instrumentation and inverse problem solving for impedance ...experimental results in the thesis...
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Instrumentation and Inverse Problem Solving for Impedance Imaging
Xiaobei Li
A dissertation submitted in partial fulfillment of the requirements for the degree of
Doctor of Philosophy
University of Washington 2006
Program Authorized to Offer Degree: Department of Electrical Engineering
University of Washington Graduate School
This is to certify that I have examined this copy of a doctoral dissertation by
Xiaobei Li
and have found it complete and satisfactory in all respects, and that any and all revisions required by the final
examining committee have been made.
Chair of the Supervisory Committee:
_________________________________________
Alexander V. Mamishev
Reading Committee:
_________________________________________
Alexander V. Mamishev
_________________________________________
Brian Otis
_________________________________________
Gunther Uhlmann
_________________________________________
Lloyd Burgess
Date: _________________
In presenting this dissertation in partial fulfillment of the requirements for the doctoral degree at the University of Washington, I agree that the Library shall make its copies freely available for inspection. I further agree that extensive copying of the dissertation is allowable only for scholarly purposes, consistent with “fair use” as prescribed in the U.S. Copyright Law. Requests for copying or reproduction of this dissertation may be referred to ProQuest Information and Learning, 300 North Zeeb Road, Ann Arbor, MI 48106-1346, 1-800-521-0600, to whom the author has granted “the right to reproduce and sell (a) copies of the manuscript in microform and/or (b) printed copies of the manuscript made from microform.”
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University of Washington
Abstract
Instrumentation and Inverse Problem Solving for Impedance Imaging
Xiaobei Li
Chair of the Supervisory Committee:
Associate Professor Alexander Mamishev Department of Electrical Engineering
Fringing electric field (FEF) sensors are widely used for non-invasive measurement
of material properties, such as moisture content, porosity, viscosity, temperature,
hardness, and degree of cure. Impedance spectroscopy using FEF sensors is a viable
process analytical technique for detecting the presence of a material or estimating the
concentration of a material within the test environment. An array of FEF sensors can be
used for electrical impedance tomography (EIT), an effective imaging method for
industrial process control as well as medical monitoring and diagnosis.
This thesis studies the instrumentation and data analysis of fringing electric field
sensing systems, and their application as non-invasive analytical tools in the food and
pharmaceutical industries. A specific emphasis is placed upon multi-channel FEE
sensors. The research work presented in this thesis has several scientific and engineering
challenges. Fringing electric field dielectrometry is a soft-field technique. Sensor
measurements are typically non-linear functions of the distribution of the dielectric or
physical properties of the material under test. The field distribution of FEF sensors is
inherently non-uniform and no generic analytical models exist for FEF sensors. Modeling
of FEF sensors typically relies on finite element simulations, which can be complex and
time-consuming. Interface circuit for FEF sensors typically requires high measurement
sensitivity, good noise immunity, and broad dynamic range. In imaging applications,
image reconstruction involves solving an ill-conditioned and under-determined non-linear
inverse problem.
The thesis provides guidelines for designing FEF sensor systems, specifically multi-
channel FEF sensor systems for industrial process imaging applications. The majority of
the impedance imaging research focuses on circular electrodes enclosing a cylinder.
Work on the instrumentation of imaging systems based on planar electrodes has been
scarce. The thesis fills in the gap by detailing the various aspects of designing a FEF
sensing system for industrial applications. General rules and principles for designing
multi-channel fringing electric field (FEF) sensors are outlined in the thesis. A non-
dimensionalized parametric modeling method is developed to model FEF sensors, thus
avoiding reconstructing a finite element simulation every time the dimension changes. A
direct non-linear image reconstruction algorithm, the layer stripping algorithm, is used in
the thesis to solve the inverse problem of one-dimensional profiling.
Experimental results in the thesis demonstrate the viability of FEF dielectrometry as
a process analytical sensing technique. FEF dielectrometry complements other traditional
process analytical techniques like NMR and NIR.
i
TABLE OF CONTENTS page
LIST OF FIGURES ........................................................................................................... iii
LIST OF TABLES............................................................................................................. vi
Chapter 1. Introduction ....................................................................................................... 1
1.1 Motivation and Problem Statement .................................................................... 1 1.2 Scientific and Engineering Challenges ............................................................... 2 1.3 Scope of the Thesis and Contributions ............................................................... 4 1.4 Thesis Outline ..................................................................................................... 6
Chapter 2. Background ....................................................................................................... 9
2.1 Background......................................................................................................... 9 2.2 Basics of Dielectrometry Sensing....................................................................... 9 2.3 Fringing Field Dielectrometry .......................................................................... 13
Chapter 3. Designing Interface Circuits for FEF Dielectrometry..................................... 19
3.1 Introduction....................................................................................................... 19 3.2 AC Impedance Divider ..................................................................................... 21 3.3 AC Current Sense Circuit ................................................................................. 23 3.4 Resonant Impedance Converter ........................................................................ 24 3.5 Charge Discharge Circuit.................................................................................. 26 3.6 Off-the-shelf Impedance Sensing Chips ........................................................... 29 3.7 MOSEFT Switch Charge Injection................................................................... 31 3.8 Excitation .......................................................................................................... 31 3.9 Conclusions....................................................................................................... 33
Chapter 4. Design Principle of Multi-channel FEF Sensors............................................. 35
4.1 Introduction....................................................................................................... 35 4.2 Figures of Merit ................................................................................................ 37 4.3 Major Design Concerns .................................................................................... 42 4.4 Major Design Trade-offs................................................................................... 43 4.5 Examples of Sensor Designs............................................................................. 44 4.6 Conclusion ........................................................................................................ 57
Chapter 5. Non-dimensionalized Parametric Modeling of FEF Sensors.......................... 58
5.1 Introduction....................................................................................................... 58 5.2 Simulation Setup............................................................................................... 59 5.3 Simulation Results ............................................................................................ 61 5.4 Non-dimensionalization.................................................................................... 62 5.5 Conclusions....................................................................................................... 71
Chapter 6. Image Reconstruction Algorithms .................................................................. 73
ii
6.1 Electrical Impedance Tomography ................................................................... 73 6.2 Background on Inverse Problems ..................................................................... 75 6.3 Modeling of Electrode ...................................................................................... 76 6.4 The Layer Stripping Algorithm ........................................................................ 77 6.5 Problem Description ......................................................................................... 78 6.6 Proof of Concept Study..................................................................................... 79 6.7 Conclusions and Future Work .......................................................................... 80
Chapter 7. Moisture Dynamics in Food Products............................................................. 81
7.1 Review of process analytical technologies in food analysis............................. 81 7.2 Definition of the Problem ................................................................................. 84 7.3 Methodology..................................................................................................... 86 7.4 Experimental Setup........................................................................................... 86 7.5 Experimental Procedure.................................................................................... 88 7.6 Experimental Result and Data Analysis ........................................................... 89 7.7 Conclusions....................................................................................................... 93
Chapter 8. Measuring Physical Properties of Pharmaceutical Samples ........................... 94
8.1 Motivation......................................................................................................... 94 8.2 Measuring Tablet Hardness and Coating Thickness......................................... 97 8.3 Measuring Tablet Coating Thickness ............................................................. 100 8.4 Acquiring Drug Signature Using a FEF Sensor.............................................. 104 8.5 Measuring API Concentration for Powder Samples....................................... 105 8.6 Conclusions..................................................................................................... 107
Chapter 9. Conclusions and Future Work....................................................................... 109
9.1 Conclusions..................................................................................................... 109 9.2 Future Work .................................................................................................... 110
References....................................................................................................................... 112
Appendix A: DiSPEC Hardware Installation Guide....................................................... 124
Appendix B: Matlab Code for the Layer Stripping Algorithm....................................... 136
iii
LIST OF FIGURES
Figure Number page
1.1. Scope of this thesis and future work. ........................................................................... 6
2.1. Flow diagram of the dielectrometry system............................................................... 11
2.2. A fringing field dielectrometry sensor can be visualized as (a) a parallel plate capacitor whose (b) electrodes open up to provide (c) a one-sided access to the material under test..................................................................................................... 14
2.3. A generic interdigital dielectrometry sensor.............................................................. 16
2.4. Half-wavelength cross-section with a superimposed equivalent π-circuit model. .... 18
2.5. A conceptual view of multi-wavelength dielectrometry............................................ 18
3.1. The fringing electric field sensors are implemented with a vast array of designs. .... 20
3.2. Circuit schematic of an AC impedance divider. ........................................................ 22
3.3. A current-sense impedance converter. ....................................................................... 24
3.4. (a) Parallel and (b) series RLC tanks. ........................................................................ 25
3.5. A resonant impedance converter circuit. ................................................................... 26
3.6. A charge/discharge circuit. ........................................................................................ 28
3.7. The timing diagram for the charge/discharge circuit................................................. 28
3.8. A voltage-divider capacitance sensing circuit. .......................................................... 31
4.1. Evaluation of the effective penetration depth γ3% of an FEF sensor.......................... 38
4.2. Cross-sectional view of a fringing electric field sensor with multiple penetration depth excitation patterns. .......................................................................................... 39
4.3. Top-down view of a concentric fringing electric field sensor head........................... 45
4.4. Top-down view of a concentric fringing field sensor head with shield electrodes between the driving and the sensing electrodes. ....................................................... 46
4.5. Simulated electric field line distribution illustrating the effect of the additional shielding electrode. ................................................................................................... 47
4.6. Layout of a test sample positioned above the unshielded concentric FEF sensor. .... 48
4.7. Layout of a test sample positioned above the shielded concentric FEF sensor ......... 49
4.8. Simulated equipotential plot of the unshielded concentric FEF sensor. .................... 49
4.9. Simulated equipotential plot of the shielded concentric FEF sensor. ........................ 50
4.10. Absolute capacitance value from both sensor designs in the FE simulation. .......... 51
iv
4.11. Normalized capacitance value from both sensor designs in the FE simulation....... 51
4.12. The effect of change in shielding electrode width on sensor signal strength. ......... 52
4.13. The effect of change in shielding electrode width on sensor penetration depths. ... 53
4.14. Absolute capacitance value of the inner channel of the unshielded sensor. ............ 55
4.15. The effect of change in substrate thickness on sensor penetration depth. ............... 55
5.1. A concentric FEF sensor and its signal conditioning circuit. .................................... 60
5.2. Simulation setup of a concentric FEF sensor in the R-Z plane.................................. 60
5.3. The electric field arrows and equipotential lines for concentric FEF sensors with separate backplanes................................................................................................... 62
5.4. The electric field arrows and equipotential lines for concentric FEF sensors with solid backplanes. ....................................................................................................... 63
5.5. Normalized capacitance between the drive electrode and sense1. ............................ 63
5.6. Normalized capacitance between sense1 and sense2................................................. 64
5.7. Normalized capacitance between sense2 and drive ................................................... 64
5.8. Normalized capacitance between the drive electrode and sense1 ............................. 65
5.9. Electrode pair sensitivity to changes in substrate thickness. ..................................... 66
5.10. Electrode pair sensitivity to changes in the dielectric permittivity of the MUT...... 66
5.11. Fitting residue of the fourth-order polynomial model for C1d/λεs. ........................... 68
5.12. Comparison between finite element simulation results and results from the parametric model. ..................................................................................................... 69
5.13. A half-wavelength portion of the interdigital sensor is shown. ............................... 70
5.14. (a) Finite element simulation results and (b) results estimated by the parametric model......................................................................................................................... 71
6.1. Side-view of the one-dimensional scanner setup....................................................... 79
7.1. Frequency ranges of various types of spectroscopic sensing techniques. ................. 81
7.2. Top and bottom view of the concentric sensor head.................................................. 86
7.3. Side view of the sensor in a voltage divider setup..................................................... 87
7.4. Sensor geometry and experimental setup. ................................................................. 88
7.5. Capacitances and phase measured at different moisture content levels. ................... 89
7.6. Capacitance measurements against the weight of added water at 10 kHz................. 90
7.7. Moisture content distribution across the radius of the sample................................... 92
8.1. Tablet sample weight and thickness against the pressure applied ............................. 98
8.2. Capacitance and phase of tablet samples measured against sample hardness. ........ 100
v
8.3. Fringing electric field sensor setup for measuring tablet coating thickness.. .......... 100
8.4. Absolute capacitance measurements of tablet samples with different coating thickness using a parallel plate sensor. ................................................................... 101
8.5. Capacitance and phase variations of samples with coating thickness.. ................... 102
8.6. Capacitance variation against sample weight using a parallel plate sensor............. 102
8.7. Absolute capacitance measurements for tablet samples with different coating thickness using a fringing electric field sensor. ...................................................... 103
8.8. Capacitance and phase variations for tablet samples of different coating thickness using a fringing electric field sensor. ...................................................................... 104
8.9. Capacitance variation against sample weight for the fringing electric field setup. . 104
8.10. Capacitance and phase measurements of three different types of tablet samples.. 105
8.11. Capacitance and phase of powder samples of various drying time. ...................... 107
8.12. Capacitance of powder samples against sample drying time measured at two separate frequencies. ............................................................................................... 107
vi
LIST OF TABLES
Table Number Page
4.1.The effect of increasing shielding electrode width.. ....................................................54
4.2. Penetration depths (mm) of the concentric sensor designs. ....................................... 56
4.3. The effect of increasing substrate thickness. ............................................................. 56
5.1. Polynomial coefficients aij for C1d / (λεs). .................................................................. 67
5.2. Polynomial coefficients aij for C12 / (λεs). ................................................................. 67
5.3. Polynomial coefficients aij for C2d / (λεs). ................................................................. 67
5.4. Polynomial coefficients aij for C12 / εs........................................................................ 71
7.1. Comparison between the actual mass of the moisture added to the sample and the mass of the moisture measured by the sensor. .......................................................... 93
8.1. Tablet sample physical properties: hardness, weight, thickness................................ 99
vii
ACKNOWLEDGEMENTS
I would like to acknowledge the support of my colleagues at Intel, Dupont,
especially, Dr. Ted Dibene and Dr. Steve Montgomery for their continuing
encouragement of and support.
To my adviser Prof. Alexander Mamishev, who has made the past five years a great
experience for me, who had been patient enough to go through twenty revisions on my
first research paper.
To my fun-loving lab mates Bing Jiang, Nels Jewell Larsen, Abhinav Mathur, Gabe
Rowe, Alanson Sample, Chih-Peng Hsu, and Kishore Sundara-Rajan at SEAL. The
experience at SEAL has been absolutely wonderful.
To the Center of Process Analytical Chemistry, Kraft Foods Inc, and the National
Science Foundation, who funded this research project and provided me with plenty of
opportunities to present my work in front of industrial research communities.
To my parents who have always valued education more than anything else, and who have
encouraged me to set high goals for myself and not to settle for anything less.
viii
DEDICATION
This thesis is dedicated to my parents Anwen Li, and Heng Zhou.
1
Chapter 1. Introduction
1.1 Motivation and Problem Statement
The work in the thesis is motivated by the omnipresent need for process control sensing
techniques in the manufacturing industries. This thesis is based on the following
publications by the author [1-7].
Process control is one of the most important tasks of analytical chemists, engineers, and
physicists in manufacturing industries. Product quality is often controlled downstream of
a process. A failed product has to be reworked, sold at a lower price, or even destroyed.
Integrated sensing techniques are the key to industrial process control. Process control
typically involves feedback and feed-forward systems that rely on accurate sensor
measurements to control the manufacturing variables within the desired limit. Variations
in the physical, chemical, and biological properties of a sample need to be closely
monitored to ensure the quality of a product. Faced with increasing cost-reduction
pressure, manufactures are pursuing wider adoption of new process analytical techniques.
Product quality and safety are critical in the food and pharmaceutical industries. Food
industry is seeing great advances in the areas of food biotechnology, food science, and
food engineering [8]. Combinations of new dehydration, food sterilization, freezing, and
separation processes lead to the improvements of safety, convenience, taste, and value of
food products [9]. Sensor technology is at the forefront of the needs of food industry [10].
Existing sensor systems should be updated and improved based on the advances in
computer technology, microelectromechanical systems (especially microfluidics), data
processing algorithms, and material science.
A similar trend is occurring in the pharmaceutical industry. FDA has been steering the
pharmaceutical industry in the direction of a quality–by–design (QbD) approach, and
away from the quality–by–testing (QbT) approach traditionally taken by the
pharmaceuticals sector. The basis of QbT is to test the finished product for quality.
Product specimens that fail to meet the specification are rejected. This approach leads to
2
a great deal of waste and thus is costly to both the manufacturer and the consumer. The
wastage caused by QbT approach can increase production costs by as much as 10
percent. The basis of the alternative approach QbD is to incorporate the knowledge of the
product and the process to ensure all critical quality parameters are adequately controlled
and that the finished product meets specifications.
FDA started a Process and Analytical Technology (PAT) initiative. Process Analytical
Technology is a system for designing, analyzing, and controlling manufacturing through
timely measurements (i.e., during processing) of critical quality and performance
attributes of raw and in-process materials and processes with the goal of ensuring final
product quality. The desired goal of the PAT framework is to design and develop
processes that can consistently ensure a predefined quality at the end of the
manufacturing process. Such procedures would be consistent with the tenet of quality by
design and could reduce risks to quality and regulatory concerns while improving
efficiency.
This thesis explores the application of fringing electric field dielectrometry to industrial
process control. FEF sensors are non-invasive and can access the sample under test from
only one side. Typically, no sample preparation is required for FEF sensor measurement.
Custom-build FEF sensing systems cost much less than other commercially available
process analytical instruments. These attributes of FEF sensing systems make them very
attractive candidates for real-time industrial process control.
1.2 Scientific and Engineering Challenges
The scientific and engineering challenges are discussed here from three aspects: sensor
modeling and design, sensor interface circuit design, and data analysis.
1.2.1 FEF sensor modeling and design
The field distribution of FEF sensors is inherently non-uniform. Sensor measurements are
typically non-linear functions of the distribution of the dielectric or physical properties of
the material under test. Modeling and design optimization are very complex due to this
inherent non-linearity. The performance of an FEF sensor is largely determined by its
3
geometry. FEF sensor design is an iterative optimization process of modeling and
redesign, where the sensor geometry is adjusted to meet the design specifications.
There are no generic analytical models for FEF sensors. Analytical models for FEF
sensors are usually based on simplified geometries and idealized assumptions, which
limits their accuracy for real world applications. Finite Element (FE) models can
typically achieve higher accuracy than analytical models. However, modeling accuracy is
heavily dependent on proper mesh and boundary conditions. Therefore, setting up the
models can be time-consuming. An efficient modeling method that is both fast and
accurate can greatly aid the design optimization process.
The design process of FEF sensors and sensor arrays relies on a good understanding of
the fundamental principles and design trade-offs. For imaging applications, the major
goal of sensor design is to achieve the optimum balance of measurement sensitivity,
signal strength, imaging resolution, and measurement speed. The finite area of sensor
head makes it impossible to achieve all design goals simultaneously. The task, therefore,
is to consider the trade-offs and determine the optimal combination of design variables
for a given application.
1.2.2 Sensor interface circuit design
FEF sensors are impedance sensors. They detect changes in the physical properties of the
MUT by measuring the sensor terminal impedance. The challenge is to design an
interface circuit that can accurately measure complex impedance across a wide frequency
range.
For imaging applications, the distribution of the physical parameter of interest is to be
recovered. Local variations in the physical properties of the MUT typically manifest as a
very small change in the terminal impedance of the sensor, while baseline impedance
value can change greatly. Achieving high measurement resolution and wide dynamic
range simultaneously is a challenge.
For industrial process control applications, the sensor measurements tend to vary within a
broad range due to variations in the process parameters such as temperature, humidity,
4
viscosity, etc. Designing an interface circuit with good dynamic range while maintaining
high measurement accuracy is a challenge. In integrated micro-sensors, the challenge is
to design circuits with good noise immunity so that high measurement sensitivity can be
delivered.
1.2.3 Image reconstruction
Electrical impedance imaging (EIT) is a soft-field imaging technique. Soft-field
modalities are different from ‘hard-field’ techniques like X-ray Computed Tomography
(CT). X-rays propagate in straight lines. Their absorption at any point inside the material
under test (MUT) is completely independent of the absorption at any other point. Image
reconstruction for X-ray CT is a linear problem involving only sparse and well-
conditioned sensitivity matrices. The term sensitivity matrix here refers to the jacobian
matrix relating the parameter of interest with sensor measurements. EIT, on the other
hand, is a low-frequency electromagnetic radiation technique, with the excitation
frequency typically lower than 100 kHz. Due to the long wavelength, an electrostatic
representation is most appropriate for such low frequency techniques. It is inherent to
soft-field methods that the field strength at any point is a non-linear function of the
distribution of the electrical properties throughout the MUT. This makes the associated
reconstruction much more difficult to solve. Image reconstruction for EIT involves
solving an ill-posed inverse problem. Advanced image reconstruction algorithms have
been developed within the mathematical context to tackle the impedance imaging
problem, but few of the advanced algorithms have been used effectively in industrial
process control. The gap between the theoretical algorithms and practical applications
needs to be bridged.
1.3 Scope of the Thesis and Contributions
Figure 1.1 shows the scope of this thesis. Design and modeling of FEF sensors is the
focus of the thesis. The trade-offs and interdependencies of multi-channel FEF sensor
design are described and the general design guidelines are presented within the same
context. In addition, a non-dimensionalized parametric model is developed for FEF
5
sensors. The model is calibrated based on finite element (FE) simulation results. Once
calibrated, no more FE simulations are necessary. This empirical modeling method
strikes a middle ground between the purely theoretical analytical methods and those
based on FE modeling.
For sensor interface circuits, typical circuit topologies used for impedance sensing are
surveyed, with special focus on those with broadband spectroscopic capabilities suitable
for real-time applications.
In the realm of parameter estimation, this thesis attempts to address the divide between
theoretical algorithms and their applications to solve real world problems. Specifically,
layer-stripping algorithm is used for one-dimensional profiling.
