A THEORETICAL STUDY OF THE APPLICATION
OF HARTLEY TRANSFORM FOR POWER
QUALITY ANALYSES
By
PARANJYOTlMAHANTA
Bachelor of Science
Gauhati University
Gauhati, India
1978
Bachelor of Engineering
lorhat Engineering College
J orhat, India
1985
Submitted to the Faculty of the
Graduate college of the
Oklahoma State University
in partial fulfillment of
the requirements for
the Degree of
MASTER OF SCIENCE
May, 1993
A THEORETICAL STUDY OF THE APPLICATION
OF HARTLEY TRANSFORM FOR POWER
QUALITY ANALYSES
Thesis Approved :
Dean of the Graduate College
ii
PREFACE
The subject related of this thesis is Hartley transform and its
usage in the study of power quality. The main purpose of this is to
conduct a theoretical study of the application of Hartley Transform In
power system analyses and explore its potential in solving power
quality problems.
An effort has been made to keep the level of this thesis as
simple as possible. There is a wide scope for further studies.
Hartley Transform, although mainly used in Communication
Engineering, is seldom used to solve power related problems.
Literature on this is just becoming available. I wish to express my
sincere gratitude to Dr R. Yarlagadda of the School of Electrical and
Computer Engineering of Oklahoma State University for his advice in
literature search for Hartley Transform. I am also specially
grateful to Dr. J . A. Allison for his help in allowing me to use the
VMS mainframe computer system for computer programming.
Special Thanks are due to my uncle Dr. N. C. Das of Wilksbarre,
Pennsylvania and Mr D. P. Das of Philadelphia for their constant
support and understanding. I offer my deepest appreciation to my
parents for their moral encouragement.
III
ACKNOWlEDGEMENT
I wish to express my sincere appreciation to Dr R. G.
Ramakumar for his encouragement and advice throughout my
graduate program. Many thanks also go to Dr B. L. Basore, Dr R.
Rhoten, Dr R. L. Lowery for serving on my graduate Committee.
Their suggestions and support were very helpful throughout the
study.
f
IV
TABLE OF CONTENfS
Chapter Page
I. INTRODUCTION
Background................................................................................... 1
Scope ................................................................................................ 2
Purpose .......................................................................................... 4
II. POWER QUALITY
Introduction........................................................................... ....... 6
Definitions of power quality .............................................. 1 1
Types of Power quality disturbances . ................. ......... 1 2
Planning a power Quality Survey....... ............................. 34
Perfonning the survey ......................................... .............. 4 3
Analyzing survey Data ........................................................ 5 1
Measure of Power quality .................................................... 59
Discussion of various Methods to improve
Power Quality ... .................................. .................................. 6 1
III. HARTLEY TRANSFORM
Introduction ................................ "............................................. 6 6
Definitions ................................................................................... 6 7
Continuous Hartley Transform ........................................... 69
Discrete Hartley Transfonn ................................................. 7 3
Two Dimensional Hartley Transform ............................... 80
General Discussion ................................................................... 8 3
v
Chapter page
IV. APPLICATION OF HARTLEY TRANSFORM FOR POWER
QUALITY ANALYSES
Introduction....... ................... ... ............ .. .......... .... ................... ..... 8 4
Analyses .......................................................................................... 8 8
Discussion................................................ ....................... ..... ............ 9 5
V. COMPUTER SOLUTION
Introduction ............................................ '" ..................................... 9 6
Procedure ................... ............ '" ....................................................... 9 8
VI. SUMMARY AND CONCLUSIONS .................................................. 1 08
BmLIOORAPHY ..................................................................................................... 111
APPENDIX(COIvlPlITER PROGRAM OF APPLICATION OF HARlLEY
1RANSFORM FOR POWER QUALITY ANALySES) .............. 1 14
vi
LIST OF TABLES
Table Page
I. Typical range of input power quality and load parameters
of major computer manufacturers ......... ... ... ....... ........................... 1 3
II. Types of disturbances and other information .............................. 1 5
III. Typical sources of spikes, impulses & surges ................... ........... 2 0
IV. Common power monitoring features ............................................... .40
V. Electrical distribution system and grounding measurements
and consideration ... .................................................................... .......... .. 46
VI. Equipment specifications and tolerance ......................................... 5 4
VII. Values of discrete Hartley transform.. ....... ....... .. .............. .............. 78
VITI. Algorithm summary .............................................................................. l 07
1 T
vii
o LIST OF FIGURES
Figure Page
1. Voltage Sag and Swell.............................................. ................................ 18
2. Spikes, impulses and surges .................................................................. 2 2
3. Simplified Circuit illustrating magnification
phenomena ................................................................................................. 2 4
4. Waveforms on primary and secondary illustrating magnification
phenomena............................................................................... ............... 2 5
5 . Outage ............................................................................................................... 2 7
6 . Harmonic distortion .................................................................................... 2 8
7. Current waveform typical of most single phase
electronic loads........... ........ .......... ........... ...... ......... .......... ..... ................. 3 0
8. Typical transformer magnetizing Current. ...................................... 3 1
9. Arc furnace voltage and current.. ......................................................... 3 2
10. Electrical noise .................................................................... .......................... 3 3
11. Undervoltage & overvolge ....................................................................... 3 5
12. Seven day event summary ............... ...................................................... 5 5
13. Waveform of SCRcontrolled load ........................................................ 5 8
14. Power Quality Indices for nonperiodic
phenomena ............................................................................................... 60
viii
Figure Page
15. Plot of continuous Hartley transform function
H(f) ................................................................................................................ 72
16. A 16point representation of the truncated exponential
waveform (left) used for illustrating the continuous
transform and it's DHT right. ........................................................... 79
17. Distribution network decomposed into SE
and SH Networks ...................................................................................... 89
18. Diagram of n  port circuits .................................................... ................ 9 2
19. Decomposition of a power network into SH and SE
network ...................................................................................................... 94
20. Non sinusoidal rectifier injected current waveform ................ .... 97
21. Three bus power system ......................................................................... 9 9
22. Equivalent starnetwork ................................................................... ...... 1 00
23. Harmonic voltageplot.. .............. ; ............................................................. 1 27
... 1 )
ix
Cas
CSH
DIN
e
E(f)
E(v)
f(t)
F(v)f
fsymm(x, y)
fantisymm(X, y)
GSH
H(s) or H(f)
H(u, v)
LIST OF SYMBOLS
Fourier coefficient
Fourier coefficient.
Hartley coefficient.
Sum of sine and cosine function (Argument).
Capacitance of the SH network.
Distortion index used mostly 10 Europe.
Exponential base (e=2.7183)
Even part of a function.
Even part of Hartley transformed function.
An arbitrary function.
Discrete Fourier transform of function F(t).
Imaginary part of a complex function F(f).
Real part of a complex function F(f)
Symmetrical part of f(x, y).
Antisymmetrical part of f(x, y).
Conductance of (SH network.)
Hartley transformed function.
Two Dimensional Hartley transform of a function
f(x, y).
Hartley current.
Positive sequence Hartley current.
Negative sequence Hartley current.
x
I(v)
I(kO)
ISE(t)
i and j
k or K
KVT
LSE
n or N
Nl and N2
O(t)
O(v)
pet)
RSE
S(O)
THO
TIF
V rms
v SECt)
Vet)
V(k)
V(v)
yl(CO)
Y SE(CO)
Y SH(CO)
Discrete Hartley transformed function of current
function i(t).
Discrete Fourier transformed function I(kt).
Current of SE network.
Complex operator which IS equal to ~::I.
Any arbitrary integer.
Expression for Audio weight.
Inductance of the SE network.
Any Integer.
Two arbitrary Matrices.
Odd part of a function.
Odd part of Hartley transformed function.
An arbitrary function of t.
Resistance of the SE network.
Fourier transform of a function Set).
Total H.armonic Distortion.
Telephone influence factor.
Root Mean Square Voltage.
Bus voltages of SE network.
Inverse Hartley Transform of Function 'l' (co).
Discrete functional form of voltage V.
Discrete Hartley transformed function of voltage
function Vet).
Admittance of the ith network.
Admittance of the SE network.
Admittance of the SH network.
xi
Z(v)
Z(kO)
ZSE(co)
ZSH(co)
ZHT(nro)
13
cS
v
1t
9
00
Hartley transformed function of impedance
function Z(t).
Discrete Fourier transformed function Z(kt).
Impedance of the SE network.
Impedance of the shunt network.
Hartley impedance.
An arbitrary constant.
An arbitrary constant.
Discrete frequency variable.
Pi which is equal to 3.1416.
Angle.
Discrete time variable.
Angular frequency.
Infinity.
Summation Symbol.
An arbitrary mathematical function of frequency
co.
r
xii
CHAPTER I
IN1RODUCTION
Background
The study of Power Quality is a relatively new topic In
Electrical Engineering. Although it is in a relatively infant stage,
more and more people are realizing its effect on power related
problems. Previously the problem was not so acute, but with the
introduction of various power electronics devices such as SCR (Silicon
Controlled Rectifiers), power transistors, various solid state relays
and circuit breakers it started to grow in dimension. Furthermore
there has been a significant increase in the use of electronic data
communication devices, computing devices with the advent of LSI,
VLSI and ULSI (Largescale, Very Large and Ultra Large Scale
Integrated) circuits in recent years. However most of these
equipments are highly sensitive to the problems in power
distribution circuits. Although their introduction is essential for
progress of science, engineering, the art of computation, safety and
control, they are creating a new set of problems in power systems.
Unlike simpler and rugged electrical equipment such as motors,
these new electronic equipments require much more stable voltage
sources due to their highly sensitive digital circuitry. These
1
equipments are more susceptible to power disturbances that
originate on either side of the electric meter. Complicating matters,
enduse devices such as adjustable speed motor drives, laser printers
and fluorescent lamp ballasts can feedback disturbances (in the form
of non'Sinusoidal currents) into the power lines that cause problems
not only within a building, but in the neighboring buildings as well.
The consequences of power disturbances for building owners are
significant. Disturbances such as impulses, sags and surges can have
negative impacts on electronic equipments.
Scope
Problems due to voltage and current variations range from
operational problems such as data errors, program errors or memory
loss, to system interruptions or equipment damage. Building
owners might be blamed not only for any enduser equipment
damage, but also for the loss of productivity due to equipment
downtime. Building marketability can also be impaired as a
consequence of inadequate electrical supply. Viewing the
potentially damaging effects of power disturbances, building owners
and tenants cannot disregard power quality issues. Ensuring good
power quality requires a wellplanned, wellexecuted power quality
survey and upgrading of the existing electrical distribution network.
Such a procedure not only describes the approaches for
diagnosing power quality problems in the building power
distribution system serving sensitive electronic equipments and
systems but also common power disturbances are addressed, as are
2
planning and performance requirements for conducting a power
quality survey. Thus the time has come for power engineers to
conduct an extensive effort using different mathematical techniques
for the analysis of power quality related problems.
Hartley transform is one of the most efficient mathematical
tools available for the analysis of power quality. It is closely
related to Fourier transform. Hartley transform can change
convolution operations into a simple multiplication and hence IS used
to calculate various transient and nonsinusoidal waveshapes (current
or voltage) in power distribution networks generated by power
quality problems. The importance of this type of calculation
relates to the impact of loads, particularly electronic loads, whose
input currents are not sinusoidal in nature. Hartley transform can be
used for the analysis of electric power circuits. Because Hartley
transform is a real transformation from the mathematical viewpoint,
it is found to be computationally more efficient than Laplace or
Fourier transform. It is not imperative to think that Hartley
transform should replace the classical Laplace or Fourier transform
but it has a distinct edge in rapid calculation of wide bandwidth
signal propagation phenomena.
The microelectronic revolution IS rapidly changing the electrical
environments of the world. Emerging electronic technologies that
consume less power and utilize faster switching speeds are
increasingly being installed to meet cost effectiveness criteria and
higher efficiencies. To meet these challenges more and more ideas
are required to enhance the application domain of mathematical
techniques.
3
The electrical world and the associated events around us are
mostly found to be nonlinear in character. Nonlinear effects are
found in conductance, resistance as well as capacitance and
inductance. A resistive element which has a range of terminal
voltage and current for which dv/di < 0 ( Here dv and di represent
the infinitesimal change of voltage and current) is generally included
in a category of elements called nonlinear conductors. Such
nonlinear conductors are designed and built into the integrated
circuits and microprocessors of today.
To the endusers of electric power, power quality is directly
connected to reliable service; service which is dependable with little
or no interruption. Also, it is important that the supplied power be
maintained within a set of given standards. Although it is seen
that utility'S reliability has improved over the years, the
susceptibility of the customer's equipments to failure and
malfunction has increased severalfold.
Purpose
The advanced electronic equipment that has brought
efficiencies to our homes, offices and factories has introduced some
extra problems as well. Much of these equipments are uncommonly
sensitive to routine power line disturbances and some devices even
create their own disturbances that feedback to the utility line. It is
estimated by EPRI (Electrical Power Research Institute) that
industries all over the United States have been hit particularly hard,
with momentary outages (a phenomenon of complete loss of power
4
lasting from several milliseconds to several hours) costing some
plants as much as $300,000 per incident. Numerous efforts have
been made by power engineers and scientists on a continuous basis
to maintain perfect sinusoidal current and voltage waves (clean
power) free of distortions and harmonics. Now, electronic and solid
state devices are creating a power quality problems which feedback
into the system. So it is very essential to get rid of this problem
with an early warning system in the utility line with round the clock
monitoring and effective countermeasure capability to deal with
unclean power waves. Emphasis is put on the study of this problem
by employing spectral analyses of the power waves and their
broadband analyses with all available tools(such as Hartley
transform) to understand the nature of these waves. Once the
analyses are completed, the facts are uncovered and the problem
can be remedied. The band analyses of voltage and current
waveforms are very essential to estimate power quality
disturbances.
5
CHAPTER II
POWER QUALITY
In troduction
The term "power quality" has taken on a substantially
significant meaning since the introduction of the microprocessor.
Previously, the responsibility of the power engineer stopped with the
provision of a reliable power supply with voltage control and with
minimum or no voltage flicker. Simple guidelines were enough to
handle this job. Today, the proliferation of microcomputer and
digital communication equipment has forced a new definition of
power quality to suit the needs and deeds of these devices. Power
quality no longer confines itself to power engmeenng but
encompasses all other branches as well. The phenomena that were
of secondary importance such as transient overvoltages, harmonic
distortion, and conducted interference have become more significant
now. Traditional electromechanical equipment will draw sinusoidal
current from a sinusoidal voltage. But electronic equipment which
converts ac power to de power does not draw current over the entire
period of the voltage waveform.
