A New Spoke PM Motor with ECC ASPs to Reduce Flux Density Harmonics

Abstract

This paper proposes new eccentric (ECC) auxiliary salient poles (ASPs) applied in spoke permanent-magnet (PM) motors. These ECC ASPs optimize flux density harmonics and further reduce torque pulsation. Then, the optimization procedure is proposed to reveal the relationship between the ECC method and ASP, and optimization steps are reduced. Firstly, the design of the ECC ASPs is based on the stairs, which are equivalent to sinusoidal flux density. Second, the expressions of offset length for maximum air gap, pole arc angle, and thickness ratio are revealed. Through theoretical analysis and the finite element method (FEM), the proposed ECC ASPs can generate low torque pulsation and improve sound and vibration features.

1. Introduction

Permanent magnet (PM) servo motors have been used for various industries. Industrial robots are popular in various industrial production methods for high operational precision and efficiency. As the main part of industrial robots, servo motors must have fast response, higher torque density, and especially lower torque pulsation [1,2,3]. In various PM motors, spoke PM motors can produce high flux density due to the concentration of the flux by the adjacent PMs [4,5]. However, an obvious disadvantage of the spoke PM motor is that the flux density is rich in harmonics, leading to high torque pulsation and high torque pulsation [6]. As a servo motor, a high torque pulsation could reduce system performance and accuracy [7,8]. Therefore, torque pulsation is the main optimization objective for spoke PM motors.

Methods to reduce torque pulsation in motor control usually include harmonic injection and harmonic current reduction [9,10]. In addition, design-based methods are generally employed [11,12,13,14,15,16,17,18,19,20,21]. Initially, a proper slot-pole combination of the motor should be determined, as it is related to the torque pulsation period [11]. After that, the optimized winding configuration effectively reduces the sub-harmonics, which obtains low torque pulsation [12]. Then, the choice of phase number could also affect the torque pulsation [13]. Furthermore, rotor-based methods to reduce torque pulsation are more common. To reduce cogging torque, the rotor segmentation skewing technique is applied in both surface-mounted and interior motors [14,15]. However, the method of axial structure design of the rotor, which requires the establishment of a 3D model and the FEM to solve the 3D model, is very time-consuming. In addition, the stair shape magnet is applied in the surface-mounted PM motor to reduce the torque ripple [16]. However, the stair PM is hard to manufacture and may produce errors. Furthermore, inverse cosine-shaped (ICS) and ICS + third-harmonic methods can be applied to both surface-mounted and interior PM motors to improve average torque [17,18]. However, the shaped rotor methods need to follow ICS and ICS + 3rd shape formula and increase the difficulty of modeling and manufacturing. To change the flux density shape, using modular PMs in the pole of the surface-mounted PM motor can reduce the harmonics of flux density and back-EMF [19]. However, it is difficult to apply in the spoke PM motor.

The ECC ASP spoke PM motor is proposed as a low power servo motor, e.g., intelligent robot joint motor or industrial robotic arm motor. For reducing torque pulsation, the ECC method has been used in the spoke PM motors [20]. However, the above ECC method does not propose an optimization procedure. In addition, a previous study has proposed that symmetric ASPs can reduce torque pulsation, but the effect is limited [21]. Hence, ECC ASPs are proposed for spoke PM motors to maintain the merit of the APS, overcome the previous drawbacks, and refine the optimization procedure. To begin with, the stair method is designed by sinusoidal flux density, then, the sinusoidal air gap flux density is equivalent to stairs, which are equal width and unequal height by using the principle of equal area. When the number of stairs is enough, connecting the midpoints of the stairs can be approximated as an arc. Theoretically, ECC ASP can reduce the flux density harmonics through equivalent sinusoidal flux density.

This paper is structured as follows. In Section 2, the optimized topologies of rotors will be presented. In Section 3, the procedure will be revealed, relating to the torque pulsation reduction through ECC ASP. In Section 4, firstly, the optimization results are given by FEM. Secondly, the performance of the ECC ASP motor is compared with the traditional motor and ASP motor. Thirdly, the sound and vibration features of the ECC ASP motor are compared with traditional ones. Finally, Section 5 concludes this paper.

