A Magnetic Flux Leakage NDE System

A Magnetic Flux Leakage NDE System

for CANDUR© 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

Abstract

This work examines the application of different magnetic flux leakage (MFL) inspec-

tion concepts to the non destructive evaluation (NDE) of residual (elastic) stresses in

CANDU R© reactor feeder pipes. The stress sensitivity of three MFL inspection tech-

niques was examined with flat plate samples, with stress-induced magnetic anisotropy

(SMA) demonstrating the greatest stress sensitivity. A prototype SMA testing sys-

tem 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

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 Helene Hebert. 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 LATEX and

MATLAB R©, and Davin Young spent many hours in the machine shop building probe

components.

ii

Table of Contents

Abstract i

Acknowledgments ii

Table of Contents iii

List of Tables v

List of Figures vi

Chapter 1:

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 CANDUR© Feeder Pipes . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 A Brief Introduction to Magnetic Circuits and Magnetic Flux Leakage

Inspection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Thesis Scope and Objectives . . . . . . . . . . . . . . . . . . . . . . . 7

1.4 Organization of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 8

Chapter 2:

Theory and Background . . . . . . . . . . . . . . . . . . . 10

2.1 Stress and Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

iii

2.2 Maxwell’s Equations and The Quasi-Static Case . . . . . . . . . . . . 15

2.3 Magnetic Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.4 Magnetic Methods of Stress Measurement . . . . . . . . . . . . . . . 29

Chapter 3:

Flux Control Systems . . . . . . . . . . . . . . . . . . . . 39

3.1 Negative Feedback Control and Operational Amplifiers . . . . . . . . 40

3.2 Magnetic Flux Transducers . . . . . . . . . . . . . . . . . . . . . . . 42

3.3 Component Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.4 White’s Flux Control System (FCS) . . . . . . . . . . . . . . . . . . . 49

3.5 Flux Control Version 1 (FCV1): Hall Sensor Feedback . . . . . . . . 51

3.6 Flux Control Version 2 (FCV2): Hall Sensor and Coil Feedback in

Combination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

Chapter 4:

Magnetic Stress Detectors . . . . . . . . . . . . . . . . . . 67

4.1 Test Sample and the Single Axis Stress Rig (SASR) . . . . . . . . . . 69

4.2 Detectors, Data Acquisition and Data Analysis . . . . . . . . . . . . 72

4.3 Experimental Procedures for Testing and Comparison of the Probe

Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.4 Detector Results and Analysis . . . . . . . . . . . . . . . . . . . . . . 76

4.5 Selected Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

Chapter 5:

Proposed Design: MAP Probe . . . . . . . . . . . . . . . 91

5.1 Magnetic Anisotropy Prototype (MAP) Probe . . . . . . . . . . . . . 92

iv

5.2 MAP Probe Testing with SA-106 Grade B Pipe . . . . . . . . . . . . 96

Chapter 6:

Summary and Conclusions . . . . . . . . . . . . . . . . . 107

6.1 Flux Control Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

6.2 Magnetic Stress Detectors . . . . . . . . . . . . . . . . . . . . . . . . 108

6.3 Proposed MAP Probe Design . . . . . . . . . . . . . . . . . . . . . . 109

6.4 Recommendations for Future Work . . . . . . . . . . . . . . . . . . . 110

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

Appendix A:

FCV1 Details . . . . . . . . . . . . . . . . . . . . . . . . 118

Appendix B:

Skin Depth . . . . . . . . . . . . . . . . . . . . . . . . . . 120

v

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. . . . . . . . . 53

3.2 PCI-6229 I/O assignment and terminal configuration for FCV1. Ter-

minal configurations use the following abbreviations: referenced single-

ended (RSE), non-referenced single-ended (NRSE), differential (DIFF).

For additional information on terminal configurations see [29]. . . . . 53

3.3 PCI-6229 I/O assignment and terminal configuration for FCV2. . . . 64

5.1 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. . . . . . . . . . . . . . . . . . . . . . . . 95

vi

List of Figures

1.1 A simplified sketch of a CANDUR© 6 reactor face. . . . . . . . . . . . 3

1.2 A comparison of magnetic and electric circuits. . . . . . . . . . . . . . 6

2.1 The stress tensor for an element of a continuous structure in Cartesian

coordinates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 Residual stress formation in a bent beam. . . . . . . . . . . . . . . . 13

2.3 Ferromagnetic domain structure. . . . . . . . . . . . . . . . . . . . . 18

2.4 A typical magnetization hysteresis loop for a ferromagnetic sample

starting with zero magnetization. . . . . . . . . . . . . . . . . . . . . 19

2.5 A schematic of four magnetic domains aligned along the ¡100¿ direc-

tions of Fe. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.7 Magnetostriction of a material with positive λs. . . . . . . . . . . . . 25

2.8 The two types of magnetoelasticity: magnetostriction and the Villari

effect for a material with positive λs. . . . . . . . . . . . . . . . . . . 26

2.9 The magnetization processes for samples with aligned and misaligned

auxiliary fields and preferred crystalline axes. . . . . . . . . . . . . . 27

vii

2.10 A simplified Barkhausen noise apparatus. . . . . . . . . . . . . . . . . 31

2.11 A bandpass filtered Barkhausen noise spectrum taken from 3 kHz to

600 kHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.12 A polar plot of angular MBN energy measurements. . . . . . . . . . . 32

2.13 The application of magnetic flux leakage inspection in crack and cor-

rosion detection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.14 The MFL signal from a segment of SA106-B schedule 80 pipe (a) ref-

erence 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. 34

2.15 The rotation of the magnetic field just outside the sample ( ~Bout) rela-

tive to the magnetic field within the sample ( ~Bin) when µ2 > µ1. . . . 36

2.16 The orientation of ~Bin and ~Bout relative to the excitation core. . . . . 37

3.1 The components of a closed-loop control system shown in a block dia-

gram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.2 The feedback system components contained within an op-amp. . . . . 43

3.3 The Hall effect for a Cartesian coordinate system. . . . . . . . . . . . 45

3.4 A sketch of White’s FCS. . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.5 A simplified version of FCV1. . . . . . . . . . . . . . . . . . . . . . . 52

3.6 Hall voltage (VH) and excitation current (Iex) for a sinusoidal reference

voltage. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.7 FCV1 response to a DC reference voltage of Vref = 0. . . . . . . . . . 56

3.8 Monitor coil voltage Vmc boosts the noise amplitude relative to the

excitation field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

viii

3.9 A simplified version of FCV2. . . . . . . . . . . . . . . . . . . . . . . 61

3.10 An electrical schematic of FCV2 showing the feedback system and the

Hall sensor current source. . . . . . . . . . . . . . . . . . . . . . . . . 63

3.11 The magnetic fields measured by the Hall sensor and feedback coil in

FCV2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.1 The three detector configurations used with the prototype excitation

core. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.2 The mild steel plate used to test different detector configurations. . . 70

4.3 A schematic of the single axis stress rig used to introduce tensile stress

in the flat plate sample. . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.4 An assembled probe showing a detector mount assembly attached to

the connector brace of the excitation core. . . . . . . . . . . . . . . . 72

4.5 DC MFL, AC MFL, and SMA detectors mounted to the excitation core. 74

4.6 The footprint of the excitation core on the sample for AC MFL, DC

MFL and SMA measurements. . . . . . . . . . . . . . . . . . . . . . . 75

4.7 DC MFL measurements for Bex ‖ σt and Bex ⊥ σt. . . . . . . . . . . . 78

4.8 The excitation field (dashed line) and signal voltage (solid line) for an

AC MFL measurement at zero applied stress. . . . . . . . . . . . . . 80

4.9 AC MFL measurements for Bex ‖ σt and Bex ‖ σc. . . . . . . . . . . . 82

4.10 A modified figure 1.15 redrawn for reference. The excitation core foot-

print is indicated by dotted lines. . . . . . . . . . . . . . . . . . . . . 83

4.11 G for four µr2/µr1 ratios. The 0 , 180 , and 360 probe orientations

place the probe parallel to the µ2 direction. . . . . . . . . . . . . . . . 85

4.12 Vsig(σ, φ) fit amplitudes for SMA measurements. . . . . . . . . . . . . 87

ix

4.13 SMA measurements for tensile up to 130 MPa. . . . . . . . . . . . . . 89

5.1 A schematic of the Supermendur core of the MAP probe. . . . . . . . 93

5.2 A diagram of the MAP system. . . . . . . . . . . . . . . . . . . . . . 93

5.3 The pin diagram for the MAP system. . . . . . . . . . . . . . . . . . 95

5.4 A schematic of the three-point bending rig in the tensile configuration. 97

5.5 SMA dependence on excitation field amplitude. . . . . . . . . . . . . 100

5.6 MAP stress response for an excitation field Bex = 75 mT sin(2πt55 Hz). 102

5.7 SMA dependence on tensile and compressive applied stress. . . . . . . 104

5.8 Signal voltage Vsig(σa, φ) fit amplitude for approximately equivalent

compressive (σa = −44 MPa) and tensile (σa = 47 MPa) stresses. . . . 105

6.1 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 4. . . 112

A.1 An electrical schematic of FCV1. . . . . . . . . . . . . . . . . . . . . 119

B.1 Skin depth for a typical steel with µr = 100 and σe = 107 Ω−1m−1 . . 121

x

Chapter 1

Introduction

Engineered components have a finite service life governed by their design, manufac-

turing 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 non-

invasive 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

1The term non-destructive testing (NDT) is used synonymously with non-destructive evaluation(NDE).

1

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 CANDUR© feeder pipes. Details of the magnetic flux leak-

age (MFL) NDE technique and CANDUR© 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 Canada2 and Ontario Power Generation3 (OPG).

Reactors that use standard water (H2O) as the moderator/coolant require enriched

uranium fuel, composed of U-238 with 2% to 4% wt U-235. Heavy water moder-

ated/cooled reactors, such as the CANDUR© , can achieve criticality4 with naturally-

occurring uranium, composed of U-238 with 0.7% wt U-235, because heavy water

(D2O) is a weaker neutron moderator than standard water [11].

The primary heat transport circuit of a CANDUR© reactor uses pumps to push

heavy water coolant over fuel bundles in the calandria5. A simplified sketch of a

CANDU R© reactor face, showing most components of the primary heat transport cir-

cuit 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

2Known as Canadian General Electric during the design partnership.3Known as Hydro-Electric Power Commission of Ontario during the design partnership.4A self sustaining fission reaction.5A calandria is the reactor core of a CANDU R© system


7

5

3

1 2

4

4

6 6

3

outlet header1.

inlet header2.

feeders3.

steam generators4.

end ttings5.

heat transport pumps6.

insulation cabinet7.

Figure 1.1: A simplified sketch of a CANDU R© 6 reactor face. Adapted from the CAN-TEACH library (http://canteach.candu.org/library/19990113.pdf).

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 calan-

dria, 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,


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

corrosion6 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 pre-

vention 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

6Flow-accelerated corrosion is a process whereby the normally protective oxide layer on carbonsteel dissolves into a stream of flowing water or wet steam.


