Eddy-Current Nondestructive Inspection with Thin Spiral Coils: Long Cracks in Steel

The impedance of a cylindrical coil and a planar circular spiral coil carrying an alternating current above (i) a defect-free conducting magnetic half-space and (ii) a conducting magnetic half-space containing an infinitely long slot with uniform depth and width is examined in detail. Closed-form expressions for the coil impedance in these cases are presented, based on the theories of Dodd and Deeds and Harfield and Bowler. The validity of these expressions is tested by measurements using steel plates over the frequency range 100 Hz-10 MHz. The theoretical predictions are in good agreement with experiment, with the best agreement for the smallest slot width. The results confirm that thin, flexible spiral coils offer some attractive features for eddy-current detection of cracks in metals, particularly in terms of sensitivity and potential for unobtrusive permanent attachment to the material being inspected. Approximate expressions for a spiral coil above a defect-free magnetic half-space are also given to allow easy calculation in limiting cases.     

Comparison of AC Losses of YBCO Circular Pancake Coils and Infinitely Long Stack Approximation

High-temperature superconducting (HTS) coils are key parts of many large-scale AC applications, such as transformers, superconducting magnetic energy storage, and motors. The estimation of AC losses of pancake coils is necessary for optimizing the design of HTS devices and cryogenic systems. To speed up the computation of AC losses, the numerical model of an infinitely long HTS stack is often utilized. An HTS stack is a good approximation of the circular coil only if the coil radius is sufficiently large, since AC losses will exhibit a stronger asymmetry along the radial direction for small values of coil radii. To assess the validation of an infinitely long stack approximation, the comparisons of AC losses between infinitely long stacks and circular coils with different radii are presented. The turn number varies from 10 to 80. We find that the AC losses of HTS circular coils will gradually increase to the same value as an infinitely long stack with increasing coil radii. A new parameter is proposed to quantitatively describe the correctness of infinitely long stack approximation. Finally, a method of AC loss estimation of HTS pancake coils is proposed.     

Reconstruction of Cracks with Multiple Eddy Current Coils Using a Database Approach

This paper presents some recent studies dealing with flaw reconstruction using eddy current testing. The locations of parallel planar cracks in an Inconel 600 plate specimen are determined and the shapes of them are reconstructed at the same time using the signals from a multi-pancake coil probe. The reconstruction strategy, consisting of a data base used fast forward solver and a first order optimization algorithm, relies on the minimization of the nonlinear least square residual function. Validity of the inverse analysis method is investigated with respect to the influence of the arrangement of scanning paths. The efficiency of the computation is also discussed.     

Inductance Calculations for Noncoaxial Coils Using Bessel Functions

A relatively simple and general method for calculating the mutual inductance and self-inductance of both coaxial and noncoaxial cylindrical coils is given. For combinations of cylindrical coils, thin solenoids, pancake coils, and simple circular loops, the mutual inductance can be reduced to a one-dimensional integral of closed form expressions involving Bessel and related functions. Coaxial and noncoaxial cases differ only by the presence of an extra Bessel factor J ₀(sp) in the noncoaxial integral, where p is the perpendicular distance separating the coil axes and s is the variable of integration. The method is related to a recently given noncoaxial generalization of Ruby's formula for a nuclear radiation source and detector system, the analogy being close but not exact. In many cases, the Bessel function integral for the inductance can be easily evaluated directly using Maple or Mathematica. In other cases, it is better to transform the integral to a more numerically friendly form. A general analytical solution is presented for the inductance of two circular loops which lie in the same plane     

A field-theoretical approach to magnetic induction heating of thin circular plates

A field-theoretical approach is used to analyze the subject of magnetic induction heating of thin circular plates by planar coils. Closed-form solutions for the electric and magnetic fields are found to the basic field problem of a single circular loop carrying current at a frequency ω in the presence of a plate characterized by a permeability μ and conductivity σ. By using these fields, expressions are derived for the complex Poynting vector at the surface of the plate, and for the induced EMF in the coil. The theory is extended to include multiturn coils and a field-dependent permeability, and a specific multiturn coil and plate combination is chosen as an example. The complex amplitude of the magnetic field and the Poynting vector are calculated along the surface of the plate using iterative methods to assure self-consistency with the field dependent permeability of the plate. By using Fourier transform techniques, the transient coil current and coil voltage waveforms are calculated under the experimental conditions used to take data on the sample coil and plate. The absorbed power is calculated from these waveforms and is found to be within 10 percent of the measured power absorption for all levels of operation from 50 to 2000 W. The calculated coil current waveform is compared with the measured waveform and is found to be in very good agreement in both shape and period.     

