Research on Synergistic Reduction of Cogging Torque and Ripple Torque of Interior Permanent Magnet Synchronous Motor Based on Magnetic Field Harmonic Offset Method

Abstract

This paper presents a method for reducing the cogging torque and ripple torque of interior permanent magnet synchronous motor (IPMSM) based on the magnetic field harmonic offset method. This method establishes the internal correlation between cogging torque harmonics and ripple torque harmonics. The suppression or cancellation of magnetic field harmonics in the rotor pole is utilized as transmission link to simultaneously weaken or eliminate lower order harmonics of cogging torque and ripple torque, which can improve operating quality of the IPMSM and obtain an acceptable total average torque. A mathematical and physical model of harmonic offset method for cogging torque is established, the distribution characteristics of permeability harmonics and field harmonics that affect cogging torque are analyzed, the analytical expression for the electromagnetic torque of the IPMSM including reluctance torque is derived, and the collaborative suppression mechanism of cogging torque and ripple torque, as well as common solutions, are studied. Finally, the suppression law of cogging torque and operating ripple torque is verified by the finite element simulation, and the compromise selection principle of permanent magnetic pole is summarized. Due to the absence of the average torque of motor in the offset method, the effects of effective pole arc of the combined rotor on the torque ripple and torque-speed characteristic curve of the IPMSM are compared and evaluated.

1. Introduction

The permanent magnet synchronous motor (PMSM) is widely used in various industrial fields due to its high efficiency, high torque density, and high power density. In particular, the IPMSM with diversified rotor structure and superior wide and constant power performance has become an important part of industrial development [1,2,3], such as the drive motor for new energy vehicles. With the continuous improvement of miniaturization and lightweight technology of motor drive system, research on vibration and noise reduction of the IPMSM with high power density has attracted wide attention [4]. Therefore, it is of great significance to study the reduction of mechanical vibration of IPMSM caused by torque ripple.

The torque ripple of the PMSM is mainly composed of cogging torque and ripple torque [5,6]. The cogging torque of the PMSM is caused by the change of magnetic field energy storage in the air gap, and the key factor causing this change is the uneven distribution of air gap permeability. Compared with the cogging torque, the ripple torque of the PMSM is caused by the interaction between the stator electrical load and the rotor magnetic load. Due to their different mechanisms, the suppression measures are different. There are three main ways to weaken cogging torque of the PMSM: stator tooth shaping, rotor magnetic pole torsional dislocation, and slot-pole combination. Stator tooth shaping essentially involves adjusting the harmonic component of air-gap magnetic conductance and reducing the influence of specific order magnetic conductance harmonics on cogging torque, such as stator tooth width design, tooth tip modification, unequal tooth width, stator skewed slot, stator auxiliary slot, etc. [7,8,9]. Among these methods, the effect of a stator skewed slot to restrain cogging torque is more prominent. However, the stator skewed slot not only increases the difficulty of wire embedding process but also easily causes axial vibration of the motor. Compared with the diversity analysis of stator tooth shape and position, the research of pole structure of permanent magnet rotor is also rich such as magnetic pole deflection, magnetic pole dislocation, magnetic pole asymmetry, unequal pole arc and unequal thickness permanent magnet (PM), etc. [10,11,12,13], in which unequal thickness PM is often used in the surface-mounted permanent magnet motors. However, this method wastes the amount of the PM while reducing the cogging torque and increases the manufacturing cost of the PM. In order to reduce the consumption and waste of permanent magnets of surface-mounting PMSM, the combination law of rotor pole arc width and stator slot opening, the structure of consequent-pole rotor, and multi-objective optimization methods were studied [14,15,16]. Combined with the flexible design characteristics of the surface-mounting PMSM and the advantages of the high torque output of the IPMSM, the axial pole shaping type of the IPMSM was proposed. The characteristic of this method is that the axial combination of the rotor and the pole arc coefficient of segmented permanent magnets are used to reduce the cogging torque and torque ripple of the motor, but the disadvantage is that the average torque is reduced [17,18].

The operating ripple torque is generated by the interaction between the stator magnetomotive force (MMF) and the rotor MMF, and the torque harmonics are 6 and 6 times. The asymmetric distribution of the d-q axis magnetic circuit of the IPMSM introduces reluctance torque, so that the superposition of permanent magnet torque and reluctance torque may further increase the torque ripple of the motor [19,20]. Reference [21] proposed an analytical method to predict and suppress the ripple torque by the d-q axis lumped loop model of the PMSM. In addition, it is also applicable to reduce the torque ripple of the PMSM by shaping the motor structure and optimizing the trajectory of control current [22,23,24]. In reference [22], the method of suppressing torque ripple of surface mounted permanent magnet motor with the third sine harmonic shape magnet was studied. It is worth noting that the cogging torque minimization of the permanent magnet motor is not necessarily the best reference for operating ripple torque suppression, so the cogging torque and torque ripple need to be traded off [25]. Due to the loss of permanent magnet utilization caused by partial overlap of magnetic pole pairs or reshaping of permanent magnet side ends, the average torque of the permanent magnet machine will be reduced with reduction of ripple torque. To improve torque output performance, the compound structure of the permanent magnet motor was studied [26], but the cost of the motor was increased due to limitation of rotor space and complexity of processing technology. Based on the above analysis, scholars have made a lot of contributions and achieved outstanding achievements in the study of cogging torque and torque ripple weakening methods of PMSMs, but there are few reports on the internal suppression between cogging torque and operating ripple torque of IPMSM and synergistic reduction methods.

In this paper, a method for synergistic reduction of cogging torque and ripple torque of IPMSM based on harmonic offset principle is proposed. The research contents are summarized as follows:

(1)

Model analysis. The mathematical and physical model of the harmonic offset method for cogging torque is established, the composition, distribution and zero crossing characteristics of permeability harmonics and air gap field harmonics are analyzed, the analytical expression of electromagnetic torque including reluctance torque of IPMSM is derived, and the internal correlation between cogging torque harmonics and running ripple torque harmonics and the suppression measures are studied.

