Comprehensive Characterization of Measurement Data Gathered by the Pressure Tube to Calandria Tube Gap Probe by Shaddy Samir Zaki Shokralla A thesis submitted to the Department of Physics, Engineering Physics and Astronomy in conformity with the requirements for the degree of Doctor of Philosophy Queen’s University Kingston, Ontario, Canada August 2016 Copyright c Shaddy Samir Zaki Shokralla, 2016
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Comprehensive Characterization of
Measurement Data Gathered by the Pressure
Tube to Calandria Tube Gap Probe
by
Shaddy Samir Zaki Shokralla
A thesis submitted to the
Department of Physics, Engineering Physics and Astronomy
a risk of hydride blister formation on the PT outer diameter with a consequent risk
of PT cracking [3]. Therefore, PT to CT contact is to be avoided, and monitoring of
PT to CT gap is required [4]. Eddy current testing is employed in monitoring PT to
CT gap [3].
Eddy current testing of conductive materials employs electromagnetic induction
for identification of material properties, including material classification, detection of
flaws, and characterization of defects [5]. Eddy current testing is typically realized
by voltage excitation of a coil, resulting in a time-varying magnetic field that is
produced around the coil. This coil can act as an eddy current test probe [5]. If the
1.1. BACKGROUND 6
Figure 1.5: Fuel channel components - garter spring spacer separates PT and CT [1].
coil is in the proximity of a conductive material, eddy currents will be induced in
the conductive material via Faraday’s law, due to the presence of the time-varying
magnetic field (see Figure 1.6). The eddy currents induced in the test material in
turn produce a magnetic field, which opposes changes in the primary magnetic field
induced into the test material, according to Lenz’s law [6]. The presence of flaws or
material changes in the proximity to the probe modifies the induced eddy currents,
that may be sensed as a change in the coil’s impedance. In general, any modification
of conducting structure geometry (e.g. flaws, variation of material thickness, edges)
in proximity of the probe will modify measured probe impedance. Eddy current
testing is a comparative technique, where measurements on a test piece are related
to measurements on a known configuration. [5]. Capacitive effects are negligible,
relative to large inductance, at the low frequencies used in this work for application
of eddy current testing for PT to CT gap measurement [5].
Since variation of PT to CT gap may affect measured probe impedance, eddy
current testing can also be used to measure PT to CT gap, from within the PT. The
separation of PT and CT is monitored by using an eddy current based technique that
1.1. BACKGROUND 7
Figure 1.6: Principle of eddy current testing. Primary magnetic field induces eddycurrents in conductive surface secondary magnetic field that opposes pri-mary field.
measures the gap between PT and CT [3]. An example of the parameters of interest
and the basic measurement configuration are shown in Figure 1.7. The technique
compensates for both wall thickness and PT inner diameter variations using normal
beam ultrasonic measurements. Since eddy current response to change in PT to
CT gap is highly sensitive to PT wall thickness, calibrated eddy current responses,
including changes in PT-CT gap and PT wall thickness are employed. The objective
of the inspection system is to measure the PT-CT gap with sufficient accuracy so that
time-to-contact between PT and CT can first be predicted and second, be avoided
[3].
Figure 1.8 depicts the general format for generating the estimate of gap. Inputs
are the calibrated multi-frequency eddy current responses that account for PT wall
1.2. NEED FOR COMPREHENSIVE CHARACTERIZATION OFMEASUREMENT DATA GATHERED BY GAP PROBE 8
Figure 1.7: PT to CT gap measurement configuration showing minimum and maxi-mum gap locations and eddy current (EC) probe that traverses the cir-cumference of the PT ID.
thickness, PT diameter and wall thickness as measured by gauging systems. Probe
response to PT wall thickness and PT-CT gap variations is established by a calibration
facility, where these parameters are varied over the range expected in the reactor.
The algorithm identified in Figure 1.8 assumes constant resistivity for PT and CT,
constant CT wall thickness and diameter and a circular PT. All of these conditions
are present in the calibration facility.
1.2 Need for Comprehensive Characterization of Measurement Data Gath-
ered by Gap Probe
In this section, motivation for a comprehensive characterization of measurement data
gathered by the gap probe is introduced. Further in-depth motivation for specific top-
ics researched in this thesis is presented in the introductions of individual manuscripts
1.2. NEED FOR COMPREHENSIVE CHARACTERIZATION OFMEASUREMENT DATA GATHERED BY GAP PROBE 9
Figure 1.8: Method for generation of estimated pressure tube to calandria tube gap.
(Chapters 5, 6, and 7).
Multi-frequency eddy current data is acquired by the gap measurement system
with helical motion of the probe within the PT. Eddy current data is gathered every
degree circumferentially and every 1 mm axially, on the surface of a 6 m long pressure
tube. The acquired data is currently only used in the measurement of pressure tube
to calandria tube gap. Given the high sensitivity of eddy current to lift-off, the
high data density employed in acquisition, and the presence of artefacts that can be
characterized as depressions or protrusions (local changes in PT diameter) on the
pressure tube surface, it is natural to assume that more inspection information can
be extracted by developing a new method for analysis of the acquired eddy current
data (as will be discussed in Section 1.2.1).
The employment of PT to CT gap measurement data towards identification of
1.2. NEED FOR COMPREHENSIVE CHARACTERIZATION OFMEASUREMENT DATA GATHERED BY GAP PROBE 10
pressure tube artefacts, as well as identification of inherent redundancies in acquired
data will serve to provide for a comprehensive characterization of relevant inspection
data gathered by the gap probe. The following subsections will provide further detail.
Additionally, the use and benefits of an analytical model for determination of effects
of PT wall thickness, and PT to CT gap change on independent frequency responses
will also be discussed (Section 1.2.3).
1.2.1 Presence of Pressure Tube Artefacts and Relationship to Gap Probe
Lift-off
In this section, the mechanics of the gap probe will be introduced, and the relationship
of acquired eddy current signals to pressure tube artefacts (described by depressions
and protrusions on the pressure tube surface) will be discussed.
Although the specifics of the probe geometry and design are proprietary, general
aspects of the probe structure may be considered. The gap probe is secured on a
fuel channel inspection head. Given an inspection head installed in the fuel channel,
transmit and receive coils are aligned in the axial direction. The area of the eddy
current probe body is approximately 50 mm x 25 mm. A spring loads the probe
against the pressure tube surface, such that it maintains contact and its area rides
the pressure tube inner diameter surface. The probe is allowed two degree-of-freedom
motion. The first degree of freedom is away from the pressure tube surface - the
direction of spring force. The second degree of motion is about the inspection head’s
rotary direction of motion. This can be described as forward and backward probe
tilt. See Figure 1.9 for an illustration of gap probe directions of motion.
Interactions with pressure tube artefacts [7, 8, 9], which can be characterized as
1.2. NEED FOR COMPREHENSIVE CHARACTERIZATION OFMEASUREMENT DATA GATHERED BY GAP PROBE 11
Figure 1.9: Eddy current (EC) probe riding over pressure tube surface protrusionwith probe degrees of motion indicated.
protrusions on the pressure tube surface, will induce lift-off of the gap probe, as
shown in Figure 1.9. The presence of lift-off, can therefore, be used as a marker
of pressure tube artefacts, since a lift-off response will be acquired by the gap mea-
surement system. The identification of coincidence of acquisition of lift-off signals
with the locations of pressure tube artefacts is a prime goal for the application of
a signal processing algorithm to extract the location and features of pressure tube
artefacts. Employment of raw eddy current data gathered by the gap measurement
system, information on probe frequency lift-off response (which is near constant for
a given PT/CT configuration) in addition to locations of acquired eddy current ac-
quisition data can all be used as inputs into an algorithm characterizing pressure
tube artefacts. Figure 1.10 gives the direction of impedance plane probe lift-off re-
sponse. Impedance plane representation of eddy current measurements, including
1.2. NEED FOR COMPREHENSIVE CHARACTERIZATION OFMEASUREMENT DATA GATHERED BY GAP PROBE 12
lift-off response is discussed in Section 3.2.
Figure 1.10: Impedance plane response of probe lift off from pressure tube surface.Direction of l gives direction of impedance plane probe lift-off responsestarting from contact.
1.2.2 Identifying Redundancies in Multifrequency Eddy Current Data
The skin depth relation [5, 10] (δ = 50√ρ/µrf , Equation 3.18, derived in Section 3.3)
gives the depth (δ) in mm at which eddy current density is decreased by 1/e, where
f represents frequency in Hz, ρ represents resistivity in µΩ · cm, and µr represents
relative permeability. It is clear from the skin depth relation that δ is dependent on
coil excitation frequency. Increased eddy current penetration occurs with a decrease
in excitation frequency. Conversely, an increase in coil excitation frequency will result
in increased sensitivity to probe lift-off and defects [5] due to decreased skin depth.
