Eddy currents in a transverse MRI gradient coil

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.     

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.     

Thermally induced directed currents in hard rod systems

We study the non equilibrium statistical properties of a one dimensional hard-rod fluid undergoing collisions and subject to a spatially non uniform Gaussian heat-bath and periodic potential. The system is able to sustain finite currents when the spatially inhomogeneous heat-bath and the periodic potential profile display an appropriate relative phase shift, ${\phi}$ . By comparison with the collisionless limit, we determine the conditions for the most efficient transport among inelastic, elastic and non interacting rods. We show that the situation is complex as, depending on shape of the temperature profile, the current of one system may outperform the others.     

Pulse distortion due to eddy currents induced in a multiply excitedcylindrical shell of finite length and thickness

The electromagnetic field due to pulse excitation of two crossed pairs of coils surrounded by a cylindrical conducting shell is determined by accounting for the eddy-current effect. Each coil is modeled as a pair of straight, parallel wires of infinite length; the surrounding conducting shell is a cylindrical tube of finite length and thickness. The problem is formulated in the frequency domain, and the resulting expressions are used to calculate the transient magnetic flux density which originates from the quadruple coil and the eddy currents induced within the metallic shell. The part due to the eddy currents represents an unwanted distortion, which is investigated through a numerical example. The analytical and numerical results are complemented by a brief discussion on compensation techniques and potential applications in magnetic resonance imaging     

The impedance of a finite length coil immersed in a circular cylindrical conducting structure

The problem of calculating the impedance of a finite length solenoid immersed in a multiregion cylindrical structure is examined. The transmission line modelling technique is introduced to deal with a large number of regions in a simple and routine way. Bessel functions are completely avoided by replacing each region equivalent T.L. by a cascade connection of elementary T-circuits. The input impedance calculated for a finite length solenoid which was either air-cored, or closely encircling an infinitely long conducting cylinder, was found to agree closely with published values.     

Eddy currents in magnetic measurements

In the field modulation technique for magnetic measurements, the voltage signalV(H) from a pickup coil surrounding a metal cylinder of permeability =1+K(H) and conductivity = ₀ [1+S(H)] in the presence of an alternating fieldh ₀ cos t superimposed on a steady fieldH (both parallel to the axis of the cylinder) provides a measure ofK(H). This technique is particularly useful for study of the de Haas-van Alphen effect, in whichK(H) oscillates withH, and the paper is mainly concerned with the effect of eddy currents on the relation betweenV(H) andK(H) and the practical implications for the use of the technique. It is also shown that if is high enough,S(H) (e.g., the oscillations of the Shubnikov-de Haas effect) and also information on the orientation dependence of can be extracted from suitable measurements onV(H) and its orientation dependence. The calculations fall into two main parts, corresponding to weak and strong modulation, i.e., according ash ₀ is small or appreciable compared with the period of the field dependence ofK andS. For weak modulation, results are obtained for arbitrary values ofK andS over the whole range of . It is shown that ifK(H) is not small,V(H) no longer faithfully reproduces the form ofK(H) once is high enough to make the skin depth of the eddy currents comparable to the radius of the cylinder. In general,V(H) contains not only the fundamental frequency , but also its harmonics, i.e.,V(H)= _n=1 V _n (H)e ^nit, but it is only for strong modulation that the higher harmonics become appreciable. The variation of theV _n with modulation current and is calculated under the special assumptions thatK andS 1 and thatK andS are simple harmonic functions ofH; for these special assumptions the weak modulation results forV ₁ obtained earlier fall out as a limiting case of the results for arbitrary modulation. Some of the theoretical predictions were tested experimentally on sodium and potassium crystals. Good agreement was found at frequencies up to such that the skin depth became comparable to or smaller than the sample radius, when appreciable discrepancies appeared. Some of these could be attributed to helicon resonances, the possibility of which was ignored in the theory, and some to sample inhomogeneity, the effect of which are briefly discussed in an appendix.     