(2)

FEA verification. Taking a 45 kW IPMSM for electric vehicle as an example, the cogging torque of the magnetic pole dislocation offset model is calculated, and the torque ripple suppression and average torque of the theoretical collaborative deflection model are checked. The combined rotor simulation model is developed using python language to parameterize the rotor auxiliary slot topology structure, and the difference between average torque and total torque ripple before and after the rotor model combination is compared.

(3)

Torque-speed characteristic analysis. The characteristics of magnetic saturation distribution and torque ripple caused by it are compared, and the torque and speed performance of the 45 kW IPMSM operating with wide and constant power is evaluated and analyzed according to the load characteristics. Finally, a conclusion is drawn, and the correlation law of using offset method to reduce the cogging torque and ripple torque of IPMSM is summarized.

2. Model Analysis

The electromagnetic torque $T_{\mathrm{e}}$ of the IPMSM is composed of the average torque $T_{\mathrm{a}\mathrm{v}\mathrm{e}}$, the cogging torque $T_{\mathrm{c}\mathrm{o}\mathrm{g}}$ and the ripple torque $T_{{\mathrm{r}}_{-}\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}$. In essence, the torque ripple of the IPMSM is caused by the distribution of some specific order and large amplitude magnetic field harmonics in the air gap. According to the spatial distribution characteristics of magnetic field harmonics, the offset method can be used to synchronically weaken the cogging torque and ripple torque of the IPMSM. Moreover, the suppression law and internal correlation of the two kinds of torque harmonics are studied.

2.1. Reduction of Cogging Torque

The cogging torque arises due to the relative displacement of stator and rotor by tangential force, which causes the variation of air-gap magnetic field energy. Based on the energy method, the expression of cogging torque is expressed as [27]:

$$\begin{array}{cc}{T}_{\mathrm{cog}}(\alpha )\hfill & =-\frac{\partial W}{\partial \alpha }\hfill \\ \hfill & =-\frac{\partial }{\partial \alpha }\left[\frac{L_{\mathrm{ef}}}{2{\mu }_0}{\displaystyle {\int }_{R_2}^{R_1}RdR{\displaystyle {\int }_0^{2\pi }{B}^2\left(\theta \right)G^2\left(\theta ,\alpha \right)}}\right]\hfill \end{array}$$

(1)

where, W is the magnetic field energy stored in motor when the winding is not energized, α is the relative position angle between stator and rotor, B(θ) is the air-gap remanent magnetic density, G(θ, α) is function of magnetic admittance, μ₀ is the air permeability, R₂ and R₁ are rotor radius and stator inner radius, respectively. $L_{\mathrm{e}\mathrm{f}}$ is the length of the core.

The Fourier series of B²(θ) and G²(θ, α) are expanded as follows [7]:

$$B^2\left(\theta \right)={\displaystyle \sum _{n=1}^{\infty }{B}_{\mathrm{r}n}\cos 2np\theta }$$

(2)

$$G^2\left(\theta ,\alpha \right)={\displaystyle \sum _{n=1}^{\infty }{G}_n\cos n{N}_{\mathrm{s}}\left(\theta +\alpha \right)}$$

(3)

where, $B_{\mathrm{r}n}$ and $G_n$ are Fourier expansion coefficients. $B_{\mathrm{r}n}=2/\left(n\pi \right)B_r^2\sin n{\alpha }_{\mathrm{p}}\pi$, $G_n=\left(2/n\pi \right)G^2\sin \left(n\pi -n{N}_{\mathrm{s}}{b}_0/2\right)$, where, α_p, b₀ and N_s are polar arc coefficient, stator slot width and stator slot number, respectively, p is number of polar pairs.

Substituting Equations (2) and (3) into Equation (1) and utilizing the orthogonality of trigonometric function, the cogging torque can be expressed as:

$$T_{\mathrm{cog}}(\alpha )=\frac{\pi N_{\mathrm{s}}{L}_{\mathrm{ef}}}{4{\mu }_0}\left(R_2^2-R_1^2\right){\displaystyle \sum _{n=1}^{\infty }n{G}_{n{N}_L}{B}_{n{N}_L}\sin n{N}_L\alpha }$$

(4)

where, N_L = LCM(N_s, 2p) is the least common multiple of N_s and 2p.

To reduce the cogging torque $T_{\mathrm{c}\mathrm{o}\mathrm{g}}$ in Equation (4) and reveal the distribution characteristics of the harmonic components of the cogging torque, the PMSM can be decomposed into m microelements along the circumference or axial direction, and the displacement angle deviation of adjacent microelements is assumed to be $\mathsf{\Delta }\alpha$. Therefore, the total cogging torque of the PMSM can be expressed as the superposition of cogging torque generated by m microelements $T_{\mathrm{c}\mathrm{o}\mathrm{g}1}$, $T_{\mathrm{c}\mathrm{o}\mathrm{g}2}$, $T_{\mathrm{c}\mathrm{o}\mathrm{g}3}$, …, $T_{\mathrm{c}\mathrm{o}\mathrm{g}\mathrm{m}}$. When $\mathsf{\Delta }\alpha =0°$, the synthetic cogging torque $T_{\mathrm{c}\mathrm{o}\mathrm{g}}$ of PMSM is the same as (4). When $\mathsf{\Delta }\alpha \ne 0°$, the motor total cogging torque $T_{\mathrm{c}\mathrm{o}\mathrm{g}}$ generated by m microelements can be expressed as:

$$T_{\mathrm{cog}}({\alpha }^{\prime })=T_{\mathrm{cog}1}\left(\alpha \right)+T_{\mathrm{cog}2}\left(\alpha +\Delta \alpha \right)+\dots +T_{\mathrm{cogm}}\left[\alpha +\left(m-1\right)\Delta \alpha \right]$$

(5)

