Analytical Modeling and GA-Based Optimization of Multi-Layered Segmented SPM Magnets

Abstract

Presented here is a 2-D analytical model for predicting the magnetic field distribution in a surface-mounted permanent magnet (SPM) rotor with multi-layered segmented permanent magnets (PMs). Each layer is treated independently, enabling the linear superposition of magnetic fields across all layers. The model employs subdomain modeling combined with the separation of variables, with the magnetic vector potential expressed as a Fourier series to derive the airgap magnetic field. The formulation is generalizable to five regions in each layer: outer airgap, optional outer inactive magnetic layer, active magnetic layer(s), optional inner inactive magnetic layer, and inner airgap. Validation against finite element analysis (FEA) shows a prediction difference of around 0.5% in airgap flux density. The model’s design utility is demonstrated through a genetic algorithm (GA) optimization, which maximizes static flux linkage and confirms performance improvements from the multi-layered configuration.

1. Introduction

PM motors are synchronous machines in which the rotor magnetic field is produced by permanent magnets, eliminating rotor windings and external excitation. This distinction offers several advantages over conventional induction motors. By removing rotor windings, PM motors eliminate rotor copper losses, leading to improved efficiency, lower thermal stress, and simpler rotor construction [1,2,3,4].

The arrangement of the rotor magnets, whether it is embedded interior permanent magnet (IPM) or surface-mounted (SPM), determines the magnetic field distribution, torque, and overall electromagnetic performance [5,6,7,8]. Among these, SPM motors are especially attractive for applications requiring well-defined airgap fields and mechanical simplicity. In such designs, the magnets are placed directly on the rotor surface, enabling direct interaction with the stator field across a narrow radial air gap. Unlike IPM motors, where the magnets are embedded within the rotor core, SPM designs rely on surface magnetization, which produces an airgap field that directly governs the motor’s torque generation, induced voltage, and harmonic behavior [9,10,11]. To improve flux distribution using the same magnet volume, SPM motors can adopt Halbach arrays (HA) or segmented PM structures. An HA is an arrangement in which the magnetization direction rotates spatially along the array [12,13]. In cylindrical geometries such as rotating motors, the HA is implemented in segmented form, each segment having a distinct magnetization angle [12,13,14,15]. This produces an asymmetric magnetic field, reinforced on the stator side and weakened toward the rotor yoke, enhancing flux utilization and reducing leakage [16,17,18,19,20]. This cylindrical adaptation of segmented PMs has been widely studied in rotor applications for SPM machines. The configuration allows designers to shape the magnetic field through controlled segmentation and magnetization patterns, which can be applied later in system-level optimization of electric machines. Several studies have explored optimizing segmented PM configurations to enhance motor performance. These efforts have focused on adjusting magnetization angles, the number of segments per pole, and geometric proportions to achieve non-uniform or tailored configurations. For example, one of the studies identified the optimal magnet ratio for torque maximization in air and iron-cored motors with Halbach structures [21]. Alternative orientation schemes were also proposed to improve airgap flux density, supported by analytical models for arbitrary magnetized angles [22]. Other works introduced unconventional geometries and magnetization topologies, such as trapezoidal or interval Halbach segments and polar anisotropic magnetization, to improve torque ripple and torque-to-weight ratio. Through engineering segmented PM configurations, these studies achieved improvements in flux density, co-energy, torque, back-EMF, power, and efficiency while minimizing harmonics, cogging torque, and saturation [23,24]. One study further explored torque ripple mitigation by analyzing both spatial and temporal harmonic fields in PM Vernier machines [25], illustrating how targeted harmonic shaping can enhance torque quality in advanced PM designs. However, most implementations rely on single-layer PMs with limited geometry and magnetization, often optimizing one parameter at a time while holding others constant. As a result, the available design freedom remains underused, limiting the ability to shape the magnetic field in more advanced or application-specific designs, a limitation directly addressed in this work.

Notably, a previous analytical study on dual-layer HA attempted to extend the single-layer concept by stacking two magnetized layers. The results showed that by carefully tuning the segment proportions and the relative orientation between layers, the airgap flux density, electromagnetic torque, and back-EMF could all be enhanced compared to the conventional single-layer design [26]. Building on this principle, the present study explores how these benefits can be further enhanced by extending the concept from a dual-layer to a multi-layer structure. To this end, a generalized rotor topology is proposed, consisting of multiple concentric layers of segmented permanent magnets, each with its own number of segments per pole, magnetization ratio, and radial position. This multi-layer segmented PM configuration offers greater flexibility in shaping the airgap field by allowing independent control of each layer’s magnetization characteristics.

The main objective of this work is to develop an analytical model that predicts the magnetic field distribution of this multi-layer configuration without relying entirely on computationally intensive finite-element simulations. The model is formulated in two dimensions, under the assumptions of radial symmetry and steady-state magnetostatics. A subdomain modeling approach is used to represent the rotor and its surrounding regions. The cross-section of the machine is divided into analytically defined subdomains, including individual magnet layers, the airgap, and the stator iron. Within each region, the governing magnetostatic equation, expressed in polar coordinates as Poisson’s or Laplace’s equation, is solved assuming linear material properties. The solution in each region is expressed as a Fourier series, and the unknown coefficients are determined by enforcing boundary conditions at the interfaces, ensuring continuity of the normal component of the flux density and the tangential component of the magnetic field. Each magnetized layer is modeled as a distinct subdomain, and the magnetic vector potential is derived using the separation of variables method, expressed as a Fourier series expansion. Finally, the superposition principle is applied to combine the contributions from all layers [16,21,27].

The validation of the analytical model is performed by comparing its predictions with two-dimensional finite-element analysis (FEA). The FEA simulations serve as a reference for assessing the accuracy of the analytically predicted field distributions, following the methodology of earlier works [16,21,22,27,28,29,30]. The comparison focuses on the model’s ability to reproduce the field magnitude, shape, and harmonic content across various configurations and magnetization ratios.

To highlight the model’s practical relevance, one representative application considered is the maximization of static flux linkage. This quantity reflects how effectively the rotor magnet configuration couples with the stator windings and, consequently, influences torque production and back-EMF. By studying how flux linkage varies with different magnet arrangements, the advantages of the proposed multi-layer design can be objectively assessed. The analytical model is integrated with a genetic algorithm (GA) to perform this optimization [27,29,31,32]. The GA systematically explores design parameters such as the number of layers, segment proportions, and magnetization ratios, using the analytical model as a fast evaluation tool. Through successive iterations, it identifies configurations that yield higher flux-linkage values while satisfying rotor geometry constraints. This hybrid approach allows wide design-space exploration while keeping computation time low, significantly reducing reliance on FEA during early-stage design.

