AP05_6.vp 1 Introduction The external partial microdischarges which appear at the insulating barriers of high voltage components are in fact of the same origin as the so-called “barrier” microdischarges studied in the field of discharge physics. The only difference is that the external partial microdischarges in high voltage technology are considered to be an undesirable phenomenon that is suppressed as much as possible, while the barrier microdischarges in discharge physics are treated as a useful phenomenon 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 is worth studying them since both the mentioned approaches (elimination versus utilisation) require a thorough knowledge of the basic physics underlying their mechanism. The fractal properties of electrical pre-breakdown phe- nomena like electrical trees in high voltage insulation are sometimes explained [1] as a consequence of competitive act- ing of positive and negative feedbacks. The positive feedback accelerates degradation of the insulation and is mainly gov- erned by an electric field. Under its influence the degradation proceeds straightforwardly through local inhomogeneities without producing “fractally” branched filamentary struc- tures. Conversely, the negative feedback tends to decelerate the degradation and to facilitate fractal branching of the filamentary structures. The negative feedback tends to equi- librate the system at low electrical fields and may even cause the 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 with the microdischarge phenomenon, the question arises wheth- er deterministic chaos also appears with the microdischarges that are in fact precursors to electrical trees. The goal of this paper is to find an answer to this question. 2 Experiment The sandwiched plane-to-plane electrode system was employed. All experiments were carried out at normal atmo- spheric conditions. Polyethylene terephthalate sheets 0.2 mm in thickness were inserted between two brass electrodes to serve as insulating barriers. The metal-insulator-metal in- terfaces created in this way were loaded with a uniform high voltage of 4 kV to study the microdischarge signal. The whole experimental arrangement is depicted in Fig. 1. Microdischarges were detected as voltage pulses across the resistor R � 10 k�. The resistor is connected in series with the electrode system and experiences electric pulses initiated by microdischarges. The pulsating signal was amplified and digi- tised in a digitiser of unique construction [3, 4] and then processed by a special software implemented on a PC. The pulsating signal has the character of a discrete time series � �Ui i N �1 with a finite number N of voltage pulses Ui registered at moments ti (see Fig.2). The number N of registered pulses amounted to about 60 000. 3 Computational model Investigations of deterministic chaos within a time series usually consist of three separate steps: determining the cor- relation coefficient (correlation dimension), specifying the Lyapunov exponent, and reconstructing the attractor. The present analysis of the discrete microdischarge series � �Ui i N �1 focuses on determining the correlation coefficient. For this purpose the series of heights (amplitudes) � �Ui i N �1 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 Prague On Deterministic Chaos in Microdischarge Phenomena T. Ficker 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. Keywords: microdischarges, deterministic chaos, fractals. Power supply HV� CF HV Sample R Preamplifier Digitiser PC Fig. 1: Experimental arrangement Fig. 2: Microdischarge time series with the interval �N � 1 between their components, i.e., the components are not successive measurements. For example, the three-dimensional (d � 3) pseudo-vectors �(3) with the component interval �N � 2 can be written as follows � � � 1( ) ( ) ( ) 3 3 3 1 3 5 2 2 4 6 2 � � � � � � � � � U U U U U U U Un n n i j k i j k i j � � � � � �U n N d N n 4 k, ( 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 N i d N d d U U U U U ( ) ( ) ( ) ( ) ( ( ) � � � � � � � � � � � � 2 2 1 � � � � �U j d N� �( ) ) .1 2 1 2 � (2) A double sum running over all the n measured values de- termines the correlation integral C(r) � C r n n H r U Ui j j j i n i n ( ) ( ) , � � � � � � �� 1 1 11 , (3) where H(x) is the Heavyside step function H x x x ( ) � � � � � 0 0 1 0 (4) Repeating the procedure for a range of r, the correlation integral in the form of the power law can be obtained C 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 random noise, � increases in proportion to d. The strange attractor with the largest value of �0 shows the greatest excursions away from its centre of attraction. Therefore, larger values of �0 correspond to a more chaotic behaviour while smaller values are related to a reduction of chaocity and to a movement towards determinicity. 4 Results By carrying the computational procedure described above for a set of d dimensions, the graphs of the functions C(r) were plotted in a bilogarithmic co-ordinate system (see Fig. 3) to obtain the correlation coefficients �. These coefficients were calculated as the slopes from the linear sections of the graphs C(r), using the least square method. The resulting depen- dences �(d) showed a monotonically increasing behaviour with an asymptote, and showed a quite satisfactory resem- blance to an exponential pattern with two parameters �0 and d0 � �� � �� � �� � � �� � � � � � �0 0 1 exp d d . (6) The asymptotes �0 are accepted as the best approximation for the correlation dimensions. The existence of the asymp- totes indicates the occurrence of deterministic chaos rather than a random noise. In Fig. 4 the corresponding value of �0 is close to 89. Such a value is a consequence of strong de- terministic chaos, i.e., pronounced chaotic behaviour of the studied discharge sequence. Similar results have been re- ported for electrical trees [1], in which microdischarges run in the very tips of the discharge branches. 