Let amplitude of the harmonic component of n_th cogging torque of $T_{\mathrm{cogm}}, T_{m\mathrm{c}\mathrm{n}}=\pi N_{\mathrm{s}}{L}_{\mathrm{e}\mathrm{f}}{G}_{n{N}_L}{B}_{n{N}_L}/\left(4m{\mu }_0\right)·\left(R_2^2-R_1^2\right)$ [28], and assume $T_{1\mathrm{c}\mathrm{n}}=T_{2\mathrm{c}\mathrm{n}}=T_{3\mathrm{c}\mathrm{n}}=\dots =T_{\mathrm{m}\mathrm{c}\mathrm{n}}$. Considering that m can be odd or even, (5) is rewritten as:

• when m is odd:

  $$\begin{array}{cc}{T}_{\mathrm{cog}}({\alpha }^{\prime })\hfill & ={\displaystyle \sum _{n=1}^{\infty }\left\{\begin{array}{l}{T}_{m\mathrm{cn}}\sin n{N}_L\left(\alpha -\frac{m-1}{2}\Delta \alpha \right)+\dots +T_{m\mathrm{cn}}\sin n{N}_L\left(\alpha -2\Delta \alpha \right)+T_{m\mathrm{cn}}\sin n{N}_L\left(\alpha -\Delta \alpha \right)+T_{m\mathrm{cn}}\sin n{N}_L\alpha \\ +T_{m\mathrm{cn}}\sin n{N}_L\left(\alpha +\frac{m-1}{2}\Delta \alpha \right)+\dots +T_{m\mathrm{cn}}\sin n{N}_L\left(\alpha +2\Delta \alpha \right)+T_{m\mathrm{cn}}\sin n{N}_L\left(\alpha +\Delta \alpha \right)\end{array}\right\}}\hfill \\ & ={\displaystyle \sum _{n=1}^{\infty }{T}_{m\mathrm{cn}}\sin n{N}_L\alpha \left\{2\cos n{N}_L\left(\frac{m-1}{2}\Delta \alpha \right)+\dots +2\cos n{N}_L\left(2\Delta \alpha \right)+2\cos n{N}_L\Delta \alpha +1\right\}}\hfill \end{array}$$

  (6)

Making $T_{\mathrm{c}\mathrm{o}\mathrm{g}}\left({\alpha }^{\prime }\right)=0,$ Let $\cos n{N}_L\left(\frac{m-1}{2}∆\alpha \right)+...+2\cos n{N}_L\left(2∆\alpha \right)+2\cos n{N}_L∆\alpha =-\frac{1}{2}$, there is:

$$\Delta \alpha =\frac{2\pi }{n{N}_{L}m}$$

(7)

when m is even:

$$\begin{array}{cc}{T}_{\mathrm{cog}}({\alpha }^{\prime })\hfill & ={\displaystyle \sum _{n=1}^{\infty }\left\{\begin{array}{l}{T}_{m\mathrm{cn}}\sin n{N}_L\left(\alpha -\frac{1}{2}\Delta \alpha \right)+T_{m\mathrm{cn}}\sin n{N}_L\left(\alpha -\frac{3}{2}\Delta \alpha \right)+\dots +T_{m\mathrm{cn}}\sin n{N}_L\left(\alpha -\frac{m-1}{2}\Delta \alpha \right)+\\ T_{m\mathrm{cn}}\sin n{N}_L\left(\alpha +\frac{1}{2}\Delta \alpha \right)+T_{m\mathrm{cn}}\sin n{N}_L\left(\alpha +\frac{3}{2}\Delta \alpha \right)+\dots +T_{m\mathrm{cn}}\sin n{N}_L\left(\alpha +\frac{m-1}{2}\Delta \alpha \right)\end{array}\right\}}\hfill \\ & =4{\displaystyle \sum _{n=1}^{\infty }{T}_{m\mathrm{cn}}\sin n{N}_L\alpha \cos n{N}_L\frac{m}{4}\Delta \alpha \left\{\begin{array}{l}\cos n{N}_L\left(\frac{m}{4}-\frac{1}{2}\right)\Delta \alpha +\cos n{N}_L\left(\frac{m}{4}-\frac{3}{2}\right)\Delta \alpha +\dots \\ +\cos n{N}_L\left(\frac{m}{4}-\frac{m-1}{2}\right)\Delta \alpha \end{array}\right\}}\hfill \end{array}$$

(8)

If $\cos n{N}_L\left(m/4\right)\mathsf{\Delta }\alpha \equiv 0$, then $T_{\mathrm{c}\mathrm{o}\mathrm{g}}\left({\alpha }^{\prime }\right)=0,$ thus:

$$\Delta \alpha =\frac{2\pi }{n{N}_{L}m}$$

(9)

Equations (7) and (9) are more general expressions for reducing the cogging torque of the motor, and no matter m is odd or even, it can be represented by a unified mathematical expression. For example, if the number of phase slots per pole q is an integer for the 8-pole 48-slot PMSM, if m is 3 or 2, and n = 1, then $\mathsf{\Delta }\alpha$ is 2.5° and 3.75°, respectively. In addition, if q = 0.5 8-pole 12-slot permanent magnet motor $\mathsf{\Delta }\alpha$ values are 5° and 7.5°, respectively. It can be seen that no matter the integer slot or fractional slot permanent magnet motor, as long as $\mathsf{\Delta }\alpha$ satisfies Equation (7) or Equation (9) can significantly reduce the cogging torque. It should be noted that the derivation and development of the cogging torque offset model is a theoretical supplement to the equivalent skew pole method of the permanent magnet motor, namely the magnetic pole dislocation model, and also provides a reference for the collaborative reduction of cogging torque harmonics and ripple torque harmonics [29]. Generally, it is more economical and effective to adopt axial permanent magnet pole dislocation or circumferential permanent magnet migration, and there is unity in the theory that the two kinds of permanent magnet deflection weaken the cogging torque. For example, the axial direction of the magnetic pole is misaligned in 2 segments and circumferential offset of magnetic pole is the same, that is, −1.875° and 1.875°, respectively. Figure 1 is the schematic diagram of axial dislocation of the rotor.