1.2. NEED FOR COMPREHENSIVE CHARACTERIZATION OFMEASUREMENT DATA GATHERED BY GAP PROBE 13
Given ZrNb pressure tube resistivity of 52 µΩ · cm, for 4 kHz, 8 kHz, and 16 kHz
frequencies, the skin depth is 5.7 mm, 4.1 mm, and 2.9 mm, respectively. Despite
differences in penetration depth and sensitivity, there is nonetheless a significant
intersection of useful penetration depth data and responses to lift-off and defects, for
different inspection frequencies (4 kHz, 8 kHz and 16 kHz) employed for PT to CT
gap inspection at Ontario Power Generation.
With respect to employment of multi-frequency eddy current data for NDE appli-
cations, a characterization of the unique (or non-redundant) data gathered by using
multiple frequencies is not presently available in the literature. As multi-frequency
eddy current is used for estimation of PT to CT gap, principal components analysis
(PCA) was investigated to assess what, if any, independent information is gathered
by the gap probe due to changes in critical parameters: pressure tube wall thickness,
and pressure tube to calandria tube gap. This will form an essential component of de-
veloping a comprehensive understanding of relevant inspection information gathered
by the gap probe, as redundancies in information gathered by multifrequency eddy
current measurements can be identified and excluded from a comprehensive model.
PCA involves employing orthogonal transformations to convert a set of observa-
tions, which are correlated, into values belonging to linearly uncorrelated (orthogonal)
variables, called principal components. The main advantage of PCA is that complex
data sets can be reduced to data sets of lower dimension to reveal unapparent and/or
simplified trends in the data [11, 12].
Multi-frequency responses to changes in gap, PT wall thickness, and PT resistiv-
ity can be represented in PCA space, where a smaller number of variables (compared
to the dimension of the original multi-frequency dataset) can be used to investigate
1.2. NEED FOR COMPREHENSIVE CHARACTERIZATION OFMEASUREMENT DATA GATHERED BY GAP PROBE 14
relationships between the identified physical variables of interest. Principal compo-
nent’s scores represent signal responses to changes in these variables. Often, a small
number of principal components (which are a linear combination of vectors, whose
entries are the original measurement data series) can be used to describe variation
in data sets when these data sets contain correlations [13]. Since multi-frequency
eddy current responses to PT to CT gap, PT wall thickness and PT resistivity are
correlated, a small number of PCA components is expected to represent the original
multi-frequency responses to variation in these parameters.
1.2.3 Employment of an Analytical Model of Responses to Factors Af-
fecting PT to CT Gap Measurement
In the previous section, the importance of identifying independence and redundancies
in multifrequency gap probe eddy current data was highlighted, especially towards
developing a comprehensive characterization of measurement data gathered by the
gap probe. Experimental data is not ideally employed in a PCA approach to repre-
senting multifrequency gap data, as a number of complications would be introduced,
compromising the efficacy of the PCA process including:
• variable signal-to-noise ratio across the frequencies employed for inspection;
• limited control of test piece geometry, due to limited machining tolerances,
resulting in multiple simultaneous parameter variation;
• variable execution of experiments, including effects of probe tilt/and possible
lift-off; and
• effects of instrumentation and setup, which can vary across different instruments
1.2. NEED FOR COMPREHENSIVE CHARACTERIZATION OFMEASUREMENT DATA GATHERED BY GAP PROBE 15
cipal components analysis has been employed on conventional eddy current testing,
to improve reliability of interpretation of eddy current signals for steam generator
tubes [37].
Particular to pulsed eddy current testing (a class of eddy current testing which
examines the transient responses to short voltage spikes applied to eddy current
coils) principal components analysis has been recently employed toward a number
of applications. PCA applied to pulsed eddy current was first shown to provide
enhanced classification of defects [38]. Further applications of PCA to pulsed eddy
current include detection of defects in multilayer aluminum lap joints [39, 40, 41],
steel [42], and aircraft structures [43, 44, 45]. Additionally, PCA was employed for
detection of cracks in multiple layers using pulsed eddy current and giant magneto
resistive sensors [46, 47]. A modified PCA technique was used to detect cracks in an
F/A-18 inner wing spar without wing skin removal [13, 48, 49]. Applied to pulsed
eddy current, PCA has also been combined with wavelet analysis [50] and smooth
nonnegative matrix factorization [51].
Typical employment of principal components analysis for pulsed eddy current ap-
plications considers principal component eigenvectors plotted as a function of time,
in classification of defects [49, 40, 50, 38]. It should be noted however, that conven-
tional eddy current data provides measurement results, while the magnetic circuit is
in steady state, and therefore, conventional eddy current measurement can be con-
sidered time invariant. Principal components analysis can therefore, be employed on
conventional eddy current data, removing the time dependence from interpretation of
results. Also, for conventional eddy current data, principal components analysis can
2.3. EMPLOYMENT OF AN ANALYTICAL MODEL OF EDDYCURRENT RESPONSES TO FACTORS AFFECTING PT TO CTGAP MEASUREMENT 22
simplify the data analysis, by relating the reduced score representation of the data to
parameters that produce variations in the signal.
Principal components analysis has not yet been applied to multifrequency eddy
current applications (e.g. [52, 24]), employing conventional eddy current testing. As
will be shown, principal components analysis will be employed to identify indepen-
dence and redundancies in multifrequency eddy current gap data.
2.3 Employment of an Analytical Model of Eddy Current Responses to
Factors Affecting PT to CT Gap Measurement
Analytic solutions to Maxwell’s equations, which model eddy current probe coils
have been developed by Dodd et al. [16, 17, 18]. Employment of these solutions
in determining independence (and redundancies) in gap probe eddy current data is
especially useful in that these models do not exhibit variable probe frequency-gain
and noise, compared to physical systems, and as such, performing PCA with model
data will provide solutions that are repeatable and free of data corruption.
In order to employ solutions to probe coils using Dodd and Deeds analytical
models, these models have to be validated. Examples of validation of analytical
models of eddy current testing methods are available [53, 54]. An important element
of the validation process is defining physical limitations for which the validation is
correct. This should also be considered in validating Dodd and Deeds solutions [16,
17, 18] for modeling pressure tube to calandria tube gap data.
Using an analytical model for predicting effects of physical parameters such as
pressure tube resistivity, pressure tube wall thickness, and pressure tube to calandria
2.3. EMPLOYMENT OF AN ANALYTICAL MODEL OF EDDYCURRENT RESPONSES TO FACTORS AFFECTING PT TO CTGAP MEASUREMENT 23
tube gap has positive side effects with respect to the process of inspection qualifi-
cation. Inspection qualification, which is a nuclear operator regulatory requirement,
[19], serves to determine an inspection system’s capabilities, as compared to its in-
spection specification requirements. Physical parameters whose change in value could
affect an inspection system, such that its specified objectives would no longer be met,
are termed essential parameters [55] and are central in the process of inspection qual-
ification. The effectiveness of modeling has been highlighted for characterizing effects
of essential parameter variation, in the context of inspection qualification [20].
24
Chapter 3
Theory
In this chapter a brief theoretical introduction to tools employed towards developing
a comprehensive model of inspection information gathered by the gap probe is laid
out. A full review of the specific theory employed for each paper is available via
review of the related theory sections in attached manuscripts (Chapters 5, 6, and 7).
Eddy current testing is employed for inspection of electrically conductive materi-
als. If an alternating current is used to drive a coil, a time-varying magnetic field will
be generated in and around the coil [5]. According to Faraday’s law, eddy currents will
be induced in the conductive material that is in the proximity of the time-varying
field. Lenz’s law specifies that induced eddy currents will have directions that in-
duce magnetic fields opposing the change in magnetic fields induced by the drive coil
current. Eddy current testing monitors coil impedance and is thereby, sensitive to
parameters that modify coupling (mutual inductance) between probe (made up of a
single combination drive/detector coil, or seperate drive/detector coils) and sample.
These parameters include the presence of flaws, and probe lift-off, as these will have
a direct impact on the impedance measured by an eddy current testing system.
3.1. MATHEMATICAL GRADIENT AND DIRECTIONALDERIVATIVE 25
3.1 Mathematical Gradient and Directional Derivative
The concepts presented in this section are further developed in the analysis of eddy
current signals in one of the papers (Manuscript I, Chapter 5) provided in this thesis
and are therefore, introduced only briefly. As laid out in Section 5.4, the directional
derivative is used to compute the projections of change in eddy current signals, in the
direction of the lift-off vector. The axial and rotary components of the directional
derivative are considered separately, in imaging pressure tube artefacts (Section 5.5).
The mathematical gradient is a well defined mathematical operator employed in
many analytical fields of study. The gradient of a function of three variables, f(x, y, z)
where i is the unit vector in the x direction, j is the unit vector in the y direction,
and k is the unit vector in the z direction is defined as [56]
∇f =∂f
∂xi+
∂f
∂yj +
∂f
∂zk. (3.1)
The gradient of a function is a vector, and represents the slope of the tangent of
a function. The direction of the gradient is in the greatest rate of increase of the
function, with respect to variables x, y and z.
