## Old but Not Obsolete: Dimensional Analysis in Nondestructive Testing and
Evaluation

Antonello Tamburrino1,2, Alessandro Sardellitti1, Filippo Milano1, Vincenzo
Mottola1, Marco Laracca3, and Luigi Ferrigno1,4

1Dept. of Electrical and Information Engineering, University of Cassino and
Southern Lazio,
03043 Cassino (FR), Italy
e-mail: {antonello.tamburrino, alessandro.sardellitti, filippo.milano,
vincenzo.mottola<EMAIL_ADDRESS>2Department of Electrical and
Computer Engineering, Michigan State University, East Lansing, MI-48824, USA
3Dept. of Astronautics, Electrical and Energy Engineering, Sapienza University
of Rome,
00186 Rome, Italy
e-mail<EMAIL_ADDRESS>4Consorzio Nazionale Interuniversitario per
le Telecomunicazioni (CNIT), Italy

Abstract

This paper proposes dimensional analysis for solving Non–Destructive Testing &
Evaluation (NDT&E) problems. This is the first time that this approach has
been adopted in the framework of NDT&E, and the paper ushers in the
development of probes and methods for simultaneously estimating several
parameters with a simple approach.

The most important theorem of dimensional analysis is Buckingham’s theorem,
based on the concept that the laws of the physics do not depend on the
particular set of units chosen. At the core of this theorem is a systematic
reduction in the number of variables describing a physical problem. This
reduction is equal to $k$, the number of fundamental dimensions required to
describe the variables of the physical problem in its original setting. This
makes the approach ideal when the number of variables in the physical problem
is not much greater than $k$.

In this paper, we demonstrate the approach’s effectiveness in the simple
problem of simultaneously estimating the thickness and electrical conductivity
of a conducting plate, via Eddy Current Testing. The approach is original,
effective, efficient, and currently patent-pending. All the aspects, from
theory to experimental validation, are provided, and it is proved that the
proposed method achieves a very good degree of accuracy over a wide range of
thicknesses and electrical conductivities. Moreover, the proposed method is
compatible with the in-line and real-time estimation of thickness and
electrical conductivity in an industrial environment.

Keywords: Dimensional analysis; Buckingham’s theorem; Non–Destructive Testing
& Evaluation (NDT&E); Eddy Current Testing (ECT); Multi-parameter simultaneous
estimation; Thickness estimation; Electrical conductivity estimation.

##  1 Introduction

Problems related to Non–Destructive Testing and Evaluation (NDT&E) generally
involve several variables. Specifically, the outcome of an NDT&E test depends
on (i) the parameters describing the probe (geometry, materials, …), (ii) the
physical and geometrical parameters of the sample being tested, (iii) the
geometrical parameters describing the position of the probe with respect to
the sample being tested and (iv) environmental factors. The number of
variables involved and the correlated nature of these variables (e.g.,
excitation frequency and thickness estimation [1, 2, 3, 4]) make an NDT&E
problem difficult to handle.

To this end, a methodology that can systematically reduce the complexity of
the analysis of NDT&E problems by decreasing the number of variables involved,
plays a very important role.

For instance, this reduction in the number of variables has a major impact
when a physical problem is modelled via either a numerical or a machine
learning approach [5, 6, 7, 8, 9, 10, 11]. In both cases there is an
exponential reduction in the number of numerical simulations required or the
size of the training database.

To this purpose, dimensional analysis is a mathematical technique for
analyzing problems involving physical quantities [12].

Dimensional analysis can be used to simplify the analysis of complex equations
by highlighting the fundamental quantities describing a problem. Specifically,
by only analyzing the physical dimensions of the variables involved in an
equation, it is possible to determine a smaller number of fundamental
quantities describing the original problem. This simplifies the computation of
the solution to the original problem [13]. Dimensional analysis is commonly
used in physics, engineering and other sciences to derive equations and verify
experimental results [14, 15, 16]. The present paper is the first systematic
study on the beauty and effectiveness of dimensional analysis, other than [2]
where dimensional analysis was merely applied in a thickness estimation
problem.

Within dimensional analysis, Buckingham’s theorem plays a key role. This
theorem is grounded on the concept that the equations of physics cannot be
affected by the choice of the physical quantities’ units [17]. It states that
any physical law can be written in terms of dimensionless groups and provides
a procedure to identify these dimensionless parameters, also called groups.
The key is that the number of groups is smaller than that of the original
variables. For instance, if a physical problem is modelled by an equation of
the type $g\left(q_{1},\ldots,q_{n}\right)=0$ and the physical dimensions of
the $q_{i}$s are expressed by a set of $k$ fundamental dimensions, then
Buckingham’s theorem allows the original physical problem to be cast as
$G\left(\pi_{1},\ldots,\pi_{p}\right)=0$, where $p=n-k$ and
$\pi_{1},\ldots,\pi_{p}$ are the dimensionless group. Buckingham’s theorem
brings a problem to its fundamental form through the groups [14], reducing the
quantities involved [18] and decreasing the complexity of the mathematical
analysis of the problem of interest.

In the scientific literature, there are many original applications where
Buckingham’s theorem has been successfully applied. Despite being proposed a
long time ago, this theorem is still applied in science and engineering (“…
old but not obsolete …”, paraphrasing the sci-fi movie Terminator Genisys)
[19]. In [18], dimensional analysis was applied to processing the biological
cells by means of microfluidic devices. In [15], using dimensionless groups,
the authors studied the characteristics of different bearing parameters as
temperature varies. Furthermore, in [16] groups were adopted for a most
effective description of the characteristic parameters of the thermal balance
for the energy demand evaluation of a high-performance non-residential
building. The creation of a rapid impedance model for proton exchange membrane
fuel cells using physical and geometric parameters was analyzed in [20]. The
latter suggests defining dimensionless groups according to Buckingham’s
theorem so that the relationships between the fundamental dimensions and the
physical variables involved in the process under discussion can be adequately
described. This strategy was helpful in solving issues where first-principles
models are unknown, challenging to build, or impossible to compute. In [21],
methods were developed based on the use of Buckingham’s theorem for optimizing
tests inside wind tunnels. In [22] a comparative study on the wire electrical
discharge machining of reinforcement materials was realized by means of
Buckingham’s theorem. The theorem was used to model the input variables and
thermophysical characteristics of wire electric discharge machining on the
material removal rate and surface roughness of aluminum and steel.

Although this approach has been widely adopted in various physical
applications, to the best of our knowledge this methodology has never been
applied in the NDT&E context. In this paper, we propose a new methodology to
simultaneously estimate the thickness and the electrical conductivity of
conductive plates by means of ECT. The proposed approach is based on
dimensionless groups derived from the celebrated Buckingham’s theorem [17].

This specific application is motivated by an acknowledgment that the
measurement of the value thickness and electrical conductivity of conductive
materials is a crucial factor in all production and manufacturing processes
(e.g., heat treatment, rolling and pressing). Indeed, these two quantities
directly affect the quality properties of finished products, such as hardness,
toughness, and tensile strength [23, 24, 25, 26]. In this scenario, accurate
and real-time monitoring of the thickness and electrical conductivity of
conductive materials are essential to improve production quality and
efficiency. In-line measurement techniques are essential because they enable
automatic quality control during the production phase at reasonable prices and
short inspection times, as required in the Industry 4.0 paradigm.

The possibility of applying ECT methods to the simultaneous estimation of
several parameters, such as thickness, electrical conductivity and lift-off,
has been widely studied in the literature. ECT methods are characterized by
low-cost hardware and experimental set-up, contactless measurements, and
insensitivity to non-conductive materials such as paints, dust, etc. In [27] a
method was proposed to simultaneously measure the thickness and electrical
conductivity of the conductive sample, based on a single-frequency ECT method
analyzing the phase of mutual impedance. ECT sensing systems using anisotropic
magnetoresistive sensors for simultaneous estimation of thickness and
electrical conductivity was proposed in [28], while in [29] a new eddy current
sensing method with a material-independent model for coupled-parameter
estimation was proposed. An improved Newton iterative method to detect
thickness, electrical conductivity, permeability, and lift-off of the
conductive sample based on multi-frequency excitation ECT was developed in
[30]. Pulsed Eddy Current (PEC) techniques were also analyzed for multi-
parametric estimation. For instance, in [31] the possibility to determine the
thickness and electrical conductivity of conductive coatings on conductive
plates was investigated using a PEC method, while in [32] a transient eddy-
current measurement approach was proposed.

