Design and Optimization of a High-Efficiency Lightweight Permanent Magnet In-Wheel Motor with Torque Performance Improvement

Abstract

In this paper, a lightweight permanent magnet in-wheel (LW-PMIW) motor is proposed. This research focuses on using a multi-modulation design to enhance the amplitude of the fundamental wave while suppressing high-order harmonics, thereby enabling the motor to achieve high output torque, a light weight, and a high efficiency. Firstly, a combined trade-off factor related to motor mass, losses, and torque is defined specifically to provide guidance for the design. Secondly, a dual-rotor structure is adopted, and a harmonic injection (HI) design is applied to the permanent magnets (PMs). By designing a targeted harmonic injection ratio coefficient, the non-working harmonics of the PM magnetomotive force (MMF) can be weakened. Then, two iron modulating blocks are introduced to asynchronously modulate the PM MMF, which can further enhance the fundamental amplitude and improve the distribution of the airgap magnetic field. Finally, to verify the effectiveness of the multi-modulation design, the electromagnetic performance of the motor is evaluated and analyzed. The analytical and simulation results show that the torque of the proposed motor can reach 35.4 Nm, which is an increase of 19.6% while the torque ripple remains unchanged compared with the initial motor. Meanwhile, the output power increased by 0.37 kW. Hence, the rationality and effectiveness of the motor design are verified.

1. Introduction

In recent years, traditional agriculture has been rapidly transitioning towards precision agriculture, and the electrification and automation of related equipment are the key driving forces of this evolution. However, traditional fuel-powered machinery has inherent limitations such as a low energy efficiency and significant pollution, which cannot meet the needs of sustainable development. In contrast, electrically driven equipment, as a typical representative of green and intelligent agriculture, has cleaner operating characteristics and is easier to adapt to precise control systems [1,2]. As the core power source in such equipment, motors directly determine operational efficiency, reliability, and overall performance [3,4], making their optimal design a pivotal factor in accelerating the electrification of agricultural machinery.

The unique and harsh operating conditions of agricultural environments impose stringent and multifaceted requirements on motor performance. For instance, deep plowing operations demand sustained high torque output to tackle tough soil [5]; precision tasks like variable-rate seeding or pesticide spraying require minimal torque ripple to ensure operational accuracy; and machinery navigating complex terrains necessitates a balance of high flexibility, rapid response, and energy efficiency [6,7]. Permanent magnet synchronous motors (PMSMs) have become a promising technology for promoting sustainable development in modern agriculture due to their superior characteristics, including high torque density and reliability [8]. However, smaller agricultural devices face additional strict weight limits [9,10]. While innovations like irregular magnet shapes and optimized iron poles have been proposed to enhance torque while reducing mass [11], traditional PMSMs still rely on heavy iron cores, which increase both weight and inertia. This not only hinders light weight goals, but also damages dynamic response—for applications that require frequent start–stop or precise control [12,13,14,15,16], a slow response may compromise operational accuracy. Ironless PMSMs offer a potential solution to these challenges. A comparative analysis shows that compared with iron core motors, ironless motors have smaller torque ripple, which is very advantageous in the context of precision operation applications [17]. In addition, ironless motors achieve a significant weight reduction by eliminating heavy iron cores, eliminating eddy current losses and hysteresis losses caused by iron cores, making them particularly suitable for scenarios that prioritize a lightweight design, rapid response, and smooth operation. However, their relatively low torque density restricts their application in heavy-duty agricultural machinery that requires an extremely high starting torque or a sustained high torque output [18,19,20,21]. This means that existing motor technologies struggle to simultaneously meet the agricultural sector’s demands for a lightweight design, high efficiency, and sufficient torque performance. Therefore, how to efficiently design ironless PMSMs that achieve a light weight and high efficiency while also having good torque performance is one of the current research hotspots in the field.

Current research on torque enhancement in ironless PMSMs has primarily centered on axial motors and linear motors. For example, Reference [22] combined a coreless stator with a Halbach rotor and introduced double-layer stacked non-overlapping windings, achieving high torque while increasing efficiency to 98.5%. This is beneficial for improving the efficiency and reliability of agricultural equipment that requires long-term operation during busy farming seasons [23,24]. Moreover, similar situations can also be found in [25,26]. These studies highlight that the magnetic focusing effect of Halbach arrays can partially compensate for the increased magnetic reluctance of the main magnetic circuit caused by the absence of an iron core, thereby enhancing torque performance. Reference [27] comprehensively reviews the research status and future development directions of ironless axial flux PM motors in key areas such as topology, performance optimization, and structural technology. Research has shown that this type of motor has no iron loss, a high efficiency, and a high power density, which can effectively balance efficiency and light weight. In addition, the symmetrical and asymmetrical segmented design of PMs has been proven to effectively improve the trapezoidal shape of the back electromotive force (EMF) and enhance electromagnetic torque [28]. Despite these advancements, existing designs have critical limitations for agricultural applications. On the one hand, linear motors require high-precision linear guides and high costs, which will to some extent affect the light weight of the motor and limit its application in small devices [29,30,31,32]. On the other hand, axial motors require sturdy bearings and mechanical structures to withstand the huge axial magnetic attraction, which again undermines the lightweight design of the motor. While symmetrical structures have been proposed to cancel axial tilting torque [33], PM position deviations can influence the synthesizing no-load back EMF and electromagnetic torque. Even hybrid axial–transverse flux PM motors with compound PM rotors, which exhibit a high torque capacity [34], retain axial force issues.