The directional derivative of f along a given unit vector u = (ux, uy, uz) at a point
(x0, y0, z0) represents the rate of change of f(x0, y0, z0) with respect to variables x, y, z,
along the direction of u. The directional derivative in the direction u is computed as
Duf(x0, y0, z0) = ∇f(x0, y0, z0) · u (3.2)
=∂f(x0, y0, z0)
∂xux +
∂f(x0, y0, z0)
∂yuy +
∂f(x0, y0, z0)
∂zuz. (3.3)
3.2. EQUIVALENT CIRCUIT AND IMPEDANCE PLANEREPRESENTATION 26
3.2 Equivalent Circuit and Impedance Plane Representation
In this section a derivation is provided for an impedance plane representation of
an eddy current coil, in proximity to a conductive test sample. The impedance
plane representation is useful, in that the changing coil impedance relates to the
sensitivity to liftoff (both PT and CT), resistivity and wall thickness. It can also be
used to explain effects due to variation in temperature, which causes changes in coil
impedance (resistance) and sample resistivity. All these factors facilitate a qualitative
interpretation of the gap probe response under various in-reactor conditions.
In the proximity of an electrically conductive test material, a coil used in eddy
current testing can be modeled as a primary winding of a transformer [5, 57]. The
eddy currents induced in the test material, conversely, can be modeled as a single
turn of a secondary winding (which sets Ns, the number of turns in the secondary
winding to 1). In the ‘send-receive’ method of eddy current testing, another coil is
used to monitor variations in received voltage (Figure 3.1). The following analysis in
this section does not consider electromagnetic interactions between the receive coil
and either the transmit coil or conductive test object.
Equivalent circuits can be used to simplify the electromagnetic relationship be-
tween a coil and test sample, and to obtain an impedance diagram, which is central
to the analysis of eddy current measurement data (see Figures 3.2 and 3.3). Note
that this treatment neglects skin effects in the conductor as described by Equation
3.17.
The equivalent parallel circuit has equivalent impedance [5]
Zp =Z1Z2
Z1 + Z2
(3.4)
3.2. EQUIVALENT CIRCUIT AND IMPEDANCE PLANEREPRESENTATION 27
Figure 3.1: Model of a transmit and receive coil with a test object.
where Z1 = N2pRs and Z2 = jX0. Np is the number of turns in the transmit coil,
Rs is the variable resistance of the conductive test object, and j is the imaginary
number. Where L0 is the probe inductance, X0 = ωL0 is the inductive reactance of
the coil in proximity to the test object. As Rp is a constant which is not affected by
test object impedance, Rp can be excluded from Zp in deriving an impedance plane
representation of an EC coil in proximity to a test object. Zp can be evaluated as
Zp =jN2
pRsX0
N2pRs + jX0
. (3.5)
Rationalizing the denominator yields [5, 57]
Zp =N2pRsX
20
(N2pRs)2 + (X0)2
+ j(N2
pRs)2X0
(N2pRs)2 + (X0)2
, (3.6)
3.2. EQUIVALENT CIRCUIT AND IMPEDANCE PLANEREPRESENTATION 28
Figure 3.2: Equivalent parallel circuit of coil and test sample represented without andwith equivalent impedance. Zp is the equivalent impedance of the parallelcircuit with N2
pRs and ωL0 components.
where RL can be used to represent the real part of the equation, and Xp can be used
to represent the imaginary part of the equation, such that [5, 57]
Zp = RL + jXp. (3.7)
Test samples have corresponding resistivities, which are represented as a parallel
resistance to the test coil inductive reactance, and can be affected by the presence of
defects [5]. Changes in test sample resistivity are reflected as changes to resistance and
reactance changes experienced by the test coil. The operating point in an impedance
3.2. EQUIVALENT CIRCUIT AND IMPEDANCE PLANEREPRESENTATION 29
Figure 3.3: Equivalent series circuit of coil and test sample.
plane representation of impedance experienced by the coil is moved up the impedance
curve (Figure 3.4 [5]. When mutual coupling between the coil and sample is decreased
(i.e. the coil experiences lift off from the test sample), smaller semi-circles are traced
on the impedance plane [5].
Figure 3.4: Impedance (reactance vs. resistance) diagram with effects of lift-off, test-object resistivity (ρ) change and test-object equivalent resistance (Rs).
3.3. MAXWELL’S EQUATIONS AND THE SKIN DEPTHEQUATION 30
3.3 Maxwell’s Equations and the Skin Depth Equation
In this section a derivation of the skin depth equation is provided, beginning with
Maxwell’s equations. The skin depth equation is discussed in relation to results
presented in Manuscripts I and II (Chapters 5 and 6).
Where E is the electric field, B is the magnetic flux, ρf is the free electric charge
density, µ is the permeability, ε is the permittivity, and σ is the conductivity, in a
linear medium. Maxwell’s equations have the form [6]
∇ ·E =ρfε
(3.8)
∇ ·B = 0 (3.9)
∇×E = −∂B∂t
(3.10)
∇×B = µσE + µε∂E
∂t(3.11)
Taking the curl of Equation 3.10, substituting into Equation 3.11, and using the
identities ∇×∇×E = ∇(∇ ·E) −∇2E and ∇(∇ ·E) = 0, we obtain, with some
manipulation,
∇2E = µσ∂E
∂t+ µε
∂2E
∂t2. (3.12)
For good conductors, where σ is very large (σ/ωε 1 [10]), and for a time
harmonic excitation, this can be written as the Diffusion Equation (for frequencies
3.3. MAXWELL’S EQUATIONS AND THE SKIN DEPTHEQUATION 31
less than 108 Hz [16]),
∇2E = µσ∂E
∂t. (3.13)
By Ohm’s Law, J = σE, the electric field drives the current
∇2J = µσ∂J
∂t. (3.14)
If the current varies sinusoidally in time, in phasor notation, J(t) = ReJ0e−jωt
where J0 is the vector having magnitude and direction of the current density, and
substituting into the diffusion equation we have [10]
∇2J0 + α2J0 = 0, (3.15)
where α2 = jµσω. Where j is the imaginary number, and since the following mathe-
matical identity holds,√j = 1√
2(1 + j),
α = (1 + j)
√µσω
2=
1 + j
δ, (3.16)
where (in MKS units) [10]
δ =
√2
σµω, (3.17)
or [5]
δ = 50
√ρ
µrf, (3.18)
where δ is the skin depth measured in mm, ρ is the resistivity measured in µΩ · cm,
3.4. DODD AND DEEDS SOLUTIONS 32
µr is the relative permeability, and f is the frequency measured in Hz. Equation 3.18
is the Skin Depth Equation, where δ is the depth at which the eddy current density
has decreased to 1/e, or 36.8% of the surface density. In general, penetration depth
decreases with increased frequency, and increases with increased resistivity as per
Equation 3.18. Eddy currents produced at and below the test surface oppose changes
in the primary magnetic field produced by the alternating current in the probe coil,
according to Lenz’s law [6, 10].
3.4 Dodd and Deeds Solutions
The following is a high-level summary of how the eddy current probe response can be
derived analytically, by approximating the test surface(s) as one or more flat plates.
The most significant points, from Ref. [18], are summarized below.
A coordinate system is employed, where the driving current and media are axially
symmetric, reflecting the coil geometry above a conducting plane (Figure 3.5). Due to
the axial symmetry, the current density J and vector potential A have only azimuthal
components, such that
J(x) = J(r, z)eφ (3.19)
and
A(x) = A(r, z)eφ, (3.20)
where eφ is an azimuthal unit vector. The magnitude of the magnetic vector potential
can be expressed as [18]
3.4. DODD AND DEEDS SOLUTIONS 33
A(r, z) =
∫∫G(r, z; r′, z′)J(r′, z′)dr′dz′, (3.21)
where G(r, z; r′, z′) is the Green’s function for a δ-function current (having infinite
current density, but total current I) at (r′, z′), as shown in Figure 3.5. G(r, z; r′, z′)
is also the magnetic vector potential at (r, z) of the δ-function current [16]. A(r, z)
is the summation of the magnetic vector potential, due to a number of δ-coils, which
together approximate real physical coils. The Green’s function satisfies the following
equation, assuming a linear, isotropic, homogeneous medium, having time-harmonic
current angular frequency ω [18],
(∂2
∂r2+
1
r
∂
∂r− 1
r2+
∂2
∂z2− jωµσ + ω2µε
)G(r, z; r′, z′) = −µδ(r−r′)δ(z−z′) (3.22)
where µ, ε, and σ are the permeability, permittivity, and conductivity of the medium,
respectively.
To find the vector potential, given in Equation 3.21, solutions of Equation 3.22
that satisfy proper boundary conditions are used. From the vector potential, all other
electromagnetic quantities of interest can be calculated [18].