In this paper, we use Buckingham’s theorem to simultaneously estimate the
thickness and electrical conductivity of conductive plates. In particular, the
major contributions of the paper are:

  * •

a relationship in term of dimensionless groups expressing the measured
quantity as a function of the physical variables affecting it, in a frequency
domain ECT experiment. We assume that the measured quantity is the self or
mutual impedance of the probe. This assumption does not restrict the
generality of the method;

  * •

a method and its algorithm counterpart for the simultaneous estimation of the
thickness and electrical conductivity by using dimensionless groups;

  * •

a versatile experimental set–up for the simultaneous estimation of the
thickness and electrical conductivity, based on single-frequency or multi-
frequency measurements.

Compared with methods already established in the literature, the methodology
proposed in this paper offers a number of advantages. First, Buckingham’s
theorem makes it possible to reduce the number of variables to be considered,
thus leading to a reduction in the problem’s computational complexity. Second,
it makes it possible to establish the structure of the relationships existing
between the variables involved. Third, the proposed procedure is compliant
with applications where the simultaneous estimation is required under in-line
and real-time industrial conditions. Fourth, the proposed approach guarantees
excellent accuracy.

The paper is organized as follows. In Section 3 we briefly summarize
Buckingham’s theorem and show an example of its application in modelling a
simple RLC circuit. In Section 4 we apply Buckingham’s theorem to the specific
problem and derive the essential structure of the relationship between the
relevant variables in terms of groups. Section 5 provides the method for the
simultaneous estimation of the thickness and electrical conductivity of a
nonmagnetic plate. Section 7 contains descriptions of the experimental set–up,
case studies and experimental results. Finally, some conclusions are drawn in
Section 8.

##  2 Notations

In this work we assume the following standard notations:

$\bar{X}$

represents a complex number;

$\dot{Z}$

(upper-case) represents a complex impedance value;

$\mathbf{v}$

represents a real valued vector.

##  3 Buckingham’s theorem

Dimensional analysis includes the set of all methods useful in reducing the
dimension of the manifold representing a physical problem, before carrying out
a quantitative analysis. Buckingham’s theorem (1914) is a fundamental “tool”
to achieve this result [17]. Its root lies in previous publications by Lord
Rayleigh (1877), J. Bertrand (1878), A. Vaschy (1892) and D. Riabouchinsky
(1911). In essence, Buckingham’s theorem states that any physical law can be
expressed in terms of dimensionless parameters, called “ groups” , since
physical laws are independent of the system of units.

Buckingham’s theorem is stated as follows.

######  Theorem 1 (Buckingham’s theorem).

Let a physical problem involving $n$ dimensional scalar variables be modeled
by a scalar equation of the type:

$g\left(q_{1},q_{2},q_{3},\ldots,q_{n}\right)=0.$ (1)

Let the physical dimensions of all variables expressed in term of a set of $k$
fundamental dimensions $D_{1},\ldots,D_{k}$:

$\dim q_{i}=D_{1}^{a_{i1}}\times\ldots\times D_{k}^{a_{ik}},\,i=1,\ldots,n,$
(2)

then there exist $p=n-k$ dimensionless groups 1, 2, $\ldots$ , p such that (1)
can be cast in the form

$G\left(\pi_{1},\pi_{2},\pi_{3},\ldots,\pi_{p}\right)=0.$ (3)

Theorem 1 can be easily extended to other cases where the laws of physics are
described by vectors (tensors) quantities and/or multiple equations.

Buckingham’s theorem does not give the explicit expression for $G$, given $g$,
but, rather, has to be derived explicitly, after the groups have been
computed.

An example of physical dimensions is given by those from the SI base units: T
(time), L (length), M (mass), I (electric current), $\Theta$ (absolute
temperature), N (amount of substance) and J (luminous intensity). However, any
set of fundamental dimensions can be used in Theorem 1.

######  Remark 1.

Buckingham’s theorem for the special case when there is one dependent
variable, i.e. $q_{1}=f\left(q_{2},\ldots,q_{n}\right)$ gives

$\pi_{1}=F\left(\pi_{2},\ldots,\pi_{p}\right).$ (4)

Similarly, Buckingham’s theorem extends can be extended to the case of two or
more dependent variables.

A possible method to find the dimensionless groups is described in detail in
Appendix A. The RLC series circuit provides an example of application to make
the concepts crystal clear. In a RLC circuit the phasor $\bar{I}$ of the
electrical current circulating in the elements (see Figure 1) is a function of
the phasor of the voltage generator and of the passive components, i.e.

$\bar{I}=f\left(\bar{E},\omega,R,L,C\right)$ (5)

that, as shown in Appendix A, can be cast in dimensionless form as:

$\bar{\pi}_{1}=F\left(\pi_{2},\pi_{3}\right)$ (6)

where

$\bar{\pi}_{1}=\frac{R\bar{I}}{\bar{E}},\ \pi_{2}=\frac{\omega L}{R},\
\pi_{3}=\frac{1}{\omega RC}.$ (7)

Figure 1: The RLC series circuit in time-harmonic operation. The angular
frequency is $\omega$.

The total number of physical variables required to describe the RLC series
circuit reduces from $n=6$ in equation (5) to $p=3$ in equation (6), because
the dimensions of $\bar{I},\bar{E},\omega,R,L,C$ can be expressed through a
set of $k=3$ physical dimensions related to time, voltage and current.

The derivation of (6) and (7) does not rely on explicit knowledge of $f$.
However, Buckingham’s theorem does not provide the explicit expression for
$F$, but rather, it must be derived from knowledge of $f$ and the related
groups. For the RLC circuit, from

$\bar{I}=\frac{\bar{E}}{R+j\omega L-j/\omega C}$ (8)

and the groups in (7), it can be easily found that

$\frac{R\bar{I}}{\bar{E}}=\frac{1}{1+j\frac{\omega L}{R}-j\frac{1}{\omega
RC}},$ (9)

i.e. $F(\pi_{2},\pi_{3})=1/\left[1+j\left(\pi_{2}-\pi_{3}\right)\right]$.

######  Remark 2.

It is worth noting that the groups are not unique. The same law of physics can
be expressed by means of many different sets of groups. Indeed, products,
powers and ratios of groups are still groups. The selection of the proper set
of groups is related to the _specific_ application and the related solution
method. This makes dimensional analysis via Buckingham’s theorem extremely
versatile. The specific procedure sketched in Appendix A is one of several
possible equivalent choices. It guarantees that all the independent variables,
apart from the repeating ones, appears in _one and only one_ group.

The main advantage of Buckingham’s theorem consists in the reduction of the
number of relevant variables describing a problem from $n$ to $p$, where
$p=n-k$. This approach is especially effective when $n$ is of the order of a
few units. This is because $k$ is of the order of few units, $k\leq 7$ in the
SI, thus making $p$ a fraction of $n$ when $n$ is of the order of few units.
For instance, by analyzing the RLC circuit we have $n=6$ and $k=3$, thus
yielding $p=3$, that is one-half of $n$. This has a major impact in reducing
the amount of experimental and/or numerical data required to make correlations
of physical variables. Indeed, in order to fully characterize the function $f$
appearing in (6), the parameters array $\left(\bar{E},\omega,R,L,C\right)$ has
to be varied in $\mathbb{C}\times\mathbb{R}^{4}$, whereas the full
characterization of the function $F$ in (6) requires the parameters array
$\left(\pi_{2},\pi_{3}\right)$ to be varied in $\mathbb{R}^{2}$. This latter
option (evaluation of $F$) is definitely less expensive, in terms of the
number of experimental tests or numerical simulations, if compared to the
former (evaluation of $f$). Summing up, dimensional analysis is an extremely
powerful technique for formulating physics problems in their most basic forms
by minimizing the degrees of freedom of the problem.

Another advantage offered by Buckingham’s theorem is that its application does
not require a priori knowledge of the law relating the key physical
quantities. It can be applied starting from only the knowledge of the physical
variables describing the phenomenon. This is very important when the laws
constraining the physical quantities are unknown, and it provides a guide for
finding such laws.