The thesis has two major contributions. First, it provides guidelines for designing FEF
sensor systems, specifically multi-channel FEF sensor systems, for industrial process
imaging applications. Such systems can be used for recovering images of the distribution
of the parameter of interest. The majority of the research on electrical impedance imaging
systems focuses on circular electrodes around an enclosed cylinder. Work on the
instrumentation of imaging systems based on planar electrodes has been scarce. The
thesis fills in the gap by detailing the various aspects of designing a FEF sensing system
for industrial applications.
The second contribution of the thesis is to demonstrate the viability of FEF
dielectrometry as a process analytical sensing technique. Till this day, FEF sensing
systems have not been explored as extensively as other analytical techniques like nuclear
magnetic resonance (NMR) or near-infrared spectroscopy. This phenomenon is
particularly evident in the food and pharmaceutical industry. The thesis presented data
which shows that FEF sensors can be used for determining physical properties of food
and pharmaceutical samples. FEF dielectrometry complements NMR and NIR because
the information it provides is inherently different. NMR and NIR typically provide
information about the molecular or chemical properties of the sample, while FEF
dielectrometry provides information about the physical properties of the sample. In
addition, instrumentation for FEF dielectrometry is less costly and more flexible.
6
Integration of FEF sensors into the manufacturing process of food and pharmaceutical
industries facilitates better control of product quality at lower cost.
Figure 1.1. Scope of this thesis and future work.
1.4 Thesis Outline
The theoretical background for the thesis work is described first before going into the
details on the instrumentation and parameter estimation for fringing electric field (FEF)
sensing/imaging. Specific applications of FEF sensing in the food and pharmaceutical
industry are then illustrated with experimental results. The outline of the thesis is listed
below:
7
Chapter 1 describes the motivation, the challenges, and the scope of this thesis.
Chapter 2 reviews the theoretical background of dielectrometry sensing and FEF sensor.
Chapter 3 reviews the state of the art of sensor interface circuits. The operating principles
of various topologies are described and the advantages and disadvantages of each
topology are explained.
Chapter 4 presents the principle of multi-channel fringing electric field sensor design for
imaging applications. The figures of merit for FEF sensor design are presented first. Then
the major design concerns and trade-offs are explained. Finally, the design principles are
illustrated through the example of a concentric FEF sensor design.
Chapter 5 deals with modeling of FEF sensors. Non-dimensionalized parametric models
are developed for both interdigital and concentric FEF sensors. Such models can be used
to estimate the terminal capacitance of a sensor with known dimensions. The accuracy of
the models is tested against finite element simulation results.
Chapter 6 describes the layer stripping algorithm in detail. The algorithm is used to
reconstruct images for a one-dimensional scanner. Numerical simulation results and
analysis are presented in this chapter.
Chapter 7 presents the experimental results of measuring moisture content distribution in
organic food materials. The experimental setup and procedures are explained and a
calibration-based sensing technology is presented. Chapter 9 presents experimental
results of measuring physical properties of pharmaceutical samples. Measurements from
FEF sensors and generic parallel plate sensors with the same set of test samples are
compared to illustrate how the choice of sensors can impact experimental results. The
rationale behind the choice of measurement sensors and experimental setup for such
experiments are dealt with in detail.
Chapter 8 reviews the state of the art of image reconstruction algorithms. The theory of
inverse problem solving is briefly touched upon. Then the various categories of
reconstruction algorithms are introduced. Major concerns such as electrode modeling,
choice of excitation patterns, etc. are also dealt with in this chapter.
8
9
Chapter 2. Background
2.1 Background
This need for real-time non-invasive sensing techniques has driven many advances in the
field of dielectrometry. Dielectrometry is widely used for determination of material
physical properties due to its non-invasiveness and wide spectrum of sensing
possibilities. Applications include agricultural products [11], soil [12], paper [13],
transformer board [14], biological sensing [15,16] and hydrophilic polymers [17].
Capacitive sensors are often used for dielectric spectroscopy. They have the advantage of
high measurement accuracy and non-invasiveness. The simplest examples of capacitive
sensors are a guard-ring parallel-plate capacitor and a coaxial cylindrical capacitor. More
complicated examples include fringing electric field sensors, which can assume various
geometries [18,19]. The penetration depth of fringing electric field sensors is proportional
to the distance between coplanar electrodes. By applying different voltage patterns to the
sensor, variable penetration depths can be achieved, thus providing FEF sensors access to
different layers of the material. This characteristic, combined with their one-sided access
capability, makes FEF sensors more flexible in use than their parallel-plate counterparts.
2.2 Basics of Dielectrometry Sensing
2.2.1 Introduction to the theory of dielectrics
Dielectric materials cover the whole spectrum of anything between conductors and
insulators. Dielectrics consist of polar molecules, or non-polar molecules, or very often
both. Due to the asymmetric configuration of polar molecules, material consisting of
these molecules has built-in dipole moments. Under an external electric field, the
polarized dipoles reorient in the electric field and neutralize some of the charges on the
electrodes. The most often used measure of material dielectric properties is the complex
10
dielectric permittivity. It is a measure of the ability of the dielectric material to reorient
and neutralize charges on the electrodes. This usually depends on how polarized the
material is and the inertial force it has to overcome to reorient. Sometimes, relative
complex dielectric permittivity is used to describe material dielectric properties. It is
defined as the ratio between the dielectric permittivity of the material and that of free
space. The dielectric permittivity of free space is 8.85×10-12 F/m.
The dielectric permittivity of most dielectric materials is frequency-dependent. In the
presence of an alternating electric field, the dipole moments inside the material oscillate
with the direction of the electric field. The higher the frequency the harder it is for the
dipole moments to catch up with the change of field direction. This results in a decreasing
ability of the material to neutralize charges on the electrodes at high frequencies. In
general, the total complex dielectric permittivity ε*(ω) is written as:
*( ) '( ) ''( )iε ω ε ω ε ω= −
(2.1)
where 'ε and ''ε are, respectively, the real permittivity and the dielectric loss factor of
the material.
Jonscher of the Chelsea Dielectric Group has been studying the problem of a universal
relaxation law [20]. Until now, no one has proven the existence of a general model to
describe the dielectric relaxation process. One of the most widely used models for fitting
dielectric relaxation data is the Havriliak-Negami (HN) function, as shown in (2.2),
where 0ε is the dielectric permittivity at dc and ε∞ is its asymptotic value at infinite high
frequency. The term 0ε ε∞− is the total dielectric relaxation strength and 0τ is the
relaxation time of the material. For 1β = , the Cole-Cole model emerges; whereas for
1α = the Davison-Cole model emerges.
( )
0
0
*( )1 i
βα
ε εε ω εωτ
∞∞
−= +
+ (2.2)
''σ ωε=
(2.3)
11
2.2.2 Principles of dielectric spectroscopy sensing
Dielectric spectroscopy involves the study of the response of a material to an applied
electric field. Typically, the technique is used 1) to gain theoretical understanding of the
dielectric properties of a material/sample, and 2) to relate the dielectric data to other
physical/chemical properties of interest.
Dielectrometry is a versatile sensing technique. Figure 2.1 shows the flow diagram of a
dielectrometry system. Material dielectric permittivity is dependent on various physical
properties such as geometry, texture, temperature, degree of cure, moisture content, and
aging status. Changes in these physical properties will be reflected as changes in such
dielectric property variables as *ε ,σ , and tanδ , where σ is defined in (2.3) and tanδ is
defined as the ratio between the real and imaginary parts of the complex dielectric
permittivity. These parameters variations, in turn, lead to changes in the impedance
measurements from the sensor. The fact that dielectric measurements are sensitive to
changes of a wide range of material physical properties makes dielectrometry sensing
technique a potential candidate for a broad spectrum of sensing applications.
Measurecapacitancesand conductances
Calibration-basedsensing
Differentialsensing
Computedistributionof dielectricproperties
Computedistributionof physicalproperties
Imaging
Faster Slower
ε σ, δ, Μ*, tan ∗
thicknesssurface texturetemperaturedegree of curemoistureporositydensityconcentrationpercolationstructural integrityaging statuscontamination .......
Figure 2.1. Flow diagram of the dielectrometry system.
2.2.2.1 Impedance spectroscopy and dielectric spectroscopy
Rather than focus on details of what happens inside dielectric materials, electrical
engineers often analyze dielectrics from a macroscopic perspective. Impedance
12
spectroscopy is one such macroscopic approach. It models dielectrics as lumped circuit
elements and uses the terminal electric impedance measurement to represent the physical,
chemical and biological processes happening inside the material.
A proper choice of a circuit model is crucial to obtaining good modeling results. The
dielectric spectra that can be represented by combinations of RC circuits are called
relaxation spectra, whereas those that can be represented by combinations of RL circuits
are called resonance spectra. To determine the choice of circuit models, the frequency
dependency of material dielectric constant should be examined. For relaxation spectra,
the dielectric constant only stays constant or falls with increasing frequency. In these
cases, RC circuit models should be used. If otherwise, RL or RLC circuit model should
be used [21]. In addition to lumped circuit models, distributed circuit models are
sometimes used to model dielectric materials as a distributed dielectric medium in
bounded or unbounded space [22,23].
Dielectric spectroscopy relates material dielectric properties with corresponding physical
properties and investigates the fundamental theoretical link between them. It is often used
for research efforts investigating material dielectric properties. For industrial
applications, where in-depth theoretical knowledge of the dielectric property of the
material is unnecessary, impedance spectroscopy is sufficient.
2.2.2.2 Calibration based sensing
Calibration based sensing works by establishing a quantitative relationship between
material physical property of interest and the resulting impedance measurements. This
functional dependence is usually empirically determined. The algorithms for such
calibration-based approaches are usually quite straightforward, yet these approaches are
not sufficient to gain insight about the physical nature of the material. For real-time
imaging applications, adaptive self-calibration mechanisms are desired to accommodate
variations in the sample or change in the setup.
2.2.2.3 Differential sensing
Differential sensing is a methodology that can be used for monitoring changes.
13
Systematic measurement errors can be greatly reduced by simply looking at the variation
of the parameters of interest. The system is typically calibrated against baseline
measurements first, and then only the deviations from the baseline measurements are
used for estimation. Examples of such applications include monitoring the aging process
of food product in storage rooms, monitoring the lung/heart activities of patients,
monitoring the drying process of pharmaceutical powder samples, etc.
2.2.2.4 Imaging – electrical impedance and capacitance
tomography (EIT/ECT)
EIT and ECT are the most widely used technologies for imaging the distribution of
material physical properties in industrial applications. Electrical impedance tomography
(EIT) is an imaging modality developed mainly for medical applications. An EIT system
works by applying current to a human body through various electrodes and measuring the
induced voltage at the electrodes. Using the different combinations of current patterns
and the corresponding voltage measurements, the inverse problem is solved to recover
the resistivity distribution of tissue being imaged. Electrical capacitance tomography
(ECT) is a similar modality developed for industrial process control applications. In an
ECT system, electrodes are typically attached to external walls of a pipeline. Similar to
EIT, current is applied and the induced voltage measured. Since the electrodes are not in
direct contact with the sample, the terminal impedance is typically reactive, thus the term
capacitance tomography.
Unlike X-ray or laser imaging, EIT and ECT are “soft field” techniques, because the field
lines that penetrate through the sample do not stay in a straight path. This greatly
increases the complexity of parameter estimation algorithms for image reconstruction.
2.3 Fringing Field Dielectrometry
A fringing field dielectrometry sensor has the same principle of operation as the more
conventional parallel-plate or coaxial cylinder dielectric sensor cell. The voltage is
applied to the electrodes, and the impedance across the electrodes is measured. However,
unlike the parallel-plate cell, the fringing field sensor does not require two-sided access to
14
material under test. Figure 2.2 shows a gradual transition from the parallel plate capacitor
to a fringing field capacitor. In all three cases, electric field lines pass through the
material under test, therefore the capacitance and conductance between the two electrodes
depends on the material dielectric properties as well as on the electrode and material
geometry.
Figure 2.2. A fringing field dielectrometry sensor can be visualized as (a) a parallel plate capacitor whose (b) electrodes open up to provide (c) a one-sided access to the material under test.
Interdigital dielectrometry is a subset of interdigital electrode sensor applications that
relies on direct measurement of dielectric properties of insulating and semi-insulating
materials from one side [24-26]. The basic idea is to apply a spatially periodic electrical
potential to the surface of the material under test. The combination of signals produced
by the variation of the spatial period of the interdigital comb electrodes combined with
the variation of electrical excitation frequency potentially provides extensive information
about the spatial profiles and dielectric spectroscopy of the material under test. Since the
changes in the dielectric properties are usually induced by changes in various physical,
chemical, or structural properties of materials, the dielectrometry measurements provide
effective means for indirect non-destructive evaluation of vital parameters in a variety of
industrial and scientific applications [27,28].
Usually, the capacitance between two co-planar strips, as shown in Figure 2.2 (c), is
comparable to the stray capacitance of the leads (conductors that connect the electrodes
with the electrical excitation source). Therefore, in order to build-up an easily measurable
electrode structure, the coplanar strip pattern may be repeated many times.
The art and science of designing sensors with multiple interdigital electrode pairs and
15
processing the output signals to gain knowledge about the material under test is the
subject of multi-wavelength interdigital frequency-wavenumber (ω-k) dielectrometry,
which has been under development for about three decades. The penetration depth of the
fringing quasistatic electric fields above the interdigital electrodes is proportional to the
spacing between the centerlines of the sensing and the driven fingers and is independent
of frequency. Overviews of important concepts related to this technology are available in
[29-35].
One of the most attractive features of multi-wavelength dielectrometry is the ability to
measure complex spatially inhomogeneous distributions of properties from one side. As
the complexity of spatial distribution grows and the number of unknown variables in each
experiment increases, the parameter estimation algorithms become more complicated,
computationally intensive, and less accurate and reliable. Ultimately, every major type of
spatial distribution of material properties requires a different parameter estimation
algorithm. The types of spatial distributions include, but are not limited to, homogeneous
materials, multiple layer materials, local discontinuities (such as cracks and electrical
trees), global discontinuities of micro-structure (such as grains or fibers forming the
material), and, finally, smoothly varying properties. On the electrical properties side,
materials under test may be purely insulating or weakly conductive. Various phenomena
may affect sensor response, including frequency dispersion, electrode polarization due to
an electrochemical double layer, quality of interfacial contact, and many others.
A conceptual schematic of an ω-k dielectrometry sensor measurement is presented in
Figure 2.3. For a homogeneous lossy dielectric medium of semi-infinite extent, periodic
variation of quasistatic electric potential along the surface in the x direction produces an
exponentially decaying pattern of electric fields penetrating into the medium in the z
direction. The variation of shade in the material under test indicates the possible
variation of material properties and thus variations in the complex dielectric permittivity
ε * with the distance z from the surface.
16
Figure 2.3. A generic interdigital dielectrometry sensor. The black stripes are the electrodes. Underneath the electrodes is the substrate of the sensor (white). At the bottom (black) is the backplane of the sensor. Above the sensor is the sample under test (SUT). The grayscale on the sidewalls of the SUT indicates the non-uniform electrical field that the SUT is subject to.
Concepts of the forward and the inverse problems are widely used in the literature related
to this technology. Here, the forward problem is defined as the task of determining the
electric field distribution and the inter-electrode admittance matrix when the geometry,
material properties, and external excitations are given. Correspondingly, the inverse
problem requires determining either material properties or associated geometry, or both,
when the imposed excitations and experimental values of the sensor admittance matrix
are available. Each application that involves theoretical modeling usually requires solving
the forward problem.
The forward problem can be solved using several approaches. One of them is to use a
continuum model [36]. From the electroquasistatic field point of view, in a homogeneous
lossy dielectric, the electric scalar potential of the field excited by the driven electrode is
a solution to Laplace's equation. At any constant z position, the electric field distribution
far away from the sensor edges is periodic in the x direction and assumed uniform in the y
direction. For a homogeneous dielectric of semi-infinite extent, the scalar potential can be
written as an infinite series of sinusoidal Fourier modes of fundamental spatial
wavelength λ that decays away in the z direction:
17
0
( sin cos )nk zn n n n n
ne A k x B k x
∞−
=
Φ = Φ +∑ (2.4)
where kn=2πn/λ is the wavenumber of each mode. For j te ω excitations at radian
frequency ω, such that ˆ j tn ne
ωΦ = ℜ Φ , the complex surface capacitance density Cn
relates ε*Êzn at a planar surface z = constant to the potential nΦ at that surface for the n-th
Fourier mode of the homogeneous dielectric of semi-infinite extent as:
* ˆˆˆ
znn
n
EC ε=
Φ (2.5)
where
)/(* ωσεε j−= (2.6)
is the complex permittivity with ε as material dielectric permittivity and σ as ohmic
conductivity. Then, the knowledge of ˆnC at the electrode surface allows calculation of the
terminal current from the potential distribution at that surface. It is also possible to solve
the forward problem with commercial finite-element software [37], with finite-difference
techniques, or by using analytical approximations [38].
Figure 2.4 shows the equivalent circuit of the sensor superimposed onto the schematic
view of a sensor half-wavelength. Note that each wavelength has an opposite conducting
guard plane at the bottom of the substrate. For each wavelength, a follower op-amp drives
the guard plane at the substrate bottom at the voltage VG = VS, thus eliminating any
current between the sensing and guard electrodes through the substrate. Therefore, the
effect of G20 and C20 on circuit response is eliminated, which simplifies response analysis
and improves device sensitivity. The concept of imposed frequency-wavenumber (ω-k)
goes beyond dielectrometry applications. Earlier studies led by J. R. Melcher employed
interdigital electrodes to study electrohydrodynamic surface waves and instabilities [39],
effects on static electrification in insulating liquids [40,41], and electromechanics of
electrochemical double layers in liquid electrolytes [42].
The penetration depth of the fringing electric fields above the interdigital electrodes is
proportional to the spacing λ/2 between the centerlines of the sensing and the driven
18
fingers. Figure 2.5 illustrates the idea of multiple penetration depths for a three-
wavelength sensor. The variation of the material properties across the thickness of the
material under test in the z direction can be approximately found by simultaneously
solving three complex equations describing this three-wavelength experimental
arrangement as a piece-wise three layer system.
Figure 2.4. Half-wavelength cross-section with a superimposed equivalent π-circuit model.
Figure 2.5. A conceptual view of multi-wavelength dielectrometry. The penetration depth of electric field lines is proportional to the electrode spatial wavelength λ.
19
Chapter 3. Designing Interface Circuits for FEF Dielectrometry
3.1 Introduction
Fringing electric field sensors are widely used for measurement of material properties in
numerous scientific and industrial applications. Generally speaking, the sensing system
always comprises three main parts: sensor head, interface electronics, and the data
acquisition/signal processing units. The sensor head is a system of electrodes that creates
the electric field for probing the material properties. The complexity of sensor heads
varies from a pair of flat metal strips to multi-electrode three-dimensional arrays. The
interface electronics also varies widely in complexity, from basic op-amp circuits to
complex bench-top instrumentation. The signal processing hardware and algorithms also
vary depending on the need of the application, ranging from an on/off signal indicating
the presence of a material to the inverse-problem solving algorithms that require complex
matrix operations. Figure 3.1 shows the conceptual description of the design domain of
fringing electric field sensing systems. The second column, Interface Electronics, is the
focus of this chapter. The subject matter of other columns is also covered, in order to put
the discussed designs in proper context – but is not the main focus of this chapter.
Extensive tutorials and reviews are available on the subject of electrode design (sensor
head design) [43-50], electrical excitation options [51], and parameter estimation
algorithms [52-55]. The details on these topics will be covered in later chapters.
20
Electrode design(sensor heads)
Interface electronicsSignal processing
Figure 3.1. The fringing electric field sensors are implemented with a vast array of designs and techniques of widely varying complexity. The connecting lines between the first and the second columns indicate typically encountered combinations. The design of dielectric sensors requires an educated selection of the electronic circuitry
that interfaces sensor electrodes with the data acquisition system. This chapter offers a
comparative review of interface circuit designs for dielectric sensors and a discussion of
their working principles, with a special focus on multi-channel interface circuit for
imaging applications.
Several review and tutorial papers appeared on this subject before. The review by Huang
et. al. [56] offers a balanced review of general principles and circuit examples, but it is
focused only on single-frequency capacitance measurements. Such circuits are not
adequate for biochemical and medical applications where samples have complex
permittivity.
These samples usually have a unique signature in the frequency domain, which can be
used to identify the presence and the concentration of samples. To recover the frequency
domain information, circuits with spectroscopic measurement capabilities are required. It
is difficult to achieve high measurement accuracy across a wide frequency range with a
single interface circuit. Different circuit topologies should be considered for the specific
frequency range of interest. This chapter reviews measurement circuitry that measures
21
complex impedances at multiple frequencies.
For many imaging applications, the sensor output has a relatively large baseline value
and small variations (variation due to change in test samples). This requires the circuit to
have a high dynamic range and measurement resolution. In addition, since the signal to
be measured is small, noise needs to be suppressed to achieve a sufficient signal to noise
ratio. Interface circuits commonly used for impedance sensing applications are detailed in
the sections below.
3.2 AC Impedance Divider
Impedance divider is one of the simplest AC impedance measurement circuits. It is
typically constructed by connecting an impedance sensor and some known reference
impedance (normally an RC tank) in series. An AC voltage is applied to the combination
of sensor and reference impedance and the gain and the phase of the divider are
measured. The terminal impedance of the sensor is then calculated based on the measured
gain, phase, and the value of the reference impedance.
Figure 3.2 shows a common implementation of impedance divider. In Figure 3.2, GX
and CX are the terminal impedances of the sensor to be measured; GREF and CREF are the
known reference impedances; GS1 and CS1 are the stray impedances at the input side; and
GS2 and CS2 are the stray impedances due to the connecting cables at the output side.
