MONITORING OF TRANSFORMER OIL
'USING MICRODIELECTRIC
SENSORS
by
Lama Mouayad
B.S., Massachusetts Institute of Technology
1983
SUBMITTED IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR
DEGREE OF
MASTER OF SCIENCE
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
February, 1985
@ Massachuset ts Institute of Technology 1985
Signature of Author
.
'
•
.
.S..
r .
.--
•
.........
Department of Electrical Engineering
and Computer science, February 1, 1985
Certified by
....... ..
.
............................. .
J. R. Melcher
Thesis Supervisor
Accepted by
'./ A. C. Smith
Chairman, Departmental Committee
on Graduate Student
MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
ARCHIVES
APR 0 11985
0w1
MONITORING OF TRANSFORMER OIL
USING MICRODIELECTRIC SENSORS
by
Lama Mouayad
Submitted to the Department of Electrical Engineering
on February 4, 1985 in partial fulfillment of the
requirements for the Degree of Master of Science in
Electrical Engineering
ABSTRACT
an
of using microdielecric sensors in
The possibility
on-line monitoring system of transformers is examined. A
numerical technique based on a transfer relation approach is
used to solve for the field in the sensor. This numerical
approach facilitates the inclusion of surface effects. The
sensor geometry has a mixed boundary at the sensor/fluid
interface, and the numerical technique used here is equally
applicable to other mixed boundary value problems.
The sensors show a clear difference between the response of
clean oil and contaminated oil obtained from a failed
transformer. The microscopic dimensions of the sensors make
them sensitive to surface phenomena. The surface phenomena
can provide additional information about the state of the
transformer not necessarily reflected in changes of the bulk
properties of transformer oil.
Experimental work using the microdielectric sensors in clean
and contaminated oil shows evidence of heterogeneity caused
by surface phenomena. The main surface phenomena involved
are the polarization of the electrodes caused by space
charge accumulation and a conduction path between the two
electrodes caused by adsorption on the interface.
increases with the
The polarization of the electrodes
Stirring of
increase of the bulk conductivity of the oil.
the fluid causes a clear reduction of the polarization of
the electrodes caused by turbulence enhanced diffusion.
Thesis Supervisor: James R. Melcher
Title:
Stratton Professor of Electrical
Physics
Engineering
and
Ac04ow
to Prof. Melcher who gave ample time
My deepest thanks
and
attention
contribution
knowledge,
to
to
gemen
g
t
guide
me
my education
to include
a
this
through
goes
thesis.
beyond the
welcomed influence
His
scientific
on my
social
awareness and moral stands.
Special thanks to Prof. Senturia for providing valuable
advice.
iscussions
Members of
with
the continuun
of technical assistance
with
Mark
Zaretsky and
helpful. Thanks
microelectronics
r.
great
electromechanics group
hels.
gave a lot
and friendly support. Conversations
Philip
von
to Norman Sheppard
laboratory for
Special thanks to my
were a
Cooke
Guggenberg were
and Mike Coln
their
very
from the
readiness to
help.
friends who gave me an essential moral
support.
Most of all, my
sisters who
and
gave me
They are
deepest gratitude to my parents and my
helped me in determining
ample encouragement
my direction in life,
throughout
always a part of everything I
my education.
do, and this thesis
would not have been possible without their support.
Table of Contents
Abstract
3
Acknowledgements
4
Table of Figures
7
Chapter I : Introduction
L.i Overview of Monitoring of Transformers13
1.2 Background on Microdielectric Sensors
Chapter II : Dielectric Properties of Transformer Oil
18
2.1 Dielectric and Loss Phenomena in Transformer Oil 19
2.2 Effect of Transformer Oil Aging on its Loss
24
2.3 Moisture in Transformers
25
Chapter
III
: Field
Solution
Using
a Transfer
Relation
Approach
27
3.1 Geometry of Sensors
27
3.2 Boundary Conditions
28
3.3 A Transfer
Relation Approach for Solving the Mixed
Boundary Value Problem
32
3.4 The Long Wave Limit
3.5 Solution
for a Two Layers
41
Geometry and an Ambient
Field
3.6 Generality of the Approach
46
48
Chapter IV : Gain and Phase Response Under Various Boundary
Conditions
51
4.1 Frequency Variant Circuit Parameters
51
4.2
Effects
of Surface
Impedance
on the
Gain-Phase
Response
54
58
4.3 Surface Contact Effect
63
Chapter V : Experimental Work
5.1 The Experimental Setup
63
5.2 Response of Clean Oil
65
5.3 Contaminated Oil
91
5.3.1 Oil With High Water Content
91
5.3.2 Oil With Additives
98
113
Chapter VI : Analysis of Results
6.1 Surface Effects Charaterized by Lumped Elements 114
6.2 The Series Capacitance
120
6.3 The Parallel Conductance
126
6.4 Additional Effects
127
Chapter VII : Suggestions for Further Research
128
7.1 Improvement of the Numerical Technique
128
7.2 Further Studies on the Interface
129
7.3
Further
Research
on
the Practicality
of
Using
Microdielectric Sensors to Monitor Transformers 130
Appendix A
132
Appendix B
149
References
151
Table of Figures
Figure 1.1 Top view of sensor
Figure 1.2 Feedback circuit
Figure 3.1 Schematic cross section of a sensor
Figure 3.2 The potential at the interface
Figure 3.3 The tangential electric field at the interface 36
Figure
3.4 Conservation
interface
Figure 3.5 Successive
interface
of charge in
a segment
along the
38
approximation to the potential at the
Figure
3.6 Snapshots of the
cycle, X/h=10
interface potential
during a
43
Figure
3.7 Snapshots of the
cycle, X/h=40
interface potential
during a
45
Figure 3.8 Geometry of a coated sensor
Figure
3.9 A bent Maxwell's
value problem
attenuator, a
mixed boundary
50
Figure 4.1 Equivalent circuit of sensor
Figure
4.2 Calculated gain-phase response at typical oil
55
permittivities with no surface effects
Figure 4.3 Effect of surface conductivity on the gain-phase
response
Figure 4.4
Effect of surface capacitance
on the gain-phase
response
Figure 4.5
Figure
4.6
Effect of zero surface conductivity at contacts
61
on the gain-phase response
Effect of zero contact conductivity on the
gain-phase response for a highly conducting
62
interface
Figure 5.1 The experimental setup
Figure 5.2 Response of clean oil stirred under vacuum
Figure 5.3 Response of reclaimed oil stirred under vacuum 70
Figure 5.4
Response of reclaimed oil stirred under vacuum
73
after it has been left exposed to air
Figure 5.5
Variation with time of the response of a clean
77
oil sample sealed in a nitrogen ambient
Figure 5.6 Effect of stirring for an oil sample left sealed
under a nitrogen ambient for more than twenty
80
days
Figure 5.7 Response of clean oil at .01 Hz as a function of
time in two consecutive nonstirring'and stirring
84
periods
Figure
5.8 Effect of stirring on
clean oil sample
frequency response
of a
86
Figure 5.9
A magnified view of the high frequency response
of the stirred clean oil sample of figure 5.8 90
Figure 5.10
Initial equivalent loss
high water content
Figure 5.11
Response of used
content
at 1 Hz of
oil sample with
oil with a
93
a high water
94
Figure 5.12 High frequency end of the Cole-Cole plot for oil
97
with a high water content
Figure 5.13 Response of a used oil sample after it was dried
in nitrogen flow
Figure 5.14
Initial equivalent loss
with .5 ppm of ASA-3
at 1 Hz
of oil sample
103
Figure 5.15 Response of oil with .5 ppm of ASA-3
Figure 5.16
Initial equivalent loss at 1
with 50 ppm ASA-3
104
Hz for oil sample
108
Figure 5.17 Response of oil sample wit 50ppm ASA-3
Figure 5.18
Figure
109
Magnified view of the low frequency end of the
response of oil sample with 50ppm ASA-3 shown in
figure 5.17
112
6.1 The lumped element model used
account of the surface effects
to get
a first
115
Figure 6.2 Cole-Cole plot for oil with 50ppm ASA-3 fitted to
116
the lumped model
Figure 6.3
The effect
of varying
on the
lumped model
12
Figure
6.4
response
118
Effect of varying Cs on the radius of
semihalf circle of the Cole-Cole plot
the
119
Figure 6.5 Cole-Cole plot for oil with .5ppm ASA-3 fitted to
the lumped model
121
Figure
6.6 Gain-phase plot for the
clear "dual-hump" shape
Figure
6.7 Value
of
lumped model
lumped parameters
showing
122
obtained from
the
response of oil sample with
function of frequency
Figure
6.8
.5ppm ASA-3
as a
123
Normalized series capacitance and equivalent
Debye length as a function of the high frequency
125
conductivity of various oil samples
10
Chapter I
Introduction
1.1 Overview of Monitoring of Transformers
power transformers
Large
their
predict
deterioration
Monitoring
the
determining
the
are
monitored regularly
and
prevent
before the
service
need for
[1]
failure
in
helps
periodically
transformers
to
transformer
reaches a critical state which will cause it to fail.
Many aging processes contribute to the deterioration of
of the transformer,
the status
damaging processes
the
of
oxidation
(i.e. decomposition and
can be chemical
insulating
These
essential.
is
levels
acceptable
above
increase
of their
and the detection
decomposition
oil,
of
the
cellulose insulation, or sludge formation), mechanical (i.e.
corrosion,
vibrations,
these
processes.
aging
phenomena
are
condition of
in
correlation often
A
discharges).
blockage
or electrical
pipes)
cooling
or
triggered
In
by
nature (i.e.
exists
common
in
the
electrical
between some
cases,
some
high moisture in the
the flow
of
various
such
causes,
of
aging
as
a
transformer causing both
the deterioration of the dielectric strength of the oil, and
the decomposition of the cellulose insulation. In some other
cases, one
the
of these phenomena can lead
decomposition
weakening
in the
of the
cellulose
mechanical support
to another, such as
insulation causing
of the
a
windings, and
leading to an increase of mechanical vibrations.
The
oil
filling
information about the
mirrors
most of
transformer
monoxide and
the degradation
events. For
the detection of these
insulation.
This
technique
of
detecting
content of the oil
by examining the gas
widely practiced by the
established [3] [4] and is
is well
example, the
in the oil reflects the degradation of
two gasses dissolved
damaging processes
important
insulation generates carbon
carbon dioxide [2], so
cellulose
carries
general state of the transformer, and
of the cellulose
thermal aging
the
the
utility companies in the industry.
In
testing the
transformer
on
the transformer.
by obtaining field samples of oil from
unit,
and
transporting
they are analyzed. This
laboratory where
relies heavily
the status of
oil to monitor
These tests are done
the
industry
the utility
general,
them
to
the
style of sampling
limits the frequency of tests on the
the oil is costly, and
oil. Fast developing events cause failure of the unit before
by
detected
being
limitation of this technique
the
required
handling
transporting cause a
technique
special
Another
of
the
time. The time delay
oil in
sampling
and
loss of critical information and limit
of the analysis [5] . An on-line continuous
the reliability
monitoring of
monitoring.
is the change of status of the
sampling time and testing
oil between
and
of
style
this
transformer oil
of fault
sensors
can provide a
detection. The
tuned
degradation processes is
to
the
more reliable
development and
detection
of
use of
various
an attractive possibility. In this
work the
use of microdielectric sensors
in transformer oil
is investigated.
1.2 Background on Microdielectric Sensors
sensors are
Microdielectric
designed
to measure
the
dielectric properties of a surrounding material. The sensors
developed by
were
Prof. S.
D.
Senturia and
coworkers at
M.I.T., and were used to measure the impedance of thin films
(6]
, and
consists
to study
of
a
the curing
pair
of planar
determined by
gate) assumes
impedance
a
between
respective impedances to
the current
which causes
question
two
depends on
electrodes
and
the
their
ground. The floating gate controls
effect transistor used
circuit (figure 1.2)
a reference FET,
assume the same
material. A
Lee [8] under
could be
manufactured on
potential as
floating gate potential
gate indicates the dielectric
surrounding
done by
the
the gate of
ratio of the
the driven
electrode (the
response signal. The floating gate potential
sensor, to
gate. The
potential which
via an interface feedback
is obtained
of the surrounding
and the other
flowing through a field
to amplify the
the
electrodes
the electrodes (the driven gate) is driven
alternating potential,
mutual
interdigitated
the dielectric properties
material. One of
floating
The sensor
, with an equivalent impedance circuit which is
(figure 1.1)
by an
[7].
of resins
finite
to that of
properties of the
difference simulation
the assumption that
modeled
the floating
as homogeneous
was
the material in
and no
surface
Driven Gate
A-
Figure 1.1
Top view of sensor [8]
14
+1volt
CF
Signal
generator
Figure 1.2
Feedback circuit [8]
phenomena
curves
are
present at
were generated
interface, and
the
for converting
potential ratio
the
dielectric constant and loss. The sensors
response into the
temperature based on the
a measurement of the
provide also
calibration
current flowing through a pn junction built into the sensor.
1.3 Overview of this Work
Current literature which
phenomena in
and associates trends
transformer oil
of transformers,
in the deterioration
in the
Moisture, a basic
aging processes is summarised.
loss with
factor
describes dielectric and loss
is discussed
potential use of the microdielectric sensors
because of the
to monitor water content in oil.
The
microscopic size
makes them
the sensor
case of
response of
the sensor can
the
surface
properties
adsorption of
the
of the
water or
sensor's
conditions is
identification
phenomena have
to
sensor/fluid
response
under
needed to be able to
sensor
and
other
medium, the
a double layer
changes of
interface due
other contaminants. The
surface-related phenomena.
microdielectric
material. In
a surrounding
be sensitive to
interface, and
at
study
of
the
surface
promise for sensing
the
to
ability to
flexible
boundary
account for surface and
With such a
becomes
sensors
occurring at the
with the surrounding
transformer oil as
forming
analyze
the microdielectric
sensitive to surface phenomena
interface of
the
of
flexible model, the
basis
for
the
phenomena.
Such
changes in the
oil not
reflected in
a sensitive way
in the bulk
conductivity and
permittivity.
A
method for
sensor
solving for
geometry based
developed.
This
on a
approach
the
electric field
transfer relation
facilitates
the
in the
approach is
inclusion
of
surface phenomena when simulating the response.
Experiments
are
performed
to
determine
the
characteristics of the sensor's response in transformer oil.
of sensors to the water
The sensitivity
is
examined.
The
effects
of
induced
content in the oil
motion
and
added
contaminants are studied.
Chapter two
of this thesis
gives a background
on the
dielectric properties of transformer oil, and their response
to aging events.
solve
for
relation
developed
Chapter three describes the method used to
the field
approach. In
in
the sensor
chapter four,
to predict
is used
the
based
the
on a
transfer
numerical method
response of
the sensor
under various bulk and surface conditions. Experimental work
is
described in
analysed
chapter
in chapter
needed to tailor
five. Results
six, and
are discussed
suggestions of
and
further work
the sensors for transformer monitoring are
provided in chapter seven.
17
Chapter II
Dielectric Properties of Transformer Oil
exact
undefined
derived
The physical,
from crude petroleum.
of the
performed
A
the
transformer.
large
number
samples
oil
These
of
[9]
from
extracted
the
in the
usually
are
an
operating
various
physical,
properties which are significant in
of the
the state
to meet
used properly
tests
measure
test
electrical and chemical
determining
to be
is
chemical, and
transformer oil need
industrial standards
transformer.
oil
Mineral transformer
composition.
electrical properties
specific
hydrocarbons, with an
is a mixture of
Transformer oil
insulating
oil, and
of the
transformer.
To be cost
to be
effective, an on-line monitoring system has
selective in the
properties it monitors.
observe is increased conduction in the
properties useful to
oil.
The microdielectric sensor
the
dielectric
frequencies.
properties
Measurements
frequencies ranging
at
especially
loss
and
low
very
permittivity
to 10 kHz can
from .005 Hz
sensor's low frequency
for measuring
is designed
of
monitoring system.
the on-line
One of the
at
be used in
But, the usefulness
of the
capacity in deducing bulk properties
remains to be determined.
