A 2D Generalized Equivalent Magnetic Network Model for Electromagnetic Performance Analysis of Surface-Mounted Permanent Magnet Electric Machines

Abstract

This paper proposes a two-dimensional (2D) generalized equivalent magnetic network (GEMN) model suitable for surface-mounted permanent magnet electric machines (SPEMs). The model divides the SPEM into eight types of regions: stator yoke, stator tooth body, stator tooth tips, stator slot body, stator slot openings, air gap, rotor permanent magnets, and rotor yoke. Each region is subdivided radially and tangentially into multiple 2D magnetic network units containing radial and tangential magnetic circuit parameters, forming a regular magnetic network covering all regions of the SPEM. The topology of this magnetic network remains unchanged during rotor rotation and can accommodate various surface-mounted permanent magnet structures including Halbach arrays, which enhances the generality of the model significantly. The proposed model can be used to calculate the 2D magnetic flux density distribution, winding electromotive force, electromagnetic torque, stator iron loss, and permanent magnet demagnetization in the influence of magnetic saturation, stator slotting, and current harmonic. Comparative analysis with the accurate subdomain method (ASDM) and finite element method (FEM) demonstrates that the GEMN model achieves a good balance between computational speed and accuracy, making it particularly suitable for efficient electromagnetic performance evaluation of SPEMs.

1. Introduction

In recent years, with the increasing demand for environmental protection and economic benefits, the transportation industry has been rapidly developing towards electrification. The electrification of power systems has been realized in the fields of electric vehicles, electric ships, and light electric aircrafts [1,2,3,4]. Surface-mounted permanent-magnet electric machines (SPEMs) possess performance advantages such as high speed, high efficiency, and high-power density, and have broad application prospects in the electrification of power systems [5,6,7,8].

Electromagnetic performance analysis is the foundation for the design and optimization of SPEMs, and has always been one of the research focuses of SPEMs. The methods of electromagnetic performance analysis for SPEMs mainly include the finite element method (FEM), subdomain method (SDM), equivalent magnetic network method (EMNM), and hybrid method (HBM).

1.1. Finite Element Method

The FEM discretizes electric machines into a large number of sub-units, and calculates the numerical solution of magnetic field distribution based on Maxwell’s equations and boundary conditions. The FEM can be used for 2D or 3D magnetic field analysis considering the influence of magnetic saturation, cogging, and harmonic currents. The calculation results of the FEM are sufficiently accurate while the computation resource requirements are high and the computation time is long. With the significant improvement in computation power and the rapid development of commercial finite element software, the FEM has been widely applied in the field of electromagnetic performance analysis for different kinds of electric machines [9,10,11,12].

1.2. Subdomain Method

The SDM divides SPEMs into several subdomains, establishes the vector magnetic potential equation for each subdomain, and then calculates the 2D magnetic field distribution based on the boundary conditions between adjacent subdomains. SDM has a relatively fast computation speed and can consider factors such as cogging and harmonic currents. However, SDM cannot account for magnetic saturation and is not suitable for the analysis of SPEM with high magnetic saturation. In [13,14,15,16], the authors proposed a subdomain method for SPEMs including a permanent-magnet subdomain and air gap subdomain, which can consider the cogging effect through conformal transformation. Based on the above research, in [17,18,19], the authors proposed an accurate subdomain method (ASDM) which adds slot subdomain and slot-opening subdomain, improving the accuracy of the magnetic field calculation influenced by the slot effect. In [20,21], the diffusion equation is introduced to consider the influence of eddy currents in permanent magnet on the magnetic field distribution, further improving the calculation accuracy of the SDM. Overall, with the development of the SDM, more factors that affect the magnetic field distribution can be considered, but the models become more complex.

1.3. Equivalent Magnetic Network Method

The EMNM divides SPEM into a magnetic network composed of equivalent magnetic potential and magnetic permeance. By applying Kirchhoff’s laws to solve node magnetic potentials and branch fluxes through iterative calculations that account for core material nonlinearity, EMNM achieves magnetic field distributions with iron saturation effects.

Developed from the traditional magnetic circuit method (MCM), which can be considered as a simplified 1D magnetic network, the EMNM has evolved significantly. In [22], a generalized magnetic circuit model for SPEMs is proposed, introducing the PM effective coefficient related to the relative position between stator and rotor, to address magnetic circuit parameter variations caused by rotor rotation and stator slotting effects. In [23], the researchers extend the above approach to the electromagnetic performance analysis of conical SPEMs, incorporating end effects via a lumped-parameter magnetic circuit model.

With the increasing demand for the accuracy of magnetic field analysis in permanent magnet (PM) electric machines, research on an improved EMNM based on 2D magnetic network elements has developed rapidly. In [23,24,25,26,27], focusing on different types of PM electric machines, hybrid methods combining the EMNM with the MCM are proposed. Specifically, the EMNMs are established in regions with complex magnetic field distributions, e.g., stator tooth tip, slot opening, and air gap, while MCMs are applied to regions with simpler flux paths, e.g., stator tooth body, stator yoke, and rotor yoke, in order to balances model complexity and computational accuracy. Regarding the dynamic change in magnetic network with the variations in the relative position between stator and rotor, different solutions have been adopted. In [23,24], the researchers update the network mapping relationships in adjacent regions between stator and rotor at each time step to achieve dynamic connectivity, while the researchers in [26,27] adjust the boundary permeances between air gap and rotor at any given moment. Further advancements include applying the improved EMNM to short-circuit fault analysis of an external-rotor SPEM in [28], expanding the EMNM framework by incorporating parallel magnetization models for SPEMs (previously dominated by radial magnetization) in [29], and avoiding dynamic stator–rotor network connections by loading equivalent magnetic flux sources derived from analytical calculations into air gaps in [30].

Overall, the EMNM possesses flexible modeling capabilities suitable for 2D or 3D electromagnetic performance analysis of various types of PM electric machines, effectively balancing computational accuracy and speed. However, it generally requires dynamic updates to the magnetic network connection topology or magnetic permeance parameters between stator and rotor based on their relative positions. Additionally, current implementations show insufficient research on PM structures with complex magnetization characteristics like Halbach arrays, and lack sufficient emphasis on accurate analysis of stator slot leakage flux, which is a critical parameter for evaluating AC copper losses in stator windings [31,32].

1.4. Hybrid Method

In recent years, HBM has received significant attention from many researchers. HBM combines various methods of electromagnetic performance analysis to fully utilize the advantages of different methods and enhance analysis efficiency. In [33,34,35], the authors proposed a hybrid method which introduces the equivalent magnetic network model of the stator core to consider stator magnetic saturation based on SDM. Compared to SDM, the above method demonstrates higher computational accuracy in the electromagnetic performance analysis for high-saturation SPEMs. In [36], a hybrid method for the electromagnetic performance analysis of SPEM is proposed, which can calculate the magnetic potential of permanent-magnets and stator windings based on the improved winding function, and consider the slot effect based on conformal transformation, also demonstrating good calculation accuracy.