The solution of Equation 3.22 for each region is given by [18]
G(n)(r, z; r′, z′) =
∫ ∞0
[Bn(α)e−αnz + Cn(α)eαnz
]J1(αr)dα (3.23)
for n = 1, 2, . . . , k−1, k, 1′, 2′, . . . , k′−1, k′, where J1(x) is a Bessel function [58] of the
first kind and the first order and α2n = α2+jωµnσn−ω2µnεn, where j is the imaginary
number. The Green’s function is finite when z approaches positive or negative infinity,
3.4. DODD AND DEEDS SOLUTIONS 34
Figure 3.5: δ-function coil between conductor plates k and k′. Conductor plates areenumerated n = 1, 2, . . . , k − 1, k, 1′, 2′, . . . , k′ − 1, k′.
and therefore B1 = 0 when z → −∞ and C1′ = 0 when z → +∞. Other coefficients
Bn, and Cn can be determined through considering boundary conditions at each
conductor interface [18]. The following boundary conditions for Green’s functions are
determined by considering tangential components of the electric field and magnetic
Principal components analysis (PCA) involves employing orthogonal transformations
to convert a set of observations, which are correlated, into a set of linearly uncor-
related variables, called principal components [12]. The main advantage of PCA is
that complex data sets can be reduced to data sets of lower dimension to reveal un-
apparent and/or simplified trends in the data [11, 12]. This can be observed through
examination of Figure 3.7, where highly correlated three dimensional data is reduced
to two dimensions. Note that in Figure 3.7, PC1 accounts for the largest variance in
the data set and PC2, which is orthogonal to it, the next largest variance.
Figure 3.7: PCA transformation, reducing the dimensionality of a data set from 3 to2 (modified from [2]).
PCA will be used to determine uniqueness and redundancy across multi-frequency
3.5. PRINCIPAL COMPONENTS ANALYSIS 39
eddy current data sets, and relate effects of physical parameter changes to principal
component scores, for PT to CT gap data.
3.5.1 Process
The objective of a principal components analysis transformation is to find a linear
combination of the original variables X = [x1, x2, . . . , xp] with maximum variance
[12]. X is first normalized such that each variable xi, i ∈ [1, 2, . . . , p] has zero mean
and unit variance.
A new set of vectors
tk(i) = x(i) · w(k) (3.33)
is defined such that individual variables of t inherit (successively) maximum possible
variance from x, with each loading vector w constrained as a unit vector. The first
loading vector w(1) is given as
w(1) = arg max
wTXTXw
wTw
, (3.34)
where arg max is defined as the argument of the maximum, which is the set of points
of the given argument for which the given function attains its maximum value. In
this case w is chosen such that wTXTXw/wTw attains its maximum value. The kth
component can be found by first subtracting the k − 1 principal components from x:
Xk = X −k−1∑s=1
Xw(s)wT(s). (3.35)
Now,
3.5. PRINCIPAL COMPONENTS ANALYSIS 40
w(k) = arg max
wT XT
k Xkw
wTw
. (3.36)
The kth principal component of vector x(i) is given as a score tk(i) = x(i) · w(k), where
the full decomposition of X is given as
T = XW. (3.37)
Equation 3.37 is used in Manuscript III (Chapter 7) to transform eddy current
responses to factors affecting PT to CT gap measurement responses (represented by
horizontal and vertical impedance plane responses for 4 kHz, and 8 kHz, and 16
kHz frequencies), to principal component scores. Scores belonging to the first and
subsequent principal components contain the majority of variance of the original data
set.
41
Chapter 4
Introduction to Manuscripts
The following journal articles, Manuscripts I, II, and III, are presented as Chapters
5, 6, and 7, respectively.
Manuscript I: S. Shokralla, T. W. Krause, and J. Morelli, “Surface profiling with high
density eddy current non-destructive examination data,” NDT & E International, vol.
62, March 2014, pp. 153-159.
Manuscript II: S. Shokralla, S. Sullivan, J. Morelli, and T. W. Krause, “Modelling
and validation of Eddy current response to changes in factors affecting pressure tube
to calandria tube gap measurement,” NDT & E International, vol. 73, July 2015,
pp. 15-21.
Manuscript III: S. Shokralla, J. E. Morelli and T. W. Krause, “Principal Components
Analysis of Multifrequency Eddy Current Data Used to Measure Pressure Tube to
Calandria Tube Gap,” IEEE Sensors Journal, vol. 16, no. 9, May 2016, pp. 3147-
3154.
42
In Manuscript I, a generalizable surface profiling technique is derived, exploiting
the interaction between the PT to CT gap probe and pressure tube inner diameter
surface. Manuscript II provides experimental validation of Dodd and Deeds solutions,
given expected in-reactor variations in physical parameters: PT wall thickness, PT
resistivity, and PT to CT gap. Using validated Dodd and Deeds solutions, PCA of
modelled changes to parameters affecting PT to CT gap measurement is employed
in Manuscript III, to relate physical parameter changes to unique information across
PT to CT gap multifrequency data. Additionally, in Manuscript III, PCA output is
validated against physical experiments, and a compressed data acquisition algorithm,
reliant on PCA fundamentals is derived and emulated using modelled PT to CT
multifrequency eddy current data.
43
Chapter 5
Surface Profiling with High Density Eddy Current
Non-Destructive Examination Data
Shaddy Shokralla, Thomas Krause, and Jordan Morelli
Abstract: The pressure tubes (PT) in CANDU (CANada Deuterium Uranium) re-
actors undergo creep induced deformation due to operating pressure, temperature
and radiation conditions. While global deformation of the tube in the form of elonga-
tion and diametral creep are well characterized and monitored by station inspection
systems, local PT deformation and the presence of inner surface artifacts due to wear
are not as directly monitored, but can still provide additional information of fuel
channel condition. A surface profiling technique for monitoring local deformation
and identification of surface wear, using an eddy current probe mounted in a small
(50 mm x 25 mm) planar probe body is presented. The sensitivity of the eddy current
probe to small lift-off variations combined with high density C-Scan information is
used to extract information on smoothly varying local deformation as well as monitor
more significant wear on the inner surface of pressure tubes. Vector separation of
components permits independent identification of axial and circumferential surface
5.1. INTRODUCTION 44
features. Analysis of this data can be used to characterize local PT deformation due
to constrictions at fuel bundle ends and loaded garter spring spacers, as well as iden-
tify areas where shallow mechanical wear has occurred. Examples of the features that
may be identified are presented.
Keywords: surface profiling, eddy current, lift-off, signal processing, data density
5.1 Introduction
Eddy current inspection is a key non-destructive examination (NDE) process em-
ployed towards characterization of defects in conductive materials. Piping and heat
exchanger (including steam generator) inspections are common applications of eddy
current non-destructive examination of nuclear power plants [27, 59, 60]. Specific to
CANDU (CANada Deuterium Uranium) reactors however, eddy current examination
has been used towards a number of non-standard fuel channel related applications
with regards to the geometry identified in Figure 5.1. These include identification
of the presence and proximity of remote structures to fuel channels [61], detection
of spacers, which maintain gap between pressure and calandria tubes [9] and direct
measurement of gap between pressure tube and calandria tube [3].
There are two standard ways in which eddy current data is interrogated. The first
is by examination of the Lissajous curve, which provides the resistance and inductive
reactance recorded by an eddy current receive probe during data acquisition. The
second is through examination of the C-Scan, in which the component orthogonal to
the direction of lift-off is plotted as a function of measurement location (points on a
two-dimensional grid).
There are many examples of employing high data density towards eddy current
5.1. INTRODUCTION 45
C-Scan imaging. Imaged data can either be minimally conditioned [21, 22], or al-
ternatively, a significant degree of signal and image processing can be employed
[23, 24, 25, 26].
For flaw detection applications, examination of lift-off in eddy current data is
normally avoided and is considered a source of noise [5, 62]. The application of paint
and metal deposition thickness measurement is well known. But its potential for
surface profiling is not as well recognized [63]. For the particular circumstance where
a large planar probe body is present, lift-off may be utilized to provide additional
information that would be available if the probe maintains surface following or surface
riding characteristics.
This study shows how high eddy current data-density, basic knowledge of mechani-
cal interaction between the probe and test-piece, and lift-off response can be employed
towards profiling a test-piece surface to a high degree of precision. The technique to
be described in detail is simple to implement, relying on a computation of partial
derivatives with respect to variables whose direction represent orthogonal image axes
(an image processing building block [32, 31]). Ease of technique implementation is
in contrast to more elaborate techniques for analysis of eddy current data, including
neural network implementation and wavelet transformation [25, 64, 65, 66].
The technique will be compared against another currently employed ultrasound
based method for detection of spacer and pressure tube artefact locations, important
activities in ensuring integrity of CANDU nuclear reactor fuel channels.
