Previously, the Dodd Deeds model has been used to evaluate the inductance change of an air-cored sensor in the presence of an infinitely large conducting plate. In practice, it can be applied to the sample relatively larger than the radius of the sensor coil (normally >3-5 times larger), which can be regarded as a plate sample with infinite planar size. However, in various practical applications, the size of the tested sample may not satisfy this condition. In this paper, a modified analytical solution based on the Dodd Deeds model is proposed, which has introduced a new initial integration point instead of 0 for the analytical inductance of the finite-size metallic plate. Theoretical derivation has been presented for the derivation of the initial integration point. Moreover, an inversely proportional relation can be observed between the initial integration point and the radius of the test sample. Simulation and experimental measurements for the thickness of several finite-size metallic samples have been carried out for the verification of the proposed method.

In this paper, an inverse method was developed which can, in principle, reconstruct arbitrary permeability, conductivity, thickness, and lift-off with a multi-frequency electromagnetic sensor from inductance spectroscopic measurements. Both the finite element method and the Dodd & Deeds formulation are used to solve the forward problem during the inversion process. For the inverse solution, a modified Newton-Raphson method was used to adjust each set of parameters (permeability, conductivity, thickness, and lift-off) to fit inductances (measured or simulated) in a least-squared sense because of its known convergence properties. The approximate Jacobian matrix (sensitivity matrix) for each set of the parameter is obtained by the perturbation method. Results from an industrial-scale multi-frequency sensor are presented including the effects of noise. The results are verified with measurements and simulations of selected cases. The findings are significant because they show for the first time that the inductance spectra can be inverted in practice to determine the key values (permeability, conductivity, thickness, and lift-off) with a relative error of less than 5% during the thermal processing of metallic plates.

Lift-off variation causes errors in the eddy current thickness measurements of metallic plates. In this paper, we have developed an algorithm that can compensate for this variation and produce an index that is linked to the thickness, but is virtually independent of lift-off. This index, termed as the compensated peak frequency, can be obtained from the measured multifrequency inductance spectral data using the algorithm we developed in this paper. This method has been derived through mathematical manipulation and verified by both the simulation and experimental data. Accuracy in the thickness measurements at different lift-offs proved to be within 2%.

Lift-off variation causes errors in eddy current measurement of nonmagnetic plates as well as magnetic plates. For nonmagnetic plates, previous work has been carried out to address the issue. In this paper, we follow a similar strategy, but try to reduce the lift-off effect on another index—zero-crossing frequency for magnetic plates. This modified index, termed as the compensated zero-crossing frequency, can be obtained from the measured multifrequency inductance spectral data using the algorithm we developed in this paper. Since the zero-crossing frequency can be compensated, the permeability of magnetic plates can finally be predicted by deriving the relation between the permeability and zero-crossing frequency from Dodd and Deeds method. We have derived the method through mathematical manipulation and verified it by both simulation and experimental data. The permeability error caused by liftoff can be reduced within 7.5%.