Torque Ripple Optimization of a Novel Cylindrical Arc Permanent Magnet Synchronous Motor Used in a Large Telescope

Abstract

This paper proposes the use of a novel cylindrical arc permanent magnet synchronous motor (CAPMSM) in a large telescope, which requires high positioning accuracy and low torque ripple. A 2D finite element method was used to analyze the cogging torque of the CAPMSM. The CAPMSM can be an alternative for a rotating motor to realize direct drive. A new method is proposed to separate the cogging torque, T_cog, into the torque, T_slot, generated by the slotted effect and the end torque, T_end, generated by the end effect. The average torque and the torque ripple are optimized considering stator center angle, the angle between two adjacent stators and the unequal thickness of a Halbach permanent magnet. The torque ripple decreased from 31.73% to 1.17%, which can satisfy the requirement of tracking accuracy for large telescopes.

1. Introduction

A traditional rotary motor integrating gear box is used to achieve limited angular motion, which causes issues such as taking up a lot of space, leading to low transmission accuracy. Therefore, an arc permanent magnet synchronous machine (APMSM) is proposed to achieve the direct drive for finite angular motion [1]. However, the application of an APMSM is subjected to reluctance force, control precision, and speed response issues. The reluctance force is generated by the slotted effect and the end effect. Chu et al. investigated the influence of skewing on the reduction of the slotted effect [2]. The slotted torque and cogging torque are reduced by the variations of permanent magnet (PM) arrangement [3,4]. Fractional slot concentrated winding is an effective way to reduce torque ripple [5,6]. Furthermore, the slotted effect can be decreased by adjusting pole pitch, optimizing magnet-pole shape and, adopting closed stator slots, etc. [7,8,9]. An auxiliary poles optimal design is adopted to reduce the end effect reluctance force of permanent magnet linear synchronous motor (PMLSM), which is caused by a finite stator length [10]. In Reference [11], the influence of the stator length on the reluctance force was analyzed, which showed that the reluctance force can be greatly decreased by optimizing the stator length. A kind of compensation winding was proposed to decrease the reluctance force in Reference [12]. Ho et al. proposed a Halbach PM array to increase in thrust force density and reduce cogging force [13]. Wang et al. showed that the dual-stator PM vernier machine with flux-reversal PM arrangement brings high torque production with low cogging torque [14]. Jin et al. proposed a combined maximum cogging torque (MCT) optimization method to optimize the MCT of surface-mounted PM machines with relatively low computer resources [15]. Du et al. proposed a new interior PM machine design, which can increase PM utilization ratio and reduce the torque ripple [16]. Liu et al. proposed a method of reducing total torque ripple by magnets shifting [17]. In Reference [18], a PM tubular linear generator with a Halbach PM arrangement was proposed to weaken the reluctance force. In Reference [19], it was shown that a ∇ + U shape topology of rotor has the least total harmonic distortion (THD) of air-gap flux density and cogging torque. Zou et al. proposed PM arc shaping to reduce cogging torque and torque ripple [20].

This paper proposes a novel cylindrical arc permanent magnet synchronous motor (CAPMSM) with three stators. The detent force of the proposed CAPMSM is optimized considering stator central angle, angle between two adjacent stators, unequal thickness of Halbach PM, and the central angle of slot pitch. The optimal stator center angle is obtained by finite element verification of the left and right end forces. The angle between two adjacent stators is optimized by Fourier analysis of the end torque of three stators. A sinusoidal distribution of air gap flux density is improved by adopting unequal thickness of Halbach PMs. The slot pitch is changed to improve the period of slot cogging force. In addition, a new method is proposed to separate the cogging torque, T_cog, into the torque, T_slot, generated by the slotted effect and the end torque, T_end, generated by the end effect for CAPMSM.

