A Magnetic Flux Leakage NDE System for CANDU R Feeder Pipes

Transcription

1 A Magnetic Flux Leakage NDE System for CANDU R Feeder Pipes by Thomas Don Mak A thesis submitted to the Department of Physics, Engineering Physics & Astronomy in conformity with the requirements for the degree of Master of Applied Science Queen s University Kingston, Ontario, Canada March 2010 Copyright c Thomas Don Mak, 2010

2 Abstract This work examines the application of different magnetic flux leakage (MFL) inspection concepts to the non destructive evaluation (NDE) of residual (elastic) stresses in CANDU R reactor feeder pipes. The stress sensitivity of three MFL inspection techniques was examined with flat plate samples, with stress-induced magnetic anisotropy (SMA) demonstrating the greatest stress sensitivity. A prototype SMA testing system was developed to apply magnetic NDE to feeders. The system consists of a flux controller that incorporates feedback from a wire coil and a Hall sensor (FCV2), and a magnetic anisotropy prototype (MAP) probe. The combination of FCV2 and the MAP probe was shown to provide SMA measurements on feeder pipe samples and predict stresses from SMA measurements with a mean accuracy of ±38 MPa. i

3 Acknowledgments First and foremost I would like to thank my supervisor, Dr. Lynann Clapham, for presenting me with this wonderful opportunity. Her guidance and expertise were greatly appreciated. This work would have been far less interesting and enjoyable without the assistance of Dr. Steven White. He acted as a teacher from the moment I began working under him as a summer student in 2006, and he provided invaluable assistance in all aspects of this project from its conception, from theory to design, data acquisition and signal processing. I would also like to thank all members of the AECL Inspection Monitoring and Dynamics Branch, in particular Hélène Hébert. She helped organize meetings with AECL and provided helpful advice and encouragement. Thanks are due to Dirk Bouma, who was consulted frequently during the design of the first flux control system (FCV1), as well as Gary Contant and Chuck Hearns for their help and supervision in the machine shop. I also thank Pat Wayman for all her help during all phases of this project. Several students provided valuable assistance: Ben Lucht helped with L A TEX and MATLAB R, and Davin Young spent many hours in the machine shop building probe components. ii

4 Table of Contents Abstract i Acknowledgments ii Table of Contents iii List of Tables v List of Figures vi Chapter 1: Introduction CANDU R Feeder Pipes A Brief Introduction to Magnetic Circuits and Magnetic Flux Leakage Inspection Thesis Scope and Objectives Organization of Thesis Chapter 2: Theory and Background Stress and Strain iii

5 2.2 Maxwell s Equations and The Quasi-Static Case Magnetic Materials Magnetic Methods of Stress Measurement Chapter 3: Flux Control Systems Negative Feedback Control and Operational Amplifiers Magnetic Flux Transducers Component Selection White s Flux Control System (FCS) Flux Control Version 1 (FCV1): Hall Sensor Feedback Flux Control Version 2 (FCV2): Hall Sensor and Coil Feedback in Combination Chapter 4: Magnetic Stress Detectors Test Sample and the Single Axis Stress Rig (SASR) Detectors, Data Acquisition and Data Analysis Experimental Procedures for Testing and Comparison of the Probe Systems Detector Results and Analysis Selected Detector Chapter 5: Proposed Design: MAP Probe Magnetic Anisotropy Prototype (MAP) Probe iv

6 5.2 MAP Probe Testing with SA-106 Grade B Pipe Chapter 6: Summary and Conclusions Flux Control Systems Magnetic Stress Detectors Proposed MAP Probe Design Recommendations for Future Work Bibliography Appendix A: FCV1 Details Appendix B: Skin Depth v

7 List of Tables 3.1 Excitation and monitor coil properties. Inductance values were recorded on-sample at 100 Hz. The monitor coil was wound around one of the core s poles, making its area the same as the pole area PCI-6229 I/O assignment and terminal configuration for FCV1. Terminal configurations use the following abbreviations: referenced singleended (RSE), non-referenced single-ended (NRSE), differential (DIFF). For additional information on terminal configurations see [29] PCI-6229 I/O assignment and terminal configuration for FCV MAP probe properties. Feedback and excitation coils were wound on an external forming rig, which is why their area differs from the Supermendur core footprint vi

8 List of Figures 1.1 A simplified sketch of a CANDU R 6 reactor face A comparison of magnetic and electric circuits The stress tensor for an element of a continuous structure in Cartesian coordinates Residual stress formation in a bent beam Ferromagnetic domain structure A typical magnetization hysteresis loop for a ferromagnetic sample starting with zero magnetization A schematic of four magnetic domains aligned along the 100 directions of Fe Demagnetizing field lines for: a) a single domain, b) two opposing domains separated by a 180 wall, and c) four domains separated by 90 and 180 walls Magnetostriction of a material with positive λ s The two types of magnetoelasticity: magnetostriction and the Villari effect for a material with positive λ s The magnetization processes for samples with aligned and misaligned auxiliary fields and preferred crystalline axes vii

9 2.10 A simplified Barkhausen noise apparatus A bandpass filtered Barkhausen noise spectrum taken from 3 khz to 600 khz A polar plot of angular MBN energy measurements The application of magnetic flux leakage inspection in crack and corrosion detection The MFL signal from a segment of SA106-B schedule 80 pipe (a) reference measurement and (b) after the introduction of residual stresses through a localized impact. Maxima correspond to red and minima correspond to blue, but no further colour scale information is available The rotation of the magnetic field just outside the sample ( B out ) relative to the magnetic field within the sample ( B in ) when µ 2 > µ The orientation of B in and B out relative to the excitation core The components of a closed-loop control system shown in a block diagram The feedback system components contained within an op-amp The Hall effect for a Cartesian coordinate system A sketch of White s FCS A simplified version of FCV Hall voltage (V H ) and excitation current (I ex ) for a sinusoidal reference voltage FCV1 response to a DC reference voltage of V ref = Monitor coil voltage V mc boosts the noise amplitude relative to the excitation field viii

10 3.9 A simplified version of FCV An electrical schematic of FCV2 showing the feedback system and the Hall sensor current source The magnetic fields measured by the Hall sensor and feedback coil in FCV The three detector configurations used with the prototype excitation core The mild steel plate used to test different detector configurations A schematic of the single axis stress rig used to introduce tensile stress in the flat plate sample An assembled probe showing a detector mount assembly attached to the connector brace of the excitation core DC MFL, AC MFL, and SMA detectors mounted to the excitation core The footprint of the excitation core on the sample for AC MFL, DC MFL and SMA measurements DC MFL measurements for B ex σ t and B ex σ t The excitation field (dashed line) and signal voltage (solid line) for an AC MFL measurement at zero applied stress AC MFL measurements for B ex σ t and B ex σ c A modified figure 1.15 redrawn for reference. The excitation core footprint is indicated by dotted lines G for four µ r2 /µ r1 ratios. The 0, 180, and 360 probe orientations place the probe parallel to the µ 2 direction V sig (σ,φ) fit amplitudes for SMA measurements ix

11 4.13 SMA measurements for tensile up to 130 MPa A schematic of the Supermendur core of the MAP probe A diagram of the MAP system The pin diagram for the MAP system A schematic of the three-point bending rig in the tensile configuration SMA dependence on excitation field amplitude MAP stress response for an excitation field B ex = 75 mt sin(2πt55 Hz) SMA dependence on tensile and compressive applied stress Signal voltage V sig (σ a,φ) fit amplitude for approximately equivalent compressive (σ a = 44 MPa) and tensile (σ a = 47 MPa) stresses The recommended system for future work. (a) Two perpendicular U- cores can rotate the magnetic field at their center by adjusting the excitation field generated by each core. Adapted from [39]. (b) The recommended anisotropy coil configuration for a tetrapole excitation system. Coils 1 and 3 are connected in series, as are coils 2 and A.1 An electrical schematic of FCV B.1 Skin depth for a typical steel with µ r = 100 and σ e = 10 7 Ω 1 m x

12 Chapter 1 Introduction Engineered components have a finite service life governed by their design, manufacturing processes, material properties and application. Components will eventually fail, terminating their service life. The causes of failure are commonly chemical or mechanical processes that alter component characteristics and material properties. When the cost of failure is sufficient, regular inspection of components becomes cost efficient: components near failure can be identified then repaired (extending their service life) or replaced (ending their service life before failure). There are many methods available for examining component degradation, but inspection techniques that do not require component disassembly or destruction are valued for their noninvasive nature; they are classified as non-destructive evaluation (NDE) 1 techniques. The risk of component failure is derived from NDE data. Components are replaced when the risk of failure reaches a threshold value, determined by: the accuracy of the NDE method, the cost of replacement and the cost of failure. Accurate NDE 1 The term non-destructive testing (NDT) is used synonymously with non-destructive evaluation (NDE). 1

13 CHAPTER 1. INTRODUCTION 2 inspection techniques reduce the cost of ownership of a system by reducing repair, replacement and failure costs. This thesis focuses on the development of a magnetic NDE method to detect regions of residual stress in CANDU R feeder pipes. Details of the magnetic flux leakage (MFL) NDE technique and CANDU R feeder pipes are provided in the following sections. 1.1 CANDU R Feeder Pipes CANDU R (CANada Deuterium Uranium) reactors are heavy water-cooled, heavy water-regulated nuclear reactors designed by Atomic Energy of Canada Ltd. (AECL) in partnership with General Electric Canada 2 and Ontario Power Generation 3 (OPG). Reactors that use standard water (H 2 O) as the moderator/coolant require enriched uranium fuel, composed of U-238 with 2% to 4% wt U-235. Heavy water moderated/cooled reactors, such as the CANDU R, can achieve criticality 4 with naturallyoccurring uranium, composed of U-238 with 0.7% wt U-235, because heavy water (D 2 O) is a weaker neutron moderator than standard water [11]. The primary heat transport circuit of a CANDU R reactor uses pumps to push heavy water coolant over fuel bundles in the calandria 5. A simplified sketch of a CANDU R reactor face, showing most components of the primary heat transport circuit is shown in figure 1.1. SA-106 grade B carbon steel feeder pipes (termed feeders and labelled 3 in figure 1.1) transport heavy water coolant from heat transport pump 2 Known as Canadian General Electric during the design partnership. 3 Known as Hydro-Electric Power Commission of Ontario during the design partnership. 4 A self sustaining fission reaction. 5 A calandria is the reactor core of a CANDU R system

14 CHAPTER 1. INTRODUCTION outlet header inlet header feeders steam generators end fittings heat transport pumps insulation cabinet 5 7 Figure 1.1: A simplified sketch of a CANDU R 6 reactor face. Adapted from the CAN- TEACH library ( input headers (labeled 2 in figure 1.1) to pressure tube inlet end fittings (labeled 5 in figure 1.1) on the reactor face. The coolant is heated as it passes through the calandria, then exits via the pressure tube outlet end fittings and is passed through feeders to outlet headers (labeled 1 in figure 1.1), where it is cooled by steam generators and returned to the heat transport pumps. There are over 700 feeders per reactor. The feeders must access the end fitting matrix at the reactor face and maintain minimum clearances of approximately 20 mm,

15 CHAPTER 1. INTRODUCTION 4 which requires a variety of feeder bending arrangements. The SA-106 grade B carbon steel feeders have schedule 80 wall thickness with nominal diameters of 2.0 or 2.5 and bend radii of 1.5 the diameter. Ovality caused during the bending process and Corrosion introduce variation in pipe wall thickness; the 2.5 diameter pipes can vary in wall thickness from 4 mm to 8 mm. The minimum tensile yield strength of SA-106 grade B carbon steel is 240 MPa [1]. An outlet feeder pipe was removed from service in 1997 following detection of a coolant leak. The leak was attributed to cracking within the pipe, which was analyzed by AECL with a variety of techniques, including neutron diffraction to determine if residual stresses contributed to the failure. The neutron diffraction data indicated that residual stresses in the vicinity of the crack were elevated. Ultimately, cracking was attributed to a combination of an elevated stress distribution and flow-accelerated corrosion 6 caused by 311 C heavy water [40]. The cracking that results from a combination of tensile stress and a corrosive environment is called stress-corrosion cracking (SCC). Following the original 1997 leak, SCC has been found in a number of outlet feeders [16]. It was further determined that the SCC found in feeders was initiated by yield strength tensile stresses on the inner pipe surface. Canadian Nuclear Safety Commission (CNSC) safety regulations require the prevention of leakage from feeder piping systems. If a feeder leak is detected in an active reactor, a shutdown leakage limit of 20 kg/h is enforced. The costs associated with forced reactor outages and the replacement of pressure boundary components are high: a minimum shutdown time of 40 h is required at a cost of approximately $20 000/h. Because of this cost, reactor operators attempt to avoid forced outages 6 Flow-accelerated corrosion is a process whereby the normally protective oxide layer on carbon steel dissolves into a stream of flowing water or wet steam.