##  4 Dimensional analysis in Eddy Current Testing

The possibility of applying ECT to the simultaneous estimation of multiple
parameters, such as thickness and electrical conductivity, has long attracted
many researchers’ interest.

An Eddy Current Probe (ECP) typically comprises a driving coil that generates
a time-varying magnetic flux density, which in turn induces a current density
in a conductive material, and a receiver coil or field sensor to sense the
reaction magnetic flux density due to the induced eddy currents. Probes of
different shapes and arrangements may also be adopted, based on a single or
multiple coils to produce the driving field and to measure the response of the
material being tested [27, 30, 31, 32]: one coil for the driving field and a
magnetic flux density sensor to sense the response [28, 29], and so on. Tests
can be performed by adopting single-frequency or multiple-frequency approaches
and analyzing the various quantities measured, such as the variation in
magnetic flux density due to the presence of the material being tested or the
variation in self-impedance (for a single coil application) or mutual-
impedance (for a multiple coil application). In all these ECT scenarios,
Buckingham’s theorem can be profitably applied for a thorough understanding of
the relationships between the variables describing the physical problem and to
simplify the model by reducing the number of degrees of freedom required to
describe the system.

The effectiveness of Buckingham’s theorem in the simultaneous evaluation of
thickness and conductivity via ECT data can be demonstrated by a specific case
study without loss of generality. This involves an ECP comprising two coaxial
coils (T/R configuration) with the upper coil used as the driving coil and the
lower coil as the pick-up coil. Coaxial coils and nonmagnetic materials have
been assumed. Figure 2 shows the geometry of the problem, with both the
conductive plate and the ECP.

In the case of interest (T/R coil configuration), the measured quantity is
$\Delta\dot{Z}_{m}=\dot{Z}_{m,plate}-\dot{Z}_{m,air}$, i.e. the difference in
mutual impedance between the coils when the ECP is located on the plate
($\dot{Z}_{m,plate}$) and in air ($\dot{Z}_{m,air}$), respectively, at a
prescribed angular frequency. Hereafter we assume coaxial coils and
nonmagnetic materials. The key physical quantities determining
$\Delta\dot{Z}_{m}(\omega)$ are (see Figure 2):

  * •

the parameters describing the geometry of the probe: the internal $r_{1}$ and
external $r_{2}$ radii of the coils, the height $h_{1}$ and number of turns
$N_{1}$ of the receiving coil, the height $h_{2}$ and number of turns $N_{2}$
of the driving coil and the separation between the coils $d$;

  * •

the angular frequency $\omega$ of the driving current applied for the test;

  * •

the thickness $\Delta h$ and the electrical conductivity $\sigma$ of the
conductive plate;

  * •

the magnetic permeability of the vacuum $\mu_{0}$ and the corresponding
magnetic reluctance $\nu_{0}=1/\mu_{0}$.

  * •

the lift-off $l_{0}$ between the plate and the ECP and the tilting of the ECP
probe w.r.t. the perpendicular to the plate.

The geometrical parameters of the ECP are normalized with respect to a length
$D=r_{2}$ and are grouped in a dimensionless vector
$\mathbf{t}=\left(r_{1}/D,h_{1}/D,h_{2}/D,d/D\right)$. The normalization
constant $D$ represents the size of the probe. Other choices for $D$ can
equally be made.

Figure 2: Representation of the axis-symmetrical ECP placed on the conductive
plate and their geometrical characteristics.

All the listed parameters (probe geometry, number of turns in the coils,
conductivity and thickness of the plate, lift-off and tilting of the ECP with
respect to the plate) affect the mutual impedance between the transmitting and
receiving coils, i.e.

$\frac{\Delta\dot{Z}_{m}}{N_{1}N_{2}}=f\left(\omega,\sigma,\nu_{0},\Delta
h,D,\textbf{t},l_{0},\theta\right).$ (10)

The evaluation of $f$, either by a numerical method or an experimental
campaign, is no simple task, because its cost/time increases exponentially
with the number of arguments.

Equation (10) involves a total of nine variables ($n=9$), with seven real-
valued and scalar independent variables $\omega,\sigma,\Delta
h,D,l_{0},\theta,\nu_{0}$, one real-valued vector independent variable
$\mathbf{t}$ and one complex-valued dependent variable
$\Delta\dot{Z}_{m}/N_{1}N_{2}$. Variables $D$ and $\mathbf{t}$ correspond to
five real-valued and scalar variables $(r_{1},r_{2},h_{1},h_{2},d)$. All
variables can be expressed in terms of three fundamental dimensions ($k=3$),
length $L$, time $T$ and impedance $\Omega$, as shown in Table
LABEL:tab:Variable_thickness.

Table 1: Dimensional variables in the multi-parameter estimation problem, expressed in terms of fundamental dimensions. Numbers | Parameters | Symbols | Fundamental dimensions
---|---|---|---
1 | impedance variation | $\Delta\dot{Z}_{m}$ | [$L^{0}T^{0}\Omega^{1}$]
2 | electric conductivity | $\sigma$ | [$L^{-1}T^{0}\Omega^{-1}$]
3 | magnetic reluctance of the vacuum | $\nu_{0}$ | [$L^{1}T^{-1}\Omega^{-1}$]
4 | angular frequency | $\omega$ | [$L^{0}T^{-1}\Omega^{0}$]
5 | thickness | $\Delta h$ | [$L^{1}T^{0}\Omega^{0}$]
6 | lift-off | $l_{0}$ | [$L^{1}T^{0}\Omega^{0}$]
7 | ECP characteristic length | D | [$L^{1}T^{0}\Omega^{0}$]
8 | ECP shape | t | [$L^{0}T^{0}\Omega^{0}$]
9 | probe orientation features | $\theta$ | [$L^{0}T^{0}\Omega^{0}$]

Buckingham’s theorem makes it possible to obtain six dimensionless groups
($p=n-k=6$) as listed in Table LABEL:tab:pi_groups_thick_final (see Appendix A
for details), where $\nu_{0}$, $\omega$ and $D$ have been assumed as repeating
variables. Each dimensionless variable of the original problem, $\mathbf{t}$
and $\theta$, is assigned to a dimensionless group, i.e.
$\mathbf{\pi}_{5}=\mathbf{t}$ and $\pi_{6}=\theta$. Consequently, (10) can be
expressed as $\pi_{1}=F\left(\pi_{2},\pi_{3},\pi_{4},\pi_{5},\pi_{6}\right)$,
that is

$\displaystyle\frac{\Delta\dot{Z}_{m}\nu_{0}}{N_{1}N_{2}\omega D}$
$\displaystyle=F\left(D\sqrt{\frac{\omega\sigma}{2\nu_{0}}},\frac{\Delta
h}{D},\frac{l_{0}}{D},\mathbf{t},\theta\right)$ (11)
$\displaystyle=F\left(\frac{D}{\delta},\frac{\Delta
h}{D},\frac{l_{0}}{D},\mathbf{t},\theta\right),$ (12)

where $F$ is a proper function and the skin-depth $\delta$ is equal to

$\delta=\sqrt{\frac{2\nu_{0}}{\omega\sigma}}.$ (13)

The choice of the groups made in Table LABEL:tab:pi_groups_thick_final is not
unique, as highlighted in Remark 2. The choice of a specific set of groups
depends on the specific application.

Table 2: List of the groups for the case of interest. Dimensionless groups
---
$\bar{\pi}_{1}=\frac{\Delta\dot{Z}_{m}\nu_{0}}{N_{1}N_{2}\omega D}$ | $\pi_{2}=D\sqrt{\frac{\omega\sigma}{2\nu_{0}}}$ | $\pi_{3}=\frac{\Delta h}{D}$
$\pi_{4}=\frac{l_{0}}{D}$ | $\pi_{5}=\mathbf{t}$ | $\pi_{6}=\theta$

By taking into account that the purpose of this study is to measure the
thickness and the electrical conductivity of a conductive plate, given the
characteristics $\mathbf{t}$ of the ECP and the lift-off $l_{0}$ and tilting
$\theta$, it is convenient to focus on groups $\bar{\pi}_{1}$, $\pi_{2}$ and
$\pi_{3}$ since $\pi_{4}$, $\pi_{5}$ and $\pi_{6}$ are known. Consequently,
the final dimensionless relationship under analysis is:

$\frac{\Delta\dot{Z}_{m}\nu_{0}}{N_{1}N_{2}\omega
D}=F\left(D\sqrt{\frac{\omega\sigma}{2\nu_{0}}},\frac{\Delta h}{D}\right),$
(14)

where, with a slight abuse of notations, the values of $\pi_{4}$, $\pi_{5}$
and $\pi_{6}$ are understood.

