A Comprehensive Analysis of the Loss Mechanism and Thermal Behavior of a High-Speed Magnetic Field-Modulated Motor for a Flywheel Energy Storage System

Abstract

This paper presents a comprehensive analytical framework for investigating loss mechanisms and thermal behavior in high-speed magnetic field-modulated motors for flywheel energy storage systems. Through systematic classification of electromagnetic, mechanical, and additional losses, we reveal that modulator components constitute approximately 45% of total system losses at rated speed. Finite element analysis demonstrates significant spatial non-uniformity in loss distribution, with peak loss densities of 5.5 × 10⁵ W/m³ occurring in the modulator region, while end-region losses exceed central-region values by 42% due to three-dimensional field effects. Our optimized design, implementing composite rotor structures, dual-material permanent magnets, and integrated thermal management solutions, achieves a 43.2% reduction in total electromagnetic losses, with permanent magnet eddy current losses decreasing by 68.7%. The maximum temperature hotspots decrease from 143 °C to 98 °C under identical operating conditions, with temperature gradients reduced by 58%. Peak efficiency increases from 92.3% to 95.8%, with the η > 90% region expanding by 42% in the speed–torque plane. Experimental validation confirms model accuracy with mean absolute percentage errors below 4.2%. The optimized design demonstrates 24.8% faster response times during charging transients while maintaining 41.7% smaller speed oscillations during sudden load changes. These quantitative improvements address critical limitations in existing systems, providing a viable pathway toward high-reliability, grid-scale energy storage solutions with extended operational lifetimes and improved round-trip efficiency.

1. Introduction

This study addresses critical challenges facing high-speed magnetic field-modulated motors in flywheel energy storage systems. Our work introduces several theoretical innovations compared to existing studies: The proposed frequency domain analytical model for eddy current loss prediction (Section 2.4, Equation (8)) surpasses time domain solutions used by Zhang et al. [1] by capturing complex spatial harmonic interactions with 70% improved computational efficiency. Our equivalent thermal circuit transformation technique (Section 2.4, Equation (9)) achieves temperature prediction accuracy within 5.8 °C for vacuum-sealed systems, compared to deviations exceeding 15 °C in conventional networks used by Liu et al. [2]. The physics-informed neural network approach (Equation (10)) embeds governing equations as constraints within deep learning architectures, enabling accurate predictions with limited experimental data. This work reveals previously unidentified phenomena, including 42% higher losses in end regions, and demonstrates optimal modulator pole arc coefficients of 0.72–0.78, diverging from the 0.65–0.70 range for conventional machines. These advancements enhance both fundamental understanding and practical design methodologies for flywheel energy storage applications.

Flywheel energy storage systems (FESSs) have emerged as a promising technology for efficient energy storage applications due to their high power density, fast response characteristics, and long operational lifetime [3]. The implementation of high-speed magnetic field-modulated motors in FESSs has significantly enhanced system performance by improving energy conversion efficiency while maintaining operational stability [4]. However, the operational challenges associated with electromagnetic losses and thermal management remain critical barriers to optimal system performance [5]. Recent advancements in multiphysics coupling analysis have facilitated comprehensive understanding of electromagnetic, thermal, and mechanical interactions in these complex systems [6].

The loss mechanisms in high-speed magnetic field-modulated motors are multifaceted, comprising core losses (hysteresis, eddy current, and excess losses), copper losses with skin and proximity effects, and mechanical losses including friction and windage [7]. Thermal management strategies must therefore address both the intensity and spatial distribution of these losses [8].

Advanced thermal modeling techniques have evolved from simplified lumped parameter thermal networks to sophisticated computational fluid dynamics (CFD) approaches capable of analyzing complex flow patterns and heat transfer mechanisms [6]. Recent research has focused on electromagnetic–thermal coupling analysis to accurately predict temperature distributions under various operational conditions [9]. This integrated approach enables optimization of cooling structures including impingement jet cooling and water cooling configurations tailored to the unique thermal challenges of FESSs [10]. Experimental verification has validated these analytical models, confirming their accuracy in predicting both electromagnetic losses and temperature distributions [11]. The ongoing development of these comprehensive analytical frameworks continues to advance both the theoretical understanding and practical implementation of high-efficiency FESS technologies [12].

2. Theoretical Model of Loss Calculation and Thermal Analysis

2.1. Motor Structure and Key Parameters

The high-speed magnetic field-modulated motor analyzed in this study features a novel topology specifically optimized for flywheel energy storage applications. Figure 1 illustrates the motor’s cross-sectional structure, highlighting the key components: a 12-slot stator with concentrated windings, a permanent magnet rotor with four pole pairs using N42SH magnets, and five ferromagnetic modulators positioned between the stator and rotor. This configuration enables effective torque transmission while maintaining mechanical robustness at high rotational speeds. The key parameters of the high-speed magnetic field-modulated motor are listed in

Table 1. Key parameters of the high-speed magnetic field-modulated motor.

Parameter	Value
Rated power	5 kW
Rated speed	30,000 rpm
Stator outer diameter	120 mm
Rotor outer diameter	60 mm
Active length	85 mm
Air gap length	0.8 mm
Number of stator slots	12
Number of rotor pole pairs	4
Number of modulators	5
Permanent magnet material	N42SH
Core material	M250-35A
Maximum flux density	1.6 T

.

The selected geometry and materials represent a carefully optimized balance between electromagnetic performance, thermal management capabilities, and mechanical integrity at high speeds. The modulator configuration creates a magnetic gearing effect that enables effective energy conversion across a wide speed range while maintaining high efficiency—a critical requirement for flywheel energy storage applications.

2.2. Electromagnetic Principles of Magnetic Field-Modulated Motors

Magnetic field-modulated motors represent an advanced class of electrical machines that leverage magnetic field modulation effects to achieve enhanced performance characteristics particularly suitable for flywheel energy storage systems [13]. The fundamental operational principle, as illustrated in Figure 1, involves the strategic interaction between permanent magnets, flux modulators, and stator windings to create effective electromagnetic coupling [14]. Unlike conventional permanent magnet synchronous machines, these motors employ a distinct configuration where flux modulators—typically ferromagnetic pole pieces—are positioned between permanent magnet arrays and armature windings [15]. This arrangement facilitates harmonic modulation of the magnetic field, enabling effective energy conversion at variable speeds while maintaining high torque density.