16 CHAPTER 1. INTRODUCTION 5 by performing regular NDE inspections of components at the reactor face. Ideally, operators would replace feeders that are at risk for developing SCC during scheduled maintenance shut-downs; however, there is currently no commercial NDE system that can evaluate the stress distribution in feeders at the reactor face, which is thought to be primary cause of feeder SCC. AECL approached the Queen s University Applied Magnetics Group (AMG) through the University Network for Excellence in Nuclear Engineering (UNENE) and proposed that the group develop a ferromagnetic NDE stress evaluation technique for the purpose of measuring residual stresses in CANDU R feeders. Two projects were proposed: a doctoral thesis focused on the use of magnetic Barkhausen noise, and a master s thesis concentrating on the adaptation of a magnetic flux leakage technique to feeders. The doctoral project was completed by Steven White in 2009 [39]. The present thesis focuses on the development of a magnetic flux leakage technique that address the unique problems associated with NDE stress evaluation of feeders. 1.2 A Brief Introduction to Magnetic Circuits and Magnetic Flux Leakage Inspection Magnetic systems make use of magnetic circuits, a concept that exploits similarities between electric and magnetic field equations and allows magnetic systems to be represented schematically. Figure 1.2 shows some analogs between electric and magnetic circuits. Just as electric circuits rely on an electric scalar potential difference (V ) to generate an electromotive force (EMF) that drives electric current (I) through a resistance (R), magnetic circuits rely on a magnetomotive force (MMF) to drive

17 CHAPTER 1. INTRODUCTION 6 current source Physical System (a) battery current I + _ light bulb (b) wire coil N turns core air gap I s flux wire (c) current I wire resistance R wire (d) flux core reluctance Circuit Schematic voltage source V + _ load bulb resistance Rload MMF source NIs N S air gap reluctance Electric Magnetic Figure 1.2: A comparison of magnetic and electric circuits. Figures (a,b) show sketches of physical systems, while the electrical and magnetic schematics of the systems are given in figures (c,d). magnetic flux (Φ) through a reluctance (R). Referring to the electric circuit case shown in figures 1.2 (a,c), a battery provides voltage V required to drive I through the light bulb load. For an equivalent magnetic circuit, the MMF of figures 1.2 (b,d) is provided by a current-carrying coil of N turns supporting current current I s. This coil generates a magnetic flux Φ, which passes through the core (R C ) and air gap (R G ). Magnetic flux leakage (MFL) inspection systems measure the magnetic flux outside of a magnetized sample, called leakage flux, and correlate it to sample properties, commonly changes in cross-section area caused by dents, gouges and pits. These measurements are conceptually quite simple: a magnetic circuit is assembled using a permanent magnet to generate a flux Φ through the magnet-sample circuit. The magnetic reluctance of sample regions with low cross-sectional area (eg. corrosion

18 CHAPTER 1. INTRODUCTION 7 pits) is increased, causing flux to leak into the surrounding environment. Once flux has left the sample it can be detected by a magnetic flux transducer, such as a Hall probe or giant magnetoresistance sensor. The transducer signal can be interpreted to determine the nature of the defect that caused the flux leakage. MFL is, as its name suggests, a measurement of leakage flux that emerges from a magnetized sample. To generate effective comparisons between different measurements, the flux Φ through different samples, or regions on a sample must be consistent. Traditionally, commercial MFL systems overcome this issue by generating flux with large permanent magnets that magnetically saturate the sample; however these magnets are large, bulky and difficult to manipulate. These commercial MFL systems are not suitable for the current application, the ferromagnetic feeder array at a CANDU R reactor face makes safe handling of large permanent magnets impossible. 1.3 Thesis Scope and Objectives As outlined earlier, this thesis project focuses on the adaptation of magnetic NDE technology, specifically flux leakage systems, to CANDU R feeder pipes. The system developed in this thesis should function as an early prototype for an industrial system. The scope was limited to the following specific project objectives: 1. design a magnetic flux leakage-based probe that can accommodate the space and geometry (lift-off) constraints imposed by the feeder pipe environment 2. conduct laboratory testing on plate samples to determine the extent of stress sensitivity of the probe designs

19 CHAPTER 1. INTRODUCTION 8 3. conduct testing on samples with feeder pipe geometry with a focus on generalized stresses 4. conduct testing on feeder pipe samples 1.4 Organization of Thesis This thesis is organized as follows: Chapter 2 presents a brief review of electrodynamic theories used to describe the stress-dependence of magnetic flux leakage and magnetic anisotropy within ferromagnetic materials. Chapter 3 outlines the two flux control designs developed with the goal of producing consistent and repeatable magnetic excitation fields in the feeder samples. Chapter 4 presents three different stress detectors (to be used with the flux control systems) and initial stress sensitivity results from those detectors. In chapter 5, a prototype system designed specifically for stress measurements on feeder pipes is presented. This system was designed based on results presented in chapters 3 and 4, and tested on a 2.5 SA-106 grade B pipe. Test results are presented in this chapter. Chapter 6 summarizes the findings of this work and provides suggestions for future system improvements.

20 CHAPTER 1. INTRODUCTION 9 All designs, figures, drawings, measurements and physics probes described in this work are the original work of the author unless otherwise noted. Exceptions include: the single axis stress rig described in section 4.1, and the three-point bending rig presented in section

21 Chapter 2 Theory and Background This chapter presents a theoretical summary of stress, strain, and quasi-static magnetic behavior to provide a basis for magnetic domain theory, design decisions, and signal analysis techniques presented in later chapters. A review of stress and strain principles is given in section 2.1. Section 2.2 begins with Maxwell s equations and leads to discussion of the quasi-static case. The classification of magnetic materials is presented in section 2.3, along with an overview of magnetic domain theory and magnetization processes. In section 2.4 different magnetic stress measurement techniques are presented. Notation in this chapter is consistent with that used in Griffiths (reference [12]). 2.1 Stress and Strain Stress is a measure of the force acting per unit area within a body. The stress state of an element within a body 1, shown in figure 2.1, can be determined by a nine 1 A body is an structure composed of a continuous distribution of elements (also known as points). 10

22 CHAPTER 2. THEORY AND BACKGROUND 11 z σ zz σ zx σ zy σ zx σ yz σ xx σ xy σ yx σ yy x y Figure 2.1: The stress tensor for an element of a continuous structure in Cartesian coordinates. component stress tensor σ, given by σ xx σ xy σ xz σ = σ yx σ yy σ yz. (2.1) σ zx σ zy σ zz Diagonal tensor elements σ xx, σ yy, and σ zz represent normal (tensile and compressive) stress components, while off-diagonal elements represent shear stress components. Stress may vary within a body, causing different elements to have different stress tensors. A complete description of the stresses within a body is therefore given by a tensor field. Ideally, each body element would have zero volume; however, all physical measurements must be performed over a sample volume, with stress averaged over that volume. The characteristics of the sample volume greatly affect the details of the stress field in a crystalline material. Consider any steel sample: a typical body will consist

23 CHAPTER 2. THEORY AND BACKGROUND 12 of numerous small crystals (called grains) in multiple orientations. A small sample volume may be less than the average grain size 2, leading to an inhomogeneous, anisotropic material at the microscopic scale. If the sample volume is large enough to enclose several million crystals, steel may be considered homogeneous, as the properties of any single crystal become insignificant. If the body is not strongly textured 3 it may also be considered isotropic. The maximum stress a material can support before undergoing plastic deformation is defined as the yield stress σ yield. Application of an external force to an object results in deformation. Deformation is elastic up to σ yield, that is, the deformation vanishes if the force is removed. External stress beyond σ yield causes irreversible plastic deformation that remains once the stress source is removed, shown in figure 2.2 for a bent beam. Removal of this external stress disrupts the stress distribution within the body, causing it to reacquire some of its initial shape via elastic deformation. Because it has been permanently deformed, it cannot return completely to its original form, thus the elastic stress distribution remains within the material. These elastic stresses are called residual stress and are often present in engineered components manufactured by plastic deformation processes, such as extruded pipes and bent beams. In addition to non-uniform plastic deformation such as that shown in figure 2.2, other sources of residual stress are welding stresses, intergranular misfit stresses, thermal expansion stresses, etc [27]. As stress is a type of force, it cannot be measured directly and must be inferred from some other physical parameter, typically geometrical deformation, typically known as engineering strain, or simply strain. The engineering strain tensor ε 2 A grain is a domain of mater that has the same structure as a single crystal. 3 Texture is the distribution of crystal orientations within a polycrystalline sample. A material is said to be strongly textured if a there is a preferential crystal orientation.

24 CHAPTER 2. THEORY AND BACKGROUND 13 (a) tensile stress (b) Force compressive stress Force compressive stress (c) tensile stress Figure 2.2: Residual stress formation in a bent beam. (a) The beam in an unstressed state. (b) A downward force applied to the ends of the beam causes plastic deformation. There is tensile stress above a neutral surface (shown with a dashed line) and compressive stress below it. (c) Once the external force is removed, internal residual stresses redistribute to elastically deform the body toward its original state.

25 CHAPTER 2. THEORY AND BACKGROUND 14 expresses geometrical deformation as a ratio of the change in dimension d to the initial dimension d 0. Diagonal components of ε are normal strains in the x, y, and z directions, and are given by ε i=j = d d 0 = d d 0 d 0, (2.2) where d is the dimension after deformation. Off diagonal components (ε i j ) are equal to one-half the engineering shear strain. 4 Each entry in ε generates a corresponding stress entry in σ. The relationship between tensor entries is defined by a fourth order stiffness tensor C ijkl, such that σ jk = kl C ijklε kl. Engineering applications generally simplify the relationship between stress and strain by assuming isotropic materials, in which case a geometrical deformation can be characterized by two parameters: Young s modulus (Y ), and Poisson s ratio (ν). 5 This simplification means the relationship between stress and strain can be expressed using a generalized Hooke s law equation as: [ σ ij = ε ij + Y 1 + ν ν 1 2ν (ε xx + ε yy + ε zz ) ]. (2.3) Strain can be measured in many ways on macroscopic and microscopic scales. Resistive strain gages are commonly used to evaluate macro-stresses, while diffraction techniques using neutrons or x-rays are can be used for micro-stress analysis. There are parameters other than strain affected by stress. Most importantly for the purpose of this thesis, magnetic properties of ferromagnetic alloys are affected by the stress field, and have the potential to be used for macroscopic stress analysis. 4 Engineering shear strain is the complement of the angle between two initially perpendicular line segments. 5 Shear modulus (G m ) is not included in this list as it is defined by G m = Y/[2(1 + ν)].

26 CHAPTER 2. THEORY AND BACKGROUND Maxwell s Equations and The Quasi-Static Case Maxwell s four equations E = ρ ǫ 0 (2.4) E = B t (2.5) B = 0 (2.6) and the Lorentz force law B = µ 0 J + µ0 ǫ 0 E t, (2.7) ( ) F = q E + v B (2.8) describe the relationship between electric and magnetic fields, and the effect these fields have on charged particles. In the above equations, t is time, B is the magnetic field (also referred to as magnetic flux density), E is an electric field, J is the current density field, F is force, q is electric charge, ρ is electric charge density, v is velocity, and ǫ 0 and µ 0 are the permittivity and permeability of free space. In most magnetic experiments, including the work presented in this thesis, fields vary at a sufficiently low rate that magnetostatics can be used to describe electric field behavior. In this quasi-static case, the displacement current term (µ 0 ǫ 0 E t ) of E equation 2.7 can be neglected because J >> ǫ 0. Thus equation 2.7 becomes t The current density J is the sum of two components: B = µ 0 J. (2.9) J = J b + J f, (2.10) where J b is the bound current due to electron spin and angular momentum, and

27 CHAPTER 2. THEORY AND BACKGROUND 16 current generated by the movement free particles is represented by J f. The magnetization field ( M) is attributed to bound currents: and the auxiliary field ( H) to free currents: M = J b, (2.11) H = J f. (2.12) Equations 2.9, 2.10, 2.11, and 2.12 can be rearranged to give B = µ 0 ( M + H ). (2.13) 2.3 Magnetic Materials Equation 2.13 can be expressed using magnetic susceptibility (χ m ) or relative permeability (µ r ) tensors as B = µ 0 µ rh = µ0 (1 + χ m ) H. (2.14) Both µ r and χ m are used to express the response of J b to J f and relate that response to magnetic flux density. For simplicity, many materials are assumed to have linear and isotropic magnetic properties, thus making the susceptibility tensor a constant (χ m ). Materials are categorized by their χ m value, the most common categories being: diamagnetic, paramagnetic, ferrimagnetic, and ferromagnetic. 6 Diamagnetism occurs when atoms or molecules have no net magnetic moment, meaning electrons constitute a closed shell. As such, nearly all organic compounds and polyatomic gases are diamagnetic [7]. Typical diamagnetic materials have a small, negative susceptibility, on the order of χ m H interacts with electrons to 6 Other varieties of magnetism are omitted for brevity.

28 CHAPTER 2. THEORY AND BACKGROUND 17 decrease B through the application of Lenz s law to the orbital rotation of electrons about nuclei. Superconductors are considered nearly perfectly diamagnetic with χ m 1, completely expelling the magnetic field from within the material. Paramagnetism is caused by atoms or molecules with a net magnetic moment generated by unpaired electrons. In the absence of an applied field, these moments are randomly oriented and cancel each other, leading to zero net magnetization of the body. When a field is applied, these moments rotate to the direction of the field; however, thermal agitation prevents atomic moments from achieving complete alignment. The end result is partial alignment with H, leading to small positive susceptibilities on the order of 10 5 to Ferro and ferrimagnetism result from a material s chemical makeup and crystal structure. As with paramagnetism, the atoms or molecules that comprise the crystal have a net magnetic moment generated by unpaired valence electrons. In a ferromagnetic material, crystalline lattice spacings are such that valence electron spins of adjacent atoms are aligned via the quantum mechanical exchange interaction. Aligned moments group together in magnetic domains, as shown in figure 2.3. Domain walls separate domains of different orientations. External magnetic fields cause shifts in the domain structure, ultimately aligning magnetic domains with the field. Because domains are composed of billions of magnetic moments, ferromagnetic materials have large magnetic susceptibilities, up to χ m Ferromagnets retain some magnetization in the absence of an H field. Ferrimagnetism is a combination of ferromagnetism and anti-ferromagnetism, which is simply the opposite of ferromagnetism. Ferrite substances are composed of iron double-oxides and at least one other metal; magnetic ions occupy different lattice

29 CHAPTER 2. THEORY AND BACKGROUND 18 Figure 2.3: Ferromagnetic domain structure. Magnetic domain orientation is shown with arrows. Domain walls appear in white. Taken from [12]. sites some of which are coupled ferromagnetically and others anti-ferromagnetically. The overall effect results in susceptibilities ranging from 10 to The isotropic and linear χ m approximation is usually valid for paramagnetic and diamagnetic substances; however, the domain structure of ferromagnetic and ferrimagnetic materials generates strong magnetic anisotropy and hysteresis effects. A typical M-H loop for a ferromagnetic material in an oscillating H field is shown in figure 2.4. The figure shows that a demagnetized sample exposed to an auxiliary field will magnetize along the initial magnetization curve (dashed line) to M s, the saturation magnetization. A subsequent decrease in H decreases the the sample s magnetization to the remnant (or residual) magnetization (M r ), defined as the magnetization of the sample at H = 0. Further decreases of H leads to the coercive field (H = H c ), defined as the auxiliary field at which the magnetization returns zero. As H continues to decrease, the sample goes into negative saturation M s. The area of

30 CHAPTER 2. THEORY AND BACKGROUND 19 magnetic Barkhausen noise M s M r Magnetization, M H c H c initial magnetization curve -M s Auxiliary Field, H Figure 2.4: A typical magnetization hysteresis loop for a ferromagnetic sample starting with zero magnetization. M increases with H along the initial magnetization curve to saturation at M s. Further variation of H changes sample magnetization as shown around the loop. The inset shows magnetic Barkausen noise, which is discussed further in section a B-H hysteresis loop corresponds to the energy lost to irreversible processes within the sample [33] Magnetic Domain Theory Since the focus of the thesis is ferromagnetic materials (specifically steel), additional discussion of their behavior is warranted, specifically with respect to their domain configuration and behavior under magnetization. As mentioned earlier, magnetic domains are groups of aligned magnetic moments found in ferromagnetic and ferrimagnetic materials. Within each domain the material is magnetized to the saturation magnetization M s, because dipoles within each domain are aligned. Even though domains are magnetically saturated, a bulk sample is generally composed of domains

31 CHAPTER 2. THEORY AND BACKGROUND 20 [100] (a) [010] 90 o wall (b) 180 o wall Figure 2.5: A schematic of four magnetic domains aligned along the [100] and [010] directions of Fe. (a) Each domain is made up of many aligned magnetic moments, but the four domains together produce no net magnetization. (b) Domain walls act as transition regions between domains of different orientation. Two types of domain wall are shown: 90 and 180. with randomly oriented magnetization vectors, producing no net sample magnetization. An example of this is illustrated in figure 2.5(a). Figure 2.5(b) shows that domains are separated by domain walls; these are transition regions in which magnetic moments gradually rotate between different orientations such that they align with domains on either side of the wall. Domain wall thickness is a function of material properties. In Fe, domain walls span approximately 120 atoms [7]. Domain walls between domains with opposite magnetization vectors are termed 180 walls. Adjacent domains with perpendicular magnetizations are separated by boundaries termed 90 walls.