Input AC voltage INV is applied across a series combination of the sensor impedance GX
//CX and the reference impedance GREF //CREF. The signal at the voltage division point
(node 1) is buffered by the voltage-follower opamp 1A . There are two reasons for using
the voltage follower. First, it decouples high-impedance node 1 from usually low-
impedance input of the measurement electronics such as a data acquisition card or a
frequency-response analyzer. Second, it provides a signal for the actively-driven shield.
In the case of the voltage divider in Figure 3.2, node 2 is the cable shielding and is
connected to the output of buffer opamp 1A . By feeding the voltage-follower signal back
to the shield, any effect of the stray impedances 2SG and 2SC is eliminated due to the
zero voltage drop across them. In addition, the stray impedance 1SG and 1SC at the input
22
side have no effect on the applied voltage. Thus, the described AC impedance divider
circuit is stray-immune.
XG
XC
1SC 1SGINV
1
1AOUTV
REFG
REFC
2SG2SC
2
Figure 3.2. Circuit schematic of an AC impedance divider. GX and CX are the unknown conductance and capacitance of the sample under test, and GREF and CREF are the reference conductance and capacitance respectively. GS1, GS2, CS1, and CS2 are the stray conductance and capacitance due to the sensor substrate and the connecting wires.
The value of GX and CX can be calculated based on equations (3.1) and (3.2).
1
Re ( ) 1( )
X INREF
OUT
G V ZV
ωω
⎧ ⎫⎪ ⎪⎛ ⎞= ⎨ ⎬−⎜ ⎟⎪ ⎪⎝ ⎠⎩ ⎭
%%
% (3.1)
1
1 Im ( ) 1( )
X INREF
OUT
C V ZV
ωω ω
⎧ ⎫⎪ ⎪⎛ ⎞= ⎨ ⎬−⎜ ⎟⎪ ⎪⎝ ⎠⎩ ⎭
%%
% (3.2)
where 1REF
REF REF
ZG j Cω
=+
% and ω is the angular frequency of the input AC signal.
The impedance divider is a non-linear circuit. The changes in the output (the complex
gain) are not linearly proportional to perturbations at the input (sensing impedance). The
dynamic range of the measured gain is between 0 and 1. The circuit is most sensitive
when the value of the reference impedance is close to the sensor impedance to be
measured. Therefore, variable capacitors and resistance are sometimes used to adjust the
reference impedance for the maximum measurement sensitivity.
23
AC impedance dividers with buffer opamps are not suitable for measuring small
capacitances (for example, XC < 0.1 pF), because state of art opamps typically have an
input capacitance greater than 1 pF. This capacitance is in parallel with the reference
capacitance REFC . Therefore, the effective reference capacitance is always greater than 1
pF, which limits the lowest measurable capacitance of the circuit.
Very low/high input frequency can be used to decouple of the conductance and the
reactance being measured. The upper frequency limit of the AC excitation is limited by
the bandwidth of the buffer opamp. Typically, there is a trade-off between the input
impedance and the frequency bandwidth (BW) of the opamp. High BW opamps tend to
have lower input impedance, while opamps with high input impedance tend to have lower
BW. Low frequency (mHz or µHz) excitation leads to long measurement time, and it
should be used only for detecting loss factors of highly resistive materials.
3.3 AC Current Sense Circuit
Another circuit commonly used for impedance measurements is the current-sense
amplifier. It is used in many commercially available RCL meters, impedance analyzers,
insulation resistance meters, and even microphones. Figure 3.3 shows a typical
implementation of a current sense impedance converter. An AC excitation voltage VIN is
applied across the impedance sensor and the resulting current is measured. An effective
way to measure small AC current is to use an I-V converter (integrator) since it is
difficult to measure current directly. The output of the current to voltage converter is
connected to a high accuracy gain/phase measurement system such as Solartron
frequency analyzer or a data acquisition card. Given the value of the reference impedance
used in the feedback loop, the value of the sensor’s impedance can be calculated from the
gain/phase measurements according to the following equations.
REFOUTXIN ZVZV // −= (3.3)
*// AZVVZZ REFOUTINREFX −=−= (3.4)
where 1REF
REF REF
ZG j Cω
=+
% , XX
X CjGZ
ω+=
1 , and INOUT VVA /* = . Note that A* is the
24
complex gain measured at the output, which contains both amplitude and phase
information.
XG
XC
1SC 1SG 2SG 2SC
1A
REFC
REFG
OUTVINV
Figure 3.3. A current-sense impedance converter. GX and CX are the unknown conductance and capacitance of the sample under test, and GREF and CREF are the reference conductance and capacitance respectively. GS1, GS2, CS1, and CS2 are the stray conductance and capacitance due to the sensor substrate and the connecting wires.
The current sense circuit is inherently stray-immune. First, the stray impedances at the
input CS1 and GS1 are in parallel with the input voltage source, therefore, they have no
effect on the VOUT; second, the sensor’s sensing electrode (node 1) is held at the virtual
ground through active feedback, thus eliminating the effect of stray impedances CS2 and
GS2 and that of the input capacitance of opamp A1.
The measurement frequency of current sense circuits typically goes from mHz to MHz.
the upper limit of the excitation frequency is determined by the bandwidth of the opamp
used.
3.4 Resonant Impedance Converter
Resonance circuits typically involve an RLC tank, which consists of a known inductance
L and an unknown conductance Gx and capacitance Cx of the test sample. Figure 3.4
shows (a) a parallel RLC tank and (b) a series RLC tank. The frequency of the AC
excitation signal is tuned until the circuit reaches resonance. The resonant frequency is
determined by (3.5). At resonance, the inductive reactance cancels the capacitive
25
reactance, and the tank becomes purely resistive. The resistance and capacitance of the
test medium are calculated based on the known inductance L and the detected resonant
frequency fR.
12Rf LCπ
= (3.5)
C GL
CGL
(a) (b)
Figure 3.4. (a) Parallel and (b) series RLC tanks. Resonance methods are capable of measuring complex impedances over a wide
frequency range with a very high accuracy. However, they are not suitable for real-time
applications because the frequency tuning is time-consuming. Circuits based on the
resonance methods are widely used for laboratory grade dielectrometry instrumentation
[57,58].
Figure 3.5 shows an example of a resonant impedance converter circuit. The frequency of
the input signal VIN is tuned until the resonance point is reached. At the resonance, the
reactance of LR cancels that of CX. We have
XR
R CLf
π21
= (3.6)
X
REF
IN
OUT
GG
VV
= (3.7)
Therefore, we can calculate the terminal impedance of the sensor based on the following:
2
1(2 )X
R R
Cf Lπ
= (3.8)
26
A
GV
VGG REF
OUT
INREFX == (3.9)
where INOUT VVA /= is the measured gain.
XG
XC
1SC 1SG 2SG 2SC
RL
REFG
1A
REFC
INVOUTV
Figure 3.5. A resonant impedance converter circuit. GX and CX are the unknown conductance and capacitance of the sample under test, and GREF and CREF are the reference conductance and capacitance respectively. GS1, GS2, CS1, and CS2 are the stray conductance and capacitance due to the sensor substrate and the connecting wires.
As in the case of the current sensing impedance converter, the circuit is immune to the
effect of stray capacitances and resistances. The stray impedances CS1, GS1, CS2, and GS2
have no effect on the measurements of the circuit.
3.5 Charge Discharge Circuit
For applications requiring the measurement of a sensor’s reactance only,
charge/discharge method is often a preferred choice. A charge/discharge circuit operates
by charging the capacitor being measured to some preset voltage and then discharging it
through a charge detector. If the period of this charge/discharge cycle is much shorter
then the time constant of the feedback loop around the charge detector op-amp (for the
voltage feedback I-V converter), then the output voltage will be smoothed to virtually
only a DC component proportional to the magnitude of the average current flowing
through the detector. This method allows an op-amp with bandwidth much lower then the
measurement frequency to be used.
27
Figure 3.6 shows an stray-immune circuit developed by S. M. Huang [59]. Electronic
switches 1S , 2S , 3S , and 4S are turned on or off according to the timing diagram in Figure
3.7. During “charge” half cycle, 1S and 4S are closed, allowing XC to charge to the
source voltage SV . At the same time, sensor’s stray capacitance 1SC , and combined
parasitic capacitance 12PC of the switches 1S and 2S , are charged to SV as well. During
“discharge” half cycle, 2S and 3S are closed, while 1S and 4S are opened up. The charge
on 1SC and 12PC flows directly to ground through 2S without affecting the measurement.
The charge transferred from the sensor’s capacitance to the detector in a single discharge
is
S XQ V C= (3.10)
The average current entering the detector (at node 1) is
AVE S XI V C f= (3.11)
where 1
C
fT
= is a charge/discharge frequency, and CT is the time of a complete
charge/discharge cycle. The average current AVEI is converted to a DC voltage by the
charge detector:
OUT S X REFV V C f G= (3.12)
It should be noted that throughout the charge/discharge virtually no current flows through
sensor’s stray capacitance 2SC and the combined parasitic capacitance 34PC of the
switches 3S and 4S , thus they have no effect on OUTV . The circuit is, therefore,
intrinsically stray-immune. Furthermore, if the switches 1S through 4S have low “on”
resistance, the circuit is immune to sensor’s conductance as well.
The measurement frequency of Huang’s implementation is in the range of 100 kHz to 5
MHz, which is typical of the charge/discharge technique. The output of the circuit is a
DC voltage proportional to the sensor’s terminal capacitance XC .
28
OUT REFX
S
V GCV f
= (3.13)
XG
XC
1 S C 1SG 2SG 2SC1 A
REF C
REF G
OUTVS V
2 S
3S
1 S
4S
P12 C
P34CGS1 GS2 CS2CS1
XG
XC
1 S C 1SG 2SG 2SC1 A
REF C
REF G
OUTVS V
2 S
3S
1 S
4S
P12 C
P34CGS1 GS2 CS2CS1
Figure 3.6. A charge/discharge circuit. GX and CX are the unknown conductance and capacitance of the sample under test, and GREF and CREF are the reference conductance and capacitance respectively. GS1, GS2, CS1, and CS2 are the stray conductance and capacitance due to the sensor substrate and the connecting wires. CP12 is the parasitic capacitance of the switches S1 and S2, and CP34 is the parasitic capacitance of the switches of S3 and S4.
S1
S2
S3
S4
On Off
Charge Discharge
Figure 3.7. The timing diagram for the charge/discharge circuit.
29
3.6 Off-the-shelf Impedance Sensing Chips
3.6.1 Analog Devices AD7745 – 7747 capacitance meter
Analog Devices released a series of single chip solutions for measuring capacitances. The
AD7745 single chip capacitance meter has an accuracy of 4 fF for floating (two-terminal)
capacitance measurements over a dynamic range of 4 pF, and can reject parasitic
capacitances to ground up to 60 pF. This device uses a 32 kHz square-wave excitation, a
second order sigma-delta modulator or charge balancing modulator, and a third order
digital filter. The maximum sampling rate is 90 Hz. A multiplexed two channel device,
the AD7746, is also available from Analog Devices. The AD7747, which is currently
only being sampled, is a single channel capacitance meter like the AD7745, however, the
capacitance being measured can be referenced to ground. The AD7747 also provides a
shield signal to drive the shield conductors to eliminate stray capacitance from cable-
shielding.
3.6.2 Analog Devices AD5933 impedance converter
The AD5933 is a single channel 1 mega-sample-per-second, 12-bit complex impedance
analyzer, operating over a range of 1 to 100 kHz, and can measure impedances between
100 Ohms to 10 MΩ at 0.5% precsion. Impedances are measured by switching into one
of the six modes to measure impedance over a set range, e.g. 100 Ohms to 1000 Ohms.
Frequency resolution is 0.1 Hz and 511 distinct frequencies can be used to measure the
impedance between 1 kHz and 100 kHz during a single sweep. The device consists of a
precise function generator, a programmable gain amplifier, a low-pass filter, a windowing
function, and a 1024 point discrete Fourier transform. As a point of reference, a
capacitance of 16 pF at 1 kHz would be measured as a reactance of approximately 10
MΩ, and at 100 kHz the 16 pF capacitance would be measured as a reactance of
approximately 100 kΩ. The real and imaginary parts of the complex impedance are
stored into separate registers which are then transferred via the I2C bus off the chip. The
frequency sweep feature offers the ability to determine resonant frequency shifts due to a
changes in an RLC network element. Analog Devices also has a single channel 250 kSPS
30
12-Bit complex impedance analyzer that does not have an internal oscillator, but has
many of the same characteristics as the AD5933, except for a quadrupled sampling
period, 4 ms instead of 1 ms for the AD5933.
3.6.3 Motorola MC33794 electric field imaging device
The Motorola MC33794 Electric Field Imaging Device is a single-chip implementation
that can interface with up to 9 sensors in an array. It was originally developed for
passenger size detection in the case of vehicular air-bag deployment. The device is
ideally suited for capacitances ranging from 10 pF to 100 pF. The device outputs two
signals: a buffered version of the AC signal at the electrode and a DC level indicating the
amplitude of that signal.
With a built-in multiplexer, the device allows the selection and measurement of one of its
nine sensor electrodes, simultaneously grounding all the rest. When an electrode is
selected, the sensor interface topology is that of a voltage divider, where the known
impedance is a 22 kΩ resistor connected to a 5 V p-p 120 kHz sinusoidal voltage source,
as shown in Figure 3.8. The sensed capacitance is that between the selected electrode and
the ground terminals. To eliminate the stray capacitance from the cable shielding, the
shielding is actively driven by the measured signal through a buffer operational amplifier.
The signal from the electrode is also fed into a rectifier followed by a low pass filter so its
amplitude can be determined. This value is then output from the device as a DC voltage,
which varies inversely with capacitance. The internal reference resistance was selected
such that the capacitance values within the range of 10 pF to 100 pF will give a nearly
linear output. For digital processing, this output must be fed to an A/D converter, like
those available in many microcontrollers. Since the output is a DC value, it requires
significantly less processing overhead than reading the AC signal directly from the
electrodes. When connected to a 10 bit A/D converter, a measurement resolution of 0.1
pF is possible.
Because of its design for automotive use, the MC33794 implements many features that
are automotive-specific, including an ISO-9141 to microcontroller translator and an
31
indicator lamp driver/sensor. In most cases these can and will be left unused. However,
one benefit of its original purpose is that while the device itself requires a 12 V source, it
includes a 5 V regulated output to drive external support circuitry, such as a
microcontroller. In summary, the MC33794 is small and suitable for single frequency
imaging applications. It is best for measuring capacitances ranging from 10 pF to 100 pF
with a resolution around 0.1 pF.
5 V AC
22 K
100 pF
Figure 3.8. A voltage-divider capacitance sensing circuit.
3.7 MOSEFT Switch Charge Injection
The problem of charge injection of MOS switches has been treated extensively in
literature. When the reset switch is active, a conduction channel extends from the source
to the drain of the transistor. When the gate voltage decreases, mobile carriers exit
through both the drain and the source end. The amount of channel charge that is injected
into the input depends on several factors, such as slope of the transient and the
input/output capacitance ratio. The problem can be generally neglected, but if the amount
of channel charge is comparable to the charge amplifier sensitivity, great care has to be
taken to avoid spurious injection of charge into the input.
3.8 Excitation
Excitation waveforms are chosen based on the desired set of excitation frequencies and
the signal to noise ratio (SNR) requirement at each frequency. To achieve a good SNR,
the total energy of the excitation signal needs to be sufficiently high, which requires
32
either greater amplitude or longer duration for the waveform. The trade-offs are that high
amplitude causes non-linearity, and longer signal duration affects the speed of
measurements. Several possible choices of excitation waveforms are considered below.
3.8.1 Impulse function
An impulse function contains spectral content at all frequencies; therefore, the frequency
characteristics of the entire spectrum of the system under test can be measured
instantaneously.
Since the duration of an impulse function is very short, the amplitude has to be large so
that the signal power is high enough to achieve a sufficient SNR. High amplitude in the
input waveform is hard to accommodate due to non-linearity. Another disadvantage is
that signal energy at the undesired frequencies is wasted.
3.8.2 Step function
Like the impulse function, the step function has spectral content at all frequencies.
However, the excitation energy is spread over the entire duration of the waveform, which
reduces the amplitude required to achieve the same SNR when compared with an impulse
function. On the other hand, unlike the flat spectrum of an impulse, step functions carry
little energy at high frequencies, which makes them unsuitable for cases when high
frequency information is required.
3.8.3 Random noise
Random noise signals also spread the excitation energy over the entire duration of the
waveform. Their spectra can be tailored to match the requirements of the specific
application. The disadvantage is that, due to its randomness, the excitation waveform has
to be sampled in addition to the output waveform, which generates significant overhead
for the analog filters and the digital computation.
3.8.4 Pseudo-random noise
To avoid sampling the excitation waveform, a pseudo-random noise signal can be used
33
instead of random noise, because pseudo-random signals are deterministic. The downside
is that the extraction of the frequency response of the system requires more computation
than the alternatives described below.
3.8.5 Frequency sweeps
Sequential measurements at frequencies of interest allows for straightforward analysis.
However, it is time consuming, and the transition from one excitation frequency to the
next causes transient that affect the following measurements.
3.8.6 A sum of sinusoids at frequencies of interest
In this case, the excitation energy is exactly matched to the frequency information desired
and the generation of the signal is straightforward. The measurement is faster than the
sequential sinusoid technique and the applied signal does not contain transients from the
transitions between successive single frequency sinusoids. In addition, system
nonlinearity can be checked by the observation of the response signals at frequencies
other than those contained in the excitation set.
3.9 Conclusions
Various aspects of interfacing circuit design for fringing electric field dielectrometry is
discussed in this chapter. Due to the high measurement sensitivity requirement of FEF
sensor applications, the noise floor of the interface circuits needs to be sufficiently
suppressed. Therefore, stray-immunity is one of the most crucial figures of merit for
these circuits. For spectroscopic applications, where the sample under test is evaluated
over a frequency band, the circuit topology optimal for that frequency band should be
used. For circuits interfacing with sensor arrays in imaging applications, parasitics due to
multiplexing need to be carefully addressed.
The future trend is to integrate the interface circuit and the controller with the sensor
electrodes on a silicon-based platform. The planar geometry of FEF sensors facilitates
easy integration into a CMOS process. The sensor electrodes can be placed at the top
metal layers, while the control and measurement circuitry are placed underneath. The
34
highly-developed and well-controlled CMOS processes can facilitate mass production of
circuits with high precision at low cost.
35
Chapter 4. Design Principle of Multi-channel FEF Sensors
4.1 Introduction
Multi-channel fringing electric field (FEF) sensors are widely used to measure material
properties as functions of position and time [60-62]. The design process of FEF sensors
and sensor arrays relies on a good understanding of the fundamental principles and
design trade-offs. The purpose of this chapter is to highlight the critical aspects of sensor
design principles and to illustrate the principles with numerical simulations and
experimental results. For imaging applications, the major goal of sensor design is to
achieve the optimum balance of measurement sensitivity, signal strength, imaging
resolution, and measurement speed. The finite area of sensor head makes it impossible to
achieve all design goals simultaneously. The task, therefore, is to consider the trade-offs
and determine the optimal combination of design variables for a given application.
Design variables include the geometry of electrodes and substrate, the choice of materials
for electrodes and substrate, the number of electrodes, and the arrangement of guard
electrodes. The optimization process can rely on either numerical simulations [60] or
analytical methods [63].
Finite element (FE) methods are used extensively for sensor modeling [64], optimization
[65], and performance evaluation [66], especially for structures that are difficult to model
analytically. The quality of the results from finite element methods depends on model
definition as well as mesh generation and refinement. When the right model and mesh are
chosen, FE simulations can generate results with high accuracy.
It is often difficult to construct an analytical model for a three-dimensional electrical
sensor. Analytical models based on conformal mapping were constructed for interdigital
structures in [49,67]. Such model was developed to evaluate the effects of design
parameters such as finger width, substrate thickness, and metallization ratio for thin film
interdigital FEF sensors [49]. The model generated solutions that match closely with
36
experimental data and FEM simulation results. In geometries where one of the three
dimensions can be considered infinite compared with the other two, a two-dimensional
model approximation can be used [50]. The effect of electric-field-bending on the
linearity of a capacitive position sensor was studied based on an analytical model [68]. A
similar model was constructed for an interdigital FEF sensor designed to detect the
presence of water on a glass surface [5]. Based on analytical calculations, [5] investigated
the effect that electrode width and the width of the gap between the electrodes have on
sensor output. The validity of the analytical model was proven by matching results from
finite difference (FD) simulations. Both the analytical calculation and FD simulation
results showed a trade-off between measurement sensitivity and signal strength. The
paper did not address, however, the penetration depth of the sensor and how it is affected
by sensor geometry. Penetration depth is an important parameter in applications where
bulk measurements (as oppose to surface measurement) of the medium under test are
required.
Among all of the design variables, electrode geometry is the major determining factor for
sensor performance. Therefore, the choice of sensor geometry is critical to meeting the
requirements of an application. Sensors of various geometries were designed previously
for profiling and imaging applications. For example, a multi-segment interdigital FEF
sensor was used for multi-phase interface detection [69]; a multi-segment cylindrical
sensor was used to image continuous flows of materials inside a pipeline [70]; a helical
wound electrode sensor and a concave electrode sensor were developed for void fraction
measurements [71]. For applications where the sample can only be accessed from one
side, FEF sensors can be used. Such applications include, among others, online
measurement of moisture content in food products [72], pharmaceutical products [73],
and paper pulp [34], as well as cure state monitoring in the resin transfer molding process
[48,74].