2. Configuration of Motor Model

The proposed CAPMSM was comprised of three stators and one rotor. The angle between two adjacent stator modules was θ_ss. The stator winding used a 12-slot/10-pole fractional-slot topology. The stator center angle was θ_s, which was the arc angle of the stator core. The optimized stator center angle was θ_sopt. The rotor module consisted of a ring yoke and an unequal thickness of Halbach PM. The finite element method (FEM) was used to analyze the electromagnetic performance of four models: the conventional APMSM (Model I), the proposed CAPMSM (before optimized, Model II), the proposed CAPMSM (after optimized, Model III), and the proposed cylindrical arc Halbach PM synchronous motor (CAHPMSM) (after optimized, Model IV) as shown in Figure 1. Figure 2 shows the partial 3D configuration of the proposed model, followed by the sectional view of the 3D model shown in Figure 3.

Table 1. Items of the initial cylindrical arc permanent magnet synchronous motor (CAPMSM) model.

Items	Symbol	Value
Stator Outer diameter	Dout (mm)	480
Air gap	g (mm)	1.2
Stator central angle	θs (°)	56.25
Angle between two adjacent stators	θss (°)	135
Central angle of pole pitch	θp (°)	4.5
Central angle of slot pitch	θsp (°)	3
Stack length	h (mm)	70
Slot number	Snum	40
Rotor pole-pair number	Pnum	64
Current	I (A)	15
Voltage	U (V)	15.82
Turns per slot	N	20
Current density	J (A/mm2)	7.5
Efficiency	η (%)	67.8%
Rated speed	nn (rpm)	11.25
PM grade	NdFeB40H
Steel grade	DW310-35

gives the key design parameters of the initial design, and the 2D finite element analysis of the single module is shown in Figure 4.

3. Investigation into Separating T_slot and T_end

For CAPMSM, the cogging torque, T_cog, included the slotted torque, T_slot, generated by the slotted effect and the end torque, T_end, generated by the end effect that can be written as:

$$T_{\mathrm{cog}}=T_{\mathrm{end}}+T_{\mathrm{slot}}$$

(1)

It is difficult to directly separate the torque, T_slot, and the torque, T_end, from the torque, T_cog. In Figure 4, a 12-slot/10-pole fractional slot structure can be seen as a one-unit motor. The torque, T_slot, of a one-unit motor can be obtained by the difference of cogging torque, T_cog, between Figure 5a,b. Therefore, the end torque, T_end, can be calculated by the difference of the total cogging torque, T_cog, and the slotting torque of K unit motors. The cogging torque can be obtained as shown in Figure 5.

$$T_{\mathrm{cogK}}=T_{\mathrm{end}}+K\cdot T_{\mathrm{slot}}$$

(2)

where K is the number of unit motors. Then, the torque T_slot and torque T_end can be separated by Equation (3).

$$\{\begin{array}{c}{T}_{\mathrm{slot}}=T_{\mathrm{cogK}+1}-T_{\mathrm{cogK}}\\ T_{\mathrm{end}}=T_{\mathrm{cogK}}-K\cdot T_{\mathrm{slot}}\end{array}$$

(3)

When considering the influence of the end effect on the torque T_slot, the more accurate the K is, the more accurate the result will be. Since rotor module circumference is finite, the value of K is taken as 6 for the CAPMSM, as shown in Figure 5. The peak-to-valley value of T_slot and T_end are 2.5 Nm and 24.78 Nm respectively, as shown in Figure 6. The separation result shows that T_end is the main component of the cogging torque.

4. Optimization Method Analysis of Torque Characteristics

The cogging torque included two aspects: one was the end effect for the arc length of the stator core and the other was the slot effect caused by the interaction between the PM on the rotor and the iron core of the stator when the winding was not energized. Therefore, the cogging torque could be reduced by the following two methods.

