Benchmarks are crucial in electrical machine research to establish reference points for evaluating performance improvements. Identifying key characteristics becomes challenging when there is no data available and measurement techniques are expensive or destructive. Solving the inverse problem to determine material properties, such as in a drone brushless DC motor (BLDC), is ill-posed when only system-level measurements are available, and deterministic methods risk converging to local solutions without indication of uniqueness. This paper proposes a probabilistic approach using Bayesian inference (BI) supported by neural network surrogate models trained on finite element simulation data, to infer the material properties of the permanent magnets, stator, and rotor core non-destructively. The framework demonstrates that BI is applicable to solve this class of inverse problem, delivering parameter estimates with credible intervals, correctly identifying magnet remanence and stator saturation flux density, while revealing the limitations of the arctangent BH model under mixed operating conditions.

This paper proposes an original integrated forward-inverse thermal framework for estimating thermal and interfacial parameters in multi-layer structure of electrical machines under ohmic heating. A nonhomogeneous heat transfer problem in multi-layer structure is studied using a permanent magnet linear synchronous motor (PMLSM). Two computationally efficient analytical forward models are developed, a separation of variables-orthogonal expansion (SOV-OE) solution for constant heat generation, and a Green's function (GF) based solution for time-varying heat generation. The analytical models are integrated with an interior-point optimization algorithm to identify effective thermal properties and delamination-related parameters. The framework is validated using synthetic finite element (FE) datasets and experimental measurements from a water-cooled single-phase motorette. The experimental results indicate that the epoxy thermal conductivity, specific heat capacity, and equivalent heat transfer coefficient can be consistently identified from limited internal temperature measurements under various current excitations, demonstrating the potential of the proposed framework for online thermal condition monitoring.

This research presents fast and closed-form analytical solutions for transient thermal modeling in multi-layer composite structures with variable internal heat generation. A five-layer segment of a water-cooled permanent magnet linear synchronous motor (PMLSM) is analyzed under both constant and time-varying ohmic heating to validate the proposed models. The separation of variables (SOV) method is employed to decouple spatial and temporal components, enabling the determination of eigenvalues. The orthogonal expansion (OE) technique is applied to compute Fourier coefficients based on the regular Sturm–Liouville theorem. For constant heat sources, an analytical solution is derived by combining the SOV method with the OE technique. To address transient heat sources and other non-homogeneous conditions including temperature-dependent thermal conductivity, a Green’s function (GF) based approach is developed. The results show that the proposed method offers significantly faster computation compared to finite element (FE) methods, while achieving even higher accuracy. This modeling framework provides an efficient tool for thermal analysis of electrical machines and a forward computational foundation for advanced applications, such as inverse modeling to detect material property variations during long-term operation.

A fast inverse heat conduction model (IHCM) is developed for estimating unknown properties of multi-layer structures considering internal heating. The model leverages a closed-form analytical forward solution to enable efficient inverse computations. It requires only a single internal temperature measurement as input, with unknown parameters estimated by minimizing an objective function using an interior-point optimization algorithm. The IHCM accurately identifies thermal properties such as thermal conductivity, specific heat capacity, density, and heat transfer coefficient in a high-precision linear motor. It also detects internal geometric variations, including the location and severity of delamination caused by thermal expansion. These predictions are validated against finite element (FE) simulations. Furthermore, a sensorless strategy is proposed, enabling non-invasive parameter estimation based on electrically inferred temperature data. The feasibility, sensitivity, and limitations of the IHCM are assessed across various scenarios. Results demonstrate its strong potential for real-time diagnostics, online defect detection, and thermal performance monitoring in multi-layer composite systems with internal heat generation, such as electrical machines.