The remainder of the manuscript is organized as follows. Section 2 introduces the mathematical framework used to model single-layer and multi-layer segmented PMs, along with the formulation employed to estimate flux linkage in surface-mounted configurations. Section 3 presents a validation study comparing analytical predictions with finite-element simulations across multiple test cases, assessing both field distribution and flux-linkage accuracy. Section 4 applies the model to a flux-linkage maximization task, where a GA explores different multi-layer configurations to evaluate the performance potential of this magnetic structure.

2. Materials and Methods

The mathematical foundation of the magnetic field distribution in a segmented magnet is first established, serving as the baseline model. Building on this case, the concept of multi-layered segmented PMs is derived by examining how stacking multiple layers with varying magnetization patterns can engineer the airgap magnetic flux field and optimize the performance or efficiency of the SPM machines based on design constraints and targets. Instead of treating the structure as a tightly coupled system, a modular approach is adopted where each layer is modeled independently. This assumption enables efficient analysis and flexible configuration of each layer, offering a balance between computational simplicity and modeling accuracy. In addition, to maintain generality and tractability for all the layers in the multi-layered rotor, each layer is defined by five subdomains: an outer airgap, a potential outer inactive magnetic layer, active magnet layer(s), a potential inner inactive magnetic layer, and an inner airgap. No stator geometry or flux distortion due to slot effect is included in the analysis; the model is restricted to the rotor and the magnetic field it generates in the airgap. Moreover, certain parameters are kept invariant. Nonetheless, the framework is fully generalizable and can accommodate variable geometric parameters or design priorities, depending on the specific requirements of the target application.

2.1. Fundamentals of Single-Layer Segmented PM

As a foundation for this study, the classical Halbach structure is considered. For modeling purposes, the HA or single-layer segmented PM is divided into three concentric regions, as shown in Figure 1. Region I represents the airgap separating the rotor from the stator, namely, an outer airgap. This region is the primary zone of electromagnetic interaction between the rotor and the stator windings, and the magnetic field distribution within it is central to performance evaluation. Region II corresponds to the magnetized layer of the rotor, where the permanent magnets are arranged with spatially rotating magnetization vectors represented by the different arrows in Figure 1. Region III in this model represents a narrow airgap separating the rotor from the internal iron core, namely an inner airgap. At the inner boundary of this region lie the iron core and the shaft, which provide mechanical support and the return path for magnetic flux. In some analytical studies, this additional airgap is introduced to simplify boundary conditions by avoiding direct flux continuity between the rotor and the core [21].

To formulate the magnetic field distribution within the segmented PM configuration, represented in Figure 1, several simplifying but physically justified assumptions are made. These assumptions help isolate the key behavior of interest while keeping the analytical model tractable:

• Magnetic Core Regions (Yoke and Stator): The structure is bounded radially by two ferromagnetic cores

For $r<R_{in}$, a rotor yoke made of soft magnetic material (e.g., iron).

For $r>R_{out}$, a stator body, also modeled as an iron core (No slot effect considered).

In both cases, the relative permeability of the iron core ${\mu }_r\gg 1$, and in the proposed analytical modeling, it was assumed to be infinite for simplicity. This implies that the tangential component of the magnetic field $H_{\alpha }$ vanishes at these boundaries, enforcing Dirichlet or Neumann-type boundary conditions depending on the formulation.

• Slotless Stator Assumption: The stator is modeled as slotless, allowing us to neglect slotting effects and assume a smooth, continuous magnetic boundary at the outer airgap. This avoids field distortions that would arise from stator teeth and simplifies the field distribution in the airgap (Region I).
• Two-Dimensional (2D) Approximation: The problem is treated as a 2D planar model in cylindrical coordinates (r,α) under the assumption of translational invariance along the axial direction z. This is valid for long machines or magnet segments with negligible axial variation.
• No Current-Carrying Conductors: The model assumes the absence of current sources inside the magnet or airgap regions. The magnetic field is purely generated by the remanent magnetization of the segmented PM. This justifies the use of scalar magnetic vector potential $A_z$ and simplifies the Maxwell equations into Laplace and Poisson equations.
• Segment Width Constraints within Each Pole: Each magnetic pole is divided into multiple magnet segments with equal angular width, except for the radial segments, which have a predefined size determined by the variable magnet ratio $Rmp$. The remaining angular space within the pole is distributed equally among the non-radial segments. This structured segmentation enables a consistent mathematical representation for any controllable number of segments per pole.
• No demagnetization or any changes in material properties are considered.

Based on the assumptions above, the magnetic vector potential $A_z$ is governed by the following differential equations in each region [16,21].

(1)

Region I and III (Laplace Equation):

$${\displaystyle \frac{{\partial }^2{A}_{zI,III}}{\partial r^2}}+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial A_{zI,III}}{\partial r}}+{\displaystyle \frac{1}{r^2}}{\displaystyle \frac{{\partial }^2{A}_{zI,III}}{\partial {\alpha }^2}}=0$$

(1)

where $A_{z,I,III}$ (Wb/m) is the magnetic vector potential in the outer and inner airgap.

(2)

Region II (Poisson Equations):

$${\displaystyle \frac{{\partial }^2{A}_{zII}}{\partial r^2}}+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial A_{zII}}{\partial r}}+{\displaystyle \frac{1}{r^2}}{\displaystyle \frac{{\partial }^2{A}_{zII}}{\partial {\alpha }^2}}={\displaystyle \frac{\nabla \overrightarrow{M}}{{\mu }_r}}={\displaystyle \frac{1}{{\mu }_r}}\left({\displaystyle \frac{M_r}{r}}+{\displaystyle \frac{\partial M_r}{\partial r}}+{\displaystyle \frac{\partial M_{\alpha }}{\partial \alpha }}\right)={\displaystyle \frac{1}{{\mu }_r}}\sum _{n=\mathrm{1,3},5\dots }^{\infty }{\displaystyle \frac{M_n}{r}}\cos \left(np\alpha \right)$$

(2)

where $A_{z,II}$ (Wb/m) is the magnetic vector potential in the segmented PM, ${\mu }_r$ is the relative permeability, $M_r$ (A/m) and $M_{\alpha }$ (A/m) are the magnetization of the segmented magnet in the radial and tangential directions.

The magnetic field can be deduced using the magnetic vector potential through the equation from [22].

$$\overrightarrow{H}[A/m]={\displaystyle \frac{\overrightarrow{B}}{\mu }}=-\overrightarrow{\nabla }{A}_z$$

(3)

Then the magnetizations in the radial and tangential directions are defined. A formulation that can express any number of segments per pole is illustrated in Figure 2:

Figure 2. Presentation in Cylindrical Coordinate System of the Segmented PM Structure from a Single-layer Configuration with Four Segments per Pole. Arrows with different colors were used to illustrate the magnetization directions of PM segments.