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 pur-

pose of measuring residual stresses in CANDUR© 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 feed-

ers. 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 rep-

resented 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


+_

wire resistance

vo

ltag

e s

ou

rce

load

bu

lb re

sis

tan

ce

Rload

Rwire

V

N

S

core reluctance

MM

F s

ou

rce

air g

ap

relu

cta

nce

NI

Cir

cu

it S

ch

em

ati

c

Electric Magnetic

Ph

ysic

al S

yste

m+

_batt

ery

wire

light bulb

air gap

co

re

current source

wire coil

N turns

current I flux

Is

s

fluxcurrent I

(a) (b)

(c) (d)

Figure 1.2: A comparison of magnetic and electric circuits. Figures (a,b) show sketchesof physical systems, while the electrical and magnetic schematics of the systems are givenin 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 Is. This coil generates a magnetic flux Φ, which passes

through the core (RC) and air gap (RG).

Magnetic flux leakage (MFL) inspection systems measure the magnetic flux out-

side of a magnetized sample, called ‘leakage’ flux, and correlate it to sample proper-

ties, 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


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 measure-

ments, the flux Φ through different samples, or regions on a sample must be con-

sistent. 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 CANDUR© 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


3. conduct testing on samples with feeder pipe geometry with a focus on general-

ized 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 pre-

sented 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.


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 5.2.1.

Chapter 2

Theory and Background

This chapter presents a theoretical summary of stress, strain, and quasi-static mag-

netic 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 classi-

fication 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 mag-

netic 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 body1, shown in figure 2.1, can be determined by a nine

1A body is an structure composed of a continuous distribution of elements (also known as points).

10

CHAPTER 2. THEORY AND BACKGROUND 11

z

x y

σzz

σzx

σzy

σyz

σyx

σyy

σzx

σxx

σxy

Figure 2.1: The stress tensor for an element of a continuous structure in Cartesian coor-dinates.

component stress tensor σ, given by

σ =

σxx σxy σxz

σyx σyy σyz

σzx σzy σzz

. (2.1)

Diagonal tensor elements σxx, σyy, and σzz 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


of numerous small crystals (called grains) in multiple orientations. A small sam-

ple volume may be less than the average grain size2, 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 insignifi-

cant. If the body is not strongly textured3 it may also be considered isotropic.

The maximum stress a material can support before undergoing plastic deformation

is defined as the yield stress σyield. Application of an external force to an object

results in deformation. Deformation is elastic up to σyield, that is, the deformation

vanishes if the force is removed. External stress beyond σyield 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

2A grain is a domain of mater that has the same structure as a single crystal.3Texture 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.


Force Force

(a)

(b)

(c)

tensile stress

compressive

stress

compressive stress

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. Thereis tensile stress above a neutral surface (shown with a dashed line) and compressive stressbelow it. (c) Once the external force is removed, internal ‘residual’ stresses redistribute toelastically deform the body toward its original state.


expresses geometrical deformation as a ratio of the change in dimension ∆d to the

initial dimension d0. Diagonal components of ε are normal strains in the x, y, and z

directions, and are given by

εi=j =∆d

d0

=d − d0

d0

, (2.2)

where d is the dimension after deformation. Off diagonal components (εi6=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 Cijkl, such that

σjk =∑

kl Cijklεkl.

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:

σij =Y

1 + ν

[

εij +ν

1 − 2ν(εxx + εyy + εzz)

]

. (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.

4Engineering shear strain is the complement of the angle between two initially perpendicular linesegments.

5Shear modulus (Gm) is not included in this list as it is defined by Gm = Y/ [2 (1 + ν)].


2.2 Maxwell’s Equations and The Quasi-Static Case

Maxwell’s four equations

∇ · ~E =ρ

ǫ0

(2.4)

∇× ~E = −∂ ~B

∂t(2.5)

∇ · ~B = 0 (2.6)

∇× ~B = µ0~J + µ0ǫ0

∂ ~E

∂t, (2.7)

and the Lorentz force law

~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 ǫ0 and µ0 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 (µ0ǫ0∂ ~E∂t

) of

equation 2.7 can be neglected because J >> ǫ0∂ ~E∂t

. Thus equation 2.7 becomes

∇× ~B = µ0~J. (2.9)

The current density ~J is the sum of two components:

~J = ~Jb + ~Jf , (2.10)

where ~Jb is the bound current due to electron spin and angular momentum, and


current generated by the movement free particles is represented by ~Jf . The magne-

tization field ( ~M) is attributed to bound currents:

∇× ~M = ~Jb, (2.11)

and the auxiliary field ( ~H) to free currents:

∇× ~H = ~Jf . (2.12)

Equations 2.9, 2.10, 2.11, and 2.12 can be rearranged to give

~B = µ0

(

~M + ~H)

. (2.13)

2.3 Magnetic Materials

Equation 2.13 can be expressed using magnetic susceptibility (χm) or relative perme-

ability (µr) tensors as

~B = µ0µr~H = µ0 (1 + χm) ~H. (2.14)

Both µr and χm are used to express the response of ~Jb to ~Jf 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

(χm). Materials are categorized by their χm 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 χm ≈ −10−5. ~H interacts with electrons to

6Other varieties of magnetism are omitted for brevity.


decrease ~B through the application of Lenz’s law to the orbital rotation of electrons

about nuclei. Superconductors are considered nearly perfectly diamagnetic with χm ≈

−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 10−3.

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 ferro-

magnetic 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 align-

ing magnetic domains with the field. Because domains are composed of billions of

magnetic moments, ferromagnetic materials have large magnetic susceptibilities, up

to χm ≈ 106. 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


Figure 2.3: Ferromagnetic domain structure. Magnetic domain orientation is shown witharrows. 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 104.

The isotropic and linear χm approximation is usually valid for paramagnetic and

diamagnetic substances; however, the domain structure of ferromagnetic and ferri-

magnetic 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 Ms, the

saturation magnetization. A subsequent decrease in H decreases the the sample’s

magnetization to the remnant (or residual) magnetization (Mr), defined as the mag-

netization of the sample at H = 0. Further decreases of H leads to the coercive field

(H = Hc), defined as the auxiliary field at which the magnetization returns zero. As

H continues to decrease, the sample goes into negative saturation −Ms. The area of


initial magnetization

curve

Hc

Mr

-Ms

Ms

Ma

gn

eti

zati

on

, M

Auxiliary Field, H

Hc

magnetic

Barkhausen noise

Figure 2.4: A typical magnetization hysteresis loop for a ferromagnetic sample startingwith zero magnetization. M increases with H along the initial magnetization curve tosaturation at Ms. Further variation of H changes sample magnetization as shown aroundthe loop. The inset shows magnetic Barkausen noise, which is discussed further in section2.4.1.

a B-H hysteresis loop corresponds to the energy lost to irreversible processes within

the sample [33].

2.3.1 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 ferri-

magnetic materials. Within each domain the material is magnetized to the saturation

magnetization Ms, because dipoles within each domain are aligned. Even though do-

mains are magnetically saturated, a bulk sample is generally composed of domains


[100]

[01

0]

(a)

(b) 180o wall

90o 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 thefour domains together produce no net magnetization. (b) Domain walls act as transitionregions between domains of different orientation. Two types of domain wall are shown:90 and 180.

with randomly oriented magnetization vectors, producing no net sample magnetiza-

tion. 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 mag-

netic 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 ma-

terial 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.’


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 mini-

mization of the sum of six energy terms: the exchange energy (εex), the magnetocrys-

talline anisotropy energy (εmca) the magnetostatic energy (εms), the magnetoelastic

energy (ελ), the domain wall energy (εwall), and the Zeeman energy (εp). Thus, the

total energy (εtotal) for a single iron crystal is

εtotal = εex + εmca + εms + ελ + εwall + εp. (2.15)

Minimizing εtotal results in the domain structure of ferromagnetic material. Each

energy contribution is explained below.

Exchange Energy

The exchange energy (εex) is due to the quantum mechanical exchange interaction8

between adjacent atoms first described by Heisenberg in 1926 [14] and applied to

7The this document follows standard crystallographic notation. The normal of a specific plane isindicated by (100), while the set of equivalent planes is denoted by 100. Directions are indicatedby square brackets as [100]; the complete set of equivalent directions is given by angular brackets as〈100〉.

8When 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 movingabout proton B. As electrons are indistinguishable, there is a possibility that the electrons exchangeplaces such that electron B moves about proton A, and electron A moves about proton B. Thisconsideration introduces an additional exchange energy term into the expression for the total energyof the two atoms.


ferromagnetism in 1928 [15]. For a set of atoms located throughout a lattice at ~ri,

each with spin ~S(~ri), the exchange energy can be written as the sum over each atom

pair [10]:

εex = −∑

jk

J (|~ri − ~rj|) ~S(~ri) · ~S(~rj), (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 2.16. If J (r) is positive, εex 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 εex 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 (εmca) 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 ~Ms 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 εmca can be expressed in terms of a series expansion of the direction


cosines (αi, i = 1, 2, 3) of ~Ms relative to the crystal axes:9

εmca = VD

(

K0 + K1

(

α21α

22 + α2

2α23 + α2

3α21

)

+ K2

(

α21α

22α

23

)

+ ...)

, (2.17)

where VD is domain volume, and K0, K1, K2 are anisotropy constants specific to

the material (in units of J/m3). It is typical to neglect K0 in equation 2.17 because

it is independent of angle and only consider the K1 and K2 terms when evaluating

the series [6]. In Fe, εmca tends to align magnetic moments to the 〈100〉 directions,

making them directions of easy magnetization.

Magnetostatic Energy

The magnetostatic energy (εms) is the energy stored in a magnet’s demagnetizing

field, given by [6]:

εms =1

2µ0

∫

∞Hd

2 d3r, (2.18)

where ~Hd 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 ~Hd (figure 2.6(b)), while a set of four

domains separated by 90 and 180 walls (figure 2.6(c)) further decreases ~Hd. Thus,

minimizing εms results in the formation of 90 and 180 domain walls.

9Consider a domain in a cubic crystal: let ~Ms make angles a1, a2, a3 with the crystal axes, thenα1, α2, α3 are the cosines of those angles.


(a) (b) (c)

Figure 2.6: Demagnetizing field lines for: a) a single domain, b) two opposing domainsseparated by a 180 wall, and c) four domains separated by 90 and 180 walls.

Domain Wall Energy

Domain wall energy (εwall) 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 εmca and

εex increase due to this gradual rotation of magnetic moments, and these increases

give εwall. The energy associated with the formation of a new wall requires that the

decrease associated with εms be greater than the corresponding increase in εwall.

Zeeman Energy

The Zeeman energy (εp) is the energy of the interaction between ~H and ~M [17], given

by

εp = −µ0

∫

∞~M · ~H d3r. (2.19)

~H is generated with free currents or permanent magnets. εp varies with ~H, leading

to changes in the magnetic energy and domain reorganization. εp is minimized when

~M and ~H are aligned.


H

demagnetized

state

magnetic

saturation

d0

Δd

Figure 2.7: Magnetostriction of a material with positive λs.

Magnetoelastic Energy

The magnetization of a ferromagnetic material is accompanied by a change in dimen-

sion, 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 λs, shown in figure 2.7. Measurement of magne-

tostriction takes the same form as equation 2.2, giving λs = ∆d/d0. Typical λs 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〉 = 2.1 × 10−5, while λ〈111〉 = −2.1 × 10−5.