4.1. Optimization of End Torque

4.1.1. Optimization of the Stator Center Angle

T_end consists of the left end torque, T_Lend, and the right end torque, T_Rend. T_Lend and T_Rend can be expressed in Fourier series as follows:

$$T_{\mathrm{Lend}}=a_0+{\displaystyle \sum _{n=1}^{\infty }{a}_{\mathrm{cn}}}\cos (\frac{2n\pi }{{\tau }_p}t)+{\displaystyle \sum _{n=1}^{\infty }{a}_{\mathrm{sn}}\sin (\frac{2n\pi }{{\tau }_p}t)}$$

(4)

$$\begin{array}{l}{T}_{\mathrm{Rend}}=-a_0+{\displaystyle \sum _{n=1}^{\infty }{a}_{\mathrm{cn}}}\cos \left(\frac{2n\pi }{{\tau }_p}\left(t+\psi \right)\right)\\ +{\displaystyle \sum _{n=1}^{\infty }{a}_{\mathrm{sn}}\sin \left(\frac{2n\pi }{{\tau }_p}\left(t+\psi \right)\right)}\end{array}$$

(5)

where φ is the phase angle difference between T_Lend and T_Rend, τ_p is pole pitch, and a_cn and a_sn are the magnitude of the nth harmonic component, respectively.

$$T_{\mathrm{end}}=T_{\mathrm{Lend}}+T_{\mathrm{Rend}}={\displaystyle \sum _{n=1}^{\infty }{a}_{\mathrm{n}}}\sin \left(\frac{2n\pi }{{\tau }_p}(t+\frac{{\tau }_p}{2})\right)$$

(6)

where a_n can be expressed as:

$$a_{\mathrm{n}}=2\left(a_{\mathrm{sn}}\cos (\frac{n\pi }{{\tau }_p}\psi )+a_{\mathrm{cn}}\sin (\frac{n\pi }{{\tau }_p}\psi )\right)$$

(7)

The amplitude of end effect force is closely related to the stator central angle. The end effect force can be reduced by optimizing the stator central angle. If a_n = 0, φ can be written as:

$$\psi =\frac{{\tau }_p}{n\pi }\arctan \left(-\frac{a_{\mathrm{sn}}}{a_{\mathrm{cn}}}\right)$$

(8)

Actually, $\psi$ is mainly dependent on the stator center angle. Thus, there will be a theoretical θ_s which enables the total torque of T_Lend and T_Rend to be zero. However, T_Lend and T_Rend cannot be ideal sinusoidal waveforms and have many harmonic components. So, the harmonic coefficients of each order cannot be equal to 0 at the same time and the end effect force cannot be eliminated completely. There is an appropriate θ_s which enables the sum torque of T_Lend and T_Rend to be at a minimum but not zero. The end force can be optimized at a minimum only when the fundamental harmonics φ_opt is eliminated.

$${\psi }_{\mathrm{opt}}=\frac{{\tau }_p}{\pi }\arctan \left(-\frac{a_{\mathrm{sn}}}{a_{\mathrm{cn}}}\right)$$

(9)

4.1.2. Optimization of the Stator Adjacent Angle

Through the above optimization analysis of the stator center angle, each stator module is equivalent to a unit motor. When the conventional splicing method is adopted, the output torque and reluctance torque of the motor are proportional to the number of stator modules of the motor.

The end torque is a function of the pole pitch period. If the end torque generated by the plurality of stators are different from each other by a certain phase, the synthetic torque of the rotor subjected to the end torque of the plurality of stators may be minimized.