A recent interesting study directly compared radial and axial motors and found that radial motors have an advantage in torque/mass ratio [35]. A common feature of axial motors that cannot be ignored is that they can achieve a higher torque density through a dual airgap design and a more compact magnetic circuit layout [36]. Inspired by this, integrating dual airgaps into the design of radial flux motors could improve the problem of low torque density [37]. In addition, scholars have continuously proposed innovative PM topology structures to improve the torque performance of motors [38,39,40]. A comparative study found that integral slot-distributed windings with interior PM structures exhibit the highest torque/power capability [41]. Meanwhile, as another element modulator that affects airgap harmonics, References [42,43] have improved the leakage flux of the motor by studying the number of modulation teeth and different modulation tooth structures, thereby increasing the torque density of the motor. However, a single design cannot simultaneously meet multiple performance improvements. For example, in [44], the use of PM shape design to reduce non-working harmonics can also affect working harmonics. Inspired by these insights, this study proposes a multi-modulation lightweight PM in-wheel (LW-PMIW) motor with a dual-rotor structure, which features an inner and outer double-layer airgap design.

The rest of this article is organized as follows. In Section 2, the LW-PMIW motor topology structure is presented, and a trade-off factor Y_total is defined based on the application background of this study. Then, as a place for exchanging energy between the armature and PM magnetic field, the airgap can intuitively reflect the performance of the motor from the perspective of airgap harmonics [45]. Thus, a theoretical analysis is conducted on torque and loss generation and improvement from the harmonic level, providing theoretical guidance for subsequent design. In Section 3, by adopting harmonic injection (HI) design and introducing modulators, a reasonable multi-modulation design is carried out on the initial prototype, effectively suppressing the 54th and 90th harmonics while increasing the 18th harmonic. In Section 4, the performances of three motor topologies are compared and analyzed. As expected, the comparative analysis results validated the effectiveness of the multi-modulation design. Finally, in Section 5, the research work of this article is summarized and specific future research directions are proposed.

2. Motor Structure and Theoretical Analysis

2.1. Motor Structure

In this article, a novel LW-PMIW motor with 27 stator slots and 36 rotor poles is taken as the research object, as illustrated in Figure 1. The major parameters of the LW-PMIW motor are listed in

Table 1. Basic design parameters of the LW-PMIW motor.

Items	Value	Unit
Outer radius of the outer rotor shell Rshell_oo	153.5	mm
Radius of the outer rotor iron ring Riron_o	150.5	mm
Radius of the outer rotor Rrotor_o	149	mm
Inner radius of the outer rotor shell Rshell_oi	143	mm
Outer radius of the inner rotor shell Rshell_io	130	mm
Radius of the inner rotor Rrotor_i	129	mm
Radius of the inner rotor iron ring Riron_i	124	mm
Inner radius of the inner rotor shell Rshell_ii	118.9	mm
Radius of stator shell Rshell_s	142	mm
Outer airgap length Rairo	1	mm
Inner airgap length Rairi	1	mm
Rated speed nr	600	rpm
Rated current Im	10	A
Stack length le	50	mm
Stator slot number Ns	27	--
Rotor pole number Pr	18	--

. The LW-PMIW motor adopts a dual-rotor design. The inner and outer rotors and the stator shell are made of stainless steel. And in which the PM are placed in the inner and outer rotors as well as the stator winding frame are made of aluminum. In order to increase the harmonic amplitude, the inner and outer rotor PM outer surfaces adopt an HI design. Two stator iron modulating blocks are placed in the middle post of the stator winding frame to increase the utilization of the magnetic field. In addition, by designing the motor winding as a flat wire, the losses generated on the winding during motor operation are reduced, thereby improving the efficiency of the motor.

2.2. Theoretical Analysis

In order to establish a simple model between the motor performance and airgap harmonics, it is necessary to make the following assumptions.

(1)

The PM permeability is the same as that of the air.