5 Conclusion Deterministic chaos seems to be one of the characteristic features of microdischarge phenomena. It manifests itself not only in electrical treeing, which is a pre-breakdown phenome- non appearing in high voltage insulations, but it is also characteristic for the microdischarges that precede treeing. The study of deterministic chaos in microdischarge time series would deserve more attention since it could provide missing information on possible connections between fractal morphology [5–7] and chaotic properties of microdischarge phenomena. 6 Acknowledgment This work was supported by the Grant Agency of the Czech 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/ 45 Czech Technical University in Prague Acta Polytechnica Vol. 45 No. 6/2005 Fig. 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 Microdischarge Pulses.” J. El. Eng., Vol. 51 (2000), p. 240–248. [4] Ficker, T., Macur, J., Pazdera, L., Kliment, M., Filip, S.: “Simplified Digital Acquisition of Microdischarge Pulses.” IEEE Trans. Diel. El. Insul., Vol. 8 (2001), p. 220–227. [5] Ficker, T.: “Electron Avalanches I – Statistics of Partial Discharges 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 Dielectric Barriers.” 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: +420 541 147 661 E-mail: fyfic@fce.vutbr.cz Department of Physics Faculty of Civil Engineering University of Technology Žižkova 17 662 37 Brno, Czech Republic 46 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 45 No. 6/2005 Czech Technical University in Prague 44 45 46 << /ASCII85EncodePages false /AllowTransparency false /AutoPositionEPSFiles true /AutoRotatePages /None /Binding /Left /CalGrayProfile (Dot Gain 20%) /CalRGBProfile (sRGB IEC61966-2.1) /CalCMYKProfile (U.S. Web Coated \050SWOP\051 v2) /sRGBProfile (sRGB IEC61966-2.1) /CannotEmbedFontPolicy /Error /CompatibilityLevel 1.4 /CompressObjects /Tags /CompressPages true /ConvertImagesToIndexed true /PassThroughJPEGImages true /CreateJobTicket false /DefaultRenderingIntent /Default /DetectBlends true /DetectCurves 0.0000 /ColorConversionStrategy /CMYK /DoThumbnails false /EmbedAllFonts true /EmbedOpenType false /ParseICCProfilesInComments true /EmbedJobOptions true /DSCReportingLevel 0 /EmitDSCWarnings false /EndPage -1 /ImageMemory 1048576 /LockDistillerParams false /MaxSubsetPct 100 /Optimize true /OPM 1 /ParseDSCComments true /ParseDSCCommentsForDocInfo true /PreserveCopyPage true /PreserveDICMYKValues true /PreserveEPSInfo true /PreserveFlatness true /PreserveHalftoneInfo false /PreserveOPIComments true /PreserveOverprintSettings true /StartPage 1 /SubsetFonts true /TransferFunctionInfo /Apply /UCRandBGInfo /Preserve /UsePrologue false /ColorSettingsFile () /AlwaysEmbed [ true ] /NeverEmbed [ true ] /AntiAliasColorImages false /CropColorImages true /ColorImageMinResolution 300 /ColorImageMinResolutionPolicy /OK /DownsampleColorImages true /ColorImageDownsampleType /Bicubic /ColorImageResolution 300 /ColorImageDepth -1 /ColorImageMinDownsampleDepth 1 /ColorImageDownsampleThreshold 1.50000 /EncodeColorImages true /ColorImageFilter /DCTEncode /AutoFilterColorImages true /ColorImageAutoFilterStrategy /JPEG /ColorACSImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /ColorImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /JPEG2000ColorACSImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /JPEG2000ColorImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /AntiAliasGrayImages false /CropGrayImages true /GrayImageMinResolution 300 /GrayImageMinResolutionPolicy /OK /DownsampleGrayImages true /GrayImageDownsampleType /Bicubic /GrayImageResolution 300 /GrayImageDepth -1 /GrayImageMinDownsampleDepth 2 /GrayImageDownsampleThreshold 1.50000 /EncodeGrayImages true /GrayImageFilter /DCTEncode /AutoFilterGrayImages true /GrayImageAutoFilterStrategy /JPEG /GrayACSImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /GrayImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /JPEG2000GrayACSImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /JPEG2000GrayImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /AntiAliasMonoImages false /CropMonoImages true /MonoImageMinResolution 1200 /MonoImageMinResolutionPolicy /OK /DownsampleMonoImages true /MonoImageDownsampleType /Bicubic /MonoImageResolution 1200 /MonoImageDepth -1 /MonoImageDownsampleThreshold 1.50000 /EncodeMonoImages true /MonoImageFilter /CCITTFaxEncode /MonoImageDict << /K -1 >> /AllowPSXObjects false /CheckCompliance [ /None ] /PDFX1aCheck false /PDFX3Check false /PDFXCompliantPDFOnly false /PDFXNoTrimBoxError true /PDFXTrimBoxToMediaBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXSetBleedBoxToMediaBox true /PDFXBleedBoxToTrimBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXOutputIntentProfile () /PDFXOutputConditionIdentifier () /PDFXOutputCondition () /PDFXRegistryName () /PDFXTrapped /False /CreateJDFFile false /Description << /ARA /BGR /CHS /CHT /CZE /DAN /DEU /ESP /ETI /FRA /GRE /HEB /HRV (Za stvaranje Adobe PDF dokumenata najpogodnijih za visokokvalitetni ispis prije tiskanja koristite ove postavke. 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