2. Topologies

Figure 1a shows the traditional spoke PM motor, and Figure 1b shows the proposed ECC ASPs spoke PM motor. For obtaining more torque pulsation cycles, the five-phase winding and 40 slots and 8 poles are chosen. Since the rotor is composed of eight identical units, it can be optimized for only one unit. Additionally, the minimum air-gap lengths of the three rotor units are the same. As shown in Figure 2, the traditional spoke PM motor, called Model 1, is selected as the optimization candidate. Compared to Model 1, Model 2 reduces torque pulsation by adding ASP to the rotor surface. Additionally, Model 3 is the ECC ASP rotor unit. Then, the main parameters of Model 1 and Model 2 are listed in

Table 1. Main Parameters of Models 1 and 2.

Items	Model 1	Model 2
Stator slots/rotor poles	40/8	40/8
Rated speed (r/min)	1500	1500
Rated current (A)	10.8	10.8
Stack length (mm)	46	46
Stator outer diameter (mm)	155	155
Stator inner diameter (mm)	98	98
Minimum air-gap length (mm)	0.5	0.5
The pole-arc ratio of the ASP	0	0.75
Depth of the ASP hm (mm)	0	2.5
Numbers of turns per slot	40	40
Phase resistance at 21 °C (Ω)	0.5	0.5
Iron core material	DW470_50	DW470_50
The thickness of the PM (mm)	3	3
Width of the PM (mm)	10	10
PM material	N35	N35
Remanence of the PM (T)	1.23	1.23

.

3. Optimization Procedure of ECC ASP

In [22], the average torque (T_avg) and torque pulsation (T_pul) of a five-phase PM motor is expressed as

$$\left\{\begin{array}{l}{T}_{avg}=\frac{{\mu }_0}{g}p{r}_{g}L\pi F_{s1}{F}_{r1}\sin ({\gamma }_d)\\ T_{pul}=-\frac{{\mu }_0}{g}p{r}_{g}L\pi {\displaystyle \sum _{\begin{array}{l}h=10m\mp 1\\ m=1,2,3\dots \end{array}}^{\infty }(h{F}_{sh}{F}_{rh}\sin ((h\pm 1)\omega t\mp {\gamma }_d)})\end{array}\right.$$

(1)

where μ₀ is the air permeability, p is pole pair, g is the air-gap length, r_g is the outer diameter of the rotor plus half the air-gap length, L is the axial length, γ_d is the current angle, h is the harmonic order, m is a positive integer, F_s1 and F_sh are first and h order stator magnetic motive force (MMF), F_r1 and F_rh are first and h order rotor MMF. From (1), 9th and 11th rotor MMF induce first order torque pulsation, and 19th and 21st rotor MMF generate second order torque pulsation. Additionally, the flux density due to PM is proportional to the rotor MMF. Hence, optimizing the harmonics of the flux density can help decrease torque pulsation.

3.1. Stair Equivalent Procedure

As shown in (2), the flux density of the traditional spoke PM motor includes many harmonics.

$$B_r(\theta )={\displaystyle \sum _{n=1}^{\infty }{B}_{r0}\cos np\theta }$$

(2)

where B_r0 is the amplitude of the flux density.

The flux density does not contain harmonics in (3).

$$B_r(\theta )=B_{r0}\cos p\theta$$

(3)

In previous studies, PWM is often used to transform a sinusoidal wave into a square wave of equal height and unequal width. However, applying this method to the rotor of spoke PM motors will raise two main problems. First, the width after segmentation is too narrow to manufacture. Second, the rib must be added to decrease the wind resistance losses, resulting in lower average torque. The proposed stair method is based on the sinusoidal flux density. Theoretically, all harmonics can be optimized. However, the accuracy of the optimization is limited due to the number of stairs. Hence, the main focus is on the harmonics contributing to the first and second-order torque pulsation.

