Moisture Measurement in Paper Pulp Using Fringing Field
Dielectrometry
Project Report 2003-F-1
K. Sundararajan, L. Byrd II, C. Wai-Mak, N. Semenyuk, and A.V. Mamishev
Sensors, Energy, and Automation Laboratory
Department of Electrical Engineering
University of Washington
Email: [email protected]
http://www.ee.washington.edu/research/seal
Abstract
Moisture control of paper pulp is a key step in paper manufacturing. To ensure
control efficiency, a fast and accurate moisture sensor is desired. Currently existing
technologies can accurately measure moisture content of the pulp with fiber content in
excess of 20%; yet for cases with lower paper percentages, challenges still remain.
Significant market potential exists for an inexpensive and reliable feed-forward moisture
control technique. Existing techniques do not yet perform adequately in the field,
although laboratory studies claim high measurement accuracy and reliability.
In this report, fringing electric field dielectric spectroscopy is used to estimate the
fiber content of laboratory-made paper pulp. The fiber content and the titanium dioxide
concentration in the paper pulp samples are varied from 100% to 3%, and 0% to 7%
respectively. The resulting dielectric measurements show a near-linear dependency on the
fiber content.
The moisture content in the pulp is estimated using a self-adapting algorithm. The
algorithm automatically selects different parameters and equations for the estimation
process based on the data sets used to train it. The fiber content was estimated, using a
linear self-learning model.
Table of Contents
List of Figures .................................................................................................................... iii
Chapter 1.
Introduction ................................................................................................. 1
Chapter 2.
State of the Art ............................................................................................ 3
2.1.
Microwave Techniques ....................................................................................... 3
2.2.
Reflection Based Techniques .............................................................................. 5
Chapter 3.
Interdigital Fringing Field Dielectrometry.................................................. 8
3.1.
Interdigital Fringing Field Sensor ....................................................................... 8
3.2.
Dielectric Spectroscopy .................................................................................... 11
3.3.
Debye Model ..................................................................................................... 12
Chapter 4.
Experimental Setup ................................................................................... 17
Chapter 5.
Experiments with High Fiber Concentration Pulp (100% to 10%) .......... 20
5.1.
Experimental Procedure .................................................................................... 20
5.2.
Experimental Results ........................................................................................ 20
5.3.
Data Analysis .................................................................................................... 22
Chapter 6.
Experiments with Low Fiber Concentration Pulp (10% to 3%) ............... 24
6.1.
Experimental Procedure .................................................................................... 24
6.2.
Experimental Results ........................................................................................ 24
6.3.
Data Analysis .................................................................................................... 27
Chapter 7.
Experiments with Titanium Dioxide......................................................... 33
7.1.
Experimental Procedure .................................................................................... 33
7.2.
Experimental Results ........................................................................................ 33
7.3.
Data Analysis .................................................................................................... 37
Chapter 8.
Experiments with Clay.............................................................................. 41
8.1.
Experimental Procedure .................................................................................... 41
8.2.
Experimental Results ........................................................................................ 41
8.3.
Data Analysis .................................................................................................... 45
i
Chapter 9.
Experiments with Calcium Carbonate ...................................................... 47
9.1.
Experimental Procedure .................................................................................... 47
9.2.
Experimental Results ........................................................................................ 47
9.3.
Data Analysis .................................................................................................... 51
Chapter 10.
Parameter Selection Algorithm ................................................................. 53
Chapter 11.
Repeatability Tests .................................................................................... 58
Chapter 12.
Reproducibility Tests ................................................................................ 61
Chapter 13.
Validation of Estimation Algorithms ........................................................ 64
Chapter 14.
Pass line Sensitivity .................................................................................. 68
14.1.
Experimental Results .................................................................................... 68
14.2.
Data Analysis ................................................................................................ 71
Chapter 15.
Disturbance factors ................................................................................... 75
Chapter 16.
Variation with Pulp ................................................................................... 77
Chapter 17.
Temperature Variations ............................................................................ 82
17.1.
Experimental Procedure ................................................................................ 82
17.2.
Experimental Results .................................................................................... 83
Chapter 18.
Future Work .............................................................................................. 86
18.1.
High Frequency Measurements: ................................................................... 86
18.2.
Temperature Variation .................................................................................. 88
18.3.
Improvements in Estimation Algorithms ...................................................... 89
18.4.
Pass line Sensitivity ...................................................................................... 89
Chapter 19.
Conclusions ............................................................................................... 90
References ......................................................................................................................... 91
Appendix ........................................................................................................................... 97
ii
List of Figures
Figure 1.1. Photograph of a paper machine. ..................................................................... 2
Figure 3.1. A fringing field dielectrometry sensor can be visualized as a parallel plate
capacitor whose electrodes open up to provide a one-sided access to material
under test [29]......................................................................................................... 8
Figure 3.2. The term “interdigital” refers to the pattern of fingers or “digits” that is
resembled by the shape and relative position of the electrodes [31]. .................... 9
Figure 3.3. A generic interdigital sensor with a periodicity λ [34,35]. ........................... 10
Figure 3.4. A conceptual view of multiple penetration depth sensor. .............................. 10
Figure 3.5. Mechanisms influencing the loss factor of a moist material over a wide range
of frequencies f (Hz). (A) DC conductivity, (B) Maxwell-Wagner polarization, (C)
dipolar polarization of water bound to the matrix of the material, (D) dipolar
polarization of free water [36]. ............................................................................ 12
Figure 3.6. Debye model for time dependent polarization [39]....................................... 15
Figure 3.7. Real and imaginary parts of complex susceptibility as a function of
frequency. [40] ...................................................................................................... 16
Figure 4.1. Photograph of experimental setup. ................................................................ 17
Figure 4.2. The top-down view of the interdigital sensor tray with the spatial periodicity
of 40 mm, finger length of 160 mm and an approximate penetration depth of 13
mm. ........................................................................................................................ 18
Figure 5.1. Measurements of paper pulp samples with 0% to 90% moisture concentration
at frequencies from 200 Hz to 100 kHz. ................................................................ 21
Figure 5.2. Fiber content as a function of the conductance of the paper pulp samples
with 0% to 90% moisture concentration at various frequencies from 200 Hz to
100 kHz. ................................................................................................................ 22
Figure 5.3. Fiber content as a function of the capacitance of the paper pulp samples
with 0% to 90% fiber concentration at various frequencies from 200 Hz to 100
kHz. ....................................................................................................................... 23
Figure 6.1. Measurements of paper pulp samples with 90% to 97% moisture
concentration at frequencies from 200 Hz to 100 kHz. ........................................ 25
Figure 6.2. The capacitance measured at 5 kHz shows separation between measurements
to be much greater than twice the standard deviation. This indicates towards the
possibility of achieving higher resolution using the sensor. ................................. 26
Figure 6.3. Cole-Cole plots from measurements of paper pulp samples with 3% to 10%
paper concentration at frequencies from 50 Hz to 100 kHz. ................................ 27
iii
Figure 6.4. Conductance plots from measurements of paper pulp samples with 90% to
96% moisture concentration at frequencies from 200 Hz to 100 kHz. ................. 29
Figure 6.5. Capacitance plots from measurements of paper pulp samples with 90% to
96% moisture concentration at frequencies from 200 Hz to 100 kHz. ................. 29
Figure 6.6. Variation of the slope, m, in (6.1) depicting the plot between percentage of
moisture content and capacitance with respect to frequency of excitation. ......... 30
Figure 6.7. Variation of the offset, k, in (6.1) depicting the plot between percentage of
moisture content and capacitance with respect to frequency of excitation. ......... 30
Figure 6.8. The data obtained experimentally at 5 kHz is in agreement with the curve
formulated in (6.1). ............................................................................................... 31
Figure 6.9. Standard deviation of the capacitance measurements at various moisture
levels is two orders of magnitude smaller than the capacitance (10-11 vs. 10-14). 32
Figure 7.1. Measurements of paper pulp samples with 0% to 7% titanium dioxide
concentration at frequencies from 200 Hz to 100 kHz. ........................................ 34
Figure 7.2. Cole-Cole plots from measurements of paper pulp samples with 0% to 7%
titanium dioxide concentration at frequencies from 200 Hz to 100 kHz. ............. 35
Figure 7.3. Conductance plots from measurements of pulp samples with 0% to 7%
titanium dioxide concentration at frequencies from 200 Hz to 100 kHz. ............. 36
Figure 7.4. Capacitance plots from measurements of pulp samples with 0% to 7%
titanium dioxide concentration at frequencies from 200 Hz to 100 kHz. ............. 36
Figure 7.5. Comparison of the estimated concentration of fiber in the pulp to the actual
concentration. ....................................................................................................... 38
Figure 7.6. Comparison of the estimated concentration of titanium dioxide in the pulp to
the actual concentration. ...................................................................................... 39
Figure 7.7. Comparison of the estimated concentration of moisture in the pulp to the
actual concentration. ............................................................................................ 39
Figure 8.1. Measurements of paper pulp samples with 10% to 5% fiber concentration
and 0% to 5% clay at frequencies from 200 Hz to 100 kHz. ................................ 42
Figure 8.2. Cole-Cole plots from measurements of paper pulp samples with 10% to 5%
paper concentration and 0% to 5% clay at frequencies from 200 Hz to 100 kHz.43
Figure 8.3. Conductance plots from measurements of pulp samples with 0% to 5% clay
concentration at frequencies from 200 Hz to 100 kHz. ........................................ 44
Figure 8.4. Capacitance plots from measurements of pulp samples with 0% to 5% clay
concentration at frequencies from 200 Hz to 100 kHz. ........................................ 44
Figure 8.5. Comparison of the estimated concentration of fiber in the pulp to the actual
concentration. ....................................................................................................... 45
iv
Figure 8.6. Comparison of the estimated concentration of clay in the pulp to the actual
concentration. ....................................................................................................... 46
Figure 8.7. Comparison of the estimated concentration of moisture in the pulp to the
actual concentration. ............................................................................................ 46
Figure 9.1. Measurements of paper pulp samples with 10% to 7.5% fiber concentration
and 0% to 2.5% calcium carbonate at frequencies from 200 Hz to 100 kHz. ...... 48
Figure 9.2. Cole-Cole plots from measurements of paper pulp samples with 10% to 7.5%
paper concentration and 0% to 2.5% calcium carbonate at frequencies from 200
Hz to 100 kHz. ....................................................................................................... 49
Figure 9.3. Conductance plots from measurements of pulp samples with 0% to 2.5%
calcium carbonate concentration at frequencies from 200 Hz to 100 kHz........... 50
Figure 9.4. Capacitance plots from measurements of pulp samples with 0% to 2.5%
calcium carbonate concentration at frequencies from 200 Hz to 100 kHz........... 50
Figure 9.5. Comparison of the estimated concentration of fiber in the pulp to the actual
concentration. ....................................................................................................... 51
Figure 9.6. Comparison of the estimated concentration of calcium carbonate in the pulp
to the actual concentration. .................................................................................. 52
Figure 9.7. Comparison of the estimated concentration of moisture in the pulp to the
actual concentration. ............................................................................................ 52
Figure 10.1. Flow chart of the training algorithm. .......................................................... 55
Figure 10.2. Flow chart for evaluation algorithm. .......................................................... 56
Figure 10.3. Plot of the estimated fiber content in pulp consisting of fibers and water.
The estimates are based on the parameter selected by the algorithm. ................. 57
Figure 11.1. Repeatability test for measurements made using pulp containing just fiber
and water. ............................................................................................................. 59
Figure 11.2. Repeatability test for measurements made using pulp containing fiber,
calcium carbonate, and water............................................................................... 60
Figure 12.1. Reproducibility test for measurements made using pulp containing just fiber
and water. ............................................................................................................. 62
Figure 12.2. Reproducibility test for measurements made using pulp containing fiber,
calcium carbonate, and water............................................................................... 63
Figure 13.1. Validation of estimation process using equation (6.1). ............................... 64
Figure 13.2. Validation of estimation process described in Section 7.3 .......................... 65
Figure 13.3. Validation of estimation process described in Section 8.3 .......................... 66
Figure 13.4. Validation of estimation process described in Section 9.3 .......................... 67
v
Figure 14.1. Measurements of paper pulp samples with 90% to 92% moisture
concentration at frequencies from 200 Hz to 100 kHz, for an air gap of 2.5 mm. 69
Figure 14.2. Measurements of paper pulp samples with 90% to 92% moisture
concentration at frequencies from 200 Hz to 100 kHz, for an air gap of 4.2 mm. 70
Figure 14.3. Cole-Cole plots from measurements of paper pulp samples with 90% to
92% moisture concentration at frequencies from 200 Hz to 100 kHz, for an air
gap of 2.5 mm. ....................................................................................................... 72
Figure 14.4. Cole-Cole plots from measurements of paper pulp samples with 90% to
92% moisture concentration at frequencies from 200 Hz to 100 kHz, for an air
gap of 4.2 mm. ....................................................................................................... 73
Figure 14.5. Normalized Capacitances measured at 7.9 kHz for an air gap of 2.5 mm,
and 4.2 mm. ........................................................................................................... 74
Figure 14.6. Pass line sensitivity of the sensor at 7.9 kHz. .............................................. 74
Figure 15.1. Photograph of the controlled environment chamber. .................................. 75
Figure 16.1. Phase measurements of 3 different types of paper pulp samples with 96% to
90% moisture concentration from 200 Hz to 100 kHz. ......................................... 78
Figure 16.2. Capacitance measurements of 3 different types of paper pulp samples with
96% to 90% moisture concentration from 200 Hz to 100 kHz. ............................ 79
Figure 16.3. Conductance measurements of 3 different types of paper pulp samples with
96% to 90% moisture concentration from 200 Hz to 100 kHz. ............................ 80
Figure 16.4. Measurements of 3 different types of paper pulp samples with 96% to 90%
moisture concentration at 600 Hz. ........................................................................ 81
Figure 17.1. Normalized measurements showing the effect of temperature variation on
various electrical parameters of a paper pulp with 95% moisture from 1kHz to
100 kHz. ................................................................................................................ 83
Figure 17.2. Normalized measurements showing the effect of temperature variation on
various electrical parameters of a paper pulp with 95% moisture at 7.9 kHz. .... 84
Figure 18.1. Cole-Cole plots from measurements of paper pulp samples with 0% to 90%
moisture concentration at frequencies from 200 Hz to 100 kHz. ......................... 87
Figure 18.2. Example Cole-Cole plots ............................................................................. 88
Figure A1.1. Plot of RMS voltage between the sensing electrodes and ground at various
water depths and frequencies................................................................................ 97
Figure A1.2. Near linear relationship was observed between the measured voltage and
water depth............................................................................................................ 98
vi
1
Chapter 1.
Introduction
One of the amazingly inconspicuous, yet indispensable articles used in everyday life
is paper. Thousands of mills worldwide churn out 312 million tons of paper annually.
Interestingly, papermaking process is still more of an art than science. More often than
not, not all of the paper rolled out of a paper machine is perfect. To ensure quality output,
it is critical to monitor and control various properties of paper such as its caliper,
thickness, opacity and, importantly, its moisture content.
Currently, all the properties of the paper are measured only at the dry end of the
paper machine; at the very end of the process cycle. Thus, the quality control is primarily
a feedback control loop [1,2]. While this control architecture offers stability, it is highly
reactive in nature. That is, the system parameters are altered only after output deviations
are registered. Hence, there is a considerable delay from the moment there is a deviation
in the output, to the moment the parameter changes are reflected in the output. This delay
is critical in the case of paper machines. A delay of even ten seconds in the control
system of a paper machine, operating at a nominal speed of 2000 m/min, will result in
more than 0.2 miles of paper of unacceptable quality. Even though this paper can be
recycled to produce lower quality paper, there is considerable monetary loss that is
incurred on counts of energy usage, machine time, and utilization of human resources.
The obvious solution to this problem is to incorporate additional feed forward
control architecture. In such architecture, the properties of the paper are measured at wet
end of the paper machine (the initial stages), and the process parameters down the
process line are modified to correct the deviations in the paper properties. Figure 1.1
shows the wire section (wet end) of the paper machine.
A major hurdle in the practical implementation of feed forward control in paper
machines is the availability of sensing technologies for the wet end. The paper pulp at the
wet end contains numerous chemical additives, and is primarily an aqueous suspension.
These additives alter the response of the paper pulp for various sensing techniques. The
changes in the response cannot be calibrated into the existing measurement techniques
2
based on microwave attenuation or infrared absorption, as the composition of the
additives change on a daily basis depending on the type of paper being produced. Thus, it
is very difficult to estimate the fiber content of the pulp accurately and reliably.
Figure 1.1. Photograph of a paper machine.
Paper manufacturers
are looking for non-invasive, non-contact
sensing
technologies that can accurately measure the fiber content of paper pulp at the wet end of
the paper machine. The moisture content of the paper pulp at the wet end ranges from
99% to 80%. This low concentration of fiber in the pulp makes it hard to detect
concentration fluctuations with adequate resolution. Fringing field dielectric spectroscopy
is a potential sensing technology that could be used to estimate the moisture content of
the paper pulp at the wet end of a paper machine. In this technique, fringing electric field
penetrates through the paper pulp. The phase and the magnitude of these field lines
change depending on the dielectric properties of the paper pulp. These changes in the
fields are studied over a frequency range of 200 Hz to 100 kHz. Based on these changes,
the fiber content of the paper pulp is estimated.
3
Chapter 2.
State of the Art
The methods currently being used to measure moisture in paper pulp are mostly
intrusive [3-5], or require certain special operating conditions such as double-sided
contact measurements [3-5].
Several patents [6-21] have proposed using an electromagnetic field perturbation
sensor for measuring the water concentration in the wet end of the paper machine. In
these patents it is assumed that all the water in the pulp is held by paper fibers and that all
of electrical conductivity is due to water molecules alone. The concentration of paper
fibers in the pulp is indirectly determined by measuring the conductivity of the pulp. The
first assumption limits the measurements to high concentrations of fiber content. At
higher moisture levels, the fiber is in suspension in water and hence the assumption is no
longer valid. The conductivity of the pulp is altered by the presence of additives such as
titanium dioxide, alkalis, and clay. Hence, this method cannot be adapted for measuring
moisture content in the pulp under realistic operating conditions.
2.1. Microwave Techniques
Microwave techniques have been used to study the subsurface moisture since 1970s
[3,22,23]. When the propagating electromagnetic wave has a frequency that is equal to
the resonance frequency of the medium of propagation, stationary waves are created.
Every medium has its own characteristic resonance frequency. Hence, in a multicomponent system, the system’s resonance frequency is a function of the resonance
frequency of the individual components and their mole fractions. This characteristic can
be used to determine the composition of materials [3,22].
Attenuation based microwave techniques have been used to estimate the moisture
content of paper pulp [3]. The attenuation factor of the signal at resonance and the
frequency shift are used to estimate the moisture content. Fiber concentration as low as
4
0.6% has been measured, with a standard deviation of 0.03% [3]. However, this method
cannot be used for on-line monitoring of fiber concentration, as it requires a closed cavity
resonator. Moreover, the method is sensitive only to fiber concentrations from 0.06% to
1%. In a paper machine, such concentrations of pulp can be found only at the headstock,
where the presence of metallic stirrers and the high entropy of the pulp can affect the
accuracy of the method.
The methods suggested in [23,24] require measuring attenuation of the material,
which is difficult to obtain [25]. The difficulty is more pronounced with low attenuation
materials as the attenuation measurements are easily influenced by multiple reflections
[25].
Most microwave techniques need at least two different types of measurements,
such as attenuation and phase [24], or attenuation and density of sample [25]. If these
techniques were to be realized, they would require at least two instruments [25] to obtain
two different parameters. This would increase the measurement complexity and the cost
of the measurement system [25].
Electromagnetic interference from other sources of radiation can affect the
accuracy of microwave techniques. The resonance frequency of pulp is around 2.6 GHz
[3], which is close to the commonly used 2.4 GHz communication channels. As the
communication signals at 2.4 GHz are random in nature, their effect on the measurements
cannot be effectively compensated. Hence, all the microwave systems have to be
electromagnetically shielded, thus rendering the open cavity measurement models [22]
impractical. The sensor reported here uses a single-sided guard plane. The proximity of
the guard plane to the sensing electrodes ensures the immunity of the sensor to stray low
frequency electromagnetic fields. The stray fields penetrate the pulp sheet and the wire to
influence the sensor output. However, the process of penetration weakens the stray fields
sufficiently, and the effect of these fields on the sensor output is negligible.
The penetration depth of the electromagnetic waves cannot be controlled. Hence, it
is not possible to study the moisture distribution in the pulp along the axis perpendicular
to the surface of the pulp sheets.
5
2.2. Reflection Based Techniques
When an electromagnetic wave encounters a discontinuity in the medium of
propagation, a part of the wave is reflected back into the incident media. The ratio of the
amplitudes of the electric field of the reflected wave to that of the incident wave is called
the reflection coefficient. The operational efficiency of reflection based microwave
techniques can be analyzed using the basic laws of electromagnetism [26-28].
Let the discontinuity in the medium of propagation be at Z = 0. Let the wave
propagate from a media with dielectric constant 1 and magnetic permeability 1 , into a
different media with dielectric constant 2 and magnetic permeability 2 . Let the angle of
incidence be i and that of refraction be r.
By Fresnel’s law, if the electric field vector of the incident wave is along the Xaxis, the reflection coefficient, R, is defined as,
R
E y1
Ey 2

