Rapid Design of a Coreless Axial Flux Motor Based on the Magnetic Charge Method

Abstract

Axial flux motors have attracted significant attention in recent years due to their advantages such as shorter axial length and high torque density. However, the optimization of axial flux motors is an extremely time-consuming process. To reduce the computational time required for motor optimization, this study employed a magnetic charge model to establish a coreless axial flux motor model and analyzed the advantages of this approach. This method is applicable to coreless axial flux motor optimizations with surface-mounted rotors and concentrated windings. Parameter optimization was subsequently performed based on the theoretical model. In terms of seeking optimal solutions, the torque obtained through the magnetic charge method (MCM) reached 99.67% of the finite element method (FEM) results. Finally, a prototype was fabricated, and a test platform was constructed based on the optimization results. The experimental torque showed a 4% deviation from simulations, validating the accuracy of the optimization.

1. Introduction

Axial flux motors have attracted significant attention in recent years due to their advantages such as shorter axial length and high torque density [1,2,3]. In 1996, Huang Surong, Luo Jian, and their collaborators established generalized dimensional equations and power density equations applicable to both radial flux motors and axial flux motors [4,5]. Their work demonstrated the superiority of axial flux motors in terms of torque density and power density. Notably, this design methodology remains a critical reference for contemporary axial flux motor design [6,7,8].

Axial flux motors can be classified into coreless and cored axial flux motors. The design of coreless axial flux motors reduces the axial attractive force between the stator and rotor [9], thereby facilitating component manufacturing. Coreless axial flux motors do not include a stator core, offering advantages such as absence of cogging torque [10], strong overload capacity, and higher torque density [11,12]. Therefore, this research will focus on coreless axial flux motors.

The dimensional design of axial flux motors constitutes a multi-objective optimization process. Numerical calculation methods utilizing finite element simulation software are predominantly employed. To maximize material utilization efficiency, particularly for permanent magnets, comprehensive parameter optimization is required to achieve desired torque output, cost-effectiveness, and other performance metrics. However, the electromagnetic torque characteristics are influenced by multiple interdependent parameters, where each additional parameter increases the required sample size exponentially. Identifying optimal solutions within such extensive sample spaces represents a computationally intensive process requiring significant computational resources.

Numerous researchers have employed finite element simulation to optimize dimensional parameters of axial flux motors. Scholars from Nanyang Technological University implemented 3D finite element simulation for structural optimization of flywheel energy storage axial flux motors [13]. Yazdi, M.A. et al. designed a novel coreless axial flux motor utilizing transient finite element simulation and conducted comparative simulations to benchmark its performance against conventional configurations [14]. Jie Mei et al. proposed a six-pole distributed-winding and double-stator-double-rotor axial flux induction motor with reduced rotor yoke for electric vehicles by 3D finite element simulation [15]. Its primary advantage lies in the high credibility of its numerical computations, though this approach requires lengthy computational duration.

To reduce computational time requirements, 3D finite element analysis is typically integrated with intelligent optimization algorithms. Zhiwei Wen et al. implemented a multi-objective optimization framework combining the 3D finite element method with the non-dominated sorting genetic algorithm (NSGA) for coreless axial flux motors with non-overlapping windings, achieving simultaneous optimization of motor efficiency, dimensions, and mass [16]. Xiaobin Xu et al. accelerated the design process of axial flux brushless DC motors through the pigeon-inspired optimization (PIO) algorithm [17]. Dong-Kyun Woo et al. significantly reduced optimization cycles for axial flux machines using modified hill-climbing algorithms [18].

In addition to intelligent optimization algorithms, researchers have explored dimensionality reduction techniques by radially unfolding 3D motor models into 2D finite element configurations. Jung Moo Seo et al. accelerated simulation speed by decomposing axial-radial hybrid flux machines into separate 2D linear motor models for axial and radial flux components during optimization [19]. Ahmed Shoeb et al. achieved a 99.5% reduction in computational time (from about 45,000 to 24 h) through 3D-to-2D model conversion, though with compromised accuracy, showing 12–18% errors in torque calculations [20]. Maarten J. Kamper’s team implemented comparative 2D-FEA optimization for coreless axial flux motors, analyzing winding configurations (overlapping vs. non-overlapping) [21].

Simulation using 2D models as substitutes for 3D finite element analysis inherently introduces modeling inaccuracies. Chang-Hoon Seok attributed these errors to flux leakage stemming from 2D models’ inability to capture axial end effects, developing an equivalent magnetic circuit method that reduced computational deviation by 22% [22]. Sung Gu Lee et al. implemented empirical correction factors (1.15–1.3 scaling coefficients) to bridge 2D/3D discrepancies in flux density calculations [23]. Seung-Hun Lee’s team achieved 8% error reduction through end-leakage compensation coefficients specifically targeting 2D model limitations in edge effect simulation [24].

To improve computational speed, analytical methods and magnetic circuit methods are used to calculate electromagnetic force or torque. Fei Zhao et al. [25] established an analytical calculation model when analyzing electromagnetic forces. In their study, the three-dimensional model was unfolded into a two-dimensional model at the mean radius, and an analytical model based on the two-dimensional model was used to describe axial electromagnetic forces. Mohammad R.A. Pahlavani et al. [26] established a three-dimensional analytical model to study the torque ripple problem in axial flux motors. During the design of axial flux motors, Attila Nyitrai et al. [27] sectioned the three-dimensional model into a two-dimensional model and subsequently analyzed the 2D model using the equivalent magnetic circuit method. Jing Zhao et al. [12] established a three-dimensional magnetic circuit model to analyze axial flux motors. Traditional analytical methods and equivalent magnetic circuit methods still face difficulties in handling end effects and often require data fitting or correction functions for compensation.

The magnetic charge method is an efficient modeling technique widely applied in static magnetic field calculations. However, limited by the applicable conditions of static magnetic fields, it has not been extensively adopted for torque calculation in axial flux motors. In terms of magnetic field calculations, the equivalent magnetic charge method can be seen in the magnetic flux leakage internal detection technology [28,29] or the verification of the magnetic field generated by Halbach permanent magnets [30,31]. Based on the magnetic charge method, the torque equation of the axial magnetic transmission mechanism is established, and the transmission mechanism is analyzed [32,33]. However, the authors did not convert the model into a static magnetic field model suitable for the magnetic charge method. In fact, while the magnetic field in motors is not static, through equivalent processing, the magnetic charge method can still be applied to motor torque calculations.

