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Compared with the time-consuming numerical method and the complex lumped parameter thermal network method to solve the steady-state heat distribution of the permanent magnet (PM) linear motor, there is no analytical method based on the thermal partial differential equations. This paper aims to propose a two-dimensional (2-D) analytical model for predicting the steady-state temperature distribution of PM linear motors to improve the prediction accuracy and speed up the calculation.

Based on the complex Fourier series theory and Cauchy’s product theorem, this paper presents for the first time a general analytical solution for 2-D temperature field in Cartesian coordinates. Then, by combining the electromagnetic field finite element model (FEM), the copper loss, iron loss and PM eddy current loss are used as the heat sources of the thermal analytical model. Finally, the solution to the temperature field is obtained by solving the system equations under boundary and interface conditions.

The analytical results are in good agreement with those from the thermal FEM, and the calculation speed is significantly faster than that of the thermal FEM.

The multilayer model proposed in this paper can consider heat conduction, convection and radiation. It is not only suitable for PM linear motors but also has significant application value for the thermal analysis of electromagnetic devices modeled in 2-D Cartesian coordinates.

Thermal analysis of electrical machines is usually performed by using numerical methods or lumped parameter thermal networks depending on the desired accuracy. The analytical prediction of temperature distribution based on the formal resolution of thermal partial differential equations (PDEs) by the harmonic modeling technique (or the Fourier method) is uncommon in electrical machines. Therefore, this paper aims to present a two-dimensional (2D) analytical model of steady-state temperature distribution for permanent-magnet (PM) synchronous machines (PMSM) operating in generator mode.

The proposed model is based on the multi-layer models with the convolution theorem (i.e. Cauchy’s product theorem) by using complex Fourier’s series and the separation of variables method. This technique takes into the different thermal conductivities of the machine parts. The heat sources are determined by calculating the different power losses in the PMSM with the finite-element method (FEM).

To validate the proposed analytical model, the analytical results are compared with those obtained by thermal FEM. The comparisons show good results of the proposed model.

A new 2D analytical model based on the PDE in steady-state for full prediction of temperature distribution in the PMSM takes into account the heat transfer by conduction, convection and radiation.

The purpose of this study is to present a novel extended hybrid analytical method (HAM) that leverages a two-dimensional (2-D) coupling between the semi-analytical Maxwell–Fourier analysis and the finite element method (FEM) in Cartesian coordinates.

The proposed model is applied to flat permanent-magnet linear electrical machines with rotor-dual. The magnetic field solution across the entire machine is established by coupling an exact analytical model (AM), designed for regions with relative magnetic permeability equal to unity, with a FEM in ferromagnetic regions. The coupling between AM and FEM occurs bidirectionally (x, y) along the edges separating teeth regions and their adjacent regions through applied boundary conditions.

The developed HAM yields accurate results concerning the magnetic flux density distribution, cogging force and induced voltage under various operating conditions, including magnetic or geometric parameters. A comparison with hybrid finite-difference and hybrid reluctance network methods demonstrates very satisfactory agreement with 2-D FEM.

The original contribution of this paper lies in establishing a direct coupling between the semi-analytical Maxwell–Fourier analysis and the FEM, particularly at the interface between adjacent regions with differing magnetic parameters.