1 sin(i) cos(i)  2 sin(r ) cos(r )
1 sin(i ) cos(i)  2 sin(r ) cos(r )
(2.1.)
where Ey1 is the amplitude of the incident electric field, and Ey2 is the amplitude of the
incident electric field. If the electric field vector of the incident wave is along the Y-axis,
we have,
R
E y1
Ey 2

1 tan(i )  2 tan(r )
1 tan(i)  2 tan(r )
(2.2)
Since the choice of axis is arbitrary, we can choose either of the equations. Let us
assume that the electric field vector of the incident wave is along the X-axis, and hence
(2.1) is valid.
By Snell’s law,
sin(i ) n2

sin(r ) n1
(2.3)
where n1 and n2 are the refractive indices of the two media. Assuming non-absorbent
mediums, the refractive indices are given by,
n1  1  1
(2.4)
6
n2   2  2
(2.5)

 
r  sin 1 sin(i ) 1 1 
 2 2 

(2.6)
From (2.3), (2.4), and (2.5),
Let the original medium of propagation be air, and the reflecting medium be
predominantly water. We have,
1  2  0
(2.7)
1  1
(2.8)
To minimize the interference between the incident and the reflected waves, let,
i  45
(2.9)
From (2.6), (2.7), (2.8), and (2.9),

 1
1
1

cos  sin 1 
 2

2
2 2
2


R

 1
1
1

cos  sin 1 
 2

2
2 2
2



 



 


(2.10)
If we consider the dielectric constant of the reflecting media to be close to that of water,
 2  80
(2.11)
R  0.72768
(2.12)
From (2.10) and (2.11),
Considering a 10% change in the dielectric constant of the reflecting media, i.e. 2 = 72,
we have,
R  0.71514
(2.13)
From (2.12) and (2.13), we can estimate the sensitivity, S, of the method, i.e. the
percentage change in the reflection coefficient for a unit change in the dielectric constant
of the reflecting medium, to be
S  0.17233
(2.14)
As seen from (2.14), the change in reflection coefficient for a unit change in
dielectric constant is very small. Hence, this method requires a setup that can measure the
7
reflection coefficient very accurately. This can be achieved by using a high power, high
amplitude incident wave. However, a high power incident wave will cause polarization in
the reflecting media.
Another important factor to be considered is the variation in the refractive indices
of the media. According to Clausius - Mosotti relation, the refractive index of a medium
vary inversely with its density.
n 1 1
 k
n2 
(2.15)
where  is the density of the medium, and k is a constant. Hence, a non-uniform
distribution of constituents of the reflecting media will adversely affect the accuracy of
the method.
8
Chapter 3.
Interdigital Fringing Field Dielectrometry
3.1. Interdigital Fringing Field Sensor
The interdigital fringing field sensor operates in a way that is very similar to a
conventional parallel plate capacitor. Figure 3.1 shows the transition from a parallel plate
capacitor to a fringing field sensor. It can be seen from Figure 3.1 that the electric field
lines always penetrate the bulk of the material under test, irrespective of the position of
the electrodes. Hence, in addition to the electrode geometry, the capacitance between the
electrodes also depends on the material’s dielectric properties and geometry.
Figure 3.1. A fringing field dielectrometry sensor can be visualized as a parallel plate
capacitor whose electrodes open up to provide a one-sided access to material under test
[29].
As seen from Figure 3.1(c), the electrodes of a fringing field sensor are coplanar.
Hence, the signal-to-noise ratio of measured capacitance will be considerably low. To
strengthen the measured signal, the electrode pattern can be repeated several times. The
resulting structure of the sensor is known as an interdigital structure. The term
“interdigital” refers to a digit-like or finger-like periodic pattern of parallel in-plane
electrodes used to build up the capacitance associated with the electric fields that
penetrate into a material sample [30]. This pattern is illustrated in Figure 3.2.
9
Figure 3.2. The term “interdigital” refers to the pattern of fingers or “digits” that is
resembled by the shape and relative position of the electrodes [31].
Figure 3.3 shows a generic interdigital sensor. The wavelength of the sensor is
defined as the distance between the centers of two adjacent electrodes of same type. For a
semi-infinite homogeneous medium placed on the surface of the sensor, the periodic
variation of the electric potential along the X-axis, creates an exponentially decaying
electric field along the Z-axis, which penetrates the medium. The possible variation in the
properties of the material under test along the Z-axis, and hence is complex dielectric
permittivity, ε*(ω), is schematically represented in Figure 3.3 by the variation in shading.
The model for analyzing such multi-layered systems is discussed in detail in [32].
There exist many definitions for penetration depth. The penetration depth of a
fringing field sensor is usually defined as the position at which the measured value varies
from the asymptotic value by 3% [33]. As the penetration depth of a sensor depends on
the position, size, and shape of the electrodes, there exists no simple mathematical
expression to compute the penetration depth. The penetration depth is often estimated to
be equal to one-third the wavelength of the sensor. Exact penetration depth of a sensor
can be estimated by using finite element analysis software. Figure 3.4 shows the cross
section of a multiple penetration depth sensor.
10
Figure 3.3. A generic interdigital sensor with a periodicity λ [34,35].
Figure 3.4. A conceptual view of multiple penetration depth sensor.
11
3.2. Dielectric Spectroscopy
All dielectric materials consist of polarized dipoles. When subjected to an external
electric field, these dipoles re-align so as to neutralize the effect of the external field. This
re-alignment of dipoles occurs to a varying extent for different materials. Thus, the
dielectric response of each dielectric material across the frequency spectrum is different,
and in most cases unique. The study of this response variation is known as dielectric
spectroscopy.
The dielectric response of a material is generally quantified in terms of its complex
dielectric permittivity. The complex dielectric permittivity  * ( ) is usually represented
as,
 * ( )   ( )  j ( )
(3.1)
where  ( ) is the real part of permittivity and  ( ) is the loss factor.
For all materials the loss factor is a function of the excitation frequency. The loss
factor mechanisms are schematically shown in Figure 3.5. For a few low-loss materials,
and non-polar materials, the variation in the loss factor with frequency is predominantly
due to distortion in the electron clouds. Hence, the magnitude of variation of loss factor is
negligibly small.
The polarization of molecules arising from their reorientation with the imposed
electric field is the most important phenomenon contributing to the loss factor in the radio
and microwave frequencies (107 to 3x1010 Hz). This includes the dipolar polarization due
to bound and free water relaxation. At infrared and visible light frequencies, the loss
mechanisms due to atomic and electronic polarization (collectively known as distortion
polarization) are the dominating loss mechanisms [36].
The description for the process for pure polar materials was developed by Debye in
1929 [37]. The Debye model is explained in detail in Section 3.3.
12
Figure 3.5. Mechanisms influencing the loss factor of a moist material over a wide range
of frequencies f (Hz). (A) DC conductivity, (B) Maxwell-Wagner polarization, (C) dipolar
polarization of water bound to the matrix of the material, (D) dipolar polarization of free
water [36].
3.3. Debye Model
The Debye dielectric relaxation model [37] is the simplest way to analyze
polarization in purely polar materials. The model assumes that the relaxation process is
governed by first order dynamics, and hence can be characterized with a single time
constant. The model can be derived using basic laws of polarization and conduction [38]
as shown in the following arguments.
When a dielectric is subjected to an electric field, it interacts in two principal ways,
re-orientation of the defects in the dielectrics with a dipole moment, and the translative
motion of the charge carriers. The resulting current in the dielectric can be written as,
J E 
D
t
(3.2)
13
where J is the current density, E is the electric field, and D is the electric
displacement. The electric displacement is defined as the total charge density on the
electrodes and is mathematically defined by,
D  0E  P
(3.3)
where ε0 is the permittivity of free space and P is the polarization vector. The Debye
model deals with the displacement current represented by the second term in (3.2).
When a dielectric is subjected to an electric field E , the resulting polarization P
comprises of two parts based on the time constant of the response. One part of the
polarization P is the instantaneous polarization P due to the displacement of electrons
with respect to the nuclei (distortion of the electron cloud). The high frequency dielectric
constant ε∞ is thus defined by,
 1 
P
0E
(3.4)
The second part of the polarization P is the time-dependent polarization P(t ) due to the
orientation of the dipoles in the electric field.
If we let the field remain in place infinitely long, the resulting total polarization
Ps defines the static dielectric constant εs as,
 s 1 
Ps
0E
(3.5)
Thus we have,
Ps  P  P(t  )
(3.6)
Let us assume that the time dependent polarization is governed by first order
kinetics with a relaxation time of τ, such that the rate at which P approaches Ps is
proportional to the difference between them. That is,
dP(t ) Ps  P

dt

When a unit step voltage u0 (t) is applied, we have,
(3.7)
14
P  Puo (t )  P(t )
(3.8)
Taking Laplace transforms of (3.7) and (3.8), and solving for the Laplace transform of

polarization P , we get,
P  
P

Ps
1
1

 p    p p  




(3.9)
where p is the complex frequency variable. Taking Inverse Laplace transform of (3.9)
and simplifying, we obtain,
P
Ps  j P
j  1
(3.10)
Assuming a conductivity of zero and a sinusoidal steady state fields of the form,
ˆ jt 
J    Je

ˆ jt 
E    Ee

(3.11)
From (3.2), (3.7), (3.10) and (3.11), we get,
 *   
 s      j   s    
2
2
1   
1   
(3.12)
The real and imaginary parts of (3.12) are known as the Debye dispersion relations, and
have remained the basic model of dielectric relaxation since their inception.
The frequency response of the Debye model corresponds exactly to the response
of the simple lumped circuit element shown if Figure 3.6 (a), for a parallel plate electrode
geometry with area A and gap d.
15
Figure 3.6. Debye model for time dependent polarization [39].
The similarity between the Debye model and the lumped circuit element is given
by (3.13), (3.14) and (3.15).
Cp 
Cs 
 0  A
d
 0  s     A
d
  RsCs
(3.13)
(3.14)
(3.15)
Even when the conductivity of a material is zero, its complex dielectric
permittivity can still have a non-zero imaginary part. The energy dissipation process due
to dipole re-orientation in the material, and energy dissipation due to translatory motion
of charge carriers introduces the imaginary part of the complex dielectric permittivity.
Equations (3.2) and (3.12) can be combined to include ohmic conductivity and Debye
polarization as,

       dipole
(3.16)
o

  dipole

(3.17)
  
 s    
2
1   
(3.18)
 s     
2
1   
(3.19)
 
