In recent times, permanent magnet synchronous motors (PMSMs) have been widely used in various industrial applications owing to their many advantages (high torque and power density, high efficiency, reliability, and stability) [1, 2]. In particular, at low-speed traction applications, PMSMs have gained widespread adoption, such as ship propulsion, lifting, mining, and oil field exploitation [3, 4]. In general, PMSMs can be categorized into two types based on the position of their permanent magnets (PMs) in the rotor structure: surface-mounted PMSM (SPMSM) and interior PMSM (IPMSM). In SPMSMs, PMs are mounted on the surface of the rotor, while in the case of IPMSMs, PMs are embedded inside the rotor. Furthermore, an SPMSM exhibits no difference in its d-axis and q-axis inductances, resulting in no reluctance torque. In other words, SPMSMs can only produce magnetic torque. In contrast, the magnetic flux distribution in IPMSMs depends on d-axis and q-axis reluctances, which occur due to the presence of flux barriers in the rotor [5]. Therefore, an IPMSM can produce both magnetic torque and reluctance torque, which results in a higher torque density and power density than an SPMSM. Along these lines, in [6], an SPMSM was examined to find that it is characterized by a better waveform and fewer harmonic components in the air gap flux density than an IPMSM due to its use of mounted PMs in the rotor configuration, resulting in no-load back electromagnetic force (EMF) that closely resembles a sinusoidal waveform.

Fractional-slot focused winding is a frequently used arrangement for achieving high torque in traction applications [7, 8]. Notably, fractional-slot winding is represented by non-overlapped coils wound around a single tooth and have the number of slots per pole in each phase less than one. Meanwhile, concentrated windings are those in which every coil is wound around a single stator tooth by itself. Such winding provides a number of benefits for low-speed direct drive transmission systems, including low copper loss, a high slot fill factor, low end-turn length, and high efficiency. Moreover, fractional-slot setups also have a few disadvantages compared to complete slot machines, such as a relatively lower winding factor and larger harmonic content in magnetomotive force (MMF) distribution [7, 8]. Therefore, depending on the intended use, the motor’s design must comprise an appropriate number of pole pairs and slots.

In this regard, a kind of torque that is frequently observed in motors that use magnets is cogging torque. Another serious challenge encountered during PMSM operations is the occurrence of a noticeable ripple in the torque waveform, which lowers the motor’s torque quality. Numerous research studies have focused on mitigating cogging torque and torque ripples in PMSM. To increase the number of cogging torque cycles, the authors in [9] first identified an appropriate pole–slot combination and then implemented modifications to the magnet arc and slot opening to obtain the ideal values for these two parameters. The application of this method lowered the cogging torque by 0.62% and raised the average torque by 3.5% compared to the initial design. Furthermore, the motor was subjected to the step rotor skewing procedure in [10, 11]. In [11], the results demonstrated a considerable reduction in both cogging torque and torque ripple compared to those calculated before using the approach. This method involved segmenting the magnet and then selecting distinct skew angles for each PM segment. Notably, the selection of the skew angles determined the motor’s back EMF, cogging torque, and torque ripple performance. Meanwhile, Zhu et al. [12] conducted an analysis of the air gap form in a V-type IPMSM using fractional-slot focused winding for compressed air system applications. Simulations were carried out on the original model (Model I) and two distinct air-gap shaping V-type IPMSMs by generating two types of unevenly distributed air gaps (Model II and Model III). Experiments were done to test characteristics of Model III and compared with the results from simulations. The proposed air-gap shaping techniques significantly minimized stator core loss and torque ripple while also slightly lowering the average torque, as evident from the results of the finite element method (FEM) and experiment test. Furthermore, in [13], four rotor topologies for PMSMs with identical motor sizes and magnet consumption were compared. The findings indicated that delta-shaped and V-shaped IPMSMs achieved maximum output torque owing to their greater salient rates. Additionally, IPMSM outperformed SPMSM in terms of efficiency, demagnetization, and torque ripple. In [14], a comparison of SPMSM and IPMSM was conducted through both numerical and experimental analyses, maintaining the same stator configuration, winding specifications, voltage limit, and pole–slot combination for both motors. The results showed that the SPMSM suffered a larger core loss and eddy current loss than the IPMSM. Moreover, based on an analysis of the no-load electromagnetic parameters, such as the total harmonic distortion (TDH) of the back EMF, cogging torque, and efficiency, among others, IPMSMs were found to be optimal for use in electric vehicle (EV) compressors compared to SPMSMs.

Although many papers have researched SPMSMs and IPMSMs, as well as compared the two motor types, there are some limitations in the calculation processes detailed for the physical dimensions of the motors. In most studies, the authors focused on providing the specifications of the motors, conducting numerical and experimental analyses to define their electromagnetic parameters, validating simulation results, and comparing the performance of the motors.

In this study, a combination of an analytical technique and FEM is employed to compare the electromagnetic parameters of an SPMSM and a V-shaped IPMSM with the same power level of 7.5 kW. First, an analytical model is employed to compute the required dimensions and parameters of the proposed machines while keeping the rotor volume, PM volume, pole–slot combination, and power supply of the two motors unchanged. Next, an FEM is introduced to validate the results of the analytical method and to compare the electromagnetic parameters of the two motors, including their back EMF waveform, output power, torque ripple, efficiency, and power factor. Finally, the rotor step skewing technique is applied to compare the results for cogging torque and torque ripple.

In this section, an IPMSM with a V-shape flux barrier is introduced for electromagnetic parameter analysis. Subsequently, a comparison of the IPMSM and the SPMSM is conducted, with the rotor and magnet volumes kept the same. The required parameters for the two motors are shown in Table 1.

