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ANALYTICAL MODELING FOR TRANSIENT PROBE RESPONSE IN EDDY CURRENT TESTING
MODELISATION ANALYTIQUE DE REPONSE TRANSITOIRE EN CONTROLE PAR COURANT DE FOUCAULT
A Thesis Submitted
to the Division of Graduate Studies of the Royal Military College of Canada
by
Daniel Desjardins, BSc Second Lieutenant
In Partial Fulfillment of the Requirements for the Degree of
Master of Physics
6 May 2011
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ABSTRACT
Thesis completed by Desjardins, Daniel, in partial fulfillment of requirements for M.Sc. in Physics from the Royal Military College of Canada on this 26 April 2011 tided Analytical Modeling for Transient Probe Response in Eddy Current Testing under the direct supervision of Dr. Thomas Krause and Dr. Napoleon Gauthier.
Fatigue cracks in multi-layer aluminum structures are prone to develop around the ferrous fasteners found on ageing Canadian Forces aircraft such as the F/A-18 Hornet, CC-130 Hercules and CP-140 Aurora. Normally, fastener removal is required for inspection by conventional eddy current testing (ET), since the conducting aluminum structure limits inspection to either surface breaking or near-surface cracks when fasteners are present. In this work the fundamental principles behind the application of transient eddy current as a means of inspecting fastened structures without fastener removal is investigated. In particular, the proposal of using ferrous fasteners as conduits to carry flux deeper into the multilayer structure is examined.
Analytical models were derived from solutions to Maxwell's equations using the magnetic vector potential and appropriate boundary conditions. The models described the progression of magnetic flux, generated by a step function current applied to a coil, 1) down the length of ferrous and nonferrous conducting rods, and 2) a hole in an infinite aluminum plate (borehole). Analytical results were validated against finite element models of the same configuration. The analytical models were then expanded to model actual transient probe responses, including feedback effects, for transmit-receive probe configurations using measured driving currents and physical probe dimensions. The model results predicted a significant shift of signal peaks to later times for the ferromagnetic rod as the distance between transmitter and receiver was increased.
A comparison of analytical and experimentally measured receive coil responses for air and conducting nonferrous rods showed excellent agreement with model predictions. Good qualitative agreement with experimental ferrous rod results was obtained, including the predicted shift in time of peaks with distance. The results are in support of experimental observations of greater depths of penetration achievable for the inspection of multilayer aluminum structures in the presence of ferrous fasteners as compared to multilayer structures with only aluminum. This work represents the first complete analytical model of transient eddy current coil response in the presence of ferrous and nonferrous conducting rods. IV
RESUME
Les structures a multiples couches en aluminium sur les aeronefs vieillissants des Forces Canadiennes tels le F/A-18 Hornet, CC-130 Hercules et CP-140 Aurora sont suscepdbles de developper des fissures dues a la tension autour des rivets de fixation. En temps normal, 1'enlevement des rivets est requis afin d'effectuer l'essai par courants de Foucault convendonnels, puisque l'influence des rivets ferrugineux presents dans ces structures restreint l'inspection aux fissures superficielles ou en proximite de la surface. Cette these developpe les principes fondamentaux qui permettraient l'application de la methode d'essai par courants de Foucault transitoires sans necessiter 1'enlevement des rivets de fixation. En particulier, l'hypothese que les rivets ferrugineux accroissent et acheminent le flux magnetique a de plus grandes profondeurs a I'interieur de la structure a multiples couches est etudiee.
Des modeles analytiques sont developpes a partir des equations de Maxwell en utilisant le potentiel magnetique vectoriel et des conditions frontieres appropriees. Les modeles decrivent revolution du champ magnetique, suivant l'application soudaine d'un courant dans une bobine electromagnetique, 1) au long de tiges conductrices ferreuses et non ferreuses, et 2) dans un trou perce dans une plaque infinie en aluminium (trou de forage). Les resultats analytiques sont valides par des modeles d'element fini decrivant la meme geometric Les modeles analytiques sont ensuite utilises afin de modeliser des signaux transitoires, en incluant les effets de reaction, pour des configurations emetteur/recepteur en utilisant la fonction decrivant le courant applique ainsi que les dimensions physiques de la sonde. Dans le cas d'une tige ferromagnedque, les resultats de la modelisation prevoient que les maximums des signaux seraient deplaces plus tard dans le temps a mesure que la distance entre l'emetteur et le recepteur est accrue.
Une comparaison effectuee entre les signaux experimentaux et modelises dans l'air et en la presence de tiges non ferreuses demontre un bon accord. Un accord qualitatif est etabli pour les tiges ferromagnetiques, qui demontre le deplacement anticipe des maximums en fonction de la distance. Ces resultats soutiennent qu'il serait possible d'utiliser les rivets ferrugineux afin de sonder plus profondement a I'interieur des structures a multiples couches en aluminium. Ce travail represente le premier modele analytique complet decrivant la reponse transitoire d'une sonde pour l'essai par courants de Foucault en la presence de tiges ferreuses et non ferreuses. V
TABLE OF CONTENTS
ACKNOWLEDGEMENTS ii ABSTRACT iii RESUME iv TABLE OF CONTENTS v LIST OF TABLES viii LIST OF FIGURES ix LIST OF ABREVIATIONS xiii LIST OF SYMBOLS xiv CHAPTER 1 -INTRODUCTION 1 1.1 The Ferrous Fastener Challenge 1 1.2 Objective 4 1.3 Research Survey 4 1.4 Scope and Methodology 7 1.5 Structure 8 CHAPTER 2 - THEORY AND BACKGROUND 9 2.1 Maxwell's Field Equations 9 2.2 Material-related Magnetic Quantities 10 2.3 Material Interface 11 2.4 Quasistationary Magnetic Fields 12 2.5 Magnetic Field Diffusion and Eddy Currents 14 2.6 Electrodynamics Problems 18 2.6.1 Problem Formulation 18 2.6.2 Solution Building 20 2.7 Feedback in Coupled Magnetic Circuits 25 2.8 Duhamel's Theorem 28 CHAPTER 3 - THEORETICAL MODEL 29 3.1 Long Rod 29 3.1.1 Stationary Solution 29 3.1.2 Transient Solution 41 3.2 Borehole 53 3.2.1 Stationary Solution 53 3.2.2 Transient Solution 55 VI
CHAPTER 4 - FINITE ELEMENT MODEL 59 4.1 Long Rod Comparison 59 4.1.1 Stationary Current Loop Results 59 4.1.2 Stationary Finite Coil Results 64 4.1.3 Transient Results 68 4.2 Borehole Comparison 70 4.2.1 Stationary Results 70 4.2.2 Transient Results 72 CHAPTER 5 - TRANSIENT PROBE RESPONSE MODEL 75 5.1 Theoretical Response 75 5.2 Modeled Response in Air 79 5.3 Modeled Response of an Aluminum Rod 82 5.4 Modeled Response of a Brass Rod 84 5.5 Modeled Response of a Steel Rod 86 CHAPTER 6 - EXPERIMENTAL WORK 89 6.1 Experimental Technique 89 6.2 Long Rod 93 6.2.1 Air 94 6.2.2 Aluminum 97 6.2.3 Brass 101 6.2.4 Steel 104 6.3 Bored Aluminum Plate 109 6.4 Rod & Bored Plate 113 CHAPTER 7 - ANALYSIS AND DISCUSSION 117 7.1 Parameters Affecting Voltage Response 117 7.2 Channelling of Magnetic Flux 121 7.3 Electromagnetic Interactions in the Borehole 122 7.4 Dual Couple Diffusion Process 123 7.5 TEC and the Ferrous Fastener Challenge 124 CHAPTER 8 - CONCLUSION 127 CHAPTER 9 - FUTURE WORK 129 9.1 Complete Ferrous-Fastener Geometry 129 9.2 Calculating Feedback 130 REFERENCES 135 ANNEX A 137 vii