 dipole
 
 dipole
16
Equations (3.16) – (3.19) can be presented as Cole-Cole plots shown in Figure 3.6
(b). The Cole-Cole plots plot ε′ vs. ε″ with frequency ω as an independent parameter and
is an exact semicircle.
Another way of studying equations (3.16) – (3.19) is shown in Figure 3.7.
Dielectric Susceptibility,  * ( ) , is related to permittivity, and can be expressed in terms
of its real and imaginary components.
 * ( )   ( )  j  (c)
(3.20)
The difference between the susceptibility and permittivity is that the term
 * ( ) refers to the sum of all permittivities between infinity and the frequency of interest,
while  * ( ) refers to the permittivity at that specific frequency [40].
Figure 3.7. Real and imaginary parts of complex susceptibility as a function of
frequency. [40]
17
Chapter 4.
Experimental Setup
The experiments reported in this report emulate the operational conditions in a
paper machine. The pulp in the wet end of the paper machine is primarily a suspension.
This pulp suspension is spread on to a semi-permeable membrane made of nylon or
similar polymer, and is hence unavailable for contact measurements. To emulate this
setup in the laboratory, the pulp is blended to a consistency of a suspension and is placed
on a tray. The tray wall prevents contact with the pulp, and hence is equivalent to the
wire on the paper machine.
The sensor used for these measurements is an interdigital sensor tray with a spatial
periodicity of 40 mm, finger length of 160 mm, and penetration depth of 7 mm. The
sensor electrodes are not in direct contact with paper pulp. Instead, the sensor is attached
to the outer side of the base of an acrylic tray with a wall thickness of 5 mm. Figure 4.1
shows a photograph of the experimental setup. A guard plane is placed underneath the
sensor electrodes to provide shielding from external electric fields. The geometry of the
sensor is shown in Figure 4.2.
Figure 4.1. Photograph of experimental setup.
18
Measurements reported here were taken using the Fluke manufactured RCL meter
(model PM 6304). It generates a one-volt sinusoidal AC voltage in the frequency range
from 50 Hz to 100 kHz. A custom designed circuit is also available for making
measurements. The circuit is capable of making measurements from three sensors
simultaneously. A data acquisition system for the circuit has been written using
LabVIEW. Advanced data analysis can be potentially integrated into the software.
However, as the measurements reported in this report require only single channel
measurements, the custom designed circuit was not used.
The interdigital sensor tray filled with paper pulp is connected to the two channels
of the RCL meter. The effective impedance between the two channels is calculated by
computing the magnitude attenuation and phase shift between the input voltage and loop
current.
4 cm
Drive
16 cm
Guard
Sense
Figure 4.2. The top-down view of the interdigital sensor tray with the spatial periodicity
of 40 mm, finger length of 160 mm and an approximate penetration depth of 13 mm.
The measurements are made at frequencies in the range of 200 Hz to 100 kHz. The
measurements made at the lower end of the frequency spectrum (below 200 Hz) have
noise due to the AC power supply. The instrumentation limits the highest viable
frequency to 100 kHz. Ten sets of measurements were taken at each frequency, and then
19
averaged to reduce the noise. It is assumed that all sources of noise have zero mean
distribution.
Initial contact with the Forestry Department of UW, and subsequent lab tours to the
paper machines at the Forestry Department of UW and at Port Townsend paper mill,
confirmed feasibility of installing these sensors on paper machines. The sensors can
potentially be installed in two positions on the paper machine depending on the need. The
sensor can be embedded in the mounts for hydrofoils in the wire section. This would be
preferable if the sensor is to be used to measure moisture in the range of 97% to 80%.
Alternatively, the sensor could also be installed just after the wire but before the press
section. Currently, the setup is very simple. Although the equipment is suitable for
portable measurements, it is too early in the project to install the sensors on a running
paper machine.
20
Chapter 5. Experiments with High Fiber Concentration Pulp
(100% to 10%)
Experiments were conducted to characterize the response of the sensor to the
variation of moisture level in pulp with very high fiber content. The moisture content of
the pulp was varied from 0% to 90%, and measurements were made using the setup
described in Chapter 2.
5.1. Experimental Procedure
When the required moisture content is relatively low, it is not possible to prepare
minced pulp in the laboratory using the currently available devices. Hence sheets of dried
pulp are used for the measurements. Sheets of dried paper pulp are cut into the size of the
sensor tray and subsequently stacked in it. The pulp sheets have to be absolutely flat,
failing which air pockets are formed between the pulp sheets. These air pockets affect the
measurements adversely. Calculated amount of water is sprinkled on to the sheets. The
pulp sheets, being relatively dry, absorb the water quickly. The key to the successful
measurement in this range is uniform distribution of water.
The interdigital sensor tray filled with paper pulp is then connected to the two
channels of the RCL meter and measurements are made.
5.2. Experimental Results
Figure 5.1(c) shows the dependence of capacitance on the moisture content and
excitation frequency. It can be seen that the capacitance initially increases with the
moisture content and then starts to decrease after moisture level of 85%.
21
Figure 5.1(b) shows the dependence of phase on the moisture content and
excitation frequency. The magnitude of the phase change is small. Hence, even a small
noise in the phase measurement leads to cross-overs at various frequencies and moisture
levels.
Figure 5.1. Measurements of paper pulp samples with 0% to 90% moisture concentration
at frequencies from 200 Hz to 100 kHz.
22
5.3. Data Analysis
Figure 5.2 and Figure 5.3 show respectively conductance and capacitance as a
function of moisture concentration of the pulp as measured. It can be observed that the
capacitance starts to decrease with increase in moisture after 85% moisture concentration.
This trend continues into lower fiber concentration region as will be shown in Chapter 6.
It can be seen from Figure 5.2 that the rate of change of moisture concentration
with conductance is very high. Hence, if conductance is used as a parameter to estimate
the paper moisture concentration, then any small error in conductance measurement will
be amplified and hence the resulting the error in the moisture concentration estimate will
be unacceptably high.
Figure 5.2. Fiber content as a function of the conductance of the paper pulp samples
with 0% to 90% moisture concentration at various frequencies from 200 Hz to 100 kHz.
As shown in Figure 5.3, the rate of change of moisture concentration with
capacitance is low, so amplification of the error in measurement of capacitance will be
23
marginal. Hence capacitance can be used as a parameter to estimate the moisture
concentration in the pulp.
Figure 5.3. Fiber content as a function of the capacitance of the paper pulp samples
with 0% to 90% fiber concentration at various frequencies from 200 Hz to 100 kHz.
24
Chapter 6. Experiments with Low Fiber Concentration Pulp
(10% to 3%)
Experiments were conducted to characterize the response of the sensor to the
variation of moisture level in pulp with very low fiber content. The mositure content of
the pulp was varied from 90% to 97% in steps of 1% and measurements were made using
the setup described in Chapter 2.
6.1. Experimental Procedure
Known quantities of paper and water are mixed in a commercial blender to obtain
the paper pulp. The pulp is then cooled to ambient temperature of 25°C. The moisture
loss due to evaporation can be neglected, as the loss is small compared to the total water
content in the pulp. The prepared pulp is then deposited in the sensor tray. The
homogeneity of spatial distribution of the pulp and reduction in the number of air pockets
in the bulk of the pulp are achieved by manually rearranging the pulp in the tray.
The interdigital sensor tray filled with paper pulp is then connected to the two
channels of the RCL meter and measurements are made.
6.2. Experimental Results
Figure 6.1(c) shows the dependence of capacitance on the moisture content and
excitation frequency. It can be seen that the variation is monotonous and strictly
decreasing with frequency and moisture content. As seen from Figure 6.2, the obtained
curves were found to be displaced by at least twice the standard deviation. Hence
inaccuracies in measurement introduce relatively smaller errors in the concentration
estimates.
25
Figure 6.1. Measurements of paper pulp samples with 90% to 97% moisture
concentration at frequencies from 200 Hz to 100 kHz.
Figure 6.1(d) shows the dependence of conductance on the moisture content and
excitation frequency. The spatial separation of the curves is not adequate to mitigate the
effect of any small inaccuracies in the measurement of conductance. This may explain the
high error percentages reported in [5,41,42].
Figure 6.1(b) shows the dependence of phase on the moisture content and
frequency. There are cross-overs in the phase plots at various frequencies and moisture
levels. This is partly due to instrumentation errors and also due to the fact that the phase
is highly sensitive to noise. Hence the phase shifts at two frequencies cannot be used with
26
the frequency range under consideration to estimate the moisture content of the pulp as
suggested in [25].
Figure 6.2. The capacitance measured at 5 kHz shows separation between measurements
to be much greater than twice the standard deviation. This indicates towards the
possibility of achieving higher resolution using the sensor.
Figure 6.3 shows the Cole-Cole Plots for the measurements made. The admittance
plot shows adequate resolution at higher frequencies to differentiate between the different
moisture content levels of the paper pulp.
27
Figure 6.3. Cole-Cole plots from measurements of paper pulp samples with 3% to 10%
paper concentration at frequencies from 50 Hz to 100 kHz.
6.3. Data Analysis
Figure 6.1(c) and Figure 6.1(d) show that the measured capacitance and
conductance decrease with increasing excitation frequency till 20 kHz and then increase.
This can be attributed to presence of multiple relaxation processes in the paper pulp. If
28
only one relaxation process was present, the trend in capacitance and conductance would
be monotonous, and strictly decreasing with frequency. The rate at which the capacitance
or conductance decreases is unique to the relaxation processes of the fiber and water
molecules. This can be exploited to estimate the moisture content in the pulp in the
presence of other fillers, which too can influence the electrical parameters. However, it
has to be first established that no other constituent of the pulp undergoes the same
relaxation processes.
Figure 6.4 shows the moisture content as a function of conductance at different
frequencies. It can be seen that the rate of change of conductance with moisture content is
small. Hence, it is not advisable to estimate the moisture content based on the
conductance and the excitation frequency. This may be the main reason the methods
suggested in [5,20,21,41] did not perform adequately for lower fiber concentrations. The
slopes of the curves are better defined at higher concentrations of paper fiber, and hence
the foresaid methods can be used to estimate the moisture content in those regions.
Figure 6.5 shows the moisture concentration as a function of capacitance at
different frequencies. It can be seen that slopes of the curves are higher than those of the
conductance plots in Figure 6.4. Hence we estimate the moisture content of the pulp
based on the measured capacitance measured and the frequency used.
The curves in Figure 6.5 can be linearized and the relationship between moisture
concentration, P and the capacitance, C can be expressed as:
P = m×C + k
(6.1)
where m is the slope of the line and k is the offset constant. It can be seen that both m and
k are related to the frequency of excitation.
29
Figure 6.4. Conductance plots from measurements of paper pulp samples with 90% to
96% moisture concentration at frequencies from 200 Hz to 100 kHz.
Figure 6.5. Capacitance plots from measurements of paper pulp samples with 90% to
96% moisture concentration at frequencies from 200 Hz to 100 kHz.
Figure 6.6 and Figure 6.7 respectively show the variation m and k with excitation
frequency. It can be observed that the resolution of this method is noticeably higher at
lower frequencies and hence these frequencies offer smaller error margins.
30
Figure 6.6. Variation of the slope, m, in (6.1) depicting the plot between percentage of
moisture content and capacitance with respect to frequency of excitation.
Figure 6.7. Variation of the offset, k, in (6.1) depicting the plot between percentage of
moisture content and capacitance with respect to frequency of excitation.
To determine the moisture content in the pulp given the capacitance of the pulp
and the frequency of excitation used, we first determine the corresponding values of m
and k from Figure 6.6 and Figure 6.7. Once these values have been determined, they are
31
substituted in (6.1) along with the measured capacitance and the moisture concentration
is estimated.
Figure 6.8 shows the conformity of the curve formulated in (6.1) to the data
obtained experimentally. The curve formulated in (6.1) will be always valid for
estimation of moisture content for all similar pulp samples if adequate reproducibility is
ensured. The reproducibility and repeatability of the measurements are discussed in
Chapter 11.
Figure 6.8. The data obtained experimentally at 5 kHz is in agreement with the curve
formulated in (6.1).
32
Figure 6.9. Standard deviation of the capacitance measurements at various moisture
levels is two orders of magnitude smaller than the capacitance (10-11 vs. 10-14).
The accuracy of this method relies on the ability to measure capacitance accurately
and the sensitivity of the method. Figure 6.9 shows the standard deviation of the
measured capacitance at 5 kHz for various moisture concentration levels. It can be seen
from Figure 6.6 and Figure 6.9 that the maximum standard deviation of the estimated
moisture content would be approximately 0.2379 %.
33
Chapter 7.
Experiments with Titanium Dioxide
Experiments were conducted to characterize the response of the sensor to the
variation of moisture level in pulp with low fiber and titanium dioxide content. The
concentration of the fiber was initially 6% with 94% moisture. The titanium dioxide
concentration was varied from 0% to 7% in steps of 1%. The measurements were made
using the setup described in Chapter 2.
7.1. Experimental Procedure
Known quantities of paper, titanium dioxide and water are mixed in a commercial
blender to obtain the paper pulp. The pulp is then cooled to ambient temperature of 25°C.
The moisture loss due to evaporation can be neglected as the loss is small compared to
the total water content in the pulp. The prepared pulp is then deposited in the sensor tray.
The homogeneity of spatial distribution of the pulp and reduction in the number of air
pockets in the bulk of the pulp are achieved by manually rearranging the pulp in the tray.
The interdigital sensor tray filled with paper pulp is then connected to the two channels of
the RCL meter and measurements are made.
7.2. Experimental Results
Figure 7.1(c) shows the dependence of capacitance on the titanium dioxide
content and excitation frequency. It can be seen that the variation is monotonous and
strictly decreasing with titanium dioxide content. The capacitance measured at 6% fiber
concentration, 94% moisture and 0% titanium dioxide is not the same as that measured
for a similar configuration in Section 6.2, as shown in Figure 6.1. This could be because
different types of pulp sheets were used to prepare the pulp for these experiments. The
34
fibers in each of the pulp sheet types are different and have unique dielectric properties. If
desired, this property can be exploited to determine the type and the quality of the fiber in
the pulp.
Figure 7.1. Measurements of paper pulp samples with 0% to 7% titanium dioxide
concentration at frequencies from 200 Hz to 100 kHz.
Figure 7.1(b) and Figure 7.1(d) shows the dependence of phase and conductance
on the titanium dioxide content and excitation frequency. There is a change in the trend
of the parameters between the 20 kHz and 100 kHz. This is due to the combination of the
different dielectric relaxation process of titanium dioxide, the paper fibers, and water.
The frequency that exhibits the minimum conduction will also depend on the composition
35
of the pulp. This frequency can potentially be used as a handle to eliminate the effects
due to additives.
Figure 7.2 shows the Cole-Cole plots for the measurements made.
Figure 7.2. Cole-Cole plots from measurements of paper pulp samples with 0% to 7%
titanium dioxide concentration at frequencies from 200 Hz to 100 kHz.
Figure 7.3 and Figure 7.4 respectively show the conductance and the capacitance
measured at different frequencies from 200 Hz to 100 kHz.
36
Figure 7.3. Conductance plots from measurements of pulp samples with 0% to 7%
titanium dioxide concentration at frequencies from 200 Hz to 100 kHz.
Figure 7.4. Capacitance plots from measurements of pulp samples with 0% to 7%
titanium dioxide concentration at frequencies from 200 Hz to 100 kHz.
37
7.3. Data Analysis
The variations in capacitance, conductance and other electrical parameters are
influenced by all the three components of the pulp, namely, paper fiber, titanium dioxide,
and moisture. Since two independent variables are involved here, it is not possible to
estimate the fiber concentration using a single parameter. So we solve the inverse
problem, by estimating any three of the electrical parameters as
 X   m11
 Y   m
   21
 Z   m31
m12
m22
m32
m13   p   C1 
m23   t   C2 
m33   w C3 
(7.1)
