Acta Polytechnica Vol. 43 No' 212003 A Note on Normalised Distributions of DC Partial Microdischarges T. Ficker, J. Macur statistical distr,ibutions (exponentfuil and. pareto) of DC partinl rnicrod,ischargu running. within sandwich electrode sJstems are discussed from 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 Introduction When studying the statistics of partial microdischarges within sandwicir eLctrode systems loading by dc voltages in excess of Paschen breakdown values, highly asymmetric dis- tributions can be encountered tll-t4l in both time and height domains. Time statistics, i.e. the densities of probability u-(l) of time intervals , between microdischarge pulses, follow an exponential distribution ,i!)=o..-ot, re(o,o), a=const. (l) while the heights U of microdischarge pulses (their peak values) 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 is nicely seen aom tfre-fottowing tlvo equations, (6) and (7)' which were obtained after logarithmic operations had been applied 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 system will 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 =a is an invariant appearing in the argument of unnormal- ised function (+). Frg. I shows an unnormalised distribution in semilogarithmic co-ordinates with the following fitting equation ) = 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 these highly asymmetric distributions should be used, especially- *iih iur.io't distribution (2), certain probability moments of which diverge The goal of this paper is to discuss problems connected with the normalisation of Pareto's statistic (2)' 2 Normalising exPonential distribution In fact there is no principal problem with normalisation of a measured (unnormalised) exponential distribution I 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' the semilogarithmic co-ordinates are usually employed' But what will haipen with the shape and position.of the graph of an .*po.r..rtial function plotted in the semilogarithmic system @ Ih S.=f zu,(r)d1=L+1. (4) Ju 0 5U 60 of time intervals t/ (ms) between microdis-l: Probability densitY charge pulses [4] Fig. 59 Acta Polytechnica Vol. 43 No. 2/2003 (18)w;(U) =0506· U- L42 (mVt'. Conclusion Assoc. Prof. RNDr. Tomáš Ficker, DrSc. phone: + 420 541 147661 e-mail: fyfic@fce.vutbr.cz Department of Physics Assoc. Prof. RNDr. Jiří Macur, CSc. phone: +420541 147249 e-mail: macur.j@fce.vutbr.cz Department of Tnformatics Faculty of Civil Engineering University of Technology Žižkova 17 662 37 Brno, Czech Republic Acknowledgement This work has been supported by the Grant Agency of the Czech 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 zero point, i.e. Ul > O. When plotting both the distributions, i.e. ex- ponentia! and Pareto, in logarithmic co-ordinates (semi- and bi- systems, respectively), normalisation procedure causes a certain shift of their graphs in the vertical direction, while their shapes and asymmetricities remain unchanged. Their functiona! character is not influenced by a normalisation procedure. In other words, the normalisation procedure can- not change the results of physical processes and this fact also supports the employment of unnormalised distributions in statistical studies realised in physics and other fields of science and technology. References [1] Ficker, T: hactal Statisties oj Partial Discharges with Poly- meric Samples.J. App!. Phys, Vol. 78, 1995, p. 5289-5295. [2] Fromm, U.: Interpretation oj Partial 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. IEEE Trans. Die!. II. Insul. Vol. 8, 2001, p. 220-227. [4] Ficker, T: Electron Avalanches I. Statistus oj Partinl Microdischarges 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) 1.0 I -- Y" -1.42x -0. 514 1 0.5o ln (wl (mVY'J -1 o -2 -3..L--~-------r------.-----.-.....J­ 1.5 ln IV I (mV)] 3 Normalising Pareto distribution The situation with the Pareto distribution (2) is more problematic. Due to the singularity at the point U = O, the integral (3) over the interval (O, O. Real measurements are lIsually performed in such intervals. In addition, the interval s in real experiments are usually fi- nite, i.e. U2 < y=k-ux, (14) lnw; =lnc-lnSp -ulnU => y' =k' -ux, (15) k'=k-lnSp, (16) similar conclusions as for exponential distribution can be drawn: in bilogarithmic systems the normalisation proce- dure changes only the vertical position of the graph of the Pareto 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 equation y = -0514 -l.42x (17) which again enables us to determine the corresponding nor- malised form of the Pareto distribution. According to (ll) we can find Fig. 2: Probability density of heights of microdischarge pulses [4) 60 Scan59 Scan60