ANNEX B 139 ANNEX C 141 ANNEX D 143 CURRICULUM VITAE 145 Vlll
LIST OF TABLES
Table 1: Sample Eigenvalues for A — 0, 10, 20 and 400, when^r = 66 and r0 = 4.8 mm 47 Table 2: Physical dimensions of the transmit and receive coils 141 Table 3: Inner wing intermediate spar lower cap specifications (source: F/A-18 Fleet Management repport) 143 IX
LIST OF FIGURES
Fig. 1: McDonnell Douglas CF-18 Hornet (source: www.aviastar.org) 2 Fig. 2: CF-18 Inner wing intermediate spar lower cap fastener cross-sectional view 3 Fig. 3: Refraction of a magnetic field line and related field vector decomposition when crossing the boundary 12 Fig. 4: Magnetic field of a current loop centered on the z-axis 14 Fig. 5: Magnetic vector potential of a current loop 16 Fig. 6: Plots of Bessel (a) and Neumann (b) functions of orders 0 - 4 23 Fig. 7: Modified Bessel functions of the first and second kind 23 Fig. 8: Transmitter coil current onset 25 Fig. 9: Simple RL circuit 26 Fig. 10: Magnetic circuit with two inductors 27 Fig. 11: Equivalent magnetic circuit with three inductors 27 Fig. 12: Diagram of the long rod model 30 Fig. 13: Axial dependence of the magnetic vector potential in a non-magnetic rod 34 Fig. 14: Radial dependence of the magnetic vector potential in a non-magnetic rod 35 Fig. 15: Axial dependence of the magnetic vector potential in a magnetic steel rod 36 Fig. 16: Radial dependence of the magnetic vector potential in a magnetic steel rod 37 Fig. 17: Axial dependence of the magnetic vector potential from a finite coil encircling magnetic and non-magnetic rods 40 Fig. 18: Radial dependence of the magnetic vector potential from a finite coil encircling magnetic and non-magnetic rods 40 Fig. 19: 3d plot of the Fourier-Bessel coefficient; lambda and m ; cross-sectional views 44 Fig. 20: Sample kernel function 49 Fig. 21: Axial dependence of the transient magnetic vector potential in a magnetic steel rod. 50 Fig. 22: Radial dependence of the transient magnetic vector potential in a magnetic steel rod. 51 Fig. 23: Diagram of the long borehole model 53 Fig. 24: Stationary magnetic vector potential in the borehole configuration 55 X
Fig. 25: Stationary magnetic vector potential of an aluminum rod and a current loop 60 Fig. 26: Axial dependence of the magnetic vector potential in a non-magnetic rod; FEA vs Analytical 61 Fig. 27: Radial dependence of the magnetic vector potential in a non-magnetic rod; FEA vs Analytical 61 Fig. 28: Stationary magnetic vector potential of a magnetic steel rod and a current loop 62 Fig. 29: Axial dependence of the magnetic vector potential in a magnetic steel rod; FEA vs Analytical 63 Fig. 30: Radial dependence of the magnetic vector potential in a magnetic steel rod; FEA vs Analytical 63 Fig. 31: Magnetic vector potential of a finite coil encircling an aluminum rod 64 Fig. 32: Analytical vs. FEA: Axial dependence of the magnetic vector potential from a finite coil encircling a non-magnetic rod 65 Fig. 33: Analytical vs. FEA: Radial dependence of the magnetic vector potential from a finite coil encircling a non-magnetic rod 65 Fig. 34: Magnetic vector potential of a finite coil 66 Fig. 35: Analytical vs. FEA: Axial dependence of the magnetic vector potential from a finite coil encircling magnetic and non-magnetic rods 67 Fig. 36: Analytical vs. FEA: Radial dependence of the magnetic vector potential from a finite coil encircling magnetic and non-magnetic rods 67 Fig. 37: Transient progression of magnetic vector potential along a magnetic steel rod 68 Fig. 38: Analytical vs. FEA: Axial dependence of the transient magnetic vector potential from a finite coil encircling magnetic and non-magnetic rods 69 Fig. 39: Analytical vs. FEA: Radial dependence of the transient magnetic vector potential from a finite coil encircling magnetic and non-magnetic rods 69 Fig. 40: Magnetic vector potential of a current loop in a non-magnetic bored structure 70 Fig. 41: Magnetic vector potential of a current loop in a magnetic bored structure 71 Fig. 42: Magnetic vector potential of a current loop in a borehole 72 Fig. 43: Transient progression of the magnetic vector potential into a bored conducting structure 73 Fig. 44: FEA: Radial dependence of the transient magnetic vector potential from a current loop inside an aluminum borehole 73 xi
Fig. 45: FEA: Axial dependence of the transient magnetic vector potential from a current loop inside an aluminum borehole 74 Fig. 46: Second-order fitted transmitter signal 80 Fig. 47: Modelled transient probe response in air at 5, 10 and 20 mm receiver positions 82 Fig. 48: Comparison of the experimental and fitted transmitter signal 83 Fig. 49: Modelled voltage response of an aluminum rod 84 Fig. 50: Modelled transient probe response in the presence of a brass rod 85 Fig. 51: Comparison of modelled brass and aluminum transient responses 85 Fig. 52: Comparison of experimental and analytical fit transmitter signals 87 Fig. 53: Modelled voltage responses in the presence of a steel rod 88 Fig. 54: Diagram of simplified experimental model 89 Fig. 55: Schematic diagrams of the different experimental configurations 90 Fig. 56: 1033-turn transmit and 675-turn receive coils 91 Fig. 57: TecScan and pulser system 91 Fig. 58: Functional flow diagram of signal generation in the experimental set-up 92 Fig. 59: Functional flow diagram of signal aquisition in the experimental set-up 92 Fig. 60: Long rod experiment configuration 94 Fig. 61: Voltage response in the absence of a conducting rod 94 Fig. 62: Peak occurence time vs. receiver position for air 96 Fig. 63: Signal peak times in air relative to the diffusion timescale 97 Fig. 64: Comparison of modelled and experimental results for an aluminum rod 98 Fig. 65: Voltage response in the presence of an aluminum rod 99 Fig. 66: Peak occurence times for an aluminum rod 100 Fig. 67: Signal peak times relative to signal duration for aluminum 101 Fig. 68: Voltage response of a brass rod 102 Fig. 69: Comparison of brass and aluminum rod signals at 5 mm distance 103 Fig. 70: Peak occurence times for a brass rod 103 Fig. 71: Signal peak times relative to the magnetic diffusion timescale 104 Fig. 72: Experimental voltage reponse in the presence of a steel rod, fir unknown 105 Fig. 73: Modelled voltage response in the presence of a steel rod, fir = 200 105 Fig. 74: Voltage respons of a steel rod 106 Fig. 75: Peak occurence times for a steel rod 107 Xll
Fig. 76: Signal peak times relative to the diffusion timescale for a steel rod 108 Fig. 77: Thick bored plate experimental configuration 109 Fig. 78: Voltage response of a bored aluminum plate 110 Fig. 79: Signal peak times relative to the diffusion timescale for a bored aluminum plate. ..Ill Fig. 80: Time-derivative of the axial magnetic flux density in the bored aluminum plate configuration 112 Fig. 81: Diagram of long half-rod & bored plate configuration 113 Fig. 82: Voltage response of a magnetic rod through a bored aluminum plate 113 Fig. 83: Voltage signal peaks versus receiver position for the rod and plate configuration.. 115 Fig. 84: Transient progression of magnetic vector potential along a magnetic steel rod shielded by an aluminum plate 115 Fig. 85: Comparison of voltage responses of air, aluminum, brass and steel 118 Fig. 86: Transmitter signals in the presence of air, aluminum and steel rods 119 Fig. 87: Normalized voltage response of air, aluminum, brass and steel 119 Fig. 88: Time-to-peak measured as a function of notch depth 125 Fig. 89: Signal peak amplitude as a function of notch depth 126 Fig. 90: Complete ferrous fastener boundary value problem 130 LIST OF ABREVIATIONS
