## 1 Introduction  
Induction heating usually is applied as the main heating means in various heat treatment processes [1; 2], for example, induction hardening, tempering, stress relieving. Besides, induction heating technology also can be used to assistant some other industrial processes, such as crack detection, depth determination [3], induction heating-assisted compaction [4], etc. In these traditional induction heating processes, the shapes of the induction coils are mostly spiral, that is, the parts to be heated are cylindrical parts. Since the electromagnetic field distribution inside the spiral coil is relatively uniform, the induction heating has the characteristics of fast heating and high efficiency [2]. However, the coil required for the planar induction heating is significantly different from the traditional spiral coils, in which the magnetic field distribution is difficult to effectively concentrated, the air impedance of the system loop is too large and the heating temperature is not uniform and difficult to control accurately [5; 6; 7]. The efficiency of the planar induction heating extensively depends on the effective conversion of the electromagnetic field of the coil and the reasonable matching of the induction heating system. As shown in Figure 1, the emergence of special magnetic flux concentrators has effectively increased the efficiency of the planar induction heating, thus expanding the application of this heating process [8; 9; 10]. In recent years, in order to apply the induction heating process more effectively, many scholars have conducted more research on induction heating [11; 12; 13; 14; 15; 16; 17; 18; 19], including numerical modeling of induction heating process and electromagnetic field conversion mechanism analysis. Numerical approaches are adopted in the power distribution and temperature prediction during induction heating process [11; 12; 13; 14; 15]. [PERSON]. [11] analyzes the machine parameters on the hardness profile heated by induction heating through simulation coupling electromagnetic and heat transfer with finite element methods. [PERSON]. [12] uses the finite element method to investigate the heating stage of a bonding process, especially the temperature evolution in the inner and outer surface of the steel tubes. [PERSON]. [13] conducts the study on the induction hardening based on the electromagnetic-thermal finite element simulation, and verifies the hardening depth and hardening pattern. [PERSON]. [14] focuses on the modeling and design of the induction coils in a brazing process by finite element method. [PERSON]. [10; 15] mainly focus on a planar induction heating process with magnetic flux concentrator, and analyzes the factors influencing the temperature profile and the heating efficiency during the coupled electro-magnetic-thermal field through the finite element simulation and experiments. Numerical simulation based on the finite element method made considerable progress in the induction heating process. However, due to the complexity of the reality physic model, it is inevitable to require the large computational cost, which will cause inconvenience and difficulty in the temperature prediction of the whole computational filed during the induction heating. In addition, analytical approaches are also adapted in the research of various fields. The analytical approaches have less computational time than finite element simulation and thus have a relatively high efficiency [16; 17]. Computational efficiency also depends on the mathematical complexity of the analytical model. [PERSON] uses the analytical model to investigate the temperature and the cutting force during the orthogonal cutting process. The analytical modeling of induction heating is rather different from other manufacturing processes. Some researchers also try to introduce analytical modeling to analyze the induction heating process. [PERSON]. [18] presents a multiple-scale perturbation method to solve the multi-physics mathematical model of induction heating process in a cylindrical coil. [PERSON].