The magnetic field distribution can be mathematically described using Fourier series analysis, where the air gap flux density contains both fundamental and modulated harmonic components [16]. These modulated harmonics are responsible for the electromechanical energy conversion through magnetic coupling, with the modulation ratio directly influencing the torque transmission capability. As depicted in Figure 1, the pole pair relationship between modulated flux components and stator windings establishes a stable electromagnetic torque production mechanism that maintains operational stability under varying load conditions. This principle is particularly valuable in flywheel applications where wide-range speed variation necessitates consistent torque production across different operational states [17]. The modulated field distribution creates a magnetic gear effect, enabling speed adaptation without mechanical transmission components, thereby reducing system complexity and potential failure points.

2.3. Comprehensive Loss Classification and Calculation Methods

The accurate prediction and quantification of power losses in high-speed magnetic field-modulated motors for flywheel energy storage systems require a methodical classification framework and precise calculation methodologies. These losses can be systematically categorized into electromagnetic, mechanical, and additional losses, each with distinct physical mechanisms and mathematical representations, as shown in

Table 2. Comprehensive classification and calculation methods for losses in high-speed magnetic field-modulated motors.

Loss Category	Loss Component	Physical Mechanism	Calculation Method	Reference
Core Losses	Hysteresis Loss	Magnetic domain wall movement	$P_h=k_{h}f{B}_m^{\alpha }$	[14]
	Eddy Current Loss	Induced currents in core material	$P_e=k_e{f}^2{B}_m^2$	[13]
	Excess Loss	Micro-eddy currents	$P_a=k_a{f}^{1.5}{B}_m^{1.5}$	[13]
Copper Losses	DC Resistive Loss	Joule heating in conductors	$P_{cu,dc}=I_{rms}^2{R}_{dc}$	[1]
	Skin Effect Loss	Current density redistribution	$P_{skin}=k_s{P}_{cu,dc}$	[15]
	Proximity Effect Loss	Adjacent conductor interaction	$P_{prox}=k_p{P}_{cu,dc}$	[10]
Mechanical Losses	Bearing Friction Loss	Contact friction in bearings	$P_{bearing}=0.5\mu F_r\omega d_m$	[4]
	Windage Loss	Air friction on rotating surfaces	$P_{windage}=k_w{\rho }_{air}{\omega }^3{r}^5$	[9]

. The comprehensive assessment of these loss components is essential for thermal analysis, efficiency optimization, and reliability enhancement of the entire system [18].

Core losses represent a significant portion of the total electromagnetic losses and consist of three principal components: hysteresis, eddy current, and excess losses [19]. The modified Bertotti equation provides an effective mathematical framework for calculating these losses:

$$P_{core}=P_h+P_e+P_a=k_{h}f{B}_m^{\alpha }+k_e{f}^2{B}_m^2+k_a{f}^{1.5}{B}_m^{1.5}$$

(1)

where $P_h$, $P_e$, and $P_a$ represent hysteresis, eddy current, and excess losses, respectively, with $f$ denoting frequency, $B_m$ signifying peak magnetic flux density, and $k_h$, $k_e$, $k_a$, and $\alpha$ serving as material-specific coefficients [20]. For high-frequency applications, this classical model requires further refinement to account for non-uniform flux distribution and magnetic saturation effects [21].

The coefficients kh, kc, and ke in Equation (1) were determined through standardized Epstein frame testing following IEEE Standard 393-1991 [22], with multiple excitation frequencies ranging from 50 Hz to 5 kHz to capture frequency-dependent effects accurately.

Copper losses in high-speed applications exhibit complex behavior due to skin and proximity effects, especially at elevated frequencies. The AC resistance factor $k_r$ characterizes the ratio of AC to DC resistance:

$$P_{cu}=k_r{I}_{rms}^2{R}_{dc}=k_r\rho l_w{J}^2{V}_w$$

(2)

where $\rho$ represents the conductor resistivity, $l_w$ is the winding length, $J$ is the current density, and $V_w$ is the winding volume [2]. The AC resistance factor can be calculated using the Dowell equation or through finite element analysis for complex winding configurations [23].

Mechanical losses primarily consist of bearing friction and windage losses. In vacuum-sealed flywheel systems, bearing losses dominate and can be approximated by the following:

$$P_{bearing}=0.5\mu F_r\omega d_m$$

(3)

where $\mu$ denotes the friction coefficient, $F_r$ is the radial load, $\omega$ is the angular velocity, and $d_m$ is the bearing mean diameter [10]. As indicated in Table 2, windage losses become significant at atmospheric pressure but are substantially reduced in vacuum environments [24].

These loss calculation formulas incorporate several assumptions that define their applicable ranges and potential limitations. The modified Bertotti Equation (1) assumes that the material’s hysteresis behavior follows a power law relationship with flux density, which is valid for flux densities below 1.7 T, where saturation effects remain moderate. Beyond this threshold, the equation may underestimate hysteresis losses by up to 15% as verified through our material testing. The AC resistance calculation in Equation (2) assumes uniform current distribution within conductor bundles and neglects the strand-level circulating currents that may occur in complex winding configurations. This simplification introduces a maximum error of 8% at the highest operating frequency (5 kHz), based on comparison with detailed finite element conductor models. For mechanical loss estimation, Equation (3) assumes laminar flow conditions in the bearing lubricant and a linear relationship between the friction coefficient and load, which holds valid for the operating temperature range of 20–120 °C. All data processing steps applied to the simulation results used standard techniques, including 5-point moving average smoothing for noise reduction in time domain waveforms and fast Fourier transform with Hanning window for harmonic analysis. These processing methods preserve the fundamental characteristics of the data while improving visualization clarity.

Additional losses encompass stray load losses, magnetic wedge losses, and harmonic-induced losses that manifest during high-speed operation [25]. These losses, while individually smaller in magnitude, collectively constitute a non-negligible portion of the total loss profile and require specialized analytical or numerical methods for accurate quantification [26]. The comprehensive calculation of all loss components, as summarized in Table 2, provides the foundation for thermal analysis and efficiency optimization in high-speed magnetic field-modulated motors for flywheel energy storage applications.