Aside from sensor geometry, sensor output also depends on the sample of interest. The
optimal design, therefore, is application dependent. The current chapter focuses on the
qualitative effect of design parameters on sensor performance for samples with low
permittivity (< 10) and low conductivity. Design of capacitance tomography sensors for
37
media with high dielectric permittivity is investigated in [69]. In paper [69], the sensor
was used to measure the media flowing in the pipe through an insulating wall. Simulation
was conducted to determine the effect of wall thickness on sensor output. Although the
simulation results apply only to tomography sensors of similar geometry, [69] provides
insights on how the permittivity of the medium under test affects the performance of
capacitive sensors in general.
The performance of FEF sensors is typically evaluated based on their penetration depth,
signal strength, measurement sensitivity, and linearity. All of these factors depend on
sensor geometry. The effect of electrode geometry on the performance of interdigital FEF
sensors was analyzed in [75], but not in the context of imaging applications. In this
chapter, we provide generalized design principles for multi-channel FEF sensors with a
focus on imaging applications. Most of the principles and results presented here can be
applied to designing multi-channel imaging sensors of other types as well.
The first part of this chapter focuses on the qualitative effect that design variables have
on sensor performance. The second part of the chapter illustrates the method of
simulation-based design optimization through the example of two multi-channel
concentric sensors. The qualitative analysis from the first part provides intuitive design
guidelines for the optimization process shown in the second part of the chapter.
4.2 Figures of Merit
Penetration depth, measurement sensitivity, dynamic range, signal strength, and noise
tolerance are the figures of merit usually used to evaluate the performance of multi-
channel FEF sensors. For imaging applications, imaging resolution and speed are also
considered. All these figures of merit are analyzed in detail in this section.
4.2.1 Penetration depth
Penetration depth is a measure of how quickly the electric field intensity decreases as the
distance from the plane of sensor electrodes increases. There is no strict definition of
penetration depth for FEF sensors. One way to evaluate the effective penetration depth is
to position a sample above the sensor head, move it away from the sensor surface, and
38
measure the terminal capacitance at each position. Penetration depth γ3% corresponds to
the position z where the difference between the capacitance at that position C(z = γ3%)
and the asymptotic capacitance C(z = ∞) equals to 3% of the difference between the
highest and the lowest values of the terminal impedance [76]. This method is illustrated
in (4.1) and Figure 4.1, where C(z = ∞) represents the sensor terminal capacitance when
the sensor is in direct contact with the sample.
3%( ) ( )100% 3%
( 0) ( )C z C z
C z C zγ= − = ∞
× == − = ∞
(4.1)
Figure 4.1. Evaluation of the effective penetration depth γ3% of an FEF sensor
For an interdigital sensor with a 50% metallization ratio (the ratio of the area of the
electrodes to the total area of the sensor surface), penetration depth γ3% is roughly one
third of its spatial wavelength λ [77]. Spatial wavelength is defined here as the distance
between the centerlines of neighboring electrodes of the same type (e.g. driving or
sensing electrodes). Figure 4.2 shows the cross-sectional view of an interdigital sensor
with electrodes extending into the plane of the paper. In Figure 4.2, the parameters l1, l2,
and l3 represent three different ways of connecting the electrodes. The letter “D”
represents the driving electrodes, “S” represents the sensing electrodes, and “G”
represents the guard electrodes. The function of driving, sensing, and guard electrodes
and their use in different electrode configurations are described in detail in [78]. In the
table below the cross-sectional view of the sensor in Figure 4.2, each row corresponds to
39
one of the three connection schemes; each column corresponds to the types of connection
used in the different connection schemes for the electrode directly above the column. For
the l1 scheme, an ac voltage signal is applied to every other electrode and the
current/voltage at the rest of the electrodes is measured. For connection schemes l2 and
l3, several fingers are chosen as guard electrodes and the current or voltage at only the
sensing electrodes is measured. The guard electrodes are either connected to ground or
kept at the same voltage potential as the sensing electrodes through unity-gain buffer
amplifiers. By using different connection schemes, the spatial wavelength of the sensor is
increased (λ3 > λ2 > λ1). As a result, sensor penetration depth is also increased. The
variable sensor penetration depth obtained from the different connection schemes
provides the sensor with access to different layers of the test specimen.
Figure 4.2. Cross-sectional view of a fringing electric field sensor with multiple
penetration depth excitation patterns.
4.2.2 Measurement sensitivity
Measurement sensitivity is defined as the ratio between the change in sensor output and
the change in the measured physical parameter of the sample. Because the electric field
of FEF sensors is non-uniform, their measurement sensitivity is position-dependent. As
illustrated in Figure 4.1, sensitivity decreases exponentially with increasing distance from
the plane of electrodes.
Measurement sensitivity also depends on the area of electrodes. For a fixed spatial
wavelength, greater electrode area means higher measurement sensitivity to changes in
40
the sample under test. In the case of multi-channel sensors, however, increasing the
electrode area will decrease the amount of measurement channels, if the spatial
wavelength for each channel is fixed.
4.2.3 The output dynamic range
The output dynamic range of a sensor is defined as the ratio of the largest to the smallest
sensor output (terminal impedance) [79]. Its value depends on both the sensor geometry
and the sample under test, and it should lie within the input range of the interface circuit.
An impedance divider circuit can take inputs of any value, because its output is always
bounded. However, this circuit is non-linear and its measurement sensitivity is
maximized only when the load and reference impedance are matched. For virtual ground
integrator circuits, adjustable gain is sometimes used to ensure linearity [80].
4.2.4 Signal strength
FEF sensors are generally made of metal strips and the capacitance between the adjacent
two strips is relatively small. This leads to low signal strength. To increase signal
strength, sensors with interdigitated periodical structures can be built. Signal strength is
improved here through adding more ‘fingers’ to a sensor.
The signal strength of an FEF sensor changes exponentially with its distance to the
sample. For capacitive measurements, if the dielectric permittivity of the sample is higher
than that of the medium, the signal strength decays with the increasing distance to the
sample. If the sensor is immersed in a medium that has finite conductivity, its signal
strength can either increase or decrease depending on the dielectric properties of the
medium and the sample.
4.2.5 Noise tolerance
Guard electrodes are usually used to shield sensing electrodes from noise. They can take
the form of a guard ring surrounding the active sensor electrodes (driving and sensing
electrodes), the guard plane beneath the sensor substrate, or a three-dimensional shield
around the sensing area. They need to be positioned properly for optimal sensor
41
performance. They should also be carefully connected to avoid stray capacitances and
ground loops. The driven-guard technique, where the guard electrodes are kept at the
same voltage potential as the sensing electrodes, is used to remove or reduce any stray
capacitances from the guard electrodes [81].
4.2.6 Imaging resolution
A straightforward approach to produce an image of a physical parameter of a sample is to
let each measurement channel of the sensor correspond to one pixel in the image. The
method has the limitation that the number of channels has to be the same as the number
of desired pixels. To generate an image with high resolution, a large number of
measurement channels is required, which is often difficult to implement. Tomography
imaging, on the other hand, reconstructs images by interpolating measurements from
different channels, and the number of pixels from such interpolation can be much greater
than the number of measurement channels. However, in tomography, it is still desirable
to have as many measurement channels as possible, because the degree of ill-posedness
in image reconstruction can be reduced by increasing the ratio of the number of
independent measurements to the number of output pixels [82].
For a sensor of fixed size, increasing the number of electrodes decreases the area of each
electrode, resulting in a reduced measurement sensitivity and signal strength. If sensor
output gets close to the minimum level measurable by the interface circuit, the resulting
measurement will lose accuracy. The maximum number of electrodes is therefore limited
by the measurement resolution and the noise floor of the interface circuit.
4.2.7 Imaging speed
The speed of imaging systems is important for real time measurement and control
applications. It depends on the total number of measurement channels, the efficiency of
the image reconstruction algorithm, and the frequency of the input driving signal. A
greater number of channels generate more data to process and require longer time for
image reconstruction. In image reconstruction, iterative algorithms are usually slower
than non-iterative ones, but they are more accurate [82]. When the algorithm is not the
42
bottleneck, imaging speed depends mostly on the frequency of the driving signal, the
faster the imaging speed. The upper limit of operating frequency is determined by the
bandwidth of circuit elements (e.g. operational amplifiers), and by the bandwidth of the
DAQ card (when A/D conversion is performed directly on ac signals).
4.3 Major Design Concerns
4.3.1 Surface contact quality
FEF sensors are highly sensitive to the composition of the volume in the immediate
vicinity of the electrodes. The smaller the spatial wavelength of the sensor, the more
pronounced is this effect. For applications involving contact measurements of solid
samples, surface contact quality between the sample and the sensor is a major source of
uncertainty. Air gaps between the sample and the electrode act as a series capacitance
with the impedance of the sample and lead to inaccurate estimate of sample impedances.
This air capacitance is difficult to determine because its value depends on the surface
roughness of the sample and the electrodes.
To improve sensor-sample contact quality, silver paint can be applied directly to the
specimen as electrodes for macro-scale samples. It conforms to the surfaces of samples,
reducing or eliminating air gaps. However, silver paint electrodes are difficult to pattern
with high resolution and difficult to remove afterwards. Another option for improving the
contact quality is the liquid immersion technique, where the sensor and the sample are
immersed in a liquid that has dielectric properties similar to that of the solid sample under
test [83]. In clinical tomography applications, saline gels are applied to patients’ skin to
improve contact with electrodes [84].
When the sensor and the sample are not in contact, its measurements are much less
sensitive to the surface roughness of the electrode and the sample. However, for micro-
sensors (spatial wavelength from 1 μm to 100 μm), it is important to keep the sensor
surface clean, so that the dust particles and other contaminants do not alter the sensor
output.
43
4.3.2 Sensor substrate and the geometry of the back plane
The distance between the backplane and the driving electrode depends on the substrate
thickness of the sensor. When the backplane is close to the driving electrodes, it affects
the field distribution pattern, thus influencing the penetration depth and signal strength.
Proper positioning and geometry design of the backplane are critical for optimizing
sensor performance. The effect of substrate thickness on sensor output characteristics is
illustrated with the example of two concentric FEF sensors in Section V.
4.3.3 Cross-talk between difference measurement channels
In general, the closer are individual sensing cells the stronger is the cross-talk between
the corresponding channels. It is therefore desirable to position the sensing cells as far
apart as possible. Cross-talk can also be reduced by inserting shielding electrodes
between neighboring sensing cells. Both of these methods, however, reduce the total
surface area of active electrodes, which in turn reduces measurement sensitivity and
signal strength.
4.4 Major Design Trade-offs
In imaging applications, electrode surface area is limited by the size of the sample. This
makes different output characteristics dependent on each other. Many trade-offs,
therefore, are present in the design process.
4.4.1 The number of channels vs. penetration depth and
measurement sensitivity
For an FEF sensor of a fixed size, increasing the number of measurement channels
decreases the gap between neighboring channels, a change that reduces sensor
penetration depth and causes stronger cross-talk. In addition, the reduced electrode
surface area for each channel results in a drop in measurement sensitivity. A secondary
effect on measurement sensitivity is caused indirectly by the decrease in penetration
depth: at smaller penetration depth, the sensor field energy is much more concentrated
44
around the sensor electrodes, thus making its output less sensitive to variations in the
sample. These trade-offs limit the number of channels that a FEF sensor can have.
4.4.2 Imaging resolution vs. measurement sensitivity and imaging
speed
Using a larger number of smaller electrodes improves imaging resolution, but leads to
two disadvantages: increased number of measurement channels reduces imaging speed;
smaller size of electrodes reduces measurement sensitivity. The loss in speed and
sensitivity can be compensated by, respectively, higher measurement sensitivity of the
interface circuit and higher system operating frequency. High operating frequency
requires, in turn, the interface circuit to have a sufficient bandwidth. Examples of circuits
with high sensitivity and bandwidth are provided in [80,85].
4.5 Examples of Sensor Designs
Two concentric FEF sensor designs are presented here to illustrate the qualitative design
principles described in the previous sections. Figure 4.3 shows a concentric sensor
designed for measuring moisture content in dough [48]. The rationale for using this type
of sensors in food manufacturing is available in [48]. The electrodes (black in the figure)
were patterned on an insulating substrate (white in the figure). The sensor is configured
as a two-channel FEF sensor, where the middle ring is used as the driving electrode and
the other two electrodes are used as the sensing electrodes. Each sensing electrode needs
individual guarding electrodes (the electrodes beneath the substrate) if the open voltage
measurement scheme is to be used. The readers who need understanding of different
measurement schemes used for this type of sensors are referred to [86].
45
Figure 4.3. Top-down view of a concentric fringing electric field sensor head. The figures were drawn to scale. The center electrode is 4 mm in diameter as marked.
The Laplace equation of the concentric FEF sensor has the following solution:
( )0 1 2( , ) ( ) z zr z J r c e c eβ βφ β − += +
(4.2)
where β denotes electric potential, r refers to the radial coordinate on the horizontal
plane, z corresponds to the vertical coordinate, J0 is the zero order Bessel function of the
first kind, and β is a scaling constant such that βr is one of the zeros of J0 [65].
The sensor has a spatial wavelength of 8 mm, which corresponds to a penetration depth
of about 2.5 mm, a value insufficient for measuring a broad variety of food products. To
increase the penetration depth, shielding electrodes are added between the driving and the
sensing electrodes, and are kept at the same voltage as their neighboring sensing
electrode. The improved design is shown in Figure 4.4. Figure 4.5 illustrates the effect of
the added shielding electrodes on the penetration depth of the sensor. Without the
shielding electrodes, the backplane draws electric field down toward itself. The shielding
electrodes counteract this effect by pushing the electric field lines upward, which
effectively increases the penetration depth of the sensor. The simulation results were
generated from the FEMLAB software package.
46
Figure 4.4. Top-down view of a concentric fringing field sensor head with additional shielding electrodes between the driving and the sensing electrodes. The figures were drawn to scale. The center electrode is 3 mm is diameter.
47
(a)
(b)
Figure 4.5. Simulated electric field line distribution illustrating the effect of the additional shielding electrode. (a) The electric field line distribution of the geometry shown in Figure 4.3. (b) The electric field line distribution of the geometry shown in Figure 4.4.
48
4.5.1 Finite element analysis
To compare the performance of the two designs, the software package Maxwell 2D
simulator by Ansoft Corp. was used for finite element (FE) simulations. Figure 4.6 and
Figure 4.7 show the layout of the simulation spaces. These spaces were defined in radial
coordinates, with the origin placed at the lower left corner of the simulation space. The
driving electrode is set to 6 volts and all other electrodes (including the backplane) are set
to 0 volts, a Dirichlet boundary condition in nature. A test sample with relative dielectric
permittivity εr = 5.0 and conductivity σ = 0 is positioned above the sensor. FR4 epoxy
with relative dielectric permittivity εr = 4.4 and conductivity σ = 0 is used for the
substrate of the sensor. The boundary of the simulation space is set as a “charge balloon.”
A charge balloon models an electrically insulated system, where the charge at infinity
balances the charge within the simulation space forcing the net charge to be zero. The
convergence criterion for total energy error is set to be within 1%. The equipotential plots
for the two concentric designs are shown in Figure 4.8 and Figure 4.9. In the simulation,
the distance of the sample to the plane of sensor electrodes is varied from 15 mm to 0
mm. The signal strength for each sensor is evaluated based on its absolute terminal
capacitance, while the penetration depth γ3% is evaluated based on normalized terminal
capacitance calculations.
Figure 4.6. Layout of a test sample positioned above the unshielded concentric FEF
sensor in the finite element simulation.
49
5.0
Figure 4.7. Layout of a test sample positioned above the shielded concentric FEF
sensor in the finite element simulation.
Figure 4.8. Simulated equipotential plot of the unshielded concentric FEF sensor.
50
Figure 4.9. Simulated equipotential plot of the shielded concentric FEF sensor.
4.5.2 The effect of the shielding electrode
Figure 4.10 and Figure 4.11 show respectively the absolute and normalized terminal
capacitance values obtained from the simulation. “Sensor 1” refers to the design without
the shielding electrodes (Figure 4.3) and “Sensor 2” refers to the design with the
shielding electrodes (Figure 4.4). For both designs, the outer channel has greater signal
strength than its respective inner channel, a difference caused by the larger sensing area
of the outer channel. When the performance of the two designs is compared, the second
design does provide greater penetration depth than the first one. This gain in penetration
depth, however, is obtained at the cost of reduced signal strength, as shown in Figure
4.10.
51
Figure 4.10. Absolute capacitance value from both sensor designs in the FE
simulation. Absolute capacitance is used here as a metric to evaluate the measurement sensitivity of the sensors. The results indicate that the original sensor design (sensor1) has higher measurement sensitivity than the shielded design (sensor2).
Figure 4.11. Normalized capacitance value from both sensor designs in the FE
simulation. Normalized capacitance is used here as a metric to evaluate the penetration depth of the sensors. The results indicate that the addition of the shielding electrodes increase the penetration depth of the shielded sensor (sensor2).
52
4.5.3 The effect of the width of the shielding electrode
The width of the shielding electrodes is varied in the second design to evaluate its effect
on sensor output characteristics. Figure 4.12 and Figure 4.13 show respectively the
absolute and normalized capacitance value of the shielded sensor when the width of the
shielding electrode is varied.
(a)
(b)
Figure 4.12. The effect of change in shielding electrode width on the signal strength of the (a) inner sensing channel and (b) outer sensing channel of the shielded sensor. The results show that the sensor signal strength decreases with increasing width of the shielding electrodes.
53
(a)
(b)
Figure 4.13. The effect of change in shielding electrode width on the penetration depths of the (a) inner sensing channel and (b) outer sensing channel of the shielded sensor. The results show that the sensor penetration depth increases with increasing width of the shielding electrodes.
The same trend exists for the capacitance value from both the inner and the outer sensing
channel: when the width of the shielding electrodes increases, the sensor signal strength
drops and its penetration depth increases. The trend can be explained with the help of
Figure 4.1 (a). Wider shielding electrode diverts electric field energy away from the
sensing electrodes, thus decreasing the signal strength; on the other hand, the field lines
are pushed further up due to the increased surface area of the shielding electrodes, which
increases the penetration depth. The simulation results presented above are summarized
54
in Table 4.1..
Table 4.1.The effect of increasing shielding electrode width on the performance of the shielded FEF sensor.
Inner2 Outer2
Penetration depth ↑ ↑
Signal strength ↓ ↓
4.5.4 The effect of substrate thickness
As illustrated in Figure 4.5, the sensor backplane draws electric field energy away from
the sensing electrodes. The closer the backplane is to the driving electrode, the more
energy is drawn away. The distance between the driving electrode and the backplane is
determined by the thickness of the substrate. The effect of substrate thickness variation is
therefore important.
The thickness of the sensor substrate is varied from 100% to 25% of its original value in
a series of FE simulations. Figure 4.14 shows the absolute capacitance value from the
inner channel of the unshielded design. The results show that the closer the backplane is
to the driving electrode, the weaker is the signal strength. This same trend exists for the
capacitance value from both channels of the two designs.
In addition to the signal strength, penetration depth is also affected by change in substrate
thickness. The results are shown in Table 4.2 and Figure 4.15. The penetration depth of
the first design decreases with increasing substrate thickness. This can again be explained
with the illustration in Figure 4.1. The farther away the backplane is positioned from the
driving electrode, the further down the electric field lines are drawn away from the top
electrodes, resulting in a decreased penetration depth. Penetration depth for the second
design is relatively stable against variation in substrate thickness because of the shielding
electrodes. There again exits a trade-off between the penetration depth and signal
strength: for greater signal strength, a thicker substrate is desirable, but this decreases the
penetration depth of the sensor. The simulation results are summarized in Table 4.3.
The sensitivity of the sensor terminal measurement to the changes in the substrate
55
thickness depends on the ratio of the dielectric permittivity of the sample and that of the
sensor substrate. For samples with a much higher dielectric permittivity than the
permittivity of the substrate, variation in substrate thickness will not affect the sensor
performance as much.
Figure 4.14. Absolute capacitance value of the inner channel of the unshielded sensor with different substrate thickness. The results indicate that increasing the substrate thickness improves the measurement sensitivity of the sensor.
Figure 4.15. The effect of change in substrate thickness on sensor penetration depth.
56
Table 4.2. Penetration depths (mm) of the concentric sensor designs with varying substrate thickness d.
d Inner1 Outer1 Inner2 Outer2 25% 2.78 3.91 3.06 4.63 50% 2.67 3.77 3.04 4.69 75% 2.54 3.67 3.09 4.66
100% 2.47 3.57 3.10 4.72
Table 4.3. The effect of increasing substrate thickness on the performance of the two FEF sensors.
Inner1 Outer1 Inner2 Outer2
Penetration depth ↓ ↓ ↑ ↑
Signal strength ↑ ↑ ↑ ↑
4.5.5 Limitation of simulation results
It is worthy to note that the optimization results presented in this chapter are application-
dependent. The optimal sensor geometry changes with respect to different samples. In the
simulations of this chapter, samples with low permittivity (< 10) and close to zero
conductivity were used. Such samples are representative for a wide range of ceramics and
plastics [64]. Design of sensors specialized for high permittivity dielectrics (10 < εr < 80)
were discussed in [34]. For highly conductive samples, resistance, instead of
capacitance, is measured to estimate sample concentration or distribution within the
sensing zone. In cases where the sample has complex permittivity and displays
frequency-dependent behavior, both the real part and the imaginary part of the complex
impedance have to be measured.
The multi-channel impedance sensors presented in this chapter are designed for imaging
applications. Electrical imaging systems typically operate in the megahertz range.
Therefore, the electrostatic model used in the FE simulations is adequate. For sensors
used in broad-band spectroscopic systems, AC simulations are necessary. Such systems
are usually used for lab-based determination of material properties of samples with
complex permittivity.
57
4.6 Conclusion
This chapter presents the design principles for multi-channel FEF sensors, with a special
focus on the analysis of figures of merit and the major trade-offs caused by various
design constraints. The effect of design variables, especially sensor geometry, on sensor
performance is analyzed. These qualitative guidelines help to understand the logic behind
the simulation-based design procedures used for the two concentric FEF sensors. The
performance of the two sensors is compared. The shielding electrodes added in the
second design were shown to increase the penetration depth of the sensor. In addition, the
effects of substrate thickness and shielding electrode width were evaluated. The
simulation results demonstrated the effects of sensor geometry on its performance and
provided insights into the design process of multi-channel FEF sensors.