32 CHAPTER 2. THEORY AND BACKGROUND 21 The domains shown in figure 2.5 are in the [100] and [010] directions; two of the easy crystallographic magnetization directions of the 100 set. 7 Magnetic saturation of iron in this easy direction is achieved at a lower field density than the 110 and 111 directions, because domains with body centered cubic structures naturally align to 100. The perpendicular arrangement of the 100 set results in strong 90 and 180 domain formation; however, the domain structure can become more complex near surfaces and inclusions. The magnetic domain structure of ferromagnetic materials results from the minimization of the sum of six energy terms: the exchange energy (ε ex ), the magnetocrystalline anisotropy energy (ε mca ) the magnetostatic energy (ε ms ), the magnetoelastic energy (ε λ ), the domain wall energy (ε wall ), and the Zeeman energy (ε p ). Thus, the total energy (ε total ) for a single iron crystal is ε total = ε ex + ε mca + ε ms + ε λ + ε wall + ε p. (2.15) Minimizing ε total results in the domain structure of ferromagnetic material. Each energy contribution is explained below. Exchange Energy The exchange energy (ε ex ) is due to the quantum mechanical exchange interaction 8 between adjacent atoms first described by Heisenberg in 1926 [14] and applied to 7 The this document follows standard crystallographic notation. The normal of a specific plane is indicated by (100), while the set of equivalent planes is denoted by {100}. Directions are indicated by square brackets as [100]; the complete set of equivalent directions is given by angular brackets as When two atoms are adjacent, there is a finite probability their electrons will exchange places, thus the term exchange energy. Consider electron A moving about proton A and electron B moving about proton B. As electrons are indistinguishable, there is a possibility that the electrons exchange places such that electron B moves about proton A, and electron A moves about proton B. This consideration introduces an additional exchange energy term into the expression for the total energy of the two atoms.

33 CHAPTER 2. THEORY AND BACKGROUND 22 ferromagnetism in 1928 [15]. For a set of atoms located throughout a lattice at r i, each with spin S( r i ), the exchange energy can be written as the sum over each atom pair [10]: ε ex = jk J ( r i r j ) S( r i ) S( r j ), (2.16) where J (r) is the exchange integral, which occurs in the calculation of the exchange effect. The magnitude of J (r) drops off rapidly for large r, meaning only the nearest neighbor spins contribute significantly to equation If J (r) is positive, ε ex is a minimum when the spins are parallel and maximum when they are anti-parallel. If J (r) < 0, the lowest energy state results from anti-parallel spins. The alignment of neighboring spins observed in ferromagnetism results from a positive exchange integral. It should be noted that the minimization of ε ex specifies only the orientation of magnetic moments relative to each other, it does not specify the orientation of the moments relative to crystallographic axes. Magnetocrystalline Anisotropy Energy The magnetocrystalline anisotropy energy (ε mca ) is the energy stored in domains aligned to the non-easy directions of a crystal. Applied fields must do work to rotate the magnetization direction M s of a domain away from an easy direction, therefore energy must be stored in domains aligned to non-easy directions. In 1929, Akulov [3] showed that ε mca can be expressed in terms of a series expansion of the direction

34 CHAPTER 2. THEORY AND BACKGROUND 23 cosines (α i,i = 1, 2, 3) of Ms relative to the crystal axes: 9 ( ( ( ) ε mca = V D K0 + K 1 α 2 1 α2 2 + α2α α3α1) K2 α 2 1 α2α3) , (2.17) where V D is domain volume, and K 0, K 1, K 2 are anisotropy constants specific to the material (in units of J/m 3 ). It is typical to neglect K 0 in equation 2.17 because it is independent of angle and only consider the K 1 and K 2 terms when evaluating the series [6]. In Fe, ε mca tends to align magnetic moments to the 100 directions, making them directions of easy magnetization. Magnetostatic Energy The magnetostatic energy (ε ms ) is the energy stored in a magnet s demagnetizing field, given by [6]: ε ms = 1 2 µ 0 H d 2 d 3 r, (2.18) where H d is the demagnetizing field and the integral is evaluated over all space. In Fe, minimization of only the exchange and magnetocrystalline energies would lead to a single magnetic domain parallel to 100 ; however, this configuration would produce a significant demagnetizing field, such as that shown shown in figure 2.6(a). The creation of an opposing domain decreases H d (figure 2.6(b)), while a set of four domains separated by 90 and 180 walls (figure 2.6(c)) further decreases H d. Thus, minimizing ε ms results in the formation of 90 and 180 domain walls. 9 Consider a domain in a cubic crystal: let M s make angles a 1, a 2, a 3 with the crystal axes, then α 1, α 2, α 3 are the cosines of those angles.

35 CHAPTER 2. THEORY AND BACKGROUND 24 (a) (b) (c) Figure 2.6: Demagnetizing field lines for: a) a single domain, b) two opposing domains separated by a 180 wall, and c) four domains separated by 90 and 180 walls. Domain Wall Energy Domain wall energy (ε wall ) is the energy associated with the formation of a single domain wall. As shown in figure 2.5(b), a domain wall consists of a region in which magnetic moments in adjacent atoms gradually change direction. Both ε mca and ε ex increase due to this gradual rotation of magnetic moments, and these increases give ε wall. The energy associated with the formation of a new wall requires that the decrease associated with ε ms be greater than the corresponding increase in ε wall. Zeeman Energy The Zeeman energy (ε p ) is the energy of the interaction between H and M [17], given by ε p = µ 0 M H d 3 r. (2.19) H is generated with free currents or permanent magnets. ε p varies with H, leading to changes in the magnetic energy and domain reorganization. ε p is minimized when M and H are aligned.

36 CHAPTER 2. THEORY AND BACKGROUND 25 demagnetized state d 0 Δd magnetic saturation H Figure 2.7: Magnetostriction of a material with positive λ s. Magnetoelastic Energy The magnetization of a ferromagnetic material is accompanied by a change in dimension, a phenomenon termed magnetostriction by Joule [18]. Conversely, external stresses applied to a ferromagnetic material result in a change in magnetic properties, a response termed the Villari effect [38]. Together, these effects are referred to as magnetoelasticity. Magnetostrictive strain (λ) is the strain tensor generated by magnetostriction. The strain at magnetic saturation parallel to the direction of magnetization is termed the saturation magnetostriction λ s, shown in figure 2.7. Measurement of magnetostriction takes the same form as equation 2.2, giving λ s = d/d 0. Typical λ s values are small, on the order of 10 5 and can be greater (positive magnetostriction) or less (negative magnetostriction) than zero. Magnetostriction is anisotropic, thus different crystalline axes have different saturation magnetostrictions, indicated by λ hkl. In iron, λ 100 = , while λ 111 = Because of magnetostrictive anisotropy, magnetoelastic energy (ε λ ) is written in terms of the saturation magnetization along specific crystalline axes; for example in

37 CHAPTER 2. THEORY AND BACKGROUND 26 (a) magnetostriction H H (b) Villari Effect σ 0 σ0 domain magnetization uniaxial stress auxiliary field Figure 2.8: The two types of magnetoelasticity: magnetostriction and the Villari effect for a material with positive λ s. (a) A change in domain structure caused by H produces magnetostrictive strains and elongation parallel to the auxiliary field. (b) Uniaxial strain results in elongation and an increased number of 180 domains. Fe, the 100 set is used as the reference direction, giving ε λ = 3 (cos 2 λ 100 σ 2 0 γ ν sin 2 γ ) d 3 r, (2.20) where σ 0 is a uniaxial stress, γ is the angle between the domain magnetization and applied stress in the sample volume, and ν is Poisson s ratio [17]. In Fe, ε λ < ε mca, meaning magnetoelastic considerations alone are insufficient to rotate the domain orientation away from the 100 axes; however, external stress may cause preferential alignment to a particular 100 direction. Figure 2.8 shows the difference between magnetostriction and the Villari effect. Increases in H cause expansion of domains parallel to the auxiliary field, leading to increased sample magnetization and eventual saturation. The sample elongates as domains grow. Conversely, the Villari effect is an increase in the number of domains (assuming positive λ s ) parallel to an applied stress σ 0.

38 CHAPTER 2. THEORY AND BACKGROUND 27 M s Magnetization, M H aligned to MCA direction H misaligned with MCA directions Auxiliary Field Magnitude, H H aligned to MCA direction H H misaligned with MCA directions reversible and irreversible wall motion irreversible wall motion and annihilation domain rotation and annihilation Figure 2.9: The magnetization processes for samples with aligned and misaligned auxiliary fields and preferred crystalline axes. H is taken from left to right. Domain configuration and corresponding H and M values are shown for demagnetized samples brought to saturation along their initial magnetization curves. Recall that MCA is magnetocrystalline anisotropy, and ε mca causes magnetic moments to align to certain crystallographic directions Magnetization Processes The domain configuration of a ferromagnet changes in response to shifts in the Zeeman energy (ε p ) produced by varying H. Increasing ε p causes domains to reconfigure to minimize the total magnetic energy of the system by increasing the average alignment between H and M. There are three process by which domains can reconfigure: domain wall motion, domain creation and annihilation, and domain rotation. Each of these processes is shown in figure 2.9 and discussed further in this section.

39 CHAPTER 2. THEORY AND BACKGROUND 28 Domain wall motion occurs in domains that are partially aligned to the auxiliary field ( M H > 0). As shown in figure 2.9, these domains increase in volume via domain wall motion at the expense of misaligned domains. Low auxiliary fields ( H) produce elastic (reversible) domain wall motion. Domain wall motion becomes irreversible at high fields. Misaligned domains become unfavorably small when ε wall > ε ms. These domains are annihilated and their moments merge with existing domains. Domain creation and annihilation are irreversible processes. If H is along an axis favored by εmca, magnetic saturation (a single domain state) will be achieved with only domain wall motion and annihilation. When H is not aligned to a favored crystalline axis, competition between ε p and ε mca results in rotation of the remaining domains toward H, until the remaining domains align leaving a single domain. This is shown at the bottom right of figure 2.9 where rotation results in the final saturated state Bulk Magnetic Anisotropy Figure 2.9 shows the domain reconfiguration processes in the order (from left to right) of the energy required to produce them. When H is aligned with a crystalline axis favored by ε ms, domain rotation - the most energy intensive reconfiguration process - is not required to achieve magnetic saturation; hence a single domain state can be achieve at the lowest ε p. These directions are referred to as magnetic easy directions, or easy axis. Bulk magnetic anisotropy occurs when an entire polycrystalline sample displays an easy direction resulting from crystallographic texture (ε mca ) or strain (ε λ ). Crystallographic texture (or simply texture ) refers to a preferred distribution of

40 CHAPTER 2. THEORY AND BACKGROUND 29 crystallographic orientations in a polycrystalline sample. A random distribution of grain orientations has no texture: no orientations are represented more than others. In the absence of any stress influences, such a sample would be magnetically isotropic. If a particular grain orientation is favored over others, the sample is said to have texture in the favored orientation. Textured ferromagnetic samples tend to exhibit bulk magnetic anisotropy, that is, they may have bulk magnetic properties that vary with orientation. For example, the easy directions in Fe are 100, and an Fe sample with 100 texture will have a bulk easy axis in the texture direction. Strain can align domains to a crystallographic orientation through minimization of ε λ. Consider a tensile stress producing a tensile strain along [100] in a single Fe crystal. ε λ is minimized by increasing the population of domains aligned to [100] and [ 100], forming a magnetic easy direction along [100]. In order to account for tensile strain effects in a polycrystalline sample, a magnetic easy direction forms along the 100 directions most closely parallel to the strain. Compressive strain in Fe generates unfavorable domain orientations, decreasing the domain population parallel to the applied strain. Thus, compression along [100] decreases the number of [100] and [ 100] domains, but increases the quantity of [010], [0 10], [001], and [00 1] domains. 2.4 Magnetic Methods of Stress Measurement There are a number different methods of magnetic non-destructive evaluation (NDE), all of which relate changes in a material s magnetic properties to structural anomalies, such as cracks, dents and pits, and localized stresses. This section reviews the theory and use of three magnetic NDE concepts: magnetic flux leakage, magnetic Barkhausen

41 CHAPTER 2. THEORY AND BACKGROUND 30 noise, and stress-induced magnetic anisotropy. Magnetic flux leakage and stressinduced magnetic anisotropy were used as the basis for sensors developed for this thesis. Magnetic Barhkausen noise measurements were used by White for his Ph.D. thesis, a parallel project to this thesis. The basics of magnetic Barkhausen noise NDE are presented to enable an appreciation for the work in this thesis when compared to White s results Magnetic Barkhausen Noise (MBN) Domain wall motion is not a continuous process, but rather motion that occurs in discrete steps, as shown earlier in figure 2.4 (inset). These discontinuities are called Barkhausen events after the physicist who discovered the effect in 1919 [4]. They occur at frequencies up to several hundred kilohertz, and as such can generate voltage pulses (called Barkhausen noise) in a search coil placed nearby. The nature of Barkhausen emissions is closely tied to the microstructure of the magnetic material and can give insight into microscopic characteristics and stress state. A symplified Barkhausen noise apparatus is shown in figure An excitation coil is driven by an AC voltage source, typically at frequencies below 1 khz so that the excitation field can be distinguished from the Barkhausen signal (> 100 khz) through bandpass or highpass filtering. Barkhausen noise signals from the sample are detected using a pickup coil 10 mounted with its axis parallel to the sample surface normal, and can be analyzed in terms of frequency content, pulse hight distribution, Barkhausen power density, and Barkhausen energy. Because of the relationship between domain wall distribution and stress state, 10 Also referred to as a search coil, pickup coil, or signal coil.

