Comparing two methods to solve the layered sphere problem, application to electromagnetic induction sensors

The layered sphere problem can be solved by using a multipole expansion in each layer and imposing the appropriate boundary conditions. This direct approach has some drawbacks because it can be numerically badly conditioned. We show that a tranmission line (TL) formalism can be used to solve the problem and that this approach is better conditioned than the direct approach. The corresponding TL is inhomogeneous because it has a characteristic impedance which varies with the radius and which also depends on the direction. We show how the classical TL expressions can be generalized to take into account such an inhomogeneous TL. Both formalisms are applied to a configuration representative of a metal detector (MD) above a magnetic soil. For such a configuration, a large number of the terms is required in the expansion and the direct approach fails in computing the high order terms because of numerical saturations. In contrast, accurate results are obtained with the TL approach.