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From
Dr.Amiya Prasad Dash
Reader in Physics, G.M.Autonomous College
Sambalpur 768004
Sambalpur
Dated:
CERTIFICATE
This is to certify that the thesis entitled "Analysis of Transformer core behavior
under transient conditions for the study of series ferroresonant circuits" being
submitted by Mrs.Pratima Rath for the award of the degree of Doctor of Philosophy
in Physics of the Sambalpur University,Burla,Sambalpur,Orissa is a record of
bonafide research work carried out by her for more than eight years in
G.M.Autonomous College,Sambalpur and University College of Engineering,Burla,
Orissa and the thesis has reached the standard fulfilling the requirements of the
regulation relating to the degree. No part of this thesis has been submitted to any
institution or university for the award of any degree or diploma.
(Amiya Prasad Dash)
2
CONTENTS
Page No
4
5-7
Acknowledgement
Synopsis
List of Principal Symbols
CHAPTER 1: INTRODUCTION
CHAPTER 2: ANALYSIS OF FERRORESONANT CIRCUIT
2.1: Preliminaries
2.2: Power system ferrroresonance
2.4: Review of work done by G.W Swift and R.J Javora etal
2.5: Conclusions
CHAPTER 3: TRUE SATURATION CHARACTERISTICS
3.1: Preliminaries
3.2: Meaning of true saturation characteristic
3.3: Assumptions for the saturation characteristic
4.4: Conclusions
CHAPTER 4: ANALYSIS OF TRANSFORMER CORE BEHAVIOUR
4.1: Preliminaries
4.2: Expression for hysteresis loop
4.3: Hysteresis loss
4.4 Suggested methods:
4.5 Comparison of suggested methods
4.6 Experimental observations
4.7 Conclusions
9-11
12-19
20-22
23-42
CHAPTER 5: SERIES FERRORESONANT CIRCUIT UNDER TRANSIENT
CONDITION
43-47
5.1 Preliminaries
5.2 Analysis of ferroresonant circuit considering core behavior
5.3 Response curve considering saturation curve as piecewise linear
5.4 Representation of B-H loop
5.5 Switching transients
5.6 Representation of magnetic characteristic of the nonlinear
inductor.
5.7 General equations for the study of transients
5.8 Conclusions
CHAPTER 6: SYSTEM UNDER INVESTIGATIONS AND FINDINGS
48-62
6.1: Preliminaries
6.2: System characteristics
6.3: True saturation characteristics curve
6.4: Transient of Series Ferroresonant circuit under study
6.5: Initial change of capacitor and switching angles
6.6: Effect of input voltage
3
6.7: Computed results
6.8: Experimental results
6.9: Study of transients
6.10: Conclusions
CHAPTER 7: GENERAL CONCLUSIONS AND SUGGESTIONS FOR FUTURE
WORK
65-68
7.1: General conclusions
7.2: Suggestions for future work
REFERENCES
69-74
APPENDICES
75-87
4
ACKNOWLEDGEMENT
I am most grateful to Dr.Amiya Prasad Dash, Reader in Physics, G.M.Autonomous
College,Sambalpur for his sincere guidance and supervision through out this work.I am
also grateful to all the staff members of G.M.Autonomous College and Electrical
Engineering Department, University college of Engineering, Burla for cooperation and
help during the thesis work.
My sincere thanks are also to Principal, U.C.E.Burla and G.M.Autonomous
College for the facilities provided at their colleges for the laboratory and computational
work.
My sincere thanks are also to all my colleagues and family members for their help
and cooperation during this research work.
I appreciate the help rendered by the laboratory staff of the Electrical Department,
U.C.E.Burla and G.M.Auto.College, Sambalpur.Lastly I thank Sri B.K.Sahu for typing
the manuscript and for various graphical works.
Pratima Rath
5
SYNOPSIS
The present thesis is an embodiment of four research papers two published
in national journals and two communicated to international publication.
The present work developed an accurate procedure for an in-depth study of
dynamic behavior of transformer core including parameters like saturation, hysteresis and
residual flux. The piece wise linear model for the magnetic characteristic of the iron core
has been utilized in the analysis. A detail investigation was done both analytical and
graphically to find the critical parameters for initiation and sustenance of ferrroresonance
phenomena. The effect of variation of phase angle of the voltage on switching in the
ferrroresonance phenomena has been investigated in details.
True saturation curve of transformer represented by two terms fifth degree
polynomial has been predetermined by a new method from the r.m.s saturation data
supplied by the manufacturer. The method needs careful selection of two points on the
r.m.s curve.The accuracy of the technique has been verified by the experimental results.
The method described here has advantage over old method since the computation for
obtaining the curve does not involve trial and error method. A computer aided method for
optimizing number of segments necessary for the true saturation characteristic has been
suggested.
The dynamic behavior of transformer core has been analyzed from the data of
two points of r.m.s saturation curve and no load loss. The weighted method of least
squares is also applied for determining the expression for hysteresis loop including both
the major and minor loops. The mathematical expression used also exhibits the hysteric
property of increasing the loop area with increase of frequency of operation. The energy
loss expression associated with hysteresis loop is derived from the expression suggested.
A new approach has been made to incorporate dynamic models of representing
the instantaneous saturation curve to get a composite model. The suggested method has
6
been utilized by optimizing it over a desired range. This technique is utilized to study
dynamic behavior of transformer core at different switching angle, different remnant
magnetism and different circuit parameters.
The contributions made within its scope are given below.
1. For fixed value of resistance and capacitance and zero degree switching angle,the
amplitude of transient current increases
the magnitudes of applied input voltage
increases.
2. The magnitude of jumps for the suddenly applied voltage is lower than that when
applied gradually.
3. The dynamic core behaviour resulting ferrroresonance is more severe when the
capacitance is increased with a fixed value of resistance in the system.
4. The effect of the increase in the value of resistance with a fixed value of capacitance
under transient conditions is less than that in steady state condition.
5. Ferrroresonance leading to high voltages and high currents occurs aacross the
transformer core for zero degree switching angles. The circuit is non resonant for
switching angle of ninety degree
6. Development of a new method to determine an analytical expression for true saturation
characteristic of transformers.
7. Development of new method to estimate the third harmonic flux linkage for correct
analysis of transformer core behavior under transient condition.
8. The development of a new and accurate method by which hysteresis loop is
determined for study of its dynamic behavior under ferroresonant conditions.
7
9. A new mathematical method hs been developed for analysis of transformer core
behavior under transient conditions. This method reveals a better understanding to the
problem then methods earlier.
The report in this thesis has been organized in eight chapters.
Chapter 1- provides a brief introduction to the ferrroresonance phenomena.
Chapter 2-is devoted to the analysis of series ferroresonant circuits and various practical
cases found in distribution network. It reviews the work of earlier workers and the
suggested methods used in the present work.
Chapter-3-deals with basic definition of the true saturation characteristic and methods
used in deriving it..
Chapter 4-is devoted to the analysis of transformer core behavior including core loss. It
has provided various methods to calculate core loss and compared with experimental
results.
Chapter 5 –is devoted to series ferroresonant circuit under transient condition. It
discusses the general equations for the study of transients..
Chapter 6- is devoted to the system under investigation and various findings considering
initial charge of capacitor, effect of input voltages and switching angles. It discusses the
various experimental results.
Chapter 7--is devoted to general conclusions and suggestions for future work in this
field.
Amiya Prasad Dash.
Pratima Rath.
8
LIST OF PRINCIPAL SYMBOLS
I
-
R.M.S. Current
Ih
-
Loss Component of no load current
Iμ
-
Magnetizing Component of no load current
i
-
Instantaneous current
ib
-
Base Quantity of Instantaneous current
K
-
Slope of a Linearised Segment for I.C
L
-
Inductance of the transformer winding
N
-
Number of zones for r.m.s curve.
N
-
Order of nonlinearity.
R
-
Resistance of the Transformer winding Number of Segments for a
Linearised I.C.
S
-
Slope of Linearised Segments for R.M.S. curve.
V
-
R.M.S. Voltage
Vb
-
Base Quantity of R.M.S Voltage
Vm
-
Maximum voltage
V
-
No load loss
α
-
Angle of the Harmonic Component
λ
-
Instantaneous Flux Linkage
λb
-
Base Quantity of Flux Linkage
λm
-
Maximum Flux Linkage
λ1m
-
Maximum flux linkage of fundamental component
λ3m
-
Maximum flux linkage of third component
w
-
Normal Angular Frequency
wb
-
Base Angular Frequency
A, B
-
coefficient of polynomial representing I.C in terms of voltage.
A', B'
-
coefficient of polynomial representing I.C in terms of Flux
linkage.
A'1B'1
-
Per Unit Coefficients Corresponding to A',B'
C', D'
-
Coefficients of Polynomial Representing the Lose Part.
C'1, D'1
-
Per Unit Coefficient Corresponding to C' and D'
9
CHAPTER 1
INTRODUCTION
The iron cored coil is a common nonlinear element in series ferroresonant circuit. The
behavior of the transformer core under transient condition in ferroresonant circuit needs
different approach then that in steady condition(2,3,4,7) The analysis of transformer core
is very much desirable in transient simulation studies such as harmonic oscillations, self
excitation, ferrroresonance phenomena, transformer core loss with and without residual
flux, effect of switching angle and circuit parameter like capacitor etc(5,10,13,.17).It is
also necessary to analyze for the predetermination of the problem and to find suitable
methods to solve it in electrical distribution system(69).
In the past literature many workers have tried to determine transient
voltages by geometrical models, equivalent circuits, and electromagnetic models etc.In
the year 1950 P.A.Abetti (2) first developed an electromagnetic model by taking care of
the demerits of other models .He combined an equivalent circuit of capacitance and
geometrical model for the self and mutual inductance. In the year 1970 L.O.Chua and
K.A.Stromasmoe (4) used analytical expression for modeling the dynamic hysteresis
loop. In the year 1972 A.Clerici (27)developd transformer model in response to the
requirement of analog and digital computers for simulating power system transient. In
1975 S.N.Talukdar (1) has suggested an algorithm for converting saturation magnetic
characteristics expressed in R.M.S.to instantaneous values. In 1980 S.Prusty and
M.V.S.Rao(21.22.23) and again in the year 1982 S.N.Bhadra(25) have given various
important direct methods to predict and predetermine magnetization characteristics of
transformer including hysteresis.They have also developed a direct saturation curve to
instantaneous curve and vice versa. Also in the literature Fourier series, power series and
exponential functions were used for modeling to give a single expression to describe the
whole saturation curve of a transformer(20,,30,33,34,35,38,42,43).S.N Bhadra(25) &
S.Prusty et al (22,23)in 1984 have raised new expression to improve the modeling of
dynamic hysteresis loop by including weightage factors to square method. In 1986
I.DMayergoyz (42) gave a mathematical model of hysteresis which is modified by S.Ray
(66) and again in 1993 for prediction of hysteresis losses in single phase transformer.
10
In 1993 G.Bertotti et al (39) gave a modified dynamic model for a three
phase power transformer core during transient condition. In 1995 F.D Leon (56) gave a
simple
representation
of
dynamic
hysteresis
losses
in
power
transformer.
S.K.Chakravarthy et al (72) extended it in that year for the three phase three limb
transformer. A Fourier description model of hysteresis loop for sinusoidal and distorted
wave forms including minor loops was given by Ismail A.Mohmmad etal in 1997(44).
Torre E.D.(46) gave a new approach to magnetic hysteresis improving the work of
M.A.Coulson et al (40) in Presiach's theory.G.W Swift(3,11) gave a new technique to
study ferrroresonance problem. Many workers after this analyzed the ferrroresonance
problem utilizing various aspects of transformer modeling (47, 4914, 15, 18)
E.P.Dick and W.Watson (10) and S.S.Udpa et al (52) calculated the transient
performance of transformer including core hysteresis. Iwaahara et al (12) and.Lamba et al
(37) gave a new approach to the effect of circuit parameters on ferroresonant circuits and
for its various solutions in the electrical networks. In 2000 M.R Iravani et al (14) gave a
detailed picture of modeling and analysis guidelines for slow transient in the study of
various magnetic core models in comparison with the laboratory test on a 33 KV voltage
transformer. In the year 2006,Li,H. et al(15) gave a detailed picture of dynamic core
behavior in their paper on chaos and ferrroresonance.For analysis of transient
phenomena.Afshin Rezaei-Zave et al in 2007(16,18) gave detailed account of analysis of
ferrroresonance modes in power transformers using Preisach type magnetic inductance.
In 2008, the same group of workers namely Afshin R-Zave et al (17) gave a modified
version to get an accurate hysteresis model for ferrroresonance analysis of a transformer
in series circuits.
The aim of the present work is to develop an accurate procedure for study of
transformer core behavior including various losses like saturation,hysteresis and eddy
current for the study of series ferroresonant circuits under transient condition.
Computation of critical voltages for various circuit parameters have been analyzed by
considering instantaneous saturation curve as piecewise linear. All the analysis have been
carried out considering switching angles, residual flux, capacitor charge etc.The present
work finds computational techniques to accurately predict and predetermine transient
11
over voltages in various electric network at no load condition and to provide suitable
steps to solve it in distribution network.
Various computer aided methods have been proposed to study the effects
carefully and suitable methods have been proposed to derive saturation characteristic in
piecewise linearised segments by optimizing it over a desired range58, 59, 60, 62). For
the loss part polynomial approximations of cubic and quintic orders have been
considerd.The technique is utilized to study transient behavior of the circuit at different
switching angle .Camera fitted oscilloscope has been proposed to take all the photographs
of the core behavior under transient condition. The various computational techniques
used for transient simulation studies are supported by experimental observations in
laboratory condition.
***********************************************************************
12
CHAPTER 2
ANALYSIS OF SERIES FERRORESONANT CIRCUIT
2.1 Preliminaries:
Ferroresonant is special types of jump phenomenon associated with electrical
circuits containing a linear capacitor and a nonlinear inductor with or without any other
element present in the circuit (fig 2.1).The magnetic core of the voltage transformer
behave as nonlinear inductor. Due to interaction between linear capacitor and nonlinear
inductor, resonance occurs. For certain circuit parameters and forcing frequency, when a
slight variation in one of them occurs, the signal is suddenly enhanced somewhere in the
circuit. There are many situations found in electrical system networks in which each case
of the above circuit parameters can give rise to the ferrroresonance phenomena. The
following sub sections discus some important situation in which ferrroresonance occurs.