Detecting defects and assessing the condition of electrical machines are critical for ensuring their reliability and efficiency. Changes in thermal properties often indicate defects such as thermal aging or delamination. This study introduces an inverse modeling approach designed to estimate internal thermal properties and detect potential defects in multi-layer permanent magnet linear synchronous motors (PMLSMs), subjected to internal heat generation from the coils. A one-dimensional (1D) analytical solution to the non-homogeneous heat equation is derived for the composite layers using the separation of variables method (SVM) and the orthogonal expansion technique (OET). The solution is validated against finite element (FE) simulations, which also serve to create reference temperature datasets required by an inverse heat transfer model (IHTM). The proposed inverse method accurately detects changes in thermal properties and identifies the location and thickness of the delamination layer. Overall, this research confirms the feasibility of monitoring internal thermal property variations with minimized temperature measurements, highlighting its potential for early detection of thermal defects in electrical machines.

In this paper, two anisotropic thermal modeling approaches based on finite element (FE) and thermal equivalent circuit (TEC) are proposed and their performance are investigated, considering anisotropic thermal conductivity and bent geometry of a racetrack foil coil. Position-dependent thermal conductivity for the bent part of the racetrack has been applied. Three-dimensional (3-D) heat conduction partial differential equations (PDEs) in both Cartesian and cylindrical coordinate systems are solved to determine the equivalent thermal resistances considering anisotropy. The results from the proposed approaches are compared with a multi-layer FE model and a homogenization FE model, exhibiting significant advantages in accounting for the temperature gradient across different materials and simplifying the pre-modeling process.

High-tech systems require precise motion control in the subnanometer position error range. With no magnetic interference and low mass, piezoelectric actuators are ideal for sensitive and high-speed environments. The proposed piezoelectric actuator, a multi-layer piezo stack made of lead zirconate titanate (PZT) material, is affected by non-linearities (e.g., hysteresis, dielectric relaxation). Therefore, to deliver high accuracy, precise mathematical modelling is essential to enable control. Current mathematical models often fail to capture all static and dynamic effects related to charge and voltage prediction. This paper introduces a comprehensive physics-based model that holistically integrates key physical phenomena, including inertia and saturation in the operational hysteresis. In this work, the model's parameters are identified and its performance is validated against experimental data across a range of operating conditions. The results demonstrate that the proposed model achieves a superior prediction accuracy, improving performance significantly, as compared to established methods. This outcome confirms the model's ability to reliably predict actuator behaviour.

This study presents fast and accurate analytical methods for transient thermal modeling in multi-layer composites with an arbitrary number of layers. The proposed approach accounts for internal heat generation and non-homogeneities in the heat diffusion equation. The separation of variables (SOV) method is employed to decouple spatial and temporal components, enabling the determination of eigenvalues. The orthogonal expansion (OE) technique is then applied to compute Fourier coefficients using 'natural' orthogonality. An analytical solution for composites with constant heat sources is developed by combining the SOV method and OE technique. Additionally, a Green's function (GF) based approach is formulated to handle transient heat sources and other non-homogeneous conditions, including temperature-dependent thermal conductivity. The results demonstrate that the proposed method offers significantly faster computations compared to finite element (FE) methods, while maintaining high accuracy. This forward modeling approach serves as an efficient basis for inverse modeling, aimed at estimating unknown material properties and geometric deformations, which are explored in Part II of this study.

In reluctance machines, repeated evaluations of complex topologies for geometric scaling make electromagnetic design optimization challenging. Accurate modeling requires capturing nonlinearities due to frequent operation in the saturated regime. This paper presents a grey-box magnetic modeling approach for variable flux reluctance machines (VFRMs). The proposed method combines the geometry-scaling flexibility and computational efficiency of the harmonic model (HM) with finite element method (FEM)-based data, to account for magnetic saturation effects in electromagnetic scaling. The proposed method constructs a dataset by computing the airgap magnetic field distribution for a range of current excitation values and selected geometric parameters that influence saturation. A non-parametric and probabilistic machine learning model, i.e. Gaussian Process Regression (GPR), is used to learn the discrepancy between the Fourier series (FS) coefficients of the airgap flux density obtained from HM and FEM. The trained model is used to correct the FS coefficients of the HM, allowing it to emulate FEM-level accuracy without requiring full simulations for each design point. The method is demonstrated on a 12-stator and 10-rotor pole VFRM. Results show that the proposed approach enables fast yet accurate performance prediction in a varying saturation behavior. The proposed grey-box modeling approach also enables training towards various magnetic behavior patterns typically encountered in electrical machine scaling.

