## 2 Formulation of the PEC Analytical Model  
The PEC analytical model, as an effective tool, has been used to analyze the change of PEC signals caused by thickness, conductivity and permeability [4; 14]. According to previous studies, it can be found that it is very difficult to directly construct a PEC analytical model. However, on the other hand, it is simpler to construct a PEC analytical model based on the harmonic eddy current analytical model. It is known that a periodic pulse signal can be expressed as a Fourier series of trigonometric functions. Therefore, in this paper, the PEC analytical model is reconstructed through using the harmonic eddy current analytical model. The main process is shown in Figure 1.  
In the process of constructing the PEC analytical model, a pulse voltage signal \(y(t)\) serves as an excitation signal of the coil sensor. The difference PEC signal \(\Delta u(t)\) is the voltage change of the coil sensor when a plate is tested. The harmonic eddy current responses \(\Delta Z(\omega)\) and \(Z_{air}(\omega)\) in the frequency domain are the basis of the PEC model construction, which have been built in previous studies [15; 16]. The case for the coil sensor above a conductive plate was considered, as illustrated in Figure 2. The PEC probe was an air-core cylindrical coil. The plate under research was assumed to be linear, homogeneous and isotropic. The electrical conductivity, permeability and thickness of the measured metallic plate are denoted as \(\sigma\), \(\mu\) and \(d\), respectively, where \(\mu=\mu_{0}\mu_{r}\) (\(\mu_{0}\) is the permeability of the vacuum, \(\mu_{r}\) is the relative permeability of the metallic plate). An artificial magnetic insulation boundary was placed at \(h\). Based on the harmonic eddy current response and spectral pulse signals, the difference PEC response can be formulated by performing Kirchhoff's Law in the frequency domain. It signifies that the algebraic sum of the voltages of the coil sensor is always equal to zero in a closed loop, which can be expressed as \(\Delta U=\left|U_{plate}-U_{air}\right|=\left|I\times(Z_{plate}-Z_{air})\right|\). Correspondingly, the difference PEC response in the frequency domain can be obtained. Where \(U_{plate}\) and \(Z_{plate}\) are the voltage and impedance signal of coil sensor when a plate is tested, \(U_{air}\) and \(Z_{air}\) are the voltage and impedance signal of the coil sensor when the coil sensor is in air, the voltage signal \(U_{air}\) and the impedance signal \(Z_{air}\) serve as a reference  
Figure 1: Flow chart of pulsed eddy current (PEC) model. Figure 2: Coil sensor above a plate. signal, \(\Delta U\) is the difference PEC response between the detection signal \(U_{plate}\) and reference signal \(U_{air}\), \(I\) is the current of coil sensor, \(\Delta Z=\left|Z_{plate}-Z_{air}\right|\) is the impedance change response when a plate is tested. Correspondingly, the difference PEC response in the frequency domain can be expressed as  
\[\Delta U(\omega)=\frac{Y(\omega)}{\Delta Z(\omega)+Z_{air}(\omega)}\times \Delta Z(\omega) \tag{1}\]  
where \(Y(\omega)\) denotes the Fourier transform of pulse voltage signal \(y(t)\), \(Z_{plate}(\omega)=\Delta Z(\omega)+Z_{air}(\omega)\), \(Y(\omega)/(\Delta Z(\omega)+Z_{air}(\omega))=I\). By using inverse Fourier transform (IFT), the difference PEC signal in the time domain can be expressed as  
\[\Delta u(t)=\text{IFT}[\Delta U(\omega)]. \tag{2}\]  
Based on the results, a numerical calculation program can be written in MATLAB software (MTLAB R2014a, Math Works, Inc, Natick, Massachusetts, United States, 2014). Correspondingly, some scientific explanations can be verified by numerical method.