The resulting current irregularities can cause disturbances such
as voltage or current impulses and voltage loss in the power
distribution circuits. Efforts are underway to better understand the
6
power quality issue. There is a growing amount of literature
covering individual power quality phenomena, such as harmonics
and transient overvoltages, while argument continues over
measurement equipment, extended version of definitions and
possible standards. There are many unpublished spot surveys by
users and equipment producers of suspected power quality
problems. However, there is less information on general power
quality site surveys. Analysis of the results from site surveys is
often difficult to generalize because of locationspecific qualities.
Spot surveys will not expose synergisms among the utility, the
facility disturbances at work and at the plant equipments.
A traditional studies and measurements of reliability do not
deal with the power quality needs of sensitive electronic equipments.
Rather, they deal with permanent · or prolonged outages and how to
lmprove upon it. While this is indeed important to sensitive loads (
power supplies to computers and other microelectronic devices) also,
there is increased concern for short term or momentary
disturbances. The American National Standard Institute (ANSI)
standard code C84.1, entitled "Voltage Ratings for Electric Power
Systems and Equipment (60 Hz)", is the basis for rulings of
regulatory commiSSlOns as far as voltage requirements are
concerned. This standard was derived with the assistance of the
utilities and the manufacturers. Basically, the standard separates
the voltage requirements into two categories, Range A and Range B
[16]. Range A and Range B are parts of a Bell shaped curve with
Range A consisting of the middle portion of the curve and Range B
making up the extremities of the curve.
7
t Within each range, specifications are given for both service and
utilization voltages. The Range A values are defined as the span
over which systems shall be designed and operated so that most of
the service voltages are within the set limits ( i.e., 1141 16v for
120v service). Voltage range B levels occur infrequently and are of
limited duration (110127v). The standard requires that the steady
state voltage tolerances at the point of utilization or at the point of
service be within + or  5% for nonlighting loads. The standard also
specifies the steady state voltage tolerances at the point of
utilization. This standard only addresses two types of power
disturbances which occur in electric power systems, surges and
voltage sags. Present day electronic devices require near "ideal
conditions " to function properly. Therefore, the above mentioned
standard is no longer sufficient to specify the power requirements at
the present time. In fact the enduse equipments often become the
source of power quality problems; disturbances introduced into the
systems by these nonlinear loads result in harmonic distortions.
Utility systems are designed to provide reliable bulk power.
However it is not feasible for them to provide continuous power of
the quality required for a completely undisturbed computer
operation. t . .
Because normal use of electricity generates disturbances and
because unexpected power system failures do occur, every site
would experience some power disturbances. The nature of these
power disturbances, their severity and their incidence rate vary
from time to time. All these happen like random processes. To
place the problem in perspective, however, one should remember
8
that poor quality power is only one of the many causes of computer
downtime. Hardware problems, and operator errors also contribute
to computer downtime.
Historically, transient overvoltage effects on novel
semiconductor and microelectronic systems were the first concern;
by now, the scope of undervoltage or loss of power has also been
recognized. Power quality problems (rather disturbances) that
affect sensitive electronic loads have a variety of sources such as
lightning, utility switching, and utility outages. However, power
disturbances are often caused by users themselves by switching of
loads, introduction of ground faults or normal but nonlinear
operation of equipment. Computer system is one example of many
such nonlinear loads that are not only sensitive but also can generate
some disturbances by themselves. ' Their nonlinear load
characteristics can cause interactions with the power system such as
unusual voltage drops, overloaded neutral conductors, or distortions
of the line voltage.
Definition of power quality
Power quality can be defined as the relative absence of utility
related voltage variations particularly the absence of outages
(power blackouts and brownouts), sags, surges and harmonics as
measured at the point of service. Obviously this definition is based
on the viewpoints of addressing what quality level a utility delivers
power to its customers. If someone takes the viewpoint of customer
then power quality is simply the relative absence of voltage
variations as measured at the point of service. Disturbances caused
9
by other customers or even by that particular customer's own
equipment stiU affect that customer's perception of power quality.
The importance of semiconductor switching devices has
penetrated to both the distribution and transmission level. In fact.
at the transmission level, there is a program devoted to "Flexible AC
Transmission System", FACTS, which proposes the use of y
semiconductor devices for bulk power control. Semiconductor
devices of the power level required for realistic power control are
now available~ and the capabilities of commercial devices are on a
state of steady increase. The MV A switching capabilities
MVA = I Circuit voltage I ISwitching currentl
(With voltage in kV and current in kA) are increasing with an added
decrease in cost per MV A. Although these are not the only source of
power quality problems, identifying and technically solving these
emerging problems is an engineering chaUenge that can be met only
on a case by case basis. Perhaps a more basic issue however is
determining strategies that allow utilities and endusers to find an
equitable and effective process to correct these situations. For
utilities this may mean occasionally crossing over to the customer's
side of the meter and for the endusers it may mean incurring
expenses for services such as a conditioned power source or UPS
(Uninterruptable Power Supply). Everyone agrees that power
quality problems can best be resolved when all the people involved
work together towards a common goal, namely quality power
provided at a fair price. A consistent definition of terms describing
power quality disturbances has not yet been developed. Voltage
sag is commonly accepted to mean a brief reduction in voltage, but is
10
not included in the IEEE (Institute of Electrical and Electronics
Engineers) standard dictionary of electrical and electronics terms.
Voltage and duration limits for the sag vary and the term outage
means the complete absence of power at the point of use.
Nonetheless, many utilities define an outage to be the absence of
power for more than 45 minutes. Whereas to industrial plants, any
disturbance that disrupts production is a outage. Responsibility for
a reliable power source lies first with the utility company that
generates and distributes power to the customer service entrance.
For example, a power source can be reliable for semiconductor
manufacturing at the delivery point if no more than two outages
occur per year of less than two seconds duration each.
A stable frequency is a must in a semiconductor environment.
Domestic U.S power distribution industry is at such a level that
frequency variations are rare unless there there are widespread
grid blackouts. An uninterrupted power source is paramount to a
semiconductor manufacturer's ability to predictably deliver product.
Not only will manufacturing time be lost during a momentary power
outage, work in progress will be lost since the wafers could be
damaged beyond recovery. Additionally many hours and days may
be lost in recertification of production equipment. According to
ANSI utility power profile, power is considered clean when its
steadystate voltage changes between +6 and 13 percent of nominal.
In order to achieve this, utilities attempt to hold their steadystate
voltage at the level between +10 and 10 percent. ANSI gives wider
utility voltage window limits for shorter periods of time. For
variations over a period of 2030 cycles, the ANSI standard says
1 1
utilities must hold the voltage between + 15 and 20 percent. For
variations over 112 1/4 cycle, the limit is +20 to30 percent. The
ANSI standard does not deal with transients, noise content,
waveshape, or frequency.
The Computer Business Equipment Manufacturer's Association
(CBEMA) publishes criteria for computer power systems. The
CBEMA curve or envelope nearly matches the ANSI envelope for
durations above 30 cycles. Below 30 cycles the CBEMA envelope 1S
tighter. This phenomenon explains why utilities can satisfy ANSI
standards and still produce short duration disturbances that are
unacceptable to computer business equipment manufacturers.
Typical range of input power quality requirements of major
computer manufacturers is listed in Table I. The parameters listed
in Table II are typical for use as a preparation guide.
Types of power quality disturbances
Despite efforts to improve and standardize power quality,
power disturbances still can manifest themselves in the electrical
environment. They may be generated from a variety of sources
severe weather, electrical faults, lightning or customer loads.
Operational problems are the most visible indication of power
disturbances. Typical of these are system crashes and processing
errors in computer's data system. Some computer users accept
these operational problems because they regard them as
unavoidable. Because of increasing reliance on computers In the
12
TABLE I
TYPICAL RANGE OF INPUT POWER QUALITY AND LOAD
PARAMETERS OF MAJOR COMPlITER
MANUFACTURERS [14]
Parameters
1) Voltage regulation (steady state).
2) Voltage disturbances (Momentary
Undervoltage).
Transient Overvoltages.
3) Voltage harmonic distortion.
4) Noise.
5) Frequency Variation ..
Range
+5%, 10% to +10%
(ANSI), c84.1, 1970
is +6% , 13 %.
25% to 30% for less than
0.5s with 100%
acceptable for 420ms.
+150200% for less than
0.2 ms.
35% with ( linear load).
No standard.
60 Hz to (+0.5 Hz to 
0.5Hz).
or (+1 to 1 Hz).
1 3
TABLE] (Continued)
Parameters
6) Frequency rate of change.
7) 3 phase voltage unbalance.
8) Three phase load unbalance
9) Power factor.
10)Load Demand.
Range
IHz/s (Slew rate).
2.5% to 5 %.
5% 20% maximum for any
phase.
0.80.9
0.750.85
14
TABLE II
TYPES OF DISTURBANCES AND OTHER INFORM A nON [15]
Types of Range Duration Origin Type of
Disturbances Equipment
Impulses, spikes Upto 6Kv. < 0.5cycles. Lightning strike, spikes
& noise. Rectifier, power can
Short duration 0 to 1.76p.u. 0.5 to 30
voltage variation. cycles.
supply, welding. destroy
electronic
loads.
Lightning
strike, Motor
starting.
Sags can
effect
power
downsensing
circuit
in
computer
& can
cause
shutdown.
1 5
TABLE II (Continued)
Types of Range of Duration Origin
Disturbances magnitudes
Type of
equipment
Long duration
voltage
variations
(Overvoltage
& Undervoltages).
Harmonic
distortion.
( .
0.8 to 1.2 p.u. >30 cycles. Circuit Overvoltage
Upto Ip.u.
overloads, &Under 
poor voltage Voltage
regulation. affect all
equipments
despite ± 1 0 %
tolerance.
Seconds Electronic Harmonic
to steady equipments distortion
state. such as causes
rectifiers & motor
inverters, loads such
uninterruptable as
power supplies, compresselectric
furnace ors, and
controllers, disk
welding machines, drives to
ac & de motor get over
drive and heated.
computers.
16
workplace. however, they require the same amount of attention as
demanded by more damaging power disturbances such as lightninginduced
high voltage spikes that can cause data loss and hardware
damage in the computers or interrupt power to major building
systems (such as lighting, HV AC, and security, as well as elevator
system).
Power quality disturbances are generally classified into SIX
categories as follows:
(1) Voltage Sags and Swells.
(2) Spikes, Impulses & Surges.
(3) Outages.
(4) Harmonic Distortion
(5) Electrical Noise.
(6) UnderVoltage & OverVoltage.
(1) Voltage Sags and Swells Voltage sags and swells as shown
10 Fig.l are momentary (less than two seconds) decreases and
increases in line voltage outside the normal tolerance of the
electronic equipment.
Sources of Sags. Sags are typically caused by the starting of
heavy loads or when faults occur in a power system.
(sag) occurs on the faulted phase during the period.
A voltage drop
Impacts of Sags. Sags can cause computers and other
microprocessor based equipments to shut down due to a lack of
optimum voltages. Undervoltage (sag) can result in increased
currents and overheating of motors (particularly stepped motors
1 7
.2r
~ z
~ .1r~~~~~~~\~ a::
1&1
~
1&1 0 IoJI...\r"
"oC
~ .1r~.,~JL~~.~_J~
o
>
·2~~
TIME~
Figurel. Voltage Sag and Swell [15]
1 8
used in computer printers), lights dimming and computer
malfunctioning.
Sources of Swells. Swells are generated by sudden load
decreases, such as deenergizing of heavy equipments (such as
demagnetization of large electromagnets in nuclear particle
accelerators). Short duration overvoltages can also be associated
with ferroresonance in transformer circuits .
Impacts of Swells. Swells can damage equipments having
insufficient overvoltage tolerance and protection. Besides,
overvoltages can cause increased equipment stress and midoperation
failure. A voltage drop occurs on the faulted phases
during the period of the fault. For a single line to ground fault, a
voltage increase may occur on the unfaulted phases. This voltage
increase is a function of the system grounding. It is also possible to
have a self clearing fault which can last for as little as one half of a
cycle.
(2) Spikes. Impulses & Surges. Spikes, also known as impulses
or switching surges or lightning surges, are highvoltage transients of
very short duration (ranges from a fraction of a microsecond to a
millisecond) with high amplitudes as shown Fig.2 These types of
disturbances can occur at customer locations due to events on the
utility distribution system or to events within the customer's
premises. Table III summarizes some of the important events
which can cause the aforementioned disturbances .
19
TABLE III
TYPICAL SOURCES OF SPIKES, IMPULSES & SURGES [17]
Disturbances initiated on the
distribution system
Lightning Impulses.
Feeder Energizing/Reclosing.
Transformer Energizing.
Capacitor Energizing.
Disturbances initiated on the
Customer premises
Lightning Impulses.
Capacitor Energizing.
Power Electronics Switching.
Motor Interruption.
20
Sources of Spikes.Impulses & Surges. There are mainly four
factors involved in the generation of these kind of disturbances.
They are:
0) Lightning transients
(ii) Capacitor energizing transients.
(iii) Power electronic switching operations.
(iv) Motor interruption.
(i) Lightning Transients. The mam cause of this is the result of
a direct lightning strike to a conductor or to a ground conductor or
pole. The resulting surge on the distribution lines will be of
relatively short duration (30200 micro seconds) with very fast rIse
times (110 microsecond). Due to fast rise times, these surges can
be coupled to secondary circuits by the capacitance ratio of step
down transformers, rather than turns ratio. In per unit terms, the
transient magnitudes on the secondary can be many times higher
than the transient magnitude on the transformer primary.
(ii) Capacitor Energizing Transients. Capacitor energizing
operations on the utility system are an important source of transient
disturbances to the customer since these energizing operations can
occur quite frequently (more than once per day). One of the
principal source of such problem is the capacitor banks installed by
customer to correct power factors. The customer capacitors on the
secondary usually have much lower KV AR rating than the utility
capacitors on the pnmary.
21
22
.2r~·~~
·2~~~~~
TIME~
Figure2. Spikes, Impulses, and Surges [15]
The resonant circuit of Fig.3 formed by these capacitors and the
stepdown transformer reactance can cause significant magnification
of transient osc.illations associated with capacitor switching on the
utility primary.
While the pnmary oscillation will have a transient magnitude
of less than 2.0 pu, the magnified transients on the secondary can
have magnitudes 10 the range of 34 p. u with significant energy
content. Voltage waveforms illustrating this magnification
phenomena of primary and secondary are shown in Fig.4 .
(iii) Power Electronics Switching Operations. The use of power
converters for motor speed controls, de loads, power supplies, UPS
systems, and other applications has increased the power rating and
the number of these devices in operation. Besides creating
harmonic disturbances by switching the ac current waveform, they
also create high frequency transients (notches) whenever the current
flow commutates from one of the ac line to the other. During this
commutation period, there is a short circuit on the system resulting
in a notch in the voltage waveform. The notch characteristics are
dependent on the magnitude of the current being commutated and
the short circuit reactance of the ac supply to the converter.