A fast inverse heat conduction model (IHCM) is developed for estimating unknown properties of multi-layer composites considering internal heat generation. This work builds on the validated analytical forward models presented in Part I. Transient temperature at a single point is used as input, with the objective function minimized through an interior-point optimization algorithm. The IHCM accurately estimates thermal properties such as thermal conductivity, specific heat capacity, density, and heat transfer coefficient. It also identifies internal geometric variations and their locations, such as delamination caused by thermal expansion or mechanical motion. These predictions are validated through finite element (FE) simulations. Additionally, a sensorless strategy is introduced, providing a non-invasive inverse modeling approach. The feasibility, sensitivity and limitations of the proposed IHCM are evaluated across various scenarios. The results demonstrate strong potential for applications such as thermal performance monitoring, online defect detection, and real-time diagnostics in multi-layer composite systems.

This article concerns the synthesis of a coil for a moving-magnet planar motor that reduces the inherent force ripple, which is present in the traditional racetrack coil. The synthesis is performed by the stream-function method (SFM), which gives extra degrees of freedom in the design and allows the coil shape to be defined based on constraints on the force production. The shape of the optimized coil is analyzed, and the harmonic content in the force production is compared with a racetrack coil. The coil synthesized by the SFM reduces the peak and root-mean-square value of the parasitic force in the y-direction by 73%, and the harmonic content in the force production in the x-, y-, and z-directions is reduced by over 71.4%.

This paper proposes a novel post-processing-based control strategy for the flux-weakening operation of variable flux reluctance machines. The proposed method achieves a high-efficiency operation at elevated speeds by determining the optimal values for both d- and q-axis currents together with the dc-field current. A 5 kW variable flux reluctance machine, which develops a continuous maximum torque of 16 Nm at a base speed of 3000 rpm, is modeled using a nonlinear magnetodynamic finite element method model. The nonlinear magnetic characteristics of the laminated rotor and stator steels are simulated in transient to calculate the torque production, back-EMF voltage, and efficiency in relation to the excitation parameters. The proposed control algorithm applies the scattered data interpolation in the post-process to obtain possible combinations of the excitation currents for a specific torque reference. An efficiency map for the analyzed variable flux reluctance machine has been generated at a rotor speed of 5000 rpm, taking both copper and iron losses into account. The findings demonstrate that the proposed control strategy ensures an efficiency exceeding 85% during the flux-weakening operation by manipulating the controllable dc-field current, thereby enabling high-efficiency performance beyond the base speed.

This research proposes a novel current reference selection algorithm based on the most critical performance constraints to achieve improved control of a variable flux reluctance machine (VFRM). The proposed strategy defines the VFRM’s torque ripple, efficiency, and power factor, which are the inherent technological challenges in the torque and efficiency profile of this machine class, as a cost function and aims to select the field and armature current combination that satisfies the torque requirement while minimizing the cost function. These cost function components of the VFRM are initially determined using a transient finite element model (FEM), and the machine representation in the electromagnetic domain is then built using regression based on the generated dataset. The paper includes the details of the representation of the machine parameters, such as the design of experiments for dataset generation and comparison of different regression analyses. Finally, the proposed current reference selection algorithm is compared with the conventional strategies, and its performance and limitations within the entire drive operation range are evaluated.