42 CHAPTER 2. THEORY AND BACKGROUND 31 excitation coil core pickup coil V sample Figure 2.10: A simplified Barkhausen noise apparatus. the Barkhausen spectrum can be used to evaluate stress in ferromagnetic materials. Studies within the Applied Magnetics Group of Queen s University frequently examine a parameter termed Barkhausen noise Energy, defined as E BN = τ 0 V 2 BN dt, (2.21) where τ is the period of the excitation signal, and V BN is the voltage of the Barkhausen noise pulses. Figure 2.11 shows a bandpass filtered Barkhausen noise spectrum for a sinusoidal excitation field. Barkhausen noise decreases as the sample moves toward saturation, with peak noise occurring in the vicinity of the coercive point. Barkhausen noise-based stress measurement methods rely on the magnetic anisotropy introduced by stress. Barkhausen spectra such as that shown in figure 2.11 are collected at regular angular intervals (typically between 5 and 15 ) about a point on the sample, and E BN is evalutated for each spectrum. Plots of E BN as a function of the angle of the excitation field relative to a reference direction, shown in figure 2.12, give insight into the magnetic easy direction and thus the surface stresses in the sample. Peak Barkhausen energy occurs along the easy direction and with proper calibration E BN can be related to stress. Depth sensitivity is limited for Barkhausen signals due to their high frequency.

43 CHAPTER 2. THEORY AND BACKGROUND Pickup Coil Voltage (mv) Barkhausen noise Time (ms) B Field Magnetic Flux Density, B (mt) Figure 2.11: A bandpass filtered Barkhausen noise spectrum taken from 3 khz to 600 khz. The excitation field amplitude is 250 mt at a frequency of 31 Hz. Taken from [39]. E BN (mv 2 s) rolling direction mv 2 s Figure 2.12: A polar plot of angular MBN energy measurements. Peak E BN values along the axis indicate the easy axis, which is in the rolling direction. Minimum E BN values give the hard axis along Taken from [25].

44 CHAPTER 2. THEORY AND BACKGROUND 33 Signal attenuation within the sample caused by eddy currents limits the maximum depth from which Barkhausen signals can be detected to between mm [39] (this attenuation is discussed further in appendix B) Magnetic Flux Leakage (MFL) The magnetic flux leakage inspection method relies on the perturbation of magnetic flux caused by defects in the sample. Localized stress may also result in MFL signals. When examining cracks and defects, the sample is magnetized to saturation using a strong DC field typically generated by a permanent magnet, shown in figure Any shifts in cross-sectional area cause magnetic flux to leak into the surrounding region. This leakage flux can then be measured with an appropriate transducer, typically a Hall probe or giant magnetoresistance sensor. Although the technique is relatively simple in application, signal analysis is problematic, and numerous studies within the Queen s Applied Magnetics Group have focused on signal interpretation for defects such as corrosion pits and generalized corrosion, dents and gouges. MFL corrosion detection systems are widely used because of their ability to characterize the size and depth of a flaw, and a matrix of scanners can be used to scan the complete surface of a specimen in one pass [26]. In addition to detecting defects, the MFL technique can be utilized to probe for regions of anomalous stress or microstructure. These regions represent localized variations in permeability. In general these regions of permeability variation will produce MFL signals of smaller magnitude than defect signals. Figure 2.14 shows the MFL signal recorded by a Hall sensor scanned over the surface of SA106-B schedule 80 pipe before and after the introduction of a region of locally high stress.

45 CHAPTER 2. THEORY AND BACKGROUND 34 N S Magnet Flux lines Sample Figure 2.13: The application of magnetic flux leakage inspection in crack and corrosion detection. (cm) (a) 0 o (cm) (b) 0 o (cm) (cm) Figure 2.14: The MFL signal from a segment of SA106-B schedule 80 pipe (a) reference measurement and (b) after the introduction of residual stresses through a localized impact. Maxima correspond to red and minima correspond to blue, but no further colour scale information is available.

46 CHAPTER 2. THEORY AND BACKGROUND Stress-Induced Magnetic Anisotropy (SMA) In the absence of stress and texture, a polycrystalline ferromagnetic material will have isotropic magnetic properties. The presence of stress introduces magnetic anisotropy through minimization of ε λ, an effect known as stress-induced magnetic anisotropy (SMA). SMA measurements were pioneered by Langman in 1981 ([21], [20], [23], [22]) in a series of experiments on mild steel samples. Langman examined the relationship between the stress state of a sample and the angle (δ) between the magnetic field within the sample ( B in ) and the field just outside the sample s surface ( B out ). Bin was determined using two perpendicular sensing coils wound through holes drilled in the sample. Bout was measured by a Hall sensor positioned directly above the sensing coils. The Hall sensor was rotated to determine the direction of B out, while the vector sum of the sensing coil signals provided the orientation of Bin. Magnetic fields were generated by an excitation core which was rotated to provide different orientation of B out and B in. This section follows the derivation of an expression for δ presented in reference [21] that will be required for SMA signal analysis in chapter 4. Within a uniaxially stressed sample there are typically two perpendicular principal magnetic directions (1 and 2) of permeability µ 1 and µ 2, and relative permeability µ r1 and µ r2. In materials with positive magnetostriction (λ s > 0), such as Fe, the greater of the two permeabilities is parallel to tensile stress, while the smaller permeability is perpendicular to it. Supposing that µ 2 > µ 1, then the magnetic field B in within the sample will be enhanced in the µ 2 direction relative to µ 1, as shown in figure When B in is applied at an angle θ relative to µ 2, the magnetic field B out just outside the sample will be rotated away from B in by an angle δ.

47 CHAPTER 2. THEORY AND BACKGROUND 36 μ 1 B in1 B out δ B in θ B in2 μ 2 Figure 2.15: The rotation of the magnetic field just outside the sample ( B out ) relative to the magnetic field within the sample ( B in ) when µ 2 > µ 1. δ can be determined from trigonometry using the ratio of relative permeabilities and angle θ. Figure 2.15 shows how B in can be resolved into the principle directions as B in1 = B sin θ (2.22) and B in2 = B cos θ. (2.23) Using B = µ o µ rh and assuming the sample is surrounded by air, the components of B out can be resolved into the principle directions as: B out1 and B out2 = B in µ r1 sin θ (2.24) = B in µ r2 cos θ. (2.25) Dividing equation 2.24 by equation 2.25 gives the ratio of external magnetic field components as B out1 B out2 = µ r2 µ r1 tan θ = tan(δ + θ); (2.26) which can be rearranged to ( ( µ r2 ) µ δ = arctan r1 1) tanθ 1 + µ. (2.27) r2 µ r1 tan 2 θ Langman found that equation 2.27 was a reasonable prediction of the behavior of

48 CHAPTER 2. THEORY AND BACKGROUND Direction of B out relative to φ Degrees Probe angle, φ (degrees) Direction of B in relative to φ -40 B out φ B in Tension μ 2 Figure 2.16: The orientation of B in and B out relative to the excitation core. The excitation core footprint is shown by dotted lines in the inset diagram. Tensile stress was used to produce µ 2 > µ 1. Taken from [21]. magnetic fields; however, it does not describe the orientation of the excitation core relative to δ and θ. The angle between the excitation core poles and µ 2, taken as φ, is not θ or δ, but between the two. This relationship is shown in figure 2.16, which shows that θ and δ deviate about from the probe angle (φ) by as much as 30. Langman s original SMA experiments were suitable for specially prepared samples that could have perpendicular sensing coils wound through them. Later SMA measurement techniques developed different sensor configurations for use on unprepared samples, such as Kishimoto s magnetic anisotropy sensor [34], which employs a sensing coil wound around a detecting core mounted perpendicular to the excitation core

49 CHAPTER 2. THEORY AND BACKGROUND 38 and the sample s surface. Modern SMA apparatus use a magnetic transducer, typically a sensing coil, oriented to measure the magnetic field perpendicular to both the excitation field and sample surface. These coils produce no signal in isotropic samples, but SMA causes shifts B out away from the excitation core so that it is detected by the coil.

50 Chapter 3 Flux Control Systems A dominant problem in magnetic NDE is ensuring that a consistent and repeatable magnetic flux is coupled into the sample. This is a problem for all sample geometries, including flat plates, where flux can be affected by surface preparation and varying sample permeability. Studies on flat plates can address the issue by inserting lift-off spacers between the magnetic field source and sample, ensuring a relatively consistent air gap: however, the curved surfaces of pipes present a more challenging geometry. Magnetic stress evaluation methods developed within the Queen s Applied Magnetics Group rely on measuring the magnetic anisotropy in the sample [19], [37], [5]. This normally implies that sensors must be physically rotated about a location to perform a measurement. The curvature of a pipe wall does not lend itself to sensor rotation: air gaps change with probe orientation, thereby altering the reluctance of the magnetic circuit for each angular measurement. For this thesis, a new magnetic flux control system was developed to compensate for the difficulties of magnetic flux leakage-based stress measurements on pipe geometries. This flux control system was developed in two stages: first using only Hall 39

51 CHAPTER 3. FLUX CONTROL SYSTEMS 40 sensor feedback (called flux control version 1 or FCV1), then expanded to both Hall sensor and coil feedback (called flux control version 2 or FCV2). In the following chapter, the basic principles of feedback control are presented in section 3.1. Section 3.2 describes the magnetic transducers used for feedback control (Hall sensors and wire coils). The components used in FCV1 and FCV2 are discussed in section 3.3. The design, performance, and shortcomings of FCV1 are presented in section 3.5. Section 3.6 discussed the design of FCV2 and presents a brief analysis of its performance. 3.1 Negative Feedback Control and Operational Amplifiers Control systems can be separated into two groups: those without feedback (termed open-loop), and those with feedback (termed closed-loop). Open-loop systems do not adjust their output to changing conditions. Applied to a magnetic circuit, any disturbance, such as changing temperature or variable magnet liftoff, causes the output to drift from the desired value. In a closed-loop system, shown in figure 3.1, the output is fed back and compared with a reference input value. The difference between the two (called an error signal) is amplified by the forward path gain (a) to minimize deviation between reference and output values. This type of system is said to have negative feedback. 1 There are three primary properties of negative feedback systems [35]: 1. They tend to maintain their output despite variations in the forward path or 1 There are also positive feedback systems, which sum the output and reference, but they will not be discussed in this thesis since they were not used.

52 CHAPTER 3. FLUX CONTROL SYSTEMS 41 input or reference + adder - error signal forward path gain a output feedback path Figure 3.1: The components of a closed-loop control system shown in a block diagram. The reference value is compared with the output, generating an error signal used to adjust the output. The forward path converts inputs to outputs with the forward path gain a. The feedback path is the mechanism through which the output is fed back for comparison with the reference. The error signal is the difference between the reference and output values. in the environment. When negative feedback is properly applied and operating stably, the output remains constant if the system is given enough time to compensate for any changes that occur. 2. They require a forward path gain which is greater than that which would be necessary to achieve the required output in the absence of feedback. Feedback decreases overall system gain, defined as the ratio of output to input. Consider two systems with the same forward path gain (a), one with feedback and the other without. The system without feedback can achieve a greater overall gain than a system with feedback. 3. The overall behavior of the system is determined by the nature of the feedback path. Since feedback systems compensate for variations in the forward path, the overall behavior of the system is determined by the feedback path. Both FCV1 and FCV2 use operation amplifiers (op-amps), shown in figure 3.2, as the adder and forward path mechanism, with some type of external negative feedback mechanism to provide flux control. The feedback mechanism, which is not shown in

53 CHAPTER 3. FLUX CONTROL SYSTEMS 42 figure 3.2, connects from the output terminal of the op-amp to the inverting input terminal. Op-amps with negative external feedback follow two Golden Rules [32]: 1. The output attempts to do whatever is necessary to make the voltage difference between the inputs zero. This rule results from the incredibly high forward path gain (a) of op-amps, typically greater than A minute voltage difference between the inverting and non-inverting terminals causes the output to saturate. The op-amp adjusts the output voltage such that the external feedback network brings the voltage difference between the two input terminals to zero. This statement is equivalent to Negative feedback is functioning properly and the output is set to its desired value. 2. The inputs draw no current. This rule results from the very high input impedance of op-amps. Typical input currents are in the sub microamp range, and this provides a convenient simplification. 3.2 Magnetic Flux Transducers Two types of magnetic flux transducers were used in this project: wire coils and Hall sensors. Both of these devices are common and well understood, and each has their own advantages. This section provides a brief overview of the function of each device, following derivations in [39] and [8] Wire Coils The force acting on a charged particle is given by equation 2.8 in chapter 2. For most substances, the free current density J f is proportional to the force per unit charge

54 CHAPTER 3. FLUX CONTROL SYSTEMS 43 V + non-inverting input (+) + adder - error signal forward path gain a output V o inverting input (-) V - Figure 3.2: The feedback system components contained within an op-amp. The op-amp is represented by a triangle with three terminals: the non-inverting input at voltage V +, the inverting input at voltage V, and the output termnial at voltage V o. The output voltage is given by the forward path gain a multiplied by the difference between the inverting and non-inverting terminals, such that V o = a(v + V ). through conductivity σ e, such that J f = σ e ( E + vd B ), (3.1) where v d is the the average velocity of particles within the material (called the drift velocity, because it is typically very small). Equation 3.1 is called Ohm s law. It is common for v d B << E, thus Ohm s law can be approximated as J f = σ e E. (3.2) Equation 3.2 is equivalent to the standard equation for resistance R in direct current (DC) circuits: R = V I, (3.3) where V is the voltage across the device, and I is the current passing through it. The resistance is a function of conductor geometry and conductivity σ e.

