1. Introduction
Due to their high power density, low loss, and small size, PMSMs are widely employed in transportation, aircraft, and industrial robots [
1,
2,
3,
4]. In order for the windings to fit into the stator slots, PMSMs are frequently designed with slotted magnetic circuit structures. However, the slotting can lead to interaction between the stator core and the magnets, which can have a significant impact on the stable operation of a PMSM. It is crucial to lower the cogging torque of the PMSM [
5,
6] because this influence is more pronounced in low-speed settings.
The cogging torque is an inherent property of slotted motors and cannot be abolished entirely. In prior research, the cogging torque was weakened primarily by motor control and construction. Regarding control, [
7] proposed a solution based on harmonic torque to counteract the cogging torque. A speed- and position-adaptive controller was created to reduce the cogging torque in [
8]. Although the aforementioned optimized controller methods could greatly reduce cogging torque, they failed to account for the effect on other motor performance characteristics.
A significant amount of research has centered on the structural design and optimization of electrical machines in an effort to enhance their output performance. In the field of structural design, researchers have altered the topology of electrical machines to enhance their electromagnetic performance [
9,
10]. In [
11], a scheme of skew-toothed stator teeth was proposed to reduce the cogging torque, however the impact of the proposed improvement on other motor performance characteristics was not explored. In [
12], a stator-tooth-notching method was used to reduce the cogging torque, but it also reduced the average torque. In [
13], the method of offset poles was utilized to lower the cogging torque of the motor; however, the average torque also decreased, and the installation perfection for this approach required high precision. In [
14], a PM radially unequal width layering technique was presented to reduce the cogging torque and torque pulsation as well as the average torque. However, the technique was too structurally changed and difficult to install. The preceding research has demonstrated that the weakening of cogging torque is frequently followed by a reduction in electromagnetic torque, and that altering a motor’s construction may make the machining and installation of components more challenging.
In the field of optimization technology for electrical machines, the objective function, constraints, and boundaries are defined based on the electrical machine’s optimization problem, and the design space is searched for the optimal combination of parameters to achieve a significant improvement in the electrical machine’s performance [
9,
10,
15]. With the advancement of optimization techniques, the study of the robustness of electrical machines design is gradually increasing [
10,
15]. Traditional approaches always optimize each structural parameter sequentially, neglecting structural parameter interactions. The Taguchi approach was proposed in [
16,
17,
18] to optimize the major structural parameters of the motor. This method takes into account the interaction of the structural parameters based on an orthogonal test designed to successfully reduce the cogging torque of the motor. However, for a broad range of structural parameter values, the Taguchi method’s optimization is limited by insufficient precision. In [
19,
20], a genetic algorithm was employed to optimize the cogging torque of the motor. However, while the cogging torque was effectively lessened, the average torque was also decreased.
In this paper, RSM and NSGA-II were applied to the structural parameter optimization design of a motor in order to optimize the torque performance of SPMSM to ensure that the average torque of the motor was not less than the average torque of the initial motor and minimize the cogging torque. This paper’s outline is as follows: The second section describes the generation causes and analytical formulae for the SPMSM cogging torque. In the third section, a parameterization model for the SPMSM was created in order to examine the structural parameters’ sensitivity. Using RSM and NSGA-II, the torque performance of the motor was optimized in the fourth section. The torque performance of the initial motor was compared to the torque performance of the optimized motor in the fifth section. Conclusions were drawn in the sixth section.
2. Mechanism and Analysis of Cogging Torque Generation in SPMSMs
Cogging torque is produced by the contact between the PM and the stator core, which is damaging to a motor’s performance. During the rotation of the rotor, the magnetic field between the PM and the stator slot is nearly constant, and the PM’s forces are balanced. However, the magnitude of magnetic conductivity along the margins of both sides of the PM is constantly changing, resulting in differential magnitudes of magnetic field strengths on the left and right sides in this region and imbalanced forces on the PM, which generate the cogging torque. The formula for calculating the cogging torque is [
21]:
where
W is the energy of the magnetic field in the motor when no current is applied to the winding and α is the relative position of the stator and rotor. Assuming that the magnetic conductivity of the armature core is infinite, the energy of the magnetic field inside the motor can be expressed as:
where
μ0 is the air magnetic conductivity,
B is the air gap magnetic density,
WPM is the magnetic field energy of the PM and
Wairgap is the magnetic field energy of the air gap.