5.2. INSPECTION REQUIREMENT 46
5.2 Inspection Requirement
Fuel channels are a critical component of CANDU power plants. They carry the fuel
bundles as they are transported across the core of the reactor, whose configuration
is shown in the top portion of Figure 5.1. Each fuel channel is made up of a six
metre long pressure tube (which directly houses twelve 0.5m long fuel bundles) and a
surrounding calandria tube. The pressure tube and calandria tube are separated by
four spacers, which are manufactured from a square cross-section Zr-alloy wire formed
into a tight helix. Further details of these Fuel Channel components are shown as a
magnification in Figure 5.1.
Figure 5.1: Configuration of a CANDU Fuel Channel.
Fuel channels are periodically inspected in order to characterize the presence and
status of different degradation mechanisms, thereby meeting their safety and licensing
requirements [4]. This is performed by insertion of an inspection head along the
length of the fuel channel during reactor shutdown periods. A number of different
degradation mechanisms and pressure tube artefacts can be identified by various NDE
5.3. MEASUREMENT TECHNIQUE 47
methods (including eddy current). These include the following.
• Localized deformation due to garter spring spacer loading: Garter spring spac-
ers serve the purpose of ensuring separation between the pressure tube and
relatively cool calandria tube. Loading is therefore induced on the pressure
tube by the spacer and supporting calandria tube. This leaves the pressure
tube slightly ovalized in the vicinity of the spacer.
• Constrictions: Locations at fuel bundle ends where radiation induced creep is
less due to lower flux densities relative to the centre of fuel bundle locations.
This leaves the diameter of the pressure tube at a constriction smaller than
for neighbouring regions, while forming a pressure tube wall protrusion on the
order of 0.1mm.
• Mechanical wear marks: Mechanical interaction between fuel bundles moving
across the fuel channel, and the supporting pressure tube can induce wear on
the pressure tube. Mechanical wear marks indicate this repeated interaction.
The wear marks have nominal depth less than 30 µm [67].
5.3 Measurement Technique
Eddy current data is gathered helically by an eddy current probe mounted on an
inspection head that is driven axially and rotationally along the pressure tube (Figure
5.2). The axial pitch of the probe’s helical travel is small (1mm), compared to its
rotational travel, allowing for approximation of the helical trajectory as a series of
circular trajectories separated axially by 1mm.
5.3. MEASUREMENT TECHNIQUE 48
The eddy current probe is mounted on the inspection head, with transmit and re-
ceive coils situated along the axial direction of the pressure tube, when the inspection
head is installed in-channel. The probe is spring loaded such that it rides the surface
of the pressure tube. In the event that the probe makes contact with an obstruction
embedded in the pressure tube interior, as shown in Figure 5.2, it is also permitted
limited motion with two degrees of freedom: about the inspection head rotary direc-
tion of motion (forward and backward tilt), and away from the pressure tube surface
in the direction of spring force.
Figure 5.2: Eddy current (EC) probe riding over exaggerated pressure tube protru-sion.
The eddy current signals are generated and received via an Olympus NDT Multi-
scan MS5800 eddy current instrument. Custom software is employed to view acquired
signals in real time as well as post-acquisition. Eddy current signals are acquired on
multiple frequencies, which include 8kHz and 16kHz. Unadjusted raw measurements
have units mV.
5.4. LIFT-OFF LOCATION EXTRACTION 49
5.4 Lift-off Location Extraction
We represent the acquired eddy current data set as a complex valued function f(x, y)
mapping rotational and axial values (x and y) to the complex plane, C. We use C
to represent the domain where eddy current signals lie. Orthogonal components of
C represent the resistance and inductive reactance components in the eddy current
impedance plane. It is worthy to note that axially adjacent values of f(x, y) (along
the y-axis) are collected 360 degrees apart, due to the helical eddy current probe
trajectory; data is collected helically at 1 rotary degree (approximately 0.9mm) per
sample.
A retraction signal is sent to the eddy current probe during its acquisition sequence
and a lift-off signal (which can be represented as a horizontally oriented vector on the
Lissajous curve) is obtained. The direction of the lift-off vector is a key component
used in the calculations and is employed to allow for surface profiling using eddy
current data. l is used to denote the complex valued unit vector lying in C with the
same direction as the direction of the lift-off response.
Since values of f(x, y) lie on the complex plane, C, f(x, y) can be written
f(x, y) = u(x, y) + i v(x, y). (5.1)
Partial derivatives of f(x, y) with respect to x and y are defined as
∂f(x, y)
∂x=∂u(x, y)
∂x+ i
∂v(x, y)
∂x(5.2)
and
5.4. LIFT-OFF LOCATION EXTRACTION 50
∂f(x, y)
∂y=∂u(x, y)
∂y+ i
∂v(x, y)
∂y. (5.3)
Where the dot product is defined for complex valued vectors [30], the scalars
gx(x, y) and gy(x, y) are defined as the components of ∂f(x, y)/∂x and ∂f(x, y)/∂y
in the direction of lift off, such that
gx(x, y) =∂f(x, y)
∂x· l, (5.4)
and
gy(x, y) =∂f(x, y)
∂y· l. (5.5)
gx(x, y) and gy(x, y) represent the change in eddy current data, in the rotary and
axial directions, respectively, in the direction of lift-off.
Examination of gx and gy reveal not only areas in the inspection space where
the eddy current probe has lifted off from the pressure tube, but also reveal key
mechanical interactions between the eddy current probe and the inspection surface.
gx represents rotary lift-off experienced by the probe. That is to say, changes in lift-off
experienced by the probe as it traverses in the rotary direction over pressure tube
obstructions extending along the axial direction of the pressure tube.
Consider Figure 5.2. The eddy current probe is not free to rotate about the axial
direction of travel. So, the physical motion of the probe as it traverses over axially
extending pressure tube obstructions is away from the surface of the pressure tube in
the direction of spring force, and is limited to one degree of freedom.
gy on the other hand, represents changes in axial lift off experienced by the probe
5.5. APPLICATIONS 51
as it traverses in the axial direction over pressure tube obstructions away from the
surface of the pressure tube. The obstructions extend in the rotary direction of the
pressure tube. Unlike the case for lift-off of the probe as it traverses in the rotary
component of travel, (again consider Figure 5.2) the probe is free to move both away
from the pressure tube wall as well as rotate (tilt) about the rotary direction of
travel. Probe motion is therefore capable of two degrees of freedom when lifting off
over rotationally extended pressure tube obstructions.
5.5 Applications
In this section a number of pressure tube artefacts are identified using the computa-
tional method introduced in the previous section.
5.5.1 Mechanical Wear Marks
Calculation of gx for a set of eddy current data obtained in-reactor determines probe
lift-off variation in its rotary component of travel. The direction of probe lift-off is
normal to the pressure tube wall.
Vertical stripes are evident in Figure 5.3, revealing a systematic phenomenon,
which possesses a lift-off component in the rotary direction of travel at the bottom of
the pressure tube (near 180 degrees) for a significant axial length along the pressure
tube (800mm). The vertical stripes represent pressure tube mechanical wear marks
that have been caused by fuel bundles sliding along the bottom of the pressure tube.
The depth of these mechanical wear marks is typically less than 30µm [67]; as such,
computation of gx for the 16kHz frequency has been used to identify features in the
pressure tube surface not evident in standard eddy current Lissajous representations.
5.5. APPLICATIONS 52
Figure 5.3: 16kHz gx revealing fuel bundle mechanical wear marks with nominal depth50µm.
Figure 5.4 shows a conventional C-Scan for the same eddy current data, repre-
sented in Figure 5.3. Vertical stripes representing positions of mechanical wear marks
are evident in the top and middle of the figure (representing lower axial positions
and rotary positions at the bottom of the pressure tube). However, it is clear that
mechanical wear marks, given in the Figure 5.4 C-Scan, are much less obvious than
those given by gx in Figure 5.3. Furthermore, it is difficult using the C-Scan alone to
differentiate between vertical stripes in the middle of the figure, representing location
of mechanical wear marks, versus vertical stripes at the sides of the figure (corre-
sponding with the top of the pressure tube), where no mechanical wear marks are
present.
5.5. APPLICATIONS 53
Figure 5.4: 16kHz C-Scan, plotting eddy current data orthogonal to direction of lift-off vector, l.
Figure 5.5 provides 16kHz C-Scan eddy current data, in the direction of lift-
off. Mechanical wear marks are more evident in this figure compared to Figure 5.4.
gx given in Figure 5.3 however still provides increased discrimination of wear mark
features. An increased number of vertical stripes revealing locations of wear marks
are apparent in Figure 5.3 over Figure 5.5. Furthermore, whereas mechanical wear
marks are not evident in axial positions between 5950mm and 6000mm in Figure 5.5,
these wear marks are visible for the same axial positions in Figure 5.3.