55 CHAPTER 3. FLUX CONTROL SYSTEMS 44 The electromotive force (EMF) around a closed path S (E S )is defined as E S = E d l, (3.4) S and the magnetic flux through the surface S (Φ S ) is defined as Φ S = B da. (3.5) S Taking S to be the closed path that bounds the surface S, equation 2.5 can be converted to integral form using Stokes theorem, giving E d B l = t d A. (3.6) Equations 3.4 and 3.5 can be applied to equation 3.6 to give S S E S = dφ S dt, (3.7) where E S is more commonly known as the back EMF. In the presence of a timevarying magnetic field, electrons in a conductive material will form free currents that oppose the existing field. These currents are generated by E S and are called eddy currents. Consider a coiled wire of resistance R carrying current I. If the coil contains N turns of area S and is subject to a time-varying magnetic flux, the voltage across the coil (V coil ) is given by V coil = RI + N dφ S dt = RI + NS db S dt, (3.8) where B S is the average magnetic flux density through area S. When considering sensing coils, the RI term in equation 3.8 is neglected, leaving only the magnetic field term. Equation 3.8 indicates that a voltage will be induced in a coil of wire if that coil surrounds a region where the magnetic flux is changing. Coil wires can be used as

56 CHAPTER 3. FLUX CONTROL SYSTEMS 45 z y B x E x J f V H side view y z x B E x E y J f V H top view Figure 3.3: The Hall effect for a Cartesian coordinate system. B is in ẑ. Electric current density J f is in ˆx, caused by the ˆx component of E, E x. Build up of electrons along the ŷ facing wall produces an electric field component in the y direction, E y. The Hall voltage (V H ) is measured across the two sides facing ±ŷ. magnetic field sensors in air, however in many applications coils are wound around specific components to measure the overall flux through the component. Sensing coils are cheap and easy to manufacture. The main drawback to sensing coils is their time dependence requirement: coils are only capable of measuring time-varying magnetic fields. Thus they are appropriate for time-varying (AC) magnetic fields, however coils cannot be used to measure flux in a permanent (DC) magnetic circuit. Hall sensors, discusses in the following section, do not suffer from this limitation Hall Sensors The Hall effect was discovered in 1879 by Edwin Hall when he attempted to determine if the force exerted on a current carrier in a magnetic field was experienced by the bulk of the material or only by the charge carriers (electrons) [13]. Hall discovered a transverse voltage across the bulk of a silver test sample, perpendicular to current flow and the magnetic field, as shown in figure 3.3. This voltage is called the Hall voltage (V H ).

57 CHAPTER 3. FLUX CONTROL SYSTEMS 46 Consider figure 3.3 in the absence of B for a metal sample: electrons flow from right to left at drift velocity v d, generating current density J f. With the introduction of B = Bẑ, moving electrons are deflected in the ŷ direction and accumulate on that side of the sample. Electrons continue to accumulate, increasing E y until E y = v d B. (3.9) Unfortunately, equation 3.9 is not particularly useful in this form. Drift velocities are rarely known and it is more convenient to measure a voltage than E y. Thus a Hall coefficient H is used to convert equation 3.9 to V H = HB, (3.10) where H is a function of the probe s charge carrier concentration, dimension, and source current. Sensors based on the Hall effect require a supply current. Thus Hall sensors typically have four terminals: two supply current (I c ) terminals for incoming and outgoing supply current, and two Hall voltage terminals for the positive (V H+ ) and negative (V H ) voltage values. The Hall voltage that appears in equation 3.10 measured across the positive and negative voltage terminals as V H = V H+ V H. (3.11) Modern Hall effect sensors are semi-conducting devices that vary in cost from a few dollars up to several hundred dollars depending on calibration qualities, response linearity and several other parameters. Hall sensors and wire coils measure magnetic fields by exploiting different electromagnetic effects, which lead to slightly different applications for the two sensors. Hall sensors measure AC and DC magnetic fields, whereas coils are insensitive to

58 CHAPTER 3. FLUX CONTROL SYSTEMS 47 DC fields, but can be used to measure flux through specific components by winding sensing coils around the region of interest. Hall sensors are enclosed devices and must be placed external to any components of interest. 2 The Hall voltage signal is therefore only proportional to the flux passing through the sensors itself. Because of this shortcoming, the placement of Hall sensors within magnetic circuits must be made carefully, in order that the Hall voltage signal accurately reflects the flux through the circuit. 3.3 Component Selection Both FCV1 and FCV2 contained similar components. The primary difference between the systems was the feedback signal. This section outlines components common to both FCV1 and FCV Data Acquisition A National Instruments R PCI-6229 Multifunction DAQ (PCI-6229) was used to generate reference voltage signals (V ref ) and record all of the feedback system s output signals. This DAQ featured 4 analog output channels (AO (0...3) ) with a ±10V range about ground and 16 bit resolution. 32 single-ended multiplexed analog input channels (AI (0...31) ) with a maximum sampling rate of 250 khz aggregate over all channels can be used for data acquisition in the PCI-6229s. Analog inputs have 16 bit resolution over each voltage range (±10V, ±5V, ±1V, ±0.2V)[28]. PCI-6229 DAQs were not purchased specifically for this project, but were used because they were available and their specifications were adequate for both FCV1 2 They cannot be placed inside a sample, and therefore cannot measure the flux inside a sample.

59 CHAPTER 3. FLUX CONTROL SYSTEMS 48 and FCV Amplifier Due to the abundance of speakers found in consumer goods, there is a large selection of low-cost solid state power amplifiers. The excitation coils of a flux-controlled circuit present similar load impedances to speakers found in audio equipment [39]. The National Semiconductor R LM4701 audio amplifier 3 was selected to act as the adder/forward path in both FCV1 and FCV2 feedback systems [31]. The LM4701 is a low noise amplifier, designed to supply 30 W into 8Ω from 20 Hz to 20 khz with total harmonic distortion + noise of 0.08% given proper heat dissipation. Supply voltage can be up to ±32 V. Typical open-loop gain is a = 110 db. These amplifiers feature National Semiconductor R s SPiKe TM protection circuitry, which protects the op-amp from voltage, current, and thermal overload Power Supply A Power-One R HCC AG regulated DC power supply with ±24 V outputs at 2.4 A was selected to provide power to amplifiers, Hall sensors, and all other components Hall Sensors F.W. Bell 4 BH-700 Hall effect sensors were used as magnetic flux density transducers for both FCV1 and FCV2. These sensors were used for their linear response and 3 The LM4701 is now obsolete. It has been replaced by LM4765 and LM4781 multi-channel amplifiers. 4 A division of Sypris Test & Measurement.

60 CHAPTER 3. FLUX CONTROL SYSTEMS 49 small size [9]. These Hall sensors were used in two different applications. The first, described in this chapter, was as part of the flux control system, where the BH-700 was used to monitor the flux density in the magnetic excitation circuit. These Hall probes were also used as MFL stress detectors, described in chapter 4, to detect the magnetic flux signal emanating from the sample itself. Note that in the latter case the Hall probes are termed detectors, in the former they are termed sensors. 3.4 White s Flux Control System (FCS) For his doctoral thesis dealing with magnetic Barkhausen noise, Steve White designed a magnetic flux control system that relied on the feedback from a coil wound near the base of an excitation core pole, termed a feedback coil, to regulate the magnetic flux generated by an excitation coil. This feedback system was termed the Flux Control System, or FCS. Figure 3.4 is a simplified sketch illustrating the basic premise of this control method. V ref is an arbitrary reference voltage waveform, and V ex and V F+ are voltage measurements referenced to ground. The ideal op-amp is configured as an adder such that the sum of the currents into the inverting input through resistors R ref, R G, and R F1 is zero V ref R ref + V F+ R F 1 + V ex R G = 0. (3.12) Resistors R F1 and R F2 were chosen to be much greater than the resistance of the feedback coil. The two resistors were placed across the coil to improve system stability. As the same current flows in R F1 and R F2, the voltage across the feedback

61 CHAPTER 3. FLUX CONTROL SYSTEMS 50 V ex V ref R ref R G R F1 V F+ R F2 Figure 3.4: A sketch of White s FCS. The resistor R G was used to limit gain to provide a stable output. A feedback coil wound around one of the poles acted as the flux feedback transducer. coil (V F ) is given by V F = Solving equations 3.12 and 3.13 for V ref gives ( ) Rref V ref = V F R F1 + R F2 ( 1 + R ) F2 V F+. (3.13) R F1 ( Rref R G ) V ex. (3.14) This system was designed to control flux independent of probe liftoff; therefore the contribution of V ex to V ref was problematic, as V ex was unknown. Ideally, the gainlimiting resistor R G could be removed (effectively setting R G ), which would nullify the contribution of V ex to equation However, in this maximum-gain configuration, any offset between the amplifier terminals would be multiplied by the full amplifier gain (a). Without any feedback to compensate for these offsets, V ex will gradually shift to the voltage supply rails. To avoid this issue, White maximized the system s performance by decreasing R G from until the circuit stabilized. The reference voltage (V ref ) and the voltage across the feedback coil (V F ) differed by the V ex term as ( ) RF1 + R F2 V ref = V F. (3.15) R G

62 CHAPTER 3. FLUX CONTROL SYSTEMS 51 The error introduced by a non-infinite R G was reduced by a digital error correction (DEC) algorithm implemented in the FCS software control and data acquisition system. 5 The DEC algorithm iteratively adjusted the reference voltage (V ref ) until the target feedback voltage (V F ) was achieved. 3.5 Flux Control Version 1 (FCV1): Hall Sensor Feedback The thesis project by White paired a flux control system (FCS) with MBN measurements to perform stress analysis on SA106-Grade B steel pipes. As described in the preceding section, White used feedback coils wound around the base of each of the excitation cores. In the present project, a Hall sensor was selector to act as the feedback path for the flux control system, as they are capable of recording both time-varying and constant magnetic fields, and MFL measurements frequently use permanent magnets to generate the excitation field FCV1 Hardware The primary advantage of a Hall sensor feedback system over a coil-based system is the ability to be used with DC excitation fields. The original concept for this project (FCV1) was to pair a Hall sensor feedback flux control system with a Hall detector for stress measurement. The control system would guarantee that a consistent flux was coupled into the sample, while the Hall detector would measure the leakage flux, which would be sensitive to variations in the stresses within the sample. 5 This DEC system designed by White is only valid for periodical reference waveforms.

63 CHAPTER 3. FLUX CONTROL SYSTEMS 52 Vref + LM F V ex, I ex excitation coil V s R s GND monitoring coil V H- ferrite excitation core V H+ V mc I c lift-off spacer sample Hall sensor Figure 3.5: A simplified version of FCV1. A Hall sensor with supply current I c is located in a plastic lift-off spacer attached to the bottom of the ferrite excitation core. This sensor measures the B component normal to the sample surface, and feeds back a Hall voltage to be compared to V ref. FCV1 was designed as the Hall sensor feedback system, shown as a simplified sketch in figure 3.5. A detailed electrical schematic of FCV1 is included in appendix A. Referring to figure 3.5, FCV1 was designed to control the flux entering the sample by measuring the flux density (B) with a Hall sensor located in an air gap between the sample and excitation core (the excitation core shown in the figure is ferrite). V ref is the user-defined reference voltage corresponding to the desired B in the air gap. Note that in figure 3.5, V H is grounded, therefore V H = V H+. The voltage V s across a 0.2 Ω series resistor (R s = 0.2 Ω) was monitored to measure excitation current I ex. The output of the LM4701 was fused at 0.5 A with a slow blow fuse (F = 0.5 A) to prevent damage to the excitation coil. A monitoring coil was wound around a pole of

64 CHAPTER 3. FLUX CONTROL SYSTEMS 53 turns 1052 excitation coil inductance 0.26 H resistance 27 Ω turns 37 monitoring coil inductance 8.7 mh resistance 1.1 Ω ferrite excitation core pole area 264 mm 2 Table 3.1: Excitation and monitor coil properties. Inductance values were recorded onsample at 100 Hz. The monitor coil was wound around one of the core s poles, making its area the same as the pole area. Variable PCI-6229 Terminal Connection Configuration V ref AO 0 RSE V H AI 0 NRSE V s AI 1 NRSE V mc AI 2 DIFF GND AI sense Table 3.2: PCI-6229 I/O assignment and terminal configuration for FCV1. Terminal configurations use the following abbreviations: referenced single-ended (RSE), non-referenced single-ended (NRSE), differential (DIFF). For additional information on terminal configurations see [29]. the ferrite core to monitor the FCV1 performance when operated in AC mode. The output voltage from this coil was designated V mc. The properties of the excitation and monitoring coils are given in table 3.1. Voltage input and output (I/O) was handled through the PCI-6229 board. I/O connections and terminal configurations are given in table 3.2.