where X, Y and Z are the electrical parameters estimated using fiber concentration p,
titanium dioxide concentration t, moisture content w, and constants m11, m12, m13…m33
and C1, C2, and C3.
Once the constants are determined, the parameters X, Y and Z can be used to
estimate the concentrations of fiber, titanium dioxide and water in the pulp using (7.2),
(7.3), (7.4), and (7.5).
w
e(  d )  a(   g )
eb  af
(7.2)
t
f (  d )  b(   g )
af  be
(7.3)
p  100  (t  w)
(7.4)
38
where,
a   m12 m21  m11m22 
b  (m12 m31  m11m32 )
d  (m12 c1  m11c2 )
e  (m13m22  m12 m23 )
f  (m13m32  m12 m33 )
(7.5)
g  (m13c2  m12 c3 )
  (m12 X  m11Y )
  (m23Y  m22 Z )
The key to the success of the estimation is in the choice of the parameters X, Y
and Z, and the constants m11, m12, m13…m33 and C1, C2, and C3.
Figure 7.5. Comparison of the estimated concentration of fiber in the pulp to the actual
concentration.
39
Figure 7.6. Comparison of the estimated concentration of titanium dioxide in the pulp to
the actual concentration.
Figure 7.7. Comparison of the estimated concentration of moisture in the pulp to the
actual concentration.
Figure 7.5, Figure 7.6 and Figure 7.7 respectively compare the concentrations of
fiber, titanium dioxide, and moisture as obtained using the method described above. The
40
estimates were based on the measured phase, capacitance and conductance. These
parameters were chosen manually. The algorithm that is introduced in Chapter 10 can be
potentially used for choosing these parameters. Currently, we are attempting to devise a
method to determine the constants and the parameters that would reduce the error.
41
Chapter 8.
Experiments with Clay
Experiments were conducted to characterize the response of the sensor to the
variation of fiber concentration the in the presence of clay and water. The concentration
of the fiber was varied from 10% to 5% and that of clay from 0% to 5% in steps of 1%.
The moisture content was maintained a constant at 90%. The measurements were made
using the setup described in the Chapter 2.
8.1. Experimental Procedure
Known quantities of paper and water are mixed in a commercial blender to obtain
the paper pulp. The pulp is then cooled to ambient temperature of 25°C. The moisture
loss due to evaporation can be neglected, as the loss is small compared to the total water
content in the pulp. The required amount of clay is dissolved in known quantity of water.
The clay solution is then added to the prepared pulp, and the mixture is thoroughly
mixed. The prepared pulp is then deposited in the sensor tray. The homogeneity of spatial
distribution of the pulp and reduction in the number of air pockets in the bulk of the pulp
are achieved by manually rearranging the pulp in the tray. The interdigital sensor tray
filled with paper pulp is then connected to the two channels of the RCL meter and
measurements are made.
8.2. Experimental Results
Figure 9.1c shows the variation of capacitance with clay content and frequency.
The change in capacitance with clay content is negligibly small. This indicates that the
reactance of the clay molecules is very close to that of the paper fiber. Hence, the
42
additional reactance due to clay compensates for the reduction in the reactance due to the
reduction in the percentage composition of the fiber molecules.
Figure 9.1b and Figure 9.1d respectively show the variation in phase and
conductance with the clay content and frequency. The conductance increases with the
clay content. This is due to the presence of free carrier molecules in clay, which increase
the conduction of the pulp. The increase in conduction, with the capacitance remaining
nearly the same is reflected in the change in phase.
Figure 8.1. Measurements of paper pulp samples with 10% to 5% fiber concentration
and 0% to 5% clay at frequencies from 200 Hz to 100 kHz.
43
Figure 8.2 shows the Cole-Cole plots for the measurements made.
Figure 8.2. Cole-Cole plots from measurements of paper pulp samples with 10% to 5%
paper concentration and 0% to 5% clay at frequencies from 200 Hz to 100 kHz.
Figure 9.3 and Figure 9.4 respectively show the conductance and the capacitance
measured at different frequencies from 200 Hz to 100 kHz.
44
Figure 8.3. Conductance plots from measurements of pulp samples with 0% to 5% clay
concentration at frequencies from 200 Hz to 100 kHz.
Figure 8.4. Capacitance plots from measurements of pulp samples with 0% to 5% clay
concentration at frequencies from 200 Hz to 100 kHz.
45
8.3. Data Analysis
The pulp with clay in it is again a three-component system similar to the one with
titanium dioxide. Hence the same estimation algorithm as used for titanium dioxide can
be used for estimating the concentrations of clay, fiber and moisture in the pulp.
The key to the success of the estimation is in the choice of the parameters X, Y
and Z, and the constants m11, m12, m13…m33 and C1, C2, and C3.
Figure 9.5, Figure 9.6, and Figure 10.3 respectively show the estimated
concentrations of fiber, clay and moisture against the respective actual concentrations.
These estimates were based on the conductance and phase measurements at 5 kHz.
Figure 8.5. Comparison of the estimated concentration of fiber in the pulp to the actual
concentration.
46
Figure 8.6. Comparison of the estimated concentration of clay in the pulp to the actual
concentration.
Figure 8.7. Comparison of the estimated concentration of moisture in the pulp to the
actual concentration.
47
Chapter 9.
Experiments with Calcium Carbonate
Experiments were conducted to characterize the response of the sensor to the
variation of fiber concentration the in the presence of calcium carbonate and water. The
concentration of the fiber was varied from 10% to 7.5% and that of calcium carbonate
from 0% to 2.5% in steps of 0.5%. The moisture content was maintained a constant at
90%. The measurements were made using the setup described in Chapter 2.
9.1. Experimental Procedure
Known quantities of paper and water are mixed in a commercial blender to obtain
the paper pulp. The pulp is then cooled to ambient temperature of 25°C. The moisture
loss due to evaporation can be neglected, as the loss is small compared to the total water
content in the pulp. The required amount of calcium carbonate is dissolved in known
quantity of water. The calcium carbonate solution is then added to the prepared pulp, and
the mixture is thoroughly mixed. The prepared pulp is then deposited in the sensor tray.
The homogeneity of spatial distribution of the pulp and reduction in the number of air
pockets in the bulk of the pulp are achieved by manually rearranging the pulp in the tray.
The interdigital sensor tray filled with paper pulp is then connected to the two channels of
the RCL meter and measurements are made.
9.2. Experimental Results
Figure 9.1c shows the variation of capacitance with calcium carbonate content
and frequency. The change in capacitance with calcium carbonate content is negligibly
small. This indicates that the reactance of the pulp is dominated by moisture
concentration.
48
Figure 9.1b and Figure 9.1d respectively show the variation in phase and
conductance with the calcium carbonate content and frequency. The conductance
increases with the calcium carbonate content. This is due to the presence of free carrier
ions in calcium carbonate, which increase the conduction of the pulp. The increase in
conduction, with the capacitance remaining nearly the same is reflected in the change in
phase.
Figure 9.1. Measurements of paper pulp samples with 10% to 7.5% fiber concentration
and 0% to 2.5% calcium carbonate at frequencies from 200 Hz to 100 kHz.
Figure 9.2 shows the Cole-Cole plots for the measurements made.
49
Figure 9.2. Cole-Cole plots from measurements of paper pulp samples with 10% to 7.5%
paper concentration and 0% to 2.5% calcium carbonate at frequencies from 200 Hz to
100 kHz.
Figure 9.3 and Figure 9.4 respectively show the conductance and the capacitance
measured at different frequencies from 200 Hz to 100 kHz.
50
Figure 9.3. Conductance plots from measurements of pulp samples with 0% to 2.5%
calcium carbonate concentration at frequencies from 200 Hz to 100 kHz.
Figure 9.4. Capacitance plots from measurements of pulp samples with 0% to 2.5%
calcium carbonate concentration at frequencies from 200 Hz to 100 kHz.
51
9.3. Data Analysis
The pulp with calcium carbonate in it is again a three-component system similar
to the one with titanium dioxide. Hence the same estimation algorithm as used for
titanium dioxide can be used for estimating the concentrations of calcium carbonate,
fiber, and moisture in the pulp.
The key to the success of the estimation is in the choice of the parameters X, Y
and Z, and the constants m11, m12, m13…m33 and C1, C2, and C3.
Figure 9.5, Figure 9.6, and Figure 10.3 respectively show the estimated
concentrations of fiber, calcium carbonate, and moisture against the respective actual
concentrations. These estimates were based on the conductance and phase measurements
at 5 kHz.
Figure 9.5. Comparison of the estimated concentration of fiber in the pulp to the actual
concentration.
52
Figure 9.6. Comparison of the estimated concentration of calcium carbonate in the pulp
to the actual concentration.
Figure 9.7. Comparison of the estimated concentration of moisture in the pulp to the
actual concentration.
53
Chapter 10. Parameter Selection Algorithm
Figure 5.1, Figure 6.1, and Figure 7.1 illustrate that all the electrical parameters
measured are dependent on the moisture content of the paper pulp. Hence, there is a
choice of parameters that can be used to estimate the moisture content. We propose an
algorithm to choose these parameters.
Figure 10.1 shows the flowchart for the training process. There are 12 basic
electrical parameters that are measured or calculated over a frequency range of 500 Hz to
100 kHz for each of the paper pulp samples. These form the base parameters for the
estimation process. Derived parameters are then obtained from a linear combination of
any two of the base parameters. To exploit the spectral data in these parameters,
parameters are further derived from a linear combination of the value of a parameter at
two distinct frequencies.
Each of the base and the derived parameters are used to derive a linear estimation
model similar to that shown in (6.1). Least square fitting technique is used to obtain these
models. Based on the model obtained, the fiber concentration of the paper pulp is
estimated for numerous training data sets. The mean estimation error is then calculated
for all of the parameters for each of the data sets. The parameters are then ranked
according to their accuracy for each individual data set. The sum total of the ranks for the
parameters is then calculated. The methods with the 10 highest total ranks are then
compared with each other based on the product of their sensitivities and the accuracies to
which they can be measured. The standard deviation of the parameters in the given
training data set is used as a measure of measurement accuracy. This product is a measure
of the inherent unbiased error component of the estimated fiber content.
Figure 10.2 shows the flowchart for the estimation process. The parameter and the
model selected during the training are used to estimate the moisture content of the given
pulp sample. At the end of measurement, the data set collected during the estimation
process is added to the database of the training set and the system is re-trained. This
improves the accuracy of the method over time.
54
Figure 10.3 shows the estimated fiber content based on the parameter selected
using the proposed algorithm on three unique training data sets. The sample pulp
consisted of only fibers and water. The parameter used in for estimation is the ratio of the
phase to the magnitude of the current flowing through the sensor at 20 kHz. The
estimates have a mean normalized error of 3.096%.
55
Figure 10.1. Flow chart of the training algorithm.
56
Figure 10.2. Flow chart for evaluation algorithm.
57
Figure 10.3. Plot of the estimated fiber content in pulp consisting of fibers and water.
The estimates are based on the parameter selected by the algorithm.
58
Chapter 11. Repeatability Tests
The ability of the sensor to repeat measurements is critical for estimation process.
The prepared pulp sample was placed in the sensor and measurements were made. The
measurements are repeated approximately every 3.24 seconds. During this process,
neither the sensor, nor the pulp, is disturbed.
The repeatability test was performed on two types of pulp samples. The first
sample had 90% moisture content with 10% fiber. Six sets of measurements were made.
Figure 11.1 shows the results of the repeatability test for capacitance measured at 7.9
kHz. The mean capacitance measured was 16.24 pF. The standard deviation was found to
be 0.03386 pF. The estimated moisture concentration showed a peak-to-peak variation of
0.86673%.
The second sample had 90% moisture content, 7.5% fiber and 2.5% calcium
carbonate. Twenty sets of measurements were made. Figure 11.2 shows the results of the
repeatability test for capacitance measured at 7.9 kHz. The mean capacitance measured
was 16.404 pF. The standard deviation was found to be 0.0075243 pF. The estimated
moisture concentration showed a peak-to-peak variation of 3.5527e-013%.
59
Figure 11.1. Repeatability test for measurements made using pulp containing just fiber
and water.
60
Figure 11.2. Repeatability test for measurements made using pulp containing fiber,
calcium carbonate, and water.
61
Chapter 12. Reproducibility Tests
The ability of the sensor to reproduce the measurements for similar pulp sample is
established by the reproducibility test. The prepared pulp sample was placed in the sensor
and measurements were made. The pulp sample is then removed from the sensor. The
sensor surface is cleaned and then the same pulp sample is deposited back into the sensor
tray.
The reproducibility tests were performed on two types of pulp samples. The first
sample had 90% moisture content with 10% fiber. Five sets of measurements were made.
Figure 12.1 shows the results of the reproducibility test for the estimated moisture
content, based on capacitance measured at 4 kHz. The mean capacitance measured was
16.569 pF. The standard deviation was found to be 3.2354e-14 F. The peak-to-peak
variation in the estimated moisture content was found to be 0.88033 %.
The second sample had 90% moisture content, 7.5% fiber and 2.5% calcium
carbonate. Twenty sets of measurements were made. Figure 12.2 shows the result of the
reproducibility test for the measured capacitance at 4 kHz. The mean capacitance
measured was 16.817 pF. The standard deviation was found to be 5.4584e-14 F. The
peak-to-peak variation in the estimated moisture content was found to be 0.1993%.
62
Figure 12.1. Reproducibility test for measurements made using pulp containing just fiber
and water.
63
Figure 12.2. Reproducibility test for measurements made using pulp containing fiber,
calcium carbonate, and water.
64
Chapter 13. Validation of Estimation Algorithms
To validate the estimation algorithms presented in so far, blind data tests were
conducted. The algorithms were trained using the data from a single experiment. One of
the data points obtained was omitted in the training data set. Hence, for the purpose of
evaluation, the omitted data point serves as a blind data point. The entire data from the
experiment is then provided to the estimation algorithms, and the estimated moisture
content is compared to the actual moisture content. Figure 13.1, Figure 13.2, Figure 13.3,
and Figure 13.4 respectively show the result of the validation tests performed on equation
(6.1), estimation algorithms in presence of titanium dioxide, clay and calcium carbonate.
Figure 13.1. Validation of estimation process using equation (6.1).
65
Figure 13.2. Validation of estimation process described in Section 7.3
66
Figure 13.3. Validation of estimation process described in Section 8.3
67
Figure 13.4. Validation of estimation process described in Section 9.3
68
Chapter 14. Pass line Sensitivity
Pass line sensitivity of the sensor is an important measure of the practical
applicability of the sensor. Pass line sensitivity can be defined as the ability of the sensor
to detect variation in the moisture content of the paper pulp in the presence of an air gap
between the sensor surface and the paper pulp. The following experiments was conducted
to demonstrate the pass line sensitivity of the fringing electric field sensor.
A sensor with a spatial periodicity of 4.3 cm, finger length of 7.9 cm, and an
approximate penetration depth of 1.43 cm is fabricated on a 5 mm thick Plexiglas
substrate. A grounded metal plate placed beneath the sensor substrate acts as a guard
plane. The paper pulp is placed in a tray of wall thickness 5 mm. The base of the tray is
elevated from the surface of the sensor electrodes by use of spacers. Two set of
experiments were conducted, one with an air gap of 2.5 mm, and other with 4.2 mm.
The pulp preparation, and measurement techniques are the same as described in
Section 6.1. The moisture content of the pulp was varied from 90% to 92% in steps of
1%.
14.1. Experimental Results
Figure 14.1(a) shows the variation in magnitude of admittance with moisture content,
and excitation frequency, for an air gap separation of 2.5 mm. A monotonous decrease in
admittance with increasing moisture content is noticed.
Figure 14.1(b) shows the variation in phase with moisture content, and excitation
frequency, for an air gap separation of 2.5 mm. The lower frequency phase measurements
are very noisy.
Figure 14.1(c) shows the variation in capacitance with moisture content, and
excitation frequency, for an air gap separation of 2.5 mm. A monotonous decrease in
admittance with increasing moisture content is noticed.
69
Figure 14.1(d) shows the variation in conductance with moisture content, and
excitation frequency, for an air gap separation of 2.5 mm. The measured conductance is
erroneous for most of the frequencies, and hence cannot used for parameter estimation
purposes.
Figure 14.1. Measurements of paper pulp samples with 90% to 92% moisture
concentration at frequencies from 200 Hz to 100 kHz, for an air gap of 2.5 mm.
Figure 14.2(a) shows the variation in magnitude of admittance with moisture content,