58
Chapter 5. Non-dimensionalized Parametric Modeling of FEF Sensors
5.1 Introduction
Fringing electric field (FEF) sensors are difficult to model analytically due to their
inherently nonlinear characteristics. Analytical models for FEF sensors are usually based
on simplified geometries and idealized assumptions, which limits their accuracy for real
world applications. For example, ideal electrodes with zero thickness are assumed in
closed-form solutions based on conformal mapping [64,65] and the continuum model
[29]. This assumption is only valid when the electrode thickness is small compared with
the spatial wavelength of the sensor [34]. Therefore, analytical models that assume
infinitesimally thin electrodes do not work well for small FEF sensors with thick
electrodes. The non-idealities of FEF sensors, such as finite finger length, finite electrode
thickness, and the non-uniform electrode surface are analyzed in [34].
Due to this lack of analytical models, FEF sensor design relies heavily on numerical
simulations. Among all numerical methods, the finite element method (FEM) is most
frequently used for FEF sensor design. At low frequency, the sensor is small compared to
the wavelength of the propagating electromagnetic wave. In such cases, electrostatic or
quasi-electrostatic simulations can be used. For sensors with uniform structures along one
dimension, a 2D simulation is sufficient. When none of the above conditions are satisfied,
a full-wave 3D simulation is required for modeling accuracy. A qualitative analysis on
multi-channel FEF sensor design is presented in [1] using 2D electrostatic simulations.
FEM modeling involves a trade-off between run time and accuracy: higher accuracy
requirement demands longer run time, and lower run time reduces modeling accuracy. A
non-parametric model based on artificial neural networks (ANN) for interdigital
capacitors is presented in [87]. To construct an accurate ANN model, a large amount of
simulation is required. Therefore, although the computation time of the ANN model
(after the model is determined) is negligible, the overall modeling throughput is not
59
reduced.
This chapter presents a parametric modeling method that can predict the output of a wide
range of FEF sensors. The model parameters are determined by polynomial fitting of
prior FEM simulation results. The simulation results were non-dimensionalized to make
the models applicable for FEF sensors with a wide range of dimensions and choice of
substrate materials. Two types of FEF sensors, interdigital sensors and concentric FEF
sensors, are considered here as examples to illustrate the non-dimensionalized modeling
method. Parametric models determined by this method demonstrate high accuracy when
compared to FEM simulation results, while greatly reducing computation time. These
models facilitate fast optimization of sensor design and quick validation of experimental
results.
5.2 Simulation Setup
Figure 5.1 shows a concentric FEF sensor designed for measuring round samples. FEF
sensors of such geometry can also be used when the orientation of the electrode matters,
as in the case of anisotropic samples [18]. The concentric FEF sensor was simulated
using the Ansoft Maxwell 2D software package. Figure 5.2 shows the simulation setup.
The left edge of the simulation space in Figure 5.2 represents the axis of symmetry in the
R-Z plane. The sensor has one driving electrode (drive), two sensing electrodes (sense1
and sense2), a guard electrode above the substrate, and several backplanes. Separate
backplanes are used here because they are each driven by a buffer operational amplifier
to stay in the same voltage as the sensing electrode immediately above. Such a
connection scheme is used to eliminate parasitic capacitance due to the backplanes and
sensing electrodes. A detailed discussion on the geometry of the backplane is presented
later in the chapter.
60
Figure 5.1. A concentric FEF sensor and a signal conditioning circuit designed for
non-invasive imaging of round samples.
Figure 5.2. Simulation setup of a concentric FEF sensor in the R-Z plane. The left
edge of the simulation space represents the axis of symmetry.
The geometry and dimensions of this sensor were chosen to balance optimal performance
and the simplicity of the design. For an FEF sensor of a fixed size, a trade-off exists
between its penetration depth and measurement sensitivity [3]. Increasing electrode width
improves the measurement sensitivity of the sensor, but reduces its penetration depth. In
the present chapter, the sensors are assumed to have 50% metallization ratio, that is, the
active electrodes (sensing and driving) and the gap between each of them have the same
width. This design is chosen to achieve a balance between penetration depth and
sensitivity. The radius of the sensor is 30 mm. All the electrodes are 0.08 mm thick. The
sensing and driving electrodes are 4 mm wide. The electrodes and backplanes are copper
and the substrate is a dielectric with a relative dielectric permittivity of 10. The relative
dielectric permittivity of the medium under test (MUT) εm varies from 4 to 100 and the
substrate thickness of the sensor d varies from 0.1 mm to 2.5 mm in the simulations.
61
The following boundary conditions are used for the simulation. All electrodes except for
the drive are set to 0 V; drive is set to 6 V. The boundary of the simulation space is set to
be a charge balloon. The balloon models an electrically insulated space, where the charge
at the infinity balances the internal charges. The simulation was electrostatic, which is
adequate for the low operating frequency of the sensing system.
5.3 Simulation Results
5.3.1 Effect of substrate thickness
Figure 5.3 (a) and (b) show respectively the electric field distribution for a sensor with a
thin substrate (d = 0.1 mm) and a sensor with a thick substrate (d = 2.5 mm). Variations
in the substrate thickness of the sensor have two effects. A thinner substrate causes more
field energy to leak through the gap in the backplane. In the simulation setup used in this
chapter Figure 5.2, the sensor is surrounded by the MUT and the effect of leakage field
on sensor output is minimal. Also, thinner substrates decrease the distance between drive
and the backplane, leading to more rapid decay of the electric field. Reduced substrate
thickness results in weaker signal strength (terminal impedance measurements), but
greater penetration depth [3,4].
5.3.2 Geometry of the backplanes
Separate backplane electrodes, instead of a solid backplane, are used for the sensor
shown in Figure 5.1. This geometry is chosen so that the effect of parasitic capacitances
between the sensing electrodes and backplanes can be eliminated through buffer
operational amplifiers. The widths of the gaps between the backplanes are optimized to
provide sufficient shielding while minimizing the leakage field between drive and the
backplanes.
Since the geometry of the backplanes used here is specific to the driven backplane
technique, we compare this geometry with the case of a solid backplane. Figure 5.4 (a)
and (b) show respectively the electric field distribution for a sensor with a thin substrate
(d = 0.1 mm) and a sensor with a thick substrate (d = 2.5 mm). Comparison of these
62
figures with Figure 5.3 (a) and (b) shows that difference between the field distribution for
the split-backplane sensor and the solid-backplane sensor is more prominent when the
substrate is thin. This is because for smaller d, more field lines leaked between the gap of
the backplanes in the case of the split-backplane sensor.
(a) (b)
Figure 5.3. The electric field arrows and equipotential lines for concentric FEF sensors with substrate thickness of (a) d = 0.1 mm and (b) d = 2.5 mm, where separate backplanes are used.
5.4 Non-dimensionalization
The normalized electric field radiation pattern of an FEF sensor depends only on the ratio
of the dielectric permittivity of the MUT, εm, to that of the sensor substrate, εs, rather than
on their absolute values. Similarly, only the ratio of the spatial wavelength λ of the sensor
to its substrate thickness d affects the normalized radiation pattern, not their absolute
values. Such properties of FEF sensors justify the use of a non-dimensionalized model to
represent sensors of different sizes.
5.4.1 Concentric ring FEF sensors
Figure 5.5 shows the normalized capacitance between the drive electrode and sense1
electrode plotted against normalized substrate thickness d/λ and normalized dielectric
permittivity of the MUT εm/εs. After normalization, the surface plot is no longer limited to
a sensor with a fixed dimension or choice of substrates, and it can be used for model
63
calibration.
(a) (b)
Figure 5.4. The electric field arrows and equipotential lines for concentric FEF sensors with substrate thickness of (a) d = 0.1 mm and (b) d = 2.5 mm, where a solid backplane is used.
Figure 5.5. Normalized capacitance between the drive electrode and sense1 plotted
against εm/εs and d/λ.
The capacitance data between other combinations of electrodes have similar trends, as
shown in Figure 5.6 and Figure 5.7 Due to the concentric geometry, the three electrodes
have different surface areas. Because of this variation in surface area, different
combinations of electrodes have different magnitude of capacitances.
64
Figure 5.6. Normalized capacitance between sense1 and sense2 plotted against εm/εs
and d/λ.
Figure 5.7. Normalized capacitance between sense2 and drive plotted against εm/εs and d/λ
A similar parametric simulation was also conducted for concentric sensors with solid
backplanes. Figure 5.8 shows the normalized capacitance between drive and sense1 of
such sensors. By comparing the results from the split-backplane geometry Figure 5.5 and
those from the solid backplane geometry Figure 5.8, one can see slight differences when
the substrate is thick and large differences when it is thin.
65
Figure 5.8. Normalized capacitance between the drive electrode and sense1 plotted
against εm/εs and d/λ for concentric sensors with solid backplanes.
5.4.1.1 Electrode pair sensitivity analysis
Figure 5.9 compares the three different pairs of electrodes in terms of their sensitivity to
changes in substrate thickness when the relative dielectric permittivity of the MUT εm is
kept constant at 52. The normalized plot was created by normalizing each value of
C/(λεs) by the minimum value of C/(λεs) (at d = 0.1 mm) as shown in (5.1).
min
%
min
100%s s
s increase
s
C CC
C
λε λελε
λε
⎛ ⎞− ⎜ ⎟
⎛ ⎞ ⎝ ⎠= ×⎜ ⎟⎛ ⎞⎝ ⎠⎜ ⎟⎝ ⎠
(5.1)
The sensitivity to substrate thickness is dependant on how much energy is absorbed from
an electrode pair by the backplanes. Figure 5.9 shows that the sense1-sense2 pair is least
sensitive to changes in substrate thickness. Sense1 and sense2 are separated by drive, and
drive is much closer to both of these electrodes than the backplanes are. Drive acts as a
shielding electrode and absorbs more energy from sense1 and sense2 than the backplanes
do and therefore the distance between the sensing electrodes and the backplanes has a
minimal effect on the capacitance between sense1 and sense2. Similarly, the sense2-
drive pair is less sensitive to substrate variation than the sense1-drive pair. The guard
electrode is in close proximity to the sense2 electrode, so it absorbs some of the energy
that would contribute to the capacitance seen between sense2 and drive. Therefore, the
66
energy absorbed by the backplanes has a smaller effect.
Figure 5.10 compares the three different electrode pairs in terms of their sensitivity to
changes in dielectric permittivity when the substrate thickness is kept at a constant value
of 1.3 mm. This was done by normalizing C/(λεs) by its initial value (when εm = 4)
according to (5.1). Figure 5.10 show that the sensitivities of different pairs of electrodes
to variations in εm/ εs do not differ much.
Figure 5.9. Electrode pair sensitivity to changes in capacitance as a result of a
changing substrate thickness (εm = 52).
Figure 5.10. Electrode pair sensitivity to changes in capacitance due to changes in the
dielectric permittivity of the MUT (d = 1.3 mm).
5.4.1.2 Determination of polynomial coefficients
The calibration surfaces are fit to the model shown in (5.2), where x = d/λ and y = εm/εs.
The fitting coefficients aij are determined in Matlab and the results are shown in Table
5.3, Table 5.1 and Table 5.2.
67
0,40,4
i jij
isj
C a x yλε =
=
= ∑ (5.2)
Figure 5.11 shows the residue from the polynomial fit. Within the simulation range, the
residue is shown to be less than 3% of the original simulation results.
Table 5.3. Polynomial coefficients aij for C1d / (λεs).
x4 x3 x2 x 1 y4 -6.436e0 1.940e2 -2.111e3 3.343e3 -1.382e3 y3 2.664e0 -7.847e1 8.267e2 -1.151e3 3.961e2 y2 -3.871e-1 1.106e1 -1.110e2 1.069e2 -1.626e1 y 2.120e-2 -5.798e-1 5.302e0 2.674e0 -1.353e0 1 -9.567e-5 1.859e-3 9.000e-3 2.225e-1 1.384e-2
Table 5.4. Polynomial coefficients aij for C12 / (λεs).
x4 x3 x2 x 1 y4 -7.385e-1 2.314e1 -2.645e2 3.011e2 -9.153e1 y3 3.090e-1 -9.441e0 1.042e2 -9.848e1 2.495e1 y2 -4.646e-2 1.370e0 -1.430e1 7.485e0 -7.787e-1 y 2.758e-3 -7.707e-2 7.249e-1 6.697e-1 -2.156e-1 1 -9.001e-6 1.470e-4 2.554e-3 6.295e-2 9.915e-4
Table 5.5. Polynomial coefficients aij for C2d / (λεs).
x4 x3 x2 X 1 y 4 -1.468e1 4.577e2 -5.132e3 7.698e3 -3.049e3 y3 6.170e0 -1.866e2 2.041e3 -2.631e3 8.510e2 y2 -9.143e-1 2.660e1 -2.715e2 2.353e2 -2.527e1 y 5.115e-2 -1.411e0 1.300e1 8.434e0 -4.019e0 1 -2.262e-4 4.246e-3 2.751e-2 7.129e-1 3.449e-2
68
Figure 5.11. Fitting residue of the fourth-order polynomial model for C1d/λεs.
5.4.1.3 An example
A practical example is provided here to illustrate how the parametric model can be used.
Suppose a concentric FEF sensor has the geometry shown in Figure 5.1. The substrate is
made from Teflon (εs = 2.08) and it is 4.8 mm thick. The spatial wavelength of the sensor
is 32 mm. The MUT is castor oil (εm = 4.5).
The first step is to calculate the non-dimensionalized parameters d/λ and εm /εs.
/ 4.8 / 32 0.15d λ = = (5.3)
/ 4.5 / 2.08 2.1625m sε ε = = (5.4)
Then, estimate the normalized capacitances based on the coefficients in Table 5.3, Table
5.6 and Table 5.7.
412 3.443 10s
Cελ
−≈ × (5.5)
31 1.994 10d
s
Cελ
−≈ × (5.6)
32 5.402 10d
s
Cελ
−≈ × (5.7)
The capacitances are found to be:
69
12 0.2029C ≈ pF (5.8)
1 1.1750dC ≈ pF (5.9)
2 3.1834dC ≈ pF (5.10)
5.4.1.4 Model evaluation
To check the validity of the parametric model, results from several test simulations were
compared with the estimated sensor output based on the model. In the test simulations,
the relative dielectric permittivity of the sensor substrate is varied while the spatial
wavelength of the sensor is kept the same as used in the original setup. Figure 5.12 shows
the residue of the estimates from the parametric model. The error is shown to be within
5%.
Figure 5.12. Comparison between finite element simulation results and results
estimated by the parametric model.
When the spatial wavelength of the sensor is varied, the sensor output is shown to scale
linearly with the size of the sensor provided that the electrodes are thin. This matches
well with the prediction of the parametric model. If the thickness of the electrodes is
comparable to the spatial wavelength of the sensor, the parasitic capacitances between the
vertical edges of the electrodes (due to the finite thickness of the electrodes) lead to
nonlinearity, and the sensor output no longer scales linearly with size.
70
5.4.2 Interdigital FEF sensors
Figure 5.13 shows a typical half-wavelength cross-section of an interdigital FEF sensor,
where εm is the dielectric permittivity of material under test of infinite thickness, and εs is
the dielectric permittivity of the substrate. The substrate thickness d was varied from 20
μm to 2000 μm, in steps of 10 μm, while the relative permittivity of the material, εm was
varied from 4 to 100, in steps of 2. The total height of the simulated region was
maintained at 4000 μm. The permittivity of the substrate, εs, was kept at 10 to permit the
ratio εm/εs to include values above and below 1. Electrode thicknesses are 80 μm.
Capacitance between the drive and sense electrodes was determined as a function of
normalized substrate thickness, d/λ, and normalized dielectric constant, εm/εs as shown in
(5.11), where x = d/λ (substrate thickness/wavelength) and y = εm/εs (material dielectric
constant/ substrate dielectric constant). Table 5.8 shows the polynomial coefficients, aij,
for x = d/λ (substrate thickness divided by the wavelength), and y = εm/εs (material
dielectric constant divided by the substrate dielectric constant).
Figure 5.13. A half-wavelength portion of the interdigital sensor is shown. An
example of a triangular mesh used for computation of capacitance and conductance matrices associated with a half-wavelength section interdigital structure.
12
=0,4=0,4
= i jij
iSj
C a x yε ∑ (5.11)
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Table 5.8. Polynomial coefficients aij for C12 / εs.
x4 x3 x2 x 1 y 4 -8.043e-1 2.131e1 -1.885e2 1.066e2 4.542e1 y3 4.865e-1 -1.275e1 1.101e2 -2.736e1 -5.412e1 y2 -9.952e-2 2.559e0 -2.113e1 -1.020e1 2.009e1 y 7.158e-3 -1.763e-1 1.272e0 4.254e0 -1.960e0 1 -1.193e-5 -3.846e-4 2.281e-2 1.004e-1 2.012e-2
(a) (b)
Figure 5.14. (a) Finite element simulation results and (b) results estimated by the parametric model. A practical example is provided here to illustrate how the parametric model can be used.
Suppose an interdigitated FEF sensor has the geometry shown in Figure 5.13. The
substrate is made from Teflon (εs = 2.08) and it is 4.8 mm thick. The spatial wavelength
of the sensor is 32 mm. The MUT is castor oil (εm = 4.5). The non-dimensionalized
parameters d/λ and εm /εs are the same as calculated in (5.3) and (5.4).
Based on the coefficients in Table 5.8, the normalized capacitance is estimated.
12 1.090s
Cε
≈ (5.12)
The capacitance is found to be:
12 2.267C ≈ pF/m (5.13)
5.5 Conclusions
Parametric models are constructed for concentric FEF sensors and interdigital FEF
72
sensors based on non-dimensionalized fitting of FEM simulation results. The models
enable direct estimation of the output of FEF sensors with a good modeling accuracy
(less than 5% error). The modeling method is not restricted to FEF sensors; its concept of
non-dimensionalization can be used for the design optimization of a wide variety of
sensors. For sensors that are difficult to model analytically, the method provides accuracy
comparable to numeric methods while greatly improves the throughput of design
optimization. The models developed by this method can also be used for quick validation
of experimental results.
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Chapter 6. Image Reconstruction Algorithms 6.1 Electrical Impedance Tomography
Electrical Impedance Tomography (EIT) produces images of the distribution of complex
electrical impedance within a closed boundary. EIT has been used in various biomedical
[88], geophysical [89], and industrial applications [90,91]. Compared with other imaging
modalities, such as X-ray computed tomography and magnetic resonance imaging (MRI),
EIT has the advantage of being non-invasive, fast, and cost-effective. The theory and
applications of electrical impedance tomography are reviewed in [92].
A typical setup of an EIT system involves an array of electrodes implanted on the surface
of a cylinder. Sinusoidal voltage/current is applied to the electrodes and the induced
voltage/current at the electrodes is measured. The distribution of the complex permittivity
of the medium under test is reconstructed from the excitation signal and electrical
measurements at the electrodes.
The forward problem in EIT is to estimate the induced electrical measurement at the
electrode given an excitation signal and permittivity distribution. The inverse problem
estimates the permittivity distribution based on the excitation signal and the terminal
electrical measurements. The inverse problem involves solving a Fredholm integral
equation of the first kind [93] and it is ill-posed. Ill-posedness means that the solution
does not depend continuously on the measured data, and that small variations in the
measured data can lead to relatively large errors in the reconstruction of the impedance
distribution. The inverse problem of EIT is also underdetermined because the number of
independent measurements is typically much smaller than the number of pixels in the
image.
Image reconstruction algorithms for EIT can be classified into linear algorithms and
nonlinear algorithms. Linear algorithms can not account for the inherent nonlinearities of
EIT and they have to be solved iteratively to generate an image with a reasonable quality.
Iterative algorithms solve both the forward and the inverse problem, where the forward
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problem is typically solved by finite element (FE) methods. Because the inverse problem
is ill-posed, the reconstructed image is very sensitive to FE simulation errors. To
minimize the error, accurate electrode modeling and meshing is needed in the FE
calculations. Although iterative linear algorithms have better accuracy than non-iterative
ones, they are more time-consuming and not suitable for real-time applications. Direct
(non-iterative) nonlinear algorithms is the key to fast and accurate image reconstruction
in EIT [94]. A thorough review on the state of art on EIT reconstruction algorithms is
presented in [51].
Image reconstruction for EIT is highly sensitive to measurement errors. Due to the finite
measurement resolution of EIT systems, the images reconstructed from ‘raw’
measurements deviates significantly from the true image. To stabilize the solution to the
ill-posed problem, Tikhonov regularization is often used. It regularizes the solution based
on the prior information about the MUT [95]. A common practice is to assume its
dielectric permittivity to be slow varying. In effect, regularization smoothes out the
highly oscillatory part of the solution caused by measurement noise. The amount of
regularization is controlled by the regularization parameter. Too much regularization
leads to large residues, while not enough regularization leads to unstable solutions. The
generalized cross-validation method, the discrepancy principle, and the L-curve method
[96] are methods for choosing regularization parameters. The ‘smooth’ prior – the
assumption that the dielectric permittivity of the sample is slow-varying – do not work
well if the MUT has sharp discontinuities in its dielectric permittivity distribution.
Regularization based on prior information about the boundary of the discontinuities is
discussed in [97,98].
In traditional 2-D EIT, the electrodes are positioned in the same plane around the cylinder
to image the cross section of the internal medium. Due to the soft-field effect of EIT, the
electric field would not stay confined in the plane of the electrodes. Leakage of electrical
energy outside of the plane leads to reconstruction errors of the permittivity distribution.
A 3-D setup is, therefore, necessary for accurate reconstruction of the interior. Compared
with 2-D systems, 3-D EIT systems require longer data acquisition time and increased
complexity in image reconstruction algorithms.