65 CHAPTER 3. FLUX CONTROL SYSTEMS FCV1 LabVIEW R Interface FCV1 was controlled through a basic LabVIEW R 8.5 user interface (UI), which was developed as part of the thesis work. V ref was controlled by an included Lab- VIEW R waveform generator (file: NI Basic Function Generator.VI ). There were four user-specified controls: signal type, amplitude, DC offset, and frequency. The signal type selected the output waveform as either sine, triangle, sawtooth, or square. Amplitude, DC offset, and frequency are self explanatory. DC signals were generated by setting the waveform amplitude to null and the DC offset to the desired value FCV1 Performance The performance of FCV1 was examined by determining how effectively it was able to produce an excitation field, and corresponding Hall voltage V H, that was equal to the reference voltage waveform V ref. The Hall sensor and monitoring coil were used to measure the excitation field. Figure 3.6 shows V H, V ref and I ex for a sinusoidal reference voltage with an amplitude of 50 mv and a frequency of 10 Hz (V ref = 50mV sin(2πt10hz)). Hall voltage (V H ) exactly matches the reference voltage (V ref ), making the two signals difficult to distinguish in the figure. However, V H contains a significant noise component that is not present in V ref. This noise is also apparent in the excitation current (I ex ) waveform. The amount of noise observed in V H and I ex was unexpected, as White s coil-based FCS system did not display this noise characteristic. It was found that the noise was independent of the V ref waveform and would occur in both AC (figure 3.6)and DC V ref (figure 3.7) system modes. Figure 3.7 indicates that noise noise in I ex peaks at approximately 2 ma. Since the ferrite core saturates at an excitation current of

66 CHAPTER 3. FLUX CONTROL SYSTEMS FCV1 Performance forv ref = 50 mv sin(2πt10 Hz) 0.06 V ref 0.04 V H 0.04 Voltage (V) I ex Excitation Current, I ex (A) Time,t(s) Figure 3.6: Hall voltage (V H ) and excitation current (I ex ) for a sinusoidal reference voltage. V H lies on V ref, making the two lines indistinguishable. 95 ma (shown in figure 3.8), this 2 ma noise signal is non-trivial. In order to further investigate the noise behavior of FCV1, the 37 turn monitoring coil shown in figure 3.5 was mounted on one of the excitation core poles. The coil voltage (V mc ) was proportional to the time-derivative of B in the excitation circuit, boosting the high-frequency noise signal relative to the low frequency (sub 50 Hz) excitation field, thereby allowing a detailed analysis of the noise component. Figure 3.8 shows V mc, I ex and V H for each of three V ref waveforms (each plot represents a different reference waveform). The noise component of V H and I ex is not obvious at this scale, but is clearly visible in V mc. A fast Fourier transform (FFT) of V mc performed in LabVIEW R using the spectral analysis tool 6 indicated a dominant noise frequency of 700 Hz. When the excitation core was removed from the sample, it 6 The Spectral Measurements express VI.

67 CHAPTER 3. FLUX CONTROL SYSTEMS 56 3 FCV1 Performance forv ref = 0mV 2 Excitation Current, Iex (ma) Time, t (ms) Figure 3.7: FCV1 response to a DC reference voltage of V ref = 0. Only the excitation current waveform is shown. V H was omitted for clarity.

68 CHAPTER 3. FLUX CONTROL SYSTEMS 57 was noted that the frequency and amplitude of the noise in the monitor coil decreased, suggesting that the noise was a function of excitation coil inductance. The noise was ultimately traced to oscillations in the excitation voltage (V ex ). The output from the LM4701 amplifier fluctuated between ±24 V (its voltage supply rails) at 700 Hz. The inductance of the excitation core decreased this 24 V amplitude voltage fluctuation to a 2 ma current oscillation FCV1 Shortcomings The instability observed in FCV1 was due to the Hall sensor feedback mechanism. The explanation for this is as follows: referring back to equation 3.10, the Hall voltage (V H ) is directly proportional to the magnetic field density (B) through the transducer. These two values are linked by a Hall constant (H) such that V H = HB. The magnetic field (B ex ) generated by an excitation coil is proportional to the current (I ex ) through the coil, giving B ex I ex. (3.16) Equation 3.16 can be derived from equation 2.9 (the quasi-static case of Ampere s law) or from the Biot-Savart law. Using equations 3.10 and 3.16, we arrive at V H I ex B ex. (3.17) Therefore, a system using Hall voltage as the feedback mechanism must control the excitation current to reliably regulate B ex. The LM4701 is a voltage amplifier, yet in FCV1 it was configured as a current controller. FCV1 would be better served with a current amplifier in place of the LM4701: however, excitation coils are generally very inductive, therefore the time derivative of the excitation current is proportional

69 CHAPTER 3. FLUX CONTROL SYSTEMS 58 Voltage (V) Voltage (V) Voltage (V) Time,t(s) (b) V ref = 95 mv sin(2πt20 Hz) Time,t(s) (c) V ref = 50 mv sin(2πt30 Hz) (a) V ref = 95 mv sin(2πt10 Hz) Time,t(s) V H I ex V mc Excitation Current, I ex (A) Excitation Current, I ex (A) Excitation Current, I ex (A) Figure 3.8: Monitor coil voltage V mc boosts the noise amplitude relative to the excitation field. Waveforms for three different sinusoidal reference voltages are shown: two 95 mv signals at 10 and 20 Hz (figures (a) and (b)), and a 50 mv signal at 30 Hz (figure (c)). A 95 mv reference voltage amplitude was enough to drive the ferrite core to saturation, indicated by the lumps at peak I ex values.

70 CHAPTER 3. FLUX CONTROL SYSTEMS 59 to the excitation voltage (V ex ) such that di ex dt V ex. (3.18) Equation 3.18 indicates that to effectively control current through an excitation coil, the current amplifier requires infinite voltage rails. These systems simply do not exist. White s FCS system relied on LM4701 amplifiers paired with coils as the feedback mechanism. The voltage across a feedback coil (V fc ) wound around the pole of the excitation core is then proportional to time-derivative of the flux through the core, giving V fc db ex dt. (3.19) The proportionality arguments of equations 3.16, 3.18 and 3.19 lead to V fc db ex dt V ex. (3.20) This direct proportionality between the feedback coil signal V fc and excitation coil voltage V ex indicates that a coil-based feedback mechanism is better suited for voltage control of a standard operational amplifier. This was the premise of the second flux control system, FCV2. It should be noted that further investigation and analysis of the dynamic properties of FCV1 may have resolved the instability observed in the system. However, due to time constraints and the fact that the FCS system functioned properly, FCV1 was abandoned in favor of a new design with coil feedback.

71 CHAPTER 3. FLUX CONTROL SYSTEMS Flux Control Version 2 (FCV2): Hall Sensor and Coil Feedback in Combination While Hall sensors are well suited to current controlled flux control systems systems, voltage controlled systems are best coupled with feedback coils. In Steven White s thesis work, the flux control system used feedback coils paired with LM4701 amplifiers to regulate the flux passing through samples. This is why the FCS excitation signals contained significantly less noise than those produced by the FCV1 system in the current study. However, relying on coil feedback only requires error correction software to compensate for DC offsets in the system. 7 The second design employed in the current project, FCV2, was designed with both coil and Hall sensor feedback to eliminate the need for error correction software and to provide a fully hardware-based magnetic flux controller FCV2 Hardware A new amplifier configuration was required to combine Hall sensor and feedback coil control. As with the FCV1 system, an LM4701 op-amp was used to power the FCV2 circuit, which is shown in figure 3.9. B was measured through the ferrite excitation core by integrating the monitor coil into the feedback loop, producing V fc, and also (as with FCV1) in the air gap between the core and sample by a Hall sensor, producing V H+ and V H. Note that both of the Hall voltage terminals were allowed to float in FCV2. FCV2 can be analyzed according to the Golden Rules given in section 3.1. When 7 See [39] p. 73.

72 CHAPTER 3. FLUX CONTROL SYSTEMS 61 R ref R G V ref V H+ V fc R H R fc V + V - + LM F V ex, I ex excitation coil V s R s GND R H V H- R G V fc I c V H+ V H- feedback coil ferrite excitation core lift-off spacer sample Hall sensor Figure 3.9: A simplified version of FCV2. There are only a few changes from figure 3.5. V fc has been integrated into the feedback circuit, and one end of the feedback coil was grounded. Neither V H+ or V H was grounded.

73 CHAPTER 3. FLUX CONTROL SYSTEMS 62 the inverting (-) and non-inverting (+) terminals to draw no current, the voltages at each terminal, V and V + respectively, are given by and V + = R G(V ref R H + V H R ref ) R H R G + R ref R G + R ref R H (3.21) V = R G(V fc R H + V H+ R fc ) R H R G + R fc R G + R fc R H. (3.22) Negative feedback was used, therefore the voltage at both input terminals must be the same, giving V + = V. (3.23) When R G = R H = R fc = R ref, equations 3.21, 3.22, and 3.23 can be solved for V ref to give V ref = V fc + V H+ V H = V fc + V H. (3.24) Equation 3.24 can be written in terms of the excitation field B ex (t), feedback coil turns (N fc ), feedback coil area (A fc ) and Hall constant H, such that V ref (t) = N fc A fc db ex (t) dt + HB ex (t). (3.25) Figure 3.9 is a simplified version of FCV2, useful for a general discussion of the feedback system and reference voltage waveform. Figure 3.10 presents a more detailed electrical schematic of the feedback system including the current source used for BH- 700 Hall sensors in FCV2. The Hall sensor control current (I c ) was supplied by a National Semiconductor LM337 negative voltage regulator run in current-control mode [30]. The 0.2 and 180 Ω resistors in series with the BH-700 sensor put V H+ and V H within the PCI-6229 s ±10 V input range. 8 Resistors in the feedback system 8 V H was always less than 1 V, but V H+ and V H had to be within ±10 V of ground for the PCI-6229 to make the differential measurement V H = V H+ V H.

74 CHAPTER 3. FLUX CONTROL SYSTEMS 63 feedback Hall sensor current supply 1μF 3 10 Ω Out LM337 Adj 1 In 2 I c +24 V 1 V 11 ex 1 kω 1 μf 10 kω 1 kω 0.2 Ω 1 kω V ref V H+ Blue +I c Red BH-700 Black -I c Yellow VH- V H- V H+ 1 kω 1 kω LM4701-3,5 4 excitation coil F = 0.5A 2 R ex L ex 1 μf L fc V s 0.2 Ω 180 Ω -24V V fc 1 kω 1 kω -24 V R fc feedback coil 10 MΩ Figure 3.10: An electrical schematic of FCV2 showing the feedback system and the Hall sensor current source. The LM4701 acts as an amplifier and adder for the feedback system. The Hall sensor current source is the LM337 voltage regulator, configured in current-control mode. Pin numbers are shown for the LM4701 and LM337, as well as BH-700 lead colors. White circles indicate external connections to voltage supplies (±24 V) or to the PCI-6229 DAQ (V ref, V H, V H+, V fc, V s, V ex ). were set to 1kΩ, while the fuse and series resistance (R s ) were unchanged from FCV1 (F = 0.5 A and R s = 0.2 Ω). A voltage divider of 10 kω and 1 kω resistors, giving a voltage divider ratio of 1/11, was used to directly measure the excitation voltage; a parameter that had not been examined in FCV1. The divider was required to bring the maximum excitation voltage V ex = 24 V into the measurement range of the PCI-6229 DAQ. The PCI-6229 terminals were reconfigured to accommodate the new feedback system. I/O connections and terminals configurations for FCV2 are given in table

75 CHAPTER 3. FLUX CONTROL SYSTEMS 64 Variable PCI-6229 Terminal Connection Configuration V ref AO 0 RSE V H AI 0 DIFF V s AI 1 NRSE V fc AI 2 DIFF V ex AI 3 NRSE V sig AI 4 DIFF GND AI sense Table 3.3: PCI-6229 I/O assignment and terminal configuration for FCV A sensor input line that recorded the magnetic detector output signal (V sig ) was added as a differential input FCV2 Software The software and user interface was rebuilt for FCV2. LabVIEW R Express VIs used for data acquisition and signal generation in FCV1 were replaced with purpose-built timing and triggering code. This improved synchronization between input and output channels, and improved the voltage and time resolution of the data acquisition system. Excitation and feedback coil parameters, such as those in table 3.1, were used to calculate V fc, V H, and subsequently V ref for a user-specified excitation magnetic field density (B ex ). The transition from the user-specified reference voltage used in FCV1 to B ex for FCV2 was done to highlight the relationship between V fc, V H, and B ex shown in equation Degaussing code was added so that any residual magnetization resulting from previous measurements or magnetic exposure could be removed from the sample prior to measurements. This ensured that measurements with FCV2 could be performed on demagnetized samples. This code was absent in FCV1 because the system never

76 CHAPTER 3. FLUX CONTROL SYSTEMS 65 matured to the point of performing a measurement FCV2 Performance The performance of FCV2 was examined using the same ferrite excitation core as FCV1. Figure 3.11(a) shows the magnetic flux density measured by the feedback coil (B fc ) and Hall sensor (B H ), and how they compare to the reference excitation magnetic field density (B ref ). The matching is excellent, with the slight offset between B H and B ref due to miscalibration of the Hall constant, which was an adjustable parameter in FCV2 s software. Offsets between B fc, B H and B ref peak at the maximum and minimum of the reference waveform, shown in figure 3.11(b). B fc is closely matched to B ref, with a maximum offset of B fc B ref = 0.15 mt. B H deviates further from the reference field, with a peak deviation of 2.5 mt at B ref = 100 mt. Additionally there is a mt shift in B H with respect to B ref. DC offsets of less than 1 mt and peak deviation of approximately 2% were considered acceptable errors in Hall sensor calibration. The 700 Hz noise observed in FCV1 was eliminated in FCV2. The FCV2 system was subsequently combined with a detector system, discussed in the following chapter, to perform several magnetic stress measurements on flat plate samples.

77 CHAPTER 3. FLUX CONTROL SYSTEMS 66 Magnetic Field Density (mt) (a) FCV2 Performance forb ref = 100 mt sin(2πt55 Hz) B ref B H B fc Time,t(ms) (b) Hall Sensor and Feedback Coil Magnetic Field Offsets Magnetic Field Density (mt) B H -B ref B fc -B ref Time,t(ms) Figure 3.11: The magnetic fields measured by the Hall sensor and feedback coil in FCV2. (a) The reference field was 100 mt in amplitude at a frequency of 55 Hz. B ref and B fc curves lie on top of each other, making them nearly indistinguishable. Offsets, shown in the bottom figure (b), were calculated by subtracting measured field density (B fc, B H ) from B ref.

