Field distribution around a long crack in a conductive half space excited by an arbitrary-frequency alternating-current-carrying coil of arbitrary shape

The authors propose a semi-analytical solution for evaluation of field distributions around a long crack in a conductive half space because of a three-dimensional (3D) current-carrying inducing coil at arbitrary frequency. The governing Helmholtz equation is solved in three dimensions by separation of variables. The solution is obtained by developing a two-dimensional (2D) Fourier series model and using exponential functions in the third dimension. To expand all possible field components in the conductor, the authors assume it as a lossy material with very large loss-tangent. In this regard, transverse electric and transverse magnetic modes are introduced to account for the contributions made by induced eddy current in the conductor as well as the displacement current passing through the crack opening. The authors use the mode matching technique and solve the resultant linear system of AX = B for the unknown coefficients. The accuracy of the proposed modelling technique is demonstrated by comparing their results with those available in the literature for the special case of a long rectangular 2D coil and with those obtained by experiment for a 3D inducing coil.