To accelerate motor optimization while preserving accuracy, this study introduces a magnetic charge modeling approach, establishing a coreless axial flux motor mathematical framework based on this theoretical model. Section 2 introduces the magnetic charge model’s conceptual framework, applicable conditions, calculation methodology, and computational advantages, which facilitated the establishment of the subsequent motor model. Section 3 describes the structural configuration of the coreless axial flux motor, including model parameters and their symbolic representations. Subsequently, magnetic charge distribution is analyzed based on the structural and magnetic circuit characteristics, leading to the establishment of a torque calculation model. Section 4 presents the optimization of the coreless axial flux motor using the magnetic charge method, guided by specified design criteria. The accuracy of this approach was verified through finite element analysis (FEA) simulations. A comparative time efficiency analysis demonstrated the accelerated optimization capability of the magnetic charge method. Section 5 provides experimental validation using a physical prototype.

2. Magnetic Charge Model and Its Technical Advantages

2.1. Concept

The equivalent magnetic charge method originates from the magnetic analogy of Coulomb’s law. During the early development of classical electromagnetism, Coulomb derived the fundamental laws governing magnetostatic fields by drawing an analogy to the operational principles of electrostatic fields. The magnetic field strength H is analogous to the electric field intensity E, and the magnetic charge $q_m$ corresponds to the electric charge q. The magnetostatic force can be calculated as the product of magnetic charge and magnetic field strength. However, since the concept of magnetic charges fails at the microscopic level and magnetic monopoles have never been experimentally observed, the molecular current hypothesis is now universally acknowledged as revealing the fundamental nature of magnetism.

Figure 1 illustrates the simplified descriptions of the magnetization process of an object using the molecular current model and the magnetic charge model. In the molecular current model, molecular currents are the source of localized magnetic fields. After applying an external magnetic field B, the elementary magnets within the region become substantially aligned in orientation, causing the entire object to exhibit magnetic properties. In the magnetic charge model, localized magnetic fields are generated by pairs of positive and negative magnetic charges. Under the influence of an external magnetic field, this similarly causes the object to exhibit magnetic properties.

Coulomb also used experiments to measure the force between ‘point magnetic charges’ and established the ‘Coulomb law of magnetism’.

$$F_m={\displaystyle \frac{K{q}_{m1}{q}_{m2}}{r^2}}$$

(1)

where K is the proportionality constant; $q_{m1}$ and $q_{m2}$ are two magnetic charges; and r is the distance between them.

The magnetic force acting on the unit magnetic charge is defined as the magnetic field strength, denoted $\mathit{H}$.

$$\mathit{H}={\displaystyle \frac{{\mathit{F}}_m}{q_m}}$$

(2)

Therefore, Equation (1) can be rewritten as

$${\mathit{F}}_{\mathit{m}}=\int {\rho }_m\mathit{H}dV$$

(3)

The magnetic charge density can be obtained from the magnetization $\mathit{M}$ of the permanent magnet.

$${\rho }_m=-{\mu }_0\nabla ·\mathit{M}$$

(4)

2.2. Applicability Criteria

The applicability conditions of the magnetic charge method are identical to those of the scalar magnetic potential method:

$$\nabla \times \mathit{H}=0$$

(5)

In time-varying electromagnetic fields, Ampère’s circuital law (with Maxwell’s addition) is expressed as

$$\nabla \times \mathit{H}={\mathit{J}}_{\mathit{f}}+{\displaystyle \frac{\partial \mathit{D}}{\partial t}}$$

(6)

Considering that $\mathit{H}$ and $\mathit{B}$ are related,

$$\mathit{H}={\displaystyle \frac{\mathit{B}}{{\mu }_0}}-\mathit{M}$$

(7)

Equation (6) can be rewritten as

$$\nabla \times {\displaystyle \frac{\mathit{B}}{{\mu }_0}}={\mathit{J}}_f+{\displaystyle \frac{\partial \mathit{D}}{\partial t}}+\nabla \times \mathit{M}$$

(8)

where the last term can be expressed as an equivalent magnetization current ${\mathit{J}}_{\mathit{M}}$.

$$\nabla \times \mathit{M}={\mathit{J}}_{\mathit{M}}$$

(9)

By comparing Equations (5) and (8), it can be concluded that the application of the magnetic charge method requires

$$\nabla \times \mathit{M}>>{\mathit{J}}_{\mathit{f}}+{\displaystyle \frac{\partial \mathit{D}}{\partial t}}$$

(10)

Thus, the equivalent magnetization current must be significantly greater than the free current:

$${\mathit{J}}_{\mathit{M}}>>{\mathit{J}}_{\mathit{f}}$$

(11)

For axial flux motors, magnetization currents exist only in magnetically conductive components such as permanent magnets, rotor yoke, and cores. Since eddy currents (which belong to free currents) may arise during motor operation, the validity condition can be interpreted as the magnetic effects of eddy currents are negligible compared to the intrinsic magnetic effects of the material.

Additionally, it should also be satisfied that the equivalent magnetization current is significantly greater than the displacement current:

$${\mathit{J}}_{\mathit{M}}>>{\displaystyle \frac{\partial \mathit{D}}{\partial t}}$$

(12)

This indicates that the magnetic charge method must be used in static magnetic fields or regions that can be treated as equivalent to static magnetic fields.

Given that magnetic monopoles have never been observed, the applicability criteria for the magnetic charge method are as follows.

(1) In regions with magnetic charges, the magnetic effects of eddy currents must be negligible compared to the material’s intrinsic magnetic response;

(2) The system must operate within a static magnetic field or its equivalent;

(3) Positive and negative magnetic charges must appear in pairs to prevent the hypothetical existence of magnetic monopoles.

2.3. Computational Method

Despite the fact that magnetic monopoles have yet to be discovered, mathematically, the force on a pair of dipoles is the same as if positive and negative monopoles were considered separately and their forces summed. Therefore, under the premise that positive and negative magnetic charges always appear in pairs within the physical model, the mechanical forces acting on magnets can be effectively calculated using the monopole model. Since the focus of this paper is on the calculation of torque, the cylindrical coordinate system is more appropriate. The electromagnetic force obtained by superposition of monopole models is

$$\begin{array}{cc}\hfill \mathit{F}=& \int {\rho }_m\mathit{H}d{V}_m\hfill \\ \hfill =& \int {\rho }_m(H_r{\mathit{e}}_r+H_{\theta }{\mathit{e}}_{\theta }+H_z{\mathit{e}}_z)d{V}_m\hfill \end{array}$$

(13)

where ${\rho }_m$ is the signed magnetic charge density, $V_m$ represents the magnetic charge region, and $d{V}_m=rdrd\theta dz$ denotes the cylindrical coordinate volume element.