AC Alternating Current AWG American Wire Gauge BC Boundary Condition BVP Boundary Value Problem CDDP Central Driver Differential Pickup CGSB Canadian General Standard Board EC Conventional Eddy Current EMF Electromotive Force FEA Finite Element Analysis FT Fourier Transform IACS International Annealed Copper Standards MAS Military Aviation Services MFL Magnetic Flux Leakage MPI Magnetic Particle Inspection MQS Magnetoquasistatic NDE Non-Destructive Evaluation NDT Non-Destructive Testing OML Outer Mold Line PDE Partial Differential Equation TEC Transient Eddy Current XIV
LIST OF SYMBOLS
A Magnetic Vector Potential (tesda • m) H Magnetic Field Strength ( ampere • m ') B Magnetic Flux Density (tesla) D Electric Flux Density ( coulomb • m2) E Electric Field Strength (volt • m1) j Total Current Density ( ampere • m 2) jf Free Current Density ( ampere • m2) j. Induced / Eddy Current Density ( ampere • m ) if Surface Free Current Density ( ampere • m :)
3 pe Volume Density of Free Electric Charges ( coulomb • m ) a Current Loop radius ( m ) R The Set of Real Numbers t Time (s) \i Magnetic Permeability ( henry • m x)
1 (l0 Magnetic Permeability of Free Space ( henry • m )
\JLR Relative Magnetic Permeability ( dimensionless )
Xm Magnetic Susceptibility (dimensionless ) a Electrical Conductivity ( Siemens • m ') p Electrical Resistivity ( ohms • ml) V Nabla Operator (m 1) V2 Laplacian Operator ( m2) c Subset of V For all G Element of fl Domain dfi. Domain Boundary
Tilde designates the Fourier transform of the function JC Subscript J-C designates the Hankel transform of the function 1
CHAPTER 1 - INTRODUCTION
The increasing demand for fast, reliable and economical aircraft inspection capabilities makes it desirable to develop novel technologies in the field of non-destructive evaluation. Transient eddy current testing is one such technology that has the potential to advance present inspection capabilities for multi-layered rivet-joint aluminum structures by using ferrous fasteners as flux-carrying conduits.
1.1 The Ferrous Fastener Challenge
In operation, aircraft aluminum structures are subjected to stresses from alternating loads, continuous vibrations, and in the case of supersonic aircraft, shock waves, all of which inevitably cause fatigue cracks. Since eddy current testing is especially sensitive for materials of high conductivity such as aluminum, it offers excellent opportunities for detecting surface-breaking and sub-surface defects, and is therefore widely used throughout the aeronautical industry [1]. Furthermore, it is easy to use, mechanically robust and is not restricted by any safety regulations [2].
In conventional eddy current non-destructive testing (NDT), eddy currents are excited in a conductor by passing an alternating current (AC) through a coil in close proximity to the conductor. AC current is time-harmonic and sinusoidal in nature and gives rise to eddy currents of a similar nature. Detection of subsurface cracks, however, is limited by skin depth considerations [3].
An alternative technique employs transient eddy currents (TEC) that are excited by means of a non-sinusoidal coil current. In most systems, a steady-state current is allowed to persist for some time before the waveform repeats. The length of this steady state period is usually made sufficiently long so that any eddy current signals have completely decayed away to undetectable levels. When present, a flaw will perturb the induced eddy current density within the conductor. Detection and analysis of the signal perturbation has the potential to permit flaw identification and characterization. 2
The sensitivity of eddy current testing systems, however, depends on the depth and volume of the target discontinuity, and its reliability is reduced by other perturbing effects such as the presence of magnetic materials or geometrical discontinuities. Ferrous fasteners strongly influence conventional eddy current NDT in multi-layered rivet-joint aluminum structures. Their presence requires specific kinds of probes which themselves are skin depth limited. Since cracks in ageing rivet-joint aluminum structures generally start from the fastener boreholes, the sensitive detection of cracks in underlying layers, separated from the strong rivet influence, demands further improvements of NDT-methods [4].
The confounding influence of the ferrous fasteners on eddy current signals may be dubbed the "ferrous fastener challenge". Because of the fastener's detrimental influences on conventional NDT systems, the current practice of performing bolt hole eddy current inspection on the CP-140 Aurora, CC-130 Hercules, and CF-18 Hornet aircraft, shown in Fig. 1, wing structures requires that ferrous fasteners be removed from the wing before inspection [5].
Fig. 1: McDonnell Douglas CF-18 Hornet (source: www.aviastar.org)
Consequently, defense aircraft experience long downtime and require several hundred man hours for inspection. For example, technicians at Canadian Forces Base Trenton report that approximately 1200 hours on average are required to inspect approximately two hundred thousand boreholes on a Hercules wing structure [5]. Military aircraft undergoing maintenance are non-operational and are therefore unavailable to meet mission objectives and national interests. Likewise, commercial aircraft undergoing maintenance do not generate revenues for their parent companies. Furthermore, repetitive removal and installation of fasteners pose additional risk of damage. Procedures for fastener 3 removal and the addition of new fasteners must be stncdy adhered to. For example, interference fit of fasteners and torque tightening is required [6].
The outer wing skin on the CF-18 must first be removed in order to access and inspect the inner wing intermediate spar lower caps, and exemplifies the challenges associated with present eddy current inspection methods [7]. A cross-sectional view of the CF-18 Hornet's inner wing structure, shown in Fig. 2, demonstrates this difficulty. The limited accessibility increases cost, downtime and risk of additional damage upon component removal and reinstallation. Therefore, the present approach using conventional eddy current is excessively invasive.
Fastener Seal groove Skin
Fatigue crack initiating at hole and Corrosion crack initiating at hole propagating in Force-Aft direction and propagating spanwise
Fig 2 CF-18 Inner wing intermediate spar lower cap fastener cross-sectional view (redrawn from F/A-18 Fleet Management report)
A unique opportunity now exists whereby the fasteners could be used to enhance the inspection's sensitivity to defects using transient eddy currents. Recent experimental work performed at the Royal Military College by P. Whalen [8] has demonstrated a mechanism which suggests that ferrous fastener magnetization enables the sensing of discontinuity- induced perturbations in the transient eddy current field at greater depths
It is proposed here that transient eddy current NDE has the potential to detect defects at much greater depths by using the ferrous fasteners as magnetic flux carrying conduits in a manner analogous to the transport of flux in magnetic circuits [9]. This technique would simultaneously address the "ferrous fastener challenge", and enable testing without removal of fasteners. With respect to the CF-18 inner wing example, stress and 4 corrosion cracks have the potential of being detected down to the required depths within the spar structure itself, and thus skin removal would no longer be required for accessibility. The inspection process would be simplified and aircraft downtime would be drastically reduced. Elsewhere, avoiding fastener removal and re-instalment would eliminate the risk of causing additional damage.