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In EIT, sensor output depends on the excitation signal as well as the permittivity
distribution. The excitation pattern can be optimized to maximize the sensor output for a
particular permittivity distribution. When compared with adjacent electrode excitation
and opposite electrode excitation, optimal excitation is shown to improve
distinguishability of permittivity distribution by 2 to 10 times, however, multi-electrode
excitation requires individual excitation source and measurement devices for each
electrode, thus greatly increasing the complexity and the cost of hardware [99]. Another
concern over multi-electrode excitation is the safety regulation in medical applications of
EIT on the maximum total current that can be applied to a patient’s body. If this
constraint is applied, optimum excitation do not always perform better than single
excitation methods [100].
6.2 Background on Inverse Problems
A problem is said to be well-posed in the Hadamard sense if: (1) the solution exists for all
data; (2) the solution is unique; (3) the solution depends continuously on the data. If a
problem fails to meet any of the above criteria, it is ill-posed. Inverse problems constitute
a broad class of ill-posed problems in real life applications. Such applications include the
inverse scattering problem in communications, image reconstruction in biomedical
sciences and industrial process control, system identification in automatic control, pattern
recognition in signal processing, and so on [93]. The inverse problem in EIT reconstructs
an image of the internal distribution of the dielectric permittivity of the medium under
test (MUT) based on boundary electrical observations. Given complete continuous data
on the boundary, the inverse problem is proven to have unique solutions [101]. Real
world systems deviate from the ideal mathematical assumption in that (1) electrodes do
not cover the entire boundary and (2) the system measurement resolution is finite. A
review of inverse problem theory for electrical impedance tomography is presented in
[102].
Algorithms using for solving ill-posed problems can be divided into statistical methods
(reference), deterministic methods based on linearization (reference), and non-linear
deterministic methods (reference). The linearized and non-linear algorithms can be
76
subdivided into direct single-step methods (reference), and iterative methods (reference).
Regularization tools are often used to determine a set of solutions for ill-posed problems
based on prior information. The underlying principle of regularization is to minimize the
difference between the measured impedance/capacitance and the calculated
impedance/capacitance, while the calculated solution of permittivity distribution is kept
reasonably close to the true solution.
Tikhonov regularization is one of the most commonly used. It can be expressed in the
general form of finding solution g, which minimizes the following function:
))ˆ((21 22 ggLSg −+− μλ (6.1)
where g is the estimated dielectric permittivity distribution according to prior
information and 2)ˆ( ggL −μ is used as a constraint for the optimization problem.
The quality of Tikhonov regularization strongly depends on the regularization parameter
μ. It is crucial to choose an optimal regularization parameter μ, so that a solution as close
to the true solution as possible can be obtained. In general, a small value of μ gives a
good approximation to the original problem but the influence of errors may make the
solution physically unacceptable. Conversely, a large value of μ suppresses the data
errors but increases the approximation error.
6.3 Modeling of Electrode
Iterative algorithms involve solving the both the forward problem and the inverse
problem. The forward problem predicts the boundary voltages induced by the injected
current for a permittivity distribution; the inverse problem estimates the permittivity
distribution based on the boundary voltage data. First, an initial permittivity distribution
is assumed. The permittivity distribution is modified based on the difference between the
boundary voltage estimated by the forward model and that measured by the sensor.
Iteration continues until the difference converges to zero. Due to the ill-posed nature of
the inverse problem, any errors from forward estimation result in large errors in the
calculated permittivity distribution. It is therefore important to construct an accurate
77
forward model.
A simple example is the gap model. It assumes constant current on the electrodes and
zero current between the electrodes. Although easy to implement, the model is not
accurate because it does not account for the shunt current and the contact impedance due
to the electrodes.
The shunt effect from the electrodes – the electrodes usually has much higher
conductivity than the material under test (MUT). When current is applies to some
electrodes, the passive electrodes behaves likes an electrical ‘short’ and shunt some of the
current, resulting in less current flowing through the MUT. The shunt current is shown to
increase with increasing metallization ration of the sensor [103].
The contact impedance – a thin high-impedance layer forms between the electrode and
the material under test (MUT). The contact impedance results in bigger boundary voltage
measurement than the actual voltage drop across the MUT, leading to image
reconstruction errors. Such errors can be avoided by using separate electrodes for current
injection and voltage measurement. By connecting the voltage-measuring electrodes to a
high input impedance operational amplifier, the current flowing through the contact
impedance and, therefore, the voltage drop across it is minimized.
The complete electrode model takes into account the effect of both the shunt current and
the contact impedance [104]. The model has demonstrated an accuracy of 0.1% [103].
Reviews on electrode modeling for EIT are presented in [103,105].
6.4 The Layer Stripping Algorithm
The layer stripping algorithm recursively identifies the media by removing the effects of
layers one at a time. It has been used to solve the inverse scattering problems in
geophysical remote sensing for 1-D discrete lossless [106] and lossy [107] media as well
as 2-D lossless media [108]. In remote sensing, layer dielectrics can be used to model
stratified soil, forest canopy, sea ice, and glaciers, where the media can be considered
both homogeneous and infinite in the horizontal plane. Under far-field approximation, the
incident field upon these layered media can be considered a plane wave normal to the
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boundary. Inverse scattering theories can only be applied to hyperbolic problems. The
inverse resistivity problem is, however, an elliptic problem. The inverse resistivity
problem is some times solved by transforming it first to an inverse scattering problem.
Layer stripping algorithms bases on this approach are presented in [109-111]. These
algorithms are developed to reconstruct the resistivity distribution in a 1D or 2D media.
Reference [112] presents a layer stripping algorithm developed to directly solve an
inverse resistivity problem for samples with a radial geometry.
The layer stripping algorithm is a direct method of obtaining an approximate solution to
the full non-linear problem, and is therefore immune to extraneous local minima which
can cause iterative methods to become stuck. In addition, the method requires far fewer
computations and storage requirements than iterative schemes.
6.5 Problem Description
The goal of this chapter is to develop a layer stripping reconstruction algorithm for a one-
dimensional imaging scanner. The scanner is designed to reconstruct the 1-dimensional
dielectric permittivity (complex for lossy dielectrics) profile of layered dielectrics based
on electrical measurements from the FEF sensor. The schematic for the scanner is shown
in Figure 6.1. A fringing electric field (FEF) sensor is positioned below a layered
dielectric sample. The sensor is attached to a stepper motor that allows the sensor to
acquire data at different positions along the ‘z’ axis. All the electrodes of the FEF sensor
are independently driven. At each given time, a low frequency ( < 10 MHz) AC electrical
current is applied at the selected pair of electrodes and the induced voltage potentials at
the rest of the electrodes are measured. The dimension of the sample and that of the
sensor along the y direction is finite. The number of electrodes and the geometry of the
backplane of the FEF sensor can be adjusted as needed. The sample under test is assumed
to be homogeneous along the x-y plane. Its dielectric property along the z axis can either
be slowly varying (continuous/smooth) or discrete (has sharp discontinuities). The
present application differs from the above mentioned examples in geophysics in that (1)
the layered dielectric sample is finite along the x-y plane as illustrated in Figure 6.1. (2)
the electromagnetic wave radiated from the FEF sensor can not be considered a plane
79
wave. These differences necessitate properly defined boundary conditions.
Figure 6.1. Side-view of the one-dimensional scanner setup.
6.6 Proof of Concept Study
As a proof of concept study the layer stripping algorithm presented in [112] is reproduced
using MATLAB. The purpose of the study is to test the validity of the algorithm with
simple test cases. The Matlab code is shown in the Appendix. The test procedure contains
the following steps:
1. A layered medium with known resistivity is constructed.
2. The Fourier basis function for the layer stripping algorithm is calculated based on
the method described in [112].
3. Using the Fourier basis function from step 2 to reconstruct the resistivity profile
of the medium.
For the layer stripping algorithm to converge, some regularization has to be used. In this
case, the algorithm is regularized by truncating higher order Fourier coefficients. It is
critical to determine the order of the coefficients to be truncated. Three different methods
are recommended in [112] to determine the order. All three methods were tried
implemented in the Matlab algorithm, but none of them produced results that are
converging.
80
6.7 Conclusions and Future Work
The layer stripping algorithm presented in [112] is used here to solve a 1D inverse
resistivity problem. A proof of concept study was done to reproduce the results presented
in the paper. However, the Matlab algorithm failed to converge.
For future work, the first step is to resolve the convergence problem by reinvestigating
the regularization methods. Secondly, the algorithm is originally developed for samples
with radial geometry. The algorithm needs to be modified to be applied to the inverse
resistivity problem in the Cartesian coordinate. Thirdly, the performance of this particular
layer stripping algorithm should be compared with the other layer stripping algorithms
proposed in [109-111].
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Chapter 7. Moisture Dynamics in Food Products 7.1 Review of process analytical technologies in food analysis Many of the real-time non-destructive sensing techniques used for food processing are
spectroscopic in nature. Spectroscopic analysis exploits the interaction of electromagnetic
radiation with atoms and molecules to provide qualitative and quantitative chemical and
physical information that is contained within the frequency spectrum of the energy that is
either absorbed or emitted. A thorough review on the various non-destructive
spectroscopic sensing techniques for the measurements of foods is available in [113].
Figure 7.1 shows the frequency spectrums of various types of spectroscopic sensing
techniques often used for food processing.
Figure 7.1. Frequency ranges of various types of spectroscopic sensing
techniques. Dielectric spectroscopy operates at lower frequency range, when compared with other spectroscopic sensing techniques.
7.1.1 Nuclear magnetic resonance spectroscopy (NMR)
In NMR analysis, a constant homogenous magnetic field is applied to the sample and RF
pulses are directed to induce changes in the nuclear spin of atoms. The various spin
responses are frequency-dependent and reflect the number and mobility of hydrogen
atoms. Low resolution instruments associated with 0.23 to 0.95 tesla (T) magnetic fields
uses RF pulses in the range of 10 MHz to 40 MHz. High-resolution instruments with field
strength of roughly 2.35 T uses RF pulses at above 100 MHz. Low-resolution food
analysis are confined to the measurements of hydrogen atom numbers and their mobility.
A wider variety of chemical species can be sensed when stronger magnetic fields are
82
used. High-resolution NMR is used to measure isotope ratios in foods. Analysis of such
data can be used to confirm authenticity and to detect the presence of adulterants. Many
high-resolution measurements, especially the isotopic ones, require complicated
extraction procedures to enrich the species under study, and therefore cannot be regarded
as truly non-destructive. The major limitation of NMR techniques is their inability to
study ferromagnetic materials and samples containing significant amounts of
paramagnetic species. A review of the applications of NMR to food measurements is
available in [114].
7.1.2 Mid-infrared (mid-IR) spectroscopy
Use of the mid-IR region of the electromagnetic spectrum involves vibrational responses
from the bonds of organic molecules, such as O-H, C-H, C-O, and N-H. When irradiated
by mid-IR electromagnetic waves, these bonds absorb energy from the radiating sources.
The energy absorption is frequency-dependent. The resulting absorption spectrum can be
used to detect the existence of these bonds. Mid-IR radiation impinges only on the
surface of optically dense food materials, which prevent wide-spread application of mid-
IR spectroscopy in food analysis.
7.1.3 Near-infrared (NIR) spectroscopy
Near-infrared (NIR) spectroscopy is based on the absorption of electromagnetic radiation
at wavelengths from 400 nm to 2500 nm. Similar to of mid-IR spectroscopy, NIR is also
based on the principles of vibration spectroscopy. NIR waves have shorter wavelengths
than mid-IR waves, and penetrate deeper into a sample. Thus, NIR can be used to obtain
spectral data of thick samples.
NIR spectroscopy is a relative method that requires calibration against a reference
method for the parameter of interest. Calibration is normally carried out using
multivariate mathematics (chemometrics). The major advantage of NIR is that it requires
no sample preparation. The analysis is, therefore, simple and fast (between 15 and 90 s)
and can be carried out online. In addition, NIR allows several parameters to be measured
concurrently. An excellent review on NIR spectroscopy is available in [115].
83
Applications of NIR spectroscopy specific to the food industry are reviewed in [116].
7.1.4 Raman spectroscopy
Another non-invasive sensing technique used widely for food process control is Raman
spectroscopy. It is a non-destructive technique that can provide information about the
concentration, structure, and interaction of biochemical molecules within intact cells and
tissues.
Although similar wavelength ranges are used for mid-IR and Raman spectroscopy,
different rules govern the principles of mid-IR absorption spectroscopy and Raman
scattering spectroscopy. The two techniques are, to a certain extent, complementary:
certain atomic bonds that do not produce strong mid-IR vibrational responses produce a
polarizability response that is measurable in Raman spectra and vice versa.
Applications of Raman spectroscopy in food industry was reviewed in [117]. Raman
scattering is a relatively weak optical effect that requires lasers for efficient excitation. In
many materials the fluorescence induced by the incident light is much more intense than
the Raman scattered light, which makes the Raman signals difficult to measure. This
problem can be avoided by using a Fourier transform (FT) Raman system with laser
excitation at 1064 nm, since negligible fluorescence occurs for this low energy excitation
wavelength. However, 1064 nm excitation results in weak Raman signals and requires
long exposure time (typically 30 min) to obtain spectra with a sufficient signal-to-noise
ratio. This approach can therefore be impractical for efficient sample analysis and not
useful for on-line purposes [118].
7.1.5 Ultrasound
Ultrasound is high frequency sound (typically around 1 MHz). An ultrasonic wave is
transmitted as a series of deformations in the medium through which it passes. The
deformations can be either normal (i.e., shear waves) or parallel (i.e., longitudinal waves)
to the direction of propagation. The motion creates alternative compression and
rarefaction of the medium particles. In low-density applications (power density around 1
W/cm2), the deformations are small enough to be within the elastic limit of the material
84
and the wave is non-destructive. However, in high-density ultrasound (power density
between 10 and 1000 W/cm2), the properties of medium under test are often changed, in
most cases permanently.
The transmission of an ultrasonic wave depends on the compression and extension of
various bonds in the medium. Therefore, ultrasonic properties are related to the number
and strengths of the bonds and hence bulk structure and composition. Ultrasound has
been used for food characterization including the measurement of the concentration of
simple solutions, lipid crystallinity, emulsion droplet size, meat composition, and
temperature [119,120]. Ultrasonic waves can be readily propagated through certain
optically opaque materials including many foods and packaging materials. Because sound
readily passes through steel, it is relatively easy to measure the properties of fluids within
pipes and other process equipment. The most significant limitation of ultrasonic devices
is that it is difficult to make measurements through air. The reflection of a wave at a
surface depends on the acoustic dissimilarity (impedance mismatch) between the two
phases. Because air is so different from food, common container materials, and even the
transducer element itself, most of the energy is reflected rather than passed into the gas.
Furthermore, small air pockets scatter sound very efficiently and the losses make
meaningful transmission measurements impossible. In some cases it is possible to make
measurements at lower frequency but the loss of spatial resolution reduces the value of
this approach [121].
Other analytical techniques that are often used for food analysis are mass spectroscopy,
gas chromatography, atomic spectroscopy, and high-performance liquid chromatography.
They are reviewed in detail in [122].
7.2 Definition of the Problem
The maintenance of food quality is gaining increasing importance as the shelf life of food
products becomes longer. Manufacturers, retailers, and consumers are demanding and
expecting longer shelf life for shelf-stable and refrigerated foods. Moisture content is an
important parameter affecting the shelf life of most food products. In some cases
excessive amounts of water will cause spoilage, while in other cases, loss of water will
85
render foods unacceptable.
Shelf-stable homogeneous and multi-layer food products represent a rapidly growing
segment of value-added products for the U.S. food industry. These types of foods may be
processed or formulated such that different regions have different moisture saturation
levels and/or different water activity levels. Moisture migration may reduce product
quality and/or decrease its stability [123].
The control of water in foods requires fast and reliable methods of evaluating the
moisture content and the moisture transport activities of foods [124]. In this chapter, a
multi-channel electrical impedance sensor is used for determining the moisture content
and relative distribution of sugar cookies. Moisture concentration is defined here as
follows:
1%
1 2100%MM
M M= ×
+ (7.1)
where M1 is the mass of the moisture contained in the unit volume, and M2 is the mass of
the dry portion of the material in the same unit volume.
Material impedance is a function of many variables, as shown in (7.2), where M% is the
moisture concentration in the material, T is the ambient temperature, D is the sample
density, and ω is the input signal frequency.
%( , , , )s ZZ f M T D ω= (7.2)
System calibration involves solving the inverse problem of determining the following
function:
% ( , , , )M sM f Z T D ω= (7.3)
or
% ( )M sM f Z= (7.4)
where the functional dependence between moisture concentration and the impedance is to
be determined. The effects from variables other than moisture content are either
eliminated or accounted for.
86
7.3 Methodology
The material contents of food products are usually complex and varying, which renders
direct determination of sample dielectric permittivity difficult and impractical. Under
these circumstances, an indirect parameter estimation approach based on quantitative
mapping between electrical measurements and the physical variable of interest can be
used. The major challenge for such an approach lies in minimizing the effect of variables
other than moisture concentration, such as ambient temperature and sample density,
which are considered here as disturbance factors. The effects of these factors should
either be eliminated or accounted for in the calibration algorithm [125].
7.4 Experimental Setup
7.4.1 The concentric sensor head
Figure 7.2 shows a concentric sensor head, designed for localized measurements. It has
three electrically separated sensing electrodes, each shielded by a guard plane on the back
of the substrate.
Channel 1 BottomTop
Channel 3Channel 2 Figure 7.2. Top and bottom view of the concentric sensor head. The center plate
is 10 mm in diameter. The outer two rings are 5 mm wide. The spacing between adjacent sensing plates is also 5 mm. The guard planes on the back are slightly wider than respective sensing electrodes.
The sensor head can be used as a fringing field sensor by applying an AC sinusoidal
voltage to the middle ring electrode and measuring the voltage at the two neighboring
electrodes. A non-linear model is needed to describe such a setup. The solution to the
87
Laplace equation of the non-linear model is:
0 1 2( , ) ( )( )z zr z J r c e c eβ βφ β − += + (7.5)
where φ iss electric potential, r is to the radial coordinate on the horizontal plane, z is the
vertical coordinate, J0 is the zero order Bessel function of the first kind and β is a scaling
constant such that βr is one of the zeros of J0 [126].
The fringing field setup provides one-sided access but has limited signal strength. It is
also susceptible to disturbances from the contact quality between the samples and the
surface of the sensors. The parallel plate setup, on the other hand, is a complement to the
fringing field setup. It lacks the one-sided access but offers greater signal strength and is
comparatively less sensitive to surface contact qualities.
Figure 7.3. Side view of the sensor in a voltage divider setup. A cookie is placed
between the sensing and driving plates.
This chapter presents experimental data obtained with the parallel plate arrangement of
Figure 7.3. A barrier made of 300 μm thick Kapton is used to avoid the Debye layer
effect [127]. The parallel plate capacitor can be modeled as a Maxwell capacitor with
three different dielectrics in series: air, polyimide (Kapton), and the material under test.
For a Maxwell capacitor like this, terminal impedance measurements are not sensitive to
vertical displacements of the polyimide and the material under test [128]. This property
makes parallel-plate sensors more robust to surface contact disturbances.
7.4.2 A voltage divider circuit
Figure 7.3 shows a voltage divider circuit, where Vi is the input voltage signal, Vs is the
sensing voltage signal, Zr is the reference impedance, and Zs is the sensing impedance.
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The effective impedance of the parallel-plate capacitor is calculated from the voltage
divider relationship 7.6. To maximize circuit sensitivity, Zr is chosen to be close to Zs. In
this investigation, Zr = 8 pF.
s r
i r s
V ZV Z Z
=+
(7.6)
Various connection schemes are available for this voltage divider setup. When there is an
electric potential difference between the sensing electrodes and their respective guard
electrodes, stray capacitances are introduced into the circuit. To prevent these stray
capacitances from affecting measurement accuracy, the guard planes are set to the same
voltage as their respective sensing electrodes by using a unity-gain voltage buffer.
Figure 7.4. Sensor geometry and experimental setup. The moisture content of the
sample is increased by adding increments of 0.2 grams of water to the center of the sample. Measurements are taken at each moisture content level. A 3-channel voltage-divider interface circuit is used to measure the complex terminal impedance of the sensor at three different radial locations.
7.5 Experimental Procedure
To calibrate the moisture sensing system, a quantitative relationship between sample
moisture content and the corresponding impedance measurements needs to be
established. The following experiment was conducted to evaluate this relationship.
1. A test specimen is placed between the sensor plates so that the center of the
specimen is aligned with channel 1 of the sensing plate.
89
2. A 6 volt, 10 Hz to 10 kHz frequency sweep signal is applied to the circuit in
Figure 7.3 and Vs is measured.
3. The moisture content of the sample is increased by adding increments of 0.2
grams of water to the center point.
4. Measurements are taken at each moisture content level.
7.6 Experimental Result and Data Analysis
Figure 7.5 (a) and Figure 7.5 (b) show respectively the capacitance and phase variations
due to moisture content increase as measured by the center sensing electrode. Change in
moisture content leads to an increase in the capacitance and phase maxima and a shift of
the curves toward higher frequencies.
102
103
1040.14
0.16
0.18
0.2
0.22
0.24
0.26
Cap
acita
nce
(pF)
Frequency (Hz)
0 g
1.0 g
0.8 g
0.6 g
0.4 g
0.2 g
10
210
310
4-12
-10
-8
-6
-4
-2
0
2
4
Frequency (Hz)
Pha
se (d
eg)
0 g
1.0 g0.8 g0.6 g0.4 g0.2 g
(a) (b) Figure 7.5. Capacitances and phase measured at different moisture content levels. The measurement results show that changes in moisture content leads to an increase in the capacitance and phase maxima.
