A recent topic of research, since 2011, Euler-Cauchy structures (ECS) are especial inhomogeneous cylindrical devices where exact closed-form analytical solutions of the wave equation can be found. This work presents a novel contribution to eddy current analysis in an ECS magnetic circuit where a coil is wound round a radially inhomogeneous magnetic core. Magnetic field and eddy current frequency-domain equations for homogeneous toroidal cores, with rectangular cross section, exhibit analytical solutions described by combinations of Bessel functions. However, in general, when the conductivity, permeability and permittivity of the core material depend on the radial coordinate an analytical solution cannot be encountered. Nonetheless, this paper shows, for the first time, that closed form solutions for the core’s electromagnetic field and coil’s impedance do actually exist provided that a specific type of radial variation of medium parameters is enforced. Analytic and computation results for general and particular ECS are presented and discussed.