In this paper, taking three stators as an example, the expression of the end torque can be written as:

$$T_{\mathrm{Rend}}=a_0+{\displaystyle \sum _{n=1}^{\infty }{a}_{\mathrm{n}}\sin (\frac{2n\pi }{{\tau }_p}t)}$$

(10)

$$T_{\mathrm{Lend}}=-a_0+{\displaystyle \sum _{n=1}^{\infty }{a}_{\mathrm{n}}\sin (\frac{2n\pi }{{\tau }_p}t+\psi )}$$

(11)

$$T_{\mathrm{end}}=T_{\mathrm{Lend}}+T_{\mathrm{Rend}}={\displaystyle \sum _{n=1}^{\infty }{a}_{\mathrm{n}}\cos (\frac{\psi }{2})\sin (\frac{2n\pi }{{\tau }_p}t+\frac{\psi }{2})}$$

(12)

$$\begin{array}{l}{T}_{\mathrm{s}\_\mathrm{end}}=T_{\mathrm{Lend}1}+T_{\mathrm{Lend}2}+T_{\mathrm{Lend}3}+T_{\mathrm{Rend}1}+T_{\mathrm{Rend}2}+T_{\mathrm{Rend}3}\\ ={\displaystyle \sum _{n=1}^{\infty }{a}_{\mathrm{n}}\sin (\frac{2n\pi }{{\tau }_p}t)}+{\displaystyle \sum _{n=1}^{\infty }{a}_{\mathrm{n}}\sin \frac{2n\pi }{{\tau }_p}(t+{\theta }_{\mathrm{ss}})}+{\displaystyle \sum _{n=1}^{\infty }{a}_{\mathrm{n}}\sin \frac{2n\pi }{{\tau }_p}(t+2{\theta }_{\mathrm{ss}})}\\ +{\displaystyle \sum _{n=1}^{\infty }{a}_{\mathrm{n}}\sin (\frac{2n\pi }{{\tau }_p}t+\psi )}+{\displaystyle \sum _{n=1}^{\infty }{a}_{\mathrm{n}}\sin [\frac{2n\pi }{{\tau }_p}(t+{\theta }_{\mathrm{ss}})+\psi ]}+{\displaystyle \sum _{n=1}^{\infty }{a}_{\mathrm{n}}\sin [\frac{2n\pi }{{\tau }_p}(t+2{\theta }_{\mathrm{ss}})+\psi ]}\\ ={\displaystyle \sum _{n=1}^{\infty }{a}_{\mathrm{n}}[\sin \frac{2n\pi }{{\tau }_p}(t+{\theta }_{\mathrm{ss}})(2\cos \frac{2n\pi }{{\tau }_p}{\theta }_{\mathrm{ss}}+1)]+}{\displaystyle \sum _{n=1}^{\infty }{a}_{\mathrm{n}}\{\sin [\frac{2n\pi }{{\tau }_p}(t+{\theta }_{\mathrm{ss}})+\psi ](2\cos \frac{2n\pi }{{\tau }_p}{\theta }_{\mathrm{ss}}+1)]\}}\end{array}$$

(13)

If the sum of the three stators end torque is minimized, then it should satisfy:

$$2\cos \frac{2n\pi }{{\tau }_p}{\theta }_{\mathrm{ss}}+1=0$$

(14)

Therefore, the angle between two adjacent stators satisfies:

$${\theta }_{\mathrm{ss}}=k{\tau }_p\pm \frac{{\tau }_p}{3n}$$

(15)

where k is an arbitrary integer and n is the harmonic order.

In order to minimize the end torque, the component with the largest proportion of end torque harmonics must be determined. It can be seen from the above analysis that the sum of the end torque is still dominated by the first harmonic.

So, the angle between two adjacent stators is:

$${\theta }_{\mathrm{ss}}=k{\tau }_p\pm \frac{{\tau }_p}{3}$$

(16)

4.2. Optimization of Slotted Torque

The slotted torque is caused by the interaction between the PM and the armature-alveolar. The slotted torque can be expressed as:

$$T_{\mathrm{slot}}=-\partial W/\partial \alpha$$

(17)

where W is the magnetic field energy, and α is the relative position angle.

The relative position of the permanent magnet and stator is shown in Figure 7. The position of the center line of the PM is represented by θ = 0. The arc length of the stator core is equal to 2π, and its practical arc length is L_s, thereby α is the angle between the center line of stator teeth and the center line of PM.