It can be seen from Figure 1a,b that the linear torsion of the rotor core segment is characterized by mechanical deflection along a certain direction of the rotating axis, and it only needs to meet the dislocation angle of the segmented core as $\mathsf{\Delta }\alpha$. However, the disadvantage is that the linear torsion of the rotor core segment is easy to produce axial electromagnetic force and cause axial vibration and noise of PMSM. In order to reduce the effect of axial electromagnetic force, the V-shaped torsional core structure as shown in Figure 1c was applied. In addition, it can also be seen from Figure 1c that the symmetry distributed structure of the V-type torsional rotor core makes its production process slightly more complicated than that of the linear torsional production process. Magnetic pole axial dislocation or circumferential magnetic pole pair migration can be collectively referred to magnetic pole deflection, which can shift the $n$-$\mathrm{t}\mathrm{h}$ harmonic component of the cogging torque by 180° to offset the influence of the cogging torque of lower order and larger amplitude. Since the deflection of permanent magnetic pole changes the effective polar arc of PM, the distribution and combination of the effective polar arc of PM can also reduce the cogging torque.

According to $B_{\mathrm{r}n}=2/\left(n\pi \right)B_r^2\sin n{\alpha }_{\mathrm{p}}\pi$, if $B_{\mathrm{r}n}=0$, it must satisfy

$${\alpha }_{\mathrm{p}}=\frac{k\pi }{n\pi }=\frac{k}{n}(k=1,2,\dots ,n)$$

(10)

where, α_p is the polar arc coefficient, and the lowest degree of n must satisfy N = LCM(N_s,2p)/2p.

As can be seen from (8), α_p = 1/N, 2/N, …, k/N, $B_{\mathrm{r}n}$ can be made constant 0, so the PMSM can choose a larger polar arc coefficient α_p to weaken the cogging torque, while the motor torque output will be effectively used. $B_{\mathrm{r}n}$ is a periodic function, for example, the period of the $n$-$\mathrm{t}\mathrm{h}$ wave in 2π radians is 2/N. When $n=kN$, the period of $B_{\mathrm{r}n}$ is $2/kN$, and $2/kN$ zero crossing occurs in 2π radian. In order to make proper use of the zero crossing of the positive and negative half waves of the cogging torque and weaken the cogging torque, the phase shift π electric radian can be realized. Usually, the amplitude of $B_{\mathrm{r}N}$ with the degree k = 1 is the largest, and the polar arc coefficient difference of adjacent magnetic pole pairs is 1/N or the offset angle difference of magnetic pole pairs is π/2pN, which can realize the phase reversal of the first-order cogging torque. Therefore, effective pole arc selection of PM and pole deflection are consistent and equally effective in reducing cogging torque of permanent magnet motor. Figure 2 shows the relationship curves of $B_{\mathrm{r}n}$, $G_n$ and $T_{\mathrm{c}\mathrm{o}\mathrm{g}}$ of the 8-pole 48-slot motor.

From Figure 2a, it can be seen that the amplitude of $B_{\mathrm{r}n}$ decreases with the increase of number n, in which B_r6 has the largest amplitude and is the main factor causing the cogging torque. To reduce the amplitude of B_r6, it is effective to select B_r6 periodic zero crossing which are 0, 1/6, 2/6… 1. It can be further inferred that by deflecting 1/6 polar arcs between adjacent permanent magnets, the positive and negative half waves of B_r6 can be cancelled out, which can weaken the cogging torque. From Figure 2b, the relatively large coefficient G₁ only has two zero crossings within a tooth distance t₁, namely, 0° and 7.5° respectively, which correspond to closed slots and fully open slots on the stator side respectively. Therefore, closed slots are used for the PMSM with the number of phase slots per pole q being an integer, and full-open slots or torsional tooth distance t₁ is more effective in reducing the cogging torque. From Figure 2c, the cogging torque can be reduced when the pole arc coefficient combinations of adjacent magnetic pole pairs are α_p1 and α_p3, α_p2 and α_p5. The reason is that B_r6 under the pole arc combination is largely offset, and the resultant cogging torque is the same as that under the polar arc coefficient of α_p4. Therefore, the essence of the offset method is to change the harmonic composition and distribution of the air-gap magnetic field of the PMSM, and it has been applied in the air-gap magnetic admittance and rotor pole suppression.

2.2. Reduction of Ripple Torque

The ripple torque of the IPMSM consists of permanent magnet torque harmonics and reluctance torque harmonics. Figure 3 shows the PMSM model in the d-q coordinate system.

According to the Bio-Safar law of electromagnetic force, the expression of permanent magnet torque of the PMSM can be expressed as [30]:

$$\begin{array}{cc}{T}_{\mathrm{pm}}\hfill & =r_{\mathrm{g}}{L}_{\mathrm{ef}}{\displaystyle \underset{2\pi p}{\int }{B}_{\mathrm{g}}d{f}_{\mathrm{s}}}\hfill \\ & =p\frac{{\mu }_0}{g}{r}_{\mathrm{g}}{L}_{\mathrm{ef}}{\displaystyle \underset{2\pi }{\int }{f}_{\mathrm{r}}\frac{d{f}_{\mathrm{s}}}{d{\theta }_{\mathrm{m}}}}d{\theta }_{\mathrm{m}}\hfill \\ & =C_{\mathrm{p}}\pi \left[f_{\mathrm{s},1}{f}_{\mathrm{r},1}\sin \left({\gamma }_{\mathrm{d}}\right)\right.-{\displaystyle \sum _{\begin{array}{l}v=6k\mp 1\\ k=1,2,3,\dots \end{array}}\left.\nu f_{\mathrm{s},\nu }{f}_{\mathrm{r},\nu }\sin \left(\left(\nu \pm 1\right)\omega t\pm {\gamma }_{\mathrm{d}}\right)\right]}\hfill \end{array}$$