For surface magnetic charges, the electromagnetic force is calculated as a surface integral:

$$\begin{array}{cc}\hfill \mathit{F}=& \int {\sigma }_m\mathit{H}d{S}_m\hfill \\ \hfill =& \int {\sigma }_m(H_r{\mathit{e}}_r+H_{\theta }{\mathit{e}}_{\theta }+H_z{\mathit{e}}_z)d{S}_m\hfill \end{array}$$

(14)

where ${\sigma }_m$ represents the signed magnetic charge density (containing both positive and negative values), and $S_m$ denotes the surface bearing these magnetic charges.

The electromagnetic force obtained through direct application of the dipole model can be expressed as

$$\begin{array}{cc}\hfill \mathit{F}=& \int (\mathit{M}·\nabla )\mathit{B}d{V}_m\hfill \\ \hfill =& \int (M_r{\displaystyle \frac{\partial B_r}{\partial r}}+{\displaystyle \frac{M_{\theta }}{r}}{\displaystyle \frac{\partial B_r}{\partial \theta }}+M_z{\displaystyle \frac{\partial B_r}{\partial z}}-{\displaystyle \frac{M_{\theta }{B}_{\theta }}{r}}){\mathit{e}}_r\hfill \\ \hfill & +(M_r{\displaystyle \frac{\partial B_{\theta }}{\partial r}}+{\displaystyle \frac{M_{\theta }}{r}}{\displaystyle \frac{\partial B_{\theta }}{\partial \theta }}+M_z{\displaystyle \frac{\partial B_{\theta }}{\partial z}}+{\displaystyle \frac{M_{\theta }{B}_r}{r}}){\mathit{e}}_{\theta }\hfill \\ \hfill & +(M_r{\displaystyle \frac{\partial B_z}{\partial r}}+M_{\theta }{\displaystyle \frac{\partial B_z}{\partial \theta }}+M_z{\displaystyle \frac{\partial B_z}{\partial z}}){\mathit{e}}_{z}d{V}_m\hfill \end{array}$$

(15)

Comparison between the two equations reveals that the simplified Equation (13) contains only 3 terms in its force calculation, whereas Equation (15) comprises 11 terms and requires differential operations. This indicates that the simplification process reduces the number of multiplication/division operations and eliminates differential computations, thereby conserving computational resources.

For coreless axial-flux motors, radially magnetized permanent magnets are generally not employed. Therefore, Equation (14) suffices for computational requirements. In practical motor models, coil dimensions are typically non-negligible, resulting in force/torque calculations involving multiple integrals that render analytical solutions intractable. Numerical integration thus emerges as a preferred computational strategy.

Simpson’s integration is a widely used numerical integration technique. Its computational complexity can be expressed as $O\left(N^d\right)$. Among them, d is the integral dimensionality, and N is the number of segments per dimension. Given that the invocation time for the derivative function is $t_f$, and each integration point requires computation of m partial derivatives, the computational complexity of the integration becomes $O\left(m{N}^d{t}_f\right)$. The time complexity using Equation (14) can be expressed as $O\left(N^5\right)$, while using Equation (15), it becomes $O\left(m{N}^6{t}_f\right)$. This demonstrates the effectiveness of the simplified method.

2.4. Computational Advantage

The three-dimensional finite element method is a commonly used optimization approach for coreless axial flux permanent magnet machines. Its high computational cost arises from the need to discretize the model into numerous elements (e.g., tetrahedrons and hexahedrons) during finite element analysis. This occurs because mathematical formulations involving ferromagnetic materials may contain computational elements such as Fredholm integral equations, making analytical solutions nonexistent and leaving only numerical solutions. These numerical solutions require discretization of the model into finite elements for computation. Field computation at each nodal point leads to massive data volumes and significant computational resource consumption, a challenge particularly pronounced in 3D finite element models.

In coreless axial flux motors where the stator lacks ferromagnetic material, the rotor yoke magnetization primarily originates from permanent magnets. Under the magnetic charge model, this configuration allows simplified treatment through magnetic charge conservation, enabling linear modeling by neglecting nonlinear factors. This implies that the torque calculation requires data representable through a unified analytical expression rather than requiring domain discretization into multiple elements. Electromagnetic forces or torque can be obtained via direct spatial integration of this expression, dramatically decreasing computational overhead. The methodological differences are summarized in

Table 1. Comparison of force calculation methods.

Method	FEM	MCM
Approach	Domain discretization	Integration
Computational Nodes	Massive	None
Data Volume	Massive	Low
Resource Consumption	High	Low

.

The comparison results of different modeling methods are summarized in

Table 2. Comparison of modeling methods.

Method	MCM	EMCM	AM
Model	Magnetic charge	Magnetic circuit	Molecular current
2D or 3D	3D	Generally 2D	2D or 3D
End effect	Support	Correction factor	Correction factor

. The approach adopted in this study employs a 3D magnetic charge model, which can directly handle end effects. In contrast, traditional analytical methods (AMs) based on molecular current models typically require correction factors to address end effects. Additionally, the time complexity of molecular current models exceeds that of magnetic charge models, resulting in longer computation times. The equivalent magnetic circuit method (EMCM) employs magnetic circuit models, which are generally implemented in 2D, though a limited number of studies have extended them to 3D. When addressing end effects, this method similarly relies on correction factors.

3. Torque Calculation

3.1. Motor Model

As shown in Figure 2, this is a coreless axial flux motor model. The electromagnetic structure of the motor consists of windings, permanent magnets, and rotor yokes. Among these, the rotor adopts a Halbach array structure, with the main magnets polarized axially and the auxiliary magnets polarized tangentially. The flux path direction has been marked in Figure 2. The concentrated windings are positioned between the two rotor discs.

For axial flux motors, the conventional torque generation mechanism is illustrated in Figure 3. Since the main magnetic poles of the rotor discs are axially magnetized, their rotation applies a periodically varying axial magnetic field to the stator coils. According to Fleming’s left-hand rule, the effective parts of the coil—specifically the regions indicated by the red arrows in Figure 3—generate tangential electromagnetic forces when current is applied.

The mechanism of torque generation in the magnetic charge model is demonstrated in Figure 4. Under this model, the function of the effective part of the coil is to generate a tangential magnetic field strength H. Assuming the current flows perpendicular into the plane (represented by ⨂), application of the right-hand rule determines the tangential magnetic field direction as rightward. Permanent magnets with uniform magnetization can be simply represented as positive and negative magnetic charges on their surfaces. Under the action of an external tangential magnetic field, positive and negative magnetic charges generate tangential forces towards the right and left, respectively. Due to the closer proximity of positive magnetic charges to the coil, resulting in greater tangential force, the net tangential force on the permanent magnet is directed to the right, thereby generating torque.