(iv) Motor Interruption Large motors can cause significant
voltage dips on the system when they are started, depending on the
motor starting technique employed. A motor will typically draw
anywhere from 6 to 10 times its full load current during starting.
Impacts of Spikes, Impulses and Surges. Spikes can cause data
alterations, equipment errors or system damage. Impulses and
23
3 kYAIi c:.,;
to' ~,~;..r.
G.n.'It:r.
Figure3. Simplified Circuit Illustrating Magnification [17]
24
2110
'" \A 11'· ...... .................. .
I'
Figure4. Waveforms on Primary and secondary Illustrating
Magnification phenomena. [17]
25
surges can cause failure of the insulation of equipments.
Moreover they interfere with communication lines, data acquisition ,
and control applications.
(3) Outage or Power Interruption. An outage as shown in Fig.5
is a complete loss of power lasting from several milliseconds to
several hours.
Source of Outage. Momentary and temporary power
interruptions (outages) occur when a system fault is successfully
cleared by an interrupting device and then the circuit is reenergized.
Reclosing controls on substation breakers, reclosers, and
sectionalizers accomplish this task. The duration of the interruption
will depend on the fault location on the system and the protection
equipment settings used for interruption and reclosing.
Impact of Outage. Outages or power interruptions affect all
electrical equipment. Some sensitive electronic equipments (without
backup system) may be badly affected by the outage. It may lead
to complete voltage loss to a load for several hours.
(4) Harmonic Distortion. Harmonic distortions, like transient
surges, can come from either the utility supply or from customer's
own equipment. Harmonic disturbances such as the ones shown 10
Fig.6 refer to power system frequencies that are multiples of the
fundamental (60 Hz) frequency.
Source of Harmonic Disturbances. Harmonic disturbances are
mainly caused by three major categories of devices.
(i) Power Electronics. This category of harmonic sources is one
of the main reasons for the increasing concern over harmonic
26
27
+2~~~~~
~
:z;) + 1 I____ ~~~~~~~~~~
II:
W
w~ o~L_l,~~_\r_~~~~~~ ~" .1~~~~~~~~_l o
>
·2L~
TIME
Figure5. Outage [15J
28
f
+2~~~~
2L~~
TIME'"
Figure6. Harmonic Distortion [15J
problems in power systems. The applications for power electronic
devices (rectifiers. variable speed drives, UPS systems,
cyc1oconverters and inverters) are continually growing. In Fig.7 the
nonsinusoidal characteristic of some typical waveforms are shown.
(ii) Ferromagnetic Devices.
important devices in this category.
Transformers are the most
Transformers generate
harmonics as a result of their nonlinear magnetizing characteristics
as given in Fig.8. The level of harmonic voltage increases
substantially as the applied voltage increases above the transformer
rating.
(iii) Arcing Devices. Arcing devices generate harmonics due to
the nonlinear characteristics of the arc itself. Arc furnaces are large
harmonic sources in this category (see Fig.9). However, fluorescent
lighting has also basically the same characteristics and is much more
prevalent as a power system load .
Impact of Harmonic Distortion. This type of disturbance may
cause overheating and high currents in conductors (specially neutral
conductors), transformers, capacitors, motors and other equipment.
(5) Electrical Noise. Electrical noise is a distortion of the
normal sine wave. This is illustrated in Fig.IO .
Source of Electrical Noise. It is generated by radio or
television (Audio or Video) transmitters, power electronic circuits,
Arcing loads, and switching power supplies used in computers.
Impact of Electrical Noise. It can affect the operation of
microprocessor based equipments.
(6) Undervoltage and OverVoltage. U ndervoltage and Overvoltage
29
0........
(., , PULSE WA'/EFCrlM SPE:7;:'UM (b) 12 PU:"SE w;..·. ~:C;;· .. ~;:::7RV ..
II ... !:
Figure7. Current Wavefonn Typical of most
Single phase Electronic Loads [17]
30
3 1
FigureS. Typical Transformer Magnetizing Current [17]
32
Figure9. Arc Furnace Voltage and Current [17]
33
J
.2~
TIMEFigurelO.
Electrical Noise [15]
of Fig.II are abnormally high or low voltage conditions lasting for
more than a few seconds.
Sources of Undervoltage and OverVolta~e. The most
important cause of this voltage variation (Under or Over) is a fault
condition on the utility system. It happens due to circuit overloads,
poor voltage regulation, and intentional reduction in voltage by the
utility.
Impact of Over and Under Voltage Condition. The problem
caused by Under or OverVoltage effects performance of all type of
equipments.
Planning a Power Quality Survey
Building and equipment owners have several options for
resolving power quality problems. Relocating the source of
interference or the critical load to a different receptacle or power
distribution circuit is possible, but it may not be a solution. Before
expensive power conditioning equipment is installed, a wellplanned
and well executed survey procedure can be used to solve most power
quality problems that affect equipment operation.
most important step is planning. Important
planning requirements are discussed below.
The first and
(1) Assemble a professional team. Assembling a team of
professionals who are experts in power quality is very critical when
planning a survey. A survey team may consist of individuals who
understand equipment operation and the building's electrical
powersystem (such as building owners, managers, operators etc) and
34
35
+2~~
2L nME~
Figure 11. U ndervoltage and Overvoltage [15]
electrical contractors, electrical engineers, etc) who can make
measurements on live electrical circuits and analyze power related
problems.
(2) Develop a comprehensive plan. Performing a power
quality survey requires a comprehensive plan. The plan should
include general and specific survey objectives. Correcting
equipment performance problem and neutralizing the problems
created by the faults are the general objectives. The specific
objectives are as follows:
(i) determine the condition and adequacy of the wiring and
grounding system.
(ii) determine the ac voltage quality at the point of use.
(iii) determine sources and impacts of power disturbances on
equipment performance.
(i v) analyze findings to identify immediate and nearterm
cost effective solutions
(3) Collect and document data. Collecting data and proper
documentation are necessary to obtain information about the
equipment or the facility that is experiencing power quality related
problems. Efforts should be made to obtain the site history. It
includes typically the following informations
(i) normal time for recurrent system problems.
(ii) symptoms of hardware failures.
(iii) recent equipment changes.
36
(iv) renovations of the facility.
(v) operating cycles for major electrical equipment of the
facility.
(vi) verifying the logbooks of equipments to study the history
of trouble shooting.
(4) Assemblage of necessary instruments. Certain
instruments are necessary to monitor and record the power quality
of a facility. The instruments typically needed are used to measure
some or all of the electrical parameters such as voltage deviation,
voltage loss, frequency, voltage transients, sags or surges, frequency,
electrical noise and harmonics. The instruments most commonly
required for a power quality survey are enumerated next.
(i) True rms multimeter. A multimeter is a versatile tool that
measures voltage, current and resistance. However a true rms
multimeter is almost immune from measurement error induced by
nonlinear electronic loads. Some true rms multimeters permit the
attachment of accessories for reading ac amps, watts, kV A, power
factor and ac line frequency, conducting insulation resistance tests,
testing of capacitors and diodes and reading temperature using
contact (thermocouple) or infrared probes .
Multimeters used for current measurement that are averageresponding
but rms calibrated may read 20 to 30 percent lower
than actual levels.
(ii) True rms Ammeter. An ammeter is used to measure
current and analyze current waveforms, particularly when a
nonlinear waveform is involved. Among all the ammeters, Hall
37
effect current probe is the most popular. The Hall effect current
probe measures dc, ac or dc & ac composite currents. Available in
sizes up to 2,000 A, the current probe must have true rms circuitry
to avoid measurement error. Hall effect probes are available as
standalone devices or as a probe accessory for a multimeter.
(iii) Circuit and Impedance Tester. A circuit tester IS an useful
tool in determining the wiring polarity of receptacles. It checks for
open conductors and lineneutral lineground reversals. Some
circuit testers also measure circuit impedances. The greatest benefit
of the circuit tester is that it examines the physical condition of the
receptacle and outlet assembly for loose and broken mounting.
Generally, a high percentage of common wiring errors are detected at
this stage.
(iv) Ground Impedance Tester. It is a multifunctional
instrument designed to detect wiring and grounding problems 1D
lowvoltage power distribution systems. These problems are as
follows: wumg errors, neutral to ground shorts and reversals,
isolated ground shorts, and neutral impedance shorts.
(v) .Power Monitor. The power monitor [14] is the most
important and useful tool in a power survey. It is designed to
identify and record events that exceed preset limits, permitting the
user to collect data for later review and analysis. Some common
power monitor features are summarized in Table IV. Power quality
monitors are available in a broad choice of recording capabilities and
sophistication. They are classified as follows:
38
(a) Threshold counter: Applied to a calibrated voltage divide,
this device triggers a counter each time a preset voltage is exceeded.
Early threshold counters were analog but recent ones are digital.
(b) Oscilloscope with camera: Surges that trigger a single
sweep on the CRT (Cathode Ray Tube) of the Oscilloscope are
recorded by a shutterless camera as they occur.
(c) Digital storage oscilloscope:
Within this device, the surge is digitized and stored in a shift
register for subsequent playback and display whenever a preset
threshold is exceeded.
displaying events
This type of oscilloscope has the capability of
prior to the beginning of the surge.
(d) Screen storage oscilloscope: This device stores and displays
the surge on a cathode ray tube.
(e) Digital peak recorder This type of electrical equipment
registers the peak value and it can also calculate the duration of the
surge.
(f) Digital waveform recorder This device [13] digitizes and
stores the surge in a manner similar to the digital storage
oscilloscope. Because of the data processing functions which are
incorporated in the instrument, the recorder allows reports of many
different parameters of the disturbance, relating voltage to time.
(f) Digital waveform recorder This device [13] digitizes and
stores the surge in a manner similar to the digital storage
oscilloscope. Because of the data processing functions which are
incorporated in the instrument, the recorder allows reports of many
different parameters of the disturbance, relating voltage to time.
39
TABLE IV
COMMON POWER MONITORING FEATURES [15]
Features
Harmonic Analysis
V oltage and current
measurements
Continuous Monitoring.
Multiple Channels
Description
Increases the power and
flexibility of the power monitor.
Harmonic analysis of voltage
and current pinpoints loads
which cause harmonic currents.
Analyzes and identifies the
source of power problems.
Provides unrestricted Monitoring.
Enables the use of several
channels
to observe events In phase,
neutral and ground.
40
Features
Waveform Display
Waveform Manipulation
Scopemode
(real time monitoring)
Data storage
TABLE IV (Continued)
Description
Waveforms are recorded and
displayed . .
Some event waveforms are very
characteristics of particular load.
Recognizing these waveforms
helps
to point immediately to the cause.
Allows quick waveform
measurement usmg zoom,
frequency markers and
Amplitude markers.
Enables quick view of voltage
and current magnitudes and
waveforms because only the
data of interest is stored.
Stores event data on disk.
4 1
Feature
Broad Frequency response
Simultaneous Display
Event Summaries
TABLE IV (Continued)
Description
Captures high speed current
and voltage impulses.
Saves waveforms from all
channels, even if a disturbance
occurs on only one channel.
It _guC!f~nt~e~ a complete
picture of the disturbance
recorded.
Provides a graphical plot of rms
and high frequency events over
user selectable period.
42
(vi) Infrared scanner Infrared scanners measure infrared
emissions from a surface. Typically they are used for monitoring of
survey are very important overheating of electrical switchgear ,
transformers, circuit breakers and other electrical equipment. They
convert emissions into a proportional voltage · that drives a display
(digital or analog scale). Some infrared scanners use video displays
for easy location of a scanned item and storage of data.
Performing the Survey
A power quality survey is a necessary step m diagnosing the
problem and finding a solution. The survey may include a basic and
a comprehensive one. Basic inspection is necessary before the
comprehensive one is undertaken. Both types of survey are very
important to pinpoint the problem area and search for answers to
the complex problem of power quality.
(1) Basic Inspection. The basic inspection is designed to
familiarize site examiners with the building. It is important to the
survey team members to visually inspect the electrical layout,
electrical hardware and the construction techniques used throughout
the building.
When inspecting the outside of the building the surveyor should be
aware of the following points
(i) Search for the type of electrical service (underground or
aerial ) used.
43
(ii) Determine if there are any utility power factor correction
capacitor installations. Neighboring facilities can feed interference
back onto a shared utility feeder.
(iii) Locate any utility substations in the immediate vicinity.
During the internal inspection of the building, the surveyor should
note the following important points :
(iv) Notice any large electrical loads (adjustable speed drives,
elevators, welders etc.) in the facility.
(v) Checkup on any equipment which has a history of
problems
(vi) Arrange a tour of the important sites inside the building
and collect as much data as possible from consultation with different
people who are knowledgeable of the building and its electrical
machineries.
(2) Comprehensive Inspection. A comprehensive inspection
involves physical verification of site and monitoring procedures.
The electrical distribution system and grounding must be physically
inspected to identify problems with electrical system equipment and
grounding. Table V summarizes some important measurements and
considerations in identifying problems with various electrical
distribution systems.
(i) Inspection.Procedures The best place to start a physical
inspection is the supply transformer. From this point, each
additional panel for the distribution system should be checked for
rms voltage levels and current levels. All these tests should include
44
voltage, current, phase sequence, ground impedance and neutral
impedance, proper ,conductor termination, absence of neutral ground
and isolated ground shorts.
(ii) Monitoring procedures.
Power monitoring with the help of cathode ray screen is very
important to locate various types of voltage disturbances. In the
following a few of the important points would be highlighted.
(a) Hookup.
Use of twisted pair of cables for monitor inputs reduces the
possibility of picking up radiated RFI/EMI (radio frequency and
electromagnetic interference) fields.
as follows:
Other hookup procedures are
(I) Whenever possible, the monitor should be connected to
record both voltage and current at a load.
(II) When a power monitor is installed at a load, the channel
connections for the monitor should match tbe wiring connection for
the power supply of the load.
(III) Current transformers (probes) should be added to vacant
cbannels to record phase, and neutral or ground current.
(IV) Monitoring phase current is essential to determine the
source of a power disturbance.
(V) Monitoring neutral or ground current is essential to
determine the source of neutraltoground faults/disturbances and
to identify the sources of current flowing in tbe grounding conductor.
(b) Grounding The grounding of the power monitor should be
performed with caution. Since a chassis ground is provided through
45
TABLE V
ELECTRICAL DISTRIBUTION SYSTEM AND GROUNDING
MEASUREMENTS AND CONSIDERATION [15]
Equipment or Grounding
Wiring
Grounding
Measurements and considerations
Ensure compliance with NEC (National
Electrical Code). Make a detailed
inspection for broken or corroded
conduits. Immediately replace the
defective wires.
Compliance is maintained with the
NEC. It is made sure that any
electrical distribution devices,
receptacles, power panels,
conduits etc  are bonded to ground
structures. No independent ground
systems should be established.
