Efficient analytical framework for transient temperature distribution in a multi-layer linear motor with variable ohmic heating

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.