IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 61, NO. 5, MAY 2014 1543
Experimental Investigation of Symmetry-Breakingin Ferroresonant Circuit
Kruno Milicevic, Member, IEEE, Dragan Vulin, and Davor Vinko
Abstract—This study focuses on the impact of circuit parame-ters on the breaking of steady-state symmetry in a ferroresonantcircuit. Symmetry-breaking occurs owing to pitchfork bifurcationand results in the ambiguousness of steady-state solution throughtwo possible waveform types. The two types are distinguishedin this study by positive and negative peak values of capacitorvoltage. Circuit parameters analyzed in detail are source voltageRMS value U and initial values of capacitor voltage and coilremanent flux. There is symmetry of occurrence in the two typesof waveform, which is expected because of the odd symmetry ofcharacteristics of all passive circuit elements. It is shown thatthe property of symmetry could be experimentally demonstratedand through simulation using a mathematical model of the ex-perimental ferroresonant circuit. In addition, measurements havebeen carried out with two further coils to investigate the impact ofthe nominal voltage and type of core material on the initiation ofthe two types of waveforms.
Index Terms—Bifurcation, circuit theory, ferroresonance, mod-eling, nonlinear circuits, symmetry.
I. INTRODUCTION
F ERRORESONANCE is a complicated nonlinear elec-trical phenomenon, which can lead to voltages several
times the normal equipment ratings. Every model of the elec-trical power network in which ferroresonance can occur couldbe reduced to the simplest physical model, as shown in Fig. 1,which consists of a linear capacitor in series with a nonlinearcoil with an iron core fed from a sinusoidal voltage source [1].In the ferroresonant circuit, six basic steady-state types have
been identified: monoharmonic (sinusoidal) steady-state (M),polyharmonic steady-state with odd harmonics only (PO), poly-harmonic steady-state with even and odd harmonics (PEO),quasi-periodic steady-state (Q), polyharmonic steady-state withsubharmonics (PS), and chaotic steady-state (CS) [2]–[5]. Inspite of the fact that most predictions of chaotic steady-stateappear to be based on unrealistically high values of sourcevoltage [6]–[8], the possible occurrence of chaotic steady-stateshould not be ignored in general during a research because ofits high interdisciplinary importance and possible applications[9]–[12].
Manuscript received January 04, 2013; revised April 23, 2013; accepted Oc-tober 01, 2013. Date of publication January 02, 2014; date of current versionApril 24, 2014. This paper was recommended by Associate Editor R. Sipahi.K. Milicevic is with the Faculty of Electrical Engineering, University
of Osijek, 31000 Osijek, Croatia (corresponding author, e-mail: [email protected]).D. Vulin and D. Vinko are with the Faculty of Electrical Engineering, Uni-
Fig. 2. Example of supercritical pitchfork bifurcation for [16].
All steady states, except monoharmonic steady state, are con-sidered ferroresonant. Ferroresonance is the change frommono-harmonic to a non-monoharmonic steady state with significantlyhigher state variable magnitudes owing to a small perturbationintroduced to a system parameter. Ferroresonance has specialimportance in power network utilities owing to the possibilityof overvoltages that can be sustained under ferroresonance con-ditions. It could lead to catastrophic damage to electrical equip-ment, affecting the reliability of power networks [13], [14].Despite numerous studies on identifying the behavior, the
problem is still very unpredictable in practical terms. This isbecause of the ostensibly stochastic nature of the ferroresonantphenomenon that is derived from the sensitivity to system pa-rameters and initial conditions that cannot be measured suffi-ciently accurately to predict a steady-state unambiguously.A sudden change in steady-state type due to a small perturba-
tion introduced to a system parameter can be defined as a bifur-cation. In such terms, symmetry-breaking corresponds to super-critical pitchfork bifurcation, where one stable solution bifur-cates into two stable solutions and one unstable solution (Fig. 2)[3], [15], [16]. Other types of bifurcations and the impact of cir-cuit parameters [1]–[8], [13], [17]–[23] as well as various con-figurations of the studied network [3], [19]–[23] have been in-vestigated in detail in previous works.Symmetry-breaking results in a polyharmonic steady state
with even and odd harmonics (PEO) that break the odd sym-metry of waveforms. Thereby, the two stable solutions appear in
1544 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 61, NO. 5, MAY 2014
Fig. 3. Waveforms and , amplitude and phase spectra of capacitorvoltage ( and ) for PEO obtained by simulation at .(a) PPV. (b) NPV.
