Eddy currents in a gradient coil, modeled as circular loops of strips

As a model for the z-coil of an MRI-scanner a set of circular loops of strips, or rings, placed on one cylinder is chosen. The current in this set of thin conducting rings is driven by an external source current. The source, and all excited fields, are time-harmonic. The frequency is low enough to allow for an electro-quasi-static approach. The rings have a thin rectangular cross-section with a thickness so small that the current can be assumed uniformly distributed in the thickness direction. Due to induction, eddy currents occur resulting in an edge-effect. Higher frequencies cause stronger edge-effects. As a consequence, the resistance of the system increases and its self-inductance decreases. The Maxwell equations imply an integral equation for the current distribution in the rings. The Galerkin method with Legendre polynomials as global basis functions is applied. This method shows fast convergence, so only a very restricted number of basis functions is needed. The general method is worked out for N (N = 1) rings, and explicit results are presented for N = 1, N = 2 and N = 24. The derived integral equation and the numerical results of the simulations show that sets of circular rings and plane strips describe the same electromagnetic behavior, thus demonstrating that inductance effects are local.