Supposing that the armature core permeability is gigantic, the magnetic field energy, W, stored in the motor can be written as:

$$W\approx W_{airgap+PM}=\frac{1}{2u_0}{\displaystyle {\int }_V{B}^2}dV$$

(18)

where B is the magnetic flux density of airgap, V is the bulk of the airgap and, PM μ₀ is the magnetic permeability.

If the influences of saturation, magnetic flux leakage, and cogging effect are ignored, and the magnetic permeability of the PM is assumed to be the same as air, then B is:

$$B=B_{\mathrm{r}}\frac{h_{\mathrm{m}}}{h_{\mathrm{m}}+{\delta }_{\mathrm{e}}}$$

(19)

where B_r is PM remanence, δ_e is the effective length of airgap and, h_m is PM thickness.

In relation to the slotted PM motor, δ_e can be written as Carter coefficient (K_δ):

$$\{\begin{array}{l}{K}_{\mathsf{\delta }}=\frac{{\tau }_1}{{\tau }_1-\gamma {\delta }^{\prime }}\\ {\delta }_{\mathrm{e}}=\delta +(K_{\mathsf{\delta }}-1){\delta }^{\prime }\end{array}$$

(20)

where ${\delta }^{\prime }=\delta +h_{\mathrm{m}}/{\mu }_{\mathrm{r}}$, τ₁ is slotting pitch, δ is the length of air-gap and, μ_r is relative permeability. The slotting coefficient γ can be written as:

$$\gamma =\frac{4}{\pi }\left(\frac{b_0}{2{\delta }^{\prime }}{\tan }^{-1}\left(\frac{b_0}{2{\delta }^{\prime }}\right)-\ln \sqrt{1+{\left(\frac{b_0}{2{\delta }^{\prime }}\right)}^2}\right)$$

(21)

where b₀ is stator notch width.

According to the expressions mentioned, W can be further expanded to:

$$W={\displaystyle {\int }_0^{2\pi R}{\displaystyle {\int }_{\frac{L}{2\pi }}^{\frac{L}{2\pi }+{\delta }_{\mathrm{e}}}{\displaystyle {\int }_0^{2\pi }{B}_{\mathrm{r}}{}^2}}}{\left(\frac{h_{\mathrm{m}}}{h_{\mathrm{m}}+{\delta }_{\mathrm{e}}}\right)}^{2}rdrd\theta dL$$

(22)

where R is the outer diameter.

According to Equation (17), the slotting torque can be expressed as:

$$T_{\mathrm{slot}}=-\frac{{\pi }^{2}Z{L}_{\mathrm{S}}}{2{\mu }_0}\left(\frac{L_{\mathrm{S}}}{\pi }{\delta }_{\mathrm{e}}+{\delta }_{\mathrm{e}}{}^2\right)\times {\displaystyle \sum _{n=1}^{\infty }n{G}_n{B}_{\mathrm{r}\left(nZ/2p\right)}}\sin \left(\frac{2\pi nZ}{L_{\mathrm{S}}}\alpha \right)$$

(23)

where B_r(nZ/2p) is (nZ/2p) the harmonic component of PM remanence, G_n is the nth harmonic magnitude, Z is the slot number and, n is an integer. At the same time, nZ/2p is also an integer. It can be seen from Equation (23) that the slotted torque has great influence on the modulus of B_r(nZ/2p). Thus, by choosing the appropriate value of nZ/2p, T_slot can be decreased.

5. Optimization Result Analysis of Torque Characteristics

5.1. Optimization of Stator Central Angle

A slotless CAPMSM model was built to avoid the influence of slot cogging force as shown in Figure 8a. In order to calculate the left end torque and the right end torque accurately, when rotating clockwise only the magnet that was closest to the left was reserved as shown in Figure 8b, when rotating anti-clockwise, only the magnet that was closest to the right was reserved which is shown in Figure 8c. In CAPMSM, the stator was maintained in a fixed place and the rotor spun clockwise and anti-clockwise at the same rotating speed of 67.5°/s, respectively. The simulation results are shown in Figure 9.

