Considering that the structure of permanent magnet synchronous motorized spindle is optimized in this paper, and the structural characteristics do not produce time harmonics, so only the eddy current loss caused by permeability harmonics and space harmonics is considered.

As shown in Figure 1 , the permeability harmonics arise because the stator core is slotted, resulting in different permeabilities between the yoke and the gap. As the rotor rotates, the air gap permeability corresponding to any segment of arc dx on the permanent magnet is constantly changing. The air gap is small and the magnetic permeability is large when it is at AB under the teeth. While at slot BC, the air gap is larger and the permeability is small. This variation is periodic in terms of slot spacing, with the permeability varying Z/λ times under each pair of poles and generating the corresponding tooth harmonics of the permeability. At this point, the air gap permeability can be written as (1) [16] :

$\Lambda ={\Lambda }_0+{\displaystyle \underset{k=1}{\overset{\infty }{\sum }}{\Lambda }_0\left(k\frac{Z}{\lambda }\omega t\right)}$ (1)

where ${\Lambda }_0$ is the constant component of the air gap permeability, ${\Lambda }_k$ is the kth harmonic amplitude of the air gap permeability, Ζ is the number of stator slots, and λ is the number of magnetic pole pairs, and ω is the electrical angular velocity.

The magnetic potential at the surface of the permanent magnet during no-load operation of the motor can be expressed as (2) [16] :

$F=F_m\cos \left(\lambda \zeta \right)$ (2)

where $F_m$ is the magnetic potential amplitude, $\lambda \zeta$ is the spatial angle of the magnetic potential of the permanent magnet concerning the reference axis.

At this point, the air-gap magnetic density at any instant is written as (3):

$B\left(\theta ,t\right)=F\left(\theta ,t\right)\cdot \Lambda \left(\theta ,t\right)=F_m\cos \left(\lambda \zeta \right)\left[\Lambda ={\Lambda }_0+{\displaystyle \underset{k=1}{\overset{\infty }{\sum }}{\Lambda }_0\left(k\frac{Z}{\lambda }\omega t\right)}\right]$ (3)

From (3), it can be seen that the number of harmonics of the magnetically guided teeth caused by slotting in the air-gap magnetic density waveform is the $kZ/\lambda \pm 1$th harmonic.

Spatial harmonic magnetomotive force is caused by non-uniform distribution of stator windings. It is divided into phase band harmonic magnetomotive force and tooth harmonic magnetomotive force. In this case, the phase-band harmonic component is since the conductor currents distributed along the periphery of the armature do not belong to the same phase of the winding, i.e. the transition between phases is not smooth enough and there are jumps; the tooth harmonic component is since the conductors are all concentrated in a finite number of slots and are not uniformly distributed along the armature, resulting in insufficient smoothing of the magnetic potential from slot to slot and jumps, as shown in Figure 2 .

When the stator winding is energized with a three-phase sinusoidal current, the νth harmonic magnetic potentials generated by the A, B, and C phase windings are added together to obtain the νth harmonic synthesized magnetic potentials of the three-phase windings as follows [16] :

$\begin{array}{c}{f}_v\left(x,t\right)=f_{Av}\left(x,t\right)+f_{Bv}\left(x,t\right)+f_{Cv}\left(x,t\right)\\ =F_{\varphi v}[\cos \frac{v\pi }{\tau }x\cos \omega t+\cos v\left(\frac{\pi }{\tau }x-120˚\right)\cos \left(\omega t-120˚\right)\\ +\cos v\left(\frac{\pi }{\tau }x+120˚\right)\cos \left(\omega t+120˚\right)]\end{array}$ (4)

where $f_{Av}$, $f_{Bv}$, and $f_{Cv}$ are the νth harmonic magnetic potentials of phases A, B, and C, respectively, x is the spatial distance coordinates, and $F_{\varphi v}$ is the νth harmonic synthesized magnetomotive force amplitude, and τ is the polar distance.

It has the following pattern:

1) When $v=3k$, $f_v\left(x,t\right)=0$, so there is no 3rd and multiples of 3rd harmonic.

2) When $v=6k\pm 1$, $f_v\left(x,t\right)=1.5F_{\varphi v}\cos \left(v\pi x/\tau \pm \omega t\right)$.