(2)

The iron core permeability is considered as infinite, and the saturation effect is ignored.

(3)

The magnetic leakage is neglected.

There is a complex interdependent relationship between the torque, quality, and efficiency of the LW-PMIW motor. Generally, it is necessary to find the optimal balance point between these conflicting targets to meet the requirements of specific applications. In order to better find the optimal balance point, this paper proposes a trade-off factor Y_total, as shown in Equation (1):

$$Y_{total}={\displaystyle \frac{{\lambda }_{tm}{T}_{e1}{m}_0}{T_{e0}{m}_1}}+{\displaystyle \frac{{\lambda }_{ml}{m}_1{P}_{cu0}}{m_0{P}_{cu1}}}$$

(1)

where T_e0 and T_e1 represent the torque before and after improvement, respectively. m₀ and m₁ are the initial and improved quality, which are closely related to the motor topology and material. P_cu0 and P_cu1 represent the copper consumption before and after improvement, respectively. λ_tm and λ_ml are weight coefficients for torque and efficiency, respectively, which need to satisfy the following relationship:

$$\left\{\begin{array}{l}0<{\lambda }_{tm}<1\\ 0<{\lambda }_{ml}<1\\ {\lambda }_{tm}+{\lambda }_{ml}=1\end{array}\right.$$

(2)

According to the realistic application background, the motor needs to meet the following design standards:

$$\left\{\begin{array}{l}{T}_e>15\mathrm{N}\mathrm{m}\\ T_{ripple}<0.5\%\\ Efficiency>90\%\end{array}\right.$$

(3)

Considering the potential research background of the motor and the fact that output torque is usually the primary factor in ensuring the stable driving operation of motors, a relatively large coefficient is required for the torque factor in the Y_total. In addition, a higher operating efficiency is often an important performance requirement in the motor design process, so losses, as an important factor affecting motor efficiency, should not be too small in the trade-off factor. Therefore, based on design experience and comprehensive considerations, the weight coefficients for torque and efficiency are selected as λ_tm as 0.7 and λ_ml as 0.3, respectively.

Then, it is necessary to theoretically analyze the relationship between the airgap harmonics and the torque performance and losses of the motor, providing theoretical guidance for targeted design. Generally, the total torque T of a rotating motor can be regarded as the sum of the torques generated by each harmonic. The torque and torque ripple generated by each harmonic order of the airgap can be calculated by Equations (4) and (5), respectively, according to the Maxwell stress tensor method.

$$T_k(t)={\displaystyle \frac{\pi R_{air}^2{L}_a}{{\mu }_0}}{B}_{rk}{B}_{tk}\mathrm{c}\mathrm{o}\mathrm{s}\left[{\theta }_{rk}(t)-{\theta }_{tk}(t)\right]$$

(4)

$$T_{kripple}={\displaystyle \frac{Max(T_k(t))-Min(T_k(t))}{Avg(T_{out}(t))}}$$

(5)

where R_air is the radius of airgap, L_a is the core length, and μ₀ is the vacuum permeability. k is the harmonic order of the airgap. In addition, B_rk and B_tk are the radial and tangential amplitudes of k-th airgap flux density harmonics, respectively. θ_rk and θ_tk are the radial and tangential phase of k-th airgap flux density harmonics, respectively.

The three main harmonics that generate torque and torque ripple can be calculated through Equations (4) and (5), as shown in Figure 2a. It can be seen that the harmonics that generate the main torque ripple of the motor are the 18th, 54th, and 90th harmonics, and the 18th is also the main harmonic that generates torque. In order to clearly reflect the frequency of harmonics, Figure 2b shows the radial and tangential phase differences in three harmonics within one cycle.

In addition, copper loss makes up the largest proportion of total loss for the LW-PMIW motor, so reducing its copper loss can effectively improve efficiency. Generally, compared with round wire windings, flat wire windings have advantages such as a higher slot filling rate, shorter end length, and better heat dissipation performance. Therefore, this article improves Y_total by using flat wire winding to reduce motor copper loss.

Actually, the AC copper loss of the winding is composed of DC copper loss P_dcl and winding eddy current loss P_ecl as follows:

$$P_{cu}=P_{dcl}+P_{ecl}=I^{2}R+{\displaystyle \frac{{\pi }^3{r}^{4}l}{\rho }}{\displaystyle \sum _k^{\infty }\left(f_{rk}^2{B}_{rk}^2+f_{tk}^2{B}_{tk}^2\right)}$$

(6)

where I is the effective value of DC current, R is the resistance, r is the radius of the conductor, l is the length of the conductor, and ρ is the resistivity of the conductor material. B_rk and B_tk are the radial and tangential amplitudes of k-th airgap flux density harmonics, respectively. f_rk and f_tk are the radial and tangential frequency, respectively.