3.2. Parameters and Calculations

This paper introduces the magnetic charge concept to simplify motor torque calculation. To meet the applicability conditions of the magnetic charge method, geometric abstraction of the model must first be performed. Since the magnetic charge method requires static magnetic field conditions, the rotor magnets can be selected as the reference frame. The stator magnetic field can be equivalently represented as a DC-energized coil rotating synchronously with the rotor, borrowing the rotating coordinate system concept from vector control.

Meanwhile, considering that the stator magnetic field shares the same pole pair count as the rotor, the stator coils can be simplified to a pair of effective parts based on symmetry principles. Permanent magnets may also retain only one pole. When calculating total torque, multiplying by the pole count suffices. Since the magnetic charge model directly selects permanent magnets as the force calculation targets, the coil model can be abstracted. A pair of effective parts suffices to reflect the stator parameters’ influence on torque.

The simplified model is shown in Figure 5. For clarity, the auxiliary PMs are configured with two poles, and the tangential length of the rotor yokes corresponds to the sum of one main PM and two auxiliary PMs. The main model parameters are labeled in Figure 5. $d_1$ represents half the gap between adjacent effective parts. $d_2-d_1$ represents the tangential thickness of the coil’s effective part. $h_c$ represents half the axial thickness of the coil. $h_1-h_c$ denotes the air gap between the permanent magnet and the coil. $h_2-h_1$ represents the axial thickness of the permanent magnet. $h_3-h_2$ represents the thickness of the rotor yokes. In addition, parameters including the main pole arc coefficient ${\alpha }_p$, along with the inner radius $R_1$ and outer radius $R_2$ of the permanent magnet and coil effective parts, are omitted from the diagram due to limitations in 2D representation.

The motor torque calculation model using the magnetic charge method can be represented as shown in Figure 5. The left side illustrates the magnetic circuit path using arrows, while the right side is divided into four regions (I–IV) demonstrating the magnetic charge distribution.

3.2.1. Region I

I represents the axially uniformly magnetized primary magnetic pole, with magnetic charges distributed on its upper and lower surfaces; the lower surface carries positive magnetic charges, while the upper surface holds negative magnetic charges. At this point, the total torque generated by the main PMs can be expressed as

$$\begin{array}{cc}\hfill T_I=& 4p{\sigma }_m\int {\rho }^2[H_t\left(h_1\right)-H_t\left(h_2\right)]d\rho d\mathsf{\Omega }\hfill \\ \hfill =& 8p{C}_1{\sigma }_m{\int }_{{\omega }_0}^{{\omega }_1}cos(p\mathsf{\Omega }+p\phi )d\mathsf{\Omega }{\int }_{R_1}^{R_2}{\rho }^2[H_m\left(h_1\right)-H_m\left(h_2\right)]d\rho \hfill \end{array}$$

(16)

where $C_1$ represents the coefficient corresponding to the main magnetic field after Fourier decomposition, ${\omega }_1$ denotes the mechanical angle at the edge of the primary magnetic pole, and ${\omega }_0$ is the mechanical angle at the central line of the permanent magnet. When the central line of the primary magnetic pole aligns with the central line of the stator’s adjacent effective part, $cos(p{\omega }_0+p\Phi )=1$. Therefore, Equation (16) can be expressed as

$$T_I=8C_1{\sigma }_{m}sin(p{\omega }_1+p\phi ){\int }_{R_1}^{R_2}{\rho }^2[H_m\left(h_1\right)-H_m\left(h_2\right)]d\rho$$

(17)

where $H_m$ can be calculated using the Biot–Savart law.

$$H_m\left(z\right)={\displaystyle \frac{J}{4\pi }}\oint {\displaystyle \frac{z-z^{\prime }}{{\left(\sqrt{{\rho }^2+{\rho }^{\prime 2}-2\rho {\rho }^{\prime }cos{\mathsf{\Omega }}^{\prime }+{(z-z^{\prime })}^2}\right)}^3}}d{V}^{\prime }$$

(18)

The portion related to dimensional information is expressed as a coefficient function:

$$\begin{array}{cc}\hfill C_I=& sin\left({\displaystyle \frac{\pi {\alpha }_p}{2}}\right){\int }_{R_1}^{R_2}{\int }_{-h_c}^{h_c}{\int }_{arcsin\frac{d_1}{\rho }}^{arcsin\frac{d_2}{\rho }}{\int }_{R_1}^{R_2}{\rho }^2[{\displaystyle \frac{h_1-z^{\prime }}{{\left(\sqrt{{\rho }^2+{\rho }^{\prime 2}-2\rho {\rho }^{\prime }cos{\mathsf{\Omega }}^{\prime }+{(h_1-z^{\prime })}^2}\right)}^3}}\hfill \\ \hfill & -{\displaystyle \frac{h_2-z^{\prime }}{{\left(\sqrt{{\rho }^2+{\rho }^{\prime 2}-2\rho {\rho }^{\prime }cos{\mathsf{\Omega }}^{\prime }+{(h_2-z^{\prime })}^2}\right)}^3}}]d\rho d{\mathsf{\Omega }}^{\prime }d{z}^{\prime }d{\rho }^{\prime }\hfill \end{array}$$

(19)

where the angle in the sine function outside the integral can be expressed in terms of the pole arc coefficient as

$$p{\omega }_1+p\Phi ={\displaystyle \frac{\pi {\alpha }_p}{2}}$$

(20)

3.2.2. Region II

$II$ denotes the auxiliary PM, where magnetic charges are distributed on both side surfaces. Based on magnetic circuit analysis, the polarity can be determined; the left side carries positive magnetic charges, while the right side bears negative charges. When calculating torque, to maintain a one-to-one correspondence between the number of main and auxiliary PMs, only the contribution from the left-side magnetic charges needs to be considered.