Complex piezoelectric systems are foundational in industrial applications. Their performance, however, is challenged by the nonlinear voltage-displacement hysteretic relationships. Efficient characterization methods are, therefore, essential for reliable design, monitoring, and maintenance. Recently proposed neural operator methods serve as surrogates for system characterization but face two pressing issues: interpretability and generalizability. State-of-the-art (SOTA) neural operators are black-boxes, providing little insight into the learned operator. Additionally, generalizing them to novel voltages and predicting displacement profiles beyond the training domain is challenging, limiting their practical use. To address these limitations, this paper proposes a neuro-symbolic operator (NSO) framework that derives the analytical operators governing hysteretic relationships. NSO first learns a Fourier neural operator mapping voltage fields to displacement profiles, followed by a library-based sparse model discovery method, generating white-box parsimonious models governing the underlying hysteresis. These models enable accurate and interpretable prediction of displacement profiles across varying and out-of-distribution voltage fields, facilitating generalizability. The potential of NSO is demonstrated by accurately predicting voltage-displacement hysteresis, including butterfly-shaped relationships. Moreover, NSO predicts displacement profiles even for noisy and low-fidelity voltage data, emphasizing its robustness. The results highlight the advantages of NSO compared to SOTA neural operators and model discovery methods on several evaluation metrics. Consequently, NSO contributes to characterizing complex piezoelectric systems while improving the interpretability and generalizability of neural operators, essential for design, monitoring, maintenance, and other real-world scenarios.

A multiple-input, multiple-output planar actuator is proposed, utilizing mechanically driven stator magnet arrays to levitate a permanent magnet mover. A state of levitation and actuation is obtained by mechanically altering the orientation of the stator magnets to control the forces and torques on the mover. A challenge in the actuator is to deal with the incapability of magnetic-field amplitude control, which inherently follows from utilizing PMs in both the stator and mover. This leads to increased sensitivity and limits actuator controllability. To address this issue, a data-driven approach is introduced to mitigate peak sensitivities within the system. The problem is simplified into a two-body interaction model between a cylindrical stator magnet and a spherical mover magnet. The magnetic potential energy interaction is calculated with the equivalent surface charge model, after which a feedforward neural network is applied to approximate the magnetic potential energy to accelerate the model. Data generation involves simulating 106 randomized stator angles and observing the resulting magnetic potential energy on the mover. The mRMR feature selection method is applied for sensitivity analysis to the established dataset. Results indicate that an equilateral triangular stator topology with a rectangular mover minimizes sensitivity, which improves the stability and control potential of the actuator, while the mRMR technique is also shown to be an effective tool for selecting control parameters for a very large stator.

Hysteresis modeling is crucial to comprehend the behavior of magnetic devices, facilitating optimal designs. Hitherto, deep learning-based methods employed to model hysteresis face challenges in generalizing to novel input magnetic fields. This article addresses the generalization challenge by proposing neural operators for modeling constitutive laws that exhibit magnetic hysteresis by learning a mapping between magnetic fields. In particular, three neural operators-deep operator network (DeepONet), Fourier neural operator (FNO), and wavelet neural operator (WNO)-are employed to predict novel first-order reversal curves and minor loops, where novel means that they are not used to train the model. In addition, a rate-independent FNO is proposed to predict material responses at sampling rates different from those used during training to incorporate the rate-independent characteristics of magnetic hysteresis. The presented numerical experiments demonstrate that neural operators efficiently model magnetic hysteresis, outperforming the traditional neural recurrent methods on various metrics and generalizing to novel magnetic fields. The findings emphasize the advantages of using neural operators for modeling hysteresis under varying magnetic conditions, underscoring their importance in characterizing magnetic material-based devices. The codes related to this article are available at https://github.com/chandratue/magnetic_hysteresis_neural_operator.