For capacitance measurements, the higher the signal frequency, the greater the
measurement sensitivity to moisture content. To achieve maximum sensitivity,
capacitance data at 10 kHz is used to calibrate the system, which here involves
establishing a quantitative mapping between capacitance values and moisture content.
7.6.1 Compensation for moisture diffusion
The triangles in Figure 7.6 show the channel 1 capacitance data at 10 kHz averaged
across different frequency sweeps. At higher moisture content level, moisture diffusion to
90
the outer channels reduces the capacitance increase between neighboring samples.
The higher the moisture content gradient between the center channel and the outer rings,
the more intensive the moisture diffusion process. This is reflected in the increasing
discrepancy between the uncompensated and compensated capacitance data as water is
being added to the center of the sample.
0.2 0.4 0.6 0.8 1 1.20
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
Moisture (g)
Cap
acita
nce
(pF)
Before compensation
After compensation
*16.4390 0.0028M CΔ = × Δ +
Figure 7.6. Capacitance measurements against the weight of added water at 10
kHz for channel 1. Saturation occurs at high water level.
To compensate for the effect of diffusion, the capacitance increase from channels 2 and 3
is measured and mapped to effective increase in channel 1. This increase, added to the
original capacitance change from channel 1, gives the new channel 1 capacitance data
after compensation, as shown in (7.7), where ∆C is the capacitance increase for each
channel, Cor is the capacitance measurement of the original sample for each channel, and
∆C1* is the channel 1 capacitance increase after compensation.
33
12
2
11
*1 C
CC
CCC
CCor
or
or
or Δ+Δ+Δ=Δ (7.7)
As indicated by the solid line in Figure 7.6, a better linear approximation is achieved after
compensation.
7.6.2 Linear regression
Assuming a linear functional dependence, the following calibration equation is
determined for channel 1 by performing linear regression on the compensated data:
91
311 108.244.6 −×+Δ×=Δ CM (7.8)
In this configuration, all sensor pixels are parallel-plate capacitors with different area.
Ideally, the functional dependence for channels 2 and 3 can be obtained by scaling (7.8)
with the respective area ratio. However, the existence of a non-uniform air gap has to be
taken into account.
7.6.3 Compensation for non-uniform air gap
The air gap between the material and the top plate of the sensor is non-uniform due to an
uneven shape of the cookie samples. To compensate for this, uniform water distribution
in the original sample is assumed and the ratios of capacitance measurements from
channels 2 and channel 3 with respect to those from channel 1 are measured. This
difference in capacitance measurements from the three channels is caused partly by the
difference in sensing plate area and partly by the non-uniformity in air gap thickness.
Taking the ratios obtained above and using them as scaling factors, the functional
dependence of capacitance measurements on water content from channels 2 and 3 can be
obtained from (7.8).
422 1022.643.1 −×+Δ×=Δ CM (7.9)
433 1095.391.0 −×+Δ×=Δ CM (7.10)
7.6.4 Moisture content distribution
Based on the calibration equations (7.8), (7.9), and (7.10), the absolute mass of moisture
contained in the portion of the sample above each ring is calculated from the capacitance
measurements. The mass of the dry portion of the sample above each ring is determined
from the ratio of the respective sensing electrode area to the area of the whole sample.
Given the absolute mass of moisture and the dry portion of the sample, moisture content
levels for all three channels can be calculated according to (7.1), enabling real-time
imaging of moisture content distribution.
Figure 7.7 shows the moisture content distribution profile of a sample at various moisture
content levels, which is obtained from fitting the moisture content data from the three
92
channels to scaled Gaussian curves.
2
2%2
)(⎟⎠⎞
⎜⎝⎛−
= σ
πσ
x
eAxM (7.11)
where x is the distance to the center of the sensing plate, σ is a measure of the width of
the curves and A is a scaling factor, which is determined by the moisture diffusion
coefficient of the diffusion process.
Figure 7.7. Moisture content distribution across the radius of the sample when different amount of water is added to the center. The moisture distribution profile is obtained through fitting of measured moisture data to Gaussian curves.
7.6.5 Evaluation of the calibration model
The calibration approach discussed above involves several approximations. To evaluate
the effectiveness of the model obtained through system calibration, the absolute masses
of the moisture measured from all three channels are summed up and compared with the
mass of the moisture added to the sample. As indicated in Table 7.1, measurement error
decreases with increasing moisture content. Further processing of experimental data is
needed to reduce the error at low moisture content levels.
93
Table 7.1. Comparison between the actual mass of the moisture added to the sample and the mass of the moisture measured by the sensor.
Moisture Added (g) Moisture Measured (g) Error 0.2 0.130 35% 0.4 0.312 22% 0.6 0.570 5% 0.8 0.776 3% 1.0 0.980 2%
7.7 Conclusions
The electrical impedance sensor is used to determine the moisture content and lateral
moisture distribution of cookie samples. The concentric sensor head used here can be
configured as a stand alone FEF sensor. The multi-channel FEF sensor allows three-
dimensional profiling of the sample under test. However, the effects of lateral variation
and those of vertical variation within in the sample could be coupled in the sensor
terminal measurements, which greatly increase the complexity of parameter estimation
algorithms. If moisture activity along only the vertical axis (the axis perpendicular to the
electrode surface) is required, a single channel FEF sensor would be sufficient.
Applications of the electrical impedance moisture sensor described in this chapter are not
limited to the study of shelf life of foods. It can be integrated into the food drying process
to control the moisture endpoint. It can also be used to moisture content of foods during
baking. The latter is a more challenging task in that many other physical parameters
(temperature, texture, porosity, and dimensional) of the sample could be changing
simultaneously. Multivariate analysis methods, such as principle component analysis and
partial least squares regression, are needed to calibrate the system.
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Chapter 8. Measuring Physical Properties of Pharmaceutical Samples 8.1 Motivation
The Food and Drug Administration (FDA) has very stringent regulations for
pharmaceutical products in such aspects as active ingredient concentration, tablet
hardness, and coating thickness. Quality control is, therefore, a top priority for
pharmaceutical companies.
A typical pharmaceutical tablet is a complex matrix containing one or more active
pharmaceutical ingredients (APIs), fillers, binders, disintegrants, lubricants, and other
materials. A basic problem in pharmaceutical manufacturing is that a relatively simple
formulation with identical ingredients can produce widely varying therapeutic
performance depending upon how the ingredients are distributed in the final matrix.
Pharmaceutical makers also are developing advanced tablets for drug dosage
management, which can provide longer, flatter, or sometimes complex bloodstream
profiles. Approaches include the use of barrier layers, cored tablets, selective release
microspheres, and even osmotic pumps. These tablets essentially are highly engineered
drug delivery systems in which the physical structure is as critical as the chemical
composition [129].
Existing analytical techniques, such as high performance liquid chromatography (HPLC)
and mass spectrometry (MS), often are used to determine the gross composition of a
dosage form, but they provide no information about the distribution of the individual
components. The manner and duration of component release is examined through
dissolution testing. These techniques require destruction of the sample, making it difficult
or impossible to trace the sources of failures or anomalies. Spectroscopic techniques
enable rapid, nondestructive analysis of samples and can be employed at a number of
points in the pharmaceutical development and manufacturing process. In particular, NIR
spectroscopy and Raman spectroscopy have gained increasing popularity for process
95
monitoring and control due to its high information content and flexibility of
implementation. It is used widely for the characterization of raw materials, and also has
been used in applications such as blend homogeneity [128-130], moisture measurement
[131-136], and the analysis of intact tablets [137-143].
IR and Raman spectra provide images of the vibrations of the atoms in a compound.
Therefore, both techniques are also referred to as vibrational spectroscopy. IR
spectroscopy is based on the absorption of electromagnetic radiation by a molecular
system, whereas Raman spectroscopy relies on inelastic scattering of electromagnetic
radiation by a molecular system. An IR spectrum is obtained by passing infrared radiation
through a sample and determining what fraction of the incident radiation is absorbed at a
particular frequency. A Raman spectrum is obtained by focusing monochromatic
radiation on a sample and analyzing the scattered light as a function of frequency [144].
Widespread application of Raman spectroscopy has been limited in part by the problem
of fluorescence, especially encountered for colored samples under visible excitation. The
fluorescence signal, when present, is usually much stronger than Raman scattering. To
avoid this problem, spectroscopists commonly employ excitation sources in the near
infrared region (NIR), where the illuminating source has insufficient frequency energy to
reach most fluorescence-producing electronic states. However, certain substances do
fluoresce when irradiated with NIR light sources. Common near infrared fluorescent
substances include iron oxide, which is often utilized in tablet coatings or incorporated in
the tablet core as a colorant and food colorants, such as Alphazurine FG. The use of these
or other NIR fluorescent materials limits the utility of Raman as a tool to support drug
research, development and quality control [142].
In recent years, NIR spectroscopy has superseded traditional methods as an analytical
tool in various fields and [115]. Near-infrared (NIR) spectroscopy is based on the
absorption of electromagnetic radiation at wavelengths in the range of 780 nm to 2500
nm. NIR offers many advantages that make it an attractive candidate for process control
and monitoring. It is a non-invasive, non-destructive technique that requires minimal
sample preparation. Its measurements are fast and relatively accurate. A single NIR
spectrum allows several analytes to be determined simultaneously. The technique allows
96
determination of non-chemical (physical) parameters such as density, viscosity, or
particle size. In addition, NIR equipment is robust and most suitable for use for process
control at production plants.
However, some characteristics of the technique restrict broader application or preclude
particular uses. NIR measurements are not selective. Therefore, chemometric techniques
have to be used to extract relevant information. Information extraction in NIR often relies
on empirical calibration models because there are no theoretical/analytical models for the
interaction between NIR light and matter. Accurate, robust calibration models are
difficult to obtain because it requires a large number of samples that encompass all
variations in physical and/or chemical properties. In addition, the incorporation of the
physical and chemical variability of samples in the calibration requires as many different
calibration models as there are sample types, and hence more than one model per analyte.
Because NIR spectroscopy is a relative method, model construction often relies on
calibration against a reference method. Measurement accuracy of the constructed NIR
model is thus limited by the accuracy of the reference method. In addition, the technique
is not very sensitive, thus, it can be applied to only major components.
Dielectric analysis methods, though less developed for pharmaceutical applications when
compared with NIR and Raman spectroscopy, have great potential for real time process
control and monitoring. Dielectric analysis is a technique involving the physical, rather
than chemical characterization of samples. Techniques like IR and NMR examine the
molecular structure of samples, while dielectric spectroscopy examines the physical
arrangement and behavior of molecules within structures. A thorough overview of
dielectric analysis for pharmaceutical systems is available in [145].
This chapter investigates the feasibility of using dielectric spectroscopy sensing for
quality control of such physical properties as tablet hardness, coating thickness, and API
content of powder samples. Tests of drug signatures are also carried out to differentiate
between unpolished, polished, and placebo tablet samples. The results show good
measurement sensitivity to parameters of interest. More extensive experiments need to be
conducted to quantify the dependencies between these physical properties and the
electrical measurements and compensate for disturbance factors. Proper choice of a
97
sensor is very crucial for achieving optimal measurement results. Both FEF sensors and
parallel plate sensors are used in the experiments. Measurements from both types of
sensors are presented and compared to give insight on choosing the right type of sensor
for a particular measurement.
8.2 Measuring Tablet Hardness and Coating Thickness
Film coating is often employed as final and separate operation in the manufacturing of
pharmaceutical or nutritional oral solid dosage forms. A coating barrier offers advantages
to the consumer and in the pharmaceutical manufacturing process. Among the most
important are the controlled release of the active pharmaceutical ingredient (API) and the
durability of the dosage form in production and on the shelf. In addition, coatings can
serve to reduce irritation associated with the exposure of the stomach to high
concentrations of medication, and increase product acceptance by improving the visual
appeal of a tablet while making it easier to swallow and enhancing its taste and odor.
Evaluating the properties of pharmaceutical coatings such as intra- and inter-tablet
thickness and uniformity is important for demonstrating adequate process controls and
for ensuring the optimal performance of the final product. Inter-tablet coating uniformity
is important to ensure that the coating is homogeneously distributed on each tablet
throughout the batch; while intra-tablet uniformity is crucial because the overall
performance of the film will likely be limited by the thinnest location on the tablet.
A number of instrumental methods have received substantial attention as potential means
for coating process monitoring. Techniques like scanning electron microscopy, atomic
force microscopy, and conventional optical microscopy have high spatial resolution.
They are typically used for intra-tablet coating uniformity measurements. These
techniques are, however, tedious and not amenable to rapid at-line analysis of a large
number of samples to obtain reliable statistics on coating uniformity. (Here ‘at-line’
refers to measuring in the manufacturing area after manual sampling. It differs from ‘in-
line’, which refers to when the sample interface is located inside the process vessel, the
chemical analysis is done in situ, thus omitting the need for transport of sample out of the
process vessel.) Laser-induced breakdown spectroscopy (LIBS) can be used as an
98
alternative technique for the study of coating thickness uniformity [146] and [147]. It has
the potential to provide both rapid at-line analysis of multiple samples as well as the
spatial resolution necessary for intra-tablet uniformity determination. The main drawback
of this approach is that it is destructive. Near infrared spectroscopy is a real-time non-
destructive technique. However, it does not provide the spatial resolution required for
intra-tablet coating uniformity analysis. Non-destructive techniques like near infrared
spectroscopy (NIR) and Raman spectroscopy are typically used for studies of inter-tablet
coating analysis [148,149] and [142].
8.2.1 Information on sample physical properties
Tablet samples of known hardness are used in this feasibility study. Other information,
such as average tablet thickness and weight, are also available. Sample pressure
(hardness) affects both weight and thickness. Figure 8.1 shows the dependence between
these physical parameters. In this case, pressure refers to the pressure applied when the
tablets are being compressed. Increase in pressure leads to increased tablet density, and,
therefore, an increased weight; at the same time, the pressure increase results in a
decrease in tablet thickness. Notice, however, the trend for weight variation is not strictly
monotonic. Table 8.1. shows the average values of hardness, weight and thickness for the
different groups of tablet samples.
Figure 8.1. Tablet sample weight and thickness against the pressure applied to the
sample when the tablets are being compressed. Higher pressure corresponds to reduced thickness and higher weight in the resulting tablets.
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Table 8.1. Tablet sample physical properties: hardness, weight, thickness.
Sample Hardness Weight Thickness Sample #1 25.6 (kp) 0.592 (mg) 5.56 (mm) Sample #2 31.5 (kp) 0.607 (mg) 5.53 (mm) Sample #3 34.3 (kp) 0.610 (mg) 5.49 (mm) Sample #4 41.8 (kp) 0.609 (mg) 5.27 (mm) Sample #5 45.1 (kp) 0.615 (mg) 5.26 (mm)
8.2.2 Experimental setup
The experimental results presented here are from a parallel plate setup. Tablet samples of
the same hardness are arranged side by side with the same orientation between the two
electrodes of the sensor. The sensor is driven by a 1 V AC voltage signal from a Fluke
RCL meter. The meter measures the loop AC current and sensor terminal impedance. The
AC signal sweeps from 50 Hz to 100 Hz.
8.2.3 Experimental results
As mentioned previously, the major variables that affect electrical measurements are
sample density and sample thickness. Density affects material dielectric permittivity εr
and conductivity σ while thickness affects d. Therefore, capacitance measurements are
sensitive to changes in both sample density and thickness. Phase measurements, on the
other hand, are determined by the relative ratio of the real and imaginary part of the
impedance. Change in sample geometry affects capacitance and conductance
measurement in the same fashion and leaves their relative ratio constant. Therefore, phase
measurements are only dependent here on density variations.
Figure 8.2 (a) and Figure 8.2 (b) show respectively the capacitance and phase
measurement of the tablet samples against hardness at various frequencies. Conductance
and current measurements are omitted because no additional information is offered.
According to Figure 8.1, increase in hardness results in a rise in sample density and a
drop in sample thickness, which affects the capacitance measurements adversely. The
resulting measurement shown in Figure 8.2 is a trade-off between these two effects,
which explains why the trend is not monotonic. Phase measurement displays a monotonic
dependence on hardness and bears information only about samples density. Using the
100
combined information from capacitance and phase measurements, samples density and
thickness can be uniquely determined.
(a) (b)
Figure 8.2. Capacitance and phase of 180 mg tablet samples measured against the hardness of the samples.
8.3 Measuring Tablet Coating Thickness
8.3.1 The experimental setup
Experiments are carried out using both a parallel plate sensor and fringing field sensor.
Figure 8.3 shows the FEF setup. For the parallel plate sensor setup, 10 samples are
arranged side by side with the same orientation between the two electrodes of the sensor.
Figure 8.3. Fringing electric field sensor setup for measuring tablet coating
thickness. The spatial wavelength of the chosen sensor is 500 μm.
101
8.3.2 The experimental results
Figure 8.4 shows the capacitance measurements of the tablet samples. To focus on the
detailed measurement variation between samples of different coating thickness, the
capacitance and phase measurements of the original uncoated sample are used as
references and subtracted from the measurements of all other samples. The results are
shown in Figure 8.5 (a) and Figure 8.5 (b). A clear dependency exists between the
capacitance variation data and sample coating thickness. Note that exact information on
coating thickness is not provided with the test samples used in these experiments. Here,
weight information, which is directly related to coating thickness, is used instead. Figure
8.9 shows the capacitance measurements acquired at 1 kHz plotted against sample
weight. A near-linear dependency is witnessed.
Figure 8.4. Absolute capacitance measurements of tablet samples with different
coating thickness using a parallel plate sensor. Tablets of various thicknesses are grouped here according to their weight. The thickness the coating, the heavier they weigh. The data ‘the original’ corresponds to the bare tablet without any coating.
102
102 103 104 105-0.5
0
0.5
1
1.5
2
2.5
3
3.5
Frequency (Hz)
Pha
se v
aria
tion
(deg
)
27 mg30 mg33 mg36 mgfinal
(a) (b) Figure 8.5. Capacitance and phase variation between samples with different
coating thickness using a parallel plate sensor. Tablets of various thicknesses are grouped here according to their weight. The thickness the coating, the heavier they weigh.
Figure 8.6. Capacitance variation against sample weight using a parallel plate
sensor. The increase in the weight of the samples is caused by increasing coating layer thickness. The thicker the coating, the heavier the sample weighs. The parallel plate sensor has a near-linear response to variations in sample weight.
Figure 8.7 shows the capacitance measurements of the samples from the fringing field
setup. A much greater difference is witnessed in this case between the measurements of
the original tablets and those of the coated ones than in the case of the parallel plate
setup. This is easily explained by the higher sensitivity of FEF sensors to the layer of
samples in direct contact with the electrodes. Again, using the measurements from the
original tablets as references, the capacitance and phase variations of the coated tablets
are calculated. The results are shown in Figure 8.8 (a) and Figure 8.8 (b). Figure 8.9
103
shows the capacitance variation data at 1 kHz plotted against sample weight. Compared
with the parallel plate result shown in Figure 8.6, the dependency between capacitance
and sample weight is non-linear in the fringing field setup. This is, however, within
expectation considering the non-uniform field distribution of a FEF sensor. As coating
thickness increases, the electrical measurement sensitivity to thickness variation
decreases. To attain an optimal sensitivity curve, a wavelength of the FEF sensor has to
be carefully chosen. The sensor used here provided three different channels with various
wavelengths. The spatial wavelength of the channel used in the experiments is 500 μm,
which corresponds to a penetration depth of around 160 μm.
102 103 104 1057.2
7.4
7.6
7.8
8
8.2
8.4
8.6
8.8
9
Frequency (Hz)
Cap
acita
nce
(pF)
Original27 mg30 mg33 mg36 mgfinal
Figure 8.7. Absolute capacitance measurements for tablet samples with different
coating thickness using a fringing electric field sensor with spatial wavelength of 500 μm. Tablets of various thicknesses are grouped here according to their weight. The thickness the coating, the heavier they weigh. The data ‘the original’ corresponds to the bare tablet without any coating.
104
102 103 104 1050.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
Frequency (Hz)
Cap
acita
nce
varia
tion
(pF) 27 mg
30 mg33 mg36 mgfinal
102 103 104 1050
0.5
1
1.5
2
2.5
3
3.5
Frequency (Hz)
Phas
e va
riatio
n (d
eg) 27 mg
30 mg33 mg36 mgfinal
(a) (b)
Figure 8.8. Capacitance and phase variation for tablet samples with different coating thickness using a fringing electric field sensor with spatial wavelength of 500 μm. Tablets of various thicknesses are grouped here according to their weight. The thickness the coating, the heavier they weigh.
26 28 30 32 34 36 38 40 42
0.44
0.46
0.48
0.5
0.52
Weight (mg)
Cap
acita
nce
var
iatio
n (p
F)
Figure 8.9. Capacitance variation against sample weight for the fringing electric
field setup. The increase in the weight of the samples is caused by increasing coating layer thickness. The thicker the coating, the heavier the sample weighs. Due to the non-uniform field distribution of FEF sensors, the sensor response to variations in tablet coating thickness in non-linear.
8.4 Acquiring Drug Signature Using a FEF Sensor
There exist the need for a non-invasive sensing technique that can differentiate different
types of drugs. One of the solutions is to look at the spectroscopy measurements of the
drug samples in the frequency domain. Information on both the absolute value and the
trend of the frequency dependencies of the electrical measurements can be used to
different types of drugs from each other.
105
This section investigates the feasibility of acquiring drug signatures using the fringing
field dielectrometry sensing technique. In the experimental setup, eight tablet samples are
positioned on a FEF sensor with spatial wavelength of 500 μm. The goal of the
experiment is to differentiate between three groups of tablet samples: the original, the
polished, and the placebos.
Figure 8.10 (a) and Figure 8.10 (b) show respectively the capacitance and phase
measurements of the three types of tablets. A great difference is witnessed between the
measurements of the original tablets and those of the other two types of tablets, which
makes the original tablets easily distinguishable. The measurement results for the
placebos are close to those of the polished samples, the difference in capacitance
measurements being in the range of 0.01 pF. High measurement resolution is necessary to
differentiate these two groups of samples.