1.5. The Method Proposed in This Article

To address the dynamic variation of magnetic network topology between the stator and rotor in SPEMs and enhance adaptability to various structural surface-mounted PM poles, this paper proposes a two-dimensional (2D) generalized equivalent magnetic network (GEMN) model characterized by the following features:

• The GEMN divides the SPEM into eight regions: stator yoke, stator tooth body, stator tooth tip, stator slot body, stator slot opening, air gap, rotor permanent magnets, and rotor yoke, each of which is composed of standardized 2D magnetic network units incorporating radial/tangential magnetic potential and permeance, with units arranged in regular patterns across different regions.
• During rotor rotation, the GEMN topology remains unchanged, with rotational effects manifested solely through updated magnetic potential parameters in PM regions, obtained via circumferential discretization of PM structures and Fourier analysis of residual magnetization intensity, ensuring applicability to diverse surface-mounted pole configurations.
• The number of magnetic network units can be flexibly controlled by adjusting radial/tangential segmentation counts in each region, enabling convenient scale regulation of the network.
• The GEMN enables calculations of electromagnetic performance for SPEMs, including 2D magnetic flux density distribution within the leakage flux of stator slot, winding electromotive force, electromagnetic torque, stator core losses, and permanent magnet demagnetization, considering factors like magnetic saturation, stator slotting, and current harmonic.

This paper elaborates on the modeling process and analytical methodology of the 2D GEMN model for SPEMs. The Section 2 details the modeling procedure and specific workflow of the 2D GEMN model. The Section 3 conducts comparative analyses of key electromagnetic performance metrics for an SPEM prototype using three approaches: ASDM, GMEN, and FEM. These comparisons validate the rationality and accuracy of the GMEN model while demonstrating its principal characteristics. The Section 4 summarizes the comprehensive findings of this paper.

2. 2D Generalized Equivalent Magnetic Network Model

The 2D model of the SPEM is shown in Figure 1, which divides the SPEM into eight regions: Region I for stator yoke, Region II for stator tooth body, Region III for stator tooth tip, Region IV for stator slot body, Region V for stator slot opening, Region VI for air gap, Region VII for permanent magnet, and Region VIII for rotor yoke. In Figure 1, R_sy is the outer radius of stator yoke, R_sb is the bottom radius of stator slot body, R_st is the top radius of stator slot body, R_g is the outer radius of air gap, R_m is the outer radius of permanent magnet, R_ry is the outer radius of rotor yoke, and R_sh is the outer radius of the shaft.

The analytical assumptions of the above model include the following:

• The model of the SPEM is 2D, without considering the axial end effect;
• The currents of windings are uniformly distributed in the region of stator slot body;
• Considering the periodicity, a unit part of SPEM is used as the analysis object to reduce computation cost.

2.1. Standard Model of Units in 2D GEMN

In the 2D GEMN model of SPEM, the standard model of the 2D magnetic network unit is illustrated in Figure 2. The standard model incorporates radial and tangential magnetic potential and permeance parameters, among which, F_r1 and F_r2 represent the radial magnetic potential of the upper and lower halves of the unit, respectively, G_r1 and G_r2 represent the radial magnetic permeance of the upper and lower halves of the unit, respectively, F_t1 and F_t2 represent the tangential magnetic potential of the left and right halves of the unit, respectively, and G_t1 and G_t2 represent the tangential magnetic permeance of the left and right halves of the unit, respectively. The calculation methods for these parameters will be discussed in subsequent chapters.

2.2. 2D GEMN Model of an SPEM

The 2D GEMN model of an SPEM is shown in Figure 3. The model uniformly divides each region into segments along the radial and tangential directions, among which N_r1, N_r2, N_r3, N_r4, N_r5, and N_r6 represent the number of radial segments for stator yoke, stator tooth body (or stator slot body), stator tooth tip (or stator slot opening), air gap, permanent magnet, and rotor yoke, respectively, while N_t1, N_t2, and N_t3 represent the tangential segments for stator slot opening, the protruding part of stator tooth tip, and the middle part of stator tooth tip, respectively. In Figure 3, the magnetic network regions with different topologies are represented by different symbols: S_y1~S_y3 represent the three types of magnetic network regions in stator yoke, S_tb represents the magnetic network region of stator tooth body, S_tt1 and S_tt2 represent the two types of magnetic network regions in stator tooth tip, S_sb1 and S_sb2 represent the two types of magnetic network regions in stator slot body, S_so represents the magnetic network region of stator slot opening, G₁~G₃ represent the three types of magnetic network regions in air gap, P₁~P₃ represent the three types of magnetic network regions in permanent magnets, and R_y1~R_y3 represent the three types of magnetic network regions in rotor yoke. It should be noted that in order to provide a more intuitive explanation of the segmented approach of the GEMN model, the model of the SPEM in the Cylindrical coordinate system in Figure 1 is transformed to the one in the Cartesian coordinate system in Figure 3, and the 2D GEMN model proposed in this article is still set up in the Cylindrical coordinate system.

For a unit part of the SPEM, the expressions for the number of total radial segments N_rz and the number of total tangential segments N_tz are as follows:

$$\left\{\begin{array}{l}{N}_{rz}=N_{r1}+N_{r2}+N_{r3}+N_{r4}+N_{r5}+N_{r6}\\ N_{tz}=Z_1(N_{t1}+2N_{t2}+N_{t3})/k\end{array}\right.$$

(1)

where Z₁ is the number of stator slots, k is the number of unit parts in one SPEM.

In the 2D GEMN model, all magnetic network units adopt identical 2D topological structures, and the magnetic network units in different regions are arranged in a regular pattern, which facilitates the simplification of magnetic network computational procedures.

2.3. Calculation of Magnetic Network Parameters

2.3.1. Calculation of Magnetic Permeance

The magnetic permeance parameters are determined by the shape, size, and magnetic permeability of the 2D magnetic network units in the GEMN. The units mainly include three kinds of shapes: sector, trapezoid, and rectangle, as shown in Figure 4.

The expressions for magnetic permeance parameters of units with different shapes are as follows [29]:

• Sector

$$\left\{\begin{array}{l}{G}_{r1}={\displaystyle \frac{{\mu }_r{\mu }_0\alpha l}{\ln (\frac{R_2}{R_{av}})}}\\ G_{r2}={\displaystyle \frac{{\mu }_r{\mu }_0\alpha l}{\ln (\frac{R_{av}}{R_1})}}\\ G_{t1}=G_{t2}={\displaystyle \frac{2{\mu }_r{\mu }_{0}l}{\alpha }}\ln ({\displaystyle \frac{R_2}{R_1}})\\ R_{av}={\displaystyle \frac{R_1+R_2}{2}}\end{array}\right.$$

(2)

• Trapezoid

$$\left\{\begin{array}{l}{G}_{r1}={\displaystyle \frac{{\mu }_r{\mu }_0(3a+b)l}{2h}}\\ G_{r2}={\displaystyle \frac{{\mu }_r{\mu }_0(a+3b)l}{2h}}\\ G_{t1}=G_{t2}={\displaystyle \frac{4{\mu }_r{\mu }_{0}hl}{a+b}}\end{array}\right.$$

(3)

• Rectangle

$$\left\{\begin{array}{l}{G}_{r1}=G_{r2}={\displaystyle \frac{2{\mu }_r{\mu }_{0}bl}{h}}\\ G_{t1}=G_{t2}={\displaystyle \frac{2{\mu }_r{\mu }_{0}hl}{b}}\end{array}\right.$$

(4)

where ${\mu }_0$ is vacuum permeability, ${\mu }_r$ is relative permeability of material, l is the effective axial length of SPEM.

2.3.2. Calculation of Magnetic Potential of Permanent Magnets

SPEMs exhibit diverse magnetization methods and shapes of PMs. In terms of magnetization methods, they primarily include relatively simple parallel magnetization and radial magnetization, along with more complex Halbach multi-directional magnetization. Regarding shapes, conventional tile-shaped configurations and truncated quasi-breadloaf configurations after pole cutting are commonly adopted. As shown in Figure 1, regardless of magnetization methods and shapes of PMs, the annular region containing permanent magnets exclusively comprises magnets and air, where their magnetic permeabilities are nearly identical. This characteristic establishes a foundation for conducting the universal analysis of magnetic potential in surface-mounted PMs through discretization and Fourier series.