Equation (14) has to be compared with the counterpart of (10) for prescribed
(understood) $\mathbf{t}$, $l_{0}$ and $\theta$:

$\frac{\Delta\dot{Z}_{m}}{N_{1}N_{2}}=f\left(\omega,\sigma,\nu_{0},\Delta
h,D\right).$ (15)

The impact of Buckingham’s Theorem is relevant. First, by starting from (15),
which involves a complex function of five real arguments, it is possible to
obtain an equivalent relationship requiring a complex function $F$ of two real
arguments, without the explicit knowledge of the original function $f$. The
new (reduced) function $F$ can be easily computed numerically or measured
experimentally, since it is defined in $\mathbb{R}^{2}$ rather than in
$\mathbb{R}^{5}$. Moreover, by combining (14), with the groups of Table
LABEL:tab:pi_groups_thick_final, it is easy to realize that $f$ and $F$ are
closely related. Indeed, it results that

$f\left(\omega,\sigma,\nu_{0},\Delta h,D\right)=\frac{\omega
D}{\nu_{0}}F\left(D\sqrt{\frac{\omega\sigma}{2\nu_{0}}},\frac{\Delta
h}{D}\right),$ (16)

thus the computation or experimental evaluation of $F$ gives the values of
$f$.

Second, Buckingham’s Theorem allows the inverse problem to be represented in
the two-dimensional $(\pi_{2},\pi_{3})$ plane, rather than in a five-
dimensional space. The latter remark underpins the method for the simultaneous
estimation of $\sigma$ and $\Delta$ via level curves, described in Section 5.

Finally, the same approach can be applied to the case of a single coil ECP or
to the case in which the reaction magnetic flux density is measured by a field
sensor.

##  5 Simultaneous estimate of thickness and electrical conductivity via
dimensionless groups

In this Section the effectiveness of the concept of dimensional analysis is
demonstrated with reference to the problem of the simultaneous estimation of
the electrical conductivity and thickness of a metallic plate. This is a
relevant problem, from the practical perspectives.

This Section is organized in two parts. The first part proposes an approach
based on level curves, which can be easily introduced, thanks to the key role
played by the groups. The second part is devoted to “translating” the method’s
physical limits in terms of level curves.

Without loss of generality, a planar geometry has been considered in order to
demonstrate the effectiveness of dimensional analysis. The same treatment can
be applied to non-planar geometries like tubes.

###  groups, level curves and estimation method

Thanks to the abstract representation of (14), where the (complex) _measured
quantity_ $\bar{\pi}_{1}$ is a function defined in the $(\pi_{2},\pi_{3})$
plane, it is possible to introduce a set of level curves with respect to
$\bar{\pi}_{1}$. This is possible because the measured quantity $\dot{Z}_{m}$
and the unknowns $\sigma$ and $\Delta h$ are not mixed in the groups.

In order to obtain a level curve in the $(\pi_{2},\pi_{3})$ plane, it is
necessary to prescribe the value of a real valued quantity. For instance, the
real part, the imaginary part, the magnitude, the phase of $\bar{\pi}_{1}$, or
any other real function of $\bar{\pi}_{1}$ can be chosen. Figure 3 shows the
level curves for the four basic quantities: $\Re{\bar{\pi}_{1}}$,
$\Im{\bar{\pi}_{1}}$, $|\bar{\pi}_{1}|$ and $\phase{\bar{\pi}_{1}}$, in the
$(\pi_{2},\pi_{3})$ plane. The parameters for the underlying ECP are provided
in Table LABEL:tab:geometrical_param.

Figure 3: Level curves for $\bar{\pi}_{1}$: (a) $\Re{\bar{\pi}_{1}}$, (b) $\Im{\bar{\pi}_{1}}$, (c) $|\bar{\pi}_{1}|$, and (d) $\phase{\bar{\pi}_{1}}$. The level curves were plotted at constant step, i.e. the difference between the values for two consecutive level curves is constant. Smaller gradients are found where the distance between the curves increases. Table 3: Values of the parameters of the ECP. Parameter | Value
---|---
$h_{1}=h_{2}$ | 6 [mm]
$d$ | 2.20 [mm]
$r_{1}$ | 23.60 [mm]
$r_{2}$ | 23.95 [mm]
$l_{0}$ | 1 [mm]
$N_{1}=N_{2}$ | 17
$\theta$ | 0 [°]

The level curves in Figure 3 were obtained from a numerical evaluation of
$F(\pi_{2},\pi_{3})$ in a range of values for $\pi_{2}$ and $\pi_{3}$. The
numerical evaluation was carried out by evaluating $\Delta\dot{Z}_{m}$ via the
the semi-analytical model by Dodd and Deeds [33]. $\pi_{2}$ was varied in the
range $\left[2.82;\ 28.2\right]$, whereas $\pi_{3}$ in the range
$\left[4.2\times 10^{-3};\ 42\times 10^{-3}\right]$. These ranges were
obtained by retaining $\sigma$ constant and varying $\Delta h$ and $\omega$,
as in Table LABEL:tab:physical_param. The thickness $\Delta h$ was varied with
increments of 10 $\mu$m, whereas the frequency was varied with increments of 2
Hz.

Table 4: The physical parameters used for the numerical evaluation of $F(\cdot,\cdot)$. Parameters | Value
---|---
$\sigma$ | 35 MS/m
$\Delta h$ | 0.1 ÷ 10 [mm]
$\omega/2\pi$ | 100 ÷ 10000 [Hz]

Level curves are a powerful tool to solve equation

$F(\pi_{2},\pi_{3})=\bar{\pi}_{1}.$ (17)

Specifically, from the measurement of $\dot{Z}_{m}$ at a prescribed angular
frequency $\omega$, it is possible to compute $\bar{\pi}_{1}$ as
$\Delta\dot{Z}_{m}\nu_{0}/N_{1}N_{2}\omega D$. Then, from the specific value
of $\bar{\pi}_{1}$, it is possible to solve (17) by finding the intersection
point between the level curves from two or more different plots of Figure 3,
as shown in Figure 4.

Figure 4: Intersection of level curves from the mutual impedance measured at
an individual prescribed frequency. The plate has an electrical conductivity
of $18$ MS/m and a thickness of $2$ mm. The frequency is 1650 Hz.

Once $\pi_{2}$ and $\pi_{3}$ have been evaluated, the unknown electrical
conductivity $\sigma$ and thickness $\Delta h$ can be evaluated as

$\displaystyle\sigma$
$\displaystyle=\frac{2\nu_{0}}{\omega}\left(\frac{\pi_{2}}{D}\right)^{2}$ (18)
$\displaystyle\Delta h$ $\displaystyle=D\pi_{3}.$ (19)

The step-by-step algorithm is:

  * •

Measure $\Delta\dot{Z}_{m}$ at a prescribed $\omega$;

  * •

compute $\bar{\pi}_{1}=\Delta\dot{Z}_{m}\nu_{0}/N_{1}N_{2}\omega D$;

  * •

find the level curves for at least two plots of Figure 3;

  * •

find the intersection point $(\pi_{2},\pi_{3})$ for the selected level curves;

  * •

compute $\sigma$ and $\Delta h$ via (18) and (19).

Summing up, dimensional analysis makes it possible to cast the problem of
retrieving $\sigma$ and $\Delta h$ in very simple terms as the intersection of
level curves in a plane. This is because the five primary parameters
$(\omega,\sigma,\nu_{0},\Delta h,D)$ influencing the measured data combine in
the very compact form given by groups $\pi_{2}$ and $\pi_{3}$, rather than
individually as in (15). For the same reason, i.e. that the influence
parameters combine in a compact form, it is computationally feasible to
numerically compute the function $F(\cdot,\cdot)$, which is a function of two
parameters rather than $f(\cdot,\cdot,\cdot,\cdot,\cdot)$, which depends on
five parameters. Finally, $F(\cdot,\cdot)$ can be pre-computed and stored once
for all, given $\mathbf{t}$, $l_{0}$ and $\theta$.