In half an electrical period, the sinusoidal flux density is equivalent to stairs of equal width and unequal height, according to the principle of equal area. As shown in Figure 3, the intersection of each stair with the sinusoidal wave is the midpoint, and stairs are symmetric at the y-axis. Then, the ratio of stairs heights can be calculated using (4), where N_seg is the number of stairs.

$$\left\{\begin{array}{l}{n}_0=\frac{\left|{\displaystyle {\int }_{\frac{i\pi }{p{N}_{seg}}}^{\frac{(i+1)\pi }{p{N}_{seg}}}\cos (p\theta )d\theta }\right|}{\frac{\pi }{2p\cdot N_{seg}}}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}(i=0)\\ n_i=\frac{\left|{\displaystyle {\int }_{\frac{i\pi }{p{N}_{seg}}}^{\frac{(i+1)\pi }{p{N}_{seg}}}\cos (p\theta )d\theta }\right|}{\frac{\pi }{p\cdot N_{seg}}}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}(i=1,2,3\dots \frac{N_{seg}-1}{2})\hspace{0.33em}\end{array}\right.$$

(4)

In addition, the fundamental amplitude B_r0 is treated as the stair fundamental height h_a in this part. Hence, the height of stairs can be expressed as

$$h_i=h_a\cdot n_i$$

(5)

3.2. ECC ASP Equivalent Procedure

ECC ASP method is used in Model 3. Unlike the traditional method of obtaining the offset, it is obtained by the stair equivalent method in Section 3.1. Hence, the ECC ASP procedure of Model 3 is also concerned with the sinusoidal flux density. Moreover, the midpoint of each stair is found and connected into a line segment. When there are sufficient numbers of stairs, these line segments can be equivalent to an arc. In this case, ECC ASP and stairs areas of flux density can also be considered equal, as shown in Figure 4. However, the above method will not only exceed the air-gap length causing torque loss, but it is also not conducive to torque pulsation reduction. Hence, stairs are added on top of the ASP.

First, the coordinates of A (0, y₁) and B (x₂, y₂) can be obtained, and point B is taken as the midpoint of the last stair. Then, EF is the perpendicular bisector, and the coordinates of point E can also be obtained in (6).

$$E(\frac{x_2}{2},y_2+\frac{y_1-y_2}{2})$$

(6)

The lengths of AD can be calculated in (7).

$$AD=\frac{y_1-y_2}{2}$$

(7)

The lengths of AE can be calculated in (8).

$$\begin{array}{l}A{E}^2=A{D}^2+(\frac{x_{{}^2}}{2}{)}^2\\ =(\frac{y_1-y_2}{2}{)}^2+(\frac{x_{{}^2}}{2}{)}^2\end{array}$$

(8)

Since ∆ADE and ∆AEG are similar, the relationship can be obtained in (9).

$$\left\{\begin{array}{l}r=AG\\ \frac{AD}{AE}=\frac{AE}{AG}\\ y_1=AG+offset\end{array}\right.$$

(9)

where r is the length of the radius of ECC circle.

Hence, the offset is obtained in (10).

$$\begin{array}{l}offset=y_1-\frac{A{E}^2}{AD}\\ =y_1-\frac{{\left(\frac{y_1-y_2}{2}\right)}^2+{(\frac{x_{{}^2}}{2})}^2}{{\left(\frac{y_1-y_2}{2}\right)}^2}\end{array}$$

(10)

Then, the length of OB is L which is calculated by (11).

$$L=R-{\delta }_{\min }-h_a(n_0-0.5*n_f)$$

(11)

where R is the stator inner diameter, δ_min is minimal air-gap length, and n_f is the height ratio of the last step.

And the absolute values of x₁ and x₂ can be obtained from L.

$$\left\{\begin{array}{l}\left|x_2\right|=L\cos {\theta }_m\\ \left|y_2\right|=L\sin {\theta }_m\end{array}\right.$$

(12)

As shown in Figure 4, θ_m is half the polar-arc angle α_p.

$${\theta }_m=0.5{\alpha }_p$$

(13)

Hence, x₁ and x₂ can be further expressed as

$$\left\{\begin{array}{l}\left|x_2\right|=L\cos (0.5{\alpha }_p)\\ \left|y_2\right|=L\sin (0.5{\alpha }_p)\end{array}\right.$$

(14)

The optimal flux density is also related to the maximum air-gap length (δ_max). Then, the relationship between δ_max and offset needs to be established. Additionally, h_b is the height of ASP and N is set as the scale factor of the total height h_m, expressed as

$$h_b=h_m\cdot N$$

(15)

The δ_max can be calculated by (16).