and excitation frequency, for an air gap separation of 4.2 mm. A monotonous decrease in
admittance with increasing moisture content is noticed.
70
Figure 14.2(b) shows the variation in phase with moisture content, and excitation
frequency, for an air gap separation of 4.2 mm. The lower frequency phase measurements
are very noisy.
Figure 14.2(c) shows the variation in capacitance with moisture content, and
excitation frequency, for an air gap separation of 4.2 mm. A monotonous decrease in
admittance with increasing moisture content is noticed.
Figure 14.2(d) shows the variation in conductance with moisture content, and
excitation frequency, for an air gap separation of 4.2 mm. The measured conductance is
erroneous for most of the frequencies, and hence cannot used for parameter estimation
purposes.
Figure 14.2. Measurements of paper pulp samples with 90% to 92% moisture
concentration at frequencies from 200 Hz to 100 kHz, for an air gap of 4.2 mm.
71
14.2. Data Analysis
Figure 14.3, and Figure 14.4 show the Cole-Cole plots obtained from the measured
data. It is clearly evident from the figures that the real part of the impedance and
admittance was incorrectly measured. This is also reflected in Figure 14.1(b), and Figure
14.2(b). The high frequency points in the Cole-Cole plots show very good resolution with
respect to moisture variation. This can be used for data extraction in the future stages of
data analysis.
As observed in Figure 14.1, and Figure 14.2, there is an offset in the absolute
values of the measured electrical parameters. This is due to the exponential decay in the
electric field intensity along the axis normal to the plane of the electrodes. The presence
of a strong dielectric, the base of the Plexiglas tray, enhances this offset variation.
As a first step towards minimizing this offset, we consider the ratio of the measured
capacitance in the presence of the tray and pulp, to that measured in the presence of only
the tray and air. Figure 14.5 shows the normalized capacitances measured at 7.9 kHz for
the two air gaps. There is still a considerable variation with the change in air gap.
Development of advanced algorithms is needed to reduce this variation.
Figure 14.6 shows the pass line sensitivity of the sensor at 7.9 kHz. The pass line
sensitivity was calculated using (14.1).
P=
Cd1 - Cd2
d1 - d 2
(14.1)
where Cd1,and Cd2 are the normalized capacitances for air gaps d1 and d2.
It can be observed from Figure 14.5 that the rate of change of normalized
capacitance is about 0.0654 for unit change in moisture concentration. It can be seen from
Figure 14.6 that the change in normalized capacitance with change in distance is about
0.2151 per millimeter. Hence, the pass line sensitivity is approximately 3.29% of
moisture per mm.
72
Figure 14.3. Cole-Cole plots from measurements of paper pulp samples with 90% to
92% moisture concentration at frequencies from 200 Hz to 100 kHz, for an air gap of 2.5
mm.
73
Figure 14.4. Cole-Cole plots from measurements of paper pulp samples with 90% to
92% moisture concentration at frequencies from 200 Hz to 100 kHz, for an air gap of 4.2
mm.
74
Figure 14.5. Normalized Capacitances measured at 7.9 kHz for an air gap of 2.5 mm,
and 4.2 mm.
Figure 14.6. Pass line sensitivity of the sensor at 7.9 kHz.
75
Chapter 15. Disturbance factors
The dielectric properties of most materials vary with temperature, which results in a
difference in the measured impedance. The experiments conducted in this paper were
performed in an air-conditioned lab with room temperature maintained around 25°C. The
paper pulp was made by mixing minced paper with water in a blender. The blending
process raises the temperature of the paper pulp. To accommodate this variation, all the
paper samples were cooled to room temperature before the experiments were conducted.
Figure 15.1 shows the controlled environmental chamber that is currently being built. The
temperature and the humidity of chamber can be digitally controlled using the same
computer that is used for data acquisition.
Figure 15.1. Photograph of the controlled environment chamber.
Paper fibers in the pulp tend to coagulate. This leads to non-homogeneous spatial
distribution. Special attention needs to taken to ensure that the distribution is as
homogeneous as possible. Uneven distribution can lead to variations in the results.
The water used to prepare the pulp samples could contain varying degrees of metal
ion concentration. This can lead to erroneous results. It is preferable to use distilled water
76
or water from a consistent source. However, if the concentration of the ions is known, the
effect of metal ions in water can potentially be calibrated out of the system.
The disturbances due to white noise and other similar noises have been cancelled
out by means of averaging.
77
Chapter 16. Variation with Pulp
The study of variation in the sensor output with change in pulp type is a critical step
towards practical on-line implementation of the sensor. The results discussed in this
section are the preliminary results obtained from the first set of experiments conducted to
study this response variation.
The experimental setup and procedure is similar to that explained in Chapter 6.1.
Paper pulp is prepared from three different types of pulp sheets, namely, Doly-HV, OlyHV, and Weycel. The moisture concentration in the pulp is varied from 90% to 96%.
Figure 16.1, Figure 16.2, and Figure 16.3 respectively show the phase, capacitance,
and the conductance measured for the three different types of pulps for frequencies from
200 Hz to 100 kHz.
Figure 16.4 compares the dissipation factor, phase, capacitance and conductance
measured for the three types of pulp at 600 Hz. It is seen that the variations in the
measurements are relatively small, and can be attributed to experimental error to a great
extent. This variation is currently being studied.
78
Figure 16.1. Phase measurements of 3 different types of paper pulp samples with 96% to
90% moisture concentration from 200 Hz to 100 kHz.
79
Figure 16.2. Capacitance measurements of 3 different types of paper pulp samples with
96% to 90% moisture concentration from 200 Hz to 100 kHz.
80
Figure 16.3. Conductance measurements of 3 different types of paper pulp samples with
96% to 90% moisture concentration from 200 Hz to 100 kHz.
81
Figure 16.4. Measurements of 3 different types of paper pulp samples with 96% to 90%
moisture concentration at 600 Hz.
82
Chapter 17. Temperature Variations
The calibration based sensing techniques using FEF sensors are sensitive to
temperature variations. The electrical properties of any material vary with temperature,
and this effect would be more pronounced in paper pulp at the wet end of the paper
machine due to very high water concentration. Thus, to accurately estimate the
concentrations of the constituents of the pulp, we have to accommodate or negate the
variations due to temperature fluctuations. The results discussed in this chapter are
preliminary in nature, and are yet to be incorporated into the estimation algorithms
discussed in Section 6.3, Section 7.3, and Chapter 10.
17.1. Experimental Procedure
The experimental setup used for the experiments reported in this section is similar to
the setup detailed in Chapter 4.
DolyHV pulp sheets were cut into small pieces and were mixed with known quantity
of water in a blender. The resulting paper pulp had 95% moisture by weight. The paper
pulp was allowed to cool down to room temperature, and then measurements were made
A part of this pulp was then placed in a preheated temperature chamber for two hours.
The remainder of the pulp is left outside the chamber for the same amount of time. At the
end of the two-hour period, the pulp samples are measured.
83
17.2. Experimental Results
Figure 17.1 shows the effect of temperature variation on various electrical
parameters of a paper pulp with 95% moisture from 1kHz to 100 kHz. The increase in
conductance with increase in temperature as seen from Figure 17.1 (d) serves as a sanity
check.
Figure 17.1. Normalized measurements showing the effect of temperature variation on
various electrical parameters of a paper pulp with 95% moisture from 1kHz to 100 kHz.
Figure 17.2 shows the effect of temperature variation on various electrical
parameters at 7.9 kHz.
84
The pulp is kept in the temperature chamber for two hours. While this wait is
required for the bulk of the pulp to attain equilibrium temperature, it also leads to settling
of pulp. The effect of pulp settlement is has a comparable effect on the measurements as
compared to that due to temperature variation.
Figure 17.2. Normalized measurements showing the effect of temperature variation on
various electrical parameters of a paper pulp with 95% moisture at 7.9 kHz.
To mitigate the effect of settling, we measure the paper pulp at room temperature
after two hours and use those data points as representative points for settling effects.
These points are shown in Figure 17.1 and Figure 17.2 as data points at 72, 73 and 74 ºF.
85
To compensate for the changes across pulp samples, all the measurements made after
two hours were normalized using the measurements made as soon as the pulp was
prepared.
86
Chapter 18. Future Work
18.1. High Frequency Measurements:
The studies presented here are based on measurements over 3 decades of
frequency. A lot more information can be extracted if higher frequency ranges are used.
One of the possible analysis methods that could be used to study the pulp over a larger
range of frequency is the Cole-Cole plot.
Figure 18.1 shows the Cole-Cole plots for the measurements made with pulp with
fiber concentration ranging from 100% to 10% and water. The Cole-Cole plots divulge
information regarding the dielectric relaxations in the material. Figure 18.2 shows a
sample Cole-Cole plot. The arc in the Cole-Cole plots is the loci of the complex
impedance and the admittance of the material under study. The real axis intercepts of the
arc are used to estimate the empirical parameters of the electrical model of the material.
In most of the materials it is not possible to obtain single arcs such as the ones shown in
Figure 18.2 due to the presence of overlapping arcs of other relaxations.
The arc in Cole-Cole plots can be used to estimate the relaxation time constant of
the relaxation process under study. The relaxation time constant R can be obtained from,
Z - Rinf º Z zarc º ( Rdc - Rinf ) I z
(18.1)
I z º [1+ ( jwt R )y ZC ]- 1
(18.2)
where
where ZC is the fractional exponent, and IZ is the normalized, dimensionless form of
Zzarc.
87
Figure 18.1. Cole-Cole plots from measurements of paper pulp samples with 0% to 90%
moisture concentration at frequencies from 200 Hz to 100 kHz.
88
Figure 18.2. Example Cole-Cole plots
The arcs in Figure 18.1 do not intercept the real axis as the phase change over the
studied frequency range is very small, and the phase does not cross 90. This is due to the
small frequency range used for the measurements. Efforts are currently on to procure a
higher frequency network analyzer to study the dielectric behavior of the paper pulp over
a wider range of frequency (possibly from 1 Hz to over 3 GHz).
18.2. Temperature Variation
Temperature of the pulp can influence the output of the sensor. We intend to
calibrate the sensor to variations in temperature of the pulp as the next immediate step. A
controlled environment chamber (shown in Figure 15.1) has been built in-house and is in
the final stages of trouble shooting and calibration process. Once the chamber is fully
operational, and the effect of temperature variations have been studied, we intend to
include the temperature as one of the input parameters for the estimation algorithms
discussed in Section 6.3, Section 7.3, and Chapter 10.
89
18.3. Improvements in Estimation Algorithms
The parameter selection algorithm suggested in Chapter 10 can be improved to
accommodate the effects due to additives. The algorithm currently uses a linear fit model
to estimate the parameters. Various non-linear, multivariable models can be added to set
of possible fits. Such non-linear and multivariable models have been used in microwave
techniques [25].
Currently only basic electrical parameters and their combinations are used as
probable parameters in the algorithms. We intend to introduce density independent and
near density independent variables to the parameter list.
18.4. Pass line Sensitivity
Detailed pass line sensitivity studies on the sensor will be carried out once the
sensor design has been fully hardened for field installation. The sensor could possibly be
installed on the paper machine at the Department of Forestry, UW, for initial pass line
sensitivity studies.
As shown in Section 14.2, the current pass line sensitivity of the sensor is about
3,29 % of moisture per mm. While, this is a significant change, it should also be noted
that the algorithms used were not meant to minimize the pass line sensitivity. The
sensitivity can be reduced by use of advanced algorithms, multiple wavelength sensor
arrays, and rigid construction.
90
Chapter 19. Conclusions
1. The ability of the sensor to accurately measure the moisture content in pulp over a
wide range of concentration (0% to 97% moisture) has been demonstrated.
2. The effect of various additives such as titanium dioxide, clay and calcium carbonate
have been studied, and the sensor has been shown to estimate the moisture content to
an acceptable accuracy in their presence.
3. Repeatability tests were carried out, and the sensor was found to have a two-sigma
variation of 67.724 fF; which is three orders of magnitude lesser than the measured
capacitance. The estimated moisture concentration showed a peak-to-peak variation
of 3.5527e-013%.
4. Reproducibility tests were carried out, and the sensor was found to have a two-sigma
variation of 0.10917 pF; which is two orders of magnitude lesser than the measured
capacitance. The peak-to-peak variation in the estimated moisture content was found
to be 0.88033 %.
5. A novel parameter estimation algorithm is in the initial stages of development and
preliminary results from it have been reported.
6. The ability of the sensor to measure various additives has been demonstrated. Hence,
in theory, we can indirectly measure the ash content of the pulp by doing a
differential measurement at two different locations on the wire.
7. A consistent drop in measured capacitance can be seen in Figure 11.1. This could be
due to sedimentation. This ability of the sensor can also be explored to study
sedimentation processes in paper pulp.
91
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[29] A. V. Mamishev, "Interdigital Dielectrometry Sensor Design and Parameter
Estimation Algorithms for Nondestructive Materials Evaluation of." Department
of Electrical Engineering and Computer Science, Massachusetts Institute of
Technology, 1999.
[30] A. V. Mamishev, "Interdigital Dielectrometry Sensor Design and Parameter
Estimation Algorithms for Nondestructive Materials Evaluation of." Department
of Electrical Engineering and Computer Science, Massachusetts Institute of
Technology, 1999.
[31] A. V. Mamishev, "Interdigital Dielectrometry Sensor Design and Parameter
Estimation Algorithms for Nondestructive Materials Evaluation of." Department
of Electrical Engineering and Computer Science, Massachusetts Institute of
Technology, 1999.
95
[32] A. V. Mamishev, "Interdigital Dielectrometry Sensor Design and Parameter
Estimation Algorithms for Nondestructive Materials Evaluation of." Department
of Electrical Engineering and Computer Science, Massachusetts Institute of
Technology, 1999.
[33] A. V. Mamishev, "Interdigital Dielectrometry Sensor Design and Parameter
Estimation Algorithms for Nondestructive Materials Evaluation of." Department
of Electrical Engineering and Computer Science, Massachusetts Institute of
Technology, 1999.
[34] M. C. Zaretsky, "Parameter Estimation Using Microdielectrometry with
Application to Transformer Monitoring." Ph.D. Thesis, Department of Electrical
Engineering and Computer Science, Massachusetts Institute of Technology, 1987.
[35] P. A. von, "Application of Interdigital Dielectrometry to Moisture and Double
Layer Measurements in Transformer Insulation." Department of Electrical
Engineering and Computer Science, Massachusetts Institute of Technology, 1993.
[36] A. W. Kraszewski, Microwave Aquametry, IEEE Press, 1996.
[37] P. Debye, Polar Molecules, Chemical Catalog Co., 1929.
[38] J. R. MacDonald, Impedance Spectroscopy : Emphasizing Solid Materials and
Systems, Wiley New York, 1987.
[39] A. V. Mamishev, "Interdigital Dielectrometry Sensor Design and Parameter
Estimation Algorithms for Nondestructive Materials Evaluation of." Department
of Electrical Engineering and Computer Science, Massachusetts Institute of
Technology, 1999.
96
[40] D. Q. M. Craig, Dielectric Analysis of Pharmaceutical Systems, Taylor & Francis
Group, 1995.
[41] S. Simula and K. Niskanen, "Electrical Properties of Viscose-Kraft Fibre
Mixtures," Nordic Pulp and Research Journal, vol. 14, no. 3, pp. 243-246, Sept.
1999.
[42] G. A. Dumont, I. M. Jonsson, M. S. Davies, F. Ordubadi, Y. Fu, K. Natarajan, C.
Lindeborg, and E. M. Heaven, "Estimation Of Moisture Variation In Paper
Machines," IEEE Transactions on Control Systems Technology, vol. 1, no. 2, pp.
101-113, June 1993.
97
Appendix
1. Water Depth
A preliminary experiment was conducted to demonstrate the ability of the sensor to
detect change in depth of a water layer in the tray. The sensor used for this experiment is
different from the sensor used for the other experiments reported in this report. A sensor
with just two electrodes, placed 3 inches apart were used. A 10 V peak-to-peak sinusoidal
voltage drove the electrodes. The results from the experiments are shown in Figure A1.1
and Figure A1.2
700
600
voltage rms (mv)
500
100kHz
400
10kHz
1kHz
300
100Hz
200
100
0
0
1
2
3
4
5
6
7
water depth (mm)
Figure A1.1. Plot of RMS voltage between the sensing electrodes and ground at various
water depths and frequencies.
98
635
y = 6.82x + 590.34
R2 = 0.968
630
625
voltage rms (mv)
100kHz
10kHz
620
Linear (100kHz)
615
610
605
600
0
1
2
3
4
5
6
7
water depth (mm)
Figure A1.2. Near linear relationship was observed between the measured voltage and
water depth.
99