The analytical results are presented in Table 2. The FEM embedded in Motor-Cad software will compute and simulate the electromagnetic parameters of the proposed machines based on these parameters. The IPMSM and SPMSM models are depicted in Fig. 3. Furthermore, the skewed PM technique for both motors, maintaining the same skew angle, is depicted in Fig. 4 and presented in the form of three steps in Table 3.

The on-load back EMF results are displayed in Figs. 5 and 6. It is evident that before implementing the skewed PM technique, the back EMF waveform of the SPMSM exhibited high harmonics, with significant amplitude achieved for the 5th and 7th components. In contrast, the IPMSM attained lower harmonic distortion with regard to its back EMF waveform, achieving high amplitude for the 5th, 7th, and 11th components. In this context, the TDH values of the back EMF for the IPMSM and the SPMSM were 6.45% and 15.12%, respectively.

After the implementation of the skewing technique, a considerable reduction in high harmonic components is observed, as shown in Figs. 5(b) and 6(b). The THD values of the two motors in this case were 1.43% and 2.7%, indicating a reduction of 5.02% and 12.42%, respectively, compared to the previous case. These outcomes demonstrate the effectiveness of the skewed PM technique.

Because the PMs are mounted on the surface of the rotor in an SPMSM, the path of the magnetic flux from its PMs to the stator magnetic core is smaller than in the case of the IPMSM. Furthermore, the combination of the convex part of magnet poles and the concave part in the middle space of the magnets enables a better waveform and greater fundamental component in its on-load air gap flux density, as shown in Fig. 7. Notably, since we studied the radial flux motor, only the radial flux density waveforms are presented.

The values obtained for flux density in various parts of the motors is presented in Table 4. Note that only a pole pair view of the finite element mesh and flux density distribution are presented owing to the symmetry of the motors. The results attained from analyzing the losses of the motors are noted in Table 5. It is observed that the IPMSM has more turns than the SPMSM, which led to a larger value of 178.7 W for its DC copper loss—more than 15.6 W compared to the DC copper loss of 163.1 W obtained for the SPMSM. As mentioned before, since the PMs of the SPMSM are mounted on the surface of the rotor, the eddy current loss on the PM is greater and more complex in the case of the SPMSM than in the IPMSM. In addition, due to the larger magnetic flux entering the stator core from the air gap, the SPMSM experiences a higher stator iron loss than the IPMSM. However, due to the high flux density in the bridge areas, the rotor iron loss of the IPMSM is preferable over that of the SPMSM. Overall, the total losses of the two motors were observed to be almost the same, exhibiting a negligible difference of 4 W.

Fig. 8 illustrates the electromagnetic parameters (efficiency, power factor, output torque, reluctance torque, torque ripple, and output power). In Fig. 8(a), the efficiency and power factor reach maximum values of 95.828% and 1 at phase advance angles of 24 and 29 degrees, respectively. This increase in the phase advance angle implies an increase in the q-axis current, which results in an upsurge in reluctance torque, as presented in Fig. 8(b). However, upon reaching a certain value of q-axis current, the magnetic flux from the PMs began to weaken, leading to a decline in the alignment torque. Moreover, the reduction in alignment torque outweighed the increase in reluctance torque, leading to a decrease in the overall output torque of the motor. Meanwhile, the cogging torque remained unchanged even when the phase advance angle varied. As a result, as depicted in Fig. 8(c), the torque ripple initially exhibited a gradual decline, followed by an increase resulting from the variation in output torque. Based on the changes mentioned above, it is evident that the best IPMSM performance could likely be achieved by controlling the appropriate value of the phase advance angle for each working condition. Finally, the output power of the motor also exhibited a gradual decline, as shown in Fig. 8(d).

The output results are provided in Table 6, indicating that with the volume of the magnet, rotor dimensions, and design parameters kept the same, IPMSM offers better output power and output torque than SPMSM. The output power of the IPMSM is 7,564 W, which is considerably more than the 7,206 W achieved by the SPMSM. Furthermore, the shaft torque attained by the IPMSM is 48.2 Nm—higher than the desirable value of 47.75 Nm—while the SPMSM does not meet the expected torque requirements. As for the power factor, the axis inductance values of the SPMSM are usually quite small, while the addition of flux barriers to the rotor in the IPMSM makes its axis inductances larger. For these reasons, the value of the power factor is larger for the SPMSM than for the IPMSM. However, by controlling the phase advance angle, the IPMSM could not only attain an increase in the power factor but also offer excellent performance for the other parameters, as investigated in this study. For the initial design with a phase advance angle of 15 degrees, the power factor of the IPMSM was 0.972, while that achieved by the SPMSM was slightly higher at 0.982.

In this research, a novel method for the analytical calculation of IPMSMs and SPMSMs is proposed. Based on the analytical technique, a V-shape IPMSM with 7.5 kW power was first examined, following which the values for an SPMSM with the same rotor dimensions, PM volume, and pole–slot combination were computed to verify the analytical parameters of the two proposed motors. Subsequently, the analytical results were considered the required parameters to simulate electromagnetic parameters for the motors via the FEM. The results showed that the SPMSM achieved lower torque and power densities, as well as higher torque ripple, than the IPMSM. Furthermore, the IPMSM exhibited the ability to change the phase advance angle—the value of the q-axis current component—to achieve better performance compared to the SPMSM. In terms of design optimization control, the IPMSM achieved excellent results with regard to its electromagnetic parameters, including output power, output torque, efficiency, power factor, and torque ripple. Meanwhile, the SPMSM attained a higher power factor, low harmonic components in the back EMF waveform, and featured a simpler structure than the IPMSM. Overall, the advantages and disadvantages of IPMSM and SPMSM should be accounted for when making a choice regarding the appropriate machine to use for specific industry applications.