Optimization of endoluminal loop radiofrequency coils for gastrointestinal wall MR imaging

In this paper, we describe the optimization of endoluminal planar coils for high-resolution magnetic resonance imaging of gastrointestinal walls. For maximizing the coil performances, electromagnetic parameters of planar rectangular radio frequency (RF) coils were simulated using the finite element method. The eddy currents were fully computed to determine the electromagnetic losses in both wires and surrounding environment. Geometric parameters of the coils (length, conductive layer section, number of layers, and turns number) were varied. Based on simulations, five loop RF coil prototypes with planar geometry were designed to fit in a 5-mm inner diameter catheter. In the immediate vicinity of single-loop coils, the signal-to-noise ratio (SNR) decreases with the length of the coil, whereas penetration depth increases with it. The double-loop coil offers a greater penetration depth in comparison to the same length single-loop coil. The multilayer coil preserves the RF field B/sub 1/ by inducing a reduction in the electrical resistance of the conductor, therefore resulting in an increase in SNR. Experimental verifications were performed on a 1.5 T clinical scanner. Simulation results were found to be in good agreement with that of MR experiments. Developed prototypes provided a dramatic increase in SNR at the region of interest.     

Integrated Planar HTS SQUID Gradiometer for NDE Application

The performance of a SQUID NDE probe with integrated planar SQUID gradiometers was compared to that with an electronic axial SQUID gradiometer. A circular excitation coil was used for the SQUID NDE probe with the planar gradiometer, whereas a differential excitation coil was adopted for the one with the axial gradiometer. These NDE systems are used to detect a buried flaw in aluminum plates, and the phase-depth relations of the buried flaw were found to show similar linear dependences. Overall, the SQUID NDE system with the planar gradiometer is relatively simple and practical for unshielded operations.     

Magnetostatic image theory for an arbitrary current loop in frontof a permeable sphere

Magnetostatic image theory recently introduced for the problem of a circular current loop in front of a permeable sphere centered on the axis of the loop is generalized to the case of an arbitrary current loop outside the sphere. The theory is derived by finding the image for a current element outside the sphere. As an application, the image for a straight line current is seen to result in a surface current in the form of a circular disk. The present theory is also seen to reproduce that for the circular loop as a special case. The analytic form of the image current is simple enough to be applicable in small-scale computation for problems involving a permeable sphere and stationary current     

Inductance Calculations for Circular Coils of Rectangular Cross Section and Parallel Axes Using Bessel and Struve Functions

A simple method for calculating the mutual and self inductances of circular coils of rectangular cross section and parallel axes is presented. The method applies to non-coaxial as well as coaxial coils, and self inductance can be calculated by considering two identical coils which coincide in space. It is assumed that current density is homogeneous in the coil windings. The inductances are given in terms of one-dimensional integrals involving Bessel and Struve functions, and an exact solution is given for one of these integrals. The remaining terms can be evaluated numerically to great accuracy using computer packages such as Mathematica. The method is compared with other exact methods for the coaxial case, and with a filamentary method for the non-coaxial case. Excellent agreement was found in all cases, and the exact method presented here agrees with another exact coaxial method to great numerical accuracy.     

Electromagnetic Forces of High-T c Superconducting Coated Conductor Coils Subjected to Sinusoidal Traveling Magnetic Fields

In order to optimize the structure of high temperature superconducting coils for linear motor applications, three separate coils with different shapes made of Re-BCO coated conductor were studied: circular shaped single pancake, circular shaped double-pancake and racetrack shaped single-pancake. The thrust and vertical forces of the three coils above a conventional flat linear three-phases winding were investigated experimentally. With the aid of the experimentally obtained values, it was found that single-pancake coil in the shape of a racetrack was the best selection for a flat single-sided linear motor system. Studies were also made on the frequency characteristics of the vertical force of the racetrack shaped single-pancake coil.     

An analysis of three-dimensional planar embedded cracks subjected to uniform tensile stress

In this paper we describe an analytical methodology for calculating Stress Intensity Factors (SIF) on planar embedded cracks with an arbitrarily shaped front. The approach is based on a first order expansion of the celebrated integral of Oore-Burns and the actual shapes of three-dimensional planar flaws are analysed in terms of homotopy transformations of a reference disk.The solution is proposed in terms of Fourier series and the first order approximation of the coefficients is given independently from the homotopy transformations.The comparison with numerical results, taken from scientific literature, indicates that the proposed equation is very accurate when the flaw presents a small deviation from the circular shape. Finally, the closed form solution is used to predict the SIF of many types of convex and non-convex planar flaws present in engineering components such as welded structures or casting components.     

Mutual Inductance and Force Calculations Between Coaxial Bitter Coils and Superconducting Coils with Rectangular Cross Section

Mutual inductance and force calculations between coaxial Bitter coils and superconducting coils with rectangular cross section in a hybrid magnet system using derived semi-analytical expressions based on two integrations were performed. The mutual inductance and force calculations are based on the assumption of the uniform current density distribution in superconducting coils. The current density distribution of a Bitter coil in radial direction, however, is inversely proportional to the radius of the Bitter coil. The influence of the current density redistribution caused by a cooling hole and an inhomogeneous temperature distribution of Bitter coil of a water-cooled magnet was not considered. The obtained expressions can be implemented by Simpson’s integration with FORTRAN programming. We confirm the validity of mutual inductance calculation by comparing it with a filament method, and give the accuracy of two methods. The mutual inductance values computed by two methods are in excellent agreement. The derived semi-analytical expressions of mutual inductance allow a low computational time compared with filament method to a specific accuracy. The force is derived by multiplying the currents of the two coils by their mutual inductance gradient.     

