Torque Ripple and Electromagnetic Vibration Suppression of Fractional Slot Distributed Winding ISG Motors by Rotor Notching and Skewing

Abstract

Torque ripple and radial electromagnetic (EM) vibration can lead to motor vibration and noise, which are crucial to the motor’s NVH (Noise, Vibration, and Harshness) performance. Researchers focus on two main aspects: motor body design and control strategy, employing various methods to optimize the motor and reduce torque ripple and radial EM vibration. Rotor notching and segmented rotor skewing are frequently used techniques. However, determining the optimal notch and skew strategy has been an ongoing challenge for researchers. In this paper, an 8-pole, 36-slot ISG motor is optimized using a combination of Q-axis and magnetic bridge notching (QMC notch) as well as segmented rotor skewing to reduce torque ripple and radial EM vibration. Three skewing strategies—step skew (SS), V-shape skew (VS), and zigzag skew (ZS)—along with four segmentation cases are thoroughly considered. The results show that the QMC notch significantly reduces torque ripple, while skewing designs greatly diminish radial EM vibrations. However, at 14 $f_e$, the EM vibration frequency is close to the motor’s third-order natural frequency, leading to mixed results in vibration reduction using skewing techniques. After a comprehensive analysis of all skewing strategies, four-segment VS and ZS are recommended as the optimal approaches.

1. Introduction

In the transportation industry, roads account for 75% of total energy use, while internal combustion engine vehicles (ICEVs) contribute 25–30% of greenhouse gas (GHG) emissions [1]. Continuing to use conventional ICEVs limits the potential for reducing emissions. Consequently, manufacturers and governments have started to develop eco-friendly modes of transportation, including hybrid electric vehicles (HEVs) and electric vehicles (EVs) [2]. Current EV technology faces issues such as limited range, high initial cost, and long recharge times. Although the limited range may not be a problem in many metropolitan areas and developing countries, the lack of sufficient fast-charging stations remains a barrier to entry even in these suitable regions. The HEV technology can be developed to overcome the aforementioned shortcomings of both ICEs and EVs [3]. These HEVs are classified into micro, mild, and full HEVs based on operating voltage and electrical power level [4].

A key technology for micro HEVs is starter/generator (SG) technology. The main feature of the SG hybrid system is its ability to stop the engine from idling. This allows the engine to be switched off frequently and restarted when necessary, effectively saving fuel during idling and deceleration. In a micro HEV system, the starter and the generator are integrated into a single electric machine called integrated starter and generator(ISG), which serves the dual purpose of starting and generating electricity [5]. The rapid advancement of permanent magnet (PM) materials over recent decades has significantly enhanced the design of high-performance permanent magnet synchronous motors (PMSMs). These motors are highly appealing due to their efficiency, minimal maintenance requirements, and impressive power and torque density. Interior permanent magnet synchronous motors (IPMSMs) have better demagnetization resistance and flux-weakening capabilities compared to surface-mounted permanent magnet synchronous motors (SPMSMs) [6]. Consequently, it is widely used in ISG machines.

However, the use of PMs in IPMSMs introduces two significant issues. The first is high torque ripple caused by tangential electromagnetic (EM) forces, and the second is intense radial EM vibration caused by radial EM forces formed by the complex air-gap magnetic field. These issues restrict the use of IPMSMs in some industries and are major concerns in current engineering practices [7].

Many scholars have conducted research on the torque ripple and radial vibration of IPMSMs. There are two main approaches to reducing torque ripple and radial EM vibration in motors. One approach is to use new EM and structural design, while another method involves utilizing advanced drive topology and control strategy [8,9]. In terms of EM and structural design, torque ripple and EM vibration are mainly suppressed by optimizing the stator and rotor, PMs, pole-slot combinations, and winding methods. Sun et al. present a novel multilevel optimization strategy for the multi-objective design of an IPMSM. By employing this multilevel approach and optimizing various parameters of the stator and rotor, the strategy effectively reduces torque ripple [10]. Fatemi et al. compared various rotor designs and pole-slot combinations, ultimately selecting a light rare earth metal V-shape PMSM for further driving cycle optimization. The design was further refined to reduce mass and torque ripple by adding inner cavities to the rotor laminations, and a multiphysics field analysis was performed [11]. The rotor designs of IPM for EVs were evaluated in [12]. Five different rotor topology configurations for automotive motors were studied, keeping the motor stator size constant while varying the rotor topology. The torque ripple performance of each configuration was then assessed. Shah et al. analyzed IPMSM vibration and noise for three winding arrangements [13]. The method of notching the rotor or stator to reduce torque ripple in IPMSM was analyzed in [14,15,16,17,18,19]. Ocak et al. proposed a method that aims to optimize the skew length and skew angle to achieve minimal torque ripple, differing from the conventional step-skew approach where both the skew length and skew angle remain constant [20]. Peng and Wang et al. present a novel rotor-segmented motor with varying pole widths to reduce EM vibration in EVs, and the results indicate that the presented motor effectively reduces EM vibration, significantly decreasing the 0th-order vibration amplitude [21,22]. Wang et al. investigated a rotor skewed strategy for PM brush DC motors, comparing the vibration effects of zigzag skew and straight skew. The results demonstrated that zigzag skew is more effective in reducing vibration [23]. Wang et al. compared the suppression effects of rotor-step straight with zigzag skewing, considering the elastoplasticity of the stator material. It highlights that rotor-step zigzag skewing can address the limitations of straight skewing in controlling EM vibrations in practical applications [24].