1.2 Objective
The aim of this thesis is to elucidate the potential of transient magnetization processes, occurring within ferrous fasteners, for increased depth of detection within multilayer aluminum structures. A theoretical model would provide a comprehensive framework to develop an understanding of the physical dynamic magnetic processes at work, and assess the ferrous fastener's ability to act as a flux-channelling conduit. The results of work in this area could have far-reaching implications for the applicability of transient eddy current non-destructive evaluation.
1.3 Research Survey
A thorough understanding of electromagnetic field theory is a pre-requisite to construct physically reasonable analytical models. In addition, a suite of mathematical tools is necessary for the formulation of the appropriate equations. Consequently, textbooks encompassing the subjects of electrodynamics, magnetic diffusion theory, boundary value problems and differential equations are of particular importance. However, transient solutions to magnetic diffusion problems in the presence of ferromagnetic materials are largely absent in the literature. In the near-neglected topic of eddy currents [10], authors have mostly developed formulae for the simplest cases which involve non-magnetic conducting materials and uniformly applied magnetic fields. Additionally, feedback effects have yet to be addressed.
Standard electromagnetics textbooks such as Jackson [10], Griffiths [11], Sadiku [9] and Plonus [12] provide fundamental knowledge of electromagnetic field theory, particularly 5 in the framework of magnetoquasistatics, required for understanding the governing physical principles. However, detailed examinations of diffusion theory must be sought elsewhere.
One of magnetic diffusion theory's earliest pioneers is Wwedensky [13], circa 1921. He developed formulae describing the time-dependent diffusion of an abruptly applied uniform magnetic field into a long circular conductor. However, his formulae can only be considered approximate because he assumes a time-independent external field. Additionally, he makes the approximation of an applied field, infinite in extent, which requires the assumption that the divergence of the magnetic induction is non-zero, thereby violating Maxwell's equations. As a result, his theoretical model does not achieve suitable agreement with experimental results at early times.
In 1959, C. Bean et al. [14], used the Wwedensky solution to develop a method for measuring the resistivity of metallic specimens. Transient magnetization for the case of ferromagnetic material and feedback effects were not incorporated into the theoretical model. Furthermore, the applied field was considered uniform which greatly simplified the problem as in the Wwedensky case. Bean et al. [14] also developed a Fourier series solution for the case of a bar with rectangular cross-section.
P. Hammond [15], in 1961, developed the response of a sinusoidally-excited magnetic conducting slab. The solutions sought in terms of the magnetic vector potential show clearly the separate effects of the exciting current and the slab. In particular, a magnetization component strengthened the vector potential outside of the slab while an opposing term acted to reduce it. This remains one of the few examples of analytical results including effects of magnetization. Furthermore, Hammond explicitly states that eddy currents modify the magnetic field outside of the metallic bodies in which they flow.
The Dodd and Deeds [16] solutions, formulated in 1968 in terms of the magnetic vector potential, describe the response of a layered half-space and of a layered rod to a non uniform poloidal excitation field. However, the solutions are only applicable to non magnetic conducting media, and the excitation field is sinusoidal, which greatly simplifies the solution to the problem. These solutions are more relevant in the framework of conventional eddy current theory than to the transient problem addressed in this thesis. However, Dodd and Deeds [16] did make extensive use of the magnetic vector potential and 6 its continuity at boundaries, which mirrors the methodology of the solutions developed in Chapter 3 of this thesis.
In 1972, R. Calleroti [17] sought to express the voltage response of a receiver coil encircling a long metallic rod excited by a long solenoid, which repeated the problem tackled by Wwedensky, but utilized Fourier transforms. The resulting expressions contained approximate expansions of partial fractions and certain equations were explicidy stated to be obtained by trial and error. Furthermore, experimental and theoretical results only agreed at later times. The implied disagreement at early times is similar to the same difficulty encountered by Wwedensky [13] as the same approximations were made.
H. Knoepfel [18] presents a comprehensive theoretical treatise on magnetic fields with particular attention to magnetic diffusion and eddy currents as of 2000. Despite their relevance, the majority of problems developed in his work are restricted to problems of uniformly applied excitation fields, and therefore only describe diffusion in one dimension. Additionally, the mathematical models are developed for non-magnetic conducting media and assume time-independent external fields.
Interestingly, the most helpful literature pertaining to the transient magnetic diffusion problem tackled in this thesis comes from the discipline of geophysics. In the Journal of Russian Geology and Geophysics, Geologiya I Geofi-^ika, G. Morozova et al. [19] thoroughly develops analytical expressions for the transient magnetic and electric fields within a hollow ferromagnetic conducting cylinder. This particular article published in 2000 is one of the few explicitly tackling magnetic diffusion due to a non-uniform excitation in the presence of ferromagnetic materials. Accordingly, it provided insight into the present problem. However, her formulae share the same lack of agreement with experimental data at early times as those of Wwedensky, Bean and Calleroti.
Recently, J. Bowler et al. [20] have sought the step-function response of a conducting cylindrical rod to a poloidal field excitation. The final solution takes the form of a sum, an approximation of an integral of products of Bessel functions. A Laplace transform is used to separate the time component, and eigenvalues are sought in the form of discrete sets of relaxation times. The overall development leading to the final solution is scarcely explained 7 and difficult to follow. However, there are points of similarity between the analytical portion of Bowler's development and the work presented in Chapter 3.
1.4 Scope and Methodology
In this thesis, analytical, finite element and experimental work are performed conjointly in order to develop, verify and test exact analytical descriptions of transient magnetic field behaviour in idealized models. That is, the models assume homogeneous, isotropic and linear media, and address infinite half-spaces such as the infinite rod and infinite borehole problems. Hence, a classical treatment of the present electrodynamics problems provides a generalized description of the electrical interactions occurring within layered conducting rivet-joint structures. Elsewhere, electromagnetic effects of material defects are discussed, but are not included in the models. The detection and characterization of target discontinuities, such as cracks, are deferred for future work.
In order to study material and geometrical influences on the magnetic diffusion processes separately, the fastened aluminum structure is subdivided into simpler idealized models. Analytical solutions for the transient step-function response expressed in terms of the magnetic vector potential are explicitly derived from Maxwell's equations for a conducting magnetic rod and borehole. They are subsequently verified with finite element results. The theoretical models, supported by finite element analysis, describe transient processes arising in the borehole and rod configurations. The analytical models are then used to elucidate the electromagnetic interactions in the complete structure incorporating a conducting magnetic rod (fastener) and a bored conducting plate (aluminum layer). Finally, a series of experiments are performed to verify the predictions made by the theoretical and finite element models. 8
1.5 Structure
This thesis will include analytical models, finite element models, and experimental work. Chapter 2 provides a comprehensive overview of the elemental concepts of magnetic field theory, magnetic diffusion theory, and introduces the mathematics required for the formulation of solutions to potential problems.
Subsequently, transient analytical solutions for conducting rods and boreholes, valid for both magnetic and non-magnetic media, are developed from first principles and verified, in Chapter 4, with Finite Element Analysis (FEA).
Chapter 5 develops a novel method for analytically modelling transient probe response in eddy current testing by applying Duhamel's theorem to the step-function solutions in conjunction with a measured transmitter signal.
The experimental work is presented in Chapter 6. Symmetries present in the multilayered aluminum and ferrous fastener structure are exploited in order to construct simplified models, such as the rod and plate geometries. These simplified models are studied and progressively combined. Accordingly, material and geometrical effects on the magnetic diffusion processes are isolated and understood separately, and the theoretical model is checked for consistency. The experimental chapter is, therefore, structured in a manner which reflects the natural progression of this piecewise approach.
In Chapter 7, the theoretical, finite element and experimental results are summarized and discussed in relation to the ferrous fastener challenge. The focus of the analysis is to explore the potential of using ferrous fasteners as flux-channelling conduits in transient testing to enable deeper detection of target discontinuities in the vicinity of ferrous fasteners.
