## 2 Theoretical Analysis
### Solution for PECT Analytical Model  
As shown in Figure 1, a ferromagnetic plate with an insulation and a cladding is modeled as a four-layered structure for simplicity. Layers from bottom to top represent the air, the plate, the insulation, and the cladding, successively. The layer over the cladding is layer 5, and it is divided into three subregions. The sensor consisting of transmitter and receiver coils with rectangular cross-sections is in section I-II. According to the reflection and transmission theory of electromagnetic waves, the expression of the reflection coefficient at the interface between layers \(k+1\) and \(k\) is:  
\[R_{k+1,k}(\alpha)=\frac{\mu_{rk}\beta_{k+1}-\mu_{r(k+1)}\beta_{k}}{\mu_{rk} \beta_{k+1}+\mu_{r(k+1)}\beta_{k}}k=1,2,3,4 \tag{1}\]  
where \(\beta_{k}=(\alpha^{2}+j\omega\mu_{0}\mu_{rk}\sigma_{k})^{1/2}\); \(\mu_{rk}\) and \(\sigma_{k}\) are the relative magnetic permeability and electrical conductivity of layer \(k\), respectively; \(j\) is the imaginary unit; \(\omega\) is the angular frequency of the sinusoidal harmonic; \(\mu_{0}\) is the permeability of vacuum; \(\alpha\) can be understood as a wavenumber [5]. Received waves are the superposition of reflection waves from multiple interfaces, thus some scholars have defined a generalized reflection coefficient to represent the superposition of reflection waves [20; 21; 22]. The generalized reflection coefficient \(R^{\prime}{}_{k+\ 1,k}\) is defined as the ratio of the electromagnetic wave reflected at all the interfaces between the layer \(1\) and \(k+1\) to the electromagnetic wave incident from the layer \(k+1\) to \(k\). It satisfies the recursive Equation (2). \[R^{\prime}{}_{k+1,k}(\alpha)=\frac{R_{k+1,k}(\alpha)+R^{\prime}{}_{k,k-1}( \alpha)e^{-2\beta_{k}(d_{k-1}-d_{k})}}{1+R_{k+1,k}(\alpha)R^{\prime}{}_{k,k-1}( \alpha)e^{-2\beta_{k}(d_{k-1}-d_{k})}}k=2,3 \tag{2}\]  
with the initial condition \(R^{\prime}{}_{2,1}(\alpha)=R_{2,1}(\alpha)\), where \((d_{k-1}-d_{k})\) denotes the thickness of layer \(k\). \(R^{\prime}{}_{5,4}(\alpha)\) is the generalized reflection coefficient of the four-layered structure which could be derived:  
\[R^{\prime}{}_{5,4}(\alpha)=\frac{R_{5,4}(\alpha)+R^{\prime}{}_{4,3}(\alpha)e^ {-2\beta_{4}(d_{3}-d_{4})}}{1+R_{5,4}(\alpha)R^{\prime}{}_{4,3}(\alpha)e^{-2 \beta_{4}(d_{3}-d_{4})}} \tag{3}\]  
For each frequency component, the induced voltage in the receiver coil could be deduced based on the Dodd-Deeds model [23]:  
\[\Delta U(\omega)=j\pi\omega\mu_{0}I(\omega)\times\int_{0}^{\infty}e^{-2\alpha I _{1}}S(\alpha)R^{\prime}{}_{5,4}(\alpha)d\alpha \tag{4}\]  
where \(I(\omega)\) is the amplitude of the harmonic excitation current; \(S(\alpha)\) is the spatial frequency spectra of the sensor which gives the amplitude of the contributions as a function of \(\alpha\)[21]. Figure 1: Schematic of a sensor over a four-layered structure. \[S(\alpha)=n_{T}n_{R}\frac{Int(\alpha r_{1T},\alpha r_{2T})}{\alpha^{2}(r_{2T}-r_{1T })}\frac{Int(\alpha r_{1R},\alpha r_{2R})}{\alpha^{2}(r_{2R}-r_{1R})}\frac{\left( e^{-\alpha(l_{2R}-l_{1})}-1\right)}{\alpha(l_{2R}-l_{1})}\frac{\left(e^{-\alpha(l_{2T} -l_{1})}-1\right)}{\alpha(l_{2T}-l_{1})} \tag{5}\]  
where \(Int(x_{1},x_{2})=\int_{x_{1}}^{x_{2}}xJ_{1}(x)\mathrm{d}x\), \(J_{1}(x)\) denotes the first-order Bessel function; \(l_{1}\) is the sensor lift-off, \(e^{-2\alpha I}\) is the lift-off coefficient; \(n\) is the coil turns number, \(r_{1}\), \(r_{2}\), and \((l_{2}-l_{1})\) are the inner radius, outer radius, and the coil height, respectively; the subscripts \(T\) and \(R\) label the transmitter and receiver coils, respectively. Substituting Equations (1)-(3) and Equation (5) into Equation (4), the induced voltage in frequency domain can be obtained. As the square-wave excitation current of PECT could be theoretically represented by superimposing a series of sinusoidal harmonics in the frequency domain.