The key point of modeling in structure health monitoring is to build the relationships between the structure damage features and the output signal. Generally, researches can be divided into two categories: forward models and reverse models. The forward models lay foundation for the reverse ones, and the former models are more convenient in conducting further study on the output signal from the mechanism. Also, the forward models are helpful in optimizing the sensor’s parameters and construction of an inversion model [30–32].

In this paper, the electromagnetic parameters of materials are equivalent to the structural damage features [33], and the damage monitoring semi-analytical model of the attached eddy current sensor is established. The model demonstrates the relationship between the output signal and the excitation frequency, sensor’s parameters, as well as the electromagnetic parameters of the structure to be monitored. Since the building of the model is dramatically complicated and cumbersome, in this section modeling process is described briefly and shown in Fig. 11. Some of the specific modeling process used here can be understood with the help of reference [34].

The sensor is placed on the top of the monitored structure. The lift-off distance is Δ*L* and the thickness of the structure is Δ*S*. The conductivity is *σ* and the permeability is *μ*. The widths of the excitation coil and of the induction coil are *p* and *s*, respectively. The length between the two coils is g and the coil thickness is Δ_{wind}. The spacing length *λ* between the adjacent excitation coils is defined as the wavelength of the sensor. The general solution of the differential equations of time-harmonic field function A under cylindrical coordinates can be expressed in the following form:

$$ \mathbf{A}={J}_1(kr)\left({c}_1{e}^{\chi z}+{c}_2{e}^{-\chi z}\right)\overrightarrow{\varphi}. $$

(1)

Here, *χ*
^{2} = *k*
^{2} + *jωμσ*, *ω* is the excitation frequency, *σ* is the conductivity of dielectric polyimide layer, *μ* is the permeability of the layer, and *k* is a constant. *J*
_{1} is a first-order Bessel function of the first type. At the cross section *Z* = 0, a Bessel function expression of current density *K* and magnetic vector *A* is obtained from Eq. (1) [35].

$$ K(r)={\displaystyle \sum_{n=1}^{\infty }{K}_{Bn}{J}_1}\left(\frac{\alpha_n}{R}r\right) $$

(2)

$$ A(r)={\displaystyle \sum_{n=1}^{\infty }{A}_{Bn}{J}_1}\left(\frac{\alpha_n}{R}r\right) $$

(3)

In the two equations above, *α*
_{n} is the forward zero sequence of *J*
_{1}, and *R* is the outer boundary of the model. Discrete allocation points and sub-domains are configured at the coil cross section, which is shown in Fig. 12. Regarding the current density of discrete points as unknown parameters, the vector expression of *K*
_{
Bn
}, which is the coefficient of *K*, will be obtained by the series formula of Bessel function according to the linear distribution assumption of current between the adjacent discrete points.

$$ {K}_{Bn}=\mathbf{M}\mathbf{K} $$

(4)

*K* is a vector composed of the unknown current density at all discrete points, and *M* is the coefficient matrix. The current density of the coil is calculated at the cross section as Eq. (4) and is substituted into Eq. (2). Meanwhile, according to the field relations at the interface between the media layer, as well as the boundaries conditions at the cross section of coils, the expression of the vector *A*
_{
Bn
} is taken, which is the coefficient of the magnetic vector *A*.

$$ {A}_{Bn}=T{K}_{Bn} $$

(5)

Then the magnetic vector *A* is obtained by substituting Eq. (5) into Eq. (3). Applying Faraday’s law of electromagnetic induction to every closed path, we can obtain:

$$ v=j\omega \left(2\pi r^{\prime}\right){A}_{\varphi}\left(r^{\prime}\right)+2\pi r^{\prime}\frac{K\left(r^{\prime}\right)}{\varDelta_{\mathrm{wind}}{\sigma}_{\mathrm{wind}}}, $$

(6)

where *v* is the voltage source, that is, the input to the excitation coil and output from the induction coil. Δ_{wind} is the thickness of the coil, and *σ*
_{wind} is the conductivity of the conductor. Each side of Eq. (6) is integrated at the defined sub-region, and the following expression is obtained:

$$ {\mathbf{M}}_{TU}\mathbf{K}=\mathbf{V} $$

(7)

In Eq. (7), *V* is the vector of the voltage source, and subscript *T* is the index of sub-region. Considering the constraint relations between the current density and current in the excitation coil as well as the induction coil, the following matrix expression is established:

$$ {\mathbf{M}}_{TD}\mathbf{K}=\mathbf{I} $$

(8)

Uniting Eqs. (7) and (8), we can obtain the matrix equation involving the unknown line current density and voltage induction coil in all discrete points. The excitation coil current is defined as 1 A, according to the theory of linear equations we can obtain the induced voltage of every channel’s induction coil, as well as the induction coil resistance value across the channel.

$$ \left[\begin{array}{cc}\hfill {\mathbf{M}}_{TU}\hfill & \hfill {\mathbf{M}}_{TL}\hfill \\ {}\hfill {\mathbf{M}}_{TD}\hfill & \hfill 0\hfill \end{array}\right]\left[\begin{array}{c}\hfill \mathbf{K}\hfill \\ {}\hfill \mathbf{V}\hfill \end{array}\right]=\left[\begin{array}{c}\hfill \mathbf{0}\hfill \\ {}\hfill \mathbf{I}\hfill \end{array}\right] $$

(9)