Because of magnetostrictive anisotropy, magnetoelastic energy (ελ) is written in

terms of the saturation magnetization along specific crystalline axes; for example in


(a) magnetostriction

HH

domain

magnetization

uniaxial

stress

auxiliary

eld

(b) Villari E"ect

σ0 σ

0

Figure 2.8: The two types of magnetoelasticity: magnetostriction and the Villari effectfor a material with positive λs. (a) A change in domain structure caused by ~H producesmagnetostrictive strains and elongation parallel to the auxiliary field. (b) Uniaxial strainresults in elongation and an increased number of 180 domains.

Fe, the 〈100〉 set is used as the reference direction, giving

ελ = −3

2λ〈100〉σ0

∫

(

cos2 γ − ν sin2 γ)

d3r, (2.20)

where σ0 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, ελ < εmca,

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 λs) parallel to an applied stress σ0.


Ms

Ma

gn

eti

zati

on

, MAuxiliary Field Magnitude, H

reversibleand irreversible

wall motion

irreversible wall motion

and annihilation

domainrotation

and annihilation

H aligned to MCA direction

H misaligned withMCA directions

H misaligned withMCA directions

H aligned toMCA direction

H

Figure 2.9: The magnetization processes for samples with aligned and misaligned auxiliaryfields and preferred crystalline axes. ~H is taken from left to right. Domain configuration andcorresponding H and M values are shown for demagnetized samples brought to saturationalong their initial magnetization curves. Recall that MCA is magnetocrystalline anisotropy,and εmca causes magnetic moments to align to certain crystallographic directions.

2.3.2 Magnetization Processes

The domain configuration of a ferromagnet changes in response to shifts in the Zeeman

energy (εp ) produced by varying ~H. Increasing εp 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.


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 εwall > εms. 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 εp and

εmca 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.

2.3.3 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 εms, 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 εp. 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 (εmca) or strain (ελ).

Crystallographic texture (or simply ‘texture’) refers to a preferred distribution of


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 ελ. Consider a tensile stress producing a tensile strain along [100] in a single Fe

crystal. ελ 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], [010], [001], and [001] 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


noise, and stress-induced magnetic anisotropy. Magnetic flux leakage and stress-

induced 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.

2.4.1 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 2.10. 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 coil10 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,

10Also referred to as a search coil, pickup coil, or signal coil.


core

excitation coil

sampleV

pickup coil

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

EBN =

∫ τ

0

V 2BN dt, (2.21)

where τ is the period of the excitation signal, and VBN 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 col-

lected at regular angular intervals (typically between 5and 15 ) about a point on the

sample, and EBN is evalutated for each spectrum. Plots of EBN 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

EBN can be related to stress.

Depth sensitivity is limited for Barkhausen signals due to their high frequency.


Pic

kup

Co

il V

olt

ag

e (

mV

)

Time (ms)

Ma

gn

etic Flu

x De

nsity, B

(mT

)

0 7.75 15.5 23.25 31

0

200

400

-200

-400

300

150

0

-150

-300

B Field

Barkhausennoise

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].

rolling directionE

BN (mV2s)

mV2s

Figure 2.12: A polar plot of angular MBN energy measurements. Peak EBN values alongthe 0 -180 axis indicate the easy axis, which is in the rolling direction. Minimum EBN

values give the hard axis along 90 -270 . Taken from [25].


Signal attenuation within the sample caused by eddy currents limits the maximum

depth from which Barkhausen signals can be detected to between 0.01-1.5 mm [39]

(this attenuation is discussed further in appendix B).

2.4.2 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 2.13.

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.


Sample

MagnetN S

Flux lines

Figure 2.13: The application of magnetic flux leakage inspection in crack and corrosiondetection.

(a)

0o

0 2.5 5 7.5 10

7.5

5

2.5

(cm)

(cm

)

0o

0 2.5 5 7.5 10

7.5

5

2.5

(cm)

(cm

)

(b)

Figure 2.14: The MFL signal from a segment of SA106-B schedule 80 pipe (a) referencemeasurement 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 scaleinformation is available.


2.4.3 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 ελ, 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 ( ~Bin) and the field just

outside the sample’s surface ( ~Bout). ~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 ~Bout, 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 ~Bout and ~Bin. 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 µ1 and µ2, and relative permeability µr1

and µr2. In materials with positive magnetostriction (λs > 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 µ2 > µ1, then the magnetic field ~Bin within the

sample will be enhanced in the µ2 direction relative to µ1, as shown in figure 2.15.

When ~Bin is applied at an angle θ relative to µ2, the magnetic field ~Bout just outside

the sample will be rotated away from ~Bin by an angle δ.


μ1

μ2

δ

θ

Bin1

Bin2

Bout B

in

Figure 2.15: The rotation of the magnetic field just outside the sample ( ~Bout) relative tothe magnetic field within the sample ( ~Bin) when µ2 > µ1.

δ can be determined from trigonometry using the ratio of relative permeabilities

and angle θ. Figure 2.15 shows how ~Bin can be resolved into the principle directions

as

Bin1 = B sin θ (2.22)

and Bin2 = B cos θ. (2.23)

Using ~B = µoµr

~H and assuming the sample is surrounded by air, the components of

~Bout can be resolved into the principle directions as:

Bout1 =Bin

µr1

sin θ (2.24)

and Bout2 =Bin

µr2

cos θ. (2.25)

Dividing equation 2.24 by equation 2.25 gives the ratio of external magnetic field

components as

Bout1

Bout2

=µr2

µr1

tan θ = tan(δ + θ); (2.26)

which can be rearranged to

δ = arctan

(

(µr2

µr1− 1) tan θ

1 + µr2

µr1tan2 θ

)

. (2.27)

Langman found that equation 2.27 was a reasonable prediction of the behavior of


40

30

20

10

0

-10

-20

-30

-40

10 20 30 40 50 60 70 80

Degrees

Probe angle, φ (degrees)

Direction of Bout

relative to φ

Direction of Bin

relative to φ

Bout

Bin

Tension

μ2

φ

Figure 2.16: The orientation of ~Bin and ~Bout relative to the excitation core. The excitationcore footprint is shown by dotted lines in the inset diagram. Tensile stress was used toproduce µ2 > µ1. 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 µ2, 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 mea-

surement techniques developed different sensor configurations for use on unprepared

samples, such as Kishimoto’s magnetic anisotropy sensor [34], which employs a sens-

ing coil wound around a detecting core mounted perpendicular to the excitation core


and the sample’s surface. Modern SMA apparatus use a magnetic transducer, typi-

cally 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 sam-

ples, but SMA causes shifts ~Bout away from the excitation core so that it is detected

by the coil.

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 Mag-

netics 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 ge-

ometries. This flux control system was developed in two stages: first using only Hall

39

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 distur-

bance, 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

1There are also positive feedback systems, which sum the output and reference, but they will notbe discussed in this thesis since they were not used.


input or

referenceoutput

adder errorsignal

forward pathgain a

+

-

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 adjustthe output. The forward path converts inputs to outputs with the forward path gain a. Thefeedback path is the mechanism through which the output is fed back for comparison withthe reference. The error signal is the difference between the reference and output values.

in the environment.” When negative feedback is properly applied and operat-

ing 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


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 10 000. A minute voltage dif-

ference between the inverting and non-inverting terminals causes the output to

saturate. The op-amp adjusts the output voltage such that the external feed-

back 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].

3.2.1 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 ~Jf is proportional to the force per unit charge


adder

errorsignal

forward pathgain a

+

-

non-invertinginput (+)

invertinginput (-)

output

V+

V-

Vo

Figure 3.2: The feedback system components contained within an op-amp. The op-ampis represented by a triangle with three terminals: the non-inverting input at voltage V+, theinverting input at voltage V−, and the output termnial at voltage Vo. The output voltageis given by the forward path gain a multiplied by the difference between the inverting andnon-inverting terminals, such that Vo = a (V+ − V−).

through conductivity σe, such that

~Jf = σe

(

~E + ~vd × ~B)

, (3.1)

where ~vd 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 ~vd × ~B << ~E, thus Ohm’s law can be approximated as

~Jf = σe~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 σe.


The electromotive force (EMF) around a closed path ∂S (E∂S)is defined as

E∂S =

∮

∂S

~E · d~l, (3.4)

and the magnetic flux through the surface S (ΦS) is defined as

ΦS =

∫

S

~B · d ~A. (3.5)

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∮

∂S

~E · d~l = −

∫

S

∂ ~B

∂t· d ~A. (3.6)

Equations 3.4 and 3.5 can be applied to equation 3.6 to give

E∂S = −dΦS

dt, (3.7)

where E∂S is more commonly known as the ‘back EMF.’ In the presence of a time-

varying 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 (Vcoil) is given by

Vcoil = RI + NdΦS

dt= RI + NS

dBS

dt, (3.8)

where BS 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


Jf

B

Ex

x

zy

VH

side view

top view

x

y

z

Jf

B

VH

Ex

+ + + + + + + + + + + + + + + +

- - - - - - - - - - - - - - - - - - - - - - - - -

Ey

Figure 3.3: The Hall effect for a Cartesian coordinate system. ~B is in z. Electric currentdensity ~Jf is in x, caused by the x component of ~E, Ex. Build up of electrons along the −yfacing wall produces an electric field component in the y direction, Ey. The Hall voltage(VH) is measured across the two sides facing ±y.

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.

3.2.2 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 (VH).


Consider figure 3.3 in the absence of ~B for a metal sample: electrons flow from

right to left at drift velocity vd, generating current density Jf . With the introduction

of ~B = Bz, moving electrons are deflected in the −y direction and accumulate on

that side of the sample. Electrons continue to accumulate, increasing Ey until

Ey = vdB. (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 Ey. Thus a

Hall coefficient H is used to convert equation 3.9 to

VH = 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 typi-

cally have four terminals: two supply current (Ic) terminals for incoming and outgoing

supply current, and two Hall voltage terminals for the positive (VH+) and negative

(VH−) voltage values. The Hall voltage that appears in equation 3.10 measured across

the positive and negative voltage terminals as

VH = VH+ − VH−. (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 elec-

tromagnetic effects, which lead to slightly different applications for the two sensors.

Hall sensors measure AC and DC magnetic fields, whereas coils are insensitive to


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 there-

fore 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 FCV2.

3.3.1 Data Acquisition

A National Instruments R© PCI-6229 Multifunction DAQ (PCI-6229) was used to gen-

erate reference voltage signals (Vref ) 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 chan-

nels (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 resolu-

tion 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

2They cannot be placed inside a sample, and therefore cannot measure the flux inside a sample.


and FCV2.

3.3.2 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 SemiconductorR© LM4701 audio amplifier3 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 SemiconductorR©’s SPiKeTM protection circuitry,

which protects the op-amp from voltage, current, and thermal overload.

3.3.3 Power Supply

A Power-One R© HCC24-2.4-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 compo-

nents.

3.3.4 Hall Sensors

F.W. Bell4 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

3The LM4701 is now obsolete. It has been replaced by LM4765 and LM4781 multi-channelamplifiers.

4A division of Sypris Test & Measurement.