Loss calculations were implemented using specialized formulations for each loss component. Core losses were calculated using the modified Bertotti equation with frequency-dependent coefficients as shown in Equation (1), with finite element field solutions providing spatially resolved flux density distributions at each time step. Copper losses incorporated both DC resistance effects and AC effects using the Dowell equation (Equation (2)) with strand-level eddy current modeling for capturing proximity effects in the end-winding regions. Permanent magnet and modulator eddy current losses were calculated directly from induced current densities in conducting regions using J²/σ integration. Mechanical losses, including bearing friction and windage losses, were calculated using analytical expressions (Equation (3)) calibrated against experimental measurements at different vacuum levels and rotation speeds. This comprehensive loss calculation approach was validated through comparison with direct calorimetric measurements, demonstrating agreement within 4.2% across the entire operating range.

2.4. Advanced Thermal Modeling Methods

Thermal modeling of high-speed magnetic field-modulated motors in flywheel energy storage systems necessitates sophisticated methodologies to accurately predict temperature distributions under dynamic operating conditions [27]. The evolution of thermal modeling approaches has progressed from simplified analytical methods to advanced multiphysics frameworks that capture the complex thermal behavior of these systems [11]. Lumped parameter thermal networks (LPTNs) remain foundational in thermal analysis due to their computational efficiency and physical interpretability [28,29]. In this approach, the motor structure is discretized into thermal nodes connected by thermal resistances and capacitances, with the heat transfer described by the following:

$$C_i{\displaystyle \frac{d{T}_i}{dt}}={\displaystyle \sum _{j=1}^n\frac{T_j-T_i}{R_{ij}}}+P_i$$

(4)

where $C_i$ represents the thermal capacitance of node $i$, $T_i$ is the temperature of node $i$, $R_{ij}$ is the thermal resistance between nodes $i$ and $j$, and $P_i$ is the heat generated at node $i$. For high-speed applications, these parameters exhibit speed dependence, particularly the convective heat transfer coefficients that scale with rotational velocity according to the following:

$$h_{conv}=k_1+k_2\cdot R{e}^{k_3}\cdot P{r}^{k_4}$$

(5)

where $Re$ and $Pr$ represent the Reynolds and Prandtl numbers, respectively, with coefficients $k_1$ through $k_4$ determined empirically or through numerical analysis.

Computational fluid dynamics (CFD) approaches provide more detailed thermal analyses by solving the governing conservation equations for mass, momentum, and energy. For vacuum-sealed flywheel systems, the Navier–Stokes equations reduce to the following:

$$\begin{array}{l}\rho {\displaystyle \frac{\partial v}{\partial t}}+\rho (v\cdot \nabla )v=-\nabla p+\mu {\nabla }^{2}v+\rho g\\ \rho c_p{\displaystyle \frac{\partial T}{\partial t}}+\rho c_{p}v\cdot \nabla T=\nabla \cdot (k\nabla T)+\dot{q}\end{array}$$

(6)

where $\rho$, $v$, $p$, $\mu$, $c_p$, $k$, and $\dot{q}$ denote fluid density, velocity vector, pressure, dynamic viscosity, specific heat capacity, thermal conductivity, and volumetric heat generation, respectively. These equations are typically solved using finite volume or finite element methods with appropriate boundary conditions and turbulence models.

The coupled electromagnetic–thermal analysis methodology represents a state-of-the-art approach, wherein electromagnetic field solutions provide spatially and temporally resolved loss distributions that serve as thermal sources in the heat transfer analysis. This bi-directional coupling accounts for temperature-dependent material properties through iterative solution procedures:

$$\begin{array}{l}\sigma (T)={\sigma }_0/[1+\alpha (T-T_0)]\\ B_r(T)=B_{r0}[1+{\alpha }_B(T-T_0)]\end{array}$$

(7)

where $\sigma (T)$ represents temperature-dependent electrical conductivity and $B_r(T)$ denotes the temperature-dependent remanent flux density of permanent magnets. This integrated approach enables accurate prediction of thermal hotspots, particularly in critical components like permanent magnets and rotor structures where cooling is limited. Recent advancements incorporate machine learning techniques to accelerate thermal model parameterization and reduce computational demands while maintaining prediction accuracy.

The thermal boundary conditions and convection coefficients used in this analysis were carefully selected based on experimental calibration. For water cooling interfaces, the convection coefficient was determined through experimental testing using the Wilson plot method, resulting in the correlation h = 2350·v^0.8W/m²K (where v is the coolant velocity in m/s), with an uncertainty of ±12% across the flow rate range of 2–10 L/min. For radiation in vacuum environments, the emissivity values were measured using infrared thermography calibration, yielding ε = 0.85 ± 0.05 for anodized aluminum surfaces and ε = 0.92 ± 0.03 for painted steel surfaces. The thermal contact resistances between components were experimentally determined using the steady-state comparative cut-bar method, resulting in values ranging from 2.1 × 10⁻⁵ to 8.4 × 10⁻⁵ m²K/W depending on interface pressure and surface finish. These experimentally validated thermal parameters ensure reliable temperature predictions across the entire operating range, with maximum temperature prediction errors below 5.8 °C as confirmed through experimental validation.

2.5. Innovative Analytical Models for Loss Estimation and Thermal Prediction

Recent advancements in loss estimation and thermal prediction for high-speed magnetic field-modulated motors have been driven by the development of innovative analytical models that balance computational efficiency with predictive accuracy. These novel approaches transcend traditional methodologies by incorporating multiphysics interactions, machine learning techniques, and hybrid analytical–numerical framework. A particularly promising development is the frequency domain analytical model for eddy current loss prediction in permanent magnets, which decomposes the time-varying magnetic field into harmonic components:

$$P_{eddy}={\displaystyle \frac{{\pi }^2\sigma d^2{l}_m}{6}}{\displaystyle \sum _{n=1}^{\infty }{\left(\frac{n\omega B_n}{2}\right)}^2}$$

(8)

where $\sigma$ represents electrical conductivity, $d$ is the magnet thickness, $l_m$ is the magnet length, $\omega$ is the angular frequency, and $B_n$ is the magnitude of the nth harmonic component. This approach significantly reduces computational complexity while maintaining accuracy for rapid design iterations.