###  Processing multiple measurements

Noise is a major issue when dealing with experimental data. Method accuracy
can be increased by processing the impedance measured at multiple frequencies.
In this case, the level curves intersection procedure has to be repeated at
each angular frequency, thus obtaining a set of points in the
$(\pi_{2},\pi_{3})$ plane, as shown in Figure 5.

Figure 5: Intersection of level curves from the mutual impedance measured at
five different frequencies. The plate has an electrical conductivity of $18$
MS/m and a thickness of $2$ mm. The frequencies are 650, 1150, 1650, 2150 and
2650 Hz.

Since $\pi_{2}$ is independent of $\omega$, whereas $\pi_{3}$ depends on
$\omega$, as it is proportional to $\sqrt{\omega}$, the intersection points
are distributed along a horizontal line. Each intersection point related to
the impedance measured at the $i-$th angular frequency $\omega_{i}$
corresponds to an estimate $\sigma_{i}$ and $\Delta h_{i}$ of the electrical
conductivity and thickness of the plate. The “final” estimate of $\sigma$ and
$\Delta h$ can be obtained by processing all the $\sigma_{i}$s and the $\Delta
h_{i}$s with an improved robustness as it combines the information from
different frequencies.

Alternatively, it is possible to plot the level curves on the $(\sigma,\Delta
h)$ plane, by means of (18) and (19). The latter strategy is extremely
convenient because _all_ the level curves intersect at the _same_ point
regardless the angular frequency, as shown in Figure 6. The intersection point
gives the estimate of $\sigma$ and $\Delta h$.

Figure 6: Representation on the normalized plane of the frequency measurements
obtained in the case of a plate with thickness of 2 mm and electrical
conductivity of 18 MS/m for the level curves $\Re{\bar{\pi}_{1}}$,
$\Im{\bar{\pi}_{1}}$, $|\bar{\pi}_{1}|$, and $\phase{\bar{\pi}_{1}}$ using
five different excitation frequencies.

###  Regions of operation

The $(\pi_{2},\pi_{3})$ plane can be divided into different regions, yielding
different information that can be inferred from the measured data.

Three basic conditions must be considered:

  1. 1.

the skin-depth $\delta(\omega)$ is sufficiently smaller than $\Delta h$. i.e.
$\pi_{2}\pi_{3}$ is sufficiently larger than 1;

  2. 2.

the size of the probe $D$ is much smaller than $\Delta h$, i.e. $\pi_{3}\gg
1$;

  3. 3.

the skin-depth $\delta(\omega)$ is much larger than $\Delta h$, i.e.
$\pi_{2}\pi_{3}\ll 1$ and the size of the probe $D$ is much larger than
$\Delta h$, i.e. $\pi_{3}\ll 1$.

In the first case (regions (c), (f) and (i) of Figure 8), the skin-depth is
smaller than $\Delta h$ and this prevents the thickness $\Delta h$ being
retrieved from the data (see [34]), i.e. the dimensionless impedance
$\bar{\pi}_{1}$. This behaviour can be easily recognized in the plots of
Figure 3, where the level curves become almost vertical, meaning that the
$\pi_{3}$ does not affect the measured data $\bar{\pi_{1}}$, i.e. $\Delta h$
cannot be retrieved from knowledge of $\bar{\pi_{1}}$. We found numerically
that the region where the level curves are almost vertical corresponds to
$\pi_{2}\pi_{3}>3$, as shown in Figure 7. However, in these regions, the
electrical conductivity can still be retrieved from $\bar{\pi}_{1}$, since a
change in the electrical conductivity, i.e. in $\pi_{2}$, determines a change
in $\bar{\pi}_{1}$.

Figure 7: The level curves above the curve $\pi_{2}\pi_{3}=3$ are almost
vertical.

In the second case (regions (a), (b) and (c) of Figure 8), the probe is
geometrically too small to interact with the bottom of the plate, regardless
of the skin-depth, i.e. regardless of the value of $\pi_{2}=D/\delta$. This
prevents the thickness $\Delta h$ being retrieved from $\bar{\pi}_{1}$, but
the electrical conductivity $\sigma$ can still be retrieved. As in the
previous case, the level curves are almost vertical in regions (a), (b) and
(c).

In the third case (region (g) of Figure 8, the probe is much larger than the
thickness $\Delta h$ and the plate is fully penetrated by the electromagnetic
field. In this case it is only possible to retrieve the surface electrical
conductivity of the plate, i.e. the $\sigma\Delta h$ product, because the
electromagnetic field only depends on this quantity, as highlighted in [2,
35].

In the remaining regions (d), (e) and (h) of Figure 8, it is possible to
retrieve both the electrical conductivity $\sigma$ and the thickness $\Delta
h$, starting from the dimensionless impedance $\bar{\pi_{1}}$. This is the so-
called _feasibility_ region, where the largest amount of information can be
retrieved from the measured data. Region (h) deserves to be highlighted,
because it opens to the possibility of measuring both the electrical
conductivity and the thickness of thin and very thin plates.

The plot of Figure 7 makes it possible to evaluate (i) whether or not a
measurement at a specific frequency has to be discarded and (ii) whether or
not the specific probe is suitable. Indeed, once the dimensionless impedance
han been measured, if the related level curve does not cross a prescribed
region(s) of interest, for instance region (d), (e) or (h) for the
$\sigma$-$\Delta h$ estimate, then the measurement has to be discarded.
Similarly, if several/many curves do not cross the region(s) of interest, it
may be the case that _all_ the frequencies are out of the proper range or the
size of the probe is not suitable for the specific plate. In any case, the key
point is that the data itself makes it possible to understand whether or not
the measurement is suitable to be processed, despite the lack of knowledge of
the unknown parameters.

The plots of the level curves provide further precious but less recognized
information. Specifically, at the higher frequencies where the thicknesses
cannot be retrieved (regions (c), (f) and (i) of Figure 8), the spacing
between the level curves increases, as shown in Figure 3. This means that the
gradient of the measured data $\bar{\pi}_{1}$ with respect to $\pi_{2}$
decreases for increasing $\pi_{2}$. In other words, whith large angular
frequencies and/or electrical conductivities, the sensitivity of the
measurement with respect to $\sigma$ decreases, even though some authors
claimed that it is convenient to estimate the electrical conductivity in such
conditions, because the data does not depend on the thickness $\Delta h$ [36].

Figure 8: The different regions of operations. The basic constraints
$\pi_{3}\gg 1$, $\pi_{2}\pi_{3}\ll 1$ and $\pi_{3}\ll 1$ are represented as
$\pi_{3}\geq k$, $\pi_{2}\pi_{3}\leq 1/k$ and $\pi_{3}\leq 1/k$, being $k\gg
1$. Here $k=10$.

##  6 The proposed measurement procedure

The measurement procedure consists of three main phases subdivided into
elementary steps, as outlined in Figure 9.

The first phase is carried out off-line, once the main parameters such as (i)
characteristic of the probe (ii), ranges of interest for the thickness and the
electrical conductivity to be estimated, and (iii) the range of interest for
the angular frequency, are defined. The first phase is characterized by three
tasks: parameters definition, numerical simulations, and experimental
calibration. This phase is performed once and for all, given the probe, and
the ranges of interest for the electrical conductivity, the thickness and the
angular frequencies.

The second phase is in-line and is characterized by two main steps: the
experimental execution of the test at a specific angular frequency; the
processing of the measured quantity to estimate the thickness and electrical
conductivity via the level curves of Figure 5 or 6. To improve the quality of
the thickness and electrical conductivity estimate, Phase 2 can be repeated at
different values of the angular frequency.

Finally, in the third phase, the final estimation of the thickness and
electrical conductivity is obtained by combining all the level curves from
data at different angular frequencies in Figure 6.

Figure 9: Operation flow-chart behind the $\Delta h-\sigma$ estimation
process.

In the following the details for each individual phase are provided.

Phase 1: off-line tasks

  1. Step # 1.1

Parameters definition In this task, the characteristics of both the SUT and
the ECP are defined. In particular, the following parameters are prescribed:
(i) geometry, dimensions and physical parameters of the ECP; (ii) ranges of
interest for both the thickness and the electrical conductivity; (iii) range
of interest for the driving angular frequency.

