400 IEEE TRANSACTIONS ON MAGNETICS, VOL. 40, NO. 2, MARCH 2004 Dyad-Based Model of the Electric Field in a Conductive Cylinder at Eddy-Current Frequencies G. Micolau, G. Pichenot, D. Prémel, D. Lesselier, Senior Member, IEEE, and M. Lambert Abstract—We propose a rigorous model of the electromagnetic field in a conductive circular cylinder illuminated by an arbitrarily shaped and oriented current-carrying coil placed nearby for non- destructive evaluation at eddy-current frequencies. The model is based on a dyadic field formulation, and its modularity enables handy generalizations to a multilayered cylindrical structure that might be affected by various defects. We give a variety of numerical results about the primary electric field within the cylinder in both canonical and practical cases in order to illustrate pros and cons of the solution method. The results agree quite well with known analytical results (which are available only in highly symmetrical cases), and fairly well with results yielded at high computational cost by an industrial finite-element code in more general cases. We discuss in detail the conditions for reaching a good numerical ac- curacy (involving some suitable regularization) while preserving both generality and small computational burden of the resulting computer code. Index Terms—Conductive cylinder, eddy currents, electric fields, Green dyad, integral equation, modeling. I. INTRODUCTION E DDY-CURRENT nondestructive evaluation of conductive materials (metal, composites, etc.) is important in many domains, for example, the characterization of defects (loss of matter, cracks, voids, and inclusions) in aircraft engines and in steam tubes in nuclear power plants. A number of models al- ready exist in order to calculate the electromagnetic field gen- erated within a structure under testing by a time-harmonic cur- rent-carrying probe placed nearby. As is well known, the most general approach in complex geo- metrical configurations is to use a finite-element method (FEM) for which many computer codes are now available. The major inconvenience, however, may be the heavy computational cost, even in so-called canonical geometries (planar, circular cylin- drical, and spherical ones). For these geometries, it is often more pertinent to develop analytical or semianalytical methods for which geometrical symmetries allow to simplify the modeling to a great extent. The most classical case in that order is the one of a circular thin or thick probe which is placed parallel to and above a planar Manuscript received March 7, 2003; revised December 12, 2003. G. Micolau was with the SISC/LCME C.E.A Saclay, 91191 Gif-sur-Yvette, France. He is now with Institut Fresnel—Faculté des Sciences et Techniques de Saint Jérôme, 13397 Marseille cedex 20, France. G. Pichenot and D. Prémel are with the SISC/LCME C.E.A Saclay, 91191 Gif-sur-Yvette, France. D. Lesselier and M. Lambert are with the Département de Recherche en Élec- tromagnétisme, Laboratoire des Signaux et Systèmes (CNRS-SUPÉLEC-UPS), 91192 Gif-sur-Yvette cedex, France. Digital Object Identifier 10.1109/TMAG.2004.824145 structure (a half space, a plate, etc.), which is investigated at length in [1] and [2] by means of an expansion of the elec- tric vector potential. Also, still in this planar geometry but now for arbitrarily shaped and arbitrarily tilted (with respect to the planar interfaces) probes, one can cite the recent developments of [3]. For the spherical geometry, expansions of the second- order vector potential (SOVP) are used with good success in [4] but the probe always lies within a surface of coordinates. A sim- ilar approach applies also in the cylindrical case as is shown in [5], the probe being a simple circular loop. Recently, a nonsym- metrical cylindrical configuration has been investigated in [6]; however, the probe is a circular loop which should remain lo- cated within a plane perpendicular to the axis of the cylinder. Few of these solution methods appear to be able to simulta- neously take care of all three-dimensional elements of interest in the configuration of study. Particularly, there are two key el- ements to account for: the probe (by which one means its con- stitutive geometry, its orientation, and its location with respect to the tested structure), and the flaws (those to be found within the tested structure). In [7], a noncircular source is considered in a circular cylindrical case (tube), but the probe and the flaw within the tube wall both keep the highly simplified cylindrical symmetry. In [8], arbitrary voluminous defects are considered but the probe displaced within the tube is a centered circular loop. Our long-range aim is to build a versatile and robust modeling tool enabling us to accurately model the interaction between an arbitrary current source at eddy-current frequencies and a circular cylindrical structure (with one or more layers) which is affected by arbitrarily shaped flaws. This should evidently be of good interest in many practical cases where, say, long enough tubes are to be tested, like in nuclear power plants, but it would also enable us to model slightly curved parts as often found in the aeronautic industry. At a necessary yet recognized preliminary stage, only a full cylinder without flaws is considered herein (extensions to a hollow tube, though theoretically straightforward, are numeri- cally more involved), but the shape, location, and orientation of the probe are arbitrary. The quantity of interest is the primary (incident) vector electric field induced inside or observed at the inner surface of the cylinder, the primary eddy currents which flow through the structure (they are directly proportional to this field) being the ones to interact with a flaw. The method is rigorous in the meaning that there are no ap- proximations on the physical quantities involved in the equa- tions. As we will discuss later, the expansion of the field is based on integrals and infinite series. In numerical practice, one has to discretize and truncate this expansion. The resulting approxima- tions are numerical approximations and not theoretical ones. 0018-9464/04$20.00 © 2004 IEEE