The SNR (signal-to-noise ratio) of mechanical wear mark indications is used to
quantitatively compare imaging capability of gx, against C-Scan representations. The
SNR for this application is defined as the rotational peak-to-peak mechanical wear
mark signal amplitude, averaged over all axial positions, divided by the background
5.5. APPLICATIONS 54
Figure 5.5: 16kHz C-Scan, plotting eddy current data in direction of lift-off vector, l.
noise, which is computed as the standard deviation of noise pixel intensities. The
SNR of mechanical wear mark indications imaged via computation of gx, shown in
Figure 5.3, is computed as 5.8, while the SNRs of mechanical wear mark indications
shown in C-Scan Figures 5.4 and 5.5 are computed as 1.5 and 1.6, respectively. This
demonstrates increased wear-mark imaging capability of gx, over C-Scan representa-
tions.
5.5.2 Pressure Tube Constrictions
Figure 5.6 represents the computation of gy for the same eddy current data set em-
ployed to calculate gx, plotted in Figure 5.3.
Large horizontal bands, which extend 360 degrees circumferentially around the
pressure tube (evident at axial positions 5500mm and 6050mm) are shown in Figure
5.5. APPLICATIONS 55
5.6. These large horizontal bands represent the locations of fuel channel constrictions.
The rotary extent (360 degrees around the pressure tube) of the bands is consistent
with the circular formation of constrictions. Furthermore, the bands are approxi-
mately 0.5m apart, consistent with the length of fuel bundles where constrictions
form (Figure 5.1).
It is worthy to note that Figures 5.4 and 5.5 give C-Scan displays for the same
eddy current data revealing constrictions in Figure 5.6. Indications of constrictions
in Figures 5.4 and 5.5 coincide, and are inseparable from indications of mechanical
wear marks, contrary to indications of constrictions in Figure 5.6.
Figure 5.6: 16kHz gy revealing locations of local pressure tube constrictions (horizon-tal stripes extending from 0 to 360).
5.5. APPLICATIONS 56
5.5.3 Spacers
Identification of spacer positioning can also be performed through identification of
the axial component of probe lift-off. Figure 5.7 shows clearly the position of the
spacer, where the spacer has locally induced a load on the pressure tube, forming an
inner-diameter pressure tube protrusion, as described in Section 5.3. As expected, the
formed protrusion is centred at the bottom of the pressure tube (near 180 degrees).
The extent of contact between pressure tube and spacers can also be inferred from
Figure 5.7. It is identified by the rotary arc for which the spacer response is made
visible, which can be approximated as 60 degrees rotationally.
Figure 5.7: 8kHz gy identifying the position of a spacer.
5.5. APPLICATIONS 57
5.5.4 Interpretation of Positive and Negative Intensities
Intensity values in images formed by the computation of gx and gy have physical
meaning. Consider Figure 5.7, which reveals the location of lift-off due to pressure
tube inner-diameter wall protrusion. Examination of the figure reveals that the spacer
location is identified by two horizontal bands having opposite intensity values. Recall-
ing Equations 5.4 and 5.5, positive intensity values correspond to probe lift-off action,
while negative intensity values correspond to return of the probe to the surface of the
pressure tube. The two horizontal bands having opposite intensity, evident in Fig-
ure 5.7 therefore correspond to locations of increase and decrease in lift-off, where
the probe first lifts off when encountering the pressure tube wall protrusion caused
by spacer loading (as in Figure 5.2), is at minimum lift-off where the protrusion is
maximized, and then extends to a surface riding configuration against the pressure
tube wall after traversing over the protrusion (in the direction of opposite tilt when
compared to initial lift-off). The ordering of positive, then negative intensities with
increased axial position is consistent with direction of probe travel (increasing in axial
position).
5.5.5 Confirmation of Spacer and Constriction Locations using Ultra-
sound Data
The inspection head used to collect eddy current data also collects ultrasonic non-
destructive examination (NDE) data. Ultrasonic NDE data can be used to confirm
position of spacers and constrictions identified through computation of gy.
Two ultrasonic probes diametrically opposed, and positioned on the inspection
5.5. APPLICATIONS 58
head, rotate at the same pitch as the eddy current probe. This allows for measure-
ments to be taken at 0 degree and 180 degree rotary positions, for all axial positions
along the pressure tube. The distance, d, between a probe face and the pressure tube
wall can be determined by the simple relationship
d =ct
2, (5.6)
where c is the speed of sound in water, and t is the travel time between ultrasound
pulse emission from, and return to, the ultrasonic probe. The pressure tube inner
diameter along the 0 degree and 180 degree vertical line can then be computed
dPT−ID = d0 + dl + d180, (5.7)
where d0 is the distance between the probe positioned at the 0 degree rotary position
and the pressure tube inner diameter wall, d180 is the distance between the probe
positioned at the 180 degree rotary position and the pressure tube inner diameter
wall, and dl is the distance between diametrically opposed probe faces.
gy for 8kHz eddy current data is shown in Figure 5.8, revealing three constriction
locations and a single spacer location. To confirm the position of these artefacts,
a comparison between locations identified through examination of gy, and locations
identified through examination of ultrasonic measurements (dPT−ID) is made. A
correction factor is introduced for axial positions of dPT−ID. This accounts for the
difference between axial position of ultrasonic probe centres and axial position centred
between transmitting and receiving eddy current coils.
Figure 5.9 a) gives gy along axial positions between 3600mm and 5000mm, but
5.5. APPLICATIONS 59
averaged for points centred at the bottom of the channel; the average is calculated for
values of gy between rotary positions 160 degrees and 200 degrees. The points where
gy crosses the x-axis in the vicinity of visible signal perturbations (each characterized
by a large positive, then negative fluctuation) marks the position of spacers and
constrictions. These points lie on x-axis positions: 3695mm, 4195mm, 4478mm, and
4698mm.
Figure 5.9 b) gives ultrasonic measurements (dPT−ID), along pressure tube axial
positions between 3600mm and 5000mm. Background variation in pressure tube inner
diameter has been removed. Positions of constrictions and spacers can be identified
by localized reductions in dPT−ID, plotted as a function of axial position. For localized
reductions, the pressure tube inner diameter is relatively smaller as compared to the
immediately surrounding regions, reflecting the characteristic protrusions at spacers
and constrictions.
The positions of lowest points in localized reductions can be used as markers for
the axial locations of spacers and constrictions, where axial locations are identified
as 3694mm, 4190mm, 4474mm, and 4702mm. Localized diameter changes can also
be estimated from Figure 5.9 b), where the first, second, and third constrictions have
associated diameter reductions 60µm, 70µm, and 90µm, respectively, while the spacer
has an associated diameter reduction of 170µm.
Comparing spacer and constriction positions determined through examination of
gy, and through ultrasonic measurement reveals consistency in artefact localization
between the two methods; measurements agree within +/- 5mm.
5.6. DISCUSSION 60
Figure 5.8: 8kHz gy identifying the positions of three constrictions and one spacer.
5.6 Discussion
5.6.1 Relationship to C-Scan Display
Eddy current data is collected rotationally every 1 degree (∼ 0.9mm) and axially
every 1mm. High data-density (typically output in C-Scan display for eddy current
applications), has been exploited in the surface profiling process.
It is worthy to note that images formed by the computation of gx (and gy) differ
from C-Scan displays due to isolation of change in eddy current data, in the direction
of lift-off. C-Scan displays reveal the component of densely collected eddy current
data in one vector direction, usually orthogonal to lift-off. Images formed through
computation of gx and gy not only differ from C-Scan data by calculation of the
component of eddy current data parallel (not orthogonal) to the lift-off direction, but
5.6. DISCUSSION 61
make use of a maximum rate of change calculation, of eddy current signals in the
direction of lift-off (Equations 5.4 and 5.5). This key feature allows the visualization
of change in acquired eddy current data in the direction of lift-off. Without this,
features such as those visible in Figure 5.6, identifying position of constrictions would
be difficult to identify. Indeed, the position of constrictions cannot be identified from
C-Scan images (Figures 5.4 and 5.5) where the same eddy current data is employed
in generation of Figure 5.6.
5.6.2 Separability of Lift-off Localization in Rotary and Axial Directions
Although eddy current data is gathered in a helical trajectory by the eddy current
probe mounted on the fuel channel inspection head, changes in probe lift-off due to
mechanical interaction between the eddy current probe and pressure tube protrusions
are decomposed into rotary and axial directions. Indications visible in figures gen-
erated via computation of gx and gy represent probe lift-off in the rotary and axial
directions, respectively. gx can therefore be used to detect axially extending obstruc-
tions, and is robust against noise due to lift-off in the axial direction of probe motion.
Conversely, gy can be be used to detect rotationally extending obstructions and is ro-
bust against noise due to lift-off in the rotary direction of probe motion. As is shown
in Figures 5.3 and 5.6, rotationally extending obstructions (e.g. constrictions) do not
appear in the figure generated through computation of gx, while axially extending
obstructions (e.g. mechanical wear marks) do not appear in the figure generated by
gy. This demonstrates the high degree of separability between localization of probe
lift-off in rotary and axial directions, through computation of gx and gy.
