Analytical and Finite Element Methods for Evaluative Electromagnetic Parameters of Inset PMSM and SPMSM

Permanent magnet synchronous motors (PMSMs), particularly surface-mounted (SPMSMs), are widely used in the fields of electric vehicles (EVs) industry due to advancements in industrialization and technology [1, 2]. In the study [3], the study highlighted the benefits of outer rotor PMSMs through two approaches: comparing torque capabilities of two hypothetical permanent magnet (PM) configurations based on their geometric volumes and evaluating their performance. In the study [4], the torque characteristics of proposed geometries were analyzed via the finite element method (FEM) with a nonlinear model. In the studies [5-7], the papers also conducted comprehensive analyses of interior PMSMs (IPMSMs) and SPMSMs with various slot and pole configurations. These studies utilized quantitative models to compare back electromotive force (EMF), torque ripple, cogging torque, total harmonic distortions (THDs), and loss characteristics. An optimized design for IPMSMs was developed to define key dimensions, followed by the FEM simulations to reduce cogging torque and enhance efficiency [8]. The FEM was proposed for SPMSMs with inner and outer rotor configurations to compare the flux density of the PM [9]. It found that while outer rotor designs produced significantly higher torque, they also exhibited higher cogging torque. Feng et al. [10] reviewed the use of SPMSMs in diverse applications such as electric drives, marine propulsion, lifting systems, and mining due to their compact size, high torque at low speeds, efficiency, and superior performance, especially with rare-earth magnet developments. In the study [11], the SPMSMs were further classified into inner and outer rotor types based on rotor positioning. In the study [12], the separate models of IPMSMs with different multilayer PM configurations were analyzed, revealing enhanced performance with double-layer PMs. Du et al. [13] focused on analysing the cogging torque distribution in PM machines via the slot and pole combinations. In the studies [14, 15], comprehensive assessments of the SPMSMs and IPMSMs from a multi-physics perspective were presented to study the optimized design for the practical machine with the output power of 60 kW and speed of 30,000 rpm. Arehpanahi and Kheiry [16] evaluated the suitability of SPMSMs and IPMSMs for high-speed applications by comparing their electromagnetic properties and demagnetization behaviors using the FEM. The conclusion was that although IPMSMs provide better cost-effectiveness and torque per permanent magnet weight, their rotor structures are less durable and more susceptible to irreversible demagnetization. Finally, Xuan [17] also used the FEM to compare four distinct rotor structures of PMSMs with the high-speed. This study evaluated the stress levels, temperature rise and sleeve thickness under various conditions.

Although there have been numerous studies on SPMSMs as mentioned above, but the combination of analytical and numerical methods to analyze and evaluate the two types of SPMSMs and inset SPMSMs with inner rotor configuration has not been addressed in previous research. In this contribution, comparative electromagnetic parameters of the SPMSM and inset PMSM are proposed via the association between the analytical model and FEM. The approach is here presented in two steps: An analytical model is first presented to design initial dimensions of both motors (including stator dimensions, rotor dimensions, and PM properties. Next, the FEM is conducted to evaluate and simulate electromagnetic parameters of the practical motors, such as magnetic fields, torque ripple, cogging torque, electromagnetic torque, and back electromotive force (EMF). The simulation results will show the highlight of advantages and disadvantages of two motor types.

The Maxwell’s equations in the frequency domain are written as [21-24]:

$\operatorname{curl} H=\mathrm{J}_s$     (12)

$\operatorname{curl} E=-j \omega \mathrm{~B}$     (13)

$\operatorname{div} B=0$     (14)

where, H, J_s, E, and B are the local magnetic field intensity (A/m), electric current density (A/m²), electric field (V/m) and local magnetic flux density (T).

The set of Maxwell’s equations from Eqs. (10) to (12) is solved with the association of the boundary conditions and behavior laws, i.e.,

$n \times\left.\mathrm{H}\right|_{\partial \Omega}=0, \mathrm{~B}=\mu \mathrm{H}, \mathrm{J}=\sigma \mathrm{E}$    (15)

where, μ is the permitability (H/m), σ is the electrical conductivity (S/m) and J is the eddy current (A/mm²).

Based on the set of equations, from Eqs. (10) to (12), the A-conformal weak formulation can be written as:

$\begin{gathered}\left(\mu^{-1} \operatorname{curl} A \cdot \operatorname{curl} A^{\prime}\right)_{\Omega}+\left(\sigma A \cdot A^{\prime}\right)_{\Omega_c}+j \omega(\sigma \operatorname{grad} \varphi \cdot \left.A^{\prime}\right)_{\Omega_c}+\left\langle n \times H \cdot A^{\prime}\right\rangle_{\Gamma_h}=\left(J_s \cdot A^{\prime}\right)_{\Omega_{S^{\prime}}} A^{\prime} \in \\ H_{\mathrm{h}}^0(\operatorname{curl} ; \Omega)\end{gathered}$     (16)

where, Ω_c and Ω_s are, respectively, the conducting and inductor domains, Ω is the studied domain (with $\partial \Omega=\Gamma=$ $\left.\Gamma_{\mathrm{h}} \cup \Gamma_{\mathrm{e}}\right)$ and $A^{\prime}$ is the test function. It should be noted that the fields of H, B, E, and J belong to the function space $H_{\mathrm{h}}^0(\operatorname{curl} ; \Omega)$.

By solving the weak form of Eq. (16), the back EMF can be defined via the post-processing, i.e., [18]:

$E=\frac{L}{S} \frac{\partial}{\partial t} \quad\left(\iint_{\Omega^{+}}^T A d \Omega^{+}-\iint_{\Omega^{-}}^{\square} A d \Omega^{-}\right)$     (17)

where, L and S are the length and cross-section area of the conductor.

The paper has proposed the analytical model and FEM to evaluate two types of practical motors (SPMSM and inset PMSM) with a power rating of 7.5 kW. The obtained results from the proposed methods have analysed and compared the electromagnetic parameters of each motor, such as the magnetic flux density, cogging torque, torque ripple, flux linkage, electromagnetic torque, back EMF, and temperature rise. Based on the achieved results, the findings offer valuable insights into the strong and weak points of each motor type, aiding researchers and designers in selecting appropriate configurations, torque density, and efficiency for electric drives and industrial applications, and their impact on overall vehicle performance as well. Furthermore, this study lays the groundwork for future research to improve the electromagnetic parameters of these motors via the optimization techniques, including genetic algorithms, swarm optimization, and other advanced methods.