(11)

where, $C_p=p\left({\mu }_0/g\right)r_g{L}_{\mathrm{e}\mathrm{f}}$, $f_{\mathrm{s},\upsilon }$ and $f_{\mathrm{r},\upsilon }$ are amplitudes of stator and rotor ν-$\mathrm{t}\mathrm{h}$ harmonic magnetomotive force, respectively, $B_{\mathrm{g}}$ is the magnetic flux density of air-gap, $r_{\mathrm{g}}$ is the air-gap radius, ω is the angular velocity, ${\gamma }_{\mathrm{d}}$ is the aangle between the current vector i_s and the d axis. $T_{\mathrm{p}\mathrm{m}}$ consists of constant torque T_ave1 and ripple torque T_r1, and is expressed as:

$$T_{\mathrm{ave}1}=C_{\mathrm{p}}\pi f_{\mathrm{s},1}{f}_{\mathrm{r},1}\sin \left({\gamma }_{\mathrm{d}}\right)$$

(12)

$$T_{\mathrm{r}1}=-\nu C_{\mathrm{p}}\pi {\displaystyle \sum _{\begin{array}{l}v=6k\mp 1\\ k=1,2,3,\dots \end{array}}{f}_{\mathrm{s},\nu }{f}_{\mathrm{r},\nu }\sin \left(\left(\nu \pm 1\right)\omega t\pm {\gamma }_{\mathrm{d}}\right)}$$

(13)

According to Equation (13), the ripple torque of the PMSM is due to the 6- and 6-times harmonics of permanent magnet torque, where ν = 6 k + 1 is generated by positive sequence magnetic field, and ν = 6 k − 1 is generated by negative sequence magnetic field.

Due to the asymmetry of magnetic admittance of the d-q axis of the rotor, the reluctance torque of the IPMSM is utilized, but it also produces the reluctance torque harmonics.

According to inductive energy of $W_{\mathrm{m}}=1/2L{i}^2$, and assuming that the A-phase winding axis is overlapped with the d axis, the magnetic co-energy of the electromechanical device can be expressed as

$$W_{\mathrm{m}}^{\prime }=\frac{1}{2}\left[L_{\mathrm{A}}\left(\theta \right){\left(i_{\max }\cos \left(\omega t-{\gamma }_{\mathrm{d}}\right)\right)}^2+L_{\mathrm{B}}\left(\theta \right){\left(i_{\max }\cos \left(\omega t-\frac{2\pi }{3}-{\gamma }_{\mathrm{d}}\right)\right)}^2\right.\left.+L_{\mathrm{C}}\left(\theta \right){\left(i_{\max }\cos \left(\omega t+\frac{2\pi }{3}-{\gamma }_{\mathrm{d}}\right)\right)}^2\right]$$

(14)

where $L_{\mathrm{A}}$, $L_{\mathrm{B}}$ and $L_{\mathrm{C}}$ are the winding inductances of phase A, B and C respectively, and are expressed as $L_{\mathrm{A}}=L_{\mathrm{aa}}+M_{\mathrm{ab}}+M_{\mathrm{ac}}$, $L_{\mathrm{B}}=L_{\mathrm{bb}}+M_{\mathrm{bc}}+M_{\mathrm{ba}}$, $L_{\mathrm{C}}=L_{\mathrm{cc}}+M_{\mathrm{cb}}+M_{\mathrm{ca}}$. Where, $L_{\mathrm{aa}}$, $L_{\mathrm{bb}}$ and $L_{\mathrm{cc}}$ are the self-inductances of phase A, B and C respectively; $M_{\mathrm{ab}}$, $M_{\mathrm{ac}}$ and $M_{\mathrm{bc}}$ are the mutual inductances of phase A and B, phase A and C, and phase B and C respectively.

After solving $\frac{\partial W_{\mathrm{m}}^{\prime }}{\partial {\gamma }_{\mathrm{d}}}$, the reluctance torque of the IPMSM is approximately expressed as:

$$T_{\mathrm{re}}=\partial W_{\mathrm{m}}^{\prime }/\partial {\gamma }_{\mathrm{d}}\approx \frac{3}{2}{i}_{\max }^2\times \frac{p}{2}\left(-L_{\mathrm{aa}2}-2M_{\mathrm{ab}2}+L_{\mathrm{aa}4}\cos 6\theta \right.\left.+2M_{\mathrm{ab}4}\cos 6\theta \right)\sin \left(2{\gamma }_{\mathrm{d}}\right)$$

(15)

From (15), it can be seen that the DC component T_ave2 of the reluctance torque T_re is generated by L_aa2 and M_ab2, while the lower order ripple torque T_r2 of the reluctance torque needs to eliminate the torque harmonic of 6 and 6 times as many as T_r1. Therefore, the total average torque and ripple torque of the IPMSM can be expressed as $T_{{\mathrm{a}\mathrm{v}\mathrm{e}}_{-}\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}=T_{\mathrm{ave}1}+T_{\mathrm{ave}2}$ and $T_{{\mathrm{r}}_{-}\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}=T_{\mathrm{r}1}+T_{\mathrm{r}2}$, respectively.

2.3. Correlation between Cogging Torque and Ripple Torque

Permanent magnet deflection or effective polar arc selection is effective for reducing cogging torque and ripple torque of the IPMSM. To reduce both cogging torque and ripple torque, and establish the internal correlation between the torque harmonics, the harmonic cancellation law of the air gap magnetic field of the IPMSM and the deflection characteristics of the permanent magnet are studied. Figure 4 shows the deflection principle diagram of permanent magnets in the PMSM.

