## 3 Methodology of Sweeping Eddy Current Pulse Thermal-Tomography
### Physical Modeling of Sweeping Eddy Current Pulse Thermal-Tomography  
The physical modeling of ECPTT is based on the thermal diffusion equation in 3D Cartesian coordinates, and the thermal diffusion equation is expressed as  
\[\rho C_{p}\frac{\partial T}{\partial t}=\lambda\left(\frac{\partial^{2}T}{ \partial x^{2}}+\frac{\partial^{2}T}{\partial y^{2}}+\frac{\partial^{2}T}{ \partial z^{2}}\right)+q \tag{1}\]  
\[q(x,y,z,t)=\frac{1}{\sigma(x,y,z)}|J(x,y,z)|^{2} \tag{2}\]  
where \(\rho\), \(C_{p}\), and \(\lambda\) are the bulk mass density, specific heat capacity, and thermal conductivity, respectively, \(q\) is the Joule heat power density, \(\sigma\) is the conductivity of the medium, and \(J\) is the current density. ECPTT workflows consist of a heating phase and cooling phase, and these phases are related to multi-physical interactions, including electromagnetic excitation, eddy current pulse heating, and thermal diffusion. Equation (1) shows that \(q\) is influenced by \(\sigma\) and \(J\) at a particular location, and the presence of defects changes \(\sigma\) and \(J\), resulting in different \(q\) values around them. In addition, the defect region also has different \(\rho\), \(C_{p}\), and \(\lambda\), and these changes will have an effect on heating conduction, as mentioned in Equation (1), and hence defects can be detected by anomalous thermal distributions or anomalous transient thermal responses utilizing both induction heating and thermal diffusion. In addition, the induction coil, excitation parameters (duration, frequency, and intensity of the heating pulse) and defect parameters (length, width, orientation, etc.) lead to different thermal responses and distributions. During the heating phase, eddy current pulse heating dominates. The ECPTT system generates eddy currents within the material under inspection by applying electromagnetic excitation, and currents primarily concentrate in the surface layer of the conductor, enhancing the Joule heating effect in this area. The conductor's surface generates more heat as the current passes through, which can be effectively detected by thermal imaging equipment. During the cooling phase, thermal diffusion dominates. The high initial temperature and the narrow shape of the defect result in a fast rate of thermal diffusion, leading to a large temperature gradient. On the other hand, the rail has a large thermal conductivity, and the spatial dimensions of the rail are much larger than the defect, resulting in a uniform cooling phase and a relatively small temperature gradient. Information about the defect can be obtained from the results of thermal diffusion in the thermal image. Figure 1: ECPTT schematic diagram. The distribution of these eddy currents is influenced by the skinfold effect. The skinfold effect refers to the phenomenon where, under the influence of an alternating current or varying electromagnetic fields, the current density inside a conductor decreases rapidly with increasing distance from the surface, causing the current to tend to concentrate near the conductor's surface. The degree of this phenomenon is determined by the skin depth, where the current density decreases to approximately 37% of its surface density. The skin depth can be mathematically expressed as  
\[\delta=\frac{1}{\sqrt{\pi\mu\sigma f}} \tag{3}\]  
where \(\delta\) is the penetration depth; \(f\) is the excitation frequency; \(\mu\) is the magnetic permeability; \(\sigma\) is the electrical conductivity. The relationship between skin depth and excitation frequency is shown in Figure 2. The skin depth is inversely proportional to the excitation frequency. When a high excitation frequency is used, the skin depth is minimized, allowing for the detection of surface defects. As the excitation frequency decreases, the skin depth increases, enabling the detection of deeper defects. Based on this relationship between frequency and skin depth, effective detection and analysis of defects at different depths can be achieved.