Measurements of
revealing
water and gas content
indicators of a
measurements.
in oil are more
transformer condition
The microdielectric
sensor is
than loss
likely
to be
oil, or
content in the
measurements of water
tailored for
dissolved gasses and other byproducts of degradation.
In
oil
transformer
are
aging phenomena with
reviewed.
The
are
for
in
the
correlates various
increase of loss of transformer oil is
of
water
content
is
also
the observed sensitivity of the sensors to
aspects of
discussed:
phenomena
background
Literature which
significance
discussed due to
humidity. Two
as
discussed
experimental analysis.
and loss
dielectric
chapter,
this
the
water content in
effect
of
water
on
transformer oil
the
dielectric
properties of oil, and the relation between water content in
oil and the general status of the transformer.
2.1 Dielectric and Loss Phenomena in Transformer Oil
The
general
is not
of
behavior
dielectric
well understood,
liquids
insulating
and
is still
in
a research
topic. Losses in insulating liquids like transformer oil are
generally attributed to three basic mechanisms, namely ionic
conduction,
orientation,
dipole
polarization [10]
space
and
charge
[11] . Because ionic space charge can link
the motion of ions to the convection of the liquid, there is
also a
complex electrohydrodynamic phenomena
mechanical-to-electrical
evident
turbulent
in
half of
our measurements
mixing
on
the
this interaction
demonstrating
diffuse
involved. The
will be
the effect
double, layer
of
existing
between an electrode and the oil.
The
loss due
to dipole orientation
19
is caused
by the
of
rotation
viscous
thermal
to
subject
dipoles
For a
an alternating field.
disorientation in
response to
single relaxation
time Debye model [12] , often represented
loss (i.e. imaginary
complex dielectric constant, the
by a
part of the dielectric constant) s"r is:
r
oa
i.+(W-C) 2
where
Es
0
frequency
free
is
static
permittivity
space eo
refraction of
time
the
relaxation
(equivalently
multiplied
the oil n2),
constant. The
(i.e. the
permittivity,
by the
and
E~
high
of
the index
of
the dipole relaxation
permittivity corresponding
real part
the
the permittivity
square of
r is
is
of the
to
a Debye
complex dielectric
constant) is given by the following equation.
E'=0" +
r ra
1+(or) 2
(2)
The tangent of loss caused by this mechanism is
tanSr =
+0(
)2
(3)
This function has a peak around the frequency w0 as given by
I (--)2
The= existence
of many types of dipoles cause the response (4)
to
The existence of many types of dipoles cause the response to
deviate from the ideal Debye model, but the general behavior
of dipole
orientation remains similar
relaxation time
model. The dipole
increases linearly
is a
the oil, and thus
of temperature [ll].
oil at room temperature
a single
relaxation time constant
with the viscosity of
strong function
the order of 10
to that of
For transformer
the frequency of maximum loss is on
Hz. If the dipole relaxation time constant
is assumed to be [12] [13]
4nn ad3
kT
-
where
n
constant,
(5)
is
the
viscosity
T
the
temperature,
of
the liquid,
and
ad
k
is the
Boltzman's
radius
of
dipoles, then the size of the dipoles is on the order of few
Angstroms.
The
dipole
orientation
degradation. Hakim
o, decrease,
rotating
hydrogen
loss
is
(13] shows that after
sensitive
oxidation of oil,
which suggests an increase in
dipoles.
bond
surroundings. At
He
suggests
develops
between
high frequencies
that
the size of the
with
the
to
oxidation,
dipole
(on the order
and
a
its
of 10 Hz)
where losses due to ion conduction are insignificant, dipole
orientation is the dominant factor.
The ionized species
causing ion migration are believed
to be ion-clusters or molecular aggregates in which ions are
surrounded
by
other
molecules.
related to conductivity by:
Ion
conduction
loss
is
(6)
=-
where
ec
is
the loss
conductivity of
due to ionic
the oil, and w is
conduction, a
is the
the angular frequency of
excitation.
The
ion
concentration
migration loss
is
of
within
impurities
introduced to the
The
natural
losses
impurities.
additional
suggests
to
those
the
oil.
charge
transformer
obtained
Hydrogen ions
that
dependent on
the
Impurities
oil produce higher ionic conduction [14].
contaminants in
similar
highly
are
carriers
oxidation
by
oil cause
adding
proposed by
in
transformer
causes
an
ionic
artificial
Hakim [12]
oil,
increase
and
of
as
he
their
concentration due to the formation of acids.
Values
of ionic
conductivity in transformer oil are
-12
typically less than 1012 mho/m. The conductivity assuming a
concentration
of ions
n, with
an
average mobility
b and
charge q is
(7)
c = nqb
Typical values
at room
for the mobility of
temperature are
ions in transformer oil
-9
m
V-sec 13].
order of 10
on the
Assuming that the ions carry a single electronic charge, the
concentration
mobility is
Assuming
of
ions
on the order
that the
based
on
the mentioned
of 10 1 6 m - 3
mobility
of ions
value
of
at room temperature.
is
given by
Stoke's
equation [14] [15]
b =
(8)
6bna
where a i is
the radius of the ion, an
oil with a viscosity
a carrier mobility of 10 -9 should have
and
of 510-2 K-sec
2
ions with a radius on the order of few Angstroms.
Ion
migration
phenomena.
The
and dipole
type
third
insulating liquids, namely
interface phenomena. In
occur
the
between
insulation,
and
of
are both
dielectric
bulk
phenomena
in
space charge polarization, is an
homogeneous liquids, the interfaces
liquid
space
orientation
the
and
charge
solid
electrodes
orientation is
a
or
strictly
surface phenomena.
The
effect
electLodes
of
(i.e.
space
the
charge
electrical
investigated thoroughly
accumulation
double layer)
in electrolytes. [16]
near
has
the
been
. Space charge
in dielectric solids and liquids creates a similar electrode
polarization
effect. Studies on
low frequencies show
dielectric solids
at very
an increased capacitance attributed to
space charge accumulation (17].
Interfaces
of
solids
with
dielectric
liquids
show
evidence of space charge effects demonstrated by an increase
of
loss
the capacitance
from
the
of the structure and the departure of
1
Commercial paper
ionic
dependence.
capacitors impregnated with
measured capacitance at
mineral oil show an increase of
low frequencies (18]. Thin films of
23
mineral oils
clear
[19) and other
space charge
insulating fluids [20]
effect, with
stronger
show a
influences with
increasing temperature or decreasing layer thickness.
Solid/liquid interfaces
of
ions
at
the
have two aspects, accumulation
interface
as
space
charge,
or
an
adsorption/absorption process at the interface [21] [22]. An
adsorbed
layer
interface.
effect
can
Such a
if an
create
surface
electric
a
conducting surface
conducting layer
field is
applied
at
the
will have
parallel to
an
the
interface.
2.2 Effect of Transformer Oil Aging on its Loss
Several
studies
have
conduction and dielectric
to
degradation. A
increase
of loss
accelerated aging
conditions.
increase
sample.
using
indicates
rather
have
molecular
change
samples of
been in
study shows
et
spectroscopy
oil obtained
various
shows an
of an
aged oil
in the
from
oil
techniques,
Crine
were
which
by impurities
et al.
[25]
transformers which
lengths of- time. Their
a tendency of increase of loss
transformer, but the
[24]
was caused
in composition.
service for
al.
changes
by
under overload
dc conduction
that the deterioration
than a
examined
by El-Sulaiman
various
of
no apparent
samples degraded
in transformers operating
no
change
[23] reveals
transformer oil
high field
However,
detected
in
the
properties of transformer oil due
study by EPRI
A study
in the
investigated
with the age of
values for samples of similar age were
scattered
over
two
of
orders
magnitude. The
of
investigated by Cesar
on transformer oil was
thermal aging
effect
[2 6 ], and an increase of loss with aging was detected.
The loss
rrequency
in most studies
measurements of
effective in
is measured at
the loss are
likely to
determining transformer oil
loss is larger
frequencies the
60 Hz. Lower
be more
condition. At low
and thus less
sensitive to
noise.
2.3 Moisture in Transformers
Moisture can leak into transformers through gaskets, or
be
generated
internally by
insulation. Being
of
insulation
transformers
a
degradation
the major
of the
cellulose
cause of deterioration
transformer,
distribution
moisture
has been investigated
of the
in recent
in
studies [27]
[281.
Water
dissolved
conductivity.
transformer
in
Guizonnier [29]
ionization centers injecting
fluid.
Furthermore,
dielectric
strength of
the
oil
indicates
increases
that water
its
forms
conduction ions throughout the
dissolved
water
transformer oil.
factor in
deterioration due to moisture
losses in
the paper insulation. Due to
reduces
the
Another important
is the increase of
the difficulty of a
direct measurement of water content in the paper insulation,
water content in the oil can
be used as an indication of it
[30-.
Water content in oil is currently determined by drawing
transformer, and analyzing them [31] [32].
samples from the
Selectively
measurement of
coated
sensors
water in
moisture sensitive
could
the transformer oil.
sensors could
the cellulose insulation
an
provide
on-line
In addition,
be covered by
samples of
and used as a direct indication of
the amount of water held by the paper insulation.
Chapter III
Field Solution Using a Transfer Relation Approach
complexity of
to the
Due
In this
phenomena.
used
is
relations
chapter an
to
sensor's geometry. The
on transfer
approach based
the
find
the
for
field solution
approach described here is generally
inhomogeneous media
and smoothly
to piece-wise
applicable
account for surface
approach is needed to
general flexible
oil, a
the surrounding transformer
sensor and
between the
solid/liquid interface
the
interfaced by the microdielectrometer. This includes diffuse
layer, adsorbed
double
surface layer,
injected into
ions
solid coatings, and other inhomogeneities of interest.
solution developed
The field
four to
calculate the equivalent circuit
sensor,
and
thus predict
the
in chapter
here is used
sensor's
parameters of the
response with
an
arbitrary load.
An analytical
of
the numerical
limit is provided to
approach. Solution
which might
prove necessary
if the
coated with
special materials
check the validity
for a
coated sensor,
to be
sensor is going
to sensitize it
to specific
gasses or other byproducts of degradation, is outlined.
3.1 Geometry of Sensors
The
sensor
consists
of
an
interdigitated
set
of
aluminium electrodes, separated from a ground substrate by a
layer
of silicon
dioxide with
27
a thickness
h as
shown in
figure 3.1.
(i.e.
from
The fluid whose
transformer oil
the sensor's
electrode
and
both
sides
the
separation
between
fluid
and has
the silicon
above
permittivity
below and
st.
The width
between
two
of each
consecutive
axis.
The
dioxide and
silicon
dioxide has
ox
the
the
has
a
fluid has
conductivity
interface the
just above the interface are
no
The interface
a surface
surface dielectric constant
sensor
At
extend to infinity on
the permittivity
and a
the
to extend
total period of the structure is X=4w.
the x
conductivity as
be measured
is assumed
structure is assumed to
of
conductivity
in this case)
surface to infinity.
electrodes is w. The
The periodic
properties are to
es. The
a
and
two surfaces
a
just
labeled "a" and "b"
respectively.
3.2 Boundary Conditions
The
driven electrode
amplitude
with
electrode
is
V0
and
an
frequency
to
assumed
is at
be
alternating potential
w,
grounded
and
the
floating
.
The
boundary
conditions are:
1) At y=0 the potential is zero.
2) At y=* the electric field goes to zero.
3)
The potential
is
continuous at
y=h
(i.e. at
the
interface between the sensor and the fluid).
Since the electrodes
represent a linear system, the
equivalent circuit obtained from the field solution with a
grounded floating gate can be used to predict the response
for any other load.
ow0 ·
m
m.
m
m.m
-
...*
m
Ia,
*C
W.
ý ýa 4mbolmodm
m oýaý
%ý
0ý
**
a
ý
ý
040M
0
ý
0 ý
0
ý
ý
29
4) At y=h the potential is constrained at the electrodes
region,
while
charge
is
constrained
to
satisfy
conservation of charge at the silicon oxide region.
The fourth
a mixed boundary.
boundary condition represents
The constrained potential at the electrodes implies:
Oa(x)=VO
for -a < x < 8
8
ea(x)=4
for
(9)
(9)
xK
38xIj
The surface charge at the silicon dioxide/fluid interface of
is:
If=E
~
a x)c5X
ax,)+
Ebb x)-%Yyk
ax
Sox
for
lI < lxi< 13-•
86
(10)
a
in the surface charge implies that surface
sx
ax
dipoles can polarize in the x direction , creating surface
Including
charges. By
conserving the surface charge
at the interface
we get:
3f
at
b
a
s
y
y
ax
The surface
=
and the
current K sJy
for I21
8
<lxi <<I
8
(11)
normal current densities Ja
and Jb are:
y
This term is negligible in
oil/sensor interface.
the
case of
the transformer
a(x)=0
y
(12)
(13)
J (x)=o Ey (x)
)
E a ( x)
(14)
Ks(x)=os
Substituting
the above
assuming steady
& as
state and uniform
are both
condition
for
equations into
equation
and
surface properties (i.e.
independent of x),
we get
conservation
charge
(10),
at
the
the boundary
fluid/silicon
dioxide interface.
jw
(
j y
EEb(x
-sEox Ea(x)
y
)
) +c
Eb(x)+ (as+jwes
X
for
a
aE (x)
8i2l<
ax)
3x
8
(15)
Ixl < 1U i
Normalizing as follows:
oOM
E OX
9
ox
=
OM
O
=
s
ox_
O$
We can write the normalized form of equation (15)
(j%-j
aEx(x)
ax
b
2
)E b(X
for IlI
Underlined
3
< lxl < IL3
(16)
-equation
numbers
indicate
a
normalized
equation.
The
solution for
conditions
in
the field has
addition
to
to satisfy
satisfying
a
these boundary
bulk
equation
(Laplace's if space charge is assumed to be zero).
3.3
A
Transfer Relation
Approach
for
Solving the
Mixed
Boundary Value Problem
The method used in solving for the field satisfying the
boundary
conditions given
in
section 3.2
is
based on
a
Fourier representation of the solution. The potential at y=h
and the normal electric field at surfaces "a" and "b" can be
written in terms of their Fourier components.
0a (x)=b(x)= z
0a(n) cos( 2nnx
E (x)=
Ea(n) cos( 2
b
E (x)=
W "b
2nnx
Ey(n) cos(..1
If the
potential at the sensor-liquid
known,
then oa(n)
)
can be
(17)
found
interface 0a(x) were
using the
Fourier series
relation.
a(n)
a(x) cos(-2
) dx
for n # 0
0
However,
the potential
electrodes only.
at the
interface
At the surface between
the potential assumes
is known
at the
the two electrodes
the appropriate functional dependence
in order
to satisfy conservation of
The interface potential
equation (16).
by
linear
charge as expressed by
with
segments
values
can be approximated
chosen
to
satisfy
conservation of charge.
The potential
is assigned to unknown
values, Vj, at k
different collocation points along the silicon dioxide/fluid
interface as shown in figure 3.2. The potential is symmetric
with respect
axis because of the
to the y
symmetry in the
geometry of the sensor. The first collocation point which is
assigned the
potential V 1 is
-k away from the
edge of the
driven electrode. All other collocation points are separated
by
intervals,
4k
Vk at
potential
which leaves
collocation
Using
A(k+23-1)
we can
equation (18)
point with
the edge
the
of the
each potential Vi corresponds to the
xj =±
point
last
away from
a distance
floating electrode. So
the
find the
along the
interface.
Fourier coefficients
0 (n) in terms of the assigned voltages V..
a
1
(44
S(0)
V
~
n 4k
0 a(n)=
1
V (
3
1 k-1
1+(V+v
k)+2k E 2
V.
n-
2 cos(--)
-2cos( nn
-(k+l)))
(k+3))-2cos( ))
k-L
nI
nn
+k-1 V (2cos( -(k+2j-l))-cos(n-(k+2j+l))
nn
-cos(nn(k+2j-3)))
SV 1 (3cos( -(k+l))-cos(
nn
Vk(3cos( nC(3k-l))-cos(
3n .
(3 k-3 ))-2cos(n))
for n # 0
(19)
Assuming that there is neither space charge in the fluid nor
3
41)
.w
c
0dc
,1
0
in
equation. Relating the
satisfies Laplace's
at
field
conductivity,
and
permittivity
uniform
each region
and that
dioxide,
the silicon
surfaces "a"
and
"b" to
the
of
is one
potential
normal electric
the
potential at
the
of the Fourier components using transfer
interface for each
relations [33] we get:
Ea(n)
y
=
-
X
X
)
a (n)
for n # 0
(0)
h
Eay (0) =
2nn •a(n)
"Oa
2
fo
E~(n)
By
coth(
(20)
differentiating
respect to
the
get the
x we
at the
potential
with
interface
at the
tangential electric field
interface.
(x) =
Ea
a
The
-
(21)
a(x)
ax
tangential electric
assumed
field
for the
interface
potential is shown in figure 3.3 (corresponding to k=5). The
tangential
field can
simple manner.
be
related to
For each segment on the
the unknown
a
Vj in
a_
interface Ex
yielding the following relations:
)
-Vj4Ex()=--(Vf
(x) 4k
j+lj+
for xj<x<xj+ 1 & 0<j<k
8k
Ex(x)=-8(V-Vl)
X
for X =< x < x
-
_V
ax,
x
X(.v,
qw
X
4.)
x
xX,<q
.-<lo
-,
4.)
c,41)
ca
ra
·
C,
X
x4
-"
4X
I
o
,LrI•N
>1>>
I
N
m)
>
I
CN
0
LA
I
v
I
"4
CO,
.9-
'41 *1C)
0)k
0'
L
CO
.4I
0)
for xk < x =< 3(22)
Vk
Ex(x)=8
The tangential field
is antisymmetric with respect to the y
on the positive side of the x
axis, and thus only the field
axis is appropriately represented by the above relations.
With
finite number
a
interface, conservation
of
points at
collocation
of charge
can not be
the
satisfied at
every value of x along the interface as required by equation
approximation can
An
(16).
As figure
within every interval.
this amounts
by dividing
3.4 demonstrates,
equation (11) between
to integrating
the
satisfying conservation of
k intervals, and
interface into
charge
obtained
be
the two
end points x i and xi+l and setting the value of the integral
to zero.
Ks(i+)-Ks(x)
s (i+l s
i
xi+l
xi (at
The points x i are chosen
at mid intervals
xj and
j
into
So
j+1 .
xj
k segments
point. By
+(
y
)
(23)
=0
at the edges of the electrodes and
between two consecutive collocation points
2 and 38 is divided
8
8
around each collocation
the distance between
± -
stretching
points along the interface
this choice of xi the
at which the tangential field is not well defined-are in the
middle
of each
introduce is
segment, so that
not more heavely
any numerical
weighted on one
error they
side of the
segment than on the other. The only exceptions happen at the
two segments
bordering the
edges of the
electrodes, where
I,)
M
44
41
0)
r.l
0
U)
2
/
I
U,
r
r
r
U)
r
I
I
I
I
I
II1
qi
IF
r
I
I+----------r
I
OcEJ
(I,
4-J
U,
0)
U
0
0
.9-4
IIl~
II
the tangential field is discontinuous at the edge of segment
next to
the electrode. It is possible to
the points of discontinuity,
use an average at
but it is closer to reality to
assume that the value of the field at the segment extends to
include the boundary points as expressed earlier by equation
(22).
This
distinction
electrode edges and
interface exists
between
discontinuities
at
the
other points of discontinuity along the
because in the first
case a discontinuity
caused by the
infinite but integrable surface charge at the
contact point
appears even in an
rest
of the
exact solution, while the
points of discontinuity
linear approximation
and would
are artifacts
not appear if
of the
the solution
was exact.
Equation (16) is integrated along each segment j of the
interface between the two points xi
+
= xj -
- and xi+ 1 = x.
and the integral is set to zero as required by equation
(23). The
electric
resulting equation, which
field
to the
Fourier
relates the tangential
components
of the
normal
field, is:
(S5-jes)s (Ex(X
xj8k +--L)-Ex
X
.
+(2n)((e-jo
) "'b
(
xj8ki
)
-- ) +4kc (s-9.yc'9,, )Eb 0
)Eb(n)-E(n)
sin(-(k+2j))-sin(4
-0
the normal
(k+2j-2))
~
(24)
By combining equations
of
A
(19) and (20) the Fourier components
field E (n)
and E (n)
terms of the unknowns Vj as shown below:
can be
expressed in
S (0) =
-
hnh
ky
Ea(n)
y
= -
nnn
V(0
)+
8k
coth(
(V+Vk)+
8k2nik
)
ni
V( (2cos(n
4
) -2cos(
4k
(k+l)))
+ V1 (3cos(nn(k+l))-cos(nn(k+3))-2cos(n
k-l
nn
nk
+ 2
V (2cos( (k+2j-l))- cos(t(k+2j+l))
nn
-cos(4-(k+2j-3)))
+ Vk( 3 cos(
(3 k-1))-cos(-(3k-3))-2cos(3-))
for n # 0
^b
Ey()
=0
Eb(n)
:
V0( 2cos(n)
8(
+ V 1 (3cos(
(k+l)))
-2cos(
(k+1))-cos(
(k+3))-2cos(n-))
+E2 k-1j (2cos( n(k+2j-l))-cos(n-(k+2j+l))
nn
-cos(4-(k+2j-3)))
VkC(3cos( n(
3 k-l))-cos(
(3 k-3))-2cos(3-))
for n # 0
Substituting
for Ey(n)
(20) into equation
and Ex(x)
from equations
(25) and
(24), and evaluating for each segment on
the interface we get a matrix
the vector
(25)
of unknowns Vj.
of the form A*V=X,
Details of the
where V is
coefficients of
the matrix A and the vector R are given in Appendix A.
One of
is the
the issues concerning
the numerical simulation
number of collocation points to
be placed along the
interface, namely k. The steepest change in the field occurs
at the edges, but
the sensor
when the conductivity of the medium above
increases, the field becomes
less steep and the
solution converges with less points. Shown in figure 3.5 are
the
successive
approximations
to
interface for a continuous medium,
at
the
potential is
in the
Also the thickness of
as shown in figure 3.6.
more uniform
potential
and h = 10. For a medium
h
conductivity the decay
a higher
with
the
the silicon dioxide layer influences the number of the terms
rate of the potential
that need
to be used since the decay
increases
with increasing
. In
used has
be larger
points
to be
X
to
general
the number
than the
of
minimum of
X
h(sJs)I and I h(
_-jg )
3.4 The Long Wave Limit
In
limit where
the
electrodes
is
thickness,
the
solution. When
much
longer
numerical
-C
the
separation
between the
than
silicon
the
solution reaches
0, the Fourier
an
two
dioxide
analytical
components of the normal
field as given in equation (20) reach the following limits:
fta
E (n)
h
h
Y
~by(n) <
<<y
E(n)
E
The above
(26)
equation shows the legitimacy
of using the quasi
one dimensional approximation , namely
Ea(x)
y
-
Assuming that I e -J3
neglect the
(27)
h
h
I is not much larger than one, we can
current on surface "b"
compared to the current
a(x)
a(x)
Oi
X
k-10
k-5
a, -
a(x)
€ (x)
x
€
k-20
k-15
Figure 3.5
Successive approximation to the potential
at the interface
-=
h
10
=1
42
alO=
9Z-
a (x)
.a.
0-(x)
[
X
3X
8
wt = 0
wt = I
4
**(x
-
-
-
*a (zr)
I
L_
1
-
--
8
Wt = 2
wt = 3n
2
Figure 3.6
TE"
Snapshots of the interface potential during a cycle.
I = 10,
= , a = o10
43
on surface "a". Substituting for Ea (x) and E(x) in terms of
x
y
second
order linear
becomes a
equation (16)
0a(x)
differential equation.
d20a(X) =
dx2
X
a(x()
(28)
h(es-ja S )
dxs
Solving the
above equation with the
the electrodes 0()1 = 1 and 0( 3 ) =
boundary conditions at
0 we get
a hyperbolic
sine dependence for the interface potential.
3
Sa(x) = sinha(- - x)
sinh-
(29)
Where o~ is defined by:
2
X
h(s -sj )
This solution
is equivalent to the
used by Garverick (61
thin
film on
geometry is
as
the sensor.
The numerical
described
by
and the
equation
different
Figure
(29).
moments
=.1,
-'
during
long wavelength
the analytical solution
between the two
medium parameters
solution obtained
approach for
in good agreement with
interface potential
four
to evaluate the surface impedance of a
the transfer relations
using
transmission line model
3.7
first
and -ss
- =1 at
half
excitation cycle. For a highly conducting surface a
the
interface
potential
electrodes. The numerical
drops
the
= 40
electrodes for h
i =0, •=0 ,
4
the
shows
linearly between
of
-
the
0, and
the
two
solution predicts accurately this
a(X))
0a(x)
I
F7 I
8
WANNOMM
v
83X
8
wt =Tr
wt =O
4
¢ a(x)
a
r
cb(x) l
-
X
-
n
|
-
--
--
3X
3X
8
8
8,
wt
Wt=iTr
T
Figure 3.7
Snapshots of the interface potential during a cycle.
= 40, : S-s.1i a
h
= 0, e = O, a = 1
-
"'
3 r
4
limit.
3.5 Solution for a Two Layers Geometry and an Ambient Field
Changing the boundary conditions requires only a simple
in solving
change
field at
sensor (figure
and
dielectric
a
infinity. The
3.8) has a
an ambient
above the
layer directly
thickness t, a
conductivity o i
above
The material
g1.
constant
we consider
this case
the sensors and
additional layer on
putting an
electric
In
the problem.
the
coating layer extends to infinity and has a conductivity ao
2
and
a dielectric
constant
is constant
field
(in space)
E~. Using
amplitude
c£2. At
infinity the
and has
transfer
electric
frequency w
relations,
the
and an
Fourier
normal electric field at surfaces "a" and
components of the
"b" can be related to oa(n) (see derivation in appendix B).
a(0 )
h
a=
y(0)
b(
yE(n)
X
X
Y
2
na
(
1
s2-_J_
siXnh2nt
2
2nt )+cosh(-)
2nnt
Ssinhsinh(
n# 0
+coth( nnt
Eby()
M2
This
E
equation is
uniform medium.
(30)
the equivalent of
When the ambient
equation (20)
field is zero
for the
and in the
limit of t-w or eil=sC2 and a9l=C£2 equation (30) reduces to
-
*
m...·I
n.~·I
*
1)..
e
O
r
r
r
w
r
CY)
II
·
Frz
e
*
0
-
mU
Sr~
equation
(20)
substituting
because the
for
the
equation (16) we get
the k unknowns Vj.
This addition
medium
normal
becomes uniform.
and
tangential
Again
fields
in
the k equations necessary to solve for
a layer
of
was
to
not only
pursued
demonstrate the generality of the approach, but also because
of
to the
sensitive
oil.
degradation of
a material
sensor with
of coating the
the possibility
of an
The addition
ambient electric field also might be necessary if the sensor
is to be used near an applied electric field.
3.6 Generality of the Approach
The
use of
general class
of mixed
interdigital
electrodes
applications, and
such a
space
bulk, the transfer
[34] can
is
the bulk.
application in
or with the
for a
presence of
charge exists
in the
relations for a medium with space charge
the formation of
can be
equation (20). Other phenomena
the space charge due to the fluid
such as the convection of
the interface
various
be equally useful
If space
a
geometry of
for
often
is of direct
method can
be used instead of
motion, or
used
nonuniform properties
charge in
applied to
be
boundary problems. The
this method
geometry. This
medium with
relations can
transfer
an electrical double
accounted for once
layer at
their appropriate
transfer relations are used.
Other
frequently,
field
and
problems
can be
with
solved
mixed
using
boundaries
appear
this approach.
For
example
the
simple
Maxwell
boundary problem if the box
and
can be
appear
motors
method
solved
frequently
becomes
approach. Mixed
in, electromechanical devices
combination
and
a
mixed
is bent as shown in figure 3.9,
using this
and generators,
in
attenuator
can be
with
the
relations in polar coordinates [35].
boundaries
such
approached using
appropriate
as
this
transfer
V0
-
-
Figure 3.9
A bent-Maxwell's attenuator, a mixed
boundary value problem.
Chapter IV
Gain and Phase Response Under Various Boundary Conditions
Since the response of
the sensor is given as the ratio
of the floating gate potential to the driven gate potential,
the
field
solution
described in
obtained
using the
the previous chapter must
numerical
method
be converted to the
equivalent circuit parameters of the sensor. In this chapter
the response of
and
the sensor is calculated under various bulk
surface conditions.
The effect
of adding
surface and
contact impedance is discussed.
4.1 Frequency Variant Circuit Parameters
The response
of the
sensor is
given as the
ratio of
floating gate potential to that of the driven gate. In terms
of equivalent
circuit parameters which are
shown in figure
4.1 the ratio of the two voltages is:
Vf
Vd
Y12
Y11+y12+Y+Y
where YL
(31)
is the load on
the driven gate, and
the equivalent admittances
Y 1 2, Y11 are
of the sensor. Having solved for
the field in the sensor for a specific set of parameters and
a short
circuit excitation (i.e. for
admittances Y..
excitation
the
and Y12 can
current
Vf=O ) the equivalent
be found. For
to
the
driven
electrodes, respectively id and if are:
a short circuit
and
floating
II
o
--
w
I
s- e
= O
0~j
I
~~.1.
I
I
I
I
I
L
mu
id
d =(Yi22 Y 1
)Vo
i,= -Y2V
J O
L
The
current
(32
to
both
the
obtained
by integrating
surface.
Assuming that
driven
and floating
the current
the
around
number of
gates
is
the electrode
repetitions of
the
periodic structure within the sensor is N and that the depth
structure is d, then the current
of the
per unit length to
each electrode is:
d =2(6 +iws)E aX
)+
((jWS +0
)E•b (x)+j•woEa(X) dX
5ab
2(as--jw~
Nd
)E
X
)+(jwo+0)E(x)+jwoxE(x))dx(33)
8
The current and the equivalent admittances are normalized as
follows:
1
=
_
jws oxNd
Y
eox Nd
Substituting for the field in equation (33) we get:
+2
if-
4h
(~ -jC
j+coth( ~)2a(n)sin(n-)
-16k(Es-jos)Vk
+2 (S-jo+coth(-2n)
a(n)sin(n )cos(n-)
(34)
The equivalent admittances can the be obtained from equation
(32).
YI = -
1
a
2hs
+k
"We
(V _V_Vk
2 nh,
9
a
4h
•a
.2,nnh,•ea
+coth( -
in((nn
(n)sin
values of V. will change
surface properties,
bulk and
n
(n)sin(--)(i+cosnn))
+16k(Fs-jis)Vk
-2," (s-jo
Since the
sk
Jo+COth(---)
+2,E(eY12= V_
CsJ)
)cos(nn)
351
for different values of
have strong
the Y parameters
frequency dependences.
4.2 Effects of Surface Impedance on the Gain-Phase Response
The ratio of the
driven
potential at the floating gate to the
presented
gate is
in terms
and phase
as
conductance
or
of gain
defined below
Gain= -201ogl(
Vf
IVd1)
d
V
Phase=
Vd
For
a
uniform
medium
and
no
surface
capacitance on the silicon dioxide/fluid interface, the gain
phase response was obtained for varying bulk parameters. The
calculated
response
agrees
simulation for the sensor
the calculated
with
the
finite
difference
done by Lee [8]. Figure 4.2 shows
gain-phase response for
54
typical transformer
o
II
a4
I
0Cf
I
0- r%
II
0
I
00I
0L
0
00
1
0
I
0)
I
-1-
04
Cu
cc
00
C)
co
Ci
444
-i
Ca
ctl
4-I
Ca
cc S.
C) 'I
'H
00
"0 I
'v
I,
Cu
Cc
'0
0)
Cu
'H
Cu
C-,
no surface effects. The intersection
oil permittivities and
of the
the
gain-phase curve with
high
frequency
the gain axis
response.
The
corresponds to
frequency
decreases
counterclockwise on the plot, corresponding to an increasing
loss.
A
layer
conducting
the
at
dioxide/fluid
silicon
interface can be formed by adsorption. The effect of surface
conductivity on
the gain-phase response is
a medium of
4.3, for
frequency
gain
zero. The
relative permittivity of
remains
conduction because at
shown in figure
at
its
value
2. The high
with
no
surface
high frequencies, the loss approaches
conductivity starts dominating
effect of surface
once it starts approaching the bulk conductivity.
An
gain
is
interesting phenomena
longer
no
the
frequency
minimum
gain
response
corresponds
high frequency
the
the
once
appreciable compared to
conductivity becomes
high
is that
to
surface
the bulk. The
the
capacitive
voltage divider relation:
Vf
Vd
CI1+C
C=
12
12
(36)
+CL
The mutual capacitance
decreases with an increasing bulk or
surface loss. This decrease of the equivalent capacitance at
higher loss
occurs because the
floating gate develops
applied
potential.
negative phase
electric field next
to the
a negative phase with respect to the
Thus,
also. The
the
conduction
imaginary part of
56
current
has
a
the conduction
0
0
0
I
I
0
I
0
I
0
0
I
0
I
0
a)
0
I
0
0
0
0
I
#
0
Co
I
nI
"
P 0
0
II*
current contributes
a negative component
to the capacitive
current to the floating gate, thus decreasing the equivalent
mutual
capacitance between the
conductance G1 2
for
compensate
reaches a
two electrodes.
part of Y12
(i.e. the imaginary
the
decrease of
and
C 12,
value less than its high
The mutual
)
does not
thus the
gain
frequency limit. As the
frequency increases the mutual capacitance rises back to its
value with no surface conductance, and the gain increases.
The surface permittivity might not be applicable to the
transformer
permittivity
oil measurement,
needs
with
surface
highly polarizable
be appropriate to model
such a
appreciable surface
molecules
to
interface in great concentrations. However,
align along the
it might
very
since an
surface
another surface phenomena
permittivity. The
permittivities
on the
medium with a permittivity of
gain
effect of
phase
various
response of
a
2 is shown in figure 4.4. The
high frequency gain increases due to the contribution of the
surface polarization current.
4.3 Surface Contact Effect
The surface
non
uniform.
conductance between the
If
the
electrically induced
the electric
surface
electrodes can be
conduction
adsorption, the lack
field causes the surface
is
caused
by
of uniformity of
conductivity to vary
along the x axis. The numerical simulation can account for a
spatially varying surface impedance.
An
accumulation
of
charges
next to
the
electrodes
0
-
I
C
I
I
-
I
I
I
I
I
0%
II
e-4
II
p4
O
03
02
0
*03
U,
Cu
.0
0.
I
C
,0
03
4
II
0.I
~rII
03
03) Ca
1.40
o~w
Cu~F
ca
03
Ca
Cu
4-
03
LW,
4-a
causes a
difference between the surface
properties next to
of the interface [36]. From the
the electrodes and the rest
edge of each electrode to a distance a on the interface, the
surface conductivity is
Ssc' The two surface
asc and the surface permittivity is
parameters
0 sc
and esc are referred to
contact parameters because they
as the
properties
next to
the contact,
but their
still surface dimensions (i.e. mhos for
The effect
esc).
of zero contact
for a contact region
figure 4.5
reflect the surface
dimensions are
sc and mhos/sec for
conductance is shown in
w
and bulk and
width a - 5
surface parameters as indicated in the figure.
For a medium with
surface
a small bulk conductivit y and a high
conductivity, the
gain-phase response
has
a dual
hump shape as shown in figure 4.6 caused by the zero contact
conductivity.
gain
Starting from the
decreases with
gain determined by
frequency
causes
hump
on the
limit, the
approaching an
asymptotic
the capacitance of the structure. As the
decreases further,
the gain
region a
frequency,
high frequency
conduction
to increase to
gain-phase
zero following
curve. The
width
determines the asymptotic gain
hump. As shown
through the
bulk
the second
of the
contact
value of the first
in figure 4.6, the asymptotic gain increases
with decreasing a.
60
o
II
So0
0
I
I
0
0
0
I
C
0
0
0 0
ac
0
0
N
0i
ua
0
0
cv.
1o
0
$.4
0
C.
a)
0
4-,
*
0I
I
0
I
I
I
0
0
I
I
riw
II
100
II
4o
(A
62
Chapter V
Experimental Work
Basic
microdielectric sensors in
response
the
characterizing
experiments
of
transformer oil are described in
this chapter. The goal of these experiments has two aspects,
one
concerning
phenomena
understanding
occurring
investigating
the
at
the
the
ability
nature of
surface,
of
the
the
the
surface
second
sensors
to
is
detect
contamination which causes a rise of conductivity.
5.1 The Experimental Setup
Microdielectric
sensors
dielectric properties
sensor, and
uniform medium
system and
IBM
SystemII
A
and
The sensor
Since the
and
the floating
on
based
the
into the memory
of the
gain and phase
of the
the permittivity and loss of the medium.
computer
monitors
interfaces
the
sensors show a sensitivity
to
the
the
The
Micromet
range
of
.005 Hz and 10 kHz.
is immersed in the oil
introduced
with
experiments.
frequencies available are between
dried
is connected to a
table
conversion
provides conversion from
personal
the
The experimental
response (i.e.
assumption is built
response signal to
study
applies a synthesized ac voltage to
measures the
potential).
electrode
to
of transformer oil.
Micromet SystemII which
An
used
shown in figure 5.1. The sensor
setup is
the
are
oil
sample in a beaker.
to humidity, they are
sample in
a
nitrogen
u
aa
o o
0 0
00
ao
0
0c
SM
nitrogen for
a flow of
exposed to
sensor is
ambient. The
about 20 minutes in an empty beaker, then a vacuum is pulled
beaker. Oil is dropped
over the
and
the beaker
nitrogen brings
flow of
a second
into the evacuated beaker,
atmospheric
pressure. The
atmosphere.
The
then sealed
sample is
response of
oils
various
back to
from the
is studied
as
described in the following sections.
In
effects
the experimental results
are evident.
extract the bulk
conversion table
Since the
can not be
phenomena, it
the surface
account for
permittivity
presented, surface
to be
-6
and loss
does not
used to
--. However, the
o
o
values obtained from the conversion table for the equivalent
permittivity
s'
and
loss
e",
while
necessarily
not
bulk parameters, can provide hints on
representative of the
the surface phenomena involved.
5.2 Response of Clean Oil
of commercial purity
Transformer oil
under
vacuum for
degassing the
5.2 were
a few
hours. After
Hz. Figure
gain phase
different sensors, and
between sensors. Figures 5.2a
and phase at
the gain
had stopped
the oil
under vacuum. The results shown in figure
obtained using two
good agreement
sample was stirred
was recorded while it
response of the sample
was still stirred
and 100
contents. The oil
35 ppm water
oil had
was studied. The
and 5.2b show
frequencies ranging between
5.2c shows the
gain vs the
response deviates from the
show a
.005 Hz
phase. The
response of a medium
Gain (dB)
0
II
__
-20
Figure 5.2a
90
.0
1248
.Ig*
-40
__
Figure 5.2
Response of clean oil
sample stirred under
Log grequency (Hz)
vacuum..
Phase
-20
Figure 5.2b
-40
-60
-2
0
4
2
Log frequency (Hz)
-40
Gain (dB)
-20
-20
Figure 5.2c
-40
-60
Phase
Log E"
Figure 5.2d
0.4
0.2
0
-2
0
2
4
Log frequency (Hz)
Log e
.0 .
Figure 5.2e
-1
-2
-3
-2
2
0
4
Log frequency (Hz)
4
Figure 5.2f
n
o
1
2
3
A
with uniform non dispersive bulk and no surface effects. The
dashed line on figure 5.2c indicates the calculated response
for a uniform medium
phase information
loss s",
and
for
is converted to
relative permittivity E'
the permittivity, as
deviates from
occurs
with no surface effects. When the gain
shown in
frequency value of
its high
frequencies
less
1
than
figure 5.2d,
2.25. Deviation
Hz. The
loss
also
deviates from the - behavior (corresponding to a slope of -1
on
the log-log
scale of
figure
5.2e) expected
for ionic
loss. The high degree of noise at higher frequencies of loss
measurement
is
due
to
the
loss
value
approaching
sensitivity range of the device. A plot of E"
Cole-Cole plot)
in figure 5.2f shows
semi circle for higher
it is
shown that the
vs e' (i.e. a
the beginning a small
frequencies followed by a curve of a
at lower frequencies. In
larger curvature
the
the next chapter
properties measured are
not the bulk
parameters but are highly influenced by surface phenomena. A
measurement
General
of the
bulk
Radio bridge
permittivity at
yields
200
a permittivity
Hz using
of 2.26.
a
The
conductivity was below the sensitivity of the bridge.
A
sample
response as
oil. The
of
reclaimed
oil
was
also
shown in figure 5.3 is similar
permittivity is again 2.2
increases for
studied.
The
to that of pure
at high frequencies and
frequencies below 1 Hz.
The small semicircle
is not clearly apparent on the Cole-Cole plot (figure 5.3f).
When the
sample was left
then connected
exposed to air for
again to the
vacuum pump and
few days, and
stirred under
Gain (dB)
I_ _
Figure 5.3a
-20
0
S *
'S·
··
0 4405m.eet@
-40
!·
-2
Figure 5.3
Log frequency (Hz)
Response of reclaimed
oil stirred under vacuum.
Phase
felt`
I0 - -.
*6
S
.
.I.
-
.
.
*
*
-20
Figure 5.3b
*
-40.
-60
I
-2
2
4
Log frequency (Hz)
40Gain
(dB)
-20
0
pp
1
*,
-20
Figure 5.3c
S-40
-60
_
_
Phase
Log Es'
0.4
H
a.
*·..
0*
*U
· ·
·
t''· "'
Figure 5.3d
0.2
·
-
0.0
I
II
II
I
-
I
-2
Log frequency (Hz)
Log E
a.
Figure 5.3e
0.
-1
-2
00
*
-3
Log frequency (Hz)
6
Figure 5.3f
2
3
F'
vacuum,
the
response
(shown
in
figure 5.4d)
showed
an
increased conductivity (at 1 Hz) from 4.6x10 - 1 2 mho/m before
-12
exposure to air to 12.4x10 2 mho/m following the exposure
to air. The
for this
high frequency semihalf circle is more apparent
case with higher
5.4f. Furthermore,
conductivity as shown
the gain-phase plot also
the
smaller lobe corresponding
The
frequency
corresponding
gain-phase and
to the
to
in figure
has two lobes,
higher frequencies.
the
breakpoint
Cole-Cole plots is around
on
the
.25 Hz. This type
of gain-phase response is explained in chapter six.
The
studied.
response of
a
The
was
sample
measurements shown in
24
hours shows
stirred
a
sample sealed
not
high similarity
under vacuum
hours of immersing the
left sealed
shown
It
is shown
obtained
in
taking
the
in figure
of the
5.2.
sample
Three of
the
5.5 were taken in the first 24
was
checked 20
days
20 days later has changed considerably.
figure 5.5
earlier. The
superimposed
conductivity at
similar nature
approaching a
to that
nitrogen and
from t 3x10-12 to 14.5x10
hump of
while
sensor in the oil sample. The sample
under
later. The response
stirred
was
figure 5.5. The response in the first
measurements shown in figure
was
under nitrogen
12
I Hz
response
has increased
. The gain-phase plot also has a
to that
shown in figure
larger gain. The hump
an asymptotic gain of -32
on the
5.4c, but
in figure 5.4c reaches
dB, while in figure 5.5c it heads
towards a gain of -17 dB. When the sample used in figure 5.5
was stirred
the asymptotic
value of
73
the hump on
the gain
Gain
A
(dB)
Figure 5.4a
-20
-40
Fiaure 5.4
Response of reclaimed
oil stirred under vacuum
after it has been left
exposed to air.
-2
0
2
4
Log frequency (Hz)
Phase
Figure 5.4b
-20
-40
-60
-2
0
2
4
Log frequency (Hz)
-40
Gain (dB)
-20
I
a
0
I
. .
U.
-20
Figure 5.4c
-40
-60
I
Phase
_
Log e
I
___
Figure 5.4d
0.4
F
005
*0'PooOp
0
0.2
0.0
I_
4
Log frequency
(Hz)
Log es"
Figure 5.4e
-1
-2
-3
-2
0
4
2
Log frequency (Hz)
I1
Figure 5.4f
1.5
t
1.0
F
0.5
@00
0.0
_
_
3
Gain (dB)
Y
Figure 5.5a
-20
t~t=20 days
E·
*::.;;*
0
I
*"f
t
v
ogggI
ls
D
oe
O
III
.
.
.
I
-40
|
i
Figure 5.5
Variation with time
of the response of
a clean oil sample
sealed in a nitrogen
ambient.
Log frequency
Phase
..
*
. . .
. .
.
..
.
. .
0
.~
•n L ..
- .
(Hz)
.
· 0
t=20 days
06
Figure 5.5b
* 0
gO
-20
a
*
M
*
o
-40
0
*
-60
I-
m
2
4
Log frequency (Hz)
-20
Gain (dB)
1
-
-40
I
- 0
-20
Figure 5.5c
*O
a
S·
-40
t=20 days
-60
Phase
Log s
o
Fiaure 5.5d
.
*
S.
0
t=20
*
days
-1
·
•
*
S
o
|i
.
S
*
e
oo
aa
e
q
. .
as *0
. .
.
.
. .•
. ..
*
-2
-3
..
p
Log frequency (Hz)
Log E
__
1.0
t=20 days
Figure 5.5e
4
0.5
*·
a
*·
a
*·
S.
EU·
0
*Or
eO
0
I
_
_
-0
S:
I
It11
t
e *169 •I:t
_
91 tt 11
.
*
... oo
__
Log frequency (Hz:
S
e..
.
t=20 days
.
Figure 5.5f
a,
S*o
0
P81*
eg
o•°.*
·'
4
*
*
10
S"
phase-plot
radius
decreased to
of the semihalf
Based on
-31 as shown
circle also
on the explanations provided
gain-phase
plot for
follow also
the nonstirred
a secondary hump,
rise after the
in chapter six, the
and the loss
expected to
is expected to
semihalf circle. So the frequency breakpoint
the gain phase plot follows
the gain
phase plot is less
this
approaching
in this chapter
sample is
at which
to
5.6c. The
decreased drastically.
other experiments described later
and based
close
in figure
may be very
than .006 Hz, and
frequency because
the
maxima. For
its local
the secondary hump on
phase
seems to
the stirred
be
sample the
frequency breakpoint is = .3 Hz.
Stirring was investigated further in a freshly immersed
sample. A sensor was placed
The response
on and
the
at .01 Hz was observed
a
response as
equivalent
reduced
permittivity
periods of stirring
permittivity
n 2.1 while
at
nonstirring was
Figure
function of
under stirring.
stirring is
5.8
and loss
The
time
Figure 5.7 shows
for two
consecutive
and nonstirring.
at
.01 Hz
with nonstirring it is
found to
be 2.1
both
as shown in
observed between frequency
were
with
n 2.5. The
with stirring
differences which
some
Both the
this frequency
permittivity at
high frequencies
shows
as stirring was turned
period of 130 hours.
off during a
alternating
in a fresh sample of clean oil.
and
figure 5.8d.
were
typically
scans with and without stirring.
The high frequency semicircle on the Cole-Cole plot as shown
in figure 5.8f is more
apparent in the case of stirring and
80
Gain(dB)
I
__
__
Figure 5.6a
-20
S
stirring
*
.4.
6064
-40
Figure 5.6
Effect of stirring for
an oil sample left sealed
under a nitrogen ambient
for more than 20 days.
s ee
S
6 4
4 4
4 i0-0
p
1
0 d 4 0
0
IS
'S(A
Log frequency (Hz)
Phase
&n
."1
Figure 5.6b
-20
stirring
I
I
I
*
0
*
f
-40
00
H
-60
,
- I9
,
mm
2
4
Log frequencY (IRz)
-40
_·
Gain(dB)
-20
_·
1-7k
2
stirring
I•
Figure 5.6c
e9
d
~-20
$4
* !
** 4
.
-~
*
..
I .*
a
,-40
S
.-60
Phase
Log E"
1.0
Figure 5.6d
0.5
0
PI
Log frequency (Hz)
Log c"
Figure 5.6e
Log frequency (Hz)
Figure 5.6f
3
t
e
a
&
U'•
S
stirring
*
iv'·
II
10
GC
Gain (dB)
0.0
· _
-1. 0e 01
-2.1 e I1
Figure 5.7a
·
_I
I
1-
t
* stirring
* non stirring
-3.1 e 91
********+ ***.I********
*s
mem
-4.1 e 91
9.9
4.
2.1 e 12
e•12
time(min)
eem
6.1 e 12
Figure 5.7
Response of clean oil at .01 Hz as a function
of time in two consecutive nonstirring and
stirring periods
Phase
8.1
**
0
~~I
************
Figure 5.7b
Seem,
-2.1 e 11 I
-4.1 e 91
I'
_
I""
-6.1 e 01
0.0
4.1 e 82
2.9 e 92
time(min)
6.9 e 92
Epsilcon
*4*4*4 *4*4
**
*
4*4*4*4*
OI0 +
SSO
sees
r
2.1
I
S
l···" ·
Figure 5.7c
1.l
I.'
0 e 92
.P4.
i
2.0 e 62
tie(in)
6.1 e 02
4.5 e 92
time(min)
6.5 e 52
***,* 4in
Loss
*~H+
*4*4*4
**++
6.0 e-61
*i
Figure 5.7d
4.
e-01
2.1 e-01
5.1
.g0
2.1 e 12
85
Gain (dB)
'0
Figure 5.8a
-20
-40
Figure 5.8
Effect of stirring on
frequency response of
a clean oil sample.
-Z
0
2
4
Log frequency (Hz)
Phase
+ indicate
stirring
Figure 5.8b
-20
-40
-60-2
0
2
Log frequency-(Hz)
-40 Gain (dB)
•
-20
h
I0
=
Ua
us
1*
1·
Figure 5.8c
-20
tc
-40
-60
Phase
Log E
I
a
0.4
Figure 5.8d
1goo
0.3
I
..... ... "... . .
" :"":
•|eo···o
*011111
· ·
· · · ·
· ·
oo eoo o
0.2
0.1
0.0
I
--|,
-2
Log frequency (Hz)
Log e
a I
I
a
#0
At
8
Figure 5.8e
-1
0
*
ra
I.
f-
&
83
*
88
a
H
a
-2
A
H
a as
U6
a
9
U
U
US
-3
4
Log frequency (Hz)
I
.0 .
-
-
-T
1.0
4.
Figure 5.8f
4.
0.5
E
SI'
58
81.
0.0
--
has a smaller radius.
The high
apparent
in
enlarged,
shown
frequency hump on the
figure
a small
in figure
5.8c,
but
gain-phase plot is not
if
the low
hump appears for
5.9. No
gain
the stirred
distinct hump
tip
is
sample as
was clear
for the
nonstirred sample.
The
basic results
presented
in this
section can
be
summerized as follows:
1)
Fresh
and
reclaimed
oil
samples
show
similar
response and close values of permittivity and loss.
2)
The equivalent permittivity
shows an
by a departure of the
low frequencies, accompanied
loss
from
permittivity
the -
dependence.
agrees with
increase at
frequency
The high
measurement at
a bridge
200 Hz.
3)
The equivalent
loss rises
appreciably
with time
after the initial preparation of a sample.
4) The Cole-Cole plot
a
smaller
radius
frequencies,
and
has a two lobed shape, one with
corresponding
to
the
higher
the
larger
the
second
with
curvature corresponding
to the
lower frequencies.
The
radius
semihalf
of
circle
curvature
of the
high
under
the
decreases
frequency
effect
of
stirring.
5) The
gain-phase plot also has
a general shape with
Gain(dB)
-35
-.-
-34
-33
0
-2
'-4
Phase
Figure 5.9
A magnified view of the high frequency response
of the stirred clean oil sample of figure 5.8
two humps. The transition between the two humps for
stirring the sample occurs
measurements done while
at
a
higher
measurements.
frequency
than
for
nonstirred
the high
value that
The asymptotic
frequency hump approaches before it follows the low
frequency
loss
hump increases with
(measured at
1 Hz
an increase
and
of the
higher frequencies).
This asymptotic value decreases under the effect of
stirring.
These findings combined with the results of experiments done
on contaminated
used to form
in
the
oil and described in
the next section, are
a picture of the mechanisms causing dispertion
equivalent
permittivity
of
the
indicate both
oil
and conductivity
sample.
5.3 Contaminated Oil
The
word contaminated
is
used to
naturally contaminated from use in a failed transformer, and
oil which has been contaminated with industrial additives to
increase
its
conductivity.
studied. Using oil
opportunity to
phase curve,
Samples
of
both
types
are
with an increased conductivity gives the
observe a more complete
compared to oil with
section of the gain
low conductivity. In the
latter, even at the lowest frequency of .005, the ionic loss
is small, and the corresponding gain is small as well.
5.3.1 Oil With High Water Content
An
water
oil sample
contents
of
from a
250
ppm
response
at 1
shown in
figure 5.10. The
loss is
Hz after
was
transformer and
studied.
the sensor is
The
with a
immediate
immersed in
oil is
initial slope of the equivalent
-13 mho/m/sec ). The
.612/min (equivalent to 5.6 10
sample was sealed
scans within
under a nitrogen ambient. Three frequency
the first 48
The sample was
hours are shown
in figure 5.11.
not stirred during the measurements shown in
this figure. The
The
failed
dual hump
response shows little stability with time.
plot is
apparent in
two of
the gain-phase
curves. The response
at t=26 hours has an asymptote heading
towards
-5
a
gain
of
gain-phase plot starts.
before
the
second
hump
of
the
The breakpoint frequency is .03 Hz.
The response at t=30 min has a local maximum of the phase at
z 20
Hz, but the asymptote
curve.
In the
details of the
Cole-Cole
is not clear on
plot shown
in
the gain phase
figure 5.11if
the
high frequency semihalf circle are not clear
because of
the dimensions of
the plot, the
portion of
the plot is shown alone in
high frequency
figure 5.12, and the
small semihalf circle is more apparent. The permittivity was
still decreasing
even at frequencies
permittivity at
varied between
had
a negative
the highest
2.86 at t=30
above 1 kHz,
available frequency of
min and 3.13 at
and the
10 kHz
t=26 hrs, and
derivative with
conductivity at 1
frequency. The equivalent
-10
mho/m at t=14
Hz varied between 1.6x10
hrs and
8.3x10 -1 0 mho/m at
for the
equivalent conductivities are much
t=26 hrs. Clearly
these values
larger than the
'ross
Loss
6
I)
0
10
oil covers sensor
a
time (minutes)
Figure 5.10
Initial equivalent loss at 1 Hz
of oil with a high water content.
I __I
_
A
Gain (dB)
o.
*·
Figure 5.11a
I3
IL
'O0
-20
5
r3e.
S.
'
Figure 5.11
Response of used oil
sample with a high
water content.
-40
'As.
·, ···
I la
F
O t= 30 minutes
A t= 14 hours
St=
26 hours-2
O
t=
26
hours
|-
Log frequency (Hz)
Phase
0
Figure 5.11b
O·
0
S....
S.O
''
-20
0
*50
D
00
0
Cl
o
o
•
-40
*
.
*
R
*.
"d
*9
0
0
-
.
.
-60
2
4
Log frequency (Hz)
Gain(dB)
-40
-20
r-
'N
Figure 5.11c
a
d
e
C..
' ··
-20
r
5
OS"-..
II
I
-40
0A.
-60
_
_I-
I _
-I
---
I_
amma
Phase
Log s
0
Figure 5.11d
E.
*4
L
44.
S.
**0.
·
•
4.
I
Log frequency (Hz)
Log E
I
Figure 5.1le
S
-.
U·
S.·
S.
*
t.
e*.
O0
*
~
S.
0'..
e
0
*
0'1..
a.
4
..
O*
Log frequency(Hz)
300
Figure 5.11f
200
100
0
0
100
200
10
5
0
0
5
10
15
Figure 5.12
High frequency end of the Cole-Cole plot
for oil sample with a high water content.
equivalent conductivities
the various
measured for clean
situations. The large
oil under all
conductivity detected is
attributed mostly to the high water contents in the oil.
A
sample of
the same
contaminated
oil was
dried by
bubbling dry nitrogen through it for few hours. Measurements
done on
were
figure
5.13.
the dried
sample, and
The
measured
loss
results are
has
decreased
shown in
for
all
frequencies, and the equivalent conductivity at 1 Hz is
-12
9.3x10-2 mho/m. The permittivity
for the dried sample
reaches
limit of
frequency
its high
higher
than .5 Hz. The
around
.02 Hz.
The radius of
frequency semihalf
the high
than for the sample
smaller for the dried sample
circle is
phase occurs
of the
local maxima
frequencies
2.0 for
with high water contents.
Basic
oil with
experiments on
results of
high water
contents are summerized below:
1)
Measurements are
much
unstable, but
than
larger
those
show
observed
for
loss values
pure
and
reclaimed oil.
2)
Water plays
an essential
role in
increasing the
measured loss values, as was demonstrated by drying
the sample.
5.3.2 Oil With Additives
An
antistatic
agent is
added
to
purity. The additive used is ASA-3 (37],
oil of
commercial
a product of Shell.
Gain(dB)
I I-
Figure 5.13a
-20
-40
Figure 5.13
Response of a used
oil sample after it was
dried in nitrogen flow.
a
-
I
4
Log frequency (Hz)
Phase
r
P
·
.···
- - ---
~ rr-
---------
C
B
Figure 5.13b
-20
I
-40
r
U.·.. g
-60
·I
I
_,
-z--
4
Log frequency (Hz).
99
-40
Gain(dB)
-20
-20
1
Figure 5.13c
. -20
o
. I
*·
* -40
-60
Phase
'·
Log
-- -"
.8
i
i
Figure 5.13d
PS....
POPS
'
Pip *P·····'
2
1.
4
Log frequency (Hz)
100
Log e'"
Figure 5.13e
II
-1
9·
-2
9O
I
i
Log frequency (Hz)
I
n
Figure 5.13f
6
0
--
-
101
-
Two
samples
of ASA-3
concentrations
with different
were
studied.
One sample was prepared with a concentration of .5 ppm.
The
of
permittivity
the
using the
sample
bridge
radio
measurement was found to be 2.26 at 200 Hz. The conductivity
of the sensitivity of the bridge and was
-11
mho/m. The immediate response at 1 Hz
found to be t 4x10
the edge
was on
after the sensor is immersed in oil is shown in figure 5.14.
the loss is .014/min, or an equivalent
-14
apparent conductivity of 1.3x0 1 4 mho/m-sec.
The initial slope of
slope for the
5.15 shows
Figure
the
The high frequency
without stirring.
stirring and
non stirring is
sample with
this
2.3 and is
in good agreement
1 Hz is =
the conductivity measured by the bridge.
about twice
vs frequency as shown in figure
The loss
and
permittivity for both
bridge measurement. The conductivity at
with the
10- 1
response of
5.15e has a slope
of -1 for frequencies above above i Hz, and below 10 Hz. The
higher frequency departure from the ideal slope is caused by
the
increase of
the
the noise
was mentioned in
signal as
response
level of
relative to
section 5.2.
the
For the
stirred sample, deviation starts at about 1 Hz, while the -I
slope continues
radius
of
the
for the nonstirred sample
high
frequency
till .15 Hz. The
semihalf circle
is
again
smaller for measurements done with stirring.
A second sample was prepared with a concentration of 50
ppm of
ASA-3. The
and permittivity
bridge measurements of
of the sample at
102
the conductivity
200 Hz were respectively
U
5 +
10
sensor covered
with oil
15
20
Time (minutes)
Figure 5.14
Initial equivalent loss at 1 Hz
of oil sample with 0.5 ppm of ASA-3.
103
Gain(dB)
I
Figure 5.15a
•e,
"1.
"! 0
I'
-20
stirring
.3r
00ga.
Figure 5.15
s1
e t)
Response of oil with
0.5 ppm of ASA-3.
fll'
!1s
.............
.....
,
-40
|
17
-2n
Log frequency (Hz
Phase
0
I'
Figure 5.15b
stirring
-20
.0 :
1
*
-40
SI
*
I,*0
I '
r
1 e
-60
I
14
Log frequency
104
I
(Hz)
Gain (dB)
-40
-20
stirring
Figure 5.15c
.-20
I.
0,
0r
I.
JAa
-.
41
ti
.o
1-40
.9
-60
Phase
Log E:
1.5
I,
*.
Figure 5.15d
1.0
*
1
et
stirring . r
.5
.
0
3
I
Log frequency(Ez)
105
Log E"
*8
IS
Figure 5.15e
.5
I
II
i.
stirring
'at
S.
I.
*'.'
(II
.
8*
If
-2
I
Log frequency
C
Figure 5.15f
15
t
.8
.d
,~
.9
0
10
I0,
F
0**
0
06
t-pI
stirring
*1
jd
106
R 00
.9-
(Hz)
10
a
mho/m
and 2.25. The initial response
immersed in
of the
oil is shown in figure
loss at 1 Hz is 1.5/min
The frequency
mho/m/sec.
stirring
conditions
frequency
is
, or equivalently 1.4x10
in
and non
stirring
figure 5.17.
The
high
2.2 for both cases, and the
-9
1 Hz is i 2x10
mho/m. The Cole-Cole plot
permittivity
conductivity at
again shows a
5.16. The initial slope
response under
shown
as the sensor is
is
smaller semihalf circle for measurements done
with stirring. The gain-phase plot does not show clearly the
difference between
the stirred
and nonstirred case,
but a
close look at the low frequency end with very small gains as
shown
in
figure
5.18
demonstrates
the
difference.
The
asymptotic gains for the stirred and nonstirred measurements
are -2.2 and -1.9 respectively.
The
experimental
in
results
this
section
can
be
summerized as follows:
1) The
high
sample with higher concentration
frequency
semi
circle
half
has a larger
and
larger
asymptotic gain value.
2) Stirring reduces
the radius of the semihalf circle
and the asymptotic gain values.
The stirring
trends are the same as noted
for the pure oil
in section 5.2. This similarity points to a common cause for
the
low frequency dispersion
of the
and conductivity.
107
measured permittivity
Loss
15
10
5
)
sensor covered
with oil
Figure 5.16
Initial equivalent loss at 1 Hz
for oil sample with 50 ppm ASA-3.
108
Gain
. (dB)
got 8411its is
I1
Figure 5.17a
-20
'sella
-40
FA
I
Figure 5.17
Response of oil
sample with 50 ppm
ASA - 3
Log frequency (Hz)
Phase
0
Figure 5.17b
-20
-40
-60
-80
0
-2.
2
4
Log frequency (Hz)
109
Gain(dB)
-40
-20
-20
Figure 5.17c
-40
-60
-80
Phase
Log e~
Figure 5.17d
-2
0
110
4
2
Log frequency (Hz)
Log E"
lee~6·r
r('
Ii
stirrIng
I.
I
i
I
Fiaure 5.17e
£
0
I
I
Il
Ut
SI
-2
-
I
I
IL
Ij
Log frequency (Hz)
S,
I
I
Figure 5.17f
100
I
I
CC
*6
I
I
trrn
U
Ii
*
f
*
I
*
S
rS
rS
C
100
111
200
Gain (dB)
-2.0
-2.5
-3.0
L
-1.5
-1.0
-
I
a
Zee·
~
*
St
am
S·
I
stirring
0
S
/
',,
I
-
**
-20
Phase
.
Figure 5.18
Magnified view of the low frequency end of
the response of oil sample with 50 ppm ASA-3
shown in figure 5.17
112
Chapter VI
Analysis of Results
Experimental
results from
that the
response of
uniform
ohmic medium
bulk parameters as
eliminated as
the sensor
above the
previous chapter
show
does not correspond
to a
sensor. Dispersion
of the
caused by hindered molecular rotation is
possible cause
the frequencies
the
of the dispersion
involved. As was discussed
because of
in chapter two,
the time constants in hindered rotations are on the order of
-7
sec.
10
Surface
the cause
two,
phenomena at
the sensor/liquid
of this dispersion.
space charge
interface are
As was discussed
accumulates at
in chapter
solid/liquid interfaces,
and polarizes such interfaces at low frequencies.
Surface
phenomena
equivalent circuit
can be
are
often
parameters [381.
quite complex depending on
involved.
These can include
well
other
as
electrochemical
described in
phenomena
reaction C39]
.
of
The equivalent circuit
the variety of phenomena
space charge
such
terms
as
polarization, as
crystallization,
The equivalent
or
parameters
help in identifying the surface phenomena involved.
In this chapter,
experimental results
of the
is
lumped parameters are used to analyze
described in chapter
parameters with conductivity,
discussed.
The
surface
motion, and frequency
phenomena
experimental measurements are identified.
113
five. Variations
affecting
the
6.1 Surface Effects Charaterized by Lumped Elements
The observed Cole-Cole
chapter
suggest
an
plots presented in the previous
equivalent
circuit
between
the
two
electrodes as shown in
figure 6.1. The elements C1 2 and G12
be the bulk conductance and capacitance. The
are assumed to
series
capacitance Cs
can explain the
half circle
on the
Cole-Cole plot. However, the departure from this half circle
to follow a secondary
path ( assumed to be perpendicular to
the permittivity axis) indicates a parallel conductance path
through Gp.
Measuring
sensor
in
measures
essence
an
an
effective
equivalent
impedance,
permittivity
and
loss
the
as
defined below.
SO =1 Real(Y eq }
s_
Imag(-Yq}
eq
Relating s'
(37)
and E"
z
to the circuit elements, we get:
2
(C 1 2 Cs+C1 2 )w +G1 2
"
(G C s +G (Cs + C
2
s 112
22
) +GpG2
(C +C ) w +GI
s 12
12
This model predicts
plots. Figure
response of
the general shape of observed Cole-Cole
6.2 shows the
oil with
fitted Cole-Cole plot
50ppm of antistatic
earlier in figure 5.7.f).
114
for the
agent (presented
>b
0
04
U
0 c
ri
41-
wca
W
0.
E-W
r_
Wa
s
(U~
v-4r
"0
0
(v=
E-4
VP
115
MM-111
l
do
a
1%
I
.!
I
%8
100
I
rring
stir
It
I'/
'I·
I
50
£
4I
0
\r'
U
LI
g
a
9*1
I a
£
.
84
AL
r
0
100
200
Figure 6.2
Cole-Cole plot for oil with 50ppm ASA-3 fitted to
the lumped model
G
C12 = 2 , XG = .01
12
nonstirring Cs= 240
stirring
116
CS
s
200
Other
data curves
minima and
do not
necessarily show
maxima of loss shown in figure
explained in
terms of the
the local
6.2. This can be
frequency breakpoints associated
with the equivalent circuit. The three frequency breakpoints
are:
G12
(Ci2Cs+C2 )
G,
012
C s+C
s 12
GG
i
2)(38)
W =(
G 2Cs+Gp(Cs+C12)
If w, is considerably smaller than w2 , the loss will exhibit
a clear local
an increase
maxima at w,,
and a local
minima at w3 . With
in the parallel conductance
Gp with respect to
G12 , the two
the bulk
frequency breakpoints become closer
G
to each other. Figure 6.3 shows the effect of increasing -12
plot, and the frequency
G
dependence of the loss. For large 0G1 the half circle is not
1i2
clear on the Cole-Cole plot, and the deviation of the loss
on
the
appears
of the
shape
as
a
frequencies.
change
At very
Cole-Cole
of
low
slope
around
frequencies (i.e.
the
w
breakpoint
<< w.)
e'
reaches C . As shown in figure 6.4,
the size of the half
P
circle is determined by
Cp.
Not all data fit this simple model as well as did the
sample with 50ppm of antistatic agent. The Cole-Cole plot of
the sample
with .5ppm antistatic
117
agent is shown
in figure
I_
10
8
6
4
0
I
7
Fi.gure0 .3a
Log
E''
I
-3
-2
0
-1
Figure 6.3b
1
Log w
2
12
Figure 6.3
The effect of varying G on the lumped model response
C1 2 2, Cs10, andI
2 is shown on each of the three curves
118
_
100
__
10.
100 .
Q•
S.
*
*
·... S
Sa
s
r aI
ee
S.
.•
solo.,
55
S *
r
H0
o,-,
*0e
U.;
-I
100
Figure 6.4
Effect of varying. C on the radius of the semihalf circle
of the Cole-Cole plot.
SG
12= 2,r7 12=. 1
119
6.5 with
the calculated values based
on the lumped element
model. One clear reason for the discrepancy is the frequency
dependence of
C1 2 and G12 as was
Combining the calculated
frequency dependent C 1 2 & G12 with
constant C. and Gp, the
various
frequencies.
gain and phase can be calculated at
The
gain
"two-humps", shape
as shown in
fitted
the
well
to
discussed in section 4.1.
phase response
shows
figure 6.6, but
actual
data,
the
can not be
because
frequency
independent values are not adequate to predict the response.
Cs and G
Shown in figure 6.7 are the
observed
gain
and phase
at
various
needed to predict the
frequencies for
oil
sample with .5ppm ASA-3.
There are three
Cs and
are
Gp. One is the
the same
reality,
as
they
phenomena on
the
aspects to the frequency dependence of
possibility
calculated with
are
influenced
no
by
the field distribution.
effect of
parameters
result of assuming that
geometry
frequency
that
in making
dependent. The
the
surface
frequency of excitation.
C 1 2 and G1i2
surface effects.
the effect
of
The second is
the equivalent
third
phenomena
surface
due to
surface
aspect is
depend
In
on
the
the
A combination of the three aspects
produces the dependence observed in figure 6.7.
6.2 The Series Capacitance
The higher
increase of
measured s'
at low frequency
stored energy. A
double layer can
indicates an
account for
this increase of stored energy. An approximate value for the
120
C1
20
15
10
5
0
0
10
20
30
40
Figure 6.5
Cole-Cole plot for oil with .5ppm ASA-3 fitted to
the lumped model
= .05
C12 = 2 ,
12
nonstirring Cs= 27
stirring
Cs= 6.5
121
-40
C;ain(dR)
.L....'"
0
-20
-20
-40
-60
Phase
Figure 6.6
Gain-phase plot for lumped model showing
clear "dual-hump" shape.
G
C
.4
s
= .1 5
Nde ox
Ndd0o
L
-
-
0
122
2.0
Normalized surface admittance
5
2
1
.5
.2
.1
.1
1.0
Frequency (Hz)
Figure 6.7
Value of lumped parameters obtained from the response
of oil sample with .5ppm ASA-3 as a function of frequency.
123
capacitance
value
involved can
of the
gain hump
be obtained
on
the gain
equivalent normalized capacitance
function
double
of the
length for
to be
For the
previous chapter, the
is shown in figure 6.8 as
Assuming the
conductivity.
layer capacitance
the asymptotic
phase plot.
described in the
various experiments
a
from
2-
,
each case can be obtained.
the
equivalent
effective Debye
As figure 6.8 shows,
the calculated values are of the expected order of magnitude
diffusion coefficient. For an oil
-10
mho/m, and assuming an ion
sample with conductivity of 10
(_ _)t
where D
is the
2
mobility of
10- 9
m
V-sec
as was discussed in section 2.2, the
calculated Debye length is n 2.2 4um.
Stirring shows an effect on the equivalent capacitance.
The
in
decrease
capacitance
is
implied
by
three
observations: a smaller half circle on the Cole-Cole plot, a
lower
asymptotic
value
on
the
gain-phase plot,
and
an
increase of the breakpoint frequency ow. The increase of the
capacitance reflects larger diffused layer thickness for the
stirred sample.
may
be due
For our
to an
case, the decrease
increase
of spatial
of capacitance
decay rate
of the
charge due to turbulence enhanced diffusion. Diffused layers
in electrolytes show a sensitivity to fluid motion [40], but
the tendency
of motion is to thin
down the diffusion layer
thickness.
Since the
Debye length (based
on molecular diffusion)
The factor of two is due to having a
of the two electrodes.
124
double layer at each
Log 0%
Cs-Z
kd (, m )
Log
I
+
-
-1
1
0
-I.
i
*-
'I
+
++
+
-1
p
p
3
Log a x10 12
(mho/m)
Figure 6.8
Normalized series capacitance and equivalent Debye.
length as a function of the high frequency conductivity
of various oil samples.
* nonstirring
+ stirring
125
for iow conductivity samples is on the order of microns, and
since
the separation between
12.5 =m, the capacitance of
same
order as
well
cause a
the measured
with the
measured
bulk. Such
A
limit agrees
corresponding to
large
explain the lack
a small
frequency gain.
high frequency
expected value
frequencies can
of the
change of high
gain in the
permittivity.
themselves is
the surface layer can be of the
the capacitance
capacitance should
But
the electrodes
the bridge
at
capacitance
of deviation in
high
the high
I
2D
diffusion skin depth of ('~-) [41] causes
frequency gain. A
the increase
of capacitance at high
impedance is
frequency dependent due to
depth [42], and such
frequency
can
be
the diffusion skin
dependence can account for part of the
dispersion apparent
conclusion
frequency. A .diffusion
made
in figure
unless the
6.6.
However, no
frequency
dependence
introduced by geometrical factors is accounted for.
6.3 The Parallel Conductance
A surface conduction
interface due
can occur at the insulation/fluid
to adsorption of
ions [43].
Silicon dioxide
shows a surface conductivity in humid atmosphere
thus
the
oil's
water
content
can
affect
[44] , and
the
surface
conductivity on the sensor.
The
time trends
observed in
figures 5.10,
5.14, and
5.16 indicate a slow adsorption process causing the increase
of equivalent loss.
Unless the polarization of the electrodes is taken into
126
account in the continuum model, the surface conductivity can
not
be
calculated. However,
using
the
lumped model,
an
estimate for its order of magnitude can be obtained from the
value
of parallel conduction
for data
with
obtained with
.5ppm
and
conductivities
surface conductivity
-17
10-17 mhos. For oil
pure oil is
50ppm
are on
Gp. The
ASA-3,
the order
the
of
estimated surface
-16
-15
1016 and 1015 mhos
respectively.
6.4 Additional Effects
The
proposed lumped
line at low
model
indicates a
perpendicular
frequencies on the Cole-Cole plot. However, the
observed data shows curvature at the low frequency limit. An
additional
this
capacitance in
curvature.
In terms
capacitance indicates
series with
of
a contact
described in section 4.3.
127
physical
Gp can
account for
phenomena, such
effect similar to
a
what is
Chapter VII
Suggestions for Further Research
Further
work is
concerns the
needed
in three
numerical method
aspects. The
used. The second
first
is further
investigation of
the surface phenomena
involved. The third
aspect
the
needed to
concerns
additional
work
use
the
sensors in an on-line monitoring system.
7.1 Improvement of the Numerical Technique
Further
technique
work
is
described
needed
in
this
collocation
points
section 3.3
is not optimum.
the
electrode
at
the
as
determining
of
the
described
The high field at
in
numerical
The choice
distances
needed at
distribution of points
improve
thesis.
equal
is critical
collocation points
to
in
the edge of
the number
the interface. Using
of
a dense
near the electrodes and a sparse one
away from them can reduce the total number of points needed,
and
improve the
efficiency of
collocation points at xj.=
of points
-
computation. Placing
i cos(&l-)
the k
reduces the number
k+1
8
4
3
required to achieve the same accuracy from k for
equally spaced collocation points to Vik [45].
The numerical technique needs to be used to account for
the polarization
This can
medium
applied
be done
of the electrodes in
by deriving
with space
field where
a continuum fashion.
the transfer relations
charge polarization in
the system
128
can
case of
be linearized.
of a
a small
It is
thin insulating layer at
assuming a
prove adequate
might
approximate solution
An
approach.
relation
to use the transfer
deal with a linear system
essential to
in
from
derived
the electrodes surface
the space
accounting for
charge
effect when combined with a surface conductivity.
7.2 Further Studies on the Interface
Further
work
is needed
and
studied.
conductivity
the
This
may
concentrations
of
ASA-3), as
(i.e.
equivalent
varying
be
of
well
magnitude
the
can
of
be
the
by
discussed
to
adding
further
in this
thesis
investigated
potential applied
be
various
of additives.
types
The
of the double
sample needs
accomplished
as other
surface
interface.
sensor/oil
the additive
capacitance
the
the
characterise the
the observed capacitance
correlation between
layer
at
appearing
phenomena
to
to
The
by
the
electrodes.
Applying a
help
DC field in the vicinity
in understanding
between the
the
nature of
electrodes. An electrically
of the sensor can
the conduction
path
induced adsorption
process will be accelerated by the applied field. The effect
of the applied field and its intensity on the time constants
observed when
provide
the sensor is initially
immersed in oil, can
about the
adsorption mechanism
useful information
causing the conduction path.
A well defined flow with a controlled flow rate need to
be used in
investigating the effects of motion further. The
129
dependence
of the
response on
the flow
rate can
lead to
identifications of critical velocities (if any exists).
Temperature
interface.
The
effects
can
balance
between
concentrations of ions
shed
further light
the
bulk
on
and
the
surface
will vary with temperature, and thus
the ratio of the surface conduction path (normalized) to the
bulk conductivity would change.
7.3
Further
Research
on
the
Practicality
of
Using
Microdielectric Sensors to Monitor Transformers
The ability of
needs to be
sensors to detect degradation processes
investigated. Oxidation of oil will probably be
detectable
with
concentrations
bare
of
sensors.
dissolved
The
water
response
needs
to
at
various
be
studied
further. A good correlation can make the sensors valuable in
measuring water contents in oil. Coatings which can increase
the sensor's
sensitivity to water, or
sensitize it to some
gasses, need to be determined.
An essential factor
in determining the practicality of
using the sensors is their ability to function in the rugged
environment
of
transformer
might prove
pieces
a
transformer.
of insulation
electrodes impeding the
The
to be too
or metal might
noise
high, or
level
in
the
some floating
get deposited
on the
operation of the sensor. Protection
mechanisms such as shielding the sensors from the noise, and
preventing
stray
pieces from
floating
into its
vicinity
should be considered. The obvious test of the ability of the
130
sensor to
function under rugged conditions
it in an operating transformer.
131
is to try using
Appendix A
The
set
equation
of
(25) for
k
equations
each of
obtained
from
the k segments
evaluating
is of
the form
A*V=X, where V is the vector of unknown V. ' s. The matrix A
can be written as a sum of four elementary matrices.
A=A1+(Se-j C)A2+A3+(cs-jCs )A4
The
vector X
can be
(Al)
written as a
sum of
four elementary
vectors.
XR=l+(c9-jaC
)X2+X3+(s-jjcs)4
(A2)
The definitions of these elementary matrices and vectors are
shown
on the
summation of
only
following page.
The only
matrices requiring
the Fourier components are A1
vectors involving
and A2, and the
such summation are
X1 and
X2. The
program "mkcoef.c" calculates these vectors and matrices for
a given
-
The
code of
provided.* The
the
program used
program "simulate.c"
their elementary components
generation
programs
of the
summation
surface
or
dimensions
used only
generates A and
was divided
"simulate.c" so
does not
need to
bulk
conductivity
of the
the simulation
is
X from
according to equation (Al). The
coefficients
"mkcoef.c" and
in
be
that
repeated for
changing the
Once
"mkcoef.c" needs
once, then "simulate.c" can
two
the Fourier
or permittivity.
sensor are set,
132
into the
the
to be
be used to calculate
the response under varying bulk and surface properties.
Solving
the
set
of
linear
equations
with
complex
coefficients requires complex numbers arithmetic subroutines
which are
circuit
provided in the file
elements
of the
subroutine provided by
"convert.c". The equivalent
sensor
are
generated using
the
"cap.c", and the gain-phase response
is calculated by the subroutine in "bode.c".
Elementary Matrices of A
=4k ZE
A[r,c]J=
n
1 coth(-2-nh
--) D(r,c]
n
=
4k Za -11
A2[r,cJ]=4Dr,c]
n
n
D[r,c]=
(sin((k+2r))-sin((k+2r-2))'
2cos(Tn(k+2c-) )-cos(
(k+2c+l))
-cos(-(k+2c-3)))
for c # I,
D[r,l]=
( sin((k+2r))
- sin(
(k+2r-2))
3
3cos(n(k+1)) - cos(n(k+3)) - 2cos(•
D[r,k]
C sin(n(k+2r)) - sin(n (k+2r-2))
)
3
3cos( (3k-1))
4k4k- cos(n(3k-3)) - 2cos(n)
43
A3[r,cJ= -7 (
8 3- 32 (8(r-l)+8(r-k))
hk
hk
A4(r,cc=
4k(28(r-c)-&(r-c~l)-8(r-c-l)+8(r-1)8(c-l)
133
k
+S(r-k)8(c-k)}
Elementary Vectors of X
4k
a
4k
Et
1
--2 coth(
2nnh
) FC[r]
L
X~T1-q
sin( nn (k+2r)) 2oav(k+l))
4k~l
X3([r
sinCT(k+2r-2))
-
0k
2cos(n)
(2k+1)x
2
32hk
X4[r] = 8k 8(r-1)
134
/*
/*
mkcoefn.c
date: 11/27/84
1*
*/
*/
*/
/* This program calculates the matrices Al and A2,
/* and the vectors X1 and X2.
/* The limit on the summation is a maximum t of n
*/
*/
*/
'include <stdio.h>
#include <math.h>
#define PI 3.141592654
main()
{
double al[40][48],a2[48][48],xl[40],x2[40],d,rw,h,pi,trml,trm2,sl,s2;
int n,k,f,r,c,t;
FILE *fp, *fopen();
fp=fopen("coef. p","w");
printf("input #of points\n");
scanf("td",&k);
printf("input normalized width\n");
scanf("tf",&rw);
h=1.8/4.8/rw;
printf("input max number of terms\n");
scanf("%t",&t);
pi=PI/4.0/k;
f=8;
r=l;
while (r<=k)
{
sl=0;
s2=0;
trml=l;
n=1;
c=l;
if(((kt2)==O)&&(r==(k/2+1))) f=l;
if (f==l)
{
al[r-l][c-l]=al[k-r][k-c];
a2[r-l][c-1]=a2[k-r][k-c];
else
while ( n<t)
trm2=1.8/n/n;
trml=trm2*(1.8/tanh(2.8*PI*n*h));
d=(sin(n*pi*(k+2.8*r))
-sin(n*pi*(k+2.0*r-2.8)))
*(3.0*cos(n*pi*(k+l.0))
-cos(n*pi*(k+3.0))
-2:0*cos(n*pi*k));
sl+=(trml*d);
s2+=(trm2*d);
135
n++;
al[r-l][8]=4.0*k/PI/PI*sl;
a2[r-l][8]=4.8*k/PI/PI*s2;
}
printf("A[%d][(]\tn=%d\n",(r-l),n);
C=2;
while(c<k)
{
if(((k%2)==l)&&(r==((k+1)/2))&&(c==((k+3)/2))) f=l;
if (f==l)
{
al[r-l][c-l]=alCk-r][k-c];
a2[r-l1][c-l]=a2[k-r][k-c];
n=1;
else
{
sl=v;
s2=8;
trml=l;
n=1;
while (n<t)
{
trm2=1.0/n/n;
trml=trm2*(1.0/tanh(2. *PI*n*h));
d=(sin(n*pi*(k+2.g*r))
-sin(n*pi*(k+2.8*r-2.0)))
*(2.8*cos(n*pi*(k+2.0*c-1.0))
-cos(n*pi*(k+2.+*c+l.0))
-cos(n*pi*(k+2.0*c-3.0)));
sl+=(trml*d);
s2+=(trm2*d);
n++;
al[r-l][c-l]=4.8*k*sl/PI/PI;
a2[r-l][c-l]=4.8*k*s2/PI/PI;
printf("A[%dl[%d]\tn=%d\n",(r-l),(c-1),n);
c++;
if(((k%2)====)&&(r==((k+l)/2))&&(c==((k+3)/2)))
if (f==l)
{
al(r-l][c-l]=al[k-r][k-c];
a2[r-l][c-1]=a2[k-r][k-c];
else
sl=8;
s2=0;
trml=l;
n=1;
136
f=l;
while (n<t)
trm2=1.8/n/n;
trml=trm2*(1.8/tanh(2.8*PI*n*h));
d=(sin(n*pi*(k+2.0*r))
-sin(n*pi*(k+2.0*r-2.8)))
*(3.0*cos(n*pi*(3.B*k-1.0))
-cos(n*pi*(3.0*k-3.0))
-2.0*cos(n*pi*3.0*k));
sl+=(trml*d);
s2+=(trm2*d);
n++;
)
al[r-l][k-l]=4.8*k/PI/PI*sl;
a2[r-1][k-l]=4.0*k/PI/PI*s2;
)
printf("A[td][Cd]\tn=td\n",(r-l),(k-1),n);
sl=0;
s2=0;
trml=l;
n=1;
while (n(t)
trm2=1.0/n/n;
trml=trm2*(1.8/tanh(2.0*PI*n*h));
d=(sin(n*pi*(k+2.0*r))-sin(n*pi*(k+2.0*r-2.0)))
*(cos(n*pi*(k+1.0))-cos(n*pi*k));
sl+=(trml*d);
s2+=(trm2*d);
n++;
xl[r-l]=8.8"k*sl/Pl/PI;
x2[r-1]=8.0*k*s2/PI/PI;
printf("X[td]\tn=td\n",(r-1),n);
r++;
fprintf(fp,"td\n",k);
fprintf(fp,"%f\n" ,h);
for(r=0;r<k;r++)
(
for(c=0;c<k;c++)
fprintf(fp,"%f\n%f\n",al[r][c],a2[r](c]);
fprintf(fp,"tf\nf\n",xl[r],x2[r]);
fclose(fp);
}
137
/*
/+
/*
/*
/*
/*
simulate.c
date: 8/3/84
/
/
This program generates the matrix A and the vector x.*/
Then it calls the appropriate subroutines to find the*/
vector V, the eqiuvalent circuit of the sensor, and */
*/
the gain-phase response.
#include <stdio.h>
#include <math.h>
#define PI 3.141592654
#include "convert.c"
#include "solv.c"
#include "cap.c"
#include "bode.c"
float fl(r,c,k)
int r,c,k;
{
if((c==0)jI(c==(k-1)))
return(3.0/32.0);
else
return(l.0/8.0);
}
float f2(r,c,k)
int r,c,k;
{
if(r==c)
{
if((r==8)1I(r==(k-1)))
return(12.0);
else
return(8.0);
}
else if (((r-c)==l)11((c-r)==l))
return(-4.0);
else
return(8.0);
main()
int k,r,c;
double al[30][30],a21[3][30],xlE[3],x21[3];
double h,rse,ise,re,rc,ryl,iyl;
char q[5],file_name(38];
struct complex ctempl,ctemp2,se,e,yl,yll,yl2;
struct complex a(38][30],x[30],y(30];
struct polar py;
FILE *fpi,*fpo,*fopen();
printf("input coefficient file name\n");
scanf("ts",file_name);
fpo=fopen("simulate.p","w");
fpi=fopen(file_name,"r");
fprintf(fpo,"output of 'sensor/simulate.c'\n\n\n");
138
printf("input normalized epsilon\n");
scanf("tf",&re);
printf("input normalized conductivity\n");
scanf("%f",&rc);
printf("input normalized surface epsilon\n");
scanf("tf",&rse);
printf("input normalized surface conductivity\n");
scanf("tf",&ise);
printf("input normalised load capacitance\n");
scanf("%f",&ryl);
printf("input normalised load conductance\n");
scanf("tf",&iyl);
e.real=re;
e.imag=(-rc);
se.real=rse;
se.imag=(-ise);
yl.real=ryl;
yl.imag=(-iyl);
fprintf(fpo,"el/eox=%f\tsigma/weox=%f\n",re,(-rc));
fprintf(fpo,"normalized surface capacitance=%f+j(tf)\n",rse,(-ise));
fprintf(fpo,"load capacitance=tf\tconductance=tf\n",ryl,(-iyl));
fscanf(fpi,"td",&k);
printf("#of points=%d\n",k);
fprintf(fpo,"#of points=td\n",k);
fscanf(fpi,"%f",&h);
printf("w/h=%f\n",(1.8/4.0/h));
fprintf(fpo,"w/h=%f\n",(1.0/4.0/h));
for(r=0;r<k;r++)
for(c=8;c<k;c++)
fscanf(fpi,"tf",&al[r][c]);
fscanf(fpi," 1'f",&a2(r][c]);
real_complex(al[r][c],&ctempl);
real_complex(a2[r][c],&ctemp2);
cmul(&ctemp2,&e,&ctemp2);
cadd(&ctempl,&ctemp2,&ctempl);
real_complex(((fl(r,c,k))/h/k/k),&ctemp2);
cadd(&ctempl,&ctemp2,&ctempl);
real_complexC((f2(r,c,k))*k),&ctemp2);
cmul(&ctemp2,&se,&ctemp2);
cadd(&ctempl,&ctemp2,&a(r][c]);
printf("a([d][%d]=tf+j(%f)\n",
r,c,a[r][c].real,atr][c].imag);
fscanf(fpi,"%f",&xl[r]);
fscanf(fpi,"tf",&x2(r]);
real_complex(xl[r],&ctempl);
real_complex(x2[r],&ctemp2);
cmul(&ctemp2,&e,&ctemp2);
cadd(&ctempl,&ctemp2,&ctempl);
realcomplex((-(2.0*k+1.0)/32.0/h/k/k),&ctemp2);
cadd(&ctempl,&ctemp2,&x[r]);
if (r==0)
139
real_complex((8.0*k),&ctempl);
cmul(&se,&ctempl,&ctempl);
cadd(&x(r],&ctempl,&x[r]);
}
printf("x[td]=tf+j(tf)\n",r,x[r].real,x[r].imag);
fclose(fpi);
solve(a,x,y,k);
for (r=O;r<k;r++)
{
for (c=g;c<k;c++)
{
fprintf(fpo,"a[%d][%d]=%f+j(tf)\n",r,c,a[r][c].real,
a[r][c].imag);
fprintf(fpo,"x[%d]=%f+j(%f)\n",r,x(r].real,xfr].imag);
for(r=0;r<k;r++)
printf("y[%d]=tf+j(%f)=",r,y[r].real,y[r].imag);
car_pol(&y[r],&py);
printf("%fexpj(%f)\n",py.mag,py.angle);
fprintf(fpo,"y[t%d]=%f+j(%f)=",r,y(r].real,y(r].imag);
fprintf(fpo,"tfexpj(%f)\n",py.mag,py.angle);
printf("do you want impedance and gain phase results?\n");
scanf("%s",q);
if(q[8]=='y')
cap(&yll,&yl2,y,&e,&se,h,k,fpo);
bode(&yll,&yl2,&yl,fpo );
}
fclose(fpo);
printf("output in file simulate.p\n");
}
140
cap. c
date: 7/26/84
- --
/*
/* This program calculates yll and y12
/* The function is cap()
/* its inputs are:
/1
V a vector of node voltages (double).
/*
pe pointer to normalized complex epsilon.
/*
pse pointer to normalized surface capacitance
/1
h normalized height.
/*
k number of nodes.
/*
fp pointer to output file.
I
*/
*/
*/
*/
*/
*/
*/
*/
*/
*/
cap(pyll,pyl2,v,pe,pse,h,k,fp)
double h;
struct complex v[];
struct complex *pyll,*pyl2,*pe,*pse;
int k;
FILE *fp;
{
double m,pi,d;
struct complex sl,s2,x,trm,templ,temp2,temp3;
int n,j;
pi=PI/4.0/k;
j=2;
cnull(&x);
while (j(k)
cadd(&x,&v[j-1],&x);
m=k;
cadd(&v[@],&v[k-1],&templ);
real complex((3.0/8.0/m),&temp2);
cmul(&templ,&temp2,&templ);
real complex((0.5/m),&temp2);
cmul(&x,&temp2,&temp2);
cadd(&temp2,&templ,&templ);
realcomplex(((2.0*m+1.8)/8.0/m),&temp2);
cadd(&temp2,&templ,&templ);
realcomplex((1.8/4.0/h*PI*PI/8.8/m),&temp2);
cmul(&templ,&temp2,&sl);
real_complex(2.0,&templ);
cmul(&sl,&templ,&s2);
cass(&s2,&trm);
cnull(&templ);
cnull(&temp2);
n=1;
while ( (n<201)&&
( (cgt(&trm,&templ)) 11 (cgt(&trm,&temp2)) ))
{
j=2;
d=(3.9*cos(n*pi*(k+1.0))
141
-cos(n*pi*(k+3.0))
-2.0*cos(n*PI/4.0));
realcomplex(d,&templ);
cmul(&templ,&v[O],&templ);
d=(3.0*cos(n*pi*(3.0*k-1.0))
-cos(n*pi*(3.0*k-3.0))
-2.0*cos(3.8*n*PI/4.0));
real complex(d,&temp2);
cmul(&temp2,&v[k-l],&temp2);
cadd(&templ,&temp2,&x);
while(j<k)
d= (2.0*cos(n*pi*(k+2.0*j-1.0))
-cos(n*pi*(k+2.0*j+1.0))
-cos(n*pi*(k+2.0*j-3.0)));
real_complex(d,&templ);
cmul(&templ,&v(j-l],&templ);
cadd(&x,&templ,&x);
j++;
d=1.,/tanh(2.0*PI*n*h);
real_complex(d,&templ);
cadd(pe,&templ,&templ);
real_complex((l.0/n/n),&temp2);
cmul(&templ,&temp2,&trm);
d=2.0*cos(n*PI/4.0)-2.0*cos(n*pi*(k+l));
real_complex(d,&templ);
cadd(&x,&templ,&templ);
cmul(&templ,&trm,&templ);
real_complex((sin(n*PI/4.0)),&temp2);
cmul(&templ,&temp2,&x);
real_complex((cos(n*PI)),&templ);
cmul(&templ,&x,&templ);
cadd(&sl,&templ,&sl);
real_complex((l.0+cos(n*PI)),&templ);
cmul(&templ,&x,&templ);
cadd(&s2,&templ,&s2);
n++;
real complex(0.001,&templ);
cmul(&s2,&templ,&temp2);
cmul(&sl, &templ,&templ);
printf("n=td\n" ,n);
fprintf(fp,"n=td\n",n);
real complex((16.0*k),&temp3);
cmul(pse,&temp3,&temp3);
/* temp3=16 kse
real complex(((-8. )*k/PI/PI),&templ);
cmul(&templ,&sl,&templ);
cmul(&v(k-l],&temp3,&temp2);
cadd(&templ,&temp2,pyl2);
real complex((8.0*k/PI/PI),&templ);
cmul(&templ,&s2,&templ);
real_complex(1.8,&temp2);
csub(&temp2,&v[g],&temp2);
142
/
csub(&temp2,&v(k-l],&temp2);
cmul(&temp2,&temp3,&temp2);
cadd(&temp2,&templ,pyll);
printf("cll=.%f\tgll/w=%f\ncl2=%f\tgl2/w=%f\n",(*pyll).real,
(-(*(*pl).mag),(yl2).real,(-(*pyl2).imag));
fprintf(fp,"cll=%f\tgll/w=%f\ncl2=tf\tgl2/w=tf\n",
(*pyll).real,(-(*pyll).imag),(*pyl2).real,
(-(*pyl2).imag));
}
143
/*
*/
*/
convert.c
date: 7/17/84
/*
*/
*/
/*This program contains complex arithmetic subroutines.*/
#include <stdio.h>
#include <math.h>
double square(x)
double x;
{
return(x*x);
/* declration of structure used */
struct complex ( double real;double imag);
struct polar ( double mag;double angle };
struct bod (double gain;double phase};
/* conversion procedures */
car_pol (pc,pp)
struct complex *pc;
struct polar *pp;
{
(*pp).mag=sqrt(square((*pc).real)+square((*pc).imag));
(*pp).angle=atan2((*pc).imag, (*pc).real);
}
pol_car (pp,pc)
struct complex *pc;
struct polar *pp;
(*pc).real=(*pp).mag*cos((*pp).angle);
(*pc).imag=(*pp).mag*sin((*pp).angle);
}
pol_bod (pp,pb)
struct bod *pb;
struct polar *pp;
{
(*pb).gain=2*1logl0((*pp).mag);
(*pb).phase=(*pp).angle*180/3.14;
}
car_bod (pc,pb)
struct bod *pb;
struct complex *pc;
struct polar p;
car_pol (pc,&p);
pol_bod (&p,pb);
}
realcomplex(a,pc)
144
double a;
struct complex *pc;
{
(*pc).real=a;
(*pc).imag=g;
}
/* arithmatic prcdures */
cmul(pcl,pc2,pc3) /* c3=c2*cl */
struct complex *pcl,*pc2,*pc3;
(
struct polar ppl,pp2,pp3;
car_pol (pcl,&ppl);
car_pol (pc2,&pp2);
pp3.mag=ppl.mag*pp2.mag;
pp3.angle=ppl.angle+pp2.angle;
pol_car (&pp3,pc3);
cdiv(pcl,pc2,pc3) /* c3=cl/c2 */
struct complex *pcl,*pc2,*pc3;
{
struct polar ppl,pp2,pp3;
car_pol (pcl,&ppl);
carpol (pc2,&pp2);
pp3.mag=ppl.mag/pp2.mag;
pp3.angle=ppl.angle-pp2.angle;
polcar (&pp3,pc3);
}
cadd(pcl,pc2,pc3) /* c3=cl+c2 */
struct complex *pcl,*pc2,*pc3;
{
(*pc3).real=(*pcl).real+(*pc2).real;
(*pc3).imag=(*pcl).imag+(*pc2).imag;
csub(pcl,pc2,pc3) /* c3=c2-cl */
struct complex *pcl,*pc2,*pc3;
{
(*pc3).real=(*pcl).real-(*pc2).real;
(*pc3).imag=(*pcl).imag-(*pc2).imag;
/* c2=cl */
cass(pcl,pc2)
struct complex *pcl,*pc2;
{
(*pc2).real=(*pcl).real;
(*pc2).imag=(*pcl).imag;
cnull(pc)
/* c=0 */
struct complex *pc;
{
145
(*pc).real=0;
(*pc).imag=0;
}
/* c=cx +/
cpow(pc,x)
struct complex *pc;
double x;
struct polar pp;
car_pol (pc,&pp);
pp.mag=pow((pp.mag),x);
pp.angle=(pp.angle)*x;
pol_car (&pp,pc);
csinh(pc)
/* c=sinh(c) */
struct complex *pc;
double r,i;
r=(*pc).real;
i=(*pc).imag;
(*pc).real=sinh(r)*cos(i);
(*pc).imag=cosh(r)*sin(i);
ccosh(pc)
/* c=cosh(c) */
struct complex *pc;
{
double r,i;
r=(*pc).real;
i=(*pc).imag;
(*pc).real=cosh(r)*cos(i);
(*pc).imag=sinh(r)*sin(i);
/* conditonals */
zero (z)
/* z==0 */
struct complex z;
{
if ((z.real==0)&&(z.imag==~)) return(l);
else return(O);
}
/* cl>c2 */
cgt(pcl,pc2)
struct complex *pcl,*pc2;
{
struct polar pl,p2;
car_pol(pcl,&pl);
car_pol(pc2,&p2);
if (pl.mag > p2.mag) return(l);
else return()i;
}
146
/*
solv.c
*/
/*
date: 7/17/84
*/
/*
*/
/* This program solves a set of linear equations with
/* complex coefficients.
*/
*/
#include <stdio.h>
#include <math.h>
solve(pa,px,py,k)
struct complex pa[]([2],px[],py1];
int k;
(
int i,r,c;
struct complex b20[20]z[2],z[2,temp(20],tempz,tempr;
for (r=8;r<k;r++)
/* initialize both b & z 9/
{
z[r].real=px[r].real;
z[r].imag=px[r].imag;
for (c=0;c<k;c++)
{
b[r][c].real=pa(r][c].real;
b[r][c].imag=pa[r][c].imag;
}
}
for (i=g;i<(k-l);i++)
{
coeff */
while(zero(b[i][i])) /* handles the case with a zero leading
{
tempz.real=z[i].real;
tempz.imag=z(i].imag;
for(c=i;c<k;c++)
{
temp(c].real=b([i[c].real;
temp[c].imag=b[i][c].imag;
}
for(r=i.;r<(k-l);r++)
{
z[r].real=z[r+l].real;
z[r].imag=z[r+l].imag;
for(c=i;c<k;c++)
{
b[r][c].real=b[r+l][c].real;
b[r][c].imag=b[r+l1[c].imag;
)
z[k-l].real=tempz.real;
z[k-l].imag=tempz.imag;
for(c=i;c<k;c++)
b[k-l][c].real=temp[c].real;
b[k-l1[c].imag=temp(c].imag;
147
}
for(r=i+l;r<k;r++) /* generates the new set of equations */
cmul(&z[i],&b[r][i],&tempz);
cdiv(&tempz,&b(i][i],&tempr);
csub(&z[r],&tempr,&tempz);
z[r].real=tempz.real;
z[rj].imag=tempz.imag;
for(c=i+l;c(k;c++)
cmul(&b[i][c],&b(r][i],&tempz);
cdiv(&tempz,&b[i)[i],&tempr);
csub(&b[rl[c],&tempr,&tempz);
b[r][c].real=tempz.real;
b[r][c].imag=tempz.imag;
}
}
for(r=k-l;r>=8;r--) /* back subtitution */
cdiv(&z[r],&b[(r][r,&py[r]);
for(c=r+l;c<k;c++)
{
cmul(&b[r][c],&py(c],&tempr);
cdiv(&tempr,&b[r][r],&tempz);
csub(&py[r],&tempz,&tempr);
py[r].real=tempr.real;
py[r].imag=tempr.imag;
}
I
148
Appendix B
Equation (30)
and
boundary
can be derived
conditions
at
using transfer relations
y=t+h.
The
potential
is
continuous at surfaces "c" and "d" shown in figure (3.8).
'd
"
0 (n)=oc(n)
The normal
(Bl)
electric field at surfaces "c"
and "d" obey the
following equation:
Id
(G 2-j£2)E y(n) =(e,1-j91 )E (n)
The normal
field can be
(B2)
written in terms
of the interface
potentials.
E (n)=
nn
c(n)
n#O
y
d
a
yEn)=
_nn
-coth(
Xnnt 0c
coth((~n)
+ ')0an)(B3)
(B3)
sinh
Combining equations
(B2) and (B3) we get
(C(n)
in terms of
0 (n).
S(a
(n)B4)
02
%2
2-nnt
2nntsinh(
)
+
cosh(
•
6 1 -Jl 1
x
The normal
field at surface "b" in
terms of the potentials
at surfaces "a" and "c" and Eo is
E (n)=2nn
ny
coth( 2nt a(n
)
)
c
sinh2nt
X
149
-
b
1cc
E (0)= E (0)=
e92 -j2(B5
Substituting for
E
(-J)
c(n) in equation (B5)
we get the normal electric
from equation (B4)
field at surface "b" as given in
equation (30).
150
References
Conf.
of
Maintenance
Preventive
Procedures of
"Diagnostic Technique and
A. M.,
11] Corvo,
Electric
High Voltage
on Large
Intl.
Large Transformers",
Systems, Paris,
September 1-9, 1982.
[21 Lawson, W. G., Simmons, M. A.,
Aging
Paper Insulation",
of Cellulose
"Thermal
and Gale, P. S.,
Trans. on
IEEE
Electr. Insul., Vol EI-12, pp61-66, February, 1977.
[3]
R.
Rogers,
Incipient
Analysis",
"IEEE
R.,
Faults
in
IEC
And
Insul.,
on Electr.
Interpert
Gas
Using
Transformers
IEEE Trans.
to
Codes
in
Oil
Vol EI-13,
pp349-354, October, 1978.
[4] Baehr,
R.,
Muller, R.,
Transformers",
Flottmeyer, F.,
Nieschwietz, H.,
and
Intl.
Conf.
Kotshingg, J.,
"Diagnostic Techniques
Procedures
Maintenance
Preventive
and
W.,
Breuer,
Large
on
for
High
Large
Voltage
Electric Systems, September 1-9, 1982.
[5] Burton, P.,
Knab,
H.
Carballeira, M.,
Markestein,
J.,
Samat, J.,
Rindfleisch, H. J.,
E.,
Soldner, K.,
P. G.,
Foschum, H., Gandillon, M.,
T.,
Praehauser,
Th.,
Schliesieng, H., Serena,
Vurachex, P.-, Whitelock, K., and Wolf,
"New Applications of Oil
Dissolved Gas Analyses
and Related Problems", Intl. Conf. on Large High Voltage
Electric Systems, Paris, August 29-September 6, 1984.
[6] Garverick S. L.,
AC Measurement of
and Senturia, S. D.,
"An MOS Device for
Surface Impedance with Application to
Moisture Monitoring",
IEEE Trans. on
151
Electron Devices,
Vol ED-29, pp90-94, January, 1982.
D. R.,
Sheppard, N. F.,
S. D.,
[7] Senturia,
H. L.,
Lee,
and Day,
Measurement of the Properties of Curing
"In-situ
J. Adhesion, Vol 15,
Systems with Microdielectrometry",
pp69-90, 1982.
[8] Lee,
H. L.,
of a Resin
"Optimization
Cure Sensor", EE
Thesis, M.I.T., 1982.
[9] ASTM D117-81
"Dielectric Loss in Insulating Liquids",
[10] Bartnikas, R.,
Electr. Insul., Vol EI-2, pp33-54, April
IEEE Trans. on
1967.
[II]
Systems -
Insulating
Losses
"Dielectric
R.,
Bartnikas,
Part I",
in
Solid-Liquid
on Electr.
IEEE Trans.
Insul., Vol EI-5, pp113-121, December 1970.
[12]
Smyth,
Behavior
Dielectric
C. P.,
and Structure
pp53-59, McGraw-Hill, New York, 1955.
[13]
Hakim,
R.
M.,
Effect
"The
of
Oxidation
on
the
Dielectric Properties of an Insulating Oil", IEEE Trans.
on Electr. Insul., Vol EI-7, pp185-195, December 1972.
[14]
Yasufuku, S.,
Umemura, T.,
Conduction
Phenomena and
Dielectric
Fluids",
and Tanii,
Carrier Mobility
IEEE
Trans. Electr.
T.,
"Electric
Behavior in
Insul.,
Vol
EI-14, pp28-35, February 1979.
[15] Adamczewiski, I., Ionization Conductivity and Breakdown
in Dielectric Liquids , ppil2, Taylor & Francis, London,
1969.
[16]
Delahay, P.,
Double
Layer and
152
Electrode Kinetics
ppl-32, Interscience Publishers, New York, 1965.
(17]
Goffaux,
R.,
"Sur
Dielectriques
Memoire
de
Solides
l'Academie
Les
Proprietes
aux
Tres
royale de
Electriques
Basses
de
Frequences",
Belgiques,
XXXVIII,
1968.
[18]
Gartner, E.,
Liquid/Solid Insulation
Trans. on
at Very Low
Electr. Insul., Vol
behavior of
"On the
R.,
and Tobazeon,
Frequencies", IEEE
EI-12, pp86-89, February
1977.
[19]
Bartnikas,
Viscosity
R.,
"Electrical
-
Oil-Paper Films
Conduction
Part I",
in
Medium
IEEE Trans.
on
Electr. Insul., Vol EI-9, pp404-413 , June 1974.
[20] Ohashi,
H., and Ueda,
a., Shimokawa,
M.,
"Dielectric
Properties of Silicone Liquid Films, IEEE Int'l Symp. on
Electr. Insul., pp245-248, 1978.
Garton, E.,
"On the Behavior of Ions
at Insulator/Liquid Interfaces
and its Consequences for
(21] Tobazeon, r.,
the
Losses in
Electrical
pp404-413,
D.C.,
[22]
and
Impregnated
Insulation
National
Insulation", Conference
and
Dielectric
Academy
of Sciences,
Phenomena
on
,
Washington,
1975.
Umemura,
T.,
Akiyam,
K.,
and
Kashiwazak,
T.,
"Dielectric Behavior of Solid/Liquid Insulation system",
IEEE
Trans. on
Electr. Insul.,
Vol
EI-17, pp276-280,
June 1982.
[23] EPRI, "Characteristics of Insulating Oil for Electrical
Application", December, 1979.
153
[24] El-Sulaiman,
A. A., Ahmed,
Spectroscopy
Quresh,
M. I., and
"High Field DC Conduction Current and
A.,
Hassan, M. M.
A. S.,
of aged Transformer
Oil", IEEE
Trans. on
PAS, Vol PAS-101, pp4358-4360, November 1982.
Transformer
"Evaluation of
Duval, M. and Lamarre, C.,
[25] Crine, J. P.,
Oil Aging Under
Chemical and Dielectric
Service Load
from Various
Measurements", IEEE Intl. Symp.
on Electr. Insul., pp35-37, 1982.
[26]
Cesar,
L.
C.,
"Aging
Oil", IEEE
Transformer
Studies
Intl. Symp.
of
Paraffin
on
Electr. Insul,
Base
pp167-171, 1983.
[27] Gussenbauer, I.,
in
"Examination of Humidity Distribution
Transformers by
Int'l
Conf.
Means of
on Large
High
Dielectric Measurements",
Voltage Electric
Systems,
August 27-September 4, 1980.
[28]
Howe,
A.
F.,
Power-Transformer
"Diffusion
Insulation",
of
Moisture
Proc.
IEE,
Through
Vol
125,
October 1978.
[29) Guizonnier, R.,
on
Conduction
"Nature des Porteurs de Charges", Conf.
Processes
in
Dielectric
Liquids,
pp251-255, September 17-19, 1968.
[30]
Oomen,
T.
V.,
"Moisture
Equilibrium
in
Paper-Oil
Insulation Systems", IEEE Int'l Symp. on Electr. Insul.,
pp162-166, 1983.
[31] ASTM D1533.
[32] Szepes, L.,
Method
Torkos, K.,
for the
and Dobo, R.,
Determination of
154
"A New Analytical
the Water
Content of
Transformer
Oil", IEEE
Trans. on
Electr.
Insul., Vol
EI-17, August 1982.
[33]
Melcher, J. R.,
, pp2.33,
Continuum Electromechanics
The M.I.T. Press, Cambridge, Massachusetts, 1981.
[34] Melcher, J.
Continuum Electromechanics , sec. 4.5,
R.,
The M.I.T. Press, Cambridge, Massachusetts, 1981.
[35] Melcher,
J. R.,
Continuum Electromechanics
, pp 2.35,
The M.I.T. Press, Cambridge, Massachusetts, 1981.
(36]
Toomer, R.,
Electroes on
and Lewis,
T. J.,
"The Effect
Insulationg Surfaces", 3rd
of Metal
Int'l Conf. on
Dielectric Materails Measurements and Applcations, 10-13
September, 1979.
[37] Shell Chemical
Company, Technical Bulletin, sc:164-77,
February 1977.
(38] Macdonald, J.
R.,
"Interface Effects in the Electrical
Response of Non-metallic Conducting Solids and Liquids",
Conf
on Electric
Insulation and
Dielectric Phenomena,
1980.
[39]
Vetter, K. J.,
Electrochemical Kinetics
, pp345-354,
Academic Press, New York, 1967
[40]
Vetter, K. J.,
Electrochemical Kinetics
, ppi88-193,
Academic Press, New York, 1967.
(41]
Melcher, J. R.,
Continuum Electromechanics
, pplO.3,
The M.I.T. Press, Cambridge, Massachusetts, 1981.
[42]
Vetter,
K.
J.,
Electrochemical
Kinetics
pp200-205,
Academic Press, New York, 1967.
[43] Davies, J. T., and Rideal, E. K.,
155
Interfacial Phenomena
, ppl08, Academic Press, New York, 1963
(44]
H.,
Koelmans,
Devices
Corrosion
"Metallization
Induced
by Moisture
in
Silicon
Electrolysis", IEEE
12th
Rel. Phys. Symp., ppl68-171, 1974.
[45]
Orszag,
Complex
S.
A.,
"Spectral
Geometries",
state of the
Proc. of
Methods for
mini
Problems
symposium on
in
the
art in computational fluid dynamics, AIAA,
pp273- 3 0 5, Boston, March 1984.
156