$${\delta }_{max}=h_m-\left(h_{m}N+0.5n_f\left(h_m-h_b\right)\right)$$

(16)

Hence, the offset can be expressed by α_p, N, and δ_max.

$$offset=R-{\delta }_{\min }-\frac{{\left[\frac{{\delta }_{\max }-{\delta }_{\min }}{(1-0.5n_f)(1-N)}\right]}^2+0.5\cos {(0.5{\alpha }_p)}^2\left[R-\frac{{\delta }_{\max }-{\delta }_{\min }}{(1-0.5n_f)(1-N)}(n_0-0.5*n_f)\right]}{\frac{{\delta }_{\max }-{\delta }_{\min }}{(1-0.5n_f)(1-N)}}$$

(17)

To sum up, the offset is related to three parameters. Among them, N can avoid the error caused by too large offset and too small h_m. Since (17) relates offset to α_p and δ_max, it can effectively simplify the optimization steps. Hence, the optimal ECC ASP can be obtained by optimized α_p, N, and δ_max.

3.3. Analysis Average Torque

The above ECC ASP method induces uneven air gap; the air-gap length is closely related to flux density. Thus, the air-gap length can influence average torque [23]. If the saturation and flux leakage are neglected, average flux density can be expressed as

$$B_{avg}= \frac{B_r}{\frac{A_g}{A_m}+{\mu }_r\frac{l_{avg}}{l_m}}$$

(18)

where B_r is remanence of PM, A_g is the cross-section area of the air gap per pole, A_m is the cross-section area of PM, μ_r is the relative permeability of PM, l_avg is the average air-gap length of ASP, l_m is the length of PM. Then, the average back-EMF can be expressed as

$$\begin{array}{l}{E}_{avg}=k_{dpn}\frac{2N}{\pi }p{\omega }_r{B}_{avg}\frac{\pi DL}{2p}\\ =k_{dpn}NDL{\omega }_r\frac{B_r}{\frac{A_g}{A_m}+{\mu }_r\frac{l_{avg}}{l_m}}\end{array}$$

(19)

where k_dpn is winding factor, D is outer rotor diameter, L is active axial length, ω_r is rotor angular speed. Then, the average torque can be expressed as

$$\begin{array}{l}{T}_{avg}=\frac{P}{{\omega }_r}=\frac{m{E}_{avg}I}{{\omega }_r}\\ =m{k}_{dpn}NDL{\omega }_r\frac{B_r}{\frac{A_g}{A_m}+{\mu }_r\frac{l_{avg}}{l_m}}\frac{I}{{\omega }_r}\end{array}$$

(20)

Through the above derivation, it can be concluded that the average air-gap length is inversely proportional to the average torque.

Furthermore, the air-gap length of ECC can be expressed as

$$R(\theta )=R-(\sqrt{D^2-r^2}\sin \theta -offset)$$

(21)

The average air gap can be expressed as

$$l_{avg}=\frac{1}{{\alpha }_p}{\displaystyle {\int }_0^{{\alpha }_p}R(\theta )d\theta }$$

(22)

Hence, calculating l_avg can compare the magnitude of the average torque. The results of the calculations will be given in the next section.

4. FEM Optimization and Verification

As described in Section 3, the optimization procedure is proposed for the ECC ASP rotor. This procedure reduces the optimization parameters, and the optimal ECC ASP can be obtained. Then, FEM models of spoke PM motors with ECC ASP are established using ANSYS Maxwell software and will be compared with the traditional and ASP motor.

4.1. Optimization Results

According to Section 3.1, the ratio of stair height can be calculated. Considering the actual machining situation, the height difference between the stairs cannot be too small. Therefore, 9, 11, and 13 segments of stairs are selected for comparison, as shown in

Table 2. The value of n_i for different segments of stairs.

Item	Nseg = 9	Nseg = 11	Nseg = 13
n0	0.995	0.997	0.998
n1	0.935	0.956	0.969
n2	0.762	0.838	0.883
n3	0.497	0.653	0.747
n4	0.173	0.414	0.567
n5		0.142	0.354
n6			0.120

. In addition, the different h_a is taken for the three stair segments to verify the optimization effect. Figure 5 shows that 13 segments gain lower torque pulsation and higher average torque. However, the torque pulsation and average torque of 13 segments need further optimization by ECC ASP equivalent method.