The existing research literature indicates that rotor notching and skewing are common methods for reducing torque ripple and radial EM vibration. However, the strategy of rotor notching and skewing significantly influences these results. Additionally, the number of skew segments and the skew angle greatly impact the outcomes, warranting further in-depth study. This paper aims to address these gaps by exploring optimized rotor designs and providing a comprehensive analysis of the effects of notching and skewing on motor performance, with the goal of offering new insights for enhancing vibration control in IPMSMs.

In this paper, the torque ripple and radial EM vibration of a 36-slot 8-pole motor are analyzed and optimized. First, the torque ripple and EM force of PMSMs are analyzed. Then, A 36-slot, 8-pole ISG machine is designed with consideration of rotor notching and skewing. The specific design of the rotor notching is detailed in [25]. For the rotor skew strategy, step skew (SS), V-shaped skew (VS), and zigzag skew (ZS) are considered, with varying numbers of segments and corresponding skew angles. Finally, a comparative analysis of the motor’s torque ripple and radial EM vibration is conducted for each configuration.

2. Torque Ripple and Radial EM Force of PMSM

During PMSM operation, the air gap EM force wave rotates relative to the stator core. Its effect on the stator teeth can be represented as a distributed EM force acting on the top of the stator teeth. The distributed force on the tooth tip simplifies into a concentrated force and a moment, with the force further divided into radial and tangential components [26,27], as depicted in Figure 1.

Typically, the tangential EM force in the motor is small, but it can cause torque ripple. In contrast, the radial EM force is large and can lead to significant radial vibration. The following sections analyze the characteristics of torque ripple and radial EM force, respectively.

2.1. Torque Ripple

For a conservative system with no energy loss, the motor’s output torque can be obtained by using the virtual work principle,

$$T={\left.{\displaystyle \frac{\partial W_{\mathrm{m}}^{\prime }}{\partial {\alpha }_{\mathrm{m}}}}\right|}_{i=\mathrm{const}}=\sum _n{i}_n{\left.{\displaystyle \frac{\partial {\mathrm{\Psi }}_n}{\partial {\alpha }_{\mathrm{m}}}}-{\displaystyle \frac{\partial W_{\mathrm{m}}}{\partial {\alpha }_{\mathrm{m}}}}\right|}_{i=\mathrm{const}}$$

(1)

where $W_{\mathrm{m}}^{\prime }$ and $W_{\mathrm{m}}$ are magnetic energy and magnetic common energy of the system, i and $\mathrm{\Psi }$ are the current and magnetic chain, ${\alpha }_{\mathrm{m}}$ is the mechanical angle of the motor, and n is the phase sequence of the winding.

For PMSMs, the magnetic chain can be divided into the PM chain and the magnetic chain generated by the windings.

$${\mathrm{\Psi }}_n={\psi }_n^{\mathrm{PM}}+{\psi }_n^i$$

(2)

where ${\psi }_n^{\mathrm{PM}}$ and ${\psi }_n^i$ are PM chains and the magnetic chains that are generated by the windings. so the Equation (1) can be written as

$$T=\sum _n{i}_n{\displaystyle \frac{\partial {\psi }_n^{PM}}{\partial {\alpha }_m}}+\sum _n{i}_n{\left.{\displaystyle \frac{\partial {\psi }_n^i}{\partial {\alpha }_m}}-{\displaystyle \frac{\partial W_m}{\partial {\alpha }_m}}\right|}_{i=\mathrm{const}}$$

(3)

In Equation (3), the first term represents the PM torque, the second term corresponds to the reluctance torque, and the third term is the cogging torque. The cogging torque expression for the motor is as follows,

$$T_{\mathrm{cog}}\left({\alpha }_{\mathrm{m}}\right)={\displaystyle \frac{\pi \mathrm{z}{L}_a}{4{\mu }_0}}\left(R_2^2-R_1^2\right)\sum _{n=1}^{\infty }n{G}_n{B}_{r{\displaystyle \frac{n\mathrm{z}}{2p}}}sinn\mathrm{z}{\alpha }_{\mathrm{m}}$$

(4)

where $L_a$ is the axial length of the armature core, and $R_1$ and $R_2$ are the outer and inner radii of the armature and stator yoke, and n is the integer ensuring $n\mathrm{z}/2p$ is an integer, p is pole pairs, ${\mu }_0$ is the permeability of vacuum, z represents the number of armature slots, $G_n$ is the Fourier decomposition coefficient for the square of the relative air-gap permeability, $B_{r{\displaystyle \frac{n\mathrm{z}}{2p}}}$ are the coefficients in the Fourier expansion of the flux density for the PM.

2.2. Radial EM Force of PMSM

The EM forces and intrinsic modes of the motor determine its harmonic vibration response. According to Maxwell’s tensor equation, the radial EM force density on the stator teeth surfaces is [28],

$$P_r(\theta ,t)={\displaystyle \frac{b_r^2(\theta ,t)-b_t^2(\theta ,t)}{2{\mu }_0}}\approx {\displaystyle \frac{b_r^2(\theta ,t)}{2{\mu }_0}}$$

(5)

where $b_r(\theta ,t)$ represents the radial flux density and $b_t(\theta ,t)$ denotes the tangential flux density. For this analysis, the tangential flux density $b_t(\theta ,t)$ is considered negligible.

