Potential Drop (PD) and Eddy Current Testing (ECT) are two common Non-Destructive Evaluation (NDE) methods, which have been used for decades. The modeling research of these problems can help in designing and improving testing technologies, explaining inspection results, and even making the more complex problem solvable, like inverse analysis of flaws and cracks in ECT and case-hardening problem with alternating current potential drop (ACPD). In this dissertation, extensive theoretical modeling research work has been developed.

For direct current potential drop (DCPD) problems, first, an analytical solution of modeling edge effects of metal plates with finite thickness has been presented. When dealing with plates somewhat thicker than the probe dimensions, a method-of-images is applied with fast convergence. For thinner ones, a Fourier series summation method, which is obtained using expressions originally developed to evaluate lattice sums in solid-state physics, can overcome traditional slow convergence problem and effectively reduce the triple infinite summation that results from the method-of-images to a double one.

Next, an analytical model of DCPD on uniformly layered conductive cylinders of finite length is developed. The solution is expressed in terms of a Green's function, which satisfies Neumann boundary conditions, and can be extended to a conductive cylinder with an arbitrary number of uniform layers, like the common case-hardening problems. This model can be used to determine the thickness or the conductivity of layered cylinders, especially helpful in monitoring wall thickness in power or chemical plants due to its high sensitivity to the thin tube thickness.

For ECT problems, first, an analytical model of an axisymmetric eddy current ferrite-cored coil above a multi-layered conductive half-space is presented by radially truncating the domain of the problem. In this work, the reflection and transmission coefficient matrices due to the end effects of the ferrite core have been introduced, then by using a recursion relationship, the reflection coefficient of a conductor with an arbitrary number of uniform layers has been determined. Furthermore, this approach can be extended to other axisymmetric ferrite core shapes, such as U-shape and E-shape. It is also possible to extend this approach to 3D problems in Cartesian coordinates using a double truncated series.

Then, a semi-analytical model of a differential bobbin coil impedance variation due to a common coaxial circular tube support plate has been developed. Within a truncated domain, the magnetic vector potential can be represented by a Fourier series and the effect of the tube support plate is evaluated theoretically. The close-form expression of the magnetic vector potential in a tube is critical for the more complicated 3D model for a support plate problem with a flaw or crack in the adjacent tube.