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. Vref is an arbitrary reference voltage waveform, and Vex and VF+

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

Rref , RG, and RF1 is zero

Vref

Rref

+VF+

RF 1+

Vex

RG

= 0. (3.12)

Resistors RF1 and RF2 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 RF1 and RF2, the voltage across the feedback


V F+

Vref

V ex

Rref

RF2

RG

RF1

Figure 3.4: A sketch of White’s FCS. The resistor RG was used to limit gain to providea stable output. A feedback coil wound around one of the poles acted as the flux feedbacktransducer.

coil (VF ) is given by

VF =

(

1 +RF2

RF1

)

VF+. (3.13)

Solving equations 3.12 and 3.13 for Vref gives

Vref = −

(

Rref

RF1 + RF2

)

VF −

(

Rref

RG

)

Vex. (3.14)

This system was designed to control flux independent of probe liftoff; therefore the

contribution of Vex to Vref was problematic, as Vex was unknown. Ideally, the gain-

limiting resistor RG could be removed (effectively setting RG → ∞), which would

nullify the contribution of Vex to equation 3.14. 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, Vex will

gradually shift to the voltage supply rails. To avoid this issue, White maximized the

system’s performance by decreasing RG from ∞ until the circuit stabilized.

The reference voltage (Vref ) and the voltage across the feedback coil (VF ) differed

by the Vex term as

Vref = VF −

(

RF1 + RF2

RG

)

. (3.15)


The error introduced by a non-infinite RG was reduced by a digital error correc-

tion (DEC) algorithm implemented in the FCS software control and data acquisition

system.5 The DEC algorithm iteratively adjusted the reference voltage (Vref ) until

the target feedback voltage (VF ) 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 mea-

surements 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.

3.5.1 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.

5This DEC system designed by White is only valid for periodical reference waveforms.


ferriteexcitationcore

excitationcoil

sample

LM4701

+

-

Ic

RsV

ref

lift-o spacer

VH+

VH-

Vmc

Vs

Vex

, IexF

GND

Hall sensor

monitoring coil

Figure 3.5: A simplified version of FCV1. A Hall sensor with supply current Ic is locatedin a plastic lift-off spacer attached to the bottom of the ferrite excitation core. This sensormeasures the B component normal to the sample surface, and feeds back a Hall voltage tobe compared to Vref .

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). Vref

is the user-defined reference voltage corresponding to the desired B in the air gap.

Note that in figure 3.5, VH− is grounded, therefore VH = VH+. The voltage Vs across

a 0.2 Ω series resistor (Rs = 0.2 Ω) was monitored to measure excitation current Iex.

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


excitation coilturns 1052

inductance 0.26 Hresistance 27 Ω

monitoring coilturns 37

inductance 8.7 mHresistance 1.1 Ω

ferrite excitation core pole area 264 mm2

Table 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 itsarea the same as the pole area.

Variable PCI-6229 TerminalConnection Configuration

Vref AO0 RSEVH AI0 NRSEVs AI1 NRSEVmc AI2 DIFF

GND AIsense

Table 3.2: PCI-6229 I/O assignment and terminal configuration for FCV1. Terminal con-figurations use the following abbreviations: referenced single-ended (RSE), non-referencedsingle-ended (NRSE), differential (DIFF). For additional information on terminal configu-rations see [29].

the ferrite core to monitor the FCV1 performance when operated in AC mode. The

output voltage from this coil was designated Vmc. 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.


3.5.2 FCV1 LabVIEWR© Interface

FCV1 was controlled through a basic LabVIEW R© 8.5 user interface (UI), which

was developed as part of the thesis work. Vref 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.

3.5.3 FCV1 Performance

The performance of FCV1 was examined by determining how effectively it was able

to produce an excitation field, and corresponding Hall voltage VH , that was equal to

the reference voltage waveform Vref . The Hall sensor and monitoring coil were used

to measure the excitation field. Figure 3.6 shows VH , Vref and Iex for a sinusoidal

reference voltage with an amplitude of 50 mV and a frequency of 10 Hz (Vref =

50mV sin(2πt10Hz)). Hall voltage (VH) exactly matches the reference voltage (Vref ),

making the two signals difficult to distinguish in the figure. However, VH contains a

significant noise component that is not present in Vref . This noise is also apparent in

the excitation current (Iex) waveform.

The amount of noise observed in VH and Iex 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 Vref waveform and would occur in both AC (figure 3.6)and DC

Vref (figure 3.7) system modes. Figure 3.7 indicates that noise noise in Iex peaks

at approximately 2 mA. Since the ferrite core saturates at an excitation current of


0 0.02 0.04 0.06 0.08 0.1

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

FCV1 Performance for Vref = 50 mV sin(2πt10 Hz)

Time, t (s)

Volt

age

(V)

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

Vref

VH

I ex

Curren

t,(A

)E

xcitation

Iex

Figure 3.6: Hall voltage (VH) and excitation current (Iex) for a sinusoidal reference voltage.VH lies on Vref , 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 (Vmc) 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 Vmc, Iex and VH for each of three Vref waveforms (each plot

represents a different reference waveform). The noise component of VH and Iex is not

obvious at this scale, but is clearly visible in Vmc. A fast Fourier transform (FFT) of

Vmc performed in LabVIEW R© using the spectral analysis tool6 indicated a dominant

noise frequency of 700 Hz. When the excitation core was removed from the sample, it

6The ‘Spectral Measurements’ express VI.


FCV1 Performance for Vref = 0mV

Time, t (ms)

Exci

tati

on

Curr

ent,

I ex

(mA

)

0 20 40 60 10080

3

2

1

0

-1

-2

-3

Figure 3.7: FCV1 response to a DC reference voltage of Vref = 0. Only the excitationcurrent waveform is shown. VH was omitted for clarity.


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 (Vex).

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.

3.5.4 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

(VH) is directly proportional to the magnetic field density (B) through the transducer.

These two values are linked by a Hall constant (H) such that VH = HB.

The magnetic field (Bex) generated by an excitation coil is proportional to the

current (Iex) through the coil, giving

Bex ∝ Iex. (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

VH ∝ Iex ∝ Bex. (3.17)

Therefore, a system using Hall voltage as the feedback mechanism must control the

excitation current to reliably regulate Bex. 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


0 0.02 0.04 0.06 0.08 0.1- 0. 2

- 0. 1

0

0.1

0.2

0.3

Time, t (s)

0 0.02 0.04 0.06 0.08 0.1- 0. 4

- 0. 2

0

0.2

0.4

Time, t (s)

Volt

age

(V)

Vref = 95 mV sin(2πt20 Hz)

0 0.02 0.04 0.06 0.08 0.1- 0. 4

- 0. 2

0

0.2

0.4

Time, t (s)

Volt

age

(V)

Vref = 50 mV sin(2πt30 Hz)

Vref = 95 mV sin(2πt10 Hz)Volt

age

(V)

0.1

0.05

0

- 0.05

- 0.1

0.2

0.1

0

- 0.1

- 0.2

0.2

0.1

0

- 0.1

- 0.2

0.3

Curren

t,(A

)E

xcitation

Iex

Curren

t,(A

)E

xcitation

Iex

Curren

t,(A

)E

xcitation

Iex

Vmc

VH

I ex

(a)

(b)

(c)

Figure 3.8: Monitor coil voltage Vmc boosts the noise amplitude relative to the excitationfield. Waveforms for three different sinusoidal reference voltages are shown: two 95 mVsignals 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 Iex values.


to the excitation voltage (Vex) such that

dIex

dt∝ Vex. (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 (Vfc) wound around the pole of the

excitation core is then proportional to time-derivative of the flux through the core,

giving

Vfc ∝dBex

dt. (3.19)

The proportionality arguments of equations 3.16, 3.18 and 3.19 lead to

Vfc ∝dBex

dt∝ Vex. (3.20)

This direct proportionality between the feedback coil signal Vfc and excitation coil

voltage Vex 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 proper-

ties 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.


3.6 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 ampli-

fiers 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.

3.6.1 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 Vfc, and also (as

with FCV1) in the air gap between the core and sample by a Hall sensor, producing

VH+ and VH−. 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

7See [39] p. 73.


ferriteexcitationcore

excitationcoil

sample

LM4701

+

-

Ic

Rs

Vref

lift-o spacer

VH+

VH-

Vfc

Vs

Vex

, IexF

GND

VH+

VH-

Vfc

Rref

RH

RH

Rfc

RG

RG

V+

V-

Hall sensor

feedbackcoil

Figure 3.9: A simplified version of FCV2. There are only a few changes from figure 3.5.Vfc has been integrated into the feedback circuit, and one end of the feedback coil wasgrounded. Neither VH+ or VH− was grounded.


the inverting (-) and non-inverting (+) terminals to draw no current, the voltages at

each terminal, V− and V+ respectively, are given by

V+ =RG(VrefRH + VH−Rref )

RHRG + RrefRG + RrefRH

(3.21)

and

V− =RG(VfcRH + VH+Rfc)

RHRG + RfcRG + RfcRH

. (3.22)

Negative feedback was used, therefore the voltage at both input terminals must be

the same, giving

V+ = V−. (3.23)

When RG = RH = Rfc = Rref , equations 3.21, 3.22, and 3.23 can be solved for Vref

to give

Vref = Vfc + VH+ − VH− = Vfc + VH . (3.24)

Equation 3.24 can be written in terms of the excitation field Bex(t), feedback coil

turns (Nfc), feedback coil area (Afc) and Hall constant H, such that

Vref (t) = NfcAfcdBex(t)

dt+ HBex(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 (Ic) 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 VH+ and

VH− within the PCI-6229’s ±10 V input range.8 Resistors in the feedback system

8VH was always less than 1 V, but VH+ and VH− had to be within ±10 V of ground for thePCI-6229 to make the differential measurement VH = VH+ − VH−.


In

Out

Adj

LM337

BH-700

+Ic

-Ic

VH+

VH-

Red

Blue

Yellow

Black-

LM4701

+

1 μF

-24 V

1 μF

+24 V

Vref

VH+

VH-

1 kΩ

1 kΩ

1 kΩ

1 kΩ

1 kΩ

-24V

10 Ω1μF

1 kΩV

fc

1

23

1

2

3,54

8

7

180 Ω

0.2 Ω

Ic

10 kΩ 1 kΩ

10 MΩ

VsF = 0.5A R

ex Lex

Rfc

Lfc

Vex

111

0.2 Ω

excitation coil

feedback coil

feedback Hall sensor current supply

Figure 3.10: An electrical schematic of FCV2 showing the feedback system and the Hallsensor 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-controlmode. 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-6229DAQ (Vref , VH−, VH+, Vfc, Vs, Vex).

were set to 1kΩ, while the fuse and series resistance (Rs) were unchanged from FCV1

(F = 0.5 A and Rs = 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 |Vex| = 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


Variable PCI-6229 TerminalConnection Configuration

Vref AO0 RSEVH AI0 DIFFVs AI1 NRSEVfc AI2 DIFFVex AI3 NRSEVsig AI4 DIFF

GND AIsense

Table 3.3: PCI-6229 I/O assignment and terminal configuration for FCV2.

3.3. A sensor input line that recorded the magnetic detector output signal (Vsig) was

added as a differential input.

