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.
3. Optimization Procedure of ECC ASP
In [
22], the average torque (
Tavg) and torque pulsation (
Tpul) of a five-phase PM motor is expressed as
where
μ0 is the air permeability,
p is pole pair,
g is the air-gap length,
rg is the outer diameter of the rotor plus half the air-gap length,
L is the axial length,
γd is the current angle,
h is the harmonic order,
m is a positive integer,
Fs1 and
Fsh are first and
h order stator magnetic motive force (MMF),
Fr1 and
Frh 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.
where
Br0 is the amplitude of the flux density.
The flux density does not contain harmonics in (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
Nseg is the number of stairs.
In addition, the fundamental amplitude
Br0 is treated as the stair fundamental height
ha in this part. Hence, the height of stairs can be expressed as
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).
The lengths of
AD can be calculated in (7).
The lengths of
AE can be calculated in (8).
Since ∆ADE and ∆AEG are similar, the relationship can be obtained in (9).
where
r is the length of the radius of ECC circle.
Hence, the
offset is obtained in (10).
Then, the length of
OB is
L which is calculated by (11).
where
R is the stator inner diameter,
δmin is minimal air-gap length, and
nf is the height ratio of the last step.
And the absolute values of
x1 and
x2 can be obtained from
L.
As shown in
Figure 4,
θm is half the polar-arc angle
αp.
Hence,
x1 and
x2 can be further expressed as
The optimal flux density is also related to the maximum air-gap length (
δmax). Then, the relationship between
δmax and
offset needs to be established. Additionally,
hb is the height of ASP and
N is set as the scale factor of the total height
hm, expressed as
The
δmax can be calculated by (16).
Hence, the
offset can be expressed by
αp,
N, and
δmax.
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 hm. Since (17) relates offset to αp and δmax, it can effectively simplify the optimization steps. Hence, the optimal ECC ASP can be obtained by optimized αp, N, and δmax.
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
where
Br is remanence of PM,
Ag is the cross-section area of the air gap per pole,
Am is the cross-section area of PM,
μr is the relative permeability of PM,
lavg is the average air-gap length of ASP,
lm is the length of PM. Then, the average back-EMF can be expressed as
where
kdpn is winding factor,
D is outer rotor diameter,
L is active axial length,
ωr is rotor angular speed. Then, the average torque can be expressed as
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
The average air gap can be expressed as
Hence, calculating lavg can compare the magnitude of the average torque. The results of the calculations will be given in the next section.
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 δmax, αp, 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.