From Equation (6), it can be seen that current has a significant impact on P_dcl. If the airgap flux density is increased while ensuring that it does not affect the output torque, the torque constant can be increased to reduce the required current. In other words, enhancing the airgap magnetic density can effectively reduce copper loss. It is also worth noting that the high-order harmonics in the airgap can generate eddy current losses, so reducing the amplitude of the high-order harmonic components can reduce P_ecl.

3. The Design Process of the LW-PMIW Motor

3.1. The PM Topology Design

The previous section analyzed the sources of torque and copper loss, and it can be seen that improving the airgap magnetic field is the key to improving Y_total. As the excitation source of the motor, the PM directly affects the distribution of the airgap magnetic field of the motor, thereby affecting performance aspects such as output torque, losses, and torque ripple. Corresponding to the LW-PMIW motor structure shown in Figure 1, Figure 3 presents the initial PM topology and its MMF waveform, where b is the width of the PM, h_pm is the thickness of the PM, and h_move is the radial distance between two PMs. For the LW-PMIW motor, the initial inner F_pmi(θ,t) and outer F_pmo(θ,t) of PM MMF can be expanded by Fourier series into the following:

$$\left\{\begin{array}{l}{F}_{pmi}(\theta ,t)={\displaystyle \sum _{k=1}^{\infty }{F}_{pmik}}\cos [k{P}_r(\theta -{\omega }_{r}t)]\\ F_{pmo}(\theta ,t)={\displaystyle \sum _{k=1}^{\infty }{F}_{pmok}}\cos [k{P}_r(\theta -{\omega }_{r}t)]\end{array}\right.$$

(7)

where P_r is the number of pole pairs of the PM, ω_r is the rotor rotation angular velocity, and k is a positive odd number. F_pmik and F_pmok, respectively, represent the amplitude of the k-th harmonic component of the MMF of the inner rotor PM and the outer rotor PM, which is closely related to the width and thickness of the PM. Hence, the size of the PM is the key factor that directly affects the waveform of PM MMF, and its size needs to be designed reasonably.

Meanwhile, the change in magnetic source not only directly affects the performance of the motor, but also affects the quality of the motor, thereby affecting Y_total. Therefore, this article selected four highly sensitive parameters, including PM area S, radial distance h_move between the inner and outer PMs, the width of PM b, and radius R_pmi of the inner rotor PM. Figure 4a presents the sensitivity analysis of four important parameters on motor performance. Correspondingly, the trend of Y_total with changes in the four parameters is shown in Figure 4b,c. Finally, the optimal values for S, h_move, b, and R_pmi are determined to be 35 mm², 19.8 mm, 8.5 mm, and 125.4 mm.

In addition, due to the dual-rotor structure of the motor, the MMF in the airgap is the result of the superposition of inner and outer PM MMFs. Figure 5a shows the MMF generated by the inner and outer PMs, respectively, and the superposition result is shown in Figure 5b. It is found that when the offset angle θ of the inner and outer PM positions changes, the superposition result of the two parts will also change.

Figure 6 shows the trend of Y_total as a function of θ. Due to the number of PM pole pairs in the motor being 18, each pole can move within a range of −10 deg to 10 deg. It is found that Y_total reaches its maximum value when θ is −4 deg. In other words, it means that the magnetic energy utilization rate is highest at this time, and the motor can achieve the best comprehensive performance.

Then, according to Equation (6), it can be seen that changing F_pmik and F_pmok can alter the airgap harmonic characteristics. Based on this, the HI functions h_pmi(x) and h_pmo(x) are introduced into the PM design specifically. In order to produce a clear description, Figure 7 shows the shape design of the PM and MMF waveform after the introduction of the HI function. It can be seen that with the introduction of the HI function, the waveform of the PM MMF of the LW-PMIW motor will be changed.

It is important to mention that the the HI functions of the inner and outer rotors can be expressed as follows:

$$\left\{\begin{array}{l}{h}_{pmi}(x)=k_1\cos (P_{r}x)+k_2\mathrm{c}\mathrm{o}\mathrm{s}(3P_{r}x)+k_3\mathrm{c}\mathrm{o}\mathrm{s}(5P_{r}x)+R_{pmi}\\ h_{pmo}(x)=k_{11}\cos (P_{r}x)+k_{22}\cos (3P_{r}x)+k_{33}\cos (5P_{r}x)+R_{pmo}\end{array}\right.$$