2.2 Power system ferrroresonance:
Ferrroresonance being a forced oscillation in a non-linear circuit, it is necessary in
order to study it ,to define the excitation voltage of the source delivering the energy
which maintains the phenomenon. The oscillation or resonance occurs between a linear
capacitance and a nonlinear inductance. The capacitance can be due to a large number of
capacitive elements. Similarly the nonlinear inductance can be that of a single element for
example the magnetic core of a voltage transformer or it may have the structure of a three
phase power transformer [69].
2.2 Power System Ferrroresonance:
2.2.1 Single-phase Ferro-resonance on a voltage transformer connected to a highvoltage line, disconnected but running alongside another energized line:
This effect can occur when two high-voltages lines are strung on the same pylons
but are operated at different voltages like for example 150 and 70 kV.[fig 2.2].As
shown in the figure line A at the higher operating voltage is energized, line B
disconnected at both ends. Only a voltage transformer Xs is still connected to line
B and possibly a current transformer and a lighting arrester. The potential on one
conductor RB of the de-energized line depends on the capacitances between these
conductors on the one hand and earth and the energized wires on the other
13
.Considering from the terminals of the voltage transformer, one can reduce the
lines to a single-phase source in series with a capacitance by applying the
Thevenin theorem. He same procedure is repeated for each phase of the
disconnected line.It is established that Ferro-resonance occurs only exceptionally
in the circuits. The possibility of its occurrence depends on the supply voltage, the
value of the capacitance and the magnetic characteristics of the non-linear
element.
2.2.2 Single-phase Ferro-resonance between voltage transformers and the h.v./m.v.
capacitance of a supply transformerThe phenomenon can occur with a supply transformer where the h.v. neutral is insulated
from earth although the h.v. system is earthed at other points. The m.v. also
insulated from earth is connected to a set of three voltage transformers [fig 2.3]
but does not supply any load.
Following a fault to earth on the h.v. system on the supply side of the
transformer, its h.v.neutral potential can be raised temporarily to a high voltage to
70 percent maximum of the phase to earth voltage. The circuit formed by this
voltage En, the capacitance Cn and on each phase the capacitance Cn in parallel
with XS can thus be brought to a state of fundamental Ferro-resonance giving rise
to an over voltage on the m.v. system. After clearance of the fault on the h.v.
system, the Ferro-resonance condition may be maintained by the normal voltage
existing on the h.v. neutral point. The normal condition is generally returned by
greatly increasing the zero-sequence capacitance C0 by the connection of a cable
to a load on the m.v. side of the circuit.
2.2.3 Single-phase Ferro -resonance between a voltage transformer and the
capacitance constituted by an open circuit breakerThe phenomenon can occurs with a circuit-breaker having a number of interrupter
heads carrying a small capacitor in parallel for voltage distributor [fig 2.4].On the
supply side of the open circuit breaker there is a set of busbars.On the other side
only a voltage transformer and an open isolator leading towards a line. There may
be in addition a current transformer and a lighting arrester. The Thevenin theorem
is used to reduce the linear circuit at the terminals of the voltage transformer to a
14
voltage in series with a capacitor. This voltage becomes zero and the phenomenon
becomes impossible when the capacitance of the open circuit breaker becomes
negligible compared with the zero-sequence capacitance of the system like for
example when the isolator which follows the system is closed on an h.v. line.
2.2.4 Three-phase Ferro-resonance with voltage transformers connected to a system
with insulated neutral and very low zero-sequence capacitanceThis concerns the classic phenomenon of Ferro-resonance and to one to which the
majority of publications[69] are devoted [fig 2.5].The figure gives only one
example of the situation where the phenomenon can occur, another situation is for
example where the voltage transformers are connected to the terminals of an
alternator. The essential conditions are that system be one with an insulated
neutral that the three voltage transformers be connected between phase and earth
and that the capacitance C0 between phase and earth be very small which is
satisfied when its impedance is of the same order of magnitude as the impedance
of the voltage transformers. If the load of the voltage transformers is ignored, the
circuit can be reduced to its simplest expression .In this case the voltage source is
represented schematically in star connection. There remains in each phase only
the non-linear inductance XS of the voltage transformer in parallel with C0. the
phenomenon is three-phase and cannot be reduced to single-phase circuits. For
example, it becomes impossible if the two neutral points are earthed that is the
neutral of the power system is earthed.
The fundamental Ferro-resonance as well as the sub-harmonic and higher
harmonic Ferro-resonance can occur in this configuration depending upon the
relative value of the impedances of XS and C0 .There is however an important
difference versus a single phase circuit. In the later there appears a sub-harmonic
or a higher harmonic which is an exact multiple or submultiples of the supply
frequency. In three phase Ferro-resonance there appears a frequency which is near
harmonic but slightly different from it so that a beat phenomenon is obtained. It is
said generally that the phenomenon only occurs on an unloaded system which is
correct insofar as the connection of a load normally increases the zero-sequence
capacitance to such an extent that the phenomenon becomes impossible.
15
2.2.6 Ferro-resonance with an unloaded power transformer supplied accidentally on
one or two phasesIf between the system and an unloaded transformer or a very lightly loaded one,
one or two phases are interrupted by a fuse blowing, the operation of a single-pole
circuit, the non-simultaneous operation of the contacts of a three-phase circuitbreaker or again a conductor breaking, very high voltages can occur on the open
conductor or conductors and this takes place on the transformer side.In addition
reversal of phase rotation is possible at the secondary terminals of the transformer
[69]. The capacitance, positive-sequence and zero-sequence, of the connection,
cable or line, between the power system and the transformer plays an essential
part [fig 2.6].as shown in the figure a star-delta transformer with three limbs is
shown. The three elements the primary and secondary winding connections and
the structure of the core play important part .The same situation is also
encountered with three single-phase transformers. The phenomenon generally
occurs when a primary winding of the transformer is insulated from earth but with
the neutral earthed.
2.3 Review of work done by G.W Swift and R. Javora etal:
G.W.Swift [3] has in his paper has studied the power transformer core behavior
under transient condition. In the paper he has stated when analyzing electric
utility networks it is sometimes necessary to stimulate the dynamic behavior of a
transformer or reactor, that is, to simulate the time dependent relationships among
current, voltage and flux for a winding on an iron core. Three examples of
problems in which nonlinear effects must be included are ferrroresonance,
subharmonics and transformer inrush current. The qualification "unloaded" is
used because this is the condition most likely to cause Ferro-resonance or
subharmonics.An unloaded transformer has one winding open-circuited and
therefore equivalent to a reactor for calculation purposes. The term iron core
referred to a laminated core of grain-oriented silicon-iron sheet, universally used
in modern power transformers and reactors.
As stated by Swift there are three non-linear effects introduced by iron
cores. They are magnetic saturation, hysteresis and eddy-currents. These three
16
effects were carefully distinguished .In the literature it is seen that some consider
saturation and hysteresis only; some consider saturation and eddy-current effect
only. There are few papers which consider all three for only in specific cases like
in isotropic iron and not in the study of transformer core behavior under transient
condition. Hence He has shown that if hysteresis is to be considered in a
particular circuit analysis, then eddy-currents must also be considered because
they represent a larger effect and a linear modeling of the eddy-current effect is
very accurate for commercial power transformers.
Three non-linear effects:
Of the three effects saturation is the most predominant not found in linear
systems. The factor which distinguishes the saturation effect from the other is that
energy loss is involved in the others, that is energy is dissipated as heat loss in the
cases of hysteresis and eddy-currents. Saturation it self does not introduce loss.
Even though saturation is the predominant effect, t is sometimes necessary to
include loss-associated effects as for example in cases where the total system loss
appears to be a major determinant as to whether or not sub harmonic oscillations
will be self-sustaining.
Display of losses:
The voltage current relationship ∫e dt and i will provide the loss per cycle. The
area bounded by the curve considering ∫e dt along Y axis and I along X axis will
provide the loss of the transformer core per cycle given by
W = ∫0T p dt
(i)
Where p is the power being dissipated and T is the time for one cycle of the
periodic voltage, current or flux.
If we take ∫e dt = y for notation convenience then the area enclosed by the contour
is given by A=∫ I dy = ∫(dy/Idt)dt = ∫Ie dt = ∫p dt
(ii)
Comparing this with (i) it is seen that A=W that is the loop area is equal to tha
energy loss per cycle.
Considering two cases (i) a resistor and (ii) an iron cored coil with negligible
winding resistance.If sinusoidal voltage is applied in each case ,the iron cored coil
loss cycle is recognizable as a hysteresis loop and this is to be expected since
17
e= N A (dB/dt)
or
B= (1/NA) ∫e dt
Where N is the number of coil turns,B is the effective core flux density, and A is
the effective core cross-sectional area and the current I is proportional to the
magnetic field intensity within the core.
Eddy-current effect:
The shape of the contour is dependent on the rate at which the contour is
traversed.For very slow traversal of the loop,the loss can be termed as pure
hysteresis loss and is attributable to domain wall movements back and forth
across
crystal
grain
boundaries
,non
magnetic
inclusion
and
imperfections.Increasing the frequency increases the loss per cycle.Any increase
in loss above the value at zero frequency has been attributed to "eddy-current
effect", and for modern commercial grain-oriented silicon-iron sheet this effect
accounts for about three times as much loss as does the pure hysteresis.It is seen
in the literature[21,23] on domain wall motion and shape indicate that the pure
hysteresis effect is frequency dependent.
The total core loss is divided in to frequency-dependent part and a nonfrequency-dependent part.The following expression [ ] gives the value of
magnetic field intensity due to eddy-currents
He= K cos(1-sinwt)
[900 < wt < 1800]
Where K is a constant for a particular core and w is the radian frequency of an
applied cosine voltage e . Let He be proportional to a component of coil current Ie
associated with it. We now have e = Em cos wt giving ∫e dt = (Em/w)sin wt
Where Em is rthe peak value of the coil voltage e and the constant of integration is
zero in the steady state.Also
Ie = K' cos wt (1-sinwt) where K' is a constant incorporating K and the linear
proportionality between He and Ie . The graph between ∫e dt and Ie gives the loss
cycle due to eddy-current.
18
Hysteresis Effect:
The modeling of hysteresis effect is more complex.The pure hysteresis loss is
only about one-third of the other loss like eddy-current loss [ ].The saturation
characterstic is given by the relation I0= α∫e dt + β[∫e dt]5
Where α and β are constants and I0 is the mean value current of the hysteresis
loop family of curves.
Hence Swift concluded that when the nonlinear aspects of an inductor
(transformer or reactor) are to be considered in a circuit theory analysis, it is
important to include the eddy current effect when the pure hysteresis effect is also
included since eddy-currents cause more loss than does hysteresis in modern
power transformer core under transient condition. The eddy-current effect is
approximated in a linear way.
Demerits: In the paper Swift has considered the three effect of the transformer
core has been considered separately which in practical cases need simultaneous
consideration as all these occurred together wlile analyzing power system network
to study ferro-resonance phenomenon.
R.Javora et al paper [6] is an improvement over the paper of Swift in that they
have considered a small resistance in parallel to the iron core inductor. They have
discussed the phenomenon in the following ways:
Javora et al have discussed the cases in which a resistor is used in parallel with the
inductor to account for the core loss. According to them a lot of articles
concerning ferrroresonance phenomenon in electric power distribution systems
consider a single-value resistor connected in parallel with a nonlinear inductor to
represent core losses of the of the transformer under study. Such a resistor cannot
properly describe the behavior of transformer in the saturated area. There better
and more complex model of the transformer core losses is necessary. They have
recommended for further improvement of the ferrroresonance studies in electric
power systems by considering a dynamic behavior of transformer core losses.
19
They have compared the results by using static (single-value resistor) and
dynamic (nonlinear resistor) representation of core losses .Javora et al have shown
that by employing only a static description of core losses there is a higher level of
instability for solutions. Measurements by them shows higher damping effect and
lower level of instability in dynamic representation of core than by using static
representation.
Demerits: Javora et al have not introduced the most accurate model of
transformer. The proposed method provides only one possible way for a
qualitative description of the effects of magnetic hysteresis in magnetic circuits.
and the consequences of assuming the static and the dynamic representation of
core losses.
2.4 Conclusion:
The present thesis has taken in to the merits and demerits of earlier workers and
more particularly the work of Swift and Javora et al and developed a new
mathematical technique to calculate the transformer core loss by putting a small
resistance in parallel with the iron core inductor.This technique is found to yield
better result in the study of transformer core loss and transformer core behavior
under transient conditions.The methods used by Swift and Javora et al needs more
in field test and tedius calculation where as the suggested method in the thesis is
more straight forward , analytical and easy to implement in the electrical
distribution system.
*************************************************************
20
CHAPTER 3
TRUE SATURATION CHARACTERISTICS
3.1 Preliminaries:
The true saturation characteristic of magnetization of iron core coil or transformer
behaves quite differently in ferrroresonance condition then that in normal condition.Its
proper determination under dynamic condition is a challenge to all workers in this field as
the core flux linkage is a nonlinear function of the current across it.The true saturation
characteristic of a transformer or an iron cored reactor gives the relationship between
peak values of flux linkage (λ) and the magnetizing current (i) when core loss is
neglected. This is shown in fig (3.1).The symmetrical nonlinear characteristic can be
represented by a polynomial
i= a1λ+ a3λ3 + a5λ5 +……..+ anλn
(3.1)
Where n is an odd integer and a1, a3, a5…….. an are constants. It has been seen by many
workers (9,13,69 ) that two term polynomial is sufficient to represent the magnetizing
characteristic for power system transient studies. Therefore the equation for true
saturation characteristic of a transformer is
i= a1λ+ anλn for n= 5, 7,9,11
(3.2)
3.2 Meaning of the true saturation characteristic:
The saturation characteristic of a transformer is in the R.M.S. form supplied by
the manufacturer.But the R.M.S. form of saturation characteristic is not useful directly to
work with the study of transformer core behavior. In general the current and flux linkage
relationship of an iron cored transformer is a nonlinear and multivalued .There may be
several flux linkage values corresponding to one current value and vice versa. It is not
practicable to tackle and use them for core studies during dynamic condition. To simplify
them to a single valued functional relationship between the flux linkage and the
magnetizing current can be used.This is the curve relating the d.c. flux linkage and the
corresponding d.c. magnetization current when the core is in demagnetized condition and
is known as true saturation curve.