the form of two different waveform types that could arise. Thewaveforms will be distinguished in the paper by positive andnegative peak values of capacitor voltage (Fig. 3) (PPV andNPV) obtained at particular RMS value of source voltage ,
. Time marked on the source voltagewaveforms is introduced and explained in Section II. ThePPV and NPV have the same amplitude spectra, but the phasespectra differ by for odd harmonics only (Fig. 3). Thereby, theDC component is equal to zero in both the cases.
Fig. 4. Simulation model of ferroresonant circuit.
There is no significant difference of phase spectra with a viewof possible damage to electrical equipment. However, the pos-sibility of two different waveforms is an important issue con-cerning the identification of ferroresonant behavior, because itcontributes to the ostensible stochastic nature and ambiguous-ness of the ferroresonant phenomenon, thereby complicating theidentification of circuit behavior (addressed in more detail inSections II and III on the example of bifurcation diagrams ob-tained by simulations and experiments).In addition, the authors are interested in this issue also be-
cause of the fact that till date, there had been no investigationon the impact of circuit parameters on the initiation of two dif-ferent waveform types.In this study, symmetry-breaking was investigated using
simulation and experiments to determine the range of sourcevoltage RMS value at which symmetry-breaking occursand the extent to which the initial conditions and phase shift ofsource voltage determine the waveform type.During the tests and simulations, the harmonic content of a
state-variable was determined by employing Fast Fourier Trans-formation (FFT). Before the implementation of FFT, a statevariable was left for a long duration to settle to a steady state.Thereby, if the PEO was identified, the sign of peak value of ca-pacitance voltage was determined and used for waveform typeidentification (PPV or NPV). The identified waveform typeswere collected and put onto a response map (Figs. 7, 13 and15) to provide a visual state for a particular case. Thereby, thewaveform types were color-coded as white and black squares inthe case of PPV and NPV, respectively.
II. MODELING AND SIMULATION
The mathematical model used in simulation is based on theexperimental ferroresonant circuit, which consists of a linear ca-pacitor of 20 and a nonlinear coil with an iron core realizedas primary winding of a toroidal iron-cored two-winding trans-former. The rated power of the transformer was 200 VA witha nominal primary voltage of 30 V. The core was made of ori-ented transformer sheets (M5 type) and was strip-wounded.A 10-kVA autotransformer with a source frequency of 50 Hz
was used as the variable voltage source. The output voltage am-plitude of this source could be varied up to an RMS value of
in steps of 0.4 V, approximately.The voltage source was modeled as an ideal sinusoidal AC
source with a frequency of 50 Hz, , andthe capacitor was a lumped linear capacitor with a value of20 (Fig. 4).
MILICEVIC et al.: EXPERIMENTAL INVESTIGATION OF SYMMETRY-BREAKING IN FERRORESONANT CIRCUIT 1545
TABLE ISTEADY-STATE TYPES OBTAINED BY SIMULATION
The nonlinear coil with an iron core was modeled based onthe actual measurements as nonlinear inductance repre-senting a saturation effect in parallel with a nonlinear resistance
presenting coil losses [24]:
(1)
(2)
The time-domain behavior of the ferroresonant circuit inFig. 4 can be described by a mathematical model composedof the second-order differential equation, which is solvednumerically:
(3)
Thereby, the resistance voltage used in (2) equals dueto the Kirchhoff’s Voltage Law and Faraday’s Law:
(4)
This kind of model presents less complication to the objectiveof this study, when compared with other model types, suchas detailed three-phase ElectroMagnetic Transients Program(EMTP) model [13], model based on Preisach theory for therepresentation of nonlinear coil [19], flux reflection model [25],etc.To determine the range of source voltage RMS values at
which the PEO occurs, preliminary numerical simulations werecarried out by increasing the source voltage RMS value con-tinuously from 0 to 60 V with a step of 0.4 V, approximately.In this way, the PEO that occurs for a range of source voltageRMS values of was determined (Table I).In general, bifurcation can be identified using Floquet theory,
i.e., calculating the Floquet multipliers that leave the unit circleif a bifurcation occurs [16]. Fig. 5 shows the Floquet multiplierscalculated for the nonlinear model (3) using the same parametervalues as those employed in the preliminary numerical simula-tions. The multipliers confirmed the results shown in Table I,i.e., a multiplier would leave the unit circle at the source voltageRMS value at which a change in steady-state type occurs.Repetition of the simulations using various values of phase
shift of the source voltage and the initial values of capacitorvoltage and remanent flux, respectively, resulted in the occur-rence of both the possible waveform types (PPV and NPV;Fig. 3).