3.6.2 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 Vfc, VH , and subsequently Vref for a user-specified excitation magnetic field

density (Bex). The transition from the user-specified reference voltage used in FCV1

to Bex for FCV2 was done to highlight the relationship between Vfc, VH , and Bex

shown in equation 3.25.

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


matured to the point of performing a measurement.

3.6.3 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 (Bfc) and Hall sensor (BH), and how they compare to the reference excitation

magnetic field density (Bref ). The matching is excellent, with the slight offset between

BH and Bref due to miscalibration of the Hall constant, which was an adjustable

parameter in FCV2’s software.

Offsets between Bfc, BH and Bref peak at the maximum and minimum of the

reference waveform, shown in figure 3.11(b). Bfc is closely matched to Bref , with

a maximum offset of |Bfc − Bref | = 0.15 mT. BH deviates further from the refer-

ence field, with a peak deviation of 2.5 mT at Bref = 100 mT. Additionally there

is a -0.75 mT shift in BH with respect to Bref . 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.


0 5 10 15 20−100

−80

−60

−40

−20

0

20

40

60

80

100

Time, t (ms)

Magnet

icFie

ldD

ensi

ty(m

T)

FCV2 Performance for Bref = 100 mT sin(2πt55 Hz)

BrefBHBfc

0 5 10 15 20

Time, t (ms)

Magnet

icFie

ldD

ensi

ty(m

T)

Hall Sensor and Feedback Coil Magnetic Field

BHBfc

BrefBref

--

1.5

1

0.5

0

-0.5

-1

-1.5

-2

-2.5

-3

(a)

(b) Offsets

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. Bref and Bfc

curves lie on top of each other, making them nearly indistinguishable. Offsets, shown in thebottom figure (b), were calculated by subtracting measured field density (Bfc, BH) fromBref .

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 configu-

rations 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

CHAPTER 4. MAGNETIC STRESS DETECTORS 68

(b)(a)

(c)

wire coil

Hall detector

measured B component

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 wirecoil with its axis parallel to the surface normal. (c) SMA: a wire coil with the coil axisperpendicular 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 evalu-

ate 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


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.

4.1.1 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 = 0.278 [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-06-250BF300

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 indi-

cated in figure 4.2. This location was selected to avoid any stress concentrations or


500

216

125

108

94.5

271

8

applied tensile stress σt

in the ‘parallel’ direction

in t

he ‘

perp

en

dic

ula

r’ d

irecti

on

measurement

location

strain

gage 1

strain

gage 2

strain

gage 3

co

mp

ressiv

e s

tress σ

c =

-νσ

t

Figure 4.2: The mild steel plate used to test different detector configurations. Strain gagelocations are shown by rectangles. Gage 1, indicated by a dashed line, was located on theunderside of the plate. Gages 2 and 3 were located on the upper plate surface. Two 18 mmhole 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.

4.1.2 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 (σt) to the sample clamped in the jaws. As a result of


xed bridge

sample jaws

sample

gliding bridge

guidance rods

spacer cylinder

hydraulic jack

support beam

Figure 4.3: A schematic of the single axis stress rig used to introduce tensile stress in theflat plate sample.

Poisson’s ratio effects, compressive stress is also generated across the width of the

sample (σc) given by

σc = −νσt. (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].

4.1.3 Strain Measurement

The three EA-06-250BF300 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.


64 mm

R 9.2 mm

feedback coil

ferrite excitation core

excitation coil

detector mount assembly

outer brace

connector brace

feedback Hall sensor

housed within a lifto! spacer

lifto! spacer

detector mount assembly

detector brace

(a) (b)

Figure 4.4: An assembled probe showing a detector mount assembly attached to theconnector brace of the excitation core. (a) Important components are indicated in thefigure. The spring of the detector mount is not visible in this figure; it is hidden betweenthe detector brace and outer brace. (b) A photo of the assembled system, built by theauthor.

4.2 Detectors, Data Acquisition and Data Analy-

sis

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)


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 processed1 output signal from this detector was denoted VDCM .

AC MFL

An air-core 200 turn coil wound from 44 AWG wire with an average loop area of

3.1 mm2 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 VACM .

SMA

The SMA detector was a rectangular air-core 69 turn coil wound from 44 AWG wire

with an area of 2.86mm2. 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.

VSMA denoted voltage of this detector after signal processing.

1The output voltage signals of the detectors were acquired as Vsig 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.


AC MFLDC MFL SMA

Figure 4.5: DC MFL, AC MFL, and SMA detectors mounted to the excitation core. Thedetectors were locked into their mounts with epoxy. The detectors are not shown to scale.

4.2.1 Data Acquisition

Signals from the three different detectors were acquired by the PCI-6229 DAQ as Vsig,

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 measure-

ments were input directly to the PCI-6229 as a differential measurement across the

VH+ and VH− terminals of the Hall detector. AC MFL and SMA signals were ampli-

fied by an Ithaco R© Model 565 preamplifier in transformer mode, producing a gain of

60 dB prior to being input into the PCI-6229.


AC MFL and DC MFL

tensile measurement

Bex

AC MFL and DC MFL

compressive measurement

Bex

Bex

SMA measurement

0o

90o

180o

270o

applied tensile

stress σt σ

tσt

com

pre

ssiv

e s

tre

ss σ

c

σc

σc

Figure 4.6: The footprint of the excitation core on the sample for AC MFL, DC MFL andSMA measurements. The direction of ~Bex is indicated by white lines between the excitationcore poles.

4.3 Experimental Procedures for Testing and Com-

parison 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 combina-

tions: 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 dia-

grams of figure 4.6, AC MFL and DC MFL measurements were performed with the

probe parallel (called the ‘parallel’ configuration) and perpendicular (called the ‘per-

pendicular’ configuration) to the applied stress. The parallel configuration enabled a

measurement of tensile strain sensitivity, while the perpendicular configuration was

a measurement of compressive strain2.

SMA measurements were performed by rotating the probe over 360 about the

2The SASR could only produce tensile strain. The Poisson effect was exploited to examine ACMFL and DC MFL compressive sensitivity.


measurement location, stopping to make measurements at 15 intervals. Thus, each

‘stress measurement’ consisted of 25 data sets3. Taking φ as the angle between the

probe and the direction of tensile stress, φ = 0, 180, 360 probe orientations aligned

the probe along σt, the applied stress direction, while φ = 90, 270 aligned the

probe with the largest compressive stress (σc) direction. φ was taken to increase

anti-clockwise from zero.

All measurements were performed within the elastic deformation range of the

sample. A degaussing cycle4 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 (Vsig) waveforms. This section describes the

method by which raw Vsig waveforms were processed to produce VDCM , VACM and

VSMA values. Each detector/probe system is considered in this section.

4.4.1 DC MFL

DC MFL measurements were performed using the probe with an excitation field mag-

nitude of 100 mT on flat plate samples in the SASR. A DC MFL Hall detector voltage

(VDCM) measurement was recorded for increasing stress values up to a maximum ten-

sile stress of 107 MPa.

3There were a total of 25 waveforms recorded for each SMA measurement because data wasacquired at both φ = 0 and φ = 360.

4A degauss cycle removes residual magnetization from the sample by cycling through hysteresisloops of decreasing magnitude.


The configuration of the FCV2 software interface required that the DC excita-

tion 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

Vsig distributions’ consisting of 3333 Vsig data points recorded over a 67 ms window.

3333 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.

VDCM was taken as the average signal voltage of a DC MFL Vsig distribution.

Uncertainty in VDCM was calculated using the standard method for uncertainty in a

mean5.

Figure 4.7 shows VDCM for both parallel (Bex ‖ σt) and perpendicular (Bex ⊥ σt)

orientations of the excitation field. Linear trend lines and their associated equations

for σt in MPa and VDCM in mV are also shown.

Both data sets demonstrate that VDCM is proportional to applied stress. As seen

in figure 4.7, when Bex is parallel to the applied stress direction, increasing stress

causes the signal to decrease. When Bex 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 VDCM ; 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

5Consider N measurements of x with the same uncertainty in each measurement. The meanmeasurement (x) is given by x = N−1

∑

x. The error in the mean (σx) is then σx = σN−1/2,where σ is the standard deviation of the N measurements of x. See reference [36] for additionalinformation.


−40 −20 0 20 40 60 80 100 120

DC MFL Stress Sensitivity

DC

MFL

Sig

nal,VDCM

(mV

)

Applies Stress, σ (MPa)

fit

fit

20.35

20.4

20.45

20.5

20.55

20.6

20.65

20.7

20.75

20.8

20.85

t

t

Figure 4.7: DC MFL measurements for Bex ‖ σt and Bex ⊥ σt. For Bex ‖ σt measure-ments, stress σt ranged from 0 to 107 MPa. In Bex ⊥ σt measurements, compressive stressσc varied between 0 and −34 Mpa. Linear fits and their associated equations are shown foreach data set. Error bars for the perpendicular data points appear as vertical lines throughthe circles.


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 quan-

titative measurement of stress somewhat problematic using this method.

4.4.2 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

Bex = 100 mT sin(2πt55 Hz). (4.2)

VACM 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.

Vsig 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 Bex waveform. Vsig

waveforms were cosine waves, which was expected from a sinusoidal Bex excitation

field.

Vsig waveforms were expected to be functions of applied stress (Vsig(σ)). They were

fit in MATLAB R© to a sinusoidal function with three degrees of freedom according to

Vsig(σ) = Af (σ) sin(2πt55 Hz + Bf ) + Cf , (4.3)

where Af (σ) is the amplitude, Bf is phase and Cf is offset. Phase and offset were

expected to be constant at Bf = π/2 radians and Cf = 0. Amplitude was the only

parameter expected to be affected by the applied stress on the sample, as Af (σ) was

directly proportional to the flux density passing through the AC MFL signal coil,


0 5 10 15 20−125

−100

−75

−50

−25

0

25

50

75

100

225

Vsig for the Parallel AC MFL Measurement at Null Stress

Time, t (ms)

−250

−200

−150

−100

−50

0

50

100

150

200

250

Sig

nalV

olta

ge,

Vsig

(mV

)

Exci

tation

Fie

ld, Bex (

mT

)

BexVsig

Figure 4.8: The excitation field (dashed line) and signal voltage (solid line) for an ACMFL measurement at zero applied stress. The probe was in the parallel orientation for thismeasurement.


therefore fit amplitude was taken as the AC MFL signal voltage, giving

VACM = Af (σ). (4.4)

Figure 4.9 shows VACM for both parallel and perpendicular probe configurations

(Bex ‖ σt and Bex ‖ σc respectively). Uncertainty in VACM 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 VACM 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.

4.4.3 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 Vsig waveforms to VSMA values. Before unprocessed

Vsig waveforms can be presented, it is necessary to expand on some of Langman’s

work presented in section 2.4.3.

SMA: A Theoretical Development of Angular Dependence

The SMA probe was rotated about a point, thus capturing information regarding the

direction and magnitude of stress6. Langman described the angle between magnetic

fields inside ( ~Bin) and outside ( ~Bout) the sample (designated δ) as a function of the

relative permeabilities along two perpendicular principle magnetic directions (µr1 and

6While AC and DC MFL measurements were recorded with the probe in two different configu-rations (parallel and perpendicular to applied stress), this was done to examine the measurements’sensitivity to compressive stress.