Mutual Impedance of Cylindrical Coils at an Arbitrary Position and Orientation Above a Planar Conductor

We derive a closed-form expression for the mutual impedance due to eddy-current induction for a pair of cylindrical air-core coils with arbitrary position and orientation above a planar conductor. By extending a recently devised model for individual coils with an arbitrary tilt with respect to the surface, we obtain a remarkably simple result. We validated our model with measurements on a conductive plate. The results should be useful for designing new probe configurations and for evaluating the signals in eddy-current inspections when driver-pickup coil configurations are utilized.     

Density of invariant disk packings for planar piecewise isometries

Iteration of a planar piecewise isometry may generate an invariant disk packing, and understanding the properties of the disk packing is helpful for estimating the Lebesgue measure of the exceptional set for the planar piecewise isometry. If the disk packing is not dense, then the Lebesgue measure of the exceptional set is positive. But it is not easy to check the density of a disk packing. In this paper, the authors present necessary and sufficient conditions for the density of a general disk packing, and discuss some properties of disk packings for planar piecewise isometries.     

Magnetic Force Calculation Between Thin Coaxial Circular Coils in Air

We present new and fast procedures for calculating magnetic forces between thin coaxial circular coaxial coils in air. The results are expressed in semianalytical form in terms of the complete elliptical integrals of the first and second kind, Heuman's Lambda function, and a term that must be solved numerically. These expressions are accurate and simple to use for several practical applications. We also describe a comparative method based on the filament technique. We discuss the computational cost and the accuracy of two methods and compare them with already published data. Results obtained by our two approaches are in excellent agreement with each other. They can be used in industrial electromagnetic applications such as electrodynamic levitation systems, linear induction launchers, linear actuators, and coil guns.     

Identification and decentralized control of a dual-actuator hard disk drive system

The dual-actuator hard disk drive with two voice coil motor actuators mounted in the diagonal corners of the baseplate allows faster data access with relatively slow spindle speed. However, the mechanical interaction between the actuators tends to affect the head positioning accuracy. In this paper, we use the circle-fit modal testing method to identify the dual-input dual-output frequency response model of the dual-actuator plant. We discuss the decentralized control scheme for the dual-actuator tracking servo. The servo loop decomposition reveals that the interaction induces narrow-band vibrations near the crossover frequency. We propose a phase-lead interaction filter with a slight change to the loop stability to suppress the vibrations. Experimental results on a dual-actuator prototype show that the position error signal 6/spl sigma/ value due to interaction vibrations is reduced by 61% with the interaction filter in combination with the decentralized control scheme.     

A minimum time seek controller for a disk drive

A time optimal controller design is proposed for the purpose of steering the read/write head of a magnetic disk drive during a seek manuever. The design includes the development of a switch-on-time (SOT) open loop slew control using a third order model of the voice coil motor. The SOT algorithm employs a look up table of switch times versus angle. A perturbation correction stage is included to tailor the switch times to adjust for variations in the operating characteristic of the disk drive (due, for example to heat buildup, wear, etc.). The design also includes a terminal stage servo control that employs an optimal feedback/feedforward controller which is used to provide precise positioning at the end of the seek maneuver     

Physics-Based Time-Domain Model of a Magnetic Induction Microgenerator

This paper presents a study of a time-domain model of a magnetic induction microgenerator for energy harvesting applications. The model is based on a simple structure for which an analytical expression of the magnetic field distribution can be computed. From this analytical expression, geometric parameters that are not taken into account in the previous literature on microgenerators are considered. Starting from the magnetic field distribution in space of a circular current loop, the paper derives the induced electromotive force in a coil depending on the distance to the magnet. Simulations give insight into the validity of linear models implicitly assumed in frequency-domain analysis of these systems     

Analysis of a fork-shaped rectangular coil facing moving sheet conductors

The electromagnetic induction method utilising eddy current plays an important role in a non-destructive material test. In testing slab-type material by electromotive force method, there is the fork-shaped coil method, which has two coils placed on both sides of the test piece. In most studies, circular coils have been analysed. However, it has been pointed out quantitatively that a rectangular coil is more useful than a circular coil for a non-destructive test. The authors derive a rigid theoretical formula with a test theory experiment.     

Piezoelectrically actuated flextensional micromachined ultrasound transducers--II: fabrication and experiments

This paper presents novel micromachined two-dimensional array piezoelectrically actuated flextensional transducers that can be used to generate sound in air or water. Micromachining techniques to fabricate these devices are also presented. Individual unimorph array elements consist of a thin piezoelectric annular disk and a thin, fully clamped, circular plate. We manufacture the transducer in two-dimensional arrays using planar silicon micromachining and demonstrate ultrasound transmission in air at 2.85 MHz with 0.15 μm/V peak displacement. The devices have a range of operating resonance frequencies starting from 450 kHz to 4.5 MHz. Such an array could be combined with on-board driving and addressing circuitry for different applications