Figure 2. Presentation in Cylindrical Coordinate System of the Segmented PM Structure from a Single-layer Configuration with Four Segments per Pole. Arrows with different colors were used to illustrate the magnetization directions of PM segments.

$$M_r={\displaystyle \frac{1}{{\mu }_0}}·\left\{\begin{array}{c}{B}_r if -{\displaystyle \frac{{\beta }_r}{2}}\le \alpha \le {\displaystyle \frac{{\beta }_r}{2}}\\ B_r\cos \left[{\displaystyle \frac{\pi }{k}}.\left(i+1\right)\right] if {\displaystyle \frac{{\beta }_r}{2}}+{\displaystyle \frac{{\beta }_m-{\beta }_r}{k-1}}·i\le \alpha \le {\displaystyle \frac{{\beta }_r}{2}}+{\displaystyle \frac{{\beta }_m-{\beta }_r}{k-1}}·\left(i+1\right), i\in ⟦0,k-2⟧ \\ -B_r if {\beta }_m-{\displaystyle \frac{{\beta }_r}{2}}\le \alpha \le {\beta }_m+{\displaystyle \frac{{\beta }_r}{2}}\\ -B_r\cos \left[{\displaystyle \frac{\pi }{k}}.\left(i+1\right)\right] if {\beta }_m+{\displaystyle \frac{{\beta }_r}{2}}+{\displaystyle \frac{{\beta }_m-{\beta }_r}{k-1}}·i\le \alpha \le {\beta }_m+{\displaystyle \frac{{\beta }_r}{2}}+{\displaystyle \frac{{\beta }_m-{\beta }_r}{k-1}}·\left(i+1\right), i\in ⟦0,k-2⟧ \end{array}\right.$$

(4)

$$M_{\alpha }={\displaystyle \frac{1}{{\mu }_0}}·\left\{\begin{array}{c}0 if -{\displaystyle \frac{{\beta }_r}{2}}\le \alpha \le {\displaystyle \frac{{\beta }_r}{2}} or {\beta }_m-{\displaystyle \frac{{\beta }_r}{2}}\le \alpha \le {\beta }_m+{\displaystyle \frac{{\beta }_r}{2}}\\ -B_r\mathit{sin}\left[{\displaystyle \frac{\pi }{k}}.\left(i+1\right)\right] if {\displaystyle \frac{{\beta }_r}{2}}+{\displaystyle \frac{{\beta }_m-{\beta }_r}{k-1}}·i\le \alpha \le {\displaystyle \frac{{\beta }_r}{2}}+{\displaystyle \frac{{\beta }_m-{\beta }_r}{k-1}}·\left(i+1\right), i\in ⟦0,k-2⟧ \\ \\ B_r\mathit{sin}\left[{\displaystyle \frac{\pi }{k}}.\left(i+1\right)\right] if {\beta }_m+{\displaystyle \frac{{\beta }_r}{2}}+{\displaystyle \frac{{\beta }_m-{\beta }_r}{k-1}}·i\le \alpha \le {\beta }_m+{\displaystyle \frac{{\beta }_r}{2}}+{\displaystyle \frac{{\beta }_m-{\beta }_r}{k-1}}·\left(i+1\right), i\in ⟦0,k-2⟧ \end{array}\right.$$

(5)

Let ${\beta }_r$ (rad) be the angular width of the radially oriented segments and ${\beta }_m$ (rad) the total width of the pole, where ${\beta }_m={\displaystyle \frac{\pi }{p}}$. The magnet ratio is then defined as $R_{mp}={\displaystyle \frac{{\beta }_r}{{\beta }_m}}$.

Using the periodicity of the segmented PM array in the tangential direction, $M_r$ and $M_{\alpha }$ can be written as a Fourier Series:

$$M_r=\sum _{n=\mathrm{1,3},5}^{\mathrm{\infty }}{M}_{rn}\cos \left(np\alpha \right) and M_{\alpha }=\sum _{n=\mathrm{1,3},5}^{\mathrm{\infty }}{M}_{r\alpha }\sin \left(np\alpha \right)$$

(6)

and $M_{rn}$ and $M_{\alpha n}$ (A/m) are obtained by dividing the segmented PM into regions, as illustrated in Figure 2, and applying the formulations from

$$\left\{\begin{array}{c}{M}_{rn}={\displaystyle \frac{p}{\pi }}\underset{-{\displaystyle \frac{{\beta }_r}{2}}}{\overset{2{\beta }_m-{\displaystyle \frac{{\beta }_r}{2}}}{\int }}{M}_r\cos \left(np\alpha \right)d\alpha ={\displaystyle \frac{p}{\pi }}\left[\underset{-{\displaystyle \frac{{\beta }_r}{2}}}{\overset{{\displaystyle \frac{{\beta }_r}{2}}}{\int }}{M}_r\cos \left(np\alpha \right)d\alpha +\underset{{\displaystyle \frac{{\beta }_r}{2}}}{\overset{{\beta }_m-{\displaystyle \frac{{\beta }_r}{2}}}{\int }}{M}_r\cos \left(np\alpha \right)d\alpha +\underset{{\beta }_m-{\displaystyle \frac{{\beta }_r}{2}}}{\overset{{\beta }_m+{\displaystyle \frac{{\beta }_r}{2}}}{\int }}{M}_r\cos \left(np\alpha \right)d\alpha +\underset{{\beta }_m+{\displaystyle \frac{{\beta }_r}{2}}}{\overset{2{\beta }_m-{\displaystyle \frac{{\beta }_r}{2}}}{\int }}{M}_r\cos \left(np\alpha \right)d\alpha \right]\\ M_{\alpha n}={\displaystyle \frac{p}{\pi }}\underset{-{\displaystyle \frac{{\beta }_r}{2}}}{\overset{2{\beta }_m-{\displaystyle \frac{{\beta }_r}{2}}}{\int }}{M}_r\sin \left(np\alpha \right)d\alpha ={\displaystyle \frac{p}{\pi }}\left[\underset{-{\displaystyle \frac{{\beta }_r}{2}}}{\overset{ {\displaystyle \frac{{\beta }_r}{2}}}{\int }}{M}_{\alpha }\sin \left(np\alpha \right)d\alpha +\underset{{\displaystyle \frac{{\beta }_r}{2}}}{\overset{{\beta }_m-{\displaystyle \frac{{\beta }_r}{2}}}{\int }}{M}_{\alpha }\sin \left(np\alpha \right)d\alpha +\underset{{\beta }_m-{\displaystyle \frac{{\beta }_r}{2}}}{\overset{{\beta }_m+{\displaystyle \frac{{\beta }_r}{2}}}{\int }}{M}_{\alpha }\sin \left(np\alpha \right)d\alpha +\underset{{\beta }_m+{\displaystyle \frac{{\beta }_r}{2}}}{\overset{2{\beta }_m-{\displaystyle \frac{{\beta }_r}{2}}}{\int }}{M}_{\alpha }\sin \left(np\alpha \right)d\alpha \right]\end{array}\right.$$

(7)

With the governing equations defined and a general expression for the magnetization pattern established, the stage for more advanced configurations is set. This initial formulation, based on a single PM layer with repeatable segment geometry, serves as the analytical foundation allowing for extending the model toward multi-layered segmented PM, with greater flexibility in engineering the airgap flux density.