Chapter 8 concludes with a summary of important results and recommendations, while ongoing and future work are discussed in Chapter 9. 9
CHAPTER 2 - THEORY AND BACKGROUND
This chapter, beginning with Maxwell's equations, presents a summary of magnetic field theory in the context of the Transient Eddy Current (TEC) technique, and reviews the mathematics necessary to formulate analytical solutions to potential problems.
2.1 Maxwell's Field Equations
Maxwell's field equations [10] in vectorial form in SI units (International System of Units) are
V • D = pe (2.1)
VB = 0 (2.2)
dB VxE = (2.3) ~~dt dD 7xH = (2.4)
where current density j and charge density pe are considered as the sources that establish the electromagnetic fields H (auxiliary magnetic field), B (magnetic flux density), E (electric field) and D (electric flux density). In order to formulate general solutions, two more equations, known as the constitutive equations, are required. They are Ohm's law [10]
j = crE, (2.5) and the relation [10]
B = nH . (2.6)
The electric conductivity a and the magnetic permeability^ characteristic of the mediums can themselves be functions of space and time. In reality, the total magnetic induction B in ferromagnetic materials exhibits a hysteretic and non-linear relationship with the auxiliary magnetic field strength H [12]. This thesis takes the classical approach of 10 treating electrodynamics problems by seeking conditions under which the relationship between B and H is approximately linear. This approximation reduces the complexity of the magnetic problem in the presence of ferromagnetic materials. Henceforth, media are assumed to be homogeneous and isotropic, and the electric conductivity a and relative magnetic permeability ^. are taken to be constants.
2.2 Material-related Magnetic Quantities
The magnetic properties of a medium can be described by the magnetization vector M [12] such that
B M = H , (2 7) Mo which only exist in a medium. It vanishes in free space where B = fJ.QH. Introducing it into
(2.1-2.4) yields B dE V x — = j + (V x M) + e0 — (2 8) Mo ot
Equation (2 8) formally shows that when material is present in a magnetic field, internal sources of bound, or Ampenan, currents appear [11]. The term
Jm = VxM (29) is defined as a material-related equivalent magnetization current density [12]. In accordance with the principle of superposition, the total local induction is a vector sum of the fields defined by the existing current densities, so that
B = ^0H + n0M = fiH (210)
It should be noted that when an auxiliary magnetic field H0 is applied to a finite body of magnetic material, a steel rod for instance, the local magnetic behaviour is not described by equation (2 10) containing the substitution H0 -» H. Specifically, the magnetic induction in the
1 1 1 1 finite body is not B = /^H0, but B = /jH , where the internal local field H is itself the superposition of the outer field with the shape-dependent magnetization component arising from the finite volume of the magnetic body [18] Likewise, the induction outside the body 11 is Be = fiHe, where the external local field He is co-determined by the magnetic effect of the body.
This argument may be extended to the transient regime, where transient eddy
currents also modify the internal and external fields [15]. This thesis has found that several authors, including Wwedensky [13], Bean [14], Callaroti [17] and Knoepfel [18] have made the
e approximation of a time-independent external field with the substitution H0 -» H . This is only true for an infinite rod within an infinite solenoid, which violates Maxwell's equation V • B = 0. Determining B1 at any point within the body and Be outside it, for both the stationary magnetic and transient cases, requires solving a well-posed potential problem as will be done in Chapter 3.
2.3 Material Interface
The fundamental boundary conditions for the tangential and normal magnetic field components between two media are [10]
~ /B2 »i\ fix — -— =i (2.11)
fi-(B2-B1) = 0 (2.12) where fi is the unit normal vector pointing from medium 1 to medium 2, i is the surface current density related to the free electric charges. In the magnetic quasistationary approximation for normally conducting media where, i vanishes, the previous boundary conditions mean that the tangential (||) and normal (l) components become
Bju=BM
Mi M2
Bi.! = Bx>2 (2.14)
Stated otherwise, the normal component of the magnetic flux density remains steady across the boundary while the tangential component may be discontinuous according to the magnetic properties of the adjoining media. For example, if the first medium is air, the discontinuity is 12
AB, = B, 2 - B,! = (g - l) B, i = *mBlu (2 15)
where ^m is the magnetic susceptibility of the adjoining medium. The magnetoquasistatic boundary conditions, in the absence of free surface currents and when the first medium is air, are summarily illustrated in Fig. 3.
Fig 3 Refraction of a magnetic field line and related field vector decomposition when crossing the boundary
Having reviewed the effects of stationary magnetic fields in the presence of matter, the next sections address the interaction between time-varying magnetic fields, in the quasistationary approximation, and the related current densities in conductors.
2.4 Quasistationary Magnetic Fields
Maxwell's equations can be simplified by considering the time rates of change that appear in the Ampere-Maxwell (2 1) and Faraday (2 2) equations: if the rate of change is sufficiently slow with respect to the dynamic phenomena of interest, a quasistationary approximation may be applied [18].
In the quasistationary approximation, certain time dependences of the fields are explicitiy allowed. Conditions for tolerable time variations must be defined with respect to the dynamic processes of interest. These processes may be compared in terms of their characteristic times. The magnetic diffusion time rm, is the characteristic time required by an electromagnetic field to diffuse into a conductor. The formal electric field or charge relaxation time in the conductor re, is a characteristic time after which a variation of the 13 electric charge density settles to steady-state conditions [18]. The magnetoquasistatic approximation requires that
Tm » Te (2.16)
where Tm is the magnetic relaxation time and rm is the charge relaxation time. This condition is generally well satisfied in electric conductors, where the effects of magnetic fields predominate over those of electrostatic fields. Therefore, the whole conductor system is subject to the same quasistationary field [18]; fields generated by current densities in existence at a given time can be considered to propagate instantaneously.
Elsewhere, Ohm's law (2.5) is only valid for time scales much larger than the average time between collisions of the free electron (with electric charge e and mass me) in the conductor [12] so that
mea Tr= -, (2.17)
where xr is the mean free time between electron collisions. For copper it is typically Tr = 2.4 • 10~14 s [12]. For the quasistationary approximation to remain valid in a conductor, it is necessary that Tm be long compared to Tr, in which case the current is in phase with the electric field and there is no lag effect in field propagation. The relaxation time rr can be taken instead of xe as a limit in most situations [18]. For typical quasistationary magnetic-
6 dominated phenomena, where rm > 10~ s [18], it follows that
Tm » rr (2.18)
As a consequence of these conditions, the displacement term in Ampere's equation can be neglected and the magnetoquasistatic equations become [10]
VxH = j (2.19)
VxE=-— (2.20) at
V • B = 0 (2.21)
The magnetoquasistationary approximation will be used to develop transient solutions for the magnetic diffusion problems associated with rivet-joint aluminum aircraft structures. 14
2.5 Magnetic Field Diffusion and Eddy Currents
The electromagnetic interaction effect in the conductor can be treated either as a magnetic diffusion process or as the induction of an eddy current distribution, the two solutions being related through Ampere's law [18]. The concentration of induced current densities in the vicinity of an abrupt change in the conductor's geometry, around a crack for instance, is the basis of magnetically sensitive non-destructive testing.
In an axisymmetnc problem, azimuthal or toroidal currents, that is, currents flowing in circular paths perpendicular to the z-axis and centered on it, generate a poloidal field, meaning B^ = 0. This is demonstrated using the curl of the magnetic flux density.