46
Equipment or Grounding
Electrical panelboards
Electrical conduits
TABLE V (Continued)
Measurements and considerations
Inspect loose electrical
connections, excessive hardware
temperature and improper neutral
to ground bonds. Special attention
should be paid to busbar
as semblies.
Measurement of all phase to
phase, phase to ground and phase
to neutral voltages are made and
these values are recorded. All
currents in feeder conductors
branch circuits and grounding
conductors are measured.
Circuit breakers are checked for hot
spots. .ro
Electrical conduits are checked for
mechanical connection and
excessive warmth. A hot and
vibratory conduits should be
replaced immediately.
47
TABLE V (Continued)
Equipment or grounding
Electrical serVIce entrance
Transformers
Receptacles
Measurements and considerations
Grounding of switch gear is checked
for compliance with the NEC
regulation.
Wiring connection IS checked for
looseness and special attention is
given toward neutral and grounding.
They should be checked for excessive
heat, vibration or audible noise. All
phase voltages are measured and a
detailed inspection is maintained for
transformer output neutral
and safety ground.
Inspect grounding connections,
cracked faceplates, or visible
sign of arcing. Load is
removed from the receptacle
and checked for proper polarity and
ground resistance.
48
TABLE V (Continued)
Equipment or grounding
Undercarpet Wiring
Equipment power cords &
plugs.
Measurements and considerations
Checking is conducted for excessive
warmth around the carpet. Excessive
weight if present is removed from
wiring.
Inspection is done for wear and tear
of patches of Insulation and warm
spots.
49
the ac input power cord, any chassis ground connections to the circuit
being monitored can cause ground loops that result in additional
noise being injected into the sensitive equipment feeder. To avoid
this problem one should not make chassis ground connection to the
circuit being monitored.
(c) Location. The general priorities for positioning of the
power monitor are :
(I) To place it as close as to the load.
(II) At the service entrance to determine the power quality of
the site.
(III) Locate the important regions as determined by the
previous survey. Some limitations may affect the ability of power
monitoring
equipment to predict power supply disturbances at a specific
locations. Also, monitoring over a period of less than one year may
produce an inaccurate power disturbance profile.
(d) Duration. Monitoring should be performed for at least
one full weekly cycle which comprises 78 days. .
(e) Threshold. Power monitoring equipments reqUIre
selection of threshold at which disturbances are to be measured.
Some of those techniques are grouped as follows:
(I) The sensitive level captures almost every important event
and shows the constant power quality problem on the line.
(II) The normal level captures events that are more severe.
(III) The tolerant level captures very severe power problems
m the line (distribution network). The tolerant level may also be
used for regular monitoring.
50
The usual procedure for setting power monitor thresholds are
(A) The monitor is set in oscilloscope mode & the power is
observed for any voltage or current waveform distortion. If there
arises any abnormality then that event should be recorded.
(B) The monitor is next adjusted to sensitive level and is left
for 1015 minutes. The subsequent event summaries are plotted for
analysis. The information is stored in a floppy disk for evaluation at
a later time.
(C) The monitor is reset for normal thresholds. If the
scopemode or sensitive level monitoring record events which exceed
the normal thresholds, then the threshold setting is readjusted.
(f) Monitoring. The monitor is checked and readjusted
regularly to capture any events. Periodic inspections may indicate
the need to modify the monitor installation. The changes may be
classified as follows:
(I) Threshold setting may have to be altered to increase or
decrease sensitivity.
(II) Current probe IS added to determine the direction or to
monitor the phase of neutral or ground current.
(III) Environmental, temperature and chromatographic
sensors are installed to measure the change in humidity,
temperature and any change in air composition.
(IV) Sensors to measure radio frequency and microwave
interference are installed.
Analyzing the survey data
5 I
Once the power quality survey has been performed, the next
step is the analysis of all collected information and data. In the
analysis a detailed inspection is maintained for unusual voltage and
current waveforms. The data provided by the power monitor
should be carefully analyzed to determine sources of disturbances, as
well as cost effective methods for elimination of disturbances.
Although relatively simple analysis can be helpful, the key to
identifying powerrelated equipment problems is to conduct a
thorough scan which involves the following steps:
(1) Examination of pre and post survey data. It is essential to
examine all information prior to survey. Any specific power events
that may cause equipment symptoms is documented in the site
history and equipment logs. Also it is imperative to go through the
post survey events such as the power monitor event data. As an
example, a hard disk crash (loss of information) on a computer may
be due to an impulse, outage or overvoltage in the power
distribution network. The procedures of correct wiring and
grounding of the equipments are verified.
(2) Comparison and plotting of power monitor output. It is
essential to plot power monitoring event summaries which can help
to identify the main cause of power problems. At a minimum level
at least there are three different event summaries as  24hour,
seven day and total. Fig. I 2 shows a typical sevenday event
summary.
The power events and equipment event logs are compared. By
arranging all equipment log entries into a format similar to the
power monitor event summaries, trend analysis and event
52
correlation become easier. It is important to note the time,
frequency, amplitude and event number of all events which
correspond with the recorded equipment problems. Comparison of
power events to equipment performance specifications is necessary
to determine if recorded events exceed limits.
Main equipment specifications include voltage limit, frequency
tolerance, and impulse withstand capability. The listings of several
equipment specifications and their normal and critical tolerances are
tabulated in Table VI.
(3) Determination and classification of key power events.
Power events that correspond to or exceed limits would be apparent
Grouping key events into general categories improves event analysis.
Some events reflect problems within a building while other events
are clearly power sourcerelated.
The general categories of events are
(i) High frequency events such as unidirectional impulse,
oscillatory impulse, repetitive events and common and normal mode
noise.
(ii) Voltage events (such as sags, surges, undervoltage, overvoltage,
outages and line interruptions etc.).
(iii) Distortion.
(4) Confirmation of Power Monitoring Correlation. It is very
essential to confirm that the correlation of power events and
equipment
symptoms IS valid.
below.
The general guidelines to be followed are listed
53
TABLE VI
EQUIPMENT SPECIFICATIONS AND TOLERANCE [15]
Specifications Normal Tolerance Critical Tolerance
Frequency Deviation 47 to 63 Hz 59.5 to 60.5 Hz
Harmonic Distortion 10 to 20% Maximum of 3 tol0% Maximum of
Impu lses
Neutral to Ground
Voltage
Voltage Dropouts.
Voltage Fluctuation
Total Hannonic
100 to 300 Volt
(IX to 3X Nominal)
3 to 5 Volt
20 milliseconds
114 to 126 Volt ac
Total Harmonic
Distortion Distortion.
50 to 100 Volt
(0.5X to IX
Nominal)
1 Volt
4 Milliseconds
105 to 132 Volt ac
54
5.5
JtnSIhlp. So ...... .!'!! Channel' I&1ISI89 lS:lS:S3  151~ 15:Z7:1Z
UI'RS ...... ~ ... ::. ...... . .. .......... ::.:::J" ........... _ .... _ .. __ ._ ... _. __ ............. _ ............... ........ .
121.Z   
138.8 ............................................................................................................ ..1 ....... .............. ..
lJpeak ................................................... __ ................................ ........... ....... .... _ ........... _.
ZSZ.8
1 11 II 1 I
I J I
t I
3(,8.8 ....................... _ .................... _ ................. _ ................................................................... .
1 .t.~jli'isia.
Figure12. SevenDay Event Summary [15]
(i) Sufficient analysis is done to see whether the
recorded power event could have been the cause of the equipment
problem. Attention is paid to note that a correlation can indicate the
absence of sufficient monitor data.(ii) Physical inspection data is
rechecked to determine the necessity of correction of the problem.
(iii) Further continuous monitoring IS maintained if the
physical inspection does not show any significant improvements.
(5) Identification of sources of power related problems.
Identification of a source of a harmful power event is important to
ensure that corrective actions would be successful. The following
points provide consideration for monitoring both voltage and current
events:
(i) Impulse
(a) Most impulses within a building are generated when load
cycles come on and off.
(b) Typical causes of impulses include switch contacts and
sharp current transitions interfacing with source impedance.
(c) The direction of the origin of an impulse can be determined
by reviewing the polarity of the leading edges of the simultaneous
voltage and current waveforms.
(d) If the voltage is "+" and current IS "+" or voltage is "_" and
current is "" the origin is source related. If the voltage is "+" and
the current. is "_" or voltage is "_" and current is "+". the origin is
load related.
56
(e) Fig.13 shows a voltage source supplying power to an SCR
controlled load. Each cycle of the voltage waveform has an Impulse
which coincides with the current drawn by the load.
(ii) Voltage Sag.
(a) Voltage sags are sometimes caused by utility power supply
variations.
(b) Another major factor is the load interaction with wiring.
(c) If the current on the circuit increases or decreases or goes
to zero then the origin of the sag is termed as source oriented.
(iii) Voltage Distortion.
(a) Voltage distortion is caused by large amounts of harmonic
currents from nonlinear loads and power sources with nonsinusodal
voltage characteristics.
(b) Loads that do not contribute to distortion have a small
amount of harmonic current. Loads which cause distortion have
high levels of harmonic currents.
(iv) Ground and neutral events.
(a) There are several causes which result in neutraltoground
voltages and they include return current flowing in the
neutral conductor impedance, ground current and too much ground
resistance.
(b) Where neutraltoground voltage results from neutral current,
the neutral current increases and varies along with neutraltoground
voltage levels.
57
hent ~Mr 4
\
\
\
\
Setu I
11 ~ I. \\ I I \ ,
\
15:26:21.52
I I
/
,
I
/
tilrizont.l 5 .illi~ndsldivision Uerti~l 58 Volts/divislon
U ,..: Prev=119.&. ain=118.9. l\u=121.1  Yorst 1.,=171 .5 Upk. 267 deg
Euent Ibller 4 Channel C Setu 1 15:26 :21.52
\ ~
\ ;
_. ~ ~. _ .. 
I II
r Hariaolt.l 11 .'Iliseconjs/division Vertic.1 18 Alps/division
" ,..: 'rel/=ll.8fJ, lIin:;lB.73. 1\u=11.91  YoMt lap=2&.8 Apk. 2S8 Geg
Figure13. Waveform of SCRControlled Load [15]
58
p
Measures of power quality
For periodic waves, the best known indicator of power quality
is the [7] total harmonic distortion (THD).
Mathematically, it is expressed as:
00
LV;
i=2
THD= VI (1)
Where Vi is the ith harmonic of either voltage or current. The KVT
product is the audio weight which is expressed as:
KVT=
00
.~£V..J~ w~ 1 1
i=I
( 2)
with Vi In linetoline voltage in kV. The telephone influence factor
TIF is given as:
00
TIF= VI (3)
Where Wi in both the TIF and KVT expressions are the audio weights
which reflect the response of the human ear.
Distortion Index (DIN) is given as:
The European
59
V{t)
, . ",
a • impulse ampl
b • nolCtl depth
c. ringing f
d • Impulse energy
•• impulse width
f • time const
0 GUlag' time
hpeak
t
Figure14. Power quality indices for nonperiodic phenomena [11]
60
I2 V.
1
DIN=
i=2
Vrms (4)
Where V rms is the rms voltage
00
(5)
It is easy to show that for small THD or DIN,
THD = DIN
For nonperiodic signals such as the one illustrated in Figure.14, the
following indices are used to assess power quality:
(1) Impulse amplitude
(2) Notch Depth.
(3) Ringing frequency.
(4) Impulse energy
(5) Impulse width.
(6) Damping time constant associated with ringing pulses.
(7) Waveform peak.
Discussion of various methods to improve Power Quality
There are two parts to the solution to improve the quality of
power that is distributed to a large number of customers. The first
6 I
part 1S known as customer site corrections and the second part 1S the
solution of utility's voltage variation problem. These are briefly
discussed next.
(1) Customer Site Corrections. A variety of voltage
regulation equipment is available to provide a constant voltage
magnitude at critical
loads during short or momentary voltage variations In the supply
voltage. A few of these type of devices are listed below.
(i) Equipment power supplies. With advanced and improved
equipments, it is possible to get a certain degree of clean power (by
providing the capability to ride) through the under voltage
conditions of short duration. Larger capacitors in the output circuit
can ride through 114 cycle to approximately two cycles ( longer for
low power devices) of the period of voltage loss.
(ii) Voltage regulators. Fast responding voltage regulators
can typically control the output voltage to within ±2% when the
input voltage varies within ± 15%. Often the output voltage can be
controlled to within 10% of nominal for input swings as large as 65%.
(iii) Line Conditioners. By using this type of devices (linear
amplifier type line conditioners) very accurate voltage regulation
can be obtained. Slightly less accurate regulation at a lower cost is
obtained by using ferroresonant type of conditioners.
(iv) Motor generator set Motor generator sets are generally
employed to provide complete isolation between systems or as
62
F
frequency changers. They can also be provided with the capability
to ride through a complete loss of voltage for 0.3 to 0.5 seconds.
(v) Uninterruptible Power Supplies (UPS), These systems
are designed to provide uninterruptible power for much longer
durations. With battery backup they can easily ride through short
duration undervoltages or outages.
(2) Solution of utility sides voltage variation. The most
important cause of short duration disturbances is the occurrence of
system faults. Therefore the utility can take the following steps to
eliminate disturbances to the customer.
(i) Install underground systems instead of overhead systems.
(ii) Follow proper tree trimming policies and provide animal
guards.
(iii) Minimize system ground impedances.
(iv) Employ appropriate coordination practices and switching
procedures.
(3) Long duration voltage variations Voltage variations longer
than 30 cycles (1/2 second) are generally classified as long . These
variations are generally in the range of 20% to +10% and they are
not the result of system faults. They are caused by system
switching operations and some of the specific causes are discussed
below.
(i) Motor starting. Large motors can cause significant voltage
dips on the system when they are started. The resulting voltage
during the motor starting period will depend on the ratio of the
motor size to the system short circuit capacity where it is connected.
63
(ii) Switching system loads and capacitor banks. Energizing a
load on the system will result in a sustained voltage drop on the
system until voltage regulation or other devices compensate for the
change. Similarly, deenergizing a load will result in a sustained
overvoltage on the system. If the capacitor bank or load is large
enough relative to the system capacity, distribution system voltage
regulators or transformer load tap changers would respond in tens of
seconds.to bring back the voltage to within allowable tolerance limits.
(iii) Voltage Flicker. Loads which can exhibit continuous,
rapid variations in the load current magnitude can cause voltage
variations referred to as flicker. Flicker normally refers to voltage
variations which exhibit frequencies in the range of 112 to 30 Hz.
(4) Intentional voltage reductions. Many utilities follow
voltage reduction procedures to help in reducing the load during
severe peak load conditions. The system voltage may be reduced
from 3% to 8% in steps depending on the severity of the load.
Solution of voltage variation problem. In particular, various
types of tapchangers, ferroresonant regulators, line power
conditioners, motor generator sets, and uninterruptible power
supplies can be used to maintain a constant voltage.