The true saturation characteristic is very much useful for study of
saturation effect of the core material and to calculate the total core loss .It may be
21
obtained experimentally using sinusoidal a.c. voltage for energization of a transformer.
The peak currents corresponding to various peak voltages are measured with the help of a
C.R.O. shown in fig (3.2).But this method is not convenient for transformers which are
already in service in the network. Hence it is desirable to develop some methods to derive
the characteristic from the R.M.S. saturation data supplied by the manufacturer.
3.3 Assumptions for the saturation characteristic:
The true saturation characteristic of a transformer can not be directly obtained
from the readings given by R.M.S.meters used in the test. The characteristic as shown in
fig( 3.1) is obtained by joining the tips of magnetization characteristic without
considering the losses in the core. Considering a transformer for which applied voltage V
produces current I and flux linkage λ , we get
V= (d λ/dt) +R I
(3.3)
Where t denotes time and R is the resistance that simulates the electrical energy
losses in the transformer.When the loss is neglected equation (4.3) becomes
V= (d λ/dt)
(3.4)
Let λ = λmsin wt
(3.5)
From equations (3.4) and (3.5) , we get
Vm = λm.w
Or V = (λmw)/sqrt 2
(3.6)
From equation (3.6) it is clear that λm is directly proportional to the applied
voltage V at a given angular frequency w. As there is functional relationship between the
maximum flux linkage of the winding of a transformer and the corresponding maximum
value of no load current, the curve relating the two quantities can be obtained with some
extra arrangements rather than with only R.M.S. meters. The same characteristic may
also be predetermined from the R.M.S. curve data. Therefore the true saturation
characteristic should satisfy the following assumptions (i) the core are in a demagnetized
state (ii) The core losses are negligible.
3.4 Conclusions:
The saturation curve data obtained from the manufacturer is of the R.M.S. form.
This data is most reliable for study of various transformer core behavior than is obtained
from test result. The experiment to be performed to obtain such data is not feasible as all
22
the practical transformers have been in the distribution network.Hence it is suggested in
the thesis to use manufacturer's data in the study related to transformer core behavior
from the available manufacturer's data. Predetermination of the phenomena and taking
suitable precaution to avoid the occurrence such ferrroresonance problem in series
ferroresonant circuits are more important.
***********************************************
23
CHAPTER 4
ANANLYSIS OF TRANSFORMER CORE BEHAVIOR INCLUDING
CORE LOSS
4.1 Preliminaries:
In the case of dynamic hysteresis loop of transformer, it can be identified that it has two
symmetrical parts. The first part is the true saturation characteristic of a transformer
which is represented as gob in fig (4.1) and which is hereafter called 'true saturation part'.
The saturation part is the locus on the mid points (abscssae) of the hysteresis loop and
represents the ideal magnetization characteristic of the transformer,.The second part is the
'loss part' because it is the main part to contribute proper shape and area to the hysteresis
loop. The periphery of the hysteresis loop, which is same as the actual hysteresis loop is
obtained by adding or subtracting 'loss part' to the 'true saturation part'. Usually for
modern transformers, true saturation characteristic is approximated by a fifth order
polynomial given by
I=A'λ+B'λ5
(4.1)
Where
λ =λmsin θ
(4.2)
and
θ=ωt
(4.3)
The nature of the loss part is that it is maximum when the true saturation
part is zero and it is zero when true saturation part is maximum. Hence the loss part is
represented by f(λ) or g(λ, λ') where λ'=dλ/dt
4.2 Expression for the hysteresis loop:
The hysteresis loop is represented by an addition or subtraction of thee
loss part to the true saturation part. Incorporating a plus or minus sign implicit in f (λ') or
g(λ, λ'), the mathematical expressions for the hysteresis loop are
I=A' λ+B' λ5+f(λ')
Or
(4.4a)
24
I=A' λ+B' λ5+f(λ,λ')
(4.4b)
For the cubic order f(λ') approximation of the loss part
f(λ')=C'(λ)+D'(λ')3
(4.5a)
=C(λm cos ωt)+D(λm cos ωt)3
(4.5b)
Where C=C'ω and D=D'ω3
Substituting eqns(4.5) and eqns(4.2) in eqn 4.4a and simplifying, we have
I=SQRT [(A' λm+5/8B' λ5m)2+(C λm+3/4D λ3m)2]
Sin (ωt+α1) +SQRT [(5/16 B' λm5)2+ (D/4 λm3)2]
Sin (3ωt+α3) + (1/16B' λm5) sin5 ωt
(4.6a)
Or
I=Im1sin (ωt+α1) + Im3 sin(ωt+α3)+ Im5 sin(5ωt) (4.6b)
Where
α1=tan-1 [C λm+3/4D λm3] /[A' λm+5/8B' λm5 ]
α3=tan-1 [D/4 λm3] / [-5/16B' λm5 ]
(4.7)
And A', B', C and D are the constants to be evaluated. The r.m.s. current is given by
I0=[1/ sqrt(2)]sqrt[(Im12+ Im32+ Im52)]
=sqrt[(Iμ2+ Ih2)]
(4.8)
(4.9)
Where Iμ and Ih are the r.m.s. magnetizing and hysteresis loss components of the no load
current which are obtained either from manufacturers data or no load test on the
transformer.
The magnetizing components of no load current are calculated as follows:
Say I0 and w0 are the no load currents and loss respectively of a given transformer,
when a voltage of V volts is applied to it.
The loss component of no load current of a transformer at the given voltage is
Ih= w0/V
(4.10)
Then,the magnetizing component of no load current is given by
Iμ=sqrt [(I02-I h2)]
(4.11)
From equation 4.6a and 4.9.separating the components, we have
Iμ=[1/sqrt (2)]/sqrt[(A' λm+5/8B' λm5)2+(5/16B' λm5)
2
+(1/16B' λm5)
2
(4.12)
25
And
Ih=[1/sqrt(2)]sqrt[(C λm+3/4D λm3)2+(D/4 λm3 ) 2 ]
(4.13)
The expressions for r.m.s. magnetizing and hysteresis loss components of current when
hysteresis loop has loss part approximation by(1) quintic order f(λ') cubic order
g(λ,λ'),and quintic order g(λ,λ') .
4.3 Hysteresis loss:
The energy loss during each cycle associated with the hysteresis loop is equal to the
area of the loop and is
W=∫idx
(4.14)
For the loss part approximation by cubic order f(λ') using eqns.(4.14.)(4.6a) and (4.2) and
simplifying
W=π λm2(C'ω+3/4D' ω3 λm2) joules/cycle
(4.15)
For the loss part approximation by cubic order f(λ')
W=π λm2(C'ω+5/8D' ω5 λm4) joules/cycle
(4.16)
Similarly for loss part approximation by cubic g(λ,λ')
W=π λm2(C'ω+D'/4 λm2)ω joules/cycle
(4.17)
Similarly for loss approximation by quintic g(λ,λ')
W=π λm2(C'+D'/8 λm4) ω joules/cycle
(4.18)
From eqns 4.15 through 4.18, it is evident that the loop area increases with the increase of
frequency of operation.
4.4 Suggested methods:
Two point data method:
The hysteresis loop is represented by a fifth order polynomial expression and
with the help of eqns (4.4a),(4.5 )and (4.2) for an approximation of the loss part by cubic
order f(λ'), it is described as
I=A'(λmsinθ) +B'(λmsinθ)5+C λmcosθ+D (λmsinθ )3
(4.19)
The coefficients of eqn. (4.19) are to be evaluated in order to model the transformer
magnetization characteristic including hysteresis loss. For the purpose,of the r.m.s.
saturation current along with no load loss is used. The method of evaluation of the
coefficients requires only two points of the data and it is as follows.
26
The hysteresis loop consists of two parts (4.12) and (4.13.) derived by the use of
eqn (4.4a) for the approximation of the loss part by cubic order f(λ') are used for the
evaluation of the coefficients A',B',C and D for the purpose, the equations are modified as
aA'2+bB'2+cA'B'=d
(4.20)
And a'C2+b'D 2+c C D = d' (4.21)
Where a = (1/2) λm2
b= (126/512) λm10
c= (5/8) λm6
d = Iμ2
(4.22)
And
Where a' = (1/2) λm2
b'= (10/32) λm10
c'= (3/4) λm6
d' = Ih2
(4.23)
The coefficient A' and B' for finding out the true saturation part is evaluated using
equation (4.20) using two point data of magnetizing component of no load current at two
different voltages on the R.M.S. curve. The coefficients C and D can also be found out
with the help of equation (4.21) using the data of loss components of no load current at
two different voltages on the R.M.S.curve. Both the equations are nonlinear in nature and
the coefficients either A' and B' or C and D at a time can be computed using two point
data.
Multi point data method:
It is clear from equations (4.20) and (4.21) that only two unknown coefficients either A'
and B' or C and D can be evaluated by using a corresponding R.M.S. saturation data and
so the predetermination of magnetization characteristic of a transformer including
hysteresis is possible with two point data but for better accuracy the entire no load test
data may also be used .However the multipoint data can not be used for the evaluation
unless a numerical algorithm like least squares method
used by many
workers[1,25,29,33 ].
The evaluation of the coefficients can be done by the least squares method using multi
point data. The transformer magnetization curve can be modeled more accurately by
27
including the hysteresis and evaluating the coefficients A',B',C' and D using multi point
data instead of two point data of the no load test. In the case of modeling with multipoint
data the coefficients are found by a method of minimization of the sum of the squares of
the errors at various data points of the saturation curve .The error it self cant become
proper basis for accounting the derivations over the entire range of the curve, the per unit
error at a data point is made as the basis for finding the modeling expression (22, 23)
For the purpose, the sum of squares of the per unit errors at different data points is
reduced to a minimum by method of least squares. Thus while calculating the per unit
error, a weighting factor, varying at different points of data, described by a function of
r.m.s. current automatically creeps into the calculations. This method of calculation is the
weighted method of least squares.
In this case the data of the set {V, Iμ} at various voltages of the
R.M.S. curve is used for computing the coefficients A' and B' and similarly the data for
calculating the coefficients C and D of the loss part of magnetization curve is the set
{V,Ih} at different voltages of the R.M.S. curve. The equations (4.10 ) and (4.11) are used
for these purpose .The unweighted and weighted method of least squares have been used
for the calculation of the transformer core behavior and for the calculation of the
coefficients to study the magnetization characteristic of transformer .
Unweighted method of least squares:
In this method the R.M.S. saturation data that is one of the sets {V, Iμ} and {V,Ih}
as required is converted in to per unit data on some base quantity with usually the rated
value. By this unweighted method of least squares the error or residual at each data point
is considered.The sum of the squares of errors for all the data points of the R.M.S. curve
is minimized by the method of least squares.
Let the true saturation part of the magnetization characteristic including
hysteresis be expressed as
Iμ = A'λ + B'λ5
(4.24)
I1μ = A1'λ1 + B1'λ15
(4.25)
But I1μ = Iμ/ Iμb and λ1 = λ/ λb with
λb = sqrt (2)Vb ) / w
28
Where w = 2πf and Iμb = A'λb + B'λb5 with λb and Iμb the base quantities of of flux
linkages and magnetizing current respectively. The coefficients of equations (4.24) and
(4.25) are related as follows
A1' = [A'λb / A'λb + B'λb5]
(4.26)
B1' = [B'λb5/ A'λb + B'λb5]
Substituting λ= λm sin wt in equation (4.24) and simplifying we get
Iμ = sqrt [(λm2 A'2)/2 + (5/8) λm6 . A'.B' + (63/256) λm10 B'2]
(4.27)
Putting w λm = sqrt (2)V in equation (4.27) we get
(V2 A'2/w2) + 5 V6 A' . B'/w6 ) + 63V10 . B'2/w10 - Iμ2 = 0
(4.28)
A relation between V1 and I1μ can be derived
p B1'2 + q B1' + r =0
(4.29)
Where p = (63/128)V110 – (5/4) V16 + V12 –(31/128) I1μ2
(4.30)
q = (5/4) V16 - 2 V12 + (3/4) I1μ2
r = V12 - I1μ2
The entire R.M.S. data for true saturation part expressed in per unit values that is the set
{V1, I1μ} can be used to evaluate the constants B1' of the equation (4.29) by the method of
least squares.
Applying the method of least squares the equation (4.29) becomes
d/d B1' (p B1'2 + q B1' + r)2 = 0
that is 2 B1'3 Σn pi2 + 3 B1'2 Σn pi qi + B1' Σn (qi2 + 2pi ri)+ Σn qi ri = 0 (4.31)
Where n is the number of points chosen from the R.M.S. data.
The positive real root less than 1 is the required solution of the equation (4.31).The other
two roots are in general complex.
Substituting B'/A' = K in equation (4.28) we get
A'2ΣVi2 /w2) + 5K Vi6 /w6) + (63/8)K2 Vi10 . B'2/w10 = Σ nIμ i 2 (4.32)
Where (Vi, Iμ i) is a point on the R.M.S. saturation curve.
A' and B' are obtained by using equation (4.32) .Then A and B are also calculated.