The equivalent thermal circuit transformation (ETCT) technique represents another innovative approach wherein complex three-dimensional thermal networks are systematically reduced to equivalent two-dimensional representations through mathematical transformations:

$$R_{eq}={\displaystyle \sum _{i=1}^n\frac{R_i}{{\lambda }_i}}\cdot {\displaystyle \frac{{\displaystyle \prod _{j=1}^n{\lambda }_j}}{{\displaystyle \sum _{k=1}^n{\displaystyle \prod _{j\ne k}{\lambda }_j}}}}$$

(9)

where $R_{eq}$ denotes the equivalent thermal resistance, $R_i$ is the original thermal resistance, and ${\lambda }_i$ is the eigenvalue of the transformation matrix. This methodology enables rapid thermal analysis with minimal compromise in accuracy, achieving computational speedups of 50–100× compared to full finite element models.

The integration of physics-informed neural networks (PINNs) with domain knowledge has emerged as a powerful paradigm for loss estimation and thermal prediction. These models leverage deep learning architectures while enforcing physical constraints through custom loss functions:

$${\mathcal{L}}_{PINN}={\mathcal{L}}_{data}+{\lambda }_1{\mathcal{L}}_{PDE}+{\lambda }_2{\mathcal{L}}_{BC}+{\lambda }_3{\mathcal{L}}_{IC}$$

(10)

where ${\mathcal{L}}_{data}$ represents the data fitting error, ${\mathcal{L}}_{PDE}$ is the residual of governing differential equations, ${\mathcal{L}}_{BC}$ is the boundary condition error, and ${\mathcal{L}}_{IC}$ is the initial condition error, with ${\lambda }_i$ denoting weighting coefficients. This hybrid approach enables accurate predictions even with limited training data, a common constraint in specialized electrical machine applications.

Modal decomposition techniques have been adapted for thermal transient analysis, wherein the temperature response is expressed as a superposition of thermal modes:

$$T(x,y,z,t)=T_{\infty }+{\displaystyle \sum _{i=1}^m{C}_i}{\varphi }_i(x,y,z)e^{-t/{\tau }_i}$$

(11)

where $T_{\infty }$ represents the steady-state temperature, ${\varphi }_i$ is the spatial modes, ${\tau }_i$ is the time constant, and $C_i$ is the amplitude coefficient determined through proper orthogonal decomposition. This approach enables accurate prediction of thermal transients during variable-speed operation, particularly crucial for flywheel energy storage applications with frequent charging and discharging cycles.

3. Finite Element Analysis of Loss Distribution

3.1. Simulation Setup and Parameter Configuration

The finite element analysis of loss distribution in the high-speed magnetic field-modulated motor was conducted using ANSYS Maxwell 3D v2023 R1 (ANSYS Inc., Canonsburg, PA, USA) for electromagnetic simulations coupled with ANSYS Thermal for thermal analysis. Figure 2 presents the three-dimensional model (Figure 2a) and finite element mesh (Figure 2b) of the motor structure developed for electromagnetic–thermal coupled analysis. The model incorporates detailed geometrical features including precise representation of stator slots, permanent magnets, flux modulators, and rotor components to ensure accurate prediction of magnetic field distribution and associated losses. The finite element mesh was generated with adaptive refinement techniques, implementing higher mesh density in critical regions such as air gaps (0.15 mm element size), permanent magnet edges (0.25 mm), and flux modulator interfaces (0.3 mm) where magnetic field gradients are most pronounced. The total mesh consisted of 1.85 million tetrahedral elements with second-order shape functions, with mesh sensitivity studies confirming that further refinement produced less than 1.2% change in calculated losses.

Material properties were implemented using manufacturer-supplied data with temperature dependence. The silicon–steel laminations (M250-35A) were modeled with nonlinear B-H characteristics and frequency-dependent core loss coefficients (kh = 132 W/m³/T²/Hz, kc = 0.76 W/m³/T²/Hz², ke = 1.85 W/m³/T^1.5/Hz^1.5). Permanent magnets (N42SH) were implemented with Br = 1.28 T, Hc = −955 kA/m, and electrical conductivity of 6.25 × 10⁵ S/m with temperature coefficients of −0.12%/°C and −0.60%/°C for remanence and coercivity, respectively. Copper windings were modeled with temperature-dependent resistivity (ρ₀ = 1.68 × 10⁻⁸ Ω·m, α = 0.00393/°C) and a fill factor of 0.65.

The simulation parameters were configured based on the actual prototype specifications, with nonlinear B-H characteristics applied to the silicon–steel laminations to account for magnetic saturation effects. The permanent magnets were modeled with temperature-dependent properties using a reversible demagnetization model valid for operational temperatures up to 150 °C. The time-stepping transient solver was employed with step sizes corresponding to 1° of mechanical rotation to accurately capture high-frequency harmonics in the magnetic field, particularly those generated at the modulator–air gap interface. Boundary conditions included zero magnetic vector potential at the outer boundaries and continuity conditions at component interfaces.

Boundary conditions were carefully implemented to accurately represent the operating environment. A perfect magnetic boundary condition (magnetic vector potential A = 0) was applied at a distance of five times the motor diameter to minimize boundary effects. For thermal analysis, radiation boundary conditions with emissivity ε = 0.85 were applied to all external surfaces to simulate the vacuum environment, while internal interfaces between components were modeled with thermal contact resistances calibrated from experimental measurements (e.g., 5.2 × 10⁻⁵ m²K/W for the interface between permanent magnets and rotor back iron). Cooling interfaces at the stator water jacket were modeled with convection boundary conditions using experimentally determined heat transfer coefficients that varied with coolant flow rate according to h = 2350·v^0.8 W/m²K, where v represents coolant velocity in m/s.

For electromagnetic loss calculation, the modified Bertotti model was implemented with frequency-dependent coefficients calibrated through material testing. The simulation incorporated mechanical rotation effects through a moving mesh technique with sliding interfaces between stationary and rotating components. The coupled thermal analysis utilized temperature-dependent material properties with radiation boundary conditions in vacuum regions and convection boundaries at cooling interfaces. Solver convergence criteria were set to 10⁻⁵ for magnetic potentials and 0.1 °C for temperature calculations to ensure solution accuracy while maintaining computational efficiency.

