The problem of an axially symmetric E-cored coil with a circular air gap inside the core column located above a two-layered conductive half-space is presented. The Truncated Region Eigenfunction Expansion method is used to derive expressions describing the magnetic vector potential of the filamentary coil. The final expressions for the impedance of the rectangular cross-section coil are obtained and calculations for various frequency values are carried out. The results are compared with those from the COMSOL package, which shows very good agreement.

Purpose Discrete eigenvalues occur in eddy current problems in which the solution domain was truncated on its edge. In case of conductive material with a hole, the eigenvalues are complex numbers. Their computation consists of finding complex roots of a complex function that satisfies the electromagnetic interface conditions. The purpose of this paper is to present a method of computing complex eigenvalues that are roots of such a function. Design/methodology/approach The proposed approach involves precise determination of regions in which the roots are found and applying sets of initial points, as well as the Cauchy argument principle to calculate them. Findings The elaborated algorithm was implemented in Matlab and the obtained results were verified using Newton’s method and the fsolve procedure. Both in the case of magnetic and nonmagnetic materials, such a solution was the only one that did not skip any of the eigenvalues, obtaining the results in the shortest time. Originality/value The paper presents a new effective method of locating complex eigenvalues for analytical solutions of eddy current problems containing a conductive material with a hole.

This work presents an axially symmetric mathematical model of an I-cored coil placed over a two-layered conductive material with a cylindrical surface hole. The problem was divided into regions for which the magnetic vector potential of a filamentary coil was established applying the truncated region eigenfunction expansion method. Then the final formula was developed to calculate impedance changes for a cylindrical coil with reference to both the air and to a material with no hole. The influence of a surface flaw in the conductive material on the components of coil impedance was examined. Calculations were made in Matlab for a hole with various radii and the results thereof were verified with the finite element method in COMSOL Multiphysics package. Very good consistency was achieved in all cases.

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This paper presents a method for calculating the air-cored coil impedance with the employment of a mathematical model of an ideal filamentary coil. The proposed algorithm enables assigning, in a very quick way, each cylindrical air-cored coil to the corresponding filamentary coil using only two equivalent parameters. The first of them is the radius of the coil, whereas the second one is the distance between the coil and the surface of the tested material. The changes both in the parameters of the system under consideration and in the tested material bring about the same change in the impedance of the air-cored coil and the corresponding filamentary coil. This property brings a lot of advantages, since it allows using simpler final formulas for the filamentary coil and performing the calculations in a much shorter time, while obtaining the same results as in the case of the air-cored coil. At the same time, the creation of the scale of the measuring instrument and its calibration becomes far simpler since it is based on only two equivalent parameters.