The procedure for the evaluation of C and D with the data of the set {V,Ih }by the
unweighted method of least squares .By using equations (4.4) and (4.6) the loss part of
the hysteresis loop is calculated by using ih= C' λ' + D' (λ')3
(4.33)
29
By using λ' = (d λ/dt) = λm w cos wt
(4.34)
Substituting equation (4.34) in equation (4.33) and solving we get
ih = [C λm +( 3/4 )D λ3m]cos wt +[ (1/4)D λ3m] cos 3 wt
(4.35)
The R.M.S. value of the loss component of no load current is
ih =sqrt [{C λm +( 3/4 )D λ3m}/sqrt 2]2 + [ { (1/4)D λ3m}/sqrt2]2
= sqrt [(1/2) (C2 λ2m + (5/8) D2 λ6m + (3/2) λ4m]
(4.36)
Substituting C=C'w and D= D'w3 and λm=sqrt(2)V/w in equation (4.36)
I2h = C' 2V2 + (5/2) D' V6 + 3 C' D' V4
(4.37)
The base quality of flux linkage, the loss component becomes
ihb = C' λ'1 b+D'(λ'1 b)3
(4.38)
Dividing the equation (4.31) by ihb we get
i1h = C1 ' λ'1 +D1 '(λ'1)3
(4.39)
The above equation describes the per unit based representation of the loss part of the
magnetic characteristic
A relation between V1 and I1h is derived in quadratic form of D'1 as given in equation
(4.38)
pD1' 2+ q D1' + r = 0
(4.40)
Where
p= (5/8V1 6-(3/2) V14 + V12-(1/8) I2 1h
q= (3/2) V1 4-2 V12 + V12+ (1/2) I2 1h
r= V12- I2 1h
The data for loss component of no load current can be used to evaluate constant D1'
By least squares method.
Minimising the sum of the squares of the errors given in equation (4.40) we get
2 D1'3Σn pi2 + 3 D1'3Σnpi qi + D1'Σn (qi2+ 2pi ri)+ Σn qi ri =0
(4.41)
Where n is the number of points chosen from the data of R.M.S. voltage versus loss
component of no load current.
The positive real root less than 1 is the required solution of equation (4.41)
Substituting (D'/C') =K in (4.41)
C'2Σn (Vi2 + (5/2) K2V6i + 3 K V4i)= Σn I2 hi = 0
(4.42)
30
Where (Vi, I hi) is a point from the data of voltage versus loss component of no load
current.
The coefficients C, D, C' and D' are obtained from equations (4.42)
The loss part or the instantaneous value of loss component of no load current I h can be
assumed to have the other approximation (i) quintic order f(λ),(ii) cubic order g (λ, λ').
Weighted method of least squares
The weighted method of least squares yields more accurate true saturation curve than that
of unweighted method of least squares. For representation of magnetization curve of a
transformer including hystresis, the coefficients A', B', C and D are to be evaluated from
the r.m.s. saturation data and no load loss. By applying the weighted method of least
squares, the evaluation of the coefficients A', B', C' and D are obtained more accuarately
than the unweighted method of least squares.
For the method of evaluation of coefficients A' and B' by the weighted method of least
squares, eqn (4.29) is to be modified conveniently. For bringing the expression in to a
convenient form so that the weighting factor can be incorporated, the eqn (4.29) is written
in a form to represent per unit error at a point as
Є'= [pB12+qB1+r'-I'2μ ]/ I12μ
(4.43)
Where p and q are given by eqn 4.40 and r'=V'2
Minimizing the sum of the squares of the per unit error given by equation (4.43)by the
method of least squares, we have
2 B1'3Σn pi2/I'
4
μ1+
3 B1'2Σnpi qi/ I'
4
μ1
+ B1'Σn (qi2+ 2pi ri)/ I'
4
μ1+
Σn qi ri / I1
4
μ1=0
(4.44)
Where n is the number of points chosen from the data R.M.S. voltage versus magnetizing
component of no load current .The eqn(4.44) contains a weighting factor 1/I12μ1 that
changes automatically through out the range of the data. The positive root less than 1 is
required solution of eqn 4.44.The other two roots in general are complex.eqns are useful
in calculating the ratio B'/A' say K. Substititing K in eqn (4.32)A' is evaluated. Then the
coefficients A and B are obtained..
For the evaluation of C and D by the weighted method of least squares, the
equation (4.40) is modified to represent the per unit error at a given data point as
31
Є'= pD'2+qD'+r'-I'2h/ I'2h
(4.43)
Where p and q are given by eqn(4.40) and
r'=V'2
Applying the method of least squares to eqn 4.43, we obtain
2 D1'3Σn pi2/I1
4
h1+
3 D1'2Σnpi qi/ I14h1 + B1'Σn(qi2+ 2pi ri)/ I1
4
h1+
Σn qi ri / I1
4
h1=
0
(4.44)
Where n is the number of points chosen from the R.M.S. data versus loss component of
no load current. The equation contains a weighing factor
1 / I1
4
h1
that automatically
changes through out the range of data. The positive root less than 1 is the required
solution for D1' .The knowledge of this will help to find out the other coefficient C' .Thus
all the coefficients of the loss part C, D,C' and D' are evaluated. The instantaneous value
of loss component of no load current can also be represented by other approximations
like quintic order and for cubic order.
4.5 Comparison of suggested methods:
The hysteresis loop currents computed for the four different forms of
approximations of loss part by the three methods are compared with the measured values
at 200V,230 V and 260 V in tables (4.2) (4.3) and (4.3) .The computed hysteresis loop
currents obtained for quintic approximation by all the three methods suggested and the
measured one at the operating voltage 230 Volts are shown for graphical comparison in
fig [4.2 ].Also the computed current wave form obtained for quintic approximation of the
loss part by all the three methods suggested and the measured one at the same voltage are
presented for comparison in fig[4.3 ] . In order to find the accuracy of the hysteresis loop
currents obtained with different approximations by the three methods suggested, R.M.S.
error in the loop currents has been computed and presented in tables (4.2),(4.3) and (4.4)
It is found from the comparison that the weighed method and the two
point methods give one for the study of transformer core loss and its behavior under
transient condition.
32
4.5 Experimental observations:
In order to apply the suggested methods to predetermine the magnetization
characteristic including hysteresis, a laboratory type single phase transformer rated at
3kVA, 230/230V, 50 Hz has been selected. The readings by varying the voltage across
the transformer up to 300 Volts are recorded. The magnetizing and the hystersis loss
components of no load current at different voltages are calculated and the data thus
obtained is used to predetermine the coefficients A', B, C and D.
Table 4.2a .Measured and computed currents (two point method) at 200 V:
Angle in
Flux
Measured
Computed current
Degree
linkage in
Current
Cubic
Quinticf(λ') Cubic
Wb.turns
in
f(λ') in A
In Ampere
g(λ, λ')
λ') in Ampere
Quintic
Ampere
0
0
0.15
0.157
0.159
0.158
0.16
18
0.223
0.21
0.228
0.227
0.2265
0.23
36
0.423
0.27
0.285
0.285
0.286
0.28
54
0.583
0.35
0.356
0.355
0.356
0.355
72
0.69
0.43
0.427
0.427
0.427
0.43
90
0.72
0.465
0.43
0.43
0.4305
0.432
108
0.69
0.34
0.3288
0.331
0.327
0.327
126
0.583
0.165
0.168
0.169
0.166
0.168
144
0.423
0
0.027
0.0277
0.027
0.0274
162
0.223
-0.1
-0.077
-0.076
-0.075
-0.074
180
0
-0.14
-0.1574
-0.1582
-0.1574
-0.1582
2.06
2.07
2.07
2.08
R.M.S.
Error in%
g(λ,
33
Table 4.2b.Measured and computed currents (two point method) at 230 V:
Angle
Flux
Measured
in
linkage in
Current
Cubic
Quinticf(λ') Cubic
Wb.turns
in
f(λ') in A
In Ampere
g(λ, λ')
in Ampere
Degree
Computed current
Quintic g(λ, λ')
Ampere
0
0
0.226
0.2265
0.227
0.226
0.227
18
0.32
0.3
0.335
0.334
0.329
0.328
36
0.61
0.426
0.475
0.474
0.476
0.475
54
0.84
0.86
0.82
0.82
0.827
0.832
72
0.98
1.34
1.3
1.3
1.31
1.3
90
1.04
1.6
1.5
1.5
1.51
1.5
108
0.99
1.1
1.16
1.16
1.15
1.15
126
0.84
0.5
0.55
0.55
0.54
0.54
144
0.61
0.1
0.098
0.099
0.096
0.098
162
0.32
-0.14
-0.112
-0.111
-0.107
-0.105
180
0
-0.226
-0.227
-0.228
-0.227
-0.228
4.65
4.65
4.4
4.34
R.M.S.Error %
34
Table 4.2c.Measured and computed currents (two point method) at 260 V:
Angle
Flux
Measured
n
linkage in
Current
Cubic
Quinticf(λ') Cubic
Wb.turns
in
f(λ') in A
In Ampere
g(λ, λ')
in Ampere
Degree
Computed current
Quintic g(λ, λ')
Ampere
0
0
0.25
0.256
0.25
0.256
0.258
18
0.36
0.35
0.38
0.381
0.36
0.35
36
0.69
0.6
0.597
0.595
0.59
0.596
54
0.945
1.2
1.21
1.21
1.22
1.23
72
1.11
2.1
2.11
2.11
2.12
2.13
90
1.17
2.5
2.52
2.52
2.52
2.53
10
1.11
1.8
1.95
1.95
1.94
1.94
126
0.945
0.9
0.91
0.91
0.89
0.89
14
0.69
0.15
0.167
0.168
0.16
0.167
162
0.36
-0.1
-0.127
-0.128
-0.12
-0.118
180
0
-0.25
-0.255
-0.257
-0.26
-0.26
4.93
4.9
4.59
4.4
R.M.S.Error %
35
Table 4.3 a. Measured and computed currents (unweighted method of least squares)
at 200 V:
Angle
Flux
Measured
in
linkage in
Current
Cubic
Quinticf(λ') Cubic
in
f(λ') in A
In Ampere
g(λ, λ')
in Ampere
Degree Wb.turns
Computed current
Quintic g(λ, λ')
Ampere
0
0
0.14
0.09
0.133
0.083
0.136
18
0.222
0.2
0.15
0.158
0.11
0.154
36
0.423
0.26
0.158
0.172
0.15
0.175
54
0.582
0.34
0.21
0.22
0.23
0.232
72
0.685
0.44
0.28
0.293
0.30
0.308
90
0.72
0.45
0.31
0.31
0..31
0.31
10
0.685
0.34
0.223
0.21
0.208
0.2
126
0.582
0.16
0.078
0.063
0.059
0.051
14
0.423
0
-0.037
-0.05
-0.03
-0.053
162
0.222
-0.1
-0.11
-0.11
-0.062
-0.106
180
0
-0.14
-0.09
-0.13
-0.08
-0.137
9.8
9.4
10.1
9.3
R.M.S.Error %
36
Table 4.3 b. Measured and computed currents (unweighted method of least squares)
at 230 V:
Angle
Flux
Measured
in
linkage in
Current
Cubic
Quinticf(λ') Cubic
in
f(λ') in A
In Ampere
g(λ, λ')
in Ampere
Degree Wb.turns
Computed current
Quintic g(λ, λ')
Ampere
0
0
0.225
0.13
0.19
0.117
0.197
18
0.31
0.3
0.28
0.26
0.174
0.227
36
0.60
0.425
0.34
0.33
0.334
0.345
54
0.83
0.35
0.68
0.69
0.753
0.75
72
0.98
1.35
1.25
1.26
1.32
1.328
90
1.03
1.6
1.53
1.53
1.53
1.53
10
0.98
1.1
1.16
1.15
1.09
1.09
126
0.83
0.5
0.47
0.46
0.408
0.41
14
0.60
0.1
-0.014
-0.01
-0.007
-0.017
162
0.31
-0.15
-0.21
-0.192
-0.098
-0.15
180
0
-0.226
-0.13
-0.191
-0.118
-0.197
8.9
9.4
8.75
6.75
R.M.S.Error %
37
Table 4.3 c. Measured and computed currents (unweighted method of least squares)
at 260 V:
Angle
Flux
Measured
in
linkage in
Current
Cubic
Quinticf(λ') Cubic
in
f(λ') in A
In Ampere
g(λ, λ')
in Ampere
Degree Wb.turns
Computed current
Quintic g(λ, λ')
Ampere
0
0
0.25
0.147
0.216
0.133
0.23
18
0.36
0.35
0.356
0.339
0.21
0.26
36
0.68
0.6
0.48
0.471
0.47
0.47
54
0.94
1.2
1.136
1.144
1.23
1.25
72
1.11
2.1
2.21
2.226
2.3
2.33
90
1.17
2.5
2.747
2.747
2.75
2.74
10
1.11
1.8
2.1
2.09
2.01
1.98
126
0.94
0.9
0.88
0.87
0.78
0.77
14
0.68
0.15
0.032
0.042
0.04
0.036
162
0.36
-0.1
-0.265
-0.25
-0.116
-0.17
180
0
-0.25
-0.147
-0.216
-0.13
-0.22
15.05
9.4
8.75
13.6
R.M.S.Error %
38
Table 4.4a. Measured and computed currents (weighted method of least squares) at
200 V:
Angle
Flux
Measured
in
linkage in
Current
Cubic
Quinticf(λ') Cubic
in
f(λ') in A
In Ampere
g(λ, λ')
in Ampere
Degree Wb.turns
Computed current
Quintic g(λ, λ')
Ampere
0
0
0.14
0.167
0.149
0.174
0.161
18
0.222
0.20
0.229
0.213
0.232
0.220
36
0.423
0.26
0.277
0.262
0.261
0.272
54
0.582
0.34
0.344
0.333
0.348
0.343
72
0.685
0.44
0.418
0.412
0.420
0.419
90
0.720
0.46
0.423
0.423
0.423
0.423
10
0.685
0.34
0.314
0.320
0.311
0.312
126
0.582
0.16
0.145
0.157
0.141
0.146
14
0.423
0
0.002
0.016
0.002
0.007
162
0.222
-0.1
-0.096
-0.080
-0.099
-0.087
180
0
-0.14
-0.167
-0.149
-0.174
-0.161
15.05
9.4
8.75
13.6
R.M.S.Error %
39
Table 4.4 a. Measured and computed currents (weighted method of least squares) at
230 V:
Angle
Flux
Measured
in
linkage in
Current
Cubic
Quinticf(λ') Cubic
in
f(λ') in A
In Ampere
g(λ, λ')
in Ampere
Degree Wb.turns
Computed current
Quintic g(λ, λ')
Ampere
0
0
0.22
0.24
0.21
0.25
0.23
18
0.32
0.3
0.337
0.33
0.33
0.32
36
0.60
0.43
0.47
0.457
0.475
0.467
54
0.837
0.85
0.84
0.824
0.846
0.856
72
0.984
1.35
1.37
1.36
1.376
1.39
90
1.035
1.6
1.598
1.59
1.598
1.59
10
0.984
1.1
1.22
1.23
1.216
1.2
126
0.837
0.5
0.55
0.566
0.544
0.535
14
0.60
0.1
0.069
0.08
0.065
0.07
162
0.32
-0.15
-0.14
-0.136
-0.14
-0.123
180
0
-0.225
-0.24
-0.214
-0.25
-0.232
4.58
4.77
4.56
3.9
R.M.S.Error %
40
Table 4.4c. Measured and computed currents (weighted method 0f least square) at
260 V:
Angle
Flux
Measured
in
linkage in
Current
Cubic
Quinticf(λ') Cubic
in
f(λ') in A
In Ampere
g(λ, λ')
in Ampere
Degree Wb.turns
Computed current
Quintic g(λ, λ')
Ampere
0
0
0.25
0.272
0.242
0.283
0.262
18
0.361
0.35
0.387
0.400
0.384
0.364
36
0.687
0.60
0.600
0.591
0.603
0.599
54
0.946
1.20
1.270
1.254
1.278
1.303
72
1.113
2.1
2.270
1.260
2.276
2.309
90
1.170
2.5
2.731
2.731
2.731
2.731
10
1.113
1.8
2.100
2.110
2.094
2.061
126
0.946
0.9
0.943
0.959
0.935
0.909
14
0.687
0.15
0.141
0.149
0.137
0.141
162
0.361
-0.1
-0.160
-0.173
-0.157
-0.137
180
0
-0.25
-0.272
-0.242
-0.283
-0.262
13.00
13.14
12.9
12.7
R.M.S.Error %
41
Table 4.5 Measured and computed energy loss per cycle (two point method)
Voltage Measured
Computed energy loss
In
energy
Cubic
Quintic
Cubicg(λ,λ')
in quintic
Volts
