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

Abstract

The z-coil of an MRI-scanner is modeled as a set of circular loops of strips, or rings, placed on one cylinder. 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 a quasi-static approximation. The rings have a thin rectangular cross-section; the thickness is so small that the current can be assumed uniformly distributed in the thickness direction. Due to induction, eddy currents occur, resulting in a so-called edge-effect. Higher frequencies cause stronger edge-effects. As a consequence, the resistance of the system increases and the self-inductance decreases. From the Maxwell equations, an integral equation for the current distribution in the rings is derived. The Galerkin method is applied, using Legendre polynomials as global basis functions, to solve this integral equation. This method shows a 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.

Eddy currents in MRI gradient coils configured as rings and patches

“…Note that for p = 1 the expansion is the same as the one obtained in the model of rings only; see [14]. Moreover, each cosine mode in (3.30) has a logarithmic singularity at z = 0, which is of the same magnitude for all p ≥ 0.…”

“…As in the model presented in [14], the set of rings is subdivided into L groups, each of which has a prescribed total current I l , l = 1, . .…”

“…Additionally, eddy currents in the rings are induced, because they experience the presence of the islands. The currents do not flow only in the tangential direction, as they do in a set of parallel rings, see [14], but also in the axial direction. This is a result of the fact that the current density is divergence free, as follows from the continuity equation.…”

“…In reality, gradient coils consist of copper strips wrapped around a cylinder. Therefore, in [14], the current distribution is determined in a parallel set of rings on a cylinder. The current then flows only in the tangential direction and distributes non-uniformly over the axial direction.…”

“…Section 5 presents numerical results for several sets of rings and islands. The geometries used in the numerical examples are chosen such that they can be compared with the ring examples presented in [14]. The examples are not meant to mimic realistic z-gradient coils.…”