(8)

where R_pmi and R_pmo are the radius from the PMs of the inner and outer rotors to the central axis of the motor, respectively. x is the absolute value of the tangential distance from each point on the HI function to the centerline of the PM. k₁, k₂, and k₃ are the proportional coefficients injected by the first-order, third-order, and fifth-order component in the HI function of the inner rotor PM, respectively. In order to enhance the first-order component and weaken the third-order and fifth-order components, the proportional coefficients need to satisfy k₁ > 0, k₂ < 0, and k₃ < 0. Meanwhile, considering that the amplitude of the first-order harmonic is the highest and the amplitude of the third-order harmonic is higher than that of the fifth-order harmonic, it is shown that |k₁| > |k₂| > |k₃|. Correspondingly, k₁₁, k₂₂, and k₃₃ are the proportional coefficients injected by the first-order, third-order, and fifth-order component in the HI function of the outer rotor PM, respectively. Similarly, the proportionality coefficient needs to satisfy k₁₁ < 0, k₂₂ > 0, k₃₃ > 0, and |k₁₁| > |k₂₂| > |k₃₃|.

In order to obtain a reasonable HI function, the influence of k₁, k₂, and k₃ on the main harmonics is studied, as shown in Figure 8. Due to the small influence of the inner rotor PM on the outer airgap, Figure 8 only shows the variation trend of the inner airgap harmonics with k₁, k₂, and k₃. Considering that reducing high-order harmonics is the main objective, it is necessary to first determine k₂ and k₃. From Figure 8b,c, it can be seen that with the introduction of k₂ and k₃, the 18th, 54th and 90th harmonics are decreased, but the magnitude of the 18th harmonic decrease is very small and can be almost ignored. In addition, by combining the trend of the 18th harmonic in Figure 8a, the final values for k₁, k₂, and k₃ can be determined to be 4.05, −0.2, and −0.05.

Similarly, Figure 9 shows the variation trend of external airgap harmonics with k₁₁, k₂₂, and k₃₃. According to the trend shown in Figure 9, the final values of the HI ratios k₁₁, k₂₂, and k₃₃ for the outer rotor PM are determined to be −0.74, 0.28, and 0.05, respectively. Finally, in order to obtain the relevant parameters of PM clearly, the parameters of optimized PM are listed in

Table 2. Optimized PM parameters of the LW-PMIW motor.

Items	Value	Unit
PM area S	35	mm2
Width of inner PM b	8.59	mm
Width of outer PM b1	8.1	mm
PM interpolar offset angle θ	−4	deg
Distance between the inner and outer PMs hmove	19.8	mm
Radius of inner PM Rpmi	125.4	mm
Radius of outer PM Rpmo	145.2	mm
k1	4.05	--
k2	−0.2	--
k3	−0.05	--
k11	−0.74	--
k22	0.28	--
k33	0.05	--

.

After introducing the HI function, the PM MMF of the inner rotor can be expressed as the superposition of the square MMF F_{pmi_downk} on the lower surface and the cosine-shaped MMF F_{pmi_upk} generated by the upper surface as follows:

$$F_{pmi}(\theta ,t)={\displaystyle \sum _{k=1,3,5\dots }^{\infty }{F}_{pmi\_downk}}\cos [k{P}_r(\theta -{\omega }_{r}t)]+{\displaystyle \sum _{k=1,3,5\dots }^{\infty }{F}_{pmi\_upk}}\cos [k{P}_r(\theta -{\omega }_{r}t)]$$

(9)

Similarly, the PM MMF of the outer rotor can be expressed as the superposition of the square MMF F_{pmo_upk} on the upper surface and the cosine-shaped MMF F_{pmo_downk} generated by the lower surface as follows:

$$F_{pmo}(\theta ,t)={\displaystyle \sum _{k=1,3,5\dots }^{\infty }{F}_{pmo\_upk}}\cos [k{P}_r(\theta -{\omega }_{r}t)]+{\displaystyle \sum _{k=1,3,5\dots }^{\infty }{F}_{pmo\_downk}}\cos [k{P}_r(\theta -{\omega }_{r}t)]$$

(10)

Among them, the various MMFs generated by the inner and outer rotor PMs after HI can be represented as follows:

$$F_{pmi\_upk}={\displaystyle \frac{2B_{pm}{h}_{pmi}(x)|x\in \left[-{\theta }_{rpm}/2,{\theta }_{rpm}/2\right]}{k\pi {\mu }_0{\mu }_r}}\cos (k\pi )\sin ({\displaystyle \frac{k{P}_r{\theta }_{rpm}}{2}})$$

(11)

$$F_{pmo\_downk}={\displaystyle \frac{2B_{pm}{h}_{pmo}(x)|x\in \left[-{\theta }_{rpm1}/2,{\theta }_{rpm1}/2\right]}{k\pi {\mu }_0{\mu }_r}}\cos (k\pi )\sin ({\displaystyle \frac{k{P}_r{\theta }_{rpm1}}{2}})$$

(12)

Finally, Figure 10 analyzes the airgap flux density generated by the PMs and change in harmonics before and after the design of the PMs. From the figure, it can be seen that the modulation design of PM can directionally suppress the 54th and 90th harmonics, providing conditions for reducing torque ripple and losses. Interestingly, the HI of PMs can also enhance the 18th harmonic, which will effectively improve the motor’s output torque.