2.2. Multi-Layered Segmented PM

The multi-layer segmented rotor consists of multiple permanent-magnet layers stacked radially, as shown in Figure 3a. Each layer is made of several magnet segments arranged around the circumference, and each segment may have its own magnetization direction. By selecting the number of segments per layer and the magnet ratio, the air-gap magnetic field can be shaped and tuned with high flexibility. This structure allows the designer to engineer flux concentration, waveform shaping, and harmonic suppression directly through the rotor configuration.

To develop the analytical field model, the rotor is not solved as a whole all at once. Instead, the magnetic contribution of each layer is computed independently. The key idea is illustrated in Figure 3b. When analyzing a particular layer l, that layer is considered the active magnetized layer, meaning its magnetization distribution is explicitly included in the formulation. The remaining layers are treated as inactive for that calculation. An inactive layer is not ignored; instead, it is modeled as a region of uniform and isotropic magnetic permeability. This allows the surrounding layers to properly influence the boundary conditions of the active layer without requiring their detailed magnetization patterns to be represented simultaneously.

To illustrate how the modeling is carried out, consider the three-layer example shown in Figure 3a. Suppose Layer 2 is selected as the active layer. In this case, the rotor is divided into radial subdomains as shown in Figure 3b. The region between the stator and the rotor forms Region I, the outer airgap. Layer 1, which lies outside the active layer, is treated as the outer inactive magnet region or Region II′. Layer 2 forms the active magnet region described as Region II″. Layer 3 is treated as the inner inactive magnet region or Region II‴. Below this appears Region III, the inner airgap, followed finally by the back-iron, which completes the magnetic circuit. This same procedure is applied for each layer in turn: if Layer 1 is being modeled, Layers 2 and 3 become inactive; if Layer 3 is being modeled, Layers 1 and 2 become inactive. For rotors with more than three layers, the exact same idea applies: layers outside the active one are grouped together as the outer inactive region, and layers inside it are grouped as the inner inactive region. After evaluating the magnetic field for each layer individually in this way, the total airgap field is obtained by superposing the contributions from all layers.

This layered segmentation enables a modular solution approach to the field equations, where the contribution of each layer is computed independently and then superposed to reconstruct the total magnetic field distribution. Additionally, the assumptions established for the single-layer segmented PM remain applicable and are extended here to support the multi-layered structure. In particular, the following additional assumptions are introduced in this multi-layer scenario:

• Layer Independence (Superposition Principle): Each magnetized layer is treated as an independent source of magnetic field. The layers are assumed to be magnetically linear, allowing the field contribution of each layer to be computed separately and superimposed to obtain the total magnetic field.
• Homogeneous Material Between Layers: The inter-layer regions not occupied by active magnets are modeled as homogeneous magnetized material with constant relative permeability ${\mu }_r$. This simplifies the field computation while retaining the magnetic influence of adjacent layers.
• No Magnetic Coupling Between Layers: Nonlinear magnetic coupling between layers is neglected. This is justified under the assumption of low magnetic loading and linear material behavior, ensuring that the field generated by one layer does not significantly alter the magnetization of others.

Given these assumptions, the governing field equations for a representative layer l are formulated by isolating the layer and treating the surrounding regions accordingly. The corresponding expressions for the magnetic vector potential in each region are then derived. For Regions I, II′, II‴, and III, the differential equation remains the same as Equation (1). Region II″, however, is governed by Equation (2).

The solutions to the mentioned differential Equations (1) and (2), considering the circumstances of each region and applying Equation (3), are:

(1)

Region I and III:

$$\left\{\begin{array}{c}{B}_{\mathrm{1,3},r}\left(r,\alpha \right)=-{\mu }_0{\displaystyle \sum _{n=\mathrm{1,3},5}^{\mathrm{\infty }}}np\left(A_{\mathrm{1,3},n}{r}^{np-1}-B_{\mathrm{1,3},n}{r}^{-np-1}\right)\cos \left(np·\alpha \right)\\ B_{1,3,\alpha }\left(r,\alpha \right)={\mu }_0{\displaystyle \sum _{n=\mathrm{1,3},5}^{\mathrm{\infty }}}np\left(A_{1,3,n}{r}^{np-1}+B_{\mathrm{1,3},n}{r}^{-np-1}\right)\sin \left(np·\alpha \right)\end{array}\right.$$

(8)

(2)

Region II′ and II‴:

$$\left\{\begin{array}{c}{B}_{2^{\prime },2‴,r}\left(r,\alpha \right)=-\mu {\displaystyle \sum _{n=\mathrm{1,3},5}^{\mathrm{\infty }}}np\left(A_{2^{\prime },2‴,n}{r}^{np-1}-B_{2^{\prime },2‴,n}{r}^{-np-1}\right)\cos \left(np·\alpha \right)\\ B_{2^{\prime },2‴,\alpha }\left(r,\alpha \right)=\mu {\displaystyle \sum _{n=\mathrm{1,3},5}^{\mathrm{\infty }}}np\left(A_{2^{\prime },2‴,n}{r}^{np-1}+B_{2^{\prime },2‴,n}{r}^{-np-1}\right)\sin \left(np·\alpha \right)\end{array}\right.$$

(9)

(3)

Region II″:

$$\left\{\begin{array}{c}{B}_{2″r}\left(r,\alpha \right)=-\mu {\displaystyle \sum _{n=\mathrm{1,3},5}^{\mathrm{\infty }}}\left[np\left(A_{2″,n}{r}^{np-1}-B_{2″,n}{r}^{-np-1}\right)+{\displaystyle \frac{M_n}{{\mu }_r(1-{\left(np\right)}^2)}}-{\displaystyle \frac{M_{rn}}{{\mu }_r}}\right]\cos \left(np·\alpha \right)\\ B_{2″\alpha }\left(r,\alpha \right)=\mu {\displaystyle \sum _{n=\mathrm{1,3},5}^{\mathrm{\infty }}}\left[np\left(A_{2″,n}{r}^{np-1}+B_{2″,n}{r}^{-np-1}\right)+{\displaystyle \frac{{np.M}_{rn}}{{\mu }_r(1-{\left(np\right)}^2)}}\right]\sin \left(np·\alpha \right)\end{array}\right.$$