2. PlotExptData.m
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%
%%%%%%
%%%%%
%%%%%%
%%%%%
Plots the basic plots for all the
%%%%%%
%%%%%
experiments
%%%%%%
%%%%%
%%%%%%
%%%%%
%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
clear all; close all;
% Set font size.
set(0, 'DefaultAxesFontSize',13)
set(0, 'DefaultTextFontSize',13)
% Initializing file names.
select = 1;
if select == 1
directname = 'C:\Data\10-24-02\'
%
filename(1,:) = '10';
filename(1,:) = '09';
filename(2,:) = '08';
filename(3,:) = '07';
filename(4,:) = '06';
filename(5,:) = '05';
filename(6,:) = '04';
%
filename(8,:) = '03';
AxisIndex = [9,8,7,6,5,4,3];
labelX = 'Moisture Content (%)';
elseif select == 2
directname = 'C:\Data\02-11-03\'
filename(1,:) = '00';
filename(2,:) = '05';
filename(3,:) = '10';
filename(4,:) = '15';
filename(5,:) = '20';
filename(6,:) = '25';
filename(7,:) = '30';
filename(8,:) = '35';
filename(9,:) = '40';
filename(10,:) = '45';
filename(11,:) = '50';
filename(12,:) = '55';
100
filename(13,:) = '60';
filename(14,:) = '65';
AxisIndex =
[0,0.5,1.0,1.5,2.0,2.5,3.0,3.5,4.0,4.5,5.0,5.5,6.0,6.5];
labelX = 'TiO_2 Content (%)';
elseif select == 3
directname = 'C:\Data\01-23-03\'
filename(1,:) = '00';
filename(2,:) = '05';
filename(3,:) = '10';
filename(4,:) = '20';
filename(5,:) = '30';
filename(6,:) = '40';
AxisIndex = [0,0.5,1.0,2.0,3.0,4.0];
labelX = 'TiO_2 Content (%)';
elseif select == 4
directname = 'C:\Data\03-24-03\'
filename(1,:) = '100';
filename(2,:) = '090';
filename(3,:) = '080';
filename(4,:) = '070';
filename(5,:) = '060';
filename(6,:) = '050';
filename(7,:) = '040';
filename(8,:) = '375';
filename(9,:) = '035';
filename(10,:) = '325';
filename(11,:) = '030';
filename(12,:) = '275';
filename(13,:) = '025';
filename(14,:) = '225';
filename(15,:) = '020';
filename(16,:) = '175';
filename(17,:) = '015';
filename(18,:) = '125';
filename(19,:) = '010';
AxisIndex =
[100,90,80,70,60,50,40,37.5,35,32.5,30,27.5,25,22.5,20,17.5,15,12.5,10]
;
labelX = 'Fiber Content (%)';
elseif select == 5
directname = 'C:\Data\04-07-03\sheet\'
filename(1,:) = '100';
filename(2,:) = '090';
filename(3,:) = '080';
filename(4,:) = '070';
filename(5,:) = '060';
filename(6,:) = '050';
filename(7,:) = '040';
filename(8,:) = '030';
filename(9,:) = '020';
101
filename(10,:) = '018';
filename(11,:) = '016';
filename(12,:) = '015';
filename(13,:) = '014';
filename(14,:) = '013';
filename(15,:) = '012';
filename(16,:) = '011';
filename(17,:) = '010';
AxisIndex = [100,90,80,70,60,50,40,30,20,18,16,15,14,13,12,11,10];
labelX = 'Fiber Content (%)';
elseif select == 6
directname = 'C:\Data\04-07-03\pulp\'
filename(1,:) = '10';
filename(2,:) = '09';
filename(3,:) = '08';
filename(4,:) = '07';
AxisIndex = [10,09,08,07];
labelX = 'Fiber Content (%)';
elseif select == 7
directname = 'C:\Data\04-07-03\TiO2\'
filename(1,:) = '000';
filename(2,:) = '001';
filename(3,:) = '002';
filename(4,:) = '003';
filename(5,:) = '004';
filename(6,:) = '005';
filename(7,:) = '006';
filename(8,:) = '007';
AxisIndex = [0,1,2,3,4,5,6,7];
labelX = 'TiO_2 Content (%)';
elseif select == 8 %%%%%%%%%%%%%% Clay
directname = 'C:\Data\06-16-03\'
filename(1,:) = '00';
filename(2,:) = '01';
filename(3,:) = '02';
filename(4,:) = '03';
filename(5,:) = '04';
filename(6,:) = '05';
AxisIndex = [0,1,2,3,4,5];
labelX = 'Clay Content (%)';
elseif select == 9 %%%%%%%%%%%% CaCO3
directname = 'C:\Data\06-23-03\CaCO3\'
filename(1,:) = '00';
filename(2,:) = '05';
filename(3,:) = '10';
filename(4,:) = '15';
filename(5,:) = '20';
filename(6,:) = '25';
AxisIndex = [0,0.5,1,1.5,2,2.5];
102
labelX = 'CaCO_3 Content (%)';
end
start = 10;
stop = 25;
% <-- the starting index.
% "Time (days)" "Frequency" "Voltage" "Current" "Phase" "Resistance"
"Capacitance"
time_ind = 1;
fre_ind = 2;
volt_ind = 3;
curr_ind = 4;
phase_ind = 5;
resis_ind = 6;
cap_ind = 7;
% Pre-Processing Each File
[no_of_files, temp] = size(filename);
for Count = 1:no_of_files
tempdata = load([directname filename(Count,:) '.txt']);
fre1 = tempdata(:, fre_ind);
volt1 = tempdata(:, volt_ind);
curr1 = tempdata(:, curr_ind);
phase1 = tempdata(:, phase_ind);
resis1 = tempdata(:, resis_ind);
cap1 = tempdata(:, cap_ind);
%
% convert the unit of capacitance from (F) to PF.
cap1 = cap1*(10^12);
% Segment into Individual Sweeps
fre1_seg = buffer(fre1, 25);
volt1_seg = buffer(volt1, 25);
curr1_seg = buffer(curr1, 25);
phase1_seg = buffer(phase1, 25);
resis1_seg = buffer(resis1, 25);
cap1_seg = buffer(cap1, 25);
% Remove the low frequency data
fre1_seg = fre1_seg(start:stop, :);
volt1_seg = volt1_seg(start:stop,:);
curr1_seg = curr1_seg(start:stop, :);
phase1_seg = phase1_seg(start:stop, :);
resis1_seg = resis1_seg(start:stop, :);
cap1_seg = cap1_seg(start:stop, :);
% Averaging the data across Sweeps
fre1_avrg = mean(fre1_seg, 2);
volt1_avrg = mean(volt1_seg, 2);
curr1_avrg = mean(curr1_seg, 2);
103
phase1_avrg = mean(phase1_seg, 2);
resis1_avrg = mean(resis1_seg, 2);
cap1_avrg = mean(cap1_seg, 2);
cap_std = std(cap1_seg,0, 2);
%Convert F to pF
cap1_avrg = cap1_avrg .* 1e12;
% Obtain Conductance from Resistance
cndut1_avrg = 1./resis1_avrg;
% Calculating ratio of Conductance and Angular Frequency
ratio_of_cndut_and_freq_avrg = cndut1_avrg./(fre1_avrg);
% Calculating tandelta
tandelta = 1./ ((2*pi*fre1_avrg) .* (cap1_avrg) .*
(resis1_avrg));
% Calculating Capacitance / Conductance
ratio_of_cap_to_cond = cap1_avrg ./ cndut1_avrg;
% Calculating Admittance
Yreal = cndut1_avrg;
w = 2*pi*fre1_avrg;
Yimg = w .* cap1_avrg;
Y = complex(Yreal,Yimg);
Ymag = abs(Y);
% Calculating Admittance
Z = 1 ./ Y;
Zimg = imag(Z);
Zreal = real(Z);
Zmag = abs(Z);
% Creating AxisMatrix
[rows,cols] = size(cap1_avrg);
for n = 1:rows
for m = 1:cols
AxisMatrix(n,m) = AxisIndex(Count);
end
end
%
% Scaled Capacitance
mincap(Count) = min(cap1_avrg);
normalised_cap = cap1_avrg - mincap(Count);
normalised_cap = normalised_cap ./ max(normalised_cap);
dynamicrange(Count) = max(normalised_cap);
% Scaled Conductance
mincond(Count) = min(ratio_of_cndut_and_freq_avrg);
normalised_condbyw = ratio_of_cndut_and_freq_avrg mincond(Count) ;
104
%
normalised_condbyw = normalised_condbyw ./
max(normalised_condbyw);
% Scaled Phase
minphase(Count) = min(phase1_avrg);
normalised_phase = phase1_avrg ./ minphase(Count) ;
% Ratio of Capacitance to Scaled Conductance
Ratio_cap_to_scaled_Cond = cap1_avrg ./ normalised_condbyw;
% Calculating the Min and Max Capacitance
cap_min = cap1_avrg - (1e12 .* min(cap1_seg,[],2));
cap_max = (1e12 .* max(cap1_seg,[],2)) - cap1_avrg;
Moisture_Content = 100 - AxisMatrix;
% Store the Processed data in a 3-D Matrix
Data(Count,:,1) = fre1_avrg;
Data(Count,:,2) = volt1_avrg;
Data(Count,:,3) = curr1_avrg;
Data(Count,:,4) = phase1_avrg;
Data(Count,:,5) = resis1_avrg;
Data(Count,:,6) = cap1_avrg;
Data(Count,:,7) = cndut1_avrg;
Data(Count,:,8) = ratio_of_cndut_and_freq_avrg;
Data(Count,:,9) = tandelta;
Data(Count,:,10) = AxisMatrix;
Data(Count,:,11) = Yimg;
Data(Count,:,12) = Yreal;
Data(Count,:,13) = Ymag;
Data(Count,:,14) = -1 .* Zimg;
Data(Count,:,15) = Zreal;
Data(Count,:,16) = Zmag;
Data(Count,:,17) = cap_std .* 1e12;
Data(Count,:,18) = cap_min;
Data(Count,:,19) = cap_max;
Data(Count,:,20) = Moisture_Content;
end
% Fitting Data and Ploting them
figurenumber = 1;
figure(figurenumber);
figurenumber = figurenumber + 1;
subplot(2,2,1);
figurenumber = surfplotsfunc(Data,1,20,13,'Frequency
(Hz)',labelX,'Admittance (S)','Admittance Plot',figurenumber);
subplot(2,2,2);
figurenumber = surfplotsfunc(Data,1,20,4,'Frequency (Hz)',labelX,'Phase
(deg)','Phase Plot',figurenumber);
subplot(2,2,3);
105
figurenumber = surfplotsfunc(Data,1,20,6,'Frequency
(Hz)',labelX,'Capacitance (F)','Capacitance Plot',figurenumber);
subplot(2,2,4);
figurenumber = surfplotsfunc(Data,1,20,8,'Frequency
(Hz)',labelX,'Conductance (mho/Hz)','Conductance Plot',figurenumber);
figure(figurenumber);
figurenumber = figurenumber + 1;
subplot(1,2,1);
figurenumber = plotallfiles(Data,filename,12,11,'Re[Y], [ohms]','Im[Y],
[ohms]','Capacitance Plot',figurenumber);
subplot(1,2,2);
figurenumber = plotallfiles(Data,filename,15,14,'Re[Z], [ohms]','Im[Z], [ohms]','Capacitance Plot',figurenumber);
FrequencyToBeUsed = [2,4,6,8,10,12];
figurenumber = plotacrossfiles(Data,FrequencyToBeUsed,6,10,'Capacitance
(pF)',labelX,'Capacitance Plot',figurenumber);
figurenumber = plotacrossfiles(Data,FrequencyToBeUsed,8,10,'Conductance
(mho/Hz)',labelX,'Conductance Plot',figurenumber);
figure(figurenumber);
figurenumber = figurenumber + 1;
FrequencyToBeUsed = 6;
errorbar( Data(:,FrequencyToBeUsed,20), Data(:,FrequencyToBeUsed,6),
Data(:,FrequencyToBeUsed,17)/2, Data(:,FrequencyToBeUsed,17)/2);
grid on;
xlabel('Moisture Concentration (%)');
ylabel('Capacitance (pF)');
axis tight;
106
3. SurfPlotFunc.m
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%
%%%%%%
%%%%%
%%%%%%
%%%%%
Function to prepare 3D Plots
%%%%%%
%%%%%
%%%%%%
%%%%%
%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [figurenumber] = surfplotsfunc
(Data,indexX,indexY,indexZ,labelX,labelY,labelZ,Title,figurenumber)
% Ploting Phase, Axisindex and Frequency in 3D
% figure(figurenumber)
colormap([0 0 0]);
mesh (Data(:,:,indexX),Data(:,:,indexY), Data(:,:,indexZ));
xlabel(labelX);
ylabel(labelY);
zlabel(labelZ);
% title(Title);
axis tight;
orient landscape;
set(gca, 'XScale', 'log');
figurenumber = figurenumber;
107
4. PlotAllFiles.m
function [figurenumber] =
plotallfiles(Data,filename,indexX,indexY,labelX,labelY,titlestr,figuren
umber)
% Ploting IndexY Vs IndexX for all the files
attrib(1,:) = 'ko- ';
attrib(2,:) = 'k.: ';
attrib(3,:) = 'kx-.';
attrib(4,:) = 'k+--';
attrib(5,:) = 'k*- ';
attrib(6,:) = 'ks: ';
attrib(7,:) = 'kd-.';
attrib(8,:) = 'kv--';
attrib(9,:) = 'kp- ';
attrib(10,:) = 'kh: ';
attrib(11,:) = 'k^-.';
attrib(12,:) = 'k<--';
attrib(13,:) = 'k>- ';
attrib(14,:) = 'k.: ';
attrib(15,:) = 'ko-.';
attrib(16,:) = 'kx--';
attrib(17,:) = 'k+- ';
attrib(18,:) = 'k*: ';
attrib(19,:) = 'kd-.';
attrib(20,:) = 'kd--';
attrib(21,:) = 'kv- ';
attrib(22,:) = 'kp: ';
attrib(23,:) = 'kh-.';
attrib(24,:) = 'k^--';
[no_of_files,no_of_freq,no_of_parameters] = size(Data)
% figure(figurenumber)
plot(Data(1,:,indexX), Data(1,:,indexY),attrib(1,:));
hold on
for Count = 2:no_of_files
plot(Data(Count,:,indexX), Data(Count,:,indexY),attrib(Count,:) );
end
hold off
xlabel(labelX);
ylabel(labelY);
% title(titlestr);
legend(filename);
axis square;
grid on;
orient landscape;
% set(gca, 'XScale', 'log');
% figurenumber = figurenumber + 1;
108
5. PlotAcrossFiles.m
function [figurenumber] =
plotacrossfiles(Data,FrequencyIndex,indexX,indexY,labelX,labelY,titlest
r,figurenumber)
% Ploting IndexY Vs IndexX for all the files
attrib(1,:) = 'ko- ';
attrib(2,:) = 'k.: ';
attrib(3,:) = 'kx-.';
attrib(4,:) = 'k+--';
attrib(5,:) = 'k*- ';
attrib(6,:) = 'ks: ';
attrib(7,:) = 'kd-.';
attrib(8,:) = 'kv--';
attrib(9,:) = 'kp- ';
attrib(10,:) = 'kh: ';
attrib(11,:) = 'k^-.';
attrib(12,:) = 'k<--';
attrib(13,:) = 'k>- ';
attrib(14,:) = 'k.: ';
attrib(15,:) = 'ko-.';
attrib(16,:) = 'kx--';
attrib(17,:) = 'k+- ';
attrib(18,:) = 'k*: ';
attrib(19,:) = 'kd-.';
attrib(20,:) = 'kd--';
attrib(21,:) = 'kv- ';
attrib(22,:) = 'kp: ';
attrib(23,:) = 'kh-.';
attrib(24,:) = 'k^--';
[no_of_files,no_of_freq,no_of_parameters] = size(Data)
[Temp,no_of_plots] = size(FrequencyIndex);
for Count = 1:no_of_plots
legendmatrix(Count,:) =
sprintf('%1.3e',Data(1,FrequencyIndex(Count),1));
end
figure(figurenumber)
plot(Data(:,FrequencyIndex(1),indexX),
Data(:,FrequencyIndex(1),indexY),attrib(1,:));
hold on;
for Count = 2:no_of_plots
plot(Data(:,FrequencyIndex(Count),indexX),
Data(:,FrequencyIndex(Count),indexY),attrib(Count,:));
legendmatrix(Count,:) =
sprintf('%1.3e',Data(1,FrequencyIndex(Count),1));
end
hold off;
109
xlabel(labelX);
ylabel(labelY);
title(titlestr);
legend(legendmatrix);
axis tight;
grid on;
orient landscape;
figurenumber = figurenumber + 1;
110
6. TitaniumDioxideCalculations.m
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%
%%%%%%
%%%%%
%%%%%%
%%%%%
This program computes the moisture
%%%%%%
%%%%%
concentration in presence of TiO2
%%%%%%
%%%%%
%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
clear all; close all;
% Set font size.
set(0, 'DefaultAxesFontSize',20)
set(0, 'DefaultTextFontSize',20)
% Initializing file names.
select = 2;
if select == 1
directname = 'C:\Data\01-23-03\'
filename(1,:) = '00';
filename(2,:) = '05';
filename(3,:) = '10';
filename(4,:) = '20';
filename(5,:) = '30';
filename(6,:) = '40';
AxisIndex = [0,0.5,1.0,2.0,3.0,4.0];
PaperConc = [10,9.5,9,8,7,6];
MoistureConc = [90,90,90,90,90,90];
elseif select == 2
directname = 'C:\Data\04-07-03\TiO2\'
filename(1,:) = '000';
filename(2,:) = '001';
filename(3,:) = '002';
filename(4,:) = '003';
filename(5,:) = '004';
filename(6,:) = '005';
filename(7,:) = '006';
filename(8,:) = '007';
AxisIndex = [0,1,2,3,4,5,6,7];
PaperConc = [7,6.93,6.86,6.79,6.72,6.65,6.58,6.51];
MoistureConc = [93,92.07,91.14,90.21,89.28,88.35,87.42,86.49];
end
111
start = 11;
stop = 25;
% <-- the starting index.
% "Time (days)" "Frequency" "Voltage" "Current" "Phase" "Resistance"
"Capacitance"
time_ind = 1;
fre_ind = 2;
volt_ind = 3;
curr_ind = 4;
phase_ind = 5;
resis_ind = 6;
cap_ind = 7;
% Pre-Processing Each File
[no_of_files, temp] = size(filename);
for Count = 1:no_of_files
tempdata = load([directname filename(Count,:) '.txt']);
fre1 = tempdata(:, fre_ind);
volt1 = tempdata(:, volt_ind);
curr1 = tempdata(:, curr_ind);
phase1 = tempdata(:, phase_ind);
resis1 = tempdata(:, resis_ind);
cap1 = tempdata(:, cap_ind);
% convert the unit of capacitance from (F) to PF.
cap1 = cap1*(10^12);
% Segment into Individual Sweeps
fre1_seg = buffer(fre1, 25);
volt1_seg = buffer(volt1, 25);
curr1_seg = buffer(curr1, 25);
phase1_seg = buffer(phase1, 25);
resis1_seg = buffer(resis1, 25);
cap1_seg = buffer(cap1, 25);
% Remove the low frequency data
fre1_seg = fre1_seg(start:stop, :);
volt1_seg = volt1_seg(start:stop,:);
curr1_seg = curr1_seg(start:stop, :);
phase1_seg = phase1_seg(start:stop, :);
resis1_seg = resis1_seg(start:stop, :);
cap1_seg = cap1_seg(start:stop, :);
% Averaging the data across Sweeps
fre1_avrg = mean(fre1_seg, 2);
volt1_avrg = mean(volt1_seg, 2);
curr1_avrg = mean(curr1_seg, 2);
phase1_avrg = mean(phase1_seg, 2);
resis1_avrg = mean(resis1_seg, 2);
cap1_avrg = mean(cap1_seg, 2);
% Obtain Conductance from Resistance
112
cndut1_avrg = 1e12 ./resis1_avrg; %coverting to pico
% Calculating ratio of Conductance and Angular Frequency
ratio_of_cndut_and_freq_avrg = cndut1_avrg./(2*pi*fre1_avrg);
% Calculating tandelta
tandelta = 1./ ((2*pi*fre1_avrg) .* (cap1_avrg) .*
(resis1_avrg));
% Calculating Capacitance / Conductance
ratio_of_cap_to_cond = cap1_avrg ./ cndut1_avrg;
% Calculatiing Imaginary part of Admittance
Yimaginary = cap1_avrg .* (2*pi*fre1_avrg);
% Creating AxisMatrix
[rows,cols] = size(cap1_avrg);
for n = 1:rows
for m = 1:cols
AxisMatrix(n,m) = AxisIndex(Count);
end
end
%
% Scaled Capacitance
mincap(Count) = min(cap1_avrg);
normalised_cap = cap1_avrg - mincap(Count);
normalised_cap = normalised_cap ./ max(normalised_cap);
dynamicrange(Count) = max(normalised_cap);
% Scaled Conductance
mincond(Count) = min(ratio_of_cndut_and_freq_avrg);
normalised_condbyw = ratio_of_cndut_and_freq_avrg mincond(Count) ;
%
normalised_condbyw = normalised_condbyw ./
max(normalised_condbyw);
% Scaled Phase
minphase(Count) = min(phase1_avrg);
normalised_phase = phase1_avrg ./ minphase(Count) ;
% Ratio of Capacitance to Scaled Conductance
Ratio_cap_to_scaled_Cond = cap1_avrg ./ normalised_condbyw;
% Store the Processed data in a 3-D Matrix
Data(Count,:,1) = fre1_avrg;
Data(Count,:,2) = volt1_avrg;
Data(Count,:,3) = curr1_avrg;
Data(Count,:,4) = phase1_avrg;
Data(Count,:,5) = resis1_avrg;
Data(Count,:,6) = cap1_avrg;
Data(Count,:,7) = cndut1_avrg;
Data(Count,:,8) = ratio_of_cndut_and_freq_avrg;
Data(Count,:,9) = Yimaginary;
113
Data(Count,:,10)
Data(Count,:,11)
Data(Count,:,12)
Data(Count,:,13)
Data(Count,:,14)
Data(Count,:,15)
Data(Count,:,16)
=
=
=
=
=
=
=
AxisMatrix;
tandelta;
ratio_of_cap_to_cond;
normalised_cap;
normalised_condbyw;
normalised_phase;
Ratio_cap_to_scaled_Cond;
end
% Fitting Data and Ploting them
figurenumber = 1;
var1 = 6;
var2 = 8;
freq = 1;
[error,constants] =
TwoVariableLinearFitFunc(Data(2:no_of_files,freq,var1),(Data(2:no_of_fi
les,freq,var2)),Data(2:no_of_files,freq,10));
y1 =
TwoVariableLinearEvalFunc(constants,Data(1:no_of_files,freq,var1),Data(
1:no_of_files,freq,var2));
if error == 1
y1 = y1 .* 0;
end
figure(figurenumber)
figurenumber = figurenumber + 1;
plot(Data(1:no_of_files,1,10),y1,'b*');
hold on; plot(Data(:,1,10),Data(:,1,10),'ro'); hold off;
grid on;
title('Tio2 Concentration'); xlabel('Actual (%)'); ylabel('Estimate
(%)');
[error,constants] =
TwoVariableLinearFitFunc(Data(2:no_of_files,freq,var1),(Data(2:no_of_fi
les,freq,var2)),MoistureConc(2:no_of_files)');
y1 =
TwoVariableLinearEvalFunc(constants,Data(1:no_of_files,freq,var1),Data(
1:no_of_files,freq,var2));
if error == 1
y1 = y1 .* 0;
end
figure(figurenumber)
figurenumber = figurenumber + 1;
plot(MoistureConc,y1,'b*');
hold on; plot(MoistureConc,MoistureConc,'ro'); hold off;
grid on;
title('Moisture Concentration'); xlabel('Actual (%)'); ylabel('Estimate
(%)');
[error,constants] =
TwoVariableLinearFitFunc(Data(2:no_of_files,freq,var1),(Data(2:no_of_fi
les,freq,var2)),PaperConc(2:no_of_files)');
114
y1 =
TwoVariableLinearEvalFunc(constants,Data(1:no_of_files,freq,var1),Data(
1:no_of_files,freq,var2));
if error == 1
y1 = y1 .* 0;
end
figure(figurenumber)
figurenumber = figurenumber + 1;
plot(PaperConc,y1,'bo');
hold on; plot(PaperConc,PaperConc,'ro'); hold off;
grid on;
title('Paper Concentration'); xlabel('Actual (%)'); ylabel('Estimate
(%)');
% figurenumber = surfplotsfunc(Data,1,10,8,'Frequency (Hz)','Fiber
Concentration (%)','Conductance (mho/rad)','Conductance
Plot',figurenumber);
% figurenumber = surfplotsfunc(Data,1,10,6,'Frequency (Hz)','Fiber
Concentration ()','Capacitance (pF)','Capacitance Plot',figurenumber);
% figurenumber = surfplotsfunc(Data,1,10,11,'Frequency (Hz)','Fiber
Concentration (%)','Dissipation Factor','Dissipation Factor
Plot',figurenumber);
% figurenumber = surfplotsfunc(Data,1,10,4,'Frequency (Hz)','Fiber
Concentration (%)','Phase (deg)','Phase Plot',figurenumber);
% figurenumber = surfplotsfunc(Data,1,10,12,'Frequency (Hz)','Fiber
Concentration (%)','Capacitance / Conductance