Taking into account the impact of stator opening slots, the radial flux density $b_r(\theta ,t)$ can be expressed as follows,

$$\left\{\begin{array}{c}{b}_r(\theta ,t)=[f_{PM}(\theta ,t)+f_{ARM}(\theta ,t)]\times {\mathrm{\Lambda }}_s\left(\theta \right)\hfill \\ f_{PM}(\theta ,t)=\sum _{\mu }{F}_{\mu }cos(\mu p\theta -\mu \omega t)\hfill \\ f_{ARM}(\theta ,t)=\sum _{\nu }{F}_{\nu }cos(\nu \alpha \theta -S_{\nu }\omega t)\hfill \\ {\mathrm{\Lambda }}_s\left(\theta \right)={\mathrm{\Lambda }}_0+\sum _k{\mathrm{\Lambda }}_{k}cos\left(kz\theta \right)\hfill \end{array}\right.$$

(6)

where $f_{PM}(\theta ,t)$ represents the PM magnetomotive force (MMF), while $f_{ARM}(\theta ,t)$ denotes the armature MMF. The function ${\mathrm{\Lambda }}_s\left(\theta \right)$ indicates the relative permeance and ${\mathrm{\Lambda }}_0$ is the mean magnetic conductivity. The term k refers to the harmonic order, with $\mu$ and $\nu$ denoting the harmonic orders for the PM and armature fields, respectively. The parameter $\alpha$ represents the number of motor units, and $S_{\nu }$ indicates the rotation direction of the $\nu$ harmonic, with values of $\pm 1$.

The radial EM force density is obtained from Equations (5) and (6)

$$\begin{array}{cc}\hfill P_{r_{\mathrm{onload}}}(\theta ,t)& \approx {\displaystyle \frac{b_r^2(\theta ,t)}{2{\mu }_0}}={\displaystyle \frac{{\left[f_{\mathrm{PM}}(\theta ,t)+f_{\mathrm{ARM}}(\theta ,t)\right]}^2\times {\mathrm{\Lambda }}_s^2\left(\theta \right)}{2{\mu }_0}}\hfill \\ & ={\displaystyle \frac{1}{2{\mu }_0}}{\left[\sum _{\mu }{F}_{\mu }cos(\mu p\theta -\mu \omega t)+\sum _{\nu }{F}_{\nu }cos(\nu \alpha \theta -S_{\nu }\omega t)\right]}^2\times {\left[{\mathrm{\Lambda }}_0+\sum _k{\mathrm{\Lambda }}_{k}cos\left(kz\theta \right)\right]}^2\hfill \\ \hfill & ={\displaystyle \frac{1}{4{\mu }_0}}\left.\left\{\begin{array}{c}\sum _{{\mu }_1}\sum _{{\mu }_2}{F}_{{\mu }_1}{F}_{{\mu }_2}cos[({\mu }_1\pm {\mu }_2)p\theta -({\mu }_1\pm {\mu }_2)\omega t]\hfill \\ +2\sum _{\mu }\sum _{\nu }{F}_{\mu }{F}_{\nu }cos[(\mu p\pm \nu \alpha )\theta -(\mu \pm S_{\nu })\omega t]\hfill \\ +\sum _{{\nu }_1}\sum _{{\nu }_2}{F}_{{\nu }_1}{F}_{{\nu }_2}cos[({\nu }_1\pm {\nu }_2)\alpha \theta -(S_{{\nu }_1}\pm S_{{\nu }_2})\omega t]\hfill \end{array}\right.\right\}\times \left\{\begin{array}{c}{\mathrm{\Lambda }}_0^2+2{\mathrm{\Lambda }}_0\sum _k{\mathrm{\Lambda }}_{k}cos\left(kz\theta \right)\\ +{\displaystyle \frac{1}{2}}\sum _{k_1}\sum _{k_2}{\mathrm{\Lambda }}_{k_1}{\mathrm{\Lambda }}_{k_2}cos\left[(k_1\pm k_2)z\theta \right]\end{array}\right\}.\hfill \end{array}$$

(7)

The radial EM force harmonics with a spatial order of $(\mu p\pm \nu \alpha )$ significantly affect vibrations. The corresponding temporal order is $(\mu \pm S_v)f$. The harmonic orders for the PM field MMF and the armature field MMF can be expressed as follows:

$$\mu =2m+1,m=0,1,2,3,\dots$$

(8)

$$v=3n+1,n=0,\pm 1,\pm 2,\dots$$

(9)

The research subject of this paper is a 36-slot, 8-pole double-layer winding ISG machine, the order of the radial EM force wave is shown in

Table 1. Harmonic order of radial EM force wave of unit motor.