102 103 104 105
6.5
6.6
6.7
6.8
6.9
7
7.1
7.2
7.3
Frequency (Hz)
Cap
acita
nce
(pF)
OriginalPolishedPlacebo
102 103 104 105
-90.5
-90
-89.5
-89
-88.5
-88
-87.5
Frequency (Hz)
Pha
se (d
eg)
OriginalPolishedPlacebo
(a) (b)
Figure 8.10. Capacitance and phase measurements of three different types of tablet samples using a fringing electric field sensor with spatial wavelength of 500 μm. The results show that the FEF sensor has high measurement sensitivity to the surface textures of the sample under test.
8.5 Measuring API Concentration for Powder Samples
Fluid bed dryers are used during the manufacture of many products, including minerals,
polymers, fertilizers, crystalline materials, and pharmaceuticals. In pharmaceutical
processing, the fluid bed drying unit operation is often used to remove water or other
solvents added to dry powder mixtures during wet granulation prior to further processing
106
(lubrication, compression, etc.). Poorly controlled moisture endpoints can affect
downstream processablilty, tablet dissolution characteristics, mass balance required for
accurate dosing, and long-term chemical stability. The pharmaceutical industry has
traditionally relied on loss on drying (LOD) and Karl Fischer (KF) titration to measure
moisture content in pharmaceutical granulation and drying processes. These techniques
are time-consuming and usually limited to analyzing only a few samples during the
process. Furthermore, sampling and preparation can lead to significant analytical errors
[132].
This section investigates the feasibility of non-invasive monitoring of the drying process
for pharmaceutical powder samples using FEF sensors. The electrical impedance of
powder samples that have been subject to different period of drying time are measured
using the impedance spectroscopy sensing system. Figure 8.11 (a) and Figure 8.11 (b)
show the capacitance and phase measurement of the various powder samples. It can be
inferred from the big difference between the measurements of ‘0 hour’ samples and ‘2
hours’ samples that most API is removed in the first two hours of the process. Figure 8.12
shows the capacitance measurements of the powder samples plotted against drying time
at two different frequencies. The exponentially decaying profile resembles that of a
diffusion process, which matches with what is expected of the drying process. This result
indicates the feasibility of the technique. More extensive experiments have to be
conducted to quantify the functional dependence between the electrical measurements
and API content.
107
(a) (b)
Figure 8.11. Capacitance and phase of powder samples of various drying time.
Figure 8.12. Capacitance of powder samples against sample drying time
measured at two separate frequencies. The trend of exponential decay approximates that of a moisture diffusion process. These measurement curves can be used to calibrate the sensor to monitor moisture endpoints during the power drying process.
8.6 Conclusions
The dielectric sensors used here demonstrated good sensitivity to the variations in the
physical properties of pharmaceutical tablet and powder samples. The feasibility of using
such sensors for rapid at-line measurements is proved. Selecting the sensor with a proper
geometry is crucial to obtaining good experimental results. Parallel plate sensors measure
a sample in bulk. Any local variations are averaged across the whole sample in the
terminal impedance measurements. The electric field distribution of FEF sensors is non-
uniform. An FEF sensor with the proper spatial wavelength can be used to detect
variations within a localized region of the sample. Measurements from parallel plate
108
sensors are, in general, more stable and less noise-prone than those from FEF sensors,
which makes parallel plate sensors good candidates for proof-of-concepts experiments.
FEF sensors are more attractive in cases where high measurement sensitivity is desired.
109
Chapter 9. Conclusions and Future Work
9.1 Conclusions
9.1.1 Parametric Modeling of FEF Sensors
Parametric models are constructed for concentric FEF sensors and interdigital FEF
sensors based on non-dimensionalized fitting of FEM simulation results. A 50%
metallization ratio is assumed when constructing the unit cell, but the methodology can
be used for modeling FEF sensors with metallization ratio other than 50% as well. Only
the terminal capacitance of the FEF sensors is considered here. To extend the results for
applications where complex impedance is of interest, the variable C can be replaced by
the complex admittance Y* and the dielectric constant ε can be replaced by the complex
dielectric permittivity ε*. In the case of interdigital FEF sensors, the model is constructed
to evaluate the terminal capacitance per unit finger length. Here the end effect of finite
finger length is ignored. For interdigital FEF sensors with finger length much greater than
its spatial wavelength (more than 10 times greater), this simplification is valid.
The modeling method is not restricted to FEF sensors; its concept of non-
dimensionalization can be used for the design optimization of a wide variety of sensors.
For sensors that are difficult to model analytically, the method provides accuracy
comparable to numeric methods while greatly improves the throughput of design
optimization.
9.1.2 Interfacing Circuits for FEF Sensors
Various aspects of interfacing circuit design for fringing electric field dielectrometry is
discussed in this chapter. Due to the high measurement sensitivity requirement of FEF
sensor applications, the noise floor of the interface circuits needs to be sufficiently
suppressed. Therefore, stray-immunity is one of the most crucial figures of merit for
these circuits. For spectroscopic applications, where the sample under test is evaluated
110
over a frequency band, the circuit topology optimal for that frequency band should be
used. For circuits interfacing with sensor arrays in imaging applications, parasitics due to
multiplexing need to be carefully addressed.
9.1.3 FEF Sensor Design Optimization
The optimal sensor geometry changes with respect to different dielectric properties of the
samples. In this thesis, the samples are assumed to have low permittivity (< 10) and close
to zero conductivity. Such samples are representative for a wide range of ceramics and
plastics [86]. Design of sensors specialized for high permittivity dielectrics (10 < εr < 80)
were discussed in [50]. For highly conductive samples, resistance, instead of
capacitance, is measured to estimate sample concentration or distribution within the
sensing zone. In cases where the sample has complex permittivity and displays
frequency-dependent behavior, both the real part and the imaginary part of the complex
impedance have to be measured.
The multi-channel impedance sensors presented in this chapter are designed for imaging
applications. Electrical imaging systems typically operate in the megahertz range.
Therefore, the electrostatic model used in the FE simulations is adequate. For sensors
used in broad-band spectroscopic systems, AC simulations are necessary. Such systems
are usually used for lab-based determination of material properties of samples with
complex permittivity.
9.2 Future Work
Instrumentation for FEF systems is developing toward custom-designed silicon-based
integrated systems. The planar geometry of FEF sensors allows easy integration into a
CMOS process. Complete systems can be fabricated with the sensor electrodes on the top
metal layers, and the interface and control circuitry underneath. The highly developed
and well-controlled CMOS processes can facilitate mass fabrication of the sensors with
high spatial resolution at low cost. The elimination of wiring and connection help to
minimize the stray impedance. In some cases, the signal conditioning circuits can be
replaced by digital signal processing units. For example in spectroscopic applications, the
111
frequency response acquired by the sensor can be analyzed by an integrated DSP unit
built-in to the sensor. Finger print sensors based on integrated FEF electrodes have been a
relative active field of research in the last decade [150-152].
As MEMS and nanotechnology reach maturity, integrated FEF sensors at the nanometer
to micrometer scale are no longer a remote reality. Due to the small electrode surface area
for FEF sensors at this scale, the sensor measurement signal can be very weak. The key to
such systems lies in highly sensitive interface circuits. Bio-chemical coatings can be used
to enhance the selectivity and sensitivity of the sensing system. Since FEF sensor
measurements are highly dependent on surface contact quality, measuring solids with
FEF sensors at the nanometer to micrometer scale may be challenging. Noise due to
surface roughness of the sample can be minimized if the test sample is placed in a
solution.
In the area of data analysis, many chemometrics methods, often used by analytical
chemists, can be applied here directly for multivariate calibration of FEF sensing
systems. Dielectric sensors typically respond to variations in several physical parameters
simultaneously. Although the dielectric properties of a sample can be measured directly
through dielectric spectroscopy, the relation between dielectric measurements and
physical properties are typically not well defined, especially when several physical
parameters are changing simultaneously. Therefore, determination of sample physical
properties typically relies on empirical calibration against results from other measurement
methods. For the FEF sensing systems described here to be used effectively in industrial
processes where many environmental variables (such as temperature during the baking
process for foods) are changing, multivariate analysis has to be used to extract useful
information from experimental data.
112
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Appendix A: DiSPEC Hardware Installation Guide
List of Instruments
The instruments listed below are necessary for the dielectric spectroscopy system: • National Instruments PCI-GPIB, NI-488.2 with Cable • Tektronix AFG310 Arbitrary Function Generator • National Instruments PCI-6035E/PCI-6036E DAQ board • National Instruments BNC-2120 terminal block • National Instruments SH-68-68-EP shielded cable • Tektronix PS280 Triple Output Power Supply
The following is a complete list of the items provided by SEAL:
• 3-channel sensor-interface circuit • Two 2-channel fringing field electric sensors with connectors attached • A K-type thermocouple • Five SMA male – BNC male cables • 5-pin male – 3 Banana plug power cable for sensor-interface circuit • 5-pin male – single wire for relay-control signal connection • Two short stripped wires for power supply lead-to-lead connection • Kapton covers for sensor heads • DiSPEC custom software CD • A user’s manual
Installing the Hardware Step 1: Install the GPIB and the DAQ Cards
Plug the NI GPIB and the NI-DAQ boards into the experimental computer and install
all recommended drivers.
Step 2: Connect BNC-2120 to the DAQ Board
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Connect BNC-2120 breakout box to the DAQ card in the computer using the shielded
SH-68-68-EP cable.
Step 3: Connect Thermocouple to BNC-2120
Plug thermocouple into the thermocouple socket of the BNC-2120 breakout box.
Check that the row switches for ACH0 and ACH1 are set to the right, the “Temp.Ref.” and “Thermocouple” positions.
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Step 4: Connect the Sensor Box to BNC-2120 Attach the BNC-ends of four BNC-SMA cables to BNC-2120’s channels ACH2-
ACH5. The switches under all ACH connectors should be in GS position.
Switch ACH3 selector (above the thermocouple connector) to BNC position.
Using the BNC-SMA cables to connect the ADC0 to ADC3 terminals on the sensor interface box to the ACH2 to ACH5 of BNC-2120.
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Place the sensor interface box and the sensor as far away as possible from the
computer, the monitor and other electronic devices to minimize noise interference.
Step 5: Connect the Relay Control Lead to BNC-2120 Connect the relay-control wire from the “Cntrl.” socket of sensor interface box to
DIO0 of BNC-2120 using a 5-pin to single wire connector.
To connect the wire to BNC-2120 DIO0 port, untighten a screw (on the right), let the wire end into the hole and re-tighten the screw.
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Step 6: Set up the Power Supply To properly set the polarity of the power supply, two short wires should be
connecting the outputs as shown in the picture below:
Set the power supply to independent mode by adjusting the two buttons in the middle
of the front panel. Set the display-mode switches to Voltage position. The current dial should be set to the twelve-o’clock position (halfway). Use voltage dials to set both outputs to 10 volts. DO NOT CHANGE THE VOLTAGE SETTINGS WHILE THE SENSOR INTERFACE IS CONNECTED TO THE POWER SUPPLY! Once the voltage is set, connect the interface box to the power supply using 5-pin to 3 banana plug cable.
The plug labeled GND can be connected to either of the two ground sockets available on the power supply. Connect the “+” plug to the “+” terminal of the right 0-20V supply and “-“ plug to the “-“ terminal of the left 0-20V supply.
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Step 7: Connect the Function Generator Using the fifth BNC-SMA cable, connect the function generator to the sensor
interface box.
Use the GPIB cable to connect the function generator to the GPIB card in the computer. Note that you don’t need to change the settings of the function generator manually. The program DiSPEC will tell it what to do. AlWAYS TURN ON THE FUNCTION GENERATOR BEFORE OPENING the DiSPEC PROGRAM.
Step 8: Connect the Sensor to the Interface Box
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The sensor is connected to the interface as follows: The cable labeled as “drive” of the sensor is connected to any of the Drive terminals of the board. The ones labeled as “inner sense” and “outer sense” are connected to the S. 1 and S. 2 terminals on the sensor interface box respectively. DiSPEC software guide Installing the Software This LABVIEW application is for viewing data in real time as well as recording data to a file. Start by opening the DiSPEC folder on the CD. Go to folder “Installer” and double click on the “setup” icon. Follow the setup guide step by step and the software will be installed on your computer. After installation, the program will show up in your computer’s start up menu as DiSPEC. Using the Software After all the hardware is installed, go to the start up menu of your computer and open the program DiSPEC. The following steps need to be followed to run the software.
1. Connect all the instruments according to the instructions in the hardware guide. 2. Open the NI software “Measure & Automation Explorer” (MAX). (The software
should be provided the DAQ board). Find out the device number for the function generator and the DAQ board. Specific instructions are available below under Device Number of the Function Generator and the DAQ Board.
3. Go to the system settings tab in the front panel and enter the device numbers found from step 1. The same values need to be manually entered each time the program is opened if they are found to be different from the default values.
4. Find out the time delay for each channel of the circuit board. For detailed instructions, go to Time Delay.
5. Enter the time delay values obtained from step 3 in the system settings tab. Note that if found to be different from the default values, these time delay constants have to be reentered each time the program is opened.
6. Go to the Controls tab, and configure the entries in the tab. For detailed instructions, go to Controls.
7. Turn on the function generator and the power supply. 8. Press the “Run” and the “Acquire” button. If an arrow appears on the top left
corner of the front panel, click on the arrow and the program will start running. If the arrow doesn’t appear, the code should already be running after the “Run” and the “Acquire” button is pressed down.
9. Go to the different tabs on the front panel to look at the data and the graphs. 10. To monitor change in sample capacitance more visually, go to the Real-time
Imaging tab. 11. To stop the program, press the “Acquire” button again.
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Configure the System In the front panel, go to the “System Settings” tab, a window like the following will show up.
Upon starting LABVIEW, the 'System settings' has all its entry values set to their defaults for the 2 channel fringing field sensor. These default values should be used unless specified otherwise below. Description of the entries can be obtained by right-clicking on an entry and selecting “Descriptions and Tips” on the menu.
Device Number of the Function Generator and the DAQ Board The device numbers of the function generator and the DAQ board are dependent on the particular setup, therefore values different from those specified as default may need to be used. To get the device numbers, open the NI software “Measurement & Automation Explorer” that is provided with the DAQ board. Double click on “Devices and Interfaces”, the DAQ board and the NI GPIB card will show up. The number listed for the DAQ board is the device number for the DAQ and the number listed for the GPIB card is the device number for the function generator.
Channel Numbers on the DAQ board
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This entry specifies the channel numbers on the DAQ board that the input and the three output signals of the circuit are connected to. The default values are “2,3,4,5”. Channels 0 and 1 on the DAQ board are saved for the thermocouple. The first channel number is the input to the DAQ board, and the following numbers are output channels, displayed in front channel as channels 1, 2, and 3 respectively. All numbers must be separated by commas. (Channel 3 is most often used in the parallel plate setup rather than the fringing field system)
Reference Capacitance The “Reference Capacitance” entries specify the reference capacitance used for each channel of the sensor circuit board. Note that these values are not the same as the values of the reference capacitors on the board. The effect of the stray capacitances (e.g. that introduced by the Op-Amp has to be accounted for.) The default values are obtained through careful calibration of the system. These values should be used unless some circuit elements are changed. Recalibration of the system is necessary if changes are made to the circuit.
Time Delay Multiplexing of the DAQ board introduces a time delay between the data stream from its different channels. The time delay causes significant phase distortion, therefore its effect has to be eliminated. Unfortunately, the time delay values are device dependent. A different computer and DAQ board will cause a change in these values, which means that these values have to be fine-tuned for each particular setup. The following procedure can be used to find the time delay constants:
1. Connect the input and all the outputs of the circuit board to the function generator. Note that since all channels are connected to the same source, ideally there should be no phase delay between the channels.
2. Run the program at the highest frequency (30 kHz). Ideally, the gain should be 1 and the phase should be zero for all channels. Adjust the time delay for each channel of the sensor in the “systems and setting” tab until the phase delays for all channels are 0.
3. The new time delay constants must be typed in manually each time the program is started.
The Settings for the Thermocouple
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Channel 0 and 1 of the DAQ board should always be used for the thermocouple as is specified in the default setting. The sampling rate for the thermocouple could be increased if an improvement in the speed of the program is desired. Otherwise, use the default value. Averaging is used here to remove noise. The number of samples for temperature averaging can be changed for different application. Controls The controls tab should look like the following window.
Mass Real-time monitoring of sample mass is not necessary at this point. This function is included for possible future applications. By pressing the “mass” button down, the
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program will acquire data from a scale that is connected to the computer through a serial port.
Temperature A K type thermocouple is connected to the computer. By pressing the “Temperature” button down, the system starts acquiring data from the thermocouple and saving the temperature data to the output file if “Saving to file” is also enabled.
Sweep When the “Sweep” button is not pressed, the system runs at a single frequency specified by the “Start Frequency” entry. The system performs frequency sweeps when the “Sweep” button is enabled. The range of the frequency sweep is defined by the start and the stop frequency. The minimum and maximum frequencies allowed by the current version of the program are 1 Hz and 30 kHz respectively. Measurements at frequencies lower than 1 Hz are comparably noisy, thus we limit the frequency to above 1 Hz. The frequency range can be easily extended for future applications.
Save to File If you wish to record the data to a file, the “Save to File” button should be pressed down. You can enter the file name and the saving directory in the “File Name” entry. If the entry is left blank, a file saving window will automatically pop up when the program starts running. The file can be saved as an Excel spreadsheet or a ‘.txt’ file.
Real-time Imaging
The following is a picture of the real-time imaging tab.
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The tab provides a profile of the sample by displaying the capacitance of all channels of the sensor. Note that in a 2-channel fringing field setup, the information shown for channel 3 should be ignored. Channel 3 is included here mainly for the consideration that a 3-channel sensor might be used in the future. The capacitance values displayed in the vertical bars are all scaled to be within 0 to 1. The “Maximum Capacitance” knobs on the right refer to the actual capacitance value in “pF” when the bar displays a value of “1”. The knobs could be adjusted for the best visual effect.
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Appendix B: Matlab Code for the Layer Stripping Algorithm
% this code generates the data that feeds the layer stripping algorithm. % vector of the radius of the concentric discs function w = data_gen(N, d, sigma) K = length(d); % in this case K=4; w = zeros(N, K-1); % calculate the w_n(1), that is the w_n for the outer boundary for n = 1:N w(n, 1) = 1/n/sigma(1); for k = 2:K-1 L0 = (d(k)/d(k+1))^(2*n); L = n*sigma(k)*w(n, k-1); H = (1-L)/(1+L); L1 = L0*H; H = (1-L1)/(1+L1); w(n, k)=1/sigma(k)/n*H; end end
% this code solve the inverse problem of electrical % impedance imaging using the layer stripping algorithm. clear all %close all clc % this code takes matrix w as the input %format long % % d1 d2 d3 d4 % d = [0, 0.08, 0.6, 1]; % sig = [1, 1, 1]; % sig2 = [2, 1, 0.5];
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% sig3 = [0.5, 0.5, 2]; % N=16; % w = data_gen(N, d, sig2) % w2= data_gen(N, d, sig2) % w3 = data_gen(N, d, sig3) d1 = [0, 0.2, 1]; d2 = [0, 0.4, 1]; d3 = [0, 0.6, 1]; d4 = [0, 0.8, 1]; sig1 = [1, 1]; sig2 = [2, 1]; sig3 = [1, 2]; sig4 = [2, 2]; N=16; w11 = data_gen(N, d1, sig1); w21= data_gen(N, d2, sig1); w31 = data_gen(N, d3, sig1); w41 = data_gen(N, d4, sig1); w12 = data_gen(N, d1, sig2); w22= data_gen(N, d2, sig2); w32 = data_gen(N, d3, sig2); w42 = data_gen(N, d4, sig2); w13 = data_gen(N, d1, sig3); w23= data_gen(N, d2, sig3); w33 = data_gen(N, d3, sig3); w43 = data_gen(N, d4, sig3); w14 = data_gen(N, d1, sig4); w24= data_gen(N, d2, sig4); w34 = data_gen(N, d3, sig4); w44 = data_gen(N, d4, sig4); w = w42; [N, K]= size(w); % step 1, choose mode-drooping radii: alpha=0.2; H=10 n=1:H; %a = alpha+(1-alpha)/(N-1).*(N-1-n); %a = alpha.^(2./(N+1-n)); a=((H+1-n)./2*alpha^2).^(1./(N+1-n)); a=[1, a,0]; delta = 1e-2; L = length(a); % reconstruction at the boundary sigma(1) = a(1)/N/w(N,K);
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% iterate to reconstruct at the subsurface. a_p = a(1); w_p = w(:,K); sigma_p= sigma(1); n = (1:N)'; for h = 2:L while a_p > a(h) delta_w=delta/a_p.*(1/sigma_p-(n.^2).*sigma_p.*(w_p.^2)); w_p=w_p-delta_w; a_p = a_p-delta; sigma_p = a_p/16/w_p(N); end sigma(h)=sigma_p; %N=N-1; end sigma for i = 1:length(sigma); if abs(sigma(i))>5; sigma(i)=5; end end sigma plot(a, sigma, 'x-') grid on hold on
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Vita
Xiaobei Li received her B.S. degree in controls theory from Northwestern Polytechnical
University, Xi’an, China, in 1999; and a M.S. in Electrical Engineering from the
University of Washington, Seattle, in 2003. She joined the Sensors, Energy and
Automation Laboratory (SEAL) at the department of Electrical Engineering, University
of Washington at 2002 as a graduate research assistant. Her research interests include
dielectric spectroscopy sensor design, sensor signal conditioning circuit design, and
image reconstruction for soft-field sensing. Xiaobei Li is currently working for the
INTEL Corporation in Dupont, WA.