Statistical distributions (exponential and Pareto) of DC partial microdischarges running within sandwich electrode systems are discussed from the viewpoint of a normalisation procedure which may influence some features of the final distribution

Ficker, T.

Macur, J.

English

T. Ficker

J. Macur

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Acta Polytechnica

A Note on Normalised Distributions of DC Partial Microdischarges

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Acta Polytechnica Vol. 43 No' 212003A Note on Normalised Distributions ofDC Partial MicrodischargesT. Ficker, J. Macurstatistical distr,ibutions (exponentfuil and. pareto) of DC partinl rnicrod,ischargu running. within sandwich electrode sJstems are discussedfrom the uieupoint ,7 " ,"lr-*t'tt.1io, ,-""auu *ht"h'*oy influence some features of the final distribution'Keywords: exponenthl and statistical d'istributions' partial mi'crod'ischarges, normalisation procedure'1 IntroductionWhen studying the statistics of partial microdischargeswithin sandwicir eLctrode systems loading by dc voltages inexcess of Paschen breakdown values, highly asymmetric dis-tributions can be encountered tll-t4l in both time and heightdomains.Time statistics, i.e. the densities of probability u-(l) oftime intervals , between microdischarge pulses, follow anexponential distribution,i!)=o..-ot, re(o,o), a=const. (l)while the heights U of microdischarge pulses (their peakvalues) obey a power law of the Pareto type (Pareto distribu-tion of the first kind) [5]wp(U)=c'[J-4, U e (0, .o), cr > 0. (2)when the normalising procedure (5) is performed? This isnicely seen aom tfre-fottowing tlvo equations, (6) and (7)'which were obtained after logarithmic operations had beenapplied to Eqs. (l) and (5)lnwr(t)=lnb-at + )=k-at' (6)tnwilt)=lnD-lns, -al + )* =k* -at' (7)k* =h-lnSr. (8)Therefore, after normalisation the graph of the exPonen-tial distribution function plotted in a semilogarithmic systemwill conserve its shape (stiaight line), but it will shift its posi-tion in the vertical direction by a constant value ln S. In fact'there is no need to look for the value of S, to obtain a normali-sation form (5), since the normalisation constant blS =ais an invariant appearing in the argument of unnormal-ised function (+). Frg. I shows an unnormalised distributionin semilogarithmic co-ordinates with the following fittingequation) = 0.0685 - 0.0895x (9)which enables us straightforwardly to determine the corre-sponding normalised exponential distribution,i1r; = o.oasr '.-o o8e5r ittt s;-r (10)ln [V(ms)'l@Some problems may arise when normalised forms of thesehighly asymmetric distributions should be used, especially-*iih iur.io't distribution (2), certain probability moments ofwhich divergeThe goal of this paper is to discuss problems connectedwith the normalisation of Pareto's statistic (2)'2 Normalising exPonentialdistributionIn fact there is no principal problem with normalisation ofa measured (unnormalised) exponential distributionI w p(U)dU = 'n .0(3)-2.5,r(t)=b-r-ot, b+a'Since there is no singular point in the interval (0' co)' nor-malisation form (l) can easily be found,i(t\=Lr-o'. (5)S", To verify the exponential character of a function' thesemilogarithmic co-ordinates are usually employed' But whatwill haipen with the shape and position.of the graph of an.*po.r..rtial function plotted in the semilogarithmic system@IhS.=f zu,(r)d1=L+1. (4)Ju05U 60of time intervalst/ (ms)between microdis-l: Probability densitYcharge pulses [4]Fig.59Acta Polytechnica Vol. 43 No. 2/2003(18)w;(U) =0506· U- L42 (mVt'.ConclusionAssoc. Prof. RNDr. Tomáš Ficker, DrSc.phone: + 420 541 147661e-mail: fyfic@fce.vutbr.czDepartment of PhysicsAssoc. Prof. RNDr. Jiří Macur, CSc.phone: +420541 147249e-mail: macur.j@fce.vutbr.czDepartment of TnformaticsFaculty of Civil EngineeringUniversity of TechnologyŽižkova 17662 37 Brno, Czech RepublicAcknowledgementThis work has been supported by the Grant Agency of theCzech Republic under the grant No. 202/03/00 ll.The Pareto distribution can be normalised in a standard,when the used interval (Ul' U2 ) does not possesses zeropoint, i.e. Ul > O. When plotting both the distributions, i.e. ex-ponentia! and Pareto, in logarithmic co-ordinates (semi- andbi- systems, respectively), normalisation procedure causes acertain shift of their graphs in the vertical direction, whiletheir shapes and asymmetricities remain unchanged. Theirfunctiona! character is not influenced by a normalisationprocedure. In other words, the normalisation procedure can-not change the results of physical processes and this fact alsosupports the employment of unnormalised distributions instatistical studies realised in physics and other fields ofscienceand technology.References[1] Ficker, T: hactal Statisties ojPartial Discharges with Poly-meric Samples.J. App!. Phys, Vol. 78, 1995, p. 5289-5295.[2] Fromm, U.: Interpretation ojPartial Discharges at DC Volt-ages. IEEE Trans. DieJ. EI. lnsul., Vol. 2, 1995,p.761-770.[3] Ficker, T, Macur, J., Pazdera, L., Kliment, M., Filip, 5.:Simplifwd Digital Acquisition ojMurodischarge Pulses. IEEETrans. Die!. II. Insul. Vol. 8, 2001, p. 220-227.[4] Ficker, T: Electron Avalanches I. Statistus oj PartinlMicrodischarges in Their Pre-Strearner Stage. TEEE Trans.Diel. El. Tnsul. (in press).[5] Johnson, N. J., Kotz, S.: Continuous Univariate Distribu-tions - I. New York: John Wiley & Sons, 1970.(ll)1.0I -- Y" -1.42x -0.514 10.5oln (wl (mVY'J-1o-2-3..L--~-------r------.-----.-.....J­1.5ln IV I (mV)]3 Normalising Pareto distributionThe situation with the Pareto distribution (2) is moreproblematic. Due to the singularity at the point U = O, theintegral (3) over the interval (O, <Xl) diverges, and normalisa-tion is not possible, in this interval. Does this mean thal nor-malisation of the Pareto distribution is not possible at all?Normalisation of Pareto's distribution is certainly possiblebut in an interval (Ul' U2) lacking any singularity, i.e. Ul > O.Real measurements are lIsually performed in such intervals.In addition, the intervals in real experiments are usually fi-nite, i.e. U2 < <Xl. For such cases the normalised distributioncan be introduced as followsU2Sp= JWp(U)dU=I~U(UJ--a-uf-a) (12)UlFor a half-infinite interval (U I, <Xl) it holdswp(U)=(u -1)Uf-1.u-a . (l3)Analysing the Pareto unnormalised (2) and normalised(II) distributions in bilogarithmic co-ordinateslnwp =Jnc-ulnU => y=k-ux, (14)lnw; =lnc-lnSp -ulnU => y' =k' -ux, (15)k'=k-lnSp, (16)similar conclusions as for exponential distribution can bedrawn: in bilogarithmic systems the normalisation proce-dure changes only the vertical position of the graph of thePareto distribution and does not influence its functional char-acter (slope or asymmetricity). Fig. 2 shows an unnorm~lise~Pareto distribution within the interval U E (05, 5) mV lil bl-logarithmic co-ordinates with the following fming equationy = -0514 -l.42x (17)which again enables us to determine the corresponding nor-malised form of the Pareto distribution. According to (ll)we can findFig. 2: Probability density of heights of microdischarge pulses [4)60

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A Note on Normalised Distributions of DC Partial Microdischarges

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