By calculation of the different segments of stairs, N_seg is chosen as 13, n₀ can be approximately set as 1, and n_f is calculated to be 0.12. As shown in Figure 6a, it is noted that the average torque and torque pulsation vary with α_p. Then, the optimal α_p is selected as 32.2 deg by FEM. Since stairs are added on top of the ASP, N related to h_a needs to be optimized. As shown in Figure 6b, the lowest torque pulsation is obtained when N is 0.47, and its average torque is less than 0.05 Nm below the maximum value, so it is acceptable. For lower torque pulsation and higher average torque, δ_max is selected as 1.22 mm through FEM, as shown in Figure 6c. Through (15) and (16), h_m can be calculated to be 2.45 mm, and h_b is 1.15 mm.

Figure 7 shows the two models differ only in the α_p. It can be seen ECC ASP obtains higher average torque and lower torque pulsation. According to Section 3.2, l_avg is inversely proportional to the average torque. The calculating results are shown in

Table 3. Torque performance comparison through l_avg.

Item	Traditional ECC	ECC ASP
lavg (mm)	8.61	10.97
Average torque (Nm)	11.56	11.91
Torque pulsation (%)	12.89	1.68

. The l_avg of traditional ECC is larger than the ECC ASP. Therefore, the FEM results are consistent with the theoretical analysis.

4.2. FEM Verification

Model 3 is established using the optimal parameters by FEM to verify the effectiveness of reducing flux density harmonics. Meanwhile, the transient performance of Model 3 is compared with Model 1 and 2.

4.2.1. Open-Circuit Flux Density Distribution

As described in Section 3, the open circuit flux density is proportional to the rotor MMF. As shown in Figure 8, the open circuit flux density can represent the rotor MMF for the three models in one electrical period. Figure 9 shows the flux density harmonics of the three models. Since rotors are symmetrical, even harmonics are low, but there are still many odd harmonics. Compared with Model 1 and 2, the 9th, 11th, 19th, and 21st flux density harmonics of Model 3, which cause torque pulsation, have decreased significantly. To clearly illustrate the optimization of the flux density, the amplitudes of primary harmonics are presented in

Table 4. The amplitude of open−circuit flux density harmonics in the three models.

Item	1st (T)	9th (T)	11th (T)	19th (T)	21st (T)
Model 1	0.685	0.052	0.034	0.002	0.006
Model 2	0.728	0.058	0.032	0.018	0.002
Model 3	0.696	0.003	0.003	0.00004	0.001

. Compared with Model 1, the 9th harmonic of Model 3 decreased by 94.2%, and the 11th harmonic decreased by 91.1%. Meanwhile, the amplitude of the 21st harmonic decreased by 83.3%. It is worth noting that the amplitude of 19th harmonic was almost zero in Model 3. Hence, the above ECC ASP method can obtain a good result for the reduction in flux density harmonics contributing to the first- and second-order torque pulsation, which also proves the correctness of the theoretical analysis in Section 3.

4.2.2. Back-EMF

The waveforms of Back-EMFs for the three models are given in Figure 10a; it can be seen that Model 3 is closer to sinusoidal. Figure 10b shows the harmonics of Back-EMFs. For the three models, the main harmonics are the 9th, 11th, 19th, and 21st, contributing to the first and second-order torque pulsation. Compared with Model 2, the 9th and 11th harmonics, which cause first order torque pulsation, are decreased by the ECC ASP method. Moreover, amplitudes of Back-EMF fundamental and other main order harmonics are listed in

Table 5. The amplitude of Back−EMFs harmonics in the three models.

Item	1st (V)	9th (V)	11th (V)	19th (V)	21st (V)
Model 1	74.68	7.05	4.97	4.33	0.02
Model 2	78.79	12.14	5.08	1.01	2.44
Model 3	70.74	1.57	0.76	0.51	0.39

. Compared with Model 2, it is noted that the 9th and 11th harmonics amplitudes of Model 3 have dropped by 87.1% and 85.1%, and the 19th and 21st have decreased by 49.5% and 84.1%, respectively. Hence, the FEM results are in agreement with the analysis in Section 3.