Loss in joules
f(λ') in
f(λ') in
joules
in joules
joules
joules
200
0.354
0.361
0.358
0.361
0.359
230
0.76
0.759
0.739
0.759
0.755
260
0.975
0.979
0.945
0.978
0.978
g(λ,λ')
Table 4.6Measured and computed energy loss per cycle (unweighted least square
method)
Voltage Measured
Computed energy loss
In
energy
Cubic
Quintic
Cubicg(λ,λ')
in quintic
Volts
Loss in joules
f(λ') in
f(λ') in
joules
in joules
joules
joules
200
0.354
0.238
0.301
0.235
0.320
230
0.75
0.759
0.622
0.601
0.734
260
0.975
1.09
0.795
0.845
1.012
g(λ,λ')
Table 4.7 Measured and computed energy loss per cycle (weighted least square
Method)
Voltage Measured
Computed energy loss
In
energy
Cubic
Quintic
Cubicg(λ,λ')
in quintic
Volts
Loss in joules
f(λ') in
f(λ') in
joules
in joules
joules
joules
200
0.355
0.386
0.337
0.396
0.370
230
0.76
0.812
0.697
0.822
0.793
260
0.975
1.049
0.891
1.053
1.044
g(λ,λ')
42
Coefficients by different methods
The coefficients A',B' are are determined by all the suggested methods namely two point
method,unweighted least square method and weighted method of least squares given in
table (4.8).Similarly the coefficients C and D of the loss part of the magnetization have
been computed as given in the table(4.8)
Table4.8 Coefficient A,B,C,and D by different methods
Method
A'
B'
Cubicf(λ)
Quintic f(λ)
Cubic f(λ,λ')
Quintic
f(λ,
λ')
C
Two
D
C
D
C
D
C
D
point 0.33 0.969 0.218 0.008 0.219 0.006 0.218 0.002 0.219 0.03
method
Unweighted 0.10 1.195 0.125 0.124 0.184 0.052 0.113 0.241 0.190 0.191
method
Weighted
0.29 1.08
0.23
0.01
0.20
0.03
0.24
0.07
0.22
0.07
method
4.6 Conclusions:
It is observed from the comparison that the two point method and the weighted
method of least squares yield better results for the calculation of energy loss per
cycle.The magnetizing characteristic of a transformer including hysteresis has been
represented in an analytical form containing two polynomial expressions.One
polymomial expression represents the true saturation part and the other represents the loss
part .The data for the R.M.S. voltage versus magnetizing component of no load current is
useful to evaluate the coefficients of the polynomial representing the true saturation part
of the magnetizing characteristic.All the expression for hysteresis loop and the energy
loss derived by the three methods with different approximations for the loss part were
found to agree well with the experimental results.
**********************************************************
43
CHAPTER 5
SERIES
FERRORESONANT
CIRCUIT
UNDER
TRANSIENT
CONDITION
5.1Preliminaries:
The behavior of the series ferroresonant circuit fig (5.1) under the transient
condition needs a different approach to the problem than that used under the steady
condition [7,69].The variation of amplitude of oscillations with frequency is shown fig
(2.1) for both linear and nonlinear system with negligible damping.It has been found in
the nonlinear system that the amplitude of the oscillation increases as the frequency is
increased till the point B is reached. A slight change in the frequency makes the
amplitude to jump to the point C. The operating point does not return back to the pre
jump conditions when the frequency is gradually decreased. When the point D is reached
during the decrease in frequency, a jump down occurs at another critical frequency and
the operating point returns to D'.
The variation of amplitude of oscillations with forcing function at the constant
frequency fig (5.2). The amplitude of oscillation jumps from A to B at the critical value
of input V01 and then moves along BL with increase in the value of forcing function.
When the forcing function is decreased, it traces the path BC instead of BA.At the point
C,it jumps back to D at another value of the critical input V01.
The operating point
follows the path DO with further decreases in the forcing function. The zone separated by
discontinuous jump points is very unstable and unrealizable.
5.2 Analysis of Ferroresonant series circuit considering transformer core behavior:
A series circuit shown in fig (5.1a) is considerd in the analysis. As shown in the
circuit the nonlinear inductor is connected in series with a resistance R and a capacitor C.
A low resistance 'r' in parallel with the inductor is modeled as the transformer core loss to
account saturation effect, hysteresis loss and eddy-current effect. The circuit is fed with
voltage E0.The current flowing in the circuit thus have two component representing the
loss component (Ir) and the magnetizing component (IL) fig (5.3).Thus from the basic
theory of electricity we write the core loss in the inductor (W) as
W = VL I cos θ
Where θ is the power factor angle.
(5.1)
44
By using the relation for the loss component and magnetizing component
Ir = I cos θ
IL = I sin θ
And
again VL = Ir . r
Or,
r = (VL/ Ir)
(5.2)
Similarly VL = IL .XL
And
XL = (VL / IL)
(5.3)
The voltage equation of the circuit is given by
E0= VL + I(R-jXC)
(5.4)
Also I = Ir – j IL
(5.5)
Ir = VL/r , XC = (1/wC) where w =1
The equation (5.4) can be written as
E0= VL + IR-j IXC
(5.6)
The equation (5.6) after substituting the value of I becomes
E0= VL + ( Ir – j IL)R-j( Ir – j IL )XC (5.7)
E0= (VL + IrR – IL XC )-j(ILR+ IrXC) (5.8)
E0= [VL + (VL R)/r – (IL XC )]-j(ILR+[(VL /r)XC]
[E0]2 = {VL + ( VL R)/r – (IL XC )}2 + j {ILR+[(VL /r)XC}2
[E0]2 = SQRT [{VL + (VL R)/r – (IL XC)}2 + j {(ILR+(VL /r)XC}]2
(5.9)
The response curve of the circuit can be plotted using the equation (5.9) with
fixed value of resistance R and capacitance C.
5.3 Response curve considering saturation curve as piecewise linear:
The magnetizing current versus the applied voltage curve derived from the data
of R.M.S. saturation and loss curve can be approximated by a finite number of straight
line segments fig(5.4) .The voltage across the non linearity VL will lie on the kth segment
if
ck <= VL < ck+1.This can be expressed as
VL = bk( IL- ak) + ck
(5.10)
Where bk is the slope of the kth segment and is a known value for the given ak and ck. The
equations (5.9) and (5.10) can be used to draw the response curve for any circuit
45
parameter. The flow diagram is shown in fig (5.5) and the computer program is given in
the appendix.
5.4 Representation of B-H loop by Hysteresis model for studying transients:
The transformer used in electrical networks has a highly nonlinear inductance
with iron core losses. A major consequence of hysteresis among other things is that when
a transformer is disconnected from the source, residual core flux is established. This
remnant magnetism in the transformer core affects among other things on the
transients.Moreever the size and type of load if any plays a major role in the final stages
of the core after the supply is disconnected. The magnitude of the residual flux depends
more importantly on the instant of switching off the supply. The transformer core flux is
taken as a function of the flux density and hence the function of voltage. This is because
to enable the hysteresis loss, eddy current loss or a combination of both losses to be
included in the study of transformer core behavior under transient condition.
5.5 Switching Transients:
The effect of switching instant on the possibility of ferrroresonance provides
enough data for the analysis of the transformer core under the transient condition. The
exact simulation of the transient phenomenon between the switching-in of a circuit and
the possible core behavior is essential [5, 13].It is in fact a characteristic of
ferrroresonance phenomena that in a given circuit, with same parameters and same
supply voltage, different transformer core behavior possible. Obtaining one or other
conditions is dependent on the initial conditions namely the remnant magnetism, the
phase angle of the voltage on switching in and the capacitor charge etc.
The voltage across the transformer rises suddenly during switching-in or
due to some other disturbance at a particular input voltage in the same power systems
containing transformers and capacitors. As shown in the figure (5.1b) when the circuit is
suddenly switched on with a voltage OG, the operating point may either be E or F
depending upon the switching instant. Hence the conditions immediately prior to the
disturbance must be accurately known.
5.6 Representation of magnetic characteristic of the nonlinear inductor:
Linear inductance:
The linear current relationship for an inductance is represented as
46
V = L (di/dt)
Or i = (1/L) ∫ V dt
(5.11)
(5.12)
For a sinusoidal excitation
V = Vm cos wt
Or, i= (1/Lw) Vm sin wt
(5.13)
Nonlinear inductance:
The variation of flux linkages with the current can be represented by few piecewise linear
segments for the analysis of transformer core behavior.
The flux linkage λ lying on the kth segment can be written as
λk = bk( i- ak) + ck
or
(5.14)
bk = [λk - ck]/[ i- ak]
(5.15)
Where bk is the slope of the kth line representing the incremental inductance ak
and ck are the co-ordinates of the lower end of the kth segments.
5.7 General equations for the study of transients of Series Ferroresonant Circuits:
The series ferroresonant circuit as shown in figure (5.3 a) is considered for the
investigation.The nonlinear inductor is connected in series with a resistance R and a
capacitor C. As the transformer core behavior including the loss is important in the
analysis and hence the coreloss of the transformer core is modeled as a small resistance
'r'in parallel with the inductor. When a sinusoidal input voltage Em sin(wt + Ф) is fed to
the circuit. Where Ф is the switching angle.Then the basic equations for the series circuit
are
Ri + Lk (diL /dt) + 1/C∫i dt = Em sin(wt + Ф)
Or,
Lk (diL /dt) – ir.r =0
(5.17)
ir = (Lk/r)( diL /dt)
(5.18)
From the circuit i = i r + iL
(5.19)
(5.16)
If the initial values of iL and Q are given as iL0 and Q0, the solution of the differential
equation (5.16) will yield (appendix)
iL = A exp –(Rt/2 Lk) sin (ν t + θ2) + [Em sin(wt + Ф + θ1)]/Z
Where Z= sqrt {(R)2 + (w LK – 1/wC)2}
(5.20)
47
tan θ1 = (1/wC – w LK ) /R
A sin θ2 = iL0 – [Em sin( θ1 + Ф )/Z]
A cos θ2 = (1/ν LkC) [-Q0- (Em cos (Ф + θ1)/ wZ]
ν = sqrt {(1/ LK C) – (R2/4 LK2 )
The equation (5.20) is used for the investigation of transients of series
ferroresonant circuit and transformer core behaviors under transient condition.The effect
of switching angle and circuit parameters on the transients are also investigated in this
work.
5.8 Conclusions: The series ferroresonant circuit is investigated to analyze the
transformer core behavior under transient conditions. In the investigation a small
resistance is connected in parallel to the iron core inductor to study the various core
losses like hysteresis and eddy-current besides saturation effect.
**************************************************************
48
CHAPTER 6
SYSTEM UNDER INVESTIGATION AND FINDINGS
6.1 Preliminaries
The transformer core behavior under transient condition in a series ferroresonant
circuit is much different than in steady state condition. There are many factors which
come in to play .Some of the important factors are like magnetizing characteristics of
nonlinear inductor, residual flux ,switching angle, capacitor effect and various core loss
like hysteresis,saturation and eddy current etc .Besides these the circuit parameters like
resistance and capacitance have significant effect in the analysis of core behavior under
transient conditions. In the transient condition the circuit exhibits high magnitudes of
currents and voltages due to the phenomena. The flux linkage versus magnetizing current
characteristic is considered to be piecewise linear unlike the polynomial form in the
calculation of loss part. This is because piecewise form is better for computational
purposes than the polynomial form in case of study of transformer core behavior. The
present investigation includes transformer core loss, residual flux, capacitor effect;
switching angle .The investigation on the analysis of transformer core behavior under
transient condition has been validated by experimental results.
6.2 System characteristics:
The laboratory study is carried on by setting up a single series ferroresonant
circuit. (Fig 6.1) The linear parameter chosen are as follows
R = 1 ohm
C= 12 Microfarads
The primary winding of an open circuited transformer with the characteristic as follows
3 KVA, 230 V/230V, 50 Hertz, single phase
The effect core loss is represented by a low variable resistance across the magnetizing
reactance. The rms excitation characteristics and the loss curves of the transformer under
transient conditions are obtained as in the circuit shown in the figures (6.1) .