As can be seen from Figure 4, although adjacent magnetic pole pairs deflected $\gamma$ mechanical angle toward each other, the distance between the center lines of each group of magnetic poles is still 2π/p. Therefore, the deflection of PMs only changes the harmonic composition of the air-gap magnetic flux density, while the q-axis of the motor does not change and maintains a symmetrical distribution. Due to the deflection of the magnetic pole pair, the partial overlap of the rotor magnetic field reduces the effective pole arc of the permanent magnet, so that the load torque output will be reduced somewhat. The Fourier series expansions of $B_{\mathrm{g}n}$ and $B_{\mathrm{r}n}$ can be expressed as:

$$\begin{array}{cc}{B}_{\mathrm{g}n}\hfill & =\frac{2}{\left(\frac{2\pi }{p}\right)}{\displaystyle {\int }_0^{\frac{2\pi }{p}}{B}_{\mathrm{g}}\cos \left(np\theta \right)}d\theta \hfill \\ & =\frac{1}{\left(\frac{\pi }{p}\right)}\left[{\displaystyle {\int }_{\frac{-\pi {\alpha }_{\mathrm{p}}}{2p}+\gamma }^{\frac{\pi {\alpha }_{\mathrm{p}}}{2p}+\gamma }{B}_{\mathrm{g}}\cos \left(np\theta \right)}\right.d\theta -\left.{\displaystyle {\int }_{\frac{\pi }{p}-\frac{\pi {\alpha }_{\mathrm{p}}}{2p}-\gamma }^{\frac{\pi }{p}+\frac{\pi {\alpha }_{\mathrm{p}}}{2p}-\gamma }{B}_{\mathrm{g}}\cos \left(np\theta \right)d\theta }\right]\hfill \\ & =\frac{4B_{\mathrm{g}}}{n\pi }\left[\sin \left(\frac{n\pi {\alpha }_{\mathrm{p}}}{2}\right)\cos \left(np\gamma \right)\right]\hfill \end{array}$$

(16)

$$\begin{array}{cc}{B}_{\mathrm{r}n}\hfill & =\frac{2}{\left(\frac{2\pi }{p}\right)}{\displaystyle {\int }_0^{\frac{2\pi }{p}}{B}_{\mathrm{r}}^2\cos \left(2np\theta \right)}d\theta \hfill \\ & =\frac{1}{\left(\frac{\pi }{p}\right)}\left[{\displaystyle {\int }_{\frac{-\pi {\alpha }_{\mathrm{p}}}{2p}+\gamma }^{\frac{\pi {\alpha }_{\mathrm{p}}}{2p}+\gamma }{B}_{\mathrm{r}}^2\cos \left(2np\theta \right)}\right.d\theta +\left.{\displaystyle {\int }_{\frac{\pi }{p}-\frac{\pi {\alpha }_{\mathrm{p}}}{2p}-\gamma }^{\frac{\pi }{p}+\frac{\pi {\alpha }_{\mathrm{p}}}{2p}-\gamma }{B}_{\mathrm{r}}^2\cos \left(2np\theta \right)d\theta }\right]\hfill \\ & =\frac{2B_{\mathrm{r}}^2}{n\pi }\left[\sin \left(n\pi {\alpha }_{\mathrm{p}}\right)\cos \left(2np\gamma \right)\right]\hfill \end{array}$$

(17)

From (16) and (17), to make $B_{\mathrm{g}n}$ and $B_{\mathrm{r}n}$ be zero, the deflection angle γ of PM must satisfy π/2np and π/4np, respectively. It can be seen that the deflection angle of the ripple torque is twice the deflection angle of the cogging torque. If the deflection angle γ decreases the ripple torque of a certain order, and the corresponding cogging torque will increase; on the contrary, a certain order of cogging torque decreases while the corresponding order of ripple torque increases. To determine the proper deflection angle of permanent magnet for torque ripple reduction of the PMSM, it is more important to compare the weight ratio of ripple torque and cogging torque. Therefore, the reduction of motor torque ripple is actually to reduce the obvious torque harmonics. To achieve the goal of coordinated reduction of different key order torque harmonics, it is essential to combine the theoretical analysis results and finite element simulation to study and verify the structure distribution and the suppression law of torque harmonics of the combined rotor the IPMSM.

3. FEA Verification

This paper takes a 45 kW ∇-shape IPMSM for electric vehicles as an example, the main parameters of the motor are shown in

Table 1. Main parameters of the IPMSM for electric vehicles.

Parameter	Unit	Value
Rated power	kW	45
Rated speed	rpm	4000
Number of pole pairs	-	4
Stator outer diameter	mm	180
Stator inner diameter	mm	132
Number of stator slots	-	48
Length of air gap	mm	0.75
PM1width	mm	20
PM1 thickness	mm	3
PM2 width	mm	2 × 17
PM remanence flux density Br_20 °C	T	1.25

. Figure 5 shows 3D model. Figure 6 and Figure 7 show the torque output and the peak-peak value of torque ripple under different deflection angles of PMs respectively.

From Figure 6a, it can be seen that stator skewing or PM deflection reduces the torque ripple of the IPMSM. Compared to stator skewing technology, the application of PM deflection angle makes the reduction effect of motor torque ripple more prominent. From Figure 6b, the torque ripple caused by a PM deflection of 1.875° is relatively small, and the proportion of torque ripple before and after deflection is 11.06% and 4.3%, respectively. It indicates that the 12-th harmonic torque of the 45 kW IPMSM is the main factor causing torque ripple. Therefore, deflecting the rotor magnetic pole by half a cycle angle, which is the mechanical angle γ = 1.875°, can achieve the cancellation of the positive and negative half waves of the 12-th harmonic, as shown in Figure 6c. From Figure 6d, it can be seen that the saliency rate of the IPMSM decreases with the increase of the permanent magnet pole deviation angle. Due to the reduction of saliency rate, the torque output performance of the motor is reduced, such as when the PM pole deviation angle γ = 3.75°, the torque output is relatively small, with an average torque value of only 103.9 Nm and a torque reduction of 6.6% compared to original permanent magnet deflection. From Figure 7, it can be seen that the peak-peak value of the torque ripple begins to rapidly decrease with the increase of the PM deflection angle, and then oscillates. Moreover, when the PM deflection angle is 1.875°, the absolute difference in torque ripple of the motor is small and only 5.0 Nm.