Similarly, the total electromagnetic torque generated by the auxiliary magnetic poles (i.e., region $II$) can be expressed as

$$\begin{array}{cc}\hfill T_{II}& =4p{\sigma }_m\int \int \rho H_t\left(z\right)d\rho dz\hfill \\ \hfill & =8p{C}_2{\sigma }_{m}cos(p{\omega }_1+p\phi ){\int }_{R_2}^{R_1}\rho {\int }_{h_1}^{H_2}{H}_m\left(z\right)dzd\rho \hfill \end{array}$$

(21)

Here, the portion related to dimensional information can be expressed as a coefficient function:

$$\begin{array}{cc}\hfill C_{II}=& {\int }_{R_1}^{R_2}{\int }_{-h_c}^{h_c}{\int }_{arcsin\frac{d_1}{\rho }}^{arcsin\frac{d_2}{\rho }}{\int }_{R_1}^{R_2}{\int }_{h_1}^{h_2}{\displaystyle \frac{\rho (z-z^{\prime })}{{\left(\sqrt{{\rho }^2+{\rho }^{\prime 2}-2\rho {\rho }^{\prime }cos{\mathsf{\Omega }}^{\prime }+{(z-z^{\prime })}^2}\right)}^3}}dzd\rho d{\mathsf{\Omega }}^{\prime }d{z}^{\prime }d{\rho }^{\prime }·pcos\left({\displaystyle \frac{\pi {\alpha }_p}{2}}\right)\hfill \end{array}$$

(22)

3.2.3. Region III

$III$ represents the rotor yoke above the auxiliary magnetic poles, magnetized in the tangential direction. Since the underlying auxiliary poles form an alternate magnetic path that diverts a portion of the magnetic flux, the magnetic charge polarity in this region becomes inverted compared to conventional magnetic circuit analysis. When equivalently distributing the magnetic charges across both lateral surfaces, the left side carries negative charges, while the right side exhibits positive charges. Consistent with the treatment of region $II$, only the left-side magnetic charges need to be considered in torque calculations.

The magnetic charges in this region arise due to the flux division caused by the auxiliary poles in region $II$. When approximating the magnetic charge distribution as uniform, the calculation method for the left-side charges follows a similar approach to region $II$, with an identical coefficient function.

$$\begin{array}{cc}\hfill C_{III}=& {\int }_{R_1}^{R_2}{\int }_{-h_c}^{h_c}{\int }_{arcsin\frac{d_1}{\rho }}^{arcsin\frac{d_2}{\rho }}{\int }_{R_1}^{R_2}{\int }_{h_2}^{h_3}{\displaystyle \frac{\rho (z-z^{\prime })}{{\left(\sqrt{{\rho }^2+{\rho }^{\prime 2}-2\rho {\rho }^{\prime }cos{\mathsf{\Omega }}^{\prime }+{(z-z^{\prime })}^2}\right)}^3}}dzd\rho d{\mathsf{\Omega }}^{\prime }d{z}^{\prime }d{\rho }^{\prime }·p{k}_{h}cos\left({\displaystyle \frac{\pi {\alpha }_p}{2}}\right)\hfill \end{array}$$

(23)

where $k_h$ represents the thickness ratio between the auxiliary pole and the rotor yoke:

$$k_h={\displaystyle \frac{h_2-h_1}{h_3-h_2}}$$

(24)

3.2.4. Region IV

$IV$ denotes the rotor yoke directly above the main pole. According to the flux continuity theorem, the axial component of magnetic flux density B remains continuous across the interface between regions I and $IV$. Consequently, the magnetization near this interface matches that of the main pole, generating opposite-polarity magnetic charges on the lower surface of $IV$ that completely neutralize the charges on the main pole’s upper surface. This cancellation explains why magnetic charges at the I–$IV$ interface are omitted in the diagram.

The magnetic circuit can be equivalently modeled as bending at the dashed line in region $IV$, where a discontinuity in magnetization generates magnetic charges. Since the flux in this region changes direction both axially and tangentially, these components can be analyzed separately.

(1) Axial flux variation: following magnetic charge conservation (i.e., the absence of magnetic monopoles), the dashed line in $IV$ exhibits negative charges with polarity opposite to the lower surface of region I.

(2) Tangential flux termination: the tangential flux component terminates at this boundary, producing positive charges.

For clarity in Figure 5, positive and negative charges are graphically separated on either side of the dashed line. The total torque generated by magnetic charges associated with axial flux in the rotor yoke can be derived by incorporating a z-direction integration into the framework of Equation (19):

$$\begin{array}{cc}\hfill C_{IV}=& {\int }_{R_1}^{R_2}{\int }_{-h_c}^{h_c}{\int }_{arcsin\frac{d_1}{\rho }}^{arcsin\frac{d_2}{\rho }}{\int }_{h_2}^{h_3}{\int }_{R_1}^{R_2}{\rho }^{2}cos\left[{\displaystyle \frac{\pi {\alpha }_p(z-h_2)}{2(h_3-h_2)}}\right]·[{\displaystyle \frac{h_1-z^{\prime }}{{\left(\sqrt{{\rho }^2+{\rho }^{\prime 2}-2\rho {\rho }^{\prime }cos{\mathsf{\Omega }}^{\prime }+{(h_1-z^{\prime })}^2}\right)}^3}}\hfill \\ \hfill & -{\displaystyle \frac{h_2-z^{\prime }}{{\left(\sqrt{{\rho }^2+{\rho }^{\prime 2}-2\rho {\rho }^{\prime }cos{\mathsf{\Omega }}^{\prime }+{(h_2-z^{\prime })}^2}\right)}^3}}]d\rho dzd{\mathsf{\Omega }}^{\prime }d{z}^{\prime }d{\rho }^{\prime }\hfill \end{array}$$

(25)

The positive magnetic charges along the dashed line in region $IV$ (associated with tangential flux in the rotor yoke) do not fully cancel the negative charges on the same dashed line but collectively satisfy magnetic charge conservation. Specifically, these positive charges balance only the negative charges on the symmetrically positioned bending dashed line (the origin of tangential flux).

To calculate the surface charge density of these positive charges, the tangential magnetization strength or tangential flux density in the rotor yoke must be determined. Assuming negligible axial leakage flux in the rotor yoke, the tangential flux originates entirely from the axial flux crossing the interface between the main pole and the rotor yoke. Consequently, a correction factor can be defined as the ratio of the average flux density at the main PM/rotor yoke interface to the remanent magnetization of the main pole. This correction factor is derivable via magnetic charge-based field computation methods.