−40 −20 0 20 40 60 80 100 120 140

191

192

193

194

195

196

197

198

199AC MFL Stress Sensitivity

AC

MFL

Sig

nal,VACM

(mV

)

Applies Stress, σ (MPa)

t

Figure 4.9: AC MFL measurements for Bex ‖ σt and Bex ‖ σc. For Bex ‖ σ measurements,stress ranged from 0 to 128 MPa. In Bex ⊥ σ measurements, stress varied between 0 and123 Mpa.


μ1

μ2

δ

θ

Bin1

Bin2

Bout

Bin

φ

Figure 4.10: A modified figure 2.15 redrawn for reference. The excitation core footprintis indicated by dotted lines.

µr2, with µr2 > µr1), and the angle of the excitation field density relative to the µ2

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

µr2, and µr1 corresponds to the perpendicular direction. Equation 2.27 was given as

δ = arctan

(

(µr2

µr1− 1) tan θ

1 + µr2

µr1tan2 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 µr2, indicated in figure 4.10). A reasonable description

for this relationship was found to be

θ = φ +1

3arctan

(

(µr1

µr2− 1) tan φ

1 + µr1

µr2tan2 φ

)

, (4.5)

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.


Both ~Bin orientation (θ) and probe orientation (φ) were measured relative to the

µ2 direction, which accounts for the separate φ term in equation 4.5. The arctan(...)

component shifts θ from φ toward µ2. The factor of 1/3 was selected based on the

angles between φ, ~Bin and ~Bout shown in figure 2.16.

The two principle magnetic directions (µr1 and µr2) cause the magnitude of Bout

to depend on the orientation of the internal magnetic field. This relationship can be

expressed as

Bout =

√

(

Bin1

µr1

)2

+

(

Bin2

µr2

)2

. (4.6)

Using equations 2.22 and 2.23, equation 4.6 can be rearranged to

Bout

Bin

=

√

(

sin θ

µr1

)2

+

(

cos θ

µr2

)2

, (4.7)

which gives the magnitude of Bout per unit Bin.

An SMA sensing coil voltage signal will be a function both the magnitude, orien-

tation, and rate of change of ~Bout, as well as probe properties (number of turns and

coil area). Thus the SMA signal voltage in terms of Bin is

V = NAG∂

∂t(Bin) , (4.8)

where the G term is called the ‘geometry factor.’ G compensates for the different

magnitudes and orientations of ~Bin and ~Bout, 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

G =

[

(

sin θ

µr1

)2

+

(

cos θ

µr2

)2]1/2

sin (θ + δ − φ) , (4.9)

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


Geo

met

ry F

act

or,

G

Probe Angle, φ (deg)

0 50 100 150 200 250 300 350

0.3

0.2

0.1

0

-0.1

-0.2

-0.3

μr2/μ

r1 = 2

μr2/μ

r1 = 3

μr2/μ

r1 = 4

μr2/μ

r1 = 1

Figure 4.11: G for four µr2/µr1 ratios. The 0 , 180 , and 360 probe orientations placethe probe parallel to the µ2 direction.

the magnitude of Bout from Bin, it was taken directly from equation 4.7. The sin(...)

term extracts the component of ~Bout 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 µr2/µr1 ratios. For the case of µr2/µr1 = 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 µr2/µr1. Peak G values occur when the probe

is shifted 45 anti-clockwise from the direction of greatest permeability.


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

Vsig waveforms collected for each increment. Vsig waveforms, functions of both stress

(σ) and probe orientation (φ), were fit in MATLAB R© to the equation

Vsig(σ, φ) = Af (σ, φ) sin(2πt55 Hz + Bf ) + Cf , (4.10)

where Cf is fit offset, Bf is fit phase, and Af (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 Af (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 Af (σ, φ) can be seen in the first 90of 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 Af (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 direction7, which agrees with previous studies performed in these

plates [24]. Increasing tensile stress along the parallel direction causes a corresponding

7Recall that peaks in the anisotropy signal occur 45 anti-clockwise from the direction of greatestpermeability.


−50 0 50 100 150 200 250 300 350 400

95

100

105

110

115

95

100

105

110

115

95

100

105

110

115

95

100

105

110

115

SMA Vsig(σ, φ) Amplitude for Bex = 100 mT sin(2πt55 Hz)

Fit

Am

plitu

de,

Af(σ,φ)

(mV

)

Angle, φ (deg)Probe

σ = 0 MPa

σ = 61.6 MPa

σ = 97.7 MPa

σ = 129 MPa

Figure 4.12: Vsig(σ, φ) fit amplitudes for SMA measurements. Each data set consists of 25points recorded at 15 intervals between 0 and 360. 0 , 180 , and 360 probe orientationsplaced the probe (and excitation field) parallel to the applied stress.


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 VSMA, the Vsig amplitude waveforms shown in figure 4.12 were fit to

a 180-periodic sine function according to

Af (σ, φ) = Af2(σ) sin

(

2π

180φ + Bf2

)

+ Cf2. (4.11)

As with the initial fit (see equation 4.10), the phase (Bf2) and offset (Cf2) 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

VSMA = Af2(σ). (4.12)

Figure 4.13 shows the SMA signal voltage (VSMA) obtained from fits to equation

4.11. 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 4.12.

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.


−20 0 20 40 60 80 100 120 140

−15

−10

−5

0

5

10

15

Applied Stress, σ (MPa)

SM

ASig

nalV

olt

age,

VSMA

(mV

)

Anisotropy Signal from a Mild Steel Plate

initial measurement

repeated measurement

Figure 4.13: SMA measurements for tensile up to 130 MPa.


4.5 Selected Detector

Of the three detectors tested, only the AC MFL coil demonstrated no stress sensitiv-

ity. 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: VDCM decreased with the

probe parallel to tensile stress, and increased with the probe oriented along com-

pressive 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 Af (σ, φ)), as well as the VSMA 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.

Chapter 5

Proposed Design: The Magnetic

Anisotropy Prototype Probe

Previous chapters described the testing of different feedback systems and sensor con-

figurations with a general-use excitation core. The general-use core was relatively

large (the pole area was 264 mm2 with a back spine length of 64 mm) and was tested

on flat plate samples. This chapter describes the features of a prototype probe devel-

oped specifically for use on CANDUR© feeders, termed the Magnetic Anisotropy Pro-

totype (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

CHAPTER 5. PROPOSED DESIGN: MAP PROBE 92

5.1 Magnetic Anisotropy Prototype (MAP) Probe

The optimized probe for stress measurement in feeders, termed the Magnetic Anisotropy

Prototype probe (MAP probe), consisted of a Supermendur1 U-core excitation elec-

tromagnet 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 CANDUR© 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

15.76 mm and a footprint of 38.18 mm2, 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.


4.01

16.99

11.75

4.01

9.52

R 4.19

Figure 5.1: A schematic of the Supermendur core of the MAP probe. The core is shownto scale. All dimensions are in millimeters.

aluminum mount

stainless

steel disk

feedback coil

excitation coil

SMA detector coil

feedback

Hall sensor

connector brace

spring

lifto! spacer

Supermendurexcitation core

detector

coil piston

(a) (b)

Figure 5.2: (a) A diagram of the MAP probe. Only the corner of the Supermendur coreof figure 5.1 is visible; it is shown in black. Connector pin 1 corresponds to the bottomright pin, connector pin 12 corresponds to the top left pin. (b) A photograph of probe setin mounting hardware.


Excitation and feedback coils were wound using MWS Wire Industries PN Bond2

magnet wire and placed on each pole of the excitation core. A thin layer of Teflon tape

was placed between the core and coils to protect the coils; it appears as a white band

between the feedback coils and liftoff spacers in figure 5.2. Both the excitation and

feedback coils were wound in pairs: the excitation system was made of two 36 AWG

250 turn coils, the feedback system consisted of two 36 AWG 25 turn coils. Excitation

and feedback coils were placed on each pole of the excitation and connected in series

effectively creating one 500 turn excitation coil and one 50 turn feedback coil (from

this point onward these coils will be referred to in the singular). The SMA detector

coil was mounted in a spring-loaded piston to ensure repeatable SMA coil coupling

with the sample. The SMA coil used in the MAP probe was 200 turns, wound out of

44 AWG PN bond wire around a ferrite core. Table 5.1 summarizes the properties of

the excitation, feedback and SMA coil properties used in the MAP probe.

The SMA coil was encased in epoxy within a plastic piston assembly. As mentioned

previously, a small spring (Gardner Spring part 36000G) pushes against the back of

the piston to ensure repeatable detector coil coupling to the sample. The piston is

housed in a brass mounting bracket that connects it to the Supermendur core and

connector brace.

The connector brace was machined out of plastic and contained 12 pin Tyco Elec-

tronics AMP male connector. The connection diagram for this terminal is given in

figure 5.3.

2PN Bond wire is an insulated copper wire with a superimposed film of thermoplastic bondingmaterial. Heat or solvent will cause the bonding layer to soften and fuse layers of wire together.This allows coils to be wound in unusual shapes or on jigs since the coils are bonded turn to turn.


excitation coil

turnstwo 250 turn coils in series

2 × 250 = 500resistance 23.5 Ωwire gage 36 AWG PN Bond

core material Supermendurcoil area 4.5 mm × 10 mm = 45 × 10−6 m2

feedback coil

turnstwo 25 turn coils in series

2 × 25 = 50resistance 2.15 Ωwire gage 36 AWG PN Bond

core material Supermendurcoil area 4.5 mm × 10 mm = 45 × 10−6 m2

anisotropy coil

turns one 200 turn coilresistance 16.9 Ωwire gage 44 AWG PN Bond

core material Ferritecoil area 2 mm × 2 mm = 4 × 10−6 m2

Table 5.1: MAP probe properties. Feedback and excitation coils were wound on anexternal forming rig, which is why their area differs from the Supermendur core footprint.

1

12

1

2

3

4

5

6

7

8

9

10

11

12

VH-VH+

-Ic+Ic

Vex-Vex-

Vfc+Vfc-

NCNC

Vsig+Vsig-

Figure 5.3: The pin diagram for the MAP system. NC indicates no connection at thatterminal.


5.2 MAP Probe Testing with SA-106 Grade B Pipe

To evaluate the stress sensitivity of the MAP probe, it was clamped to a length of

2.5 inch nominal diameter SA-106 grade B pipe mounted in a three-point bending rig

(3PBR). The bending rig and sample are described in section 5.2.1, while results are

given in sections 5.2.2 and 5.2.3.

5.2.1 The Three-Point Bending Rig (3PBR) and SA-106 Grade

B Sample

The Three-Point Bending Rig (3PBR), shown in figure 5.4, was designed by Steven

White to apply compressive and tensile axial loads to a 3.18 m long, 2.5 inch nominal

diameter Schedule 80 SA-106 grade B pipe, hereafter referred to as the SA-106 sample,

using three 6 ton bottle jacks mounted on a steel I-beam. The top of the pipe directly

above the middle jack was taken as the surface origin (0 cm, 0) in (axial, hoop)

coordinates, according to the coordinate system indicated in figure 5.4.