The main parameters used in this contribution are summarized in Tables
LABEL:tab:geometrical_param and LABEL:tab:physical_param.

  2. Task # 1.2

Numerical simulations This task entails the numerical computation of the
dimensionless impedance, with a view to the calculation of the level curves.
This numerical calculation is carried out with reference to the prescribed
parameters defined in Task # 1.1. In detail, for each thickness and frequency
value, the corresponding simulated value of $\Delta\dot{Z}_{m}$ is obtained
using the semi-analytic models developed by Dodd and Deeds [33]. Then, the
resulting values of the dimensionless groups ($\bar{\pi}_{1}$, $\pi_{2}$,
$\pi_{3}$) are calculated, according to their definitions shown in Table
LABEL:tab:pi_groups_thick_final.

During this phase, the key is to avoid under sampling $F(\cdot,\cdot)$ in the
region of interest, as guaranteed in the present contribution.

  3. Step # 1.3

Experimental calibration To process the experimental data via the numerically
computed level curves of Step # 1.2, a suitable calibration is needed. The aim
is to get a proper fit between the numerically computed data and the
experimentally measured data. As is well known, a mismatch between the
numerically computed and the experimental data is due to many factors, such as
the specific numerical model, the uncertainty in the knowledge of the
geometrical and physical characteristics of the ECP, experimental noise,
measurement uncertainty, and so on. In the specific case, by considering a
number of different reference plates with known electrical conductivity and
thickness, it was found that the ratio $c$ between the numerically computed
data and the experimentally measured data depends mainly upon the angular
frequency, i.e. $c=c(\omega)$.

Phase 2: in-line tasks

  1. Task # 2.1

Experimental execution of the test at a defined excitation frequency This task
consists of the experimental measurement of the mutual impedances
$\Delta\dot{Z}_{m,k}^{exp}$, at the $k-$th prescribed angular frequency
$\omega_{k}$. The value of $\omega$ is selected from the range defined in Task
# 1.1. This step can be carried out either at a single angular frequency or
repeated at other angular frequencies, to improve the accuracy of the estimate
of the unknown thickness and electrical conductivity. The SUT is assumed to
have an unknown thickness and electrical conductivity within the ranges of
Task # 1.1.

  2. Task # 2.2

Data processing to estimate thickness and electrical conductivity at each
individual frequency First, the calibration factor $c(\omega_{k})$ is applied
to the mutual impedance $\Delta\dot{Z}_{m,k}^{exp}$ measured at $\omega_{k}$.
Then, the corresponding dimensionless impedance $\bar{\pi}_{1,k}^{exp}$ is
evaluated together with the corresponding level curves for
$\real\\{\bar{\pi}_{1,k}^{exp}\\}$, $\imaginary\\{\bar{\pi}_{1,k}^{exp}\\}$,
$\real\\{\bar{\pi}_{1,k}^{exp}\\}$ and $|\bar{\pi}_{1,k}^{exp}|$, and the
intersection point $(\pi_{2,k}^{exp},\pi_{3,k}^{exp})$ between the curves. If
the position of the intersection point is within one of the regions of
interest (d), (e) or (h) of Figure 8, then the $k-$th estimate of the
electrical conductivity and thickness is given by $\Delta
h_{k}=D\pi_{3,k}^{exp}$ and
$\sigma_{k}=\frac{2\nu_{0}}{\omega_{k}}\left(\frac{\pi_{2,k}^{exp}}{D}\right)^{2}$.

Task 2 is repeated at a different angular frequencies if either
$(\pi_{2,k}^{exp},\pi_{3,k}^{exp})$ is not within one of the region of
interest (d), (e) or (h) of Figure 8, or to improve the accuracy of the
estimate of $\sigma$ and $\Delta h$, by leveraging measurements from different
frequencies.

Phase 3: Data processing for thickness and electrical conductivity estimation
in a multi-frequency approach

  1. This last task is carried out only for measurements at multiple angular frequencies to improve the accuracy of the estimate of $\sigma$ and $\Delta$. The final estimate can be achieved by combining level curves from different angular frequencies or, more simply, by averaging the $\sigma_{k}$s and $\Delta h_{k}$s estimated at each individual angular frequency $\omega_{k}$. Hereafter, for the sake of simplicity, the results arising from the latter choice are shown.

##  7 Experimental characterization of the proposed measurement method

###  Experimental set–up

The experimental set-up consists of an ECP, a waveform generator, a current
probe, two signal amplifiers, a data acquisition board and a Personal Computer
(PC). A schematic block diagram of the experimental set-up is shown in Figure
10.

Figure 10: Schematic block diagram of the experimental set-up.

The ECP consists of two coaxial coils, a driving coil and a receiver coil. The
geometry of the coils and their dimensions are provided in Figure 2 and Table
LABEL:tab:geometrical_param, respectively. An Agilent 33120A waveform
generator provides the source current to the ECP driving coil, which is sensed
by means of a Tektronix TCP202A current probe. Both the output of the current
probe, proportional to the driving current, and the output voltage at the
receiver coil, are conditioned by means of two SR560 Stanford Research System
low-noise signal amplifiers. A TIE-PIE Engineering Handyscope
$HS5-540XMS-W5^{TM}$ data acquisition board is adopted to digitize the output
signals provided by the two conditioning units (the signal amplifiers).

The experimental tests were carried out at several angular frequencies, to
perform a metrological characterization of the method. Specifically, the
frequencies from $300$ Hz to $3$ kHz, with a frequency step equal to $50$ Hz,
were considered. The RMS value of the applied current at each angular
frequency was $115$ mA. The conditioning unit consists of a bandpass filter
from $30$ Hz to $10$ kHz, a gain of $200$ for the output of the current probe
and a gain of $100$ for the output measured at the receiver coil. To optimize
the data processing in the time domain, signal digitization is performed by
adopting a sampling frequency $1000$ times the considered signal frequency and
acquiring $4$ periods of each signal. A script developed in the MATLABTM,
running on a dual-core PC, manages the automation of the measurement station
and performs the signal processing. The final output of the measurement
station is the mutual impedance $\Delta\dot{Z}_{m,k}^{exp}$.

###  Results and discussion

The experimental set-up described in Section 7.1 is used to make both the off-
line tasks (Step # 1.3) and the in-line tasks (Step # 2.1). As regards Step #
2.1, the tests were carried out on six plates with known thicknesses and
electrical conductivities. The main characteristics of the plates are listed
in Table LABEL:tab:plates_param.

Table 5: Main characteristics of analyzed plates.

Name code Metal alloy Electrical conductivity ($\widetilde{\sigma}$) [MS/m]
Thickness ($\widetilde{\Delta h}$) [mm] Plate dimensions [mm x mm] $\\#$a
Aluminium (2024-T3) 17.66 2.03 200 x 200 $\\#$b Copper 58.50 0.98 200 x 200
$\\#$c Aluminium (6061-T6) 28.23 1.97 200 x 200 $\\#$d Aluminium (AW-1050A)
35.27 1.03 250 x 250 $\\#$e Aluminium (AW-1050A) 35.44 2.93 250 x 250 $\\#$f
Aluminium (AW-1050A) 35.91 3.98 250 x 250

For each specific plate and driving angular frequency, 20 repeated
measurements were carried out to investigate repeatability in the estimation
of both electrical conductivity and the thickness.

(a)  (b)

(c)  (d)

(e)  (f)

Figure 11: Level curves corresponding to the experimental data for all six
plates in the $(\pi_{2},\pi_{3})$ plane, at the five angular frequencies
corresponding to the lowest estimation errors.

(a)  (b)

(c)  (d)

(e)  (f)

Figure 12: Level curves corresponding to the experimental data for all six
plates in the $(\sigma,\Delta h)$ plane, at the five angular frequencies
corresponding to the lowest estimation errors.