(10)

where $B_X\left(\mathrm{T}\right)$ refers to the magnetic field in the $X^{th}$ region. $\mu$ (H/m) is the permeability of the magnet material, and ${\mu }_0$ (H/m) is the vacuum permeability. $A_{X,n}$ (A/$m^{np}$) and $B_{X,n}$ (A.$m^{-(np+2)}$) are expressions to be deduced using the boundary conditions. $M_n$ is expressed as in [21] by:

$$M_n=M_{rn}+{np·M}_{\alpha n}$$

(11)

As mentioned in the assumptions made, each layer was isolated within the multi-layered structure, as shown in Figure 4, and its surrounding domains were defined accordingly. For a given layer $l$, the rotor is partitioned into five radial regions, each bounded by predefined radii. These boundaries provide the framework for applying field continuity conditions across regions and form the basis for solving the magnetic vector potential in closed form.

The next step is to apply continuity conditions for the magnetic field and potential at each radial interface [16,21,22]. These conditions ensure consistency across regions and allow for solving the unknown coefficients in the vector potential expression.

$$\left\{\begin{array}{c}Region I: H_{1\alpha }\left(R_{out},\alpha \right)=0\\ Region I/{II}^{\prime }: \left\{\begin{array}{c}{B}_{1r}\left(R_0,\alpha \right)=B_{2r}(R_0,\alpha )\\ H_{1\alpha }\left(R_0,\alpha \right)=H_{2\alpha }(R_0,\alpha )\end{array}\right.\\ Region {II}^{\prime }/{II}^{\prime \prime }: \left\{\begin{array}{c}{H}_{2r}\left(R_{l-1},\alpha \right)=H_{3r}\left(R_{l-1},\alpha \right)\\ H_{2\alpha }\left(R_{l-1},\alpha \right)=H_{3\alpha }\left(R_{l-1},\alpha \right)\end{array}\right.\\ Region {II}^{\prime \prime }/{II}^{\prime \prime \prime }: \left\{\begin{array}{c}{H}_{3r}\left(R_l,\alpha \right)=H_{4r}\left(R_l,\alpha \right)\\ H_{3\alpha }\left(R_l,\alpha \right)=H_{4\alpha }\left(R_l,\alpha \right)\end{array}\right.\\ Region {II}^{\prime \prime \prime }/III: \left\{\begin{array}{c}{B}_{4r}\left(R_N,\alpha \right)=B_{5r}\left(R_N,\alpha \right)\\ H_{4\alpha }\left(R_N,\alpha \right)=H_{5\alpha }\left(R_N,\alpha \right)\end{array}\right.\\ Region III: H_{5\alpha }\left(R_{in},\alpha \right)=0\end{array}\right.$$

(12)

Since the outer airgap is the region of electromagnetic interaction with the stator, the determination of the field in this region is of primary interest. As stated in the Region I boundary condition in (12), the tangential component of the magnetic field must be zero due to the high-permeability assumption applied at the stator boundary. When this condition is applied to the general form of the magnetic vector potential in Region I, we get:

$$B_{1,n}=-A_{1,n}{R}_{out}^{2np}$$

(13)

Therefore, the magnetic flux density in Region I may be fully expressed in terms of $A_{1,n}$ and solving the continuity conditions across the radial interfaces leads to the following closed-form expression:

$$A_{1,n}=-{\displaystyle \frac{\mu R_{23}\left(N_{1,n}-N_{2,n}+N_{3,n}-N_{4,n}\right)}{D_n}}{\left({\displaystyle \frac{R_{l-1}}{R_{out}^2}}\right)}^{np}$$

(14)

where

$$N_{1,n}=M_{1,n}\mu \left[1+{\left({\displaystyle \frac{R_N}{R_{l-1}}}\right)}^{2np}-{\displaystyle \frac{R_l}{R_{l-1}}}\left({\left({\displaystyle \frac{R_l}{R_{l-1}}}\right)}^{np}+{\left({\displaystyle \frac{R_N^2}{R_{l-1}{R}_l}}\right)}^{np}\right)\right]\left(1-{\left({\displaystyle \frac{R_{in}}{R_N}}\right)}^{2np}\right)$$

(15)

$$N_{2,n}=M_{2,n}\mu \left[1-{\left({\displaystyle \frac{R_N}{R_{l-1}}}\right)}^{2np}-{\displaystyle \frac{R_l}{R_{l-1}}}\left({\left({\displaystyle \frac{R_l}{R_{l-1}}}\right)}^{np}-{\left({\displaystyle \frac{R_N^2}{R_{l-1}{R}_l}}\right)}^{np}\right)\right]\left(1-{\left({\displaystyle \frac{R_{in}}{R_N}}\right)}^{2np}\right)$$

(16)

$$N_{3,n}=M_{1,n}{\mu }_0\left[1-{\left({\displaystyle \frac{R_N}{R_{l-1}}}\right)}^{2np}-{\displaystyle \frac{R_l}{R_{l-1}}}\left({\left({\displaystyle \frac{R_l}{R_{l-1}}}\right)}^{np}-{\left({\displaystyle \frac{R_N^2}{R_{l-1}{R}_l}}\right)}^{np}\right)\right]\left(1+{\left({\displaystyle \frac{R_{in}}{R_N}}\right)}^{2np}\right)$$

(17)

$$N_{4,n}=M_{2,n}{\mu }_0\left[1+{\left({\displaystyle \frac{R_N}{R_{l-1}}}\right)}^{2np}-{\displaystyle \frac{R_l}{R_{l-1}}}\left({\left({\displaystyle \frac{R_l}{R_{l-1}}}\right)}^{np}+{\left({\displaystyle \frac{R_N^2}{R_{l-1}{R}_l}}\right)}^{np}\right)\right]\left(1+{\left({\displaystyle \frac{R_{in}}{R_N}}\right)}^{2np}\right)$$