The total torque generated by magnetic charges associated with tangential flux in the rotor yoke can be derived by incorporating a tangential-direction integration into the framework of Equation (22)

$$\begin{array}{cc}\hfill C_V=& {\int }_{R_1}^{R_2}{\int }_{-h_c}^{h_c}{\int }_{arcsin\frac{d_1}{\rho }}^{arcsin\frac{d_2}{\rho }}{\int }_{R_1}^{R_2}{\int }_{h_1}^{h_2}{\int }_0^{{\alpha }_p}p{k}_{M}cos\left({\displaystyle \frac{\pi \mathsf{\Omega }}{2}}\right)·{\displaystyle \frac{\rho (z-z^{\prime })}{{\left(\sqrt{{\rho }^2+{\rho }^{\prime 2}-2\rho {\rho }^{\prime }cos{\mathsf{\Omega }}^{\prime }+{(z-z^{\prime })}^2}\right)}^3}}d\mathsf{\Omega }dzd\rho d{\mathsf{\Omega }}^{\prime }d{z}^{\prime }d{\rho }^{\prime }\hfill \end{array}$$

(26)

The correction factor $k_M$ can be determined from the magnetic fields generated by the magnetic charges on the upper and lower surfaces of the main pole at the rotor yoke surface. Here, the correction factor for the upper surface is 1, while that for the lower surface is denoted $k_M^{\prime }$. Similarly, the influence of all neighboring permanent magnets on the magnetic charge distribution in this region can be calculated using this method.

$$k_M=1-k_M^{\prime }$$

(27)

$$\begin{array}{cc}\hfill k_M^{\prime }=& {\int }_{-\frac{\pi {\alpha }_p}{2}}^{\frac{\pi {\alpha }_p}{2}}{\int }_{R_1}^{R_2}{\int }_{-\frac{\pi {\alpha }_p}{2}}^{\frac{\pi {\alpha }_p}{2}}{\int }_{R_1}^{R_2}{\displaystyle \frac{2{\rho }^{\prime }(h_2-h_1)}{4\pi {\left(\sqrt{{\rho }^2+{\rho }^{\prime 2}+2\rho {\rho }^{\prime }cos(\mathsf{\Omega }-{\mathsf{\Omega }}^{\prime })+{(h_2-h_1)}^2}\right)}^3}}·d\rho d\mathsf{\Omega }d{\rho }^{\prime }d{\mathsf{\Omega }}^{\prime }\hfill \end{array}$$

(28)

The influence of dimensional parameters on electromagnetic torque can be calculated analytically through the coefficient functions $C_I$ to $C_V$. However, due to the complexity of the expressions involving multiple nested integrals, directly solving them is not feasible. Therefore, under assumed parameter values, numerical integration can be applied to compute these coefficient functions, enabling systematic evaluation of dimensional parameter effects and optimization of the design.

4. Parameter Optimization

4.1. Design Requirements

The design requirements for the motor are specified in

Table 3. Design requirements.

Attribute	Specification
Maximum radial dimension of motor	370 mm
Maximum axial dimension of motor	110 mm
Maximum voltage	700 Vrms
Maximum current	180 Arms
Peak torque	650 Nm
Maximum speed	1000 rpm
Pole–slot combination	10-pole; 12-slot
PM material	N48SH

. The maximum diameter and maximum axial length of the motor, including the housing, are 370 mm and 110 mm, respectively. The inverter has a maximum voltage RMS of 700 V and a maximum current RMS of 180 A. The motor must deliver a peak torque of 650 Nm and a maximum speed of 1500 rpm. The pole–slot combination of the motor is restricted to 10-pole/12-slot.

4.2. Optimization Process

The optimization process using the magnetic charge method is shown in Figure 6. The symbol legend for Figure 6 is presented in

Table 4. Symbol legend.

Symbol	Meaning
$R_{max}$	Maximum usable radius obtained from given radius
$h_{max}$	Maximum usable height obtained from given height
$h_{yoke}$	Rotor yoke thickness
$h_{gap}$	Air gap thickness
$d_{coil}$	Tangential thickness of the coil
$R_2$	Outer radius of effective part of the coil
$R_1$	Inner radius of effective part of the coil
$Area$	Area of effective part of the coil
$h_{PM}$	PM thickness
$h_{coil}$	Axial thickness of the coil
${\alpha }_p$	Pole arc coefficient of main PM
$h_{total}$	Total axial length of electromagnetic materials
$C_I$ to $C_V$	Torque coefficient by MCM
$MCM$	Magnetic charge method

. To visually demonstrate the optimization results, this study’s optimization process explicitly selects parameter ranges encompassing extremum points ($h_{PM}, h_{coil}, {\alpha }_p$).

Given that motors involve numerous parameters, multi-parameter optimization encompassing all variables is time-consuming. Therefore, in this paper’s optimization process, the initial step involves simplifying optimization variables. Based on dimensional constraints, machining conditions, and mechanical properties, $R_{max}, h_{max}, h_{yoke}, h_{gap}$ are determined first. Considering the interdependencies among certain variables, establishing relational tables for $d_{coil}, R_2,Area$ from $R_1$ reduces optimization variables and narrows down the optimization scope. Finally, by leveraging the magnetic charge method, coefficients $C_I$ to $C_V$ can select the optimal solution, and the optimization process can be completed.

4.2.1. Determining $R_{max}, h_{max}, h_{yoke}, h_{gap}$

The maximum diameter of the motor is 370 mm. After deducting spatial allocation for the motor housing and cooling oil channels, the maximum permissible dimension for electromagnetic components is set to 340 mm in diameter (i.e., $R_{max}=170$ mm).

Given the axial length constraint of 110 mm for the motor assembly, the maximum allowable axial dimension for electromagnetic components is determined as $h_{max}$ = 80 mm to accommodate the spatial requirements of motor end shields and other structural components.

The rotor yoke is required to fulfill dual functions: establishing magnetic circuits and providing structural support. Due to the NS configuration of permanent magnets generating significant attractive forces between rotor discs, the rotor yoke axial thickness must maintain sufficient mechanical integrity. An initial thickness setting of 8 mm is recommended [34] (i.e., $h_{yoke}=h_3-h_2$ = 8 mm).

Considering the practical difficulty in achieving perfect alignment during coil installation, a mechanical clearance of 0.5 mm is preliminarily specified between the 12 coils. To address the significant attractive forces between dual rotor discs in axial flux machines that pose challenges during assembly, the air gap between coils and permanent magnets should be configured as 1.75 mm (i.e., $h_{gap}=h_1-h_c$ = 1.75 mm).

4.2.2. Creat Table of $d_{coil}, R_2, Area$ from $R_1$

In this paper, PM thickness ($h_{PM}=h_2-h_1$), main PM arc coefficient (${\alpha }_p$), coil inner radius ($R_1$), and coil thickness ($h_{coil}=2h_c$) are defined as independent variables. Tangential thickness ($d_{coil}=d_2-d_1$) and effective part outer radius ($R_2$) are dependent variables.