Homonic Order	Space/Temporal Order
	$\mathbf{\nu }$
$\mu$	1	−2	4	−5	7	−8	10
1	$0/0$	$-1/2$	$-3/0$	$-4/2$	$-6/0$	$-7/2$	$-9/0$
	$2/2$	$3/0$	$5/2$	$6/0$	$8/2$	$9/0$	$11/2$
3	$2/2$	$1/4$	$-1/2$	$-2/4$	$-4/2$	$-5/4$	$-7/2$
	$4/4$	$5/2$	$7/4$	$8/2$	$10/4$	$11/2$	$13/4$
5	$4/4$	$3/6$	$1/4$	$0/6$	$-2/4$	$-3/6$	$-5/4$
	$6/6$	$7/4$	$9/6$	$10/4$	$12/6$	$14/4$	$15/6$
7	$6/6$	$5/8$	$3/6$	$2/8$	$0/6$	$-1/8$	$-3/6$
	$8/8$	$9/6$	$11/8$	$12/6$	$14/8$	$15/6$	$17/8$
9	$8/8$	$7/10$	$5/8$	$4/10$	$2/8$	$1/10$	$-1/8$
	$10/10$	$11/8$	$13/10$	$14/8$	$16/10$	$17/8$	$19/10$
11	$10/10$	$9/12$	$7/10$	$6/12$	$4/10$	$3/12$	$1/10$
	$12/12$	$13/10$	$15/12$	$16/10$	$18/12$	$19/10$	$21/12$
13	$12/12$	$11/14$	$9/12$	$8/14$	$6/12$	$5/14$	$3/12$
	$14/14$	$15/12$	$17/14$	$18/12$	$20/14$	$21/12$	$23/14$
15	$14/14$	$13/16$	$11/14$	$10/16$	$8/14$	$7/16$	$5/14$
	$16/16$	$17/14$	$19/16$	$20/14$	$22/16$	$23/14$	$25/16$

, and the negative sign in the table indicates that the harmonic shift is reversed. The motor is composed of four unit motors, and for the results shown in Table 1, both the PM harmonic order number and the EM force spatial order are multiplied by four, while the temporal order remains unchanged. The lowest nonzero spatial order of the radial force wave harmonics is four.

3. ISG Motor Model with Rotor Notching and Skewing Configurations

3.1. Initial Design of the ISG Motor

An exploded view of the prototype model is shown in Figure 2.

Table 2. Initial parameters of ISG machine.

Parameters	Value	Unit
Stator Outer Diameter	130	mm
Stator Inner Diameter	81.4	mm
Rotor Outer Diameter	80	mm
Rotor Inner Diameter	30.9375	mm
Air-gap	0.7	mm
Stack Length	78	mm
Rate Speed	4200	rpm
Max Speed	15,000	rpm
PM Width	14.44	mm
PW Thickness	3.5	mm
Phase	3
Permanent Magnet	N36Z
Silicon Steel Sheet	50PN470

shows the main parameters of the model. The initial ISG machine studied in this paper is a 36-slot, 8-pole IPMSM with a double winding layer and distributed winding, as shown in Figure 3. Figure 3 on the left illustrates the three-dimensional configuration of the motor’s stator and rotor. To better utilize the power characteristics of the motor, PMs are arranged in Bar-shape. To ensure that the strength of the rotor meets the requirements at high rotational speeds, the permanent magnet is divided into two pieces. The upper diagram on the right displays the motor’s armature winding design, configured as a three-phase short-pitch winding with a winding pitch of four. While the lower diagram depicts the rotor notching schematic, it should be noted that the initial motor does not have the notching setup as shown in the figure.

3.2. Rotor Notching Design

According to Equation (4), reducing the amplitude of $G_n$ decreases the cogging torque. Additionally, incorporating notches in the motor rotor reduces magnetic leakage, thereby diminishing torque pulsation. The notch strategy is categorized into three types, Q-axis notch, magnetic bridge notch(MB notch), and Q-axis and magnetic bridge notch combination(QMC notch). The specific rotor notching configuration is illustrated in the lower right side of Figure 3. Through analysis and validation, the QMC notch was found to be preferred. The main variables include the magnetic bridge notch offset angle $\gamma$, notch depth $d_1$, notch angles ${\alpha }_1$, ${\alpha }_2$, Q-axis notch depth $d_2$, and notch angles ${\beta }_1$, ${\beta }_2$, respectively. The Q-axis notches are designed symmetrically, ${\beta }_1={\beta }_2$. Due to the large number of parameters, the parameter values that minimize torque ripple are determined using the non-dominated sorting genetic algorithm-II (NSGA-II) multi-objective optimization algorithm, as presented in

Table 3. QMC notch design parameter values.

Parameter	$\mathit{\gamma }$ (°)	${\mathit{d}}_1$ (mm)	${\mathit{\alpha }}_1$ (°)	${\mathit{\alpha }}_2$ (°)	${\mathit{\beta }}_2$ (°)	${\mathit{d}}_2$ (mm)
Value	139	0.23	18.065	1.195	1.23	0.92

.

3.3. Rotor Skew Design

The rotor-skewed structure can effectively suppress torque ripple and radial EM vibration. This often results in reduced motor torque and increased manufacturing costs as trade-offs [29]. The rotor segmented skewed (RSS) structure comprises various types, as illustrated in Figure 4. These include SS, VS, and ZS. The yellow dotted line in the figure indicates the skewed angle for clarity. In the axial direction, the rotor is divided into multiple segments (five segments in Figure 4), and in the circumferential direction, each segment is staggered at a specific angle according to the form of the skewed.

The selection of segment number and skewed angle directly affects the suppression of torque ripple and radial EM vibration. Selecting the appropriate number of segments and skewed angles is a topic worth discussing. Increasing the number of segments not only raises costs incrementally but also potentially increases axial leakage and introduces other harmonics. Therefore, having more segments is not necessarily better. For the skewed angle, given that the harmonic components of the radial and tangential EMF waves are identical [26], the skewed angle is determined by adopting a strategy aimed at suppressing cogging torque, the methodology is as follows [30].