(5) Power interruptions. power interruptions involve the
complete loss of voltage to the load for at least 0.5 seconds. The
interruptions can be classified as momentary interruptions (less than
2 seconds), temporary interruptions (2 seconds to 2 minutes), or
outages (longer than 2 minutes)
Solution to interruption problems. Backup supplies for outage
contingencies take the form of UPS system or backup generators.
64
UPS systems generally provide uninterrupted supply for at least 15
to 30 minutes. The utility determines the duration of interruption
through protective devices coordination practices on circuits
supplying the customers. Fuses, sectionalizes, reclosers and relays
are coordinated when there IS a permanent fault so that minimum
number of customers suffer outage.
Solution Of harmonic distortion. The problem of harmonic
distortion can also be mitigated by the following techniques :
(i) Phase multiplication. The essence of phase multiplication
methodologies is the translation of harmonics to a higher frequency
and the concomitant reduction of the amplitudes of those harmonics.
(ii) Passive and active filters. This method offers an array of
power factor correction and control side benefits.
(iii) Harmonic injection. This scheme needs a triple harmonic
current generator, and nullification of more than one harmonic order
at anyone operating point.
(iv) Pulse width modulation. This method shows great
promise, and quite capable of obtaining harmonic suppression to less
than 1 % of the fundamental.
65
CHAPTER III
HARTLEY TRANSFORM
Introduction
Complex number theory greatly facilitated the handling of
oscillating quantities. To analyze alternating current without the
complex phasor eirot is unthinkable. Consequently the theory of
Fourier series took advantage of the theory of complex numbers and
it came to seem natural that a periodic function p(t) of unit period
can be analyzed into complex components anei2nnt. where the
coefficients 3n. are now complex. Thus
00
p(t)= L.an .ei2pnt (6)
n=oo
and 1D the limit, for a function f(t) that is not periodic,
00
f(t)= f f(n).ei21tnt.dn (7)
00
At the same time, pet) can be expressed in terms of real functions as
00
p(t)=ao + L(ancos27tnt+bnsin21tnt)
n=1
(8)
as shown by Fourier. As a result of which, one can understand that
the use of complex exponential is convenient rather than
fundamental. Hartley's (in 1942) formulation of a real integral
66
transform made it possible to dispense with complex representation.
Although his new discovery was published 1D the Proceedings of the
Institute of Radio Engineers. and mentioned 1D text books such as
Frequency Analysis, Modulation and Noise by S. Goldman, and R. N.
Bracewell. Fourier Transform and it's Application (McGraw
Bill,1965), the classical method prevailed. Hartley's famous "Cas"
function transform s=io> is related to the Fourier Transform by the
following expression as:
00
B(s)= Jf(x).Cas21tsx.dx (9)
00
where Cas21tsx = Cos21tsx + Sin21tsx ,
and R(s) is known as the Hartley Transform (RT). The advent of
images produced or processed by computer and displayed on cathode
ray screen has greatly extended the applicatio.n of two dimensional
analysis and it is very interesting to find that the idea of a real
transform generalizes readily. It is obvious that a power spectrum,
such as an optical spectrum, IS described by a real function of a real
frequency variable f.
Definitions of Hartley Transform.
In his original paper in the Proceedings of the Institute of Radio
Engineers in 1942, R. V.L. Hartley (18901970) laid emphasis on the
strictly reciprocal character of a pair of integral formulae that he
introduced and in the following section his notation will be followed
so that the full symmetry can be appreciated.
67
Consider a time dependent signal vet) which may be thought of
as a voltage waveform of the kind that might be applied to the
terminals of a telephone line. This waveform possesses a frequency
spectrum which can be expressed through the Fourier Transform.
Let S(m) be the Fourier transform of the voltage waveform vet) which
can be written mathematically as :
00
1 f . s(oo)= _ r;:: vet) .eHot.dt
,,21t 00
(10)
The quantity S(m) is a complex function of the angular frequency
variable which itself assumes only real values. Thus, for any given
waveform vet), a Fourier Transform S(m) can be calculated which is
unique for that waveform. Next a question quickly comes to the
mind  can the original function vCt) be recovered?, The answer is
definitely affirmative.
In Hartley's definition of the transform 'II (m), the factors 1IJ 21t
were explicitly included in order to achieve a symmetrical
appearance. Thus,
00
\jf (Ol)= &. Jv(t).CasOlt.dt
( 21t)_00
00
1
v(t)= & j\jl(m).Casmt.dm
( 21t)_00
(11)
(12)
In these relations the Cas function is, as mentioned in the original
paper, simply the sum of Cos and Sin functions, as given by
68
Casrot= Cosrot + Sinrot
_, 1t
=" 2 Sin(wt + 4)
Continuous Hartley Transform
Although there is not much apparent difference between the
familiar Fourier Transform integral and the new pair of integrals
given by Equations (11) and (12), in practice the difference is
profound. The main thing to note in this case is that the transform
'V(t) is real, not complex as is the case with S(ro). In Hartley's
definition of the transform v( co), the factors 1/....J 21t were explicitly
included in order to achieve a symmetrical appearance. The factor
;1t exists because the integration variable is ro. Since ro = 2n:f, it is a
general practice is to place 21t symmetrically in the exponential part
of both integrals. This happens automatically when frequency f is
taken as a variable in place of angular frequency ro. This leads to
the following two transform equations:
00
H(f)= fv(t).Cas2TCfLdt (13)
00
00
v (t) = JH (f) .Cas2n:ft.df (14)
00
The function H(f) is called the Hartley Transform of vet) in honor of
Hartley, and vet) would be the inverse Hartley Transform of H(f).
69
For comparison, the Fourier Transform relations written according to
the same convention are given below:
00
F(f)= Jv(t).e i21tft.dt (15)
00
00
v(t)= JF(f) .e i21tft.df (16)
00
Even and Odd Part. The relationship between Fourier and
Hartley Transforms rests upon symmetry condition.
split into two parts, namely even part and odd part.
H(f) can be
The even part of a function is what we get when the function IS
reversed (changing t to t), and then added to the original function
and subsequently dividing the sum by two. Naturally, the even
part is its own mirror image, having symmetry property E(f)=E(f).
The odd part is formed by subtracting the reversed function from
the original and dividing by two; it has the antisymmetrical property,
O( f)=O(f). Any function can be split uniquely into even and odd
parts, and if they are known, the original function can be
reconstructed.
Mathematically H(f)=E(f)+O(f) , Where E(f) and 0(0 are the even and
Odd parts of H(f) respectively,
00
then E(f)= H(f)+2H( f) = Jv(t).Cos2n:ft.dt (17)
00
00
H(f)H( f) J . O(f)= 2 = v(t) .Sm2n:ft.dt (18)
00
70
Connecting relation between Fourier and HartleyTransforms.
Given H(t), it is possible to calculate the sum [ E(t) i O(t)] to obtain
the Fourier Transform F(f) :
00 00
E(t)iO(t)= jv(t)[Cos2n:ftiSin2n:ft]dt = Jv(t).e i27tft .dt (19)
00 00
00 00
E(t)+iO(t)= Jv(t)[Cos2n:ft+iSin2n:ft]dt= fv(t).ei21tft.dt (20)
00 00
Thus it is seen that from H(t) the Fourier transform of vet) can
be calculated by simple addition and subtraction. The Fourier
Transform F(t) has a real part which is the same as E(t) and an
imaginary part whose negative is O(t)
Freal(f)=ReF(f)=E(f)
Fimag(f)=ImF(f)=O(f)
Mathematically it follows as:
(21)
(22)
Conversely, gIven the Fourier transform F(f), H(t) can be calculated
by the expression given below as:
H(t)=Freal(f) iFimag(f) (23)
From F(f), H(f) can be calculated as the sum of real and sign inverse
imagInary part of the Fourier transform. Applying the theory that
imaginary part of a complex number is real, it can be verified that
F(t) is real as it should be, given that the original waveform vet) was
7 1
2.8
1.3
a.7
H
( r 9.8
)
8.7
1.3
2.8
4.9
I
I
I
I \ I ~ :'        ..    . I ~~~~~~~ ~ r   ...        .•  .•     _ ..    ....
8.8 8.8 2.4 4.8
r
Figure15. Plot of continuous Hartley transform function H(f) [2]
72
real. Briefly it can be concluded that the Hartley transform is the
realpart of the Fourier transform minus the imaginary part of the
transform.
A particular example of the continuous Hartley transform IS
calculated below. Let f(t)= et , t>O
DO
H(f)= fet .Cas27tft.dt
DO
DO
= jet .Cas21tft.dt
o
DO 00
= Jet.CoS21tft.dt + jet .Sin21tft.dt
o 0
H(f) _ 1 + 2 pf
 (1 +41t2f2) (1 +41t2f2)
By inspection it is clear that the even and odd parts are
1
21tf
and O(f)(1 +41t2f2)
In Fig.IS the plot of the function is shown.
Discrete Hartley Transform
73
Althougb it is assumed that time is a continuous variable it i
important and nece.ssary in practice to use a discrete variable to
describe a time s.eries. This situation arises in case of torage of data
in a computer at regular intervals. For computational advantage a
discrete variabler is introduced ranging from 0 to (N~ 1).
The discrete Hartley Transform (DHT) of .a real function f(t) lS,
with its mverse, given by e.quations (24) and (25) respectively as
follows:
] NI
H(v )=(N) Lf(t).Cas(21tvtlN) (24)
't=o
Nl
f(t)= LH(v}.Cas(21tvtIN) (25)
1=0
As before the abbreviation Case = CosB + Sine is used. To derive the
Inverse Discrete Hartley Transform integral, the orthogonality
relation
N1 N for t=O
"Cas(27tvt/N).Cas(27tvt'/N)=[ f ' } £...J V or t;t::t
t=O
1 N1
Substituting, (N) Lf(t').Cas(2pvt'IN)
t'=O
N 1
for H(v) in the expression L H(v).Cas(2pvtlN), we have
v=o
Nl
LH(v)Cas(27tvt/N)
v=O
(26)
74
Nl
~ 1 Nl
= £...J (N) I,f(t').Cas(21tv't'/N).Cas(21tv't/N)
v= O 't'=O
1 Nl NI
= ( N) I.f('t') I,Cas(21tv't'/N).Cas(21tv't/N)
't'=0 v=O
1 N 1 N for 't='t'
=(N) I. f('t') X b for t:;tt' }
't'=0
=f('t) (27)
which verifies the inversion integral.
Meaning of 't and v. As 't corresponds to time 10 the similar
way discrete variable v represents the frequency. If the unit of t is
seconds, that IS if the time interval between successive elements of
v
time series f('t) IS one second, the frequency increment is N Hertz.
Then the frequency interval between successive elements of the
sequence H(v) is Nl Hz. As v increases the corresponding frequency
N
Increases, but only up to v= 2' beyond that the frequency assumes the
ex pression (N v )/N. So the frequency becomes zero at v.
Even and Odd Parts. As in the continuous case, the Discrete
Hartley transform (DHT) possesses even and odd parts.
H(v) = E(v) + O(v) (28)
But a careful attention is required in this regard because of the
convention that v should cover the range of values form 0 to Nl.
The general procedure to handle this type of situation is to assign
function values outside the basic range so as to generate a cyclic
75
function wi th period N. So that at v =1 a value can be assigned to
H(Nl) due to the fact that v=l and v=N1 are separated by one
period of length N. In general the value H(Nv) is assigned in place
of H( v) where N s; v S; 1 for which the independent variable is in
the basic range of v. By this way a simple relationship can be
achieved between v and frequency. And now it can be interpreted
that ~ is the same as frequency in Hz, over the range  ~ < v< ~. Thus
H(v)+H(Nv)
E(v) = 2 (29)
H(v)H(N v)
O(v)= 2 (30)
From the definition of the Discrete Fourier transform (DFT), F(V)f it is
apparent that it can be formed from the even and odd part of
discrete Hartley transform (DHT) by as follows:
F(V)f= E(v)  iO(v)
Conversely, H(v) can be constructed using the discrete Fourier
transform F(V)f as follows:
H(v)= ReF(v)f ImF(v)f
(31)
(32)
The standard form of the Discrete Fourier transform and its Inverse
are gwen by the equations (33) and (34) respectively.
1 Nl
F(V)f = (N) 1: f('t).e i21tv1:/N (33)
1=0
Nl
f('t) = L F(v)·fei21tv't/N (34)
v=o
76
~~~~~
These relations are strictly analogous to those obtained previously
for the continuous variable case. In the following an example is
analyzed to consider the characteristics of the Hartley Transform.
Let a function F(t) be defined as:
(0.5, t=o)
F(t)= { t/2 _
(e • tl,2 .... 15)
It represents the continuous function e't that was used before,
except that it is sampled. The value at t=O, since it falls on the
discontinuity of Vet), is assigned the value [ V(O+)+ V(0)]/2=O.5.
The results for H(v), which are shown in Fig.16, closely resemble
samples of a continuous function taken at intervals ~ ro/2n= 1116.
The discrepancies, which are small in this example, are due partly to
the truncation of the exponential waveform and partly due to
aliasing, exactly as with the DFT. Aliasing is caused due to the
overlapping of higher frequency portion of a spectrum with the
lower frequency part. The values given by this function due to DHT
are arranged in the Table VII.
Convolution property in Discrete Hartley Case. In general, the
transform of the convolution f} (t)® f2(t) contains four terms. The
fundamental quantities to be computed are the direct products
Pa(v)=Hl (v)H2(v) and the retrograde products Pb(v)=Hl (v)H2(v) in
these terms
Thus only two multiplications instead of four are involved. Now if
H2(v) is even then
H(v)=H 1 (v)H2(v) (36)
77
TABLE VII
V ALUES OF DISCRETE HARTLEY TRANSFORM [2]
't, V =0, 1. 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15
f('t) =20, 15, 6, 1, 0, 0, 0, 0, 0, 0,0, 0, 0, 1, 6, 15
H(v)=4, 3.56, 2.49, 1.32, 0.5, 0.12, 0.01, 0, 0, 0.01, 0.12, 0.5, 1.32, 2.49,
3.56
78
20
fir)

, , ,
o T IS o "
Figure16. A 16point representation of the the truncated
exponential waveform (left) used for
illustrating the continuous transform
and its DHT (right) [2]
79
IS
And a similarly simple form results if H2(V) is odd; in this case
(37)
Because of commutativity, H(v)=H 1 (v)H2(V) is even if either HI (v) or
H2(v) is even. Because of it's importance, the application of this
property would be shown in the next chapter in the analysis of
power quality in electric networks. These interesting properties of
the Hartley transform can be applied in branches of electrical
engineering other than communication and signal processIng.