3) When $v=kZ/\lambda \pm 1$, there are magnetomotive force tooth harmonics of the same order as the permeability tooth harmonics.

From Equation (4), the amplitude of the harmonic magnetic potential varies with time and space, while the air gap permeability varies synchronously, and the product of the two is the flux $\varphi$. There are two types of flux variations, one of which is generated by time-varying currents, i.e. the flux is a function of time t, and the other is generated by the relative motion between the permanent magnet and the magnetic field, i.e. the flux is a function of position variable x.

From the law of electromagnetic induction [16] :

$e=-\frac{\text{d}\psi }{\text{d}t}=-\frac{\text{d}\Phi }{\text{d}t}$ (5)

where e is the induced electromotive force in the loop, ψ is the magnetic chain of the crosslinked loop, and Φ is the loop area flux.

From Equation (5), the change of harmonic magnetic potential will produce induced electromotive force on the surface of the conductor of the stator and rotor, and due to the different flux densities or magnetic field variation strengths in each part of the conductor, resulting in different induced electromotive forces in each part, which in turn form harmonic currents of different sizes inside the conductor and consume in the conductor’s internal resistance to form eddy current losses, as shown in Figure 3 .

Air gap harmonics (magnetic conductivity harmonics and space harmonics, etc.) generate eddy current losses in the permanent magnets on the surface of the rotor of a surface-mounted permanent magnet synchronous motor. During rotation, the air gap density and the magnetic density of the permanent magnets change accordingly, and the changing density induces eddy currents in the permanent magnets.

The variation of the air-gap flux density at a slot distance can be written in the form of a Fourier series in terms of $\alpha \in \left[0,2\pi /Z\right]$, which is obtained when the slot is in the middle position [17] :

$B_{\delta }(\alpha )=B_{av}\left[1-{\displaystyle {\sum }_{k=1}^{\infty }{\left(-1\right)}^k\beta k_c{\alpha }_{1k}\cos \left(kZ\alpha \right)}\right]$ (6)

where

${\alpha }_{1k}=\frac{2\sin \left(k\pi \frac{{b^{\prime }}_1}{{\tau }_u}\right)}{k\pi \left[1-{\left(k\frac{{b^{\prime }}_1}{{\tau }_u}\right)}^2\right]},{b^{\prime }}_1=\gamma \frac{\delta }{\beta },\gamma =\frac{{\left(\frac{b_1}{\delta }\right)}^2}{5+\frac{b_1}{\delta }}$

$\beta =\frac{B_{\max }-B_{\min }}{2B_{\max }}=\frac{1+u^2-2u}{2\left(1+u^2\right)},\frac{B_{\min }}{B_{\max }}=\frac{2u}{1+u^2},u=\frac{b_1}{2\delta }+\sqrt{1+{\left(\frac{b_1}{2\delta }\right)}^2}$

where $B_{av}$ is the average flux density; ${\tau }_u$ is the slot spacing; u is the voltage specification; ${b^{\prime }}_1$ is the total width of the region; and $\delta$ is the air gap length, using the equivalent air gap length ${\delta }_e$ proposed by Carter.

From the above equation, it can be seen that the ratio of notch width to air gap length affects the change of air gap flux, i.e. keeping the notch width constant, the air gap flux can be reduced by increasing the air gap length, and chipping the permanent magnets can change the length of the equivalent air gap.

Based on the above principle, this paper proposes to cut the corners of the permanent magnets to increase the equivalent air gap length by changing the structural shape of the permanent magnets to reduce the eddy current loss. The tile-type permanent magnet pole chipping scheme is shown in Figure 4 , where the structural shape of the permanent magnet with right angles at both ends as shown in (a) is changed to a chipped structural shape with angles at both ends of the permanent magnet as shown in (b) without reducing the volume of the permanent magnet.

After chipping, the air gap length in Equation (6) changes and the surface of the permanent magnet poles is no longer isotropic and A is corrected to [18] :

$K_{\alpha }=1-1/2\left(1-h_{M1}/h_M\right)\left(1-{\theta }_3/{\theta }_2\right)$ (7)

According to the clipping angle scheme in this paper, where $h_{M1}$ is taken as 0 in Equation (7).

