# FEA-Based Design Procedure for IPMSM and IM for a Hybrid Electric Vehicle
## 1 Introduction
### Background and Literature Review  
Sizing equations [2], subdomain models (SDMs) [3], magnetic equivalent circuits (MECs) [4], and FEA [5] are common methods used for the electromagnetic design of electric machines. Sizing equations are based on empirical equations developed by engineers. These equations are usually suitable for a specific geometry and material. Hence, they cannot predict electric machine performance parameters with a high accuracy, especially with a wide range of dimensions and machine sizes [6,7]. In SDMs, the Maxwell equations are solved in different parts of an electric machine. The obtained magnetic vector potentials are utilized to find the flux density and, hence, estimate the performance parameters [8; 9; 10]. In the past decade, researchers have proposed SDMs for induction machines [10], switched reluctance machines [9], and permanent magnet machines [8]. It was found that SDMs for IMs require slip and induced current modeling, those for SRMs require nonlinear magnetic saturation modeling, and those for PM motors require complex magnet interaction and torque ripple modeling. Researchers have demonstrated the accuracy of SDMs in the prediction of the performance parameters of electric machines. The implementation of SDMs is mathematically complex and needs a deep understanding of the electric machines being modeled. Hence, they have not become popular in engineering practice. In MEC models, the electromagnetic reluctance and magnetic motive force (mmf) are obtained under a pole. Then, the MEC model is solved to find the mmf and flux in each region. This estimation enables a prediction of the performance parameters [11; 12; 13]. MECs are inaccurate if limited flux paths are investigated in the MEC model [11]. They can become more accurate by increasing the number of equivalent circuit elements, but this increases the computational burden [12; 13]. FEA is one of the most precise tools for the prediction of electric machine performance parameters [14; 15]. FEA is computationally expensive, which limits its application in designing machines for a given geometry [3]. On the other hand, the appearance of commercial and open-source software has led to the popularity of FEA. The literature has described the design procedure for IPMSMs. The available literature discusses the optimization of IPMSMs with different objective functions to design a machine with the desired performance characteristics [16; 17; 18; 19]. These studies normally use available commercialized geometries as the baseline. Some studies in the literature have optimized rotor or stator parameters to improve their performance based on the results of sensitivity analyses [20]. The effect of the winding configuration on improving the design has been another subject of study [21]. Rotor design and PM location and structure have become a popular method for improving IPMSM performance. The rotor of an IPMSM can include permanent magnets arranged in various patterns, including single-V-shape, double-V-shape, delta-shape, W-shape, U-shape, and spoke-type patterns. [PERSON] et al. introduced an electromagnetic design procedure for a double-V-shaped IPMSM [22]. They used the results of a sensitivity analysis of different parameters of the stator and rotor to prepare a baseline design for their optimization study. The temperature effect, which is a crucial factor for design of electric machines, was ignored in Ref. [22]. A comprehensive comparative analysis was conducted in Ref. [23] to determine the optimum shape of an IPMSM rotor between single-V-shape, double-V-shape, and delta-shape configurations. The authors concluded that the single-V shape not only has the highest efficiency over a wide range of torques and speeds but also has a lower risk of demagnetization. A comparison of the V-shape, U-shape, and spoke-type IPMSMs revealed that the spoke type offered better performance at higher speeds, but it had the lowest torque density [24].