Figures 11 and 12 provide some examples of level curves corresponding to the
experimental data for all six plates, in both the $(\pi_{2},\pi_{3})$ and the
$(\sigma,\Delta h)$ planes. To each frequency corresponds one single
intersection of the level curves for the real part, imaginary part, magnitude
and phase of the dimensionless impedance. The position of the intersections of
the curves in the $(\pi_{2},\pi_{3})$ plane makes it possible to establish
whether or not data has to be retained. Indeed, when estimating $\sigma$ and
$\Delta h$, if the intersection does not fall in region (d), (e) or (h) of
Figure 8, the data has to be discarded. This criterion is very powerful
because it can be applied without _any_ knowledge of the unknown parameters
$\sigma$ and $\Delta h$. The position of the intersections in the
$(\sigma,\Delta h)$ makes it possible to evaluate the accuracy of the
estimate. Ideally, all intersections should be located in a single point. When
this is not the case, the spreading of the intersections gives the accuracy of
the estimate of the unknown quantities. Points that are “far” from the main
cluster, can be retained as outliers and, therefore, safely discarded.

Hereafter, we define $\sigma_{k}^{i}$ as the $i-th$ estimate of the electrical
conductivity at the $k-th$ angular frequency, where $i=1,\ldots,20$ (20
repeated measurements). Similarly, we define $\Delta h_{k}^{i}$ as the $i-th$
estimate of the thickness at the $k-th$ angular frequency.

To quantitatively analyze the performances levels of the proposed method, the
following figures of merit have been considered:

  * •

The average electrical conductivity $\overline{\sigma}_{k}$ and the average
thickness $\overline{\Delta h}_{k}$, obtained from the 20 repeated
measurements at the $k-$th angular frequency.

  * •

The absolute relative errors $\epsilon_{\sigma,k}$ and $\epsilon_{\Delta h,k}$
for the mean value of electrical conductivity and thickness, respectively:

$\epsilon_{\sigma,k}=\frac{|\overline{\sigma}_{k}-\widetilde{\sigma}|}{\widetilde{\sigma}}\cdot
100,\ \epsilon_{\Delta h,k}=\frac{|\overline{\Delta h}_{k}-\widetilde{\Delta
h}|}{\widetilde{\Delta h}}\cdot 100.$ (20)

In (20) $\widetilde{\sigma}$ and $\widetilde{\Delta h}$ are the known values
of electrical conductivity and thickness, respectively.

  * •

The standard deviations $std_{\epsilon_{\sigma,k}}$ and $std_{\epsilon_{\Delta
h,k}}$ for the sets
$\\{\epsilon_{\sigma,k}^{1},\ldots,\epsilon_{\sigma,k}^{20}\\}$ and
$\\{\epsilon_{\Delta h,k}^{1},\ldots,\epsilon_{\Delta h,k}^{20}\\}$, where

$\epsilon_{\sigma,k}^{i}=\frac{|{\sigma}_{k}^{i}-\widetilde{\sigma}|}{\widetilde{\sigma}}\cdot
100,\ \epsilon_{\Delta h,k}^{i}=\frac{|{\Delta h}_{k}^{i}-\widetilde{\Delta
h}|}{\widetilde{\Delta h}}\cdot 100.$ (21)

  * •

The final estimate of electrical conductivity $\overline{\sigma}$ and of
thickness $\overline{\Delta h}$, obtained by averaging the
$\overline{\sigma}_{k}$s and the $\overline{\Delta h}_{k}$s, respectively.

  * •

The relative error for the final estimate $\overline{\sigma}$ and
$\overline{\Delta h}$, with respect to $\widetilde{\sigma}$ and
$\widetilde{\Delta h}$:

$\epsilon_{\sigma}=\frac{|\overline{\sigma}-\widetilde{\sigma}|}{\widetilde{\sigma}}\cdot
100,\ \epsilon_{\Delta h}=\frac{|\overline{\Delta h}-\widetilde{\Delta
h}|}{\widetilde{\Delta h}}\cdot 100$ (22)

  * •

The standard deviations $std_{\sigma}$ and $std_{\Delta h}$ for the
$\overline{\sigma}_{k}$s and $\overline{\Delta h}_{k}$s.

  * •

The standard deviations $std_{\epsilon_{\sigma}}$ and $std_{\epsilon_{\Delta
h}}$ for the $\epsilon_{\sigma,k}$s and the $\epsilon_{\Delta h,k}$s.

Figure 13 shows the behaviour $\epsilon_{\Delta h,k}$ and
$std_{\epsilon_{\Delta h,k}}$, related to the thickness estimate. The figures
of merit for estimating electrical conductivity are shown in Figure 14. The
frequency range for different plates varies, to guarantee operations within
regions (d), (e) and (h), as discussed in Section 5.3 and Figure 7.

Figure 13: (a) Behaviour of the mean relative error ($\epsilon_{\Delta h,k}$)
and (b) the corresponding relative standard deviation ($std_{\epsilon_{\Delta
h,k}}$) for the estimated thicknesses at different frequencies and for all the
considered metallic plates. Figure 14: (a) Behaviour of the mean relative
error ($\epsilon_{\sigma,k}$) and (b) the corresponding relative standard
deviation ($std_{\epsilon_{\sigma,k}}$) for the estimated electrical
conductivities at different frequencies and for all the considered metallic
plates.

As expected, for all the analysed plates, both the error and the standard
deviations are generally more significant at low excitation frequencies for
both $\Delta h$ and $\sigma$, due to the weakness of the intensity of the eddy
currents at those angular frequencies. As the driving frequency is increased,
the relative error assumes suitable values lower than 5 % for thickness and 4
% for electrical conductivity. Similar behaviors can be observed for the
standard deviations (see Figure 13 (b) and Figure 14 (b)). These results prove
the effectiveness of the proposed method to measure both thickness and
electrical conductivity with a single frequency measurement. This is confirmed
by the minimum values obtained for the errors and standard deviations of both
thickness and conductivity that can reach values lower than 0.1 %.

To complete the analysis for this first experimental validation, Tables
LABEL:tab:results_thickness and LABEL:tab:results_sigma show the figures of
merit on the estimation of electrical conductivity and thickness obtained by
averaging the individual estimates at each angular frequency. This is a simple
scheme that can be replaced by more advanced schemes combining all the level
curves from different frequencies in the final estimate.

Table 6: Summary of the obtained results for the thickness estimation considering the overall executed tests. Name code |  $\widetilde{\Delta h}$ [mm] |  $\overline{\Delta h}$ [mm] |  $std_{\Delta h}$ [mm] |  $\epsilon_{\Delta h}$ [%] |  $std_{\epsilon_{\Delta h}}$ [%]
---|---|---|---|---|---
#a | 2.03 | 2.10 | 0.18 | 3.78 | 0.97
#b | 0.984 | 0.981 | 0.091 | 0.40 | 1.34
#c | 1.97 | 1.98 | 0.17 | 0.51 | 0.71
#d | 1.03 | 1.02 | 0.14 | 0.91 | 1.45
#e | 2.93 | 2.90 | 0.20 | 0.96 | 1.55
#f | 3.98 | 3.87 | 0.19 | 2.77 | 0.97
Table 7: Summary of the obtained results for the electrical conductivity estimation considering the overall executed tests. Name code |  $\widetilde{\sigma}$ [MS/m] |  $\overline{\sigma}$ [MS/m] |  $std_{\sigma}$ [MS/m] |  $\epsilon_{\sigma}$ [%] |  $std_{\epsilon_{\sigma}}$ [%]
---|---|---|---|---|---
#a | 17.66 | 17.27 | 2.3 | 2.18 | 0.86
#b | 58.50 | 58.44 | 6.8 | 0.10 | 2.2
#c | 28.23 | 28.26 | 4.8 | 0.11 | 1.1
#d | 35.27 | 35.77 | 4.2 | 1.44 | 2.1
#e | 35.44 | 35.33 | 2.1 | 0.31 | 1.1
#f | 35.91 | 36.76 | 2.3 | 2.38 | 0.86

Finally, Table LABEL:tab:comparison shows a comparison between the performance
levels achieved in different methods. Some methods provide a single estimate
of either thickness or electrical conductivity, while other methods provide
the estimate of both quantities. The table shows the range of thicknesses
and/or electrical conductivities investigated in the related paper, together
with the metrological performance levels expressed in terms of average
relative error ($\overline{\epsilon}_{\Delta h}$,
$\overline{\epsilon}_{\sigma}$) and maximum relative error ($\epsilon_{\Delta
h}^{max}$, $\epsilon_{\sigma}^{max}$). From Table LABEL:tab:comparison, it
follows that the proposed method outperforms all the other methods estimating
both electrical conductivity and thickness [29, 37, 27, 28], while being in
line with the performance levels of those methods retrieving one single
parameter only [38, 1, 3, 4, 39, 40].