有限长圆柱绕流气动噪声源特性分析

气动噪声源的能级、分布特性及其产生根源还不够清晰。以有限长三维圆柱绕流为研究对象,基于声源方程分析气动噪声源的种类构成及其与气动参数的关系,通过数值计算得到可压非定常流场,利用气动参数定量计算圆柱顶部、中部和底部的声源大小分布,研究声源的分布特性和产生根源。结果表明,在有限长圆柱绕流场中,以偶极子声源为主,单极子声源可以忽略不计,四极子源项的值比偶极子小1～2个数量级。偶极子主要分布在来流分离点及圆柱后壁面湍流涡二次碰撞区域,四极子主要分布在来流分离点及其向后拖曳区域。偶极子声源主要由于圆柱两侧涡脱落处的脉动压力在横向(y方向)上的二阶梯度引起。以上结果为气动噪声控制的进一步研究提供了借鉴和参考。     

Eddy currents in conducting magnetic recording media

Use of metallic magnetic recording media raises the possibility that medium eddy currents will influence the recording process. The mathematical analog to diffusion of heat allows exact solutions for cases approximating those of interest. Eddy current effects are shown to be negligible for usual thicknesses of longitudinal and perpendicular media.     

Eddy current probe sensitivity as a function of coil construction parameters

We report the results of the first phase of a study designed to quantify the relationship between eddy current coil construction and the performance of these coils used in nondestructive evaluation (NDE) inspections. The ferrite core coils wound for this study are small but typical of the sizes commonly used in commercially manufactured eddy current probes. Coil diameters range from 1 mm to 7 mm with lengths from 0.5 mm to 4 mm. Seven parameters were studied and included ferrite diameter, ferrite permeability, coil aspect ratio, number of turns, distance of the windings from the inspection end of the ferrite, wire gauge, and length of the ferrite beyond the end of the windings. Additionally, the coil set was designed to provide some indication of the repeatability of identical constructions, what we have called winding inhomogeneity. The coils were incorporated into surface probes for scanning defects in flat plate specimens. The measure of sensitivity was the change of probe impedance (Z) as the probe was scanned from an unflawed area to the flawed area of the test specimen. Measurements were also made of the component of Z perpendicular to the liftoff vector.The data reported here were produced from a set of 27 probes scanned over a single defect. The defect was an electrical-discharge-machined (EDM) notch in a 19 mm thick 7075-T6 aluminum alloy specimen. The part-circular EDM notch was 9 mm long and 3 mm deep and 0.1 mm wide.Analysis of the data shows that the number of turns, the winding distance, the coil aspect ratio, and the backside ferrite length all affect the coil sensitivity. Winding inhomogeneity is significant for coils having many winding layers and can be considerably larger than the contribution made by variations in some of the construction factors. Wire gauge, ferrite diameter, and permeability showed no significant effects on our measure of sensitivity in this study.     

Torsional oscillations of a rod in a layered medium of simple fluids

Free surface flow of a layered medium of simple fluids driven by a torsionally oscillating, cylindrical rod is investigated. The stress, expressed as a series of multiple integrals of polynomials in the strain histories, is expanded into a Fréchet series in terms of the amplitude of the oscillation of the rod. A domain perturbation method is used to perturb the rest state. The mean shapes and deviations from flatness at the rod of the interfaces between layers are determined together with the flow field up to and including second order in the amplitude of the oscillation.     

The chord length distribution of a triangular rod

A triangular rod (edge length of the basic triangle is a) is considered. Analytic expressions of the chord length distribution density A(l,a) and other structure functions of small-angle scattering are calculated. Applications of A(l,a) in the field of SAS are discussed. The three-dimensional correlation function and the asymptotic behaviour of the SAS intensity I(h) are analyzed. Here, the Kirste–Porod term is missing which correlates with the fact that parallel interfaces do not exist.     

A simple evaluation of magnetic fields and eddy currents inferromagnetic straps of finite length

A simple analytical method to evaluate the electromagnetic fields in a ferromagnetic strap of finite dimensions, excited by a short coil, is presented. The method can be extended to ferromagnetic cylindrical parts of finite length, and also to nonlinear ferromagnetic materials by assuming an appropriate change of magnetic permeability with material depth. The original configuration of the finite-length strap and short excitation coil is replaced by an arrangement of many straps glued together, and magnetized by a suitable succession of excitation coils. This last configuration can be solved analytically according to the Fourier transform method. Results obtained by the proposed method are in good agreement with results obtained by a finite-element method, and also with experimental results     

Eddy currents in anisotropic composites applied to pulsed machinery

Pulsed electrical machines such as the compensated pulsed alternator (compulsator) are increasingly being considered for mobile applications. Size and weight reduction in these machines is therefore very important. Composite materials, such as carbon fiber and glass fiber epoxy composites, with high strength and low density are a natural choice for the various structural components of these electrical machines. The carbon fiber-epoxy composite (CFC) is particularly useful because of its high strength and modulus. However, this composite (CFC) is electrically conductive with greatly different electrical conductivities along the fibers and transverse to the fibers. When these materials are used in pulsed machines, it is important that the designer be cognizant of the conductivity and treat the anisotropy adequately. This paper illustrates how the eddy current distribution can be determined in carbon fiber composites in light of the anisotropy     

Revising the formula of thermal focal length in a side-pumped laser rod by experiments

The correct formula of the thermal focal length in a side-pumped Nd:YAG laser rod is discussed and confirmed by experimental results. It is shown that thermally induced stresses that cause a distortion of flatness occur within the region of the whole rod. The presented calculations are in agreement with the experimental observations. The results reveal that the temperature-dependent variation of the refractive index and the distortion caused by the thermally induced stresses constitute the major contributions to thermal lensing.     