5.7. CONCLUSIONS 62
5.6.3 Generalization of Results
Application of the techniques described in this study have focused on localization of
CANDU fuel channel component features. However, the surface profiling methods
provided can be generalized to other applications where the following requirements
are met: eddy current data with high data density is collected; direction of the lift-off
vector can be identified; the surface area of the eddy current probe is larger than the
dimensions of surface protrusions to be imaged; and, mechanical interaction between
the probe body and profiled surface is understood.
Separability of lift-off localization, allowing identification of protrusion orientation
relative to direction of probe motion is dependent on the number of degrees of freedom
of the probe. In the case of the eddy current probe used towards results presented in
this study, two degrees of freedom are allowed, and therefore, separability of lift-off
localization is available in two directions (axial and rotary directions of probe travel).
An increased number of degrees of freedom (e.g. tilt about the axial direction of probe
travel) will allow for increased characterization of protrusion orientation relative to
direction of probe motion.
5.7 Conclusions
Surface profiling of CANDU fuel channel pressure tube artefacts has been realized
through the methods laid out in this study. Mechanical wear marks, constrictions,
and local pressure tube deformation due to spacer loading can be identified through a
simple computational manipulation of high density eddy current data. These artefacts
range in size; mechanical wear marks have depressions ranging from 50µm to 100µm,
constrictions have protrusions in the same range, while garter spring spacers have
5.7. CONCLUSIONS 63
protrusions on the order of 200µm.
Detection capability of mechanical wear mark and constriction locations using
the presented surface profiling technique was shown to increase as compared to em-
ployment of C-Scan imaging. Mechanical wear marks were made clearly discernible
from background noise (SNR increased by over a factor of 3), while both mechani-
cal wear marks and constrictions could be independently localized, despite remaining
inseparable using C-Scan imaging.
Ultrasonic inspection data was used to independently verify the axial location
of garter spring spacers and pressure tube constrictions, against their location as
determined from the surface profile technique provided in this study.
Generalization of the presented technique towards profiling of surfaces other than
CANDU fuel channels has been considered. A base number of requirements for exten-
sion of the current set of technique applications have been identified. These are: high
data density of acquired eddy current inspection data; identification of lift-off vector
direction; greater surface area of the eddy current probe body, compared to dimen-
sions of surface protrusions to be imaged; and well understood mechanical interaction
between the probe body and the profiled surface.
Acknowledgements
The authors would like to thank, from Ontario Power Generation: Tulchand Harduwar
for supporting this work; as well as Andrew Hong and Jerry Piskorski for facilitating
access to source material.
5.7. CONCLUSIONS 64
Figure 5.9: a) gy computed from 8kHz eddy current data, averaged at the bottomof the pressure tube. b) dPT−ID computed from ultrasonic measurementdata.
65
Chapter 6
Modelling and Validation of Eddy Current
Response to Changes in Factors Affecting Pressure
Tube to Calandria Tube Gap Measurement
Shaddy Shokralla, Sean Sullivan, Jordan Morelli, and Thomas W. Krause
Abstract: Procedures employed to non-destructively examine nuclear power plants
must undergo inspection qualification to ensure that they meet their respective in-
spection specification requirements. Modelling is a powerful tool that can be exploited
in the inspection qualification process. The gap between pressure tubes (PTs) and
calandria tubes (CTs) in CANDU (CANada Deuterium Uranium) fuel channels is
periodically measured, as contact can result in localized cooling and potential crack-
ing. This work shows how an analytical model can be employed to characterize the
effects of PT wall thickness and resistivity variation on gap measurement, and details
Figure 6.7 shows the gap probe response rate of change with resistivity as a func-
tion of frequency, obtained from the slope at 55 µΩ · cm in the scaled model curve
in Figure 6.6 as well as slopes for 8kHz and 16kHz frequencies. Slope values are
shown in the second last column of Table 6.4. These results indicate that the effects
of resistivity on gap response are greater at higher frequencies, again consistent with
Equation 6.2.
The last column in Table 6.4 shows the ratio of rate of change of gap amplitude
with resistivity relative to that of wall thickness as obtained from model results. The
values show that the effect of resistivity, relative to wall thickness, on gap measure-
ment decreases with increasing frequency. The largest ratio arises at 4 kHz frequency
(25 µΩ · cm/mm) indicating that this frequency is the most impacted by changes in
resistivity when compared with response due to PT wall thickness variations. The
next largest, with similar order of magnitude (17 µΩ · cm/mm), is the 8 kHz ampli-
tude response. These results are consistent with the relative % error, presented in
the second column of Table 6.4. The relative % error shows a progressive increase
in % standard deviation with increasing frequency that results in increased scatter,
attributed to the small wall thickness variations in this tube.
6.5 Discussion
6.5.1 Modelling Parameter Continuity
Although experimental results have been employed to show agreement with a discrete
number of modeled results, an advantage to modelling effect of parameter variation
is the ability to model all values within a range of an essential parameter (or ranges
of essential parameters). This is obvious, considering voltage responses are through
6.5. DISCUSSION 82
Figure 6.6: Amplitude as a function of resistivity for (a) 4kHz, (b) 8kHz, (c) 16kHzfrequencies for scaled model and experimental data from composite tubes.
6.5. DISCUSSION 83
Figure 6.7: Variation of scaled model based on gap probe response amplitude (0.5to 16 mm gap) with resistivity at 55 µΩ · cm plotted as a function offrequency.
Dodds and Deeds equations [18, 16, 17], which are analytic and closed form. However,
this presents clear advantages to examining effects of essential parameters in the
context of inspection qualification. Effects of essential parameter variation can be
observed to an arbitrary level of specificity, which can be performed rapidly and
inexpensively, compared to performing a large number of experiments.
In the model results presented here, the relative sensitivity of the eddy current
based gap response to bounding pressure tube wall thickness and resistivity variations
has been studied. The model results have assisted in the identification of frequencies
most sensitive to either the effects of wall thickness or resistivity, relative to the desired
PT to CT gap response. This information can be applied to select the most robust
frequency, that which is most independent of the essential parameter variations, for
in-reactor gap measurement.
6.6. CONCLUSIONS 84
6.5.2 Parallel Plate Approximation
Voltage responses to wall thickness and gap have been modeled by Dodds and Deeds
equations [18, 16, 17], where the pressure tube and calandria tube geometries (relative
to the eddy current probe) have been approximated as parallel conductive plates. For
the probe configuration employed in this study, this has shown not to be a significant
source of error. However, if select probe parameters are modified, the parallel plate
approximation may cease to approximate pressure tube to calandria tube geometries
to sufficient accuracy. If, for example, frequency is lowered well below 4kHz, there is
potential for increased eddy current penetration for larger PT and CT surface areas,
which, relative to the probe surface, no longer approximate parallel plates. Similarly,
if the probe was situated away from the surface of the pressure tube, curvature of PT
and CT components relative to the probe surface would discount their approximation
as paraellel plates.
6.6 Conclusions
Validation of a model based on Dodds and Deeds equations [18, 16, 17] used to es-
timate eddy current voltage response to PT wall thickness and resistivity changes
(with varying PT to CT gap) has been performed via comparison with physical mea-
surements.
For wall thickness values between 3.48mm and 4.36mm, and gap values between
1mm and 16mm, average agreement exceeded 95 % for all frequencies used, despite
small machining deviations (microns) in the wall thickness of test pieces, which re-
sulted in an exaggeration of disagreement between measured and modeled results.
Standard deviation of the agreement ranged between 7.9 % (4kHz signal) and 18.5
6.6. CONCLUSIONS 85
% (16kHz signal). Increased standard deviation with frequency is a result of larger
sensitivity to wall thickness variations with higher freuquency. This follows from an
inverse relationship between eddy current skin depth and frequency.
Gap measurements performed on composite pressure tube (PT) sections with dif-
ferent resistivities (52.7, 55.0, 57.2 and 57.4 µΩ·cm) were compared with 2D modelling
results obtained with equivalent gap probe parameters. Gap signal amplitudes for a
0.4 mm minimum to 16.3 mm maximum gap were evaluated at axial positions in
composite PT sample sections where pressure tube wall thickness was in the range of
4.0 to 4.1 mm and differences in wall thickness between 0.4 and 16.3 mm gap were
at a minimum (0.01 mm). The minimum wall thickness variation criterion was main-
tained for the probe, with its sensing area of 12mm x 11mm, giving good agreement
between measured and scaled model results. Results demonstrated consistent agree-
ment between modeled and laboratory measured dependence of gap signal amplitude
variation with resistivity over all frequencies. The observed agreement validates the
2D analytical models for the range of resistivities tested at a nominal wall thickness
of 4 mm.
Validated employment of an analytical model to measure the effect of essential
parameter variation on inspection system output has significant implications for the
inspection qualification process. Modelling can serve as a versatile and efficient tool
in quantifying the effect of essential parameter variation on system performance. This
is necessary to determine if a system meets its inspection specification requirements.