Figure 8 shows the distribution of the cogging torque under different PM deflection angles. Increasing the deflection angle of the PM alters the phase and peak-peak value of the cogging torque of the IPMSM. When the deflection angle of the PM γ = 1.875°, the offset reduction of B_r6 significantly weakens the cogging torque of the motor. From Figure 8, it can also be seen that the most effective method to reduce the cogging torque is to twist the stator by one tooth pitch to obtain a cogging torque close to zero, but the complexity of the motor production process has increased. From Figure 9, it is obvious that the peak-peak value of cogging torque is even symmetric, where the minimum cogging torque corresponds to PM deflection angles of 1.875°, 5.625°, …, (2n − 1) × 1.875°, respectively. Therefore, the magnetic pole dislocation angle (2n − 1)$\mathsf{\Delta }\alpha$ can effectively weaken the permanent magnet motor cogging torque. To meet the cogging torque and ripple torque are reduced at the same time, it is necessary to combine the magnetic pole dislocation angle of the rotor structure with the effective pole arc coefficient, such as the 45 kW IPMSM permanent magnet pole deflection angle 1.875° and the rotor pole arc coefficient 0.75 combination.

The offset method aims to balance key-order harmonic compositions of air-gap magnetic field to achieve more reasonable path of magnetic flux in the motor. In addition, the cancellation and suppression of magnetic field harmonics are not limited to combination and arrangement of one or two motor structural parameters. The optimal combination of multi-dimensional and multi-spatial structural parameters is more beneficial for the suppression of motor torque harmonics. Therefore, the structural parameters and torque harmonic suppression law of composite IPMSM with auxiliary slots have been further studied. Due to the competition between stator slots, it is difficult to flexibly open auxiliary slots on the top of the teeth. However, the design and research of rotor auxiliary slots, as well as the summary and summarization of their distribution patterns, are more convenient and effective, including shape, position, and quantity of rotor auxiliary slots. Since the current electromagnetic field commercial software is not perfect in the parameterization of the rotor auxiliary slot model, this paper uses python language to compile the rotor geometric modeling and embed it into Maxwell 2D, which better improves the accuracy of the calculation performance of the IPMSM and reliability of conclusion summary. Figure 10 and Figure 11 show the comparison results of the customized rotor auxiliary slot model and torque output, respectively.

From Figure 11a, it can be seen that the variation range of torque ripple of the IPMSM with a combined auxiliary slot on the rotor side is 2.36–4.68% under θ₁ = 80° and θ₂ = 30°–40°. The more important reason for the difference in motor torque ripple is the matching relationship between the position angles of the two auxiliary slots. When θ₁ = 80° and θ₂ = 35°–40°, the torque ripple of the motor varies between 2.36–3.41%, while when θ₂ ≤ 30, torque ripple is slightly greater than 4%. Therefore it is more suitable to design the auxiliary slot of the IPMSM rotor in the range of θ₁/θ₂ = 2–2.28. In addition to the correlation of the position angle of the auxiliary slot of the rotor, the shape of the auxiliary slot also has an influence on the torque ripple of the IPMSM. From Figure 11b, with the increase of θ₂₁/θ₂₂, the trend line of torque ripple shows an increasing and decreasing trend respectively, and the increasing rate of torque ripple curve of θ₁/θ₂ = 2.28 is more prominent than the decreasing rate of θ₁/θ₂ = 2, since the relative position and shape of the combined rotor auxiliary slot can effectively suppress the torque harmonics of specific order of the IPMSM. Additionally, the smooth design of the auxiliary slot also reduces the possibility of harmonic distortion of the air gap magnetic field. From Figure 11c, the torque performance of the IPMSM with combined auxiliary slot rotor structure is superior, in which the IPMSM with single auxiliary slot angle θ₁ = 95° takes into account both the torque performance index and the processing economy. By comparing the torque output performance of the motor with the three rotor types, when the average torque is about 109 Nm, the torque ripple is 2.36%, 4.3%, and 5.71%, respectively, which indicates that design of auxiliary rotor slot only changes the harmonic composition and distribution of specific times of air-gap magnetic flux density of the IPMSM. However, it has little influence on the fundamental wave component and magnetic circuit structure of air-gap magnetic field.

To reduce the torque ripple of the IPMSM and improve the mechanical vibration caused by electromagnetic harmonics, in addition to optimal design of motor structure, it can also increase the current lead angle in the control strategy. Figure 12 and Figure 13 show torque output and torque ripple at different internal power factor angles, respectively.

As can be seen from Figure 12, with the increase of the internal power factor angle, the torque output of the IPMSM first increases and then decreases, among which the constant torque output reaches maximum within the range of 35–40°. In addition, for the same internal power factor angle, the torque output of the IPMSM increases with the increase of stator current. Figure 13 shows that the torque ripple of the IPMSM decreases with the increase of internal power factor angle under different load currents, and the torque ripple becomes closer with the increase of internal power factor angle. When the internal power factor angle is the same, the larger the stator current is, the smaller the torque ripple of the IPMSM will be. This is due to the fact that the increase of the average torque generated by the stator armature current is higher than the variation of the peak-peak value of the torque. For example, when the internal power factor angle is 20°, the torque ripple difference under different stator currents is large. Figure 14 compares the torque of the IPMSM with control current lead angle of 50° and PM deflection of 1.875°, respectively.

From Figure 14, when the stator rated current is 175 A, the torque ripple of the IPMSM with current lead angle of 50° is higher than that of the PM with deflection angle of 1.875°, and the torque ripple is 6.83% and 4.3%, respectively. The reason for this is that although the superposition of PM torque T_pm1 and reluctance torque T_re1 with current lead angle of 50° reduces the torque ripple, the difference of harmonic amplitude and phase between T_pm1 and T_re1 makes the torque ripple larger. In addition, the electromagnetic torque at the current lead angle of 50° is lower than that at the PM deflection angle of 1.875°, whose torque values are 105.4 Nm and 109.3 Nm, respectively. To meet the load torque requirements, the stator current of the motor should be increased at the current control angle of 50°, resulting in a slight increase in copper consumption of the motor. Therefore, compared with controlling the current lead angle, the research on the PM deflection angle reducing the torque ripple of the IPMSM is superior, and the reduction of torque is acceptable. Compared with the traditional non-skewed permanent magnet pole, the total torque ripple of the motor is significantly reduced regardless of the combination rotor structure or d-q axis current control mode.

Table 2. Torque performance of 45 kW IPMSM.

