Current distribution in a parallel set of conducting strips

Abstract

The current distribution in a parallel set of thin conducting sheets due to an external applied source is investigated. All sheets are placed in one plane. The source, and all excited fields, are time harmonic. The frequency is low enough to allow for an electro quasi-static approximation (neglecting the displacement current). The conducting sheets are infinitely long and the current is uniform in the longitudinal direction of the sheets. The sheets have a thin rectangular cross-section, so thin that the current can be assumed uniform in the thickness-direction. Hence, the current distribution only depends on the transverse coordinate. Due to the mutual induction between the sheets, the current distribution over the width of the cross-section becomes non-uniform: it accumulates at the edges of the sheets. It is especially this so-called edge-effect, and its dependence on the applied frequency and the distances between the sheets, that is the aim of this investigation. From the Maxwell equations, a set of integral equations for the current distribution in the sheets is derived. These integral equations are solved, as far as possible by analytical means, by writing the current distribution in each sheet as a series of Legendre polynomials. The general method is worked out for N (N ≥ 1) sheets, but explicit results are presented for N = 1 and N = 3. It turns out that the edge-effect becomes stronger for increasing frequencies. For this solution, only a very restricted number of Legendre polynomials is needed.

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

“…This model was extended in [31], and forms the basis of the present work. The eddy currents in a set of rings have been determined by the author [32], [33] and the characteristics of the magnetic field induced by the resulting currents are investigated by Tas [34].…”

Eddy currents in MRI gradient coils configured as rings and patches

“…For low frequencies (quasi-static range), the behavior of j ϕ near the edges is regular, as described in [13]. In case of high frequencies, a square-root singular edge-effect will occur in the current near the edges.…”

“…One central line of each island is parallel to the rings. Since the strips are very thin, the current distribution in the r -direction can be considered uniform; see [13,Sect. 2].…”

“…In [13], as a starting model, the current distribution is determined in a parallel set of plane rectangular strips. This study describes the derivation of a Fredholm integral equation of the second kind with logarithmically singular kernel.…”

“…The current then flows only in the tangential direction and distributes non-uniformly over the axial direction. The solution method is similar to the approach in [13], but allowing currents to flow around the cylinder. This model can be used to determine the current distribution as well as the corresponding magnetic field of the so-called z-gradient coil.…”