To facilitate a more intuitive presentation of the calculation approach for the magnetic potential of PMs, the PM region in the range of a pair of poles in the Cylindrical coordinate system is expanded and then transformed to the one in the Cartesian coordinate system, as shown in Figure 5. Based on discretization concepts, the PM region in the range of a pair of poles is equally divided into N_pm segments along the tangential direction, where j is the numbering of PM segments (j = 1, 2, 3, …, N_pm), α_pj is the ratio between the tangential spatial angular range of the j-th PM segment and the tangential spatial angular range of the entire PM region under a pair of poles, θ_r is the tangential mechanical angular coordinate in the rotor coordinate system, while h_m represents the radial thickness of the PM region. Obviously, we have

$${\displaystyle \sum _{j=1}^{N_{pm}}{\alpha }_{pj}=1}$$

(5)

After discretizing the PM region, each PM segment can be considered as parallel magnetized, with the magnetization angles uniformly referenced to the positive direction of the radial central axis of each segment, with counterclockwise being positive, as shown in Figure 6. Here, θ_pmj is the magnetization direction of the j-th PM segment, θ_j is the tangential mechanical angular coordinate of any point within the j-th PM segment relative to its radial central axis, α_pmj is the angle between the magnetization direction and the radial direction of any point in the j-th PM segment. Obviously, we have

$$\left\{\begin{array}{l}{\alpha }_{pmj}={\theta }_{pmj}-{\theta }_j={\theta }_{pmj}-({\theta }_r-{\theta }_{j0})\\ {\theta }_{j0}={\displaystyle \frac{\pi }{p}}({\displaystyle \sum _{z=0}^j{\alpha }_{pz}}+{\displaystyle \sum _{z=0}^{j-1}{\alpha }_{pz}})\end{array}\right.$$

(6)

where θ_j0 is the tangential mechanical angular coordinate of the central axis in the j-th permanent magnet segment and p is the number of pole pairs in the SPEM.

The spatial distribution expressions of the residual magnetization of the PM region in the range of a pair of poles are as follows:

$$\left\{\begin{array}{l}{M}_r({\theta }_r)=M_{rmj}\cos ({\alpha }_{pmj})=M_{rmje}\cos [{\theta }_{pmj}-({\theta }_r-{\theta }_{j0})]\hspace{1em}j=1,2,\dots \dots ,N_{pm}\hspace{1em}{\theta }_r\subseteq [{\displaystyle \frac{2\pi }{p}}{\displaystyle \sum _{z=0}^{j-1}{\alpha }_{pz}},{\displaystyle \frac{2\pi }{p}}{\displaystyle \sum _{z=0}^j{\alpha }_{pz}}]\\ M_t({\theta }_r)=M_{rmj}\sin ({\alpha }_{pmj})=M_{rmj}\sin [{\theta }_{pmj}-({\theta }_r-{\theta }_{j0})]\hspace{1em}j=1,2,\dots \dots ,N_{pm}\hspace{1em}{\theta }_r\subseteq [{\displaystyle \frac{2\pi }{p}}{\displaystyle \sum _{z=0}^{j-1}{\alpha }_{pz}},{\displaystyle \frac{2\pi }{p}}{\displaystyle \sum _{z=0}^j{\alpha }_{pz}}]\\ M_{rmje}={\displaystyle \frac{h_{mj}}{h_m}}{M}_{rmj}\end{array}\right.$$

(7)

where M_r, M_t are the radial and tangential spatial distribution of the residual magnetization of the PM region respectively, M_rj is the residual magnetization of the j-th PM segment, h_mj is the radial thickness of the j-th PM segment.

Based on Equation (7), we can derive the expressions in the Fourier series of M_r and M_t in the rotor rotating coordinate system as follows:

$$\left\{\begin{array}{l}{M}_r({\theta }_r)=M_{r0}+{\displaystyle \sum _{n=1}^{\infty }{M}_{rn11}\cos (np{\theta }_r)+M_{rn22}\sin (np{\theta }_r)}\\ M_t({\theta }_r)=M_{t0}+{\displaystyle \sum _{n=1}^{\infty }{M}_{tn11}\cos (np{\theta }_r)+M_{tn22}\sin (np{\theta }_r)}\end{array}\right.$$

(8)

in which

$$\left\{\begin{array}{l}{M}_{r0}={\displaystyle \frac{p}{\pi }}{\displaystyle \sum _{j=1}^{N_{pm}}{M}_{rmje}\cos ({\theta }_{pmj})\sin (\frac{\pi }{p}{\alpha }_{pj})}\\ M_{t0}={\displaystyle \frac{p}{\pi }}{\displaystyle \sum _{j=1}^{N_{pm}}{M}_{rmj}\sin ({\theta }_{pmj})\sin (\frac{\pi }{p}{\alpha }_{pj})}\\ M_{rn11}={\displaystyle \frac{p}{\pi (1+np)}}{\displaystyle \sum _{j=1}^{N_{pm}}{M}_{rmje}\cos }[n\pi ({\displaystyle \sum _{z=0}^j{\alpha }_{pz}}+{\displaystyle \sum _{z=0}^{j-1}{\alpha }_{pz}})-{\theta }_{pmj}]\sin [(1+np){\alpha }_{pj}{\displaystyle \frac{\pi }{p}}]+{\displaystyle \frac{p}{\pi (1-np)}}{\displaystyle \sum _{j=1}^{N_{pm}}{M}_{rmje}\cos }[n\pi ({\displaystyle \sum _{z=0}^j{\alpha }_{pz}}+{\displaystyle \sum _{z=0}^{j-1}{\alpha }_{pz}})+{\theta }_{pmj}]\sin [(1-np){\alpha }_{pj}{\displaystyle \frac{\pi }{p}}]\\ M_{tn11}=-{\displaystyle \frac{p}{\pi (1+np)}}{\displaystyle \sum _{j=1}^{N_{pm}}{M}_{rmj}\sin }[n\pi ({\displaystyle \sum _{z=0}^j{\alpha }_{pz}}+{\displaystyle \sum _{z=0}^{j-1}{\alpha }_{pz}})-{\theta }_{pmj}]\sin [(1+np){\alpha }_{pj}{\displaystyle \frac{\pi }{p}}]+{\displaystyle \frac{p}{\pi (1-np)}}{\displaystyle \sum _{j=1}^{N_{pm}}{M}_{rmj}\sin }[n\pi ({\displaystyle \sum _{z=0}^j{\alpha }_{pz}}+{\displaystyle \sum _{z=0}^{j-1}{\alpha }_{pz}})+{\theta }_{pmj}]\sin [(1-np){\alpha }_{pj}{\displaystyle \frac{\pi }{p}}]\\ M_{rn22}={\displaystyle \frac{p}{\pi (1+np)}}{\displaystyle \sum _{j=1}^{N_{pm}}{M}_{rmje}\sin [n\pi ({\displaystyle \sum _{z=0}^j{\alpha }_{pz}}+{\displaystyle \sum _{z=0}^{j-1}{\alpha }_{pz}})-{\theta }_{pmj})]}\sin [(1+np){\alpha }_{pj}{\displaystyle \frac{\pi }{p}}]+{\displaystyle \frac{p}{\pi (1-np)}}{\displaystyle \sum _{j=1}^{N_{pm}}{M}_{rmje}\sin [n\pi ({\displaystyle \sum _{z=0}^j{\alpha }_{pz}}+{\displaystyle \sum _{z=0}^{j-1}{\alpha }_{pz}})+{\theta }_{pmj})]}\sin [(1-np){\alpha }_{pj}{\displaystyle \frac{\pi }{p}}]\\ M_{tn22}={\displaystyle \frac{p}{\pi (1+np)}}{\displaystyle \sum _{j=1}^{N_{pm}}{M}_{rmj}\cos [n\pi ({\displaystyle \sum _{z=0}^j{\alpha }_{pz}}+{\displaystyle \sum _{z=0}^{j-1}{\alpha }_{pz}})-{\theta }_{pmj})]}\sin [(1+np){\alpha }_{pj}{\displaystyle \frac{\pi }{p}}]-{\displaystyle \frac{p}{\pi (1-np)}}{\displaystyle \sum _{j=1}^{N_{pm}}{M}_{rmj}\cos [n\pi ({\displaystyle \sum _{z=0}^j{\alpha }_{pz}}+{\displaystyle \sum _{z=0}^{j-1}{\alpha }_{pz}})+{\theta }_{pmj})]}\sin [(1-np){\alpha }_{pj}{\displaystyle \frac{\pi }{p}}]\end{array}\right.$$