This ambiguousness is also obvious in bifurcation diagrams(Fig. 6). These diagrams were constructed by sampling thecapacitance voltage at the source voltage frequency with thesource voltage RMS value as the bifurcation parameter in twosimulations carried out in the same way as the preliminaryone, but using different values of phase shift of the sourcevoltage ( and ). The sampling was carried outfor 60 cycles using the sampling period of afterthe initial transients had died out. Time marked on thesource voltage waveforms presented in Fig. 3 is the firstinstant of periodic sampling carried out for 60 times after thewaveform had reached a steady state. For example, the 60sampled values of capacitance voltage shown in Fig. 3 arethe ones used for bifurcation diagrams presented in Fig. 6 at thesource voltage RMS value of 40.0 V. Thereby, the 60 sampledvalues of capacitance voltage form a single point, shownin Fig. 6, for a particular source voltage RMS value, becausethe sampled waveform has the period equal to that of sampling( ).It is obvious that the difference in the appearance of bifurca-
tion diagrams is determined at the moment of initiation of PEO,i.e., it is caused at by the initiation of differentwaveform types of PEO. Consequently, the two different bi-furcation branches for (the PEO (PPV) branchin Fig. 6(a) and the PEO (NPV) branch in Fig. 6(b)) representthe two stable solutions that have resulted from supercriticalpitchfork bifurcation. However, if one would identify the be-havior using the bifurcation diagrams without this foreknowl-edge, a wrong conclusion that different steady-state types arisefor could be made.
1546 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 61, NO. 5, MAY 2014
Fig. 6. Bifurcation diagrams in two simulations using different values of phaseshift of the source voltage. (a) Initiated PPV . (b) Initiated NPV
.
To determine the impact of initial conditions and phase shiftof source voltage more precisely, additional simulations werecarried out for a source voltage RMS of 40.0 V by varying theparameters according to:• , , , ,• ;• ;
This combination resulted in 51005 different sets of parametervalues, and hence, as many simulation results were obtained, asshown in Fig. 7.The response maps show significant impact of the parame-
ters. Thereby, there is an odd symmetry of occurrence of twowaveform types, i.e., the different waveform types occur if theinitial conditions differ by their sign and phase shift differs by. For instance, if the PPV type occurs for parameter values
, , and , then the NPV typeoccurs for the values , , and
, as can be seen by comparing Fig. 7(a) and (e), as wellas the corresponding transient states given in Fig. 8(a) and (b).Fig. 8(a) and (c) shows that the duration and shape of transientstate strongly depend on initial conditions. In future, more de-tailed analysis of transient state, including the investigation ofwaveform shape in transient state, with a view to the initiationof particular waveform type of PEO, will be carried out.
Symmetry-breaking was additionally investigated by usingthe physical model of the ferroresonant circuit and corre-sponding measurements.
MILICEVIC et al.: EXPERIMENTAL INVESTIGATION OF SYMMETRY-BREAKING IN FERRORESONANT CIRCUIT 1547
Fig. 8. Transient states obtained by simulation. (a) , ,and . (b) , , and .(c) , , and .