It can be seen from Figure 9 that T_Lend was positive while T_Rend was negative. Furthermore, they both used pole pitch θ_p as the minimum period.

$${\theta }_{\mathrm{p}}=\nu \times T={67.5}^{\circ }/\mathrm{s}\times 0.066 \mathrm{s}={4.5}^{\circ }$$

(24)

where v is the rotating speed and T is a period. When T_Lend achieves the maximum amplitude, T_Rend cannot achieve the minimum amplitude, and there is a time delay of Δt = 32 ms. If the time delay is Δt = 0 ms, the synthetic end torque will reach its smallest measure.

$$\Delta \theta =\nu \times \Delta t={67.5}^{\circ }/\mathrm{s}\times 0.032 \mathrm{s}={2.16}^{\circ }$$

(25)

The left and right end torque were expanded into Fourier series as seen in Equations (4)–(9), which fluctuate with the pole pitch as the period. The phase difference is φ, and a_n changes with φ. When the left end torque reaches the maximum amplitude and the right end torque reaches the minimum amplitude, the synthetic end torque will reach its smallest measure. The appropriate angle is then obtained by calculating the phase of the two end torque. In the simulation of CAPMSM, if T_Lend is at the same place, while T_Rend is close to T_Lend, then the curve of T_Rend displayed in Figure 9 will move towards the left. The moved angle is Δθ.

After the stator central angle is optimized, the central angle of the stator can be calculated by:

$${\theta }_{\mathrm{sopt}}={\theta }_{\mathrm{s}}-\Delta \theta ={52.075}^{°}-{2.16}^{°}={49.915}^{°}$$

(26)

It can be seen from Figure 10 that after the stator central angle was optimized, the synthetic end torque was smaller than 3 Nm. The synthetic end torque was reduced when the stator central angle was optimized.

5.2. Optimization of the Adjacent Angle Between Two Stators

The end torque, T_end, is the cycle of the pole pitch, θ_p. When the phase angle difference between the adjacent stator end torque is 180°, the ideal total end torque generated by the adjacent stator end effect is zero. However, since T_end is not a desirable sinusoidal waveform that includes a large amount of high harmonic components, T_end cannot be reduced to zero.

For the purpose of offsetting the end torque, T_end, between the adjacent stator, the phase angle difference between the adjacent stator can be expressed as:

$${\theta }_{\mathrm{ss}}=k{\tau }_p\pm \frac{{\tau }_p}{3n}$$

(27)

Consequently, the phase angle difference between stator modules A and B is maintained using θ_ss, meanwhile the adjacent angle between stator modules B and C is maintained using the same value:

$${\theta }_{\mathrm{ss}}=27{\tau }_p-\frac{{\tau }_p}{3}={120}^{°}$$

(28)

It can be seen from Figure 11 that, after the stator adjacent angle is optimized, the peak-to-valley value of end torque decreased from 35.35 Nm to 4.36 Nm.

5.3. Unequal Thickness Halbach PM

Since the unequal thickness of Halbach PM is used, the flux density distribution of the air gap is closer to sinusoidal, which can weaken the cogging torque. The magnetic field on one side is enhanced, while it is weakened on the other side. Therefore, both the radial thickness and the amount of PM can be reduced. In Model III, the radial thickness of the PM was 2 mm and, in Model IV, the radial thickness of the PM was labeled as shown in Figure 12. For the same output torque, Model IV had a lower PM usage than that of Model III. Figure 13 shows that the peak-to-valley value of output torque decreased from 3.96 Nm to 1.47 Nm.