(9)

Defining the tangential mechanical angular coordinate in the stator stationary coordinate system as θ_s, we have

$${\theta }_r={\theta }_s-{\displaystyle \frac{{\omega }_1}{p}}t-{\theta }_{sr0}$$

(10)

where ω₁ is the fundamental electrical angular frequency and θ_sr0 is the initial tangential mechanical angle difference between the stator stationary coordinate system and the rotor rotating coordinate system.

By substituting expression (10) into expression (8), we can obtain the expressions in the Fourier series of M_r and M_t in the stator stationary coordinate system as follows:

$$\left\{\begin{array}{l}{M}_r({\theta }_s,t)=M_{r0}+{\displaystyle \sum _{n=1}^{\infty }[M_{rn1}\cos (np{\theta }_s)+M_{rn2}}\sin (np{\theta }_s)]\\ M_t({\theta }_s,t)=M_{t0}+{\displaystyle \sum _{n=1}^{\infty }[M_{tn1}\cos (np{\theta }_s)+M_{tn2}}\sin (np{\theta }_s)]\end{array}\right.$$

(11)

in which

$$\left\{\begin{array}{l}{M}_{rn1}=M_{rn11}\cos (n{\omega }_{1}t+np{\theta }_{sr0})-M_{rn22}\sin (n{\omega }_{1}t+np{\theta }_{sr0})\\ M_{rn2}=M_{rn11}\sin (n{\omega }_{1}t+np{\theta }_{sr0})+M_{rn22}\cos (n{\omega }_{1}t+np{\theta }_{sr0})\\ M_{tn1}=M_{tn11}\cos (n{\omega }_{1}t+np{\theta }_{sr0})-M_{tn22}\sin (n{\omega }_{1}t+np{\theta }_{sr0})\\ M_{tn2}=M_{tn11}\sin (n{\omega }_{1}t+np{\theta }_{sr0})+M_{tn22}\cos (n{\omega }_{1}t+np{\theta }_{sr0})\end{array}\right.$$

(12)

In the 2D GEMN model of the SPEM shown in Figure 3, the magnetic network units in the PM region are sectorial, as illustrated in Figure 4a. The residual magnetization intensity of one unit can be characterized by the residual magnetization intensity at the geometric center of the sector. Taking a certain sectorial unit U in the PM region as an example, the expressions for the radial magnetic potential and tangential magnetic potential parameters of unit U are as follows:

$$\left\{\begin{array}{l}{F}_{r1}=F_{r2}={\displaystyle \frac{M_r({\theta }_{spmu},t)h_m}{2N_{r5}}}\\ F_{t1}=F_{t2}={\displaystyle \frac{M_t({\theta }_{spmu},t)R_{pmu}{\alpha }_{pmu}}{2}}\end{array}\right.$$

(13)

where θ_spmu is the tangential mechanical angular coordinate of the geometric center of the sectorial unit U in the stator stationary coordinate system which can be determined based on the size parameters and the tangential segmentation settings of the permanent magnet region, R_pmu is the radius of the geometric center of the sectorial unit U, and α_pmu is the central angle of the sectorial unit U.

This section obtains spatiotemporal expressions of magnetic potential parameters for magnetic network units in the PM region in the stator coordinate system through discretization and Fourier series. When the rotor rotates, dynamic updates of magnetic network connection topology between stator and rotor are no longer required—only the magnetic potential parameters of the PM region network units need refreshing. The above method considers parameters including residual magnetization intensity, magnetization orientation, and geometric dimensions of each tangential PM segment in the PM region. This approach possesses universal applicability to various magnetization patterns, including radial magnetization, parallel magnetization, Halbach multi-directional magnetization, and to various shapes of PM poles, including tile-shaped and breadloaf-shaped configurations.

2.3.3. Calculation of Magnetic Potential Parameters of Stator Windings

The expressions for the magnetic potential generated by the currents of the windings in the h-th slot are as follows:

• Single-layer winding

$$\left\{\begin{array}{l}{F}_{sh}(t)=i_h(t)N_s\\ i_h(t)={\displaystyle \sum _v{i}_{hv}\cos (v{\omega }_{1}t+{\phi }_{hv})}\end{array}\right.$$

(14)

where v is the order of harmonic current, i_hv is the amplitude of the v-th harmonic current of the winding in the h-th slot, φ_hv is the initial phase angle of the vth harmonic current of the winding in the h-th slot, and N_s is the number of turns in a single-layer winding.

• Double-layer winding

$$\left\{\begin{array}{l}{F}_{sh}(t)=[i_{h1}(t)+i_{h2}(t)]N_s\\ i_{h1}(t)={\displaystyle \sum _v{i}_{hv1}\cos (v{\omega }_{1}t+{\phi }_{hv1})}\\ i_{h2}(t)={\displaystyle \sum _v{i}_{hv2}\cos (v{\omega }_{1}t+{\phi }_{hv2})}\end{array}\right.$$

(15)

where i_hv1, i_hv2 are the amplitude of the v-th harmonic currents in the different layer of the winding in the h-th slot, φ_hv1, φ_hv2 are the initial phase angle of the v-th harmonic currents in different layer of the winding of the h-th slot.

According to Ampere’s circuital theorem, the magnetic potential drop of each stator tooth in a unit part of one SPEM satisfies the following relationship:

$$\left\{\begin{array}{l}{F}_{t2}(t)-F_{t1}(t)=F_{s1}(t)\\ \dots \\ F_{t(h+1)}(t)-F_{th}(t)=F_{sh}(t)\\ \dots \\ F_{t1}(t)-F_{t{Z}_k}(t)=F_{s{Z}_k}(t)\end{array}\right.$$

(16)

where Fth is the magnetic potential drop of the h-th tooth, Z_k is the number of stator teeth in a unit part of one SPEM, and Z_k = Z₁/k.