6.3 True saturation characteristics curve:
In stead of the rms saturation curve available from the manufacturer of the
transformer the true saturation curve has been considered to represent the magnetic
49
characteristic of the transformer under study. An integrator is used as shown in the fig
(3.2) for the determination of true saturation curve. As shown in the figure the nonlinear
inductor is fed with a sinusoidal voltage which after integration yields flux
linkage(λ).This flux linkage is then fed to the Y channel of a cathode ray
oscilloscope(CRO).The different voltage signal proportional to corresponding inductor
current is fed to the X channel of the CRO.The hysteresis loop of various shape and sizes
are displayed on the screen of CRO as the input voltage to inductor is varied .The tips of
the hysteresis loops corresponding to different voltages are then joined to give the i L
versus λ characteristics. The curve obtained under laboratory test condition is shown in
the figure (4.3)
Experimental data for the true saturation curve:
Table (6.1)
Sl No.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Flux linkage (λ)
Wb-turn
0.126605
0.218423
0.308466
0.410176
0.511214
0.673511
0.731042
0.750456
0.7711
0.81042
0.86102
0.9004
1.0102
1.04484
1.0882
1.1505
1.17833
1.26212
1.35244
1.6452
Inductance current (iL)
In Ampere
0.027
0.0584
0.1114
0.1512
0.26030
0.4703
0.56132
0.583402
0.6524
0.7688
0.934
1.05096
1.30425
1.4184
1.7824
2.118
2.521
3.682
5.6882
6.0242
50
The value of inductance is assumed to remain constant for the entire length of each
segment. The value of the inductance for each segment remains same for two limiting
values for current in each segment and it changes as the current value considered for the
segment considered crosses the limiting value.
6.4. Transients of series ferroresonant circuit under study:
The transients obtained with the help of following equation (6.5.1) (Appendix-I) .The
computed response curve has been checked with the experimentally obtained curve.
IL= A exp – (Rt/2Lk) Sin (ν t +Ө2) + [EmSin (wt +ф+Ө1)]/Z (6.5.1)
The current is computed in small steps with the variation of time using the
equation (6.5.1).If the current calculated in small steps on the kth linear segment of the
magnetizing characteristic crosses either of the limiting values of the segment .It is
always essential to find the exact instant at which this occurs (66)
Let the limiting current value be I, and then the time is obtained by solving equation
(6.5.1) as
G (t) = [EmSin (wt +ф+Ө1)]/Z + A exp – (Rt/2Lk) Sin (ν t +Ө2) = I
(6.5.2)
The value of t in the above equation is solved using Newton-Raphson iteration method in
the following ways:
The function G'(t) = [w EmCos (wt +ф+Ө1)]/Z] + A exp – (Rt/2Lk) {ν Cos (ν t +Ө2) (R/2Lk) Sin (ν t +Ө2)
(6.5.3)
Initially the value of t is assumed as the time step just before the current crosses the
limiting value t'. Then
Δt = [I – G(t')]/G'(t)
(6.5.4)
The value of t is now modified as
t=t' + Δt
(6.5.5)
With the above value taken as t', G (t') is again computed and another value of t is
obtained by relation (6.5.3) and (6.5.4).This method is repeated till Δt is negligibly small.
The final value of t then become the solution of equation (6.5.2)
Finding out the solution:
For kth segment of the characteristic, the current at small interval is computed as soon as
it crosses any of the limiting value of current for the segment. The exact time at which it
occurs is determined by the technique mentioned. The current and charge at this instant
51
becomes iL(0) and Q0 with k = k+1 or k-1 . The time t measured from this instant for
calculation of next linear segment of magnetizing characteristic.
6.5. Intial charge of capacitor and switching angles
The initial charge in the capacitor is obtained by using the relation Q= ∫ iLdt
The value of iL is obtained by using equation (6.5.2) and the initial value of i L (0) and Q0
are taken as zero for the first linear segment. The initial charge for the segment is
calculated by the integration of current (iL) in the time interval of first segment when it
crosses the limiting value. The initial value of current for the second segment is the
limiting value for the first segment. In this way all the linear segments of the
characteristic are used for the computation of the initial charge in the capacitor.
The phase angle of voltage on switching play one of the important effect of transformer
core behavior under transient condition .For this purpose a triac circuit is used to
determine the exact instant of switching in.The analysis is done for two switching angles
zero and 90 degree.
52
[CRITICAL VOLTAGE AND CURRENT UNDER STEADY STATE]
[R=1 OHM C = 12 MICROFARAD, VOTAGE PEAK=480 V, CURRENT PEAK =
750 MILLIVOLT]
53
CURRENT UNDER STEADY AND STATE
[ R=1 OHM, C=12 MICROFARAD]
54
EFFECT OF INPUT TRANSIENT VOLTAGE TOP-200 V, MIDDLE-230 V,
BOTTOM-260 V, R=1 OHM, C = 12 MICROFARAD,SWITCHING ANGLE=0
55
6.6. Effect of input voltage:
The effect of input voltages namely 200 volts, 230 volts and 260 volts across the iron
core inductor of the transformer under transient condition have been studied. The
computed values are given in tables (6.1) and (6.2) respectively. The computed and
experimental plots have been presented in figures (6.2).The curves found to show that the
computed and experimental data found to be in close agreement.
TRANSIENTS FOR 200 VOLTS
AND ZERO DEGREE SWITCHING
ANGLE
CURRENT IN
AMPERES
10
5
0
-5
1
5
9
13 17 21 25 29 33 37 41 45 49 53
-10
-15
TIME IN MILLISECONDS
TRANSIENTS AT 230 VOLTS AND
ZERO DEGREE SWITCHING
ANLES
CURRENT IN
AMPERES
15
10
5
0
-5
1
5
9
13 17 21 25 29 33 37 41 45 49 53
-10
-15
TIME IN MILLISECONDS
CURRENT IN
AMPERES
TRANSIENTS AT 260 VOLTS AND
ZERO DEGREE SWITCHING
ANGLE
15
10
5
0
-5 1
-10
-15
-20
6 11 16 21 26 31 36 41 46 51
TIME IN MILLISECONDS
COMPUTED TRANSIENTS FOR SWITCHING ANGLE= 0 DEGREE FOR
VOLTAGE 200V, 230 V AND 260 V
56
CURRENT IN AMPERES
TRANSIENTS AT 200 VOLTS AND
90 DEGREE SWITCHING ANGLE
8
6
4
2
0
-2
-4
-6
-8
1
5
9
13 17 21 25 29 33 37 41 45 49
TIME IN MILLISECONDS
CURRENT IN AMPERES
TRANSIENTS AT 230 VOLTS AND
90 DEGREE SWITCHING ANGLES
10
5
0
1
5
9 13 17 21 25 29 33 37 41 45 49 53
-5
-10
TIME IN MILLISECONDS
CURRENT IN AMPERES
TRANSIENTS AT 260 VOLTS AND
90 DEGREE SWITCHING ANGLE
10
5
0
1
5
9 13 17 21 25 29 33 37 41 45 49 53
-5
-10
TIME IN MILLISECONDS
EFFECT OF INPUT TRANSIENT VOLTAGE TOP-200 V, MIDDLE-230 V,
BOTTOM-260 V, R=1 OHM, C = 12 MICROFARAD, SWITCHING ANGLE=90
57
EFFECT OF VARIATION OF CAPACITANCE TOP = 24 MIDDLE= 18
BOTTOM = 12 MICROFARADPEAK CURRENT = 30 A, 21 A AND 14.5 A
58
HYSTERESIS LOOP UNDER NORMAL CONDITION
59
6.7. Computed results.
Table (6.2)
Applied input voltage
E0 in Volts
0
22.5
40.1
44.1
55.5
64.1
70.8
77.1
82.5
86.2
90.1
89.5
86.8
85.1
80.5
71.3
65.8
53.5
46.9
38.0
45.5
133.5
165
206
307
Voltage across nonlinear
Inductor VL in volts
0
40
70
80
100
110
120
130
140
150
160
170
180
190
200
210
220
230
240
250
260
270
280
290
300
Current through nonlinear
Inductor iLin Amperes
0
0.101
0.16
0.18
0.22
0.29
0.24
0.26
0.28
0.31
0.34
0.39
0.45
0.51
0.58
0.67
0.76
0.88
0.98
1.15
1.35
1.59
1.92
2.3
2.83
60
6.8 Experimental results.
(R= 1 ohm C = 12 microfarad)
Table (6.3)
Input voltage
(E0 ) In volts
0
20
30
40
50
60
70
80
90
100
104
104
110
120
130
140
150
160
180
200
230
260
230
200
180
160
140
120
100
70
50
46
46
40
30
20
0
Voltage
nonlinear
Inductor
volts(VL)
0
24
37.5
52
61.5
81
96
109
126
142
148
280
282
285
285
286
287
291
293
295
298
301
298
295
293
291
286
285
278
270
262
254
58
52
38
25
0
across Current through
Nonlinear inductor
in (iL )In Amperes
0
0.025
0.05
0.08
0.1
0.125
0.14
0.156
0.18
0.226
0.25
1.9
1.95
1.95
1.98
1.985
2.02
2.1
2.16
2.26
2.52
3.16
2.85
2.62
2.43
2.36
2.02
1.98
1.86
1.64
1.36
1.26
0.1
0.08
0.06
0.03
0
61
6.9 Study of Transients:
(A) Different input voltages and fixed values of R = 1 ohm, C = 12 ohm and
Switching angle = 0 degree:
Table (6.4)
Time (t) in
Current through nonlinear inductor ( iL)
Millisecond Applied input
Applied input voltage
voltage
(200 (230 volts)
volts)
0
0
0
2
0.1
0.124
4
0.155
0.19
6
0.132
0.1604
8
0.038
0.038
10
-0.12
-0.136
12
-0.25
-0.294
14
-0.305
-0.371
16
-0.275
-0.332
18
-0.15
-0.185
20
0.035
0.036
22
0.205
0.318
24
0.836
1.0
26
-9.78
-11.71
28
-6.34
-4.68
30
3.94
5.3
32
8.26
10.45
34
0.025
0.072
36
-5.606
-7.34
38
-3.82
-4.62
40
2.31
3.25
42
6.04
8.24
44
-0.28
-0.584
46
-3.88
-5.12
48
-1.35
-1.682
50
3.44
4.52
52
3.82
4.88
54
-0.612
-0.972
56
-3.868
-5.02
58
-2.02
-2.45
60
1.7
2.38
62
2.1
2.68
64
-0.796
-1.18
66
-2.31
-3.05
68
-0.086
-0.064
Applied
input
voltage(260 volts)
0
0.13
0.201
0.431
0.495
0.295
-0.155
-0.691
-1.08
-1.12
-0.732
0.005
1.1
-13.85
-5.25
6.05
12.326
-0.002
-8.32
-5.12
3.81
9.92
-0.751
-5.76
-1.88
5.14
5.42
-1.16
-5.66
-2.68
2.78
3.04
-1.37
-3.48
-0.732
62
70
72
74
76
78
80
82
84
86
88
90
92
94
96
98
100
2.78
2.28
-0.85
-2.6
-1.12
1.05
0.904
-0.86
-1.27
0.556
2.18
1.42
-0.808
-1.74
-1.36
0.052
3.615
2.78
-1.14
-3.35
-1.285
1.76
1.152
-1.206
-1.676
0.732
2.81
1.718
-1.12
-2.214
-0.758
0.742
4.065
3.212
-1.424
-3.752
-1.372
1.735
1.306
-1.310
-1.92
0.808
3.121
1.882
-1.276
-2.462
0.782
0.892
(B) different input voltages and fixed values of R = 1 ohm, C = 15 ohm and
Switching angle = 90 degree
Table (6.5)
Time (t) in
Current through nonlinear inductor ( iL)
Millisecond Applied input
Applied input voltage Applied
voltage (200 volts)
(230 volts)
inputvoltage(260 volts)
0
0
0
0
2
0.1
0.124
0.13
4
0.155
0.19
0.201
6
0.132
0.1604
0.431
8
0.038
0.038
0.495
10
-0.12
-0.136
0.295
12
-0.25
-0.294
-0.155
14
-0.305
-0.371
-0.691
16
-0.275
-0.332
-1.08
18
-0.15
-0.185
-1.12
20
0.035
0.036
-0.732
22
0.205
0.318
0.005
24
0.836
1.0
1.1
26
-5.78
-7.71
-8.85
28
-3.34
-4.68
-5.25
30
3.94
5.3
6.05
32
6.26
8.05
9.026
34
0.025
0.072
-0.002
36
-5.606
-7.34
-8.32
63
38
40
42
44
46
48
50
52
54
56
58
60
62
64
66
68
70
72
74
76
78
80
82
84
86
88
90
92
94
96
98
100
-3.82
2.31
4.04
-0.28
-3.88
-1.35
3.44
3.82
-0.612
-3.868
-2.02
1.7
2.1
-0.796
-2.31
-0.086
2.78
2.28
-0.85
-2.6
-1.12
1.05
0.904
-0.86
-1.27
0.556
2.18
1.42
-0.808
-1.74
-1.36
0.052
-4.62
3.25
5.24
-0.584
-5.12
-1.682
4.52
4.88
-0.972
-5.02
-2.45
2.38
2.68
-1.18
-3.05
-0.064
3.615
2.78
-1.14
-3.35
-1.285
1.76
1.152
-1.206
-1.676
0.732
2.81
1.718
-1.12
-2.214
-0.758
0.742
-5.12
3.81
5.92
-0.751
-5.76
-1.88
5.14
5.42
-1.16
-5.66
-2.68
2.78
3.04
-1.37
-3.48
-0.732
4.065
3.212
-1.424
-3.752
-1.372
1.735
1.306
-1.310
-1.92
0.808
3.121
1.882
-1.276
-2.462
0.782
0.892
7.10. Conclusions:
Investigations of transients for study of transformer core behavior have been done in
detail for a single phase series Ferro resonant circuit. The behaviors of the circuit under
transient conditions have been analyzed both theoretically and experimentally. This
present chapter gives a detail account of the results obtained theoretically and
experimentally. A brief comparison of both the results of all the findings described in the
previous chapters have been done in tabular form and also in graphical form for easy
understanding and analysis of the problem and its solution suggested. The CRO reports
are shown for necessary comparison with analytically computed curves.
64
CHAPTER 7
GENERAL CONCLUSIONS AND SUGGESTION FOR FUTURE
WORK
7.1 General Conclusions:
The analyses of magnetizing characteristics of a transformer for study of
transformer core behavior under transient conditions have been studied in depth in the
present thesis. The magnetization characteristic of a transformer with negligible iron loss
is modelled by its magnetizing branch. A generalized technique has been developed to
derive its true saturation characteristic in piecewise linearised segments .The suggested
method to derive the instantaneous curve is accurate over any extended range of I.C,
including very high voltages in te saturated air core region.
