Time series of pulsating microdischarges were analysed. The results showed that deterministic chaos is present in these series. The estimated values of the correlation coefficients indicated a strong chaotic discharge behaviour.

Ficker, T.

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Time series of pulsating microdischarges were analysed. The results showed that deterministic chaos is present in these series. The estimated values of the correlation coefficients indicated a strong chaotic discharge behaviour

T. Ficker

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Acta Polytechnica

On Deterministic Chaos in Microdischarge Phenomena

CTU Open Journal Systems (Czech Technical University, Prague / České vysoké učení technické v Praze)

1 IntroductionThe external partial microdischarges which appear at theinsulating barriers of high voltage components are in fact ofthe same origin as the so-called “barrier” microdischargesstudied in the field of discharge physics. The only differenceis that the external partial microdischarges in high voltagetechnology are considered to be an undesirable phenomenonthat is suppressed as much as possible, while the barriermicrodischarges in discharge physics are treated as a usefulphenomenon with many industrial applications (ozone gen-erators, modern flat plasma displays, excimer lamps, etc.).Regardless of the diverse approaches to the microdis-charges running in the close vicinity of insulation barriers, it isworth studying them since both the mentioned approaches(elimination versus utilisation) require a thorough knowledgeof the basic physics underlying their mechanism.The fractal properties of electrical pre-breakdown phe-nomena like electrical trees in high voltage insulation aresometimes explained [1] as a consequence of competitive act-ing of positive and negative feedbacks. The positive feedbackaccelerates degradation of the insulation and is mainly gov-erned by an electric field. Under its influence the degradationproceeds straightforwardly through local inhomogeneitieswithout producing “fractally” branched filamentary struc-tures. Conversely, the negative feedback tends to deceleratethe degradation and to facilitate fractal branching of thefilamentary structures. The negative feedback tends to equi-librate the system at low electrical fields and may even causethe system to attain the regime of deterministic chaos [2].Since a similar situation, i.e. competition between accelerat-ing and decelerating factors, may also be encountered withthe microdischarge phenomenon, the question arises wheth-er deterministic chaos also appears with the microdischargesthat are in fact precursors to electrical trees. The goal of thispaper is to find an answer to this question.2 ExperimentThe sandwiched plane-to-plane electrode system wasemployed. All experiments were carried out at normal atmo-spheric conditions. Polyethylene terephthalate sheets 0.2 mmin thickness were inserted between two brass electrodes toserve as insulating barriers. The metal-insulator-metal in-terfaces created in this way were loaded with a uniform highvoltage of 4 kV to study the microdischarge signal. The wholeexperimental arrangement is depicted in Fig. 1.Microdischarges were detected as voltage pulses across theresistor R  10 k. The resistor is connected in series with theelectrode system and experiences electric pulses initiated bymicrodischarges. The pulsating signal was amplified and digi-tised in a digitiser of unique construction [3, 4] and thenprocessed by a special software implemented on a PC. Thepulsating signal has the character of a discrete time series Ui iN1 with a finite number N of voltage pulses Ui registeredat moments ti (see Fig.2). The number N of registered pulsesamounted to about 60 000.3 Computational modelInvestigations of deterministic chaos within a time seriesusually consist of three separate steps: determining the cor-relation coefficient (correlation dimension), specifying theLyapunov exponent, and reconstructing the attractor. Thepresent analysis of the discrete microdischarge series Ui iN1focuses on determining the correlation coefficient. For thispurpose the series of heights (amplitudes) Ui iN1 was rear-ranged into a sequence of d-dimensional pseudo-vectors (d)44 ©  Czech Technical University Publishing House http://ctn.cvut.cz/ap/Acta Polytechnica Vol. 45  No. 6/2005 Czech Technical University in PragueOn Deterministic Chaos inMicrodischarge PhenomenaT. FickerTime series of pulsating microdischarges were analysed. The results showed that deterministic chaos is present in these series. The estimatedvalues of the correlation coefficients indicated a strong chaotic discharge behaviour.Keywords: microdischarges, deterministic chaos, fractals.PowersupplyHVCFHVSampleR Preamplifier DigitiserPCFig. 1: Experimental arrangementFig. 