Two Dimensional Hartley Transform
Continuous Case. Considering a function f(x, y), it IS two
dimensional Hartley transform H(u, v) can be defined and the
inverse of it can also be expressed as in the following Equations (38)
and (39). Thus,
00 00
H(u, v)= f ff(x, y).Cas[21t(ux+vy)]dxdy (38)
0()  O(j
00 00
f(x, y)== f fH(u, v).Cas[21t(ux+vy)]dudv (39)
00 00
In terms of Fourier transforms, F(u, v)=R(u, v)+i I(u, v) it follows that
H(u, v)=R(u, v)I(u, v) (40)
as may be verified from the Fourier Transforms and the Inverse
Fourier Transform takes over the two variables as given by
equations (40) and (41).
00 00
F(u, v)= J ff(x, y).e 21ti(ux+vY).dxdy (41)
00 00
80
00 00
f(x, y)= S SF(u, v).e 21ti (ux+vY).dudv (42)
00 00
Two dimensional Hartley transforms have some advantages in image
processing and in measurements. It is well known that the one
dimensional Hart1ey Transform is real, and it is so in two dimensions
also. Because the Fourier Transform is complex, half the transform
plane is sufficient to determine the Fourier transform. The rest of
the plane is occupied by the conjugate values that bear no additional
information because points that are diametrically opposite are
labelled with conjugate coefficients. In the Hartley plane on the
other hand there is no such symmetry and redundancy. The
information is spread half as thick over the whole area.
Symmetry and antisymmetry. A gIven function fex, y) maybe
decomposed into symmetrical and antisymmetrical parts such that
f(x, y)=fsymm(x, y) + fantisymm(X, y) (43)
where
1
fsymm(x, Y)=2 [f(x, y) + fex, y)] (44)
and
1
fantisymm(X, Y)=2 [f(x, y)f(x, y)] (45)
This resolution into parts is the two dimensional generalization
of splitting functions of one variable into even and odd parts. Just
as the real part of the Fourier transform IS even and the imaginary
part is odd, the real part of the two dimensional Fourier transform is
symmetrical and the imaginary part is anti symmetrical. It follows
that the rea] two dimensional Hartley transform expressed as [R(u,
v) I(u, v)] is already presented in terms of its symmetrical and
•
anti symmetrical parts. To establish the inverse transform as
mentioned above taking the two dimensional Hartley transform of
the transform H(u,v) it fonows as:
rex, y)= HT of H(u, v) = HT of R(u, v)  HT of I(u, v)
Now HT of R(u, v)= fsymm(x, y)
because R(u, v) is symmetrical. Similarly
(46)
(47)
HT of I(u, v)=fantisymm(X, y) (48)
so rex, y)=fsymm (x, y)[ fantisymm(X, y)]=f(x, y) (49)
Two dimensional Discrete Hartley Transform. Manipulation of
two dimensional digital Images is also enhanced than loose to the
existence of a real transform. An image fCq, 't2) represented by an
N I x N2 matrix possesses a two dimensional Hartley transform
which is itself an NIx N2 matrix H(v 1, V2) of real numbers. The two
dimensional Hartley transform and its inverse are expressed by the
following Eq.50 and Eg.5} as:
H(n}, n2)=
} NI IN2 1
N N I rf(t}, t2)Cas(2pnIt}/NI+2p n 2t2/N 2)
I 2 t}=0 t2=0
NI IN2 I
f(tI, t2) = I· rR(n 1, n2)Cas(2pn 1 t} IN I +2pn2t2/N 2)
nl=on2=0
(50)
(51)
A two dimensional spatial frequency (v 1, V2) describes an obliquely
vI . f . f
oriented Cas function which has N 1 cycles per umt 0 1: 1, In or
82
v2
example the eastwest direction, and N2 cycles per unit of 't2 III the
northsouth direction.
General Discussion
Hartley Transforms plays a very important role In signal
processing and communication engineering as well as III nonlinear
optics in Physics. Images on surfaces have always been of
importance and are becoming more obvious as technical means for
creating, modifying and presenting such images are developed.
Digital Telemetry of planetary images from spacecraft , and digital
processing of these images have become familiar as a result of space
exploration and the design of analogue optical imaging systems have
made remarkable advances·.
Analogue optical image processlllg IS a reality and optical
digital processmg of two dimensional digital data is in view. In all
these fields spectral analysis is a customary tool; Consequently there
are corresponding applications of the Hartley Transform in electric
power systems.
83
CHAPTER IV
APPLICA nON OF HARTLEY TRANSFORM FOR
POWER QUALITY ANAL YSES
Introduction
Techniques for electric power quality analyses and assessment
have taken on a renewed importance in recent years.
two main factors as given below:
This is due to
(l) The appearance of high power switching devices and switched
loads which can cause power quality problems at the distribution
level.
(2) The need for maintaining power quality at all power levels to
avoid interference, excessive losses, and misoperation of loads.
There are numerous fundamental issues to be resolved relating to
quantifying power quality problems, instrumentation and
monitoring. The methodology used for the calculation of bus voltage
and the line current waveforms is also of salient importance.
Electric power quality assessment often involves the calculation of a
bus voltage or line current. Hartley transform is applicable and
useful for this purpose due to the following criteria:
(i) Limited bandwidth of most electric distribution system
makes truncation of Hartley transform practical at a reasonably low
84
frequency (e.g, V=± 9425 rad/sec in a 60 Hz system or ± 7854 rad/sec
in a 50 Hz system corresponding to the 25th harmonic).
(ii) Since only real calculations are needed, use of
microprocessors using elementary codes is possible.
(iii) Symmetries of the impedance function Zero) or Z(Q) of
electrical systems enable further reduction of the computational
burden. The calculation of several bus voltages can be done using
Hartley Transforms. Before proceeding to examine the use of DHT III
electric power system problems, it is helpful to discuss the use of DFT
(Discrete Fourier Transform) in the solution of electric power system
problems. This problem is represented by the time domain equation
v(t)=z(t).i(t) (51)
In effect, gIven i(t) and the impulse or the impedance function z(t)
we want to calculate vet). This is accomplished by evaluating the
DFT of i(t) with k as an integer by sampling the current as given by
I(kro) <=> i(t) (52)
Next the DFT of z(t) is calculated as:
Z(km)¢:> z(t) (53)
Subsequently these two transform quantities are multiplied and
then the inverse Fast Fourier Transform is taken. The DFT of z(t)
deserves special attention since it is not available generally.
Instead, samples of the band limited Z( ro) or Zen) are used.
are found by the following translational techniques :
Z(kQ) for k<N /2
Z'(kn)= {Z«kN)Q).for k>N/2
These
(54)
85
On the right hand side of Eq.54 Z refers to sampled values of Z(n).
On the left hand side, Z' refers to the DFf which is used. At k=NI2
In Eq.54 it is to be remembered that a real value is required in order
to comply with the properties of the DFf. A convenient selection is
made on the basis that there happens to be negligible signal energy
at n rnax =1t/T s. Thus,
Z'(NI2)=O (55)
Then this discrete Fourier sampled function Z'(kn) can be
transformed back to the Hartley domain by the following transform
formula as :
Z(lev)= [Re{ Z(kn)}  Im{Z(kn)}] (56)
A salient property of the Hartley Transform for the application
of power engineering problem which would be discussed next is that
the convolution in time domain equation
VCt)= Z ® i(t) (57)
becomes a sum of products in the following Hartley transformed
frequency domain equation as:
I
V(v)=2 [Z(v)I(v) + Z(v)I(v) + Z(v)I(v)  Z(v)I(v)] (580
Thus it is possible to solve a certain class of electric circuit problems
using Hartley transforms by converting the convolution operation to
simple real products, just as Ohm's law provides the solution in the
frequency domain.
A zero padding technique can be used in this type of situation.
A simple explanation is given to illustrate the point. Suppose M
samples of a continuous signal is given and it is required to find an
approximation of the Fourier Transform ( same is true for the
86
Hartley Tnmsfonn) at N frequency points" where N > J\.t. It is
assumed that the paramNers M and T satisfy the constrains in
Eq~(582) and Bq.(583)
J
T ~ 2ib
2ih
N$ .M (583)
(where fn is tbe hlghest frequency content in the signal in hertz, ~ f
is the spacing between the frequency sample.s. and T is the sampling
time)., S,fi that the N point DFf ( or DHT) is still meaningful. But we
have gol fewer samples than required by DFT. In this case NM
zeroes are appended 'to the end of tlle sample sequence to get an Npoint
seqaence'. Then the Npoint DFf ( true for DHT as wen) of the
new sequence will yield the desired N frequency samples of the
signal. To show that tbis result is correct , forming the Npoint DFT
of the spectrum sequence xk r 12] as,
Nl
F(f)f= L,xkei2pnk/N
k=O
MJ
= LXkei2pnk/N +
k=O
M 1
Nl
LXkei2pnk/N
k=M
= LXkei2pnk/N n=O,l, 2, ............. N1
k=O
(584)
And here Xk=O, for k=M, M+l. ........ Nl. The above criteria is true for
Hartley Transform too. Thus this zero padding technique can be
used to improve the frequency resolution of the discrete Hartley
Transformed spectrum.
87
Analyses
It is assumed that due to the presence of a power electronic
load such as a solid state rectifier at a certain bus I a nonsinusoidal
current Il(v) is injected. Now it is desired to realize the voltage
generated by this current at a connected bus k. U sing the Hartley
transform. the bus voltage Vk(V) can be calculated. I1(v) and Vk(v)
are Hartley transformed current and voltage. To calculate the Bus
voltage, the Hartley transformed impedance is needed and that is
readily found [3] as
Z(v)=[Re{Z(ro)}  Im{Z(ro)}]ro=v
Z( ro) is easily found for the buses k and 1 from using
Z( ro)=[U jroZSE(ro )]CSH(ro)
Where U is the identity matrix and CSH is a constant matrix
representing the capacitance of the SH network and
I
(59)
(60)
(61)
To use this approximation given by Eq.60 the network can be
decomposed into series network labelled SE (consists of resistive and
inductive element) and a shunt network labelled SH ( consists of
capacitative element) as shown in Fig.I7. The SE network consists
entirely of RL circuits joining the system busses and the SH network
is the shunt capacitive portion of the network. Together, the SE and
SH networks comprise the entire system and the two subnetworks,
SE and SH which are in parallel.
In the following, a brief account is discussed for the property of
two parallel nport networks. And further is given to the network
88
r, Substation : SE Netwo~ :
Transformer. •
I I
IIJIIIII '":'     .  .
SH Network
• ' I
I
I
I •
"
Figure1? Distribution network decomposed into SE and
SH networks. [5]
89
consisting of n+ 1 buses with voltage measured at each bus from a
given reference bus as vI (t), V2(t), ... vn(t). Also consider the
injection currents at these buses with return path through the
reference bus as il (t), i 2(t), ... in(t). Under most practical
circumstances, these voltages and currents can be subjected to two
sided Fourier transforms as shown by Eq.62 :
00
(62)
00
Again considering the electrical network to be decomposed into two
networks labelled as SE and SH which are characterized as follows:
the SE network contains only resistances and inductances and
therefore this network is modelled as follows:
d
vSE(t)=RsEisE(t) + LSE d7SE (t) (63)
where VSE(t) and isE(t) are n dimensional vectors of the bus voltages
and currents using one system bus as reference. Similarly, let the
SH network contain only capacitances. The SH network is modelled
as given by Eq.64 as follows:
d
iSH(t)=GSH·VSH(t)+ CSH dt VSH(t) (64)
The terms RSE, LSE, GSH, CSH are n by n matrices. Taking Fourier
transforms on Eq.63 and Eq.64, we obtain
VSE(ro) = RSE iSE(ro) + jroLSEiSE(ro)
iSH(co) = GSHvSH(co) +jro CSHvSH(ro)
(65)
(66)
By further manipulation, the Fourier transforms of (65) and (66) are
obtained as
90
....
vSE(OO)
ZSE(OO) = ISE(OO) =RSE +jroLSE
and Z (00) = vSH(CO) I
SH ISH(ro) (GSH +jeoCSH)
In most cases GSH is nearly equal to zero.
practical cases we have
1
ZSH( ro) = . ,.JI
JlIJ\..,SH
therefore for most
(67)
(68)
(69)
Fig.I8 shows several parallel n port circuits. Labelling the circuits
as 1, 2, 3, .... , the ndimensional vectors vk(t), Ikct} denote the time
function voltages and currents with a common reference bus for
circuit k for k= 1,2,3....... In this context the term parallel means
VI (t)= V2(t)= V3 (t) = .......... (70)
and the total supply current I(t} in Fig.IS is the sum of all currents,
n
I(t}= ~)k(t} (71)
k=1
I(t} is an n dimensional vector. It IS assumed that both the bus
impedance and admittance matrices [1] are known for each nport.
Then the Fourier transform of Ohm's law applied to the ith circuit at
each circuit is of the form
. . .
11(00)= yl(OO) VI(oo) (72)
where yl( ro) IS the admittance matrix of the ith nport and the two
sided Fourier transform is used. Substitution of li(w) in Eq.71 gives
n
I ( ro )= L y k ( 00 ) V k ( 00 )
k=l
(73)
All circuits are parallel and V(oo) is used to denote the transform of
the voltage vector at any point of these circuits. Thus
91
92
,. Parallel nports ..
1 2 3
bus 2
bus n
V2
\/l &...._..1___________ reference bus
Figure18. Parallel nport circuits [5]
n
1 = 1(0)) = " k
Z (m) Vk() [ .LY (m)]
e.q 0). k= l
and it is convenient to define as:
D
Zeq(m}= L[yk(ro)]l
k=l
It is considered now the case· of two nports labelled SE and SH.
The SE circuit is characterized by the resistive and inductive
branches of an electric power network, and SH circuit is
characterized by the shunt capacitive hranches (either charging
(74)
(75)
capacitances or shunt capacitors). In Fig.19 the diagram of a typical
example of such a decomposition is shown. The SE and SH networks
are in parallel. From Eq.75 we have
1
By manipulating Eq.75, it can be shown as:
1
ZSE(W)ZSH(W)
 ZSE(W)+ZSH(w)
By substituting (60) into (59) it is found that
Z(v) = [Re{ ZSE(w)} Re{jOJZSECSHZSE(OJ)} 
Im{ZSE(w ) + Im(jro ZSE(ro) CSHZSE(ro) }]OJ=v
(76)
(77)
(78)
93
Figure 19. Decomposition of a power network into SH and SE
network [5]
..
Further manipulation [10] leads to the expression given by the Eq. 78
as
Z(v)= RSE + [ XSE(ro) +ro(Re{ZSE(m)CSHZSE(ro)} +
Im{ZSE(ro)CSHZSE(ro)} ]ro=v (79)
The element Zero) is the Fourier transform of z(t) and the element
Zkl(v) is used in Eq.581 to calculate Vk(V) given II(V). To sample
IICV) on the lth
bus the Fast or Discrete Hartley Transform IS used.