Unsteady heat transfer in a system of three coaxial finite cylinders

An integral transformation with the aid of which a solution of the problem of unsteady-state heat transfer in a system of three coaxial finite cylinders with different boundary conditions on their surfaces depending on space and time is presented. Each of the cylinders evolves heat of a certain intensity, depending on time and coordinates. A numerical solution of one variant of the boundary conditions is given and illustrated by figures. The method of transforming the solution of the problem with other boundary conditions is shown. __________ Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 80, No. 1, pp. 140–148, January–February, 2007.     

Convection induced by vibrating rod in fine-powder bed

In this study, vibration-induced convection was studied experimentally using a fine powder with a mass median particle diameter of 8 μm. A cylindrical rod arranged vertically in a powder container was vibrated horizontally with simple harmonic motion at a frequency of 300 Hz using a piezoelectric vibrator. For a vibration amplitude of 10 μm, particles around the cylindrical rod were consolidated to a certain extent due to gravity; however, for a vibration amplitude of 70 μm or more, a pair of convection rolls formed on both sides of the vibrating rod. The strength of the convection was quantified from the particle velocity distribution in the convection rolls, and the relationship between the convection strength and vibration amplitude was elucidated. In addition, the air-pressure distribution in the powder bed was measured showing that the convection strength correlates with the characteristic positive pressure, i.e., the average value of positive pressure measurements. Elliptical motion and circular motion as well as simple harmonic motion were applied to the cylindrical rod by adding two harmonic motions in directions orthogonally crossing each other with a phase difference of π/2 rad. The convection of the particles varied according to the Lissajous trajectory of the cylindrical rod. Even for simple harmonic motion, heaps of a pair of convection cells overlapped each other. In the case of elliptical motion, the overlapping portion of the heaps became larger. In the case of circular motion, the two heaps were integrated into one circular heap, and there were no effects of the circumferential angle on the particle velocity and the characteristic positive pressure.     

Magnetization self-oscillations induced by spin-polarized currents

The effect of a spin-polarized current on magnetization dynamics is described by the Landau-Lifshitz-Gilbert (LLG) equation with the addition of Slonczewski spin-transfer term. Detailed predictions about the existence, position, and stability of magnetization self-oscillations induced by the spin-polarized current are obtained by an analytical perturbative method for different levels of current injection and externally applied field. The two cases of opposite direction of current flow are discussed.     

Eddy current profiling of magnetic coercivity in layered ferromagnetic materials

The eddy current response of a ferromagnetic material having a surface modified layer is analyzed for the case where an external magnetic field is swept from negative to positive saturation. Using a plane wave model, it is shown that the impedance of the layered material can display either one or two peaks as a function of field depending on the shape of the hysteresis loops and hence of the reversible permeabilities of core and surface materials. By varying the eddy current frequency different thicknesses of material are probed, resulting in changes in the relative contribution of core and surface material. This causes changes in both the amplitude and position of the peaks with frequency, a feature that can be used for characterizing surface layers. The case of single peaked impedance curves is analyzed in detail as it corresponds to the often encountered practical case of a hardened surface layer. In particular it is shown that as the frequency is varied from a low value corresponding to penetration of the bulk material to a high value corresponding to penetration of the surface layer only, the peak in impedance shifts gradually from the coercivity,H _c , in the core toH _c of the surface. However, at intermediate frequencies, effective values ofH _c which are greater than either that of the core or the surface can occur. This is explained in terms of reflections at the core/surface layer interface. Experimental results are then presented for carburized materials of various geometries and shown to be in agreement with the calculations. The frequencyf _max at which the coercivity reaches its maximum value, and hence where reflection effects are a maximum is also shown to be a useful feature for nondestructively determining the thickness of carburized layers and nugget penetration in resistance welds.This article is dedicated to Professor Dr. Paul Höller on the occasion of his 65th birthday.