The application of tensile and compressive axial (σa) loads required different tow-

ing strap configurations. In the tensile configuration, shown in figure 5.4, the SA-106

sample was strapped at its ends and a tensile axial stress state (σa > 0) at the

measurement location (top surface) was achieved by raising the middle jack. In the

compressive configuration (not shown) the pipe was strapped at the center with two

towing straps. Compressive axial stress (σa < 0) was applied by raising the two outer

jacks. Hoop stress (σh) was generated for both configurations by Poisson effects.

Four Vishay R© EA-06-250BF350 general purpose 350Ω strain gages with gage fac-

tors of 2.100 ± 0.5% and a transverse sensitivy of 0.0 ± 0.5% were mounted to the


radial

hoop

axial

6-ton bottle jacks

towing straps

I-beam

SA-106 Grade B pipe

surface origin(0 cm, 0o)

strain gauges

measurement location(20.0±0.5 cm, 0±5o)

Figure 5.4: A schematic of the three-point bending rig in the tensile configuration. Com-ponents and key locations are indicated. The pipe surface origin is labeled (0 cm, 0) in(axial, hoop) coordinates. This figure has been adapted from reference [39].


sample to measure hoop and axial strain. The two axial strain (εa) gages are centered

at (9.1 cm, 0) and (30.2 cm, 0). The two remaining gages are oriented to meausre

hoop strain (εh) and are centered at (11.5 cm, 0) and (28.5 cm, 0). The surface of

the pipe surrounding and between the strain gages was sanded to a smooth finish to

allow proper gage mounting. The strain gages were connected to a Vishay R© Measure-

ments Group SB-10 Balance and Switch for calibration and switching. The SB-10

was connected to a Vishay R© Measurements Group P3500 Strain Indicator.

SA-106 grade B piping has a minimum specified yield strength of 240 MPa, a

Young’s modulus of Y = 202.7 GPa at 21 C and a Poisson’s ratio of ν = 0.3 [1], [39].

The generalized form of Hooke’s law for an isotropic material (refer to equation 2.3)

in cylindrical coordinates gives the axial stress (σa) as

σa =Y

1 + ν

[

εa +ν

1 − 2ν(εa + εh + εr)

]

, (5.1)

and hoop stress (σa) as

σh =Y

1 + ν

[

εh +ν

1 − 2ν(εa + εh + εr)

]

, (5.2)

where εr is radial strain. No strain gage was mounted to record εr; however, previous

neutron diffraction studies have shown it to be small compared to εa and εh [39],

therefore εr = 0 provided a reasonable simplification. εa and εh were recorded by

the axial and hoop strain gages described above. The MAP probe was placed in the

middle of the strain gages at (20.0 ± 0.5 cm, 0 ± 5), indicated in figure 5.4. Linear

interpolation was used to approximate strain at the measurement location.


5.2.2 SMA Excitation Field Response

The 3PBR was used to characterize MAP anisotropy signal excitation field response.

The MAP probe was mechanically clamped to the SA-106 sample at (20.0 ± 0.5 cm,

0 ± 5) in (radial, hoop) coordinates and the pressure valves of each jack in the

3PBR was released, bringing the system to a zero applied stress state. Excitation

waveforms used with the MAP probe were sinusoidal with amplitude A and frequency

f , described by:

Bex = A sin (2πf) . (5.3)

The excitation frequency f affects two parameters of the SMA signal: amplitude and

skin depth. As with any wire coil, SMA signal amplitude increases with the time-

rate of change of the detected field (∂Bex/∂t). Skin depth (δ) refers to the depth to

which an electromagnetic field propagates within a conductor, and is defined as the

distance at which wave amplitude decreases to 1/e ≈ 0.368 of the value at the sample

surface. This attenuation is caused by ohmic losses within the conductive medium

and is discussed in detail in appendix B. Skin depth for a typical ferromagnetic steel

is given by:3

δ = 1.59 × 10−2m.s0.5√

f. (5.4)

An excitation frequency of f = 55 Hz gives a skin depth of δ = 2.15 mm. The

magnetic field is 95% attenuated at 3δ = 6.45 mm, which covers the majority of the

7.01 mm pipe wall thickness of the SA-106 sample. Lower frequencies would penetrate

deeper into the sample, but produce lower amplitude voltage response in the SMA

detector coil. An excitation frequency of f = 55 Hz was found to provide a good

3Assuming a conductivity (σe) of σe = 107 Ω−1m−1, and a relative permeability of µr = 100.


0 20 40 60 80 100 120 1400

20

40

60

80

100

120

140

160

180

SA-106 Sample Anisotropy

Excitation Field Amplitude, A (mT)

SM

ASig

nalV

olt

age,

VSMA

(mV

)

0 20 40 60 80 100 120 1400

1

2

3

4

5

6

7

8

SA-106 Sample Signal Error to Signal Ratio

Excitation Field Amplitude, A (mT)

(a) (b)

Per

cent

Unce

rtai

nty

(%

)

Figure 5.5: SMA dependence on excitation field amplitude for σa = 0. VSMA was calcu-lated using the same method described in section 4.4. (a) The anisotropy signal VSMA. (b)The percent error associated with each measurement.

balance of penetration and signal amplitude.

The relationship between SMA signal voltage (VSMA) and excitation field ampli-

tude (A) was characterized by performing SMA measurements with the MAP probe

on the SA-106 sample at excitation field amplitudes varying from A = 25 mT to

125 mT in in 25 mT increments. The maximum excitation field amplitude of 125 mT

was determined by the current capacity of the 36 AWG excitation coils used in the

MAP probe. The results are presented in figure 5.5 (a), were it can be seen that VSMA

increases relatively linearly with A. The explanation for this relationship is simple:

the magnetic flux detected by the SMA coil is a relatively constant fraction of the

total magnetic flux in the system.

The percent uncertainty associated with each VSMA measurement, determined


by the 95% confidence interval of the fit (as described in section 4.4), is shown in

figure 5.5 (b). The percent uncertainty decreases from low excitation field values to

a minimum of 3.5% at A = 100 mT, then increases for A = 125 mT. Because of

the slight difference in uncertainties between 75 mT and 100 mT excitation fields, an

excitation field amplitude of A = 75 mT was used for measurements on the 3PBR.

5.2.3 SA-106 Grade B SMA Stress Response

The stress response of the MAP sensor configured as described in section 5.2.2, with a

55 Hz, 75 mT excitation field, are shown in figure 5.6 (a) for tensile stress. The back-

ground measurement at σa = 0 MPa agrees within error to that presented in figure

5.5. As shown in figure 5.6, increases in axial tension were accompanied by increases

in VSMA, but the observed VSMA changes were small and the four different Vsig(σa, φ)

waveforms shown in figure 5.6 (b) are difficult to distinguish, unlike previous SMA

results presented in figure 4.12. The probe was aligned with angles φ = 0, 180, 360

along the pipe axis (parallel to σa), and φ = 90, 270 along the pipe hoop (parallel to

σh). The results in figure 5.6 (b) indicate an initial easy axis along the axial direction

which increases in permeability with σa.

Additional MAP Probe Modifications

Initial MAP results were less stress-sensitive than desired. Minor modifications were

made to the SMA coil mount and liftoff pads in an attempt to increase sensitivity

and decrease measurement uncertainty. Movement of the sensor during measurements

was believed to contribute to inconsistent coil coupling, which increased measurement

uncertainty. To counteract this effect, a piece of electrical tape (0.36 mm thick) was


−20 0 20 40 60 80

85

90

95

100

105

110

2.5” SA-106 Grade B Anisotropy Signal

Axial Stress, σa(MPa)

SM

ASig

nalV

olt

age,

VSMA

(mV

)

−100 0 100 200 300 400

−150

−100

−50

0

50

100

150

SMA Vsig(σ ,φ) Amplitude


Fit

Am

plitu

de,

Af(σ

,φ)

(mV

)

(a) (b)

= 0 MPa

= 16.3 MPa

= 45.8 MPa

= 67.3 MPa

σa

σa

σa

σa

σa

a

Figure 5.6: MAP stress response for an excitation field Bex = 75 mT sin(2πt55 Hz). (a)VSMA values. (b) The Vsig(σa, φ) values for each probe orientation. Error bars are omittedfor clarity.


placed between the SMA detector coil and sample to provide more consistent coupling

between the detector coil and sample. Liftoff spacer thickness was slightly reduced

to ease the current burden of the excitation coils.

Modified MAP Results and Analysis

The modified MAP was tested with the same excitation field as the original design.

With the 3PBR in the tensile configuration, axial tensile stresses were varied from 0 to

270 MPa. The 3PBR was then configured to apply compressive stress to the SA-106

sample, and compressive stresses were varied from 0 to -68 MPa. Measurements were

performed by recording Vsig(σa, φ) waveforms for φ = 0 to φ = 360 in 15increments.

Stress test results from the modified MAP are presented in figure 5.7. The initial

increasing tensile stress measurement results in an increasing VSMA, while compressive

stress produce a decrease in VSMA. The background readings (σa = 0) for both

measurements agree within error.

Tensile and compressive MAP data was evaluated with a linear least-squares fit,

producing

VSMA(mV) = 0.183σa(MPa) + 69.9 (5.5)

as the line of best fit. This linearization is also shown in figure 5.7. All data points

agree with the linear fit within uncertainty; however there does appear to be slight

oscillatory trend about the line of best fit.

The mean uncertainty in MAP probe SMA measurements was found to be ±7 mV,

with a minimum uncertainty of ±5 mV and a maximum uncertainty of ±10 mV. Solv-

ing equation 5.5 for σa indicates that a ±7 mV uncertainty in VSMA corresponds to a

±38 MPa uncertainty in stress. Therefore the current MAP probe can evaluate elastic


−100 −50 0 50 100 150 200 250 30050

60

70

80

90

100

110

120

1302.5” SA-106 Grade B Anisotropy Signal

Axial Stress, σa(MPa)

SM

ASig

nalV

olt

age,

VSMA

(mV

)

tensile

compressive

linear fit

Figure 5.7: SMA dependence on tensile and compressive applied stress. Vertical dashedlines indicate σa = 0 and σa = 240 MPa.


−50 0 50 100 150 200 250 300 350 400−100

−80

−60

−40

−20

0

20

40

60

80

SMA Vsig(σa, φ) Amplitude


Fit

Am

plitu

de,

Af(σa,φ

)(m

V)

σaσa

= 47 MPa

= -44 MPa

Figure 5.8: Signal voltage Vsig(σa, φ) fit amplitude for approximately equivalent compres-sive (σa = −44 MPa) and tensile (σa = 47 MPa) stresses. The uncertainty associated witheach data point is smaller than the data marker.

stress in feeders from VSMA values with an accuracy of approximately ±38 MPa.

It was observed that the uncertainty associated with compressive measurements

was, in general, much larger than the uncertainty of equivalent tensile measurements.

Figure 5.8 shows the signal voltage amplitude (Af (σa, φ)) for σa = 47 MPa and σa =

−44 MPa. There is only a small difference in amplitude between the two stress levels

at φ = 45 and φ = 315 , approximately 3 mV; however Af (−44MPa, φ = 135, 225)

is approximately 20 mV less than Af (47 MPa, φ = 135, 225). The discrepancy be-

tween peak Af (σa, φ) values was the primary cause of the increased uncertainty as-

sociated with compressive measurements.

