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.
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.
The z-coil of an MRI-scanner is modelled as a set of circular loops of strips, or rings. Due to induction eddy currents occur which lead to the so-called edge-effect. The edgeeffect depends on the applied frequency and the distances between the strips, and affects the impedances. The current distribution in the rings is determined and from that the total resistance and self-inductance of each ring separately, all as far as possible by analytical means. From the Maxwell equations, an integral equation for the current distribution in the strips is derived. The Galerkin method is applied, using global basis functions to solve this integral equation. It turns out that Legendre polynomials as basis functions are an appropriate choice. They provoke an analytical expression for the integrals with a singular kernel function, and bring about a fast convergence; only a very restricted number of Legendre polynomials is needed.
A transverse gradient coil (x-or y-coil) of an MRI-scanner is modeled as a network of curved circular strips placed at the surface of a cylinder. The current in this network is driven by a time-harmonic source current. The low frequency applied allows for an electro-quasi-static approach. The strips are thin and the current is assumed to be uniformly distributed in the thickness direction. For the current distribution in the width direction of the strips, an integral equation is derived. Its logarithmically singular kernel represents inductive effects related to the occurrence of eddy currents. For curved circular strips of width much smaller than the radius of the cylinder one may locally replace the curved circular strip by a tangent plane circular strip. This plane geometry preserves the main characteristics of the transverse current distribution through the strips. The current distribution depends strongly on the in-plane curvature of the strips. The Petrov-Galerkin method, using Legendre polynomials, is applied to solve the integral equation and shows fast convergence. Explicit results are presented for two examples: a set of 1 strip and one of 10 strips. The results show that the current distributions are concentrated near the inner edges and that resulting edge-effects, both local and global, are non-symmetric. This behavior is more apparent for higher frequencies and larger in-plane curvatures. Results have been verified by comparison with finite-element results.
Research on wave propagation in liquid filled vessels is often motivated by the need to understand arterial blood flows. Theoretical and experimental investigation of the propagation of waves in flexible tubes has been studied by many researchers. The analytical one-dimensional frequency domain wave theory has a great advantage of providing accurate results without the additional computational cost related to the modern time domain simulation models. For assessing the validity of analytical and numerical models, well defined in vitro experiments are of great importance. The objective of this paper is to present a frequency domain analytical model based on the one-dimensional wave propagation theory and validate it against experimental data obtained for aortic analogs. The elastic and viscoelastic properties of the wall are included in the analytical model. The pressure, volumetric flow rate, and wall distention obtained from the analytical model are compared with experimental data in two straight tubes with aortic relevance. The analytical results and the experimental measurements were found to be in good agreement when the viscoelastic properties of the wall are taken into account.
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