78 Chapter 4 Magnetic Stress Detectors The magnetic excitation system based on FCV2 provided an effective, consistent and repeatable method of coupling magnetic flux into samples. The next stage in the process involved adding a detector to measure the magnetic signal emanating from the sample when an excitation field was generated by FCV2. A this point it is convenient to define some important terms. The detector refers to the magnetic flux transducer used to measure stress-induced leakage flux emanating from the sample. This detector is located between the poles of the excitation magnet and can be either a wire coil or a Hall probe. Detectors are not to be confused with feedback sensors (referring to the feedback Hall sensor and feedback coil sensor) used in the excitation flux control system. Finally, the term probe refers to the entire physical device, consisting of the detector, excitation coil, core, feedback Hall sensor and feedback coil. Three detector configurations were tested with the FCV2 probe. These configurations are shown in figure 4.1. The first, figure 4.1 (a), was a Hall detector aligned parallel to the sample surface normal, termed the DC MFL detector. The second, 67

79 CHAPTER 4. MAGNETIC STRESS DETECTORS 68 (a) (b) (c) measured B component wire coil Hall detector Figure 4.1: The three detector configurations used with the prototype excitation core. (a) DC MFL: a Hall detector oriented parallel to the surface normal. (b) AC MFL: a wire coil with its axis parallel to the surface normal. (c) SMA: a wire coil with the coil axis perpendicular to a line between the excitation core poles and the surface normal. shown in figure 4.1 (b), was a wire coil with its axis aligned to the sample surface normal, termed the AC MFL detector. Finally, figure 4.1 (c) shows a wire coil with its axis perpendicular to both the sample surface and a line joining the poles of the excitation core. This was termed the SMA detector and was used for stress-induced anisotropy measurements. A flat plate sample subjected to a variable uniaxial applied load was used to evaluate the stress sensitivity of the probe for each of the the three detector configurations. The remainder of this chapter is organized in the following sections: Section 4.1 provides an overview of the steel sample and the Single Axis Stress Rig - the apparatus used to apply stress to the sample. Section 4.2 describes in detail the three detector configurations shown in figure 4.1, as well as the data acquisition system. Section 4.3 outlines the experimental procedures used to test the three detector

80 CHAPTER 4. MAGNETIC STRESS DETECTORS 69 configurations Section 4.4 presents the results for each of the detector tests. 4.1 Test Sample and the Single Axis Stress Rig (SASR) To determine the effectiveness of different detector configurations, a flat plate sample was subjected to a uniaxial tensile stress via a single axis stress rig (SASR). Details of the test sample and SASR are described here Test Sample Measurements were performed on a 2.8 mm thick hot-rolled mild steel plate, 500 mm long and 216 mm wide, shown in figure 4.2. Tensile strength tests on these samples indicated a yield strength of 291 MPa, a Young s modulus of Y = 219 GPa [2], and Poisson s ratio to be ν = [24]. The sample was used in previous Ph.D. thesis work by Catalin Mandache [24]. For this work, two electrochemically milled 18 mm diameter holes were located in the center of the plate. A total of three Vishay R Measurements Group EA BF300 strain gages were mounted at different locations on the plate, as indicated in figure 4.2. Strain gage 1 measured the strain along the length of the plate, which corresponded to the applied stress direction. Gages 2 and 3, located on the opposite side of gage 1, were used to determine the uniformity of the applied strain. Measurements using the prototype probes were performed at the location indicated in figure 4.2. This location was selected to avoid any stress concentrations or

81 CHAPTER 4. MAGNETIC STRESS DETECTORS strain gage measurement location strain gage 2 strain gage 3 compressive stress σ c = -νσ t in the perpendicular direction applied tensile stress σ t in the parallel direction Figure 4.2: The mild steel plate used to test different detector configurations. Strain gage locations are shown by rectangles. Gage 1, indicated by a dashed line, was located on the underside of the plate. Gages 2 and 3 were located on the upper plate surface. Two 18 mm hole defects were at the center of the plate. All dimensions are in mm. other localized stress effects resulting from the sample edges or hole defects The Single Axis Stress Rig (SASR) The single axis stress rig (SASR) was designed and built within the Applied Magnetics Group to serve as a general purpose stressing device for the application of tensile stresses. It was configured to apply to single axis tensile loads along the length of flat plate samples, such as that indicated in figure 4.2. A schematic diagram of the SASR is shown in figure 4.3. Two sets of steel jaws clamp down on either end of the sample. One set of jaws is connected to a fixed bridge, while the other is connected to a gliding bridge that moves along guidance rods. Two hydraulic jacks extend when pressurized by a manual pump. Extension of the pistons within the hydraulic jacks pushes the gliding bridge along the guide rods, applying tensile stress (σ t ) to the sample clamped in the jaws. As a result of

82 CHAPTER 4. MAGNETIC STRESS DETECTORS 71 sample jaws gliding bridge fixed bridge sample guidance rods support beam hydraulic jack spacer cylinder Figure 4.3: A schematic of the single axis stress rig used to introduce tensile stress in the flat plate sample. Poisson s ratio effects, compressive stress is also generated across the width of the sample (σ c ) given by σ c = νσ t. (4.1) The pressure in hydraulic lines was monitored by an Omega Engineering Inc. PX302-10KGV pressure transducer connected to an Omega Engineering Inc. DP25-S digital meter. A more detailed description of the SASR and its operation is provided in reference [24] Strain Measurement The three EA BF300 strain gages mounted on the sample were connected to a Vishay R Measurements Group SB-10 Balance and Switch used to calibrate the strain gages and sequentially switch between the output of each gage. The SB-10 was connected to a Vishay R Measurements Group P3500 Strain Indicator, which provided a direct indication of the strain measured by the gages.

83 CHAPTER 4. MAGNETIC STRESS DETECTORS 72 detector mount assembly detector brace feedback Hall sensor housed within a liftoff spacer detector mount assembly outer brace liftoff spacer R 9.2 mm feedback coil connector brace excitation coil ferrite excitation core 64 mm (a) (b) Figure 4.4: An assembled probe showing a detector mount assembly attached to the connector brace of the excitation core. (a) Important components are indicated in the figure. The spring of the detector mount is not visible in this figure; it is hidden between the detector brace and outer brace. (b) A photo of the assembled system, built by the author. 4.2 Detectors, Data Acquisition and Data Analysis The three different detectors - DC MFL, AC MFL, and SMA - were attached to the excitation core with a detector mount assembly, shown in figure 4.4(a). The detector mount assembly consisted of three primary parts: a detector brace that housed the detector, a spring (not shown) to push the detector brace against the sample, and an outer brace that housed the detector and spring system and attached it to the excitation core. Each of the detectors was fixed to its own mounting assembly, which could be quickly connected to the excitation core. The assembled core and detector system is called a probe. A photograph of the assembled probe is shown in figure 4.4(b)

84 CHAPTER 4. MAGNETIC STRESS DETECTORS 73 Figure 4.5 shows a plan, underside view of each detector mounted to the excitation core. The details of each detector are outlined below. DC MFL The Hall sensor used for DC MFL measurements was a F.W. Bell BH-700 sensor; this was the same Hall sensor used in both FCV1 and FCV2 flux control systems. The processed 1 output signal from this detector was denoted V DCM. AC MFL An air-core 200 turn coil wound from 44 AWG wire with an average loop area of 3.1 mm 2 was used for the AC MFL detector. This coil was circular with an inner diameter of 0.98 mm and an outer diameter of 2.99 mm. The processed output voltage of this detector was denoted V ACM. SMA The SMA detector was a rectangular air-core 69 turn coil wound from 44 AWG wire with an area of 2.86mm 2. The coil was 2.15 mm long and 1.35 mm high. The 2.15 mm length dimension lay along the sample s surface, while the height dimension projected away from it. The coil was constructed as a rectangle, and oriented as described, in order to maximize the measurement region in close proximity to the sample s surface. V SMA denoted voltage of this detector after signal processing. 1 The output voltage signals of the detectors were acquired as V sig by the LabVIEW R software: however, each detector required different signal processing methods, such as averaging and fitting, to produce a useful signal. The signal processing for each detector is outlined in section 4.4.

85 CHAPTER 4. MAGNETIC STRESS DETECTORS 74 DC MFL AC MFL SMA Figure 4.5: DC MFL, AC MFL, and SMA detectors mounted to the excitation core. The detectors were locked into their mounts with epoxy. The detectors are not shown to scale Data Acquisition Signals from the three different detectors were acquired by the PCI-6229 DAQ as V sig, configured as indicated in table 3.3 (located in chapter 3). The DC MFL, AC MFL, and SMA detector signals were conditioned differently depending on the output voltage magnitude of each transducer. DC MFL measurements were input directly to the PCI-6229 as a differential measurement across the V H+ and V H terminals of the Hall detector. AC MFL and SMA signals were amplified by an Ithaco R Model 565 preamplifier in transformer mode, producing a gain of 60 db prior to being input into the PCI-6229.

86 CHAPTER 4. MAGNETIC STRESS DETECTORS 75 applied tensile stress σ t σ t σ t compressive stress σ c B ex AC MFL and DC MFL tensile measurement σ c B ex AC MFL and DC MFL compressive measurement σ c 90 o 180 o 0 o 270 o Bex SMA measurement Figure 4.6: The footprint of the excitation core on the sample for AC MFL, DC MFL and SMA measurements. The direction of B ex is indicated by white lines between the excitation core poles. 4.3 Experimental Procedures for Testing and Comparison of the Probe Systems Tensile stress was applied to the sample by pressurizing the hydraulic jacks of the SASR. Measurements were performed with each of the three detector/probe combinations: DC MFL, AC MFL, and SMA. These measurements were recorded at different stress levels to determine which detector was most sensitive to stress effects. Figure 4.6 shows the orientation of the probe excitation core relative to the stress direction for DC MFL, AC MFL and SMA detectors. As shown in the first two diagrams of figure 4.6, AC MFL and DC MFL measurements were performed with the probe parallel (called the parallel configuration) and perpendicular (called the perpendicular configuration) to the applied stress. The parallel configuration enabled a measurement of tensile strain sensitivity, while the perpendicular configuration was a measurement of compressive strain 2. SMA measurements were performed by rotating the probe over 360 about the 2 The SASR could only produce tensile strain. The Poisson effect was exploited to examine AC MFL and DC MFL compressive sensitivity.

87 CHAPTER 4. MAGNETIC STRESS DETECTORS 76 measurement location, stopping to make measurements at 15 intervals. Thus, each stress measurement consisted of 25 data sets 3. Taking φ as the angle between the probe and the direction of tensile stress, φ = 0, 180, 360 probe orientations aligned the probe along σ t, the applied stress direction, while φ = 90, 270 aligned the probe with the largest compressive stress (σ c ) direction. φ was taken to increase anti-clockwise from zero. All measurements were performed within the elastic deformation range of the sample. A degaussing cycle 4 was completed before each measurement to remove any residual magnetization from the sample. 4.4 Detector Results and Analysis Voltage signals from the DC MFL, AC MFL, and SMA detectors were recorded in the FCV2 software as signal voltage (V sig ) waveforms. This section describes the method by which raw V sig waveforms were processed to produce V DCM, V ACM and V SMA values. Each detector/probe system is considered in this section DC MFL DC MFL measurements were performed using the probe with an excitation field magnitude of 100 mt on flat plate samples in the SASR. A DC MFL Hall detector voltage (V DCM ) measurement was recorded for increasing stress values up to a maximum tensile stress of 107 MPa. 3 There were a total of 25 waveforms recorded for each SMA measurement because data was acquired at both φ = 0 and φ = A degauss cycle removes residual magnetization from the sample by cycling through hysteresis loops of decreasing magnitude.

88 CHAPTER 4. MAGNETIC STRESS DETECTORS 77 The configuration of the FCV2 software interface required that the DC excitation fields used in DC MFL measurements were input as sine functions with null amplitude, a 15 Hz frequency, and a 100 mt offset term. This resulted in DC MFL V sig distributions consisting of 3333 V sig data points recorded over a 67 ms window data points for a 15 Hz signal corresponds to the system s sampling frequency of 50 KHz, while 67 ms is simply the period of a 15 Hz wave. V DCM was taken as the average signal voltage of a DC MFL V sig distribution. Uncertainty in V DCM was calculated using the standard method for uncertainty in a mean 5. Figure 4.7 shows V DCM for both parallel (B ex σ t ) and perpendicular (B ex σ t ) orientations of the excitation field. Linear trend lines and their associated equations for σ t in MPa and V DCM in mv are also shown. Both data sets demonstrate that V DCM is proportional to applied stress. As seen in figure 4.7, when B ex is parallel to the applied stress direction, increasing stress causes the signal to decrease. When B ex lies along the direction of compressing stress, higher values of compressive stress cause the signal to increase. This relationship can be explained by the positive magnetostriction of Fe: tensile stress increases sample permeability in the direction of applied stress, which causes less flux to be forced out of the sample. The compressive stress resulting from Poisson s effect produced the opposite outcome in V DCM ; the decrease in permeability caused more flux to be forced out of the sample, increasing signal magnitude. The large difference between parallel and perpendicular signals in the absence 5 Consider N measurements of x with the same uncertainty in each measurement. The mean measurement ( x) is given by x = N 1 x. The error in the mean (σ x ) is then σ x = σn 1/2, where σ is the standard deviation of the N measurements of x. See reference [36] for additional information.

89 CHAPTER 4. MAGNETIC STRESS DETECTORS DC MFL Stress Sensitivity DC MFL Signal,VDCM (mv) t t fit fit Applies Stress, σ (MPa) Figure 4.7: DC MFL measurements for B ex σ t and B ex σ t. For B ex σ t measurements, stress σ t ranged from 0 to 107 MPa. In B ex σ t measurements, compressive stress σ c varied between 0 and 34 Mpa. Linear fits and their associated equations are shown for each data set. Error bars for the perpendicular data points appear as vertical lines through the circles.