The outer diameter of the effective part is related to the dimensions of the outer end portion, specifically the coil’s tangential thickness. The relationship between the outer diameter of the effective part $R_2$ and the tangential thickness $d_2-d_1$ is given by

$$R_2=170-(d_2-d_1).$$

(29)

Since a single tangential thickness of the effective part may correspond to multiple inner diameters, the inner diameter is selected as the independent variable. The interrelationships of these variables under the condition of maximizing stator fill (i.e., achieving the highest possible slot fill factor) are shown in

Table 5. The relationship between tangential thickness, outer radius, and inner radius.

${\mathit{R}}_1$/mm	${\mathit{d}}_2-{\mathit{d}}_1$/mm	${\mathit{R}}_2$/mm	${\mathit{R}}_2-{\mathit{R}}_1$/mm	Area/${\mathbf{mm}}^2$
85	17	153	68	1156
84	17	153	69	1173
83	16	154	71	1136
82	16	154	72	1152
81	16	154	73	1168
80	16	154	74	1184
79	15	155	76	1140
78	15	155	77	1155
77	15	155	78	1170
76	14	156	80	1120
75	14	156	81	1134
74	14	156	82	1148
73	14	156	83	1162
72	13	157	85	1105
71	13	157	86	1118
70	13	157	87	1131

.

The minimum dimensional variation is 1 mm. The inner radius variation range of the effective part shown in the table is 70–85 mm. Further expanding this range does not increase the effective part area; thus, only the tabulated portion is displayed.

The area serves to quantify the slot fill factor, expressed as

$$Area=(R_2-R_1)\ast (d_2-d_1)$$

(30)

Table 5 can be used to determine the scope of the enumeration method.

4.2.3. Calculating the Torque Density Coefficient

We select the parameter set with the maximum area from Table 5 as the basis for optimization ($R_1$ = 80 mm). The results of both single-variable and multi-variable optimization are presented to facilitate demonstration.

For single-variable optimization, the main pole arc coefficient can initially be assumed to be ${\alpha }_p=0.75$, with the axial thickness of the coil $2h_c$ = 24.5 mm, where an additional 0.5 mm is allocated for winding clearance.

The torque coefficient based on the magnetic charge method (MCM) can be obtained using $C_I$ to $C_V$:

$$C_{torque}=C_I+C_{II}+C_{III}+C_{IV}+C_V$$

(31)

The coefficient of torque density can be derived from the ratio of torque coefficient to weight m:

$$C_{density}={\displaystyle \frac{C_I+C_{II}+C_{III}+C_{IV}+C_V}{m}}$$

(32)

As derived from previous theoretical analysis, the torque density coefficient $C_{density}$ does not include Fourier coefficients, current density, or magnetic charge density but solely represents the dimensional integral with units of ${\mathrm{m}}^4$.

Figure 7 shows the comparison results between the magnetic charge method coefficient function and finite element analysis using the currently assumed parameters. The current density in the finite element method is 20 A/${\mathrm{mm}}^2$. The current density remains constant throughout the optimization process. The circular markers represent finite element method (FEM) calculation results, while the red curve shows the outcomes obtained using the magnetic charge method (MCM).

From Figure 7, it can be observed that results of FEA and magnetic charge method exhibit essentially consistent trends, with the torque density reaching its maximum value around a 17 mm thickness. Regarding permanent magnet thickness, the magnetic charge method demonstrates significant reference value for estimating optimal parameters.

A comparative analysis of torque computation results between the magnetic charge method and finite element method can be found in Figure A1 in Appendix A.

Building upon this foundation, the reference value of dual-parameter optimization can be verified by selecting the axial thickness of both permanent magnets and coils as variables. As shown in Figure 8, the torque density is calculated via the finite element method, while Figure 9 displays the torque density computed using the magnetic charge method.

Comparative analysis of Figure 8 and Figure 9 reveals that the finite element calculation attains maximum torque density at approximately 32 mm coil thickness and 19 mm magnet thickness, while the magnetic charge method predicts peak torque density at 31 mm coil thickness and 19 mm magnet thickness. Although minor deviations exist between the two methods, these discrepancies remain operationally significant for parameter optimization.

Furthermore, the multi-parameter optimization results for permanent magnet axial thickness, pole arc coefficient, and coil axial thickness are presented in

Table 6. Multi-parameter optimization results of magnetic charge method and finite element method.

	MCM	FEM
Main PM arc coefficient	0.75	0.72
PM thickness/mm	19	19
Coil thickness/mm	31	33

.

Based on this foundation, the optimization results incorporating all parameters can be calculated. As shown in Figure 10 and Figure 11, the relationship between torque density and inner diameter of the effective part calculated using the finite element method (FEM) and magnetic charge method (MCM) under unrestricted axial length conditions is presented. Herein, the torque density at each point represents the optimal result under its corresponding inner radius. A comparison of the two methods of torque density calculation is shown in Figure A2.

As evidenced by the comparative analysis of Figure 10 and Figure 11, the torque density exhibits an overall decreasing trend with reduced inner diameter. This phenomenon arises because increased magnet usage at smaller radii contributes less to torque generation, following a cubic relationship with the outer radius (proportional to $R_2^3$), thereby diminishing torque density. Notably, minor fluctuations in torque density align with variations in the projected area of effective parts. Table 5 reveals that both the projected area fluctuations and torque density variations stem from changes in turn count. Under identical turn counts, smaller inner diameters reduce the inter-segment clearance between adjacent effective parts, resulting in higher torque density.

This analysis confirms that under identical turn counts, the inter-segment clearance between adjacent effective parts exerts a dominant influence on torque density. This conclusion enables narrowing the optimization scope. Given axial length constraints, five inner diameters (84 mm, 80 mm, 77 mm, 73 mm, and 70 mm) can be selected for parameter optimization to determine the optimal solution. This indicates that when compiling Table 5, the optimization range of $R_1$ can directly utilize the points with the largest areas. The larger optimization range was adopted in this paper solely for illustrative convenience regarding this scenario.

4.2.4. Axial Length Comparison

Due to the limited axial length of the motor, the total axial length of electromagnetic materials should not exceed 80 mm (i.e., $h_{total}\le h_{max}$). The total axial length of optimization results under varying inner radius in Figure 11 is illustrated in Figure 12. The blue curve represents the total axial length, while the red curve indicates the axial length constraint.

Figure 12 indicates that the total axial length exceeds the constraint. It is necessary to reduce the optimization range for the axial lengths of both permanent magnets and coils.