3.2. Comparative Analysis of Loss Distribution in Different Components

The finite element analysis reveals distinct patterns of electromagnetic losses across the magnetic field-modulated motor components, with significant implications for thermal management and efficiency optimization. Figure 3 presents a comprehensive visualization of the loss distribution patterns across various components of the high-speed magnetic field-modulated motor used in flywheel energy storage systems. The total electromagnetic loss distribution shown in Figure 3a reveals significant spatial non-uniformity, with peak loss densities of 5.5 × 10⁵ W/m³ occurring predominantly in the modulator region due to the intense flux modulation effect. Figure 3b illustrates the permanent magnet eddy current losses, which exhibit pronounced concentration at the magnet edges and corners where flux density gradients are steepest, with localized hotspots forming at regions aligned with modulator teeth. The modulator component losses depicted in Figure 3c demonstrate the highest magnitude among all components, constituting approximately 45% of total system losses at rated speed, with distinctive periodic patterns corresponding directly to the modulator pole geometry. Armature winding losses visualized in Figure 3d show a complex distribution characterized by slot-specific concentration patterns and significant axial variation, with end-region losses exceeding central-region values by 42% due to enhanced three-dimensional field effects. The end-region loss concentration illustrated in Figure 3e highlights the pronounced axial non-uniformity, with substantially higher loss densities at the axial extremities where magnetic flux fringing creates complex eddy current paths in both active and structural components. The axial cross-sectional view in Figure 3f further confirms this axial variation, demonstrating that two-dimensional analyses would substantially underestimate total losses by neglecting these end effects. Comparative analysis reveals a distinct operational speed dependency, with permanent magnet losses increasing more rapidly than other components due to their approximately quadratic relationship with frequency, suggesting that they become dominant at extreme speeds. These spatially resolved loss distributions provide essential insights for targeted thermal management strategies, particularly for the modulator components and permanent magnets where cooling pathways are most constrained, and for optimized material selection to enhance system efficiency across diverse operating conditions.

3.3. Impact of Operating Speed on Loss Characteristics

The loss–speed relationship curves presented in Figure 4 were obtained through extensive parametric simulations across the entire speed range (0–30,000 rpm) with 2000 rpm increments. For each operating point, steady-state electromagnetic simulations were performed with a time-step corresponding to 1° of mechanical rotation to ensure accurate capture of high-frequency harmonics. The raw simulation data were processed using synchronous demodulation to eliminate numerical noise while preserving the physical characteristics of loss variation. The power law relationships between losses and speed were established through least-squares regression analysis, with goodness-of-fit metrics exceeding R² > 0.98 for all components, confirming the validity of the derived scaling exponents. The minor discrepancies from theoretical scaling factors (e.g., ω^1.8 versus theoretical ω² for core losses) were confirmed through targeted experiments at selected operating points, validating that these deviations represent physical phenomena rather than numerical artifacts.

The relationship between operating speed and loss characteristics in magnetic field-modulated motors exhibits complex nonlinear behavior that significantly impacts system efficiency and thermal management requirements. Analysis reveals distinctive scaling patterns across various loss components that dictate the overall performance of flywheel energy storage systems. Core losses demonstrate a speed dependency of approximately ω^1.8 rather than the theoretical ω² relationship due to material saturation effects at higher flux densities and frequencies. This deviation from classical theory is attributed to microstructural changes in soft magnetic materials under high-frequency excitation. Permanent magnet eddy current losses exhibit even more pronounced scaling behavior, increasing at approximately ω^2.3 due to the amplification of high-order spatial harmonics that penetrate deeper into conductive regions at elevated speeds. Notably, the relative contribution of these components shifts dramatically across the operating range, with core losses dominating at 57% of total losses during high-speed operation compared to just 28% at base speed. Copper losses follow a more complex pattern—initially decreasing with speed in the field-weakening region but subsequently increasing at very high speeds due to enhanced proximity effects. Mechanical losses, though reduced in vacuum environments, still scale approximately as ω^2.8 and account for up to 12% of total losses at maximum speed. As illustrated in Figure 5, the composite loss distribution creates a pronounced efficiency curve, with peak values occurring near rated speed and significant degradation at both extremes. The differential scaling behavior creates distinct thermal challenges, particularly at modulator interfaces where loss density increases by 45% between moderate and high-speed operation. These findings highlight the importance of component-specific loss analysis rather than simplified scaling models when designing thermal management systems for wide-speed-range flywheel energy storage applications.

3.4. Sensitivity Analysis of Loss to Design Parameters

The sensitivity of electromagnetic losses to key design parameters exhibits pronounced nonlinear relationships that significantly impact the thermal behavior and efficiency of high-speed magnetic field-modulated motors in flywheel energy storage systems. As illustrated in Figure 6, parametric analysis reveals that air gap length demonstrates an inverse quadratic relationship with modulator losses but exhibits a more complex influence on permanent magnet eddy current losses, where initial reductions in air gap length decrease losses until reaching a critical threshold of approximately 0.5 mm, beyond which further reductions dramatically increase losses due to enhanced spatial harmonic penetration. As shown in Figure 7, permanent magnet segmentation emerge as a crucial design parameter, with circumferential segmentation providing substantially greater loss reduction (up to 73% with eight segments) compared to axial segmentation (42% reduction with equivalent segmentation) due to fundamental differences in eddy current path disruption mechanisms. A consolidated sensitivity profile is provided in Figure 8. Modulator pole geometry, quantified through the pole arc coefficient, exhibits a parabolic relationship with total losses, achieving minimum values at 0.72–0.78 due to the optimal balance between magnetic loading and harmonic content. Material selection analysis demonstrates that while higher grade silicon steel reduces core losses by up to 35%, this benefit diminishes at extreme speeds where proximity to saturation negates the advantage of improved B-H characteristics. Temperature sensitivity analysis reveals that permanent magnet losses increase by approximately 18% when operating temperature rises from 25 °C to 100 °C due to reduced material resistivity, creating a positive feedback mechanism that can precipitate thermal runaway under inadequate cooling conditions. Rotor skew angle demonstrates particular sensitivity for loss reduction, with an optimal value of 0.8 slot pitches reducing harmonic losses by 68% while maintaining 96% of the rated torque capability. These complex parametric relationships necessitate multi-objective optimization approaches that consider the interdependence of design parameters and their collective impact on both electromagnetic and thermal performance across diverse operating conditions.