Since the sum of the winding currents in the stator slots in a unit part of SPEM is 0, the following relationship exists:

$$\left\{\begin{array}{l}{\displaystyle \sum _{h=1}^{Z_k}{F}_{sh}}(t)=0\\ {\displaystyle \sum _{h=1}^{Z_k}{F}_{th}}(t)=0\end{array}\right.$$

(17)

Based on Equations (16) and (17), we can obtain the magnetic potential drop of each stator tooth of a unit part in SPEM as follows [26]:

$$\left\{\begin{array}{l}{F}_{t{Z}_k}(t)={\displaystyle \frac{1}{Z_k-1}}({\displaystyle \sum _{h=1}^{Z_k-1}{F}_{sh}}(t)+{\displaystyle \sum _{h=2}^{Z_k-1}{F}_{sh}}(t)+\dots +{\displaystyle \sum _{h=Z_k-1}^{Z_k-1}{F}_{sh}}(t))={\displaystyle \frac{1}{Z_k-1}}{\displaystyle \sum _{h=1}^{Z_k-1}h}{F}_{sh}(t)\\ F_{t(Z_k-1)}(t)=F_{t{Z}_k}(t)-F_{s(Z_k-1)}(t)\\ \dots \\ F_{t1}(t)=F_{t2}(t)-F_{s1}(t)\end{array}\right.$$

(18)

In the 2D GEMN model shown in Figure 3, the magnetic potential of the stator tooth is applied to the radial branches in the stator tooth body region S_tb. The expression of the radial magnetic potential parameter for the magnetic network units in the stator tooth body region S_tb is

$$F_{r1}=F_{r2}={\displaystyle \frac{F_{th}(t)}{2N_{r2}}}$$

(19)

2.4. Iterative Computation of the 2D GEMN Model

Magnetic networks share similarities with electrical circuit networks, and this article employs the node magnetic potential method to solve an equivalent magnetic network.

In the 2D GEMN model shown in Figure 3, only magnetic potential and magnetic permeance parameters are contained, and its node magnetic potential equation is [37]

$${\mathit{A}}_{\mathbf{[}\mathbf{(}{\mathit{N}}_{\mathit{n}\mathit{d}}\mathbf{-}\mathbf{1}\mathbf{)}\mathbf{\times }{\mathit{N}}_{\mathit{b}\mathit{r}}\mathbf{]}}{\mathit{G}}_{\mathit{m}\mathbf{[}{\mathit{N}}_{\mathit{b}\mathit{r}}\mathbf{\times }{\mathit{N}}_{\mathit{b}\mathit{r}}\mathbf{]}}{\mathit{A}}_{\mathbf{[}{\mathit{N}}_{\mathit{b}\mathit{r}}\mathbf{\times }\mathbf{(}{\mathit{N}}_{\mathit{n}\mathit{d}}\mathbf{-}\mathbf{1}\mathbf{)}\mathbf{]}}^{\mathit{T}}{\mathit{U}}_{\mathit{n}\mathbf{[}\mathbf{(}{\mathit{N}}_{\mathit{n}\mathit{d}}\mathbf{-}\mathbf{1}\mathbf{)}\mathbf{\times }\mathbf{1}\mathbf{]}}\mathbf{=}\mathbf{-}{\mathit{A}}_{\mathbf{[}\mathbf{(}{\mathit{N}}_{\mathit{n}\mathit{d}}\mathbf{-}\mathbf{1}\mathbf{)}\mathbf{\times }{\mathit{N}}_{\mathit{b}\mathit{r}}\mathbf{]}}{\mathit{G}}_{\mathit{m}\mathbf{[}{\mathit{N}}_{\mathit{b}\mathit{r}}\mathbf{\times }{\mathit{N}}_{\mathit{b}\mathit{r}}\mathbf{]}}{\mathit{U}}_{\mathit{s}\mathbf{[}{\mathit{N}}_{\mathit{b}\mathit{r}}\mathbf{\times }\mathbf{1}\mathbf{]}}$$

(20)

where N_nd is the number of nodes in the GEMN, N_br is the number of branches in the GEMN, ${\mathit{A}}_{[({\mathit{N}}_{\mathit{n}\mathit{d}}-1)\times {\mathit{N}}_{\mathit{b}\mathit{r}}]}$ is derived from the incidence matrix ${\mathit{A}}_{[{\mathit{N}}_{\mathit{n}\mathit{d}}\times {\mathit{N}}_{\mathit{b}\mathit{r}}]}$ of the GEMN by reducing order, ${\mathit{G}}_{\mathit{m}[{\mathit{N}}_{\mathit{b}\mathit{r}}\times {\mathit{N}}_{\mathit{b}\mathit{r}}]}$ is the matrix of magnetic permeance, ${\mathit{U}}_{\mathit{n}[({\mathit{N}}_{\mathit{n}\mathit{d}}-1)\times 1]}$ is the matrix of node magnetic potentials, and ${\mathit{U}}_{\mathit{s}[{\mathit{N}}_{\mathit{b}\mathit{r}}\times 1]}$ is the matrix of branch magnetic potential sources.

Based on Equation (20), the matrix of node magnetic potential can be expressed as

$${\mathit{U}}_{\mathit{n}\mathbf{[}\mathbf{(}{\mathit{N}}_{\mathit{n}\mathit{d}}\mathbf{-}\mathbf{1}\mathbf{)}\mathbf{\times }\mathbf{1}\mathbf{]}}\mathbf{=}\mathbf{-}{\mathbf{(}{\mathit{A}}_{\mathbf{[}\mathbf{(}{\mathit{N}}_{\mathit{n}\mathit{d}}\mathbf{-}\mathbf{1}\mathbf{)}\mathbf{\times }{\mathit{N}}_{\mathit{b}\mathit{r}}\mathbf{]}}{\mathit{G}}_{\mathit{m}\mathbf{[}{\mathit{N}}_{\mathit{b}\mathit{r}}\mathbf{\times }{\mathit{N}}_{\mathit{b}\mathit{r}}\mathbf{]}}{\mathit{A}}_{\mathbf{[}{\mathit{N}}_{\mathit{b}\mathit{r}}\mathbf{\times }\mathbf{(}{\mathit{N}}_{\mathit{n}\mathit{d}}\mathbf{-}\mathbf{1}\mathbf{)}\mathbf{]}}^{\mathit{T}}\mathbf{)}}^{\mathbf{-}\mathbf{1}}{\mathit{A}}_{\mathbf{[}\mathbf{(}{\mathit{N}}_{\mathit{n}\mathit{d}}\mathbf{-}\mathbf{1}\mathbf{)}\mathbf{\times }{\mathit{N}}_{\mathit{b}\mathit{r}}\mathbf{]}}{\mathit{G}}_{\mathit{m}\mathbf{[}{\mathit{N}}_{\mathit{b}\mathit{r}}\mathbf{\times }{\mathit{N}}_{\mathit{b}\mathit{r}}\mathbf{]}}{\mathit{U}}_{\mathit{s}\mathbf{[}{\mathit{N}}_{\mathit{b}\mathit{r}}\mathbf{\times }\mathbf{1}\mathbf{]}}$$

(21)

According to the topology of the GEMN, we can obtain the matrix of branch magnetic potential sources as follows:

$${\mathit{U}}_{\mathbf{[}{\mathit{N}}_{\mathit{b}\mathit{r}}\mathbf{\times }\mathbf{1}\mathbf{]}}\mathbf{=}{\mathit{A}}_{\mathbf{[}{\mathit{N}}_{\mathit{b}\mathit{r}}\mathbf{\times }\mathbf{(}{\mathit{N}}_{\mathit{n}\mathit{d}}\mathbf{-}\mathbf{1}\mathbf{)}\mathbf{]}}^{\mathit{T}}{\mathit{U}}_{\mathit{n}\mathbf{[}\mathbf{(}{\mathit{N}}_{\mathit{n}\mathit{d}}\mathbf{-}\mathbf{1}\mathbf{)}\mathbf{\times }\mathbf{1}\mathbf{]}}$$

