivation of inductance and iron losses, etc.