(pF/S)','Capacitance/Conducatance Plot',figurenumber);
% figurenumber = plotallfiles(Data,filename,1,6,'Frequency, f
(Hz)','Capacitance, C (pF)','Capacitance Plot',figurenumber);
% figurenumber = plotallfiles(Data,filename,1,8,'Frequency, f
(Hz)','Conductance, S (mho/rad)','Conductance Plot',figurenumber);
% figurenumber = plotallfiles(Data,filename,1,4,'Frequency (Hz)','Phase
(Deg)','Phase Plot',figurenumber);
% [temp_matrix,error,figurenumber] =
datafittingfunc(Data,7,AxisIndex,figurenumber,'Ratio of
Conductances','Percentage (%)',' ',5);
% [temp_matrix,error,figurenumber] =
datafittingfunc(Data,6,AxisIndex,figurenumber,'Ratio of
Capacitance','Percentage (%)',' ',5);%
% [temp_matrix,error,figurenumber] =
datafittingfunc(Data,11,AxisIndex,figurenumber,'Tan delta','Percentage
(%)',' ',5);
% [temp_matrix,error,figurenumber] =
datafittingfunc(Data,4,AxisIndex,figurenumber,'Phase (deg)','Percentage
(%)',' ',5);
% [temp_matrix,error,figurenumber] =
datafittingfunc(Data,12,AxisIndex,figurenumber,'Ratio of Capacitance to
Condunctance (pf/S)','Percentage (%)',' ',5);
% FrequencyToBeUsed = [1];
% figurenumber = plotacrossfiles(Data,FrequencyToBeUsed,10,6,'Paper
Content (%)','Capacitance (pf)','Capacitance Plot',figurenumber);
115
% figurenumber = plotacrossfiles(Data,FrequencyToBeUsed,10,8,'Paper
Content (%)','Conductance (mho/rad)','Conductance Plot',figurenumber);
% figurenumber = plotacrossfiles(Data,FrequencyToBeUsed,10,4,'Paper
Content (%)','Tan Delta','Tan delta Plot',figurenumber);
% figurenumber = linearfitfunc(Data,10,5,figurenumber,'Actual Fiber
Content(%)','Estimated Fiber Content(%)','Fit Using Capacitance');
% figurenumber = linearfitfunc(Data,10,4,figurenumber,'Actual Fiber
Content(%)','Estimated Fiber Content(%)','Fit Using Scaled Cond');
% figurenumber = linearfitfunc(Data,10,15,figurenumber,'Actual Fiber
Content(%)','Estimated Fiber Content(%)','Fit Using Phase');
116
7. ClayCalculations.m
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%
%%%%%%
%%%%%
%%%%%%
%%%%%
This program computes the moisture
%%%%%%
%%%%%
concentration in presence of Clay
%%%%%%
%%%%%
%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
clear all; close all;
% Set font size.
set(0, 'DefaultAxesFontSize',20)
set(0, 'DefaultTextFontSize',20)
% Initializing file names.
select = 1;
if select == 1
directname = 'C:\Data\06-16-03\'
filename(1,:) = '00';
filename(2,:) = '01';
filename(3,:) = '02';
filename(4,:) = '03';
filename(5,:) = '04';
filename(6,:) = '05';
AxisIndex = [0.0,1.0,2.0,3.0,4.0,5.0];
PaperConc = [10,9,8,7,6,5];
MoistureConc = [90,90,90,90,90,90];
end
start = 11;
stop = 25;
% <-- the starting index.
% "Time (days)" "Frequency" "Voltage" "Current" "Phase" "Resistance"
"Capacitance"
time_ind = 1;
fre_ind = 2;
volt_ind = 3;
curr_ind = 4;
phase_ind = 5;
117
resis_ind = 6;
cap_ind = 7;
% Pre-Processing Each File
[no_of_files, temp] = size(filename);
for Count = 1:no_of_files
tempdata = load([directname filename(Count,:) '.txt']);
fre1 = tempdata(:, fre_ind);
volt1 = tempdata(:, volt_ind);
curr1 = tempdata(:, curr_ind);
phase1 = tempdata(:, phase_ind);
resis1 = tempdata(:, resis_ind);
cap1 = tempdata(:, cap_ind);
% convert the unit of capacitance from (F) to PF.
cap1 = cap1*(10^12);
% Segment into Individual Sweeps
fre1_seg = buffer(fre1, 25);
volt1_seg = buffer(volt1, 25);
curr1_seg = buffer(curr1, 25);
phase1_seg = buffer(phase1, 25);
resis1_seg = buffer(resis1, 25);
cap1_seg = buffer(cap1, 25);
% Remove the low frequency data
fre1_seg = fre1_seg(start:stop, :);
volt1_seg = volt1_seg(start:stop,:);
curr1_seg = curr1_seg(start:stop, :);
phase1_seg = phase1_seg(start:stop, :);
resis1_seg = resis1_seg(start:stop, :);
cap1_seg = cap1_seg(start:stop, :);
% Averaging the data across Sweeps
fre1_avrg = mean(fre1_seg, 2);
volt1_avrg = mean(volt1_seg, 2);
curr1_avrg = mean(curr1_seg, 2);
phase1_avrg = mean(phase1_seg, 2);
resis1_avrg = mean(resis1_seg, 2);
cap1_avrg = mean(cap1_seg, 2);
% Obtain Conductance from Resistance
cndut1_avrg = 1e12 ./resis1_avrg; %coverting to pico
% Calculating ratio of Conductance and Angular Frequency
ratio_of_cndut_and_freq_avrg = cndut1_avrg./(2*pi*fre1_avrg);
% Calculating tandelta
tandelta = 1./ ((2*pi*fre1_avrg) .* (cap1_avrg) .*
(resis1_avrg));
% Calculating Capacitance / Conductance
118
ratio_of_cap_to_cond = cap1_avrg ./ cndut1_avrg;
% Calculatiing Imaginary part of Admittance
Yimaginary = cap1_avrg .* (2*pi*fre1_avrg);
% Creating AxisMatrix
[rows,cols] = size(cap1_avrg);
for n = 1:rows
for m = 1:cols
AxisMatrix(n,m) = AxisIndex(Count);
end
end
%
% Scaled Capacitance
mincap(Count) = min(cap1_avrg);
normalised_cap = cap1_avrg - mincap(Count);
normalised_cap = normalised_cap ./ max(normalised_cap);
dynamicrange(Count) = max(normalised_cap);
% Scaled Conductance
mincond(Count) = min(ratio_of_cndut_and_freq_avrg);
normalised_condbyw = ratio_of_cndut_and_freq_avrg mincond(Count) ;
%
normalised_condbyw = normalised_condbyw ./
max(normalised_condbyw);
% Scaled Phase
minphase(Count) = min(phase1_avrg);
normalised_phase = phase1_avrg ./ minphase(Count) ;
% Ratio of Capacitance to Scaled Conductance
Ratio_cap_to_scaled_Cond = cap1_avrg ./ normalised_condbyw;
% Store the Processed data in a 3-D Matrix
Data(Count,:,1) = fre1_avrg;
Data(Count,:,2) = volt1_avrg;
Data(Count,:,3) = curr1_avrg;
Data(Count,:,4) = phase1_avrg;
Data(Count,:,5) = resis1_avrg;
Data(Count,:,6) = cap1_avrg;
Data(Count,:,7) = cndut1_avrg;
Data(Count,:,8) = ratio_of_cndut_and_freq_avrg;
Data(Count,:,9) = Yimaginary;
Data(Count,:,10) = AxisMatrix;
Data(Count,:,11) = tandelta;
Data(Count,:,12) = ratio_of_cap_to_cond;
Data(Count,:,13) = normalised_cap;
Data(Count,:,14) = normalised_condbyw;
Data(Count,:,15) = normalised_phase;
Data(Count,:,16) = Ratio_cap_to_scaled_Cond;
end
% Fitting Data and Ploting them
119
figurenumber = 1;
var1 = 4;
var2 = 8;
freq = 8;
[error,constants] =
TwoVariableLinearFitFunc(Data(2:no_of_files,freq,var1),(Data(2:no_of_fi
les,freq,var2)),Data(2:no_of_files,freq,10));
y1 =
TwoVariableLinearEvalFunc(constants,Data(1:no_of_files,freq,var1),Data(
1:no_of_files,freq,var2));
if error == 1
y1 = y1 .* 0;
end
figure(figurenumber)
figurenumber = figurenumber + 1;
plot(Data(1:no_of_files,1,10),y1,'k*');
hold on; plot(Data(:,1,10),Data(:,1,10),'ko-'); hold off;
grid on;
title('Clay Concentration'); xlabel('Actual Clay Concentration (%)');
ylabel('Estimated Clay Concentration (%)');
[error,constants] =
TwoVariableLinearFitFunc(Data(2:no_of_files,freq,var1),(Data(2:no_of_fi
les,freq,var2)),MoistureConc(2:no_of_files)');
y1 =
TwoVariableLinearEvalFunc(constants,Data(1:no_of_files,freq,var1),Data(
1:no_of_files,freq,var2));
if error == 1
y1 = y1 .* 0;
end
figure(figurenumber)
figurenumber = figurenumber + 1;
plot(MoistureConc,y1,'k*');
hold on; plot(MoistureConc,MoistureConc,'ro'); hold off;
grid on;
title('Moisture Concentration'); xlabel('Actual Mositure Concentration
(%)'); ylabel('Estimated Moisture Concentration (%)');
[error,constants] =
TwoVariableLinearFitFunc(Data(2:no_of_files,freq,var1),(Data(2:no_of_fi
les,freq,var2)),PaperConc(2:no_of_files)');
y1 =
TwoVariableLinearEvalFunc(constants,Data(1:no_of_files,freq,var1),Data(
1:no_of_files,freq,var2));
if error == 1
y1 = y1 .* 0;
end
figure(figurenumber)
figurenumber = figurenumber + 1;
plot(PaperConc,y1,'ko');
hold on; plot(PaperConc,PaperConc,'ro'); hold off;
grid on;
120
title('Paper Concentration'); xlabel('Actual Fiber Concentration (%)');
ylabel('Estimated Fiber Concentration (%)');
121
8. CalciumCarbonateCalculations.m
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%
%%%%%%
%%%%%
%%%%%%
%%%%%
This program computes the moisture
%%%%%%
%%%%%
concentration in presence of CaCO3
%%%%%%
%%%%%
%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
clear all; close all;
% Set font size.
set(0, 'DefaultAxesFontSize',20)
set(0, 'DefaultTextFontSize',20)
% Initializing file names.
select = 1;
if select == 1
directname = 'C:\Data\06-23-03\CaCO3\'
filename(1,:) = '00';
filename(2,:) = '05';
filename(3,:) = '10';
filename(4,:) = '15';
filename(5,:) = '20';
filename(6,:) = '25';
AxisIndex = [0.0,0.5,1.0,1.5,2.0,2.5];
PaperConc = [10,9.5,9,8.5,8,7.5];
MoistureConc = [90,90,90,90,90,90];
end
start = 11;
stop = 25;
% <-- the starting index.
% "Time (days)" "Frequency" "Voltage" "Current" "Phase" "Resistance"
"Capacitance"
time_ind = 1;
fre_ind = 2;
volt_ind = 3;
curr_ind = 4;
phase_ind = 5;
122
resis_ind = 6;
cap_ind = 7;
% Pre-Processing Each File
[no_of_files, temp] = size(filename);
for Count = 1:no_of_files
tempdata = load([directname filename(Count,:) '.txt']);
fre1 = tempdata(:, fre_ind);
volt1 = tempdata(:, volt_ind);
curr1 = tempdata(:, curr_ind);
phase1 = tempdata(:, phase_ind);
resis1 = tempdata(:, resis_ind);
cap1 = tempdata(:, cap_ind);
% convert the unit of capacitance from (F) to PF.
cap1 = cap1*(10^12);
% Segment into Individual Sweeps
fre1_seg = buffer(fre1, 25);
volt1_seg = buffer(volt1, 25);
curr1_seg = buffer(curr1, 25);
phase1_seg = buffer(phase1, 25);
resis1_seg = buffer(resis1, 25);
cap1_seg = buffer(cap1, 25);
% Remove the low frequency data
fre1_seg = fre1_seg(start:stop, :);
volt1_seg = volt1_seg(start:stop,:);
curr1_seg = curr1_seg(start:stop, :);
phase1_seg = phase1_seg(start:stop, :);
resis1_seg = resis1_seg(start:stop, :);
cap1_seg = cap1_seg(start:stop, :);
% Averaging the data across Sweeps
fre1_avrg = mean(fre1_seg, 2);
volt1_avrg = mean(volt1_seg, 2);
curr1_avrg = mean(curr1_seg, 2);
phase1_avrg = mean(phase1_seg, 2);
resis1_avrg = mean(resis1_seg, 2);
cap1_avrg = mean(cap1_seg, 2);
% Obtain Conductance from Resistance
cndut1_avrg = 1e12 ./resis1_avrg; %coverting to pico
% Calculating ratio of Conductance and Angular Frequency
ratio_of_cndut_and_freq_avrg = cndut1_avrg./(2*pi*fre1_avrg);
% Calculating tandelta
tandelta = 1./ ((2*pi*fre1_avrg) .* (cap1_avrg) .*
(resis1_avrg));
% Calculating Capacitance / Conductance
123
ratio_of_cap_to_cond = cap1_avrg ./ cndut1_avrg;
% Calculatiing Imaginary part of Admittance
Yimaginary = cap1_avrg .* (2*pi*fre1_avrg);
% Creating AxisMatrix
[rows,cols] = size(cap1_avrg);
for n = 1:rows
for m = 1:cols
AxisMatrix(n,m) = AxisIndex(Count);
end
end
%
% Scaled Capacitance
mincap(Count) = min(cap1_avrg);
normalised_cap = cap1_avrg - mincap(Count);
normalised_cap = normalised_cap ./ max(normalised_cap);
dynamicrange(Count) = max(normalised_cap);
% Scaled Conductance
mincond(Count) = min(ratio_of_cndut_and_freq_avrg);
normalised_condbyw = ratio_of_cndut_and_freq_avrg mincond(Count) ;
%
normalised_condbyw = normalised_condbyw ./
max(normalised_condbyw);
% Scaled Phase
minphase(Count) = min(phase1_avrg);
normalised_phase = phase1_avrg ./ minphase(Count) ;
% Ratio of Capacitance to Scaled Conductance
Ratio_cap_to_scaled_Cond = cap1_avrg ./ normalised_condbyw;
% Store the Processed data in a 3-D Matrix
Data(Count,:,1) = fre1_avrg;
Data(Count,:,2) = volt1_avrg;
Data(Count,:,3) = curr1_avrg;
Data(Count,:,4) = phase1_avrg;
Data(Count,:,5) = resis1_avrg;
Data(Count,:,6) = cap1_avrg;
Data(Count,:,7) = cndut1_avrg;
Data(Count,:,8) = ratio_of_cndut_and_freq_avrg;
Data(Count,:,9) = Yimaginary;
Data(Count,:,10) = AxisMatrix;
Data(Count,:,11) = tandelta;
Data(Count,:,12) = ratio_of_cap_to_cond;
Data(Count,:,13) = normalised_cap;
Data(Count,:,14) = normalised_condbyw;
Data(Count,:,15) = normalised_phase;
Data(Count,:,16) = Ratio_cap_to_scaled_Cond;
end
% Fitting Data and Ploting them
124
figurenumber = 1;
var1 = 4;
var2 = 8;
freq = 8;
[error,constants] =
TwoVariableLinearFitFunc(Data(2:no_of_files,freq,var1),(Data(2:no_of_fi
les,freq,var2)),Data(2:no_of_files,freq,10));
y1 =
TwoVariableLinearEvalFunc(constants,Data(1:no_of_files,freq,var1),Data(
1:no_of_files,freq,var2));
if error == 1
y1 = y1 .* 0;
end
figure(figurenumber)
figurenumber = figurenumber + 1;
plot(Data(1:no_of_files,1,10),y1,'k*');
hold on; plot(Data(:,1,10),Data(:,1,10),'ko-'); hold off;
grid on;
title('CaCO_3 Concentration'); xlabel('Actual CaCO_3 Concentration
(%)'); ylabel('Estimated CaCO_3 Concentration (%)');
[error,constants] =
TwoVariableLinearFitFunc(Data(2:no_of_files,freq,var1),(Data(2:no_of_fi
les,freq,var2)),PaperConc(2:no_of_files)');
y1 =
TwoVariableLinearEvalFunc(constants,Data(1:no_of_files,freq,var1),Data(
1:no_of_files,freq,var2));
if error == 1
y1 = y1 .* 0;
end
figure(figurenumber)
figurenumber = figurenumber + 1;
plot(PaperConc,y1,'ko');
hold on; plot(PaperConc,PaperConc,'ro'); hold off;
grid on;
title('Paper Concentration'); xlabel('Actual Fiber Concentration (%)');
ylabel('Estimated Fiber Concentration (%)');
[error,constants] =
TwoVariableLinearFitFunc(Data(2:no_of_files,freq,var1),(Data(2:no_of_fi
les,freq,var2)),MoistureConc(2:no_of_files)');
y1 =
TwoVariableLinearEvalFunc(constants,Data(1:no_of_files,freq,var1),Data(
1:no_of_files,freq,var2));
if error == 1
y1 = y1 .* 0;
end
figure(figurenumber)
figurenumber = figurenumber + 1;
plot(MoistureConc,y1,'k*');
125
hold on; plot(MoistureConc,MoistureConc,'ro'); hold off;
grid on;
title('Moisture Concentration'); xlabel('Actual Mositure Concentration
(%)'); ylabel('Estimated Moisture Concentration (%)');
126
9. Repeatability.m
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%
%%%%%%
%%%%%
%%%%%%
%%%%%
Program to test the Repeatability
%%%%%%
%%%%%
%%%%%%
%%%%%
%%%%%%
%%%%%
%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
clear all; close all;
% Set font size.
set(0, 'DefaultAxesFontSize',13)
set(0, 'DefaultTextFontSize',13)
% set(0, 'DefaultLineThick',2.5)
% Initializing file names.
select = 1;
if select == 1
directname = 'C:\Data\06-22-03\'
filename(1,:) = 'run2';
filename(2,:) = 'run3';
filename(3,:) = 'run4';
filename(4,:) = 'run5';
filename(5,:) = 'run6';
AxisIndex = [1,2,3,4,5];
elseif select == 2
directname = 'C:\Data\06-23-03\Repeatability\'
filename(1,:) = '01';
filename(2,:) = '02';
filename(3,:) = '03';
filename(4,:) = '04';
filename(5,:) = '05';
filename(6,:) = '06';
filename(7,:) = '07';
filename(8,:) = '08';
filename(9,:) = '09';
filename(10,:) = '10';
filename(11,:) = '11';
filename(12,:) = '12';
127
filename(13,:) = '13';
filename(14,:) = '14';
filename(15,:) = '15';
filename(16,:) = '16';
filename(17,:) = '17';
filename(18,:) = '18';
filename(19,:) = '19';
filename(20,:) = '20';
AxisIndex = [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20];
end
start = 13;
stop = 25;
% <-- the starting index.
% "Time (days)" "Frequency" "Voltage" "Current" "Phase" "Resistance"
"Capacitance"
time_ind = 1;
fre_ind = 2;
volt_ind = 3;
curr_ind = 4;
phase_ind = 5;
resis_ind = 6;
cap_ind = 7;
% Pre-Processing Each File
[no_of_files, temp] = size(filename);
for Count = 1:no_of_files
tempdata = load([directname filename(Count,:) '.txt']);
fre1 = tempdata(:, fre_ind);
volt1 = tempdata(:, volt_ind);
curr1 = tempdata(:, curr_ind);
phase1 = tempdata(:, phase_ind);
resis1 = tempdata(:, resis_ind);
cap1 = tempdata(:, cap_ind);
% Segment into Individual Sweeps
fre1_seg = buffer(fre1, 25);
volt1_seg = buffer(volt1, 25);
curr1_seg = buffer(curr1, 25);
phase1_seg = buffer(phase1, 25);
resis1_seg = buffer(resis1, 25);
cap1_seg = buffer(cap1, 25);
% Remove the low frequency data
fre1_seg = fre1_seg(start:stop, :);
volt1_seg = volt1_seg(start:stop,:);
curr1_seg = curr1_seg(start:stop, :);
phase1_seg = phase1_seg(start:stop, :);
resis1_seg = resis1_seg(start:stop, :);
128
cap1_seg = cap1_seg(start:stop, :);
% Averaging the data across Sweeps
fre1_avrg = mean(fre1_seg, 2);
volt1_avrg = mean(volt1_seg, 2);
curr1_avrg = mean(curr1_seg, 2);
phase1_avrg = mean(phase1_seg, 2);
resis1_avrg = mean(resis1_seg, 2);
cap1_avrg = mean(cap1_seg, 2);
% Obtain Conductance from Resistance
cndut1_avrg = 1./resis1_avrg;
% Calculating ratio of Conductance and Angular Frequency
ratio_of_cndut_and_freq_avrg = cndut1_avrg./(fre1_avrg);
% Calculating tandelta
tandelta = 1./ ((2*pi*fre1_avrg) .* (cap1_avrg) .*
(resis1_avrg));
% Calculating Capacitance / Conductance
ratio_of_cap_to_cond = cap1_avrg ./ cndut1_avrg;
% Calculating Admittance
Yreal = cndut1_avrg;
w = 2*pi*fre1_avrg;
Yimg = w .* cap1_avrg;
Y = complex(Yreal,Yimg);
Ymag = abs(Y);
% Calculating Admittance
Z = 1 ./ Y;
Zimg = imag(Z);
Zreal = real(Z);
Zmag = abs(Z);
% Creating AxisMatrix
[rows,cols] = size(cap1_avrg);
for n = 1:rows
for m = 1:cols
AxisMatrix(n,m) = AxisIndex(Count);
end
end
% Calculating the Min, Max and Std. Deviation of Capacitance
cap_min = cap1_avrg - min(cap1_seg,[],2);
cap_max = max(cap1_seg,[],2) - cap1_avrg;
cap_std = std(cap1_seg,0, 2);
% Store the Processed data in a 3-D Matrix
Data(Count,:,1) = fre1_avrg;
Data(Count,:,2) = volt1_avrg;
129
Data(Count,:,3) = curr1_avrg;
Data(Count,:,4) = phase1_avrg;
Data(Count,:,5) = resis1_avrg;
Data(Count,:,6) = cap1_avrg;
Data(Count,:,7) = cndut1_avrg;
Data(Count,:,8) = ratio_of_cndut_and_freq_avrg;
Data(Count,:,9) = tandelta;
Data(Count,:,10) = AxisMatrix;
Data(Count,:,11) = Yimg;
Data(Count,:,12) = Yreal;
Data(Count,:,13) = Ymag;
Data(Count,:,14) = -1 .* Zimg;
Data(Count,:,15) = Zreal;
Data(Count,:,16) = Zmag;
Data(Count,:,17) = cap_std;
Data(Count,:,18) = cap_min;
Data(Count,:,19) = cap_max;
end
Capacitance = cap1_seg(8,:)';
std_dev = std(Capacitance,0, 1);
Mean_cap = mean(Capacitance, 1);
[no_of_runs,temp] = size(Capacitance);
for n = 1:no_of_runs
low(n,1) = Mean_cap - (std_dev/2);
low(n,2) = n;
end
for n = 1:no_of_runs
high(n,1) = Mean_cap + (std_dev/2);
high(n,2) = n;
end
for n = 1:no_of_runs
mean(n,1) = Mean_cap;
mean(n,2) = n;
end
for n = 1:no_of_runs
delta(n,1) = Mean_cap - Capacitance(n,1);
mean(n,2) = n;
end
[n,temp] = size(Capacitance);
figure(1);
subplot(2,1,1);
bar(1:n,Capacitance,'kd-');
xlabel('Trial Number');
Ylabel('Capacitance (F)');
hold on;
plot(low(:,2),low(:,1),'k--');
plot(high(:,2),high(:,1),'k--');
plot(mean(:,2),mean(:,1),'k-');
hold off;
130
Grid on;
axis([1,no_of_runs,(Mean_cap - (std_dev*2)),(Mean_cap + (std_dev*2))]);
subplot(2,1,2);
bar(1:n,delta,'k');
xlabel('Trial Number');
Ylabel('Deviation from Mean (F)');
hold on;
plot(mean(:,2),(mean(:,1).*0),'k-');
plot(low(:,2),low(:,1)-mean(:,1),'k-');
plot(high(:,2),high(:,1)-mean(:,1),'k-');
hold off;
axis([1,no_of_runs,(0 - (std_dev*2)),(0 + (std_dev*2))]);
Grid on;
131
10.Reproducability.m
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%
%%%%%%
%%%%%
%%%%%%
%%%%%
Reproducability Test for Fiber
%%%%%%
%%%%%
%%%%%%
%%%%%
%%%%%%
%%%%%
%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
clear all; close all;
% Set font size.
set(0, 'DefaultAxesFontSize',13)