This paper presents an extended design methodology for variable flux reluctance machines to investigate their potential for heavy-duty vehicle applications. The proposed strategy considers both constant-torque and flux-weakening operations in the design stage to maximize the efficiency, torque density, and power factor and to minimize the torque ripple. The nonlinear magnetostatic finite element analysis is coupled with the single-valued curve of the laminated soft-magnetic rotor and stator for the constant-torque region, while the nonlinear magnetodynamic model, including the classical eddy current and excess fields in the laminated steel, is utilized for the maximum speed operation. A 3-ton fully electric tractor is selected as the benchmark for the feasibility study, where the maximum continuous torque is 200 Nm at a nominal speed of 1600 rpm. It is demonstrated that the optimal high-torque variable flux reluctance machine achieves a torque density of 20 Nm/L and a power factor above 0.7 in both constant-torque and flux-weakening regions. Torque ripple is 10% during continuous operation, with efficiency reaching 90% at the nominal speed and 95% in flux weakening due to reduced copper loss by adjusting the dc-field excitation. Additionally, the optimal design is capable of sustaining 400 Nm overload for up to 65 seconds before reaching the 100 degrees C winding temperature limit.

Hysteresis is a ubiquitous phenomenon in magnetic materials; its modeling and identification are crucial for understanding and optimizing the behavior of electrical machines. Such machines often operate under uncertain conditions, necessitating modeling methods that can generalize across unobserved scenarios. Traditional recurrent neural architectures struggle to generalize hysteresis patterns beyond their training domains. This paper mitigates the generalization challenge by introducing a physics-aware recurrent neural network approach to model and generalize the hysteresis manifesting in sequentiality and history-dependence. The proposed method leverages ordinary differential equations (ODEs) governing the phenomenological hysteresis models to update hidden recurrent states. The effectiveness of the proposed method is evaluated by predicting generalized scenarios, including first-order reversal curves and minor loops. The results demonstrate robust generalization to previously untrained regions, even with noisy data, an essential feature that hysteresis models must have. The results highlight the advantages of integrating physics-based ODEs into recurrent architectures, including superior performance over traditional methods in capturing the complex, nonlinear hysteresis behaviors in magnetic materials. The codes and data related to the paper are at github.com/chandratue/HystRNN.

In high-precision motion systems, a position controller generates a reference signal that power amplifiers utilize to drive actuators, thereby translating the reference into the desired motion. Common actuators include piezoelectric actuators and electromagnetic actuators. Compared to electromagnetic actuators, piezoelectric actuators have a higher force density, which can be one to two orders of magnitude greater than that of equivalently sized electromagnetic actuators. In addition, they do not generate electromagnetic interference. However, their mechanical stiffness is often thousands of times higher than the electromagnetic actuators and they exhibit inherent material nonlinearities. These distinct piezoelectric actuator characteristics significantly influence the dynamic behavior and achievable precision of the overall motion systems. In such systems, the performance of the actuator drivers is particularly critical. Any imperfection in the power amplifier is passed on to these actuators, adversely affecting overall system accuracy. Hence, to improve piezoelectric motion system performance, this paper investigates the effects and contributions of power amplifier characteristics when coupled with such high-stiffness, nonlinear actuators.

The phase composition, density, microhardness and fracture toughness of (ZrO2)1-x(R2O3)х crystals (where R = Y, Sm and Gd) for x = 0.02–0.04 have been compared. The crystals have been grown using directional melt crystallization in a cold crucible. The phase composition of the crystals has been studied using X-ray diffraction and Raman spectroscopy. The microhardness and fracture toughness of the crystals have been evaluated by means of indentation. At stabilizing oxide concentrations of ≥ 2.8 mol.% for Y2O3 and Gd2O3 and ≥ 3.7 mol.% for Sm2O3 the crystals have densities close to the theoretical ones and contain two tetragonal phases. At lower stabilizing oxide concentrations the crystals contain the monoclinic phase. The fracture toughness of the tetragonal crystals increases with the ionic radius of the stabilizer. The highest fracture toughness values achieved when stabilized by a specific oxide are 11.0, 13.0 and 14.3 MPa·m1/2 for the 2.8YSZ, 2.8GdSZ and 3.7SmSZ crystals, respectively. The fracture toughness proves to depend on the crystallographic orientation of the crystals. The results of this work can be used in the design and fabrication of various structural components and devices.