The curl operator in circular cylindrical coordinates is given by the expression
(1 dBz dflrfA. (dBr dBz\ ~ 1 / d , . dBr\ "XB=^ = {r^-^)f+{^-^ + r{^rB^-^)2- (222)
However, given a stationary azimuthal current density) = J
(dBr dBz\ „
As a result, azimuthal currents will give rise to a poloidal magnetic field, that is, a field with axial and radial components only as illustrated in Fig 4.
Fig 4 Magnetic field of a current loop centered on the z-axis
To better address the magnetic diffusion problems presented in Chapter 3, it is convenient to introduce the magnetic vector potential A. The vector potential is useful because it applies to both current-carrying materials and to empty space. Additionally, it is 15 particularly convenient because the axial symmetry of the problems treated in this thesis eliminates two of its three vector components. The magnetic vector potential is defined as
B = Vx A (2.24) and is in agreement with (2.1-3), which requires B to be the curl of some vector field; that is, the divergence of B vanishes according to the differential relation V • (V x A) = 0 [11].
The expression for the diffusion of the magnetic vector potential is sought by making use of the constitutive relations and Maxwell's equations under the magnetoquasistationary (MQS) approximation.
Vx H = Vx — = j (2.25)
With the assumption that the medium of interest is isotropic, n is approximated as spatially constant, and equation (2.25) can be written (Ampere's law)
VxB=/jj. (2.26)
Substituting equation (2.24) into (2.26), using the known vector calculus identity [11]
V x (V x A) = V(V • A) - V2A (2.27) and the coulomb gauge
V-A = 0, (2.28)
Poisson's equation is obtained, and is written as
VZA = -ii\. (2.29) where V2 is the vector Laplacian given by
d2 Id2 d2 Id __2__3_ _Ar dr r2d(p2 dz2 r dr r2 d(j) 1 32 d2 1 d 2 d V2A _ i! :A + A* + ^r^A* + —^-Aa, + ~T^rA - -^r (2.30) lh 2 2 * r2d
Due to the axial symmetry, there is no (^-dependence, and given azimuthal stationary currents, the vector Laplacian appropriately reduces to
1 d ™ = [a-2A* + wA* + rd?A**l + (2.31)
As a result, azimuthal stationary currents will give rise to an azimuthal magnetic vector potential field, that is, with an azimuthal component only as illustrated in Fig. 5.
Fig. 5: Magnetic vector potential of a current loop.
Equation (2.29), Poisson's equation, defines the magnetic vector potential according to the existing free current densities. For the transient state, Faraday's law in MQS conditions
dB (2.32)
is combined with Ohm's law to describe the induced current density j; as a function of the time-derivative of a magnetic flux density, such that
Vx!i = -^. (2.33) a dt
Assuming that electrical conductivity is spatially invariant, equation (2.33) becomes
Vxji = -a^. (2.34) dt
Expressing B as the curl of A yields 17
aVxA dA _„. Vxji = -ff— =-VXff —, (2.35) at at and an expression for the induced current density which, arises naturally from a time- dependent potential, is written as
In a source-free region, where stationary and bound current densities are absent, the substitution of (2.36) into Poisson's equation yields the diffusion equation for the magnetic vector potential under quasistatic conditions:
dA V2A = na—. (2.37) at
Together with the appropriate initial, final and boundary conditions, the partial differential equation in (2.37) defines the transient magnetic processes which occur in dynamic potential problems. Boundary conditions for the magnetic vector potential A can be derived from the conditions on B given previously in (2.11) and (2.12). One of the physical meanings of A is related to the magnetic flux
ct> =
At the boundary, B may be discontinuous due to Amperian currents and even induced currents. However, for an infinitesimal increment of distance crossing the boundary dCl, the flux
Ai = A2 (2.39)
Additional boundary conditions for A can be developed from the boundary conditions on B; their derivation is shown in ANNEX A. They are
A , dAi . dA2 A r A r i + dr = 2+ dr (2.40) Ml i"2 18
dz dz
Having reviewed the fundamental aspects of magnetic diffusion theory, the following sections present some of the mathematics used to solve magnetic diffusion problems.
2.6 Electrodynamics Problems
Problems involving the diffusion equation, such as the determination of magnetic potentials, are often stated as boundary value problems. A boundary value problem is a partial differential equation (PDE) together with a set of additional constraints, called boundary conditions. A solution to a boundary value problem is a solution of the differential equation which also satisfies the boundary conditions.
2.6.1 Problem Formulation
In order to achieve a physically meaningful solution, an electrodynamics problem must be well-posed. The correct partial differential equation must be solved for each region of a system in the appropriate coordinate system. Subsequently, the solutions must be matched at the interfaces using the correct boundary conditions. These conditions impose values of the solution and/or of its derivatives at two or more points. The number of conditions imposed is equal to the order of the differential equation [21].
Dirichlet boundary Condition
The Dirichlet (first-type) boundary condition specifies the values taken by a solution on the boundary of the domain. If fi is the domain on which the partial differential equation of the form
V2u(x) + u(x) = 0 (2.42) is to be solved and dQ. denotes its boundary, a Dirichlet boundary condition takes the form
[21]
u(x) - f{x) v x 6 an (2.43) 19 where u{x) is a solution of (2.42) defined on H.
For example, Wwedensky [13], Callaroti [17] and Knoepfel [18] have maintained in their work that when a uniform applied excitation field H0 is applied along an infinitely long non-magnetic conducting rod, the value of the solution along the surface of the conductor will correspond to H0. However, the infinite solenoid required to generate a perfecdy uniform field is unphysical. Maxwell's equation (2.3) is violated since it assumes an infinite solenoid where V • B =£ 0. Under experimental conditions, induced current densities generate a field outside of the conductor [15] when the applied field is non-uniform or when the rod is finite. The reaction field in turn modifies the external field He that is necessarily
e different from H0, and the boundary condition H = H0 does not hold, especially at early times, due to the transient eddy current field. As discussed previously in section 2.2, the internal and external fields are codetermined and can only be obtained from a well-posed potential problem.
As induced current densities diffuse further into the conductor, the value of the field on the boundary rapidly assumes H0 and a Dirichlet condition may be approximately correct. Coincidently, Wwedensky [13], Callaroti [17] and Morozova [19] have explicitly noted disagreement between theory and experiment specifically at early times. This thesis proposes that the initial disagreement arises from the misuse of the Dirichlet boundary condition, valid for infinite systems, as an approximation in finite experimental systems. In magnetic materials, the boundaries become time-dependent because of magnetization processes as well as transient eddy-current fields.
Neumann Boundary Condition
The Neumann (second-type) boundary condition specifies the values of the solution function's normal derivative along the boundary of the domain. It is expressed mathematically [21] as
Vu(x) • nix) = f(x) VxSdD.. (2.44)
Physically, it is related to the flux of the function at the boundary. Non-homogeneous (non zero) Neumann boundary conditions represent external (or internal) sources impressing normal flux densities on an outer (or inner) boundary, where the sign is indicative of the 20 direction of flow [21]. A homogeneous Neumann boundary condition is said to be insulating because it specifies that flux is neither gained nor lost at the boundary.
Robin Boundary Condition
The Robin (third type) boundary condition is a specification of a linear combination of the values of a function and the values of its normal derivative on the boundary of the domain. It is a general form of the insulating boundary condition for convection—diffusion equations [22]. The Robin boundary condition is written
a u(x) + b —=^ = f(x) Vxeda (2.45) on for some non-zero coefficients a and b and a given function / defined on dft. More generally, a and b may be functions rather than constants [21].