90 CHAPTER 4. MAGNETIC STRESS DETECTORS 79 of stress (at σ = 0) is likely due to significant anisotropy within the sample in its unstressed state, likely a result of manufacturing and previous experiments. Although the data clearly indicate a trend, the scatter in data would make quantitative measurement of stress somewhat problematic using this method AC MFL As with DC MFL measurements, AC MFL data was recorded in both parallel and perpendicular probe orientations. The excitation field used was a 55 Hz sine wave with an amplitude of 100 mt described by B ex = 100 mt sin(2πt55 Hz). (4.2) V ACM readings were recorded with this excitation field up to a maximum SASR tensile stress of 128 MPa and a maximum compressive stress of 35 MPa. V sig was sampled at a frequency of 50 KHz over one complete 18 ms period, shown in figure 4.8. Also shown in this diagram is the corresponding B ex waveform. V sig waveforms were cosine waves, which was expected from a sinusoidal B ex excitation field. V sig waveforms were expected to be functions of applied stress (V sig (σ)). They were fit in MATLAB R to a sinusoidal function with three degrees of freedom according to V sig (σ) = A f (σ) sin(2πt55 Hz + B f ) + C f, (4.3) where A f (σ) is the amplitude, B f is phase and C f is offset. Phase and offset were expected to be constant at B f = π/2 radians and C f = 0. Amplitude was the only parameter expected to be affected by the applied stress on the sample, as A f (σ) was directly proportional to the flux density passing through the AC MFL signal coil,

91 CHAPTER 4. MAGNETIC STRESS DETECTORS V sig for the Parallel AC MFL Measurement at Null Stress Excitation Field, Bex (mt) Signal Voltage, Vsig (mv) B ex V sig Time,t(ms) Figure 4.8: The excitation field (dashed line) and signal voltage (solid line) for an AC MFL measurement at zero applied stress. The probe was in the parallel orientation for this measurement.

92 CHAPTER 4. MAGNETIC STRESS DETECTORS 81 therefore fit amplitude was taken as the AC MFL signal voltage, giving V ACM = A f (σ). (4.4) Figure 4.9 shows V ACM for both parallel and perpendicular probe configurations (B ex σ t and B ex σ c respectively). Uncertainty in V ACM was determined by the 95% confidence interval of the fit to equation 4.3. Most of the measurements in figure 4.9 agree within error, indicating no significant relationship between V ACM and σ beyond uncertainty. While there may be a trend in the data, it is not one which is understandable. As such, this method was deemed to be of little use for any practical application SMA SMA measurements were the most demanding of the three measurement types, in terms of both the time required to perform each measurement and the amount of data analysis needed to convert V sig waveforms to V SMA values. Before unprocessed V sig waveforms can be presented, it is necessary to expand on some of Langman s work presented in section SMA: A Theoretical Development of Angular Dependence The SMA probe was rotated about a point, thus capturing information regarding the direction and magnitude of stress 6. Langman described the angle between magnetic fields inside ( B in ) and outside ( B out ) the sample (designated δ) as a function of the relative permeabilities along two perpendicular principle magnetic directions (µ r1 and 6 While AC and DC MFL measurements were recorded with the probe in two different configurations (parallel and perpendicular to applied stress), this was done to examine the measurements sensitivity to compressive stress.

93 CHAPTER 4. MAGNETIC STRESS DETECTORS AC MFL Stress Sensitivity 198 t AC MFL Signal,VACM (mv) Applies Stress, σ (MPa) Figure 4.9: AC MFL measurements for B ex σ t and B ex σ c. For B ex σ measurements, stress ranged from 0 to 128 MPa. In B ex σ measurements, stress varied between 0 and 123 Mpa.

94 CHAPTER 4. MAGNETIC STRESS DETECTORS 83 μ 1 B in1 B out δ θ B in φ B in2 μ 2 Figure 4.10: A modified figure 2.15 redrawn for reference. The excitation core footprint is indicated by dotted lines. µ r2, with µ r2 > µ r1 ), and the angle of the excitation field density relative to the µ 2 direction (θ). A modified figure 2.15 is presented here as figure 4.10 for convenient reference. In the present application, the direction parallel to applied stress corresponds to µ r2, and µ r1 corresponds to the perpendicular direction. Equation 2.27 was given as ( ( µ r2 ) µ δ = arctan r1 1) tanθ 1 + µ. r2 µ r1 tan 2 θ This equation provided accurate δ values in Langman s earlier study; however, to use this equation in its current form requires knowledge of the magnetic field orientation within the sample, something that is not possible in the present application. However, it is possible to develop an alternate relationship by expressing θ in terms of φ (recall that φ is probe angle relative to µ r2, indicated in figure 4.10). A reasonable description for this relationship was found to be ( θ = φ + 1 ( µ r1 ) 3 arctan µ r2 1) tanφ 1 + µ, (4.5) r1 µ r2 tan 2 φ based on data presented in figure 2.16 and reference [21]. The relationship between equation 2.27 and equation 4.5 can be explained by considering the two terms in equation 4.5 independently.

95 CHAPTER 4. MAGNETIC STRESS DETECTORS 84 Both B in orientation (θ) and probe orientation (φ) were measured relative to the µ 2 direction, which accounts for the separate φ term in equation 4.5. The arctan(...) component shifts θ from φ toward µ 2. The factor of 1/3 was selected based on the angles between φ, B in and B out shown in figure The two principle magnetic directions (µ r1 and µ r2 ) cause the magnitude of B out to depend on the orientation of the internal magnetic field. This relationship can be expressed as B out = (Bin1 µ r1 ) 2 + ( Bin2 µ r2 ) 2. (4.6) Using equations 2.22 and 2.23, equation 4.6 can be rearranged to (sin ) 2 ( ) 2 B out θ cos θ = +, (4.7) B in µ r1 which gives the magnitude of B out per unit B in. An SMA sensing coil voltage signal will be a function both the magnitude, orientation, and rate of change of B out, as well as probe properties (number of turns and µ r2 coil area). Thus the SMA signal voltage in terms of B in is V = NAG t (B in), (4.8) where the G term is called the geometry factor. G compensates for the different magnitudes and orientations of B in and B out, as well probe orientation. For the SMA detector described in this thesis, which is a coil rotated 90 from the probe angle φ, the geometry factor takes the form [ (sin ) 2 ( ) ] 2 1/2 θ cos θ G = + sin (θ + δ φ), (4.9) µ r1 µ r2 where θ is given by equation 4.5 and δ is determined by equation 2.27, both of which are functions of probe angle φ. The square-root term, shown in square brackets, gives

96 CHAPTER 4. MAGNETIC STRESS DETECTORS μ r2 /μ r1 = 4 μ r2 /μ r1 = Geometry Factor, G μ r2 /μ r1 = 2 μ r2 /μ r1 = Probe Angle, φ (deg) Figure 4.11: G for four µ r2 /µ r1 ratios. The 0, 180, and 360 probe orientations place the probe parallel to the µ 2 direction. the magnitude of B out from B in, it was taken directly from equation 4.7. The sin(...) term extracts the component of B out parallel to the coil s axis. Figure 4.11 shows the geometry factor G over a complete probe rotation (φ = 0 to 360 ) for several theoretical µ r2 /µ r1 ratios. For the case of µ r2 /µ r1 = 1 the geometry factor is zero, indicating that isotropic samples would produce no signal in the SMA coil. Other relative permeability ratios produce sinusoidal, 180 periodic geometry factors, with amplitude increasing with µ r2 /µ r1. Peak G values occur when the probe is shifted 45 anti-clockwise from the direction of greatest permeability.

97 CHAPTER 4. MAGNETIC STRESS DETECTORS 86 SMA Results and Analysis The excitation field used in SMA measurements was a 55 Hz sine wave with a 100 mt amplitude, identical to the field used for AC MFL measurements (refer to equation 4.2). Measurements were performed up to a maximum sample tensile stress of 130 MPa. For each SMA measurement the probe was rotated 360 in 15 increments, with V sig waveforms collected for each increment. V sig waveforms, functions of both stress (σ) and probe orientation (φ), were fit in MATLAB R to the equation V sig (σ,φ) = A f (σ,φ) sin(2πt55 Hz + B f ) + C f, (4.10) where C f is fit offset, B f is fit phase, and A f (σ,φ) is fit amplitude, which was expected to be a function of both applied stress and probe angle. Both offset and phase were expected to be constant. Figure 4.12 shows A f (σ,φ) for four stress levels (σ = 0 MPa, 61.6 MPa, 97.7 MPa, 129 MPa). Uncertainty values were taken as the 95% confidence intervals of the fit. The effects of stress on A f (σ,φ) can be seen in the first 90 of rotation, where the signal follows an inverted sinusoidal line for σ = 0 MPa, decreases in amplitude as stress increases to σ = 61.6 MPa, then inverts to a standard sine waveform at σ = 97.7 MPa, and finally increases in amplitude for σ = 129 MPa. Based on the amplitude of A f (σ,φ) and anisotropy analysis presented earlier in this section, it can be seen that the initial bulk magnetic easy axis of the sample is in the perpendicular direction 7, which agrees with previous studies performed in these plates [24]. Increasing tensile stress along the parallel direction causes a corresponding 7 Recall that peaks in the anisotropy signal occur 45 anti-clockwise from the direction of greatest permeability.

98 CHAPTER 4. MAGNETIC STRESS DETECTORS 87 SMAV sig (σ, φ) Amplitude forb ex = 100 mt sin(2πt55 Hz) 115 σ = 0 MPa σ = 61.6 MPa Fit Amplitude, Af(σ, φ) (mv) σ = 97.7 MPa σ = 129 MPa Probe Angle, φ (deg) Figure 4.12: V sig (σ, φ) fit amplitudes for SMA measurements. Each data set consists of 25 points recorded at 15 intervals between 0 and , 180, and 360 probe orientations placed the probe (and excitation field) parallel to the applied stress.

99 CHAPTER 4. MAGNETIC STRESS DETECTORS 88 increase in magnetic permeability, bringing the sample close to an isotropic state for σ = 66.1 MPa. At σ = 91.7 MPa, the easy axis has shifted to the parallel orientation. To arrive at V SMA, the V sig amplitude waveforms shown in figure 4.12 were fit to a 180 -periodic sine function according to A f (σ,φ) = A f2 (σ) sin ( ) 2π 180 φ + B f2 + C f2. (4.11) As with the initial fit (see equation 4.10), the phase (B f2 ) and offset (C f2 ) parameters were expected to be constant: though they were fit in MATLAB, they were confirmed to remain relatively constant. The amplitude was expected to be proportional to the stress applied to the sample, thus V SMA = A f2 (σ). (4.12) Figure 4.13 shows the SMA signal voltage (V SMA ) obtained from fits to equation The data shows a linear increase in SMA signal amplitude with applied stress and sufficiently low uncertainties for data points to be clearly distinguished. The initial measurement data set corresponds to the data presented earlier in figure The repeated measurement data set was acquired after the initial measurement at approximately equivalent applied stress levels. Between measurements the probe was removed from the sample and replaced at the same location. The purpose of the repeated measurement was to evaluate the repeatability of SMA measurements. The two data sets agree within uncertainty, although the repeated measurement has consistently higher uncertainty than the initial measurement.

100 CHAPTER 4. MAGNETIC STRESS DETECTORS Anisotropy Signal from a Mild Steel Plate 10 initial measurement repeated measurement SMA Signal Voltage, VSMA (mv) Applied Stress, σ (MPa) Figure 4.13: SMA measurements for tensile up to 130 MPa.

101 CHAPTER 4. MAGNETIC STRESS DETECTORS Selected Detector Of the three detectors tested, only the AC MFL coil demonstrated no stress sensitivity. This could have been due to the apparatus or data processing methods, as the measurement was of the same nature as DC MFL tests. DC MFL measurements behaved generally as expected: V DCM decreased with the probe parallel to tensile stress, and increased with the probe oriented along compressive stress. However, significant scatter in the data suggested that quantitative measurements may be difficult with a DC MFL-based probe. The SMA detector indicated a strong relationship between applied stress and both the initial measurement (as A f (σ,φ)), as well as the V SMA value. SMA measurements are also capable of providing additional information about the orientation of stresses within the sample, and were demonstrated to be repeatable. For these reasons, an SMA sensing coil was selected as the magnetic flux transducer for the second probe design.

102 Chapter 5 Proposed Design: The Magnetic Anisotropy Prototype Probe Previous chapters described the testing of different feedback systems and sensor configurations with a general-use excitation core. The general-use core was relatively large (the pole area was 264 mm 2 with a back spine length of 64 mm) and was tested on flat plate samples. This chapter describes the features of a prototype probe developed specifically for use on CANDU R feeders, termed the Magnetic Anisotropy Prototype (MAP) probe, that combined a smaller, Supermendur-cored excitation core with the FCV2 flux control system and an SMA detector coil. Section 5.1 outlines the design characteristics of the MAP probe, while section 5.2 describes experiments conducted on a section of pipe similar to the feeders found in CANDU R reactors. 91

103 CHAPTER 5. PROPOSED DESIGN: MAP PROBE Magnetic Anisotropy Prototype (MAP) Probe The optimized probe for stress measurement in feeders, termed the Magnetic Anisotropy Prototype probe (MAP probe), consisted of a Supermendur 1 U-core excitation electromagnet coupled with a 200 turn SMA coil. A BH-700 feedback Hall sensor and a feedback coil were integrated into the MAP probe so that it could be used with FCV2. It should be noted that the MAP probe will not fit within the clearances of a CANDU reactor face: it is too tall. However, the components selected for the probe were chosen so that modification to certain parts, such as the connector assembly, would allow the system to be used in a CANDU R feeder pipe environment. The Supermendur core, shown in figure 5.1, consists of thin layers of a 49% Co, 49% Fe, 2% V alloy. The layers are held together with a non-conductive epoxy that limits the formation of eddy currents, which decreases power loss within the core, making it ideal for AC magnetic applications. The core is small, with a height of mm and a footprint of mm 2, and was integrated into a housing assembly appropriate for SMA measurements. The MAP assembly, shown in figure 5.2(a), was built around the Supermendur core. For the probe to function with FCV2, a feedback Hall sensor, feedback coils and excitation coils were mounted to the core. These coils, and other important MAP components are shown in figure 5.2. The connector brace, shown in white, fits tightly into a stainless steel disk (shown in figure 5.2 (b)) which is free to rotate in a larger aluminum mount that can be clamped to samples. 1 Supermendur is the product name of a discontinued layered magnetic alloy from Carpenter Technologies.

104 CHAPTER 5. PROPOSED DESIGN: MAP PROBE R Figure 5.1: A schematic of the Supermendur core of the MAP probe. The core is shown to scale. All dimensions are in millimeters detector coil piston feedback Hall sensor SMA detector coil liftoff spacer feedback coil stainless steel disk excitation coil Supermendur excitation core spring connector brace aluminum mount (a) (b) Figure 5.2: (a) A diagram of the MAP probe. Only the corner of the Supermendur core of figure 5.1 is visible; it is shown in black. Connector pin 1 corresponds to the bottom right pin, connector pin 12 corresponds to the top left pin. (b) A photograph of probe set in mounting hardware.