###### Abstract  
Temperature rise of the tubular permanent magnet linear generator (TPMLG) might lead to insulation failure and demagnetization of permanent magnets, affecting the safe and stable operation of other equipment and the entire system. Herein, a bidirectional electromagnetic-thermal coupling method for analyzing the electromagnetic loss and thermal characteristics of a TPMLG considering the effect of increased temperature on the permanent magnet was proposed. To study the electromagnetic-thermal characteristics of the TPMLG under stable power generation, a two-dimensional electromagnetic field model and a three-dimensional temperature field model were established and coupled. The temperature field of the TPMLG was numerically calculated using computational fluid dynamics over finite volume method under natural air cooling and forced air cooling conditions. Effects of loss and air flow velocity on the steady temperature field were investigated. Results indicated that copper loss increased by 24.5% considering the influence of temperature rise. The windings' top central position in the TPMLG was the spot with the highest temperature of 127.8 \({}^{\circ}\)C and there was a potential demagnetization risk for the permanent magnets. Some reference for future research of clarifying thermal characteristics and cooling design was provided. tubular permanent magnet linear motor; electromagnetic-thermal bidirectional coupling; finite volume method; temperature field +
Footnote †: journal: Applied Sciences  
temperature rise in windings [13]. For PMLGs with long-term reciprocating motion, the temperature rise is particularly serious, bringing about a series of problems, such as irreversible demagnetization of permanent magnets, reduced efficiency, shortened lifespan, and insulation deterioration. The high-temperature problems of LGs severely restrict improvement in power generation performance [14]. Obviously, accurately clarifying thermal characteristics is of increasing significance in the design, manufacture, optimization, and application of linear generators. Up till now, many researchers have been making efforts in the analysis of the LG temperature field [14; 15], and mainstream approaches include a lumped-parameter thermal network [16; 17], numerical techniques [18; 19; 20], and hybrid thermal analysis combining two methods [21; 22]. Among the numerical methods, finite element method (FEM) and finite volume method (FVM) [23; 24] are most used. Computational fluid dynamics (CFD) over FVM is commonly used to calculate convection heat transfer coefficients and FEM can model heat conduction of solid components more accurately than the thermal network method. The application occasions of the thermal network and numerical methods are different. All the methods have both advantages and disadvantages, so the methodology selection is a trade-off between cost and value. [PERSON] et al. [11] proposed a multi-physics model of a permanent magnet linear motor by analyzing a one-direction coupling of electromagnetic and thermal fields, and obtained two-dimensional axisymmetric finite element steady-state and transient solutions for the permanent magnet linear motor prototype. The iron loss of magnetic field results was used as the main heat generation source for a three-dimensional finite element thermal simulation to obtain the temperature distribution of the permanent magnet linear motor. [PERSON] et al. [19] conducted a comprehensive study on a double-sided linear induction motor as a braking system with different displacement speeds. Considering both the edge effect and multi-physics factors of the linear motor, a three-dimensional FEM thermal model of the studied linear motor was built. A temperature field model and an electromagnetic field model were two-way coupled and startup thrust force was calculated and analyzed. [PERSON] et al. [21] studied the thermal performance of a 12-slot and 14-pole permanent magnet linear motor under water cooling conditions.