If the motor is divided into k segments, the total cogging torque produced by these k segments is:

$$\begin{array}{cc}\hfill T_{\mathrm{cog}}\left({\alpha }_m\right)& =\sum _{i=1}^k{T}_{\mathrm{cog}i}\hfill \\ \hfill & ={\displaystyle \frac{\pi z{L}_a}{4{\mu }_{0}k}}(R_2^2-R_1^2)\sum _{n=1}^{\infty }n{G}_n{B}_{r{\displaystyle \frac{nz}{2p}}}\left\{\sum _{i=1}^{k}sinnz\left[{\alpha }_m+(i-1)N_s{\theta }_{s1}\right]\right\}\hfill \\ \hfill & =\left\{\begin{array}{cc}{\displaystyle \frac{\pi z{L}_a}{4{\mu }_{0}k}}(R_2^2-R_1^2)\sum _{n=1}^{\infty }n{G}_n{B}_{r{\displaystyle \frac{nz}{2p}}}{\displaystyle \frac{sin\frac{nkz{N}_s{\theta }_{s1}}{2}}{sin\frac{nz{N}_s{\theta }_{s1}}{2}}}sinnz\left({\alpha }_m+{\displaystyle \frac{k-1}{2}}{N}_s{\theta }_{s1}\right)\hfill & \mathrm{when}n{N}_s\ne 1,2,3,\cdots \hfill \\ {\displaystyle \frac{\pi z{L}_a}{4{\mu }_0}}(R_2^2-R_1^2)\sum _{n=1}^{\infty }n{G}_n{B}_{r{\displaystyle \frac{nz}{2p}}}sinnz{\alpha }_m\hfill & \mathrm{when}n{N}_s=1,2,3,\cdots \hfill \end{array}\right.\hfill \end{array}$$

(10)

where $N_s{\theta }_{s1}$ is the skew angle between two adjacent segments of the rotor core in the circumferential direction, $N_s$ is the number of stator skew slots, and ${\theta }_{s1}$ is the tooth pitch between two stator teeth expressed in radians.

It can be seen that when $n{N}_s\ne 1,2,3$, as long as $sin{\displaystyle \frac{nkz{N}_s{\theta }_{s1}}{2}}=0$, the cogging torque can be made zero; as long as n is the smallest value $N_p={\displaystyle \frac{2p}{\mathrm{GCD}(z,2p)}}$(greatest common divisor short as GCD), satisfying $sin{\displaystyle \frac{nkz{N}_s{\theta }_{s1}}{2}}=0$, then the cogging torque will also be zero for other values of n. Therefore, to make the total cogging torque zero, it should satisfy:

$$sin{\displaystyle \frac{N_{p}kz{N}_s{\theta }_{s1}}{2}}=0$$

(11)

So,

$${\displaystyle \frac{N_{p}kz{N}_s{\theta }_{s1}}{2}}=180°$$

(12)

It can be deduced that,

$$N_s={\displaystyle \frac{360°}{{\theta }_{s1}{N}_{p}kz}}={\displaystyle \frac{1}{N_{p}k}}$$

(13)

When a k-segment rotor is skewed, the angle ${\theta }_k$ at which the k-th rotor skewed slice is staggered with respect to the first skewed slice is given by:

$${\theta }_k={\displaystyle \frac{k-1}{N_{p}k}}{\theta }_{s1}$$

(14)

When $n{N}_s=1,2,3,\cdots$(namely n is an integer multiple of $N_{p}k$), the cogging torque expression remains the same as in Equation (4), indicating that the rotor skew method has no attenuating effect on the cogging torque harmonics during these order.

For the 36-slot, 8-pole motor studied in this paper, $N_p=2$, ${\theta }_{s1}=10°$, ${\theta }_k={\displaystyle \frac{5(k-1)}{k}}$. The number of segments, ranging from 3 to 6, was selected as the skewing strategy. The skewing angles for each segmentation strategy are presented in

Table 4. Number of segments and the i-th segment rotation angle.

Segments Number/Length	SS (deg)	VS (deg)	ZS (deg)
3/26 (mm)	$0;1.67;3.33$	$0;1.67;0$	$0;1.67;0$
4/19.5 (mm)	$0;1.25;2.5;3.75$	$0;1.25;1.25;0$	$0;1.25;0;1.25$
5/15.6 (mm)	$0;1;2;3;4$	$0;1;2;1;0$	$0;1;0;1;0$
6/13 (mm)	$0;0.83;1.67;2.5;3.33;4.17$	$0;0.83;1.67;1.67;0.83;0$	$0;0.83;0;0.83;0;0.83$

.

4. Analysis of Torque Ripple and EM Vibrations

An accurate predictive model is urgently required for a comprehensive analysis of the vibration properties of PMSMs. The calculation of EM vibration is a complex multiphysics challenge, involving the EM field, structural modes, and vibration response. To accurately compute the radial EM vibration in motors, three main methods can be employed: analytical methods, semi-analytical methods, and numerical methods [31]. For detailed and precise analysis, numerical methods must be used [32], such as the finite element analysis (FEA) method employed in this study. The technology roadmap for ISG motors, from original design to torque performance and vibration response analysis, is shown in Figure 5. Initially, a structural model is generated from the original design, followed by modal analysis to evaluate the vibration modes. Next, torque performance is analyzed using the EM field, and EM force harmonics are obtained. The EM field is then coupled to the structural domain to analyze the harmonic vibration response. During this process, motor performance is optimized to reduce torque ripple and radial EM vibration by improving the structural design, primarily through rotor notching and rotor segmented skew.