III. MEASUREMENTS
The experimental ferroresonant circuit had been described inthe beginning of Section II. The same type of circuit can be ap-plied as a physical model to the Manitoba Hydro’s, Winnipeg,Canada 230 kV Dorsey converter station [1]. There, ferroreso-nance had been reported to have occurred when a power trans-former, connected to a busbar in an air-insulated substation, wasde-energized by opening a circuit breaker and leaving the non-linear transformer core connected to the supply via the circuitbreaker grading capacitance [13]. Important modeling issues,including the reasons and consequences of disagreement of pa-rameter values of the original and model are discussed in [1].Preliminary measurements were carried out in the same way
as preliminary numerical simulations to validate the model.Comparison, given in Tables I and II, revealed that the simu-lation does not result in more complex behavior identified inthe experiment. The term “more complex behavior” (MCB)
Fig. 9. Waveforms and , amplitude and phase spectra of capacitorvoltage ( and ) for PEO obtained by using measurements at
. (a) PPV. (b) NPV.
is used for all identified steady states with harmonic contentdenser than the PEO.1 However, in this study, the steady state ofinterest is the PEO that occurs in measurements and simulation.Conclusively, the mathematical model presented in Section II
1The term “more complex behavior” (MCB) is used for all identified steadystates with harmonic content denser than the PEO because it is not possible toidentify chaotic steady state unambiguously by using the FFT; at the same time,the distinction between the polyharmonic steady states with subharmonics andchaotic steady state is not important for the subject of this study.
1548 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 61, NO. 5, MAY 2014
Fig. 10. Bifurcation diagrams in two subsequent measurements. (a) InitiatedPPV. (b) Initiated NPV.
and the physical model used in this section can be used toinvestigate the symmetry-breaking phenomenon, in spite of thenoticed disagreements.As the source voltage RMS value was adjusted manually
via the potentiometer of the autotransformer, at the momentof variation, it was not possible to set the phase shift of thesource voltage and the initial values of capacitor voltage andremanent flux, respectively. Therefore, repetition of measure-ments resulted in the occurrence of both possible waveformtypes (PPV and NPV), ostensibly in a stochastic manner. Forinstance, the waveforms shown in Fig. 9 were obtained throughtwo subsequent measurements at the same source voltage RMSvalue . The harmonic spectra of waveformsshown in Fig. 9 have the same properties as the spectra obtainedby simulation (Fig. 3; the PPV and NPV have same amplitudespectra, but phase spectra differ by for odd harmonics only;the DC component is equal to zero).The bifurcation diagrams shown in Fig. 10 were obtained
experimentally and illustrate the ambiguousness in the sameway as the diagrams obtained by simulation (Fig. 6). Thecomparison of the diagrams confirmed the results presented inTables I and II, i.e., the PEO occurs at somewhat higher sourcevoltage RMS value in the experiment (34.8 V in Table II,when compared with 33.6 V in Table I); for a chosen range
Fig. 11. Waveforms of PEO obtained at . (a) PPV. (b) NPV.
of source voltage RMS values, the MCB appeared in ex-periment only. Thereby, the deviations between the bifurca-tion diagrams obtained by simulation (Fig. 6) and experiment(Fig. 10) directly resulted from those of waveforms that havebeen sampled ( ) to obtain the bifurcation diagrams.For example, in the case of NPV at the source voltage RMSvalue of 40 V, in the instant of sampling , a negative value ofcapacitor voltage (Fig. 3(b)) and a positive value (Fig. 9(b))were sampled. Consequently, the branch of PEO (NPV) inFig. 6(b) has negative values of sampled capacitor voltage incontrast to the positive values of sampled capacitor voltage inFig. 10(b). The deviations in waveforms are likely occurringdue to incomplete modeling. This issue will be addressed inmore detail after the experimental investigation of impact ofinitial conditions presented hereinafter.The benefits and appropriateness of a bifurcation diagram in
the analysis of ferroresonance has been widely reported in pre-vious publications [20]–[22]. To a large extent, this is true; how-ever, limitations of the experimental approach could result infalse conclusions based only on bifurcation diagrams obtained
MILICEVIC et al.: EXPERIMENTAL INVESTIGATION OF SYMMETRY-BREAKING IN FERRORESONANT CIRCUIT 1549
Fig. 12. Measurement circuit.