Substituting $K_{\alpha }$ into the equivalent air gap length equation, the equivalent air gap after clipping the angle poles is obtained as:

${\delta }_e=K_{st}{K}_{\delta }\delta /K_{\alpha }$ (8)

Because the magnetic permeability of permanent magnets and air permeability is close, it can be considered that the change rule of the magnetic density of permanent magnets and air gap magnetic density change rule is basically the same, and the introduction of the skin effect penetration depth ${\delta }_{PM}$ to consider the influence of the permanent magnet vortex reaction, can be calculated that the air gap harmonics in the permanent magnet internal sensing of the vortex density $J_M$ is [6] :

$J_m=\sigma \frac{\partial B_{\delta }\left(\alpha \right)}{{\partial }_t}{\text{e}}^{-Z/{\delta }_{PM}}$ (9)

where the rate of attenuation of the magnetic field in the conductor is expressed in terms of the depth of penetration [19] :

${\delta }_{PM}=\frac{1}{\sqrt{\pi f\sigma \mu }}$ (10)

where ${\delta }_{PM}$ is the harmonic penetration depth in the permanent magnet, μ is the magnetic permeability of the material, σ is the conductivity of the material, and f is the harmonic frequency.

According to the Poincarin vector [20] , the power penetrating the closed surface S from within the volume V is:

$P={\displaystyle {\oint }_S\left(E\times H\right)\text{d}S}={\displaystyle {\int }_V\nabla \cdot \left(E\times H\right)\text{d}V}$ (11)

where $E$ is the electric field strength, $J$ is the eddy current density, $H$ is the magnetic field strength.

Since the permanent magnet hysteresis loss exists only under special conditions, the permanent magnet loss is almost exclusively eddy current loss, so the formula for calculating the permanent magnet eddy current loss can be simplified as [21] :

$P={\displaystyle {\int }_{V}E\cdot J\text{d}V}={\displaystyle {\int }_V\frac{J^2}{\sigma }\text{d}V}={\displaystyle {\int }_V\rho J^2\text{d}V}$ (12)

where V is the volume of the PM, ρ is the permanent magnet resistivity.

According to the integral form of Faraday’s law of electromagnetic induction, we get:

${\displaystyle {\oint }_{l}E\text{d}l}=-\frac{\text{d}\varphi }{\text{d}t}=-\frac{\text{d}}{\text{d}t}{\displaystyle \underset{S}{\iint }B\text{d}S}$ (13)

where $\varphi$ is the total magnetic flux in the eddy current path, S is the total area in the vortex path.

Substituting $E=\rho J$ into the above equation gives:

${\displaystyle {\oint }_{l}J\text{d}l}=\frac{1}{\rho }{\displaystyle {\oint }_{l}E\text{d}l}=-\frac{1}{\rho }\frac{\text{d}}{\text{d}t}{\displaystyle {\iint }_{S}B\text{d}S}$ (14)

From Equations (6)-(12), it can be seen that the clipping angle can increase the equivalent air gap length to reduce the harmonic flux rate of change, and the eddy current density decreases with the decrease of the flux rate of change, so that the eddy current loss becomes smaller. Then, from Equation (12) and Equation (14), it can be seen that the permanent magnet eddy current loss is determined by the permanent magnet resistivity and eddy current density, in the case of the permanent magnet resistivity is unchanged, the magnetic flux density on the permanent magnet can reflect the size of the eddy current density.

In this paper, a 25 kW table-applied permanent magnet synchronous motor is taken as an example, it has no metal sheath on the surface of the permanent magnet and is made of carbon fiber tied, and its detailed parameters are shown in Table 1 . According to the parameters of Table 1 , the finite element simulation two-dimensional model of the surface-mounted permanent magnet synchronous motor is drawn, as shown in Figure 5 . Finite element calculation mesh using triangular mesh, taking into account the main study in this paper the eddy current loss of permanent magnets, so the permanent magnets, and permanent magnets near the carbon fiber protective sleeve and air gap in the mesh partition to do encrypted processing, set up the triangular mesh length of not more than 0.4 mm, the permanent magnets a little far away from the windings part of the mesh set up the mesh length of not more than 0.5 mm, far away from the permanent magnets of the stator and rotor part of the mesh set up the mesh length of not more than 5 mm. The mesh length is not more than 5 mm, and the meshing results are shown in Figure 6 .

To simplify the analysis, the following assumptions are made:

1) Neglecting the end effect of the eddy current, only the axial component of the rotor eddy current is considered.