The distribution,
B (
θ,α), of the airgap magnetic density along the armature surface of the motor in (2) can be expressed as:
From Equations (2) and (3), it can be seen that:
where
hm(
θ) is the length of the PM in the magnetization direction, also known as the thickness of the PM,
hg(
θ,
α) is the effective airgap length and
Br(
θ) is the residual magnetic induction of the PM. To further calculate the magnetic field energy in the motor, a Fourier expansion of
and
(
θ) is required and can be expressed as:
where,
Br0 =
αp,
,
αp is the pole arc coefficient,
p is the number of pole pairs,
Brn is the Fourier decomposition coefficient of the square of the air-gap magnetic density generated by PM,
Gn is the Fourier decomposition coefficient of the square of the relative air gap permeability and
Br0 and
G0 are the constant terms of
(
θ) and
after the Fourier decomposition.
G0 and
Gn in Equation (6) can be expressed as:
where,
θso is the width of the slot opening of the stator expressed in radians and
s is the number of slots.
Substituting Equations (5) and (6) into (4), which is then juggled with (1), the cogging torque,
Tc, can be expressed as:
where
La is the axial length of the stator core,
D2 and
Di2 are the inner radius of the armature and the outer radius of the rotor, respectively, and
n is the integer that makes
ns/2
p an integer.
The following equation can be used to calculate the electromagnetic torque of the SPMSM:
where
Tem is the electromagnetic torque,
Pout is the output power of the motor,
wr is the angular speed of the rotor and
TL is the load torque, where the output torque power can be expressed as:
where
m is the number of phases of the motor,
Iph is the effective value of the phase current and
E1 is the effective value of a counter electromotive force.
E1 can be expressed as [
22]:
where
Kw1 is the fundamental winding factor,
f is the current frequency of the stator,
Nph is the number of turns in series per winding,
B1 is the magnetic induction,
Ns is the number of conductors per slot and
ns is the total number of slots. As
Ns is related to the area of the stator slot and the wire gauge of the winding, it can be expressed as [
23]:
where
As and
Ai can be expressed as:
where
Ci is the insulation thickness of the slot,
Aef is the effective area of the slot,
ACu is the wire gauge of the estimated winding,
As and
Ai are the area of the slot and the insulation occupied area of the slot, respectively,
Bs1 is the width of the slot,
Bs2 is the radius of the slot and
Hs2 is the depth of the slot.
Substituting (12) and (13) into (11), the expression for
Pout is obtained as:
Letting
, (15) and (16) are thus substituted into (14), which is then juggled with (10), (11) and (12) to derive the expression for the electromagnetic torque
Tem:
The average torque magnitude (
Ta) of the SPMSM can be defined as the peak-to-peak average of the electromagnetic torque (
Tem) during steady operation of the motor as follows:
The torque ripple,
Tr, can be defined as the ratio of the peak-to-peak difference in the electromagnetic torque to the average torque, as shown below:
where
avg means the calculated average,
Tmax is the maximum peak of the electromagnetic torque and
Tmin is the minimum peak of the electromagnetic torque.
According to Equation (9), the cogging torque is dominated by the magnitudes of Gn and Brn. It can be shown from Equation (5) that the pole arc coefficient has a direct effect on Brn. The width of the slot opening of the stator and the thickness of the PM have a direct effect on Gn, as shown by Equation (8). The structural parameters to be optimized are, therefore, the pole arc coefficient, the width of the slot aperture and the thickness of the PM. Changes in the other structural parameters of the stator slot in the SPMSM will likewise impact the magnetic field distribution in the stator’s teeth and yoke as well as the motor’s electromagnetic torque. Therefore, the stator-slot-related structural characteristics were chosen as the structural parameters to be optimized. It can be shown from Equation (18) that variations in the depth, width and radius of the slot will affect the electromagnetic torque capabilities. Changes in the thickness of the PM will also impact the distribution of the flux density in the airgap, thus affecting the electromagnetic torque capability. Consequently, the electromagnetic torque performance must be taken into account when reducing the cogging torque so that the electromagnetic torque performance is not severely diminished.