4. Loss Reduction Optimization Design

4.1. Proposed Innovative Design Improvements

Several innovative design improvements have been developed to minimize electromagnetic losses and enhance thermal performance in high-speed magnetic field-modulated motors for flywheel energy storage systems. (1) Advanced composite rotor structures incorporating carbon fiber reinforcement with embedded copper shielding layers effectively reduce rotor eddy current losses by up to 67% while maintaining mechanical integrity at extreme rotational speeds. The layered configuration creates destructive interference patterns for induced eddy currents while preserving structural strength characteristics essential for high-speed operation. (2) Optimized modulator geometry utilizing non-uniform pole distribution with sinusoidal profile variation along the axial direction substantially mitigates spatial harmonic components responsible for eddy current losses. This configuration redistributes the magnetic flux more uniformly, reducing peak loss density by 42% compared to conventional uniform modulators. (3) Dual-material permanent magnet configuration combining high-remanence NdFeB segments with high-resistivity SmCo segments strategically positioned at regions of maximum harmonic concentration achieves a 38% reduction in magnet losses compared to homogeneous magnet arrangements. The tailored material distribution effectively addresses location-specific loss generation mechanisms. (4) Integrated thermal management solutions featuring direct-contact liquid cooling channels embedded within modulator structures substantially improve heat dissipation from critical loss hotspots. The microchannel configuration enables precision cooling with minimal impact on electromagnetic performance by maintaining optimal operating temperatures throughout the entire speed range. (5) Adaptively controlled field weakening strategies utilizing real-time loss estimation models dynamically optimize excitation patterns based on instantaneous operating conditions. This approach enables proactive loss management across diverse speed and load profiles, substantially improving system efficiency during transient operations. Implementation of these improvements collectively yields a 29% reduction in total electromagnetic losses while enhancing thermal stability by maintaining temperature gradients below 25 °C across all active components.

4.2. Comparative Analysis of Losses Before and After Optimization

Comprehensive evaluation of the electromagnetic losses before and after implementation of the proposed design improvements reveals significant performance enhancements across the entire operational envelope of the high-speed magnetic field-modulated motor. The optimization strategies collectively achieve a 43.2% reduction in total electromagnetic losses at rated speed, with component-specific improvements varying according to the targeted loss mechanisms, as illustrated in Figure 9. The most substantial improvement occurs in permanent magnet eddy current losses, which decrease by 68.7% due to the combined effects of segmentation, material hybridization, and magnetic shielding. Core losses demonstrate a 37.4% reduction primarily attributable to the optimized modulator geometry and improved material selection, with the greatest benefits observed in the intermediate speed range where the modified flux patterns effectively mitigate harmonic components. Copper losses exhibit a more modest 26.5% reduction through enhanced winding configuration and proximity effect management, remaining largely dependent on current density requirements dictated by torque production. Mechanical losses show similar behavior between original and optimized designs at lower speeds but diverge substantially above 80% of rated speed, with the optimized design maintaining nearly linear scaling compared to the exponential increase observed in the original configuration. The comparative efficiency maps reveal a significant expansion of the high-efficiency operational region, with peak efficiency increasing from 92.3% to 95.8% and the η > 90% region expanding by 42% in the speed–torque plane. Temperature distribution analysis demonstrates that maximum temperature hotspots decrease from 143 °C to 98 °C under identical operating conditions, with temperature gradients reduced by 58% across active components. Transient performance evaluation indicates that the optimized design maintains efficiency above 93% during rapid speed transitions, whereas the original design experiences substantial efficiency degradation to approximately 86% during similar operating profiles. A full three-dimensional comparison is given in Figure 10. These improvements collectively enhance the viability of high-speed magnetic field-modulated motors for flywheel energy storage applications requiring extended operational durability and enhanced thermal stability.

4.3. Thermal Performance Improvement Assessment

The thermal performance improvement assessment of the optimized high-speed magnetic field-modulated motor reveals significant enhancements in heat dissipation capabilities and temperature distribution characteristics across all critical components. Comprehensive thermal analysis demonstrates that the implementation of advanced cooling strategies and optimized electromagnetic design resulted in a 45.3% reduction in maximum temperature rise under identical operating conditions. The temperature distribution exhibits substantially improved uniformity, with maximum temperature gradients decreasing from 38.7 °C to 16.2 °C across active components, effectively mitigating the risk of thermal stress-induced demagnetization and material degradation. Transient thermal response testing verifies enhanced thermal stability during rapid charging and discharging cycles, with temperature rise rates decreasing by 52.7% compared to the original design. Three-dimensional thermal mapping reveals that the innovative dual-material permanent magnet configuration dramatically reduced hotspot formation at the magnet–modulator interface, maintaining temperatures below the critical threshold throughout the entire operating envelope. The strategic implementation of integrated micro-channel cooling structures within the modulator assembly demonstrates exceptional effectiveness, reducing thermal resistance by 63.8% while minimizing the impact on electromagnetic performance.

As illustrated in Figure 11a, axial Temperature Distribution shows the significant reduction in peak temperatures from 140 °C to 95 °C along the motor’s axial length, with substantially reduced temperature gradients. The shaded regions represent measurement uncertainty bounds, demonstrating improved stability in the optimized design. As shown in Figure 11b, dynamic thermal impedance measurements confirm the reduction in thermal time constants from 387 s to 142 s, enabling significantly faster thermal stabilization following load transients. Infrared thermography conducted during extended operation verifies the computational predictions, with measured temperature distributions closely matching simulated profiles within a 5.3% margin. The enhanced thermal performance directly contributed to efficiency improvements by maintaining optimal material properties across diverse operating conditions, particularly evident during high-speed operation where the temperature-dependent performance degradation was substantially mitigated. This comprehensive thermal improvement enables reliable extended operation of the flywheel energy storage system across the full operational spectrum, effectively addressing one of the critical limitations of high-speed magnetic field modulated motors in energy storage applications.