2) The material is isotropic and the hysteresis effect of the ferromagnetic material is neglected.

3) Neglect the temperature effects of the magnetic permeability μ and the electric conductivity σ, which are only spatial functions.

4) Negligible eddy currents in the stator-rotor laminated core and current source current region.

Describing the field in terms of the magnetic field vector magnetic potential A, the problem of the fixed solution of the transient electromagnetic field can be expressed as [20] :

$\{\begin{array}{l}\frac{\partial }{\partial x}\left(\nu \frac{\partial A}{\partial x}\right)+\frac{\partial }{\partial y}\left(\nu \frac{\partial A}{\partial y}\right)=-\left(J_Z-\sigma \frac{\text{d}A}{\text{d}t}\right)\\ {\nu \frac{\partial A}{\partial n}|}_{l^{-}}-{\nu \frac{\partial A}{\partial n}|}_{l^{+}}=J_s\\ {A|}_{ABCD}={A|}_{EFGH}=0\end{array}$ (15)

where A is the internal magnetic field vector potential of the motor, ν is the magnetoresistivity, $J_Z$ is the source current density, and $J_s$ is the permanent magnet equivalent surface current density, and $\sigma \text{d}A/\text{d}t$ is the eddy current density, and ABCD and EFGH are motor boundaries.

Where the magnetic field vector potential A is given by the following equation [17] :

$B=\nabla \times A$ (16)

From the Coulomb condition, the magnetic field vector potential is defined as follows [17] :

$\nabla \cdot A=0$ (17)

In the two-dimensional case, A has only Z-axis components, in the case of a two-dimensional field containing eddy currents [17] :

$-\nabla \cdot \left(\frac{1}{\mu }\nabla A_Z\right)+\sigma \frac{\partial A_Z}{\partial t}=0$ (18)

The following component form is derived from the definition of the flux density vector [17] :

$Bx=\frac{\partial A_Z}{\partial y},By=-\frac{\partial A_Z}{\partial x}$ (19)

$B=\sqrt{{\left(\frac{\partial A_Z}{\partial x}\right)}^2+{\left(\frac{\partial A_Z}{\partial y}\right)}^2}$ (20)

According to Stokes’ theorem, the magnetic flux Φ [17] :

$\Phi ={\displaystyle {\int }_{a}B\cdot \text{d}a}={\displaystyle {\int }_{a}rotA\cdot \text{d}a}={\displaystyle {\oint }_{\Gamma }A\cdot \text{d}l}$ (21)

The equation for the magnetic lines of force in a plane field is [17] :

$B\times \text{d}l=\left(\frac{\partial A_Z}{\partial y}i-\frac{\partial A_Z}{\partial x}j\right)\times \left(\text{d}xi+\text{d}yj\right)=\left(\frac{\partial A_Z}{\partial y}\text{d}y+\frac{\partial A_Z}{\partial x}\text{d}x\right)k=0$ (22)

The full differential of the isovector A line is [17] :

$\text{d}A=\frac{\partial A_Z}{\partial x}\text{d}x+\frac{\partial A_Z}{\partial y}\text{d}y=0$ (23)

So, the magnetic lines of force are then isovectorial A lines.

In this paper, no-load and load simulations of permanent magnet synchronous motors are carried out. Figures 7-10 show the magnetic induction intensity cloud diagram and magnetic force line cloud diagram of the permanent magnet synchronous motor at no load and load, respectively. According to the simulation cloud diagrams, the magnetic lines of force and magnetic induction intensity of the prototype are symmetric along the center of the magnetic poles and are uniformly distributed at no load, but at load the magnetic field is distorted due to the armature reaction, resulting in that the magnetic lines of force and magnetic induction intensity are no longer uniformly distributed. In the no-load state, because of zero current in the winding, does not produce three-phase winding magnetic potential, so there is only the magnetic conduction harmonics due to slotting in the motor; in the load state, the winding is a non-zero three-phase sinusoidal current, not only the magnetic induction intensity of the permanent magnets is increased compared to the no-load state, but there are more magnetic potential harmonics due to three-phase winding in the motor. Whether in the no-load state or load state, the magnetic induction intensity and the size of the magnetic lines of force at both ends of each permanent magnet are maximum, according to Equation (5), the induced electromotive force generated on both sides of the permanent magnet is the maximum, which produces the most eddy current losses.