3.2. The Modulator Design

According to the general airgap field modulation theory, the PM generates the initial PM MMF. The flux density harmonics of the PM magnetic fields are achieved by modulating MMF through a modulator function. Thus, in addition to the magnetic source, the modulator is also the key to changing the airgap harmonics. As shown in Figure 11, placing two iron modulating blocks inside the winding frame on the stator can modulate the PM MMF. Two iron modulating blocks are symmetrically distributed and both are rectangular, with a length of l_tie and a width of h_tie. Figure 11b shows the corresponding airgap permeance model, and it can be seen that the length and width of the iron modulating blocks will affect the peak height and waveform width, respectively.

At present, the permeance modulation function can be written as follows:

$$\begin{array}{cc}\hfill M_{stie}(\theta )& ={\displaystyle \sum _{m=0,\pm 1,\pm 2\dots }^{\infty }{M}_{sm}\cos (m{N}_s\theta +\frac{{\theta }_1+{\theta }_2}{2})}+{\displaystyle \sum _{m=0,\pm 1,\pm 2\dots }^{\infty }{M}_{sm}\cos (m{N}_s\theta -\frac{{\theta }_1+{\theta }_2}{2})}\hfill \\ & =2\cos ({\displaystyle \frac{{\theta }_1+{\theta }_2}{2}}){\displaystyle \sum _{m=0,\pm 1,\pm 2\dots }^{\infty }{M}_{stiem}\cos (m{N}_s\theta )}\hfill \end{array}$$

(13)

where N_s is the number of stator teeth, θ is the rotor position, m is an integer, θ₁ is the radians occupied by the distance between the iron modulating blocks and the column in the winding frame, and θ_tie is the radians occupied by the modulating blocks. M_stiem is the amplitude of the m-th harmonic component of the permeance modulation function, which can be expressed as follows:

$$M_{stiem}={\displaystyle \frac{2(R_{si}+h_{tie})}{m\pi N_s{R}_{air}^2}}\left\{\sin \left[m\pi -{\displaystyle \frac{m{N}_s(l_{tie}/{\theta }_s)}{4}}\right]-\sin {\displaystyle \frac{m{N}_s(l_{tie}/{\theta }_s)}{4}}\right\}$$

(14)

Combining Equations (9), (10) and (13), the magnetic density of the inner and outer airgaps can be obtained as follows:

$$\begin{array}{cc}\hfill B_{pmi}(\theta ,t)& =F_{pmi}(\theta ,t)M_{stie}(\theta )\hfill \\ & =\left[{\displaystyle \sum _{k=1,3,5\dots }^{\infty }{F}_{pmi\_upk}}\cos [k{P}_r(\theta -{\omega }_{r}t)]+{\displaystyle \sum _{k=1,3,5\dots }^{\infty }{F}_{pmi\_downk}}\cos [k{P}_r(\theta -{\omega }_{r}t)]\right]\hfill \\ & \times 2\cos ({\displaystyle \frac{{\theta }_1+{\theta }_2}{2}}){\displaystyle \sum _{m=0,\pm 1,\pm 2\dots }^{\infty }{M}_{stiem}\cos (m{N}_s\theta )}\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill B_{pmo}(\theta ,t)& =F_{pmo}(\theta ,t)M_{stie}(\theta )\hfill \\ & ={\displaystyle \sum _{k=1,3,5\dots }^{\infty }{F}_{pmo\_upk}}\cos [k{P}_r(\theta -{\omega }_{r}t)]+{\displaystyle \sum _{k=1,3,5\dots }^{\infty }{F}_{pmo\_downk}}\cos [k{P}_r(\theta -{\omega }_{r}t)]\hfill \\ & \times 2\cos ({\displaystyle \frac{{\theta }_1+{\theta }_2}{2}}){\displaystyle \sum _{m=0,\pm 1,\pm 2\dots }^{\infty }{M}_{stiem}\cos (m{N}_s\theta )}\hfill \end{array}$$

(16)

