·Table of Contents
·Methods and Instrumentation
Electromagnetic Device for Recovery of Electric Conductivity Profile in Layered Flat and Tubular Products
C. E. GONZALEZ
Departamento de Física Aplicada, Facultad de Ingeniería, Universidad Central de Venezuela
Apartado 47533, Caracas 1041A, Venezuela.
A devise for the recovery of the conductivity profile of flat and tubular products is developed. This devise is based in the processing of the electromagnetic field near a circular coil placed above the surface of the sample. The coil is placed parallel to the sample for flat products and it is placed encircling the sample for tubular products. The coil is driven by a current controlled harmonic source. The resulting electromagnetic filed is picked up by two Hall effect detectors, one for the radial component of the magnetic field and the other for the normal one, for flat products. In the case of tubular products, axial and radial components are detected. The two detectors scans radially, for flat products and axially for tubular ones. Stationary arrangements of detector could be used instead of the scanner mechanism. Appropriate multiplexing of the acquisition system should be used in the later case. Recovery of the conductivity profile is possible by means of several strategies for solving the inverse problem, including Monte Carlo methods and direct comparison with reference pieces .
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] .
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 Js 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.
The schematic design of the system is shown in Figure 3. The signal for every Hall effect detector is digitised. Signal processing is carried out in order to obtain the amplitude and the face of the magnetic field. The components of the electric field is readily obtained from the magnetic field components. For a fixed surface to coil distance, the plot of the unique non vanishing component of the electric field, as a function of the radial coordinate for fixed z (flat specimen), is compared with a reference plot. This reference plot could be obtained from a reference specimen or is calculated solving the direct problem.
Fig 3: Schematic design of the system|
- Thickness measurements of test pieces (known conductivity).
- Thickness measurements of test pieces (unknown conductivity).
- Simultaneous measurements of thickness and conductivity.
- Measurement of non conductive coating thickness for pieces of known conductivity.
- Determination of conductivity profile for multylayered conductive materials, including non conductive interleave layers.
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.
The developed devise is a low cost probe for determination of conductivity profile of flat and tubular products. It could be used as a thickness measurement devise for multylayered materials where ultrasonic measurements are difficult. Conductivity measurements are possible even when a conductive coating is present. The technique is limited to thin overall thickness compared with the standard depth of penetration.
Ferromagnetic material can be tested if magnetic saturation of the sample is achieved.
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