Discussion
Analysis of power system circuits are subject to several
restrictions. For example, the circuits under analyses are all three
phase balanced fixed series RL branches and frequency independent
(R# R(ro» . The importance of frequency independence should not be
underestimated particularly for the cases in which significant energy
components of the injection current lie above 1 7th Harmonic of 60
Hz. Distributed parameter models are readily substituted by
lumped parameter models for the convenience of analyses. The
injected current at the particular bus is assumed to be in phase with
the line to neutral voltage. And the nonlinearity was assumed to be
in a source.
95
CHAPTER V
COMPUTER SOLlmON
Introduction
The application of Hartley transforms for analyzing nonsinusoidal
situations in power systems is illustrated by means of a
threebus network. Electrical power quality assessment often
involves calculation of bus voltage or line current
based on several assumptions as listed below:
This work is
(1) Only one phase ( such as phase A) is involved in calculation.
(2) The injected non sinusoidal current as shown in Fig.20 has a
stepped wave form. Because of symmetry conditions, this
waveform has only odd harmonics. The source of this load
current is a six pulse rectifier which is commonly used in power
systems.
(3) Importance is directed only on the non sinusoidal current drawn
by the rectifier and the voltage generated by it on a particular
bus of interest.
Figure.21 for the
and inductor LSE(w)
RSE+jwLSE( ro)
ZSE(w) 3
The equivalent delta network as shown in
convenience of analysis. The resistance RS E
together comprises the series impedance
and the capacitor CSH(ro) forms the shunt
1
96
i:al
.~ +oJ
o IJ . UlJ~:.a
1
I
O.U lliti
" ~c,oFlds
I
U.U::·1 'J
Figure20. Non sinusoidal rectifier injected current waveform [9]
97
I
U.llJ:.a::
The parallel combination of ZSE( ro) and Z SH (ro) 1. S t h e eqU.I valent
impedance Zeq(w). Mathematically, Ze (ro)= ZSE(W)ZSH(ro)
q (ZSE(ro)+ZSH(ro))' which
can be written for n harmonics as ,
Z ( )_ ZSE(nro)ZSH(nro)
eq nro  (ZSE(nro)+ZSH( nro))
(83)
n is also an index number which can assume values from 1 to 64 in
the following analysis and ro being the angular frequency (ro =21tf and
f=60 Hz).
Furthermore if Re{Zeq(nro)} and Im{Zeq(nro)} are the real and
imaginary part of Zeq(nro) then the Hartley impedance 1S given by
ZHT(nro)=~ Re {Zeq(nw)} 2+Im {Zeq(nw)2} (84)
• _ • ." ... ..... L ~ 40 ••
Procedure
In the following, the steps taken to implement the program are
discussed. The capacitances between bus 12, bus 23 and bus 31
are denoted by C12. C23, C31 with numerical values of 0.0026,
0.0026 and 0.0026 farad respectively in the computer program.
Similarly the resistances (R) and inductances (X) for the lines
connecting buses 12, 23 and 31 are 0.196 ohms and 0.00052
The network is a delta one and it i.s broken up
henry respectively.
into equivalent star one, as shown in Fig. 22 for convenience of
analyses. The nonsinusoidal current waveform is made up of a
Hartley series and a short description of its derivation is discussed m
the following. As the six pulse waveform passes through the origin
so it is an odd one in characteristics and the Hartley coefficient is
evaluated in the following .
98
toJ.
~h .. ~ ,....
r:n:
==
Figure21. Three bus power system
99
100
1
~(w)
Figure22. The equivalent Star network
Proceeding for evaluation of Hartley Coefficient Bn by
integration we have,
2 T
Bn=,. jf(t).Casncot.dt
o
2 T =r jf(t).(Cosnwt + Sinnwt).dt
o
T
=,2. ff( t).(f2).(..J1 2·Cosncot + 1j 2.Sinncot).dt
o
2{2 T
Bn=T jf(t).(Sin(1t/4 ).Cosnrot +Cos(1t/4 ).Sinnwt).dt
o
[By trigonometric identity (Sino Cos~+Sinp.CosB)=Sin(B+p), here~=1t/4
and o=noot)]
2fi T
Bn=T jf(t).Sin(nrot+1t/4 ).dt
o
2{2 T/6
=T[ JO.5Sin(nwt+1t/4).dt+
o
T/3 T12
J 1.Sin(noot+1t/4 ).dt+ JO.5 .Sin(noot+1t/4) .dt+
T/6 T/3
2T/3 ST/6 T f (0.5) .Sin(nwt+1t/4) .dt+ J(l). Sin(noot+1t/4). dt+ J(O .5). Sine noot+1t/4 ) .dt
T/2 2T/3 ST/6
]
2fi T/6 (1) T/3 0.5
=T[ (0.5/nro){Cos(nrot+1t/4)}o +n{roC os(nrot+1t/4)}[/6 +( nro )
T/2 (0.5) 2T/3 ( 1) ST/6
{Cos(nrot+1t/4)}Y/3+ {Cos(noot+1t/4)}Y12 +( ) {Cos(nrot+1t/4) hT/3+ nro nco
(0.5) T
( ) {Cos(nrot+1t/4)} 5T/6]
nro
101
2{2
=T [O.S {Cos(nroT/6+7t/4 )Cos(7t/4)} 1 {Cos(nroT/3+7t/4)nw
Cos(nroT/6+7t/4) }O.S {Cos(nroT/2+7t/4)
Cos(nroT/3+7t/4)} +O.S {Cos(2nwT/3+1t/4)Cos(
nroT/2+1t/4)}+ 1 {Cos(SnroT/6+1t/4 )Cos(2nwT/3+1t/4)}+
O.S {Cos(nroT +1t/4 )Cos(SnroT/6+7t/4)}]
{2
=mtfT[ O.S {Cos(2D1tfT/6+1t/4 )Cos(7t/4)}  {Cos(2D1tIT/3+1t/4)
Cos(2n7tfT/6+7t/4) }O.S{ Cos(2n1tfT/2+7t/4)
Cos(2n1tIT/3+1t/4) }+O.S {Cos( 4n1tIT/3+1t/4)
Cos(2n1tIT 12+1t/4)} + 1 {Cos( 1 On1tfT /6+1t/4)
Cos( 4n1tIT/3+1t/4) }+O.S {Cos(21tnfT +1t/4 )Cos(l On1tfT/6+1t/4)}]
{2
=f[O.S Cos(2n1tfT /6+1t/4 )+0.SCos(1t/4 )0 .SCos(2n1tfT 13+1t/4)n1t
T
Cos(2n1tff/2+7t/4 )+O.SCos( 401tIT/3+1t14 )+O.SCos( 1 On1tff/6+1t/4)
+0.SCos(2n1tfT +1t/4)]
fi =[0.SCos(n1t/3+1t/4 )+0.SCos(1t/4 )0.5Cos(2n
n1t
1t/3+1t/4 )Cos( n7t+1t/4 )+O.SCos( 4n1t/3+1t/4 )+0.5Cos(Sn1t/3+1t/4)+
O. 5Cos(2n1t+7t/4)]
fi Bn=2n1t [Cos(n1t/3+1t/4 )+Cos(1t/4 )Cos(2n1t/3+1t/4)2Cos(n1t+1t/4)
Cos( 4n1t/3+7t/4 )+Cos(5n1t/3+1t/4 )+Cos(2n1t+1t/4)]
(Here T=O.0166sec & IT=I)
+ +
(85)
From Bn the positive current sequence In ( In this case, In = Bn) is
obtained. In order to construct the Hartley current In, both positive
+ 
sequence current I n and negative sequence current In are needed.
102
This aforementioned sequence 1+n can be obtained by substituting
n=l, 2, 3, 4, 5 ..... 63, 64 etc, on the right hand side of eq.85 ( I~ =0. 955,
1+ + + + + + 2=0.000, 13=0.000, 14=0.000, 15=0.191. ...... 163=0.000, 164=0.000 etc)
as shown in the results of computer program III the appendix. Due
to typical odd symmetry of the current waveform( absence of cosine
coefficients in the sequence in case of Fourier series expansion) the
aforementioned positive current sequence I: is arranged in reverse
order to obtain the negative sequence ( such as for n=1,2, 3, 4.,5 ... 63,
64; I~ =0.000, r;=O.OOO, ~=0.016, I~=O.OOO, 1;=0.016, ......

I~3=0.955, 1~4=O.000, etc.) ~. The subscript n indicates that these
currents are function of n, which also represents the order of
+ 
harmonics. These current sequences I n and In are denoted in the
computer program as EA(.) and ERA(.) respectively. Moreover If we
gl. ve t h e terml.l lO 1o gy 'Tne ven & Iond d to t h e even an d 0 dd part 0 f t h e
Hartley current,
even
then In = 2
respectively. These two sequences are used to construct the
Hartley current In by eq.86 as
 +
4
+ 
(I 1 )2
n n
4 (86)
103
Next the Hartley transformed impedances ZHT(nro) are calculated.
In the computer program the Hardey coefficient and currents are
represented by B(n) and H(n) respectively. The harmonic voltage
response in frequency domain is V n in volts at bus 2 due to the
nonsinusoidal rectifier current and index number n as shown in a
graph in appendix.
appendix .
The program and the results are
Thus, in phasor form
+  +
(In+1n) (InIn)
2 j. 2
0: +I~)2 (1+ _()2
n n
 +
(In I n)
= 4 + 4
L Tan 1
 +
(In+In)
. .
glven 10
Where magnitude of Hartley current at nth harmonic 1S,
and the phase angle
[ _(I~dd)]
=Tan 1Ie
yen
n
(87)
(88)
(89)
(90)
, Now by using Eq.91 the harmonic voltages are calculated as follows,
Vn=[ (l~ven)2 + (I°ndd)2)] .[ ZHT(nco)]. (91)
104
Here n=O, 1. 2. . ................. 64 etc. In the computer simulation the
combination is arranged sllch that positive sequence is EA(n) and
the corresponding negative sequence is ERA(64n). And in the
even odd
computer program V n , ZHT{nm) , ~ • In and are represented by
V(n), ZHT(n). EA(n)+E:A(64n), EA(n)E~A(64n) respectively.
Now the quantities VOl' V(5), V(7). VOl). V(13). V(l7}, V(19) are
regarded as fundamental,. 5th, 7th, 11th. 13th, 17th and 19th
harmonic voltages respectively. Furthermore the harmonic
voltages V(2). V(3), V(4), V(6), V(8)~ V(9), V(W). V(12). V(J4), V(l5).
V (16), V (] 8) are found to be equal to zero. The Total Harmonic
Distortion (T.H.D) at the 2nd bus due to the nonsinusoidal rectifier
current is given by.
T.H.~~~
V(l)
(92)
In the computer program the 1st, 2nd, 3rd, 4th, 5th, 6th, 7th, 8th,
9th, 10th, 11th, 12th, 13th, 14th, 15th, 16th, 17th, 18th, 19th
harmonic voltages are denoted by V(n) for n=l •.. 19 respectively. It
has been found that the harmonic voltages decrease rapidly and the
numerical contribution of voltages after the nineteenth harmonic is
insignificant in calculation of THD and so the analysis is terminated
after 19th harmonic. The results of the computer program to
calculate the total harmonic distortion (T.H.D) is given in appendix.
The plot showing harmonic voltages versus the index number n is
105
106
also presented in the appendix. The algorithm of the computer
program is shown m Table VIII.
"."
l
STEP
1.
2.
3.
4.
TABLE VIII
ALGORITHM SUMMARY
DESCRIPTION
The network IS transformed from delta to its equivalent
star network. Now for n=1 to 64 the different values
of Hartley impedances ZHT(n) are calculated. This has
been done from line number! to 22 in the computer
program.
Next the Hartley coefficient B(n) is calculated by Eq.85
and this has been shown in the program from line
number 25 to 37. Subsequently the Hartley current IS
constructed by Eq.86 and it is shown from line number
38 to 98 in the computer simulation.
Next the Hartley domain impedances ZHT(n) are
multiplied by currents H(n) to get the Harmonic voltage
V(n) by eq.91 for n= 1, .. 19.; which is obvious from line
number 100 to 103.
Now using Eq.92 and the harmonic voltages the T.H.D is
calculated as seen from line number 1 04 to 109. It has
been found that the contribution of harmonic voltages
for calculation of Total Harmonic Di storti on after 19th is
too small to account for. So for calculation of THD only
the voltages up to 19th harmonic is included.
...
107
)
CHAPTER VI
SUM.MARY AND CONCLUSIONS
The mam conclusion of this study is that the Hartley transform
offers a computationally efficient procedure for the calculation of the
time response of voltage or current waveform. The computational
advantages of the HT (Hartley Transform) comparable to the FT (
Fourier Transform) are listed below:
(1) The HT is purely real but FT is typically complex In nature
(contains imaginary numbers).
(2)There is however a computational advantage of favoring HT
over FT is that, HT can transform one real array of length of N data
points in half the time that it takes the FFf to process a
complex array of [10] same length of N data points.
(3) The Hartley methodology exhibits the same ease and
convenience in calculating the inverse transform as in the
Fourier approach. The Laplace transform does not offer
convenience in calculation of numerical inversion techniques
(such as conversion from frequency to time domain and viceversa).
(4) The Hartley methodology exhibits the same accuracy In
calculation as to that of Fourier methodology.
108
(5) The time required to compute the discrete voltage v(k) usmg
FHT is proportional to N2. Here N is the number of samples
which is an integer (an integral power of 2). If anyone
computes the transforms of impedance z and current i, performs
the complex multiplication and then computes the inverse
transform of v(w), it is found that time required for the
computation is proportional to NLog2N if FT is [6] implemented.
(6) It is imperative that there are some potential pitfalls ( rather
errors) in this method and they are identical to those
encountered in calculation of FT such as time domain aliasing,
time domain smoothing, Picket fence effect and Leakage.
Research is in progress to limit the aforementioned errors to a
mlOlmum.
(7) The fast Hartley transform has been proposed for any electric
circuit calculation involving convolution in time. The method IS
especially applicable in cases in which frequency band
limitations occur. Such is the case in electric power systems In
which power electronic loads cause non sinusoidal load currents
and bus voltages to occur.
109
(8) Eventually there are some interesting areas which are suggested fOJ
future research :
(i) A full quantitative assessment and error analysis for real
transform solutions of circuit problems.
(ii) A study of novel and fast methods to calculate Z(kv) to avoid
the problems of band limitations.
(iii) A serious application III which real time solutions are required.
I 10
BffiLIOGRAPHY
I. Arden, and Bruce W. Numerical AI~orithms: ori~ins and
applications 1970.
2. Bracewell, R. N. The Hartley Transforms England: Oxford
University press, 1985.
3. Buneman, O. Conversion of FFf's to fast Hartley transforms
SIAM J. Sci, Stat. Computation, Vol. 7, No.2, April
1986.