5.4. Change the Central Angle of Slot Pitch

From the above analysis, it can be seen that by reasonably selecting the number of armature slots and the number of poles, changing the B_rn plays a major role in the slotted torque and can weaken the slotted torque. The cycle of the slotted torque γ is the least common multiple (LCM) of slots, Z, and pole pairs, p. The amplitude will decrease with the increasing of γ:

$$\gamma =LCM\left[{\rm Z},2p\right]$$

(29)

When the pole pairs, p, of the rotor are constant, the number of slots of the stator can be changed by changing the central angle of slot pitch, θ_sp, the LCM of stator slots, and the rotor poles can be increased, thereby the fundamental frequency of the slotted torque is increased. The amplitude can be reduced by increasing the fundamental frequency of the slotted torque.

In the initial design of the CAPMSM, the single rotor was selected to be 12 slots and 8 poles, with the central angle of slot pitch at 3°. The motor had 120 slots and 80 poles. Then γ₁ was:

$${\gamma }_1=LCM\left[{\rm Z},2p\right]=240$$

(30)

It was necessary to increase the LCM of stator slots and poles to reduce the slotted torque. The central angle of slot pitch was changed to 3.75°. The motor had 96 slots and 80 poles. Then γ₂ was:

$${\gamma }_2=LCM\left[{\rm Z},2p\right]=480$$

(31)

It can be seen from Figure 14 that after the central angle of slot pitch was optimized, the magnitude decreased from 8.42 Nm to 2.21 Nm. It can be seen from Figure 15 that the peak-to-valley value of output torque decreased from 35.35 Nm to 1.47 Nm after adopting the optimized stator central angle, the angle between two adjacent stators, the unequal thickness of Halbach PM, and the central angle of slot pitch.

Figure 16 shows the air gap flux density distributions of the four models. Because of the 4mm radial thickness of the permanent magnet of Model I, the peak value of air gap flux density of Model I was greater than those of other models. Figure 16 shows that the sinusoidal property of air gap flux density in Model II was the worst, but that of Model IV was the best. Further, harmonic spectra analysis of air-gap flux density of the four models was carried out, which is shown in Figure 17. The 5th and 7th harmonic components occupied the main proportion, and the 3rd, 5th and, 7th harmonic components of Model II were larger than those of Model IV.

From Figure 18, the flux density of the air gap side can be increased, and the flux density of the rotor yoke can be reduced by using Halbach PMs with different thicknesses. Therefore, the radial thickness and PM consumption can be reduced.

6. Conclusions

This paper proposes techniques to reduce torque ripple of CAPMSM including optimizing stator central angle, changing angle between two adjacent stators, adopting unequal thickness of Halbach PMs, and adjusting the central angle of slot pitch. Fourier analysis and FEM simulation were used to obtain an optimal stator central, θ_sopt, a suitable angle between two adjacent stators, θ_ss, and an appropriate central angle of slot pitch, θ_sp. The method was proposed to separate the cogging torque, T_cog, into the torque, T_slot, caused by slotting and the end torque, T_end, caused by the end effect of stator. In

Table 2. Comparisons of the four models.

Items	Symbol	Model I	Model II	Model III	Model IV
Stator Outer diameter	Dout (mm)	480
Air gap	g (mm)	1.2
Stator central angle	θs (°)	56.25	52.075	49.915	49.915
Angle between two adjacent stators	θss (°)	90	135	120	120
Stack length	h (mm)	50	70	70	70
Central angle of pole pitch	θp (°)	5.625	4.5
Central angle of slot pitch	θsp (°)	5	3	3.75	3.75
Rated output torque	Tm (Nm)	120	126.65	126.56	126.04
Torque ripple	κ	4%	31.73%	3.13%	1.17%

, the torque ripple of the proposed CAPMSM (Model III) and CAHPMSM, CAPMSM with Halbach (Model IV), was 0.87% and 2.83% smaller than that of Model I, respectively. When the optimization model was established, the torque ripples decreased from 31.73% to 1.17%. The optimized CAPMSM can meet the requirements of tracking accuracy for a large telescope, whose torque ripple is below 2%.