·Table of Contents ·Methods and Instrumentation | Electromagnetic Device for Recovery of Electric Conductivity Profile in Layered Flat and Tubular ProductsC. E. GONZALEZDepartamento de Física Aplicada, Facultad de Ingeniería, Universidad Central de Venezuela Apartado 47533, Caracas 1041A, Venezuela. E-Mail: esculpigonzales@cantv.net Contact |
There is much more information in the electromagnetic field around a coil, than the information that we could gather from the impedance measurement used in conventional Eddy Current applications. The devise presented in this paper could be used empirically for comparison of the thickness or conductivity between the test sample and reference pieces, but more applications could be obtained using the solution to the direct electromagnetic problem. The so called direct problem, in this case, consists in the determination of the electromagnetic field given the source and space distribution of the conductivity and magnetic permeability. This problem have been solved analytically and numerical solution for finite differences and finite elements are easy to obtain [1, 2, 3].
Using Maxwell equations for the harmonic time dependence case and for axial symmetry bring to the following equation in cylindrical coordinates [2, 4, 5] .
(1) |
Where Ej is the only non vanishing component of the electric field and ¶Ej/¶ j = 0. w is the angular frequency, s the conductivity, m the magnetic permeability, supposed to be nearly equal to that for air, and J_{s} is the current density in the coil.
Analytical solution of equation (1) is obtained by means of the Fourier-Bessel transform (Tsiboukis 1992). Finite elements as well as finite difference methods are also feasible as boundary conditions and initial conditions are very simple to state in both flat and tubular cases. For example, finite difference method with a 100´100 square mesh, give rise to results in good correspondence with experimental measurements. Discretization region includes the radial interval 0<r<2R where R is the inducting coil radius, for the case of flat products. The plane z = 0 is the surface of the specimen tested. Boundary conditions and values for the field and its derivatives at r = 0 are readily obtained. Mesh parameter h is taken as h=d /5 where d is the average standard skin depth [4, 6] .
Figure 1 shows the corresponding coordinates and geometry for recovery of the electromagnetic field in amplitude and fase. Figure 2 shows the arrangement for tubular samples.
Fig 1: Arrangement for multilayered flat sample |
Fig 2: Cross section of tubular sample. |
Once the direct problem is solved, the inverse problem could be attempted by stochastic methods like Simulated Annealing. Under certain conditions, deterministic solutions to the inverse problem are also feasible.
Fig 3: Schematic design of the system |
The sensitivity for this applications depends on the number of detectors. For critical applications, best results are obtained using mechanical scanning, with small distance increments and detectors of relatively small effective area. Resolution of the conversion is another critical parameter. In flat products, if dimensions of the test pieces is relatively large, the radius of the driving coil could be increased in order to improve the sensitivity of the system. Low conductivity materials give rise to poor sensitivity.
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