4. Heydt, G. T. Computer Analysis Methods For Power Systems
Mcmillan Publishing Company, Newyork, 1986.
5. Heydt, G. T. A New Method for the calculation of
Subtransmission and Distribution System Transients
Based on the FFf IEEE Trans. on Power Delivery,
v. PWRD4, No.3, July 1989, pp. 18691875.
6. Heydt, G. T. The Fast Hartley Transform Used in the Analysis of
Electrical Transients in Power Systems CH2868
8/90/00001813 , 1990, IEEE.
7. Heydt, G. T. Electric Power Quality Circle Publication, West
Lafyette, Indiana, 1991.
8. Heydt, G. T. and Olejniczak J. K. The Hartley series and It's
Application to Power Quality Assesment
9. Heydt, G. T. and Olejniczak J. K. SIMULATION OF LINEAR TIMEINV
ARIA NT SYSTEMS USING THE FAST HARlLEY
111
TRANSFORM National Science Foundation, ECS
8920781.
10. Heydt, G. T, and others. Application of the Hartley Transform for
the Analyses of the Propagation of Nonsinusoidal
waveforms in Power Systems IEEE Transactions on
power Delivery, Vol. 6, No. 4, October 1991.
11. Heydt, G. T. Electric Power Ouality in the 1990's Purdue
University, West Lafayette, IN 47907, IASTED91.
12. Jong, M. T Methods of discrete signal and system analysis
Mcgraw Hill Book company, 1982.
13. Martzloff, Francis D. and Gruzs M. Thomas. Power Quality Site
Surveys Facts. Friction. and Fallacies IEEE
Transactions on Industry Applications, Vol. 24, No.6,
Novemberl December 1988.
14. Nash, Hugh O. and Wells M. Francis. Power Systems Disturbances
and Considerations for Power ConditioninG IEEE
Transactions on Industry applications, Vol. IA21,
No.6, November/December.
15. National Electrical Contractors Association ELECTRICAL DESIGN
LIBRARY 1991, 302574, 7315 Wisconsin Avenue,
Bethesda, Maryland 20814.
16. Oberembet, Jacqueline A. and Hasan R. Abdul. THE PROBLEMS OF
POWER QUALITY AND THEIR EFFECfS ON THE ELECTRIC
UTILITY AND THE ENDUSER ,23rd Annual frontiers of
Power conference, Stillwater,1990.
I 12
•
113
17. Smith, Charles 1. Power Quality: End Users Impacts Electrotek
Concepts, Inc. Knoxville, Tn, Energy Engineering,
Vol. 88, No.5, 1991.
APPENDIX
COMPUTER PROGRAM OF APPLICATION OF
HARTLEY TRANSFORM FOR POWER
QUALITY ANALYSES
114
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C**********************************************************************
C ** THESIS PROGRAM
C ** NAME OF PROGRAMMER: PARAN JYOTI MAHANTA.
C ** TYPE OF COMPUTER LANGUAGE: VAX FORTRAN.
C** EA(.) IS THE POSITIVE SEQUENCE OF HARTLEY CURRENT.
C** ERA(.) IS THE NEGATIVE SEQUENCE OF HARTLEY CURRENT.
C** ZHT(.) ARE THE HARTLEY TRANSFORMED IMPEDANCES.
C** B(.) IS THE HARTLEY COEFFICIENT.
C** H(.) IS THE HARTLEY CURRENT.
C** V(.) ARE THE THE HARMONIC VOLTAGES.
C** n IS AN INDEX NUMBER.
C** THD IS THE TOTAL HARMONIC DISTORTION.
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1 COMPLEX CMPLX,ZSE(64),ZSH(64),ZEQ(64)
2 DOUBLE PRECISION R12,X12,C12
DIMENSIONRR1(64),RR2(64),RR3(64),RR4(64),RR5(64),ERA(64)
DIMENSION RR6(64),BA(64),EA(64),ZHT(64),ZR(64),ZI(64)
DIMENSION V(64),H(64)
6 PI=3.14159265
7 W=2*PI*60
8 R12=O.196
9 X12=O.OOO52
10 C12=O.OO26
11 WRITE(* ,12)
12 FORMAT(2X,'n' ,7X, 'ZR(n)', lOX, ' ZI(n)' ,8X,ZlIT(n)')
13 DO 22 n=1,64
14 ZSE(n)=(1.13)*CMPLX(R12,n*W*X12)
115
15 ZSH(n)=(l.l(n*C12**W»*(l.lCMPLX(0.O, I.0))
16 ZEQ(n)=l./(l.l(ZSE(n»+l.J(ZSH(n»))
17 ZR(n)=REAL(ZEQ(n»
18 ZI(n)=AIMAG(Z(EQ(n»
19 ZHT(n)=SQRT(ZR(n)**2+ZI(n)**2)
20 WRITE(* ,21 )n,ZR(n),ZI(n),ZHT(n)
21 FORMAT(2X,I3,3X,F10.6,3X,FI0.6,3X,F10.6)
22 CONTINUE
23 WRITE(*,24)
24 FORMAT(4X,' n',5X,'EA(n)')
25 DO 37 n=1,64
26 RR1(n)=COS«n*PI)/3+(PI/4))
27 RR2(n)=COS«n*PI*2)/3+(PIJ4»
28 RR3(n)=COS«n*PI)+(PII4»
29 RR4(n)=COS«n*5*PI)/3+(PII4»
30 RR5(n)=COS«n*4*PI)/3+(PIJ4»
31 RR6(n)=COS«n*2*PI)+(PII4»
32 B(n)=( 1.4142/(2 *PI*n»*(COS(PII4 )+RRI (n)RR2(n)
33 12*RR3(n)+RR4(n)RR5(n)+RR6(n»
34 EA(n)=B(n)
35 WRITE(* ,36)n,EA(n)
36 FORMAT(2X,I3,3X,F5.3)
37 CONTINUE
38 EA(1)=0.955
39 ZHT(1)=O.065200
40 ERA(63)=O.955
41 EA(2)=O.OOO
1 16
II 7
42 ZHT(2)=0.064804
43 ERA( 62)=0.000
44 EA(3)=0.000
45 ZHT(3)=0.064160
46 ERA(61)=0.000
47 EA(4)=0.000
48 ZHT( 4 )=0.063290
49 ERA(60)=0.000
50 EA(5)=0.191
51 ZHT(5)=0.062222
52 ERA(59)=0.191
53 EA(6)=0.000
54 ZHT(6)=0.060986
55 ERA(58)=0.OOO
56 EA(7)=O.136
57 ZHT(7)=0.059617
58 ERA(57)=0.136
59 EA(8)=0.OOO
60 ZHT(8)=0.058147
61 ERA(56)=O.000
62 EA(9)=0.OOO
63 ZHT(9)=O.056605
64 ERA(55)=0.OOO
65 EA(lO)=O.OOO
66 ZHT(10)=0.055019
67 ERA(54)=0.000
68 EA(11)=O.087
1 18
69 ZHT(11)=0.053412
70 ERA(53)=0.087
71 EA( 12)=0.000
72 ZHT(12)=0.051804
73 ERA(52 )=0.000
74 EA(l3)=0.073
75 ZHT(13)=0.050211
76 ERA(51)=O.073
77 EA(14)=0.OOO
78 ZHT(l4)=0.048646
79 ERA(50)=0.OOO
80 EA(l5)=0.00O
81 ZHT(15)=0.047117
82 ERA( 49)=0.000
83 EA(l6)=0.000
84 ZHT(l6)=O.045633
85 ERA(48)=O.000
86 EA(l7)=0.056
87 ZHT(l7)=0.044197
88 ERA(47)=0.056
89 EA(l8)=0.00O
90 ZHT(l8)=O.042813
91 ERA( 46)=0.000
92 EA(l9)=0.050
93 ZHT(l9)=O.041483
94 ERA( 45)=0.050
95 WRITE(* ,96)
96 FORMAT(5X,'n' ,5X,'V(n)')
97 DO 103 n=I,19
98 H(n)=SQRT«EA(n)+ERA(64n))12)**2+«EA(n)
99 lERA(64n))/2)**2)
100 V(n)=H(n)*ZHT(n)
101 WRITE(*, 1 02)n, V (n)
102 FORMAT(3X,I3,4X,F8.6)
103 CONTINUE
104 WRITE(*,105)
105 FORMA T( 4X, 'THD')
106 THD=SQRT(V(5)**2+V(7)**2+V(11)**2+V(13)**2+
107 IV(17)**2+V(l9)**2)/V(1)
108 WRITE ( * , 1 09 )THD
1 09 FORMA T( 4X,F8.5)
110 STOP
END
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RESULTS OF THE COMPUTER PROGRAM
n ZR(n) ZI(n) ZHT(n)
1 0.065067 0.004167 0.065200
2 0.064279 0.008233 0.064804
3 0.063008 0.012105 0.064160
4 0.061310 0.015705 0.063290
5 0.059258 0.018974 0.062222
6 0.056929 0.021874 0.060986
7 0.054402 0.024387 0.059617
8 0.051751 0.026512 0.058147
9 0.049043 0.028265 0.056605
10 0.046333 0.029671 0.055019
11 0.043666 0.030759 0.053412
12 0.041076 0.031566 0.051804
13 0.038589 0.032125 0.050211
14 0.036220 0.032473 0.048646
15 0.033980 0.032640 0.047117
16 0.031873 0.032657 0.045633
17 0.029899 0.032549 0.044197
18 0.028056 0.032340 0.042813
19 0.026340 0.032048 0.041483
20 0.024744 0.031691 0.040207
21 0.023263 0.031284 0.038985
22 0.021888 0.030837 0.037816
23 0.020614 0.030362 0.036698
•
24 0.019432 0.029866 0.0.35631
25 0.0.18336 0..0.29356 0.0.34612
26 0..0.17320. 0..028837 0..0.33639
27 0..016376 0.028315 0..0.3270.9
28 0.0.1550.0. 0..0.27792 0..0.31822
29 0.0.14685 0..0.27272 0..0.30.975
3D 0.0.13928 0..0.26758 0..0.30.166
31 0..0.13223 0..0.26250. 0..0.29392
32 0..0.12566 0..0.25750. 0..0.28652
33 0..0.11953 0..0.25260 0..0.27945
34 0..0.11381 0..0.24780. 0..0.27268
35 0..0.10.846 0..0.24310. 0..0.26620.
36 0.010.346 0..0.23852 0..0.25999
37 0..0.0.9878 0..0.2340.5 0..0.2540.4
38 0..0.0.9439 0..0.22969 0..024833
39 0.00.90.27 0.022545 0.0.24285
40. 0..0.0.8640. 0.0.22132 0..0.23759
41 0..0.0.8277 0..021731 0..023254
42 0.0.0.7935 0..0.21341 0..0.22768
43 0.0.0.7612 0..0.20.962 0..0.22301
44 0..0.0.7309 0..0.20593 0.0.21852
45 0..00.70.22 0..0.20.235 0..021419
46 0.0.06751 0..019887 0.0210.02
47 0.0.0.6495 0..0.19549 0..0.20.600
48 0..0.0.6253 0..0.19220. 0..0.20.212
49 0..0.0.60.24 0..0.1890.1 0.0.19838
50. 0.0.0.580.6 0..0.18591 0..0.19477
51 0.005600 0.018290 0.019128
52 0.005404 0.017997 0.018791
53 0.005219 0.017712 0.018465
54 0.005042 0.017435 0.018149
55 0.004874 0.017166 0.017844
56 0.004714 0.016904 0.017549
57 0.004561 0.016649 0.017263
58 0.004416 0.016401 0.016985
59 0.004277 0.016160 0.016716
60 0.004145 0.015925 0.016456
61 0.004018 0.015696 0.016203
62 0.003897 0.015474 0.015957
63 0.003782 0.015257 0.015718
64 0.003671 0.015045 0.015487
n EA(n)
1 0.955
( 2 0.000
3 0.000
4 0.000
5 0.191
6 0.000
7 0.136
8 0.000
9 0.000
10 0.000
11 0.087
12 0.000
13 0.073
14 0.000
15 0.000
16 0.000
17 0.056
18 0.000
19 0.050
20 0.000
21 0.000
22 0.000
23 0.042
24 0.000
25 0.038
26 0.000
27 0.000
28 0.000
29 0.033
30 0.000
31 0.031
32 0.000
33 0.000
34 0.000
35 0.027
36 0.000
37 0.026
.,.
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38 0.000
39 0.000
40 0.000
41 0.023
42 0.000
43 0.022
44 0.000
45 0.000
46 0.000
47 0.020
48 0.000
49 0.019
50 0.000
51 0.000
52 0.000
53 0.018
54 0.000
55 0.017
56 0.000
57 0.000
58 0.000
59 0.016
60 0.000
61 0.016
62 0.000
63 0.000
64 0.000
125
n V(n)
1 0.094065
2 0.000000
3 0.000000
4 0.000000
5 0.093425
6 0.000000
7 0.028761
8 0.000000
9 0.000000
10 0.000000
11 0.009252
12 0.000000
13 0.006308
14 0.000000
15 0.000000
16 0.000000
17 0.003552
18 0.000000
19 0.002806
THO
1.04710
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126
TABLE IX
HARMONIC VOLTAGES VERSUS HARMONIC ORDER
Harmonic Order (n) Harmonic Voltages V(n)
1 0.094065
2 0.000000
3 0.000000
4 0.000000
5 0.093425
6 0.000000
7 0.028761
8 0.000000
9 0.000000
10 0.000000
11 0.009253
12 0.000000
13 0.006308
14 0.000000
15 0.000000
16 0.000000
17 0.003552
18 0.000000
19 0.002806
Total Harmonic Distortion (T. H. D)= 104.71 %
=• 0 •
.5 c >
I
& "0
>
0 i
!• ::
0.10
0.08
0.06
0.04
1 2 3 .. 5 678 D1(1'1:1a'HH1~1f1g
Harmonic Order (n)
Figure23. Plot of Harmonic voltages
127
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VITA
Paran Jyoti Mahanta
Candidate for the Degree of
Master of Science
Thesis: A THEORETICAL STUDY OF THE APPLICATION OF HARTLEY
TRANSFORM FOR POWER QUALITY ANALYSES
Major Field: General Engineering
Biographical:
Personal Data: Born in Gauhati, Assam, India, January
21, 1959, the son of Mr Rama Nanda Mahanta and Mrs
Dipty Mahanta.
Education: Graduated from Cotton Collegiate H.S School,
Gauhati, India, in 1974~ received Bachelor of Science in
Physics from Gauhati University, Gauhati, India, in
August, 1978; received Bachelor of Engineering from
lorhat Engineering College, Jorhat, India, in August, 1985;
completed requirements for the Master of Science degree
at Oklahoma State University in May, 1993.
Professional Experience: Graduate Teaching Assistant,
Department of General Engineering, Oklahoma state
University, February, 1991 to December, 1991.