Since it is the various harmonics that lead to the generation of eddy current losses in the permanent magnets, and the air-gap radial magnetization waveform contains the various harmonics, the air-gap radial magnetization waveform of the motor under the rated load is extracted as shown in Figure 11 . It can be seen that the prototype air gap radial magnetization jumps up and down, and the sine wave tends to flatten. This is due to the cogging effect, which makes the air gap magnetomagnetism drop significantly at the position of the slot.

The use of Fourier analysis of the motor air gap radial magnetic field decomposition can be obtained 1, 3, 5, 7, and other odd-order harmonic components, there are no even-order harmonic components because the magnetic field waveform is symmetrical concerning the centerline of the magnetic poles. Figure 12 shows the magnitude of the magnetic induction of each odd-order harmonic component after Fourier decomposition, and it can be seen that in addition to the space harmonics (phase band harmonics) and the notch harmonics, there are also the 3rd, 9th, 15th and 21st harmonics, which are saturated harmonics due to the material properties.

According to the flute law, the eddy currents in the permanent magnet in turn generate an induced magnetic field that impedes the variation of this harmonic field, and this induced magnetic field causes the high-frequency harmonic field to decay in the axial direction of the permanent magnet. The farther the permanent magnet is from the air gap, the smaller the eddy current loss inside the permanent magnet.

The frequency of the vth harmonic is a multiple of the fundamental frequency, and as the number of harmonics rises, the depth of skin penetration decreases, resulting in higher current densities in the rotor sheath as well as on the permanent magnets.

When the prototype is operated at rated condition, ${\delta }_9=4.27$ mm, ${\delta }_{11}=3.86$ mm, it can be seen that the high harmonic penetration depth of 11 and above is less than the thickness of the permanent magnet, and the skin effect can not be ignored, so it is necessary to set up an encrypted mesh of the skin layer of the permanent magnet during the simulation of the permanent magnet’s eddy current loss.

For this paper chipping program, the chipping angle θ is calculated to know that the chipping angle θ is less than 25˚ when the motor permanent magnets will be intersected together, so the minimum value of the chipping angle θ is set to 30˚. θ angle in addition to the initial 90˚, the other selected angle is equal to 60˚, 45˚, and 30˚, to study the influence of the chipping angle of the permanent magnets for permanent magnet eddy current loss.

Based on the selected motor parameters and set boundary conditions in this paper, the magnetic line cloud diagrams of permanent magnets with four cutting angles are obtained by finite simulation as shown in Figure 13 , where the magnitude of the magnetic lines represents the magnetic induction intensity per unit area. As can be seen from the magnetic line cloud diagram in Figure 13 , the maximum value of the magnetic lines of force appears at both sides of the permanent magnet, that is, the place where the permanent magnet is clipped, and the maximum value of the permanent magnet’s magnetic lines of force decreases gradually with the decrease of the clipping angle θ, decreasing from 0.0185 Wb/m at $\theta =90˚$ to 0.0178 Wb/m at $\theta =30˚$. According to Equation (14), it can be seen that the eddy current density and the size of the magnetic induction strength are proportional to the relationship, that is, the eddy current loss and the size of the magnetic lines of force are also proportional to the relationship, so the permanent magnets for the chipping process can reduce its eddy current loss, and the smaller the permanent magnet chipping eddy current loss will be less.

The values of average flux density and eddy current loss of permanent magnet with different clipping angles θ extracted from the simulation results are shown in Table 2 , from which it can be seen that as the clipping angle θ decreases, the average flux density and eddy current loss of the permanent magnet is gradually reduced. The permanent magnet eddy current loss decreases from 15.81 W for $\theta =90˚$ to 14.26 W for $\theta =30˚$, which is a reduction of 9.8%. The size of the permanent magnet flux density not only affects the size of the eddy current loss but also affects the performance of the motor, according to the simulation results, the permanent magnet chipping angle $\theta =30˚$ motor torque is only 0.51% lower than that of the unchipped angle ( $\theta =90˚$), which has a negligible effect on the performance of the motor. Therefore, the simulation results show that the eddy current loss of permanent magnets can be significantly reduced by chipping the end face of permanent magnets to change its structural shape under the condition of hardly affecting the performance of the motor.