set(0, 'DefaultTextFontSize',13)
% Initializing file names.
directname = 'C:\Data\06-22-03\'
filename(1,:) = 'run2';
filename(2,:) = 'run3';
filename(3,:) = 'run4';
filename(4,:) = 'run5';
filename(5,:) = 'run6';
AxisIndex = [1,2,3,4,5];
start = 13;
stop = 25;
% <-- the starting index.
% "Time (days)" "Frequency" "Voltage" "Current" "Phase" "Resistance"
"Capacitance"
time_ind = 1;
fre_ind = 2;
volt_ind = 3;
curr_ind = 4;
phase_ind = 5;
resis_ind = 6;
cap_ind = 7;
% Pre-Processing Each File
[no_of_files, temp] = size(filename);
for Count = 1:no_of_files
tempdata = load([directname filename(Count,:) '.txt']);
fre1 = tempdata(:, fre_ind);
volt1 = tempdata(:, volt_ind);
132
curr1 = tempdata(:, curr_ind);
phase1 = tempdata(:, phase_ind);
resis1 = tempdata(:, resis_ind);
cap1 = tempdata(:, cap_ind);
% Segment into Individual Sweeps
fre1_seg = buffer(fre1, 25);
volt1_seg = buffer(volt1, 25);
curr1_seg = buffer(curr1, 25);
phase1_seg = buffer(phase1, 25);
resis1_seg = buffer(resis1, 25);
cap1_seg = buffer(cap1, 25);
% Remove the low frequency data
fre1_seg = fre1_seg(start:stop, :);
volt1_seg = volt1_seg(start:stop,:);
curr1_seg = curr1_seg(start:stop, :);
phase1_seg = phase1_seg(start:stop, :);
resis1_seg = resis1_seg(start:stop, :);
cap1_seg = cap1_seg(start:stop, :);
% Averaging the data across Sweeps
fre1_avrg = mean(fre1_seg, 2);
volt1_avrg = mean(volt1_seg, 2);
curr1_avrg = mean(curr1_seg, 2);
phase1_avrg = mean(phase1_seg, 2);
resis1_avrg = mean(resis1_seg, 2);
cap1_avrg = mean(cap1_seg, 2);
% Obtain Conductance from Resistance
cndut1_avrg = 1./resis1_avrg;
% Calculating ratio of Conductance and Angular Frequency
ratio_of_cndut_and_freq_avrg = cndut1_avrg./(fre1_avrg);
% Calculating tandelta
tandelta = 1./ ((2*pi*fre1_avrg) .* (cap1_avrg) .*
(resis1_avrg));
% Calculating Capacitance / Conductance
ratio_of_cap_to_cond = cap1_avrg ./ cndut1_avrg;
% Calculating Admittance
Yreal = cndut1_avrg;
w = 2*pi*fre1_avrg;
Yimg = w .* cap1_avrg;
Y = complex(Yreal,Yimg);
Ymag = abs(Y);
% Calculating Admittance
Z = 1 ./ Y;
133
Zimg = imag(Z);
Zreal = real(Z);
Zmag = abs(Z);
% Creating AxisMatrix
[rows,cols] = size(cap1_avrg);
for n = 1:rows
for m = 1:cols
AxisMatrix(n,m) = AxisIndex(Count);
end
end
% Calculating the Min, Max and Std. Deviation of Capacitance
cap_min = cap1_avrg - min(cap1_seg,[],2);
cap_max = max(cap1_seg,[],2) - cap1_avrg;
cap_std = std(cap1_seg,0, 2);
% Store the Processed data in a 3-D Matrix
Data(Count,:,1) = fre1_avrg;
Data(Count,:,2) = volt1_avrg;
Data(Count,:,3) = curr1_avrg;
Data(Count,:,4) = phase1_avrg;
Data(Count,:,5) = resis1_avrg;
Data(Count,:,6) = cap1_avrg;
Data(Count,:,7) = cndut1_avrg;
Data(Count,:,8) = ratio_of_cndut_and_freq_avrg;
Data(Count,:,9) = tandelta;
Data(Count,:,10) = AxisMatrix;
Data(Count,:,11) = Yimg;
Data(Count,:,12) = Yreal;
Data(Count,:,13) = Ymag;
Data(Count,:,14) = -1 .* Zimg;
Data(Count,:,15) = Zreal;
Data(Count,:,16) = Zmag;
Data(Count,:,17) = cap_std;
Data(Count,:,18) = cap_min;
Data(Count,:,19) = cap_max;
end
Capacitance = Data(:,5,6);
m = 12.399*(10^12);
k = -195.45;
Moisture = Capacitance .* m + k;
std_dev = std(Moisture,0, 1);
Mean_cap = mean(Moisture, 1);
[no_of_runs,temp] = size(Moisture);
for n = 1:no_of_runs
low(n,1) = Mean_cap - (std_dev/2);
low(n,2) = n;
end
for n = 1:no_of_runs
134
high(n,1) = Mean_cap + (std_dev/2);
high(n,2) = n;
end
for n = 1:no_of_runs
mean(n,1) = Mean_cap;
mean(n,2) = n;
end
for n = 1:no_of_runs
delta(n,1) = Mean_cap - Moisture(n,1);
mean(n,2) = n;
end
figure(1);
subplot(2,1,1);
bar(AxisIndex,Moisture,'kd-');
xlabel('Trial Number');
Ylabel('Moisture (%)');
hold on;
plot(low(:,2),low(:,1),'k--');
plot(high(:,2),high(:,1),'k--');
plot(mean(:,2),mean(:,1),'k-');
hold off;
Grid on;
axis([1,no_of_runs,(Mean_cap - (std_dev*2)),(Mean_cap + (std_dev*2))]);
subplot(2,1,2);
bar(AxisIndex,delta,'k');
xlabel('Trial Number');
Ylabel('Deviation from Mean (%)');
hold on;
plot(mean(:,2),(mean(:,1).*0),'k-');
plot(low(:,2),low(:,1)-mean(:,1),'k-');
plot(high(:,2),high(:,1)-mean(:,1),'k-');
hold off;
axis([1,no_of_runs,(0 - (std_dev*2)),(0 + (std_dev*2))]);
Grid on;
Peak_to_Peak_Variation = max(delta) - min(delta);
135
11.LinearFitFunc.m
function [figurenumber] =
linearfitfunc(Data,XAxis,YAxis,figurenumber,labelX,labelY,Title)
[no_of_files,no_of_freq,no_of_parameters] = size(Data);
for Count = 1:no_of_freq
[p_cap(Count,:),s_cap(Count,:)] =
polyfit(Data(:,Count,YAxis),Data(:,Count,XAxis),1);
Fitted_axis(Count,:) =
polyval(p_cap(Count,:),Data(:,Count,YAxis))';
error = Fitted_axis(Count,:) - Data(:,Count,XAxis)';
abs_error = abs(error);
percentage_error = abs_error * 100 ./ Data(:,Count,XAxis)'
errormatrix(Count,1) = Count;
errormatrix(Count,2) = mean(percentage_error);
end
errormatrix = sortrows(errormatrix,2);
Count
Text1
Text2
Text3
Text4
=
=
=
=
=
errormatrix(1,1);
strcat('Frequency = ',sprintf('%1.3e',Data(1,Count,1)));
strcat('Error = ',sprintf('%1.3e',errormatrix(1,2)));
strcat('m = ',sprintf('%1.3e',p_cap(Count,1)));
strcat('k = ',sprintf('%1.3e',p_cap(Count,2)));
for Counter = 1:1
Count = errormatrix(Counter,1);
figurenumber = figurenumber + 1;
figure(figurenumber);
plot(Data(:,Count,XAxis), Fitted_axis(Count,:), 'ko');
hold on;
plot(Data(:,Count,XAxis),Data(:,Count,XAxis), 'k:');
xlabel(labelX);
ylabel(labelY);
title(Title);
grid on;
space = ( max(Fitted_axis(Count,:)) - min(Fitted_axis(Count,:)) ) /
10;
text((space+min(Fitted_axis(Count,:))),(1*space +
min(Fitted_axis(Count,:))),Text1);
text((space+min(Fitted_axis(Count,:))),(2*space +
min(Fitted_axis(Count,:))),Text2);
text((space+min(Fitted_axis(Count,:))),(3*space +
min(Fitted_axis(Count,:))),Text3);
text((space+min(Fitted_axis(Count,:))),(4*space +
min(Fitted_axis(Count,:))),Text4);
end
136
12.DataFittingFunc.m
function [temp_matrix,error,figurenum] =
datafittingfunc(Data,index,AxisIndex,figurenumber,labelX,labelY,Title,n
o_of_plots)
% Calculating all Possible Ratios of Conductances
[no_of_files,no_of_freq,no_of_parameters] = size(Data);
Count = 0;
for freq1 = 1:(no_of_freq-1)
for freq2 = (freq1+1):no_of_freq
Count = Count + 1;
ratio_of_conductance(Count,:) = (Data(:,freq1,index) ./
Data(:,freq2,index))';
freq_used(Count,1) = Data(1,freq1,1);
freq_used(Count,2) = Data(1,freq2,1);
end
end
% Linear Fitting the Ratio of Conductances
[no_of_ratios,no_of_files] = size(ratio_of_conductance);
for Count = 1:no_of_ratios
temp1_matrix(:,1) = ratio_of_conductance(Count,:)';
temp1_matrix(:,2) = AxisIndex(:);
temp1_matrix = sortrows(temp1_matrix);
temp_matrix(Count,:,1) = temp1_matrix(:,1);
temp_matrix(Count,:,2) = temp1_matrix(:,2);
[p_cond(Count,:),s_cond(Count,:),mu] =
polyfit(temp_matrix(Count,:,1)',temp_matrix(Count,:,2)',1);
newyvalues =
polyval(p_cond(Count,:),temp_matrix(Count,:,1)',[],mu);
temp_matrix(Count,:,3) = newyvalues;
end
% Calculating the Normalized absolute error in curver fitting
[no_of_ratios,no_of_files,no_of_parameters] = size(temp_matrix);
for Count = 1:no_of_ratios
local_min = min(temp_matrix(Count,:,1));
local_max = max(temp_matrix(Count,:,1));
local_range = local_max - local_min;
local_error = ( temp_matrix(Count,:,2) - temp_matrix(Count,:,3)
).^2;
local_error = local_error .* local_range;
error(Count,1) = sum(local_error,2);
error(Count,2) = Count;
end
error = sortrows(error);
% Plotting The Top 5 ratios
for counter = 1:no_of_plots
137
figurenum = figurenumber + counter - 1;
figure(figurenum)
Count = error(counter,2);
hold on;
freq1_string = num2str(freq_used(Count,1));
freq2_string = num2str(freq_used(Count,2));
plot(temp_matrix(Count,:,1),temp_matrix(Count,:,2),'bo' );
plot(temp_matrix(Count,:,1),temp_matrix(Count,:,3),'rp-' );
xlabel(labelX);
ylabel(labelY);
title_str = strcat(Title,' - ',freq1_string,'Hz /
',freq2_string,'Hz');
title(title_str);
orient landscape;
grid on;
axis tight;
hold off;
end
figurenum = figurenum + 1;
138
13.TwoVariableLinearFitFunc.m
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%
Least Square Fitting of Y = aX1 + bX2 + C
%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [error, constants] = TwoVariableLinearFitFunc(x2,x1,y)
% Determining the Number of Elements
[temp,n] = size(x1);
% Computing Sums
sumX1 = sum(x1);
sumX2 = sum(x2);
sumY = sum(y);
sumX1sqr = sum(x1 .* x1);
sumX2sqr = sum(x2 .* x2);
sumX1X2 = sum(x1 .* x2);
sumYX1 = sum(x1 .* y);
sumYX2 = sum(x2 .* y);
% Forming the Coefficient Matrix
coeff(1,1) = sumX1X2;
coeff(1,2) = sumX1sqr;
coeff(1,3) = sumX1;
coeff(2,1) = sumX2sqr;
coeff(2,2) = sumX1X2;
coeff(2,3) = sumX2;
coeff(3,1) = sumX2;
coeff(3,2) = sumX1;
coeff(3,3) = n;
% Forming the RHS
RHS(1) = sumYX1;
RHS(2) = sumYX2;
RHS(3) = sumY;
% Scaling the Matrix
min_ele = min(coeff,[],2);
coeff(1,:) = coeff(1,:) ./ min_ele(1);
coeff(2,:) = coeff(2,:) ./ min_ele(2);
coeff(3,:) = coeff(3,:) ./ min_ele(3);
RHS(1) = RHS(1) ./ min_ele(1);
RHS(2) = RHS(2) ./ min_ele(2);
RHS(3) = RHS(3) ./ min_ele(3);
% Testing the condition of Coeffecient Matrix
139
error = 0;
if (coeff == coeff') % Checking for Singular Matrix
error = 1;
elseif (cond(coeff) > 1e15)
error = 1;
end
% Computing the Inverse of Coeff Matrix
coeff_inverse = inv(coeff);
% Solving for Constants
constants = coeff_inverse * RHS';
140
14.TwoVariableLinearEvalFunc.m
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%
Least Square Fitting of Y = aX1 + bX2 + C
%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [y] = TwoVariableLinearEvalFunc(constants,x1,x2)
y = (constants(1) .* x1) + (constants(2) .* x2) + constants(3);
141
15.ParameterEstimation.m
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%
%%%%%%%
%%%%%%%
%%%%%%%
%%%%%%% This program tries to evaluate the most useful
%%%%%%%
%%%%%%% parameters. It uses the all_comb_func to generate all
%%%%%%%
%%%%%%% possible combination of parameters and ranks them
%%%%%%%
%%%%%%% according to the normalized errors of the fits.
%%%%%%%
%%%%%%% This is done of each of the data sets.
%%%%%%%
%%%%%%% The parameter fit that best fits all the data
%%%%%%%
%%%%%%% sets is then chosen
%%%%%%%
%%%%%%%
%%%%%%%
%%%%%%%
%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
clear all; close all;
% Set font size.
set(0, 'DefaultAxesFontSize',20)
set(0, 'DefaultTextFontSize',20)
start = 5;
stop = 25;
% <-- the starting index.
for select = 1:2
% Initializing file names.
if select == 2
directname = 'C:\Data\10-24-02\'
filename(1,:) = '10';
filename(2,:) = '09';
filename(3,:) = '08';
filename(4,:) = '07';
filename(5,:) = '06';
filename(6,:) = '05';
filename(7,:) = '04';
filename(8,:) = '03';
AxisIndex = [10,9,8,7,6,5,4,3];
elseif select == 1
directname = 'C:\Data\04-07-03\pulp\'
filename(1,:) = '10';
filename(2,:) = '09';
filename(3,:) = '08';
filename(4,:) = '07';
AxisIndex = [10,09,08,07];
142
end
[a,Data,Parameter_matrix] =
all_comb_func(directname,filename,AxisIndex,start,stop,select);
sortederrormatrix(select,:,:) = a;
end
[no_of_datasets,no_of_parameters,temp] = size(sortederrormatrix);
for n = 1:no_of_parameters
rankmatrix(n,1) = n;
rankmatrix(n,2) = 0;
rankmatrix(n,3) = 0;
rankmatrix(n,4) = 0;
for m = 1:no_of_datasets
rankmatrix(n,2) = rankmatrix(n,2) + sortederrormatrix(m,n,6);
rankmatrix(n,3) = rankmatrix(n,3) + sortederrormatrix(m,n,1);
end
end
sortedrankmatrix = sortrows(rankmatrix,2);
% for n = 1:no_of_parameters
%
rankmatrix(n,4) = rankmatrix(n,4) + n;
% end
%
% sortedrankmatrix = sortrows(rankmatrix,3);
% for n = 1:no_of_parameters
%
rankmatrix(n,4) = rankmatrix(n,4) + n;
% end
% sortedrankmatrix = sortrows(rankmatrix,4);
% Plotting the best fits
figurenumber = 1;
for counter = 1:15
parameter = sortedrankmatrix(counter,1);
figurenumber = linearfitfunc(Data,10,parameter,figurenumber,'Actual
Fiber Content(%)','Estimated Fiber Content(%)',parameter);
end
143
16.all_comb_func.m
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%
%%%%%%%
%%%%%%%
%%%%%%%
%%%%%%% This Function generates all possible combination
%%%%%%%
%%%%%%% of parameters and does linear fit on them. It then
%%%%%%%
%%%%%%% evaluates the normalized error for each of the fits
%%%%%%%
%%%%%%% and ranks the parameters according to the errors.
%%%%%%%
%%%%%%% The function requires an array containing the
%%%%%%%
%%%%%%% filenames, a string with the path and an Axis index
%%%%%%%
%%%%%%% detailing the actual Y Values for the Fit. The
%%%%%%%
%%%%%%% function returns an array with all the details of the %%%%%%%
%%%%%%% parameters and their ranks.
%%%%%%%
%%%%%%%
%%%%%%%
%%%%%%%
%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [sortederrormatrix,Data,parameter_matrix] =
all_comb_func(directname,filename,AxisIndex,start,stop,select)
% "Time (days)" "Frequency" "Voltage" "Current" "Phase" "Resistance"
"Capacitance"
time_ind = 1;
fre_ind = 2;
volt_ind = 3;
curr_ind = 4;
phase_ind = 5;
resis_ind = 6;
cap_ind = 7;
% Pre-Processing Each File
[no_of_files, temp] = size(filename);
for Count = 1:no_of_files
tempdata = load([directname filename(Count,:) '.txt']);
fre1 = tempdata(:, fre_ind);
volt1 = tempdata(:, volt_ind);
curr1 = tempdata(:, curr_ind);
phase1 = tempdata(:, phase_ind);
resis1 = tempdata(:, resis_ind);
cap1 = tempdata(:, cap_ind);
% Segment into Individual Sweeps
fre1_seg = buffer(fre1, 25);
volt1_seg = buffer(volt1, 25);
144
curr1_seg = buffer(curr1, 25);
phase1_seg = buffer(phase1, 25);
resis1_seg = buffer(resis1, 25);
cap1_seg = buffer(cap1, 25);
% Remove the low frequency data
fre1_seg = fre1_seg(start:stop, :);
volt1_seg = volt1_seg(start:stop,:);
curr1_seg = curr1_seg(start:stop, :);
phase1_seg = phase1_seg(start:stop, :);
resis1_seg = resis1_seg(start:stop, :);
cap1_seg = cap1_seg(start:stop, :);
% Averaging the data across Sweeps
fre1_avrg = mean(fre1_seg, 2);
volt1_avrg = mean(volt1_seg, 2);
curr1_avrg = mean(curr1_seg, 2);
phase1_avrg = mean(phase1_seg, 2);
resis1_avrg = mean(resis1_seg, 2);
cap1_avrg = mean(cap1_seg, 2);
% Obtain Conductance from Resistance
cndut1_avrg = 1./resis1_avrg;
% Calculating ratio of Conductance and Angular Frequency
ratio_of_cndut_and_freq_avrg = cndut1_avrg./(2*pi*fre1_avrg);
% Calculating tandelta
tandelta = 1./ ((2*pi*fre1_avrg) .* (cap1_avrg) .*
(resis1_avrg));
% Creating AxisMatrix
[rows,cols] = size(cap1_avrg);
for n = 1:rows
for m = 1:cols
AxisMatrix(n,m) = AxisIndex(Count);
end
end
% Calculating Admittance
Yreal = cndut1_avrg;
w = 2*pi*fre1_avrg;
Yimg = w .* cap1_avrg;
Y = complex(Yreal,Yimg);
Ymag = abs(Y);
% Calculating Admittance
Z = 1 ./ Y;
Zimg = imag(Z);
Zreal = real(Z);
Zmag = abs(Z);
% Store the Processed data in a 3-D Matrix
145
Data(Count,:,1) = fre1_avrg;
Data(Count,:,2) = volt1_avrg;
Data(Count,:,3) = curr1_avrg;
Data(Count,:,4) = phase1_avrg;
Data(Count,:,5) = resis1_avrg;
Data(Count,:,6) = cap1_avrg;
Data(Count,:,7) = cndut1_avrg;
Data(Count,:,8) = ratio_of_cndut_and_freq_avrg;
Data(Count,:,9) = tandelta;
Data(Count,:,10) = AxisMatrix;
Data(Count,:,11) = Yimg;
Data(Count,:,12) = Yreal;
Data(Count,:,13) = Ymag;
Data(Count,:,14) = Zimg;
Data(Count,:,15) = Zreal;
Data(Count,:,16) = Zmag;
end
[no_of_files,no_of_freq,no_of_parameters] = size(Data);
New_Parameter = no_of_parameters;
% The parameter_matrix has 6 cols. parameter number, parameter1,
% parameter2, frequency1, subtract, and divide.
counter = 0;
for counter = 1:no_of_parameters
parameter_matrix(1,counter) = counter;
parameter_matrix(2,counter) = counter;
parameter_matrix(3,counter) = 0;
parameter_matrix(4,counter) = 0;
parameter_matrix(5,counter) = 0;
parameter_matrix(6,counter) = 0;
end
origparameters = no_of_parameters + 1;
% Difference of all Parameters
for parameter1 = 1:no_of_parameters
if parameter1 ~= 10 % Avoiding the Index
for parameter2 = 1:no_of_parameters
if parameter1 ~= parameter2 % Avoiding all Zeros
if parameter2 ~= 10
New_Parameter = New_Parameter + 1
counter = counter + 1;
parameter_matrix(1,counter) = New_Parameter;
parameter_matrix(2,counter) = parameter1;
parameter_matrix(3,counter) = parameter2;
parameter_matrix(4,counter) = 0;
parameter_matrix(5,counter) = 1;
parameter_matrix(6,counter) = 0;
Data(:,:,New_Parameter) = Data(:,:,parameter1) Data(:,:,parameter2);
end
end
end
146
end
end
% Ratio of all Parameters
for parameter1 = 1:no_of_parameters
if parameter1 ~= 10
for parameter2 = 1:no_of_parameters
if parameter1 ~= parameter2
if parameter2 ~= 10
New_Parameter = New_Parameter + 1
counter = counter + 1;
parameter_matrix(1,counter) = New_Parameter;
parameter_matrix(2,counter) = parameter1;
parameter_matrix(3,counter) = parameter2;
parameter_matrix(4,counter) = 0;
parameter_matrix(5,counter) = 0;
parameter_matrix(6,counter) = 1;
Data(:,:,New_Parameter) = Data(:,:,parameter1) ./
Data(:,:,parameter2);
end
end
end
end
end
parameters_after_P = New_Parameter + 1;
% Difference of all Frequency
for freq1 = 1:no_of_freq
for parameter = 2:no_of_parameters
if parameter ~= 10
New_Parameter = New_Parameter + 1
counter = counter + 1;
parameter_matrix(1,counter) = New_Parameter;
parameter_matrix(2,counter) = parameter;
parameter_matrix(3,counter) = 0;
parameter_matrix(4,counter) = freq1;
parameter_matrix(5,counter) = 1;
parameter_matrix(6,counter) = 0;
for freq2 = 1:no_of_freq
%
for files = 1:no_of_files
Data(:,freq2,New_Parameter) =
Data(:,freq1,parameter) - Data(:,freq2,parameter);
%
end
end
end
end
end
% Ratio of all Frequency
for freq1 = 1:no_of_freq
147
for parameter = 2:no_of_parameters
if parameter ~= 10
New_Parameter = New_Parameter + 1
counter = counter + 1;
parameter_matrix(1,counter) = New_Parameter;
parameter_matrix(2,counter) = parameter;
parameter_matrix(3,counter) = 0;
parameter_matrix(4,counter) = freq1;
parameter_matrix(5,counter) = 0;
parameter_matrix(6,counter) = 1;
for freq2 = 1:no_of_freq
%
for files = 1:no_of_files
Data(:,freq2,New_Parameter) =
Data(:,freq1,parameter) ./ Data(:,freq2,parameter);
%
end
end
end
end
end
parameters_after_F = New_Parameter;
Text1 = strcat('Select',sprintf('%d',select),'level1');
save(Text1);
% Fitting the data
[no_of_files,no_of_freq,no_of_parameters] = size(Data);
Count = 0;
for parameter = 2:no_of_parameters
if parameter ~= 10
Count = Count + 1;
[errormatrix(Count,1),errormatrix(Count,2),errormatrix(Count,3),errorma
trix(Count,4)] = linearfitfuncwitherror(Data,10,parameter);
errormatrix(Count,5) = parameter;
end
end
% sorting the fits to determine the best fits
sortederrormatrix = sortrows(errormatrix,1);
[no_of_parameters,temp] = size(sortederrormatrix);
for count = 1:no_of_parameters
sortederrormatrix(count,6) = count; % Assigning ranks by error
end
sortederrormatrix = sortrows(sortederrormatrix,5); % Rearrange by
parameter