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The aim of this paper is to provide a simple yet accurate and efficient geometric method for thermal homogenization of impregnated and non-impregnated coil winding technologies based on the concept of thermal resistance.

For regular windings, the periodic microscopic cell in the winding space is identified. Also, for irregular windings, the average microscopic cell of the winding is determined. An approximation is used to calculate the thermal resistance of the winding cell. Based on this approximation, the winding insulation is considered as a circular ring around the wire. Mathematical equations are obtained to calculate the equivalent thermal resistance of the cell. The equivalent thermal conductivity of the winding is calculated using equivalent thermal resistance of the cell. Winding thermal homogenization is completed by determining the equivalent thermal properties of the cell.

The thermal pattern of different windings is simulated and compared with the results of different homogenization methods. The results show that the proposed method is applicable for a wide range of windings in terms of winding scheme, packing factor and winding insulation. Also, the results show that the proposed method is more accurate than other winding homogenization methods in calculating the equivalent thermal conductivity of the winding.

In this paper, the change of electrical resistance of the winding with temperature and thermal contact between the sub-components are ignored. Also, liquid insulators, such as oils, and rectangular wires were not investigated. Research in these topics is considered as future work.

Unlike other homogenization methods, the proposed method can be applied to non-impregnated and irregular windings. Also, compared to other homogenization methods, the proposed method has a simpler formulation that makes it easier to program and implement. All of these indicate the efficiency of the proposed method in the thermal analysis of the winding.

To review and discuss recently proposed homogenization methods for laminated magnetic cores and multi‐turn windings in FE models of electromagnetic devices.

The frequency‐domain homogenization is based on the adoption of complex and frequency‐dependent material characteristics (e.g. reluctivity) in the homogenized domain. The value of the complex quantity is obtained analytically or by means of a simple 2D FE model. The time‐domain counterpart requires the introduction of additional unknowns and equation.

The homogenization methods allow to take into account the global eddy current effect in the individual laminations and wires, with a reasonable precision and computational cost.

The homogenization methods have been validated numerically, i.e. by comparison with brute‐force FE computations where the eddy current effects are directly and accurately taken into account. Experimental validation should follow.

The analogy between the homogenization of laminated cores and windings has been evidenced.

The purpose of this paper is to propose a numerical procedure for the extraction of RL equivalent circuits of high frequency multi‐winding transformers with a low computational time.

Rigorous RL equivalent circuits of multi‐winding transformers can be obtained by performing open and short‐circuit tests. In this work, the finite element method (FEM) is employed as a virtual laboratory in order to derive such circuits. However, an accurate modeling of skin and proximity effects in the windings requires extremely dense meshes at high frequencies. Therefore, a 2D frequency‐domain homogenization of the windings, which conducts to coarser meshes, is applied in order to decrease the computational burden. The fine and homogenized models are compared in terms of simulation time as well as accuracy.

A significant decrease in simulation times is observed with the homogenized model (one order of magnitude at high frequencies for 2D models), while keeping acceptable relative error values (below 8 percent in the worst case, taking the fine model as reference). Furthermore, it is shown that the skin effect could contribute in a significant way to the total values of the circuit parameters, especially for high frequencies and for small fill factors. It should therefore not be neglected compared to the proximity effect when gathering such conditions, as commonly assumed in the literature.

Equivalent circuits which capture the skin and proximity effects are obtained at an acceptable computational cost, thanks to the use of homogenization techniques in FE simulations. To the best of the authors knowledge, such a procedure has not yet been published.

This paper aims to derive a simple and effective but still a reasonably accurate model for electromagnetic problems with hysteretic magnetic properties and/or induced currents in heterogeneous regions in 2D, meant particularly for non‐destructive testing (NDT) of steel cables by eddy‐currents.

It is assumed that the diffusion of electromagnetic fields in a heterogeneous cable, which consists of many strands, can be described by the Maxwell equations with periodically oscillating coefficients. A computationally efficient model can then be derived. The idea behind this is to replace the heterogeneous material in the cross‐section by a fictitious homogeneous one, whose behaviour at the macroscopic level is a good approximation of the one of the composite material. Such a homogenized model is obtained by employing the two‐scale convergence.

The model is validated based on experimental electromagnetic data from a steel cable (measured magnetic hysteresis loops) to show that the model is applicable for NDT of cables. The model is useful for studying NDT of cables, both for excitation at low frequency (where changes in magnetic properties are investigated) and at higher frequency (eddy current testing). It is valid for a wide range of amplitudes and frequencies.

From the mathematical point of view the model incorporated a non‐local boundary condition that has to be included in the analysis. From the engineering point of view, by solving an inverse problem based on this model and on measured hysteresis loops at several frequencies, a broader range of defects in the cable can be detected.

The authors aim to search for a practical and accurate way to get good loss estimates for coil filaments in electrical machines, for example transformers. At the moment including loss estimations into standard finite element computations is prohibitively expensive for large coils.

A low-dimensional function space for finite element method (FEM) is introduced on the filament-air interface and then extended into the filament to significantly reduce the number of unknowns per filament. Careful choice of these extensions enables good loss estimate accuracy. The result is a system matrix assembly block that can be used verbatim for all filaments, further reducing the cost. Both net current and voltage per length of the filament are readily available in the problem formulation.