(18)

$$\begin{gathered} D_n={\mu }^2\left(1-{\left({\displaystyle \frac{R_N}{R_0}}\right)}^{2np}\right)\left(1-{\left({\displaystyle \frac{R_{in}}{R_N}}\right)}^{2np}\right)\left({\left({\displaystyle \frac{R_0}{R_{out}}}\right)}^{2np}-1\right) \\ -{\mu }_0^2\left(1-{\left({\displaystyle \frac{R_N}{R_0}}\right)}^{2np}\right)\left(1+{\left({\displaystyle \frac{R_{in}}{R_N}}\right)}^{2np}\right)\left({\left({\displaystyle \frac{R_0}{R_{out}}}\right)}^{2np}+1\right) \\ +2\mu {\mu }_0\left(1+{\left({\displaystyle \frac{R_N}{R_0}}\right)}^{2np}\right)\left({\left({\displaystyle \frac{R_0{R}_{in}}{R_{out}{R}_N}}\right)}^{2np}-1\right) \end{gathered}$$

(19)

$$M_{1,n}={\displaystyle \frac{M_n}{np.{\mu }_r(1-{\left(np\right)}^2)}}-{\displaystyle \frac{M_{rn}}{{np·\mu }_r}}$$

(20)

$$M_{2,n}={\displaystyle \frac{M_n}{{\mu }_r(1-{\left(np\right)}^2)}}$$

(21)

A key advantage of this approach is that all coefficients can be expressed as functions of radius ratios raised to powers of $n\times p$, making the solution compact, scalable, and easily programmable.

2.3. Simplified Model for Multi-Layered Segmented PMs: Approach and Comparison

In practice, many commercial-grade magnet materials, namely NdFeB and SmCo, exhibit relative permeability close to that of air (${\mu }_r\approx 1)$. This observation enables a simplification that reduces the computational complexity of the proposed model. Instead of modeling five distinct regions per layer (as in the full formulation), the simplified approach considers only three: the outer airgap, the active magnetized region, and the inner airgap. Since ${\mu }_r\approx 1$, the potential inactive material can be treated as extensions of the outer and inner airgaps, respectively.

This reduction leverages the negligible contrast in magnetic properties between adjacent magnet layers and their surroundings. As a result, each layer’s contribution can be modeled using a linear combination of pre-characterized solutions from the single-layered configuration. To express the airgap magnetic field in analytical form, we follow the scalar magnetic potential approach used in Shen et al. [21] for Halbach magnetized rotors. In that formulation, the airgap flux density is expanded into spatial harmonics, where each harmonic corresponds to a term in the magnetization distribution. This yields the general expression for the radial magnetic field in Region I:

$$B_{r1}\left(r,\alpha \right)={\displaystyle \frac{1}{2}}\sum _{n=\mathrm{1,3},5,\dots }^{\infty }{K}_B\left(n\right)f_{B_r}\left(n,r\right)\mathit{cos}\left(np\alpha \right)$$

(22)

In this expression, $K_B\left(n\right)$ captures the harmonic amplitude and explicitly depends on the magnetization coefficients $M_n$ and $M_{rn}$, which are functions of the layer segmentation k and magnet ratio $Rmp$. The term $f_{B_r}\left(n,r\right)$, on the other hand, characterizes how this harmonic varies with radius within the airgap. Together, these two terms describe both the strength and radial behavior of each spatial harmonic of the magnetic field.

In the present work, this harmonic representation is extended to a multi-layer segmented configuration. The modification appears in the harmonic amplitude coefficient $K_B\left(n\right)$, which is reformulated so that it reflects the radii and magnetization of each individual layer. The radial dependence $f_{B_r}\left(n,r\right)$ remains consistent with the classical solution and the resulting expressions are therefore given by:

$$K_B\left(n\right)=-{\mu }_0{\displaystyle \frac{\left(\frac{M_n}{1+np}-M_{rn}\right)\left[1-{\left(\frac{R_l}{R_{l-1}}\right)}^{np+1}\right]-\left(\frac{M_n}{1-np}-M_{rn}\right)\left[\left({\left(\frac{R_l}{R_{l-1}}\right)}^{2np}-{\left(\frac{R_l}{R_{l-1}}\right)}^{np+1}\right){\left(\frac{R_{in}}{R_l}\right)}^{2np}\right]}{1-{\left(\frac{R_{in}}{R_{out}}\right)}^{2np}}}$$

(23)

$$f_{B_r}\left(n,r\right)={\left({\displaystyle \frac{r}{R_{out}}}\right)}^{np-1}{\left({\displaystyle \frac{R_{l-1}}{R_{out}}}\right)}^{np+1}+{\left({\displaystyle \frac{R_{l-1}}{r}}\right)}^{np+1}$$

(24)

This simplified model preserves the accuracy in capturing the essential airgap field behavior while substantially reducing the mathematical complexity of layer-by-layer evaluation. It forms the core analytical framework employed throughout the remainder of this study to simulate field distributions and calculate flux linkage.

To verify the effectiveness of this simplification, a comparative analysis was performed between the full proposed analytical model and the simplified Model adapted from Shen et al. [21]. Two configurations, a single-layer case and a two-layer case, are used to evaluate how closely the simplified model replicates the original field behavior.

Table 1. Original and Simplified Models Comparison Study Cases.

	N	${\mathit{\mu }}_{\mathit{r}}$	Br (T)	K (Outer to Inner)	Rmp (Outer to Inner)	Rin (mm)	Rout (mm)	Radii (mm) (Outer to Inner)
Case a	1	1.0446	1.2	2	0.7	22.275	28.5	[27.5, 22.275]
Case b	2	1.0446	1.1	[4, 2]	[0.25, 0.75]	16	26	[25, 20, 15]

summarizes the geometric and material configurations of the rotors.

The results in Figure 5a,b demonstrate a strong agreement between the full analytical model and the simplified model. In both the single-layer and two-layer test cases, the radial magnetic field components closely match across the entire angular domain. Minor deviations are observed, particularly near peak values, but the overall accuracy of the simplified model is sufficient for capturing key magnetic behaviors. These results confirm that the reduced formulation can serve as a reliable alternative for simulation and design, reducing computational complexity without compromising fidelity.

2.4. Flux Linkage Calculation

To evaluate the interaction between the magnetic field and the stator windings, the flux linkage produced by the PM was calculated using a hypothetical winding coil spanning the width of a single pole. The flux linkage was only evaluated when the coil was angularly aligned with a rotor pole. In the absence of a detailed stator/slot design, this simplified calculation serves as a useful proxy for torque generation and EMF estimation.

For simplification, the winding coil width was assumed equal to the pole width ${\beta }_m$. The instantaneous flux linkage of a single coil was calculated when the coil was aligned with the corresponding rotor pole.

$$\psi =L·r\underset{{\alpha }_1}{\overset{{\alpha }_2}{\int }}{B}_{r1}\left(r,\alpha \right)d\alpha$$

(25)

where $L$ is the machine axial length, $r$ is the radial position within the airgap (typically taken at the midpoint), and ${\alpha }_1$ and ${\alpha }_2$ define the angular span of a single magnetic pole. For consistency, the radial evaluation point is set to $r={\displaystyle \frac{R_{12}+R_{out}}{2}}$. This integral is used throughout the study to quantify the flux contribution of each configuration and serves as a basis for comparing performance across different layer setups.