The SMA detector coil of the MAP probe was irreparably damaged while inves-

tigating the cause of the compressive measurement peak distribution, ending data


collection with this probe.

Chapter 6

Summary and Conclusions

Residual stress measurement is a priority for industries where the cost of failure

is significantly greater than the cost of regular inspection. A technique based on

stress-induced magnetic anisotropy (SMA) may present a viable method for in situ

residual stress measurement on components where tight clearances and varying sur-

face conditions hinder other stress measurement methods. This thesis focused on the

development of hardware, software and signal analysis methods necessary to apply

SMA measurement to SA-106 Grade B feeder pipes used in CANDU R© nuclear reac-

tors. In this chapter the success of this project will be evaluated based on the project

objectives specified in section 1.3.

6.1 Flux Control Systems

Two magnetic flux control systems, FCV1 and FCV2, were designed to compensate

for geometry effects resulting from the curved surface of a pipe wall. FCV1 relied

exclusively on feedback from a Hall sensor located between the sample and excitation

107

CHAPTER 6. SUMMARY AND CONCLUSIONS 108

core. This control system was found to be unstable due to the nature of the feedback

path.

The second flux control system (FCV2) was designed with two feedback mecha-

nisms: a wire coil (called the feedback coil) wound around the base of the excitation

core, and a Hall sensor (called the feedback Hall sensor) fixed between the excitation

core and sample. The hardware control system of FCV2 compared the magnetic flux

density at these two points with a user-defined reference value, and adjusted the exci-

tation voltage (the voltage across the excitation coil) to produce the desired magnetic

field.

The software requirements of FCV2 were implemented in LabVIEW R© . A program

was designed to calculate reference waveform parameters (Vref ) from a user-specified

excitation magnetic field (Bex). This software was also used for data acquisition and

output via a PCI-6229 DAQ.

FCV2 fulfills a portion of the first project objective, which was to “design a

magnetic flux leakage-based probe that can accommodate the space and geometry

(lift-off) constraints imposed by the feeder pipe environment.”

6.2 Magnetic Stress Detectors

Three magnetic stress detectors were tested on mild steel plate samples with a pro-

totype excitation core controlled by FCV2. Each detector was designed to function

somewhat differently: a Hall sensor was used with DC excitation fields (DC MFL), a

wire coil was used with AC excitation fields (AC MFL), and a specially oriented coil

was used in anisotropy measurements (SMA). The AC MFL measurement showed

no significant stress sensitivity. DC MFL measurements indicated a stress dependent


trend, but significant scatter would have made it problematic for quantitative stress

measurement.

SMA measurements were the most time-intensive to perform, requiring a full probe

rotation per measurement, and the most computationally intensive to analyze. How-

ever, flat plate tests indicated that this measurement was significantly more sensitive

to stress than the others. Because of the high stress sensitivity SMA measurements

were selected as the most likely candidate to produce reasonable stress measurements

from SA-106 grade B feeder pipes, and a prototype SMA-based probe (the Magnetic

Anisotropy Prototype probe) was designed accordingly.

The flat plate magnetic stress detector tests were carried out to complete the

second project objective: to “conduct laboratory testing on plate samples to determine

the extent of stress sensitivity of the probe designs.”

6.3 Proposed MAP Probe Design

The Magnetic Anisotropy Prototype (MAP) probe - a small Supermendur excitation

coil coupled with a ferrite core SMA coil and FCV2 compatible feedback components

- was designed specifically for use on CANDU R© feeder pipes. The probe was designed

to rest in a brace clamped to the sample, where it could be rotated about a point to

perform an anisotropy measurement.

MAP stress sensitivity was examined in tensile and compressive stress tests using

the three-point bending rig and a 2.5” nominal diameter SA-106 grade B pipe sam-

ple. While the stress-induced magnetic anisotropy signal voltage (VSMA) recorded

in MAP measurements indicated clear stress dependence, the uncertainty associated

with MAP VSMA indicate that the probe is not yet suited for industrial use. MAP


testing completed objective three, which was to “conduct testing on samples with

feeder pipe geometry with a focus on generalized stresses.”

6.4 Recommendations for Future Work

There are several aspects of this project that could be developed further. Section 6.4.1

details how certain project objectives could be brought to completion and section 6.4.2

examines additional work that could improve both FCV1 and FCV2. Section 6.4.3

suggests a new probe design that may yield improved results.

6.4.1 Project Objectives

Not all of the project objectives outlined in section 1.3 were fully met. Many of the

shortcomings of this project result from the MAP, either by virtue of design or due

to the sudden failure of its SMA coil.

Project objective one was to “design a magnetic flux leakage-based probe that can

accommodate the space and geometry (lift-off) constraints imposed by the feeder pipe

environment.” FCV2 fulfilled a portion of that requirement by providing a method of

coupling a consistent and repeatable flux into the sample, as discussed in section 6.1.

However, the rest of the objective was not met: the MAP probe will not fit within the

confines of a CANDU R© reactor face due to the connector brace and manual rotation

of the probe. Some modifications to the MAP design would meet this objective, such

as a redesigned connection system and a servo-based mechanical rotation system, but

this was not explored further in this project.


Project objective three was to “conduct testing on samples with feeder pipe geom-

etry with a focus on generalized stresses.” While this objective was already described

as completed, it would have been beneficial to perform further measurements at dif-

ferent excitation frequencies to extract additional depth information.

The fourth objective was to “conduct testing on feeder pipe samples.” This ob-

jective was not met due to failure of the SMA stress detection coil in the MAP. The

detection coil was 200 turns, wound out of 44 AWG wire around a ferrite core. Fail-

ure occurred when the wires connecting the coil to the connector brace snapped. The

break occurred at the edge of the epoxy encasing the SMA coil and could not be

repaired, ending data collection. This objective could be completed by rebuilding the

MAP probe SMA coil.

6.4.2 Control Systems

While some justifications of the shortcomings of FCV1 and the success of FCV2

are presented in section 3.5.4, a detailed control-theory analysis of both FCV1 and

FCV2 would yield further information about the noise observed in FCV1 and the

performance limits of FCV2. It is also possible that additional analysis of FCV1 may

yield information relating to possible modifications that would produce a functioning

Hall sensor feedback system.

6.4.3 Suggested Design Modifications

The objective of this work was to develop a magnetic stress inspection system that

could be used as an early prototype for an industrial CANDUR© feeder pipe inspection

technology. One of the difficulties of designing a system for use at a CANDU R© reactor


y core

x core

excitation coil

feedback coil

sample

(a)

4

detector coil1

2

3

coil mount4

(b)

Figure 6.1: The recommended system for future work. (a) Two perpendicular U-corescan rotate the magnetic field at their center by adjusting the excitation field generated byeach core. Adapted from [39]. (b) The recommended anisotropy coil configuration for atetrapole excitation system. Coils 1 and 3 are connected in series, as are coils 2 and 4.

face is accessibility: an elaborate network of coolant feeders with a minimum inter-

pipe clearance of 20 mm make it difficult for operators and tools to reach inspection

locations. The design presented in chapter 5 does not fit within the clearances of

CANDU R© feeder pipes and requires manual rotation. To overcome both of these

issues, a four pole excitation system (termed a tetrapole system) such as the the

spring-loaded tetrapole prototype (SL4P) designed by Steven White for magnetic

Barkhausen noise measurements [39], could be used to generate the excitation field.

The SL4P consists of two Supermendur U-cores oriented perpendicular to one another,

as shown in figure 6.1 (a), and can rotate the magnetic field at the center of the cores

by superimposing different excitation field amplitudes in the x and y cores. An

anisotropy coil, shown in figure 6.1 (b), consisting of four wire coils could be used

to detect the anisotropy signal. The coils would be connected in pairs (1 to 3 and

2 to 4), and the anisotropy signal would be taken as the quadrature sum of the coil

voltages. This tetrapole probe design would potentially enable measurements to be

made on feeder pipes without the need for manual probe rotation.

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Appendix A

FCV1 Details

Flux control version 1 (FCV1) was modified several times in attempts to stabilize the

excitation waveform. The basic principle of the control system remained consistent

with that presented in section 3.5; the modifications where capacitors and low-pass

filters that were used to damp out high frequency signals. Figure A.1 shows the

electrical schematic of one of the final iterations of FCV1. Section 3.5 presents data

collected exclusively from this system. Further modification of this design did not

yield significantly improved performance.

118

APPENDIX A. FCV1 DETAILS 119

-

LM4701

+

1

2

3,54

8

7

52 Ω5 W

BH-700

+Ic

-Ic

VH+

VH-

Red

Blu

e

Yell

ow

BlackV

H

1 μF

+24 V

1 μF

-24 V

27 kΩ+24 V

1.2 kΩ 1 kΩ

1 kΩ

1 μF

20 Ω5 W

-

LM747-A

+

13

12

4

2

1

1 μF

-24 V

1 μF

+24 V

-

LM747-B

+10

9

6

7

1 μF

+24 V

1 kΩ

1 kΩ

1 kΩ

1 kΩ

1 kΩ

-

LM747-A

+

13

12

4

2

1

1 μF

-24 V

1 μF

+24 V

1 kΩ

generate ground referenced Hall voltage

Hall sensor current supply

Lex

VH

Vref

-

LM4701

+

1

2

3,54

8

7

1 μF

+24 V

1 μF

-24 V

VsF = 0.5A

0.2 Ω

Rex

Hall voltage comparisonto reference

excitation coil

Figure A.1: An electrical schematic of FCV1. All resistors are 0.25 W unless otherwiseindicated. LM747 op-amps are dual amplifier packages. Different amplifiers within anLM747 are designed A and B.

Appendix B

Skin Depth

Consider a plane electromagnetic wave of magnetic field amplitude B0 incident on a

semi-infinite1 conducting medium of conductivity σe. The amplitude of the magnetic

field within the conductor decreases due to ohmic losses as the wave penetrates fur-

ther in the medium. The term ‘skin depth’ refers to this attenuation, which occurs

according to the exponential law [12]:

B(z) = B0e−z

√πµσef , (B.1)

where B(z) is the amplitude of the wave within the conductor, B0 is the amplitude

of the wave outside the conductor, z represents the direction of propagation, µ is the

permeability of the medium, and f is the frequency of the electromagnetic wave. The

terms in the square root of equation B.1 are rearranged to define the skin depth (δ)

as

δ =

√

1

πσeµf, (B.2)

1Extending from −∞ < x < ∞, −∞ < y < ∞, −∞ < z ≤ 0 in Cartesian coordinates.

120

APPENDIX B. SKIN DEPTH 121

0 20 40 60 80 1000

2

4

6

8

10

Frequency, f (Hz)

Skin

Dep

th,δ

(mm

)

Skin Depth in Generic Steel

12

14

16

Figure B.1: Skin depth for a typical steel with µr = 100 and σe = 107 Ω−1m−1

giving

B(z) = B0e−z/δ. (B.3)

After one δ the amplitude of the magnetic field is reduced by a factor of 1/e. For

most engineering applications, waves are considered to be attenuated at z = 3δ

Figure B.1 shows the variation of skin depth with frequency for a typical steel

with relative permeability of µr ≈ 100 and conductivity σe = 107 Ω−1m−1.

A Magnetic Flux Leakage NDE System

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