4.1. Analysis of Torque Ripple in ISG Motors

The torque curves for the original motor design, the QMC notching design, and the three types of rotor skew designs at rated speed (4200 r/min) were obtained using FEA software Ansys Maxwell. A comparison of torque curves is presented in Figure 6, the small inset in the figure shows the comparison of the FFT of torque. A comparison of No-load back-electromotive force (back-EMF) and Fast Fourier transform (FFT) is shown in Figure 7. Specific data are provided in

Table 5. Torque performance and THD of back-EMF comparison (number of rotor segments is 5).

	${\mathbf{T}}_{\mathbf{avg}}\mathbf{(}\mathbf{N}\mathbf{·}\mathbf{m}\mathbf{)}$	${\mathbf{T}}_{\mathbf{Rip}}\mathbf{(}\mathbf{\%}\mathbf{)}$	THD of No-Load Back-EMF (%)
Original design	12.48	2.85	13.04
QMC notch design	12.58	1.71	9.70
QMC notch with SS	12.49	1.45	9.31
QMC notch with VS	12.55	1.49	9.57
QMC notch with ZS	12.57	1.49	9.64

, including total harmonic distortion (THD) data of no-load back-EMF. To clarify, all comparisons utilize identical meshing strategies and consistent time steps.

It can be seen that the QMC notch design not only improves the average torque but also reduces the torque ripple by about 40% compared to the initial no-notch design. With the rotor segmented skewed, the average torque decreases while the torque ripple is reduced. For a rotor segmented skew number of 5, the largest decrease in average torque and the most significant reduction in torque ripple are achieved with the SS design. The torque performance is nearly the same when compared with VS and ZS designs. However, in practice, the difference in torque performance among the three-segmented skew strategies is not very significant. Comparisons of no-load back-EMF and FFT analysis also indicate minimal differences between them.

Using the three skew strategies, the number of segments is selected from 3 to 6, as shown in Table 4. The torque performance for all the cases listed therein is calculated at rated speed and presented in

Table 6. Performance Comparison for Different Skew Types and Segmentation.

Segments	Notch with SS	Notch with VS	Notch with ZS
3	${\mathrm{T}}_{\mathrm{avg}}$ = 12.50 Nm ${\mathrm{T}}_{\mathrm{rip}}$ = 1.49%	${\mathrm{T}}_{\mathrm{avg}}$ = 12.55 Nm ${\mathrm{T}}_{\mathrm{rip}}$ = 1.64%	${\mathrm{T}}_{\mathrm{avg}}$ = 12.55 Nm ${\mathrm{T}}_{\mathrm{rip}}$ = 1.64%
4	${\mathrm{T}}_{\mathrm{avg}}$ = 12.49 Nm ${\mathrm{T}}_{\mathrm{rip}}$ = 1.45%	${\mathrm{T}}_{\mathrm{avg}}$ = 12.56 Nm ${\mathrm{T}}_{\mathrm{rip}}$ = 1.60%	${\mathrm{T}}_{\mathrm{avg}}$ = 12.56 Nm ${\mathrm{T}}_{\mathrm{rip}}$ = 1.60%
5	${\mathrm{T}}_{\mathrm{avg}}$ = 12.49 Nm ${\mathrm{T}}_{\mathrm{rip}}$ = 1.45%	${\mathrm{T}}_{\mathrm{avg}}$ = 12.55 Nm ${\mathrm{T}}_{\mathrm{rip}}$ = 1.49%	${\mathrm{T}}_{\mathrm{avg}}$ = 12.57 Nm ${\mathrm{T}}_{\mathrm{rip}}$ = 1.49%
6	${\mathrm{T}}_{\mathrm{avg}}$ = 12.49 Nm ${\mathrm{T}}_{\mathrm{rip}}$ = 1.42%	${\mathrm{T}}_{\mathrm{avg}}$ = 12.56 Nm ${\mathrm{T}}_{\mathrm{rip}}$ = 1.51%	${\mathrm{T}}_{\mathrm{avg}}$ = 12.57 Nm ${\mathrm{T}}_{\mathrm{rip}}$ = 1.55%

. It can be seen that the average torque does not change significantly with different numbers of segments for each skew strategy. As the number of segments increases, the torque ripple reduces; however, when divided into six segments, the torque ripple of the VS design and ZS design increases compared to the division into five segments.

Additionally, by skewing without notching, the torque performance obtained through simulation is presented in

Table 7. Performance comparison for different skew types and segmentation (no notch).

Segments	SS	VS	ZS
3	${\mathrm{T}}_{\mathrm{avg}}$ = 12.39 Nm ${\mathrm{T}}_{\mathrm{rip}}$ = 2.11%	${\mathrm{T}}_{\mathrm{avg}}$ = 12.45 Nm ${\mathrm{T}}_{\mathrm{rip}}$ = 2.45%	${\mathrm{T}}_{\mathrm{avg}}$ = 12.45 Nm ${\mathrm{T}}_{\mathrm{rip}}$ = 2.45%
4	${\mathrm{T}}_{\mathrm{avg}}$ = 12.39 Nm ${\mathrm{T}}_{\mathrm{rip}}$ = 1.90%	${\mathrm{T}}_{\mathrm{avg}}$ = 12.46 Nm ${\mathrm{T}}_{\mathrm{rip}}$ = 2.60%	${\mathrm{T}}_{\mathrm{avg}}$ = 12.46 Nm ${\mathrm{T}}_{\mathrm{rip}}$ = 2.60%
5	${\mathrm{T}}_{\mathrm{avg}}$ = 12.38 Nm ${\mathrm{T}}_{\mathrm{rip}}$ = 1.92%	${\mathrm{T}}_{\mathrm{avg}}$ = 12.45 Nm ${\mathrm{T}}_{\mathrm{rip}}$ = 2.52%	${\mathrm{T}}_{\mathrm{avg}}$ = 12.46 Nm ${\mathrm{T}}_{\mathrm{rip}}$ = 2.64%
6	${\mathrm{T}}_{\mathrm{avg}}$ = 12.38 Nm ${\mathrm{T}}_{\mathrm{rip}}$ = 1.89%	${\mathrm{T}}_{\mathrm{avg}}$ = 12.45 Nm ${\mathrm{T}}_{\mathrm{rip}}$ = 2.51%	${\mathrm{T}}_{\mathrm{avg}}$ = 12.47 Nm ${\mathrm{T}}_{\mathrm{rip}}$ = 2.66%