4.2.5. Reducing the Optimization Range of $h_{PM}, h_{coil}$

Given the fixed axial lengths of the rotor yoke and air gap, the axial lengths of the coil and permanent magnet must satisfy the following constraints:

$$h_{PM}+h_{coil}\le 60.5\mathrm{mm}$$

(33)

4.2.6. Select the Optimal Solution

The thermal management of dual external rotor coreless axial flux motors is constrained by the challenging design of cooling oil channels in this configuration. Consequently, electromagnetic optimization prioritizes maximizing torque under constant current density. As inner diameter optimization imposes no additional constraints, while axial dimensions are subject to length limitations, the torque–current ratio becomes the preferred optimization target for balancing magnet–copper allocation. This ratio increases with magnet usage but exhibits saturation characteristics, making the inflection point of the curve the optimal selection.

By processing each inner diameter value accordingly, the relationship between torque and magnet inner diameter can be obtained, as shown in Figure 13. The blue curves represent finite element method (FEM) results, while red curves indicate magnetic charge method results. The optimal solution obtained through finite element method (FEM) optimization is located at an inner radius of 77 mm, with a maximum torque of 703.3 Nm. The optimal solution derived from magnetic charge method (MCM) optimization is positioned at an inner radius of 80 mm, achieving a maximum torque of 701 Nm. Using FEM as the benchmark, the torque obtained via MCM reaches 99.67% of the FEM result in torque optimization. The key design parameters are summarized in

Table 7. Coreless axial flux parameter design results.

Property	Parameter
Effective part outer radius (mm)	155
Effective part inner radius (mm)	77
Coil axial thickness (mm)	24.5
Coil tangential thickness (mm)	15
Coil turns	15
Air gap thickness (mm)	1.75
Magnet thickness (mm)	18
Magnet outer radius (mm)	152
Magnet inner radius (mm)	81
Main pole arc coefficient	0.78
Rotor yoke thickness (mm)	8

.

4.3. Computation Time Comparison

The computation time comparison between the finite element method and magnetic charge method demonstrates significant differences. The 3D finite element method takes 53 min per point, while the magnetic charge method requires only 1.65 s, demonstrating the latter’s superiority in computational speed. Additionally, the magnetic charge method supports parallel computing, further enhancing its computational efficiency.

5. Experimental Validation

To validate the accuracy of the motor design proposed in this paper, a functional prototype was fabricated according to the design parameters, and a dedicated test rig was constructed.

In Figure 14, the back-to-back test rig for the prototype evaluation is illustrated. The prototype corresponds to the coreless axial flux motor designed in this study, whose current is controlled by a dedicated controller. The load machine employs a radial flux motor configuration, with its current regulated by a separate test motor controller. The results demonstrate that the axial flux motor achieves comparable torque output while exhibiting significantly more compact dimensions than the radial flux motor. The controller can supply AC power with a maximum voltage of 700 Vrms and a maximum current of 180 Arms. Both control systems can receive real-time control commands from the host computer. The torque sensor provides real-time torque measurement. The torque sensor has a maximum torque range of 2000 Nm and a maximum rotational speed range of 2500 rpm, supporting experiments within the given specification limits. Voltage measurements can be achieved using a voltage probe and oscilloscope, with a maximum range of 1000 V. Current measurements can be achieved using a current probe and oscilloscope, with a maximum range of 200 Arms. Both voltage and current measurements are capable of meeting experimental requirements.

The experimental back electromotive forces (back-EMFs) at 1000 rpm between phases A, B, and C are shown in Figure 15. The symmetry of three-phase back-EMFs and sinusoidal waveforms demonstrate the absence of machining asymmetry in the motor manufacturing process.

The back-EMF comparison at 1000 rpm is depicted in Figure 16. Blue data are simulation results, while red data are experimental measurements. The fundamental amplitude of the simulation reaches 409.6 V, compared with 416.0 V in the experimental results. Moreover, this voltage is below the specified maximum voltage of 700 Vrms. The harmonic content remains minimal, with the fifth harmonic being the most prominent component at 2.8 V (simulation) and 3.1 V (experiment), respectively.

For coreless axial flux motors, phase current waveform distortion is typically more severe under low-speed high-torque operating conditions. The phase current waveform for full-load at 200 rpm is illustrated in Figure 17. As evidenced by Figure 17, the phase current waveform maintains a sinusoidal profile without noticeable distortion even under low-speed high-torque operating conditions.

Figure 18 shows the torque comparison between simulation and experimental results. The abscissa represents the phase current amplitude, and the ordinate denotes the average torque. The red curve represents the simulation results, and the circular markers denote the test data. Under full-load conditions (650 Nm), the experimental phase current amplitude is 210 A, corresponding to a simulated torque of 676 Nm. Moreover, this current is below the specified maximum current of 180 A RMS (with a peak value of approximately 255 A).

The error analysis results are shown in Figure 19. The blue data indicates the errors between the finite element method (FEM) and actual measurement data, while the red data represents the errors between the magnetic charge method (MCM) and actual measurement data. The maximum errors are 5% and 4.8%, respectively. Figure 19 data indicates that the error is fluctuating, which may be caused by torque measurement errors. Meanwhile, the experimental torque is generally lower than the FEM simulation torque. The simulation torque represents the maximum torque achievable with the current control method. The lower measured torque is likely due to d-q axis current calculation errors resulting from current sampling errors, leading to torque reduction.

6. Conclusions

This study achieved accelerated optimization of the coreless axial flux motor through implementation of the magnetic charge model. Comprehensive verification via simulation and experimental methods led to the following conclusions.

(1) The application of the magnetic charge model for motor parameter optimization reduces single-point simulation time from 53 min using finite element method (FEM) to 1.65 s. This methodology is specifically applicable to coreless axial flux motors, optimizing the issue of excessive computational time in FEM simulations. Through refinement, this method can also be extended to the rapid optimization of cored axial flux permanent magnet machines.

(2) The optimization results obtained using the magnetic charge model show close alignment with simulation data, with the optimal parameter’s torque reaching 99.67% of the simulated value.

(3) Comparative analysis of experimental and simulated torque results demonstrates a minor deviation of approximately 4%, validating the accuracy of simulation-based optimization and indirectly confirming the precision of the magnetic charge method in parameter optimization.

(4) Since the model employed in this paper relies on magnetic circuit equivalence, it does not account for magnetic saturation in the rotor yoke. For complex-structure motors such as interior permanent magnet (IPM) motors, further research remains necessary. Since the model is established based on the fundamental magnetic field, research on torque ripple will require supplementary analysis and computation of harmonic magnetic fields.