Table 8: Estimation performance comparison between the proposed method and
some methods proposed in the scientific literature.

Estimated parameters and analyzed ranges Estimation performance Method
Thickness ($\Delta h$) [mm] Electrical conductivity ($\sigma$) [MS/m]
$\overline{\epsilon}_{\Delta h}$[%] $\epsilon_{\Delta h}^{max}$[%]
$\overline{\epsilon}_{\sigma}$[%] $\epsilon_{\sigma}^{max}$ [%] Yin et al.
[38] 1 ÷ 5 N/A 0.88 1.48 N/A Sardellitti et al. [1] 0.5 ÷ 4 N/A 1.59 3.19 N/A
Sardellitti et al. [2] N/A 1.10 2.43 N/A Sardellitti et al. [4] 1 ÷ 4 N/A 1.26
2.28 N/A Xie et al. [39] N/A 0.58 ÷ 58.2 N/A 1.05 1.8 Wang et al. [40] N/A
14.5 ÷ 51.5 N/A 1.00 2.79 Lee et al. [29] 1 ÷ 3.5 0.56 4.69 8.72 1.87 N/A
Huang et al. [37] 0.2 ÷ 1.5 0.6 ÷ 58 3.16 6.70 3.81 4.40 Huang et al. [27] 1 ÷
4 0.6 ÷ 1.3 3.70 5.40 3.68 5.20 Lin et al. [28] 0.4 ÷ 2.6 18.1 ÷ 58.9 3.89
17.25 5.74 21.14 Proposed method 1 ÷ 4 17.7 ÷ 58.5 1.55 3.78 1.09 2.38

##  8 Conclusions

This paper introduces dimensional analysis in Non–Destructive Testing &
Evaluation problems for the first time.

The use of dimensional analysis makes it possible to represent a physical
system via a minimal set of variables, the so-called groups. The groups are
dimensionless quantities. Moreover, they can be found from the knowledge of
only the physical dimensions of the variables describing the original problem.
This makes it possible (i) to reduce the computational cost when numerically
simulating the physical system of interest, as required for quantitative
inversion of the data, training of Artificial Intelligence algorithms,
repeated simulations to design a new probe, etc., and (ii) to represent an
inverse problem in a reduced dimensional space, as in our application where a
simple inverse problem was represented in a plane.

In order to present the approach in as clear a manner as possible, an ECT
method was proposed for the simultaneous estimation of the thickness and the
electrical conductivity of conductive plates. The method has been presented
from the underlying concept to a successful experimental validation carried
out on metallic plates with different thicknesses (from 1 to 4 mm) and
electrical conductivities (from 17 to 58 MS/m). The method can be applied to
either single or multi-frequency data and its negligible computational cost
makes it suitable for industrial in-line inspections.

From a general perspective, the proposed approach is highly effective for
treating _all_ NDT problems where the number of unknowns is limited, as in
many practical situations. Future work includes a complete metrological
characterization of the proposed method, its extension to magnetic materials,
where the key parameters are $\sigma$, $\Delta h$ and $\mu$, the optimization
of the frequency range for the inspection and introduction of multi-frequency
excitation signals.

##  Appendix A Construction of the dimensionless groups

Here the method to compute the dimensionless group is outlined. We refer to
[16] and [41] for details.

From a general perspective, each group can be expressed as

$\pi_{i}=q_{1}^{\alpha_{i1}}\cdots q_{j}^{\alpha_{ij}}\cdots
q_{n}^{\alpha_{in}}$ (23)

where the exponents $\alpha_{ij}$, $j=1,2,\ldots,n$, are rational numbers such
that $\pi_{i}$ is a dimensionless quantity.

To find the $\alpha_{ij}$s, a set of $k$ so called “repeating variables” (see
[41]), chosen among the $n$ dimensional quantities $q_{1},\ldots,q_{n}$, has
to be defined. The repeating variables must satisfy the following constraints:
(i) their products, with proper exponents, provide all the physical dimensions
of the underlying problem; (ii) they are independent; (iii) their arbitrary
nontrivial products do not generate a dimensionless quantity; (iv) they should
not be dependent variables of the problem, if any. Assuming the $k$ repeating
variables are $q_{1},\ldots,q_{k}$, each group is expressed as

$\pi_{i}=q_{1}^{\alpha_{i1}}\times\ldots\times q_{1}^{\alpha_{ik}}\,q_{k+i},\
i=1,\ldots,n-k,$ (24)

and coefficients $\alpha_{ij}$s are found by imposing each group to be
dimensionless.

As an example of application, the procedure for the construction of the
dimensionless groups for the simple RLC series circuit in Figure 1, carrid out
in the frequency domain, is proposed. The physical quantities are $n=6$ and
listed in Table 9.

Table 9: Dimensional variables in the RLC circuit, expressed in terms of fundamental dimensions. Number | Parameter | Symbol | Fundamental dimensions
---|---|---|---
1 | angular frequency | $\omega$ | [$A^{0}$ $V^{0}$ $T^{-1}$]
2 | voltage | $\bar{E}$ | [$A^{0}$ $V^{1}$ $T^{0}$]
3 | electrical resistance | R | [$A^{-1}$ $V^{1}$ $T^{0}$]
4 | current | $\bar{I}$ | [$A^{1}$ $V^{0}$ $T^{0}$]
5 | inductance | L | [$A^{-1}$ $V^{1}$ $T^{1}$]
6 | capacitance | C | [$A^{1}$ $V^{-1}$ $T^{1}$]

The electrical current $\bar{I}$ is assumed to depend on the other quantities,
i.e.

$\bar{I}=f\left(\bar{E},R,L,C,\omega\right).$ (25)

A set of possible repeating variables is ${R,\bar{E},\omega}$. Indeed, (i) the
fundamental dimensions can be expressed in term of monomial products of the
repeating variables (see Table 10), (ii) the repeating variables are
independent and (iii) arbitrary nontrivial monomial products do not generate
dimensionless quantities and (iv) they are not dependent variables. Condition
(iii) can be checked algebraically. Indeed, this condition is satisfied if and
only if the kernel of the matrix made by the coefficients of the fundamental
dimensions for the repeating variables is $\\{\mathbf{0}\\}$. In this specific
case, the exponents of the fundamental dimensions for $\omega$, $\bar{E}$ and
$R$ give the matrix

$\begin{bmatrix}0&0&-1\\\ 0&1&0\\\ -1&1&0\end{bmatrix},$

that which is invertible and, therefore, its kernel is $\\{\mathbf{0}\\}$.

Table 10: Fundamental dimensions from monomial products of the repeating variables. Product | Dimension
---|---
$E/R$ | $\left[A\right]$
$E$ | $\left[V\right]$
$\omega$ | $\left[T\right]$

The groups arising from these choices, are in the form

$\displaystyle\pi_{1}$
$\displaystyle=R^{\alpha_{11}}E^{\alpha_{12}}\omega^{\alpha_{13}}I$ (26)
$\displaystyle\pi_{2}$
$\displaystyle=R^{\alpha_{21}}E^{\alpha_{22}}\omega^{\alpha_{23}}L$ (27)
$\displaystyle\pi_{3}$
$\displaystyle=R^{\alpha_{31}}E^{\alpha_{32}}\omega^{\alpha_{33}}C.$ (28)

To compute coefficients $\alpha_{ij}$, it suffices to write a system of
equations imposing that each group is dimensionless. For instance, with
reference to $\pi_{1}$, this condition is

$A^{0}V^{0}T^{0}=\left(A^{-1}V^{1}T^{0}\right)^{\alpha_{11}}\left(A^{0}V^{1}T^{0}\right)^{\alpha_{12}}\left(A^{0}V^{0}T^{-1}\right)^{\alpha_{13}}\left(A^{1}V^{0}T^{0}\right),$
(29)

which gives $\alpha_{11}=1$, $\alpha_{12}=-1$ and $\alpha_{13}=0$ as solution
of (30).

$\begin{cases}0&=-\alpha_{11}+1\\\ 0&=+\alpha_{11}+\alpha_{12}\\\
0&=-\alpha_{13}\end{cases}$ (30)

The complete list of groups is given in (7).

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