Mathematically, equation (2.45) describes a concentration-dependent flux. Thus, the Robin boundary condition can be interpreted as a radiative boundary condition [21]. In the context of potential diffusion problem, behaviour of the magnetic vector potential across a cylindrical boundary at a material interface is given by a Robin boundary condition as shown in (2.40). Physically, magnetized materials will contain a greater flux line density than an adjoining region of lower permeability. Since divergence of B is zero, the field lines inevitably curl out of the boundary. Therefore, greater concentrations of magnetic flux within a geometrically finite region generate larger concentrations of magnetic flux density at the boundary, hence the radiative nature of the boundary.
Once a potential problem is correctly posed with the appropriate PDE and boundary conditions, the solution can be sought
2.6.2 Solution Building
Solutions of potential problems can be constructed using expansions in terms of orthogonal functions which are solutions to the PDE. The particular orthogonal set chosen depends on the coordinate system of choice. Consider an interval (a, b) in a variable z with a set of real or complex functions Un(x), n = 1,2,..., square integrable and orthogonal on the interval (a, b). The orthogonality condition on the functions Un(z) is expressed by [10] 21
r (0 if m =£ n w(x)UXx)Um(x)dx = 8nmPn = (2.46) J tPn > 0 if m = n a where w(x) is the weight factor obtained from the PDE. An arbitrary function f(x) can be expanded in a series of the orthonormal functions Un(x). The series representation
oo
f(x) = J] anUn(x) (2.47) 71=1
with weighted coefficients an converging in the mean to f(x) [10].
Rectangular Coordinates
An expansion in terms of sines and cosines is a Fourier series, which is well-suited for constructing a solution in rectangular coordinates. The series equivalent to (2.47) on the interval (- a, a) is
oo
f(x)=-A0+2^ [Am cos (—^—J + Bm sin (-^—Jj, (2.48)
where
2fu/i (2nmx\ Am=-\ /(x)cos dx,
(2.49) 2 fa/2 (2nmx\ Bm=-\ /(x)sinl——Jdx. aJ_a/2 \ a y
The interval (a, fe) becomes infinite or semi-infinite when the solution exists at infinity, which it often does in the framework of potential problems. In this case, the set of orthogonal functions Un(x) becomes a continuum of functions, rather than a denumerable set [10]. The Kronecker delta symbol becomes a Dirac delta function, and the series expansion takes the form of an integral. An important generalization is the Fourier transform. The Fourier integral, equivalent to (2.48), is written
1 r°° /(*)=— A(X)eax dA, (2.50) V27T J-oa
where 22
A(A)=-=— f e-'^/OOdA. (2.51)
Additionally, if the solution is known to be symmetric or anti-symmetric about the origin, a Fourier cosine or sine integral representation may be used, respectively [21]. They are written
/.GO f(x)= I A(A) cos(Ax) dA,
(2.52) i r
A(X) = - f(x) cos(Ax) cU, for the case of a solution symmetric in x, and
/•CO /(x) = I B(A)sin(Ax)dA, •' — CO (2.53) l r°° fi(A)=- /(x) sin(Ax) dA, nJ-oo for an anti-symmetric solution. Generally, Fourier coefficients are determined from initial, stationary and boundary conditions by exploiting the orthogonality relation (2.46).
Cylindrical Coordinates
Bessel functions are also known as cylinder functions or cylindrical harmonics because they are found in the solution to Laplace's equation in cylindrical coordinates. They arise naturally in solutions developed for rods and for boreholes which will be presented in Chapter 3. Bessel's differential equation is written [10]
r2 , , +r—r^+(a2r2-v2)u(r) = 0 (2-54) drz dr where v is the order of the Bessel function. The solutions to (2.54) are [10]
u(r) - A)v(ar) + BYv(ar) (2.55)
where A, B are constants. Jv(ar) are Bessel functions of the first kind. Bessel functions of the second kind Yv(ar) are also known as Neumann functions. Fig. 6 illustrates plots of Bessel and Neumann functions of integer orders 0 to 4. 23
Fig 6 Plots of Bessel (a) and Neumann (b) functions of orders 0 — 4
Similarly, the modified Bessel Functions Iv and Kv, sometimes referred to as the hyperbolic Bessel functions, are solutions to the following differential equation
d2u(r) du(r) + r —-— + {-a2r2 - v2)u(r) = 0, (2 56)
which differs from equation (2 54) by a negative sign in front of the third term. Plots of Iv and Kv are drawn for orders 0 to 2 in
Fig 7 Modified Bessel functions of the first and second kind
Bessel and modified Bessel functions of the first and second kind form pairs of linearly independent solutions to the radial component of Laplace's equation [10]. The 24 ordinary Bessel functions form an orthogonal set of functions which can expand an arbitrary function in cylindrical coordinates on the interval 0 < r < a in a Fourier-Bessel series [10]
CO
fir) = ^T Avn ]v (xvn £) where Jv(xvn) = 0 for n 6 M (2.57) n=l
and coefficient Av„ = — r r /(r) Jv (xvn -) dr (2.58) a x a I v+1K vn)Jo
given the orthogonality relation
2 J r }v (xvn -) Jv (xv?n -) dr = — []v+1(xvn)] Smn (2.59)
with weight function r.
The Fourier-Bessel series is appropriate for a finite interval 0 < r < a. However, when the solution exists at infinity as it often does in potential problems, the series becomes an integral in a manner entirely analogous to the transition from a trigonometric Fourier series to a Fourier integral [10]. The resulting radial integral is known as a Hankel transform, or a Fourier-Bessel transform [10]
/•CO Fv(m) = /(r) Jv(mr) r dr , •>o (2.60) ^•00 f(r) = I Fv(m) Jv(mr) m dm . Jo
These transforms are verified using the orthogonality relationship
/•CO S(m — n) I Jv(mr) Jv(nr) r dr = (2.61) Jo m
In order to elucidate the progression of flux in ferrous fasteners in aluminum wing structures, the mathematics presented above can be employed to construct solutions to the magnetic conducting rod and borehole diffusion problems. Historically, authors have sought transient solutions in the form of step-function responses, which are often less difficult to obtain. In reality, the applied field H0 is generated by a transmitter coil, which, by 25 virtue of Faraday's Law, is subject to mutual coupling with the conducting sample. The
coupling effects, herein referred to as feedback, cause a delay in the onset of the applied field.
2.7 Feedback in Coupled Magnetic Circuits
As discussed in the literature survey, theoretical models are often formulated for the
application of a step-function excitation field [18]. However, although voltage can be
abrupdy applied, the current flowing through the inductor, which defines the excitation field, has an exponential form as shown in Fig 8.
Step-function
Actual transmitter signal /(t) 05-
10
Fig 8 Transmitter coil current onset
The non-zero rise-time is a result of induced electromotive forces (EMF) which oppose the establishment of the steady-state field. Back-EMF, also called feedback, arise from self-inductance effects in addition to mutual inductance effects between the transmitter coil, the receiver coil and the conducting sample. Formally, the complete diffusion problem becomes quite complex.
The inductance L is a constant of proportionality which relates the magnetic flux to the current, and is defined in Griffiths [11] as
O (2 62)
The value of L of a conducting sample is often unknown and presumably time- dependent, since it depends on the diffusion of the magnetic field and accompanying 26 currents, both of which have complex dependencies on material and boundary conditions. Circuit components such as inductors may be approximated as having a constant inductance and treated as lumped circuit elements [11].
Any circuit which contains an inductor, such as a solenoid, has a self-inductance which prevents the current from increasing or decreasing instantaneously. Consider the simple circuit shown in Fig. 9 corresponding to a coil (inductor) in air.
Fig. 9: Simple RL circuit.