. Clearly, the torque performance achieved by skewing alone without notching is not as effective as when both notching and skewing are applied.

4.2. Analysis of Radial EM Forces in ISG Motors

The excitation source of EM vibration in PMSMs is the EM force wave. Generally, the radial vibration on the motor surface is the primary source of this EM vibration. Therefore, from the perspective of analyzing the excitation source, the main focus is on the vibration caused by the radial EM force. The radial EM force of the original design ISG machine at a rated speed of 4200 r/m is obtained by FEA, as shown in Figure 8a. Given that the radial EM force is a function of both time and space and satisfies the Dirichlet conditions, the spatial and temporal distribution of the radial EMF waves in the frequency domain can be obtained using a two-dimensional Fourier transform, as shown in Figure 8b. The spatial order is selected from 0 to 50, and the temporal order is selected from 0 to 14, where a negative spatial order indicates harmonic inversion. For clarity, EM force densities with amplitudes less than 1000 N/m² are excluded from the figure. The spatial and temporal harmonic orders with larger magnitudes are labeled in the figure.

Combining Figure 8b and Table 1 reveals certain characteristics and patterns in the EM force density distribution. This comparison also indirectly verifies the correctness of the previous radial EM force theory analysis. The EM force density is distributed diagonally, with the larger amplitudes primarily located along three diagonal lines. The first diagonal line is generated by the interaction between the armature’s fundamental magnetic field and the magnetic field produced by the permanent magnet, which is (0, 0), (8, 2 $f_e$), (16, 4 $f_e$), (24, 6 $f_e$), (32, 8 $f_e$), (40, 10 $f_e$), (48, 12 $f_e$). The other two diagonal lines arise from the interaction between the armature’s first-order toothed harmonic magnetic field and the magnetic field produced by the PM, which is (−36, 0), (−28, 2 $f_e$), (−20, 4 $f_e$), (−4, 8 $f_e$), (4, 10 $f_e$), (12, 12 $f_e$), (20, 14 $f_e$), (36, 0), (44, 2 $f_e$). Additionally, a significant portion of the larger amplitude EM force density is produced by the interaction between the permanent magnet’s fundamental wave magnetic field and the armature magnetic field.

After analyzing the initial design, the notched design, and three kinds of skew design, the EM force density magnitudes in the spatiotemporal distribution were obtained, as shown in

Table 8. Amplitude of the main order radial force components (number of rotor segments is 5).

Time Order	Space Order	Electromagnetic Force Density Amplitude (N/m²)
		Original	Notch	Notch with SS	Notch with VS	Notch with ZS
0	0	131,638.13	128,180.61	127,853.64	128,394.58	128,301.41
0	36	68,078.03	66,731.04	66,517.37	66,260.75	66,469.52
0	−36	68,078.03	66,731.04	66,517.37	66,260.75	66,469.52
$2f_e$	−4	5340.94	5185.46	5087.87	5059.66	5103.88
$2f_e$	8	168,268.39	162,046.25	158,610.26	161,456.88	161,828.05
$2f_e$	−28	48,022.29	46,959.22	46,368.14	46,553.48	46,750.13
$2f_e$	44	33,989.96	33,254.56	32,412.25	32,731.30	32,989.59
$4f_e$	16	49,143.17	46,657.48	43,159.87	45,775.31	46,300.18
$4f_e$	−20	9414.52	10,174.06	9511.67	9906.00	10,052.87
$6f_e$	24	16,897.49	19,245.34	16,813.19	18,605.81	18,985.44
$8f_e$	−4	2105.29	2814.72	2046.35	2559.99	2708.35
$8f_e$	32	20,699.69	21,333.41	15,912.21	19,812.11	20,679.35
$10f_e$	4	4506.94	3785.27	2390.15	3355.33	3596.44
$10f_e$	40	21,690.71	17,778.95	10,830.79	15,729.13	16,880.03
$12f_e$	12	4233.29	3279.49	1654.69	2763.66	3048.51
$12f_e$	48	16,489.92	10,271.28	4869.42	8603.97	9526.76
$14f_e$	20	2082.36	1761.19	645.54	1390.30	1593.09

. For the skewed design, the spatiotemporal distribution of the EM force density was determined by averaging the superimposed EM force density waveforms of all segments, followed by a two-dimensional Fourier transform.

It can be seen that the notch design effectively reduces the amplitude of most EM force harmonics in the spatiotemporal distribution compared to the original design. Additionally, the skew designs further reduce the amplitudes of these harmonics to a certain extent. Among the three types of skew designs with five rotor segments, the SS design achieves the greatest reduction, followed by the VS design, and finally the ZS design.