4.2.3. Torque Performance

The cogging torque waveforms of the three models are shown in Figure 11, and they have 10 cogging torque periods. The peak-to-peak value of cogging torque are 1.29 Nm and 1.23 Nm for Model 1 and Model 2, respectively. It is noted that the proposed ECC ASP method can reduce the peak-to-peak value of the cogging torque. Additionally, the peak-to-peak value of Model 3 has decreased to 0.06 Nm due to the uneven air gap.

Figure 12 shows the torque characteristics of the three models at the current angle of maximum torque. As shown in Figure 12a, the torque pulsation in Model 3 is reduced significantly due to the ECC ASP method. As shown in Figure 12b, first order (h ± 1 = 10) torque pulsation harmonics are below 0.001 Nm in Model 3, the second-order torque pulsation (h ± 1 = 20) harmonics are below 0.1 Nm, and the torque pulsation of Model 3 is 1.68%. Additionally, the torque performance for the three models is shown in

Table 6. The amplitude of torque harmonics in the three models.

Item	Model 1	Model 2	Model 3
Average torque (Nm)	14.38	13.41	11.91
10th (Nm)	5.44	3.58	0.0005
20th (Nm)	0.32	0.66	0.04
Torque pulsation (%)	70.0	51.0	1.7
Torque Loss (%)	0	6.7	17.1

. Compared with Model 1, the 10th harmonic of Model 3 has been reduced by 99.9%, and the 20th harmonic has been reduced by 87.5%, which is why the torque pulsation of Model 3 is decreased.

In addition, the loss of average torque of Model 3 is 17.1%. The reason is as follows. If the saturation and flux leakage are neglected, the average air-gap length is inversely proportional to the average torque. Figure 2 shows that the average air-gap length is larger than the traditional spoke PM motor with the addition of ECC ASPs, which will reduce the average torque, although the minimum air gap is the same for all three models.

Figure 13 shows the average torque and torque pulsation vary with current from 1 A to the rated value. The torque pulsation is highest at a current of 1 A, which is still much less than the traditional spoke PM motor. Additionally, average torque rises steadily with increasing current. Therefore, the torque response is good under different loads.

4.3. Sound and Vibration

The radial force was calculated using the 2D FEM with a two-dimensional Fourier decomposition to obtain the vibration features. Furthermore, the radial force is mainly concentrated in the stator tooth, and the space order is related to the number of stator slots, which is a multiple relationship. In previous studies, the concept of equal zero order radial force is revealed, and the lowest non-zero space order is no longer concerned at this point [24]. In this paper, the number of stator slots is 40, thus, the 40th and 80th space order non-zero frequency radial forces are more important. Figure 14 shows the distribution of the radial force orders for Model 1 and 3. Model 3 optimizes the 40th and 80th space order at 10f_r and 20f_r; the fundamental value of f_r is 100 Hz. Then, the results of sound and vibration are obtained using Maxwell and Workbench combined simulation. Figure 15 compares the simulated accelerations of Model 1 and 3 at 1500 r/min. The maximum acceleration is mainly caused by the 40th and 80th space order radial force at 10f_r and 20f_r, and the amplitude is decreased from 0.63 to 0.22. Figure 16 shows the simulated sound power level (SWL) from 0 to 60f_r. The maximum sound of Model 1 and 3 is related to the acceleration at 20f_r, which is 42.8 dB and 33.9 dB, respectively. As for total SWL, Model 3 is 7.4 dB smaller than Model 1.

5. Conclusions

This paper proposes a new type of ECC ASP for spoke PM motor. With ECC ASPs, the flux density harmonics can be effectively optimized. The optimization procedure of ECC ASPs is revealed as well. First, the ECC ASP is obtained by stairs, which can equivalently generate sinusoidal flux density by area equivalence. Second, the relationship between δ_max, α_p, and offset is deduced. Based on this relationship, the optimization procedure can be conducted with FEM. Finally, it is verified that the proposed ECC ASPs produce a low torque pulsation by reducing flux density harmonics, contributing to the first and second order torque pulsation. Meanwhile, the simulated sound power level (SWL) can also be optimized by decreasing the equivalent zero-order radial pressure.