Suppose the switch S is closed at t — 0 (step function application of voltage). The current begins to increase and, because the current is time-dependent, the inductor L generates a back-EMF which opposes the increasing current as it tends to equilibrium. The expression for the self-induced back-EMF'is given in most electromagnetic textbooks [11] as
dl (2.63) e, = -L di
Applying Kirchhoff s loop equation to the circuit and solving the resulting partial differential equation, assuming that that inductor is the system's sole source of inductance, yields the expression for the time-dependent current in the circuit.
/(t)=i(l-e-S') (2.64)
When a conducting material is placed in proximity to the transmitter coil, eddy current densities are induced within its volume by virtue of Faraday's law. Transient fields generated by the eddy currents induce an additional back-EMF in the transmitter coil. The conductor can be understood as a lumped inductor, and the system, illustrated in Fig. 10, can be represented as coupled magnetic circuits. 27
Fig. 10: Magnetic circuit with two inductors.
Although the solution for a system of two coupled inductors is known [23], no solution currendy exists for a system containing a solid inductor.
When a receiver coil is introduced into the system, additional coupling effects between the transmitter and the receiver, and between the receiver and the sample, further increase the complexity of the problem.
Fig. 11: Equivalent magnetic circuit with three inductors.
Exact analytical models which describe the behaviour of magnetic fields in the physical system represented in Fig. 11 do not exist. This thesis, however, develops two methods of incorporating all feedback effects into transient solutions. The first method is to recover the physical system response from the fundamental solution, or step-function response, via Duhamel's Theorem. 28
2.8 Duhamel's Theorem
Duhamel's integral is a technique of calculating the response of a linear system to an arbitrary time-varying external excitation. Given the response of a linear system with a zero initial condition to a single, constant non-homogeneous term with magnitude of unity (referred to as the fundamental solution or step-function response), then the response of the same system to a single, time-varying non-homogeneous term with magnitude T(t) can be obtained from the fundamental solution [24]. Mathematically, the integral is written
f* dT(t') A(r, z,t) = Af(r, z, t - t') —V" dt' + Vt=0 • A/(r, z, t) (2.65) J0 at where A(r,z, t) is the response of the system to a time-varying input excitation T(t) given the fundamental solution Af(r,z,t). A transient analytical expression, for the step-function response of a system, is achieved by solving the corresponding potential problem as is done in the following chapter.
The second proposed method of addressing coupling effects is to incorporate back- EMF direcdy into the governing partial differential equation (PDE). A correctly-posed initial-value boundary-value problem includes all electromagnetic interactions of the system. Solution of the resulting PDE, by means of integral transforms, simultaneously integrates all mutually inductive and self-inductive effects. An analytical model obtained in this fashion should describe the physical system exactly. Exact details of this solution are beyond the scope of this thesis and are left for future work. 29
CHAPTER 3 - THEORETICAL MODEL
Magnetized ferrous fasteners are hypothesised to act as conduits carrying magnetic flux deeper into aircraft wing structures. As a result, target discontinuities within the aluminum adjacent to the fastener may become detectable at greater depths. In order to elucidate the potential for this effect, an analytical model describing time-dependent flux progression into and along a magnetic conducting rod, excited by a current loop, is developed.
3.1 Long Rod
A loop carrying a direct current I in the azimuthal direction 0 is located in a plane normal to and centered on the axis z of a vertical ferromagnetic conducting rigid rod as shown in Fig. 12. The current is abruptly turned on at / = 0. The solution describing transient magnetic diffusion processes within the media is sought using a separation of variables technique and corresponds to the step-function response of the system. This approach requires knowledge of the stationary solution which is, therefore, developed first.
3.1.1 Stationary Solution
The regions of the model shown in Fig. 12 are region I, the cylindrical conducting ferromagnetic rod 0 < r < r0, and region II, the non-conducting non-magnetic region outside the rod r0 < r < oo. The transmitter loops' axis coincides with the cylindrical rod's axis of symmetry. The boundary value problem is most conveniently treated in circular cylindrical coordinates using the magnetic vector potential. 30 0 u
JI
^0 t^fiil " a<,
Ha Mo
Fig. 12: Diagram of the long rod model.
The conductivity a is assumed to be constant and uniform throughout the volume of the rod. In reality, the permeability \i of a ferromagnetic body is field-dependent and possesses hysteresis. It is non-linearly related to the external magnetic field, except for small applied fields where it can be assumed to be linear.
Poisson's equation describes the stationary state of the magnetic vector potential. It is a parabolic non-homogeneous differential equation, which is written
V^2A - -Ms (3.1)
In the equation above, subscript i designates the cylindrical region of interest, I or II, as illustrated in Fig. 12. A is the magnetic vector potential which, for the given geometry, always circulates in the azimuthal direction (p. The solution of a non-homogeneous partial differential equation is the superposition of the homogenous solution with a particular solution [25]. Region I does not include any stationary current densities within its volume, except for Amperian - or magnetization - currents on its surface, therefore js = 0 and equation (3.1) becomes
V2A = 0. (3.2) 31
The vector Laplacian in cylindrical coordinates and basis was stated in equation (2.30). Here we need only the
Equation (3.2) above is known as Laplace's equation [21]. Here, the vector Laplacian is recast into its differential operator form for the appropriate cylindrical geometry where
( d2 Id 1 d2\ \^ + rTr--2+^rr-Z) = Q- (33)
Next, the z-coordinate is separated by means of the Fourier cosine integral transform as discussed in sub-section 2.6.2. The transform parameter, A, transforms the second order z- derivative, such that A{r, A) = J™ cos(Az) A(r, z)dz, where
(JF_ l_d__£_ 2\ -( \dr2 r dr r2 J
Equation (3.4) is Bessel's modified differential equation and its solutions are the modified Bessel functions of the first and second kind of order 1, written
A(r,X) = CliCAr) + DKx(Ar) . (3.5)
The general solution is required to be finite everywhere inside of the rod. However, the modified Bessel function of the second kind diverges as the argument approaches zero, as shown in Fig. 7. It must, therefore, be omitted from the solution inside the rod, which then becomes
A\r,X) = e\x{kr) . (3.6)
The steady-state solution for the magnetic vector potential outside of the rod is the superposition of the contributions from the magnetized rod (solution to Laplace's homogeneous equation), with the contribution from the current loop denoted \p(r,z) (solution to Poisson's non-homogeneous equation). However, the modified Bessel function of the first kind diverges as its argument tends to infinity, which is unphysical. Consequendy, its coefficient is necessarily zero and the external solution (Region II) is
i"(r,A) = BK^Ar) + ${r.X) . (3.7) where i/i(r,A) is the Fourier cosine transform of rp(r, z). 32
The magnetic vector potential attributed to the current loop, xp(r, z), is solved for explicitly in what follows. Returning to equation (3.1), a current density I is confined, using delta Dirac functions, to a loop of radius a on the plane z = 0. Accordingly, Poisson's equation becomes
a2 Id 1 d2\ d^ + -g;-^+Q^)^,z) = -n0I8(r- a)S{z) . (3.8)
The z-coordinate in the expression above is separated by means of a Fourier cosine transform. Equation (3.8) transforms according to
+ A2 ^ 7a7~" )^,X) = -n0lS{r-d) . (3.9)
A first-order Hankel-transform, with radial transform parameter y, separates the r- coordinate as discussed in Section 2.6.2, such that
2 2 (-K - A )^(y, X) = -j ix0I8{r - ^rJ^yOdr . (3.10)
The expression for the spatially transformed magnetic vector potential is isolated as
Ji(ya) and the inverse first-order Hankel-transform can be applied to recover the Fourier transformed magnetic vector potential. Integral transform tables (Erdelyi, 1954, Ch. 8, Sec. 11, eq. (lo)) provide the following piecewise solution, which is, however, singular at the radius of the current loop: