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Department of Electrical and Computer
Engineering Technical Reports
Department of Electrical and Computer
Engineering
7-1-1989
Digital Measurement of Partial Discharge
James Patrick Steiner
Purdue University
W. L. Weeks
Purdue University
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Steiner, James Patrick and Weeks, W. L., "Digital Measurement of Partial Discharge" (1989). Department of Electrical and Computer
Engineering Technical Reports. Paper 671.
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Digital Measurement of
Partial Discharge
J. P. Steiner
W. L. Weeks
TR-EE 89-40
July, 1989
XvXvXvXvXvXvI-IvIvXv
School of Electrical Engineering
Purdue University
West Lafayette, Indiana 47907

DIGITAL MEASUREMENT OF PARTIAL DISCHARGE
A Thesis
Submittedto th e Faculty
of
Purdue University
by
James Patrick Steiner
In Partial Fulfillment of the
Requirements for the Degree
of
Doctor of Philosophy
May 1988

ii
This thesis is dedicated to my wife, Rebecca and my family.

iii
ACKNOWLEDGMENTS
I would like to express my deepest appreciation to Prof. Walter L.
Weeks for his support, encouragement and direction in preparing this thesis.
Also I would like to thank the members of my committee, Michael Zoltowski,
Edward Coyle and Vaughan Weston, for their invaluable comments, sugges
tions and discussions. Special thanks is due to Edward K. Duffy, Wilbur A.
Starrett and William J. Schmidt (BF) of Essex (later Cablec) for their help
in the cable experiments. I would also like to especially thank Eric S. Fur-
gason for his help and suggestions. I would also like to acknowledge the
support of Essex , Alcoa , Square D Company, Cablec and the Army Corps
of Engineers (CERL).

iv
TABLE OF CONTENTS
. ■Page
.LIST-OF'FIGURES'........................................................................ Yi
ABSTRACT
CHAPTER ONE- PARTIAL DISCHARGE .........
1.1 Introduction.... .......... .....................
1.2 Background............................ .................
1.3 Previous Work.... .........................
1.4 Contributions.................................... .. ii
CHAPTER TWO - LUMPED SPECIMENS AND MODELS 14 -
2,1 Introduction..............................
2.2 Amplitude and Occurrence Time Estimation..... ...........
2.3 Poisson Shot Noise Process .................... . .
2.4 Higher Order Models................ ...................
2.5 Experimental Study ................. ..........
2.6 Ultra Low Level Measurements................ ......
CHAPTER THREE - DELAY ESTIMATION ..................
3.1 Introduction...........................................
3.2 A Non-correlation Based Location Technique....,..........
3.3 Generalized Cross-correlation....
3.4 BiasofDelayEstimates........ ....................
3.5 Variance of Delay Estimates ...........................
3.6 Modified PHAT Processor.... ..................... .
3.7 Robust Deconvolution of Correlation Functions........ ‘ .......108

Page
CHAPTER FOUR - PARTIAL DISCHARGE IN C A B L E S .................12]
4.1 Introduction.............. ,........ ........... ................................... ..................121
4.2 Cable System M o d e l ...............................................122
4.3 Measurement Calibration.............................................. ........ .......135
4.4 Location of Partial Discharge..................... ...............139
4.5 Equalization................ ............149
4.6 Impulsive Interference............... ......154
4.7 Narrow Band Interference..................... ......................... ....................157
4.8 Signal Sub-space Processing..................... .....163
4.9 Mark Space Division........ ................... .......192
CHAPTER FIVE - TRANSFORMER MEASUREMENTS..........................198
5.1 Introduction................................ ...198
5.2 Experimental Apparatus ....................... ........199
5.3 Discrimination of Interference in Transformer Measurements........206
5.4 RF Magnetic Measurements............... .211
5.5 Signal Sub-space Processing................. .......................................231
5.6 Acoustic Models................................ ...234
5.7 Acoustic Coupling.................................... 238
5.8 Location Using Attenuation .................... .243
5.9 Acoustic Delay Estimation ."...... .........................251
CHAPTER SIX - SUMMARY AND CONCLUSIONS
6.1 Summary and Conclusions....
LIST OF REFERENCES.............
VITA ......................................... ............................................................ .......... 286
270
270
276

UST OK FIGUHPS
Figure Page
1.1 Voltage-life Characteristics of polyethylene. .............___........................ 4
1.2 Slowly developing partial discharge............................................... ...............6
1.3 Rapidly developing partial discharge.............................................................7
2.1 Nonlinear transfer function of measurement system. .............................21
2.2 Performance of estimators versus digitizer input level..............................2.3
2.3 Performance of estimators for low level measurements....................... 25
2.4 Generation of signal using the m o d e l . ___,...,,...,.,.,32
2.5 Test cell.......___ .....48
2.6 Acoustic response of test c e l l . ......________________ ,,,..,„.,„.49
2.7 Relative energy versus excitation voltage phase..................................,..,..50
2.8 Histogram of time between partial discharges........................................... 52
2.9 Histogram of acoustic energy from partial discharges__........................53
2.10 Mean energy versus gap s i z e . .......,.,,,.,.,.,.....„...„,..,.......54
2.11 Mean partial discharge energy vs. surface conductivity. ..„..,.,.....„......56
2.12 Typical sample function of Trichel pulses______________________..,.58
2.13 Statistical description of Trichel pulses...........„ ,.,.„ „ ,.,..,.,,,,....,.,,...,...„ 5 9
2.14 Serial correlation functions, X = 5.7xl03. ......„61

2.15 Serial correlation functions, X —8.OxlO5. ......................................... . ....62
2.16 Serial correlation functions, X = .1.43x10®. •
............................................63
2.17 Typical pooled records for Trichel pulses........ ......................................65
2.18 Typical histogram of times between events for pooled records.............67
■
■ :■
. ■
■
■ - vii
Figure Page
2.19. Low level partial discharge pulse..............................................................76
2.20 Results of level crossing algorithm................................... ........... ........... .78
2.21 Histograms of the processor output under the two hypothesis;,...........80
3.1 Block diagram of generalized cross-correlator...........................................88
3.2 Output ESD using the ML GCC weighting...................... ......................109
3.2 Output ESD using the modified PHAT GGC weighting.. ..........HO
3.4 Illustration of delay bias. ....................................... .................................112
3.5 Block diagram of minimax GCG. .............................................. ................117
3.6 Original cross-correlation function__ __ _________________________119
3.7 Output of minimax GCC.,......................................................................120
4.1 Theoretical cable diagram. __,___ ___ ......................................128
4.2 Block diagram of experimental system..._____________ .....................140
4.3 Cable termination circuit. .................................__.......................... „142
4.4 Typical "ambers" partial discharge signals........................ ....................143
4.5 Typical cross-correlation function for "ambers".............. ........ ..............144
4.6 Fault generated signals.................................................... ........... . ...........147
4.7 Low frequency fault generated signals...................... ..............................148

Viii
4.8 Attenuation characteristics of a power cable.............. ..... .... ..............]5 1
4.f Phase characteristics of a power cable.... ............................................... 152
4,10' Illustration of iterative interference removal........................................ 161
4.11 Normalized error vs, iteration: number...................................................162
4.M lkstration of Wiener processor results......—....................................... 164
4.13 Typical low level partial discharge...................... ................ ................. 176
Figure . Page
4.14 Cross-correlation function for a single low level partial
d i s c h a r g e _____ • • • • • • • • • • • • * • • • • • • • • . . « • •. .177
4.15 ©veriaid cross-correlation functions of single and double
partial: discharge sites. ___ __________________ 178:
4.16 Cross-correlation function and output from minimax GCC................. 179
4.17 Typical impulsive interference signals............................. ......................180
4,18: ESD’s corresponding to Fig. 4.16.............................................................181
4-19 Typical low level partial discharge__________ _______ _____ ____182
4.20 Calibration signals. ..............................................................................184
4-21 MUSIC simulation; single site, low noise. .......................................185
4.22: MUSIC simulation; single site, high noise....................................... .......186
4.23 MUSIC simulation; two sites, low noise.......................................
4.24 MUSIC simulation; two sites, moderate noise. _____ __ ___
4.25 Experimental trial of MUSIC; single site........ .............................
4.26 Experimental trial of MUSIC; two sites.......................................
4.27 Experimental trial of MUSIC; cable with a suspected "pip".....
..188
..189
.190
..191
,.193
k;:, / I ' ' : -

4.28 Histogram of delay estimates from multiple partial discharge sites....196
5.1 Partial discharge soirce.................................................... ........................ 205
5.2 Partial discharge source with extra capacitance................ .................... 207
5.3 Typical transformer signals....... ............................................. ......... .....209
5.4 Bimodal histogram measured from two distinct partial discharge
sites. .................... .................................. ..........................................212
5.5 Magnetic field plot for calibration signals............................................... 214
5.6 Magnetic field plot for partial discharge signals ................................215
5.7 PSDVof magnetic probe signals.................................................... ..........217
5.8 Results of ML algorithm applied to the magnetic probe signals...........228
5.9 Position of partial discharge source. .....................................................230
5.10 Output from MUSIC algorithm..............................................................233
5.11 Diagram of solid cast transformer.......................... ................................235
5.12 Diagram of trailing pulses. ........... ......... ............................. 237
: = ■ ■; ..................
5.13 Various coupling methods........................................................................240
5.14 Loss versus distance for various coupling methods............................... 241
5.15 Trailing pulses.................................. ........... ......................................... 244
5.16 Location using attenuation data.............................................. ..............250
5.17 Time limited signals......................... ...................................................... 263
5.18 Acoustic-acoustic cross-correlation function.................................... ......264
5.19 Original electro-acoustic cross-correlation function.............................267
ix
Figure Page

5.20 Deconvolved electro-acoustic cross-correlation function.................... 269
Figure Page

ABSTRACT
Steiner, James Patrick. Ph.D., Purdue University. May 1988. Digital Meas
urement of Partial Discharge. Major Professor: W. L. Weeks.
Various new measurement techniques have been developed for a high
voltage phenomenon referred to as partial discharge. Partial discharge is a
localized breakdown of the high voltage insulation system which is observed
as low level, random emissions. Both electrical and acoustic emissions have
been measured in underground power cables, solid cast power transformers
and in lumped specimens. Typical problems complicating the measurements
are the randomness of the emission, high levels of interference and extreme
distortion of the signal by the propagation path.
Various signal processing techniques have been adapted to the measure-
ment of partial discharge. The techniques investigated are capable of reduc-
ing noise in the measurements and have provided orders of magnitude
improvement in sensitivity over ordinary methods. Some of the techniques
studied are capable of providing information about the location of the par-
tial discharge site.

I
CHAPTER ONE
PARTIAL DISCHARGE
1.1 Introduction
This study is concerned with the measurement of partial discharge in
high voltage equipment. Partial discharge is a localized breakdown of the
high voltage insulation system and is observed as extremely low level,
random emissions that can be measured using electrical, acoustical, optical
or thermal techniques. The most popular and, in most cases, the most
sensitive methods are the electrical techniques. One common thread tying
the different measurement methods together is that the observed signals are
very low level, random emissions that are corrupted by. noise. The
randomness of the emissions complicates measurements but the noise is the
primary difficulty associated with partial discharge measurements.
Various types of signal processing techniques are used to improve the
sensitivity of the measurements and in some cases provide additional
information that was not previously obtainable. The techniques that will be
discussed are, for the most part, standard approaches that have been used
for many years in different applications and have been successfully adapted
to partial discharge measurements in this study.
Successful adaptation requires that the observed signals be modeled so
that the necessary changes can be made in the techniques. Modeling of
partial discharge is an extremely difficult task from the point of view of the
observed signal properties. The difficulties arise because partial discharge is
a complex physical process that is not completely understood at this time.
Rather than trying to derive signal properties based on first principles, a
phenomenological approach is taken because of the difficulties associated
with developing a satisfactory physical model. The measurement techniques
will use models based on experimentally observed behavior and in some cases
the models will not be completely accurate but will lead to tractable results.

In other cases the models will be chosen so that existing solutions to other
problems can be utilized.
There are different problems that are encountered in the measurement
of partial discharge and each situation has its own problems. A lumped
specimen was studied in a test cell that provides high integrity signals with
little attenuation and minimum interference. In general, the attribute of
lumped specimens is that the signal is attenuated and requires that a large
amount of gain be used in the measurement system. Large gains always
cause problems with the thermal noise in the amplifier, however, this is the
simplest noise to deal with. Also associated with high gain is susceptibility
to external interference. External interference can take many forms and in
some cases it is easily dealt with in the same manner as the thermal noise
but there are situations where the interference is difficult to eliminate.
Distributed systems, such as cables, suffer from the same type of
difficulties with noise as lumped specimens. The difference in the
measurement is that the propagation of the signals must now be taken into
account. The presence of a well defined propagation path allows the use of
estimafidn techniques so that the partial discharge site can be located.
Various difficulties are encountered in estimating the position of the partial
discharge site. These difficulties are associated with dispersion in the cable,
reflections in the cable and the presence of multiple partial discharge sites.
Another type of distributed parameter system is the transformer and it
is certainly the most difficult to measure and analyze. Transformer
measurements suffer from the same type of noise and interference problems
as mentioned above. The new element in the problem is that the
propagation paths are more difficult to define and describe. From an
electrical standpoint it has been possible to construct linear circuit models
as discrete approximations to the distributed parameter system but these
models appear to be valid only across narrow frequency bands and at lower
frequencies. Acoustically, the propagation paths escape description except

3
for very localized simple models. This difficulty can be traced to the highly
nonhomogeneous nature of the materials though, which the acoustic signals
must propagate. ... .
1.2 Background
•
• Q-
In the manufacture, installation, or operation of high voltage equipment
small occlusions may develop in the insulation. These occlusions enclose a
different dielectric material than that used for insulation, and it is often a
gas whose dielectric constant is much less than material around it. This
causes an abnormally high electrical stress in the void and partial discharges
may occur. Discharges in voids, enclosed by dielectric or conductor and
dielectric, cause degradation of the insulation. Damage resulting from these
discharges may lead to total failure of the insulation system. Studies on
time to failure have been made for various types of defects [I]. This study
investigated the life time of dielectrics with cavities and Fig. 1.1 shows its
results as a double logarithmic plot of field strength vs, life. Gonsider a
typical cable system for which field strengths are on the order of 0.5 to 2
kV/mm. As can be seen, for relatively minor defects, the lifetime of a cable,
under normal operating conditions, can extend into thousands of hours.
To relate the measurements made to the discharge process certain
fundamentals need to be discussed. In a void with dielectric or conductor-
dielectric boundaries two basic mechanisms are believed to cause
breakdown. The first is the Townsend mechanism and the second type,
which is believed to start initially as a Townsend mechanism, is the streamer
mechanism [24,25,2,3,4].
The Townsend mechanism is a release of an avalanche of electrons.
When the surface of a material is highly stressed and an energetic particle
or photon of sufficient energy strikes this surface an electron is emitted.
This electron collides with molecules in the void and releases further
electrons. These electrons in turn release further electrons thus creating an
avalanche. The molecules that released the electrons are now positive ions.
As the positive ions accumulate, a positive space charge develops which
enhances the field and accelerates the electrons causing further electron
release. As the discharge continues the space charge migrates h^vay from
the surface. As the space charge migrates, the field, which %ailbn£ihally
enhanced by the positive ion cloud, is reduced and this quenches the electron
avalanche. 'J T:
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r r . ' -* i ■
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5
Tbe scenario above is greatly simplified and two points need
emphasizing. The initializing event that starts the avalanche is random.
For example, it can in be generated by background radiation which has
events that are often modeled as a Poisson process. In fact radioactive
illumination of the void drastically alters the observed behavior of the
partial discharge [67]. The second point is that the space charge eventually
detracts from the field and most of the positive ions must be driven into the
Opposite electrode before the probability for another discharge becomes
appreciable. In other words there is a mechanism which inhibits further
events from occurring. A point not brought but in the above discussion is
that typical partial discharge pulses have random energies (amplitudes). It
is not clear what mechanisms cause the randomness but this characteristic
has been recognized for several decades.
The second type, the streamer mechanism, begins with the formation of
electron avalanches, as in the Townsend mechanism. Essentially, the
positive space charge causes auxiliary electron avalanches that leads to a
conductive plasma channel that bridges the void. It is believed that
streamer development is preceeded by a buildup of roughly IO6 avalanches
[3].
An elaborate and accurate analysis of partial discharges in voids has
been developed in [24,25]. That work takes a rigorous mathematical
approach using Maxwell’s equations along with the continuity equations for
the various charged species in the void to describe the propagation of the
avalanche across the void. There appears to be excellent agreement
between the theoretically predicted shape and the experimentally measured
shape. One signal type observed in this study was referred to as a slowly
developing partial discharge (Townsend mechanism). An example of this
type of discharge is shown in Fig. 1.2. Another type of discharge that was
observed was referred to as a rapidly developing partial discharge (streamer
mechanism). An example of this type of discharge is shown in Fig. 1.3. The
figures depicting these two types of partial discharge were band limited to 20
MHz and do not indicate the broad band nature of the signals. Typical
pulse rise times can be less than I ns with pulse widths as short as a few
nanoseconds. However, for large void dimensions the pulse width of the
ionic component can be as large as thousands of nanoseconds.
In many typical measurement situations the the received signal is so
distorted that the original pulse shape is inconsequential. For example in
transformer measurements there are so many self resonances in the

6
1200 2400 3600
Time, seconds (x IO"9)
4800
Figure 1.2. Slowly developing partial discharge.

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uu u pry-;
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.;-.- -U
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Figure 1.3, ,Pppidty developing partial discharge.

transformer that the observed signals bear no resemblance to the original
pulse shape. Another example is given by URD cable which is so lossy that
the maximum received bandwidth after propagation of only 500 meters will
be a few MHz. In both of these cases the original waveform was Completely
obscured. In these cases the only partially distinguishable characteristics of
the original waveform are the time of occurrence and the energy. The most
important factor influencing the observed signal is then the propagation
path which in most cases will not be known.
1.3 Previous W ork
The literature on partial discharge is extensive and it would be
impractical to try to give an account of the various techniques that are
available. However, there is an excellent reference that provides a
comprehensive review of most of the currently accepted measurement
techniques [3]. This reference provides a large variety of techniques covering
measurements in both lumped parameter systems or distributed parameter
systems that are either capacitive or inductive in nature. The difference
between the work in this thesis and others is that most methods do, not take
into account the randomness of the measurement; in previous literature the
observed signals are treated as deterministic quantities. The noiseTs treated
as an inevitable consequence of trying to measure low level signals and few
attempts are made to minimize its effects. There are some exceptions but
usually little effort is exerted in combating the noise and interference.
Harrold [3] has performed extensive investigations into the acoustic
behavior of partial discharge. The references contained in [3] is not an
exhaustive list of Harrold’s work but does provide a reasonably complete
list of investigations made into the acoustic behavior of partial discharge.
The thesis is different from others because acoustic methods are used
for locating partial discharge in solid cast transformers. These efforts are
detailed in chapter four where a rigorous development of the measurement
techniques is presented. A slightly different approach was taken and the
main similarity between this study and others is that acoustic signals are
utilized.
A most notable work is [21] in which the fundamental limitations of
partial discharge measurements are discussed. This paper discusses the
matched filtering approach to amplitude estimation in white noise for ultra

9
wide band measurements. Results are given relating the sensitivity of the
measurement to the bandwidth of the measuring system. Further results are
given relating the minimum sensitivity of partial discharge measurements to
the length of the cable through which the partial discharge is observed.
This was derived using the bandwidth relationship and the measured
transfer characteristics of a typical lossy power cable.
Another method, [9], by the same authors, describes a partial discharge
location technique using a correlation based system. This method is intended
for ultra wide band measurements and only uses one or two bits of
amplitude information. As a consequence the results are highly nonlinear
and it would appear that subsequent processing might be impeded by the
presence of the nonlinearity. One of the attributes of correlation analysis as
used in time delay estimation is that the correlation functions can be
optimally filtered. This aspect of time delay estimation is discussed in
chapter two.
Another method, [82], attempts to optimize the temporal resolution of
the measurement using a Wiener filter. The formulation in [82] assumes
that the signal is stationary which is not the case for transient pulses.
Narrow band examples are used as illustrations of the technique and the
estimate of the charge (amplitude) is provided by performing the integration
O
O
Q = / U O ‘it
-OO
where i0(t) is the optimally filtered current waveform. The problem with this
solution is that the original observed waveform does not appear to have a
DC component and therefore the optimaly filtered version cannot have a DC
component. The integral used to estimate the charge therefore evaluates to
zero.
Another paper, [10], discusses the advantages of signal processing for
making partial discharge measurements. The authors state that full wave
rectification of the acoustic signals and electrical signals can be used to
locate the source of acoustic emissions in transformers. The paper is
extremely vague and it is difficult to see what methods they are using and
how the signals are modeled and utilized in the subsequent processing.

Acoustic location methods are put on solid mathematical foundation in
chapter four of this thesis in which a similar method is used for locating
acoustic emission sites.
; A partial discharge location scheme is described in [8] but it is not
completely clear how the location is determined. The method is described
using a block diagram with the critical blocks stating what the block output
is without any mention of the manner in which the output is determined.
The critical element in the block diagram is the time of arrival section and
it is never stated how the time of arrival is measured. The purported
accuracy is described but there is no way of analytically verifying this.
An energy measurement scheme is described in [83] using a cross-
correlator. This is a novel approach for partial discharge measurements
that is closely related to some techniques described in this chapter.
Unfortunately, the authors failed to recognize the tremendous power
afforded using correlation techniques. Use of similar techniques can provide
vast improvements in system sensitivity and experimental simulations have
indicated that measurements sensitivities can be improved by at least three
orders of magnitude.
An intriguing new approach to partial discharge measurements has been
developed in [84]. This technique appears to have great promise if analytical
methods can be found for interpreting the data. A short discussion
concerning a mathematical framework for this technique is contained in
chapter two. Measurements such as those found in [84] will probably lead to
the next generation of partial discharge measurements techniques.
Another fascinating new method is contained in [13] in which X-rays are
used to excite the partial discharge process. The technique involves
illuminating the partial discharge site with a modulated X-ray source (via a
chopper). The modulation waveform is then cross-correlated with the output
of a standard partial discharge detection circuit using a lock-in amplifier. In
essence the modulation source correlates with the changing statistics of the
random point process describing the partial discharge. This technique is
used to locate partial discharge site by sweeping the X-ray across the device
under test. If no partial discharge site is encountered then the correlator
does not respond. However, if the X-ray source illuminates a partial
discharge site then the correlator responds which is the indication that there

11
is a site at that location. .
G ontributions
.-.This thesis applies the theory of digital signal processing to partial
discharge metrology. As seen in section 1.3, which is a limited search of
current literature, very little work has been done in formulating partial
discharge measurements in terms of modern signal processing techniques.
"!Phis thesis develops a signal processing framework by presenting a signal
model and adapting a variety of signal processing techniques to partial
discharge measurements.
The title of this thesis (Digital Measurement of Partial Discharge) was
chosen because the techniques that are presented measure well defined
parameters associated with the partial discharge, such as its location. These
parameters are fundamental to any method aimed at interpreting partial
discharge. The information presented in this thesis might be thought of as
the front end processing of any technique aimed at interpreting partial
discharge because it provides accurate methods for measuring the data.
Analysis of data requires that a model be used so that inferences about the
partial discharge can be made in terms of the model. Analysis and
interpretation of partial discharge is a difficult task and is only briefly
investigated in chapter two. The model presented in chapter two and
similar models are believed to provide the proper mathematical tools for any
such investigation.
' - ' . ' . . . .-y}' '
Paramount to the development and analysis of a signal processing
algorithm is the use of an accurate signal model. A signal model is
presented in chapter two which is believed to be new to partial discharge
measurements. The model is very natural for this application and is widely
used in many disciplines. In essence this model takes into account the
random variations of both the time of occurrence and the amplitude of the
partial discharge. The models presented are only first order models and
more elaborate and accurate models could be developed with further work.
However, experimental work shows the consistency of the measurements
With the simple, first order models.
Some fundamental measurement techniques are presented in chapter
two which can be of use in practice. The techniques applied are standard
methods in signal processing but have not been used in partial discharge

12
measurements before. Also a more accurate lower bound on the
measurability of partial discharge is discussed in terms of a well known
result. Finally, a detection technique for measuring ultra low level partial
discharge is discussed along with a technique which greatly reduces the
amount of data processing required.
Chapter three discusses general techniques for estimating the locations
of partial discharges. There are several contributions in this chapter related
to the generalized cross-correlator. A detailed analysis of the bias of delay
estimators is presented and this material is believed to be new. The bia,s
expressions are developed from techniques that appeared recently in signal
processing literature and are an extension of these results to practical
measurement situations. Two new generalized cross-correlators are
presented one of which is a minimax version. These generalized cross-
correlators are used to enhance the resolution of the delay estimation
measurements so that multiple partial discharge sites are more easily
located.
Partial discharge measurements in cables are discussed in chapter four
and several contributions are presented in this chapter. Several standard
signal processing techniques are adapted to the partial discharge
measurement in cables for the first time; system identification, equalization
and generalized cross-correlation. Also a discrimination technique for
eliminating impulsive interference in partial discharge measurements is used
for the first time and is implemented using a generalized likelihood ratio
test. A new application of the Gerchberg-Papoulis algorithm is presented
which is capable of iteratively removing the effects of narrow band
interference. Super-resolution is investigated using an ad hoc
implementation of the MUSIC algorithm. The MUSIC algorithm is applied
to the analysis of multiple partial discharge sites in short cables when data
is collected with a low bandwidth system. Finally, the peculiarities of the
signal model are exploited to provide a simple high resolution method for
delay estimation in cables.
Partial discharge measurements in solid cast transformers are presented
in chapter five. One technique that is investigated is the acoustic method
which has been used for several years by the transformer industry; however,
this is the first time it has been applied to solid cast transformers.
Furthermore* the technique has been refined in a manner which is believed
to be new. Current literature on acoustic measurement techniques lack the
mathematical details presented in chapter five and this development is also

13
believed to be new. A new asymptotic maximum likelihood measurement of
the attenuation of the acoustic energy is developed which provides a new
method for locating partial discharge. A jnew, unique measurement
technique, based on the rf magnetic fields in the transformer windings, is
presented. Together with an asymptotic maximum likelihood algorithm, the
magnetic measurements provide a new method to locate partial discharge in
transformers. Finally, the same ad hoc formulation of MUSIC, used in
chapter four, is applied to the signature analysis of partial discharge
waveforms in transformers. This new technique provides another highly
effective method for locating partial discharge in transformers.

14
CH APTER TW O
LUM PED SPECIM ENS A ND M ODELS
2.1 Introduction
Good models make it possible for digital signal processing to improve
measurements of partial discharge. In the past, when partial discharge has
been modeled, the approach has been to develop expressions that describe
the physical behavior of the individual partial djscharge pulse. While this is
of great interest, it does not provide the information necessary for the signal
analysis used in this thesis. In contrast, the work in this thesis requires
statistical models of the entire observed signal process so that signal
processing algorithms can be developed. Since tljie necessary physical models
required to develop statistical descriptions of the observed signal process are
not yet available the approach to modeling in this thesis is
phenomenological.
The chief advantage of using signal models is that many results that
are available in signal processing literature will be available for adaptation
to partial discharge measurements. Depending upon the situation different
types of models will be used. If the observed partial discharge is very active
and the pulses are not easily separated then the appropriate model would be
a stochastic process. In many cases there are only a few pulses and single
events are easily observed. The appropriate model in this case is a
nonstationary transient model in which only the single pulse is considered.
Individual partial discharges are random, time localized events. When
random events are time localized they are often modeled as a stochastic
point process. There are various forms which a point process can take and
the general type which will be used for partial discharge can be called a shot
noise process. To characterize this type of process the times of occurrence of
the partial discharge and the amplitudes must be measured. Section 2.2
considers various estimates of these quantities.

In general, mathematical analysis of point processes is difficult unless
very special models are used. Section 2.3 describes, in detail, a special point
process which is referred to as the Poisson' process. The reason for treating
this case in detail is the amount of literature available for this model. Also,
under suitable conditioiis or interpretation, the Poisson model appears to be
a valid model for the analysis of partial discharge.
In more accurate analysis it may be desirable to employ more
sophisticated models which use more of the available information. Section
2.4 will consider another special model referred to as the renewal model in
which the memory of the process is restricted to the previous event and the
intensity is homogeneousi The distribution of the inter-arrival times
between partial discharge pulses will be the important parameter in this
case.
Point process models can be of use when trying to estimate various
quantities that may be of interest. Section 2.5 gives the details of some
experimental work which Uses these models to estimate fundamental
quantities associated with the geometry of the gap in which the partial
discharge is occurring. Measurements are made which can be used to infer
the size of the gap. Further measurements are made which estimate the
conductivity of the surface of the gap for high conductivities. Similar types
of arguments will be used in the analysis in later chapters when the
locations of the partial discharge is inferred.
Techniques used to increase the signal to noise ratio are presented in
section 2.6. One technique is similar to a detection technique in that a
threshold is selected to provide the detection of an event. The mean number
of level crossings of the signal process are used to approximate the increase
in SNR. This technique is capable 6f detecting pulses buried in thermal
noise and has provided gains in the SNR of the measurements. The
advantages that this technique has over a rigorous detection approach is
that this method can be easily implemented with standard equipment and,
more importantly, the data rate is slow which reduces the computational
load. .
2.2 A m plitude and O ccurrence Tim e E stim ation
One of the most fundamental quantities that is of interest in partial
discharge measurements is the charge transferred by the discharge current.

16
That charge is defined as
'; "
f;'
Q = I W dt
where i(t) is the current in the partial discharge. It is easy to see that this
integral is equivalent, to evaluating I(w) at u
>= 0 where I(o
j) is the Fourier
transform of the current pulse. Unfortunately, this equation is not of much
use in practical situations because the measurement circuits used are almost
neverrDC coupled to the partia,l discharge site. The usual manner, in which
this problem is circumvented is through calibration of the system under test.
In this nianner the voltage amplitude of the output from the detection
circuit is compared with a calibrated reference value. The problem is then
to estimate the amplitude of the voltage in the best manner possible and
some solutions to this problem will be presented below.
Another fundamental quantity of interest is the occurrence time of the
partial discharge pulse. If statistical analysis of the partial discharge
process is desired then both the charge and time of occurrence are necessary
pieces of information. If it is assumed that the pulse shape is known then
the occurrence time is easily estimated using some very classical results.
These results will be discussed in conjunction with the LMMSE estimator
described below because the solutions are intimately related.
The signals to be discussed will be general in nature; they could be the
output from either a narrow band or wide band partial discharge detector.
The observed signal will be modeled as
x(t) = a; s( t —T
j ) + n(t)
where s(t) is the output of the particular partial discharge detector being
used. The pulse shape, s(t), will be assumed known and is easily measured
using the calibration signal. The amplitude, aj and the time of occurrence, T
j
df the signal, s(t), will be considered to be random. The noise term, n(t),
Will be assumed to be stationary and have a power spectral density N(f).
The noise term, n(t), will model the thermal noise in the amplifiers of the
detector and the quantization noise of the digitizer. Any external interfering
signal that is from a stationary random process can also be modeled by the

17
noise term, n(t). A small amount of white Gaussian (thermal) noise will be
assumed to be present at the input to the quantizer and as a consequence
the noise from the quantizer will be assumed to be uniformly distributed
white noise. Nonstationary interference such as intermittent or impulsive
signals not related to the desired signal, s(t), will not be considered in the
ensuing analysis. Other techniques, discussed in later chapters, will be used
to discriminate between the desired signal and interference.
The first estimator to be discussed is the LMMSE filter, [42], and takes
the form of a linear filter, H(c
j), which is given by
H(w)
a S (w) exp( —j27rrj )
N(c
j) .
where S (w) is the complex conjugate of the Fourier transform of s(t).
constant a is given by
The
a =
1 + A }
N(cj)
where A is the mean square value of a;. It should be noted that the filter
attenuates the observed signal at those frequencies at which the noise level
is high and passes those frequencies where the noise is low. This could be
particularly useful when combating narrow band interference, for example
the interference due to an AM broadcast. The estimate of the amplitude is
then given by
a: 9 - '( X(W)H(W) }[_„
As can be seen, to implement this filter/ it is necessary to know the arrival
time, T1
, of the partial discharge pulse. Since the optimal linear filter is in
essence a correlation operation the estimate of the arrival time is the time at
which the output of the filter achieves its maximum. This estimate provides
the second piece of information necessary to statistically characterize the
partial discharge process. Other pieces of information that are required are

18
the mean square value of the fandom amplitudes. It is impossible to know
this quantity a priori but satisfactory results have been obtained by simply
using the reference value from the calibration. It is also possible to
adaptively adjust the mean square value as data becomes available,
however, this aspect was not pursued.
There are other implementations of The LMMSE estimator which
address different aspects of the amplitude estimation problem. One typical
problem, encountered in practice, is the estimation of the amplitudes of
closely spaced partial discharge pulses. When this occurs, superposition of
the pulses can cause errors in the amplitude estimate. As mentioned in
chapter one, a solution to, this problem was given in [82]. However, the
formulation was incorrect and the amplitude estimate for most cases (non-
DC coupied) is zero* The proper solution can be found in [42] and is given
by . ... . •
•
H(C) = = ■
$a2 S (c)
a2 X IS(c)|2 + N(c) ’ H(°) 0
where j3 is defined as
1 + f "f
—
inf a
a2 |S*(cVl2
X |S(c) p -|- N(c)
dc
-I
This solution assumes that the input signal process is a homogeneous
Poisson shot process with intensity parameter X Uhd that the signal is not
D€ coupled. The estimate of the amplitude is given by
a ,= $ x(c)H(c) }|t=ti
where t; is determined as the peak of the filter’s output. In essence, this
filter performs a deconvolution of the input signal process. In some cases it
will be necessary to constrain the solution to reduce the variance of the
estimate. Some typical constraints for this type of problem are discussed in
later chapters. In general, deconvolution schemes of this type provide
increases in the pulse resolution on the order of 3 to 5 times before the

19
variance of the estimate becomes excessive.
The second method, the autocorrelation (AC) estimator, is an ad hoc
technique and is very simple to implement. This technique is based upon a
maximum likelihood argument used in a different context in chapter 5 but it
should be noted that this estimator is not a maximum likelihood estimator
for this case. The estimate is formed by evaluating the autocorrelation
function at zero delay, removing the bias due to the noise, and taking the
square root of the result. The value of the time autocorrelation function at
zero delay is defined by
'■T/2 ;
■
> y • / X
i(I).!.
■ . ; —T/2 . ■
; •
where the partial discharge pulse is assumed to be contained in the interval
(—
-T/2 —
T/2 ). It was more convenient in this analysis to offiit the
factor from the usual definition of the time autocorrelation function. The
resulting value will be an estimate of the sum of the noise energy and the
energy in the partial discharge pulse. Taking the expected value 6ne obtains
E{ y j J
E
L
i } = a;2? + T o2;
where a2 is the variance of the noise and % is the energy in the signal pulse,
s(t). The term, Td2, is an unwanted bias term and is removed from the
calculation before further processing. The final step is to calculate the
square root of the result after removal of the bias. The estimate of the
amplitude is then
ai
V y - T o 2
‘■
-
r
=
-
*
: • ‘.....
a calib
where aca!ib is the reference level obtained through calibration.
One interesting sidelight is that the AC method can provide meaningful
results when ^he partia,l discharge rate is too high to resolve the individual
pulses. Consider the case in which the observed signal is
. I; : -'

where there are now Nx pulses in the interval ( -T /2 , T/2 ). If the process
is a homogeneous Poisson process with intensity parameter X then the
estimate of the the autocorrelation function at zero delay becomes
y = XA % + To2
where A is the mean square value of the amplitude defined above. The
output of the estimator is slightly different because it now gives an estimate
of the square root of A. We no longer get an estimate of the magnitude of
the charge but an estimate of its average rms value.
Three estimators were compared experimentally; the peak value,
LMMSE and AC estimators. Measurements were made using a Data 6000
waveform analyzer which has an 8 bit A/D converter and a general purpose
microprocessor. The measurements were made using a precision calibration
source, calibrated attenuators and a noise generator. Two different signals
types where investigated: a, broad band pulse and a narrow band pulse.
These signals were chosen because they are representative of outputs from
typical partial discharge detectors. The broad band pulse had a band width
of 2 MHz while the narrow band pulse had a bandwidth of 10 KHz and a
center frequency of 185 KHz. The signal was fed through an attenuator into
one input of a summing amplifier and the noise source was fed through an
attenuator into the other input of the summing amplifier. In this manner
various input signal levels and noise levels could be measured.
The first measurement represents a typical calibration curve obtained
by injecting a calibration pulse into the measurement system. The
calibration signal was varied from 1% to 1000% of the full scale input level
of the digitizer. The curve in Fig. 2.1 represents the nonlinear transfer
function of the measurement system, with the noise bias removed, for
various input levels using the AC estimator. The curve is linear until the
full scale input level is reached and after this point the measurements begin
being compressed. This calibration curve can be used to undo the
nonlinearity by mapping the measured value into its actual value. The
calibration curve will be unique to the measurement system, the calibrating

1000
Output Estimate, pC
Figure 2.1. Nonlinear transfer function of measurement system

22
signal and noise level. The removal of the bias, Tcr, in the nonlinear region
is an ad hoc approach which is not correct. The correct approach uses a
measured calibration curve to perform the mapping, however, this requires a
calibration curve for each noise level. An ad hoc approach of subtracting an
equivalent noise bias for signals operating in the nonlinear region can be
used if a calibration measurement is made. The calibration would measure
a time, feq, which would be the time that the digitizer operated in its linear
region for a given input level. The bias that is removed is then Te which
still is not quite correct but leads to satisfactory results. This particular
nonlinearity is referred to as a soft limiter and these results were incidental
to the measurements that were made. There is an entire branch of signal
processing devoted to optimal nonlinearities for use in nonlinear quantization
schemes. These were not pursued but undoubtably would yield improved
quantization schemes for partial discharge.
The second set of measurements illustrate the relative performance of
the three different estimators obtained using an 8 bit quantizer. The
measure of performance will be a signal to noise ratio which is defined as
SNR = 20 log [ ---- —■meaD of estimate ,■
root mean square error of estimate ^ ^
For input signals less than the full scale input level of the digitizer the rms
error is due to the variance of the estimator. When the signal exceeds the
fuff scale input level of the digitizer the rms error for the peak value
estimator includes both the bias due to the clipping of the signal and the
variance of the estimate. This error term was included in the peak value
estimator because there is not a method to compensate for measurements
exceeding the full scale input level of the digitizer. However, since it is
possible, corrections for the nonlinearity and noise bias were made for the
LMMSE and AC estimators where appropriate. Fig. 2.2 illustrates the
performance of the various estimators for different input levels to the
digitizer. A sampling rate of 2.5 times the Nyquist rate was used and the
waveforms contained 128 points. The rms input noise was approximately
1/2 lsb of the quantizer. As can be seen, in the linear region of the digitizer
both the LMMSE and AC estimator perform almost 20 dB better than the
peak value estimator. In the nonlinear region, the peak measurement has
large errors because there is not a method to compensate for the

24
nonlinearity. However, the LMMSE and AC estimator maintain their
performance because there is a technique to map the measurements into
their correct values.
The final set of measurements show the relative performance of the
three estimators under varying noise conditions. In these measurements the
rms error is due only to the variance of the estimator because the digitizer
was operated in its linear region. The input signal was maintained at a
fixed level of about 10 % of the full scale input level of the digitizer. The
input noise level was adjustable so that the input SNR could be varied
between -10 dB and 30 dB. Fig. 2.3 illustrates the effectiveness of both the
LMMSE and AC estimators in low level measurements while the peak value
estimator tracks the input SNR. The AC estimate performed better than
the LMMSE estimate at low SNR’s and it is believed that losses in the
performance were observed because the arrival time also had to be
estimated.
As was illustrated by the experimental measurements the LMMSE and
AC estimators out perform the simple peak value estimator. However,
implementation of the LMMSE technique requires relatively sophisticated
computations that are time consuming. Special purpose hardware is
available to accelerate the calculations, but is expensive. The AC method
provides estimates, that in these measurements, were as good as the LMMSE
estimate. However, the major advantage of the AC estimator is that it is
much simpler to implement. The hardware implementation of the AC
estimator is particularly simple and could easily operate in real time. The
basic configuration could use an inexpensive 8 bit A/D flash converter. A
pre-estimate could be calculated using a single chip multiplier-accumulator.
After rapid collection, the data could then be properly rescaled by removing
the bias due to the noise, correcting for the nonlinearity, if necessary, and
taking the square root. This Could be easily performed on any general
purpose microprocessor in an expedient fashion for display purposes.
One final variation that should be mentioned is that a modification of
these techniques would be useful in narrow band partial discharge detectors
that use a diode envelope detector. This technique was not pursued but the
implementation would amount to simply integrating the output of the
envelope detector to develop the estimate. From some preliminary results it
appears that similar gains in performance can be obtained.

Output
SNR,
dB
AC Estimator
LMMSE Estimator
Peak Value Estimator
Input SNR, dB
Flfiife■&
& Petfotaismeetef esiliMtefs f&
?,tew level meastiremeuts.

26
The results discussed above provide, solutions to the amplitude
estimation problem when the second order statistics of the noise are known.
The LMMSE solution is the optimal solution when the noise is Gaussian
(assuming the arrival time is known) and is the optimal linear solution when
the noise is from a wide sense stationary random process (not necessarily
Gaussian). The important question that needs to be answered is what is the
performance of the best estimator. This question can be answered in terms
of a bound on the variance of the estimate. The bound is the Crarner-Rao
lower bound and gives the minimum variance of the amplitude estimate. To
calculate the bound assumptions about the statistical nature of the noise
must be made. If the noise is from a stationary Gaussian random process
with power spectral density, N(w), then the bound on the variance of the
amplitude estimate is given by [41]
var( a —a ) ^ ISMl2
NM
where it is assumed that the observation interval is infinite and that SM is
the Fourier transform of the unit amplitude pulse, s(t).
The actual quantity of interest is the charge and these results need to
be stated in terms of bounds on estimating the charge. Following Boggs [21]
the pulse will be assumed to have a Gaussian shape
_ j i
s(t) = a e 2r°
where a is the amplitude to be estimated. The rms duration of the pulse is
— The noise will be
V 87r2rO
. . . N0
assumed to be white Gaussian noise with the variance ----. For purposes of
2
analysis and comparison to the charge, a DC coupled system will be used.
Also, to simplify interpretation, a relatively simple case of a power cable will
be considered. The charge measured from one end of the cable is
V 27rrOwhere Z0 is the characteristic impedance of the cable. The
2 A0
given by —-j= and the rms bandwidth is
v2

output SNR of the estimator is given by
: . 27
SNRout
a2&
where $ s is the energy of the unit amplitude pulse and is */t
tT
q- The
noise, Np, is defined as IcTeq where k is Boltzmann’s constant and Teq is the
effective noise temperature of the entire system. The effective noise
temperature includes the contributions to the thermal noise from cable,
measurement impedance and amplifier. The minimum measurable charge,
with the estimate having the (subjectively) chosen SNRout, is
N0 '
Qmin
(
_
8
_ *' i
'SNRput- - :
T
T.
> B
N0 has the
where B is the rms bandwidth of the Gaussian pulse and
approximate value of 4 x IO-21 at the temperature 290°K . As an example
consider a typical half kilometer cable with Z0 = 35 and having an rms
bandwidth of B = 5 MHz.
16 x IO^2v then Qm
in
If the system has a NF = 4 so that
0.014 pC for an output SNR of 20 dB.
is
This
particular number represents the theoretical minimum measurable charge if
the optimal estimator were used. Since an optimal estimator is not available
an additional loss will be incurred from the estimator used but will be small
(on the order of a few dB).
2.3 Poisson Shot Noise Process
Observed partial discharge signals are random in nature and difficult to
characterize. Current technology in partial discharge measurements tacitly
assumes a deterministic model for.' .-parti-al" discharge. However, to take full
advantage of signal processing techniques a stochastic model should be used
when modeling a signal that has random qualities. Uflfortunately, present
lack of understanding of the mechanisms of partial discharge makes a

28
stochastic model developed from first principles still beyond reach. Models
for partial discharge signals can, however, be constructed based on
experimental observations. The goal of this section will be to develop a first
order theoretical model for partial discharge that will allow easy application
of the theory to practical measurement situations. This development is
taken directly from Snyder [61] but only the salient points are mentioned.
Typically, individual partial discharges are time localized and are
randomly spaced in time with respect to each other which suggests modeling
partial discharge in terms of a stochastic point process. The simplest
example of a point process is the time homogeneous Poisson process for
which each localized event is statistically independent of the others. There
are several methods for deriving the Poisson process. The most informative
approach is the constructive method in which the time intervals between
partial discharges are considered to be exponentially distributed with mean
l/A ? The resultant process is then, by definition, a time homogeneous
Poisson process with the intensity function equal to a constant, X
. The
intensity function X has the interpretation of being the mean number of
partial discharges per unit time. For small At the probability of having a
single partial discharge in the interval (t, t + At) is then
h
>
T
t, t+At = I ) = XAt + o(At)
where
o(At)
approaches zero as At approaches zero.
Work by Devins, [12], showed that a model for the statistical time lag
associated with the epochs of the individual discharges under DC conditions
is exponential. If the probability density of the time between partial
discharges is truly exponential, then by definition, it must be a Poisson
process.
Most of the experimental results that will be discussed were made under
AC conditions and the voltage across the defect is therefore a function of
time. Consequently, the epochs of the partial discharges will be related to
the phase of the high voltage excitation (e.g., 60 Hz). If the process is still
modeled as Poisson, this variation can be taken into account by introducing
a time varying intensity function, X
t, which is how a function of the
excitation voltage. The introduction of a time varying intensity function
means that the process is now nonstationary. The interpretation of this

nonstationarity is that the statistical quantities describing the process are
now functions of time. For example, the average number of partial
discharges occurring in the time interval (tj, tj + At) will not be the same
as the average number of partial discharges occurring in a different time
interval (t2, t2 + At) because these averages are functions of the times tj
and t2. For small At the probability of having a single partial discharge in
the interval (t, t + At) is
•P-r( Ntf t+zit —
' I ) — At + o(At)
where o(At) is defined above. This nonstationarity will complicate the
analysis if a further simplification cannot be found.
To simplify the problems associated with the above nonstationarity,
further assumptions about the nature of the nonstationarity will be made.
The excitation voltage is a sinusoid and therefore it will be expected that
tlie intensity function, X
t, will also be periodic with the same period as the
excitation voltage. Based on the assumption of a periodically varying
intensity function, the process is now called periodic nonstationary. This
interpretation is appropriate if observations are made during a short time
interval, T, located at a particular phase of the excitation voltage. If the
process is observed for many cycles of the excitation voltage and the process
is periodic nonstationary, then it is expected that on the average there will
be N = X
1
^T partial discharges per cycle, where X^ = X
t is now
approximately a constant. The intensity function will be approximately a
constant for any chosen, small interval at a given fixed phase, however, if
the positioning of the measurement interval is moved to another phase of
the excitation voltage this constant can be expected to change.
To complicate the matter further, the partial discharge process
undergoes a slow metamorphosis. For example, Luczynski [25] observed that
in his specimens the process initially consisted mainly of partial discharge
pulses with very short durations and large energies. These high energy
partial discharges are referred to as rapidly developing partial discharges.
Rapidly developing partial discharges occurred relatively infrequently with
only a few partial discharges p e r cycle of excitation voltage and therefore
having relatively small values of X
t . Luczynski [25] also observed that after
a period of many hours the process changed to one with a high incidence of
slowly developing partial discharges. Slowly developing partial discharges

30
have longer durations and low energy. The number of slowly developing
partial discharges per cycle of excitation voltage was large implying a large
X
t . This change represents another type of nonstationarity and is referred
to as a slowly varying nonstationarity. If the measurement interval is
confined to a small enough time period in the entire history of the process,
then it can still be considered to be periodic nonstationary. In the following
description the measurements intervals will be assumed to be sufficiently
short so that the slowly varying nonstationarity can be ignored.
To develop the model further, some measure of the energy must be
associated with the occurrence times of the partial discharges. The manner
in which this is included is to use a marked point process [61j. Now each
epoch has associated with it a mark, Ui, indicating its energy, which also
can be described statistically. The assumptions about the stationarity of the
process with intensity function X
t can also be made about the time varying
nature of the parameter(s) describing the mark distribution. For the Poisson
model the mark process will be considered to be independent of the
occurrence times and that the individual marks, Ui, will be independent of
each other.
The reason for discussing the Poisson process in detail is that this
model is the easiest to manipulate mathematically. Literature abounds with
results concerning the Poisson process and several results will be presented
which apply to the partial discharge measurements made in this thesis. The
assumptions required for the partial discharge process to be a Poisson
process are very restrictive and this model may not apply in all cases.
However, when a more complete model is developed for partial discharge a
limit theorem will be presented. Using this limit theorem, a method to
sample the partial discharge process will be discussed which allows one to
interpret it as a Poisson process.
There are various assumptions which can be used to define a marked
Poisson process and the following conditions are an example of one set of
conditions which have the advantage that they can be easily modified to
admit non-Poisson models. The first assumption that is made is that the
process is conditionally orderly which means that multiple events can not
occur simultaneously. ■This is a reasonable assumption because it is highly
unlikely that two partial discharges can occur simultaneously (the pulses
could be separated by a pico-second or femto-second) The second
assumption that is made is that the process develops without aftereffects

31 ;.v
which means that the past history of the partial discharges is independent of
the future behavior of the partial discharges. In other words all previous
partial discharges will not influence the behavior of any partial discharges in
the future. Also a finite, integrable function, X
1, called the intensity of the
process, is assumed to exist. The third assumption that is made is that
Pr( bJt(i = 0 )•== I where t0 is the beginning of the interval on which the
Poisson process is defined. This simply means that the number of partial
discharge pulses counted at the beginning instance of observation is zero.
The final assumption is that the marks are mutually independent and
independent of the Poisson process.
As mentioned these conditions are restrictive and it is informa,tive to
consider the appropriateness of the assumptions. The conditional orderliness
restriction can be removed by treating multiple occurrences as a mark. This
might be useful if a single discharge were considered to be a multiplicity of
simultaneous events, for example the number of electrons in the partial
discharge. Evolution without aftereffects is an assumption that could be
difficult to justify in some situations. In various cases there appears to be
an influence of previous partial discharge behavior of future behavior. The
independence of the marks is also an assumption that may not-always be
accurate. In some cases there appears to be a dependency between the
counting process and the marks. If these cases are considered then highly
complex models are required.
The simplest model that can be used is the time homogeneous Poisson
shot hpise process [61]. In the case of a time homogeneous Poisson model
the time between the occurrences of the partial discharges will be
exponentially distributed with density
P( O = Y e X u( r )
where the average number of events per second is X
. Each of these events
can be considered to be delta functions occurring at random times, with
randpm areas passing through a linear filter, h(t), as depicted in Fig. 2.4. In
this case the observed signal, s(t), is given by
" . . V t -

32
fa)
(b)
(c)
v t
Figure 2.4. Generation of signal using the model, a) Marked point process,
b) Linear filter, c) Resulting filtered process.

33
Nt
s(t) = E ai ${ t 't t j ) * h(t) ; -
; V V . ■
" . ■ ■
■ i - l ' .. . - / , : /V
where A1 is the random area of the delta function (the mark) and * is the
convolution operator. The linear filter, for most practical purposes, can be
coiisidered to be the cascade of the impulse responses of the propagation
path and the detection circuit. In general this will not be true because the
physical behavior of the partial discharge itself determines the shape of the
pulse. An excellent example of such a case is provided by the slowly
developing partial discharge whose shape is determined by the physical and
geometrical properties of the void. However, in most cases the true shape
will be masked by the impulse response of the filters (medium) through
which it is observed. A more complete model could include the random
variation of the impulse response. This random variation could arise In a
number of ways, for example, the randomness of the individual discharge
itself could cause fluctuations in the the pulse shape. Another manner in
which the impulse shape could become random is if the partial discharges
are emanating from different positions in the device under test. In this case
the random variation will be due to the impulse response of the different
propagation paths.
As mentioned above a further complication can be included by assigning
a mark to each random event. /These marks need not be restricted to just
the random amplitude, charge or energy of the individual partial discharge
pulse. In general, a mark is any auxiliary random variable associated with
the epoch of the partial discharge. Suppose that there are two separate
partial discharge sites then the mark could be the position of the observed
partial discharge interior to the device under test. Both of the amplitude
and position marks can be combined into a single vector mark. Another
example would be td use the partial discharge type as the mark, it could be
either a slowly or rapidly developing partial discharge. The signature of a
particular partial discharge might be used if extremely complex structures
are being measured (e.g., a transformer). This case would also correspond to
partial discharges located at different positions.
There are various quantities that are of interest which can be obtained
from the modeling. For complex (non-Poisson) models it is sometimes only
possible to calculate some of the moments but for the Poisson shot noise
model it is possible to derive a complete statistical description. This

34
description is in terms of the characteristic function of a marked, filtered
Poisson process and is given by [61]
Mt(j w) exp
t
f E
t<
)
exp( j wh( t, r : u )) I ] dr
where u is the mark vector and h is the pulse shape. A similar expression
for the joint characteristic function for yt at times tlr and t2 is given by
Mt,,1,0 coV J aV
)
exp -
to
J E exp[ j W1 h( t1( T : u ) + j W
2h( t2, r : u )] —I dr
These expressions are useful for calculating the moments of the process for
example the average charge is given by
O
O
Qavg = / E{ aM) Mt) Mt) dt
-OO
where both the amplitude and intensity are considered to be periodic
functions of time.
In la t e r c h a p te r s h ig h e r o r d e r m o m e n ts of t h e filte r e d p r o c e s s will b e
used in the analysis of different techniques. Second order moments are the
basis of many signal processing algorithms because these moments are easily
utilized in the analysis. Furthermore, second order moments are easily
measured using standard (highly developed) spectral estimation techniques.
The autocorrelation function of the shot noise is given by

minf t,, to)
R( t1( t2 ) = J K E I h( t1
? T : u ) h( t2, r : u ) J dr
1
1 ' . -to
+ / Xr E [ h( tl5 T : u ) ] dr J X. E [ h( t2, r : u ) j dr
t(>
where the expectation is taken with respect to the mark. The case that is of
most interest in the following chapters is the cross-correlation function of
the shot noise passing through two dissimilar, time-invariant, linear filters.
Typical observation windows are short with respect to the period of the
process (5 /is vs. 16.66 ms). In this respect the intensity function will be
considered to be a constant. For two different filters, h^t) and h2(t) the
cross-correlation function is
R# ) = (a,)2 X| / hj(t) dt / h2(t) dt
' —
O
C —
oo
+ I h 1(')) h 2( ' + o) d'y
where it is assumed that the statistics of the amplitude are functions of 4
>
but constant in the observed window. The subscript <
f>is used to indicate
that these results apply only to the small window located at phase <
jh The
power spectral density of the partial discharge is then easily found by letting
hj = h2 and taking the Fourier transform of Rt^r) yielding S^(w).
Other disciplines have found the Poisson model to be of great use in
statistical inference of the underlying nature of the mechanisms generating
the process. One application of the Poisson model is for making a
statistically based decision about the type of process being observed. For
example a hypothesis testing procedure could be used to determine what
type of partial discharge process is being observed. Something similar to
this is presently being performed on a regular basis by operators of
commercial partial discharge detection equipment. A GIGRE report, [87],
gives list of expected detector outputs for different partial discharge types.
Two papers, [88,89], were recently presented which describe the use of expert
systems, based on the CIGRE report, for determining the type of partial

36
discharge being observed. Using a hypothesis testing procedure, based on
the Poisson model, could lead to a similar system. Another prominent
application in statistical inference is the estimation of various parameters
associated with factors influencing the behavior of the point process. This
particular application could be used for investigating the influence of
physical changes on the partial discharge process. If the situation had 'been
catalogued then the reverse problem is possible; determine the values of the
variables controlling the partial discharge. Some work, other than that
contained in this thesis, has been performed on this problem. In the past
few years a new measurement technique has been developed, [84] which in
essence makes the measurements of partial discharge in the manner to be
presented below. However, the measurements are not analyzed using this
framework.
Both of the applications, discussed above, require the use of a likelihood
ratio for the observed process. The few results presented thus far were
concerned with the integrated response observed through a linear filter. In
some cases this is the desired approach for analyzing partial discharge. Now
suppose that the estimation procedures discussed in section 2.2 are used to
estimate both the occurrence time and mark of each event. The result is a
sequence of points, ( tj, U
; ) which can be used in the analysis of the point
process via the likelihood function. In the analysis that follows it will be
assumed that only the sequence of points is available for use.
An important function associated with the analysis of the Poisson
process is a function referred to as the sample function density [61]. The
sample function density can be roughly described as the probability of
observing a particular realization of the point process. For a univariate
Poisson process (without marks) the sample function density is given by
P [ { N„; t0 ^ <
7< t } I a J=
/
exp
t t
I M a ) + I
to t()
where the second integral has the evaluation

37
J In[Xf7(O)] XiNf7 = V In[X
tJ
■' ' t() " . ‘ -i=l
'■' ' . i
- ■ ' ■
' ' ’ '■
The variable, o, is a non-random parameter that is associated with (and
may control in some sense) the particular process being observed. In the
estimation problem o will be the variable that is under study and will be
estimated. In the hypothesis testing problem o can be thought of as an
index to the process being observed and will be an integer; in the binary
problem o = 0, I.
For a marked Poisson process the sample function density is given by
Xjt; .to ^ C
r < t } I a
exp ^ - J X
jt
(Q) do -k J :In[X
jt
(O)] dN* + £ In[ P( Ui J a ) JNt(Ui)
t,, t0 : i= l
For a marked Poisson process the sample function density can be also be
written as
= i l P f{ Nj7
(Ui); t0 ^ cr < t } I o
i=1 L -
where the process has been quantized and split along the level Ui so that
there are now sub-processes each having it’s own sample function density
given by
P [{ X fj; t 0 ^ a < t I
p [ ( N jr(U i); t0;< }=
IW-J-
exp
t • t
- J PM I a)X„(a) d<r + J ln[P( Ui I a )X„(a)| JN1
7
(Ui)
to to

38
This representation follows from an extremely important theorem discussed
in [61] in which a marked PbiSson process can be represented as a
Superposition of sub-processes. Each of these sub-processes is a Poisson
process with it’s intensity function given by
' ^ ' . ... X„(rt;i)_P(U, :
where there are now M sub-processes for ( i = I, 2, .A4 ) for M quantization
Ieyels. This characterization can also be extended to the continuous case,
however, measurements by nature are quantized so it seems that the
denumerable case is the most practical to discuss. The factorization of the
marked Poisson process into M separate processes is an extremely important
and valuable tool for analysis.
Another type of analysis can be performed if a count rate histogram is
utilized. The above formulations assume that all information is available;
the exact time of occurrence of each point in order of occurrence. Collecting
data can be difficult in situations where the data rate is high and the
observation interval long. However, there are instrumentation techniques
available that allow high data rates and long observation intervals while
remaining cost effective. Using these techniques, only the number of counts
in a particular quantization segment, (^i, Ui) would be recorded in terms of a
histogram using a histograming memory (which are commercially available).
The data collection system would collect the number of counts in each bin
where each bin would be assigned to a small phase window and charge
window. Analysis of the measurements would then proceed in terms of the
number of counts in each bin. The histogram will be at least a two
dimensional histogram with the number of dimensions being equal to p+1
where p is the dimension of the mark space. For a one dimensional mark
space the probability of observing a particular histogram contour ( a contour
along Ui) is given by
Pr{ NtjJ1
,t/Uj) = O
j; j - I, k I,a }.=
?.

39
k
<

tJ
nJ
b
11 •
j=i
[(H j) ! ] - 1 J P(U i | < * ) X » d a
tj I
exp - / P(U , I a )X ,(„ ) d a
tj !
where it is assum ed th at the counts nj were collected in k sub-intervals of
the interval (t0, T). T he interval, (t0, T) would be an integral num ber of
periods of the excitation voltage and the sub-intervals, (tj_1, tj), would be
the phase intervals <
/>
j) observed over the integral num ber of periods
used. Since the m arks are assum ed to be independent of the counting
process and independent from each other the probability of observing the
num ber of counts in the sub-intervals (IjJ1, tj) (the phase bin) becom es
P r{ N tj1
Ztj ” nj! J = U k j O
' } =
M Pr{ Ntj „ t/u i) - nj; i = i, -r k I « } ■
i= l
U sing these equations the log likelihood function is easily seen to be
- M
■ * » = E ^i(a)
i= I
where
T k tj
e |(«) - - / p(Uj I « ) V ( « ) d<7 + ■2 7 H p (u i I « ) M “ ) <•" I
to J= I tj_,
It is interesting to note that this likelihood function is the sum of the
likelihood functions for each charge bin.
Now that the basic formulas have been presented their use will be
described for different situations. The sample function density is the more
useful of the two characterizations because it uses all of the information
available. The likelihood function for the count rate data is more restrictive
because it ignores potentially useful data (the occurrence times). However,

40
as pointed out, data collection requirements are greatly reduced using the
count rate approach. Furthermore, the amount of numerical analysis is
reduced because of the compression in data. For these reasons, emphasis
will be placed on the count rate analysis but the techniques are completely
analogous for the sample function density analysis. In the following,
reference will be made to the likelihpod function which can be either the
count rate likelihood function or the sample function density because the
sample function density is the likelihood function for the time data.
The first case to be discussed is the estimation problem in which a
variable, a, influencing the partial discharge process, is estimated. The
value of cx which maximizes the likelihood function is the value of d which
maximizes the probability of observing the recorded data. In other words
maximization of the likelihood function (or it’s logarithm) with respect to a
gives the maximum likelihood estimate, aml, of cx. Solution of the likelihood
equation will yield an estimate of a
d C (a)
d cx "=S-m
l
The solution can easily be extended to a parameter vector a. As an
example, consider the estimation of the intensity of the process in terms of
the count rate data.
d € jX( >Ui )]
d X
( (Pj , Ui )
= 0
The m axim um likelihood estim ate is then
I; ;• ■
* .v . njK ) ■ .
KA c
P
i, Ui ) =
where it is assumed that each phase bin was observed an equal amount of
time and t0 = 0 •; :
Estimation of parameters influencing the partial discharge process
requires a priori knowledge of the functional form of X(a). These functional
forms could be theoretically predicted but as already mentioned sufficient

understanding of the physics of the process is lacking for this approach to be
fruitful. The alternative procedure is the phenomenological approach in
which particular situations are measured while carefully controlling the
variable to be examined. The desired end result is a functional form for the
dependence of the intensity on the Variable in question. Two examples of
this procedure are investigated in section 2.5. Once this parameterization
has been accomplished, then the next time that this situation is
encountered, the variable can be estimated based upon the previously
measured functional form. The measurements made in [84] attempt to
parameterize the behavior of partial discharge histograms similar to the type
developed here for voids with different geometries.
The proposed procedure presents an insurmountable task in attempting
to parameterize all partial discharge types for all possible variables. The
results from this impossible task would certainly be a desirable data base,
however, the purpose of this exposition is not to propose the developihent of
such a data base. Rather, it is to present a mathematical framework, under
which partial discharge can be studied. Study of partial discharge in this
manner may lead to a greater understanding of the mechanisms that
influence it’s behavior.
2.4 Higher Order M odels
The Poisson model contained in the previous section assumes that all of
the events are statistically independent of each other with exponentially
distributed inter-ocCurrence times and in certain situations this may be a
reasonable assumption. There are instances in which the inter-occurrence
times are clearly not exponentially distributed and the Poisson model will be
incorrect. Furthermore, there are situations where there appears to be a
dependence between events and in these case it is more appropriate to
describe the partial discharge process with more elaborate models. Use of
more elaborate models for statistical inference will give better results
(assuming the models are accurate) because the information is being used
correctly. However, when the models for a point process become more
elaborate the analysis soon becomes intractable. There are some models and
special cases in which results can be obtained. One point process which
models the dependence between the events and is somewhat tractable is the
self exciting point process. Self exciting processes model dependencies

42
betw een events in the process and this model seem s well suited for m odeling
partial discharge. Referring back to the conditions for the partial discharge
process to be Poisson the condition th at is relaxed for this case is evolution
w ithout aftereffects.
A self-exciting point process is one in which the intensity of the process
is now allow ed to be a function of the past events. The intensity function
for the process is given by
U;
P (U i) t O < t = W1
■
P ( U i) Mt( N t; W1, r .wNr), w Nt < t ^ w Ntm
where the marks are assum ed to be independent of the counting process and
of each other. The variables, Wj, are the occurrence tim es of the partial
discharge pulses. The equation states that the present intensity at tim e, t,
is a function of the number of previous events as well as the tim es at w hich
they occurred. In other words the probability of observing a partial
discharge in ( t, t + A t ) is
p ( N t,t+At = I I N t; W1, ,..W ^ i-Ui ) = Xt u . A t + o(A t)
for sm all A t. The sam ple function density can be w ritten down for this case
and can be found in [61]. In general this process is still difficult to analyze
and further sim plifications can be used.
A w idely used model w hich is sim ilar to the Poisson m odel is the
renewal m odel. In this case the interarrival tim es are still considered to be
independent, identically distributed random variables but they no longer
have an exponential distribution. The renewal process has inter-occurrence
tim es th at can have an arbitrary distribution. The renewal process is a
special case of the self exciting point process in w hich the intensity function
is hom ogeneous and dependent on only the previous event: In this case the
the process is described as a hom ogeneous, m em ory one, self exciting point
process w ith
-
j

for some function h(t) which is referred to as the hazard function.
To use this model for delay estimation it will be necessary to investigate
the second order moments of the integrated process. For a point process
with independent marks the correlation function can be written as
R(ti>t2)= ^2/ d r,/ dr2g(tj-T 1)g(t2- r 2)^ r 11T
2)
■ 0 ■ / ' 0
■
■ _
_
_
_
_ min(t,,to)
.+ a2 J drg(tj —7
’
)g(t2—r)fj(r)
' > t ( n U w L r-WN,.) = h( t ~ WN |. )
where the functions T1(r) and I2(TljT
2) are referred to as the first and second
order product densities, respectively [62]. The function g(t) is the transfer
characteristic of the linear filter through which the process is observed.
First order product densities have the interpretation that T1(t) At is the
probability of observing an event in the infinitesimal interval (t, t + At)
regardless of the points occurring elsewhere. Second order product densities
have the interpretation that f2(t1( t2) At1At2 is the probability of
simultaneously observing events in the infinitesimal intervals (t1
? tj + At1)
and (t2, t2 + At2) regardless of the points Occurring elsewhere. These
product densities are usually difficult to determine but there are a few cases
for which: they are easily found. For a stationary Poisson process the
product densities are
and
“ A2
so the correlation function becomes those presented in the previous section.
It is also possible to write down the correlation functions for a renewal
process and the form depends on the actual description of the time between

events. The second ;order product density for a renewal process takes the
form/ V1 .' :
^ (rLr2) = fl(rlKl('2 - T
l)
so if the first order product density can be found the second order product-
density can be easily calculated. The first order product density is found
from the solution of the integral equation
fi(t) = p(t) + J fj(u)p(t - a) da
0
where p(t) is the probability density function of the interarrival times of the
events. This integral equation is easily solved using the Laplace transform
and the solution is the inverse Laplace transform of
FlW
PM
i - PM
This transform is particularly easy to calculate for the gamma density
fuhdtioji; This solution will apply for the memory one self exciting point
process (non-homogeneous renewal process) if the periodic nonstationarity
assumption is made. In this case the observations would be made in a small
phase window for which the inter-occurrence times would be homogeneous.
There is a further assumption that can be made in many cases when
the intensity function is small enough, The correlation function for a
renewal process takes into account the correlation existing between
subsequent events, (e.g., partial discharges). Typically, the small time
intervals, in which the measurement is made, contain either one partial
discharge or none. Therefore, when calculating the correlation function from
the data, there are no subsequent partial discharge events entering into the
calculation. Since there are not any subsequent events, the correlation
function is the same as for a Poisson process, justifying the Poisson
assumption for the case in which the intensity is small enough. This
particular case is actually very common considering the small observation
windows used in the implementation of the delay estimation schemes to be

45
discussed.
There is a more general model which encompasses all the cases
discussed and is referred to as the general marked point process. In this
case the intensity of the process can now be a function of the marks and
dependencies between the counting process and the marks is allowed. The
form of the intensity will be
lh , U;(0 )> t 0 < t g Wj
I'l, u,( N t; w i> V-WN,, Uj, ,..UNt) WN.r < t ^ Wn tm
Consideration of a complex model such as this leads to intractable equations
which are of little use in the analysis. There is method by which such a
process can be analyzed using a very simple approach. The approach
throws out a tremendous amount of information that could be potentially
useful but the resulting simplicity justifies the method to be proposed. The
method is a technique by which the general point process is sampled so that
the resulting point process can be interpreted as a Poisson process. Current
statistical research on point processes is concerned with an operation on
point processes referred to as pooling. Pooling of point processes means that
individual processes are overlaid one on top of another and the statistics of
the resulting process are considered. Under the proper conditions the pooled
process converges to a Poisson process [61]. This result can be shown to
hold for univariate point processes as well as marked point processes. The
drawback of using the pooling operation to simplify data collection and
analysis is that this technique ignores potentially useful information
contained in the unppoled sequence of events.
The method for sampling partial discharge involves pooling the point
processes from small phase windows on different cycles of the excitation.
The assumption of periodic nonstationarity is still used and each component
of the pooled process comes from a phase window located at the same phase
but from a different cycle. The phase windows are of an extent such that
each component process pooled into the result contributes no more than one
point. The basic assumption that is made for this technique to work is that
the partial discharges occurring at sufficiently separated times are
statistically independent. It is presently not clear if separation of the

46
processes by only a single cycle is sufficient for independence. The resulting
pooled process in each window will be approximately a marked Poisson
process with intensity X^. While there is no mathematical proof that true
partial discharge sampled in this manner converges to a marked Poisson
process, if the assumptions made about the process are true then it will be
approximately a marked Poisson process. Using this sampling method and
assuming that the resultant pooled process is Poisson then the analysis in
the previous section applies. Questions about the residual dependencies
between adjacent phase bins are still unanswered. If statistical analysis
indicates the existence of these dependencies then the sampling method can
be modified. The modification would involve sampling in a manner which
would make the constituent point processes to be pooled more sparse. This
could be accomplished, for example, by sampling only a few phase bins per
cycle which would make the (per cycle) point processes more sparse.
In actual practice when partial discharge with large intensities are
observed it is likely that the large intensities are due to multiple discharge
sites that are simultaneously active. When this is the case then these
multiple sites will be independent if they are sufficiently separated.
Sufficient separation means that there is little physical communication of
charged species and photons between the sites. In this case the process is
already pooled and it is likely that a Poisson model may be accurate.
2.5 E xperim ental Study
The following experiments demonstrate the consistency of the point
process model discussed In the previous sections. The results are not
definitive in the sense that they prove that the model is correct but rather
the results demonstrate that the model is consistent with the observations.
Only a limited number of experiments were performed for this particular
aspect of the thesis.
High voltage systems, by necessity, are large to prevent unwanted
discharge and breakdown. Typical high voltage systems also have complex
structures, in which the signals of interest may propagate through many
poorly defined paths. Subtle details of the signals may then be masked.
The structures considered in the following sections, concerned with
estimating the location of the partial discharge site, are too large for a
controlled study of the characteristics of the partial discharge process. To

47
properly study the partial discharge process with as many variables as
possible under control, some type of test cell must be used.
The test cell used in this work is a slight modification of the test cell
used by Luczynski [25]. The principle modification was the inclusion of an
acoustic transducer to permit the acoustic emission to be measured
simultaneously with the electric signal. The test cell, depicted in Fig, 2.5,
provides high fidelity signals with high sensitivity and allows the Control of
parameters influencing the discharge process. In particular the variables of
interest are the materials surrounding the discharge gap and the actual
discharge gap geometry. When a high voltage is applied to the high voltage
electrode a discharge occurs in the region labeled E in Fig. 2.5. The acoustic
transducer is a Physical Acoustics integral preamp transducer with a center
frequency of 150 KHz. The insulating material used on both faces of the
discharge gap was Plexiglas.
The test cell is small compared to the electrical wavelengths studied,
(bandwidth of 20 MHz), however, it is still large with respect to the acoustic
wavelengths. Unwanted multipath effects will be present for the acoustic
signals and therefore a characterization of the typical acoustic signal will be
necessary in order to extract information about the waveforms. The
acoustic signature of the test cell was studied by fracturing a pencil lead in
the center of the test cell. A typical response to this impulsive force is
shown in Fig. 2.6. The peaks in the envelope of the response are due to
reverberations in the test cell. The first peak is the desired signal, the
second is due to a reflection from the outer edge of the central measuring
electrode and the third peak is from a reflection from, the outer edge of the
guard electrode.
The signal model discussed above makes assumptions about the time
varying-nature of the process and the type of stationarity that will be used.
The relative energy vs. the phase of the excitation voltage is shown in Fig.
2.7. The relationship of the energy to the phase of the excitation voltage
was developed by averaging over many cycles of the excitation voltage a
quantity referred to as the mark accumulator process [61] which for this
application will be defined as ,
V
;'"
’
V
/l/*
■
?
'A
,''J
'
* .(4 ) =
i=l

48
H
Vf
^ /
■
■ '■
%
re 2.5. Test cell. A: acoustic transducer; B: central measuring
electrode; C: coaxial connection; D: guard electrode; Er
discharge region; F: high voltage electrode; G: isolation resistor;
H: micrometer adjustment.

20 40 60 80 100 120 140 160
Time, seconds (X10'®)
Figure 2,6;. Acoustic response of test cell.

50
Time, seconds (x IO-3)
Figure 2.7 Relative energy versus excitation voltage phase.

51
which is defined slightly different than in [61]. In this case the m ark is the
energy and the m ark accum ulator function is defined for each phase bin. As
can be seen from Fig. 2.7 the energy is distributed across the excitation
voltage in a cyclic m anner.
A m odel w as developed in w hich assum ptions about the station arity of
the process were m ade. To verify the assum ptions about the station arity of
the process, it is necessary to test the station arity of the com ponents of the
m odel. It w as stated above in the section about the signal m odel th at if the
m easurem ents were confined to a sm all periodic interval, (w ith the sam e
period as the excitation voltage), then the param eters describing the process
would be constant. In particular it is necessary to test the station arity of
the tim es betw een partial discharges and their energies. The m ethod th at
w as used to test the hypothesis of station arity w as a nonparam etric test
called the run test, w hich can be used to test for trends in tim e, [15]. The
test used the m ean and variance of the sam ples over 20 second intervals and
checked the station arity over a h alf hour period at a significance level of
0.05. The run test w as applied to sim ilar sam ple paths to test for trends in
tim e. R esults of the test indicate w hether or not the sam ple paths cam e
from a stationary process; usually, the d ata w as accepted as stationary.
A typical histogram of the tim e betw een discharges is shown in Fig. 2.8.
The histogram is not exponential, ruling out a Poisson process. The next
order approxim ation would be a renewal process where the tim es betw een
partial discharges can assum e an arbitrary distribution. A typical histogram
of the energies for the acoustic em ission is depicted in Fig. 2.9 for a fixed
phase window. A n interesting relation betw een the gap size, x, and the
energy em erged from these m easurem ents. Fig. 2.10 shows the relationship
betw een the m ean energy of the discharge and the gap size for a fixed phase
window. W hen plotted on a sem ilog scale the resulting curve suggests using
b(x) = k ex p (m x ) ; k > 0, m > 0, x > 0
as a m odel for the energy/gap relationship, where b(x) is the param eter
controlling the energy distribution. T he constan ts k and m can be
determ ined by perform ing a sim ple least squares fit on the data in Fig. 2.10.
Com bining this w ith the histogram s of the energy, a sim ple m odel can be
constructed by assum ing the histogram s can be m odeled as gam m a density
functions. This yields a m odel for the m ark distribution as a function of gap

Number
of
Intervals
(x
lO
2)
52
Time, seconds (XlO"6)
Figure 2.8. Histogram of time between partial discharges.
1600

2000
90 100 1500
Energy, W
as (XIO"9)
Figure 2.9. Histogram of acoustic energy from partial discharges.

Log
Energy,
Ws
54
x, mm
Figure 2.10. M ean energy versus gap size.

size
55
Kih1M
P(u;x) — unexp(—b(x) u)
n!
This particular representation has a mean value of
m(x):
n + 1
b(x)
and a variance of
O
2(X)
n+1
b2(x)
It should be remarked that the constants k and m are actually time varying
quantities, As mentioned above, the parameters controlling the mark
distribution will be assumed to be periodic nonstationary. Therefore, for
any small time interval located at a fixed phase of the excitation voltage,
the quantities k and m will be constants, (which will change as the time
interval is moved to another phase of the excitation voltage).
Another experiment which was performed involved the study of partial
discharge when the surface conductivity, ps, is changed. The same type of
electrode configuration was used as in the energy/gap experiment except
that the surface conductivity of the discharge region was controlled. The
gap separation was held constant and the surface conductivity was
controlled by coating the faces of the partial discharge region with resistive
paint. The results of the experiment are illustrated in Fig. 2.11 where the
mean partial discharge energy versus surface conductivity is plotted. Only a
few points are plotted because of the difficulty of obtaining controlled
surface conductivities on the samples. On a log-log scale there appears to be
a linear relationship between the mean energy and the surface conductivity.
Further study of the energy relation to the surface conductivity is needed to
validate this model.
A classic example of partial discharge with non-exponential occurrences
times and a complex dependency structure is the DC point to plane process
'4

56
10 000
lo
g resistivity. O
K
M
S/SC
U
A
R
E
Figure 2.11. M ean partial discharge energy vs. surface conductivity.

57
referred to as Trichel pulses. Fig. 2.12 illustrates a typical sample function
from a DC point to plane partial discharge in which the events seem to form
a periodic sequence. Trichel pulses seem very regular but there is a small
randomness in the behavior of discharge processes like these. Referring to
Fig. 2.13 a histogram of the time between events and a histogram of the
relative charge in the events clearly show that even a seemingly stable
partial discharge process like DC Trichel pulses have random characteristics.
There are several reasons for considering DC Trichel pulses as an
experimental beginning for detailed statistical analysis under a point process
framework. The most important reason is that this partial discharge has
been extensively studied and is probably the best understood partial
discharge type. Also, the experimental conditions are relatively simple to
reproduce without having to resort to exotic sample preparation techniques.
Another reason is that the excitation voltage is constant removing one
source of nonstationarity in the process. There are similar types of partial
discharge that can occur under AC conditions and this beginning point
would lend understanding to the AC case. Finally, Trichel pulses appear to
be so regular that if a reasonable model can be found for this, situation,
which appears to have a highly complex dependency structure, then
modeling of other more random appearing partial discharges would seem,
intuitively, a more tractable task.
A single model for the simple case of Trichel pulses is difficult to
develop because the statistical characteristics change as the applied
potential is changed. At lower potentials for which the repetition rate
remains below several hundred KHz a renewal model appears to be an
accurate model for the occurrence times. At higher repetition rates a
complex dependency structure begins to become apparent with the memory
of the process extending over large numbers of events in the'series. The
following analysis used observation times that ranged from 300 /rs to 15 ms.
Slowly varying nonstationarities from changes in the materials due to aging
were observed; the measurements changed slightly when repeated with a gap
that had been allowed to age.
A simple experiment was performed using the same test cell as used in
the other experiments except that the high voltage electrode was replaced by
a phonograph needle. The experiments were performed at 766 mm Hg, 22°
C and 60% relative humidity. The spacing of the point to plane was 16 mm
and the applied voltage was varied between 5 KV and 8.5 KV with the point
being kept at the most negative potential. A single recording provided the

BLfF. XO
V O U T B
rrm rIH -
B U F . XO
V O U T S
-
B U F . XO
V O U T B
V O U T B
V ^ il
B U F . XO
SE C O N D S
SE C O N D S
C
n
O
O
SE C O N D S
•snsw
SE C O N D S
Figure 2.12. Typical sample function of Trichel pulses.

59
OSTOMA
OSTDXO
VOLTS
Figure 2.13. Statistical description of Trichel pulses, a) Histogram of time
between partial discharges, b) Histogram of partial discharge
energies.

60
data (30,000 points) and the techniques in section 2.2 were used to measure
the data points for the sequence. Two sequences were formed; an inter-
occurrence time sequence and an amplitude sequence. Typical sequences
contained between 100 and 400 data points. The sampling rate was varied
between 2 MHz and 100 MHz so that an adequate number of data points
could be collected for different repetition rates. The input bandwidth to the
recorder was varied between 500 KHz and 10 MHz depending upon the
sampling rate.
Statis analysis of the series of events is difficult without a specific
model to use. Since the inter-occurrence time histogram is not exponential
the Poisson process is ruled out. The next easiest model is the renewal
model where the inter-occurrence times can have an arbitrary distribution.
T he‘general characteristic of renewal processes is that the inter-occurrence
times form an independent series of events. Since the series forms an iid
sequence of events the serial correlation coefficient should be be zero
(ideally) for lags other than zero. A natural first step is to check the serial
correlation coefficient to ensure thht there is not correlation between the
events; a typical example of the correlation coefficient for different lags is
depicted in Fig. 2.14. A hypothesis test was performed which tests for serial
correlation in the sequence [85]. This test was performed at a significance
level of 0.05 and the data always passed; this test if the repetition rate was
low enough. As can be seen from Fig. 2.14 the amplitude data has a serial
correlation which is similar to the inter-occurrence time’s. Another aspect
that is apparent is that there is a strong correlation between the inter
occurrence times and the amplitudes. The assumption that the counting
process and the mark process are independent is ruled out by this result.
More sophisticated models need to be considered to handle this type of
dependence for example a mutually self exciting process. It is interesting to
examine similar sample serial correlation functions obtained from high
repetition rate Trichel pulses. Fig. 2.15 and 2.16 illustrate two different
situations; one in which the serial correlation is just beginning to become
apparent and the other in which there is clearly a significant memory
reaching far into the past of the process. It is interesting to note the highly
structured form of the serial correlation coefficient for the highest repetition
rate data. It seems that this serial correlation function might be easily
described with a simple parametric model. Also note the similarity between
the correlation functions for the data set; even the cross-correlation has the
same form.

61
ACOWXO
u a - -
•>4’ - ■
MA P u
Aa /A a v A A z j i I AA- -I A a A /
XJ vv y y y V /y W.Vs^ v/
ACOfIXl
U i--
E I
■-* A A fA , / s ^ j iAA I IIa J w A A A .A /
' V r vV vA / Vvvs V vV V /
CCbWXD
I .....
1 it t "• “■
I l A /^ a 7 A a /
iy. y. — j..t ..n — y .— V -
Figure 2.14. Serial correlation functions, X= 5.7xl03. a) Autocorrelation of
time sequence, b) Autocorrelation of energy sequence, c)
Cross-correlation of time and energy sequences.

Figure 2.15. Serial correlation functions, X= 8.0x10s a) Autocorrelation of
Gross-correlation of time and energy sequences.

Figure 2.16. Serial correlation functions, X= 1.43xl06 a) Autocorrelation of
Cross-correlation of time and energy sequences.

64
Checking for serial correlation is important as a first step but this does
not guarantee independence. More powerful tests which test for serial
independence must be used on the sequence. The preferable manner for
testing a series of events for the general renewal hypothesis would be to use
a nonparametric test for serial independence in the sequence. The run test
[l5] is a nonparametric test which can be used for testing the hypothesis of
serial independence. There are other tests which can also be used and can
be found in [85]. The results from applying the run test to the data, which
did not exhibit serial correlation, were that the sequences consist of serially
independent events at the 0.05 significance level. These results are not as
definitive as would be desired because the run test has a low power.
The concept of pooling the data to form a Poisson process was also
investigated using a Trichel pulse process. In this experiment multiple
records were superposed to form the pooled process. Figure 2.17 illustrates
the results of pooling pulses from a Trichel pulse process which had a
repetition rate of 28130 pulses per second. The pooled record length was
20.44 /is and contained the superposition of sixteen records. On the average,
the pooled records contain about 9 pulses each. There are several different
tests that can be used to test the data to see if it was generated by a
Poisson process [85]. The statistical test that was used is called a dispersion
test [85] and uses a test statistic, d, which is given by
k (Hi - S ) 2
Z j —
where there are k observation intervals of equal length with m events
contained in the itn observation interval. The term, n is the mean number
of events and is given by
For large enough k a chi-squared distribution with k-1 degrees of freedom is
a good approximation to the distribution of d under the null hypothesis.
Data similar to that shown in Fig. 2.17 was tested using 40 pooled records
and the Poisson hypothesis was accepted at the 0.05 significance level. A

AVO. Xl
Figure 2.17. Typical pooled records for Trichel pulses.

66
typical histogram for the tim es betw een events in a pooled process is shown
in Fig. 2.18.
2.6 U ltra Low Level M easurem ents
It is of great interest to measure partial discharge signals with
extremely low levels of charge. This section will deal with a technique which
can aid in the measurement of low level pulses buried in noise with small
repetition rates. The basic approach will use cross-correlation and averaging
to reduce the noise level in the estimate. Cross-correlation will be used to
provide a coherent averaging method. To increase the output signal to noise
ratio an ad hoc detection technique based on the level crossings of a
Gaussian random process will be used. The Gaussian noise assumption will
be used because it is an excellent model for the thermal noise in the
amplifiers and is the dominant source of noise in the application: to be
discussed. The final estimate will not be the charge in a single pulse but will
be an average value that is difficult to interpret. This technique applies to
partial discharge pulses that have peak values either greater than or less
than the standard deviation of the Gaussian noise. The most interesting
case is that in which the peak value of the pulse is on the order of the
standard deviation of the noise. In this case the SNR is too low and the
individual partial discharge pulse is not directly observable.
The most desirable approach would be to setup up a detection
procedure to test the hypothesis that the signal is present. However, there
are several unwanted parameters in the formulation for which the a priori
probability densities are unknown. Implementation of such a scheme is
difficult in this case and approximate procedures are needed, Another
reason for using the method to be discussed is that is provides a simple, real
time alternative to a true detection algorithm which can become numerically
intensive. If a digitally implemented detection scheme were used and the
signals were observed for a one minute interval using an eight bit quantizer
with a 100 MHz sampling rate then over 10 giga-bytes of data would have
to be processed. This would translate into an excessive amount of data
processing for even a super-mini computer to perform. However, once the
preprocessing has been performed by the algorithm to be presented then it is
desirable to perform a hypothesis test using, for example, the generalized
likelihood ratio test discussed in chapter four. Follow up processing by more
exact tests will further increase the output SNR of the measurements.

67
SE C O N D S
Figure 2.18. Typical histogram of times between events for pooled records.

68
The measuring system will consist of two inputs; these two inputs can
be at physically separated locations or at a single location (two amplifiers
connected to the same point). The input signals are modeled as
,. , ' Nt
x(i ) = v; ajs( t - Ii ) + H1(I)
i= l
and
: Ny
y ( 0 = > ] a is( i ~ t S
i=l
D ) + n2(t)
where D is the delay between the signal s(t) at the two input points and is
zero if the amplifiers are connected to the same point. The number of
events in the interval, T, is N
<p and will be assumed to be Poisson
distributed. The noise terms P1
,(t)l and n2(t) 'will be assumed to be Gaussian
and independent from each other and from the signal s(t). The input SNR
for either channel is easily shown to be
Xa2 JF |S(c
j)|2 dw
SNR;,
- O O
OO
J N(o
j) dw
- O O
where S(g
j) is the Fourier transform of the partial discharge pulse, N(a,’
) is
the noise power spectral density, the partial discharge process is assumed to
be homogeneous with intensity parameter X
, the pulse is not DC coupled and
the mean square value of a, is a2. The Poisson process is assumed to be
homogeneous because no a priori knowledge is available about the intensity
function X
t. The homogeneous intensity, X
, will have the interpretation that
it is the average number of partial discharge pulses per second even though
the true process will not be homogeneous.
One approach to the problem of measuring low level partial discharge
would be brute force averaging of the cross-correlation function. This
approach has limitations for low repetition rate (small X
) partial discharge

69
because the output SNR is still low when X is small. There are tremendous
difficulties associated with retrieving these pulses from the noise if there are
only a few pulses per cycle of the excitation voltage because the variance of
the estimate will be large compared to the partial discharge energy. An
obvious modification would be to limit the processing to a small phase
window, of extent <
/>
, to a region where the probability of measuring a partial
discharge is the highest. The intensity, X was assumed to be constant
because there is not a method for knowing the true function X
t before
measuring it. In actual practice some information is known; the discharges
are most likely to occur near the voltage peaks. If the correct phase window
is chosen which contains all of the partial discharge pulses then the
equivalent homogeneous intensity will be
'eq
360
<
P
X
That is to say that the intensity parameter increases giving a higher input
SNR which yields an improved output SNR. Furthermore, if an accurate
trigger is available so that the processing can be limited to regions in the
near vicinity of the pulse then the output SNR increases dramatically. The
problem with using only the regions in the near vicinity of the pulse is that
these regions are difficult to determine unless the pulse has sufficient
amplitude. In this case the pulse itself would be required to provide the
trigger.
There is an alternative which can provide increased performance by
locating the regions (on the average) in the vicinity of the partial discharge
pulse. The algorithm locates data segments that have a significant
probability of containing a partial discharge pulse and then processes only
those segments. This approach yields better results because the data has an
effective higher input SNR. This technique is not perfect because it will miss
pulses and indicate that a pulse is present when it is not. These deficiencies
amount to a loss in the output SNR but the improvement of the output SNR
can be almost as great as if the region in the vicinity of the pulse were
actually known. This technique relies on the level crossings of a random
process and works by using the trigger on the digital recorder as the
detection level. If the noise level is high then a trigger set at a proper level
will initiate recording a few times per second (when the noise crosses the

70
trigger threshold). The basic idea lies in the fact th at when a sm all signal is
added to the noise then the recorder will trigger more often than if the noise
alone were present. Proper selection of the trigger level will im prove the
output SN R because there is an increased probability that there is a partial
discharge and not noise alone.
To analyze the algorithm consider w hat happens when noise alone is
present at the input to the recorder. The noise will be G aussian w ith zero
m ean and having a power spectral density, N (cj). The trigger level will be set
at a level u so th at any upcrossing of the level u by the noise will initiate
recording. The m ean num ber of triggers (upcrossings) in the interval T is
given by ,[86]
E { Cu(OjT) }
V
z __h
1
e 2n"
where the orj term s are the spectral m om ents of the noise and are given by
aO= Rn(O
)
«2 = - R n (O
)
w here R n is the autocorrelation function of the noise and the prime refers to
differentiation. The second spectral m om ent, a 2, is required to be finite;
w hite noise is not allow ed. W hen a signal is added to the noise then the
m ean num ber of triggers is given by [86]

I
E{ Cm(t)(0,T) } I
V ^o o
<
/>
m (t)
V « o
2 ^
V «2
+ m (t)
w here <f> and <
}> refer to the G aussian probability density and error function
respectively. T he expression above gives the crossings of a curve m (t) by the
G aussian noise. T he particular curve th a t is of interest is the level shifted
signal m (t) = a s(t) — u where s(t) is the partial discharge pulse w ith
am plitude a and u is the trigger level of the transient recorder. The
integrals defining the expected num ber of triggers, for realistic signals, are
difficult to calculate and further sim plifications need to be found.
Furtherm ore, the signal is not determ inistic and is actu ally a random
process w hich further com plicates the analysis.
F ortunately, som e assum ptions can lead to a usable im plem entation of
this technique. One assum ption is th a t the point process has a sm all
in ten sity so th a t the probability of tw o partial discharge pulses having
significant overlap is sm all. T he next assum ption is th a t the partial
discharge pulse will be m odeled as a rectangular pulse w ith the pulse w idth
being equal to the duration of the true pulse. T hese tw o assum ptions lead
to a sim ple approxim ate solution in w hich the level crossings of the noise can
be used. It should be noted th a t even if the tw o assum ptions were true then
the solution to be presented w ould still be approxim ate. The solution to be
presented is based on a relative frequency approach to the probabilities and
is a very sim ple approach to an extrem ely com plex problem . H owever, the
approach works and provides a sim ple technique to m easure low level partial
discharge.
Several definitions need to be sta ted before the solution can be given.
T hese definitions are
A T ^ recorded observation interval for a single trigger

72
N pd = num ber o f triggers due partial discharge plus noise
N n = num ber of triggers due noise only
D = duration of a single partial discharge pulse
T = -total observation interval
N t = total num ber of triggers
X = average num ber of partial discharge pulses per second
By assum ption the process is a hom ogeneous Poisson process so the
probability th a t an event is observed in (t, t + A t) is approxim ately XAt.
Since it w as assum ed th at the partial discharge pulses do not overlap the
coverage of the interval, (0,T), by the square pulses is X T D where D is the
w idth of the square pulse. A lso, the coverage of (0,T) by noise only is
T — X T D . The m ean num ber of triggers in the interval (0,T) due to the
noise only is approxim ately given by
E{ C„(0, T - XTD) } = (T ^ TD)
a2
ao
%
e
u~
2oo
The m ean num ber of triggers in the interval (0,T) due to partial discharge
pulses plus noise is approxim ately given by
E { Cu_ a(0, X T D ) } =
XTD
27T
a“—2au
0 p(a) da
where p(a) is the probability density of the partial discharge amplitude.
The amplitude, a, is assumed to be of one polarity and if the discharge has a
negative polarity then the same expression applies if a negative trigger is
used (downcrossing). The integral in the above expression is easily
evaluated for special probability densities such as the gamma density. If the
amplitudes are gamma distributed with density given by

then the mean number of triggers is approximately given by
E{ Cu_.a((), XTB) }
I
(XTD) a2 2 n IO77rv I1
X
- 2 n „
e
2t
t aO
M
n I277 aO] |'(n+1) ( U - nu-i)-
2 n ()
e
The constant, //„, is found from evaluating the expression
ft, = I
0 V 27raO
[ a - ( u - OoB ) f
2o0
Using these formula, Nn, Npd and Nt are approximately given by
Nn = E{Cu(0, T —XT AT }
Npd = E{Cu_a(0, XTAT }
Nt = Nn + Nr
The input SNR can also be written as
SNR;in,I
X T ^ p
Ton
2
where ^ pd and T o2 are the average partial discharge pulse energy and noise
energy respectively. The level, u, is chosen so that Nt AT < T which means
that no more than one trigger due to noise occurs in AT. If this condition is
not met then the following approximations for the increase in SNR will be
pessimistic. Also the noise bandwidth is assumed to be comparable to the
pulse bandwidth so that Cn_a(0, D) < I. This means that each partial

74.
discharge pulse will provide only one trigger rather than multiple triggers.
As a practical matter, this choice of bandwidth is preferred because it
provides the best input SNR. In fact, to achieve better performance, an
analog matched filter should be used on the incoming data which Would
force; the signal and noise to have comparable bandwidths. Provided that
the$e conditions are met then the input SNR for the reduced data set is
SNRin,2
Npd %pd
Nt:AT 4
The gain in the input SNR is
r-
SNRill2
SNRjJljl
Npd T I ‘
Nt AT XT
In the noiseless case where Npd = Nt = XT the gain in the SNR is for
which it is assumed that the only triggers generated are generated by the
partial discharge pulses themselves. The SNR gain can be put into a more
suggestive form using the definitions X
p
ciT = Npd and X
nT = N1
1which gives
C X ' . : 5 ; ' s :
■
■
■ .
GSNR
i
XAT
X.
^nd + i
where the first term represents the ideal gain and the term in brackets
represents the loss in gain due to the nonideal behavior of the algorithm.
The above formulas represent the gain in the input SNR using the
algorithm and it is of interest to find a similar expression for the Output
SNR. The estimate being used is the cross-corfelation between the "two
inputs and expressions for the variance of a cross-correlation estimate can be
found in [15]. Ignoring the contribution to the variance from the
randomness of the partial discharge amplitude and considering only the
contribution from the noise gives the output SNR for direct averaging

75
SNRoutjj
( XT % pd )2
T &
The T e r m , i s defined by
§>
OO
I K lU
lIr) K , n i r) dr
where RniU
t(r) and Rno„.,(7) are the autocorrelation functions of the noise in
the two channels. Using the level crossing algorithm gives an output SNR
given by
SNR,out,2
(Npjgy)2
Nt AT 9?
The gain in the output SNR is given by
g SNR
I
XAT
L
>d ^pd
X ; d + Xn
where the first term is the ideal gain in the SNR and the term in brackets is
the loss due to the algorithm.
It should be pointed out that this analysis is only approximate and
gives a simple closed form expression for a highly complex problem. These
expressions might best be treated as upper bounds on the possible increase
in the SNR. Even though these expressions are approximate and the
technique ad hoc, the results of using this technique are excellent. The
following experimental simulation illustrates the effectiveness of the
algorithm for measuring partial discharge buried in noise. The rms level of
the noise was 97 mv with a peak pulse height of 200mv. Fig. 2.19 illustrates
a typical data record with a pulse buried in noise. The noise had a
bandwidth of 2MHz, the pulse had a duration of 500 ns and Xwas 60 . The
observation time, T, was 600 seconds, the sampling rate was 10 MHz and

VQUTBf
c
SeCONOS
Figure 2.19. Low level partial discharge pulse, a) Pulse buried in noise, b)
Actual pulse shown to scale of a), c) Magnified (XlO) pulse.

77
AT was 6.4 /rs. The input SNR for a single data segment containing a pulse
was about -6 dB while the input SNR for the entire process was about -74
dB. The trigger level was 500 mv and the residual mean at the digitizer
input was 1.8 mv giving an effective trigger level of 498.2 mv. The measured
value of Nn was 2230 while the theoretical value was 2244. The measured
Value of Npd was 335 while the theoretical value was 319. The results of the
processing are shown in Fig. 2.20. The theoretical gain in the input SNR
was 25 dB, the theoretical gain in the output SNR was 5 dB and the
theoretical output SNR was 15 dB. The average energy in the partial
discharge pulses is calculated from
Nt
g D
d = --------1Tr— R(O)AT
Nt - Nn
theor • '
where R is the estimated cross-correlation function. The estimate for the
partial discharge was 16.9 nv2 and the actual value was 17.5 nv , which is
within the expected accuracy based on the theoretical output SNR. This
result corresponds to measuring partial pulses with a charge of about 0.0045
pC that occurs only once per cycle at 60 Hz. The output had a modest gain
in SNR as compared to direct averaging but this figure does not reflect the
true performance of the algorithm. To accurately assess the algorithm both
the gain in output SNR and compression of the data need to be examined.
For direct averaging 12 giga-bytes of data would need to have been collected
and processed. Using the level crossing algorithm only 328 kilo-bytes were
processed which is a compression factor of 36550. One final comment is in
Order about the algorithm. The trigger level is in the exponent of the
formulas and consequently the algorithm is very sensitive to small changes in
this level. r
To improve the output SNR of the measurement further, more
sophisticated processing could be performed. The generalized likelihood
ratio test could be used as a second screening algorithm before the data is
included in the estimate. This test is discussed in chapter four where an
approximation is presented for the tests implementation. Using some
definitions, the performance gain from the second test can be stated in terms
of the previous results. These definitions are

78
AVEGX2
S E C O N D S
AVECXa
S E C O N D S
Figure 2.20. Results of level crossing algorithm, a) Ideal output of
algorithm, b) Output of algorithm for experimental data.

79
P0 =
=Pr(deciding a pulse is present | pulse is present)
Pf = Pr(deciding a pulse is present j pulse is not present)
When a hypothesis test is performed on the reduced data set containing a
total of N1 records then new values for the counts of detected and falsely
detected partial discharge pulses must be used. These new numbers arc
given by
Mt = Nt ( P0 + P f )
Mpd = Nt Pd
Replacing Nt, Npd and Nn with Mt, Mpd and Mn in the above expressions
then gives the performance of the the two algorithms used in conjunction.
The only motivation for using these algorithms in conjunction is to reduce
the amount of data that needs to be processed. If the processing resources
were available then only a detection algorithm would be necessary (and
desirable). An ad hoc detection scheme, based on the cross-correlation
function, can be used and will respond to any correlated input (both partial
discharge or interference). The test would be ,
Hu
I xItMt) dt ^ T -
where 'y would be the threshold of the test. Histograms under the two
alternatives using this cross-correlation based test are shown in Fig. 2.21 for
the same data discussed above. The H0 alternative is signal plus noise and
the H1 alternative is noise only. It is easy to see from Fig. 2.21 that a
decision region can be chosen such that Pf is small which will improve the
output SNR if the detection algorithm is used.

□ . O l
Figure 2.21. Histograms of the processor output under the two hypothesis.
H0 is to the right of the threshold and H1 is to the left of the
threshold.
O
O
o

CHAPTER THREE
DELAY ESTIM ATION
3.1 Introduction
This chapter discusses techniques for locating partial discharge sites in
high voltage equipment using digital signal processing techniques to estimate
the time of arrival of the signals. There is a vast body of literature
pertinent to delay estimation including passive delay estimation techniques
[16]. Passive delay estimation is required to locate a signal source, (signal
emitter), when the measuring device has no knowledge about the absolute
temporal characteristics of the signal, (ie; its absolute time origin). For
example, passive sonar systems use the sound emitted by enemy submarines
to deduce their locations without any information about the absolute time
at which the sound was produced. The time delay information is deduced
using the relative time delays between the signals, (multiple receiver
approach), or the signal and its reflection, (single receiver approach). There
are so many delay estimation techniques that it would be impractical to
discuss each in detail so several will be outlined but only a few specific
approaches will be consider in detail.
Cross-correlation techniques have several advantages over other
techniques that are used to determine delay differences. The primary
advantage that cross-correlation has is that it maps a signal with a random
time variation into a time (time delay) stable function. This allows the use
of a 100% coherent averaging. In other words by averaging cross-correlation
functions one is assured that each element of the average is perfectly lined
up with the others. Direct averaging of the signals can never provide this
perfect alignment. The next advantage is that uncorrelated noise averages
to zero so that signals buried in thermal noise can be easily extracted. The
filial advantage that cross-correlation has for this application is that
physically separated partial discharge sites are uncorrelated with each other.

82
This says that a cross-correlation function for a device with multiple
discharge sites will simply be the superposition of the individual cross
correlation functions of each site.
A non-correlation based technique for locating the partial discharge
sites is discussed in section 3.2. This method seems to predominate the
current techniques for locating partial discharge even though this method
has been abandoned by other disciplines because of its inaccuracies; This
method is included for comparison and an error analysis is performed later
in the chapter.
The generalized cross-correlation (GCC) method for locating a passive
emitter is discussed in section 3.3. This technique is the preferred method
for source localization because of the accuracy that it provides. This
technique uses prefilters that are optimal in some prescribed sense. The
different variations that will be discussed are useful in solving some of the
problems encountered in measuring time delay in high voltage systems. High
accuracy results are possible using the maximum likelihood (ML) weighting
function and this method is the preferred method for small data records. If
the measurements are made in a system with narrow band interference the
Wiener processor (WP) implementation seems to provide the best results.
Bias errors that can occur when making delay estimates are discussed
in section 3.4. In the ideal case many delay estimators are unbiased,
however, in the general measurement situation various types of bias errors
can corrupt the estimate. These errors can arise because of the lack of
temporal resolution when overlapping pulses are encountered. Other errors
occur when the channel contains a phase distortion which distorts the
correlation function causing a shift in it’s maximum. Bias errors can occur
through the processing methods that are chosen to estimate the delay. For
example, in certain situations autocorrelation estimates can be highly
biased. Section 3.5 discusses the variance of delay estimates and illustrates
the advantage of correlation based estimators over time of arrival
estimators. The Cramer-Rao lower bound is discussed along with a local
analysis of the variance for wide sense stationary processes and transient
nonstationary processes.
The methods that are used to reduce the delay bias due to overlapping
correlation functions are referred to as deconvolution. Section 3.6 discusses
an ad hoc generalized cross-correlator, referred to as the modified PHAT
(PHAse Transform) processor, which provides greater range resolution in the

83
measurements. The filter is a combination of the PHAT GCC and the ML
GCC which uses a naive approach to deconvolution for the frequencies with
high signal to noise ratios and uses the optimal ML solution for the
frequency regions with a low signal to noise ratio. Section 3.7 discusses a
new GCC based on a minimax equalizer that recently appeared in literature.
This particular implementation provides improved range resolution at the
cost of a greater variance of the estimate. The unusual aspect of this
processor is that the necessary signal spectral characteristics are estimated
and the estimation error is incorporated into the solution.
3.2 A Non-correlation Based Location Technique
The, first method uses only the estimated arrival times at a single
receiver and there are several approaches to estimate the time of arrival of a
signal. The simplest method is to use the time at which the signal crosses
some predetermined threshold (the threshold is often chosen to be the
signal’s maximum). This approach is used in TDR systems and has limited
application because of the difficulty in determining the appropriate threshold
for distorted, noisy or unknown waveforms. Furthermore, if the discharge
site is extremely active or there are multiple discharge sites, active at nearly
the same time, the number of discharges present on the TDR screen
prevents a correct delay estimate from being made. This technique has been
used in factory testing in attempts to locate partial discharge sites.
Typically, a wide band coupling device is placed at one end of the cable and
the pulses are observed on an oscilloscope. The observed pattern consists of
a directly arriving pulse and a pulse reflected back from the opposite end of
the cable. There are other terms due to further reflections as the pulse rings
back and forth on the cable but these are of little interest. The location is
deduced from an estimate of the time difference between the first two pulses
appearing on the screen. The time difference will be T = T + b —a, where
T is the total electrical length of the cable, a is the propagation time from
the site to the cable end with the coupler and b is the propagation time
from the site to the other end of the cable. The position of the site, € , is
then given by
e = y2 [L - ( r - T ) u
where L is the length of the cable and v the propagation velocity.

84
As mentioned above, there are several difficulties with this approach.
The cable is dispersive and consequently neither the front edge of the pulse
or the pulse shape is preserved. The second pulse is less in amplitude than
the first. The amplitudes are random, causing a jitter on the screen. All
these factors contribute to the difficulty of obtaining a delay estimate using
either the peak or some predetermined amplitude threshold as a reference on
the two pulse. These problems can easily cause large errors in cables of
substantial length. Another problem occurs when the discharge is low level,
or if there is a large amount of interference because it is very difficult to
obtain a stable enough display in order to make a measurement. A further
problem occurs when there is a large amount of discharge activity present in
the cable. In the case where there is only one site, which is very active,
confusing results can be obtained. The problem occurs when the reciprocal
of the count rate intensity function is on the order of the electrical length of
the cable. In this case the second pulse on the screen is not necessarily due
to the reflection from the opposite end of the cable but may be due to a
second discharge occurring. The same phenomena happens when the
reciprocal of the combined count rate intensity function for multiple partial
discharge sites is on the order of the electrical length of the cable.
An improvement in this technique can be made by using matched
filtering. There are numerous articles on this subject, most of v/hich apply
to active delay estimation schemes. This technique cross-correlates a known
waveform with the incoming signal; the times of arrival are then the times
at which the output correlation function attains its maximums.
Mathematically this operation is represented by
R(r) - / x(t) s(t-f) dt
x(t) = s(t) + n(t)
where x(t) is the signal received at a single location on the cable , n(t) is a
white noise process, and s(t) is the known partial discharge waveform. This
approach would measure the times of arrival of the signal and its reflections.
Matched filtering yields excellent delay estimates provided that the signal
shape is accurately known before hand. The same approach used for the

85
TDR would then be used to locate the site, the only difference being the
method of choosing the arrival time.
This approach has fewer problems than the TDR approach but it also
fails in certain situations. The problems associated with selecting a proper
threshold are now eliminated and this technique performs very well in
situations where there is high levels of noise. However, there is now the
problem of selecting a proper known waveform to use for the matched filter.
A calibration experiment can provide an adequate waveform for this
purpose. However, because the signal will travel though an unknown
amount of cable the shape will be slightly different than The calibrating
signal potentially causing an error in the delay estimate. The major
problem is due multiple discharge waveforms arriving in close temporal
proximity to each other. In this case the matched filter will not provide any
better result than the TDR did.
There are generalizations to the matched filter approach which take
into account non-white noise case [42]. The resulting correlation function
will then become ■
R (-) = & -1
XM s*M
: NM
whera X(cc’
) and S ( c
<
j) are the Fourier transforms of the received data and
known waveform respectively. The term, N(o
j), is the power spectral density
of the non-white wide sense stationary noise corrupting the measurement.
This particular implementation is effective in eliminating narrow band
interference from the measurements which is a particularly troublesome
problem in these measurements. Another problem that can be addressed
using this approaches the problem associated with closely spaced pulses. If
the spacings are exponentially distributed random variables (a Poisson
process) with a mean value then the matched filter can be modified to
R (r) _1
x m s*M
.X|SMI2 + nH .

86
This particular approach essentially deconvolves the input waveform,
however, the variance rapidly increases and typical resolution performance
can not be expected to exceed a factor of two if a reliable estimate is
desired. '
3.3 G eneralized C ross-correlation
Correlation methods are widely used in delay estimation problems and
there is vast literature on various correlation techniques. In the past decade
a new type of correlation scheme emerged and is referred to as generalized
correlation, [18,39]. The basic idea is to prefilter the signals in some optimal
manner and then cross-correlate. The optimization criteria is dictated by
the needs and constraints of the particular problem at hand.
Once a measurement model is developed the correlation techniques can
be easily implemented on a computer using an FFT approach. The received
signals are as
x(t) = s(t) + nx(t) + I1(I)
y(t) = s(t-D) + n2(t) + i2(t)
where the signal s(t) and a delayed version, s(t-D), are the signals from the
partial discharge site to be located. The parameter to be estimated is the
relative delay, D, between the two signals. For simplicity, the distortion due
to the propagation through different path lengths will not be included. The
signals will be generated by either a wide sense stationary random process or
by random transient events. The thermal noise from the amplifiers is
modeled as. wide sense stationary noise with power spectral density N(cv’
).
The quantization noise from the digitizer, will be assumed to be zero mean,
independentj identically distributed, (iid), random variables. The
measurement noise, n^t) and n2(t), is a combination of both the thermal
noise and quantization noise. The signals .ii;(t) and i2(t) are associated with
signals from other sources not presently of interest and interfering with the
signals from the current site under investigation. These signals can come
from a number of different sources for example reflections of the signal from
discontinuities in the device, other partial discharge sites, signal sources
exterior to the device under test. With the exception of the reflected signals

87
the interfering signals, I1(t) and i2(t), will be considered to be independent of
s(t) but correlated w ith each other.
T he delay is determ ined from the cross-correlation function which is
estim ated as
I T/2
R(r) = Tf I M
t)y(t+r) dt
1 --T/2
T he delay estim ate, D, betw een the tw o signals is the value of r at which the
function, R(r), has its m axim um
D = arg m ax R(r)
C ross-correlation is an effective m anner for determ ining the delay betw een
signals and has been used extensively for this purpose by m any disciplines.
The perform ance of the cross-correlator can be im proved upon by
optim ally prefiltering the input signals before cross-correlating. This
operation is referred to as generalized cross-correlation [18,37,38,39,59] and
its im plem entation is shown in Fig. 3.1. The filters h x(t) and h2(t), or
equivalently, a w eighting function, w (t), are chosen to optim ize the
m easurem ent of D in som e chosen m anner. The resultant correlation
function, is given by
R(r) - 9 -■{ H1(W
) Hj*(w) Sxy(W
) }
= S ' 1( W(V) Sxy(w) }
T he term Sx y C
w) is the cross-spectral density of the signals x(t) and y(t) and
can be expressed as
Sxy(w) = Gpi(w) Gu(w) Gp2(w) Gt2(^) S(w)

88
GENERALIZED CROSS CORRELATOR
x(t) = s(t) + I l 1( I )
y(t) = s(t-D ) +n2(t)
K f f c M = T 1(G^If)W (O )
Figure 3.1, Block diagram of generalized cross-correlator.

89
where S(o
j) is the spectral density of the signal before being filtered. The
filters acting on the signal will be the cascade of the filter, GD
(w), describing
the propagation paths through the device and the filter, Gt(o>), describing
the transfer characteristics of the measuring circuits. Once the relative
delay, D, between the two signals is determined, then along with the known
length, L, between measuring points and a measured velocity of propagation,
the partial discharge site can be located. The location, C , will be given
by ; ^
, V ’ I = ( L - z/D )
Some of the generalized cross-correlation techniques available can be
adapted to autocorrelation processing. In this case the correlator would use
the form of the generalized correlator that utilizes the weighting function,
W(w). The inputs x(t) and y(t) would be the same signal and would come
from a single receiver. In this case, the interfering signals, due to reflections,
would be necessary to determine the relative delay. The correct delay would
be the second largest peak in the correlation function, (since the
autocorrelation function, by definition, always has its maximum at the
origin). The location, ( , will be given by
e L
uD
2
A few of the more useful approaches are mentioned below. The
Hannan-Thompson processor, (HT), is the maximum likelihood
implementation of the generalized cross-correlator [39], The HT processor
attains the theoretically best performance of any .delay estimator, (it
achieves the Cramer-Rao lower, bound on the variance). The constrained
deconvolution processor, (CD), solves the deconvolution problem with the
constraint that the output noise be held to a prescribed level or that the
gain bandwidth of the weighting be held at a prescribed level. This filter is
then used to prefilter the signals before correlating. This approach yields
the best resolution of the correlation techniques for a given output SNR.
This particular processor is developed in section 3.7 along with a minimax
version. The Wiener processor, (WP), makes a best, (linearly constrained),

90
estimate of the input signals before correlating [38]. The WP approach can
be useful in situations where there is a large amount of interference or
distortion. There are other generalized cross-correlators but these are the
most useful in the present study.
3.4 Bias of D elay E stim ates
When processing signals to determine where the sources of the emissions
are located one is naturally concerned about the validity of the estimate.
For example, assume that the measurements will be made on a cable of
length L and that the cable has matched impedances at both ends. A
measurement is then made of a signal arriving at both ends of the cable.
Furthermore, assume that a perfect parametric model of both the signal and
noise are available. The question is how accurately, based on a record of
length T, can the position of the signal be located. Typical measures of
performance are the variance, bias and probability of error of the delay
estimate. These calculations will depend on the parametric model of the
signal and the noise and the technique used to determine the delay. Many
times these calculations are intractable so certain assumptions and
approximations may be necessary.
One factor affecting the accuracy is the error due to the bias of the
estimate. Typically, estimates are sought which are either unbiased or
asymptotically unbiased, (as the the record length increases the bias goes to
zero). Fortunately, for two input delay estimators the bias of the delay
estimate is zero in the ideal case. However, when a Situation develops in
Which the peaks are not highly resolved there is the possibility that this lack
of resolution will cause the estimate to be biased. The bias is due to the
overlapping of adjacent individual correlation functions constituting the
entire correlation function. Asymmetry in the differential channel will also
contribute to the bias by skewing the correlation functions. : •• /
The following discussion on bias is an extension of an analysis given in
[37] in which assumptions about the integration length are used as
justification for some of the derivation. The purpose of the analysis in [37]
was to show that the delay bias was zero but as will be seen below this only
happens in special cases. To develop expressions for the bias it is necessary
to consider a particular signal model. The measurements from two locations
will be modeled as

91
x(t) = £ Si( I ) ^ n 1( I ) (3.4.1a)
i=l
y(t) = E M t ) *si(t—Di) + n2(t) (3.4.1b)
i—
I
The signals, Si, will be considered to be either wide sense stationary or
random transient events with relative delays, Di. If the signals are
considered to be wide sense stationary then the expectation will be taken
with respect to the appropriate random process. If random transient events
are considered then the expectations will be taken with respect to the
random amplitude. The transient events will be considered to be entirely
contained within the observation interval. If necessary the pulse shapes of
the transients could be considered random but this will not be done. The
filter, hj, is a differential filter describing the difference in the channel
characteristic due to factors other than the linear delay. These filters will
describe the nonlinear phase distortion and the spectral shaping of the
signals. The two noise terms, nj and n2 are zero mean, stationary random
processes, (or wide sense stationary if desired). The noise terms are
independent of each other and independent of the signals, Si. Depending
upon the situation, the signal terms, Si, may or may not be independent of
each other. To estimate the delays, Di , the function </>(r) is used
T/2
<
H
r)= f I S i M M t )] I Eitt) * y(1+ r ) ] dt
- w .......
where the filters, gj(t) and g2(t), are the optimal filters used in the
generalized cross-correlator. For simplicity, these filters will not be included
in the following analysis, (it is a simple matter to include them if desired).
The generalized cross-correlation prefilters are zero phase and it will be seen
later that zero phase filters have no effect on the bias. The estimate of the
delays, Di , are the values of r at which <
j)(r) has its local maximums. This
function consists of two terms
= + f3-4-2)

92
The first term is comprised of the cross products of the signals and is given
by
M T)
N ’ N '
Y
x : Si( I )
i=l . H i
dt (3.4.3a)
The second term is comprised of cross products of the signals and and noises
and is given by
M r ) n i ( t ) E Si(t + r)
i=l
d t (3.4.3b)
T/2
+ I n2(V)
-T/2
N
E-hi(t)'* si(t -I-T--Di)
i=i
dt
. ! T/2 '
+ / n1(t)n 2(t + r ) dt
-T/2
Different assum ptions about the signals, Sj, w ill be m ade depending on w h i c h
effect is under consideration. H ow ever, for each case the bias of t h e
estim ate will be defined as
B(D) = E{D -D }
where D will refer to the estimate and D will refer to the true value and will
not be subscripted unless necessary. To arrive at a form amenable to
calculation of the bias it will be necessary to express j r —D] in terms of the
functions in Eq. 3.4,2. To arrive at this form it is necessary to differentiate
Eq. 3.4.2 which yields

93
M ( T )
=
M s (T)
+
^ M t ) I
d r
7-D d r ; r=D d r
(3.4.4)
O
Expanding 4>
s{T) into a Taylor series about the true delay* D, yields
^sw (D)
M T)
OC
E
H
==O
r- D (3.4.5)
where <f>s^X D ) refers to the n*^ derivative of (j>§ with respect to T evaluated
at the delay D. Taking the derivative of Eq'. 3.4.5 and inserting it into Eq.
3.4.4 yields
OC 1VD) ,
< h ' ( T ) + 0 S (D ) = - S V I ^ - D I
n=l
To arrive at the desired relation it is necessary to use a reversion of series
[75] which yields
CO
V - D ] = V An [<%'(r) + ^s(U )]"
- I l = = I !
where the coefficients An are the standard coefficients in a reversion of series.
Retaining only the terms in the Taylor series up to the second derivative
and performing the reversion yields
(D -D )
4>
s{D ) + <
t>
n(D )
<^s(D)
(3.4.6)
where t was evaluated at D. This analysis is a local analysis of the bias
whose validity holds near the point of expansion. If the range of validity
needs to be extended higher order terms in the Taylor expansion can be
retained however the analysis is extremely tedious. Taking the expectation
of Eq. 3.4.6 results in

94
E {D —D } = E
^s(D)
^s(D )
/
+ E
4 > N ( D)
^s'(D)
(3.4.7)
where the second exp ectation evalu ates to zero. This reduces the bias to the
expression
B ( D ) = E
^s(D)
tf s'( D)
This expectation is still difficult to evalu ate so another T aylor expansion will
be used. The functions, ^ g( D) and <
^
>
g ( D ) are both random variables,
defining these as
V = ^s( D )
and
Z = ^ ( D )
T hese random variables will have m ean values
and
E{ V } M
v
dRSS{r)
d r
E{ Z } = ^
T ^2r ss(r )
d i 2
T-=D
E xpanding the function
F ( V , Z )
V
Z
into a T aylor series about the m ean values /iv and /iz ,[6], retaining only the
term s up to and including the second order partial derivatives, yields

95
F(VfZ) -
/^
'V I /^
'V 9 I
— ■+ — V + — Z2 + - lT V Z
I1Z V z K v l
(3.4.8)
T aking the expected value of Eq. 3.4.8 and inserting the expressions in for V
and Z gives
(3.4.9)
4 T /2 T /2 -
+ ~ T J dx J dt E { s (t)s(t+ r-D )s (x )s(x + r - D ) }
Vz dr _t/2 _t/2
I T /2 T /2
+ — - — r- J dx f dt E{ s(t)s(t + r —D )s(x)s(x + r —D) }
Vz - t /2 -T /2
r=D
T= D
To simplify these expressions any further will involve the calculation of the
fourth order moments inside the integrals which can be calculated if specific
models are considered.
The first case to be considered is the bias due to the measurement
itself. To separate the individual effects the only signal that will be used in
this case will be s^ t). Furthermore, the filter, h ^ t) will be assumed to be
zero phase so that only the spectral magnitude is altered. In most analysis
the first derivative of <
/>
$ is approximately zero because <
/>
s is assumed to be
equal to Rss, the autocorrelation function, for large record lengths, T. The
bias is also zero for the shorter record lengths due to the symmetry of the
moments used in the calculations. In fact one can continue the Taylor
expansion and the resulting terms will either have an odd number of
derivatives with respect to r or be multiplied by /iv which evaluates to zero.
Only taking an odd number of derivatives will give a result of zero provided
that the higher order moments are symmetric about the true delay, D. The
filter has no effect because it is a zero phase filter and will always be an
even function.

96
The second case to be considered is the bias due to nonlinear phase
distortion in the channel. This distortion can com e from a variety of sources
and in a practical sense can never be elim inated. One exam ple is the
nonlinear phase of the m easuring devices; it is difficult to exactly m atch the
two receivers which will give a sm all nonlinear distortion. A nother exam ple
is the dispersion in the propagation path which in the case of a cable is a
sm all effect. A nother exam ple is given by reflected w aves which reflect from
discontinuities th a t have com plex im pedances. To separate the individual
affects the only signal th at will be used in this case will be s 2( t ). In m ost
analysis the first derivative of <
/>
s is approxim ately zero. This is due to the
fact that ( J) s is an even function about the delay D. H owever, in nonlinear
phase system s <
/>
s is no longer sym m etric about D and consequently the
derivative is no longer zero. It should be noted that the definition of the
delay, D, for this analysis refers to the delay due to the linear phase shift
only. One of the purposes of including h j(t) in the y( t ) channel was to
easily include the effect of a nonsym m etric correlation function. The filter,
h(t),is a filter describing the difference betw een the two channels and can be
decom posed into its even and odd com ponents.
'M O = he( t ) + h0( t )
_ h( t ) -f' h( —t ) h ( t ) — h( —t )
~ 2 2
Considering only the first term of Eq. 3.4.9 and expressing it in term s of its
F o u riertra n sfo rm g iv es
V1
V 1
OC
f wH( o j ) Ss( o j ) d u j
OC
J < & . H( C d ) Sg( ( j ) d c j
—
oo

97
. CO ' '
j —
j caH0(w)S(cu)dw ■
.■ _ 0 -
CO • .
; J J t i e { u j ) S ( o j ) dco
It should be noted that H0 is purely imaginary and as a consequence the
term —j cancels. The magnitude of this bias is a function of the degree of
asymmetry of the correlation function about its true linear, phase delay, D.
The third case will develop the bias error due to overlapping correlation
functions. In this case the signal will consist of the sum of two individual
signals with relative delays, D^and D2- The true delay being sought will be
D,. The correlation functions due to each individual signal will be assumed
to be symmetric about its corresponding delay. This case is slightly different
than the preceding cases and the change is incorporated by modifying Eq.
3.4.6 to be
(D -D ) = -
,(D i) + <
j>
si D1) + <%(D1)
A similar equation can be developed that replaces Eq. 3.4.9 and it’s first
terms are given by
^S1
(Di) + ^So(Di)
4',(Di )
„ I1
V + /iI
— 2 ------------
M
Z
(3.4.10)
where the terms /iy and /U
z are defined as above. The term /^1 is the
interfering term and is given by the expression
■
« s.! '
Using Eq 3.4.10 the delay bias can be expressed as

98
2 f —
j W So( w) dw
by + /-tI o
I S1(W) d
where S 2 ( c j ) is the odd portion of the power spectral density of s2(t) shifted
by the delay term
exP[ - J ^ ( D 1- D 2)]
If the correlation function has a finite duration less than the delay
difference, D1- D 2, then as the delay difference becomes larger the first and
second derivatives of the correlation function at that point approach zero.
In other words as the functions become more separated the bias due to the
overlap goes to zero. This result can also be expressed in the time domain in
terms of the autocorrelation functions R1 and R2 of the signals S1 and s2.
The delay bias for overlapping correlation functions is given by
B(D1)
2R2(D2 —D1)
Rr(O)
The interpretation of this bias is that the interfering correlation function
tilts the desired correlation in such a manner that it’s maximum is shifted.
The tilt is affected by the slope of the interfering term at the location of the
desired peak. Other factors affecting the bias are the bandwidth and power
of the desired correlation function. The term in the denominator is the rrns
bandwidth squared (for low pass signals) times the power in the correlation
function. If the correlation function has a large power or has a large
bandwidth then the bias is reduced.
This bias expression has other practical implications other than giving
the bias for overlapping correlation functions from adjacent (but desired)
correlation functions. If a narrow band signal is interfering with the
measurements then the resulting correlation function will be the
superposition of the desired correlation functions and a damped sinusoid.
Depending on the exact relationship between the phase of the sinusoid and

99
the desired peaks there can be significant bias in the desired estimates due
the superposition. Another problem can be caused by low frequency high
energy impulsive interference. This is a common interfering signal in partial
discharge measurements and is caused by SCR noise, switching surges and
the like. The superposition of these pulses into the measurement has an
adverse effect because the correlation time of this interference is typically in
excess of the total propagation time of the device under test. In other words
the presence of this interference will bias all the delay measurements.
The next case will be concerned with the inherent bias in
autocorrelation processing. The above analysis assumed that the noise in
the two channels was independent. In contrast now the noise terms,
U1 and n2 will be assumed to be equal. Furthermore, the signal model has to
be adjusted so that reflections of the signal are included. Making these
changes gives the new model
x ( t ) = £ S j f t - D i ) + Di( t )
i=l
y(t) = E Si( I - D i) - W t )
i=l
The new terms for the correlation output are
T/2 N- N I
M t)= I ■ E Si( I - P i ) •
E s j( t + r — D j )
-T /2 i=l . j= i J J
and
M t )
.T/2
I n ^ t )
-T /2
S si(t + r -D )
i=l
dt

100
T/2
+ / n , ( t )
- T / 2
E si( t + t—Di)
i-l
T/2
+ / n,(t)iii(t + r) dt
T/2
To simplify the problem and make the effect clear consider the simple case
in which the observed signal consists of only the direct signal and one
reflection. Furthermore, assume that the channel is distortion free and that
the reflected signal is delayed from the direct signal by a delay D. The
signal model then becomes
x(t) = s(t)-f s(t—
D) + n(t)
In this case the autocorrelation function R(f) is given by
R(r) —2Rss(r) + Rss(t —D) + Rss(r + D) + Rnn(7)
where Rss and R^n are autocorrelation function of the signal and noise.
In essence this problem is identical to the problem of overlapping echos
discussed above. Suppose that interest is focused on the peak located at D,
then the other terms are overlapping pulses and comprise the interfering
term 1(f) where
I(r) = 2RSs(f) + Rss(r + D) + Rnn(7)
The same solution applies as for overlapping correlation functions and in
this case the bias in the time domain is given by
: B(D) ^
Rss(O)
This bias can cause difficulties in the location of the partial discharge site if
the delay, D, is small. Consider for example the case of an underground

101
power cable. If the received signal has a bandwidth of 4 Mfiz then if D is
less than 250 ns then the estimate can have significant bias. This
correspond to a partial discharge site located within 125 feet of either end of
a typical 1500 foot URD Cable. In fact, if located too Close to the end of the
cable the desired secondary peak in the Cable may not be discernible.
The analysis given in this section was concerned with the bias errors
associated with correlation analysis. A similar analysis can be carried out
for time of arrival measurement schemes for determining the bias. The
results are the same as given above except .that the autocorrelation
functions (spectral densities) are replaced by the signal (magnitude
spectrum). In other words the same errors associated with the time delay
bias estimates obtained from correlation functions will affect time of arrival
methods.
3.5 V ariance of D elay E stim ates
Another commonly used measure of the performance of an estimate is
its. variance which is defined as
Var1
1D ] =E{ (D - D )2 } - [e {D - D } J 2
This is a difficult quantity to calculate so a similar approach will be used as
for the bias fiowever, before calculating the variance, it is interesting to
ask what is the best possible performance that could be attained. This
question is answered in terms of a lower bound on the variance of all
possible delay estimators. To obtain a bound it is necessary to consider
specific signal and noise models and for this type of problem the only
tractable model is the Gaussian model. The bound is referred to as the
Cramer-Rao lower bound [41] and is given by
- I
d2 £ np(X I Q, t )
I ** J T = D
where X is the observation vector containing both x(t) and y(t), Q is
spectral density matrix containing all the spectral densities describing the

102
Gaussian probability density function p(X | Q, r ). For Gaussian signals
and noise this quantity is calculated in [39] and is given by
minimum var[ D T I M 2
h ’i2(f)l2
I - I'Yi2COI2
df
-I
where |"/12(f) P is the magnitude squared coherence function. This bound is
achieved by the ML generalized cross-correlator discussed in [37]. In
general, bounds like this simply state the best performance and give no
indication of how to achieve them.
For the general wide sense stationary case or for random transient
pulses contained within the observation interval the following analysis gives
the local variance of the delay estimate, D. Only the unbiased case will be
considered so using Eq. 3.4.6 as a starting point the variance is given by
var( D ) = E
^s(D) + ^(D )
^
2
4 (D )
This can be treated as function of two variables V and Z
F(V1
Z)
where V and Z are given by
V = </>s(D
) + 4(D )
Z-—4 (D)
and

103
Follow ing the sam e line of reasoning as in section 3.4 a Taylor series
expansion about the m ean values //v and /Iz will be used. R etaining all
term s up to the second partial derivatives and deleting the term s whose
exp ectation s evalu ate to zero yields
F(V,Z) ~ y!.
/4
w here
I1Y ~ 0
and
fl z — .E{</>s (D)}
This is the sam e result given in [37] except th at the random variation of the
signal term was not ignored in this analysis. The analysis given in [37]
assum es a large observation tim e, T, and argues th a t the random variations
of the signal can be ignored; this analysis does not m ake th at assum ption.
The variance can be shown to be [37]
T CO
J u ? S(c<;) dcu
OO
2
OO' .. ■
I
- C O
S(o
j) N 1(Cj) -f- S(g
j) N 2(cj) -f- N 1(Cj) N 2( cj) dec
w here S(cj), N 1(C
tj) and N 2(cj) are the spectral densities of the signal and
noise respectively. If the generalized cross-correlator is used then the
variance becom es

104
var[ D ]
T
I
J J W (w)S(a;) dw
- C O
.CC
J J W2(w)
—
oo
S(CJ)N1(C
J) + S(G
J) N2(c
j) + N1(Cj)N2(Cj) dcj
where W(c
j) is the optimal weighting function for the generalized cross
correlator’s implementation.
Since a popular m ethod of locating partial discharge is to use the peak
value of the received w aveform s a sim ilar analysis will be carried out for this
case. Consider the tw o received signals th a t are m odeled as
x(t) = Sj(t - D 1) + ni(t)
and
y(t) = s2(t - D2) + n2(t)
where it will be assumed that the delay estimate will be made from transient
type signals that are corrupted by additive wide sense stationary noise. The
signals may be different from each other because of frequency dependent
attenuation but the phases will be assumed to be undistorted. The desired
delay difference is D1 —D2 and the variance of the delay estimate is given
by VarfD1 - D 2]. Using the same reasoning that resulted in Eq. 3.4.6 the
delay difference D1 - D1is given by
—n,
D 1 - D 1 = — ; i
S1
where the signal Sj(t - D 1) has it’s maximum at t = D 1. The variance is
found by calculating the expected value of

105
V a r p D 1 A2]. —E{(Dj D2)(D2 —D2))
Expanding this expression gives
VarjD1 — D 2] = E
- Hj(D1)
si (0 )
~ n2(Pl)
s 2 ( 0 )
where the cross term s were ignored because the noise term s n t(t) and n2(t)
are considered to be independent. The resulting variance expression is then
given by
VarjD1 - D 2]
- R j f(Q
) - R H 0)
[sT(Q)]2 . [sT(o)]2
where the term s R 1 and R 2 refer to the autocorrelation functions of the noise
n r(t) and n2(t) respectively. The im portant difference to notice betw een the
correlation based delay estim ator and the peak location based delay
estim ator is the scaling factor . As the observation interval is increased
the variance of the correlation based estim ator decreases. There is no
m ethod by which th e peak location estim ator can be im proved. In essence
w h at this result says is th at collecting additional data will im prove the
cross-correlation based estim ate substantially. H owever, it should be noted
th a t excellent results can be obtained from a single record w ith one partial
discharge pulse of sufficient energy using cross-correlation. A typical partial
discharge w ith a received bandw idth of 10 MHz and a charge of 0.025 pC
will have an EN R of 14 dB for a record length of 2.56 /is w hen m easured
w ith an am plifier w ith an 8 dB noise figure. In this particular case the delay
estim ate has a standard deviation of about 7 ns.

106
3.6 Modified PHAT Processor
In situations where multipath signals are present it is necessary to
increase the resolution of the resulting correlation functions. The following
GCC is an ad hoc type of filtering which increases the bandwidth of the
measurement [51]. The following filter is based upon ML and PHAT [39]
generalized cross-correlators. The maximum likelihood, ML, implementation
[39,18] of the generalized cross-correlator uses a weighting function of the
form
W(cu) =
K(w)
C H =
r a K(w)
= CM
i - c m
ISxyM P
SxxM SyyM
where C(oJ) is the magnitude squared coherence function. As can be seen
from the above equation, the first operation involved in the weighting is to
set the cross-spectral magnitude equal to one. This is accomplished by
dividing the cross-spectral density by its magnitude and if only this
operation is performed then the GCC is referred to as the PHAT processor.
If only this operation is performed, (assuming perfect knowledge of the
cross-spectral density), the correlation function would have a single impulse
at the delay D. In practice this leads to a correlation function which consists
of a random collection of impulses with little relation to the true value D.
The basic problem with this technique is that only estimates of the cross-
spectral density are available. Furthermore, those frequencies where the
cross-spectral estimates are poor have been given the same weight as those
regions where the estimate is good. The ML weighting attenuates those
frequencies at which the estimate of the phase of the cross-spectrum is poor
while accentuating those frequencies where the estimate is good.
To interpret the ML weighting function, K(c
j) will be rewritten as

107
SNRxM SNRy(Q
J)
M SNRX(,,') + SNRyM + 1
The shaping of the cross-spectrum provided by K M can be described by
considering the cases where there is' large amount of signal energy at the
frequencies of interest and where there is not. When there is a large amount
of signal energy the SNR is high and K M weights the cross-spectrum
according to the combined SNR in both channels. If the noise in each
channel has a constant spectral density across those frequencies then the
cross-spectrum essentially retains its original shape. For frequencies where
there is little signal energy and the SNR is low the weighting, KM ,
attenuates the cross spectrum as the product of the SNR in each channel.
The acoustic measurements that were made in this study used
transducers that were narrow band and consequently the time resolution of
the cross-correlation function is poor. The time resolution refers to the
ability to unambiguously resolve individual peaks in the correlation function.
In the structures studied, multiple propagation paths were present and
consequently the correlation functions exhibit multiple peaks. It is therefore
necessary to maintain an adequate time resolution in order to identify the
individual peaks and measure the correct delay. Unfortunately, when using
narrow band transducers, the resulting resolution of the measurements using
a ML weighting are not adequate to separate the multiple peaks that are
Usually present in these measurements. The reason for this problem is that
as mentioned above the ML weighting essentially retains the spectral shape
at those frequencies where the SNR is high.
Using only the phase information by setting the magnitude of the
cross-spectrum equal to one would provide the greatest time resolution, but
as mentioned above this approach has other deficiencies. However, if the
frequency range where the magnitude is set equal to one is limited to those
frequencies where the SNR is high enough, then there is a resulting increase
in resolution without the noise problems that cause the spurious peaks. The
question remains how to choose those frequencies where the signal energy is
large enough to prevent the occurrence of the spurious peaks. Intuitively it
makes sense to use some measure of the SNR at the various frequencies to
decide which frequencies to include. The function, K(w), described above
provides a convenient measure on which to base the decisions. So as a
purely ad hoc method the weighting function, W(cu), is used to filter the

108
waveforms [53] and is given by the expression
7 S N R x (w) S N R Y(w)
IS x y (CO) I SN R x (w) + SN R t (W) + I
W(W) =
K(w) SNRx(W
) SNRy(w)
Is Xy M I SNRx(w) + SNRy(w) + I
This can also be written more compactly as
W (w ) V
2 7 [Sgn(K (w) - 7)4-1] + V
2 K (w )[ S gn (7 —K (w )) + l]
When this weighting is used there is a resulting increase in resolution
without the spurious spikes in the time domain. This weighting replaces the
portion of the spectrum having a high S N R with a constant, 7, and the low
SN R portion with the same weighting as used by the ML processor. A
typical example of the cross-spectral shaping provided by the ML filter and
the filter discussed above are shown in Fig. 3.2 and Fig. 3.3.
3.7 R obust D econvolution of C orrelation Functions
A generalized cross-correlator is discussed in this section which can
provide improved resolution in delay estimation measurements [52]. The
prefilters for this GCC are robust (in the minimax sense) linear filters which
have found use in linear equalization, [40]. The presence of the prefilters, in
effect, deconvolves the cross-correlation function. This GCC is used
primarily for locating acoustic emissions from partial discharge in high
voltage, solid cast transformers. Both the electrical and acoustic emissions
are used in the time delay estimation technique for locating the emission
sites. Improved resolution is needed in power transformers because of the
complicated multipath signals that are present in the transformer winding
apd this GCC is effective in combating the problems associated with the

109
Frequency, KHz
Figure 3.2. Output ESD using the ML GCC weighting.

HO
V O
120
Frequency, KHz
Figure 3.3. Output ESD using the modified PH AT GCC weighting.

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