The loss estimates from the developed model agree well with traditional FEM and the computation times are faster.

To produce accurate loss estimates in large coils, the low-dimensional function space is constricted on the filament boundaries. The proposed method enables electrical engineers to compute the ohmic losses of individual conductors.

To prevent the coil from burning or getting damaged, it is necessary to estimate the duration of its operation as long as its temperature does not exceed the permissible limit. This paper aims to investigate the effect of switching on the thermal behavior of impregnated and nonimpregnated windings. Also, the safe operating time for each winding is determined.

The power loss of the winding is expressed as a function of the winding specifications. Using homogenization techniques, the equivalent thermal properties for the homogenized winding are calculated and used in a proposed thermal equivalent circuit for winding modeling and analysis. The validity and accuracy of the proposed model are determined by comparing its analysis results and simulation and measurement results.

The results show that copper windings have better thermal behavior and lower temperature compared to aluminum windings. On the other hand, by impregnating or increasing the packing factor of the winding, the thermal behavior is improved. Also, by choosing the right duty cycle for the winding current source, it is possible to prevent the burning or damage of the winding and increase its lifespan. Comparing the measurement results with the analysis results shows that the proposed equivalent circuit has an error of less than 4% in the calculation of the winding center temperature.

In this paper, the effect of temperature on the electrical resistance of the coil is ignored. Also, rectangular wires were not investigated. Research in these topics are considered as future work.

By calculating the thermal time constant of the winding, its safe operation time can be calculated so that its temperature does not exceed the tolerable value (150 °C). The proposed method analyzes both impregnated and nonimpregnated windings with various schemes. It investigates the effects of switching on their thermal behavior. Additionally, it determines the safe operating time for each type of winding.

The purpose of this paper is to present a homogenization approach to model mechanical structures with multiple scales and periodicity, as they occur, e.g. in power transformer windings, subjected to magnetic forces.

The idea is based on the framework of generalized finite element methods (GFEM), where the normal polynomial finite element basis functions are enriched by problem dependent basis functions, which are, in this case, the eigenmodes of a quasi‐periodic unit cell setup. These eigenmodes are used to enrich the standard polynomial basis functions of higher order on a coarse grid modeling the whole periodic structure.

It is shown that heterogeneous magnetomechanical structures can be homogenized with the developed method, as demonstrated by homogenization of a transformer coil setup.

An efficient homogenization procedure is proposed on the basis of the GFEM, which is extended using a special set of enrichment functions, i.e. the mechanic eigenmodes of a generalized eigenvalue problem.

The purpose of this paper is to propose a general approach for the frequency‐domain homogenization of electromagnetic periodic structures. The method allows calculating macroscopic equivalent properties including local effects. It is based on the equivalence of active and reactive electromagnetic powers on an elementary cell. This work is applied to the modelling of eddy current losses in windings, by the use of the finite element method in 2D and 3D.

The approach is based on an homogenization technique, allowing describing local properties (permeability and conductivity) and local effects (eddy currents) of periodical structures, through macroscopic homogenized behaviour laws.

It was found that the presence of local loops of eddy currents at the local scale implies that the average values of the electric and magnetic field are different from the macroscopic fields. This implies some precautions to implement the homogenization. Furthermore, the question of the coupling of the macroscopic laws has been clarified.

The proposed method is limited to the frequency domain. Some additional work is necessary to extend the researches in the time domain.

The proposed methodology is applied for determining losses in coils with the finite element method. The major interest of the method is that it allows taking into account local effects (losses in particular), with a reduced computational time.

The method proposed in this paper is general and clarifies the principle of homogenization in the case of periodical structure in presence of local eddy currents (local loops of current).

The purpose of this paper is to determine the broadband frequency response of the impedance matrix of wireless power transfer (WPT) systems comprising litz wire coils.

A finite-element (FE)-based method is proposed which treats the microstructure of litz wires by an auxiliary cell problem. In the macroscopic model, litz wires are represented by a block with a homogeneous, artificial material whose properties are derived from the cell problem. As the frequency characteristics of the material closely resemble a Debye relaxation, it is possible to convert the macroscopic model to polynomial form, which enables the application of model reduction techniques of moment-matching type.

FE-based model-order reduction using litz wire homogenization provides an efficient approach to the broadband analysis of WPT systems. The error of the reduced-order model (ROM) is comparable to that of the underlying original model and can be controlled by varying the ROM dimension.

Since the present model does not account for displacement currents, the operating frequency of the system must lie well below its first self-resonance frequency.

The proposed method is well-suited for the computer-aided design of WPT systems. It outperforms traditional FE analysis in computational efficiency.

The presented homogenization method employs a new formulation for the cell problem which combines the benefits of several existing approaches. Its incorporation into an order-reduction method enables the fast computation of broadband frequency sweeps.

This paper aims to present a 3D numerical analysis of the load noise generation associated with large, oil immersed three‐phase power transformers.

After studying the mechanical behavior of the winding structures of transformers, the results of coupled magneto‐mechanical simulations are presented.

An appropriate modeling strategy of the vibratory winding structures of transformers is necessary to reduce complexity and computational resources.

The presented model setup describes a fully transient, 3D coupled magneto‐mechanical simulation of the vibratory winding structure of large power transformers.