in this way. For instance, one may assume that the discontinuitypoint at of 40.0 V is a bifurcation. However, comparison ofwaveforms at a source voltage RMS value of 40.0 V, shown inFig. 9, with those measured at a source voltage RMS value of39.6 V (Fig. 11) revealed that the waveforms are of the sametype (PEO). Conclusively, the discontinuity point does not rep-resent a bifurcation, rather it is caused by a finite step of sourcevoltage increase ( ) and would not be emphasizedin a case of a smaller step of source voltage increase.The experimental investigation of the impact of initial con-
ditions was carried out in the same way as the final numericalsimulation described in the previous section, i.e., for a sourcevoltage RMS of 40 V, but for lower number of phase shifts andwith lower resolution of response maps:• ,• ;• ;That is, it is far less cumbersome to increase the resolution in
simulation than in measurements, and therefore, more detailedresponse maps were obtained using simulation, as described inSection II.The experiments presented in this study were carried out by
varying the circuit parameters in a controlled manner, includingthe phase shift of the source voltage and the initial values of ca-pacitor voltage and remanent flux, respectively. This was doneto determine the unambiguous dependence of the symmetry-breaking type on the parameter values of the circuit. The sameapproach can also be used for other types of bifurcations to de-termine the extent to which the initiation of a particular bifur-cation depends on the initial conditions and phase shift underexperimental conditions (e.g., see [26]).The following parameters were set using the measurement
circuit with electronically controlled switches (Fig. 12):— the moment of closing of switch determines the phaseshift of the source voltage .
— the remanent flux was set by using an additional ACvoltage source .
— the initial condition of the capacitor voltage wasvaried by using an additional DC voltage source .
The voltage source was applied directly and the voltagesource was applied through a resistor (4.7 kOhm; 5 W)using switches and .Switch was closed at the most 20 ms after opening of
the control switches and such that the remanent flux andcapacitor voltage did not drop significantly. This ensured that atthe moment of the closure of switch , the value of remanentflux was approximately equal to the flux applied by the voltagesource in the moment of the opening of switch and thatthe value of capacitor voltage was approximately equalto the applied DC voltage , and hence, its initial condition.The combination used in experiment results in 1250 different
sets of parameter values, and hence, as many measurement re-sults were obtained, as shown in Fig. 13.Comparison of the results shown in Fig. 13(a) and (b) con-
firms the property of odd symmetry in 89% of the measure-ments. It must be noted that the axis presented in Fig. 13(b) areinverted to observe the odd symmetry easily.Fig. 14 shows the transient state leading to a particular wave-
form type for three pairs of initial values marked in Fig. 13 bywhite and black x’s, respectively. The transient states obtainedexperimentally are also symmetric (Fig. 14(a) and (b)), showingthe impact of initial conditions on duration and shape of tran-sient state (Fig. 14(a) and (c)).
1550 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 61, NO. 5, MAY 2014
Fig. 14. Transient states obtained by experiment. (a) , ,and . (b) , , and . (c)
, , and .
Disagreement in the shape of bifurcation diagrams, areas ofresponse maps, and waveforms during steady state and transientstate (Figs. 6, 7, 3 and 8) in comparison with simulation results(Figs. 10, 13, 9 and 14) is caused by assumptions made duringmodeling. For instance, the switches are supposed to be ideal,the coil losses are modeled as resistance that does not takeinto account the frequency dependence of eddy currents [23],and the copper losses as well as internal resistance of the sourceare completely neglected.Contrary to the maps obtained experimentally, the response
maps obtained by simulation form shapes with clear outlines,indicating that the measurement noise had the most impact atthe shape boundaries. However, considering that the main goalof simulation was to determine the symmetry of occurrence oftwo waveform types depending on initial conditions and phaseshift of source voltage, these discrepancies are considered to betolerable.
Fig. 15. Response maps obtained by experiments with: (a) coil A , (b) coil B.