2: Microdischarge time serieswith the interval N 1 between their components, i.e., thecomponents are not successive measurements. For example,the three-dimensional (d  3) pseudo-vectors (3) with thecomponent interval N  2 can be written as follows1( )( )( )3331 3 52 2 4 62     ~~U U UU U UU Un n ni j ki j ki j  Un N d Nn 4k,( 1) .(1)Between the i-th and j-th pseudo-vectors i(d) and j(d)there is a scalar separation i j i j i N j Ni d Nd d U U U UU( ) ( ) ( ) ( )( ( )     ~~   2 21 U j d N ( ) ) .1212(2)A double sum running over all the n measured values de-termines the correlation integral C(r) 	C rn nH r U Ui jj j inin( )( ),  1111, (3)where H(x) is the Heavyside step functionH xxx( ) 0 01 0(4)Repeating the procedure for a range of r, the correlationintegral in the form of the power law can be obtainedC r r( )  . (5)By carrying the whole procedure for increasing values of d,deterministic chaos – if present – will manifest itself in a con-stant value of . This constant value is just the fractal correla-tion dimension 0. If the sequence is the result of randomnoise,  increases in proportion to d.The strange attractor with the largest value of 0 showsthe greatest excursions away from its centre of attraction.Therefore, larger values of 0 correspond to a more chaoticbehaviour while smaller values are related to a reduction ofchaocity and to a movement towards determinicity.4 ResultsBy carrying the computational procedure described abovefor a set of d dimensions, the graphs of the functions C(r) wereplotted in a bilogarithmic co-ordinate system (see Fig. 3) toobtain the correlation coefficients . These coefficients werecalculated as the slopes from the linear sections of the graphsC(r), using the least square method. The resulting depen-dences (d) showed a monotonically increasing behaviourwith an asymptote, and showed a quite satisfactory resem-blance to an exponential pattern with two parameters 0 andd0  001 expdd. (6)The asymptotes 0 are accepted as the best approximationfor the correlation dimensions. The existence of the asymp-totes indicates the occurrence of deterministic chaos ratherthan a random noise. In Fig. 4 the corresponding value of 0is close to 89. Such a value is a consequence of strong de-terministic chaos, i.e., pronounced chaotic behaviour of thestudied discharge sequence. Similar results have been re-ported for electrical trees [1], in which microdischarges runin the very tips of the discharge branches.5 ConclusionDeterministic chaos seems to be one of the characteristicfeatures of microdischarge phenomena. It manifests itself notonly in electrical treeing, which is a pre-breakdown phenome-non appearing in high voltage insulations, but it is alsocharacteristic for the microdischarges that precede treeing.The study of deterministic chaos in microdischarge timeseries would deserve more attention since it could providemissing information on possible connections between fractalmorphology [5–7] and chaotic properties of microdischargephenomena.6 AcknowledgmentThis work was supported by the Grant Agency of theCzech Republic under the Grant No. 202/03/0011.References[1] Dissado, L. A., Dodd, S. J., Champion, J. V., Williams,P. I., Alison, J. M: “Propagation of Electrical Tree Struc-tures in Solid Polymeric Insulation.” IEEE Trans. Diel. El.Insul., Vol. 4 (1997), p. 259–279.[2] Zaslavsky, G. M.: Physics of Chaos in Hamiltonian System.Imperial College Press, London, 1998.©  Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 45Czech Technical University in Prague Acta Polytechnica Vol. 45  No. 6/2005Fig. 3: Graphs of correlation integrals C(r)Fig. 4: Correlation coefficients  in dependence on d-dimensions[3] Ficker, T., Macur, J., Pazdera, L., Kliment, M., Filip, S.:“New System for Digital Acquisition of MicrodischargePulses.” J. El. Eng., Vol. 51 (2000), p. 240–248.[4] Ficker, T., Macur, J., Pazdera, L., Kliment, M., Filip, S.:“Simplified Digital Acquisition of MicrodischargePulses.” IEEE Trans. Diel. El. Insul., Vol. 8 (2001),p. 220–227.[5] Ficker, T.: “Electron Avalanches I – Statistics of PartialDischarges in their Pre-Streamer Stage.” IEEE Trans.Diel. El. Insul., Vol. 10 (2003), p. 689–699.[6] Ficker, T.: ”Electron Avalanches II – Fractal Mor-phology of Partial Microdischarge Spots on DielectricBarriers.” IEEE Trans. Diel. El. Insul., Vol. 10 (2003),p. 700–707.[7] Ficker, T.: “A Note on Pareto Statistics of Partial Micro-discharge Spots.” IEEE Trans. Diel. El. Insul., Vol. 11(2004), p. 136–138.Doc. RNDr. Tomáš Ficker, DrSc.phone: +420 541 147 661E-mail: fyfic@fce.vutbr.czDepartment of PhysicsFaculty of Civil EngineeringUniversity of TechnologyŽižkova 17662 37 Brno, Czech Republic46 ©  Czech Technical University Publishing House http://ctn.cvut.cz/ap/Acta Polytechnica Vol. 45  No. 6/2005 Czech Technical University in Prague

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On Deterministic Chaos in Microdischarge Phenomena

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