# Flaw Size Quantification for Cable Flaw Inspection System with Inductive Search Coil Sensor
## 2 Methodology  
### Model-Based Principle  
Using the basic governing equations, the self-inductance \(L\) (H) of a coil driven by an excitation current \(I\) (A) can be described as [34]:  
\[L=\frac{2W_{m}}{I^{2}} \tag{1}\]  
where \(W_{m}\) (J) represents the intrinsic magnetic energy of the coil. The magnetic energy is the work done by the power source on the entire current loop during the establishment of the magnetic field [35], which can be expressed using the Maxwell equations as follows:  
\[ \begin{split} W_{m}&=\frac{1}{2}\int_{\Omega}\vec{B} \cdot\vec{H}\,\mathrm{d}\Omega=\frac{1}{2}\int_{\Omega}\frac{1}{\mu}|B|^{2}\, \mathrm{d}\Omega\\ &=\frac{1}{2}\int_{\Omega}\frac{1}{\mu}\Bigg{[}\left(\frac{ \partial A}{\partial x}\right)^{2}+\left(\frac{\partial A}{\partial y}\right)^ {2}+\left(\frac{\partial A}{\partial z}\right)^{2}\Bigg{]}\mathrm{d}\Omega \end{split}  \tag{2}\]  
where \(B\) and \(H\) represent the induction and the intensity of the magnetic field respectively, and \(\mu\) denotes the permeability. Specially, \(\frac{1}{2}\vec{B}\cdot\vec{H}\) represents the energy density of the magnetic field, which denotes the magnetic field stored per unit volume, with the unit of J/m\({}^{2}\). Accordingly, the magnetic energy of each element in the Finite Element Model (FEM) could be indicated by the following:\[W_{m}=\frac{1}{2}\sum_{e=1}^{M}\int_{\Omega_{e}}\frac{1}{\mu}\Bigg{[}\bigg{(}\frac{ \partial A}{\partial x}\bigg{)}^{2}+\bigg{(}\frac{\partial A}{\partial y}\bigg{)} ^{2}+\bigg{(}\frac{\partial A}{\partial z}\bigg{)}^{2}\Bigg{]}\mathrm{d}\Omega \tag{3}\]  
Accordingly, the self-inductance \(L\) of the coil is presented:  
\[L=\frac{1}{I^{2}}\sum_{e=1}^{M}\int_{\Omega_{e}}\frac{1}{\mu}\Bigg{[}\bigg{(} \frac{\partial A}{\partial x}\bigg{)}^{2}+\bigg{(}\frac{\partial A}{\partial y }\bigg{)}^{2}+\bigg{(}\frac{\partial A}{\partial z}\bigg{)}^{2}\Bigg{]}\mathrm{d}\Omega \tag{4}\]  
where \(A\) denotes the magnetic vector potential, \(M\) refers to the number of the FEM elements, and \(\Omega_{e}\) represents a sub-region corresponding to an element in space. The aforementioned equations demonstrate that, in addition to the factors of excitation current and permeability, the self-inductance of the coil is influenced by the variation of the magnetic vector potential \(A\). In the finite element model, the magnetic vector potential \(A\) can be solved under the constraint of \(\
abla\cdot A=0\), by using the governing equation extended from Maxwell equations:  
\[\
abla\times(\mu^{-1}\
abla\times A)=J \tag{5}\]  
where \(J\) (A) represents the excitation current. Based on the theory of the above partial differential equations and boundary conditions, the magnetic potential \(A\) is solvable in COMSOL. Accordingly, the coil inductance can be simulated. For the purpose of verifying the feasibility and correctness of the proposed approach, a 3-dimensional numerical simulation has been developed to analyze the variation of coil self-inductance along the axial direction of a cable with parallel wires. The effects of the flaw axial length and cross-sectional area were evaluated respectively. This simulation utilizes COMSOL Multiphysics with the AD/DC module and frequency domain analysis. The simulated three-dimensional finite element model (FEM) based on COMSOL is illustrated in Figure 1, including a three-layer cable, polyethylene (PE) sheaths and a coil. The self-inductance coil features an inner diameter of 60 mm and an outer diameter of 70 mm. It is wound 300 times and excited by a 1 V AC signal at 1000 Hz. The axial scanning length of the coil along the cable is set as 150 mm. The cable's inner material property is set as \"steel\", with a relative permeability of \(\mu_{steel}=200\). A 1 mm PE sheath surrounds the cable, with a relative permeability of \(\mu_{PE}=1\). The material property of the remaining domain inside the PE sheath is also set as \"PE\". The flaws are separately modeled in the middle of the cable, consisting of multiple wire segments.