Item	Conventional Rotor	Combined Rotor			Current Control
	PM No Skew	No Auxiliary Slot	One Auxiliary Slot	Two Auxiliary Slots	Lead Angle 50°
Average torque (Nm)	112.7	109.1	109.3	109.8	105.4
Torque ripple (%)	11.06	5.71	4.3	2.36	6.83

shows the torque performance comparison of the 45 kW IPMSM.

As can be seen from Table 2, the harmonics synergistic suppression method of cogging torque and ripple torque of IPMSM proposed in this paper is relatively effective and has been applied in the permanent magnet combined rotor model, and the maximum reduction of total torque ripple is reduced from the initial 11.06% to 2.36%. However, the slight disadvantage is that the average torque also decreases and is 112.7 Nm and 109.8 Nm, respectively.

4. Torque-Speed Characteristic Analysis

The application of the integrated rotor structure of the IPMSM effectively reduces the key order torque harmonics of the PM torque, reluctance torque and cogging torque, and better improves the operation quality of the IPMSM. However, limited to the design requirements of the IPMSM with high torque density and high-power density, and considering the influence of the difference in the structure of the interior rotor on magnetic circuit saturation, the torque output characteristics of the IPMSM with combined rotor are studied. Figure 15 and Figure 16 respectively show the torque ripple and the simulation results of the external characteristics of the 45 kW IPMSM with a deflection of 1.875°.

As can be seen from Figure 15a,b, there are both magnetizing and demagnetizing regions within the polar arc range of PM of the IPMSM under the load field. The reason is that the common magnetic circuit of the q axis and the d axis in the d-q coordinate system is coupled, saturated and cross-saturated. For example, when the current lead angle is 30° and the axis of the A-phase winding is reconnected with d axis, the synthesized current vector coincides with the axis of the C-phase winding, so the q-axis magnetic field of the IPMSM is superimposed and weakened with the excitation field provided by the PM, and the saturation degree of the magnetic circuit is intensified with the increase of the load rate. By comparing the magnetic density distribution of Model I and Model II, it can be seen that the magnetic density saturation degree of the magnetizing region in the core of Model II is higher than that of Model I. It indicates that increasing the pole arc of the PM is more conducive to improve the utilization rate of the air-gap synthetic magnetic field when the effective material remains unchanged, and thus increases the average torque of the IPMSM in the constant torque interval, as shown in Figure 15c. It can also be seen from Figure 15c that the increase of torque ripple in Model II is higher than that in Model I due to the improvement of magnetic properties, which are 5.49% and 4.3%, respectively.

Figure 16a,b show that in the constant torque interval, Model II exhibits a higher electromagnetic torque (224.9 Nm) compared to Model I (220.7 Nm). Among them, the PM torque of Model II is the largest, while the reluctance torque is small, since the larger polar arc coefficient of PM improves the torque output performance of the motor. As the speed range of the IPMSM for electric vehicles increases, the reluctance torques of Model I and II are larger than the corresponding PM torques. In addition, the electromagnetic torque and electromagnetic power of Model II are gradually smaller than that of Model I, since the high rotor excitation field with wide constant power needs to provide larger direct axis demagnetization current, and its q-axis current is reduced by the limitation of the current limit circle. As can be seen from Figure 16c, the $L_{\mathrm{d}}$ of the IPMSM is less affected by the load rate, while the $L_{\mathrm{q}}$ of the q-axis decreases significantly with the increase of the load rate. For example, when the load rates of model I are 0.93 and 2.06, the $L_{\mathrm{q}}$ are 1.11 and 0.63, respectively. Therefore, the salient pole rate of the motor increases with the intensification of magnetic weakening depth. Therefore, it is more crucial to select a reasonable polar arc coefficient of PM when the IPMSM is running in a wide range of constant power. For instance, although the designed polar arc coefficient of PM is larger, the average torque in the constant torque interval is improved, but the power output in the constant power interval is reduced.

5. Conclusions

In this paper, the synergistic reduction method of cogging torque and ripple torque of IPMSM is studied based on harmonic offset principle. The composition and distribution of magnetic field harmonics of the IPMSM and the effective suppression measures of torque harmonics are analysed, and the torque-power characteristics of a 45 kW IPMSM for an electric vehicle is studied by using the finite element method. The summary of our findings is as follows:

(1)

The magnetic field harmonic offset method is more effective in weakening or eliminating some specific times of torque harmonics of PMSM, and has broad applicability. The deflection angles of permanent magnet to reduce ripple torque and cogging torque are π/2np and π/4np, respectively, and there is a two-fold relationship between the two deflection angles.

(2)

When the pole dislocation angle is (2n − 1) $\mathsf{\Delta }\alpha$, the first order tooth harmonics and the 6 or 12 ripple torque of the PMSM with larger amplitude and lower order can be effectively reduced. For example, when the number of slots per pole and phase q are 0.5 and 2, the deflection angle of the permanent magnet is 7.5° and 1.875° can weaken the 6 and 12 ripple torque harmonics, respectively. At the same time, the cogging torque is greatly reduced.

(3)

The average torque of PMSM in constant torque range is improved by increasing the rotor arc coefficient, but the torque ripple is also increased. With the increase of rotor pole arc coefficient, the speed regulation performance of wide constant power range decreases slightly. The trade off of average torque and torque ripple should be considered in combination with the load characteristics of the motor and the economy of the motor.

(4)

When the electric angle ratio of the rotor auxiliary slot of the combined rotor IPMSM θ₁/θ₂ is between 2–2.28, the torque ripple of the motor decreases significantly, such as $T_{{\mathrm{r}\mathrm{i}\mathrm{p}\mathrm{p}}_{-}\mathrm{m}\mathrm{i}\mathrm{n}}=2.36\%$, and reveals that the effective air gap harmonic suppression only changes the torque harmonics of some specific orders, but has little effect on the average torque.

(5)

Due to the fact that the torque ripple of the motor under load has a high requirement on the test accuracy, and the high precision instruments and equipment are also relatively expensive, the finite element analysis in this paper is effective in verifying the reduction of the cogging torque and ripple torque of the IPMSM by the offset method, and the comparative analysis of the diversity of the rotor structure is also relatively scientific.