(22)

Based on Equation (22), we can obtain the matrix of the branch magnetic flux as follows:

$${\mathit{I}}_{\mathbf{[}{\mathit{N}}_{\mathit{b}\mathit{r}}\mathbf{\times }\mathbf{1}\mathbf{]}}\mathbf{=}{\mathit{G}}_{\mathit{m}\mathbf{[}{\mathit{N}}_{\mathit{b}\mathit{r}}\mathbf{\times }{\mathit{N}}_{\mathit{b}\mathit{r}}\mathbf{]}}\mathbf{(}{\mathit{U}}_{\mathbf{[}{\mathit{N}}_{\mathit{b}\mathit{r}}\mathbf{\times }\mathbf{1}\mathbf{]}}\mathbf{+}{\mathit{U}}_{\mathit{s}\mathbf{[}{\mathit{N}}_{\mathit{b}\mathit{r}}\mathbf{\times }\mathbf{1}\mathbf{]}}\mathbf{)}$$

(23)

The radial and tangential magnetic flux of each unit in the GEMN can be calculated by the above method, and the radial and tangential magnetic flux density can be further obtained by combining the size parameters of the GEMN units.

The aforementioned method can calculate two radial branch magnetic fluxes and two tangential branch magnetic fluxes for each magnetic network unit. The magnetic flux characteristics of each magnetic network unit are characterized by the average values of its radial branch magnetic flux and tangential magnetic flux. Furthermore, combined with the dimensional parameters of the magnetic network units, both radial and tangential magnetic flux densities can be obtained for each unit.

Stator yoke, stator tooth, and rotor yoke all utilize nonlinear ferromagnetic materials, and therefore, iteration of the permeability of ferromagnetic materials must be introduced in the process of solving the 2D GEMN model. Let B_r1q and B_t1q represent the radial and tangential magnetic flux density of any magnetic network unit in the nonlinear ferromagnetic region obtained from the q-th iteration calculation of the 2D GEMN model, and the total magnetic flux density of the magnetic network unit is

$$B_{z1q}=\sqrt{B_{r1q}^2+B_{t1q}^2}$$

(24)

Based on the nonlinear B-H curve of ferromagnetic materials, we can calculate the permeability corresponding to B_r1z by interpolate:

$${\mu }_{fe0}^q=f_{BH}(B_{z1q})$$

(25)

If $\left|{\mu }_{fe0}^q-{\mu }_{fe}^{q-1}\right|/\left|{\mu }_{fe}^{q-1}\right|>\epsilon$, we update the permeability according to the following formula and then proceed to the (q + 1)-th iteration calculation:

$${\mu }_{fe}^q=k_p{\mu }_{fe0}^q+(1-k_p){\mu }_{fe}^{q-1}$$

(26)

where ε is the allowable error limit, and k_p is the iteration coefficient, satisfying k_p ∈ (0, 1).

If $\left|{\mu }_{fe0}^q-{\mu }_{fe}^{q-1}\right|/\left|{\mu }_{fe}^{q-1}\right|<\epsilon$, it indicates that the calculation results of the 2D GEMN model meet the requirements, and the calculation is complete.

The modeling and computational process of the 2D GEMN model proposed in this paper primarily includes the following stages: establishment of the GEMN topology, calculation of magnetic potential and magnetic permeance parameters in the GEMN, solution of the GEMN, iteration of nonlinear ferromagnetic material permeability, and post-processing of magnetic field data. The specific workflow is illustrated in Figure 7.

3. Results and Discussion

The ASDM and FEM are mature analytical methods for evaluating the electromagnetic performance of SPEMs. For SPEMs with large equivalent electromagnetic air gaps, the ASDM demonstrates satisfactory analytical accuracy, while the FEM’s superior precision has become an industry consensus. To verify the rationality and accuracy of the 2D GEMN model, this section conducts comparative analyses of electromagnetic performance at no-load and load conditions for an SPEM prototype (hereafter referred to as “prototype M”) using three approaches: the 2D GEMN model, the ASDM, and the FEM. The finite element model of prototype M is shown in Figure 8, with key parameters listed in

Table 1. The main parameters of prototype M.

Parameter		Parameter
Pole pair number	3	Stator slot number	36
Winding phase number	6	Electrical fundamental frequency	1000 Hz
Outer stator radius	270 mm	Inner stator radius	154 mm
Stator tooth body height	41.05 mm	Stator tooth body width	7.1 mm
Stator tooth tip height	3 mm	Stator tooth tip width	7.1 mm
Stator yoke thickness	13.95 mm	Axial length	180 mm
Outer rotor radius	114 mm	Inner rotor radius	98 mm
Permanent magnet thickness	14.5 mm	Rotor yoke thickness	8 mm
Winding layer number	2	Turns in one layer of windings	6

. Prototype M employs double-layer full-pitch windings and Halbach-array surface-mounted poles, and the geometric configuration and magnetization angles of Halbach poles are illustrated in Figure 9. The phase currents of stator windings at load condition are specified in

Table 2. The phase currents of windings at load condition.

Frequency [Hz]	Amplitude [A]	Initial Phase Angle [°]
1000	420.6	−6.6
10,500	14.0	97.4
14,500	11.2	59.8

.

The nonlinear ferromagnetic material used for the stator core of prototype M is USW35085, and its BH curve and iron loss curve are shown in Figure 10. The nonlinear ferromagnetic material used for the rotor core of prototype M is 25Cr2Ni4Mov, and its BH curve is shown in Figure 11. The PM material used in prototype M is SmCo30H, and its magnetic properties are shown in Figure 12, in which the red dots are the knee points of demagnetization curves at different temperatures.

The calculation configuration of the 2D GEMN model for electromagnetic performance analysis of prototype M is detailed in

Table 3. The calculation configuration of the 2D GEMN model.

Parameter		Parameter		Parameter
Nr1	1	Nr2	3	Nr3	1
Nr4	3	Nr5	3	Nr6	1
Nt1	3	Nt2	0	Nt3	3
ε	0.025	kp	0.15	Temperature of PMs	120 °C

, while the geometric configurations of magnetic network units in different regions are specified in

Table 4. The geometric configurations of magnetic network units in different regions.

Region	Geometric Configuration	Region	Geometric Configuration
Stator yoke	Sector	Stator tooth body	Trapezoid
Stator tooth tip	Trapezoid	Stator slot body	Rectangle
Stator slot opening	Rectangle	Air gap	Sector
PM	Sector	Rotor yoke	Sector

.

The ASDM is referenced from the literature [17,18,19], and its detailed theoretical framework is not elaborated here. The primary computational parameter configurations for the ASDM are listed in

Table 5. The calculation configuration of the ASDM.

Parameter		Parameter		Parameter
Nsd1	100	Nsd2	100	Nsd3	100

, where N_sd1, N_sd2, and N_sd3 represent the maximum spatial harmonic order for the PM and air gap subdomain, stator slot body subdomain, and stator slot opening subdomain, respectively.

3.1. The Distribution of Magnetic Field

The spatial distributions of magnetic flux density in the air gap and PM region of prototype M at no-load and load conditions are calculated using the GEMN, ASDM, and FEM, as shown in Figure 13 and Figure 14. Using the FEM results as the benchmark, the computational errors of the spatial fundamental harmonic for GEMN and ASDM are summarized in

Table 6. The amplitude of the spatial fundamental harmonic magnetic flux density in the air gap and PM regions and the associated computational errors.