6.6. CONCLUSIONS 86
Acknowledgements
The authors wish to thank: Ontario Power Generation management for supporting
this work; as well as Stuart Craig from Atomic Energy of Canada Limited and John
Sedo for valuable discussions.
87
Chapter 7
Principal Components Analysis of Multi-frequency
Eddy Current Data Used to Measure Pressure
Tube to Calandria Tube Gap
Shaddy Shokralla, Jordan Morelli, and Thomas Krause
Abstract: Principal components analysis (PCA) involves transforming a set of cor-
related observations into a set of linearly uncorrelated variables, which can reveal
simplified trends in data. Multifrequency eddy current testing contains correlations
across different test frequencies. In this work, PCA is used to extract unique informa-
tion from multifrequency eddy current data sets, used to measure the pressure tube
(PT) to calandria tube (CT) gap, in CANDU (CANada Deurterium Uranium) fuel
channels. Advantages include compressed data acquisition, allowing for increased in-
spection speed, and monitoring for variation in physical parameters using a reduced
number of variables. PCA employing analytical input model data is validated against
PCA employing data from physical experiments.
Keywords: principal components analysis, multi-frequency, eddy current, pressure
tube, gap
7.1. INTRODUCTION 88
7.1 Introduction
Multi-frequency eddy current non-destructive evaluation (NDE) is employed to pro-
vide enhanced sensitivity to test surface condition, over single frequency eddy current
NDE [52, 76]. Advantages are: different depths of penetration for each frequency,
varying sensitivity to lift-off or fill factor, variable noise sensitivity, ability to vary rel-
ative drive amplitudes and capability to perform multi-frequency mixing for removal
of signal artefacts.
Measurement of pressure tube (PT) to calandria tube (CT) gap, necessary for
monitoring the separation between PT and CT for CANDU fuel channels is one
application that has exploited multi-frequency eddy current NDE to provide for col-
lectively greater sensitivity to changes in PT to CT gap [3]. Different excitation
frequencies (4 kHz, 8 kHz, and 16 kHz) respond with varying sensitivity to changes
in PT to CT gap, in addition to other confounding factors, and collectively provide
enhanced response over application of only an individual frequency, in addition to
providing additional measurement redundancy. Factors affecting pressure tube to
calandria tube gap measurement have been analytically modelled using Dodd and
Deeds equations [16, 77]. This has been shown to provide practical benefit to the
inspection qualification process [20, 77], including the ability to model eddy current
response to any combination of relevant factors (PT wall thickness, PT resistivity,
and PT to CT gap) affecting gap measurement.
Principal components analysis (PCA) involves employing orthogonal transforma-
tions to convert a set of observations, which are correlated, into a set of linearly
uncorrelated variables, called principal components [11, 12]. The main advantage of
PCA is that complex data sets can be reduced to data sets of lower dimension to
7.1. INTRODUCTION 89
reveal unapparent and/or simplified trends in the data [11, 12]. Particular to conven-
tional eddy current testing, principal components analysis (PCA) has been employed
to increase reliable interpretation of steam generator tube signals [37]. PCA has also
been applied to pulsed eddy current, and was first shown to provide enhanced classi-
fication of defects [38]. Further applications of PCA to pulsed eddy current include
detection of defects in multilayer aluminum lap joints [39, 40, 49], steel [42], and
Two sets of samples with changing pressure tube wall thickness, pressure tube resis-
tivity, and pressure tube to calandria tube gap were used for experiments discussed in
this paper. The first sample varied PT to CT gap continuously between 0.8 mm and
16 mm. Pressure tube to calandria tube wall thickness also varied, as two PT sections
had their inner diameters machined to different depths. To summarize, this sample
was made up of three sections with wall thicknesses 3.48 mm, 3.84 mm, and 4.36 mm
(resistivity of the PT was ∼ 53.8 µΩ · cm at 20 C). The experimental configuration
is described in detail elsewhere [77].
The other sample set that was used included changes to PT resistivity and gap.
7.6. EXPERIMENTAL VALIDATION 103
Two different pressure tube samples, each with differing resistivity (51.38 µΩ · cm
and 55.95 µΩ · cm) were cut and welded, and their respective wall thicknesses were
machined such that they were equal within a few microns. A composite pressure tube
was housed within a calandria tube such that the gap varied continuously, from just
above 0 mm to just below 16 mm [77].
7.6.1 Wall Thickness Variation
Validation of model responses to changes in wall thickness was performed by compar-
ison of modeled responses to change in both wall thickness and gap, against physical
measurements, where both wall thickness and gap were varied. Physical measure-
ments of eddy current responses to changes in wall thickness and gap employed in
the comparison made use of gap values (listed in Table 7.1) of 1.0 mm and greater,
as well as wall thickness values of 3.48 mm and 4.36 mm.
PCA and Regression
Weighted PCA (where the inverse variable variances are used as weights) was per-
formed on physical measurements of eddy current responses to changes in PT wall
thickness and PT to CT gap (detailed in Section 7.6). The percentage of the total
variance explained by each principal component (ordered in terms of significance) is
as follows: 87.94%, 11.66%, 0.36%, 0.03%, 0.00%, and 0.00%.
A polynomial surface of the form
f(x, y) = p00 + p10x+ p01y + p20x2 + p11xy + p02y
2, (7.8)
where x represents PT to CT gap (mm), and y represents PT resistivity (µΩ ·cm),
7.6. EXPERIMENTAL VALIDATION 104
is fit to the first principal component score values shown in Figure 7.6 (blue circles),
such that f(x, y) = 48.82 + 0.91x− 22.87y − 0.01694x2 − 0.1077xy + 2.439y2.
Towards validating the model data against experimental data, PCA was performed
on a subset of the model data, where parameters were restricted to those with values
near the physical parameter values present in the physical experiment, where PT wall
thickness and PT to CT gap were varied. More specifically, PT resistivity was fixed at
52.4 µΩ ·cm, while CT resistivity was fixed at 74 µΩ ·cm. The percentage of the total
variance explained by each principal component (ordered in terms of significance) is
as follows: 87.98%, 11.90%, 0.09%, 0.03%, 0.00%, and 0.00%.
A polynomial surface of the same form used in surface regression for PCA of
experimental data (Equation 7.8), is fit to the first principal component score values
for the model data, shown in Figure 7.6 (yellow data), such that f(x, y) = 52.05 +
1.862x− 24.91y − 0.02486x2 − 0.3097xy + 2.784y2.
Examination of Figure 7.6 reveals that polynomial fits with respect to the exper-
imental and model data (restricted to approximate physical parameter values) are
similar, serving to validate the PCA approach for PT wall thickness and PT to CT
gap variations.
7.6.2 Resistivity Variation
Eddy current scans of a composite pressure tube, having two sections with different
resistivities (51.38 µΩ · cm and 55.95 µΩ · cm) were taken. A calandria tube enclosed
the pressure tubes during the scan, to simulate in-channel conditions. For each axial
position along the pressure tube, gap varied uniformly between minimum and max-
imum (i.e. cross sectional gap profiles are identical along the pressure tube in the
7.6. EXPERIMENTAL VALIDATION 105
Figure 7.6: Component 1 score with regression surfaces, for variations in PT wallthickness and PT to CT gap. Experimental data is shown in blue whileanalytical model data is shown in yellow.
axial direction).
PCA and Regression
Weighted PCA (where the inverse variable variances are used as weights) was per-
formed on physical measurements of eddy current responses to changes in PT resis-
tivity and PT to CT gap (detailed in Section 7.6.2). The percentage of the total
variance explained by each principal component (ordered in terms of significance) is
as follows: 81.46%, 18.05%, 0.43%, 0.05%, 0.01%, and 0.00%.
A polynomial of the same form as Equation 7.8, where x represents PT to CT
7.6. EXPERIMENTAL VALIDATION 106
gap (µΩ · cm), and y represents PT wall thickness (mm), is fit to the first princi-
pal component score values shown in Figure 7.7 (blue circles), such that f(x, y) =
Figure 7.7: Component 1 score with regression surfaces, for variations in PT resis-tivity and PT to CT gap. Experimental data is shown in blue whileanalytical model data is shown in yellow.
Towards validating the model data, PCA was performed on a subset of the model
data. Parameters were restricted to those with values near physical parameter values
present in the physical experiment, where PT resistivity and PT to CT gap were
varied. More specifically, PT wall thickness was fixed at 4.36 mm, while CT resistivity
was fixed at 74 µΩ · cm. The percentage of the total variance explained by each
principal component (ordered in terms of significance) is as follows: 88.32%, 11.11%,
0.54%, 0.03%, 0.00%, and 0.00%.
7.7. DISCUSSION 107
A polynomial surface of the form of Equation 7.8 is fit to the first principal
component score values shown in Figure 7.7 (yellow circles), such that f(x, y) =