3. FEA Validation

To assess the reliability of the proposed analytical model, a series of FEA simulations were conducted using ANSYS Electronic Desktop 2024 R2 under 2D steady-state magnetic conditions. These simulations replicate various multi-layer segmented PM configurations under conditions representative of SPM rotor design. The 2-D geometry was modeled in FEA, with adaptive mesh settings and a 0.001 percent energy error allowed for the convergence criterion. The total mesh could be up to 0.5 million for the multi-layer PM rotor designs.

This five-layer case is shown as a representative example of the FEA setup. The other validation cases were modeled in the same manner, with the number of magnetized layers, segmentation patterns, and magnet ratios being adjusted.

The validation focuses on two key quantities: the magnetic flux density distribution across the central line of the air gap, as simulated by FEA, particularly the radial component, and the computed flux linkage. A total of four test cases were studied, each with different combinations of layer count, number of pole segments, and magnet ratios. For each case, results from the analytical model were compared with FEA outputs using pointwise field evaluation and integrated flux linkage. This comparison provides a direct measure of the model’s accuracy relative to high-resolution numerical simulation.

Table 2 and Table 3 summarize the fixed and case-specific parameters used in the simulations. The resulting magnetic field plots and flux linkage comparisons are shown below, demonstrating the accuracy of the analytical model across different design configurations, with Case 3 having the same five-layer rotor architecture as depicted earlier in Figure 6.

Table 2. Multi-Layered Segmented SPM Physical entity configuration.

Vacuum Permeability ${\mathit{\mu }}_0$(H/m)	Relative Permeability ${\mathit{\mu }}_{\mathit{r}}$	Magnet Remanence ${\mathit{B}}_{\mathit{r}}$(T)	Number of Pole Pairs in Magnet p	Thickness L (mm)
$4\pi ·{10}^{-7}$	1	1.1	2	20

Table 2. Multi-Layered Segmented SPM Physical entity configuration.

Vacuum Permeability ${\mathit{\mu }}_0$(H/m)	Relative Permeability ${\mathit{\mu }}_{\mathit{r}}$	Magnet Remanence ${\mathit{B}}_{\mathit{r}}$(T)	Number of Pole Pairs in Magnet p	Thickness L (mm)
$4\pi ·{10}^{-7}$	1	1.1	2	20

And for each case, we have:

Table 3. Study Cases Breakdown: Number of Layers k, the Magnet Ratio Rmp, The Radii for different Layers and the Outer and inner airgap Radii (Rin and Rout).

Cases	N	k (Outer to Inner)	Rmp (Outer to Inner)	Radii (mm) (Outer to Inner)	Rin (mm)	Rout (mm)
Case 1	3	[2, 2, 2]	[0.75, 0.75, 0.75]	[30, 25, 20, 15]	14	31
Case 2	2	[4, 2]	[0.75, 0.25]	[25, 20, 15]	14	26
Case 3	5	[2, 4, 4, 4, 2]	[0.8, 0.7, 0.5, 0.7, 0.8]	[25, 23, 21, 19, 17, 15]	14	26
Case 4	2	[4, 2]	[0.75, 0.25]	[25, 23, 15]	14	26

Table 3. Study Cases Breakdown: Number of Layers k, the Magnet Ratio Rmp, The Radii for different Layers and the Outer and inner airgap Radii (Rin and Rout).

Cases	N	k (Outer to Inner)	Rmp (Outer to Inner)	Radii (mm) (Outer to Inner)	Rin (mm)	Rout (mm)
Case 1	3	[2, 2, 2]	[0.75, 0.75, 0.75]	[30, 25, 20, 15]	14	31
Case 2	2	[4, 2]	[0.75, 0.25]	[25, 20, 15]	14	26
Case 3	5	[2, 4, 4, 4, 2]	[0.8, 0.7, 0.5, 0.7, 0.8]	[25, 23, 21, 19, 17, 15]	14	26
Case 4	2	[4, 2]	[0.75, 0.25]	[25, 23, 15]	14	26

Across all validation cases, the analytical model exhibits a high level of agreement with the FEA results, accurately reproducing the shape and amplitude of the radial magnetic field $B_r$. The field distribution aligns closely with the FEA benchmark across the entire pole span, including regions near the peaks and zero crossings (see Figure 7).

Beyond the visual comparison of the field distribution, a quantitative assessment was conducted by evaluating the flux linkage error between the analytical model and FEA results. The deviation in flux linkage is computed for each case as:

$$Error\left(\mathrm{\%}\right)={\displaystyle \frac{\left|{\psi }_{FEA}-{\psi }_{Analytical}\right|}{{\psi }_{FEA}}}$$

(26)

Table 4. Simulation Summary for the four cases: Flux linkage and Deviation in total flux linkage calculated using the Formulas (25) and (26).

Cases	Case 1	Case 2	Case 3	Case 4
FEA flux Linkage (Wb)	0.000746	0.000597	0.000616	0.000585
Computed flux linkage (Wb)	0.000750	0.000600	0.000620	0.000589
Deviation %	0.50	0.43	0.69	0.67

summarizes the computed flux linkage values for each test case, showing the FEA result, the analytical prediction, and the resulting deviation percentage.

Across all validation cases, the analytical predictions closely match the FEA benchmarks. The flux-linkage deviations are consistently near 0.5% (≈0.4–0.7% across cases), while the airgap field shape and amplitude are accurately reproduced over the full pole span. These results support using the model as a fast, reliable surrogate for early-stage design and control-oriented studies, delivering substantial computational savings with minimal loss in accuracy.

4. Applications: Flux Linkage Maximization

This section presents a case study applying the proposed analytical model of the multi-layer segmented PM for performance optimization. It focuses on maximizing the static flux linkage by optimizing the rotor’s segmented PM configuration. The study is conducted under a simplified setting in which the stator is assumed to be slotless and equipped with distributed windings, each spanning an angular width equal to that of a rotor pole. This assumption allows for direct computation of the flux linkage from the magnetic field distribution in the airgap, avoiding the complexities introduced by slotting effects or winding overhangs. The rotor is considered at an instantaneous angular position with one coil matching a rotor pole, and the optimization is performed to maximize the flux linkage, focusing solely on the magnetic interaction between the rotor and the equivalent stator winding at that instant.