To further investigate the EM vibration generated by the EM force, the equivalent concentrated EM force and torque are coupled to the motor’s structural field, as shown in Figure 9. Figure 9a depicts the schematic of the design without skew, while Figure 9b shows the schematic of the rotor divided into five segments with skew.

4.3. Motor Modal Analysis

Modal analysis can be utilized to preliminarily predict the possibility of resonance and provide valuable guidance for motor structure design [33]. As the rotor and shaft have minimal impact on the modal results [34], the established three-dimensional prototype model includes only the housing and stator core to analyze the modal characteristics of the ISG machine. To more closely simulate the actual use case, fixed support constraints are added to both sides of the housing. The FEA software calculates the natural frequencies and mode shapes of the stator core and housing, with the results presented in

Table 9. Modal shapes and frequencies.

Modal order	0	2	3
Modal shape
Modal frequency	8835.2 Hz	1366.8 Hz	3842.4 Hz
Modal order	4	5	6
Modal shape
Modal frequency	6175.5 Hz	7675 Hz	9094.9 Hz

.

4.4. EM Vibration Response Analysis of ISG Machines

Using finite element software and the modal superposition method, the vibration acceleration curves on the outer casing of the motor at the rated speed were obtained. The results for the initial design, QMC notch design, and three skew strategies are all plotted on a linear coordinate. The comparison of these curves is shown in Figure 10.

From the figure, it can be seen that the vibration acceleration of the initial motor is relatively high. Using the formula

$$f_e={\displaystyle \frac{p\times n_1}{60}}$$

(15)

where $f_e$ is the fundamental frequency, $n_1$ is the speed of the motor.

The fundamental frequency of the motor at the rated speed of 4200 r/min is calculated to be $f_e=280$ Hz. Significant peaks in vibration acceleration are observed at 1680 Hz, 2800 Hz, 3920 Hz, 5040 Hz, and 5600 Hz, corresponding to 6 $f_e$, 10 $f_e$, 14 $f_e$, 18 $f_e$, and 20 $f_e$, respectively, with the highest peak occurring at 2800 Hz. After using the QMC notch design, most radial EM vibrations are reduced to some extent. However, at 14 $f_e$, the vibrations increase instead. With rotor skew, motor vibrations are significantly suppressed. Except for a specific amount of vibration at 14 $f_e$, vibrations at other frequencies are reduced to very low levels. The difficulty in suppressing the vibration at 14 $f_e$ arises because 3920 Hz is very close to the motor’s intrinsic third-order modal frequency of 3842.4 Hz. For suppressing the 14 $f_e$ vibration, the five-segment skewed rotor performs most effectively with the SS design, followed by the VS design, and lastly the ZS design. From the comparison of motor housing deformations shown in Figure 11, it is evident that the housing deformation is significantly reduced after skewing, both at 10 $f_e$ and 14 $f_e$.

To examine the relationship between the number of segments and EM vibration, vibration response curves for three rotor segment skew strategies with segment counts ranging from three to six were simulated and are presented in Figure 12.

It is evident that different segmentations have varying effects on vibration suppression, particularly for the vibration at 14 $f_e$. Overall, the three-segment and six-segment configurations are less effective in suppressing vibrations compared to the four-segment and five-segment configurations. Specifically, for the SS design, the five-segment configuration is the most effective, while for the VS design and ZS design, the four-segment configuration performs the best.

4.5. Result Discussion

To further analyze and determine the optimal rotor skewing scheme, the torque ripple and radial EM vibration results at 14 $f_e$ for all notching and skewing designs are listed in

Table 10. Comparison of analysis results.

Segments	Torque Ripple (%)			Radial EM Vibration at 14 ${\mathit{f}}_{\mathit{e}}$ (mm/s2)
	Notch with SS	Notch with VS	Notch with ZS	Notch with SS	Notch with VS	Notch with ZS
3	1.49	1.64	1.64	99.62	93.3	93.3
4	1.45	1.60	1.60	31.61	25.90	24.89
5	1.45	1.49	1.49	16.37	42.65	70.05
6	1.42	1.51	1.55	87.62	96.95	102.00

.

From the table, it can be observed that, in terms of optimizing both torque ripple and radial EM vibration at 14 $f_e$, the best scheme is the rotor with a five-segment SS design. Following closely are the four-segment VS design and ZS design. However, the SS design introduces axial EM forces, and the cost of a five-segment configuration is higher. Therefore, the four-segment VS and ZS options are likely the optimal choices.

5. Conclusions

This article investigates the optimization of torque ripple and radial EM vibration in an ISG machine through rotor notching and skewing methods. First, the initial motor is optimized for torque ripple using the QMC notch strategy. Then, the notched motor is further optimized for torque performance by applying rotor skewing techniques, comparing three skewing strategies: SS, VS, and ZS, with the number of segments ranging from 3 to 6. The radial EM forces before and after optimization are calculated and coupled with the motor’s structural field to obtain the radial EM vibration through modal analysis. The analysis indicates that the QMC notch design effectively diminishes torque ripple and has a modest effect on increasing average torque. With the introduction of rotor skewing, the average torque experiences a slight decrease, and torque ripple is further minimized. The combined notch and skew designs generally reduce radial EM forces, with skewing being particularly effective in significantly lowering radial EM vibrations. Upon analyzing the optimization results, it is concluded that the five-segment SS design offers the best optimization performance. However, considering axial force and cost, the four-segment VS and ZS designs are the optimal choices. In future research, further studies will be conducted on additional skewing strategies and the relationship between skewing and noise.