The accuracy of the method has been improved by optimizing the number of
segments representing the I.C over a desired range thus the computer aided method for
deriving I.C with optimum number of segments from the r.m.s. saturation data yields an
optimal true saturation characteristics with prescribed degree of accuracy.
A new concept of r.m.s error has been introduced in the thesis for deriving
magnetization and hysteresis loss characteristics of transformers. This has resulted in
more accurate results when compared to average error approach.
Moreover the method computation of I.C. does not involve any trial and error
procedure whereas the method suggested by other earlier workers involve cut and trial
process to obtain a point on the instantaneous curve.
The true saturation characteristic has been represented by a two term fifth degree
polynomial and it is shown that it can be predetermined using r.m.s. data available from
the manufacturer.The methods
suggested for predetermination of I.C are two point
method unweighted least square method. The two point method is suitable where the
computation for modeling should be simple and quick. This is a new approach for the
predetermination of an I.C in an analytical form from the manufacturer's data in r.m.s
form.By this method,the process of conversion from r.m.s curve to I.C. and I.C. to r.m.s.
form is possible through two analytical equations where as the earlier workers suggested
an analytical method of conversion. Out of various methods for predetermination of I.C,
65
the method of using multiple point data and weighted least square method yields more
accurate results.
Each of the methods suggested for predermination of I.C using a polynomial has
also been applied to evaluate the resultants true saturation characteristic of two reactors in
parallel or series.
It has been found that the true saturation characteristic of individual reactors need
only be available in one form ,namely, i=f1(λ) whereas the method suggested by Talukdar
et al, requires these characteristics in two forms , i=f1(λ) and λ=f(i).
A procedure is developed for quantitative determination of third harmonic flux
linkage and consequent over voltage magnitudes on the secondary side of the Y/∆
transformer bank under certain internal fault conditions.The computed results validated
by experiment establish the existence of over voltage phenomenon so that the distribution
engineers in future to make necessary provisions for proper coordination of protective
devices.
Generally,it is adequate to model the transformer saturation characteristic by a
single valued nonlinear inductance,neglecting the hysteresis phenomenon.However, for
improved accuracy it is desirable to include hysteresis phenomenon.In this thesis the
magnetization characteristic of a transformer including hysteresis has been represented in
an analytical form containing two polynomial expressions one polynomial expression
representing the true saturation part and the other representing the loss part of the
magnetization characteristic. Each of the two polynomial expressions has been formed
with two terms only. The data of r.m.s voltages versus magnetizing component of no load
current are useful to evaluate the coefficient of polynomial expression representing the
true saturation part of the magnetization characteristic. Similarly the data of r.m.s voltage
versus the loss component of no load current is useful for finding the coefficients of the
loss part of the magnetization characteristic. Both the data are obtained either from the
manufacturer or a test on the transformer.
Modelling of a transformer including hysteresis is desirable in transient
simulation studies such as the inrush current drawn by a transformer possessing a residual
flux when it is energized;the determination of residual flux remaining in a transformer
when it is deenergised and examination of phenomena etc.For the purpose,the modeling
66
of hysteresis loop has been presented in the existing literature by either a differential
equation or some mathematical expressions based on fourier series or arational function
approximation.All these suggested methods require experimental hysteresis loops t for
determination of magnetization characteristic where as the methods suggested in the
thesis do not require the experimental hysteresis loop for its determination but require
only the r.m.s data including no load loss. The expressions suggested for describing the
hysteresis loop exhibits the hysteretic property of increasing the loop area with increase
of frequency of operating voltage.All the expressions describing the hysteresis loop under
various approximations of loss part have been used to derive the corresponding
expressions for hysteresis energy loss.
The present thesis is an in-depth investigation of transformer core behavior under
transient conditions applied to series ferroresonant circuits in this study; a mathematical
model was developed to analyze transient phenomena under different circuit parameters
and for different switching angles. The mathematical model developed in the thesis for
different transformer core losses is simple for computational purposes than other methods
discussed by earlier workers in this area. The results obtained in the suggested method
are found to agree well with the experimental findings. The new method is based on
mathematical model based on piecewise linearization of magnetic characteristic of the
iron core inductor.
It was found that a voltage applied suddenly to the system shows more severe
jump in the system parameters like current or voltage then that of input voltages applied
gradually.In the later case the circuit attains ferroresonant conditions for particular input
voltage which is a function of system resistance and capacitance.The initial input voltage
for depends on the value of resistance ,capacitance and switching angle.This condition is
achieved when the magnitude of suddenly applied voltage is more than the critical value
corresponding to particular circuit condition.The ferroresonant conditions giving rise to
very high system voltage and current occurs when the phase angle on switching in is zero
degree.the transformer core loss in this condition is the minimum but at switching angle
90 degree the jumps are very less and the transformer core loss is maximum.An increase
in the value of circuit capacitance requires a higher input voltage for the onset of under
the transient condition.The peak value of current through the inductor increases with a
67
decrease in the value of circuit résistance. The effect of core loss by way of putting
resistance in parallel to the inductor circuit gives better result than calculating the loss
separately in to consideration.
7.2. Suggestions for future work:
The present investigation can be potentially extendable to three phase
ferroresonant circuits.The coreloss which is modeled by a constant resistance across the
nonlinear reactor can be improved by considering the three nonlinear effects found in the
iron cores viz saturation,hysteresis and eddy currents. The leakage reactance and
resistance of primary winding so far neglected in this analysis can be taken into account
for more correct investigation of the core behavior under transient conditions. The effect
of residual flux on transients can be further improved by better modeling the hysteresis
loss and of including dielectric loss of the capacitor. Inclusion of three terms in the
polynomial expression for magnetizing saturation curve is so far not included in the
investigation which can be done in future work.Multivalued modeling having the major
and minor loops of magnetization extending the methods suggested can be taken in the
further work.
68
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70
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**********************************************************
APPENDIX
Computation of current under transient condition
74
The equation (4.6.9) is a second order differential equation can be solved by use of
Laplace transform method. The time response differential equation
LkdiL/dt + RiL+ (1/C) ∫iLdt = Em Sin (wt+ф)
(1)
Taking the Laplace transform of equation (1) we get
–
LkSiL(S)
LkiL(0)
+
RiL(S)
+
iL(S)/C.S+Q0/C.S
ф)(S2+w2)]+[(S.EmSinф)/(S2+w2)]
S2iL(S)-SiL(0)
+
(R/Lk)
S
iL(S)
=[(w.EmCos
(2)
+
[iL(S)]/[Lk
Cosф]/[LK(S2+w2)]+[SEmSinф]/[Lk(S2+w2)]
C]+Q0/(Lk.C)
=
[Emw
S
(3)
Rearranging the equation (3) we get
IL(S)[S2 +(R/Lk)S + 1/Lk.C]-S iL(0) + Q0/Lk.C = [Em w S Cos ф]/[Lk(S2 +w2)] + [ S2
EmSinф]/[Lk.(S2+w2)]
(4)
IL(S) = ([S iL(0)]/[S2+(R/Lk)S + 1/Lk.C])- Q0/Lk.C [S2 +(R/Lk)S + 1/Lk.C])+ [Em w S Cos
ф]/[ LK(S2+[ R/Lk)S+1/Lk.C)( S2+w2)]+ [ S2 EmSinф] /[Lk.(S2+w2)]
1/Lk.C]
S2 +(R/Lk)S +
(5)
The time domain equation of current is obtained by taking inverse Laplace of both side of the
equation (5)
Considering each term of the equation (5) inverse Laplace of
(1)
iL(S) = iL
(2)
[S iL(0)]/[S2+(R/Lk)S + 1/Lk.C] is breaked to [S iL(0)]/ [S2+(2 R/2Lk)S++[
2
R/2Lk) +( 1/Lk.C)]
(6)
(7)
2
2
= [iL(0)].S/[(S+R/2Lk) +[( 1/Lk.C)-( R/2Lk) ]
= [ iL(0)].S/[( S+a)2+b2]
Where
a= R/2Lk
b= [(1/Lk.C)-(R2/4Lk2)]
a2+b2=1/Lk.C =w02
(8)
75
Now inverse Laplace of equation (8) is
= iL(0) [exp(-at).Cos(bt)-(a/b)exp(-at)Sin(bt)]
(3)
(9)
–Q0/[ Lk.C(S2+( R/Lk)S+1/Lk.C)]
= - (Q0/ Lk.C) [1/(S+a)2 +b2]
(10)
Inverse Laplace of equation (15) is
= - (Q0/b Lk.C) exp (-at) Sin (bt)
(11)
(4) (Em w S Cosф)/(Lk(S2 +w2)(S2 + [R/Lk] S +1/Lk.C] = [(Em Cosф)/Lk ][w.S/(S2+
w2)[(S+a)2+b2]}
(12)
The partial fraction of equation (12) is
= [(Em Cosф)/Lk][B0/( S2 +w2) +B1S/( S2 +w2)+C0/)[(S+a)2+b2] + C1S/)[(S+a)2+b2}]
(13)
The values of B0 , B0 , C0 ,C1 can be found out
B0= 2 aw3/D
(14)
B1= -w (w2 –w02)/D
(15)
C0 = - (2aw w02)/D
C1 = w (CD
(16)
(17)
Where D= (C + 4 a2w2
