Computation of the Eigenvalues for Bounded Domain Eddy-Current Models With Coupled Regions

2021

J Nondestruct Eval

This article presents an innovative technique for detecting flaws in conductive materials by using an ideal filamentary coil. To characterize such a coil accurately and explicitly, it is sufficient to be in possession of merely two parameters: the radius of the circle within which all the turns are located and the distance of the coil from the tested surface. The mathematical model derived using the truncated region eigenfunction expansion method enables the calculation of the changes in the components of the filamentary coil impedance that are the result of positioning the coil close to the conductive material with a hole. Because of this, the air-cored coil model can be replaced with a much simpler filamentary coil model. This solution makes it is possible to detect various types of holes (internal, surface, subsurface or through) occurring in both multilayer magnetic and non-magnetic materials. The derived results were verified by means of measurements and numerical calculations based on the finite element method. Very good agreement was observed in both cases. The paper contains the source code implemented in Matlab, which is used to for calculations.

Section: Methodsmentioning

confidence: 99%

Locating Defects in Conductive Materials Using the Eddy Current Model of the Filamentary Coil

2021

J Nondestruct Eval

“…The problem was analysed in a cylindrical coordinate system, and the solution domain was divided into 9 regions and limited to the value of parameter b ( Figure 2 ). Bounding the solution domain, i.e., limiting the range of a coordinate, results in discrete eigenvalues for that coordinate direction [ 47 ]. The discrete eigenvalues q of regions with a homogeneous structure (1, 5–8) are the positive real roots of the Bessel function of the first kind J 1 ( x ) and are calculated from equation J 1 ( q b ) = 0.…”

Section: Analytical Modelmentioning

confidence: 99%

Eddy Current Testing of Conductive Coatings Using a Pot-Core Sensor

2023

Sensors

Conductors consisting of thin layers are commonly used in many industries as protective, insulating or thermal barrier coatings (TBC). Nondestructive testing of these types of structures allows one to determine their dimensions and technical condition, while also detecting defects, which significantly reduces the risk of failures and accidents. This work presents an eddy current system for testing thin layers and coatings, which has never been presented before. It consists of an analytical model and a pot-core sensor. The analytical model was derived through the employment of the truncated region eigenfunction expansion (TREE) method. The final formulas for the sensor impedance have been presented in a closed form and implemented in Matlab. The results of the calculations of the pot-core sensor impedance for thin layers with a thickness above 0.1 mm were compared with the measurement results. The calculations made for the TBC were verified with a numerical model created using the finite element method (FEM) in Comsol Multiphysics. In all the cases, the error in determining changes in the components of the pot-core sensor impedance was less than 4%. At the same time, it was shown that the sensitivity of the applied pot-core sensor in the case of thin-layer testing is much higher than the sensitivity of the air-core sensor and the I-core sensor.

“…At the same time, attempts to develop a reliable algorithm based on iterative methods, such as the Newton–Raphson method, have been unsuccessful so far, even with the employment of a very small step. More promising seems to be the approach based on Cauchy’s argument principle [ 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 ] the solution domain is divided into small parts where the roots are found with contour integration. Unfortunately, this is a time-consuming method that requires carrying out numerous integrations, and the roots located too close to the contour edge are omitted anyway.…”

Section: Introductionmentioning

confidence: 99%

Computation of Eigenvalues and Eigenfunctions in the Solution of Eddy Current Problems

Theodoulidis

Skarlatos

2023

Sensors

The solution of the eigenvalue problem in bounded domains with planar and cylindrical stratification is a necessary preliminary task for the construction of modal solutions to canonical problems with discontinuities. The computation of the complex eigenvalue spectrum must be very accurate since losing or misplacing one of the thereto linked modes will have an important impact on the field solution. The approach followed in a number of previous works is to construct the corresponding transcendental equation and locate its roots in the complex plane using the Newton–Raphson method or Cauchy-integral-based techniques. Nevertheless, this approach is cumbersome, and its numerical stability decreases dramatically with the number of layers. An alternative, approach consists in the numerical evaluation of the matrix eigenvalues for the weak formulation for the respective 1D Sturm–Liouville problem using linear algebra tools. An arbitrary number of layers can thus be easily and robustly treated, with continuous material gradients being a limiting case. Although this approach is often used in high frequency studies involving wave propagation, this is the first time that has been used for the induction problem arising in an eddy current inspection situation. The developed method is implemented in Matlab and is used to deal with the following problems: magnetic material with a hole, a magnetic cylinder, and a magnetic ring. In all the conducted tests, the results are obtained in a very short time, without missing a single eigenvalue.