Based on the above analysis, in order to obtain a reasonable design of the modulator, four key parameters were studied, including the length l_tie and width h_tie of the modulating block, the radian occupied by the modulating block width θ_tie, and the radius R_tie of the modulation block from the motor center axis. Considering the synchronous trend of the influence of introducing modulating blocks on the inner and outer airgaps of the motor and the high amplitude of the external airgap harmonics, Figure 12 only shows the influence trend of each parameter on the outer airgap harmonics. In addition, through the design of PMs, the 54th and 90th harmonics have been effectively weakened, so the effect on the 18th harmonic will be the main basis for selection. From Figure 12a, it can be seen that the amplitude of the 18th harmonic decreases with the increase of h_tie and R_tie, and increases with the increase of l_tie and θ_tie. At the same time, taking into account the changing trends of the 54th and 90th harmonics, the optimal values for h_tie, l_tie, R_tie, and θ_tie are determined to be 7 mm, 0.7 mm, 132.5 mm, and 0.3 deg, respectively. In order to obtain the relevant parameters of the modulator clearly, the parameters of the optimized modulator are listed in

Table 3. Optimized modulator parameters of the LW-PMIW motor.

Items	Value	Unit
The length of modulator ltie	0.7	mm2
The width of modulator htie	7	mm
The radius of modulator Rtie	132.5	mm
The radian occupied by modulator width θtie	0.3	deg

.

Finally, Figure 13 presents a comparison of the airgap flux density waveform and main harmonic changes before and after introducing the iron modulating blocks. From the figure, there are two new components (iP_r + jN_s) and (iP_r − jN_s) produced by iron modulation. It is worth noting that the number of pole pairs in the armature winding is nine, which means that the newly introduced 9th harmonic will be beneficial for improving torque performance. In addition, it can be seen from Figure 12 that the 18th harmonic can be added while keeping the 54th and 90th harmonics unchanged. In other words, theoretically introducing a modulator would improve Y_total.

Furthermore, to clearly illustrate the distribution of the three design objectives—torque, efficiency, and the trade-off factor Y_total—Figure 14 displays the solution set and Pareto front derived from the multi-objective optimization, which subsequently determined the final design parameters.

In summary, through multi-modulation design, the 54th and 90th harmonics in the inner and outer airgap magnetic density are directionally weakened; in particular, the amplitude of the 90th harmonic is significantly reduced. In addition, the 18th harmonic is raised, which will increase the output torque. Meanwhile, due to the asynchronous modulation of the PM MMF by the iron modulating block, rich harmonics are generated. For example, the 63rd harmonic is generated by modulating the fundamental wave with third-order magnetic conductance, and it increases with the increase in the 18th harmonic, resulting in some torque ripple. In fact, the amplitude of other harmonics generated by high-order magnetic conductance harmonic modulation usually decreases with increasing magnetic conductance order. Considering that the introduction of a modulator can increase the amplitude of the 18th harmonic, which mainly generates harmonic components of torque, the positive influence of introducing a modulator outweigh their negative effects. Ultimately, the Y_total of the motor was effectively improved.

4. Electromagnetic Performance Evaluation

In order to verify the effectiveness of the multi-modulation design, this section compares and analyzes the motor performance of three motor topologies: initial rectangular PM, HI design PM, and optimized LW-PMIW motor with a multi-modulation design. For ease of presentation, the three topological structures are referred to as motor 1, motor 2, and motor 3, respectively, as shown in Figure 15.

The no-load back EMF of the three topology motors at the same speed is shown in Figure 16, and the difference in back EMF amplitude means the difference in torque. Although the no-load back EMF of all three topologies has good sinuosity, compared with the other two motors, motor 3 has a significant advantage in amplitude.

Then, generally speaking, the cogging torque T_cog of the motor is the derivative of the magnetic energy W_pm to the rotation angle θ of magnetic field, and its equivalent formula is as follows:

$$T_{cog}={\displaystyle \frac{d{W}_{pm}}{d\theta }}={\displaystyle \frac{d(W_{pmi}+W_{pmo})}{d\theta }}={\displaystyle \frac{d{\displaystyle \int \left[\frac{1}{2{\mu }_0}{B}_{pmi}^2(\theta ,t)dV+\frac{1}{2{\mu }_0}{B}_{pmo}^2(\theta ,t)dV\right]}}{d\theta }}$$

(17)

The cogging torque is proportional to the sum of the squares of the magnetic flux density amplitudes based on Equation (17). The cogging torque of the three motor topologies is shown in Figure 17, with motor 3 having the lowest peak to peak cogging torque, which indirectly validates the effectiveness of the multi-modulation design.

Furthermore, the torque of the three motor topologies at the rated current is shown in Figure 18. From the figure, it can be seen that compared with motor 1, the introduction of HI increases the torque of motor 2 by 1.7%, which is consistent with the variation in the 18th harmonic in Figure 10. In addition, the comparison between motor 2 and motor 3 demonstrates the advantage of introducing a modulator, which is consistent with the harmonic changes shown in Figure 13. In summary, due to the multi-modulation design, the harmonics related to motor torque have been improved, and the torque value of motor 3 is 35.4 Nm, an increase of 19.6% compared with motor 1. It is worth mentioning that the three motors maintain the same amount of PM, which means that motor 3 improves the utilization rate of PMs.