Putting the values c in equation (13)
=[(Em Cosф)/Lk][(2aw3)/D(S2 + w02)- w ((w2 –w02)S/D. (S2 +w02)+ (-2aww02)/(D[(Sa2)+b2])+[ w(w2 –w02)S]/D[(S+a)2+b2]]
(18)
Rearranging the equation (18)
= [(Em Cosф)/Lk ][(2aw3)/D(S2+w02) - w(w2 –w02 )/D] (S/( S2 +w2))+[ w(w2 –
w02)(S+a)]/D[(S+a)2+b2]]
(19)
Taking the inverse Laplace of equation (19)
= (Em
Cosф)/Lk ][( 2aw2/D)Sin (wt)-[w( w2 –w02)/D] Cos (wt) + [w(w2 –
w02)/D]exp(-at) Cos (bt)[aw (w2 +w02) Db] exp (-at) Sin (bt)]]
(20)
(5)Laplace inverse of
[S2Em Sinф]/[ Lk ( S2 +w2) (S2 +R/ Lk ).S + 1/Lk.C]
= [(Em Sinф]/ Lk ][ S2/( S2 +w2).[( S+a )2)+b2]] (21)
76
Taking the partial fraction of equation (21)
=[(Em Sinф]/ Lk ][B0/( S2 +w2) +B1S/( S2 +w2) +C0[( S+a )2)+b2]+C1S / [ ( S+a )2)+b2
]]
The value of B0 , B0 , C0 ,C1 can be found out as
B0= w2(w2 –w02)/D
(22)
B1= 2aw2/D
(23)
C0 =-w0 2(w2 –w02)/D
(24)
C1= -2aw2/D
(25)
Putting the values of B0 , B0 , C0 ,C1 in equation (21)
= [(Em Sinф]/ Lk] [ w(w2 –w02)/ D(S2 +w2) + 2aw2S/D(S2 +w2) - w0 2(w2 –w02)/D{(
S+a )2)+b2} - 2aw2S2/D [( S+a )2)+b2]]
(26)
=[(Em Sinф]/ Lk] [ w(w2 –w02)/ D(S2 +w2) + 2aw2S/D(S2 +w2)- 2aw2(S+a)/D [( S+a
)2)+b2}+ B {2aw2- w0 2(w2 –w02)/D b{(( S+a )2)+b2}
(27)
The inverse Laplace equation (27) is
= [(Em Sinф]/ Lk] [ w(w2 –w02)Sin (wt)/D + (2aw2/D).Cos (wt)-( 2aw2/D)exp(-bt)
Cos (bt)) +[2aw2- w0 2(w2 –w02)/D b}[ exp(-at) Sin (bt)]]]
(28)
Now adding the equations (9), (11), (20), (28)
IL = IL(0)[exp Cos (bt) – (a/b)exp (-at) Sin (bt)] – [Q0/b Lk. C ][exp(-at) Sin(bt)] +[Em
Cosф)/Lk ]. [(2aw2/D).Sin (wt)-{w/D} {(w2 –w02)}}Cos(wt)+ {w(w2 –w02)/D}exp(at)Cos (bt)-[aw/bD]( w2 –w02)exp(-at) Sin(bt)] +[(Em Sinф]/ Lk] [w(w2 –w02)Sin
(wt)/D] Sin wt + [(2aw2/D).Cos (wt) +[2aw2(bt)]
(29)
w0 2(w2 –w02)/D b}[ exp(-at) Sin
77
The equation (29) can be solved as
IL(1) = [(Em Sinф]/ Lk][{ 2aw2/D).Cos (wt)}Sin (wt)-(w/D)(w2 –w02)}}Cos(wt)+ {[Em
Sinф]/ Lk] [w(w2 –w02)/D Sin(wt) + [(2aw2/D).Cos (wt)]]
(30)
= [[(Em / Lk].{ { 2aw2/D).Cos (wt)}Sin (wt)-(w/D)(w2 –w02)}}Cos ф.Cos(wt)+ w02(w2
–w02)/D}Sin(wt)Sinф+[2aw2/D].Cos (wt)}Sinф]
=[
Em
/
Lk
][[
w(w2
–w02)/D)[
(31)
Sin(wt).Sinф
+(2aw2/D)[Cosф.Sin(wt)+Cos(wt).Sinф]]
–
(34)
Where
B SinӨ = -w (w2 –w02)/D
(35)
B Cos Ө= 2aw2/D
(36)
And B = SQRT [[-w (w2 –w02)/D]2 + [2aw2/D]2]
= SQRT [w2(w2 –w02)2/D2 + [4a2 w4/D2]
= (w/D) SQRT [((w2 –w02)2 + 4a2 w2]
= (w/D) SQRT D
= w/ SQRT D
(37)
Putting the values of D in equation (37)
B= w/ SQRT [4a2 w2 + (w2 –w02)2]
= (w/ SQRT [4R2/4 Lkw2) + (w2 –1/LkC)2]
= w/ SQRT [R2w2./ Lk +(w2 –1/LkC)2]
(wt).Cosф]]
(32)
= [Em / Lk ][[ -w(w2 –w02)/D)[ Cos(wt+ф) +( 2aw2/D). Sin (wt+ф)
=( Em / Lk) B Sin (wt+ф+Ө)
Cos
(33)
78
= Lk / Z
(38)
Putting the values of B in equation (34) and solving we get
= (Em / Z) [Sin (wt+ф+Ө)]
(39)
Now TanӨ= - [w (w2 –w02)/D] 2 + [2aw2/D]
= - (w2 –w02)/2 aw
(40)
Putting the value of a and w0 , we have
= (1/ LkC – w2) / R w/Lk
(41)
= (1/wC – wLk)/R
(42)
(b) The other parts can be solved by taking the rest of the terms
IL(2) = IL(0)[exp Cos (bt) – (a/b)exp (-at) Sin (bt)] – [Q0/b Lk. C ][exp(-at) Sin(bt)]
+[Em Cosф)/Lk ].[ w(w2 –w02)/D exp (-at) Cos (bt)]-[aw/bD]( w2 +w02 ) exp(-at) Sin
(bt)] + [Em Sinф]/ Lk][(-2aw2/D) exp (-at) Cos (bt)]+{ 2aw2exp(-at) Sin(bt)]
w0 2(w2 –w02)/D b}[
(43)
= exp (-at) Sin (bt)[-(a/b) IL(0)- Q0/bLkC – [Em Cosф)/Lk ](a/b) w(w2 +w02 )/ D +[[ Em
/ Lk] Sinф{2a2 w2- w0 2(w2 –w02)/D b] + exp (-at) Cos (bt)]+{[ IL(0+ Em / Lk) Cosф
(w/D) ((w2 –w02)-( 2aw2/D] Em Sinф]/ Lk]
= exp (-at) [A Sin (bt) + Ө2]
(44)
(45)
Where
A Cos(Ө2) = [(-a/b) IL(0)[ - Q0/bLkC – (Em Cosф)/Lk) (a/b) (w2 +w02 )/ D +[[ Em/ Lk]
Sinф{2a2 w2-w 2(w2 –w02)/D b]
(46)
And
A SinӨ2 = [IL(0+ (Em / Lk) Cosф(w/D) ((w2 –w02)-( 2aw2/D) (Em/ Lk ) Sinф]
(47)
79
The equation (46) and (47) are simplified further as
A Sin Ө2 = IL(0+ (Em / Lk) Cosф(w/D) ((w2 –w02)-( 2aw2/D) (Em/ Lk ) Sinф]
= IL(0)- (Em / Lk) Cosф(-w/D) (w2 –w02)-( 2aw2/D) (Em/ Lk ) Sinф]
= IL(0)- (Em / Lk) [w S (-w/D) ((w2 –w02)+ Sinф ( 2aw2/D)]
(48)
Now Let us assume
(-w/D) (w2 –w02) = Y SinӨ
(49)
And 2aw2/D = YCosӨ
And
Y = SQRT [(w2/D2) (w2 –w02)2 + [4a2 w4/D2]
= (w/D) SQRT [(w2 –w02)2 + 4a2 w4]
= (w/D) SQRT (D)
(50)
Putting the value of D
Y = (w/D) SQRT (D) = Lk/Z
(51)
Now A SinӨ2 = IL(0)- (Em / Lk) (Lk/Z) Sin (Ө +ф)
(52)
Similarly,
A Cos(Ө2) =[[(-a/b) IL(0)[ - Q0/bLkC - (Em / Lk).aw [(w2 +w02 )/ Db]Cosф+(Em/ Lk ) {
2aw2-( w02/Db).(w2 –w02) Sinф]
= (-a/b) IL(0) + 1/ bLkC[-Q0- Em a C (w/D).( w2 +w02 )Cosф + Em C{2aw2-(w02(w2
–w02)}/Db Sinф]]
=(-a/b) IL(0) - Q0/ bLkC – [E// bLkC][aCw(w2 +w02 )}/D Sinф]]
Now again let us assume
P CosӨ = (aCw/D) (w2 +w02 )
(55)
(54)
80
P SinӨ = C {2a2w2-(w2–w02)}/D
(56)
And
P= SQRT [{(aCw/D) (w2 +w02)}2 + {2a 2w2C-C w02 (w2–w02)}/D]2]
= (C/D) SQRT {( w2–w02)2(a2 w2+ w04 ) + 4 a2 w2(a2 w2+ w04 )}
=( C/D) SQRT [(a2 w2+ w04 ).D]
=(C/SQRT D) SQRT [(a2 w2+ w04 ).
(60)
Now the equation (54) can be written as follows
= (-a/b) IL(0) - Q0/ bLkC – Em / bLkC[P Cos (Ө+ф)]
Substituting the value of P in (61)
We get
= (-a/b) IL(0) - Q0/ bLkC – Em / bLkC[(C/D) SQRT [(a2 w2+ w04 ). Cos (Ө+ф))
(62)
On simplification and assumption of small resistance the first term of equation
(62) can be neglected.
= - Q0/ bLkC – (Em / bLkC)[ Cos (Ө+ф)/wZ]
(63)
= (- 1/ bLkC)[ Q0+ (Em / wZ)[ Cos (Ө+ф)]
(64)
Now the complete equation can be written as
iL = iL(1) + iL (2)
(65)
iL = (Em / Z) Sin (wt +ф+Ө1) + exp –(Rt /2Lk) A Sin (bt +Ө2)
(66)
COMPUTER PROGRAMS
1. COMPUTATION FOR CURRENT OF SERIES FERRORESONANT CIRCUIT
UNDER TRANSIENT CONDITION:
81
**********************************************************************
C
R1-SERIES RESISTANCE
C
AL-NONLINEAR INDUCTANCE
C
C1-CAPACITANCE
C
E-INPUT APPLIED VOLTAGE
C
Q-INITIAL CAPACITOR CHARGE
C
S-INITIAL CURRENT
C
PHI-SWITCHING ANGLE
C
T-TIME
C
R2-LOSS RESISTANCE MODELLED PARALLEL TO
C
INDUCTOR
WRITE(*,*)R1,AL1,C1,W,E,Q,S,T,TF,EPS,PHI
READ(*,*)R1,AL1,C1,W,E,Q,S,T,TF,EPS,PHI
R2=1470.0
AL=AL1+ (AL1*R1/R2)
R=R1+AL1/ (C1*R2)
C=C1
XL= (W*AL)-(1.0/ (W*C))
Z=SQRT(R*R+XL*XL)
XL=-XL
THETA1=ATAN (XL/R)
V1=(1.0/(AL*C))-(R/(2.0*AL))**2
V=SQRT (V1)
PHI2=PHI+THETA1
B=S-(E*SIN (PHI2))/Z
D=(1.0/(V*AL*C))*(-Q-(E*COS(PHI2))/(W*Z))
50
PHI1+0.314*T+PHI+THETA1
AI1= (E*SIN (PHI1))/Z
F1= (-(R*T))/ (2.0*AL))
F2=F1/1000
82
F=EXP (F2)
AI2=F*B*COS (.001*T*V)+F*D*SIN(.001*T*V)
AI=AI1+AI2
VL=AI*XL
WRITE (*,*) T, AI, VL
T=T+EPS
IF (T.GT.TF) GO TO 100
GO TO 50
100
STOP
END
COMPUTER PROGRAM –2
NEWTON-RAPHSON METHOD FOR DETERMINATION OF EXACT TIME
C
R1-SERIES RESISTANCE
C
AL1-NONLINEAR INDUCTANCE
C
C1-CAPACITANCE
C
E-INPUT VOLTAGE APPLIED SUDDENLY
C
Q-INITIAL CHARGE IN THE CAPACITOR
C
S-INITIAL CURRENT
C
PHI-SWITCHING ANGLE
C
G=LIMITING VALUE OF CURRENT
WRITE(*,*)R1,AL1,C1,W,E,Q,S,T,G,EPS,PHI
READ(*,*)R1,AL1,C1,W,E,Q,S,T,G,EPS,PHI,AI
R2=1470.0
AL=AL1+ (AL1*R1/R2)
R=R1+AL1/ (C1*R2)
C=C1
XL= (W*AL)-(1.0/(W*C))
Z=SQRT(R*R+XL*XL)
XL=-XL
83
THETA1=ATAN (XL1/R)
V1=(1.0/(AL*C))-(R/(2.0*AL))**2
V=SQRT (V1)
PHI2=PHI+THETA1
B=S-(E*SIN (PHI2))/Z
D=(1.0/(V*AL*C))*(-Q-(E*COS(PHI2))/(W*Z))
100
PHI1=0.314*T+PHI+THETA1
AI1= (E*SIN (PHI1))/Z
F1= (-(R*T)/ (2.0*AL))
F2=F1/1000
F=EXP (F2)
AI2=F*B*COS (.001*T*V) +F*D*SIN (.001*T*V)
DI1= (W*E*COS (PHI1))/Z
DI2= (F*V*COS)(.001*V*T)*D)-(F*V*SIN(.01*V*T)*B)
DI3= ((R/(2.0*AL))*F)*(SIN(.001*V*T)D+COS(.001*V*T)*B)
DI=DI1+DI2-DI3
DELT= (G-AI)/DI
IF (ABS (G-AI).LE.EPS GOTO 50
T1=T+DELT
T=T1
GOTO 100
WRITE (*.*) T, AI
50
STOP
51
END
***************************************************************
COMPUTER PROGRAM -3
TO DERIVE I.C WITH OPTIMUM NO. OF SEGMENTS
84
C
PIECEWISE IC OPTIMISATION
DIM V(30),L(30),S(30),K(30),T(30).G(30),J(30),D(30)
DIM P(30),Q(30),A(30),B(30),C(30),M(30),V(30),I(30)
INPUT E5
FOR N=1 TO13
READ U (N)
DATA 0,100,140,160,180,200,240,250,280,290,300
NEXT N
FOR N=1 TO 13
READ M (N)
DATA 0, 0.135, 0.215, 0.275, 0.374, 0.505, 0.68, 0.94, 1.125, 1.325
DATA 1.85, 2.25, 2.85
NEXT N
V (1) =0
L (1) =0
J (1) =0
P (1) =0
T (1) =0
Q (1) =0
G (1) =0
R=2
N=2
330
340
360
420
500
S (N) = (M (N)-M (N-1))/ (U (N)-U (N-1))
V(R) =U (N)
I(R) =M (N)
GOSUB 650
GOSUB 780
IF R>2 THEN 420
R=R+1
N=N+1
GOTO 330
E1=V(R-1) + (1/2.0)*(V(R)-V(R-1))
V(R) =E1
GOSUB 650
GOSUB 870
E3=S (N)*(V(R)-V(R-1)) +I(R-1)
E4= (E2-E3)/E3
IF ABS (E4) <E5 THEN 500
GOTO 550
V(R) =2.0*E1-V(R-1)
85
550
580
580
610
650
770
770
700
800
I(R) =S (N)*(V(R)-V(R-1)) +I(R-1)
L(R) = (SQRT (2.0)*V(R))/ (2200.0/7.0)
J(R) =K(R)*(L(R)-L(R-1)) +J(R-1)
IF ABS (V(R)-U (N)) <1E-2 THEN 580
R=R+1
GO TO 340
V(R) =E1
I(R) =E3
GO TO 360
V(R) =E1
I(R) =E3
GO TO 360
N=N+1
IF N>13 THEN 610
R=R+1
GO TO 330
PRINT "NO OF SEGMENTS="(R-1)
PRINT "ERROR TOLERANCE="E5
PRINT END.V.RMS*.*"SLOPE*,*LAMDA*,*I.CURRENT*,*
FOR N=2 TO R
PRINT V (H), K (H), L (H), J (H)
NEXT H
STOP
L(R)=(SQRT(2.0)*V(R))/(2200.0/7.0)
IF R>2 THEN 700
IF V(R) > U (N) THEN 700
K(R) = (SQRT (2.0)*I)) /L(R)
PRINT K (R)
GOTO 770
GOSUB 950
T (R) = (11.0/7.0) –P(R-1)))
G (R) = - (1/2.0)* (SIN (2 * P(R-1)))
G (R) =- COS (P(R-1))
D (R) = ((J(R-1)*2)*T(R)
A5 = (((L (R) *2)/2.0)*(T(R)-Q(R))
A6= 2.0 *L(R) *L(R-1)* G(R)* ((L(R -1)) **2) * T(R))
A(R) =A5*A6
B(R )=-2.0 * (J(R-1)) * (L( R )*G( R )*L( R )* L (R-1)*T( R ))
RETURN
IF R>2 THEN 800
GOTO 860
F3=0
FOR H=2 TO (N-1)
F4 = (K (H)*2)4A (H)*K (H) * B (H)*D (H)
F3 = F3*F4
86
NEXT N
C(R) =F3*D(R)-(11.0/7.0)*I(R) **2
K( R ) = (-B( R )*SQRT(B( R )*B(R )-( 4.D* A (R )*I( R))))/(2.0*A( R ))
860
RETURN
870
F5=0
FOR H=2 TO R
F6= ((K (H) **2))*A (H) +K (H)*B (H) +D (H)
F5=F5+F6
NEXT H
E2=SQRT ((7.0/11.0)*F5)
RETURN
950
FOR I =2 TO (R-1)
P(X) =ATAN ((L(X)/L(R))/SQRT (1-((L(X)/L(R)) **2))))
T(X) = P(X)-P(X-1)
G(X)=(1.0/2.0)*SIN(2.0*P(X))-(1.0/2.0)*SIN(2.0*P(X-1))
G(X) =COS (P(X))-COS (P(X-1)
J(X) =K(X)*(L(X)-L(X-1)) +J(X-1)
D(X) =J(X-1)*J(X-1)*T(X)
A1= ((L(R)*L(R))/2.0)*(T(X)-Q(X))
A2=2.0*L(R)*L(X-1)*G(X)
A3=L(X-1) * L(X-1)*T(X)
A(X) =A1+A2+A3
B(X) =-2*J(X-1)*(L(R)*G(X) +L(X-1)*t(X))
NEXT X
RETURN
***************************************************************
87