4.4. Efficiency Enhancement Under Various Operating Conditions

The comprehensive efficiency enhancement analysis of the optimized high-speed magnetic-field-modulated motor reveals substantial performance improvements across diverse operating conditions, essential for flywheel energy storage applications. As shown in Figure 12a, the original design demonstrates baseline efficiency with a peak of 92.3% in a relatively narrow operational range, with significant degradation at high speeds and partial loads. In contrast, as illustrated in Figure 12b, the optimized design achieves a peak efficiency of 96.2% and maintains excellent performance (η > 93%) over a much wider region, expanding by 37.8% in the speed–torque domain.

The efficiency enhancement is particularly pronounced at high speeds, where the original design deteriorates rapidly due to escalating core and mechanical losses, whereas the optimized design retains high efficiency even at 110% rated speed. As quantified in Figure 12c, efficiency improvement maps reveal a maximum gain of 7.8 percentage points, especially in the field-weakening region, attributed mainly to the redesigned modulator geometry that suppresses harmonic losses. Figure 12d further compares the field-weakening performance, showing progressively larger improvements in efficiency as speed decreases from rated values.

Transient operation analysis in Figure 12e confirms that the optimized design delivers superior performance during acceleration and deceleration phases, with efficiency degradation limited to only 1.8 percentage points relative to steady-state values, as opposed to 5.3 points in the original design. In addition, Figure 12f highlights improvements in partial-load characteristics, where the optimized machine maintains efficiency above 90% down to 20% of rated load, compared to only 60% in the original motor, thus extending the usable operating range for energy storage under variable load conditions.

5. Experimental Verification

Rigorous experimental validation forms a critical component of this comprehensive analysis, establishing the reliability of the theoretical frameworks and simulation methodologies presented in previous sections. The experimental verification process employed a purpose-built prototype incorporating the optimized design features, tested under conditions that accurately replicate the operating environment of flywheel energy storage systems. This section details the experimental setup, measurement methodologies, and validation results that confirm the accuracy of our loss prediction and thermal modeling approaches.

5.1. Prototype Design and Testing Platform

The experimental validation of the analytical model and finite element results was conducted using a purpose-built prototype and comprehensive testing apparatus at the High-Speed Electrical Machines Laboratory. The prototype motor was manufactured according to the optimized design specifications with key parameters: four pole pairs, 12 stator slots, five modulator segments, 30,000 rpm rated speed, 5 kW rated power, and a 0.8 mm air gap length. Figure 1 displays the key sub-assemblies of the high-speed magnetic field-modulated motor together with the integrated test bench that was built for model verification. The modulator ring in Figure 13a is a high-strength alloy sleeve in which several equally spaced ferromagnetic segments are press-fitted; this segmented construction provides the required spatial harmonics while withstanding the mechanical stress at the rated 30 krpm. Figure 13b shows the laminated silicon–steel stator core (M250-35A, 0.35 mm) whose slot openings and tooth-tip fillets were refined through finite element optimization to mitigate flux leakage and iron loss. The permanent magnet rotor, illustrated in Figure 13c, adopts surface-mounted NdFeB (N40UH) magnets that are circumferentially constrained by a pre-stressed T700 carbon fiber sleeve, guaranteeing mechanical integrity under the high peripheral speed. A flywheel hub with dual inertia discs is presented in Figure 13d; the discs are shrink-fitted to a Ti-6Al-4V shaft, supplying a 0.05 kg·m² moment of inertia that emulates the storage rotor required by the flywheel energy system. The complete experimental rig is shown in Figure 13e: the prototype is connected to a 22 kW prime mover through zero-backlash diaphragm coupling, while a Magtrol HD-815 torque–speed transducer (±0.1% FS, ±1 r min⁻¹) and a Yokogawa WT5000 power analyser (0.01% reading) record mechanical and electrical outputs, respectively.

All tests were conducted inside a 10⁻² Pa vacuum chamber at an ambient temperature of 22 ± 1 °C. Sixteen Neoptix T1 fibre-optic probes (±0.2 °C) were embedded in the stator windings, stator core, permanent magnets, and modulator, and surface temperatures were tracked with a FLIR A655sc infrared camera (FLIR Systems, Wilsonville, OR, USA, ±2 °C). A National Instruments PXIe-1075 system sampled all electrical, mechanical, and thermal channels synchronously at 20 kHz.

The fabricated prototype retains the electromagnetic specifications adopted in the analytical study—four pole pairs, twelve stator slots, five modulator segments, and a 0.8 mm air gap—and is capable of delivering 5 kW at the rated speed of 30,000 r min⁻¹, thereby providing a faithful platform for validating the coupled electromagnetic–mechanical–thermal models developed in this work.

5.2. Measurement Methodology for Different Loss Components

Accurate quantification of individual loss components in the high-speed magnetic field-modulated motor requires a systematic measurement approach with specialized instrumentation and decoupling techniques. The comprehensive loss measurement methodology employs the segregated loss determination principle, wherein total losses are first measured through an input–output power differential using precision power analyzers with sampling rates exceeding 1 MHz to capture high-frequency harmonics. Core losses are isolated through specialized no-load tests with compensated friction and windage contributions determined via deceleration tests at various vacuum levels. Copper losses are measured using a combination of DC resistance tests and AC impedance analysis that accounts for skin and proximity effects at operating frequencies up to 20 kHz. Mechanical losses, particularly challenging in high-speed applications, are quantified through custom-designed coast-down tests with temperature-controlled environments to ensure reproducibility. Rotor losses in permanent magnets and conducting components are indirectly measured using calorimetric methods with carefully calibrated heat flow sensors that can detect power losses as low as 0.5 W. Additional losses are extracted by subtracting all identified loss components from the total measured loss, with uncertainty analysis conducted through Monte Carlo simulations to establish measurement confidence intervals. This comprehensive methodology enables accurate validation of the theoretical loss models and finite element predictions across the entire operating range of the flywheel energy storage system.

5.3. Experimental Results and Validation of Theoretical and Simulation Models

Rigorous experimental validation confirms the remarkable accuracy of both theoretical and finite element models in predicting the electromagnetic loss mechanisms and thermal behavior of high-speed magnetic-field-modulated motors for flywheel energy storage systems. As shown in Figure 14a, total-loss validation across the entire speed range demonstrates strong agreement between analytical predictions, FEA simulations, and experimental measurements, with a mean absolute percentage error of only 4.2%. The experimental error band (±5%) encompasses most predicted values, confirming model credibility.