			GEMN		ASDM		FEM
			Amplitude	Error	Amplitude	Error	Amplitude
Air gap	No-load	Radial	0.8670 T	0.70%	0.8814 T	2.37%	0.8610 T
		Tangential	0.1278 T	8.31%	0.1182 T	0.17%	0.1180 T
	Load	Radial	0.8458 T	1.11%	0.8508 T	1.71%	0.8365 T
		Tangential	0.1987 T	2.21%	0.2009 T	3.34%	0.1944 T
PM	No-load	Radial	0.7181 T	0.50%	0.7329 T	2.58%	0.7145 T
		Tangential	0.9253 T	0.88%	0.9363 T	2.08%	0.9172 T
	Load	Radial	0.7006 T	1.79%	0.7070 T	2.72%	0.6883 T
		Tangential	0.9127 T	1.20%	0.9201 T	2.02%	0.9019 T

. The results demonstrate excellent agreement in both magnetic flux density waveforms and primary spatial harmonics across all three methods. The amplitude of fundamental spatial harmonic calculated by the GEMN and ASDM is slightly higher than that obtained from the FEM, which stems from the GEMN’s relatively sparse mesh discretization due to computational efficiency constraint and ASDM’s inability to account for a nonlinear core saturation effect.

The ASDM, which does not consider core saturation as a prerequisite for analysis, lacks direct computational capability for magnetic flux density distribution in stator tooth, slot, and yoke regions. Therefore, the GEMN and FEM are employed to analyze the time-varying magnetic flux density at sampling points: Point A in the stator teeth, Point B in the stator slot, and Point C in the stator yoke, as shown in Figure 15, Figure 16 and Figure 17. Using FEM results as the benchmark, the calculation errors of the time fundamental harmonic magnetic flux density for GEMN are summarized in

Table 7. The amplitude of the time fundamental harmonic magnetic flux density in the stator tooth, slot, and yoke regions and the associated computational errors.

			GEMN		FEM
			Amplitude	Error	Amplitude
Point A (Stator tooth)	No-load	Radial	1.0567 T	1.84%	1.0376 T
		Tangential	0.0309 T	94.34%	0.0159 T
	Load	Radial	1.0765 T	3.34%	1.0417 T
		Tangential	0.1396 T	3.18%	0.1353 T
Point B (Stator slot)	No-load	Radial	0.0020 T	1076.47%	0.0002 T
		Tangential	0.0053 T	−2.22%	0.0054 T
	Load	Radial	0.0059 T	1645.56%	0.0003 T
		Tangential	0.1377 T	−7.02%	0.1481 T
Point C (Stator yoke)	No-load	Radial	0.1043 T	−22.63%	0.1348 T
		Tangential	1.4900 T	0.56%	1.4817 T
	Load	Radial	0.1007 T	−23.83%	0.1322 T
		Tangential	1.5292 T	−0.50%	1.5369 T

. The results demonstrate that for primary magnetic flux density components in stator regions—including radial flux density in stator teeth, tangential flux density in stator slots, and tangential flux density in the stator yoke—the GEMN and FEM exhibit strong consistency in waveforms and primary time harmonic. However, larger deviations occur in non-primary components such as tangential flux density in stator teeth, radial flux density in stator slots, and radial flux density in the stator yoke. These non-primary components generally exhibit low amplitudes, rendering their computational errors negligible for electromagnetic performance analysis of SPEMs. The primary error sources include the following: (1) relatively sparse mesh discretization in the GEMN model causing inaccuracies in tangential flux density calculations for stator teeth and radial flux density in the yoke; (2) discrepancies in slot field modeling due to the GEMN model’s equivalent loading of stator winding magnetomotive force on stator teeth versus the actual placement of stator windings within the stator slots.

The comparative analysis of magnetic flux density calculations using the GEMN, ASDM, and FEM demonstrates that the 2D GEMN model can accurately calculate the spatiotemporal magnetic field distribution in SPEMs, including stator slot leakage flux effect, which lays the foundation for conducting comprehensive electromagnetic performance analysis of SPEMs based on the 2D GEMN model.

3.2. Flux Linkage and Back-EMF of Stator Windings

In the 2D GEMN model, using the A1-phase winding as an example, the formula for calculating the winding flux linkage is expressed as follows:

$$\psi (t)=N_s{\displaystyle \sum _{h=1}^{N_w}[{\psi }_{th}(t)+{\psi }_{sh}(t)]-{\psi }_{sl}(t)}$$

(27)

where Φ_th is the radial magnetic flux at the base of the h-th stator tooth within the distribution range of a single branch of the A1-phase winding, Ψ_sh is the radial magnetic flux at the base of the h-th stator slot within the same branch distribution range, Ψ_sl is the tangential leakage flux in the slot containing the A1-phase winding coil, and N_w is the number of stator teeth within the distribution range of one branch of A1-phase winding.

Based on Equation (27), the back electromotive force (back-EMF) of one phase winding can be further calculated by the following equation:

$$E(t)=-{\displaystyle \frac{\mathrm{d}\psi (t)}{\mathrm{d}t}}$$

(28)

The calculation results of winding flux linkage and back-EMF using the GEMN, ASDM, and FEM are shown in Figure 18 and Figure 19. With FEM results as the benchmark, the computational errors of fundamental magnetic flux and fundamental back-EMF for the GEMN and ASDM are summarized in

Table 8. The amplitude of the fundamental component for magnetic flux, back-EMF, and the associated computational errors.

		GEMN		ASDM		FEM
		Amplitude	Error	Amplitude	Error	Amplitude
Flux linkage	No-load	0.0903 Wb	0.89%	0.0925 Wb	3.35%	0.0895 Wb
	Load	0.0884 Wb	−1.65%	0.0920 Wb	2.30%	0.0899 Wb
Back-EMF	No-load	569.53 V	0.47%	586.56 V	3.48%	566.85 V
	Load	559.44 V	−1.05%	584.98 V	3.47%	565.36 V

. The results demonstrate that, whether in winding flux linkage, back electromotive force waveforms, or fundamental component amplitudes, the GEMN exhibits strong consistency with the ASDM and FEM. Notably, the computational accuracy of the GEMN surpasses that of ASDM, a benefit derived from GEMN model’s ability to account for core saturation and stator slot leakage flux.

3.3. Electromagnetic Torque

In Section 3.1, the spatiotemporal distribution of magnetic flux density in the air gap region has been obtained by the 2D GEMN model, and electromagnetic torque can be further calculated by Maxwell tensor method as follows:

$$T_m(t)=l{r}_g^2{\displaystyle {\int }_0^{2\pi }\frac{B_r(r_g,\theta ,t)B_t(r_g,\theta ,t)}{{\mu }_0}}\mathrm{d}\theta$$

(29)

where r_g is the radius of one point in the air gap region.

The electromagnetic torque calculation results of the GEMN, ASDM, and FEM are shown in Figure 20, with the computational errors of the main torque components illustrated in

Table 9. The amplitude of the main torque components and the associated computational errors.

		GEMN		ASDM		FEM
		Amplitude	Error	Amplitude	Error	Amplitude
No-load	12f1	0.1479 Nm	−1.33%	0.0005 Nm	−99.70%	0.1499 Nm
Load	0f1	324.24 Nm	−1.55%	341.07 Nm	3.56%	329.33 Nm

. The results demonstrate that, whether at no-load or load conditions, GEMN achieves superior computational accuracy compared to ASDM, thereby effectively supporting the evaluation of electromagnetic torque performance in SPEMs. In contrast, ASDM exhibits relatively larger errors in no-load torque calculations, primarily due to its inherent limitation in accounting for stator core saturation effects.