In addition, the mechanical stress of the proposed LW-PMIW motor at the maximum speed of 3000 rpm is shown in Figure 19. The maximum stress part of the inner rotor structure is about 4.71 MPa and the outer rotor structure is about 6.8648 MPa, which is located in the PM shape of the rotor. From the figure, it can be seen that the maximum stress of the proposed motor exists on the outer rotor and is relatively low, which is capable of meeting the stress requirements of the material.

Then, under the condition of ensuring the same input power, Figure 20 shows the output power of the three motor topologies. It can be seen that with the continuous improvement of motor topology, the output power of the motor continues to increase. This means that the efficiency of the motor has been improved.

In order to address threats to validity or alternative hypotheses,

Table 4. Comparative analysis of different design methods.

Method	Harmonic Variation	Output Torque	Torque Ripple	Output Power
HI	54th and 90th decrease; 18th increases	30.1 Nm	0.27%	1.89 kW
HI + Modulation	18th increases; (iPr ± jNs) generation	35.4 Nm	0.24%	2.23 kW

presents a comparative analysis of different design methods. From Table 4, the changes in motor harmonics and various performances under different design methods can be intuitively observed, highlighting the necessity of multi-modulation design methods and quantitatively verifying the effectiveness of this study.

Finally, as shown in

Table 5. Comparison between proposed method and other existing methods.

Comparison	Reference [25]	Reference [29]	Reference [36]	Proposed Method
Research object	AMFM-BDRM	CS-MGDRM	V-PMV	LW-PMIW
Main technical methods	Theoretical analysis; simulation.	Ironless stator; finite element analysis	Rotor shape design, star–delta hybrid connection winding	Multi-modulation design, definition of comprehensive trade-off factors
High output torque	×	×	×	√
Low torque ripple	-	√	√	√
Iron loss	√	×	√	×
High efficiency	-	√	√	√

, compared with existing techniques, the proposed design idea can achieve multi-objective collaborative optimization capability, and can simultaneously improve torque, power, weight, and efficiency. By quantifying factor Y_total and innovating modulation strategies, the design becomes more scientific, targeted, and has comprehensive advantages.

5. Conclusions

This study proposes a multi-modulation design strategy for an LW-PMIW motor, which achieves a comprehensive improvement in torque performance, quality, and efficiency through the collaborative optimization of a PM and a modulator. The core contributions and innovative value are as follows:

(1)

A novel trade-off factor Y_total is defined to quantify the balance between torque, efficiency, and airgap harmonics, breaking through the limitations of optimizing single performance indicators and providing generalizable theoretical guidance for multi-objective collaborative design.

(2)

The HI design adopted by the PM is different from the traditional harmonic suppression approach, achieving the dual goals of high-order harmonic suppression and 18th harmonic enhancement through directional regulation. The proposed HI ratio coefficient provides a quantitative basis for the magnetic field design of the PM. In addition, the modulator design actively introduces new harmonics and enhances the amplitude of the 18th harmonic, forming a synergistic modulation with the PM, breaking through the limitations of optimizing the modulator or PM topology separately in existing research and significantly improving the coupling efficiency with the armature magnetic field.

(3)

Through the multi-modulation design of PM source and modulator, the torque of the improved LW-PMIW motor is increased by 19.6% while the torque ripple remains unchanged. At the same time, the output power increased by 0.37 kW. The research results directly verify the engineering value of the motor in low-speed and high-torque scenarios, providing a feasible technical path for the performance improvement of this type of motor in agricultural driving, new energy equipment, and other fields.

(4)

The framework of “performance quantification factor—harmonic directional control—multi-modulation design” established in this study enriches the theory of harmonic optimization for PM motors and provides new ideas for the multi-objective design of similar modulation motors. It can be extended to the design of other motors with modulation structures.

For the LW-PMIW motor, the motor is suitable for low-speed, high-torque situations, and further research is needed for its application in higher speed situations. For high-speed scenarios, the focus will be on studying the mechanism of harmonic losses at high frequencies. By improving the modulator materials and optimizing the segmented structure of PMs, eddy current losses at high speeds will be suppressed. In addition, dynamic constraints such as temperature and vibration will be introduced to establish a multi-physics field collaborative optimization model over a wide speed range. Finally, carrying out prototype manufacturing and long-term operation experiments will verify the stability and durability of the motor in actual working conditions, and promote the transformation from theoretical design to engineering applications.