At rated conditions, Figure 14b details the component-specific breakdown of losses into copper, core, and permanent magnet parts, revealing excellent agreement: 3.1% error for copper, 4.2% for core, and 6.8% for PM eddy-current losses compared to experimental results. Figure 14c further illustrates axial temperature distribution along the stator, with predicted temperature profiles closely matching measurements—maximum deviation limited to 5.8 °C—confirming hotspot prediction accuracy at both ends and center.

Figure 14d validates the electromagnetic model’s capability to capture harmonic behavior in the air-gap flux density, with a correlation coefficient of 0.967 between experimental and theoretical harmonic magnitudes, thus verifying precise implementation of the magnetic field modulation principle. Efficiency consistency across various load levels is demonstrated in Figure 14e, where deviations remain below 1.2 percentage points throughout the entire operating range, confirming the reliability of the model for optimization tasks.

Finally, Figure 14f shows transient thermal response behavior, in which the proposed modal-based prediction method achieves exceptional accuracy—temperature time constants deviate by only 7.3% compared to measured values—supporting the model’s suitability for operational planning in vacuum-sealed environments. Together, these results comprehensively verify the integrated electromagnetic–thermal framework’s predictive reliability and establish a robust basis for high-fidelity performance analysis and optimization in flywheel energy storage systems.

This comprehensive validation establishes the reliability of the integrated multiphysics modeling framework, providing a solid foundation for optimizing high-speed magnetic field-modulated motors in flywheel energy storage applications.

While the analytical and finite element models demonstrate remarkable agreement with experimental measurements, several factors contribute to the observed discrepancies in loss component prediction. The slightly higher deviation in permanent magnet eddy current losses (6.8%) compared to core and copper losses can be attributed to three primary factors. First, manufacturing tolerances in magnet positioning and segmentation create variations in flux paths not fully captured in the idealized simulation model, particularly affecting harmonic content at high frequencies. Second, the temperature-dependent properties of permanent magnet materials introduce additional complexity, as local temperature gradients during operation differ from the uniform temperature assumptions in the simulation, affecting both conductivity and magnetic properties. Third, the measurement methodology for isolating rotor losses relies on indirect calorimetric methods with inherent uncertainty margins of approximately ±3%. For core losses, the deviation of 4.2% stems primarily from the simplified representation of material nonlinearity in the modified Bertotti model, particularly at the steep transition regions of the B-H curve where minor hysteresis loops occur during rapid field variations. The superior accuracy observed in copper loss prediction (3.1%) reflects the more deterministic nature of resistive losses, though skin and proximity effects in the end-winding regions still present challenges for precise modeling. These identified discrepancy sources have informed our ongoing model refinement process, particularly through the implementation of stochastic material property variations and enhanced end-effect modeling techniques that promise to further reduce the gap between simulation and experimental results in future iterations.

5.4. Performance Comparison Under Different Operating Conditions

Comprehensive performance evaluation of both original and optimized high-speed magnetic-field-modulated motors under diverse operating conditions reveals substantial enhancements across critical parameters that significantly influence flywheel energy storage system viability. As shown in Figure 15a, systematic comparison of steady-state performance characteristics demonstrates that the optimized design achieves a 3.9 percentage-point higher peak efficiency at rated load while maintaining over 91% efficiency down to 20% load—offering superior performance across the entire operational range.

As illustrated in Figure 15b, thermal measurements under vacuum operation confirm that the optimized motor achieves a 23.7% lower temperature rise at rated load and remains below critical thresholds even at 120% overload, thanks to the improved thermal management structure. Meanwhile, Figure 15c shows that the optimized design simultaneously reduces acoustic noise by 7.2 dB (A) and vibration amplitude by 64.5% under rated conditions, enhancing mechanical robustness and operational quietness.

Partial-load performance evaluation, as shown in Figure 15d, demonstrates that the optimized design sustains a power factor above 0.92 down to 20% load, while the original motor experiences degradation below 0.85 under 40% load—substantially improving grid interaction characteristics for energy storage scenarios. Environmental robustness assessments further confirm that the optimized design maintains consistent efficiency across ambient temperature variations from –20 °C to +60 °C, with variations of only 1.3 percentage points, compared to 4.8 in the original design.

Acceleration/deceleration performance analysis, as depicted in Figure 15e, indicates that the optimized design achieves 24.8% faster response times during charging transients, reducing the time to reach 90% target speed from 2.76 s to 2.07 s, while maintaining 41.7% smaller speed oscillations during sudden load changes due to better electromagnetic damping.

The performance advantages become particularly pronounced during continuous cycling operations, where the optimized motor maintains thermal stability within ±3.5 °C after reaching steady-state conditions—compared to the greater temperature swings observed in the original design. Figure 15f further validates this robustness through endurance testing: the optimized motor maintains stable thermal and electromagnetic performance after 72 h of continuous operation under rated load, with losses increasing by only 2.1%, significantly lower than the 8.7% rise in the original system.

Together, these comprehensive performance improvements across the entire operational spectrum substantially enhance the practical viability of high-speed magnetic-field-modulated motors for advanced flywheel energy storage applications.

6. Conclusions

This comprehensive investigation into the loss mechanisms and thermal behavior of high-speed magnetic field-modulated motors for flywheel energy storage systems has yielded significant insights and technological advances. Through systematic analysis and innovative optimization approaches, we have successfully identified and quantified the complex interactions between electromagnetic phenomena and thermal responses that influence system performance. The novel loss classification framework and advanced thermal modeling methodologies developed in this study provide a solid foundation for accurate performance prediction across diverse operating conditions. Our optimized motor design, featuring composite rotor structures, dual-material permanent magnets, and integrated cooling channels, achieves substantial improvements with a 3.9% higher efficiency, a 23.7% lower temperature rise, and a 64.5% reduced vibration amplitude. Experimental validation confirms the remarkable accuracy of both theoretical and finite element models, with prediction errors below 4.2%. The enhanced dynamic response characteristics and superior partial load performance significantly expand the practical operational window for flywheel energy storage applications. These comprehensive improvements collectively address critical limitations in existing systems, providing a viable pathway toward high-reliability, grid-scale energy storage solutions with extended operational lifetimes and improved round-trip efficiency. This study establishes a new benchmark for high-speed electromagnetic design in energy storage applications and provides valuable design guidelines for future advancements.