IV. CONCLUSIONS
It has been shown experimentally that the initial conditions(initial values of capacitor voltage and coil remanent flux) andphase shift of the source voltage have a clear and definite impacton the occurrence of two possible waveform types that resultedfrom symmetry-breaking, i.e., pitchfork bifurcation.Thereby, there is an odd symmetry of occurrence of two
waveform types: the different waveform types occur wheneach initial condition differs by its sign and when phase shiftdiffers by 180 degrees. This experimentally obtained result isexpected because of the odd symmetry of characteristics of allpassive circuit elements. The experimental findings have beensupplemented with a mathematical model. The impact of thechosen circuit parameters on the occurrence of two possiblewaveform types has been shown by simulation for significantlyhigher resolution of response maps and for additional threephase shift values. Future research will carry out detailedanalysis of transient state, with a view of initiating particularwaveform type of polyharmonic steady-state with even andodd harmonics.
APPENDIX
Measurements were carried out with two additional coils toinvestigate the impact of the nominal voltage and type of corematerial on the initiation of the two types of waveforms:
MILICEVIC et al.: EXPERIMENTAL INVESTIGATION OF SYMMETRY-BREAKING IN FERRORESONANT CIRCUIT 1551
— coil A – nominal primary voltage of 36 V with the coremade of oriented transformer sheets (M5-type) and
— coil B – nominal primary voltage of 30 V with the coremade of Ni-Fe alloy (Trafoperm N3).
Thereby, owing to its higher nominal primary voltage, coil Awas investigated for a wider range of values of remanent flux:• ;Preliminary measurements revealed that symmetry-breaking,
i.e., PEO, occurs for a ferroresonant circuit made with these twocoils as well (Table III). Thereby, the symmetry of waveformsbreaks at higher values of RMS source voltage in the case ofcoil with the core made of Ni-Fe alloy (coil B), when comparedwith the core made of oriented transformer sheets (coil A andcoil described in Section II of this study), as shown in Tables Iand III.Conclusively, measurements at the source voltage RMS value
of , as described at the end of Section II, were alsocarried out, but for one phase shift value only (Fig. 15).Figs. 15 and 13(a) are significantly correlated in terms of
the shapes of areas of PPV (white) and NPV (black) waveformtypes of the PEO. This result is expected because of similar mag-netization characteristics of cores made of the Ni-Fe alloy (coilB) and cores made of oriented transformer sheets (coil A andcoil described in Section II of this study) [27].
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Kruno Milicevic (M’05) was born in Lampertheim,Germany. He received the M.Sc. and Ph.D. degreesin 2003 and 2008, respectively, from the Univer-sity of Osijek, Osijek, Croatia, where he worksas associate professor at the Faculty of ElectricalEngineering. He is engaged in teaching courses onmeasurements in electrical engineering, analysis oflinear and nonlinear electrical circuits, and transientphenomena.His specific area of interest in scientific research
covers nonlinear electrical circuits, including circuitmodeling and analysis of pre-chaotic and chaotic behavior in electrical circuits.
1552 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 61, NO. 5, MAY 2014
Dragan Vulin was born in 1986 in Osijek, Croatia.He received the Bachelor degree and the Master de-gree in the field of power engineering, both at the Fac-ulty of Electrical Engineering, University of Osijek,Osijek, Croatia, in 2008 and 2010, respectively.After finishing his studies, Mr. Vulin was em-
ployed at the Faculty of Electrical EngineeringOsijek as a scientific novice in the Department forElectromechanical Engineering. His specific areaof interest in scientific research covers nonlinearelectrical circuits, renewable energy sources and
power electronics.
Davor Vinko was born in 1980 in Cakovec, Croatia.He received the M.Sc. and Ph.D. degrees from J. J.Strossmayer University of Osijek, Faculty of Elec-trical Engineering, Osijek, Croatia, in 2005 and 2012,respectively.Since 2005, he has been affiliated with the Depart-
ment of Communication, Faculty of Electrical En-gineering, University of Osijek, where he reached arank of Assistant Professor. His main research inter-ests include charge pumps, energy harvesting circuitsand wirelessly powered devices.