To explore the large and highly nonlinear design space defined by the multi-layer structure, a genetic algorithm is employed as the optimization strategy. This approach, described in Figure 8, systematically adjusts layer-specific parameters, including the number of magnet segments, radial proportions, and magnetization ratios, to identify configurations that maximize the static flux linkage. Its flexibility enables efficient navigation of the configuration space without requiring exhaustive search or manual parameter tuning, demonstrating the practical utility of the analytical model in supporting performance-driven rotor design. The genetic algorithm begins by setting the initial parameters and initializing a population with random configurations. During each generation, individuals undergo adaptive mutation and are selected based on constraint satisfaction and performance. Their fitness is then evaluated and compared to previous results. If an improvement is found, the stagnation counter is reset. If no improvement is observed over ten generations, a portion of the population is replaced to encourage exploration. If the lack of improvement persists for fifteen generations, the process is terminated early. Otherwise, the loop continues until the maximum number of generations is reached.

To formalize the optimization process, the goal is to maximize the magnetic flux linkage ${\psi }_{HA}$ generated by a multi-layer segmented PM:

$$\begin{array}{c}\max \psi \left(r,L\right)\\ R,K,Rmp\end{array}=L.r.\underset{{\alpha }_1}{\overset{{\alpha }_2}{\int }}{B}_{r1}\left(r,\alpha ;R,K,Rmp\right)d\alpha$$

where

• $R=\left\{R_1,R_2,\dots ,R_{N-1}\right\}$: radii of intermediate magnet layers (outer and inner bounds fixed).
• $K=\left\{k_1,k_2,\dots ,k_{N-1}\right\}:$ number of segments per pole per layer.
• $Rmp=\left\{{Rmp}_1,{Rmp}_2,\dots ,{Rmp}_{N-1}\right\}:$ magnet ratios in each layer.
• L: magnet axial length.
• r: midpoint in the outer airgap.

And the chromosome of each individual in the genetic algorithm represents a design candidate:

$$\mathsf{{\rm X}}=\left\{R_1,,\dots ,R_{N-1},k_1,,\dots ,k_{N-1},{Rmp}_1,\dots ,{Rmp}_{N-1}\right\}$$

Several test cases were conducted to evaluate the effectiveness of the proposed algorithm, each corresponding to a different number of magnetic layers ranging from 2 to 10. After optimization, the performance of each configuration was compared against a baseline case consisting of a single-layer segmented PM with two segments per pole and a magnet ratio of 0.5. This comparison highlights the performance gains introduced by the multi-layered model. The configuration parameters of the genetic algorithm and the primary design constraints are summarized in

Table 5. Flux Linkage Algorithm Setup.

Parameter	Population Size	Generations	Crossover Rate	Radii (mm)	k	Rmp
Details	300	200	0.3	$15 \mathrm{t}\mathrm{o} 30$	2 to 8	0.1 to 0.9

.

Figure 9 illustrates the normalized flux linkage as a function of the number of rotor layers, with all values referenced to the 10-layer configuration. The results demonstrate that the majority of the improvement occurs with the first few layers, particularly between the baseline and two layers, where the gain is most pronounced. Beyond four layers, the additional improvement becomes marginal, indicating diminishing returns as complexity increases.

In particular, two optimized configurations, the 2-layer and the 10-layer optimal cases, were selected for detailed evaluation. These cases represent the lower and upper bounds of the layered design space considered in this study.

Table 6. 2-layered and 10-layered cases detailed configurations.

	Radii (mm) (Outer to Inner)	Segment Count k (Outer to Inner)	Magnet Ratios Rmp (Outer to Inner)	Optimal Linkage Flux (Wb)	Improvement %
Case 5	[30, 27.3, 15]	[8, 8]	[0.8, 0.5]	0.000823	11.88
Case 6	[30, 28.3, 27, 25.4, 23.5, 22.9, 20.1, 18.5, 16.7, 15.8, 15]	[8, 8, 8, 8, 8, 8, 8, 8, 8, 8]	[0.8, 0.7, 0.6, 0.5, 0.5, 0.4, 0.4, 0.5, 0.6, 0.7]	0.000826	12.22

summarizes the optimal design parameters for each case, including the radial positions of the magnet layers, the number of segments per pole per layer, and the corresponding magnetization ratios. It also reports the resulting flux linkage values and the percentage improvement relative to the baseline configuration.

Figure 10 shows the radial magnetic field distribution for the baseline, 2-layer, and 10-layer configurations described in Table 6. As expected, the most substantial improvement occurs between the baseline and the 2-layer case, confirming the earlier observation that most performance gains are achieved with only a few magnetized layers. The difference between the 2-layer and 10-layer configurations is comparatively minor, suggesting that additional layering beyond the second has a limited impact on the overall field shape in this case. The similarity between the 2-layer and 10-layer results also underscores the efficiency of low-complexity designs when guided by a focused optimization problem.

While the present study focuses on static flux linkage as the primary optimization objective, the same framework can be extended to other design targets. Potential directions include maximizing electromagnetic torque, back-EMF, and flux linkage under transient conditions, minimizing cogging torque and torque ripple to improve smoothness, and optimizing stator geometry in conjunction with rotor design. Another possible route is reducing the input current required to achieve a given torque, thereby minimizing energy losses and improving overall efficiency. These extensions would allow the analytical model, combined with optimization methods, to address a broader set of performance criteria relevant to practical machine applications.

5. Conclusions

This study introduced an analytical framework for modeling multi-layered segmented SPMs. The model treats each magnetized layer as an independent subdomain and combines their contributions through superposition, making it both general and computationally efficient. Using a hybrid formulation based on separation of variables and Fourier series, the airgap field distribution was derived and then validated against two-dimensional FEA. The comparison showed good agreement, with deviations around 0.5%, confirming both the accuracy and efficiency of the proposed approach. The framework was then used to examine how adding magnetized layers affects the static flux linkage in a representative scenario. This test case, carried out with the support of a genetic algorithm, demonstrated how the analytical model can be integrated into an optimization process to explore design variations systematically. The results showed that most of the performance improvement is achieved with only a few layers, while additional layering yields only marginal gains. Although this work focused on static flux linkage as a demonstration, the same methodology can be extended to other performance objectives. It can also serve as an early-stage design tool that precedes full machine prototyping. The resulting analytical–optimization framework can then be carried forward once complete rotor–stator geometries are available for experimental evaluation.

Future work will also consider nonlinear material properties, thermal effects, and manufacturing tolerances to strengthen the model’s applicability to practical machine designs. In addition, we are currently extending this framework to model the full machine, rather than the rotor alone. This includes incorporating the stator slot geometry, winding layout, and the current excitation. With these additions, we will be able to evaluate performance measures such as torque density and torque ripple in a unified way and optimize them alongside the rotor structure using the same GA-based approach. The results of this complete machine-level model will be presented in future work.