3.4. Stator Core Loss

As derived in Section 3.1, the spatiotemporal magnetic flux density distribution in the stator core region can be obtained through the 2D GEMN model, and the time-harmonic magnetic flux density components of each magnetic network unit in the stator core region can be further decomposed via Fourier analysis.

This section employs an improved Bertotti model considering time-harmonic magnetic flux density and rotational magnetization effect for the calculation of stator iron loss, with the improved Bertotti model expressed as follows [38,39]:

$$\mathsf{\Delta }{P}_k={\displaystyle \sum _u[K_h{f}_u(B_{kur}^2+B_{kut}^2)+K_c{f}_u^2(B_{kur}^2+B_{kut}^2)+K_e{f}_u^{1.5}(B_{kur}^{1.5}+B_{kut}^{1.5})]}$$

(30)

where △P_k is the iron loss density of the k-th magnetic network unit in the stator core region, K_h is the hysteresis loss coefficient, K_c is the eddy current loss coefficient, K_e is the additional loss coefficient, f_u is the frequency of the u-th time harmonic magnetic density, and B_kur, B_kut are the radial and tangential components of the u-th time harmonic magnetic flux density of the k-th magnetic network unit, respectively.

Stator iron loss can be calculated based on the following formula:

$$P_{fe}={\rho }_{fe}{\displaystyle \sum _{k=1}^{N_{core}}\mathsf{\Delta }{P}_k{S}_{k}l{k}_{fe1}}$$

(31)

where ρ_fe is the material density of stator core, S_k is the radial cross-sectional area of the k-th magnetic network unit in the stator core region, and k_fe1 is the stacking factor of the stator core, N_core is the total number of magnetic network units in the stator core region.

The calculation results of stator core loss at no-load and load conditions using the GEMN and FEM are shown in Figure 21. With FEM results as the benchmark, the computational errors of the stator core loss for the GEMN are summarized in

Table 10. The calculation results of stator core loss and the associated computational errors.

	GEMN		FEM
	Value	Error	Value
No-load	1918.8	−0.49%	1928.2
Load	2046.2	−2.18%	2091.8

. In addition, the no-load stator core losses at different rotational speeds are calculated by means of the GEMN and FEM, as shown in Figure 22. The above results indicate that the calculations from both methods are fundamentally consistent, demonstrating that the 2D GEMN model possesses the analysis capabilities of stator core loss for SPEMs comparable to FEM.

3.5. Demagnetization of Permanent Magnets

The analysis of permanent magnet demagnetization is enabled by obtaining the spatiotemporal magnetic flux density distribution in the PM region through the 2D GEMN model, as derived in Section 3.1. The demagnetization criterion is defined as follows: when the magnetic flux density along the magnetization direction in any magnetic network unit of the PM region falls below the knee point of the demagnetization curve, that specific magnetic network unit undergoes irreversible demagnetization.

In this article, the demagnetization coefficient is introduced to characterize the demagnetization degree of permanent magnets. The definition of demagnetization coefficient is as follows:

$$K_{dm}={\displaystyle \sum _{w=1}^{N_{dm}}\frac{B_{dmw}}{B_k}\frac{S_w}{S_{PM}}}$$

(32)

where N_dm is the total number of demagnetized magnetic network units in the permanent magnet region, and B_dmw is the magnetic flux density along the magnetization direction in the w-th demagnetized magnetic network unit, B_k is the magnetic flux density at the knee point of the demagnetization curve in the operating temperature, S_w is the cross-sectional area of the w-th demagnetized magnetic network unit, and S_PM is the total cross-sectional area of the permanent magnet region.

In order to reflect the analysis ability of the 2D GEMN model for permanent magnet demagnetization of SPEMs, we only consider demagnetization current at the d-axis, and define the demagnetization current coefficient according to the following formula:

$$F_{dm}={\displaystyle \frac{I_{dm}}{I_1}}$$

(33)

where I_dm is the amplitude of demagnetization phase current, and I₁ is the amplitude of phase current at rated operating condition.

Based on the 2D GEMN model, the demagnetization in permanent magnet region at different F_dm was analyzed, as shown in Figure 23. We can obtain that when F_dm > 2.75, the demagnetization occurs, and as the demagnetization current coefficient increases, the demagnetization coefficient and the number of demagnetized magnetic network units increases.

3.6. Application Characteristic

The main configuration of the computer used in the research of this article is shown in

Table 11. The main configuration of the computer used in this article.

Parameter
CPU	i9-13900H
RAM	96 GB (DDR5 5600 MHz)
Hard disk	1024 GB (Solid State)

.

A comparative analysis of the GEMN, ASDM and FEM at load condition for prototype M is summarized in Table 12, which reveals the following:

• Compared to the FEM, the GEMN requires fewer sub-elements and computational resources while maintaining advantages in calculation time.
• Compared to the ASDM, the GEMN requires comparable computational resources but longer calculation time, yet achieves higher accuracy. Its elimination of complex analytical formula derivations makes it more intuitive in physical interpretation, and its regular arrangement of magnetic network units facilitates programming implementation. Consequently, the GEMN demonstrates advantages in both model complexity and usability over the ASDM, proving more suitable for efficient electromagnetic performance analysis of SPEMs.

In the aforementioned analysis process, the magnetic field distributions of SPEMs at different time steps were calculated using a serial computation mode. If grouped parallel computation is applied to the magnetic field distributions at multiple time steps, the computational efficiency of the GEMN would be significantly improved, which constitutes a future research direction.

Table 12. Comparison of characteristics for different methods.

	ASDM	GEMN	FEM
Spatial calculation range	a unit part of prototype M
Temporal calculation range	0~1/f1
Sampling rate of time	100f1
Number of units	——	864 (12 × 72)	14,962
Computation time	122 s	162 s	186 s
CPU utilization	24%	23%	69%
Memory occupied	3400 MB	3100 MB	5700 MB
Error of back-EMF	3.47%	−1.05%	——
Error of electromagnetic torque	3.56%	−1.55%	——

Table 12. Comparison of characteristics for different methods.

	ASDM	GEMN	FEM
Spatial calculation range	a unit part of prototype M
Temporal calculation range	0~1/f1
Sampling rate of time	100f1
Number of units	——	864 (12 × 72)	14,962
Computation time	122 s	162 s	186 s
CPU utilization	24%	23%	69%
Memory occupied	3400 MB	3100 MB	5700 MB
Error of back-EMF	3.47%	−1.05%	——
Error of electromagnetic torque	3.56%	−1.55%	——

4. Conclusions

This paper proposes a 2D GEMN model for SPEMs, which enables regular arrangement of magnetic network units across different regions to simplify the magnetic network topology. The topology of the GEMN remains unchanged during rotor movement by updating the magnetic potential parameters of magnetic network units in the PM region to reflect rotor rotation effects, and adapts to various surface-mounted PM pole structures, which effectively enhances the generality of the 2D GEMN model.

The 2D GEMN model can effectively analyze electromagnetic performance of SPEMs including spatiotemporal magnetic flux density distribution within the leakage flux of stator slot, winding electromotive force, electromagnetic torque, stator iron loss, and PM demagnetization in the influences of magnetic saturation, stator slotting, and current harmonic.

Comparative analyses with ASDM and FEM validate the 2D GEMN model’s rationality and effectiveness, demonstrating its balanced capability in computational accuracy and speed. This approach can provide support for efficient design, optimization, and performance evaluation of SPEMs.